LMFDB - Embedded newform 112.6.p.a.31.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 293 x^{10} + 336 x^{9} + 60118 x^{8} + 67616 x^{7} + 5772748 x^{6} + \cdots + 37147165696 $ x^12 - 3x^11 + 293x^10 + 336x^9 + 60118x^8 + 67616x^7 + 5772748x^6 + 13469792x^5 + 396106192x^4 + 376848768x^3 + 4277005888x^2 - 2508651776*x + 37147165696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Root			$-5.11675 + 8.86248i$ of defining polynomial
Character	$\chi$	$=$	112.31
Dual form			112.6.p.a.47.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.74078 + 8.21127i) q^{3} +(45.4281 - 26.2279i) q^{5} +(-128.021 + 20.4331i) q^{7} +(76.5500 + 132.588i) q^{9} +(-21.1688 - 12.2218i) q^{11} +58.8178i q^{13} +497.363i q^{15} +(334.006 + 192.839i) q^{17} +(292.463 + 506.560i) q^{19} +(439.140 - 1148.09i) q^{21} +(-2812.19 + 1623.62i) q^{23} +(-186.693 + 323.362i) q^{25} -3755.65 q^{27} -6058.46 q^{29} +(-4755.55 + 8236.85i) q^{31} +(200.714 - 115.882i) q^{33} +(-5279.85 + 4285.97i) q^{35} +(2894.31 + 5013.10i) q^{37} +(-482.969 - 278.842i) q^{39} -8858.89i q^{41} +7390.78i q^{43} +(6955.04 + 4015.49i) q^{45} +(-1315.66 - 2278.78i) q^{47} +(15972.0 - 5231.75i) q^{49} +(-3166.90 + 1828.41i) q^{51} +(-6075.73 + 10523.5i) q^{53} -1282.21 q^{55} -5546.01 q^{57} +(-7267.76 + 12588.1i) q^{59} +(-18180.8 + 10496.7i) q^{61} +(-12509.2 - 15410.0i) q^{63} +(1542.67 + 2671.98i) q^{65} +(36597.2 + 21129.4i) q^{67} -30788.8i q^{69} -75175.4i q^{71} +(-8333.30 - 4811.23i) q^{73} +(-1770.14 - 3065.98i) q^{75} +(2959.80 + 1132.11i) q^{77} +(71853.0 - 41484.4i) q^{79} +(-796.954 + 1380.36i) q^{81} +81002.0 q^{83} +20231.0 q^{85} +(28721.8 - 49747.7i) q^{87} +(-26817.6 + 15483.2i) q^{89} +(-1201.83 - 7529.93i) q^{91} +(-45090.0 - 78098.2i) q^{93} +(26572.0 + 15341.4i) q^{95} +131483. i q^{97} -3742.33i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 66 q^{5} - 544 q^{9} - 1602 q^{17} + 2674 q^{21} - 2460 q^{25} + 22512 q^{29} - 15582 q^{33} - 7938 q^{37} + 3072 q^{45} - 23772 q^{49} + 22854 q^{53} - 25900 q^{57} - 12870 q^{61} - 62220 q^{65}+ \cdots - 273182 q^{93}+O(q^{100}) $ 12 * q - 66 * q^5 - 544 * q^9 - 1602 * q^17 + 2674 * q^21 - 2460 * q^25 + 22512 * q^29 - 15582 * q^33 - 7938 * q^37 + 3072 * q^45 - 23772 * q^49 + 22854 * q^53 - 25900 * q^57 - 12870 * q^61 - 62220 * q^65 + 232938 * q^73 + 154098 * q^77 - 93082 * q^81 + 70116 * q^85 - 328542 * q^89 - 273182 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$	−4.74078	+	8.21127i	−0.304121	+	0.526753i	−0.977065	−	0.212940i	$-0.931696\pi$
$3$	−4.74078	+	8.21127i	−0.304121	+	0.526753i	0.672944	+	0.739693i	$0.265029\pi$
$4$		0			0
$5$	45.4281	−	26.2279i	0.812642	−	0.469179i	−0.0352304	−	0.999379i	$-0.511217\pi$
$5$	45.4281	−	26.2279i	0.812642	−	0.469179i	0.847873	+	0.530200i	$0.177883\pi$
$6$		0			0
$7$	−128.021	+	20.4331i	−0.987501	+	0.157612i
$7$	−128.021	+	20.4331i	−0.987501	+	0.157612i
$8$		0			0
$9$	76.5500	+	132.588i	0.315021	+	0.545632i
$10$		0			0
$11$	−21.1688	−	12.2218i	−0.0527492	−	0.0304547i	0.473393	−	0.880851i	$-0.343029\pi$
$11$	−21.1688	−	12.2218i	−0.0527492	−	0.0304547i	−0.526143	+	0.850396i	$0.676362\pi$
$12$		0			0
$13$			58.8178i			0.0965273i	0.998835	+	0.0482636i	$0.0153688\pi$
$13$			58.8178i			0.0965273i	−0.998835	+	0.0482636i	$0.984631\pi$
$14$		0			0
$15$			497.363i			0.570749i
$16$		0			0
$17$	334.006	+	192.839i	0.280306	+	0.161835i	0.633562	−	0.773692i	$-0.281592\pi$
$17$	334.006	+	192.839i	0.280306	+	0.161835i	−0.353256	+	0.935527i	$0.614926\pi$
$18$		0			0
$19$	292.463	+	506.560i	0.185860	+	0.321919i	0.943866	−	0.330328i	$-0.107159\pi$
$19$	292.463	+	506.560i	0.185860	+	0.321919i	−0.758006	+	0.652248i	$0.773826\pi$
$20$		0			0
$21$	439.140	−	1148.09i	0.217297	−	0.568103i
$22$		0			0
$23$	−2812.19	+	1623.62i	−1.10847	+	0.639976i	−0.938433	−	0.345461i	$-0.887722\pi$
$23$	−2812.19	+	1623.62i	−1.10847	+	0.639976i	−0.170038	+	0.985437i	$0.554389\pi$
$24$		0			0
$25$	−186.693	+	323.362i	−0.0597418	+	0.103476i
$26$		0			0
$27$	−3755.65			−0.991460
$28$		0			0
$29$	−6058.46			−1.33773			−0.668863	−	0.743386i	$-0.733219\pi$
$29$	−6058.46			−1.33773			−0.668863	+	0.743386i	$0.733219\pi$
$30$		0			0
$31$	−4755.55	+	8236.85i	−0.888785	+	1.53942i	−0.0474710	+	0.998873i	$0.515116\pi$
$31$	−4755.55	+	8236.85i	−0.888785	+	1.53942i	−0.841314	+	0.540547i	$0.818217\pi$
$32$		0			0
$33$	200.714	−	115.882i	0.0320843	−	0.0185239i
$34$		0			0
$35$	−5279.85	+	4285.97i	−0.728537	+	0.591397i
$36$		0			0
$37$	2894.31	+	5013.10i	0.347569	+	0.602007i	0.985817	−	0.167824i	$-0.0536739\pi$
$37$	2894.31	+	5013.10i	0.347569	+	0.602007i	−0.638248	+	0.769831i	$0.720341\pi$
$38$		0			0
$39$	−482.969	−	278.842i	−0.0508461	−	0.0293560i
$40$		0			0
$41$		−	8858.89i		−	0.823037i	−0.911401	−	0.411519i	$-0.864999\pi$
$41$		−	8858.89i		−	0.823037i	0.911401	−	0.411519i	$-0.135001\pi$
$42$		0			0
$43$			7390.78i			0.609563i	0.952422	+	0.304782i	$0.0985835\pi$
$43$			7390.78i			0.609563i	−0.952422	+	0.304782i	$0.901416\pi$
$44$		0			0
$45$	6955.04	+	4015.49i	0.511998	+	0.295602i
$46$		0			0
$47$	−1315.66	−	2278.78i	−0.0868755	−	0.150473i	0.819313	−	0.573346i	$-0.194355\pi$
$47$	−1315.66	−	2278.78i	−0.0868755	−	0.150473i	−0.906189	+	0.422873i	$0.861022\pi$
$48$		0			0
$49$	15972.0	−	5231.75i	0.950317	−	0.311284i
$50$		0			0
$51$	−3166.90	+	1828.41i	−0.170494	+	0.0984347i
$52$		0			0
$53$	−6075.73	+	10523.5i	−0.297104	+	0.514599i	−0.975472	−	0.220123i	$-0.929354\pi$
$53$	−6075.73	+	10523.5i	−0.297104	+	0.514599i	0.678368	+	0.734722i	$0.262687\pi$
$54$		0			0
$55$	−1282.21			−0.0571549
$56$		0			0
$57$	−5546.01			−0.226096
$58$		0			0
$59$	−7267.76	+	12588.1i	−0.271813	+	0.470794i	−0.969326	−	0.245778i	$-0.920957\pi$
$59$	−7267.76	+	12588.1i	−0.271813	+	0.470794i	0.697513	+	0.716572i	$0.254290\pi$
$60$		0			0
$61$	−18180.8	+	10496.7i	−0.625589	+	0.361184i	−0.779042	−	0.626972i	$-0.784294\pi$
$61$	−18180.8	+	10496.7i	−0.625589	+	0.361184i	0.153453	+	0.988156i	$0.450961\pi$
$62$		0			0
$63$	−12509.2	−	15410.0i	−0.397081	−	0.489161i
$64$		0			0
$65$	1542.67	+	2671.98i	0.0452886	+	0.0784422i
$66$		0			0
$67$	36597.2	+	21129.4i	0.996003	+	0.575042i	0.907063	−	0.420995i	$-0.138319\pi$
$67$	36597.2	+	21129.4i	0.996003	+	0.575042i	0.0889396	+	0.996037i	$0.471652\pi$
$68$		0			0
$69$		−	30788.8i		−	0.778521i
$70$		0			0
$71$		−	75175.4i		−	1.76982i	−0.465760	−	0.884911i	$-0.654219\pi$
$71$		−	75175.4i		−	1.76982i	0.465760	−	0.884911i	$-0.345781\pi$
$72$		0			0
$73$	−8333.30	−	4811.23i	−0.183025	−	0.105669i	0.405688	−	0.914012i	$-0.367032\pi$
$73$	−8333.30	−	4811.23i	−0.183025	−	0.105669i	−0.588713	+	0.808342i	$0.700365\pi$
