LMFDB - Embedded newform 60.6.i.a.53.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(17,60)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("60.17");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.i (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$9.62302918878$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} + \cdots + 48\!\cdots\!00 $ x^20 - 2x^19 + 2x^18 - 382x^17 + 117610x^16 - 661518x^15 + 1160778x^14 - 161859774x^13 + 12872525097x^12 - 60314430972x^11 + 238998177000x^10 - 23647959230160x^9 + 244333990200544x^8 + 369253960052800x^7 + 17753999958800000x^6 - 195554595639827200x^5 + 1195920778734496000x^4 - 5202205084237824000x^3 + 36839424090977280000x^2 - 600638984600463360000*x + 4896482487496796160000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{14}\cdot 5^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.8
Root			$-16.3862 - 9.05032i$ of defining polynomial
Character	$\chi$	$=$	60.53
Dual form			60.6.i.a.17.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.91146 - 12.0317i) q^{3} +(26.7203 - 49.1022i) q^{5} +(1.70680 - 1.70680i) q^{7} +(-46.5259 - 238.504i) q^{9} +O(q^{10})$	\(q+(9.91146 - 12.0317i) q^{3} +(26.7203 - 49.1022i) q^{5} +(1.70680 - 1.70680i) q^{7} +(-46.5259 - 238.504i) q^{9} +159.189i q^{11} +(-348.051 - 348.051i) q^{13} +(-325.947 - 808.167i) q^{15} +(286.676 + 286.676i) q^{17} -2120.89i q^{19} +(-3.61890 - 37.4527i) q^{21} +(238.492 - 238.492i) q^{23} +(-1697.05 - 2624.05i) q^{25} +(-3330.76 - 1804.14i) q^{27} +5308.79 q^{29} +4749.30 q^{31} +(1915.32 + 1577.79i) q^{33} +(-38.2013 - 129.414i) q^{35} +(-7540.18 + 7540.18i) q^{37} +(-7637.36 + 737.968i) q^{39} +15461.0i q^{41} +(10753.2 + 10753.2i) q^{43} +(-12954.3 - 4088.40i) q^{45} +(6192.52 + 6192.52i) q^{47} +16801.2i q^{49} +(6290.58 - 607.835i) q^{51} +(16971.1 - 16971.1i) q^{53} +(7816.52 + 4253.58i) q^{55} +(-25518.0 - 21021.1i) q^{57} +42879.1 q^{59} -24298.8 q^{61} +(-486.489 - 327.669i) q^{63} +(-26390.1 + 7790.03i) q^{65} +(38389.1 - 38389.1i) q^{67} +(-505.672 - 5233.29i) q^{69} +72926.9i q^{71} +(-57492.8 - 57492.8i) q^{73} +(-48392.2 - 5589.76i) q^{75} +(271.703 + 271.703i) q^{77} -37982.7i q^{79} +(-54719.7 + 22193.3i) q^{81} +(60200.5 - 60200.5i) q^{83} +(21736.5 - 6416.33i) q^{85} +(52617.9 - 63874.0i) q^{87} +58847.5 q^{89} -1188.11 q^{91} +(47072.5 - 57142.4i) q^{93} +(-104140. - 56670.9i) q^{95} +(-46768.7 + 46768.7i) q^{97} +(37967.2 - 7406.40i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) $ 20 * q + 2 * q^3 + 76 * q^7	$ 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 76 * q^7 + 1068 * q^13 - 130 * q^15 + 2180 * q^21 + 4060 * q^25 + 1454 * q^27 - 4720 * q^31 - 460 * q^33 - 612 * q^37 - 24012 * q^43 - 18860 * q^45 - 31700 * q^51 + 19200 * q^55 + 33476 * q^57 + 59880 * q^61 + 67208 * q^63 - 80804 * q^67 - 56956 * q^73 - 102470 * q^75 - 9980 * q^81 + 239260 * q^85 + 71540 * q^87 + 218520 * q^91 + 307928 * q^93 - 151164 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	9.91146	−	12.0317i	0.635821	−	0.771837i
$3$	9.91146	−	12.0317i	0.635821	−	0.771837i
$4$		0			0
$5$	26.7203	−	49.1022i	0.477988	−	0.878366i
$5$	26.7203	−	49.1022i	0.477988	−	0.878366i
$6$		0			0
$7$	1.70680	−	1.70680i	0.0131655	−	0.0131655i	−0.700493	−	0.713659i	$-0.747037\pi$
$7$	1.70680	−	1.70680i	0.0131655	−	0.0131655i	0.713659	+	0.700493i	$0.247037\pi$
$8$		0			0
$9$	−46.5259	−	238.504i	−0.191465	−	0.981500i
$10$		0			0
$11$			159.189i			0.396671i	0.980134	+	0.198336i	$0.0635536\pi$
$11$			159.189i			0.396671i	−0.980134	+	0.198336i	$0.936446\pi$
$12$		0			0
$13$	−348.051	−	348.051i	−0.571196	−	0.571196i	0.361267	−	0.932462i	$-0.382344\pi$
$13$	−348.051	−	348.051i	−0.571196	−	0.571196i	−0.932462	+	0.361267i	$0.882344\pi$
$14$		0			0
$15$	−325.947	−	808.167i	−0.374041	−	0.927412i
$16$		0			0
$17$	286.676	+	286.676i	0.240585	+	0.240585i	0.817092	−	0.576507i	$-0.195585\pi$
$17$	286.676	+	286.676i	0.240585	+	0.240585i	−0.576507	+	0.817092i	$0.695585\pi$
$18$		0			0
$19$		−	2120.89i		−	1.34783i	−0.738811	−	0.673913i	$-0.764612\pi$
$19$		−	2120.89i		−	1.34783i	0.738811	−	0.673913i	$-0.235388\pi$
$20$		0			0
$21$	−3.61890	−	37.4527i	−0.00179072	−	0.0185325i
$22$		0			0
$23$	238.492	−	238.492i	0.0940058	−	0.0940058i	−0.658540	−	0.752546i	$-0.728826\pi$
$23$	238.492	−	238.492i	0.0940058	−	0.0940058i	0.752546	+	0.658540i	$0.228826\pi$
$24$		0			0
$25$	−1697.05	−	2624.05i	−0.543055	−	0.839697i
$26$		0			0
$27$	−3330.76	−	1804.14i	−0.879295	−	0.476278i
$28$		0			0
$29$	5308.79			1.17220			0.586098	−	0.810240i	$-0.300663\pi$
$29$	5308.79			1.17220			0.586098	+	0.810240i	$0.300663\pi$
$30$		0			0
$31$	4749.30			0.887617			0.443809	−	0.896122i	$-0.353627\pi$
$31$	4749.30			0.887617			0.443809	+	0.896122i	$0.353627\pi$
$32$		0			0
$33$	1915.32	+	1577.79i	0.306166	+	0.252212i
$34$		0			0
$35$	−38.2013	−	129.414i	−0.00527118	−	0.0178571i
$36$		0			0
$37$	−7540.18	+	7540.18i	−0.905477	+	0.905477i	−0.995903	−	0.0904264i	$-0.971177\pi$
$37$	−7540.18	+	7540.18i	−0.905477	+	0.905477i	0.0904264	+	0.995903i	$0.471177\pi$
$38$		0			0
$39$	−7637.36	+	737.968i	−0.804048	+	0.0776920i
$40$		0			0
$41$			15461.0i			1.43641i	0.695831	+	0.718205i	$0.255036\pi$
$41$			15461.0i			1.43641i	−0.695831	+	0.718205i	$0.744964\pi$
$42$		0			0
$43$	10753.2	+	10753.2i	0.886884	+	0.886884i	0.994223	−	0.107339i	$-0.0342329\pi$
$43$	10753.2	+	10753.2i	0.886884	+	0.886884i	−0.107339	+	0.994223i	$0.534233\pi$
$44$		0			0
$45$	−12954.3	−	4088.40i	−0.953634	−	0.300969i
$46$		0			0
$47$	6192.52	+	6192.52i	0.408905	+	0.408905i	0.881357	−	0.472451i	$-0.156631\pi$
$47$	6192.52	+	6192.52i	0.408905	+	0.408905i	−0.472451	+	0.881357i	$0.656631\pi$
$48$		0			0
$49$			16801.2i			0.999653i
$50$		0			0
$51$	6290.58	−	607.835i	0.338661	−	0.0327235i
$52$		0			0
$53$	16971.1	−	16971.1i	0.829889	−	0.829889i	−0.157612	−	0.987501i	$-0.550380\pi$
$53$	16971.1	−	16971.1i	0.829889	−	0.829889i	0.987501	+	0.157612i	$0.0503796\pi$
$54$		0			0
$55$	7816.52	+	4253.58i	0.348423	+	0.189604i
$56$		0			0
$57$	−25518.0	−	21021.1i	−1.04030	−	0.856975i
$58$		0			0
$59$	42879.1			1.60367			0.801836	−	0.597544i	$-0.203857\pi$
$59$	42879.1			1.60367			0.801836	+	0.597544i	$0.203857\pi$
$60$		0			0
$61$	−24298.8			−0.836105			−0.418052	−	0.908423i	$-0.637287\pi$
$61$	−24298.8			−0.836105			−0.418052	+	0.908423i	$0.637287\pi$
$62$		0			0
$63$	−486.489	−	327.669i	−0.0154427	−	0.0104012i
$64$		0			0
$65$	−26390.1	+	7790.03i	−0.774743	+	0.228694i
$66$		0			0
$67$	38389.1	−	38389.1i	1.04477	−	1.04477i	0.0458212	−	0.998950i	$-0.485410\pi$
$67$	38389.1	−	38389.1i	1.04477	−	1.04477i	0.998950	−	0.0458212i	$-0.0145904\pi$
$68$		0			0
