LMFDB - Embedded newform 256.10.b.l.129.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,10,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-5568,0,-60788] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$131.849174058$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{106})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2809 $ x^4 + 2809
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$5.14782 - 5.14782i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.10.b.l.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+252.730i q^{3} -2019.84i q^{5} -3039.30 q^{7} -44189.5 q^{9} +42489.0i q^{11} -74173.8i q^{13} +510474. q^{15} -607725. q^{17} -164849. i q^{19} -768123. i q^{21} -2.08911e6 q^{23} -2.12663e6 q^{25} -6.19353e6i q^{27} +1.87705e6i q^{29} -669635. q^{31} -1.07383e7 q^{33} +6.13890e6i q^{35} +5.06145e6i q^{37} +1.87459e7 q^{39} +1.46245e7 q^{41} +1.15906e7i q^{43} +8.92557e7i q^{45} +3.35490e7 q^{47} -3.11163e7 q^{49} -1.53590e8i q^{51} +2.03950e7i q^{53} +8.58211e7 q^{55} +4.16623e7 q^{57} -1.19399e8i q^{59} -9.81307e7i q^{61} +1.34305e8 q^{63} -1.49819e8 q^{65} +1.01247e8i q^{67} -5.27981e8i q^{69} +3.11299e8 q^{71} +6.82495e6 q^{73} -5.37464e8i q^{75} -1.29137e8i q^{77} +5.23080e8 q^{79} +6.95509e8 q^{81} -2.37668e8i q^{83} +1.22751e9i q^{85} -4.74388e8 q^{87} +6.21070e8 q^{89} +2.25436e8i q^{91} -1.69237e8i q^{93} -3.32969e8 q^{95} +1.11708e9 q^{97} -1.87757e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5568 q^{7} - 60788 q^{9} + 1115456 q^{15} - 944376 q^{17} - 7767360 q^{23} - 1105532 q^{25} + 9714432 q^{31} - 26938624 q^{33} + 41883584 q^{39} + 64718040 q^{41} - 41886080 q^{47} - 142809372 q^{49}+ \cdots + 3701486920 q^{97}+O(q^{100}) $ 4 * q - 5568 * q^7 - 60788 * q^9 + 1115456 * q^15 - 944376 * q^17 - 7767360 * q^23 - 1105532 * q^25 + 9714432 * q^31 - 26938624 * q^33 + 41883584 * q^39 + 64718040 * q^41 - 41886080 * q^47 - 142809372 * q^49 + 215481408 * q^55 + 228461312 * q^57 + 275654336 * q^63 - 334886448 * q^65 - 13610944 * q^71 + 46434840 * q^73 + 1453197696 * q^79 + 1539882020 * q^81 - 636858944 * q^87 + 142858200 * q^89 - 1830814144 * q^95 + 3701486920 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$3$	252.730i	1.80140i	0.434437	+	0.900702i	$0.356947\pi$
$3$	252.730i	1.80140i	−0.434437	+	0.900702i	$0.643053\pi$
$4$	0	0
$5$	− 2019.84i	− 1.44528i	−0.691224	−	0.722640i	$-0.742928\pi$
$5$	− 2019.84i	− 1.44528i	0.691224	−	0.722640i	$-0.257072\pi$
$6$	0	0
$7$	−3039.30	−0.478446	−0.239223	−	0.970965i	$-0.576893\pi$
$7$	−3039.30	−0.478446	−0.239223	+	0.970965i	$0.576893\pi$
$8$	0	0
$9$	−44189.5	−2.24506
$10$	0	0
$11$	42489.0i	0.875003i	0.899218	+	0.437502i	$0.144137\pi$
$11$	42489.0i	0.875003i	−0.899218	+	0.437502i	$0.855863\pi$
$12$	0	0
$13$	− 74173.8i	− 0.720287i	−0.932897	−	0.360143i	$-0.882728\pi$
$13$	− 74173.8i	− 0.720287i	0.932897	−	0.360143i	$-0.117272\pi$
$14$	0	0
$15$	510474.	2.60353
$16$	0	0
$17$	−607725.	−1.76477	−0.882383	−	0.470532i	$-0.844062\pi$
$17$	−607725.	−1.76477	−0.882383	+	0.470532i	$0.844062\pi$
$18$	0	0
$19$	− 164849.i	− 0.290199i	−0.989417	−	0.145099i	$-0.953650\pi$
$19$	− 164849.i	− 0.290199i	0.989417	−	0.145099i	$-0.0463502\pi$
$20$	0	0
$21$	− 768123.i	− 0.861874i
$22$	0	0
$23$	−2.08911e6	−1.55663	−0.778316	−	0.627873i	$-0.783926\pi$
$23$	−2.08911e6	−1.55663	−0.778316	+	0.627873i	$0.783926\pi$
$24$	0	0
$25$	−2.12663e6	−1.08884
$26$	0	0
$27$	− 6.19353e6i	− 2.24285i
$28$	0	0
$29$	1.87705e6i	0.492817i	0.969166	+	0.246408i	$0.0792505\pi$
$29$	1.87705e6i	0.492817i	−0.969166	+	0.246408i	$0.920750\pi$
$30$	0	0
$31$	−669635.	−0.130230	−0.0651150	−	0.997878i	$-0.520741\pi$
$31$	−669635.	−0.130230	−0.0651150	+	0.997878i	$0.520741\pi$
$32$	0	0