$74$		0			0
$75$	−1770.14	−	3065.98i	−0.0363375	−	0.0629384i
$76$		0			0
$77$	2959.80	+	1132.11i	0.0568899	+	0.0217602i
$78$		0			0
$79$	71853.0	−	41484.4i	1.29532	−	0.747854i	0.315729	−	0.948850i	$-0.397751\pi$
$79$	71853.0	−	41484.4i	1.29532	−	0.747854i	0.979592	+	0.200996i	$0.0644178\pi$
$80$		0			0
$81$	−796.954	+	1380.36i	−0.0134965	+	0.0233766i
$82$		0			0
$83$	81002.0			1.29063			0.645313	−	0.763918i	$-0.276727\pi$
$83$	81002.0			1.29063			0.645313	+	0.763918i	$0.276727\pi$
$84$		0			0
$85$	20231.0			0.303718
$86$		0			0
$87$	28721.8	−	49747.7i	0.406831	−	0.704652i
$88$		0			0
$89$	−26817.6	+	15483.2i	−0.358877	+	0.207197i	−0.668588	−	0.743633i	$-0.733101\pi$
$89$	−26817.6	+	15483.2i	−0.358877	+	0.207197i	0.309711	+	0.950831i	$0.399768\pi$
$90$		0			0
$91$	−1201.83	−	7529.93i	−0.0152139	−	0.0953208i
$92$		0			0
$93$	−45090.0	−	78098.2i	−0.540596	−	0.936341i
$94$		0			0
$95$	26572.0	+	15341.4i	0.302076	+	0.174404i
$96$		0			0
$97$			131483.i			1.41886i	0.704775	+	0.709431i	$0.251048\pi$
$97$			131483.i			1.41886i	−0.704775	+	0.709431i	$0.748952\pi$
$98$		0			0
$99$		−	3742.33i		−	0.0383755i
$100$		0			0
$101$	98317.7	+	56763.7i	0.959021	+	0.553691i	0.895872	−	0.444313i	$-0.146552\pi$
$101$	98317.7	+	56763.7i	0.959021	+	0.553691i	0.0631495	+	0.998004i	$0.479886\pi$
$102$		0			0
$103$	−52916.3	−	91653.7i	−0.491469	−	0.851250i	0.508482	−	0.861072i	$-0.330207\pi$
$103$	−52916.3	−	91653.7i	−0.491469	−	0.851250i	−0.999952	+	0.00982251i	$0.996873\pi$
$104$		0			0
$105$	−10162.7	−	63673.1i	−0.0899570	−	0.563616i
$106$		0			0
$107$	29305.1	−	16919.3i	0.247448	−	0.142864i	−0.371147	−	0.928574i	$-0.621035\pi$
$107$	29305.1	−	16919.3i	0.247448	−	0.142864i	0.618595	+	0.785710i	$0.287702\pi$
$108$		0			0
$109$	28521.5	−	49400.7i	0.229935	−	0.398260i	−0.727853	−	0.685733i	$-0.759482\pi$
$109$	28521.5	−	49400.7i	0.229935	−	0.398260i	0.957789	+	0.287473i	$0.0928152\pi$
$110$		0			0
$111$	−54885.2			−0.422812
$112$		0			0
$113$	118518.			0.873148			0.436574	−	0.899668i	$-0.356192\pi$
$113$	118518.			0.873148			0.436574	+	0.899668i	$0.356192\pi$
$114$		0			0
$115$	−85168.1	+	147515.i	−0.600527	+	1.04014i
$116$		0			0
$117$	−7798.56	+	4502.50i	−0.0526683	+	0.0304081i
$118$		0			0
$119$	−46700.2	−	17862.7i	−0.302309	−	0.115632i
$120$		0			0
$121$	−80226.8	−	138957.i	−0.498145	−	0.862812i
$122$		0			0
$123$	72742.7	+	41998.0i	0.433538	+	0.250303i
$124$		0			0
$125$			183511.i			1.05048i
$126$		0			0
$127$		−	47301.9i		−	0.260237i	−0.991498	−	0.130119i	$-0.958464\pi$
$127$		−	47301.9i		−	0.260237i	0.991498	−	0.130119i	$-0.0415358\pi$
$128$		0			0
$129$	−60687.7	−	35038.0i	−0.321090	−	0.185381i
$130$		0			0
$131$	118503.	+	205253.i	0.603325	+	1.04499i	0.992314	+	0.123747i	$0.0394912\pi$
$131$	118503.	+	205253.i	0.603325	+	1.04499i	−0.388989	+	0.921243i	$0.627175\pi$
$132$		0			0
$133$	−47792.1	−	58874.7i	−0.234276	−	0.288602i
$134$		0			0
$135$	−170612.	+	98502.8i	−0.805702	+	0.465172i
$136$		0			0
$137$	−44673.5	+	77376.8i	−0.203352	+	0.352216i	−0.949606	−	0.313445i	$-0.898517\pi$
$137$	−44673.5	+	77376.8i	−0.203352	+	0.352216i	0.746254	+	0.665661i	$0.231850\pi$
$138$		0			0
$139$	−146398.			−0.642685			−0.321342	−	0.946963i	$-0.604134\pi$
$139$	−146398.			−0.642685			−0.321342	+	0.946963i	$0.604134\pi$
$140$		0			0
$141$	24948.9			0.105683
$142$		0			0
$143$	718.861	−	1245.10i	0.00293971	−	0.00509173i
$144$		0			0
$145$	−275224.	+	158901.i	−1.08709	+	0.627633i
$146$		0			0
$147$	−32760.3	+	155953.i	−0.125042	+	0.595251i
$148$		0			0
$149$	−208817.	−	361681.i	−0.770548	−	1.33463i	−0.937263	−	0.348624i	$-0.886649\pi$
$149$	−208817.	−	361681.i	−0.770548	−	1.33463i	0.166714	−	0.986005i	$-0.446684\pi$
$150$		0			0
$151$	−370440.	−	213874.i	−1.32213	−	0.763334i	−0.338065	−	0.941123i	$-0.609772\pi$
$151$	−370440.	−	213874.i	−1.32213	−	0.763334i	−0.984069	+	0.177789i	$0.943106\pi$
$152$		0			0
$153$			59047.2i			0.203925i
$154$		0			0
$155$			498913.i			1.66800i
$156$		0			0
$157$	413174.	+	238546.i	1.33778	+	0.772367i	0.986478	−	0.163895i	$-0.0524060\pi$
$157$	413174.	+	238546.i	1.33778	+	0.772367i	0.351301	+	0.936262i	$0.385739\pi$
$158$		0			0
$159$	−57607.4	−	99778.9i	−0.180711	−	0.313001i
$160$		0			0
$161$	326844.	−	265319.i	0.993748	−	0.806685i
$162$		0			0
$163$	−396470.	+	228902.i	−1.16880	+	0.674808i	−0.953398	−	0.301717i	$-0.902440\pi$
$163$	−396470.	+	228902.i	−1.16880	+	0.674808i	−0.215404	+	0.976525i	$0.569107\pi$
$164$		0			0
$165$	6078.69	−	10528.6i	0.0173820	−	0.0301065i
$166$		0			0
$167$	1763.27			0.00489247			0.00244623	−	0.999997i	$-0.499221\pi$
$167$	1763.27			0.00489247			0.00244623	+	0.999997i	$0.499221\pi$
$168$		0			0
$169$	367833.			0.990682
$170$		0			0
$171$	−44776.0	+	77554.4i	−0.117100	+	0.202822i
$172$		0			0
$173$	−525979.	+	303674.i	−1.33614	+	0.771423i	−0.986233	−	0.165360i	$-0.947121\pi$
$173$	−525979.	+	303674.i	−1.33614	+	0.771423i	−0.349911	+	0.936783i	$0.613788\pi$
$174$		0			0
$175$	17293.4	−	45212.0i	0.0426861	−	0.111599i
$176$		0			0
$177$	−68909.7	−	119355.i	−0.165328	−	0.286357i
$178$		0			0
$179$	231557.	+	133689.i	0.540163	+	0.311863i	0.745145	−	0.666903i	$-0.232380\pi$
$179$	231557.	+	133689.i	0.540163	+	0.311863i	−0.204982	+	0.978766i	$0.565714\pi$
$180$		0			0
$181$		−	170105.i		−	0.385941i	−0.981205	−	0.192971i	$-0.938188\pi$
$181$		−	170105.i		−	0.385941i	0.981205	−	0.192971i	$-0.0618122\pi$
$182$		0			0
$183$		−	199050.i		−	0.439375i
$184$		0			0
$185$	262966.	+	151824.i	0.564898	+	0.326144i
$186$		0			0
$187$	−4713.68	−	8164.34i	−0.00985727	−	0.0170733i
$188$		0			0
$189$	480803.	−	76739.6i	0.979068	−	0.156266i
$190$		0			0
$191$	630495.	−	364016.i	1.25054	−	0.722000i	0.279324	−	0.960197i	$-0.409890\pi$
$191$	630495.	−	364016.i	1.25054	−	0.722000i	0.971217	+	0.238196i	$0.0765562\pi$
$192$		0			0
$193$	−237753.	+	411800.i	−0.459443	+	0.795779i	−0.998932	−	0.0462138i	$-0.985284\pi$
$193$	−237753.	+	411800.i	−0.459443	+	0.795779i	0.539488	+	0.841993i	$0.318618\pi$
$194$		0			0
$195$	−29253.8			−0.0550929
$196$		0			0
$197$	−411672.			−0.755763			−0.377881	−	0.925854i	$-0.623347\pi$
$197$	−411672.			−0.755763			−0.377881	+	0.925854i	$0.623347\pi$
$198$		0			0
$199$	363719.	−	629981.i	0.651079	−	1.12770i	−0.331782	−	0.943356i	$-0.607650\pi$
$199$	363719.	−	629981.i	0.651079	−	1.12770i	0.982861	−	0.184346i	$-0.0590168\pi$
$200$		0			0
$201$	−346998.	+	200340.i	−0.605811	+	0.349765i
$202$		0			0
$203$	775613.	−	123793.i	1.32101	−	0.210842i
$204$		0			0
$205$	−232350.	−	402442.i	−0.386152	−	0.668835i
$206$		0			0
$207$	−430546.	−	248576.i	−0.698382	−	0.403211i
$208$		0			0
$209$		−	14297.7i		−	0.0226413i
$210$		0			0