$69$	−505.672	−	5233.29i	−0.0127863	−	0.132328i
$70$		0			0
$71$			72926.9i			1.71689i	0.512908	+	0.858444i	$0.328568\pi$
$71$			72926.9i			1.71689i	−0.512908	+	0.858444i	$0.671432\pi$
$72$		0			0
$73$	−57492.8	−	57492.8i	−1.26272	−	1.26272i	−0.949770	−	0.312948i	$-0.898683\pi$
$73$	−57492.8	−	57492.8i	−1.26272	−	1.26272i	−0.312948	−	0.949770i	$-0.601317\pi$
$74$		0			0
$75$	−48392.2	−	5589.76i	−0.993395	−	0.114747i
$76$		0			0
$77$	271.703	+	271.703i	0.00522238	+	0.00522238i
$78$		0			0
$79$		−	37982.7i		−	0.684729i	−0.939567	−	0.342364i	$-0.888772\pi$
$79$		−	37982.7i		−	0.684729i	0.939567	−	0.342364i	$-0.111228\pi$
$80$		0			0
$81$	−54719.7	+	22193.3i	−0.926683	+	0.375845i
$82$		0			0
$83$	60200.5	−	60200.5i	0.959190	−	0.959190i	−0.0400092	−	0.999199i	$-0.512739\pi$
$83$	60200.5	−	60200.5i	0.959190	−	0.959190i	0.999199	+	0.0400092i	$0.0127387\pi$
$84$		0			0
$85$	21736.5	−	6416.33i	0.326318	−	0.0963250i
$86$		0			0
$87$	52617.9	−	63874.0i	0.745307	−	0.904745i
$88$		0			0
$89$	58847.5			0.787504			0.393752	−	0.919217i	$-0.371177\pi$
$89$	58847.5			0.787504			0.393752	+	0.919217i	$0.371177\pi$
$90$		0			0
$91$	−1188.11			−0.0150401
$92$		0			0
$93$	47072.5	−	57142.4i	0.564365	−	0.685096i
$94$		0			0
$95$	−104140.	−	56670.9i	−1.18389	−	0.644245i
$96$		0			0
$97$	−46768.7	+	46768.7i	−0.504691	+	0.504691i	−0.912892	−	0.408201i	$-0.866156\pi$
$97$	−46768.7	+	46768.7i	−0.504691	+	0.504691i	0.408201	+	0.912892i	$0.366156\pi$
$98$		0			0
$99$	37967.2	−	7406.40i	0.389333	−	0.0759485i
$100$		0			0
$101$			99697.8i			0.972483i	0.873825	+	0.486241i	$0.161632\pi$
$101$			99697.8i			0.972483i	−0.873825	+	0.486241i	$0.838368\pi$
$102$		0			0
$103$	61353.7	+	61353.7i	0.569834	+	0.569834i	0.932082	−	0.362248i	$-0.117991\pi$
$103$	61353.7	+	61353.7i	0.569834	+	0.569834i	−0.362248	+	0.932082i	$0.617991\pi$
$104$		0			0
$105$	−1935.71	−	823.052i	−0.0171343	−	0.00728541i
$106$		0			0
$107$	−92648.6	−	92648.6i	−0.782311	−	0.782311i	0.197909	−	0.980220i	$-0.436585\pi$
$107$	−92648.6	−	92648.6i	−0.782311	−	0.782311i	−0.980220	+	0.197909i	$0.936585\pi$
$108$		0			0
$109$			151720.i			1.22314i	0.791189	+	0.611572i	$0.209463\pi$
$109$			151720.i			1.22314i	−0.791189	+	0.611572i	$0.790537\pi$
$110$		0			0
$111$	15987.3	+	165456.i	0.123160	+	1.27460i
$112$		0			0
$113$	−8322.98	+	8322.98i	−0.0613172	+	0.0613172i	−0.737100	−	0.675783i	$-0.763806\pi$
$113$	−8322.98	+	8322.98i	−0.0613172	+	0.0613172i	0.675783	+	0.737100i	$0.263806\pi$
$114$		0			0
$115$	−5337.89	−	18083.1i	−0.0376379	−	0.127505i
$116$		0			0
$117$	−66818.3	+	99205.1i	−0.451264	+	0.669992i
$118$		0			0
$119$	978.596			0.00633484
$120$		0			0
$121$	135710.			0.842652
$122$		0			0
$123$	186023.	+	153241.i	1.10867	+	0.913299i
$124$		0			0
$125$	−174192.	+	13213.1i	−0.997135	+	0.0756362i
$126$		0			0
$127$	−109942.	+	109942.i	−0.604857	+	0.604857i	−0.941597	−	0.336741i	$-0.890675\pi$
$127$	−109942.	+	109942.i	−0.604857	+	0.604857i	0.336741	+	0.941597i	$0.390675\pi$
$128$		0			0
$129$	235960.	−	22799.9i	1.24843	−	0.120631i
$130$		0			0
$131$		−	9892.58i		−	0.0503653i	−0.999683	−	0.0251826i	$-0.991983\pi$
$131$		−	9892.58i		−	0.0503653i	0.999683	−	0.0251826i	$-0.00801673\pi$
$132$		0			0
$133$	−3619.93	−	3619.93i	−0.0177448	−	0.0177448i
$134$		0			0
$135$	−177586.	+	115341.i	−0.838639	+	0.544688i
$136$		0			0
$137$	−184071.	−	184071.i	−0.837883	−	0.837883i	0.150697	−	0.988580i	$-0.451848\pi$
$137$	−184071.	−	184071.i	−0.837883	−	0.837883i	−0.988580	+	0.150697i	$0.951848\pi$
$138$		0			0
$139$			1426.02i			0.00626021i	0.999995	+	0.00313011i	$0.000996345\pi$
$139$			1426.02i			0.00626021i	−0.999995	+	0.00313011i	$0.999004\pi$
$140$		0			0
$141$	135884.	−	13129.9i	0.575599	−	0.0556179i
$142$		0			0
$143$	55405.9	−	55405.9i	0.226577	−	0.226577i
$144$		0			0
$145$	141853.	−	260673.i	0.560296	−	1.02962i
$146$		0			0
$147$	202147.	+	166524.i	0.771569	+	0.635600i
$148$		0			0
$149$	−202862.			−0.748574			−0.374287	−	0.927313i	$-0.622113\pi$
$149$	−202862.			−0.748574			−0.374287	+	0.927313i	$0.622113\pi$
$150$		0			0
$151$	−83433.4			−0.297781			−0.148891	−	0.988854i	$-0.547570\pi$
$151$	−83433.4			−0.297781			−0.148891	+	0.988854i	$0.547570\pi$
$152$		0			0
$153$	55035.6	−	81711.2i	0.190071	−	0.282198i
$154$		0			0
$155$	126903.	−	233201.i	0.424270	−	0.779653i
$156$		0			0
$157$	−10332.3	+	10332.3i	−0.0334541	+	0.0334541i	−0.723636	−	0.690182i	$-0.757531\pi$
$157$	−10332.3	+	10332.3i	−0.0334541	+	0.0334541i	0.690182	+	0.723636i	$0.257531\pi$
$158$		0			0
$159$	−35983.6	−	372400.i	−0.112879	−	1.16820i
$160$		0			0
$161$		−	814.117i		−	0.00247527i
$162$		0			0
$163$	−286026.	−	286026.i	−0.843212	−	0.843212i	0.146063	−	0.989275i	$-0.453340\pi$
$163$	−286026.	−	286026.i	−0.843212	−	0.843212i	−0.989275	+	0.146063i	$0.953340\pi$
$164$		0			0
$165$	128651.	−	51887.2i	0.367878	−	0.148371i
$166$		0			0
$167$	108045.	+	108045.i	0.299789	+	0.299789i	0.840931	−	0.541142i	$-0.182008\pi$
$167$	108045.	+	108045.i	0.299789	+	0.299789i	−0.541142	+	0.840931i	$0.682008\pi$
$168$		0			0
$169$		−	129014.i		−	0.347471i
$170$		0			0
$171$	−505841.	+	98676.2i	−1.32289	+	0.258061i
$172$		0			0
$173$	−190692.	+	190692.i	−0.484416	+	0.484416i	−0.906539	−	0.422123i	$-0.861285\pi$
$173$	−190692.	+	190692.i	−0.484416	+	0.484416i	0.422123	+	0.906539i	$0.361285\pi$
$174$		0			0
$175$	−7375.25	−	1582.21i	−0.0182046	−	0.00390544i
$176$		0			0
$177$	424995.	−	515911.i	1.01965	−	1.23777i
$178$		0			0
$179$	−225167.			−0.525257			−0.262628	−	0.964897i	$-0.584589\pi$
$179$	−225167.			−0.525257			−0.262628	+	0.964897i	$0.584589\pi$
$180$		0			0
$181$	247482.			0.561496			0.280748	−	0.959782i	$-0.409418\pi$
$181$	247482.			0.561496			0.280748	+	0.959782i	$0.409418\pi$
$182$		0			0
$183$	−240837.	+	292357.i	−0.531612	+	0.645336i
$184$		0			0
$185$	168763.	+	571715.i	0.362533	+	1.22815i
$186$		0			0
$187$	−45635.6	+	45635.6i	−0.0954332	+	0.0954332i
$188$		0			0
$189$	−8764.25	+	2605.64i	−0.0178468	+	0.00530592i
$190$		0			0
$191$		−	787664.i		−	1.56228i	−0.624359	−	0.781138i	$-0.714640\pi$
$191$		−	787664.i		−	1.56228i	0.624359	−	0.781138i	$-0.285360\pi$
$192$		0			0
$193$	−483197.	−	483197.i	−0.933751	−	0.933751i	0.0641873	−	0.997938i	$-0.479554\pi$
$193$	−483197.	−	483197.i	−0.933751	−	0.933751i	−0.997938	+	0.0641873i	$0.979554\pi$
$194$		0			0
$195$	−167837.	+	394730.i	−0.316083	+	0.743384i
$196$		0			0
$197$	264653.	+	264653.i	0.485860	+	0.485860i	0.906997	−	0.421137i	$-0.138369\pi$
$197$	264653.	+	264653.i	0.485860	+	0.485860i	−0.421137	+	0.906997i	$0.638369\pi$
$198$		0			0
$199$			609515.i			1.09107i	0.838089	+	0.545534i	$0.183673\pi$
$199$			609515.i			1.09107i	−0.838089	+	0.545534i	$0.816327\pi$