$33$	−1.07383e7	−1.57624
$34$	0	0
$35$	6.13890e6i	0.691488i
$36$	0	0
$37$	5.06145e6i	0.443984i	0.975049	+	0.221992i	$0.0712558\pi$
$37$	5.06145e6i	0.443984i	−0.975049	+	0.221992i	$0.928744\pi$
$38$	0	0
$39$	1.87459e7	1.29753
$40$	0	0
$41$	1.46245e7	0.808262	0.404131	−	0.914701i	$-0.367574\pi$
$41$	1.46245e7	0.808262	0.404131	+	0.914701i	$0.367574\pi$
$42$	0	0
$43$	1.15906e7i	0.517009i	0.966010	+	0.258505i	$0.0832297\pi$
$43$	1.15906e7i	0.517009i	−0.966010	+	0.258505i	$0.916770\pi$
$44$	0	0
$45$	8.92557e7i	3.24474i
$46$	0	0
$47$	3.35490e7	1.00286	0.501428	−	0.865199i	$-0.332808\pi$
$47$	3.35490e7	1.00286	0.501428	+	0.865199i	$0.332808\pi$
$48$	0	0
$49$	−3.11163e7	−0.771090
$50$	0	0
$51$	− 1.53590e8i	− 3.17906i
$52$	0	0
$53$	2.03950e7i	0.355045i	0.984117	+	0.177522i	$0.0568082\pi$
$53$	2.03950e7i	0.355045i	−0.984117	+	0.177522i	$0.943192\pi$
$54$	0	0
$55$	8.58211e7	1.26463
$56$	0	0
$57$	4.16623e7	0.522765
$58$	0	0
$59$	− 1.19399e8i	− 1.28282i	−0.767196	−	0.641412i	$-0.778349\pi$
$59$	− 1.19399e8i	− 1.28282i	0.767196	−	0.641412i	$-0.221651\pi$
$60$	0	0
$61$	− 9.81307e7i	− 0.907446i	−0.891143	−	0.453723i	$-0.850096\pi$
$61$	− 9.81307e7i	− 0.907446i	0.891143	−	0.453723i	$-0.149904\pi$
$62$	0	0
$63$	1.34305e8	1.07414
$64$	0	0
$65$	−1.49819e8	−1.04102
$66$	0	0
$67$	1.01247e8i	0.613824i	0.951738	+	0.306912i	$0.0992957\pi$
$67$	1.01247e8i	0.613824i	−0.951738	+	0.306912i	$0.900704\pi$
$68$	0	0
$69$	− 5.27981e8i	− 2.80412i
$70$	0	0
$71$	3.11299e8	1.45383	0.726917	−	0.686726i	$-0.240953\pi$
$71$	3.11299e8	1.45383	0.726917	+	0.686726i	$0.240953\pi$
$72$	0	0
$73$	6.82495e6	0.0281285	0.0140642	−	0.999901i	$-0.495523\pi$
$73$	6.82495e6	0.0281285	0.0140642	+	0.999901i	$0.495523\pi$
$74$	0	0
$75$	− 5.37464e8i	− 1.96143i
$76$	0	0
$77$	− 1.29137e8i	− 0.418641i
$78$	0	0
$79$	5.23080e8	1.51094	0.755469	−	0.655185i	$-0.227409\pi$
$79$	5.23080e8	1.51094	0.755469	+	0.655185i	$0.227409\pi$
$80$	0	0
$81$	6.95509e8	1.79523
$82$	0	0
$83$	− 2.37668e8i	− 0.549691i	−0.961488	−	0.274845i	$-0.911373\pi$
$83$	− 2.37668e8i	− 0.549691i	0.961488	−	0.274845i	$-0.0886267\pi$
$84$	0	0
$85$	1.22751e9i	2.55058i
$86$	0	0
$87$	−4.74388e8	−0.887763
$88$	0	0
$89$	6.21070e8	1.04927	0.524633	−	0.851328i	$-0.324202\pi$
$89$	6.21070e8	1.04927	0.524633	+	0.851328i	$0.324202\pi$
$90$	0	0
$91$	2.25436e8i	0.344618i
$92$	0	0
$93$	− 1.69237e8i	− 0.234597i
$94$	0	0
$95$	−3.32969e8	−0.419418
$96$	0	0
$97$	1.11708e9	1.28118	0.640591	−	0.767882i	$-0.278690\pi$
$97$	1.11708e9	1.28118	0.640591	+	0.767882i	$0.278690\pi$
$98$	0	0
$99$	− 1.87757e9i	− 1.96443i
$100$	0	0
$101$	− 2.84480e8i	− 0.272023i	−0.990707	−	0.136011i	$-0.956572\pi$
$101$	− 2.84480e8i	− 0.272023i	0.990707	−	0.136011i	$-0.0434284\pi$
$102$	0	0
$103$	5.93311e8	0.519416	0.259708	−	0.965687i	$-0.416374\pi$
$103$	5.93311e8	0.519416	0.259708	+	0.965687i	$0.416374\pi$
$104$	0	0
$105$	−1.55149e9	−1.24565
$106$	0	0
$107$	6.44578e8i	0.475388i	0.971340	+	0.237694i	$0.0763916\pi$
$107$	6.44578e8i	0.475388i	−0.971340	+	0.237694i	$0.923608\pi$
$108$	0	0
$109$	5.64540e8i	0.383067i	0.981486	+	0.191534i	$0.0613461\pi$
$109$	5.64540e8i	0.383067i	−0.981486	+	0.191534i	$0.938654\pi$
$110$	0	0
$111$	−1.27918e9	−0.799794
$112$	0	0
$113$	−3.95050e8	−0.227929	−0.113964	−	0.993485i	$-0.536355\pi$
$113$	−3.95050e8	−0.227929	−0.113964	+	0.993485i	$0.536355\pi$
$114$	0	0
$115$	4.21967e9i	2.24977i
$116$	0	0
$117$	3.27770e9i	1.61709i
$118$	0	0
$119$	1.84706e9	0.844344
$120$	0	0
$121$	5.52630e8	0.234369
$122$	0	0
$123$	3.69604e9i	1.45601i
$124$	0	0
$125$	3.50455e8i	0.128392i
$126$	0	0
$127$	−3.41306e9	−1.16420	−0.582099	−	0.813118i	$-0.697769\pi$
$127$	−3.41306e9	−1.16420	−0.582099	+	0.813118i	$0.697769\pi$
$128$	0	0
$129$	−2.92930e9	−0.931343
