LMFDB - Embedded newform 126.6.g.e.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,6,Mod(37,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.g (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-8,0,-32,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.2083612964$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{130})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 130x^{2} + 16900 $ x^4 + 130*x^2 + 16900
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-5.70088 + 9.87421i$ of defining polynomial
Character	$\chi$	$=$	126.109
Dual form			126.6.g.e.37.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 - 18.1865i) q^{5} +(-10.4105 + 129.223i) q^{7} +64.0000 q^{8} +(-42.0000 + 72.7461i) q^{10} +(165.937 - 287.411i) q^{11} +66.8211 q^{13} +(468.463 - 222.383i) q^{14} +(-128.000 - 221.703i) q^{16} +(120.268 - 208.311i) q^{17} +(-220.989 - 382.765i) q^{19} +336.000 q^{20} -1327.49 q^{22} +(-535.684 - 927.832i) q^{23} +(1342.00 - 2324.41i) q^{25} +(-133.642 - 231.475i) q^{26} +(-1707.28 - 1178.04i) q^{28} -1791.75 q^{29} +(2844.37 - 4926.59i) q^{31} +(-512.000 + 886.810i) q^{32} -962.147 q^{34} +(2459.43 - 1167.51i) q^{35} +(-5605.36 - 9708.77i) q^{37} +(-883.958 + 1531.06i) q^{38} +(-672.000 - 1163.94i) q^{40} -12077.9 q^{41} -9921.98 q^{43} +(2654.99 + 4598.58i) q^{44} +(-2142.74 + 3711.33i) q^{46} +(-8434.62 - 14609.2i) q^{47} +(-16590.2 - 2690.56i) q^{49} -10736.0 q^{50} +(-534.568 + 925.900i) q^{52} +(2649.38 - 4588.87i) q^{53} -6969.35 q^{55} +(-666.274 + 8270.28i) q^{56} +(3583.49 + 6206.79i) q^{58} +(20692.4 - 35840.3i) q^{59} +(10761.6 + 18639.7i) q^{61} -22754.9 q^{62} +4096.00 q^{64} +(-701.621 - 1215.24i) q^{65} +(13309.4 - 23052.5i) q^{67} +(1924.29 + 3332.98i) q^{68} +(-8963.24 - 6184.70i) q^{70} +58096.5 q^{71} +(19993.8 - 34630.2i) q^{73} +(-22421.5 + 38835.1i) q^{74} +7071.66 q^{76} +(35412.7 + 24435.0i) q^{77} +(21974.9 + 38061.7i) q^{79} +(-2688.00 + 4655.75i) q^{80} +(24155.7 + 41838.9i) q^{82} -22421.4 q^{83} -5051.27 q^{85} +(19844.0 + 34370.7i) q^{86} +(10620.0 - 18394.3i) q^{88} +(12031.0 + 20838.3i) q^{89} +(-695.642 + 8634.83i) q^{91} +17141.9 q^{92} +(-33738.5 + 58436.8i) q^{94} +(-4640.78 + 8038.06i) q^{95} +71896.4 q^{97} +(23860.1 + 62851.4i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 32 q^{4} - 42 q^{5} + 232 q^{7} + 256 q^{8} - 168 q^{10} - 294 q^{11} - 280 q^{13} + 232 q^{14} - 512 q^{16} + 1302 q^{17} + 1442 q^{19} + 1344 q^{20} + 2352 q^{22} + 2646 q^{23} + 5368 q^{25}+ \cdots - 63272 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 - 32 * q^4 - 42 * q^5 + 232 * q^7 + 256 * q^8 - 168 * q^10 - 294 * q^11 - 280 * q^13 + 232 * q^14 - 512 * q^16 + 1302 * q^17 + 1442 * q^19 + 1344 * q^20 + 2352 * q^22 + 2646 * q^23 + 5368 * q^25 + 560 * q^26 - 4640 * q^28 - 3336 * q^29 + 14798 * q^31 - 2048 * q^32 - 10416 * q^34 + 1218 * q^35 - 5182 * q^37 + 5768 * q^38 - 2688 * q^40 + 10248 * q^41 - 9040 * q^43 - 4704 * q^44 + 10584 * q^46 - 14994 * q^47 - 2876 * q^49 - 42944 * q^50 + 2240 * q^52 + 24006 * q^53 + 12348 * q^55 + 14848 * q^56 + 6672 * q^58 + 38850 * q^59 + 23618 * q^61 - 118384 * q^62 + 16384 * q^64 + 2940 * q^65 - 32002 * q^67 + 20832 * q^68 - 24360 * q^70 + 178752 * q^71 + 47138 * q^73 - 20728 * q^74 - 46144 * q^76 + 22890 * q^77 + 40970 * q^79 - 10752 * q^80 - 20496 * q^82 - 136752 * q^83 - 54684 * q^85 + 18080 * q^86 - 18816 * q^88 + 123102 * q^89 - 53680 * q^91 - 84672 * q^92 - 59976 * q^94 + 30282 * q^95 - 87304 * q^97 - 63272 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−8.00000	+	13.8564i	−0.250000	+	0.433013i
$5$	−10.5000	−	18.1865i	−0.187830	−	0.325331i	0.756697	−	0.653766i	$-0.226812\pi$
$5$	−10.5000	−	18.1865i	−0.187830	−	0.325331i	−0.944526	+	0.328436i	$0.893479\pi$
$6$		0			0
$7$	−10.4105	+	129.223i	−0.0803022	+	0.996771i
$7$	−10.4105	+	129.223i	−0.0803022	+	0.996771i
$8$	64.0000			0.353553
$9$		0			0
$10$	−42.0000	+	72.7461i	−0.132816	+	0.230043i
$11$	165.937	−	287.411i	0.413486	−	0.716179i	−0.581782	−	0.813345i	$-0.697644\pi$
$11$	165.937	−	287.411i	0.413486	−	0.716179i	0.995268	+	0.0971656i	$0.0309776\pi$
$12$		0			0
$13$	66.8211			0.109662			0.0548308	−	0.998496i	$-0.482538\pi$
$13$	66.8211			0.109662			0.0548308	+	0.998496i	$0.482538\pi$
$14$	468.463	−	222.383i	0.638786	−	0.303237i
$15$		0			0
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	120.268	−	208.311i	0.100932	−	0.174820i	−0.811137	−	0.584856i	$-0.801151\pi$
$17$	120.268	−	208.311i	0.100932	−	0.174820i	0.912069	+	0.410037i	$0.134484\pi$
$18$		0			0
$19$	−220.989	−	382.765i	−0.140439	−	0.243247i	0.787223	−	0.616668i	$-0.211518\pi$
$19$	−220.989	−	382.765i	−0.140439	−	0.243247i	−0.927662	+	0.373421i	$0.878185\pi$
$20$	336.000			0.187830
$21$		0			0
$22$	−1327.49			−0.584758
$23$	−535.684	−	927.832i	−0.211149	−	0.365721i	0.740925	−	0.671587i	$-0.234387\pi$
$23$	−535.684	−	927.832i	−0.211149	−	0.365721i	−0.952074	+	0.305866i	$0.901054\pi$
$24$		0			0
$25$	1342.00	−	2324.41i	0.429440	−	0.743812i
$26$	−133.642	−	231.475i	−0.0387713	−	0.0671538i
$27$		0			0
$28$	−1707.28	−	1178.04i	−0.411539	−	0.283965i
$29$	−1791.75			−0.395623			−0.197812	−	0.980240i	$-0.563383\pi$
$29$	−1791.75			−0.395623			−0.197812	+	0.980240i	$0.563383\pi$
$30$		0			0
$31$	2844.37	−	4926.59i	0.531596	−	0.920751i	−0.467724	−	0.883875i	$-0.654926\pi$
$31$	2844.37	−	4926.59i	0.531596	−	0.920751i	0.999320	−	0.0368765i	$-0.0117408\pi$
$32$	−512.000	+	886.810i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−962.147			−0.142740
$35$	2459.43	−	1167.51i	0.339363	−	0.161098i
$36$		0			0
$37$	−5605.36	−	9708.77i	−0.673131	−	1.16590i	−0.977012	−	0.213187i	$-0.931616\pi$
$37$	−5605.36	−	9708.77i	−0.673131	−	1.16590i	0.303881	−	0.952710i	$-0.401718\pi$
$38$	−883.958	+	1531.06i	−0.0993053	+	0.172002i
$39$		0			0
$40$	−672.000	−	1163.94i	−0.0664078	−	0.115022i
$41$	−12077.9			−1.12210			−0.561048	−	0.827783i	$-0.689602\pi$
$41$	−12077.9			−1.12210			−0.561048	+	0.827783i	$0.689602\pi$
$42$		0			0
$43$	−9921.98			−0.818328			−0.409164	−	0.912461i	$-0.634180\pi$
$43$	−9921.98			−0.818328			−0.409164	+	0.912461i	$0.634180\pi$
$44$	2654.99	+	4598.58i	0.206743	+	0.358090i
$45$		0			0
$46$	−2142.74	+	3711.33i	−0.149305	+	0.258604i
$47$	−8434.62	−	14609.2i	−0.556956	−	0.964676i	−0.997748	−	0.0670671i	$-0.978636\pi$
$47$	−8434.62	−	14609.2i	−0.556956	−	0.964676i	0.440792	−	0.897609i	$-0.354697\pi$
$48$		0			0
$49$	−16590.2	−	2690.56i	−0.987103	−	0.160086i
$50$	−10736.0			−0.607320
$51$		0			0
$52$	−534.568	+	925.900i	−0.0274154	+	0.0474849i
$53$	2649.38	−	4588.87i	0.129555	−	0.224396i	−0.793949	−	0.607984i	$-0.791978\pi$
$53$	2649.38	−	4588.87i	0.129555	−	0.224396i	0.923504	+	0.383588i	$0.125312\pi$
$54$		0			0
$55$	−6969.35			−0.310660
$56$	−666.274	+	8270.28i	−0.0283911	+	0.352412i
$57$		0			0
$58$	3583.49	+	6206.79i	0.139874	+	0.242269i
$59$	20692.4	−	35840.3i	0.773892	−	1.34042i	−0.161522	−	0.986869i	$-0.551640\pi$
$59$	20692.4	−	35840.3i	0.773892	−	1.34042i	0.935415	−	0.353552i	$-0.115026\pi$
$60$		0			0
$61$	10761.6	+	18639.7i	0.370300	+	0.641379i	0.989612	−	0.143766i	$-0.0459214\pi$
$61$	10761.6	+	18639.7i	0.370300	+	0.641379i	−0.619311	+	0.785146i	$0.712588\pi$
$62$	−22754.9			−0.751790
$63$		0			0
$64$	4096.00			0.125000
$65$	−701.621	−	1215.24i	−0.0205977	−	0.0356763i
$66$		0			0
$67$	13309.4	−	23052.5i	0.362219	−	0.627381i	−0.626107	−	0.779737i	$-0.715353\pi$
$67$	13309.4	−	23052.5i	0.362219	−	0.627381i	0.988326	+	0.152356i	$0.0486860\pi$