$200$		0			0
$201$	−81395.9	−	842380.i	−0.142106	−	1.47068i
$202$		0			0
$203$	9061.04	−	9061.04i	0.0154326	−	0.0154326i
$204$		0			0
$205$	759170.	+	413124.i	1.26169	+	0.686587i
$206$		0			0
$207$	−67977.5	−	45785.4i	−0.110265	−	0.0742679i
$208$		0			0
$209$	337622.			0.534644
$210$		0			0
$211$	48108.3			0.0743899			0.0371950	−	0.999308i	$-0.488158\pi$
$211$	48108.3			0.0743899			0.0371950	+	0.999308i	$0.488158\pi$
$212$		0			0
$213$	877438.	+	722812.i	1.32516	+	1.09163i
$214$		0			0
$215$	815335.	−	240676.i	1.20293	−	0.355089i
$216$		0			0
$217$	8106.11	−	8106.11i	0.0116859	−	0.0116859i
$218$		0			0
$219$	−1.26158e6	+	121901.i	−1.77748	+	0.171750i
$220$		0			0
$221$		−	199556.i		−	0.274842i
$222$		0			0
$223$	−209536.	−	209536.i	−0.282161	−	0.282161i	0.551809	−	0.833970i	$-0.313938\pi$
$223$	−209536.	−	209536.i	−0.282161	−	0.282161i	−0.833970	+	0.551809i	$0.813938\pi$
$224$		0			0
$225$	−546892.	+	526840.i	−0.720186	+	0.693781i
$226$		0			0
$227$	738208.	+	738208.i	0.950855	+	0.950855i	0.998848	−	0.0479930i	$-0.0152825\pi$
$227$	738208.	+	738208.i	0.950855	+	0.950855i	−0.0479930	+	0.998848i	$0.515283\pi$
$228$		0			0
$229$			1.03498e6i			1.30419i	0.758136	+	0.652096i	$0.226110\pi$
$229$			1.03498e6i			1.30419i	−0.758136	+	0.652096i	$0.773890\pi$
$230$		0			0
$231$	5962.04	−	576.089i	0.00735132	−	0.000710329i
$232$		0			0
$233$	−1.02815e6	+	1.02815e6i	−1.24070	+	1.24070i	−0.280984	+	0.959712i	$0.590661\pi$
$233$	−1.02815e6	+	1.02815e6i	−1.24070	+	1.24070i	−0.959712	+	0.280984i	$0.909339\pi$
$234$		0			0
$235$	469533.	−	138600.i	0.554621	−	0.163717i
$236$		0			0
$237$	−456999.	−	376464.i	−0.528499	−	0.435365i
$238$		0			0
$239$	−676145.			−0.765677			−0.382838	−	0.923815i	$-0.625053\pi$
$239$	−676145.			−0.765677			−0.382838	+	0.923815i	$0.625053\pi$
$240$		0			0
$241$	185935.			0.206214			0.103107	−	0.994670i	$-0.467122\pi$
$241$	185935.			0.206214			0.103107	+	0.994670i	$0.467122\pi$
$242$		0			0
$243$	−275328.	+	878341.i	−0.299113	+	0.954218i
$244$		0			0
$245$	824974.	+	448933.i	0.878062	+	0.477822i
$246$		0			0
$247$	−738178.	+	738178.i	−0.769872	+	0.769872i
$248$		0			0
$249$	−127642.	−	1.32099e6i	−0.130466	−	1.35021i
$250$		0			0
$251$		−	559.689i		−	0.000560741i	−1.00000		0.000280371i	$-0.999911\pi$
$251$		−	559.689i		−	0.000560741i	1.00000		0.000280371i	$-8.92447e-5\pi$
$252$		0			0
$253$	37965.3	+	37965.3i	0.0372894	+	0.0372894i
$254$		0			0
$255$	138241.	−	325123.i	0.133133	−	0.313110i
$256$		0			0
$257$	419945.	+	419945.i	0.396606	+	0.396606i	0.877034	−	0.480428i	$-0.159519\pi$
$257$	419945.	+	419945.i	0.396606	+	0.396606i	−0.480428	+	0.877034i	$0.659519\pi$
$258$		0			0
$259$			25739.1i			0.0238421i
$260$		0			0
$261$	−246996.	−	1.26617e6i	−0.224434	−	1.15051i
$262$		0			0
$263$	1.05942e6	−	1.05942e6i	0.944449	−	0.944449i	−0.0540873	−	0.998536i	$-0.517225\pi$
$263$	1.05942e6	−	1.05942e6i	0.944449	−	0.944449i	0.998536	+	0.0540873i	$0.0172249\pi$
$264$		0			0
$265$	−379844.	−	1.28679e6i	−0.332270	−	1.12562i
$266$		0			0
$267$	583265.	−	708038.i	0.500711	−	0.607825i
$268$		0			0
$269$	583920.			0.492008			0.246004	−	0.969269i	$-0.420882\pi$
$269$	583920.			0.492008			0.246004	+	0.969269i	$0.420882\pi$
$270$		0			0
$271$	1.33412e6			1.10350			0.551750	−	0.834010i	$-0.313960\pi$
$271$	1.33412e6			1.10350			0.551750	+	0.834010i	$0.313960\pi$
$272$		0			0
$273$	−11775.9	+	14295.0i	−0.00956284	+	0.0116085i
$274$		0			0
$275$	417720.	−	270151.i	0.333084	−	0.215414i
$276$		0			0
$277$	−870629.	+	870629.i	−0.681763	+	0.681763i	−0.960397	−	0.278634i	$-0.910118\pi$
$277$	−870629.	+	870629.i	−0.681763	+	0.681763i	0.278634	+	0.960397i	$0.410118\pi$
$278$		0			0
$279$	−220966.	−	1.13273e6i	−0.169947	−	0.871196i
$280$		0			0
$281$		−	1.35673e6i		−	1.02501i	−0.858684	−	0.512506i	$-0.828717\pi$
$281$		−	1.35673e6i		−	1.02501i	0.858684	−	0.512506i	$-0.171283\pi$
$282$		0			0
$283$	−1.20786e6	−	1.20786e6i	−0.896504	−	0.896504i	0.0986213	−	0.995125i	$-0.468557\pi$
$283$	−1.20786e6	−	1.20786e6i	−0.896504	−	0.896504i	−0.995125	+	0.0986213i	$0.968557\pi$
$284$		0			0
$285$	−1.71403e6	+	691298.i	−1.24999	+	0.504142i
$286$		0			0
$287$	26388.9	+	26388.9i	0.0189111	+	0.0189111i
$288$		0			0
$289$		−	1.25549e6i		−	0.884238i
$290$		0			0
$291$	99163.1	+	1.02626e6i	0.0686463	+	0.710433i
$292$		0			0
$293$	−1.26013e6	+	1.26013e6i	−0.857527	+	0.857527i	−0.991046	−	0.133519i	$-0.957372\pi$
$293$	−1.26013e6	+	1.26013e6i	−0.857527	+	0.857527i	0.133519	+	0.991046i	$0.457372\pi$
$294$		0			0
$295$	1.14574e6	−	2.10546e6i	0.766536	−	1.40861i
$296$		0			0
$297$	287199.	−	530220.i	0.188926	−	0.348791i
$298$		0			0
$299$	−166015.			−0.107391
$300$		0			0
$301$	36707.1			0.0233525
$302$		0			0
$303$	1.19954e6	+	988150.i	0.750598	+	0.618325i
$304$		0			0
$305$	−649273.	+	1.19312e6i	−0.399648	+	0.734406i
$306$		0			0
$307$	−274503.	+	274503.i	−0.166227	+	0.166227i	−0.785319	−	0.619092i	$-0.787501\pi$
$307$	−274503.	+	274503.i	−0.166227	+	0.166227i	0.619092	+	0.785319i	$0.287501\pi$
$308$		0			0
$309$	1.34630e6	−	130088.i	0.802130	−	0.0775067i
$310$		0			0
$311$			1.68608e6i			0.988503i	0.869319	+	0.494252i	$0.164558\pi$
$311$			1.68608e6i			0.988503i	−0.869319	+	0.494252i	$0.835442\pi$
$312$		0			0
$313$	−1.16841e6	−	1.16841e6i	−0.674116	−	0.674116i	0.284546	−	0.958662i	$-0.408157\pi$
$313$	−1.16841e6	−	1.16841e6i	−0.674116	−	0.674116i	−0.958662	+	0.284546i	$0.908157\pi$
$314$		0			0
$315$	−29088.4	+	15132.3i	−0.0165175	+	0.00859266i
$316$		0			0
$317$	−2.07096e6	−	2.07096e6i	−1.15751	−	1.15751i	−0.985010	−	0.172497i	$-0.944816\pi$
$317$	−2.07096e6	−	2.07096e6i	−1.15751	−	1.15751i	−0.172497	−	0.985010i	$-0.555184\pi$
$318$		0			0
$319$			845100.i			0.464977i
$320$		0			0
$321$	−2.03301e6	+	196442.i	−1.10123	+	0.106407i
$322$		0			0
$323$	608007.	−	608007.i	0.324267	−	0.324267i
$324$		0			0
$325$	−322646.	+	1.50396e6i	−0.169441	+	0.789822i
$326$		0			0
$327$	1.82546e6	+	1.50377e6i	0.944068	+	0.777700i
$328$		0			0
$329$	21138.8			0.0107669
$330$		0			0
$331$	1.19564e6			0.599833			0.299917	−	0.953965i	$-0.403041\pi$
$331$	1.19564e6			0.599833			0.299917	+	0.953965i	$0.403041\pi$
$332$		0			0
$333$	2.14918e6	+	1.44755e6i	1.06209	+	0.715358i
$334$		0			0
$335$	−859219.	−	2.91076e6i	−0.418304	−	1.41708i
$336$		0			0
$337$	303742.	−	303742.i	0.145690	−	0.145690i	−0.630499	−	0.776190i	$-0.717150\pi$
$337$	303742.	−	303742.i	0.145690	−	0.145690i	0.776190	+	0.630499i	$0.217150\pi$
$338$		0			0
$339$	17647.1	+	182633.i	0.00834015	+	0.0863137i
$340$		0			0
$341$			756036.i			0.352092i
$342$		0			0
$343$	57362.4	+	57362.4i	0.0263264	+	0.0263264i
$344$		0			0
$345$	−270477.	−	115006.i	−0.122344	−	0.0520201i