$130$	0	0
$131$	5.95732e9i	1.76738i	0.468071	+	0.883691i	$0.344949\pi$
$131$	5.95732e9i	1.76738i	−0.468071	+	0.883691i	$0.655051\pi$
$132$	0	0
$133$	5.01026e8i	0.138844i
$134$	0	0
$135$	−1.25099e10	−3.24155
$136$	0	0
$137$	−3.18592e9	−0.772666	−0.386333	−	0.922359i	$-0.626258\pi$
$137$	−3.18592e9	−0.772666	−0.386333	+	0.922359i	$0.626258\pi$
$138$	0	0
$139$	− 1.31882e9i	− 0.299652i	−0.988712	−	0.149826i	$-0.952129\pi$
$139$	− 1.31882e9i	− 0.299652i	0.988712	−	0.149826i	$-0.0478714\pi$
$140$	0	0
$141$	8.47883e9i	1.80655i
$142$	0	0
$143$	3.15157e9	0.630253
$144$	0	0
$145$	3.79135e9	0.712259
$146$	0	0
$147$	− 7.86401e9i	− 1.38904i
$148$	0	0
$149$	− 6.79472e9i	− 1.12936i	−0.825309	−	0.564681i	$-0.808999\pi$
$149$	− 6.79472e9i	− 1.12936i	0.825309	−	0.564681i	$-0.191001\pi$
$150$	0	0
$151$	−4.19125e9	−0.656065	−0.328032	−	0.944666i	$-0.606386\pi$
$151$	−4.19125e9	−0.656065	−0.328032	+	0.944666i	$0.606386\pi$
$152$	0	0
$153$	2.68551e10	3.96200
$154$	0	0
$155$	1.35256e9i	0.188219i
$156$	0	0
$157$	8.30545e9i	1.09097i	0.838119	+	0.545487i	$0.183655\pi$
$157$	8.30545e9i	1.09097i	−0.838119	+	0.545487i	$0.816345\pi$
$158$	0	0
$159$	−5.15444e9	−0.639579
$160$	0	0
$161$	6.34943e9	0.744763
$162$	0	0
$163$	1.02083e10i	1.13269i	0.824170	+	0.566343i	$0.191642\pi$
$163$	1.02083e10i	1.13269i	−0.824170	+	0.566343i	$0.808358\pi$
$164$	0	0
$165$	2.16896e10i	2.27810i
$166$	0	0
$167$	1.78819e10	1.77906	0.889529	−	0.456880i	$-0.151033\pi$
$167$	1.78819e10	1.77906	0.889529	+	0.456880i	$0.151033\pi$
$168$	0	0
$169$	5.10275e9	0.481187
$170$	0	0
$171$	7.28460e9i	0.651513i
$172$	0	0
$173$	− 4.35787e9i	− 0.369885i	−0.982749	−	0.184943i	$-0.940790\pi$
$173$	− 4.35787e9i	− 0.369885i	0.982749	−	0.184943i	$-0.0592100\pi$
$174$	0	0
$175$	6.46347e9	0.520948
$176$	0	0
$177$	3.01758e10	2.31089
$178$	0	0
$179$	− 1.08673e10i	− 0.791193i	−0.918424	−	0.395597i	$-0.870538\pi$
$179$	− 1.08673e10i	− 0.791193i	0.918424	−	0.395597i	$-0.129462\pi$
$180$	0	0
$181$	− 2.14725e10i	− 1.48706i	−0.668703	−	0.743530i	$-0.733150\pi$
$181$	− 2.14725e10i	− 1.48706i	0.668703	−	0.743530i	$-0.266850\pi$
$182$	0	0
$183$	2.48006e10	1.63468
$184$	0	0
$185$	1.02233e10	0.641681
$186$	0	0
$187$	− 2.58216e10i	− 1.54418i
$188$	0	0
$189$	1.88240e10i	1.07308i
$190$	0	0
$191$	−2.77603e10	−1.50930	−0.754648	−	0.656130i	$-0.772192\pi$
$191$	−2.77603e10	−1.50930	−0.754648	+	0.656130i	$0.772192\pi$
$192$	0	0
$193$	−1.74020e10	−0.902801	−0.451400	−	0.892322i	$-0.649075\pi$
$193$	−1.74020e10	−0.902801	−0.451400	+	0.892322i	$0.649075\pi$
$194$	0	0
$195$	− 3.78638e10i	− 1.87529i
$196$	0	0
$197$	2.25126e10i	1.06495i	0.846447	+	0.532473i	$0.178737\pi$
$197$	2.25126e10i	1.06495i	−0.846447	+	0.532473i	$0.821263\pi$
$198$	0	0
$199$	−3.06974e10	−1.38760	−0.693798	−	0.720170i	$-0.744064\pi$
$199$	−3.06974e10	−1.38760	−0.693798	+	0.720170i	$0.744064\pi$
$200$	0	0
$201$	−2.55881e10	−1.10575
$202$	0	0
$203$	− 5.70493e9i	− 0.235786i
$204$	0	0
$205$	− 2.95391e10i	− 1.16817i
$206$	0	0
$207$	9.23167e10	3.49473
$208$	0	0
$209$	7.00428e9	0.253925
$210$	0	0
$211$	2.20646e10i	0.766348i	0.923676	+	0.383174i	$0.125169\pi$
$211$	2.20646e10i	0.766348i	−0.923676	+	0.383174i	$0.874831\pi$
$212$	0	0
$213$	7.86745e10i	2.61894i
$214$	0	0
$215$	2.34112e10	0.747223
$216$	0	0
$217$	2.03522e9	0.0623079
$218$	0	0
$219$	1.72487e9i	0.0506708i
$220$	0	0
$221$	4.50773e10i	1.27114i
$222$	0	0
$223$	1.51667e10	0.410695	0.205348	−	0.978689i	$-0.434167\pi$
$223$	1.51667e10	0.410695	0.205348	+	0.978689i	$0.434167\pi$
$224$	0	0
$225$	9.39748e10	2.44450
$226$	0	0
$227$	− 2.19810e10i	− 0.549455i	−0.961522	−	0.274727i	$-0.911412\pi$
$227$	− 2.19810e10i	− 0.549455i	0.961522	−	0.274727i	$-0.0885876\pi$
$228$	0	0
$229$	3.45174e10i	0.829427i	0.909952	+	0.414714i	$0.136118\pi$