$211$		−	88502.1i		−	0.136851i	−0.997656	−	0.0684254i	$-0.978202\pi$
$211$		−	88502.1i		−	0.136851i	0.997656	−	0.0684254i	$-0.0217975\pi$
$212$		0			0
$213$	617285.	+	356390.i	0.932260	+	0.538241i
$214$		0			0
$215$	193845.	+	335749.i	0.285994	+	0.495357i
$216$		0			0
$217$	440508.	−	1.15166e6i	0.635045	−	1.66026i
$218$		0			0
$219$	79012.7	−	45618.0i	0.111323	−	0.0642726i
$220$		0			0
$221$	−11342.3	+	19645.5i	−0.0156215	+	0.0270572i
$222$		0			0
$223$	145641.			0.196120			0.0980602	−	0.995180i	$-0.468736\pi$
$223$	145641.			0.196120			0.0980602	+	0.995180i	$0.468736\pi$
$224$		0			0
$225$	−57165.4			−0.0752796
$226$		0			0
$227$	485173.	−	840344.i	0.624930	−	1.08241i	−0.363624	−	0.931546i	$-0.618461\pi$
$227$	485173.	−	840344.i	0.624930	−	1.08241i	0.988554	−	0.150865i	$-0.0482060\pi$
$228$		0			0
$229$	−1.13344e6	+	654390.i	−1.42827	+	0.824609i	−0.996984	−	0.0776082i	$-0.975272\pi$
$229$	−1.13344e6	+	654390.i	−1.42827	+	0.824609i	−0.431281	+	0.902218i	$0.641938\pi$
$230$		0			0
$231$	−23327.8	+	18936.6i	−0.0287637	+	0.0233492i
$232$		0			0
$233$	−547913.	−	949012.i	−0.661183	−	1.14520i	−0.980305	−	0.197489i	$-0.936721\pi$
$233$	−547913.	−	949012.i	−0.661183	−	1.14520i	0.319123	−	0.947713i	$-0.396612\pi$
$234$		0			0
$235$	−119535.	−	69013.8i	−0.141197	−	0.0815204i
$236$		0			0
$237$			786673.i			0.909753i
$238$		0			0
$239$		−	48553.5i		−	0.0549826i	−0.999622	−	0.0274913i	$-0.991248\pi$
$239$		−	48553.5i		−	0.0549826i	0.999622	−	0.0274913i	$-0.00875186\pi$
$240$		0			0
$241$	729752.	+	421323.i	0.809344	+	0.467275i	0.846728	−	0.532026i	$-0.178569\pi$
$241$	729752.	+	421323.i	0.809344	+	0.467275i	−0.0373842	+	0.999301i	$0.511903\pi$
$242$		0			0
$243$	−463867.	−	803442.i	−0.503939	−	0.872848i
$244$		0			0
$245$	588358.	−	656580.i	0.626220	−	0.698832i
$246$		0			0
$247$	−29794.7	+	17202.0i	−0.0310740	+	0.0179406i
$248$		0			0
$249$	−384013.	+	665130.i	−0.392507	+	0.679842i
$250$		0			0
$251$	1.42770e6			1.43039			0.715193	−	0.698927i	$-0.246339\pi$
$251$	1.42770e6			1.43039			0.715193	+	0.698927i	$0.246339\pi$
$252$		0			0
$253$	79374.3			0.0779612
$254$		0			0
$255$	−95910.8	+	166122.i	−0.0923670	+	0.159984i
$256$		0			0
$257$	−398669.	+	230172.i	−0.376513	+	0.217380i	−0.676300	−	0.736626i	$-0.736418\pi$
$257$	−398669.	+	230172.i	−0.376513	+	0.217380i	0.299787	+	0.954006i	$0.403084\pi$
$258$		0			0
$259$	−472967.	−	582644.i	−0.438108	−	0.539702i
$260$		0			0
$261$	−463775.	−	803282.i	−0.421411	−	0.729906i
$262$		0			0
$263$	962880.	+	555919.i	0.858387	+	0.495590i	0.863472	−	0.504397i	$-0.168285\pi$
$263$	962880.	+	555919.i	0.858387	+	0.495590i	−0.00508508	+	0.999987i	$0.501619\pi$
$264$		0			0
$265$			637414.i			0.557580i
$266$		0			0
$267$		−	293609.i		−	0.252053i
$268$		0			0
$269$	1.10870e6	+	640110.i	0.934189	+	0.539354i	0.888134	−	0.459585i	$-0.152002\pi$
$269$	1.10870e6	+	640110.i	0.934189	+	0.539354i	0.0460552	+	0.998939i	$0.485335\pi$
$270$		0			0
$271$	487763.	+	844830.i	0.403446	+	0.698789i	0.994139	−	0.108107i	$-0.0344790\pi$
$271$	487763.	+	844830.i	0.403446	+	0.698789i	−0.590693	+	0.806896i	$0.701146\pi$
$272$		0			0
$273$	67528.0	+	25829.2i	0.0548374	+	0.0209751i
$274$		0			0
$275$	7904.16	−	4563.47i	0.00630266	−	0.00363884i
$276$		0			0
$277$	737950.	−	1.27817e6i	0.577866	−	1.00089i	−0.417858	−	0.908513i	$-0.637219\pi$
$277$	737950.	−	1.27817e6i	0.577866	−	1.00089i	0.995724	−	0.0923809i	$-0.0294478\pi$
$278$		0			0
$279$	−1.45615e6			−1.11994
$280$		0			0
$281$	1.59868e6			1.20780			0.603902	−	0.797058i	$-0.293612\pi$
$281$	1.59868e6			1.20780			0.603902	+	0.797058i	$0.293612\pi$
$282$		0			0
$283$	−554736.	+	960832.i	−0.411738	+	0.713150i	−0.995080	−	0.0990760i	$-0.968411\pi$
$283$	−554736.	+	960832.i	−0.411738	+	0.713150i	0.583342	+	0.812226i	$0.301745\pi$
$284$		0			0
$285$	−251944.	+	145460.i	−0.183735	+	0.106080i
$286$		0			0
$287$	181015.	+	1.13413e6i	0.129721	+	0.812750i
$288$		0			0
$289$	−635555.	−	1.10081e6i	−0.447619	−	0.775299i
$290$		0			0
$291$	−1.07964e6	−	623332.i	−0.747390	−	0.431506i
$292$		0			0
$293$			1.42205e6i			0.967710i	0.875148	+	0.483855i	$0.160764\pi$
$293$			1.42205e6i			0.967710i	−0.875148	+	0.483855i	$0.839236\pi$
$294$		0			0
$295$			762472.i			0.510116i
$296$		0			0
$297$	79502.7	+	45900.9i	0.0522987	+	0.0301947i
$298$		0			0
$299$	−95497.5	−	165406.i	−0.0617752	−	0.106998i
$300$		0			0
$301$	−151017.	−	946178.i	−0.0960745	−	0.601945i
$302$		0			0
$303$	−932205.	+	538209.i	−0.583317	+	0.336778i
$304$		0			0
$305$	−550613.	+	953690.i	−0.338920	+	0.587027i
$306$		0			0
$307$	−2.54564e6			−1.54153			−0.770764	−	0.637121i	$-0.780125\pi$
$307$	−2.54564e6			−1.54153			−0.770764	+	0.637121i	$0.780125\pi$
$308$		0			0
$309$	1.00346e6			0.597865
$310$		0			0
$311$	−1.20916e6	+	2.09432e6i	−0.708895	+	1.22784i	0.256373	+	0.966578i	$0.417472\pi$
$311$	−1.20916e6	+	2.09432e6i	−0.708895	+	1.22784i	−0.965268	+	0.261263i	$0.915861\pi$
$312$		0			0
$313$	−2.99487e6	+	1.72909e6i	−1.72790	+	0.997601i	−0.829338	+	0.558747i	$0.811282\pi$
$313$	−2.99487e6	+	1.72909e6i	−1.72790	+	0.997601i	−0.898558	+	0.438854i	$0.855384\pi$
$314$		0			0
$315$	−972443.	−	371956.i	−0.552189	−	0.211210i
$316$		0			0
$317$	−1.56117e6	−	2.70402e6i	−0.872572	−	1.51134i	−0.859327	−	0.511427i	$-0.829117\pi$
$317$	−1.56117e6	−	2.70402e6i	−0.872572	−	1.51134i	−0.0132455	−	0.999912i	$-0.504216\pi$
$318$		0			0
$319$	128251.	+	74045.5i	0.0705639	+	0.0407401i
$320$		0			0
$321$			320843.i			0.173792i
$322$		0			0
$323$			225592.i			0.120315i
$324$		0			0
$325$	−19019.4	−	10980.9i	−0.00998824	−	0.00576671i
$326$		0			0
$327$	270428.	+	468395.i	0.139857	+	0.242239i
$328$		0			0
$329$	214995.	+	264850.i	0.109506	+	0.134899i
$330$		0			0
$331$	−752986.	+	434737.i	−0.377761	+	0.218100i	−0.676844	−	0.736127i	$-0.736653\pi$
$331$	−752986.	+	434737.i	−0.377761	+	0.218100i	0.299083	+	0.954227i	$0.403319\pi$
$332$		0			0
$333$	−443119.	+	767505.i	−0.218983	+	0.379289i
$334$		0			0
$335$	2.21672e6			1.07919
$336$		0			0
$337$	1.87504e6			0.899365			0.449682	−	0.893188i	$-0.351537\pi$
$337$	1.87504e6			0.899365			0.449682	+	0.893188i	$0.351537\pi$
$338$		0			0
$339$	−561867.	+	973182.i	−0.265543	+	0.459934i
$340$		0			0
$341$	201339.	−	116243.i	0.0937653	−	0.0541354i
$342$		0			0
$343$	−1.93785e6	+	996134.i	−0.889377	+	0.457175i
$344$		0			0
$345$	−807527.	−	1.39868e6i	−0.365266	−	0.632659i
$346$		0			0
$347$	3.15389e6	+	1.82090e6i	1.40612	+	0.811826i	0.995012	−	0.0997590i	$-0.0318072\pi$
$347$	3.15389e6	+	1.82090e6i	1.40612	+	0.811826i	0.411112	+	0.911585i	$0.365141\pi$
$348$		0			0
$349$			665541.i			0.292490i	0.989248	+	0.146245i	$0.0467188\pi$