$346$		0			0
$347$	315076.	+	315076.i	0.140473	+	0.140473i	0.773846	−	0.633373i	$-0.218330\pi$
$347$	315076.	+	315076.i	0.140473	+	0.140473i	−0.633373	+	0.773846i	$0.718330\pi$
$348$		0			0
$349$			4.27223e6i			1.87755i	0.344531	+	0.938775i	$0.388038\pi$
$349$			4.27223e6i			1.87755i	−0.344531	+	0.938775i	$0.611962\pi$
$350$		0			0
$351$	531344.	+	1.78721e6i	0.230201	+	0.774297i
$352$		0			0
$353$	2.18140e6	−	2.18140e6i	0.931747	−	0.931747i	−0.0660681	−	0.997815i	$-0.521045\pi$
$353$	2.18140e6	−	2.18140e6i	0.931747	−	0.931747i	0.997815	+	0.0660681i	$0.0210455\pi$
$354$		0			0
$355$	3.58087e6	+	1.94863e6i	1.50806	+	0.820651i
$356$		0			0
$357$	9699.31	−	11774.2i	0.00402782	−	0.00488947i
$358$		0			0
$359$	−3.07492e6			−1.25921			−0.629605	−	0.776915i	$-0.716783\pi$
$359$	−3.07492e6			−1.25921			−0.629605	+	0.776915i	$0.716783\pi$
$360$		0			0
$361$	−2.02207e6			−0.816635
$362$		0			0
$363$	1.34508e6	−	1.63283e6i	0.535775	−	0.650390i
$364$		0			0
$365$	−4.35925e6	+	1.28680e6i	−1.71269	+	0.505565i
$366$		0			0
$367$	−668839.	+	668839.i	−0.259213	+	0.259213i	−0.824734	−	0.565521i	$-0.808675\pi$
$367$	−668839.	+	668839.i	−0.259213	+	0.259213i	0.565521	+	0.824734i	$0.308675\pi$
$368$		0			0
$369$	3.68752e6	−	719338.i	1.40984	−	0.275022i
$370$		0			0
$371$		−	57932.5i		−	0.0218518i
$372$		0			0
$373$	179707.	+	179707.i	0.0668794	+	0.0668794i	0.739755	−	0.672876i	$-0.234941\pi$
$373$	179707.	+	179707.i	0.0668794	+	0.0668794i	−0.672876	+	0.739755i	$0.734941\pi$
$374$		0			0
$375$	−1.56752e6	+	2.22680e6i	−0.575620	+	0.817717i
$376$		0			0
$377$	−1.84773e6	−	1.84773e6i	−0.669553	−	0.669553i
$378$		0			0
$379$			2.12353e6i			0.759383i	0.925113	+	0.379691i	$0.123970\pi$
$379$			2.12353e6i			0.759383i	−0.925113	+	0.379691i	$0.876030\pi$
$380$		0			0
$381$	233108.	+	2.41247e6i	0.0822705	+	0.851431i
$382$		0			0
$383$	−865082.	+	865082.i	−0.301342	+	0.301342i	−0.841539	−	0.540197i	$-0.818350\pi$
$383$	−865082.	+	865082.i	−0.301342	+	0.301342i	0.540197	+	0.841539i	$0.318350\pi$
$384$		0			0
$385$	20601.2	−	6081.22i	0.00708339	−	0.00209093i
$386$		0			0
$387$	2.06438e6	−	3.06499e6i	0.700669	−	1.04028i
$388$		0			0
$389$	5.57596e6			1.86829			0.934147	−	0.356888i	$-0.116162\pi$
$389$	5.57596e6			1.86829			0.934147	+	0.356888i	$0.116162\pi$
$390$		0			0
$391$	136740.			0.0452328
$392$		0			0
$393$	−119025.	−	98049.9i	−0.0388738	−	0.0320233i
$394$		0			0
$395$	−1.86504e6	−	1.01491e6i	−0.601443	−	0.327292i
$396$		0			0
$397$	918981.	−	918981.i	0.292638	−	0.292638i	−0.545484	−	0.838121i	$-0.683654\pi$
$397$	918981.	−	918981.i	0.292638	−	0.292638i	0.838121	+	0.545484i	$0.183654\pi$
$398$		0			0
$399$	−79432.9	+	7675.29i	−0.0249786	+	0.00241359i
$400$		0			0
$401$			2.31260e6i			0.718189i	0.933301	+	0.359095i	$0.116914\pi$
$401$			2.31260e6i			0.718189i	−0.933301	+	0.359095i	$0.883086\pi$
$402$		0			0
$403$	−1.65300e6	−	1.65300e6i	−0.507003	−	0.507003i
$404$		0			0
$405$	−372391.	+	3.27987e6i	−0.112814	+	0.993616i
$406$		0			0
$407$	−1.20031e6	−	1.20031e6i	−0.359177	−	0.359177i
$408$		0			0
$409$		−	2.08308e6i		−	0.615742i	−0.951428	−	0.307871i	$-0.900384\pi$
$409$		−	2.08308e6i		−	0.615742i	0.951428	−	0.307871i	$-0.0996164\pi$
$410$		0			0
$411$	−4.03911e6	+	390283.i	−1.17945	+	0.113966i
$412$		0			0
$413$	73186.0	−	73186.0i	0.0211132	−	0.0211132i
$414$		0			0
$415$	−1.34740e6	−	4.56455e6i	−0.384039	−	1.30100i
$416$		0			0
$417$	17157.5	+	14134.0i	0.00483186	+	0.00398037i
$418$		0			0
$419$	−1.14299e6			−0.318059			−0.159029	−	0.987274i	$-0.550836\pi$
$419$	−1.14299e6			−0.318059			−0.159029	+	0.987274i	$0.550836\pi$
$420$		0			0
$421$	560751.			0.154193			0.0770965	−	0.997024i	$-0.475435\pi$
$421$	560751.			0.154193			0.0770965	+	0.997024i	$0.475435\pi$
$422$		0			0
$423$	1.18883e6	−	1.76506e6i	0.323050	−	0.479631i
$424$		0			0
$425$	265750.	−	1.23875e6i	0.0713676	−	0.332669i
$426$		0			0
$427$	−41473.2	+	41473.2i	−0.0110077	+	0.0110077i
$428$		0			0
$429$	−117476.	−	1.21578e6i	−0.0308182	−	0.318943i
$430$		0			0
$431$			6.08563e6i			1.57802i	0.614381	+	0.789009i	$0.289406\pi$
$431$			6.08563e6i			1.57802i	−0.614381	+	0.789009i	$0.710594\pi$
$432$		0			0
$433$	−1.86084e6	−	1.86084e6i	−0.476969	−	0.476969i	0.427192	−	0.904161i	$-0.359503\pi$
$433$	−1.86084e6	−	1.86084e6i	−0.476969	−	0.476969i	−0.904161	+	0.427192i	$0.859503\pi$
$434$		0			0
$435$	−1.73039e6	−	4.29039e6i	−0.438450	−	1.08711i
$436$		0			0
$437$	−505816.	−	505816.i	−0.126703	−	0.126703i
$438$		0			0
$439$		−	4.65060e6i		−	1.15172i	−0.817547	−	0.575861i	$-0.804667\pi$
$439$		−	4.65060e6i		−	1.15172i	0.817547	−	0.575861i	$-0.195333\pi$
$440$		0			0
$441$	4.00715e6	−	781690.i	0.981159	−	0.191398i
$442$		0			0
$443$	5.02519e6	−	5.02519e6i	1.21659	−	1.21659i	0.247767	−	0.968820i	$-0.420303\pi$
$443$	5.02519e6	−	5.02519e6i	1.21659	−	1.21659i	0.968820	−	0.247767i	$-0.0796968\pi$
$444$		0			0
$445$	1.57242e6	−	2.88954e6i	0.376418	−	0.691717i
$446$		0			0
$447$	−2.01066e6	+	2.44078e6i	−0.475959	+	0.577777i
$448$		0			0
$449$	3.01321e6			0.705364			0.352682	−	0.935743i	$-0.385270\pi$
$449$	3.01321e6			0.705364			0.352682	+	0.935743i	$0.385270\pi$
$450$		0			0
$451$	−2.46122e6			−0.569783
$452$		0			0
$453$	−826947.	+	1.00385e6i	−0.189336	+	0.229839i
$454$		0			0
$455$	−31746.6	+	58338.6i	−0.00718901	+	0.0132108i
$456$		0			0
$457$	1.48634e6	−	1.48634e6i	0.332911	−	0.332911i	−0.520780	−	0.853691i	$-0.674359\pi$
$457$	1.48634e6	−	1.48634e6i	0.332911	−	0.332911i	0.853691	+	0.520780i	$0.174359\pi$
$458$		0			0
$459$	−437646.	−	1.47205e6i	−0.0969597	−	0.326130i
$460$		0			0
$461$		−	5.60249e6i		−	1.22780i	−0.789383	−	0.613901i	$-0.789599\pi$
$461$		−	5.60249e6i		−	1.22780i	0.789383	−	0.613901i	$-0.210401\pi$
$462$		0			0
$463$	2.95717e6	+	2.95717e6i	0.641096	+	0.641096i	0.950825	−	0.309729i	$-0.100238\pi$
$463$	2.95717e6	+	2.95717e6i	0.641096	+	0.641096i	−0.309729	+	0.950825i	$0.600238\pi$
$464$		0			0
$465$	−1.54802e6	−	3.83823e6i	−0.332005	−	0.823187i
$466$		0			0
$467$	4.14460e6	+	4.14460e6i	0.879408	+	0.879408i	0.993473	−	0.114065i	$-0.0363874\pi$
$467$	4.14460e6	+	4.14460e6i	0.879408	+	0.879408i	−0.114065	+	0.993473i	$0.536387\pi$
$468$		0			0
$469$		−	131045.i		−	0.0275099i
$470$		0			0
$471$	21907.5	+	226724.i	0.00455030	+	0.0470919i
$472$		0			0
$473$	−1.71179e6	+	1.71179e6i	−0.351802	+	0.351802i
$474$		0			0
$475$	−5.56533e6	+	3.59925e6i	−1.13177	+	0.731944i
$476$		0			0
$477$	−4.83727e6	−	3.25808e6i	−0.973430	−	0.655641i
$478$		0			0
$479$	1.43812e6			0.286388			0.143194	−	0.989695i	$-0.454263\pi$
$479$	1.43812e6			0.286388			0.143194	+	0.989695i	$0.454263\pi$
$480$		0			0
$481$	5.24874e6			1.03441
$482$		0			0