$68$	1924.29	+	3332.98i	0.0504661	+	0.0874098i
$69$		0			0
$70$	−8963.24	−	6184.70i	−0.218635	−	0.150860i
$71$	58096.5			1.36774			0.683870	−	0.729603i	$-0.260295\pi$
$71$	58096.5			1.36774			0.683870	+	0.729603i	$0.260295\pi$
$72$		0			0
$73$	19993.8	−	34630.2i	0.439124	−	0.760585i	−0.558498	−	0.829506i	$-0.688622\pi$
$73$	19993.8	−	34630.2i	0.439124	−	0.760585i	0.997622	+	0.0689204i	$0.0219555\pi$
$74$	−22421.5	+	38835.1i	−0.475975	+	0.824413i
$75$		0			0
$76$	7071.66			0.140439
$77$	35412.7	+	24435.0i	0.680663	+	0.469662i
$78$		0			0
$79$	21974.9	+	38061.7i	0.396150	+	0.686151i	0.993247	−	0.116017i	$-0.0370128\pi$
$79$	21974.9	+	38061.7i	0.396150	+	0.686151i	−0.597098	+	0.802169i	$0.703679\pi$
$80$	−2688.00	+	4655.75i	−0.0469574	+	0.0813327i
$81$		0			0
$82$	24155.7	+	41838.9i	0.396721	+	0.687141i
$83$	−22421.4			−0.357246			−0.178623	−	0.983918i	$-0.557164\pi$
$83$	−22421.4			−0.357246			−0.178623	+	0.983918i	$0.557164\pi$
$84$		0			0
$85$	−5051.27			−0.0758322
$86$	19844.0	+	34370.7i	0.289322	+	0.501121i
$87$		0			0
$88$	10620.0	−	18394.3i	0.146189	−	0.253208i
$89$	12031.0	+	20838.3i	0.161001	+	0.278861i	0.935228	−	0.354047i	$-0.115195\pi$
$89$	12031.0	+	20838.3i	0.161001	+	0.278861i	−0.774227	+	0.632908i	$0.781861\pi$
$90$		0			0
$91$	−695.642	+	8634.83i	−0.00880608	+	0.109308i
$92$	17141.9			0.211149
$93$		0			0
$94$	−33738.5	+	58436.8i	−0.393827	+	0.682129i
$95$	−4640.78	+	8038.06i	−0.0527572	+	0.0913782i
$96$		0			0
$97$	71896.4			0.775850			0.387925	−	0.921691i	$-0.373192\pi$
$97$	71896.4			0.775850			0.387925	+	0.921691i	$0.373192\pi$
$98$	23860.1	+	62851.4i	0.250962	+	0.661074i
$99$		0			0
$100$	21472.0	+	37190.6i	0.214720	+	0.371906i
$101$	−7932.73	+	13739.9i	−0.0773783	+	0.134023i	−0.902118	−	0.431489i	$-0.857988\pi$
$101$	−7932.73	+	13739.9i	−0.0773783	+	0.134023i	0.824740	+	0.565512i	$0.191322\pi$
$102$		0			0
$103$	80286.8	+	139061.i	0.745677	+	1.29155i	0.949878	+	0.312622i	$0.101207\pi$
$103$	80286.8	+	139061.i	0.745677	+	1.29155i	−0.204200	+	0.978929i	$0.565459\pi$
$104$	4276.55			0.0387713
$105$		0			0
$106$	−21195.1			−0.183219
$107$	−102523.	−	177576.i	−0.865693	−	1.49942i	−0.866357	−	0.499425i	$-0.833545\pi$
$107$	−102523.	−	177576.i	−0.865693	−	1.49942i	0.000664234	−	1.00000i	$-0.499789\pi$
$108$		0			0
$109$	56272.1	−	97466.2i	0.453657	−	0.785756i	−0.544953	−	0.838466i	$-0.683453\pi$
$109$	56272.1	−	97466.2i	0.453657	−	0.785756i	0.998610	+	0.0527102i	$0.0167860\pi$
$110$	13938.7	+	24142.5i	0.109835	+	0.190240i
$111$		0			0
$112$	29981.6	−	14232.5i	0.225845	−	0.107210i
$113$	118332.			0.871782			0.435891	−	0.900000i	$-0.356433\pi$
$113$	118332.			0.871782			0.435891	+	0.900000i	$0.356433\pi$
$114$		0			0
$115$	−11249.4	+	19484.5i	−0.0793202	+	0.137387i
$116$	14334.0	−	24827.2i	0.0989058	−	0.171310i
$117$		0			0
$118$	−165539.			−1.09445
$119$	25666.5	+	17710.1i	0.166150	+	0.114645i
$120$		0			0
$121$	25455.4	+	44090.1i	0.158058	+	0.273765i
$122$	43046.6	−	74558.9i	0.261842	−	0.453524i
$123$		0			0
$124$	45509.9	+	78825.5i	0.265798	+	0.460376i
$125$	−121989.			−0.698306
$126$		0			0
$127$	−245195.			−1.34897			−0.674485	−	0.738289i	$-0.735634\pi$
$127$	−245195.			−1.34897			−0.674485	+	0.738289i	$0.735634\pi$
$128$	−8192.00	−	14189.0i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−2806.48	+	4860.97i	−0.0145648	+	0.0252270i
$131$	34996.9	+	60616.3i	0.178177	+	0.308611i	0.941256	−	0.337694i	$-0.109647\pi$
$131$	34996.9	+	60616.3i	0.178177	+	0.308611i	−0.763079	+	0.646305i	$0.776313\pi$
$132$		0			0
$133$	51762.7	−	24572.2i	0.253739	−	0.120452i
$134$	−106475.			−0.512254
$135$		0			0
$136$	7697.18	−	13331.9i	0.0356849	−	0.0618080i
$137$	−93831.9	+	162522.i	−0.427119	+	0.739792i	−0.996616	−	0.0822015i	$-0.973805\pi$
$137$	−93831.9	+	162522.i	−0.427119	+	0.739792i	0.569496	+	0.821994i	$0.307138\pi$
$138$		0			0
$139$	−78272.7			−0.343616			−0.171808	−	0.985130i	$-0.554961\pi$
$139$	−78272.7			−0.343616			−0.171808	+	0.985130i	$0.554961\pi$
$140$	−3497.94	+	43419.0i	−0.0150831	+	0.187223i
$141$		0			0
$142$	−116193.	−	201252.i	−0.483569	−	0.837567i
$143$	11088.1	−	19205.1i	0.0453436	−	0.0785374i
$144$		0			0
$145$	18813.3	+	32585.7i	0.0743098	+	0.128708i
$146$	−159950.			−0.621015
$147$		0			0
$148$	179372.			0.673131
$149$	−46083.2	−	79818.5i	−0.170050	−	0.294536i	0.768387	−	0.639985i	$-0.221060\pi$
$149$	−46083.2	−	79818.5i	−0.170050	−	0.294536i	−0.938437	+	0.345450i	$0.887726\pi$
$150$		0			0
$151$	−26708.0	+	46259.6i	−0.0953232	+	0.165105i	−0.909743	−	0.415171i	$-0.863722\pi$
$151$	−26708.0	+	46259.6i	−0.0953232	+	0.165105i	0.814420	+	0.580276i	$0.197055\pi$
$152$	−14143.3	−	24497.0i	−0.0496527	−	0.0860009i
$153$		0			0
$154$	13819.9	−	171543.i	0.0469574	−	0.582869i
$155$	−119463.			−0.399398
$156$		0			0
$157$	140250.	−	242921.i	0.454103	−	0.786530i	−0.544533	−	0.838739i	$-0.683293\pi$
$157$	140250.	−	242921.i	0.454103	−	0.786530i	0.998636	+	0.0522096i	$0.0166264\pi$
$158$	87899.6	−	152247.i	0.280120	−	0.485182i
$159$		0			0
$160$	21504.0			0.0664078
$161$	125474.	−	59563.6i	0.381496	−	0.181099i
$162$		0			0
$163$	−313405.	−	542833.i	−0.923925	−	1.60028i	−0.793281	−	0.608855i	$-0.791629\pi$
$163$	−313405.	−	542833.i	−0.923925	−	1.60028i	−0.130643	−	0.991429i	$-0.541704\pi$
$164$	96622.8	−	167356.i	0.280524	−	0.485882i
$165$		0			0
$166$	44842.8	+	77670.0i	0.126306	+	0.218768i
$167$	−293997.			−0.815741			−0.407870	−	0.913040i	$-0.633728\pi$
$167$	−293997.			−0.815741			−0.407870	+	0.913040i	$0.633728\pi$
$168$		0			0
$169$	−366828.			−0.987974
$170$	10102.5	+	17498.1i	0.0268107	+	0.0464375i
$171$		0			0
$172$	79375.8	−	137483.i	0.204582	−	0.354346i
$173$	357093.	+	618504.i	0.907124	+	1.57118i	0.818041	+	0.575160i	$0.195060\pi$
$173$	357093.	+	618504.i	0.907124	+	1.57118i	0.0890831	+	0.996024i	$0.471606\pi$
$174$		0			0
$175$	286397.	+	197616.i	0.706925	+	0.487783i
$176$	−84959.7			−0.206743
$177$		0			0
$178$	48124.1	−	83353.3i	0.113845	−	0.197185i
$179$	290025.	−	502338.i	0.676554	−	1.17183i	−0.299458	−	0.954110i	$-0.596806\pi$
$179$	290025.	−	502338.i	0.676554	−	1.17183i	0.976012	−	0.217717i	$-0.0698609\pi$
$180$		0			0
$181$	−308046.			−0.698907			−0.349454	−	0.936954i	$-0.613633\pi$
$181$	−308046.			−0.698907			−0.349454	+	0.936954i	$0.613633\pi$
$182$	31303.2	−	14859.9i	0.0700503	−	0.0332535i
$183$		0			0
$184$	−34283.8	−	59381.3i	−0.0746525	−	0.129302i
$185$	−117713.	+	203884.i	−0.252868	+	0.437980i
$186$		0			0
$187$	−39913.9	−	69132.9i	−0.0834681	−	0.144571i
$188$	269908.			0.556956
$189$		0			0
$190$	37126.2			0.0746100
$191$	−101395.	−	175621.i	−0.201109	−	0.348331i	0.747777	−	0.663950i	$-0.231121\pi$
$191$	−101395.	−	175621.i	−0.201109	−	0.348331i	−0.948886	+	0.315619i	$0.897788\pi$
$192$		0			0
$193$	−302994.	+	524801.i	−0.585519	+	1.01415i	0.409292	+	0.912403i	$0.365776\pi$
$193$	−302994.	+	524801.i	−0.585519	+	1.01415i	−0.994811	+	0.101744i	$0.967558\pi$
$194$	−143793.	−	249057.i	−0.274305	−	0.475109i
$195$		0			0
$196$	170003.	−	208357.i	0.316095	−	0.387407i
$197$	310.819			0.000570614			0.000285307	−	1.00000i	$-0.499909\pi$
$197$	310.819			0.000570614			0.000285307		1.00000i	$0.499909\pi$
$198$		0			0
$199$	204833.	−	354782.i	0.366664	−	0.635080i	−0.622378	−	0.782717i	$-0.713833\pi$
$199$	204833.	−	354782.i	0.366664	−	0.635080i	0.989042	+	0.147637i	$0.0471667\pi$
$200$	85888.0	−	148762.i	0.151830	−	0.262977i
$201$		0			0
$202$	63461.9			0.109429
$203$	18653.0	−	231535.i	0.0317694	−	0.394346i
$204$		0			0
$205$	126817.	+	219654.i	0.210763	+	0.365052i