$349$			665541.i			0.292490i	−0.989248	+	0.146245i	$0.953281\pi$
$350$		0			0
$351$		−	220899.i		−	0.0957030i
$352$		0			0
$353$	376510.	+	217378.i	0.160820	+	0.0928493i	0.578250	−	0.815860i	$-0.303736\pi$
$353$	376510.	+	217378.i	0.160820	+	0.0928493i	−0.417430	+	0.908709i	$0.637069\pi$
$354$		0			0
$355$	−1.97169e6	−	3.41507e6i	−0.830364	−	1.43823i
$356$		0			0
$357$	368071.	−	298785.i	0.152848	−	0.124076i
$358$		0			0
$359$	1.83391e6	−	1.05881e6i	0.751005	−	0.433593i	−0.0750521	−	0.997180i	$-0.523912\pi$
$359$	1.83391e6	−	1.05881e6i	0.751005	−	0.433593i	0.826057	+	0.563587i	$0.190579\pi$
$360$		0			0
$361$	1.06698e6	−	1.84806e6i	0.430912	−	0.746361i
$362$		0			0
$363$	1.52135e6			0.605986
$364$		0			0
$365$	−504754.			−0.198312
$366$		0			0
$367$	−122836.	+	212757.i	−0.0476057	+	0.0824555i	−0.888846	−	0.458205i	$-0.848492\pi$
$367$	−122836.	+	212757.i	−0.0476057	+	0.0824555i	0.841241	+	0.540661i	$0.181826\pi$
$368$		0			0
$369$	1.17459e6	−	678148.i	0.449075	−	0.259274i
$370$		0			0
$371$	562796.	−	1.47138e6i	0.212283	−	0.554994i
$372$		0			0
$373$	1.74188e6	+	3.01702e6i	0.648254	+	1.12281i	0.983540	+	0.180692i	$0.0578339\pi$
$373$	1.74188e6	+	3.01702e6i	0.648254	+	1.12281i	−0.335286	+	0.942117i	$0.608833\pi$
$374$		0			0
$375$	−1.50686e6	−	869984.i	−0.553342	−	0.319472i
$376$		0			0
$377$		−	356345.i		−	0.129127i
$378$		0			0
$379$		−	3.07711e6i		−	1.10039i	−0.835037	−	0.550194i	$-0.814554\pi$
$379$		−	3.07711e6i		−	1.10039i	0.835037	−	0.550194i	$-0.185446\pi$
$380$		0			0
$381$	388409.	+	224248.i	0.137081	+	0.0791436i
$382$		0			0
$383$	2.41980e6	+	4.19121e6i	0.842912	+	1.45997i	0.887423	+	0.460957i	$0.152494\pi$
$383$	2.41980e6	+	4.19121e6i	0.842912	+	1.45997i	−0.0445109	+	0.999009i	$0.514173\pi$
$384$		0			0
$385$	164151.	−	26199.6i	0.0564405	−	0.00900831i
$386$		0			0
$387$	−979932.	+	565764.i	−0.332597	+	0.192025i
$388$		0			0
$389$	−1.08742e6	+	1.88347e6i	−0.364354	+	0.631079i	−0.988672	−	0.150091i	$-0.952043\pi$
$389$	−1.08742e6	+	1.88347e6i	−0.364354	+	0.631079i	0.624319	+	0.781170i	$0.285377\pi$
$390$		0			0
$391$	−1.25238e6			−0.414281
$392$		0			0
$393$	−2.24719e6			−0.733936
$394$		0			0
$395$	2.17610e6	−	3.76911e6i	0.701755	−	1.21547i
$396$		0			0
$397$	−372640.	+	215144.i	−0.118662	+	0.0685098i	−0.558156	−	0.829736i	$-0.688491\pi$
$397$	−372640.	+	215144.i	−0.118662	+	0.0685098i	0.439494	+	0.898246i	$0.355158\pi$
$398$		0			0
$399$	710008.	−	113322.i	0.223270	−	0.0356355i
$400$		0			0
$401$	44992.6	+	77929.5i	0.0139727	+	0.0242014i	0.872927	−	0.487850i	$-0.162219\pi$
$401$	44992.6	+	77929.5i	0.0139727	+	0.0242014i	−0.858955	+	0.512052i	$0.828886\pi$
$402$		0			0
$403$	−484473.	−	279711.i	−0.148596	−	0.0857920i
$404$		0			0
$405$			83609.7i			0.0253291i
$406$		0			0
$407$		−	141495.i		−	0.0423405i
$408$		0			0
$409$	2.06212e6	+	1.19057e6i	0.609546	+	0.351921i	0.772788	−	0.634665i	$-0.218862\pi$
$409$	2.06212e6	+	1.19057e6i	0.609546	+	0.351921i	−0.163242	+	0.986586i	$0.552195\pi$
$410$		0			0
$411$	−423575.	−	733653.i	−0.123687	−	0.214233i
$412$		0			0
$413$	673214.	−	1.76005e6i	0.194213	−	0.507751i
$414$		0			0
$415$	3.67977e6	−	2.12451e6i	1.04882	−	0.605535i
$416$		0			0
$417$	694041.	−	1.20211e6i	0.195454	−	0.338536i
$418$		0			0
$419$	943020.			0.262413			0.131207	−	0.991355i	$-0.458115\pi$
$419$	943020.			0.262413			0.131207	+	0.991355i	$0.458115\pi$
$420$		0			0
$421$	−2.41477e6			−0.664005			−0.332002	−	0.943279i	$-0.607724\pi$
$421$	−2.41477e6			−0.664005			−0.332002	+	0.943279i	$0.607724\pi$
$422$		0			0
$423$	201427.	−	348881.i	0.0547351	−	0.0948041i
$424$		0			0
$425$	−124713.	+	72003.3i	−0.0334920	+	0.0193366i
$426$		0			0
$427$	2.11306e6	−	1.71529e6i	0.560843	−	0.455270i
$428$		0			0
$429$	6815.93	+	11805.5i	0.00178806	+	0.00309701i
$430$		0			0
$431$	3.50960e6	+	2.02627e6i	0.910048	+	0.525417i	0.880447	−	0.474145i	$-0.157243\pi$
$431$	3.50960e6	+	2.02627e6i	0.910048	+	0.525417i	0.0296015	+	0.999562i	$0.490576\pi$
$432$		0			0
$433$		−	4.08792e6i		−	1.04781i	−0.851777	−	0.523905i	$-0.824475\pi$
$433$		−	4.08792e6i		−	1.04781i	0.851777	−	0.523905i	$-0.175525\pi$
$434$		0			0
$435$		−	3.01325e6i		−	0.763506i
$436$		0			0
$437$	−1.64492e6	−	949694.i	−0.412041	−	0.237892i
$438$		0			0
$439$	−2.04135e6	−	3.53573e6i	−0.505541	−	0.875623i	−0.999979	−	0.00641057i	$-0.997959\pi$
$439$	−2.04135e6	−	3.53573e6i	−0.505541	−	0.875623i	0.494438	−	0.869213i	$-0.335374\pi$
$440$		0			0
$441$	1.91632e6	+	1.71721e6i	0.469216	+	0.420462i
$442$		0			0
$443$	3.64580e6	−	2.10491e6i	0.882641	−	0.509593i	0.0111124	−	0.999938i	$-0.496463\pi$
$443$	3.64580e6	−	2.10491e6i	0.882641	−	0.509593i	0.871528	+	0.490345i	$0.163129\pi$
$444$		0			0
$445$	−812182.	+	1.40674e6i	−0.194425	+	0.336755i
$446$		0			0
$447$	3.95982e6			0.937361
$448$		0			0
$449$	−3.96898e6			−0.929102			−0.464551	−	0.885546i	$-0.653784\pi$
$449$	−3.96898e6			−0.929102			−0.464551	+	0.885546i	$0.653784\pi$
$450$		0			0
$451$	−108272.	+	187532.i	−0.0250654	+	0.0434145i
$452$		0			0
$453$	3.51235e6	−	2.02785e6i	0.804178	−	0.464292i
$454$		0			0
$455$	−252091.	−	310549.i	−0.0570860	−	0.0703237i
$456$		0			0
$457$	3.40079e6	+	5.89033e6i	0.761708	+	1.31932i	0.941969	+	0.335699i	$0.108972\pi$
$457$	3.40079e6	+	5.89033e6i	0.761708	+	1.31932i	−0.180261	+	0.983619i	$0.557694\pi$
$458$		0			0
$459$	−1.25441e6	−	724233.i	−0.277912	−	0.160453i
$460$		0			0
$461$		−	1.05680e6i		−	0.231601i	−0.993272	−	0.115801i	$-0.963057\pi$
$461$		−	1.05680e6i		−	0.231601i	0.993272	−	0.115801i	$-0.0369433\pi$
$462$		0			0
$463$			3.32704e6i			0.721283i	0.932704	+	0.360642i	$0.117442\pi$
$463$			3.32704e6i			0.721283i	−0.932704	+	0.360642i	$0.882558\pi$
$464$		0			0
$465$	−4.09671e6	−	2.36524e6i	−0.878623	−	0.507273i
$466$		0			0
$467$	296405.	+	513388.i	0.0628917	+	0.108932i	0.895757	−	0.444544i	$-0.146634\pi$
$467$	296405.	+	513388.i	0.0628917	+	0.108932i	−0.832865	+	0.553476i	$0.813301\pi$
$468$		0			0
$469$	−5.11696e6	−	1.95722e6i	−1.07419	−	0.410873i
$470$		0			0
$471$	−3.91754e6	+	2.26179e6i	−0.813694	+	0.469786i
$472$		0			0
$473$	90328.9	−	156454.i	0.0185641	−	0.0321540i
$474$		0			0
$475$	−218403.			−0.0444145
$476$		0			0
$477$	−1.86039e6			−0.374375
$478$		0			0
$479$	1.78128e6	−	3.08527e6i	0.354726	−	0.614404i	−0.632345	−	0.774687i	$-0.717907\pi$
$479$	1.78128e6	−	3.08527e6i	0.354726	−	0.614404i	0.987071	+	0.160283i	$0.0512407\pi$
$480$		0			0
$481$	−294859.	+	170237.i	−0.0581101	+	0.0335499i
$482$		0			0
$483$	629112.	+	3.94163e6i	0.122704	+	0.768790i
$484$		0			0
$485$	3.44852e6	+	5.97302e6i	0.665700	+	1.15303i
$486$		0			0