$483$	−9795.25	−	8069.09i	−0.00191050	−	0.00157383i
$484$		0			0
$485$	1.04677e6	+	3.54612e6i	0.202068	+	0.684540i
$486$		0			0
$487$	−5.84038e6	+	5.84038e6i	−1.11588	+	1.11588i	−0.123544	+	0.992339i	$0.539426\pi$
$487$	−5.84038e6	+	5.84038e6i	−1.11588	+	1.11588i	−0.992339	+	0.123544i	$0.960574\pi$
$488$		0			0
$489$	−6.27633e6	+	606457.i	−1.18695	+	0.114691i
$490$		0			0
$491$			8.77431e6i			1.64251i	0.570558	+	0.821257i	$0.306727\pi$
$491$			8.77431e6i			1.64251i	−0.570558	+	0.821257i	$0.693273\pi$
$492$		0			0
$493$	1.52190e6	+	1.52190e6i	0.282013	+	0.282013i
$494$		0			0
$495$	650827.	−	2.06218e6i	0.119386	−	0.378279i
$496$		0			0
$497$	124472.	+	124472.i	0.0226037	+	0.0226037i
$498$		0			0
$499$			1.68446e6i			0.302837i	0.988470	+	0.151418i	$0.0483841\pi$
$499$			1.68446e6i			0.302837i	−0.988470	+	0.151418i	$0.951616\pi$
$500$		0			0
$501$	2.37086e6	−	229087.i	0.422000	−	0.0407762i
$502$		0			0
$503$	−4.23036e6	+	4.23036e6i	−0.745516	+	0.745516i	−0.973634	−	0.228117i	$-0.926743\pi$
$503$	−4.23036e6	+	4.23036e6i	−0.745516	+	0.745516i	0.228117	+	0.973634i	$0.426743\pi$
$504$		0			0
$505$	4.89538e6	+	2.66396e6i	0.854196	+	0.464835i
$506$		0			0
$507$	−1.55226e6	−	1.27871e6i	−0.268191	−	0.220929i
$508$		0			0
$509$	1.80043e6			0.308023			0.154011	−	0.988069i	$-0.450781\pi$
$509$	1.80043e6			0.308023			0.154011	+	0.988069i	$0.450781\pi$
$510$		0			0
$511$	−196257.			−0.0332486
$512$		0			0
$513$	−3.82638e6	+	7.06418e6i	−0.641940	+	1.18514i
$514$		0			0
$515$	4.65199e6	−	1.37321e6i	0.772896	−	0.228149i
$516$		0			0
$517$	−985780.	+	985780.i	−0.162201	+	0.162201i
$518$		0			0
$519$	404323.	+	4.18440e6i	0.0658885	+	0.681891i
$520$		0			0
$521$		−	4.93499e6i		−	0.796512i	−0.917274	−	0.398256i	$-0.869616\pi$
$521$		−	4.93499e6i		−	0.796512i	0.917274	−	0.398256i	$-0.130384\pi$
$522$		0			0
$523$	4.22362e6	+	4.22362e6i	0.675197	+	0.675197i	0.958909	−	0.283712i	$-0.0915660\pi$
$523$	4.22362e6	+	4.22362e6i	0.675197	+	0.675197i	−0.283712	+	0.958909i	$0.591566\pi$
$524$		0			0
$525$	−92136.3	+	73055.1i	−0.0145892	+	0.0115678i
$526$		0			0
$527$	1.36151e6	+	1.36151e6i	0.213547	+	0.213547i
$528$		0			0
$529$			6.32259e6i			0.982326i
$530$		0			0
$531$	−1.99499e6	−	1.02269e7i	−0.307046	−	1.57400i
$532$		0			0
$533$	5.38123e6	−	5.38123e6i	0.820471	−	0.820471i
$534$		0			0
$535$	−7.02485e6	+	2.07365e6i	−1.06109	+	0.313220i
$536$		0			0
$537$	−2.23173e6	+	2.70915e6i	−0.333969	+	0.405413i
$538$		0			0
$539$	−2.67456e6			−0.396534
$540$		0			0
$541$	3.71467e6			0.545666			0.272833	−	0.962061i	$-0.412039\pi$
$541$	3.71467e6			0.545666			0.272833	+	0.962061i	$0.412039\pi$
$542$		0			0
$543$	2.45290e6	−	2.97764e6i	0.357011	−	0.433383i
$544$		0			0
$545$	7.44980e6	+	4.05402e6i	1.07437	+	0.584648i
$546$		0			0
$547$	6.16600e6	−	6.16600e6i	0.881121	−	0.881121i	−0.112528	−	0.993649i	$-0.535895\pi$
$547$	6.16600e6	−	6.16600e6i	0.881121	−	0.881121i	0.993649	+	0.112528i	$0.0358948\pi$
$548$		0			0
$549$	1.13052e6	+	5.79538e6i	0.160084	+	0.820636i
$550$		0			0
$551$		−	1.12593e7i		−	1.57992i
$552$		0			0
$553$	−64828.9	−	64828.9i	−0.00901480	−	0.00901480i
$554$		0			0
$555$	8.55142e6	+	3.63602e6i	1.17844	+	0.501065i
$556$		0			0
$557$	2.71742e6	+	2.71742e6i	0.371123	+	0.371123i	0.867886	−	0.496763i	$-0.165478\pi$
$557$	2.71742e6	+	2.71742e6i	0.371123	+	0.371123i	−0.496763	+	0.867886i	$0.665478\pi$
$558$		0			0
$559$		−	7.48533e6i		−	1.01317i
$560$		0			0
$561$	96760.5	+	1.00139e6i	0.0129805	+	0.134337i
$562$		0			0
$563$	−8.53542e6	+	8.53542e6i	−1.13489	+	1.13489i	−0.145537	+	0.989353i	$0.546491\pi$
$563$	−8.53542e6	+	8.53542e6i	−1.13489	+	1.13489i	−0.989353	+	0.145537i	$0.953509\pi$
$564$		0			0
$565$	186284.	+	631069.i	0.0245501	+	0.0831679i
$566$		0			0
$567$	−55516.1	+	131275.i	−0.00725206	+	0.0171484i
$568$		0			0
$569$	−1.03664e6			−0.134229			−0.0671144	−	0.997745i	$-0.521379\pi$
$569$	−1.03664e6			−0.134229			−0.0671144	+	0.997745i	$0.521379\pi$
$570$		0			0
$571$	−9.40785e6			−1.20754			−0.603768	−	0.797160i	$-0.706335\pi$
$571$	−9.40785e6			−1.20754			−0.603768	+	0.797160i	$0.706335\pi$
$572$		0			0
$573$	−9.47697e6	−	7.80690e6i	−1.20582	−	0.993327i
$574$		0			0
$575$	−1.03055e6	−	221084.i	−0.129987	−	0.0278861i
$576$		0			0
$577$	2.93464e6	−	2.93464e6i	0.366958	−	0.366958i	−0.499409	−	0.866366i	$-0.666449\pi$
$577$	2.93464e6	−	2.93464e6i	0.366958	−	0.366958i	0.866366	+	0.499409i	$0.166449\pi$
$578$		0			0
$579$	−1.06029e7	+	1.02452e6i	−1.31440	+	0.127005i
$580$		0			0
$581$		−	205500.i		−	0.0252564i
$582$		0			0
$583$	2.70161e6	+	2.70161e6i	0.329193	+	0.329193i
$584$		0			0
$585$	3.08578e6	+	5.93172e6i	0.372799	+	0.716624i
$586$		0			0
$587$	6.11826e6	+	6.11826e6i	0.732880	+	0.732880i	0.971189	−	0.238309i	$-0.0765932\pi$
$587$	6.11826e6	+	6.11826e6i	0.732880	+	0.732880i	−0.238309	+	0.971189i	$0.576593\pi$
$588$		0			0
$589$		−	1.00727e7i		−	1.19635i
$590$		0			0
$591$	5.80733e6	−	561140.i	0.683924	−	0.0660849i
$592$		0			0
$593$	−3.00015e6	+	3.00015e6i	−0.350353	+	0.350353i	−0.860241	−	0.509888i	$-0.829687\pi$
$593$	−3.00015e6	+	3.00015e6i	−0.350353	+	0.350353i	0.509888	+	0.860241i	$0.329687\pi$
$594$		0			0
$595$	26148.4	−	48051.2i	0.00302798	−	0.00556431i
$596$		0			0
$597$	7.33353e6	+	6.04118e6i	0.842126	+	0.693723i
$598$		0			0
$599$	−1.34031e7			−1.52630			−0.763150	−	0.646222i	$-0.776348\pi$
$599$	−1.34031e7			−1.52630			−0.763150	+	0.646222i	$0.776348\pi$
$600$		0			0
$601$	−1.33813e7			−1.51116			−0.755580	−	0.655056i	$-0.772645\pi$
$601$	−1.33813e7			−1.51116			−0.755580	+	0.655056i	$0.772645\pi$
$602$		0			0
$603$	−1.09421e7	−	7.36988e6i	−1.22548	−	0.825405i
$604$		0			0
$605$	3.62621e6	−	6.66365e6i	0.402777	−	0.740157i
$606$		0			0
$607$	5.90305e6	−	5.90305e6i	0.650286	−	0.650286i	−0.302776	−	0.953062i	$-0.597913\pi$
$607$	5.90305e6	−	5.90305e6i	0.650286	−	0.650286i	0.953062	+	0.302776i	$0.0979133\pi$
$608$		0			0
$609$	−19212.0	−	198828.i	−0.00209908	−	0.0217238i
$610$		0			0
$611$		−	4.31063e6i		−	0.467130i
$612$		0			0
$613$	−6.34586e6	−	6.34586e6i	−0.682086	−	0.682086i	0.278384	−	0.960470i	$-0.410201\pi$
$613$	−6.34586e6	−	6.34586e6i	−0.682086	−	0.682086i	−0.960470	+	0.278384i	$0.910201\pi$
$614$		0			0
$615$	1.24951e7	−	5.03948e6i	1.33214	−	0.537277i
$616$		0			0
$617$	−1.47580e6	−	1.47580e6i	−0.156068	−	0.156068i	0.624754	−	0.780822i	$-0.285199\pi$
$617$	−1.47580e6	−	1.47580e6i	−0.156068	−	0.156068i	−0.780822	+	0.624754i	$0.785199\pi$
$618$		0			0
$619$		−	5.76985e6i		−	0.605254i	−0.953109	−	0.302627i	$-0.902136\pi$
$619$		−	5.76985e6i		−	0.605254i	0.953109	−	0.302627i	$-0.0978636\pi$
$620$		0			0
$621$	−1.22463e6	+	364088.i	−0.127432	+	0.0378859i
$622$		0			0