$229$	3.45174e10i	0.829427i	−0.909952	+	0.414714i	$0.863882\pi$
$230$	0	0
$231$	3.26368e10	0.754143
$232$	0	0
$233$	4.06497e10	0.903558	0.451779	−	0.892130i	$-0.350790\pi$
$233$	4.06497e10	0.903558	0.451779	+	0.892130i	$0.350790\pi$
$234$	0	0
$235$	− 6.77636e10i	− 1.44941i
$236$	0	0
$237$	1.32198e11i	2.72181i
$238$	0	0
$239$	3.09648e10	0.613872	0.306936	−	0.951730i	$-0.400696\pi$
$239$	3.09648e10	0.613872	0.306936	+	0.951730i	$0.400696\pi$
$240$	0	0
$241$	3.55827e10	0.679457	0.339728	−	0.940524i	$-0.389665\pi$
$241$	3.55827e10	0.679457	0.339728	+	0.940524i	$0.389665\pi$
$242$	0	0
$243$	5.38689e10i	0.991082i
$244$	0	0
$245$	6.28499e10i	1.11444i
$246$	0	0
$247$	−1.22275e10	−0.209026
$248$	0	0
$249$	6.00657e10	0.990216
$250$	0	0
$251$	2.11617e10i	0.336525i	0.985742	+	0.168263i	$0.0538157\pi$
$251$	2.11617e10i	0.336525i	−0.985742	+	0.168263i	$0.946184\pi$
$252$	0	0
$253$	− 8.87642e10i	− 1.36206i
$254$	0	0
$255$	−3.10228e11	−4.59463
$256$	0	0
$257$	−1.03338e11	−1.47761	−0.738804	−	0.673921i	$-0.764609\pi$
$257$	−1.03338e11	−1.47761	−0.738804	+	0.673921i	$0.764609\pi$
$258$	0	0
$259$	− 1.53833e10i	− 0.212422i
$260$	0	0
$261$	− 8.29460e10i	− 1.10640i
$262$	0	0
$263$	−9.63331e10	−1.24158	−0.620790	−	0.783977i	$-0.713188\pi$
$263$	−9.63331e10	−1.24158	−0.620790	+	0.783977i	$0.713188\pi$
$264$	0	0
$265$	4.11947e10	0.513139
$266$	0	0
$267$	1.56963e11i	1.89015i
$268$	0	0
$269$	7.17446e10i	0.835419i	0.908581	+	0.417709i	$0.137167\pi$
$269$	7.17446e10i	0.835419i	−0.908581	+	0.417709i	$0.862833\pi$
$270$	0	0
$271$	1.31827e11	1.48471	0.742356	−	0.670006i	$-0.233708\pi$
$271$	1.31827e11	1.48471	0.742356	+	0.670006i	$0.233708\pi$
$272$	0	0
$273$	−5.69746e10	−0.620796
$274$	0	0
$275$	− 9.03585e10i	− 0.952734i
$276$	0	0
$277$	− 9.13454e10i	− 0.932240i	−0.884722	−	0.466120i	$-0.845652\pi$
$277$	− 9.13454e10i	− 0.932240i	0.884722	−	0.466120i	$-0.154348\pi$
$278$	0	0
$279$	2.95908e10	0.292374
$280$	0	0
$281$	−7.50850e10	−0.718414	−0.359207	−	0.933258i	$-0.616953\pi$
$281$	−7.50850e10	−0.718414	−0.359207	+	0.933258i	$0.616953\pi$
$282$	0	0
$283$	9.96369e10i	0.923381i	0.887041	+	0.461691i	$0.152757\pi$
$283$	9.96369e10i	0.923381i	−0.887041	+	0.461691i	$0.847243\pi$
$284$	0	0
$285$	− 8.41513e10i	− 0.755542i
$286$	0	0
$287$	−4.44481e10	−0.386710
$288$	0	0
$289$	2.50742e11	2.11440
$290$	0	0
$291$	2.82319e11i	2.30793i
$292$	0	0
$293$	− 2.18507e11i	− 1.73205i	−0.499999	−	0.866026i	$-0.666666\pi$
$293$	− 2.18507e11i	− 1.73205i	0.499999	−	0.866026i	$-0.333334\pi$
$294$	0	0
$295$	−2.41167e11	−1.85404
$296$	0	0
$297$	2.63157e11	1.96251
$298$	0	0
$299$	1.54957e11i	1.12122i
$300$	0	0
$301$	− 3.52274e10i	− 0.247361i
$302$	0	0
$303$	7.18966e10	0.490023
$304$	0	0
$305$	−1.98208e11	−1.31151
$306$	0	0
$307$	4.39947e10i	0.282669i	0.989962	+	0.141334i	$0.0451393\pi$
$307$	4.39947e10i	0.282669i	−0.989962	+	0.141334i	$0.954861\pi$
$308$	0	0
$309$	1.49948e11i	0.935678i
$310$	0	0
$311$	1.89071e11	1.14605	0.573023	−	0.819539i	$-0.305770\pi$
$311$	1.89071e11	1.14605	0.573023	+	0.819539i	$0.305770\pi$
$312$	0	0
$313$	2.35414e10	0.138638	0.0693191	−	0.997595i	$-0.477917\pi$
$313$	2.35414e10	0.138638	0.0693191	+	0.997595i	$0.477917\pi$
$314$	0	0
$315$	− 2.71275e11i	− 1.55243i
$316$	0	0
$317$	1.91969e11i	1.06774i	0.845567	+	0.533869i	$0.179262\pi$
$317$	1.91969e11i	1.06774i	−0.845567	+	0.533869i	$0.820738\pi$
$318$	0	0
$319$	−7.97542e10	−0.431216
$320$	0	0
$321$	−1.62904e11	−0.856366
$322$	0	0
$323$	1.00183e11i	0.512133i
$324$	0	0
$325$	1.57740e11i	0.784273i
$326$	0	0
$327$	−1.42676e11	−0.690059
$328$	0	0
$329$	−1.01965e11	−0.479812
$330$	0	0
$331$	− 4.23565e11i	− 1.93952i	−0.244063	−	0.969759i	$-0.578481\pi$