$487$	−5.36910e6	−	3.09985e6i	−1.02584	−	0.592268i	−0.110049	−	0.993926i	$-0.535101\pi$
$487$	−5.36910e6	−	3.09985e6i	−1.02584	−	0.592268i	−0.915790	+	0.401658i	$0.868434\pi$
$488$		0			0
$489$		−	4.34069e6i		−	0.820894i
$490$		0			0
$491$			8.52991e6i			1.59676i	0.602152	+	0.798382i	$0.294310\pi$
$491$			8.52991e6i			1.59676i	−0.602152	+	0.798382i	$0.705690\pi$
$492$		0			0
$493$	−2.02356e6	−	1.16830e6i	−0.374972	−	0.216490i
$494$		0			0
$495$	−98153.4	−	170007.i	−0.0180050	−	0.0311855i
$496$		0			0
$497$	1.53607e6	+	9.62406e6i	0.278945	+	1.74770i
$498$		0			0
$499$	−5.79813e6	+	3.34755e6i	−1.04241	+	0.601833i	−0.920514	−	0.390710i	$-0.872229\pi$
$499$	−5.79813e6	+	3.34755e6i	−1.04241	+	0.601833i	−0.121892	+	0.992543i	$0.538896\pi$
$500$		0			0
$501$	−8359.28	+	14478.7i	−0.00148790	+	0.00257712i
$502$		0			0
$503$	−8.34500e6			−1.47064			−0.735320	−	0.677720i	$-0.762968\pi$
$503$	−8.34500e6			−1.47064			−0.735320	+	0.677720i	$0.762968\pi$
$504$		0			0
$505$	5.95518e6			1.03912
$506$		0			0
$507$	−1.74382e6	+	3.02038e6i	−0.301288	+	0.521845i
$508$		0			0
$509$	9.27824e6	−	5.35679e6i	1.58734	−	0.916453i	0.593601	−	0.804760i	$-0.297706\pi$
$509$	9.27824e6	−	5.35679e6i	1.58734	−	0.916453i	0.993743	−	0.111694i	$-0.0356276\pi$
$510$		0			0
$511$	1.16515e6	+	445666.i	0.197392	+	0.0755017i
$512$		0			0
$513$	−1.09839e6	−	1.90246e6i	−0.184273	−	0.319170i
$514$		0			0
$515$	−4.80777e6	−	2.77577e6i	−0.798777	−	0.461174i
$516$		0			0
$517$			64318.9i			0.0105831i
$518$		0			0
$519$		−	5.75861e6i		−	0.938425i
$520$		0			0
$521$	−1.93965e6	−	1.11985e6i	−0.313060	−	0.180745i	0.335235	−	0.942135i	$-0.391184\pi$
$521$	−1.93965e6	−	1.11985e6i	−0.313060	−	0.180745i	−0.648295	+	0.761389i	$0.724518\pi$
$522$		0			0
$523$	6.00615e6	+	1.04030e7i	0.960156	+	1.66304i	0.722102	+	0.691786i	$0.243176\pi$
$523$	6.00615e6	+	1.04030e7i	0.960156	+	1.66304i	0.238054	+	0.971252i	$0.423491\pi$
$524$		0			0
$525$	289264.	+	356341.i	0.0458032	+	0.0564245i
$526$		0			0
$527$	−3.17677e6	+	1.83411e6i	−0.498263	+	0.287672i
$528$		0			0
$529$	2.05409e6	−	3.55778e6i	0.319139	−	0.552764i
$530$		0			0
$531$	−2.22539e6			−0.342507
$532$		0			0
$533$	521060.			0.0794456
$534$		0			0
$535$	887516.	−	1.53722e6i	0.134058	−	0.232195i
$536$		0			0
$537$	−2.19552e6	+	1.26758e6i	−0.328550	+	0.189688i
$538$		0			0
$539$	−402050.	−	84456.8i	−0.0596085	−	0.0125217i
$540$		0			0
$541$	2.36003e6	+	4.08769e6i	0.346677	+	0.600461i	0.985657	−	0.168762i	$-0.0539768\pi$
$541$	2.36003e6	+	4.08769e6i	0.346677	+	0.600461i	−0.638980	+	0.769223i	$0.720643\pi$
$542$		0			0
$543$	1.39678e6	+	806432.i	0.203296	+	0.117373i
$544$		0			0
$545$		−	2.99224e6i		−	0.431524i
$546$		0			0
$547$			8.42384e6i			1.20377i	0.798584	+	0.601883i	$0.205583\pi$
$547$			8.42384e6i			1.20377i	−0.798584	+	0.601883i	$0.794417\pi$
$548$		0			0
$549$	−2.78348e6	−	1.60705e6i	−0.394147	−	0.227561i
$550$		0			0
$551$	−1.77187e6	−	3.06897e6i	−0.248630	−	0.430640i
$552$		0			0
$553$	−8.35107e6	+	6.77907e6i	−1.16126	+	0.942665i
$554$		0			0
$555$	−2.49333e6	+	1.43952e6i	−0.343595	+	0.198375i
$556$		0			0
$557$	596846.	−	1.03377e6i	0.0815125	−	0.141184i	−0.822387	−	0.568928i	$-0.807358\pi$
$557$	596846.	−	1.03377e6i	0.0815125	−	0.141184i	0.903900	+	0.427744i	$0.140692\pi$
$558$		0			0
$559$	−434709.			−0.0588395
$560$		0			0
$561$	89386.2			0.0119912
$562$		0			0
$563$	−3.72901e6	+	6.45884e6i	−0.495818	+	0.858783i	−0.999988	−	0.00482184i	$-0.998465\pi$
$563$	−3.72901e6	+	6.45884e6i	−0.495818	+	0.858783i	0.504170	+	0.863604i	$0.331798\pi$
$564$		0			0
$565$	5.38404e6	−	3.10848e6i	0.709557	−	0.409663i
$566$		0			0
$567$	73822.0	−	193000.i	0.00964336	−	0.0252116i
$568$		0			0
$569$	874103.	+	1.51399e6i	0.113183	+	0.196039i	0.917052	−	0.398768i	$-0.130562\pi$
$569$	874103.	+	1.51399e6i	0.113183	+	0.196039i	−0.803869	+	0.594807i	$0.797229\pi$
$570$		0			0
$571$	8.18258e6	+	4.72422e6i	1.05027	+	0.606372i	0.922724	−	0.385461i	$-0.125957\pi$
$571$	8.18258e6	+	4.72422e6i	1.05027	+	0.606372i	0.127544	+	0.991833i	$0.459291\pi$
$572$		0			0
$573$			6.90289e6i			0.878303i
$574$		0			0
$575$		−	1.21247e6i		−	0.152933i
$576$		0			0
$577$	−1.01751e7	−	5.87459e6i	−1.27233	−	0.734578i	−0.296901	−	0.954908i	$-0.595953\pi$
$577$	−1.01751e7	−	5.87459e6i	−1.27233	−	0.734578i	−0.975425	+	0.220330i	$0.929287\pi$
$578$		0			0
$579$	−2.25427e6	−	3.90450e6i	−0.279453	−	0.484027i
$580$		0			0
$581$	−1.03700e7	+	1.65512e6i	−1.27449	+	0.203418i
$582$		0			0
$583$	257232.	−	148513.i	0.0313440	−	0.0180965i
$584$		0			0
$585$	−236182.	+	409080.i	−0.0285337	+	0.0494218i
$586$		0			0
$587$	−137068.			−0.0164188			−0.00820941	−	0.999966i	$-0.502613\pi$
$587$	−137068.			−0.0164188			−0.00820941	+	0.999966i	$0.502613\pi$
$588$		0			0
$589$	−5.56328e6			−0.660759
$590$		0			0
$591$	1.95165e6	−	3.38035e6i	0.229844	−	0.398101i
$592$		0			0
$593$	1.04876e7	−	6.05503e6i	1.22473	−	0.707097i	0.258806	−	0.965929i	$-0.416671\pi$
$593$	1.04876e7	−	6.05503e6i	1.22473	−	0.707097i	0.965922	+	0.258832i	$0.0833376\pi$
$594$		0			0
$595$	−2.59000e6	+	413383.i	−0.299922	+	0.0478696i
$596$		0			0
$597$	3.44863e6	+	5.97320e6i	0.396014	+	0.685916i
$598$		0			0
$599$	−1.36113e7	−	7.85848e6i	−1.55000	−	0.894893i	−0.998140	−	0.0609570i	$-0.980585\pi$
$599$	−1.36113e7	−	7.85848e6i	−1.55000	−	0.894893i	−0.551860	−	0.833936i	$-0.686082\pi$
$600$		0			0
$601$		−	4.02817e6i		−	0.454906i	−0.973789	−	0.227453i	$-0.926960\pi$
$601$		−	4.02817e6i		−	0.454906i	0.973789	−	0.227453i	$-0.0730398\pi$
$602$		0			0
$603$			6.46982e6i			0.724601i
$604$		0			0
$605$	−7.28909e6	−	4.20836e6i	−0.809627	−	0.467439i
$606$		0			0
$607$	−220931.	−	382664.i	−0.0243380	−	0.0421546i	0.853600	−	0.520929i	$-0.174414\pi$
$607$	−220931.	−	382664.i	−0.0243380	−	0.0421546i	−0.877938	+	0.478775i	$0.841081\pi$
$608$		0			0
$609$	−2.66051e6	+	6.95564e6i	−0.290684	+	0.759966i
$610$		0			0
$611$	134033.	−	77383.9i	0.0145247	−	0.00838586i
$612$		0			0
$613$	6.48760e6	−	1.12369e7i	0.697321	−	1.20780i	−0.272071	−	0.962277i	$-0.587709\pi$
$613$	6.48760e6	−	1.12369e7i	0.697321	−	1.20780i	0.969392	−	0.245518i	$-0.0789582\pi$
$614$		0			0
$615$	4.40608e6			0.469748
$616$		0			0
$617$	2.98260e6			0.315415			0.157707	−	0.987486i	$-0.449590\pi$
$617$	2.98260e6			0.315415			0.157707	+	0.987486i	$0.449590\pi$
$618$		0			0
$619$	4.28183e6	−	7.41634e6i	0.449161	−	0.777971i	−0.549170	−	0.835711i	$-0.685056\pi$
$619$	4.28183e6	−	7.41634e6i	0.449161	−	0.777971i	0.998332	+	0.0577400i	$0.0183894\pi$
$620$		0			0
$621$	1.05616e7	−	6.09773e6i	1.09900	−	0.634511i
$622$		0			0
$623$	3.11686e6	−	2.53014e6i	0.321734	−	0.261171i
$624$		0			0
$625$	4.22969e6	+	7.32603e6i	0.433120	+	0.750186i