$623$	100441.	−	100441.i	0.0103679	−	0.0103679i
$624$		0			0
$625$	−4.00569e6	+	8.90628e6i	−0.410182	+	0.912003i
$626$		0			0
$627$	3.34633e6	−	4.06218e6i	0.339938	−	0.412658i
$628$		0			0
$629$	−4.32317e6			−0.435688
$630$		0			0
$631$	1.11410e7			1.11392			0.556958	−	0.830541i	$-0.311969\pi$
$631$	1.11410e7			1.11392			0.556958	+	0.830541i	$0.311969\pi$
$632$		0			0
$633$	476824.	−	578827.i	0.0472987	−	0.0574169i
$634$		0			0
$635$	2.46069e6	+	8.33604e6i	0.242172	+	0.820400i
$636$		0			0
$637$	5.84767e6	−	5.84767e6i	0.570997	−	0.570997i
$638$		0			0
$639$	1.73934e7	−	3.39299e6i	1.68512	−	0.328723i
$640$		0			0
$641$		−	2.02277e7i		−	1.94448i	−0.233996	−	0.972238i	$-0.575180\pi$
$641$		−	2.02277e7i		−	1.94448i	0.233996	−	0.972238i	$-0.424820\pi$
$642$		0			0
$643$	1.55327e6	+	1.55327e6i	0.148156	+	0.148156i	0.777294	−	0.629138i	$-0.216592\pi$
$643$	1.55327e6	+	1.55327e6i	0.148156	+	0.148156i	−0.629138	+	0.777294i	$0.716592\pi$
$644$		0			0
$645$	5.18540e6	−	1.21954e7i	0.490776	−	1.15424i
$646$		0			0
$647$	−6.43585e6	−	6.43585e6i	−0.604429	−	0.604429i	0.337056	−	0.941485i	$-0.390569\pi$
$647$	−6.43585e6	−	6.43585e6i	−0.604429	−	0.604429i	−0.941485	+	0.337056i	$0.890569\pi$
$648$		0			0
$649$			6.82588e6i			0.636131i
$650$		0			0
$651$	−17187.3	−	177874.i	−0.00158948	−	0.0164498i
$652$		0			0
$653$	−4.13970e6	+	4.13970e6i	−0.379915	+	0.379915i	−0.871071	−	0.491157i	$-0.836574\pi$
$653$	−4.13970e6	+	4.13970e6i	−0.379915	+	0.379915i	0.491157	+	0.871071i	$0.336574\pi$
$654$		0			0
$655$	−485747.	−	264333.i	−0.0442392	−	0.0240740i
$656$		0			0
$657$	−1.10374e7	+	1.63872e7i	−0.997592	+	1.48112i
$658$		0			0
$659$	4.36670e6			0.391687			0.195844	−	0.980635i	$-0.437255\pi$
$659$	4.36670e6			0.391687			0.195844	+	0.980635i	$0.437255\pi$
$660$		0			0
$661$	−1.79540e7			−1.59830			−0.799151	−	0.601131i	$-0.794717\pi$
$661$	−1.79540e7			−1.59830			−0.799151	+	0.601131i	$0.794717\pi$
$662$		0			0
$663$	−2.40100e6	−	1.97789e6i	−0.212133	−	0.174750i
$664$		0			0
$665$	−274472.	+	81020.7i	−0.0240682	+	0.00710464i
$666$		0			0
$667$	1.26611e6	−	1.26611e6i	0.110193	−	0.110193i
$668$		0			0
$669$	−4.59789e6	+	444277.i	−0.397186	+	0.0383785i
$670$		0			0
$671$		−	3.86810e6i		−	0.331659i
$672$		0			0
$673$	9.66735e6	+	9.66735e6i	0.822754	+	0.822754i	0.986502	−	0.163748i	$-0.0523585\pi$
$673$	9.66735e6	+	9.66735e6i	0.822754	+	0.822754i	−0.163748	+	0.986502i	$0.552359\pi$
$674$		0			0
$675$	918306.	+	1.18018e7i	0.0775761	+	0.996986i
$676$		0			0
$677$	−8.44751e6	−	8.44751e6i	−0.708365	−	0.708365i	0.257826	−	0.966191i	$-0.416994\pi$
$677$	−8.44751e6	−	8.44751e6i	−0.708365	−	0.708365i	−0.966191	+	0.257826i	$0.916994\pi$
$678$		0			0
$679$			159650.i			0.0132890i
$680$		0			0
$681$	1.61987e7	−	1.56521e6i	1.33848	−	0.129332i
$682$		0			0
$683$	−3.17200e6	+	3.17200e6i	−0.260185	+	0.260185i	−0.825129	−	0.564944i	$-0.808898\pi$
$683$	−3.17200e6	+	3.17200e6i	−0.260185	+	0.260185i	0.564944	+	0.825129i	$0.308898\pi$
$684$		0			0
$685$	−1.39567e7	+	4.11984e6i	−1.13647	+	0.335470i
$686$		0			0
$687$	1.24526e7	+	1.02581e7i	1.00662	+	0.829232i
$688$		0			0
$689$	−1.18136e7			−0.948057
$690$		0			0
$691$	1.82167e7			1.45136			0.725678	−	0.688034i	$-0.241526\pi$
$691$	1.82167e7			1.45136			0.725678	+	0.688034i	$0.241526\pi$
$692$		0			0
$693$	52161.2	−	77443.7i	0.00412586	−	0.00612566i
$694$		0			0
$695$	70020.7	+	38103.8i	0.00549876	+	0.00299231i
$696$		0			0
$697$	−4.43230e6	+	4.43230e6i	−0.345579	+	0.345579i
$698$		0			0
$699$	2.17997e6	+	2.25609e7i	0.168755	+	1.74648i
$700$		0			0
$701$			1.46631e7i			1.12702i	0.826111	+	0.563508i	$0.190549\pi$
$701$			1.46631e7i			1.12702i	−0.826111	+	0.563508i	$0.809451\pi$
$702$		0			0
$703$	1.59919e7	+	1.59919e7i	1.22043	+	1.22043i
$704$		0			0
$705$	2.98615e6	−	7.02303e6i	0.226276	−	0.532171i
$706$		0			0
$707$	170164.	+	170164.i	0.0128032	+	0.0128032i
$708$		0			0
$709$		−	1.49044e7i		−	1.11352i	−0.830673	−	0.556760i	$-0.812044\pi$
$709$		−	1.49044e7i		−	1.11352i	0.830673	−	0.556760i	$-0.187956\pi$
$710$		0			0
$711$	−9.05905e6	+	1.76718e6i	−0.672061	+	0.131101i
$712$		0			0
$713$	1.13267e6	−	1.13267e6i	0.0834412	−	0.0834412i
$714$		0			0
$715$	−1.24009e6	−	4.20101e6i	−0.0907165	−	0.307319i
$716$		0			0
$717$	−6.70159e6	+	8.13521e6i	−0.486833	+	0.590977i
$718$		0			0
$719$	−6.69566e6			−0.483027			−0.241514	−	0.970397i	$-0.577644\pi$
$719$	−6.69566e6			−0.483027			−0.241514	+	0.970397i	$0.577644\pi$
$720$		0			0
$721$	209437.			0.0150043
$722$		0			0
$723$	1.84289e6	−	2.23712e6i	0.131115	−	0.159164i
$724$		0			0
$725$	−9.00926e6	−	1.39305e7i	−0.636567	−	0.984290i
$726$		0			0
$727$	−1.68120e7	+	1.68120e7i	−1.17974	+	1.17974i	−0.199924	+	0.979811i	$0.564069\pi$
$727$	−1.68120e7	+	1.68120e7i	−1.17974	+	1.17974i	−0.979811	+	0.199924i	$0.935931\pi$
$728$		0			0
$729$	7.83907e6	+	1.20183e7i	0.546318	+	0.837578i
$730$		0			0
$731$			6.16536e6i			0.426742i
$732$		0			0
$733$	6.28485e6	+	6.28485e6i	0.432051	+	0.432051i	0.889326	−	0.457274i	$-0.151174\pi$
$733$	6.28485e6	+	6.28485e6i	0.432051	+	0.432051i	−0.457274	+	0.889326i	$0.651174\pi$
$734$		0			0
$735$	1.35781e7	−	5.47630e6i	0.927091	−	0.373911i
$736$		0			0
$737$	6.11112e6	+	6.11112e6i	0.414431	+	0.414431i
$738$		0			0
$739$		−	1.03783e7i		−	0.699061i	−0.936925	−	0.349530i	$-0.886341\pi$
$739$		−	1.03783e7i		−	0.699061i	0.936925	−	0.349530i	$-0.113659\pi$
$740$		0			0
$741$	1.56515e6	+	1.61980e7i	0.104715	+	1.08372i
$742$		0			0
$743$	3.14751e6	−	3.14751e6i	0.209168	−	0.209168i	−0.594746	−	0.803914i	$-0.702747\pi$
$743$	3.14751e6	−	3.14751e6i	0.209168	−	0.209168i	0.803914	+	0.594746i	$0.202747\pi$
$744$		0			0
$745$	−5.42054e6	+	9.96096e6i	−0.357810	+	0.657523i
$746$		0			0
$747$	−1.71590e7	−	1.15572e7i	−1.12510	−	0.757794i
$748$		0			0
$749$	−316265.			−0.0205990
$750$		0			0
$751$	1.31733e7			0.852304			0.426152	−	0.904652i	$-0.359869\pi$
$751$	1.31733e7			0.852304			0.426152	+	0.904652i	$0.359869\pi$
$752$		0			0
$753$	−6734.04	−	5547.34i	−0.000432801	−	0.000356531i
$754$		0			0
$755$	−2.22937e6	+	4.09676e6i	−0.142336	+	0.261561i
$756$		0			0
$757$	7.69115e6	−	7.69115e6i	0.487811	−	0.487811i	−0.419804	−	0.907615i	$-0.637901\pi$
$757$	7.69115e6	−	7.69115e6i	0.487811	−	0.487811i	0.907615	+	0.419804i	$0.137901\pi$
$758$		0			0
$759$	833081.	−	80497.3i	0.0524907	−	0.00507197i
$760$		0			0
$761$			8.06267e6i			0.504681i	0.967638	+	0.252341i	$0.0812003\pi$
$761$			8.06267e6i			0.504681i	−0.967638	+	0.252341i	$0.918800\pi$
$762$		0			0
$763$	258956.	+	258956.i	0.0161033	+	0.0161033i
$764$		0			0
$765$	−2.54163e6	−	4.88572e6i	−0.157021	−	0.301839i
$766$		0			0
$767$	−1.49241e7	−	1.49241e7i	−0.916011	−	0.916011i
$768$		0			0