$206$	321147.	−	556243.i	0.527274	−	0.913265i
$207$		0			0
$208$	−8553.09	−	14814.4i	−0.0137077	−	0.0237425i
$209$	−146681.			−0.232278
$210$		0			0
$211$	−441339.			−0.682442			−0.341221	−	0.939983i	$-0.610840\pi$
$211$	−441339.			−0.682442			−0.341221	+	0.939983i	$0.610840\pi$
$212$	42390.1	+	73421.9i	0.0647777	+	0.112198i
$213$		0			0
$214$	−410094.	+	710304.i	−0.612137	+	1.06025i
$215$	104181.	+	180446.i	0.153706	+	0.266227i
$216$		0			0
$217$	607018.	+	418847.i	0.875089	+	0.603817i
$218$	−450177.			−0.641567
$219$		0			0
$220$	55754.8	−	96570.1i	0.0776650	−	0.134520i
$221$	8036.46	−	13919.6i	0.0110684	−	0.0191710i
$222$		0			0
$223$	265133.			0.357028			0.178514	−	0.983937i	$-0.442871\pi$
$223$	265133.			0.357028			0.178514	+	0.983937i	$0.442871\pi$
$224$	−109266.	−	75394.4i	−0.145501	−	0.100397i
$225$		0			0
$226$	−236665.	−	409916.i	−0.308221	−	0.533855i
$227$	−715220.	+	1.23880e6i	−0.921244	+	1.59564i	−0.123751	+	0.992313i	$0.539492\pi$
$227$	−715220.	+	1.23880e6i	−0.921244	+	1.59564i	−0.797493	+	0.603328i	$0.793841\pi$
$228$		0			0
$229$	629025.	+	1.08950e6i	0.792646	+	1.37290i	0.924323	+	0.381610i	$0.124630\pi$
$229$	629025.	+	1.08950e6i	0.792646	+	1.37290i	−0.131678	+	0.991293i	$0.542036\pi$
$230$	89994.9			0.112176
$231$		0			0
$232$	−114672.			−0.139874
$233$	112362.	+	194616.i	0.135590	+	0.234849i	0.925823	−	0.377958i	$-0.123374\pi$
$233$	112362.	+	194616.i	0.135590	+	0.234849i	−0.790233	+	0.612807i	$0.790040\pi$
$234$		0			0
$235$	−177127.	+	306793.i	−0.209226	+	0.362390i
$236$	331078.	+	573444.i	0.386946	+	0.670211i
$237$		0			0
$238$	10016.5	−	124332.i	0.0114623	−	0.142279i
$239$	−1.28193e6			−1.45167			−0.725837	−	0.687867i	$-0.758547\pi$
$239$	−1.28193e6			−1.45167			−0.725837	+	0.687867i	$0.758547\pi$
$240$		0			0
$241$	288062.	−	498939.i	0.319480	−	0.553356i	−0.660899	−	0.750475i	$-0.729825\pi$
$241$	288062.	−	498939.i	0.319480	−	0.553356i	0.980380	+	0.197118i	$0.0631583\pi$
$242$	101822.	−	176360.i	0.111764	−	0.193581i
$243$		0			0
$244$	−344373.			−0.370300
$245$	125266.	+	329970.i	0.133326	+	0.351204i
$246$		0			0
$247$	−14766.7	−	25576.8i	−0.0154008	−	0.0266749i
$248$	182040.	−	315302.i	0.187948	−	0.325535i
$249$		0			0
$250$	243978.	+	422582.i	0.246888	+	0.427623i
$251$	609040.			0.610185			0.305092	−	0.952323i	$-0.401313\pi$
$251$	609040.			0.610185			0.305092	+	0.952323i	$0.401313\pi$
$252$		0			0
$253$	−355559.			−0.349229
$254$	490390.	+	849380.i	0.476933	+	0.826072i
$255$		0			0
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	515335.	+	892586.i	0.486695	+	0.842980i	0.999883	−	0.0152961i	$-0.00486909\pi$
$257$	515335.	+	892586.i	0.486695	+	0.842980i	−0.513188	+	0.858276i	$0.671536\pi$
$258$		0			0
$259$	1.31295e6	−	623269.i	1.21619	−	0.577333i
$260$	22451.9			0.0205977
$261$		0			0
$262$	139987.	−	242465.i	0.125990	−	0.218221i
$263$	−455293.	+	788590.i	−0.405883	+	0.703011i	−0.994424	−	0.105457i	$-0.966369\pi$
$263$	−455293.	+	788590.i	−0.405883	+	0.703011i	0.588540	+	0.808468i	$0.299703\pi$
$264$		0			0
$265$	−111274.			−0.0973374
$266$	−188646.	−	130167.i	−0.163472	−	0.112797i
$267$		0			0
$268$	212950.	+	368840.i	0.181109	+	0.313691i
$269$	710499.	−	1.23062e6i	0.598663	−	1.03692i	−0.394355	−	0.918958i	$-0.629032\pi$
$269$	710499.	−	1.23062e6i	0.598663	−	1.03692i	0.993019	−	0.117957i	$-0.0376346\pi$
$270$		0			0
$271$	148765.	+	257668.i	0.123049	+	0.213127i	0.920969	−	0.389637i	$-0.127400\pi$
$271$	148765.	+	257668.i	0.123049	+	0.213127i	−0.797920	+	0.602764i	$0.794066\pi$
$272$	−61577.4			−0.0504661
$273$		0			0
$274$	750655.			0.604038
$275$	−445374.	−	771411.i	−0.355135	−	0.615112i
$276$		0			0
$277$	−424806.	+	735785.i	−0.332653	+	0.576171i	−0.983031	−	0.183439i	$-0.941277\pi$
$277$	−424806.	+	735785.i	−0.332653	+	0.576171i	0.650378	+	0.759610i	$0.274610\pi$
$278$	156545.	+	271144.i	0.121487	+	0.210421i
$279$		0			0
$280$	157404.	−	74720.7i	0.119983	−	0.0569569i
$281$	7680.49			0.00580261			0.00290130	−	0.999996i	$-0.499076\pi$
$281$	7680.49			0.00580261			0.00290130	+	0.999996i	$0.499076\pi$
$282$		0			0
$283$	492684.	−	853353.i	0.365681	−	0.633378i	−0.623204	−	0.782059i	$-0.714170\pi$
$283$	492684.	−	853353.i	0.365681	−	0.633378i	0.988885	+	0.148681i	$0.0475029\pi$
$284$	−464772.	+	805008.i	−0.341935	+	0.592249i
$285$		0			0
$286$	−88704.6			−0.0641255
$287$	125737.	−	1.56074e6i	0.0901068	−	1.11847i
$288$		0			0
$289$	681000.	+	1.17953e6i	0.479625	+	0.830736i
$290$	75253.4	−	130343.i	0.0525450	−	0.0910105i
$291$		0			0
$292$	319900.	+	554083.i	0.219562	+	0.380293i
$293$	2.16934e6			1.47625			0.738124	−	0.674665i	$-0.235712\pi$
$293$	2.16934e6			1.47625			0.738124	+	0.674665i	$0.235712\pi$
$294$		0			0
$295$	−869080.			−0.581440
$296$	−358743.	−	621362.i	−0.237988	−	0.412207i
$297$		0			0
$298$	−184333.	+	319274.i	−0.120244	+	0.208268i
$299$	−35795.0	−	61998.7i	−0.0231550	−	0.0401056i
$300$		0			0
$301$	103293.	−	1.28215e6i	0.0657135	−	0.815685i
$302$	213664.			0.134807
$303$		0			0
$304$	−56573.3	+	97987.8i	−0.0351097	+	0.0608118i
$305$	225995.	−	391434.i	0.139107	−	0.240940i
$306$		0			0
$307$	−1.39093e6			−0.842287			−0.421143	−	0.906994i	$-0.638371\pi$
$307$	−1.39093e6			−0.842287			−0.421143	+	0.906994i	$0.638371\pi$
$308$	−621882.	+	295212.i	−0.373535	+	0.177320i
$309$		0			0
$310$	238927.	+	413834.i	0.141209	+	0.244580i
$311$	1.01897e6	−	1.76491e6i	0.597394	−	1.03472i	−0.395810	−	0.918332i	$-0.629536\pi$
$311$	1.01897e6	−	1.76491e6i	0.597394	−	1.03472i	0.993204	−	0.116385i	$-0.0371306\pi$
$312$		0			0
$313$	584789.	+	1.01288e6i	0.337395	+	0.584385i	0.983942	−	0.178489i	$-0.0571210\pi$
$313$	584789.	+	1.01288e6i	0.337395	+	0.584385i	−0.646547	+	0.762874i	$0.723788\pi$
$314$	−1.12200e6			−0.642199
$315$		0			0
$316$	−703197.			−0.396150
$317$	400568.	+	693805.i	0.223887	+	0.387783i	0.955985	−	0.293416i	$-0.0947921\pi$
$317$	400568.	+	693805.i	0.223887	+	0.387783i	−0.732098	+	0.681199i	$0.761459\pi$
$318$		0			0
$319$	−297317.	+	514968.i	−0.163585	+	0.283337i
$320$	−43008.0	−	74492.0i	−0.0234787	−	0.0406663i
$321$		0			0
$322$	−457283.	−	315528.i	−0.245779	−	0.169589i
$323$	−106312.			−0.0566992
$324$		0			0
$325$	89673.9	−	155320.i	0.0470931	−	0.0815677i
$326$	−1.25362e6	+	2.17133e6i	−0.653313	+	1.13157i
$327$		0			0
$328$	−772983.			−0.396721
$329$	1.97565e6	−	937859.i	1.00629	−	0.477692i
$330$		0			0
$331$	−1.32476e6	−	2.29454e6i	−0.664608	−	1.15114i	−0.979391	−	0.201972i	$-0.935265\pi$
$331$	−1.32476e6	−	2.29454e6i	−0.664608	−	1.15114i	0.314783	−	0.949164i	$-0.398068\pi$
$332$	179371.	−	310680.i	0.0893115	−	0.154692i
$333$		0			0
$334$	587994.	+	1.01844e6i	0.288408	+	0.499537i
$335$	−558994.			−0.272142
$336$		0			0
$337$	−3.40056e6			−1.63108			−0.815541	−	0.578699i	$-0.803560\pi$
$337$	−3.40056e6			−1.63108			−0.815541	+	0.578699i	$0.803560\pi$
$338$	733656.	+	1.27073e6i	0.349302	+	0.605008i
$339$		0			0
$340$	40410.2	−	69992.5i	0.0189580	−	0.0328363i
$341$	−943971.	−	1.63501e6i	−0.439615	−	0.761436i
$342$		0			0
$343$	520396.	−	2.11583e6i	0.238835	−	0.971060i
$344$	−635007.			−0.289322
$345$		0			0
$346$	1.42837e6	−	2.47402e6i	0.641433	−	1.11100i
$347$	−308855.	+	534952.i	−0.137699	+	0.238501i	−0.926625	−	0.375986i	$-0.877304\pi$
$347$	−308855.	+	534952.i	−0.137699	+	0.238501i	0.788926	+	0.614488i	$0.210637\pi$
$348$		0			0
$349$	2.70539e6			1.18896			0.594479	−	0.804111i	$-0.297358\pi$
$349$	2.70539e6			1.18896			0.594479	+	0.804111i	$0.297358\pi$
$350$	111767.	−	1.38734e6i	0.0487691	−	0.605359i
$351$		0			0