$626$		0			0
$627$	117403.	+	67782.4i	0.0119264	+	0.00688570i
$628$		0			0
$629$			2.23254e6i			0.224995i
$630$		0			0
$631$		−	3.19204e6i		−	0.319150i	−0.987186	−	0.159575i	$-0.948988\pi$
$631$		−	3.19204e6i		−	0.319150i	0.987186	−	0.159575i	$-0.0510124\pi$
$632$		0			0
$633$	726715.	+	419569.i	0.0720866	+	0.0416192i
$634$		0			0
$635$	−1.24063e6	−	2.14883e6i	−0.122098	−	0.211480i
$636$		0			0
$637$	307720.	+	939436.i	0.0300474	+	0.0917315i
$638$		0			0
$639$	9.96739e6	−	5.75467e6i	0.965671	−	0.557530i
$640$		0			0
$641$	−3.40490e6	+	5.89745e6i	−0.327310	+	0.566917i	−0.981977	−	0.189000i	$-0.939475\pi$
$641$	−3.40490e6	+	5.89745e6i	−0.327310	+	0.566917i	0.654667	+	0.755917i	$0.272809\pi$
$642$		0			0
$643$	5.81654e6			0.554801			0.277401	−	0.960754i	$-0.410527\pi$
$643$	5.81654e6			0.554801			0.277401	+	0.960754i	$0.410527\pi$
$644$		0			0
$645$	−3.67590e6			−0.347908
$646$		0			0
$647$	−446222.	+	772879.i	−0.0419074	+	0.0725857i	−0.886218	−	0.463268i	$-0.846677\pi$
$647$	−446222.	+	772879.i	−0.0419074	+	0.0725857i	0.844311	+	0.535854i	$0.180010\pi$
$648$		0			0
$649$	307700.	−	177651.i	0.0286758	−	0.0165560i
$650$		0			0
$651$	7.36828e6	+	9.07692e6i	0.681418	+	0.839433i
$652$		0			0
$653$	1.39446e6	+	2.41528e6i	0.127974	+	0.221658i	0.922892	−	0.385060i	$-0.125819\pi$
$653$	1.39446e6	+	2.41528e6i	0.127974	+	0.221658i	−0.794917	+	0.606718i	$0.792486\pi$
$654$		0			0
$655$	1.07667e7	+	6.21618e6i	0.980575	+	0.566135i
$656$		0			0
$657$		−	1.47320e6i		−	0.133152i
$658$		0			0
$659$		−	2.06628e7i		−	1.85342i	−0.375771	−	0.926712i	$-0.622622\pi$
$659$		−	2.06628e7i		−	1.85342i	0.375771	−	0.926712i	$-0.377378\pi$
$660$		0			0
$661$	−842350.	−	486331.i	−0.0749875	−	0.0432941i	0.462037	−	0.886860i	$-0.347119\pi$
$661$	−842350.	−	486331.i	−0.0749875	−	0.0432941i	−0.537025	+	0.843566i	$0.680452\pi$
$662$		0			0
$663$	−107543.	−	186270.i	−0.00950164	−	0.0164573i
$664$		0			0
$665$	−3.71526e6	−	1.42108e6i	−0.325788	−	0.124613i
$666$		0			0
$667$	1.70375e7	−	9.83661e6i	1.48283	−	0.856113i
$668$		0			0
$669$	−690454.	+	1.19590e6i	−0.0596443	+	0.103307i
$670$		0			0
$671$	513156.			0.0439991
$672$		0			0
$673$	4.79868e6			0.408399			0.204199	−	0.978929i	$-0.434541\pi$
$673$	4.79868e6			0.408399			0.204199	+	0.978929i	$0.434541\pi$
$674$		0			0
$675$	701153.	−	1.21443e6i	0.0592316	−	0.102592i
$676$		0			0
$677$	4.45819e6	−	2.57394e6i	0.373841	−	0.215837i	−0.301294	−	0.953531i	$-0.597419\pi$
$677$	4.45819e6	−	2.57394e6i	0.373841	−	0.215837i	0.675135	+	0.737694i	$0.264085\pi$
$678$		0			0
$679$	−2.68661e6	−	1.68326e7i	−0.223630	−	1.40113i
$680$		0			0
$681$	4.60019e6	+	7.96777e6i	0.380109	+	0.658368i
$682$		0			0
$683$	8.53426e6	+	4.92726e6i	0.700026	+	0.404160i	0.807357	−	0.590063i	$-0.200897\pi$
$683$	8.53426e6	+	4.92726e6i	0.700026	+	0.404160i	−0.107331	+	0.994223i	$0.534231\pi$
$684$		0			0
$685$			4.68677e6i			0.381634i
$686$		0			0
$687$		−	1.24093e7i		−	1.00312i
$688$		0			0
$689$	−618967.	−	357361.i	−0.0496729	−	0.0286786i
$690$		0			0
$691$	3.80600e6	+	6.59219e6i	0.303231	+	0.525212i	0.976866	−	0.213853i	$-0.0686012\pi$
$691$	3.80600e6	+	6.59219e6i	0.303231	+	0.525212i	−0.673635	+	0.739064i	$0.735268\pi$
$692$		0			0
$693$	76467.4	+	479098.i	0.00604844	+	0.0378958i
$694$		0			0
$695$	−6.65058e6	+	3.83971e6i	−0.522273	+	0.301534i
$696$		0			0
$697$	1.70833e6	−	2.95892e6i	0.133196	−	0.230702i
$698$		0			0
$699$	1.03901e7			0.804319
$700$		0			0
$701$	−2.19372e7			−1.68611			−0.843055	−	0.537827i	$-0.819245\pi$
$701$	−2.19372e7			−1.68611			−0.843055	+	0.537827i	$0.819245\pi$
$702$		0			0
$703$	−1.69296e6	+	2.93229e6i	−0.129199	+	0.223778i
$704$		0			0
$705$	1.13338e6	−	654358.i	0.0858823	−	0.0495841i
$706$		0			0
$707$	−1.37466e7	−	5.25804e6i	−1.03430	−	0.395617i
$708$		0			0
$709$	5.67874e6	+	9.83586e6i	0.424264	+	0.734847i	0.996351	−	0.0853458i	$-0.0271995\pi$
$709$	5.67874e6	+	9.83586e6i	0.424264	+	0.734847i	−0.572087	+	0.820193i	$0.693866\pi$
$710$		0			0
$711$	1.10007e7	+	6.35126e6i	0.816105	+	0.471179i
$712$		0			0
$713$		−	3.08847e7i		−	2.27520i
$714$		0			0
$715$		−	75416.9i		−	0.00551701i
$716$		0			0
$717$	398686.	+	230181.i	0.0289623	+	0.0167214i
$718$		0			0
$719$	−9.79576e6	−	1.69667e7i	−0.706669	−	1.22399i	−0.966086	−	0.258221i	$-0.916864\pi$
$719$	−9.79576e6	−	1.69667e7i	−0.706669	−	1.22399i	0.259417	−	0.965765i	$-0.416469\pi$
$720$		0			0
$721$	8.64719e6	+	1.06524e7i	0.619494	+	0.763149i
$722$		0			0
$723$	−6.91919e6	+	3.99480e6i	−0.492277	+	0.284216i
$724$		0			0
$725$	1.13107e6	−	1.95908e6i	0.0799182	−	0.138422i
$726$		0			0
$727$	−386859.			−0.0271467			−0.0135733	−	0.999908i	$-0.504321\pi$
$727$	−386859.			−0.0271467			−0.0135733	+	0.999908i	$0.504321\pi$
$728$		0			0
$729$	8.40905e6			0.586041
$730$		0			0
$731$	−1.42523e6	+	2.46856e6i	−0.0986485	+	0.170864i
$732$		0			0
$733$	7.18824e6	−	4.15013e6i	0.494154	−	0.285300i	−0.232142	−	0.972682i	$-0.574573\pi$
$733$	7.18824e6	−	4.15013e6i	0.494154	−	0.285300i	0.726296	+	0.687382i	$0.241240\pi$
$734$		0			0
$735$	2.60208e6	+	7.94387e6i	0.177665	+	0.542393i
$736$		0			0
$737$	−516480.	−	894570.i	−0.0350255	−	0.0606660i
$738$		0			0
$739$	−1.15579e7	−	6.67298e6i	−0.778520	−	0.449478i	0.0573858	−	0.998352i	$-0.481723\pi$
$739$	−1.15579e7	−	6.67298e6i	−0.778520	−	0.449478i	−0.835905	+	0.548874i	$0.815057\pi$
$740$		0			0
$741$		−	326204.i		−	0.0218245i
$742$		0			0
$743$			2.31448e7i			1.53809i	0.639196	+	0.769044i	$0.279267\pi$
$743$			2.31448e7i			1.53809i	−0.639196	+	0.769044i	$0.720733\pi$
$744$		0			0
$745$	−1.89723e7	−	1.09537e7i	−1.25236	−	0.723051i
$746$		0			0
$747$	6.20070e6	+	1.07399e7i	0.406574	+	0.704207i
$748$		0			0
$749$	−3.40597e6	+	2.76483e6i	−0.221838	+	0.180079i
$750$		0			0
$751$	−818115.	+	472339.i	−0.0529315	+	0.0305600i	−0.526232	−	0.850341i	$-0.676396\pi$
$751$	−818115.	+	472339.i	−0.0529315	+	0.0305600i	0.473301	+	0.880901i	$0.343062\pi$
$752$		0			0
$753$	−6.76842e6	+	1.17233e7i	−0.435011	+	0.753461i
$754$		0			0
$755$	−2.24378e7			−1.43256
$756$		0			0
$757$	−2.40741e7			−1.52690			−0.763448	−	0.645869i	$-0.776495\pi$
$757$	−2.40741e7			−1.52690			−0.763448	+	0.645869i	$0.776495\pi$
$758$		0			0
$759$	−376296.	+	651764.i	−0.0237097	+	0.0410663i
$760$		0			0
$761$	−1.39052e7	+	8.02818e6i	−0.870394	+	0.502522i	−0.867479	−	0.497474i	$-0.834261\pi$
$761$	−1.39052e7	+	8.02818e6i	−0.870394	+	0.502522i	−0.00291487	+	0.999996i	$0.500928\pi$
$762$		0			0
$763$	−2.64195e6	+	6.90713e6i	−0.164291	+	0.429523i
$764$		0			0
$765$	1.54868e6	+	2.68240e6i	0.0956774	+	0.165718i
$766$		0			0
$767$	−740405.	−	427473.i	−0.0454445	−	0.0262374i
$768$		0			0