$769$			7.04385e6i			0.429531i	0.976666	+	0.214765i	$0.0688987\pi$
$769$			7.04385e6i			0.429531i	−0.976666	+	0.214765i	$0.931101\pi$
$770$		0			0
$771$	9.21494e6	−	890404.i	0.558286	−	0.0539450i
$772$		0			0
$773$	1.28966e7	−	1.28966e7i	0.776295	−	0.776295i	−0.202904	−	0.979199i	$-0.565038\pi$
$773$	1.28966e7	−	1.28966e7i	0.776295	−	0.776295i	0.979199	+	0.202904i	$0.0650379\pi$
$774$		0			0
$775$	−8.05979e6	−	1.24624e7i	−0.482025	−	0.745329i
$776$		0			0
$777$	309687.	+	255112.i	0.0184022	+	0.0151593i
$778$		0			0
$779$	3.27911e7			1.93603
$780$		0			0
$781$	−1.16091e7			−0.681040
$782$		0			0
$783$	−1.76823e7	−	9.57779e6i	−1.03071	−	0.558292i
$784$		0			0
$785$	231256.	+	783423.i	0.0133943	+	0.0453756i
$786$		0			0
$787$	3.29667e6	−	3.29667e6i	0.189731	−	0.189731i	−0.605849	−	0.795580i	$-0.707166\pi$
$787$	3.29667e6	−	3.29667e6i	0.189731	−	0.189731i	0.795580	+	0.605849i	$0.207166\pi$
$788$		0			0
$789$	−2.24627e6	−	2.32471e7i	−0.128461	−	1.32946i
$790$		0			0
$791$			28411.3i			0.00161454i
$792$		0			0
$793$	8.45723e6	+	8.45723e6i	0.477579	+	0.477579i
$794$		0			0
$795$	−1.92471e7	−	8.18378e6i	−1.08006	−	0.459236i
$796$		0			0
$797$	1.05205e7	+	1.05205e7i	0.586664	+	0.586664i	0.936726	−	0.350062i	$-0.113840\pi$
$797$	1.05205e7	+	1.05205e7i	0.586664	+	0.586664i	−0.350062	+	0.936726i	$0.613840\pi$
$798$		0			0
$799$			3.55049e6i			0.196753i
$800$		0			0
$801$	−2.73793e6	−	1.40354e7i	−0.150779	−	0.772935i
$802$		0			0
$803$	9.15222e6	−	9.15222e6i	0.500884	−	0.500884i
$804$		0			0
$805$	−39974.9	−	21753.5i	−0.00217419	−	0.00118315i
$806$		0			0
$807$	5.78750e6	−	7.02557e6i	0.312829	−	0.379750i
$808$		0			0
$809$	5.87579e6			0.315642			0.157821	−	0.987468i	$-0.449553\pi$
$809$	5.87579e6			0.315642			0.157821	+	0.987468i	$0.449553\pi$
$810$		0			0
$811$	−3.38892e7			−1.80929			−0.904646	−	0.426164i	$-0.859865\pi$
$811$	−3.38892e7			−1.80929			−0.904646	+	0.426164i	$0.859865\pi$
$812$		0			0
$813$	1.32231e7	−	1.60518e7i	0.701627	−	0.851721i
$814$		0			0
$815$	−2.16872e7	+	6.40179e6i	−1.14369	+	0.337604i
$816$		0			0
$817$	2.28064e7	−	2.28064e7i	1.19537	−	1.19537i
$818$		0			0
$819$	55277.7	+	283369.i	0.00287966	+	0.0147619i
$820$		0			0
$821$			3.59758e7i			1.86274i	0.364072	+	0.931371i	$0.381386\pi$
$821$			3.59758e7i			1.86274i	−0.364072	+	0.931371i	$0.618614\pi$
$822$		0			0
$823$	−1.55143e7	−	1.55143e7i	−0.798424	−	0.798424i	0.184423	−	0.982847i	$-0.440958\pi$
$823$	−1.55143e7	−	1.55143e7i	−0.798424	−	0.798424i	−0.982847	+	0.184423i	$0.940958\pi$
$824$		0			0
$825$	889828.	−	7.70349e6i	0.0455167	−	0.394051i
$826$		0			0
$827$	−2.87490e6	−	2.87490e6i	−0.146170	−	0.146170i	0.630235	−	0.776405i	$-0.282959\pi$
$827$	−2.87490e6	−	2.87490e6i	−0.146170	−	0.146170i	−0.776405	+	0.630235i	$0.782959\pi$
$828$		0			0
$829$		−	1.31575e7i		−	0.664945i	−0.943113	−	0.332472i	$-0.892117\pi$
$829$		−	1.31575e7i		−	0.664945i	0.943113	−	0.332472i	$-0.107883\pi$
$830$		0			0
$831$	1.84598e6	+	1.91044e7i	0.0927310	+	0.959689i
$832$		0			0
$833$	−4.81649e6	+	4.81649e6i	−0.240502	+	0.240502i
$834$		0			0
$835$	8.19228e6	−	2.41826e6i	0.406620	−	0.120029i
$836$		0			0
$837$	−1.58188e7	−	8.56840e6i	−0.780477	−	0.422753i
$838$		0			0
$839$	2.80055e7			1.37353			0.686764	−	0.726880i	$-0.259030\pi$
$839$	2.80055e7			1.37353			0.686764	+	0.726880i	$0.259030\pi$
$840$		0			0
$841$	7.67209e6			0.374045
$842$		0			0
$843$	−1.63239e7	−	1.34472e7i	−0.791142	−	0.651724i
$844$		0			0
$845$	−6.33485e6	−	3.44729e6i	−0.305207	−	0.166087i
$846$		0			0
$847$	231630.	−	231630.i	0.0110939	−	0.0110939i
$848$		0			0
$849$	−2.65044e7	+	2.56102e6i	−1.26197	+	0.121939i
$850$		0			0
$851$			3.59655e6i			0.170240i
$852$		0			0
$853$	−1.35022e7	−	1.35022e7i	−0.635376	−	0.635376i	0.314035	−	0.949411i	$-0.398319\pi$
$853$	−1.35022e7	−	1.35022e7i	−0.635376	−	0.635376i	−0.949411	+	0.314035i	$0.898319\pi$
$854$		0			0
$855$	−8.67103e6	+	2.74746e7i	−0.405654	+	1.28533i
$856$		0			0
$857$	1.61749e7	+	1.61749e7i	0.752298	+	0.752298i	0.974908	−	0.222610i	$-0.0714576\pi$
$857$	1.61749e7	+	1.61749e7i	0.752298	+	0.752298i	−0.222610	+	0.974908i	$0.571458\pi$
$858$		0			0
$859$		−	1.91209e6i		−	0.0884149i	−0.999022	−	0.0442075i	$-0.985924\pi$
$859$		−	1.91209e6i		−	0.0884149i	0.999022	−	0.0442075i	$-0.0140763\pi$
$860$		0			0
$861$	579056.	−	55952.0i	0.0266203	−	0.00257222i
$862$		0			0
$863$	4.65774e6	−	4.65774e6i	0.212887	−	0.212887i	−0.592606	−	0.805493i	$-0.701901\pi$
$863$	4.65774e6	−	4.65774e6i	0.212887	−	0.212887i	0.805493	+	0.592606i	$0.201901\pi$
$864$		0			0
$865$	4.26805e6	+	1.44588e7i	0.193950	+	0.657039i
$866$		0			0
$867$	−1.51058e7	−	1.24438e7i	−0.682487	−	0.562216i
$868$		0			0
$869$	6.04643e6			0.271612
$870$		0			0
$871$	−2.67228e7			−1.19354
$872$		0			0
$873$	1.33305e7	+	8.97858e6i	0.591985	+	0.398724i
$874$		0			0
$875$	−274759.	+	319864.i	−0.0121320	+	0.0141236i
$876$		0			0
$877$	2.80145e7	−	2.80145e7i	1.22994	−	1.22994i	0.265952	−	0.963986i	$-0.414314\pi$
$877$	2.80145e7	−	2.80145e7i	1.22994	−	1.22994i	0.963986	−	0.265952i	$-0.0856862\pi$
$878$		0			0
$879$	2.67185e6	+	2.76514e7i	0.116638	+	1.20710i
$880$		0			0
$881$			1.69086e7i			0.733952i	0.930230	+	0.366976i	$0.119607\pi$
$881$			1.69086e7i			0.733952i	−0.930230	+	0.366976i	$0.880393\pi$
$882$		0			0
$883$	6.32751e6	+	6.32751e6i	0.273106	+	0.273106i	0.830349	−	0.557243i	$-0.188141\pi$
$883$	6.32751e6	+	6.32751e6i	0.273106	+	0.273106i	−0.557243	+	0.830349i	$0.688141\pi$
$884$		0			0
$885$	−1.39763e7	−	3.46535e7i	−0.599839	−	1.48727i
$886$		0			0
$887$	−7.73038e6	−	7.73038e6i	−0.329907	−	0.329907i	0.522644	−	0.852551i	$-0.324946\pi$
$887$	−7.73038e6	−	7.73038e6i	−0.329907	−	0.329907i	−0.852551	+	0.522644i	$0.824946\pi$
$888$		0			0
$889$			375296.i			0.0159265i
$890$		0			0
$891$	−3.53292e6	−	8.71076e6i	−0.149087	−	0.367589i
$892$		0			0
$893$	1.31336e7	−	1.31336e7i	0.551133	−	0.551133i
$894$		0			0
$895$	−6.01653e6	+	1.10562e7i	−0.251066	+	0.461368i
$896$		0			0
$897$	−1.64545e6	+	1.99745e6i	−0.0682816	+	0.0828886i
$898$		0			0
$899$	2.52130e7			1.04046
$900$		0			0
$901$	9.73039e6			0.399318
$902$		0			0
$903$	363821.	−	441651.i	0.0148480	−	0.0180244i
$904$		0			0
$905$	6.61279e6	−	1.21519e7i	0.268388	−	0.493199i
$906$		0			0
$907$	−2.25983e7	+	2.25983e7i	−0.912131	+	0.912131i	−0.996440	−	0.0843087i	$-0.973132\pi$
$907$	−2.25983e7	+	2.25983e7i	−0.912131	+	0.912131i	0.0843087	+	0.996440i	$0.473132\pi$
$908$		0			0
$909$	2.37784e7	−	4.63853e6i	0.954491	−	0.186196i
$910$		0			0
$911$		−	33960.8i		−	0.00135576i	−1.00000		0.000677880i	$-0.999784\pi$
$911$		−	33960.8i		−	0.00135576i	1.00000		0.000677880i	$-0.000215776\pi$
$912$		0			0
$913$	9.58324e6	+	9.58324e6i	0.380483	+	0.380483i