$352$	169919.	+	294309.i	0.0730947	+	0.126604i
$353$	−1.59032e6	+	2.75452e6i	−0.679279	+	1.17655i	0.295919	+	0.955213i	$0.404374\pi$
$353$	−1.59032e6	+	2.75452e6i	−0.679279	+	1.17655i	−0.975198	+	0.221333i	$0.928959\pi$
$354$		0			0
$355$	−610013.	−	1.05657e6i	−0.256902	−	0.444968i
$356$	−384993.			−0.161001
$357$		0			0
$358$	−2.32020e6			−0.956792
$359$	−2.22985e6	−	3.86222e6i	−0.913145	−	1.58161i	−0.809595	−	0.586989i	$-0.800313\pi$
$359$	−2.22985e6	−	3.86222e6i	−0.913145	−	1.58161i	−0.103550	−	0.994624i	$-0.533020\pi$
$360$		0			0
$361$	1.14038e6	−	1.97519e6i	0.460554	−	0.797703i
$362$	616093.	+	1.06710e6i	0.247101	+	0.427992i
$363$		0			0
$364$	−114083.	−	78717.7i	−0.0451300	−	0.0311400i
$365$	−839738.			−0.329922
$366$		0			0
$367$	−230623.	+	399451.i	−0.0893795	+	0.154810i	−0.907249	−	0.420594i	$-0.861822\pi$
$367$	−230623.	+	399451.i	−0.0893795	+	0.154810i	0.817869	+	0.575404i	$0.195155\pi$
$368$	−137135.	+	237525.i	−0.0527873	+	0.0914302i
$369$		0			0
$370$	941701.			0.357609
$371$	565406.	+	390134.i	0.213268	+	0.147156i
$372$		0			0
$373$	−1.70977e6	−	2.96141e6i	−0.636305	−	1.10211i	−0.986237	−	0.165338i	$-0.947129\pi$
$373$	−1.70977e6	−	2.96141e6i	−0.636305	−	1.10211i	0.349932	−	0.936775i	$-0.386205\pi$
$374$	−159656.	+	276532.i	−0.0590208	+	0.102227i
$375$		0			0
$376$	−539816.	−	934988.i	−0.196914	−	0.341065i
$377$	−119726.			−0.0433847
$378$		0			0
$379$	16355.6			0.00584882			0.00292441	−	0.999996i	$-0.499069\pi$
$379$	16355.6			0.00584882			0.00292441	+	0.999996i	$0.499069\pi$
$380$	−74252.5	−	128609.i	−0.0263786	−	0.0456891i
$381$		0			0
$382$	−405579.	+	702483.i	−0.142206	+	0.246308i
$383$	−1.71822e6	−	2.97604e6i	−0.598523	−	1.03667i	−0.993039	−	0.117784i	$-0.962421\pi$
$383$	−1.71822e6	−	2.97604e6i	−0.598523	−	1.03667i	0.394516	−	0.918889i	$-0.370912\pi$
$384$		0			0
$385$	72554.6	−	900601.i	0.0249467	−	0.309657i
$386$	2.42395e6			0.828048
$387$		0			0
$388$	−575171.	+	996226.i	−0.193963	+	0.335953i
$389$	−2.40526e6	+	4.16604e6i	−0.805914	+	1.39588i	0.109759	+	0.993958i	$0.464992\pi$
$389$	−2.40526e6	+	4.16604e6i	−0.805914	+	1.39588i	−0.915673	+	0.401925i	$0.868341\pi$
$390$		0			0
$391$	−257704.			−0.0852469
$392$	−1.06178e6	−	172196.i	−0.348994	−	0.0565989i
$393$		0			0
$394$	−621.638	−	1076.71i	−0.000201742	−	0.000349428i
$395$	461473.	−	799295.i	0.148817	−	0.257759i
$396$		0			0
$397$	1.64924e6	+	2.85657e6i	0.525180	+	0.909638i	0.999570	+	0.0293232i	$0.00933522\pi$
$397$	1.64924e6	+	2.85657e6i	0.525180	+	0.909638i	−0.474390	+	0.880315i	$0.657331\pi$
$398$	−1.63867e6			−0.518540
$399$		0			0
$400$	−687104.			−0.214720
$401$	68082.0	+	117921.i	0.0211432	+	0.0366211i	0.876403	−	0.481578i	$-0.159936\pi$
$401$	68082.0	+	117921.i	0.0211432	+	0.0366211i	−0.855260	+	0.518199i	$0.826603\pi$
$402$		0			0
$403$	190064.	−	329200.i	0.0582957	−	0.100971i
$404$	−126924.	−	219838.i	−0.0386892	−	0.0670116i
$405$		0			0
$406$	−839368.	+	398454.i	−0.252719	+	0.119967i
$407$	−3.72054e6			−1.11332
$408$		0			0
$409$	−1.37435e6	+	2.38044e6i	−0.406245	+	0.703637i	−0.994466	−	0.105064i	$-0.966495\pi$
$409$	−1.37435e6	+	2.38044e6i	−0.406245	+	0.703637i	0.588220	+	0.808701i	$0.299829\pi$
$410$	507270.	−	878617.i	0.149032	−	0.258131i
$411$		0			0
$412$	−2.56918e6			−0.745677
$413$	4.41597e6	+	3.04705e6i	1.27395	+	0.879032i
$414$		0			0
$415$	235425.	+	407767.i	0.0671014	+	0.116223i
$416$	−34212.4	+	59257.6i	−0.00969282	+	0.0167884i
$417$		0			0
$418$	293362.	+	508118.i	0.0821228	+	0.142241i
$419$	1.87219e6			0.520973			0.260486	−	0.965478i	$-0.416117\pi$
$419$	1.87219e6			0.520973			0.260486	+	0.965478i	$0.416117\pi$
$420$		0			0
$421$	655225.			0.180171			0.0900856	−	0.995934i	$-0.471286\pi$
$421$	655225.			0.180171			0.0900856	+	0.995934i	$0.471286\pi$
$422$	882677.	+	1.52884e6i	0.241280	+	0.417909i
$423$		0			0
$424$	169561.	−	293688.i	0.0458047	−	0.0793361i
$425$	−322800.	−	559107.i	−0.0866886	−	0.150149i
$426$		0			0
$427$	−2.52072e6	+	1.19660e6i	−0.669044	+	0.317600i
$428$	3.28075e6			0.865693
$429$		0			0
$430$	416723.	−	721786.i	0.108687	−	0.188251i
$431$	3.18845e6	−	5.52256e6i	0.826773	−	1.43201i	−0.0737832	−	0.997274i	$-0.523507\pi$
$431$	3.18845e6	−	5.52256e6i	0.826773	−	1.43201i	0.900556	−	0.434739i	$-0.143159\pi$
$432$		0			0
$433$	4.80003e6			1.23034			0.615168	−	0.788396i	$-0.289088\pi$
$433$	4.80003e6			1.23034			0.615168	+	0.788396i	$0.289088\pi$
$434$	236891.	−	2.94047e6i	0.0603704	−	0.749362i
$435$		0			0
$436$	900354.	+	1.55946e6i	0.226828	+	0.392878i
$437$	−236761.	+	410082.i	−0.0593071	+	0.102723i
$438$		0			0
$439$	2.79754e6	+	4.84548e6i	0.692811	+	1.19998i	0.970913	+	0.239432i	$0.0769612\pi$
$439$	2.79754e6	+	4.84548e6i	0.692811	+	1.19998i	−0.278102	+	0.960551i	$0.589705\pi$
$440$	−446038.			−0.109835
$441$		0			0
$442$	−64291.7			−0.0156531
$443$	2.07005e6	+	3.58543e6i	0.501154	+	0.868024i	0.999999	+	0.00133297i	$0.000424298\pi$
$443$	2.07005e6	+	3.58543e6i	0.501154	+	0.868024i	−0.498845	+	0.866691i	$0.666242\pi$
$444$		0			0
$445$	252651.	−	437605.i	0.0604814	−	0.104757i
$446$	−530267.	−	918449.i	−0.126228	−	0.218634i
$447$		0			0
$448$	−42641.5	+	529298.i	−0.0100378	+	0.124596i
$449$	305966.			0.0716239			0.0358119	−	0.999359i	$-0.488598\pi$
$449$	305966.			0.0716239			0.0358119	+	0.999359i	$0.488598\pi$
$450$		0			0
$451$	−2.00416e6	+	3.47131e6i	−0.463971	+	0.803622i
$452$	−946660.	+	1.63966e6i	−0.217945	+	0.377493i
$453$		0			0
$454$	5.72176e6			1.30284
$455$	164342.	−	78014.4i	0.0372151	−	0.0176663i
$456$		0			0
$457$	−446304.	−	773021.i	−0.0999632	−	0.173141i	0.811706	−	0.584066i	$-0.198539\pi$
$457$	−446304.	−	773021.i	−0.0999632	−	0.173141i	−0.911669	+	0.410925i	$0.865206\pi$
$458$	2.51610e6	−	4.35801e6i	0.560485	−	0.970789i
$459$		0			0
$460$	−179990.	−	311752.i	−0.0396601	−	0.0686933i
$461$	3.01465e6			0.660670			0.330335	−	0.943864i	$-0.392838\pi$
$461$	3.01465e6			0.660670			0.330335	+	0.943864i	$0.392838\pi$
$462$		0			0
$463$	3.45497e6			0.749017			0.374508	−	0.927224i	$-0.377812\pi$
$463$	3.45497e6			0.749017			0.374508	+	0.927224i	$0.377812\pi$
$464$	229344.	+	397235.i	0.0494529	+	0.0856549i
$465$		0			0
$466$	449446.	−	778464.i	0.0958767	−	0.166063i
$467$	1.41888e6	+	2.45757e6i	0.301060	+	0.521451i	0.976376	−	0.216077i	$-0.0693263\pi$
$467$	1.41888e6	+	2.45757e6i	0.301060	+	0.521451i	−0.675316	+	0.737528i	$0.735993\pi$
$468$		0			0
$469$	2.84036e6	+	1.95987e6i	0.596268	+	0.411429i
$470$	1.41702e6			0.295890
$471$		0			0
$472$	1.32431e6	−	2.29378e6i	0.273612	−	0.473910i
$473$	−1.64642e6	+	2.85169e6i	−0.338367	+	0.586069i
$474$		0			0
$475$	−1.18627e6			−0.241240
$476$	−450731.	+	213965.i	−0.0911800	+	0.0432839i
$477$		0			0
$478$	2.56386e6	+	4.44073e6i	0.513244	+	0.888965i
$479$	−2.35221e6	+	4.07415e6i	−0.468422	+	0.811331i	−0.999349	−	0.0360866i	$-0.988511\pi$
$479$	−2.35221e6	+	4.07415e6i	−0.468422	+	0.811331i	0.530926	+	0.847418i	$0.321844\pi$
$480$		0			0
$481$	−374556.	−	648750.i	−0.0738167	−	0.127854i
$482$	−2.30450e6			−0.451813
$483$		0			0
$484$	−814574.			−0.158058
$485$	−754912.	−	1.30755e6i	−0.145728	−	0.252408i
$486$		0			0
$487$	−2.08317e6	+	3.60816e6i	−0.398018	+	0.689387i	−0.993481	−	0.113995i	$-0.963635\pi$
$487$	−2.08317e6	+	3.60816e6i	−0.398018	+	0.689387i	0.595463	+	0.803383i	$0.296969\pi$
$488$	688745.	+	1.19294e6i	0.130921	+	0.226762i
$489$		0			0
$490$	892518.	−	1.09387e6i	0.167929	−	0.205815i
$491$	−1.57876e6			−0.295537			−0.147768	−	0.989022i	$-0.547209\pi$
$491$	−1.57876e6			−0.295537			−0.147768	+	0.989022i	$0.547209\pi$