$769$		−	1.52779e7i		−	0.931641i	−0.884879	−	0.465820i	$-0.845759\pi$
$769$		−	1.52779e7i		−	0.931641i	0.884879	−	0.465820i	$-0.154241\pi$
$770$		0			0
$771$		−	4.36478e6i		−	0.264439i
$772$		0			0
$773$	1.62363e7	+	9.37402e6i	0.977322	+	0.564257i	0.901461	−	0.432861i	$-0.142496\pi$
$773$	1.62363e7	+	9.37402e6i	0.977322	+	0.564257i	0.0758615	+	0.997118i	$0.475829\pi$
$774$		0			0
$775$	−1.77566e6	−	3.07553e6i	−0.106195	−	0.183935i
$776$		0			0
$777$	7.02648e6	−	1.12148e6i	0.417528	−	0.0666403i
$778$		0			0
$779$	4.48756e6	−	2.59089e6i	0.264952	−	0.152970i
$780$		0			0
$781$	−918781.	+	1.59138e6i	−0.0538995	+	0.0933566i
$782$		0			0
$783$	2.27534e7			1.32630
$784$		0			0
$785$	2.50263e7			1.44951
$786$		0			0
$787$	−7.47326e6	+	1.29441e7i	−0.430104	+	0.744962i	−0.996882	−	0.0789087i	$-0.974856\pi$
$787$	−7.47326e6	+	1.29441e7i	−0.430104	+	0.744962i	0.566778	+	0.823871i	$0.308190\pi$
$788$		0			0
$789$	−9.12961e6	+	5.27098e6i	−0.522107	+	0.301439i
$790$		0			0
$791$	−1.51728e7	+	2.42169e6i	−0.862234	+	0.137619i
$792$		0			0
$793$	−617393.	−	1.06936e6i	−0.0348641	−	0.0603864i
$794$		0			0
$795$	−5.23398e6	−	3.02184e6i	−0.293707	−	0.169572i
$796$		0			0
$797$			1.23503e7i			0.688702i	0.938841	+	0.344351i	$0.111901\pi$
$797$			1.23503e7i			0.688702i	−0.938841	+	0.344351i	$0.888099\pi$
$798$		0			0
$799$		−	1.01484e6i		−	0.0562379i
$800$		0			0
$801$	−4.10578e6	−	2.37047e6i	−0.226107	−	0.130543i
$802$		0			0
$803$	117604.	+	203697.i	0.00643627	+	0.0111479i
$804$		0			0
$805$	7.88914e6	−	2.06254e7i	0.429082	−	1.12179i
$806$		0			0
$807$	−1.05122e7	+	6.06925e6i	−0.568214	+	0.328058i
$808$		0			0
$809$	−5.24379e6	+	9.08250e6i	−0.281691	+	0.487904i	−0.971801	−	0.235801i	$-0.924229\pi$
$809$	−5.24379e6	+	9.08250e6i	−0.281691	+	0.487904i	0.690110	+	0.723704i	$0.257562\pi$
$810$		0			0
$811$	−1.19421e7			−0.637570			−0.318785	−	0.947827i	$-0.603275\pi$
$811$	−1.19421e7			−0.637570			−0.318785	+	0.947827i	$0.603275\pi$
$812$		0			0
$813$	−9.24951e6			−0.490786
$814$		0			0
$815$	−1.20072e7	+	2.07971e7i	−0.633212	+	1.09675i
$816$		0			0
$817$	−3.74387e6	+	2.16153e6i	−0.196230	+	0.113294i
$818$		0			0
$819$	906383.	−	735765.i	0.0472174	−	0.0383292i
$820$		0			0
$821$	−199392.	−	345358.i	−0.0103241	−	0.0178818i	0.860817	−	0.508914i	$-0.169953\pi$
$821$	−199392.	−	345358.i	−0.0103241	−	0.0178818i	−0.871141	+	0.491032i	$0.836620\pi$
$822$		0			0
$823$	2.15862e7	+	1.24628e7i	1.11090	+	0.641380i	0.939063	−	0.343745i	$-0.111696\pi$
$823$	2.15862e7	+	1.24628e7i	1.11090	+	0.641380i	0.171840	+	0.985125i	$0.445029\pi$
$824$		0			0
$825$			86537.6i			0.00442660i
$826$		0			0
$827$		−	4.53496e6i		−	0.230573i	−0.993332	−	0.115287i	$-0.963221\pi$
$827$		−	4.53496e6i		−	0.230573i	0.993332	−	0.115287i	$-0.0367787\pi$
$828$		0			0
$829$	2.34324e7	+	1.35287e7i	1.18422	+	0.683708i	0.956986	−	0.290133i	$-0.0936996\pi$
$829$	2.34324e7	+	1.35287e7i	1.18422	+	0.683708i	0.227230	+	0.973841i	$0.427033\pi$
$830$		0			0
$831$	6.99691e6	+	1.21190e7i	0.351483	+	0.608786i
$832$		0			0
$833$	6.34362e6	+	1.33257e6i	0.316756	+	0.0665394i
$834$		0			0
$835$	80102.0	−	46246.9i	0.00397582	−	0.00229544i
$836$		0			0
$837$	1.78602e7	−	3.09347e7i	0.881194	−	1.52627i
$838$		0			0
$839$	−3.66855e7			−1.79924			−0.899620	−	0.436673i	$-0.856157\pi$
$839$	−3.66855e7			−1.79924			−0.899620	+	0.436673i	$0.856157\pi$
$840$		0			0
$841$	1.61938e7			0.789511
$842$		0			0
$843$	−7.57901e6	+	1.31272e7i	−0.367319	+	0.636215i
$844$		0			0
$845$	1.67100e7	−	9.64750e6i	0.805070	−	0.464808i
$846$		0			0
$847$	1.31101e7	+	1.61502e7i	0.627908	+	0.773515i
$848$		0			0
$849$	−5.25977e6	−	9.11018e6i	−0.250436	−	0.433768i
$850$		0			0
$851$	−1.62787e7	−	9.39850e6i	−0.770540	−	0.444872i
$852$		0			0
$853$			2.04043e7i			0.960174i	0.877221	+	0.480087i	$0.159395\pi$
$853$			2.04043e7i			0.960174i	−0.877221	+	0.480087i	$0.840605\pi$
$854$		0			0
$855$			4.69753e6i			0.219763i
$856$		0			0
$857$	2.10554e6	+	1.21564e6i	0.0979292	+	0.0565395i	0.548165	−	0.836370i	$-0.315327\pi$
$857$	2.10554e6	+	1.21564e6i	0.0979292	+	0.0565395i	−0.450236	+	0.892910i	$0.648660\pi$
$858$		0			0
$859$	−8.31785e6	−	1.44069e7i	−0.384617	−	0.666176i	0.607099	−	0.794626i	$-0.292333\pi$
$859$	−8.31785e6	−	1.44069e7i	−0.384617	−	0.666176i	−0.991716	+	0.128450i	$0.959000\pi$
$860$		0			0
$861$	−1.01708e7	−	3.89029e6i	−0.467570	−	0.178844i
$862$		0			0
$863$	−2.63064e7	+	1.51880e7i	−1.20236	+	0.694183i	−0.961079	−	0.276273i	$-0.910901\pi$
$863$	−2.63064e7	+	1.51880e7i	−1.20236	+	0.694183i	−0.241280	+	0.970455i	$0.577567\pi$
$864$		0			0
$865$	−1.59295e7	+	2.75907e7i	−0.723871	+	1.25378i
$866$		0			0
$867$	1.20521e7			0.544522
$868$		0			0
$869$	−2.02806e6			−0.0911028
$870$		0			0
$871$	−1.24278e6	+	2.15256e6i	−0.0555073	+	0.0961415i
$872$		0			0
$873$	−1.74331e7	+	1.00650e7i	−0.774176	+	0.446970i
$874$		0			0
$875$	−3.74970e6	−	2.34933e7i	−0.165568	−	1.03735i
$876$		0			0
$877$	−1.50861e7	−	2.61299e7i	−0.662336	−	1.14720i	−0.980000	−	0.198997i	$-0.936232\pi$
$877$	−1.50861e7	−	2.61299e7i	−0.662336	−	1.14720i	0.317664	−	0.948203i	$-0.397102\pi$
$878$		0			0
$879$	−1.16768e7	−	6.74162e6i	−0.509744	−	0.294301i
$880$		0			0
$881$			1.00893e7i			0.437947i	0.975731	+	0.218973i	$0.0702708\pi$
$881$			1.00893e7i			0.437947i	−0.975731	+	0.218973i	$0.929729\pi$
$882$		0			0
$883$			1.36170e7i			0.587733i	0.955846	+	0.293867i	$0.0949422\pi$
$883$			1.36170e7i			0.587733i	−0.955846	+	0.293867i	$0.905058\pi$
$884$		0			0
$885$	−6.26087e6	−	3.61471e6i	−0.268705	−	0.155137i
$886$		0			0
$887$	5.09687e6	+	8.82804e6i	0.217518	+	0.376752i	0.954048	−	0.299652i	$-0.0968707\pi$
$887$	5.09687e6	+	8.82804e6i	0.217518	+	0.376752i	−0.736531	+	0.676404i	$0.763537\pi$
$888$		0			0
$889$	966525.	+	6.05566e6i	0.0410165	+	0.256984i
$890$		0			0
$891$	33741.2	−	19480.5i	0.00142386	−	0.000822064i
$892$		0			0
$893$	769560.	−	1.33292e6i	0.0322934	−	0.0559338i
$894$		0			0
$895$	1.40256e7			0.585279
$896$		0			0
$897$	1.81093e6			0.0751485
$898$		0			0
$899$	2.88113e7	−	4.99026e7i	1.18895	−	2.05932i
$900$		0			0
$901$	−4.05866e6	+	2.34327e6i	−0.166560	+	0.0961634i
$902$		0			0
$903$	8.48526e6	+	3.24558e6i	0.346295	+	0.132457i
$904$		0			0
$905$	−4.46151e6	−	7.72756e6i	−0.181076	−	0.313632i
$906$		0			0
$907$	−3.97148e6	−	2.29294e6i	−0.160300	−	0.0925495i	0.417704	−	0.908583i	$-0.362835\pi$
$907$	−3.97148e6	−	2.29294e6i	−0.160300	−	0.0925495i	−0.578004	+	0.816034i	$0.696168\pi$
$908$		0			0
$909$			1.73811e7i			0.697696i
$910$		0			0
$911$			2.52740e7i			1.00897i	0.863421	+	0.504484i	$0.168317\pi$
$911$			2.52740e7i			1.00897i	−0.863421	+	0.504484i	$0.831683\pi$
$912$		0			0