$914$		0			0
$915$	7.92014e6	+	1.96375e7i	0.312737	+	0.775414i
$916$		0			0
$917$	−16884.6	−	16884.6i	−0.000663084	−	0.000663084i
$918$		0			0
$919$			1.29040e7i			0.504007i	0.967726	+	0.252004i	$0.0810895\pi$
$919$			1.29040e7i			0.504007i	−0.967726	+	0.252004i	$0.918911\pi$
$920$		0			0
$921$	582026.	+	6.02348e6i	0.0226096	+	0.233991i
$922$		0			0
$923$	2.53823e7	−	2.53823e7i	0.980678	−	0.980678i
$924$		0			0
$925$	3.25819e7	+	6.98979e6i	1.25205	+	0.268602i
$926$		0			0
$927$	1.17786e7	−	1.74877e7i	0.450188	−	0.668394i
$928$		0			0
$929$	−4.22853e7			−1.60750			−0.803749	−	0.594969i	$-0.797164\pi$
$929$	−4.22853e7			−1.60750			−0.803749	+	0.594969i	$0.797164\pi$
$930$		0			0
$931$	3.56334e7			1.34736
$932$		0			0
$933$	2.02865e7	+	1.67116e7i	0.762963	+	0.628511i
$934$		0			0
$935$	1.02141e6	+	3.46020e6i	0.0382094	+	0.129441i
$936$		0			0
$937$	−2.21532e7	+	2.21532e7i	−0.824305	+	0.824305i	−0.986722	−	0.162417i	$-0.948071\pi$
$937$	−2.21532e7	+	2.21532e7i	−0.824305	+	0.824305i	0.162417	+	0.986722i	$0.448071\pi$
$938$		0			0
$939$	−2.56387e7	+	2.47737e6i	−0.948925	+	0.0916909i
$940$		0			0
$941$			2.34896e7i			0.864771i	0.901689	+	0.432385i	$0.142328\pi$
$941$			2.34896e7i			0.864771i	−0.901689	+	0.432385i	$0.857672\pi$
$942$		0			0
$943$	3.68734e6	+	3.68734e6i	0.135031	+	0.135031i
$944$		0			0
$945$	−106241.	+	499967.i	−0.00387001	+	0.0182122i
$946$		0			0
$947$	1.36672e7	+	1.36672e7i	0.495228	+	0.495228i	0.909949	−	0.414721i	$-0.136121\pi$
$947$	1.36672e7	+	1.36672e7i	0.495228	+	0.495228i	−0.414721	+	0.909949i	$0.636121\pi$
$948$		0			0
$949$			4.00209e7i			1.44252i
$950$		0			0
$951$	−4.54435e7	+	4.39103e6i	−1.62937	+	0.157440i
$952$		0			0
$953$	−1.45787e7	+	1.45787e7i	−0.519981	+	0.519981i	−0.917566	−	0.397584i	$-0.869849\pi$
$953$	−1.45787e7	+	1.45787e7i	−0.519981	+	0.519981i	0.397584	+	0.917566i	$0.369849\pi$
$954$		0			0
$955$	−3.86760e7	−	2.10466e7i	−1.37225	−	0.746749i
$956$		0			0
$957$	1.01680e7	+	8.37617e6i	0.358886	+	0.295642i
$958$		0			0
$959$	−628344.			−0.0220623
$960$		0			0
$961$	−6.07327e6			−0.212136
$962$		0			0
$963$	−1.77865e7	+	2.64077e7i	−0.618053	+	0.917623i
$964$		0			0
$965$	−3.66372e7	+	1.08148e7i	−1.26650	+	0.373854i
$966$		0			0
$967$	−1.59614e7	+	1.59614e7i	−0.548914	+	0.548914i	−0.926127	−	0.377213i	$-0.876883\pi$
$967$	−1.59614e7	+	1.59614e7i	−0.548914	+	0.548914i	0.377213	+	0.926127i	$0.376883\pi$
$968$		0			0
$969$	−1.28915e6	−	1.33416e7i	−0.0441056	−	0.456456i
$970$		0			0
$971$		−	3.08866e7i		−	1.05129i	−0.850705	−	0.525644i	$-0.823825\pi$
$971$		−	3.08866e7i		−	1.05129i	0.850705	−	0.525644i	$-0.176175\pi$
$972$		0			0
$973$	2433.93	+	2433.93i	8.24188e−5	+	8.24188e-5i
$974$		0			0
$975$	1.48974e7	+	1.87885e7i	0.501880	+	0.632965i
$976$		0			0
$977$	−2.02775e6	−	2.02775e6i	−0.0679637	−	0.0679637i	0.672308	−	0.740272i	$-0.265303\pi$
$977$	−2.02775e6	−	2.02775e6i	−0.0679637	−	0.0679637i	−0.740272	+	0.672308i	$0.765303\pi$
$978$		0			0
$979$			9.36786e6i			0.312380i
$980$		0			0
$981$	3.61860e7	−	7.05893e6i	1.20052	−	0.234189i
$982$		0			0
$983$	−3.95587e7	+	3.95587e7i	−1.30574	+	1.30574i	−0.381286	+	0.924457i	$0.624519\pi$
$983$	−3.95587e7	+	3.95587e7i	−1.30574	+	1.30574i	−0.924457	+	0.381286i	$0.875481\pi$
$984$		0			0
$985$	2.00666e7	−	5.92342e6i	0.658998	−	0.194528i
$986$		0			0
$987$	209516.	−	254337.i	0.00684581	−	0.00831028i
$988$		0			0
$989$	5.12911e6			0.166744
$990$		0			0
$991$	4.23122e7			1.36862			0.684308	−	0.729193i	$-0.260104\pi$
$991$	4.23122e7			1.36862			0.684308	+	0.729193i	$0.260104\pi$
$992$		0			0
$993$	1.18505e7	−	1.43856e7i	0.381386	−	0.462973i
$994$		0			0
$995$	2.99285e7	+	1.62864e7i	0.958357	+	0.521517i
$996$		0			0
$997$	−1.76141e6	+	1.76141e6i	−0.0561208	+	0.0561208i	−0.734610	−	0.678489i	$-0.762635\pi$
$997$	−1.76141e6	+	1.76141e6i	−0.0561208	+	0.0561208i	0.678489	+	0.734610i	$0.262635\pi$
$998$		0			0
$999$	3.87181e7	−	1.15110e7i	1.22744	−	0.364922i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.i.a.53.8	yes	20
3.2	odd	2	inner	60.6.i.a.53.3	yes	20
5.2	odd	4	inner	60.6.i.a.17.3	✓	20
5.3	odd	4		300.6.i.d.257.8		20
5.4	even	2		300.6.i.d.293.3		20
15.2	even	4	inner	60.6.i.a.17.8	yes	20
15.8	even	4		300.6.i.d.257.3		20
15.14	odd	2		300.6.i.d.293.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.i.a.17.3	✓	20	5.2	odd	4	inner
60.6.i.a.17.8	yes	20	15.2	even	4	inner
60.6.i.a.53.3	yes	20	3.2	odd	2	inner
60.6.i.a.53.8	yes	20	1.1	even	1	trivial
300.6.i.d.257.3		20	15.8	even	4
300.6.i.d.257.8		20	5.3	odd	4
300.6.i.d.293.3		20	5.4	even	2
300.6.i.d.293.8		20	15.14	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(9.91146 - 12.0317i) q^{3} +(26.7203 - 49.1022i) q^{5} +(1.70680 - 1.70680i) q^{7} +(-46.5259 - 238.504i) q^{9} +O(q^{10})\)	\(q+(9.91146 - 12.0317i) q^{3} +(26.7203 - 49.1022i) q^{5} +(1.70680 - 1.70680i) q^{7} +(-46.5259 - 238.504i) q^{9} +159.189i q^{11} +(-348.051 - 348.051i) q^{13} +(-325.947 - 808.167i) q^{15} +(286.676 + 286.676i) q^{17} -2120.89i q^{19} +(-3.61890 - 37.4527i) q^{21} +(238.492 - 238.492i) q^{23} +(-1697.05 - 2624.05i) q^{25} +(-3330.76 - 1804.14i) q^{27} +5308.79 q^{29} +4749.30 q^{31} +(1915.32 + 1577.79i) q^{33} +(-38.2013 - 129.414i) q^{35} +(-7540.18 + 7540.18i) q^{37} +(-7637.36 + 737.968i) q^{39} +15461.0i q^{41} +(10753.2 + 10753.2i) q^{43} +(-12954.3 - 4088.40i) q^{45} +(6192.52 + 6192.52i) q^{47} +16801.2i q^{49} +(6290.58 - 607.835i) q^{51} +(16971.1 - 16971.1i) q^{53} +(7816.52 + 4253.58i) q^{55} +(-25518.0 - 21021.1i) q^{57} +42879.1 q^{59} -24298.8 q^{61} +(-486.489 - 327.669i) q^{63} +(-26390.1 + 7790.03i) q^{65} +(38389.1 - 38389.1i) q^{67} +(-505.672 - 5233.29i) q^{69} +72926.9i q^{71} +(-57492.8 - 57492.8i) q^{73} +(-48392.2 - 5589.76i) q^{75} +(271.703 + 271.703i) q^{77} -37982.7i q^{79} +(-54719.7 + 22193.3i) q^{81} +(60200.5 - 60200.5i) q^{83} +(21736.5 - 6416.33i) q^{85} +(52617.9 - 63874.0i) q^{87} +58847.5 q^{89} -1188.11 q^{91} +(47072.5 - 57142.4i) q^{93} +(-104140. - 56670.9i) q^{95} +(-46768.7 + 46768.7i) q^{97} +(37967.2 - 7406.40i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) \) 20 * q + 2 * q^3 + 76 * q^7	\( 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 76 * q^7 + 1068 * q^13 - 130 * q^15 + 2180 * q^21 + 4060 * q^25 + 1454 * q^27 - 4720 * q^31 - 460 * q^33 - 612 * q^37 - 24012 * q^43 - 18860 * q^45 - 31700 * q^51 + 19200 * q^55 + 33476 * q^57 + 59880 * q^61 + 67208 * q^63 - 80804 * q^67 - 56956 * q^73 - 102470 * q^75 - 9980 * q^81 + 239260 * q^85 + 71540 * q^87 + 218520 * q^91 + 307928 * q^93 - 151164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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