$492$		0			0
$493$	−215491.	+	373241.i	−0.0399311	+	0.0691627i
$494$	−59067.0	+	102307.i	−0.0108900	+	0.0188620i
$495$		0			0
$496$	−1.45632e6			−0.265798
$497$	−604815.	+	7.50741e6i	−0.109833	+	1.36332i
$498$		0			0
$499$	−778559.	−	1.34850e6i	−0.139972	−	0.242438i	0.787514	−	0.616297i	$-0.211368\pi$
$499$	−778559.	−	1.34850e6i	−0.139972	−	0.242438i	−0.927486	+	0.373859i	$0.878034\pi$
$500$	975912.	−	1.69033e6i	0.174576	−	0.302375i
$501$		0			0
$502$	−1.21808e6	−	2.10978e6i	−0.215733	−	0.373660i
$503$	1.19459e6			0.210523			0.105261	−	0.994445i	$-0.466432\pi$
$503$	1.19459e6			0.210523			0.105261	+	0.994445i	$0.466432\pi$
$504$		0			0
$505$	333175.			0.0581358
$506$	711118.	+	1.23169e6i	0.123471	+	0.213858i
$507$		0			0
$508$	1.96156e6	−	3.39752e6i	0.337242	−	0.584121i
$509$	1.58428e6	+	2.74406e6i	0.271043	+	0.469461i	0.969129	−	0.246553i	$-0.0792980\pi$
$509$	1.58428e6	+	2.74406e6i	0.271043	+	0.469461i	−0.698086	+	0.716014i	$0.745965\pi$
$510$		0			0
$511$	4.26688e6	+	2.94418e6i	0.722867	+	0.498783i
$512$	262144.			0.0441942
$513$		0			0
$514$	2.06134e6	−	3.57034e6i	0.344145	−	0.596077i
$515$	1.68602e6	−	2.92028e6i	0.280121	−	0.485183i
$516$		0			0
$517$	−5.59846e6			−0.921175
$518$	−4.78497e6	−	3.30166e6i	−0.783529	−	0.540640i
$519$		0			0
$520$	−44903.7	−	77775.6i	−0.00728239	−	0.0126135i
$521$	−503289.	+	871722.i	−0.0812312	+	0.140697i	−0.903779	−	0.427999i	$-0.859219\pi$
$521$	−503289.	+	871722.i	−0.0812312	+	0.140697i	0.822548	+	0.568696i	$0.192552\pi$
$522$		0			0
$523$	−4.39493e6	−	7.61224e6i	−0.702583	−	1.21691i	−0.967557	−	0.252653i	$-0.918697\pi$
$523$	−4.39493e6	−	7.61224e6i	−0.702583	−	1.21691i	0.264974	−	0.964255i	$-0.414637\pi$
$524$	−1.11990e6			−0.178177
$525$		0			0
$526$	3.64234e6			0.574006
$527$	−684175.	−	1.18503e6i	−0.107310	−	0.185867i
$528$		0			0
$529$	2.64426e6	−	4.57999e6i	0.410832	−	0.711582i
$530$	222548.	+	385465.i	0.0344140	+	0.0596067i
$531$		0			0
$532$	−73619.7	+	913823.i	−0.0112776	+	0.139985i
$533$	−807055.			−0.123051
$534$		0			0
$535$	−2.15299e6	+	3.72909e6i	−0.325206	+	0.563273i
$536$	851800.	−	1.47536e6i	0.128064	−	0.221813i
$537$		0			0
$538$	−5.68399e6			−0.846638
$539$	−3.52623e6	+	4.32176e6i	−0.522804	+	0.640749i
$540$		0			0
$541$	1.77432e6	+	3.07321e6i	0.260639	+	0.451439i	0.966412	−	0.256999i	$-0.0827336\pi$
$541$	1.77432e6	+	3.07321e6i	0.260639	+	0.451439i	−0.705773	+	0.708438i	$0.749400\pi$
$542$	595059.	−	1.03067e6i	0.0870086	−	0.150703i
$543$		0			0
$544$	123155.	+	213310.i	0.0178424	+	0.0309040i
$545$	−2.36343e6			−0.340841
$546$		0			0
$547$	4.68179e6			0.669027			0.334513	−	0.942391i	$-0.391428\pi$
$547$	4.68179e6			0.669027			0.334513	+	0.942391i	$0.391428\pi$
$548$	−1.50131e6	−	2.60035e6i	−0.213560	−	0.369896i
$549$		0			0
$550$	−1.78150e6	+	3.08564e6i	−0.251118	+	0.434950i
$551$	395957.	+	685818.i	0.0555609	+	0.0962343i
$552$		0			0
$553$	−5.14722e6	+	2.44342e6i	−0.715747	+	0.339771i
$554$	3.39845e6			0.470442
$555$		0			0
$556$	626181.	−	1.08458e6i	0.0859040	−	0.148790i
$557$	2.89754e6	−	5.01868e6i	0.395723	−	0.685412i	−0.597470	−	0.801891i	$-0.703827\pi$
$557$	2.89754e6	−	5.01868e6i	0.395723	−	0.685412i	0.993193	+	0.116479i	$0.0371608\pi$
$558$		0			0
$559$	−662997.			−0.0897392
$560$	−573647.	−	395821.i	−0.0772992	−	0.0533370i
$561$		0			0
$562$	−15361.0	−	26606.0i	−0.00205153	−	0.00355336i
$563$	−3.79852e6	+	6.57922e6i	−0.505060	+	0.874789i	0.494923	+	0.868937i	$0.335196\pi$
$563$	−3.79852e6	+	6.57922e6i	−0.505060	+	0.874789i	−0.999983	+	0.00585257i	$0.998137\pi$
$564$		0			0
$565$	−1.24249e6	−	2.15206e6i	−0.163747	−	0.283617i
$566$	−3.94147e6			−0.517151
$567$		0			0
$568$	3.71817e6			0.483569
$569$	−4.90850e6	−	8.50176e6i	−0.635576	−	1.10085i	−0.986393	−	0.164406i	$-0.947429\pi$
$569$	−4.90850e6	−	8.50176e6i	−0.635576	−	1.10085i	0.350816	−	0.936444i	$-0.385904\pi$
$570$		0			0
$571$	−2.62944e6	+	4.55432e6i	−0.337499	+	0.584566i	−0.983962	−	0.178380i	$-0.942914\pi$
$571$	−2.62944e6	+	4.55432e6i	−0.337499	+	0.584566i	0.646462	+	0.762946i	$0.276248\pi$
$572$	177409.	+	307282.i	0.0226718	+	0.0392687i
$573$		0			0
$574$	−5.65803e6	+	2.68591e6i	−0.716779	+	0.340261i
$575$	−2.87555e6			−0.362703
$576$		0			0
$577$	−4.31734e6	+	7.47785e6i	−0.539855	+	0.935056i	0.459057	+	0.888407i	$0.348187\pi$
$577$	−4.31734e6	+	7.47785e6i	−0.539855	+	0.935056i	−0.998911	+	0.0466486i	$0.985146\pi$
$578$	2.72400e6	−	4.71810e6i	0.339146	−	0.587419i
$579$		0			0
$580$	−602027.			−0.0743098
$581$	233418.	−	2.89736e6i	0.0286876	−	0.356092i
$582$		0			0
$583$	−879261.	−	1.52292e6i	−0.107139	−	0.185570i
$584$	1.27960e6	−	2.21633e6i	0.155254	−	0.268908i
$585$		0			0
$586$	−4.33869e6	−	7.51483e6i	−0.521932	−	0.904014i
$587$	3.32014e6			0.397705			0.198852	−	0.980029i	$-0.436279\pi$
$587$	3.32014e6			0.397705			0.198852	+	0.980029i	$0.436279\pi$
$588$		0			0
$589$	−2.51430e6			−0.298627
$590$	1.73816e6	+	3.01058e6i	0.205570	+	0.356058i
$591$		0			0
$592$	−1.43497e6	+	2.48545e6i	−0.168283	+	0.291474i
$593$	−3.20001e6	−	5.54257e6i	−0.373692	−	0.647254i	0.616438	−	0.787403i	$-0.288575\pi$
$593$	−3.20001e6	−	5.54257e6i	−0.373692	−	0.647254i	−0.990130	+	0.140149i	$0.955242\pi$
$594$		0			0
$595$	52586.4	−	652741.i	0.00608949	−	0.0755873i
$596$	1.47466e6			0.170050
$597$		0			0
$598$	−143180.	+	247995.i	−0.0163730	+	0.0283589i
$599$	3.74931e6	−	6.49400e6i	0.426957	−	0.739512i	−0.569644	−	0.821892i	$-0.692919\pi$
$599$	3.74931e6	−	6.49400e6i	0.426957	−	0.739512i	0.996601	+	0.0823800i	$0.0262521\pi$
$600$		0			0
$601$	−227052.			−0.0256412			−0.0128206	−	0.999918i	$-0.504081\pi$
$601$	−227052.			−0.0256412			−0.0128206	+	0.999918i	$0.504081\pi$
$602$	−4.64808e6	+	2.20648e6i	−0.522736	+	0.248147i
$603$		0			0
$604$	−427328.	−	740153.i	−0.0476616	−	0.0825523i
$605$	534564.	−	925892.i	0.0593761	−	0.102842i
$606$		0			0
$607$	−8.15183e6	−	1.41194e7i	−0.898015	−	1.55541i	−0.830029	−	0.557721i	$-0.811676\pi$
$607$	−8.15183e6	−	1.41194e7i	−0.898015	−	1.55541i	−0.0679862	−	0.997686i	$-0.521657\pi$
$608$	452586.			0.0496527
$609$		0			0
$610$	−1.80796e6			−0.196727
$611$	−563610.	−	976202.i	−0.0610767	−	0.105788i
$612$		0			0
$613$	5.20312e6	−	9.01207e6i	0.559259	−	0.968664i	−0.438300	−	0.898829i	$-0.644419\pi$
$613$	5.20312e6	−	9.01207e6i	0.559259	−	0.968664i	0.997559	−	0.0698355i	$-0.0222474\pi$
$614$	2.78186e6	+	4.81833e6i	0.297793	+	0.515793i
$615$		0			0
$616$	2.26641e6	+	1.56384e6i	0.240651	+	0.166050i
$617$	−4.74140e6			−0.501411			−0.250705	−	0.968063i	$-0.580663\pi$
$617$	−4.74140e6			−0.501411			−0.250705	+	0.968063i	$0.580663\pi$
$618$		0			0
$619$	5.42668e6	−	9.39929e6i	0.569256	−	0.985981i	−0.427383	−	0.904070i	$-0.640565\pi$
$619$	5.42668e6	−	9.39929e6i	0.569256	−	0.985981i	0.996640	−	0.0819103i	$-0.0261021\pi$
$620$	955708.	−	1.65533e6i	0.0998495	−	0.172944i
$621$		0			0
$622$	−8.15177e6			−0.844843
$623$	−2.81804e6	+	1.33775e6i	−0.290889	+	0.138087i
$624$		0			0
$625$	−2.91287e6	−	5.04523e6i	−0.298277	−	0.516632i
$626$	2.33915e6	−	4.05154e6i	0.238574	−	0.413222i
$627$		0			0
$628$	2.24400e6	+	3.88673e6i	0.227052	+	0.393265i
$629$	−2.69659e6			−0.271762
$630$		0			0
$631$	1.58978e7			1.58951			0.794757	−	0.606928i	$-0.207598\pi$
$631$	1.58978e7			1.58951			0.794757	+	0.606928i	$0.207598\pi$
$632$	1.40639e6	+	2.43595e6i	0.140060	+	0.242591i
$633$		0			0
$634$	1.60227e6	−	2.77522e6i	0.158312	−	0.274204i
$635$	2.57455e6	+	4.45925e6i	0.253377	+	0.438861i
$636$		0			0
$637$	−1.10858e6	−	179786.i	−0.108247	−	0.0175553i