$913$	−1.71472e6	−	989994.i	−0.0680795	−	0.0393057i
$914$		0			0
$915$	−5.22068e6	−	9.04247e6i	−0.206146	−	0.357055i
$916$		0			0
$917$	−1.93649e7	−	2.38554e7i	−0.760487	−	0.936837i
$918$		0			0
$919$	2.46259e7	−	1.42178e7i	0.961842	−	0.555320i	0.0651028	−	0.997879i	$-0.479262\pi$
$919$	2.46259e7	−	1.42178e7i	0.961842	−	0.555320i	0.896740	+	0.442559i	$0.145929\pi$
$920$		0			0
$921$	1.20683e7	−	2.09030e7i	0.468811	−	0.812005i
$922$		0			0
$923$	4.42165e6			0.170836
$924$		0			0
$925$	−2.16139e6			−0.0830576
$926$		0			0
$927$	8.10149e6	−	1.40322e7i	0.309646	−	0.536322i
$928$		0			0
$929$	−3.23329e7	+	1.86674e7i	−1.22915	+	0.709650i	−0.966852	−	0.255337i	$-0.917814\pi$
$929$	−3.23329e7	+	1.86674e7i	−1.22915	+	0.709650i	−0.262298	+	0.964987i	$0.584480\pi$
$930$		0			0
$931$	7.32141e6	+	6.56068e6i	0.276835	+	0.248070i
$932$		0			0
$933$	−1.14647e7	−	1.98574e7i	−0.431180	−	0.746825i
$934$		0			0
$935$	−428267.	−	247260.i	−0.0160209	−	0.00924965i
$936$		0			0
$937$			2.42023e7i			0.900548i	0.892890	+	0.450274i	$0.148674\pi$
$937$			2.42023e7i			0.900548i	−0.892890	+	0.450274i	$0.851326\pi$
$938$		0			0
$939$		−	3.27890e7i		−	1.21357i
$940$		0			0
$941$	3.86426e7	+	2.23103e7i	1.42263	+	0.821356i	0.996523	−	0.0833160i	$-0.0265511\pi$
$941$	3.86426e7	+	2.23103e7i	1.42263	+	0.821356i	0.426108	+	0.904672i	$0.359884\pi$
$942$		0			0
$943$	1.43834e7	+	2.49128e7i	0.526724	+	0.912313i
$944$		0			0
$945$	1.98292e7	−	1.60966e7i	0.722315	−	0.586347i
$946$		0			0
$947$	2.29406e7	−	1.32447e7i	0.831246	−	0.479920i	−0.0230333	−	0.999735i	$-0.507332\pi$
$947$	2.29406e7	−	1.32447e7i	0.831246	−	0.479920i	0.854279	+	0.519815i	$0.173999\pi$
$948$		0			0
$949$	282986.	−	490146.i	0.0102000	−	0.0176669i
$950$		0			0
$951$	2.96046e7			1.06147
$952$		0			0
$953$	3.59018e7			1.28051			0.640257	−	0.768161i	$-0.278828\pi$
$953$	3.59018e7			1.28051			0.640257	+	0.768161i	$0.278828\pi$
$954$		0			0
$955$	1.90948e7	−	3.30731e7i	0.677495	−	1.17346i
$956$		0			0
$957$	−1.21602e6	+	702067.i	−0.0429200	+	0.0247799i
$958$		0			0
$959$	4.13812e6	−	1.08187e7i	0.145297	−	0.379865i
$960$		0			0
$961$	−3.09159e7	−	5.35480e7i	−1.07988	−	1.87040i
$962$		0			0
$963$	4.48661e6	+	2.59035e6i	0.155902	+	0.0900103i
$964$		0			0
$965$			2.49430e7i			0.862245i
$966$		0			0
$967$			5.76157e6i			0.198141i	0.995080	+	0.0990705i	$0.0315869\pi$
$967$			5.76157e6i			0.198141i	−0.995080	+	0.0990705i	$0.968413\pi$
$968$		0			0
$969$	−1.85240e6	−	1.06948e6i	−0.0633761	−	0.0365902i
$970$		0			0
$971$	−4.61284e6	−	7.98967e6i	−0.157007	−	0.271945i	0.776781	−	0.629771i	$-0.216851\pi$
$971$	−4.61284e6	−	7.98967e6i	−0.157007	−	0.271945i	−0.933788	+	0.357826i	$0.883518\pi$
$972$		0			0
$973$	1.87421e7	−	2.99137e6i	0.634652	−	0.101295i
$974$		0			0
$975$	180334.	−	104116.i	0.00607527	−	0.00350756i
$976$		0			0
$977$	−5.97202e6	+	1.03438e7i	−0.200163	+	0.346693i	−0.948581	−	0.316535i	$-0.897481\pi$
$977$	−5.97202e6	+	1.03438e7i	−0.200163	+	0.346693i	0.748418	+	0.663228i	$0.230814\pi$
$978$		0			0
$979$	756931.			0.0252406
$980$		0			0
$981$	8.73328e6			0.289738
$982$		0			0
$983$	−3.13440e6	+	5.42895e6i	−0.103460	+	0.179197i	−0.913108	−	0.407718i	$-0.866325\pi$
$983$	−3.13440e6	+	5.42895e6i	−0.103460	+	0.179197i	0.809648	+	0.586916i	$0.199658\pi$
$984$		0			0
$985$	−1.87015e7	+	1.07973e7i	−0.614165	+	0.354588i
$986$		0			0
$987$	−3.19400e6	+	509784.i	−0.104362	+	0.0166569i
$988$		0			0
$989$	−1.19998e7	−	2.07842e7i	−0.390106	−	0.675683i
$990$		0			0
$991$	4.51836e6	+	2.60867e6i	0.146149	+	0.0843792i	0.571291	−	0.820747i	$-0.306443\pi$
$991$	4.51836e6	+	2.60867e6i	0.146149	+	0.0843792i	−0.425142	+	0.905127i	$0.639776\pi$
$992$		0			0
$993$		−	8.24397e6i		−	0.265316i
$994$		0			0
$995$		−	3.81584e7i		−	1.22189i
$996$		0			0
$997$	2.64151e7	+	1.52508e7i	0.841617	+	0.485908i	0.857814	−	0.513961i	$-0.171822\pi$
$997$	2.64151e7	+	1.52508e7i	0.841617	+	0.485908i	−0.0161963	+	0.999869i	$0.505156\pi$
$998$		0			0
$999$	−1.08700e7	−	1.88274e7i	−0.344601	−	0.596866i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.p.a.31.3	✓	12
4.3	odd	2	inner	112.6.p.a.31.4	yes	12
7.3	odd	6		784.6.f.b.783.5		12
7.4	even	3		784.6.f.b.783.8		12
7.5	odd	6	inner	112.6.p.a.47.4	yes	12
28.3	even	6		784.6.f.b.783.7		12
28.11	odd	6		784.6.f.b.783.6		12
28.19	even	6	inner	112.6.p.a.47.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.a.31.3	✓	12	1.1	even	1	trivial
112.6.p.a.31.4	yes	12	4.3	odd	2	inner
112.6.p.a.47.3	yes	12	28.19	even	6	inner
112.6.p.a.47.4	yes	12	7.5	odd	6	inner
784.6.f.b.783.5		12	7.3	odd	6
784.6.f.b.783.6		12	28.11	odd	6
784.6.f.b.783.7		12	28.3	even	6
784.6.f.b.783.8		12	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.74078 + 8.21127i) q^{3} +(45.4281 - 26.2279i) q^{5} +(-128.021 + 20.4331i) q^{7} +(76.5500 + 132.588i) q^{9} +(-21.1688 - 12.2218i) q^{11} +58.8178i q^{13} +497.363i q^{15} +(334.006 + 192.839i) q^{17} +(292.463 + 506.560i) q^{19} +(439.140 - 1148.09i) q^{21} +(-2812.19 + 1623.62i) q^{23} +(-186.693 + 323.362i) q^{25} -3755.65 q^{27} -6058.46 q^{29} +(-4755.55 + 8236.85i) q^{31} +(200.714 - 115.882i) q^{33} +(-5279.85 + 4285.97i) q^{35} +(2894.31 + 5013.10i) q^{37} +(-482.969 - 278.842i) q^{39} -8858.89i q^{41} +7390.78i q^{43} +(6955.04 + 4015.49i) q^{45} +(-1315.66 - 2278.78i) q^{47} +(15972.0 - 5231.75i) q^{49} +(-3166.90 + 1828.41i) q^{51} +(-6075.73 + 10523.5i) q^{53} -1282.21 q^{55} -5546.01 q^{57} +(-7267.76 + 12588.1i) q^{59} +(-18180.8 + 10496.7i) q^{61} +(-12509.2 - 15410.0i) q^{63} +(1542.67 + 2671.98i) q^{65} +(36597.2 + 21129.4i) q^{67} -30788.8i q^{69} -75175.4i q^{71} +(-8333.30 - 4811.23i) q^{73} +(-1770.14 - 3065.98i) q^{75} +(2959.80 + 1132.11i) q^{77} +(71853.0 - 41484.4i) q^{79} +(-796.954 + 1380.36i) q^{81} +81002.0 q^{83} +20231.0 q^{85} +(28721.8 - 49747.7i) q^{87} +(-26817.6 + 15483.2i) q^{89} +(-1201.83 - 7529.93i) q^{91} +(-45090.0 - 78098.2i) q^{93} +(26572.0 + 15341.4i) q^{95} +131483. i q^{97} -3742.33i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 66 q^{5} - 544 q^{9} - 1602 q^{17} + 2674 q^{21} - 2460 q^{25} + 22512 q^{29} - 15582 q^{33} - 7938 q^{37} + 3072 q^{45} - 23772 q^{49} + 22854 q^{53} - 25900 q^{57} - 12870 q^{61} - 62220 q^{65}+ \cdots - 273182 q^{93}+O(q^{100}) \) 12 * q - 66 * q^5 - 544 * q^9 - 1602 * q^17 + 2674 * q^21 - 2460 * q^25 + 22512 * q^29 - 15582 * q^33 - 7938 * q^37 + 3072 * q^45 - 23772 * q^49 + 22854 * q^53 - 25900 * q^57 - 12870 * q^61 - 62220 * q^65 + 232938 * q^73 + 154098 * q^77 - 93082 * q^81 + 70116 * q^85 - 328542 * q^89 - 273182 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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