$331$	− 4.23565e11i	− 1.93952i	0.244063	−	0.969759i	$-0.421519\pi$
$332$	0	0
$333$	− 2.23663e11i	− 0.996769i
$334$	0	0
$335$	2.04502e11	0.887148
$336$	0	0
$337$	3.28269e11	1.38642	0.693212	−	0.720734i	$-0.256195\pi$
$337$	3.28269e11	1.38642	0.693212	+	0.720734i	$0.256195\pi$
$338$	0	0
$339$	− 9.98410e10i	− 0.410592i
$340$	0	0
$341$	− 2.84522e10i	− 0.113952i
$342$	0	0
$343$	2.17218e11	0.847370
$344$	0	0
$345$	−1.06644e12	−4.05274
$346$	0	0
$347$	2.25945e11i	0.836604i	0.908308	+	0.418302i	$0.137375\pi$
$347$	2.25945e11i	0.836604i	−0.908308	+	0.418302i	$0.862625\pi$
$348$	0	0
$349$	− 5.81052e10i	− 0.209653i	−0.994491	−	0.104826i	$-0.966571\pi$
$349$	− 5.81052e10i	− 0.209653i	0.994491	−	0.104826i	$-0.0334287\pi$
$350$	0	0
$351$	−4.59397e11	−1.61550
$352$	0	0
$353$	−1.00599e11	−0.344832	−0.172416	−	0.985024i	$-0.555157\pi$
$353$	−1.00599e11	−0.344832	−0.172416	+	0.985024i	$0.555157\pi$
$354$	0	0
$355$	− 6.28774e11i	− 2.10120i
$356$	0	0
$357$	4.66807e11i	1.52101i
$358$	0	0
$359$	1.42379e11	0.452397	0.226198	−	0.974081i	$-0.427370\pi$
$359$	1.42379e11	0.452397	0.226198	+	0.974081i	$0.427370\pi$
$360$	0	0
$361$	2.95512e11	0.915785
$362$	0	0
$363$	1.39666e11i	0.422194i
$364$	0	0
$365$	− 1.37853e10i	− 0.0406536i
$366$	0	0
$367$	5.83998e11	1.68041	0.840203	−	0.542272i	$-0.182436\pi$
$367$	5.83998e11	1.68041	0.840203	+	0.542272i	$0.182436\pi$
$368$	0	0
$369$	−6.46247e11	−1.81460
$370$	0	0
$371$	− 6.19866e10i	− 0.169869i
$372$	0	0
$373$	− 7.04616e11i	− 1.88479i	−0.334505	−	0.942394i	$-0.608569\pi$
$373$	− 7.04616e11i	− 1.88479i	0.334505	−	0.942394i	$-0.391431\pi$
$374$	0	0
$375$	−8.85705e10	−0.231286
$376$	0	0
$377$	1.39228e11	0.354969
$378$	0	0
$379$	4.54881e11i	1.13246i	0.824248	+	0.566228i	$0.191598\pi$
$379$	4.54881e11i	1.13246i	−0.824248	+	0.566228i	$0.808402\pi$
$380$	0	0
$381$	− 8.62583e11i	− 2.09719i
$382$	0	0
$383$	6.11355e11	1.45177	0.725887	−	0.687814i	$-0.241429\pi$
$383$	6.11355e11	1.45177	0.725887	+	0.687814i	$0.241429\pi$
$384$	0	0
$385$	−2.60836e11	−0.605054
$386$	0	0
$387$	− 5.12183e11i	− 1.16072i
$388$	0	0
$389$	− 9.28801e10i	− 0.205660i	−0.994699	−	0.102830i	$-0.967210\pi$
$389$	− 9.28801e10i	− 0.205660i	0.994699	−	0.102830i	$-0.0327897\pi$
$390$	0	0
$391$	1.26960e12	2.74709
$392$	0	0
$393$	−1.50559e12	−3.18377
$394$	0	0
$395$	− 1.05654e12i	− 2.18373i
$396$	0	0
$397$	− 6.58291e11i	− 1.33003i	−0.746831	−	0.665014i	$-0.768426\pi$
$397$	− 6.58291e11i	− 1.33003i	0.746831	−	0.665014i	$-0.231574\pi$
$398$	0	0
$399$	−1.26624e11	−0.250115
$400$	0	0
$401$	−1.93836e11	−0.374356	−0.187178	−	0.982326i	$-0.559934\pi$
$401$	−1.93836e11	−0.374356	−0.187178	+	0.982326i	$0.559934\pi$
$402$	0	0
$403$	4.96694e10i	0.0938029i
$404$	0	0
$405$	− 1.40482e12i	− 2.59461i
$406$	0	0
$407$	−2.15056e11	−0.388487
$408$	0	0
$409$	3.09099e11	0.546188	0.273094	−	0.961987i	$-0.411953\pi$
$409$	3.09099e11	0.546188	0.273094	+	0.961987i	$0.411953\pi$
$410$	0	0
$411$	− 8.05177e11i	− 1.39188i
$412$	0	0
$413$	3.62890e11i	0.613762i
$414$	0	0
$415$	−4.80051e11	−0.794457
$416$	0	0
$417$	3.33304e11	0.539795
$418$	0	0
$419$	6.55357e11i	1.03876i	0.854544	+	0.519380i	$0.173837\pi$
$419$	6.55357e11i	1.03876i	−0.854544	+	0.519380i	$0.826163\pi$
$420$	0	0
$421$	2.00415e11i	0.310929i	0.987841	+	0.155465i	$0.0496875\pi$
$421$	2.00415e11i	0.310929i	−0.987841	+	0.155465i	$0.950313\pi$
$422$	0	0
$423$	−1.48251e12	−2.25147
$424$	0	0
$425$	1.29241e12	1.92154
$426$	0	0
$427$	2.98249e11i	0.434163i
$428$	0	0
$429$	7.96497e11i	1.13534i
$430$	0	0
$431$	−5.21200e11	−0.727540	−0.363770	−	0.931489i	$-0.618511\pi$
$431$	−5.21200e11	−0.727540	−0.363770	+	0.931489i	$0.618511\pi$
$432$	0	0
$433$	3.23838e11	0.442723	0.221361	−	0.975192i	$-0.428950\pi$
$433$	3.23838e11	0.442723	0.221361	+	0.975192i	$0.428950\pi$