$638$	2.37854e6			0.231344
$639$		0			0
$640$	−172032.	+	297968.i	−0.0166020	+	0.0287554i
$641$	5.59139e6	−	9.68457e6i	0.537496	−	0.930970i	−0.461543	−	0.887118i	$-0.652704\pi$
$641$	5.59139e6	−	9.68457e6i	0.537496	−	0.930970i	0.999038	−	0.0438515i	$-0.0139628\pi$
$642$		0			0
$643$	1.55983e6			0.148782			0.0743909	−	0.997229i	$-0.476299\pi$
$643$	1.55983e6			0.148782			0.0743909	+	0.997229i	$0.476299\pi$
$644$	−178456.	+	2.21513e6i	−0.0169557	+	0.210467i
$645$		0			0
$646$	212624.	+	368276.i	0.0200462	+	0.0347210i
$647$	−7.74118e6	+	1.34081e7i	−0.727020	+	1.25924i	0.231117	+	0.972926i	$0.425762\pi$
$647$	−7.74118e6	+	1.34081e7i	−0.727020	+	1.25924i	−0.958137	+	0.286310i	$0.907571\pi$
$648$		0			0
$649$	−6.86726e6	−	1.18944e7i	−0.639988	−	1.10849i
$650$	−717391.			−0.0665997
$651$		0			0
$652$	1.00290e7			0.923925
$653$	307376.	+	532391.i	0.0282090	+	0.0488593i	0.879785	−	0.475371i	$-0.157686\pi$
$653$	307376.	+	532391.i	0.0282090	+	0.0488593i	−0.851576	+	0.524231i	$0.824353\pi$
$654$		0			0
$655$	734934.	−	1.27294e6i	0.0669337	−	0.115933i
$656$	1.54597e6	+	2.67769e6i	0.140262	+	0.242941i
$657$		0			0
$658$	−7.20015e6	−	4.96815e6i	−0.648301	−	0.447332i
$659$	1.32697e7			1.19028			0.595139	−	0.803623i	$-0.297097\pi$
$659$	1.32697e7			1.19028			0.595139	+	0.803623i	$0.297097\pi$
$660$		0			0
$661$	−2.51792e6	+	4.36116e6i	−0.224150	+	0.388238i	−0.956064	−	0.293158i	$-0.905294\pi$
$661$	−2.51792e6	+	4.36116e6i	−0.224150	+	0.388238i	0.731914	+	0.681397i	$0.238627\pi$
$662$	−5.29902e6	+	9.17817e6i	−0.469949	+	0.813975i
$663$		0			0
$664$	−1.43497e6			−0.126306
$665$	−990391.	−	683377.i	−0.0868466	−	0.0599247i
$666$		0			0
$667$	959811.	+	1.66244e6i	0.0835355	+	0.144688i
$668$	2.35198e6	−	4.07375e6i	0.203935	−	0.353226i
$669$		0			0
$670$	1.11799e6	+	1.93641e6i	0.0962166	+	0.166652i
$671$	7.14301e6			0.612457
$672$		0			0
$673$	8.28068e6			0.704739			0.352369	−	0.935861i	$-0.385376\pi$
$673$	8.28068e6			0.704739			0.352369	+	0.935861i	$0.385376\pi$
$674$	6.80112e6	+	1.17799e7i	0.576675	+	0.998830i
$675$		0			0
$676$	2.93462e6	−	5.08292e6i	0.246994	−	0.427805i
$677$	−7.62563e6	−	1.32080e7i	−0.639446	−	1.10755i	−0.985554	−	0.169359i	$-0.945830\pi$
$677$	−7.62563e6	−	1.32080e7i	−0.639446	−	1.10755i	0.346108	−	0.938195i	$-0.387503\pi$
$678$		0			0
$679$	−748480.	+	9.29068e6i	−0.0623025	+	0.773345i
$680$	−323282.			−0.0268107
$681$		0			0
$682$	−3.77588e6	+	6.54002e6i	−0.310855	+	0.538416i
$683$	3.94337e6	−	6.83011e6i	0.323456	−	0.560243i	−0.657743	−	0.753243i	$-0.728489\pi$
$683$	3.94337e6	−	6.83011e6i	0.323456	−	0.560243i	0.981199	+	0.193000i	$0.0618218\pi$
$684$		0			0
$685$	3.94094e6			0.320903
$686$	−8.37025e6	+	2.42896e6i	−0.679091	+	0.197065i
$687$		0			0
$688$	1.27001e6	+	2.19973e6i	0.102291	+	0.177173i
$689$	177035.	−	306633.i	0.0142073	−	0.0246077i
$690$		0			0
$691$	830950.	+	1.43925e6i	0.0662033	+	0.114667i	0.897227	−	0.441569i	$-0.145578\pi$
$691$	830950.	+	1.43925e6i	0.0662033	+	0.114667i	−0.831024	+	0.556237i	$0.812245\pi$
$692$	−1.14270e7			−0.907124
$693$		0			0
$694$	2.47084e6			0.194736
$695$	821863.	+	1.42351e6i	0.0645413	+	0.111789i
$696$		0			0
$697$	−1.45258e6	+	2.51595e6i	−0.113256	+	0.196164i
$698$	−5.41078e6	−	9.37175e6i	−0.420360	−	0.728085i
$699$		0			0
$700$	−5.02942e6	+	2.38751e6i	−0.387947	+	0.184162i
$701$	1.39364e7			1.07117			0.535583	−	0.844483i	$-0.320092\pi$
$701$	1.39364e7			1.07117			0.535583	+	0.844483i	$0.320092\pi$
$702$		0			0
$703$	−2.47745e6	+	4.29107e6i	−0.189068	+	0.327475i
$704$	679677.	−	1.17724e6i	0.0516858	−	0.0895224i
$705$		0			0
$706$	1.27226e7			0.960646
$707$	−1.69293e6	−	1.16813e6i	−0.127377	−	0.0878908i
$708$		0			0
$709$	5.09368e6	+	8.82252e6i	0.380554	+	0.659139i	0.991142	−	0.132810i	$-0.0424000\pi$
$709$	5.09368e6	+	8.82252e6i	0.380554	+	0.659139i	−0.610587	+	0.791949i	$0.709067\pi$
$710$	−2.44005e6	+	4.22629e6i	−0.181657	+	0.314640i
$711$		0			0
$712$	769985.	+	1.33365e6i	0.0569223	+	0.0985923i
$713$	−6.09473e6			−0.448984
$714$		0			0
$715$	−465699.			−0.0340675
$716$	4.64040e6	+	8.03740e6i	0.338277	+	0.585913i
$717$		0			0
$718$	−8.91940e6	+	1.54489e7i	−0.645691	+	1.11837i
$719$	6.95844e6	+	1.20524e7i	0.501983	+	0.869461i	0.999997	+	0.00229184i	$0.000729516\pi$
$719$	6.95844e6	+	1.20524e7i	0.501983	+	0.869461i	−0.498014	+	0.867169i	$0.665937\pi$
$720$		0			0
$721$	−1.88057e7	+	8.92721e6i	−1.34726	+	0.639555i
$722$	−9.12301e6			−0.651321
$723$		0			0
$724$	2.46437e6	−	4.26841e6i	0.174727	−	0.302636i
$725$	−2.40452e6	+	4.16476e6i	−0.169896	+	0.294269i
$726$		0			0
$727$	3.42063e6			0.240033			0.120016	−	0.992772i	$-0.461705\pi$
$727$	3.42063e6			0.240033			0.120016	+	0.992772i	$0.461705\pi$
$728$	−44521.1	+	552629.i	−0.00311342	+	0.0386461i
$729$		0			0
$730$	1.67948e6	+	2.90894e6i	0.116645	+	0.202035i
$731$	−1.19330e6	+	2.06686e6i	−0.0825955	+	0.143060i
$732$		0			0
$733$	5.15810e6	+	8.93410e6i	0.354593	+	0.614173i	0.987048	−	0.160424i	$-0.0512861\pi$
$733$	5.15810e6	+	8.93410e6i	0.354593	+	0.614173i	−0.632455	+	0.774597i	$0.717953\pi$
$734$	1.84499e6			0.126402
$735$		0			0
$736$	1.09708e6			0.0746525
$737$	−4.41703e6	−	7.65052e6i	−0.299545	−	0.518827i
$738$		0			0
$739$	5.19448e6	−	8.99710e6i	0.349889	−	0.606026i	−0.636340	−	0.771409i	$-0.719553\pi$
$739$	5.19448e6	−	8.99710e6i	0.349889	−	0.606026i	0.986229	+	0.165383i	$0.0528859\pi$
$740$	−1.88340e6	−	3.26215e6i	−0.126434	−	0.218990i
$741$		0			0
$742$	220652.	−	2.73889e6i	0.0147129	−	0.182627i
$743$	1.04738e7			0.696035			0.348018	−	0.937488i	$-0.386855\pi$
$743$	1.04738e7			0.696035			0.348018	+	0.937488i	$0.386855\pi$
$744$		0			0
$745$	−967748.	+	1.67619e6i	−0.0638810	+	0.110645i
$746$	−6.83908e6	+	1.18456e7i	−0.449936	+	0.779312i
$747$		0			0
$748$	1.27725e6			0.0834681
$749$	2.40142e7	−	1.13997e7i	1.56410	−	0.742490i
$750$		0			0
$751$	4.70223e6	+	8.14449e6i	0.304231	+	0.526944i	0.977090	−	0.212827i	$-0.0682671\pi$
$751$	4.70223e6	+	8.14449e6i	0.304231	+	0.526944i	−0.672859	+	0.739771i	$0.734934\pi$
$752$	−2.15926e6	+	3.73995e6i	−0.139239	+	0.241169i
$753$		0			0
$754$	239453.	+	414745.i	0.0153388	+	0.0265676i
$755$	1.12174e6			0.0716181
$756$		0			0
$757$	−1.33677e7			−0.847848			−0.423924	−	0.905698i	$-0.639348\pi$
$757$	−1.33677e7			−0.847848			−0.423924	+	0.905698i	$0.639348\pi$
$758$	−32711.2	−	56657.5i	−0.00206787	−	0.00358166i
$759$		0			0
$760$	−297010.	+	514436.i	−0.0186525	+	0.0323071i
$761$	−1.11312e7	−	1.92797e7i	−0.696752	−	1.20681i	−0.969586	−	0.244749i	$-0.921294\pi$
$761$	−1.11312e7	−	1.92797e7i	−0.696752	−	1.20681i	0.272834	−	0.962061i	$-0.412039\pi$
$762$		0			0
$763$	1.20091e7	+	8.28634e6i	0.746789	+	0.515289i
$764$	3.24463e6			0.201109
$765$		0			0
$766$	−6.87287e6	+	1.19042e7i	−0.423220	+	0.733038i
$767$	1.38269e6	−	2.39488e6i	0.0848664	−	0.146993i
$768$		0			0
$769$	9.65833e6			0.588961			0.294480	−	0.955658i	$-0.404853\pi$
$769$	9.65833e6			0.588961			0.294480	+	0.955658i	$0.404853\pi$
$770$	−3.26488e6	+	1.54987e6i	−0.198445	+	0.0942035i
$771$		0			0
$772$	−4.84790e6	−	8.39681e6i	−0.292759	−	0.507074i
$773$	−2.55476e6	+	4.42497e6i	−0.153780	+	0.266355i	−0.932614	−	0.360875i	$-0.882478\pi$
$773$	−2.55476e6	+	4.42497e6i	−0.153780	+	0.266355i	0.778834	+	0.627230i	$0.215811\pi$
$774$		0			0
$775$	−7.63428e6	−	1.32230e7i	−0.456577	−	0.790815i
$776$	4.60137e6			0.274305
$777$		0			0
$778$	1.92421e7			1.13973
$779$	2.66908e6	+	4.62298e6i	0.157586	+	0.272947i
$780$		0			0
$781$	9.64034e6	−	1.66976e7i	0.565542	−	0.979548i
$782$	515407.	+	892711.i	0.0301393	+	0.0522028i