$434$	0	0
$435$	9.58188e11i	1.28307i
$436$	0	0
$437$	3.44388e11i	0.451732i
$438$	0	0
$439$	1.59945e11	0.205532	0.102766	−	0.994706i	$-0.467231\pi$
$439$	1.59945e11	0.205532	0.102766	+	0.994706i	$0.467231\pi$
$440$	0	0
$441$	1.37501e12	1.73114
$442$	0	0
$443$	− 2.33953e11i	− 0.288610i	−0.989533	−	0.144305i	$-0.953905\pi$
$443$	− 2.33953e11i	− 0.288610i	0.989533	−	0.144305i	$-0.0460946\pi$
$444$	0	0
$445$	− 1.25446e12i	− 1.51648i
$446$	0	0
$447$	1.71723e12	2.03444
$448$	0	0
$449$	−1.44728e12	−1.68052	−0.840262	−	0.542180i	$-0.817599\pi$
$449$	−1.44728e12	−1.68052	−0.840262	+	0.542180i	$0.817599\pi$
$450$	0	0
$451$	6.21379e11i	0.707232i
$452$	0	0
$453$	− 1.05925e12i	− 1.18184i
$454$	0	0
$455$	4.55346e11	0.498070
$456$	0	0
$457$	−1.09012e12	−1.16909	−0.584547	−	0.811359i	$-0.698728\pi$
$457$	−1.09012e12	−1.16909	−0.584547	+	0.811359i	$0.698728\pi$
$458$	0	0
$459$	3.76396e12i	3.95811i
$460$	0	0
$461$	1.66397e12i	1.71589i	0.513738	+	0.857947i	$0.328260\pi$
$461$	1.66397e12i	1.71589i	−0.513738	+	0.857947i	$0.671740\pi$
$462$	0	0
$463$	−1.55834e12	−1.57597	−0.787985	−	0.615695i	$-0.788875\pi$
$463$	−1.55834e12	−1.57597	−0.787985	+	0.615695i	$0.788875\pi$
$464$	0	0
$465$	−3.41832e11	−0.339058
$466$	0	0
$467$	− 1.17880e12i	− 1.14687i	−0.819251	−	0.573436i	$-0.805610\pi$
$467$	− 1.17880e12i	− 1.14687i	0.819251	−	0.573436i	$-0.194390\pi$
$468$	0	0
$469$	− 3.07719e11i	− 0.293681i
$470$	0	0
$471$	−2.09904e12	−1.96529
$472$	0	0
$473$	−4.92474e11	−0.452385
$474$	0	0
$475$	3.50573e11i	0.315979i
$476$	0	0
$477$	− 9.01246e11i	− 0.797096i
$478$	0	0
$479$	−1.05266e11	−0.0913645	−0.0456822	−	0.998956i	$-0.514546\pi$
$479$	−1.05266e11	−0.0913645	−0.0456822	+	0.998956i	$0.514546\pi$
$480$	0	0
$481$	3.75427e11	0.319795
$482$	0	0
$483$	1.60469e12i	1.34162i
$484$	0	0
$485$	− 2.25632e12i	− 1.85167i
$486$	0	0
$487$	2.92582e11	0.235704	0.117852	−	0.993031i	$-0.462399\pi$
$487$	2.92582e11	0.235704	0.117852	+	0.993031i	$0.462399\pi$
$488$	0	0
$489$	−2.57995e12	−2.04043
$490$	0	0
$491$	− 1.72770e12i	− 1.34154i	−0.741667	−	0.670768i	$-0.765965\pi$
$491$	− 1.72770e12i	− 1.34154i	0.741667	−	0.670768i	$-0.234035\pi$
$492$	0	0
$493$	− 1.14073e12i	− 0.869706i
$494$	0	0
$495$	−3.79239e12	−2.83916
$496$	0	0
$497$	−9.46130e11	−0.695580
$498$	0	0
$499$	− 2.08491e12i	− 1.50534i	−0.658396	−	0.752672i	$-0.728765\pi$
$499$	− 2.08491e12i	− 1.50534i	0.658396	−	0.752672i	$-0.271235\pi$
$500$	0	0
$501$	4.51930e12i	3.20480i
$502$	0	0
$503$	−2.67459e11	−0.186295	−0.0931474	−	0.995652i	$-0.529693\pi$
$503$	−2.67459e11	−0.186295	−0.0931474	+	0.995652i	$0.529693\pi$
$504$	0	0
$505$	−5.74604e11	−0.393149
$506$	0	0
$507$	1.28962e12i	0.866813i
$508$	0	0
$509$	− 7.85816e11i	− 0.518909i	−0.965755	−	0.259454i	$-0.916457\pi$
$509$	− 7.85816e11i	− 0.518909i	0.965755	−	0.259454i	$-0.0835427\pi$
$510$	0	0
$511$	−2.07431e10	−0.0134580
$512$	0	0
$513$	−1.02100e12	−0.650873
$514$	0	0
$515$	− 1.19839e12i	− 0.750701i
$516$	0	0
$517$	1.42546e12i	0.877503i
$518$	0	0
$519$	1.10137e12	0.666313
$520$	0	0
$521$	4.69385e11	0.279100	0.139550	−	0.990215i	$-0.455434\pi$
$521$	4.69385e11	0.279100	0.139550	+	0.990215i	$0.455434\pi$
$522$	0	0
$523$	− 2.40838e12i	− 1.40756i	−0.710418	−	0.703780i	$-0.751494\pi$
$523$	− 2.40838e12i	− 1.40756i	0.710418	−	0.703780i	$-0.248506\pi$
$524$	0	0
$525$	1.63351e12i	0.938439i
$526$	0	0
$527$	4.06954e11	0.229825
$528$	0	0
$529$	2.56322e12	1.42310
$530$	0	0
$531$	5.27619e12i	2.88002i
$532$	0	0
$533$	− 1.08475e12i	− 0.582181i
$534$	0	0
$535$	1.30194e12	0.687069
$536$	0	0
$537$	2.74649e12	1.42526
$538$	0	0
$539$	− 1.32210e12i	− 0.674706i
$540$	0	0
$541$	2.43978e12i	1.22451i	0.790660	+	0.612255i	$0.209737\pi$
$541$	2.43978e12i	1.22451i	−0.790660	+	0.612255i	$0.790263\pi$
$542$	0	0
$543$	5.42674e12	2.67880
$544$	0	0
$545$	1.14028e12	0.553640
$546$	0	0
$547$	− 8.95402e11i	− 0.427637i	−0.976873	−	0.213818i	$-0.931410\pi$
$547$	− 8.95402e11i	− 0.427637i	0.976873	−	0.213818i	$-0.0685901\pi$
$548$	0	0
$549$	4.33635e12i	2.03727i
$550$	0	0
$551$	3.09431e11	0.143015
$552$	0	0
$553$	−1.58980e12	−0.722901
$554$	0	0
$555$	2.58374e12i	1.15593i
$556$	0	0
$557$	3.89951e12i	1.71657i	0.513171	+	0.858286i	$0.328471\pi$
$557$	3.89951e12i	1.71657i	−0.513171	+	0.858286i	$0.671529\pi$
$558$	0	0
$559$	8.59720e11	0.372395
$560$	0	0
$561$	6.52591e12	2.78169
$562$	0	0
$563$	− 3.25127e11i	− 0.136384i	−0.997672	−	0.0681922i	$-0.978277\pi$
$563$	− 3.25127e11i	− 0.136384i	0.997672	−	0.0681922i	$-0.0217231\pi$
$564$	0	0
$565$	7.97938e11i	0.329421i
$566$	0	0
$567$	−2.11386e12	−0.858920
$568$	0	0
$569$	7.30167e11	0.292023	0.146012	−	0.989283i	$-0.453356\pi$
$569$	7.30167e11	0.292023	0.146012	+	0.989283i	$0.453356\pi$
$570$	0	0
$571$	1.59452e11i	0.0627723i	0.999507	+	0.0313861i	$0.00999216\pi$
$571$	1.59452e11i	0.0627723i	−0.999507	+	0.0313861i	$0.990008\pi$
$572$	0	0
$573$	− 7.01587e12i	− 2.71885i
$574$	0	0
$575$	4.44276e12	1.69492
$576$	0	0
$577$	1.82973e12	0.687218	0.343609	−	0.939113i	$-0.388350\pi$
$577$	1.82973e12	0.687218	0.343609	+	0.939113i	$0.388350\pi$
$578$	0	0
$579$	− 4.39802e12i	− 1.62631i
$580$	0	0
$581$	7.22343e11i	0.262997i
$582$	0	0
$583$	−8.66565e11	−0.310665
$584$	0	0
$585$	6.62044e12	2.33714
$586$	0	0
$587$	− 3.74725e12i	− 1.30269i	−0.758782	−	0.651345i	$-0.774205\pi$
$587$	− 3.74725e12i	− 1.30269i	0.758782	−	0.651345i	$-0.225795\pi$
$588$	0	0
$589$	1.10389e11i	0.0377925i
$590$	0	0
$591$	−5.68961e12	−1.91840
$592$	0	0
$593$	−4.51432e12	−1.49916	−0.749578	−	0.661916i	$-0.769743\pi$
$593$	−4.51432e12	−1.49916	−0.749578	+	0.661916i	$0.769743\pi$
$594$	0	0
$595$	− 3.73077e12i	− 1.22031i
$596$	0	0
$597$	− 7.75816e12i	− 2.49962i
$598$	0	0
$599$	2.95283e12	0.937168	0.468584	−	0.883419i	$-0.344764\pi$
$599$	2.95283e12	0.937168	0.468584	+	0.883419i	$0.344764\pi$
$600$	0	0
$601$	3.34262e12	1.04509	0.522543	−	0.852613i	$-0.324984\pi$
$601$	3.34262e12	1.04509	0.522543	+	0.852613i	$0.324984\pi$
$602$	0	0
$603$	− 4.47404e12i	− 1.37807i
$604$	0	0
$605$	− 1.11622e12i	− 0.338729i
$606$	0	0
$607$	4.79377e12	1.43327	0.716636	−	0.697448i	$-0.245681\pi$
$607$	4.79377e12	1.43327	0.716636	+	0.697448i	$0.245681\pi$
$608$	0	0
$609$	1.44181e12	0.424746
$610$	0	0
$611$	− 2.48845e12i	− 0.722344i
$612$	0	0
$613$	3.04095e12i	0.869837i	0.900470	+	0.434918i	$0.143223\pi$
$613$	3.04095e12i	0.869837i	−0.900470	+	0.434918i	$0.856777\pi$
$614$	0	0
$615$	7.46541e12	2.10434
$616$	0	0
$617$	1.62097e12	0.450288	0.225144	−	0.974325i	$-0.427715\pi$
$617$	1.62097e12	0.450288	0.225144	+	0.974325i	$0.427715\pi$
$618$	0	0
$619$	− 1.34898e12i	− 0.369315i	−0.982803	−	0.184658i	$-0.940882\pi$
$619$	− 1.34898e12i	− 0.369315i	0.982803	−	0.184658i	$-0.0591176\pi$
$620$	0	0
$621$	1.29390e13i	3.49130i
$622$	0	0
$623$	−1.88762e12	−0.502017
$624$	0	0
$625$	−3.44571e12	−0.903273
$626$	0	0
$627$	1.77019e12i	0.457421i
$628$	0	0
$629$	− 3.07597e12i	− 0.783527i
$630$	0	0
$631$	8.33673e11	0.209346	0.104673	−	0.994507i	$-0.466620\pi$
$631$	8.33673e11	0.209346	0.104673	+	0.994507i	$0.466620\pi$
$632$	0	0
$633$	−5.57640e12	−1.38050
$634$	0	0
$635$	6.89384e12i	1.68259i
$636$	0	0
$637$	2.30801e12i	0.555406i
$638$	0	0
$639$	−1.37561e13	−3.26394
$640$	0	0
$641$	5.03354e12	1.17764	0.588820	−	0.808264i	$-0.299593\pi$
$641$	5.03354e12	1.17764	0.588820	+	0.808264i	$0.299593\pi$
$642$	0	0
$643$	5.08235e12i	1.17251i	0.810128	+	0.586253i	$0.199398\pi$
$643$	5.08235e12i	1.17251i	−0.810128	+	0.586253i	$0.800602\pi$
$644$	0	0
$645$	5.91671e12i	1.34605i
$646$	0	0