$783$		0			0
$784$	1.52705e6	+	4.02249e6i	0.0887283	+	0.233725i
$785$	−5.89051e6			−0.341176
$786$		0			0
$787$	−1.25796e7	+	2.17885e7i	−0.723984	+	1.25398i	0.235406	+	0.971897i	$0.424358\pi$
$787$	−1.25796e7	+	2.17885e7i	−0.723984	+	1.25398i	−0.959391	+	0.282081i	$0.908975\pi$
$788$	−2486.55	+	4306.84i	−0.000142653	+	0.000247083i
$789$		0			0
$790$	−3.69178e6			−0.210460
$791$	−1.23190e6	+	1.52913e7i	−0.0700060	+	0.868966i
$792$		0			0
$793$	719105.	+	1.24553e6i	0.0406078	+	0.0703347i
$794$	6.59696e6	−	1.14263e7i	0.371358	−	0.643211i
$795$		0			0
$796$	3.27733e6	+	5.67650e6i	0.183332	+	0.317540i
$797$	3.35976e6			0.187354			0.0936768	−	0.995603i	$-0.470138\pi$
$797$	3.35976e6			0.187354			0.0936768	+	0.995603i	$0.470138\pi$
$798$		0			0
$799$	−4.05767e6			−0.224859
$800$	1.37421e6	+	2.38020e6i	0.0759150	+	0.131489i
$801$		0			0
$802$	272328.	−	471686.i	0.0149505	−	0.0258951i
$803$	−6.63540e6	−	1.14929e7i	−0.363144	−	0.628983i
$804$		0			0
$805$	−2.40073e6	−	1.65652e6i	−0.130573	−	0.0900964i
$806$	−1.52051e6			−0.0824426
$807$		0			0
$808$	−507695.	+	879353.i	−0.0273574	+	0.0473844i
$809$	−1.48525e7	+	2.57253e7i	−0.797862	+	1.38194i	0.123143	+	0.992389i	$0.460703\pi$
$809$	−1.48525e7	+	2.57253e7i	−0.797862	+	1.38194i	−0.921006	+	0.389549i	$0.872631\pi$
$810$		0			0
$811$	1.26386e7			0.674757			0.337379	−	0.941369i	$-0.390460\pi$
$811$	1.26386e7			0.674757			0.337379	+	0.941369i	$0.390460\pi$
$812$	3.05902e6	+	2.11075e6i	0.162814	+	0.112343i
$813$		0			0
$814$	7.44109e6	+	1.28883e7i	0.393619	+	0.681767i
$815$	−6.58150e6	+	1.13995e7i	−0.347081	+	0.601162i
$816$		0			0
$817$	2.19265e6	+	3.79779e6i	0.114925	+	0.199056i
$818$	1.09948e7			0.574517
$819$		0			0
$820$	−4.05816e6			−0.210763
$821$	−1.30880e7	−	2.26691e7i	−0.677666	−	1.17375i	−0.975682	−	0.219190i	$-0.929658\pi$
$821$	−1.30880e7	−	2.26691e7i	−0.677666	−	1.17375i	0.298017	−	0.954561i	$-0.403675\pi$
$822$		0			0
$823$	3.64922e6	−	6.32063e6i	0.187802	−	0.325283i	−0.756715	−	0.653745i	$-0.773197\pi$
$823$	3.64922e6	−	6.32063e6i	0.187802	−	0.325283i	0.944517	+	0.328462i	$0.106530\pi$
$824$	5.13835e6	+	8.89989e6i	0.263637	+	0.456632i
$825$		0			0
$826$	1.72335e6	−	2.13915e7i	0.0878867	−	1.09091i
$827$	−1.18681e7			−0.603418			−0.301709	−	0.953400i	$-0.597557\pi$
$827$	−1.18681e7			−0.603418			−0.301709	+	0.953400i	$0.597557\pi$
$828$		0			0
$829$	5.21689e6	−	9.03593e6i	0.263649	−	0.456653i	−0.703560	−	0.710636i	$-0.748407\pi$
$829$	5.21689e6	−	9.03593e6i	0.263649	−	0.456653i	0.967209	+	0.253983i	$0.0817407\pi$
$830$	941698.	−	1.63107e6i	0.0474479	−	0.0821821i
$831$		0			0
$832$	273699.			0.0137077
$833$	−2.55576e6	+	3.13234e6i	−0.127617	+	0.156407i
$834$		0			0
$835$	3.08697e6	+	5.34679e6i	0.153220	+	0.265385i
$836$	1.17345e6	−	2.03247e6i	0.0580696	−	0.100579i
$837$		0			0
$838$	−3.74438e6	−	6.48546e6i	−0.184192	−	0.319029i
$839$	2.91444e7			1.42939			0.714695	−	0.699436i	$-0.246565\pi$
$839$	2.91444e7			1.42939			0.714695	+	0.699436i	$0.246565\pi$
$840$		0			0
$841$	−1.73008e7			−0.843482
$842$	−1.31045e6	−	2.26977e6i	−0.0637001	−	0.110332i
$843$		0			0
$844$	3.53071e6	−	6.11537e6i	0.170610	−	0.295506i
$845$	3.85169e6	+	6.67133e6i	0.185571	+	0.321418i
$846$		0			0
$847$	−5.96247e6	+	2.83043e6i	−0.285573	+	0.135564i
$848$	−1.35648e6			−0.0647777
$849$		0			0
$850$	−1.29120e6	+	2.23643e6i	−0.0612981	+	0.106171i
$851$	−6.00541e6	+	1.04017e7i	−0.284262	+	0.492356i
$852$		0			0
$853$	2.63032e7			1.23776			0.618879	−	0.785487i	$-0.287587\pi$
$853$	2.63032e7			1.23776			0.618879	+	0.785487i	$0.287587\pi$
$854$	9.18660e6	+	6.33881e6i	0.431032	+	0.297415i
$855$		0			0
$856$	−6.56150e6	−	1.13649e7i	−0.306069	−	0.530127i
$857$	1.71799e6	−	2.97565e6i	0.0799040	−	0.138398i	−0.823304	−	0.567600i	$-0.807872\pi$
$857$	1.71799e6	−	2.97565e6i	0.0799040	−	0.138398i	0.903208	+	0.429202i	$0.141205\pi$
$858$		0			0
$859$	−5.45696e6	−	9.45173e6i	−0.252329	−	0.437047i	0.711837	−	0.702344i	$-0.247863\pi$
$859$	−5.45696e6	−	9.45173e6i	−0.252329	−	0.437047i	−0.964167	+	0.265297i	$0.914530\pi$
$860$	−3.33378e6			−0.153706
$861$		0			0
$862$	−2.55076e7			−1.16923
$863$	1.09529e7	+	1.89710e7i	0.500614	+	0.867088i	1.00000		0.000708873i	$0.000225641\pi$
$863$	1.09529e7	+	1.89710e7i	0.500614	+	0.867088i	−0.499386	+	0.866380i	$0.666441\pi$
$864$		0			0
$865$	7.49896e6	−	1.29886e7i	0.340770	−	0.590230i
$866$	−9.60005e6	−	1.66278e7i	−0.434990	−	0.753424i
$867$		0			0
$868$	−1.06599e7	+	5.06032e6i	−0.480233	+	0.227970i
$869$	1.45858e7			0.655210
$870$		0			0
$871$	889347.	−	1.54039e6i	0.0397215	−	0.0687997i
$872$	3.60142e6	−	6.23784e6i	0.160392	−	0.277807i
$873$		0			0
$874$	1.89409e6			0.0838729
$875$	1.26997e6	−	1.57638e7i	0.0560755	−	0.696051i
$876$		0			0
$877$	−3.23464e6	−	5.60256e6i	−0.142012	−	0.245973i	0.786242	−	0.617919i	$-0.212024\pi$
$877$	−3.23464e6	−	5.60256e6i	−0.142012	−	0.245973i	−0.928254	+	0.371946i	$0.878691\pi$
$878$	1.11902e7	−	1.93819e7i	0.489891	−	0.848516i
$879$		0			0
$880$	892076.	+	1.54512e6i	0.0388325	+	0.0672599i
$881$	−2.03983e7			−0.885430			−0.442715	−	0.896662i	$-0.645985\pi$
$881$	−2.03983e7			−0.885430			−0.442715	+	0.896662i	$0.645985\pi$
$882$		0			0
$883$	1.62381e7			0.700862			0.350431	−	0.936589i	$-0.386035\pi$
$883$	1.62381e7			0.700862			0.350431	+	0.936589i	$0.386035\pi$
$884$	128583.	+	222713.i	0.00553419	+	0.00958550i
$885$		0			0
$886$	8.28019e6	−	1.43417e7i	0.354369	−	0.613786i
$887$	−3.39513e6	−	5.88055e6i	−0.144893	−	0.250962i	0.784440	−	0.620205i	$-0.212951\pi$
$887$	−3.39513e6	−	5.88055e6i	−0.144893	−	0.250962i	−0.929333	+	0.369242i	$0.879617\pi$
$888$		0			0
$889$	2.55261e6	−	3.16849e7i	0.108325	−	1.34461i
$890$	−2.02121e6			−0.0855336
$891$		0			0
$892$	−2.12107e6	+	3.67380e6i	−0.0892570	+	0.154598i
$893$	−3.72792e6	+	6.45696e6i	−0.156437	+	0.270956i
$894$		0			0
$895$	−1.21810e7			−0.508308
$896$	1.91883e6	−	910881.i	0.0798482	−	0.0379046i
$897$		0			0
$898$	−611933.	−	1.05990e6i	−0.0253229	−	0.0438605i
$899$	−5.09639e6	+	8.82721e6i	−0.210312	+	0.364271i
$900$		0			0
$901$	−637275.	−	1.10379e6i	−0.0261526	−	0.0452976i
$902$	1.60333e7			0.656155
$903$		0			0
$904$	7.57328e6			0.308221
$905$	3.23449e6	+	5.60229e6i	0.131276	+	0.227376i
$906$		0			0
$907$	−1.71071e7	+	2.96304e7i	−0.690492	+	1.19597i	0.281185	+	0.959654i	$0.409273\pi$
$907$	−1.71071e7	+	2.96304e7i	−0.690492	+	1.19597i	−0.971677	+	0.236313i	$0.924061\pi$
$908$	−1.14435e7	−	1.98207e7i	−0.460622	−	0.797821i
$909$		0			0
$910$	−598933.	−	413268.i	−0.0239759	−	0.0165435i
$911$	4.47390e6			0.178604			0.0893019	−	0.996005i	$-0.471536\pi$
$911$	4.47390e6			0.178604			0.0893019	+	0.996005i	$0.471536\pi$
$912$		0			0
$913$	−3.72053e6	+	6.44415e6i	−0.147716	+	0.255852i
$914$	−1.78522e6	+	3.09208e6i	−0.0706846	+	0.122429i
$915$		0			0
$916$	−2.01288e7			−0.792646
$917$	−8.19737e6	+	3.89136e6i	−0.321922	+	0.152819i
$918$		0			0
$919$	268950.	+	465835.i	0.0105047	+	0.0181946i	0.871230	−	0.490875i	$-0.163323\pi$
$919$	268950.	+	465835.i	0.0105047	+	0.0181946i	−0.860725	+	0.509070i	$0.829990\pi$
$920$	−719960.	+	1.24701e6i	−0.0280439	+	0.0485735i
$921$		0			0
$922$	−6.02930e6	−	1.04431e7i	−0.233582	−	0.404576i
$923$	3.88207e6			0.149989
$924$		0			0
$925$	−3.00896e7			−1.15628
$926$	−6.90993e6	−	1.19684e7i	−0.264817	−	0.458677i
$927$		0			0
$928$	917375.	−	1.58894e6i	0.0349685	−	0.0605672i
$929$	−8.86206e6	−	1.53495e7i	−0.336896	−	0.583521i	0.646951	−	0.762531i	$-0.276044\pi$
$929$	−8.86206e6	−	1.53495e7i	−0.336896	−	0.583521i	−0.983847	+	0.179011i	$0.942710\pi$
$930$		0			0
$931$	2.63642e6	+	6.94475e6i	0.0996873	+	0.262593i
$932$	−3.59557e6			−0.135590
$933$		0			0
$934$	5.67551e6	−	9.83028e6i	0.212881	−	0.368722i
$935$	−838192.	+	1.45179e6i	−0.0313556	+	0.0543094i
$936$		0			0
$937$	−3.32444e7			−1.23700			−0.618499	−	0.785786i	$-0.712259\pi$
$937$	−3.32444e7			−1.23700			−0.618499	+	0.785786i	$0.712259\pi$
$938$	1.10846e6	−	1.37590e7i	0.0411352	−	0.510600i
$939$		0			0
$940$	−2.83403e6	−	4.90869e6i	−0.104613	−	0.181195i
$941$	1.39240e7	−	2.41170e7i	0.512613	−	0.887871i	−0.487281	−	0.873245i	$-0.662011\pi$
$941$	1.39240e7	−	2.41170e7i	0.512613	−	0.887871i	0.999893	−	0.0146254i	$-0.00465559\pi$
$942$		0			0
$943$	6.46991e6	+	1.12062e7i	0.236930	+	0.410374i
$944$	−1.05945e7			−0.386946
$945$		0			0
$946$	1.31714e7			0.478523
$947$	−1.15284e7	−	1.99678e7i	−0.417729	−	0.723529i	0.577981	−	0.816050i	$-0.303841\pi$
$947$	−1.15284e7	−	1.99678e7i	−0.417729	−	0.723529i	−0.995711	+	0.0925215i	$0.970507\pi$
$948$		0			0
$949$	1.33600e6	−	2.31403e6i	0.0481551	−	0.0834071i
$950$	2.37254e6	+	4.10936e6i	0.0852914	+	0.147729i
$951$		0			0
$952$	1.64266e6	+	1.13345e6i	0.0587429	+	0.0405330i
$953$	−2.85536e7			−1.01843			−0.509213	−	0.860641i	$-0.670063\pi$
$953$	−2.85536e7			−1.01843			−0.509213	+	0.860641i	$0.670063\pi$
$954$		0			0
$955$	−2.12929e6	+	3.68804e6i	−0.0755486	+	0.130854i
$956$	1.02554e7	−	1.77629e7i	0.362919	−	0.628593i
$957$		0			0
$958$	1.88177e7			0.662449
$959$	−2.00247e7	−	1.38172e7i	−0.703105	−	0.485147i
$960$		0			0
$961$	−1.86629e6	−	3.23251e6i	−0.0651884	−	0.112910i
$962$	−1.49823e6	+	2.59500e6i	−0.0521963	+	0.0904066i
$963$		0			0
$964$	4.60900e6	+	7.98302e6i	0.159740	+	0.276678i
$965$	1.27257e7			0.439911
$966$		0			0
$967$	−3.45310e7			−1.18753			−0.593763	−	0.804640i	$-0.702358\pi$
$967$	−3.45310e7			−1.18753			−0.593763	+	0.804640i	$0.702358\pi$
$968$	1.62915e6	+	2.82177e6i	0.0558820	+	0.0967905i
$969$		0			0
$970$	−3.01965e6	+	5.23019e6i	−0.103045	+	0.178479i
$971$	6.24666e6	+	1.08195e7i	0.212618	+	0.368265i	0.952533	−	0.304435i	$-0.0984677\pi$
$971$	6.24666e6	+	1.08195e7i	0.212618	+	0.368265i	−0.739915	+	0.672700i	$0.765134\pi$
$972$		0			0
$973$	814860.	−	1.01146e7i	0.0275931	−	0.342506i
$974$	1.66654e7			0.562883
$975$		0			0
$976$	2.75498e6	−	4.77177e6i	0.0925751	−	0.160345i
$977$	−1.94720e7	+	3.37265e7i	−0.652641	+	1.13041i	0.329839	+	0.944037i	$0.393006\pi$
$977$	−1.94720e7	+	3.37265e7i	−0.652641	+	1.13041i	−0.982480	+	0.186370i	$0.940328\pi$
$978$		0			0
$979$	7.98556e6			0.266286
$980$	−5.57432e6	−	904029.i	−0.185407	−	0.0300689i
$981$		0			0
$982$	3.15751e6	+	5.46897e6i	0.104488	+	0.180979i
$983$	139950.	−	242401.i	0.00461945	−	0.00800112i	−0.863706	−	0.503995i	$-0.831863\pi$
$983$	139950.	−	242401.i	0.00461945	−	0.00800112i	0.868326	+	0.495994i	$0.165196\pi$
$984$		0			0
$985$	−3263.60	−	5652.72i	−0.000107178	−	0.000185638i
$986$	1.72393e6			0.0564711
$987$		0			0
$988$	472536.			0.0154008
$989$	5.31505e6	+	9.20593e6i	0.172789	+	0.299280i
$990$		0			0
$991$	−9.81715e6	+	1.70038e7i	−0.317542	+	0.549999i	−0.979975	−	0.199122i	$-0.936191\pi$
$991$	−9.81715e6	+	1.70038e7i	−0.317542	+	0.549999i	0.662432	+	0.749122i	$0.269524\pi$
$992$	2.91263e6	+	5.04483e6i	0.0939738	+	0.162767i
$993$		0			0
$994$	2.72161e7	−	1.29197e7i	0.873694	−	0.414749i
$995$	−8.60299e6			−0.275481
$996$		0			0
$997$	1.86687e7	−	3.23351e7i	0.594807	−	1.03024i	−0.398767	−	0.917052i	$-0.630562\pi$
$997$	1.86687e7	−	3.23351e7i	0.594807	−	1.03024i	0.993574	−	0.113184i	$-0.0361050\pi$
$998$	−3.11423e6	+	5.39401e6i	−0.0989749	+	0.171430i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.g.e.109.1		4
3.2	odd	2		14.6.c.b.11.2	yes	4
7.2	even	3	inner	126.6.g.e.37.1		4
7.3	odd	6		882.6.a.bl.1.1		2
7.4	even	3		882.6.a.bt.1.1		2
12.11	even	2		112.6.i.b.81.1		4
21.2	odd	6		14.6.c.b.9.2	✓	4
21.5	even	6		98.6.c.f.79.1		4
21.11	odd	6		98.6.a.c.1.1		2
21.17	even	6		98.6.a.f.1.2		2
21.20	even	2		98.6.c.f.67.1		4
84.11	even	6		784.6.a.bc.1.2		2
84.23	even	6		112.6.i.b.65.1		4
84.59	odd	6		784.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.2	✓	4	21.2	odd	6
14.6.c.b.11.2	yes	4	3.2	odd	2
98.6.a.c.1.1		2	21.11	odd	6
98.6.a.f.1.2		2	21.17	even	6
98.6.c.f.67.1		4	21.20	even	2
98.6.c.f.79.1		4	21.5	even	6
112.6.i.b.65.1		4	84.23	even	6
112.6.i.b.81.1		4	12.11	even	2
126.6.g.e.37.1		4	7.2	even	3	inner
126.6.g.e.109.1		4	1.1	even	1	trivial
784.6.a.r.1.1		2	84.59	odd	6
784.6.a.bc.1.2		2	84.11	even	6
882.6.a.bl.1.1		2	7.3	odd	6
882.6.a.bt.1.1		2	7.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 - 18.1865i) q^{5} +(-10.4105 + 129.223i) q^{7} +64.0000 q^{8} +(-42.0000 + 72.7461i) q^{10} +(165.937 - 287.411i) q^{11} +66.8211 q^{13} +(468.463 - 222.383i) q^{14} +(-128.000 - 221.703i) q^{16} +(120.268 - 208.311i) q^{17} +(-220.989 - 382.765i) q^{19} +336.000 q^{20} -1327.49 q^{22} +(-535.684 - 927.832i) q^{23} +(1342.00 - 2324.41i) q^{25} +(-133.642 - 231.475i) q^{26} +(-1707.28 - 1178.04i) q^{28} -1791.75 q^{29} +(2844.37 - 4926.59i) q^{31} +(-512.000 + 886.810i) q^{32} -962.147 q^{34} +(2459.43 - 1167.51i) q^{35} +(-5605.36 - 9708.77i) q^{37} +(-883.958 + 1531.06i) q^{38} +(-672.000 - 1163.94i) q^{40} -12077.9 q^{41} -9921.98 q^{43} +(2654.99 + 4598.58i) q^{44} +(-2142.74 + 3711.33i) q^{46} +(-8434.62 - 14609.2i) q^{47} +(-16590.2 - 2690.56i) q^{49} -10736.0 q^{50} +(-534.568 + 925.900i) q^{52} +(2649.38 - 4588.87i) q^{53} -6969.35 q^{55} +(-666.274 + 8270.28i) q^{56} +(3583.49 + 6206.79i) q^{58} +(20692.4 - 35840.3i) q^{59} +(10761.6 + 18639.7i) q^{61} -22754.9 q^{62} +4096.00 q^{64} +(-701.621 - 1215.24i) q^{65} +(13309.4 - 23052.5i) q^{67} +(1924.29 + 3332.98i) q^{68} +(-8963.24 - 6184.70i) q^{70} +58096.5 q^{71} +(19993.8 - 34630.2i) q^{73} +(-22421.5 + 38835.1i) q^{74} +7071.66 q^{76} +(35412.7 + 24435.0i) q^{77} +(21974.9 + 38061.7i) q^{79} +(-2688.00 + 4655.75i) q^{80} +(24155.7 + 41838.9i) q^{82} -22421.4 q^{83} -5051.27 q^{85} +(19844.0 + 34370.7i) q^{86} +(10620.0 - 18394.3i) q^{88} +(12031.0 + 20838.3i) q^{89} +(-695.642 + 8634.83i) q^{91} +17141.9 q^{92} +(-33738.5 + 58436.8i) q^{94} +(-4640.78 + 8038.06i) q^{95} +71896.4 q^{97} +(23860.1 + 62851.4i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 32 q^{4} - 42 q^{5} + 232 q^{7} + 256 q^{8} - 168 q^{10} - 294 q^{11} - 280 q^{13} + 232 q^{14} - 512 q^{16} + 1302 q^{17} + 1442 q^{19} + 1344 q^{20} + 2352 q^{22} + 2646 q^{23} + 5368 q^{25}+ \cdots - 63272 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 - 32 * q^4 - 42 * q^5 + 232 * q^7 + 256 * q^8 - 168 * q^10 - 294 * q^11 - 280 * q^13 + 232 * q^14 - 512 * q^16 + 1302 * q^17 + 1442 * q^19 + 1344 * q^20 + 2352 * q^22 + 2646 * q^23 + 5368 * q^25 + 560 * q^26 - 4640 * q^28 - 3336 * q^29 + 14798 * q^31 - 2048 * q^32 - 10416 * q^34 + 1218 * q^35 - 5182 * q^37 + 5768 * q^38 - 2688 * q^40 + 10248 * q^41 - 9040 * q^43 - 4704 * q^44 + 10584 * q^46 - 14994 * q^47 - 2876 * q^49 - 42944 * q^50 + 2240 * q^52 + 24006 * q^53 + 12348 * q^55 + 14848 * q^56 + 6672 * q^58 + 38850 * q^59 + 23618 * q^61 - 118384 * q^62 + 16384 * q^64 + 2940 * q^65 - 32002 * q^67 + 20832 * q^68 - 24360 * q^70 + 178752 * q^71 + 47138 * q^73 - 20728 * q^74 - 46144 * q^76 + 22890 * q^77 + 40970 * q^79 - 10752 * q^80 - 20496 * q^82 - 136752 * q^83 - 54684 * q^85 + 18080 * q^86 - 18816 * q^88 + 123102 * q^89 - 53680 * q^91 - 84672 * q^92 - 59976 * q^94 + 30282 * q^95 - 87304 * q^97 - 63272 * q^98

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