$647$	3.34349e12	0.750120	0.375060	−	0.927001i	$-0.377622\pi$
$647$	3.34349e12	0.750120	0.375060	+	0.927001i	$0.377622\pi$
$648$	0	0
$649$	5.07316e12	1.12248
$650$	0	0
$651$	5.14362e11i	0.112242i
$652$	0	0
$653$	− 5.29726e12i	− 1.14010i	−0.821611	−	0.570049i	$-0.806924\pi$
$653$	− 5.29726e12i	− 1.14010i	0.821611	−	0.570049i	$-0.193076\pi$
$654$	0	0
$655$	1.20328e13	2.55436
$656$	0	0
$657$	−3.01591e11	−0.0631501
$658$	0	0
$659$	− 7.48186e12i	− 1.54534i	−0.634805	−	0.772672i	$-0.718920\pi$
$659$	− 7.48186e12i	− 1.54534i	0.634805	−	0.772672i	$-0.281080\pi$
$660$	0	0
$661$	− 1.03230e12i	− 0.210330i	−0.994455	−	0.105165i	$-0.966463\pi$
$661$	− 1.03230e12i	− 0.210330i	0.994455	−	0.105165i	$-0.0335371\pi$
$662$	0	0
$663$	−1.13924e13	−2.28983
$664$	0	0
$665$	1.01199e12	0.200669
$666$	0	0
$667$	− 3.92137e12i	− 0.767134i
$668$	0	0
$669$	3.83309e12i	0.739828i
$670$	0	0
$671$	4.16948e12	0.794018
$672$	0	0
$673$	−8.13048e12	−1.52774	−0.763868	−	0.645373i	$-0.776702\pi$
$673$	−8.13048e12	−1.52774	−0.763868	+	0.645373i	$0.776702\pi$
$674$	0	0
$675$	1.31714e13i	2.44210i
$676$	0	0
$677$	7.61157e12i	1.39260i	0.717753	+	0.696298i	$0.245171\pi$
$677$	7.61157e12i	1.39260i	−0.717753	+	0.696298i	$0.754829\pi$
$678$	0	0
$679$	−3.39514e12	−0.612976
$680$	0	0
$681$	5.55527e12	0.989790
$682$	0	0
$683$	7.40992e12i	1.30293i	0.758680	+	0.651464i	$0.225845\pi$
$683$	7.40992e12i	1.30293i	−0.758680	+	0.651464i	$0.774155\pi$
$684$	0	0
$685$	6.43504e12i	1.11672i
$686$	0	0
$687$	−8.72358e12	−1.49413
$688$	0	0
$689$	1.51278e12	0.255734
$690$	0	0
$691$	− 3.64791e11i	− 0.0608685i	−0.999537	−	0.0304343i	$-0.990311\pi$
$691$	− 3.64791e11i	− 0.0608685i	0.999537	−	0.0304343i	$-0.00968902\pi$
$692$	0	0
$693$	5.70650e12i	0.939875i
$694$	0	0
$695$	−2.66380e12	−0.433081
$696$	0	0
$697$	−8.88765e12	−1.42639
$698$	0	0
$699$	1.02734e13i	1.62767i
$700$	0	0
$701$	− 2.33617e12i	− 0.365404i	−0.983168	−	0.182702i	$-0.941516\pi$
$701$	− 2.33617e12i	− 0.365404i	0.983168	−	0.182702i	$-0.0584844\pi$
$702$	0	0
$703$	8.34375e11	0.128843
$704$	0	0
$705$	1.71259e13	2.61097
$706$	0	0
$707$	8.64620e11i	0.130148i
$708$	0	0
$709$	2.17052e12i	0.322594i	0.986906	+	0.161297i	$0.0515677\pi$
$709$	2.17052e12i	0.322594i	−0.986906	+	0.161297i	$0.948432\pi$
$710$	0	0
$711$	−2.31147e13	−3.39214
$712$	0	0
$713$	1.39894e12	0.202720
$714$	0	0
$715$	− 6.36567e12i	− 0.910893i
$716$	0	0
$717$	7.82574e12i	1.10583i
$718$	0	0
$719$	2.66823e10	0.00372342	0.00186171	−	0.999998i	$-0.499407\pi$
$719$	2.66823e10	0.00372342	0.00186171	+	0.999998i	$0.499407\pi$
$720$	0	0
$721$	−1.80325e12	−0.248512
$722$	0	0
$723$	8.99281e12i	1.22398i
$724$	0	0
$725$	− 3.99180e12i	− 0.536596i
$726$	0	0
$727$	8.81984e12	1.17100	0.585499	−	0.810673i	$-0.300899\pi$
$727$	8.81984e12	1.17100	0.585499	+	0.810673i	$0.300899\pi$
$728$	0	0
$729$	7.54253e10	0.00989107
$730$	0	0
$731$	− 7.04391e12i	− 0.912400i
$732$	0	0
$733$	− 6.94718e12i	− 0.888874i	−0.895810	−	0.444437i	$-0.853404\pi$
$733$	− 6.94718e12i	− 0.888874i	0.895810	−	0.444437i	$-0.146596\pi$
$734$	0	0
$735$	−1.58841e13	−2.00756
$736$	0	0
$737$	−4.30187e12	−0.537098
$738$	0	0
$739$	1.65991e12i	0.204732i	0.994747	+	0.102366i	$0.0326413\pi$
$739$	1.65991e12i	0.204732i	−0.994747	+	0.102366i	$0.967359\pi$
$740$	0	0
$741$	− 3.09025e12i	− 0.376541i
$742$	0	0
$743$	6.92810e12	0.833997	0.416998	−	0.908907i	$-0.363082\pi$
$743$	6.92810e12	0.833997	0.416998	+	0.908907i	$0.363082\pi$
$744$	0	0
$745$	−1.37242e13	−1.63224
$746$	0	0
$747$	1.05024e13i	1.23409i
$748$	0	0
$749$	− 1.95907e12i	− 0.227447i
$750$	0	0
$751$	1.09479e13	1.25589	0.627943	−	0.778259i	$-0.283897\pi$
$751$	1.09479e13	1.25589	0.627943	+	0.778259i	$0.283897\pi$
$752$	0	0
$753$	−5.34819e12	−0.606219
$754$	0	0
$755$	8.46565e12i	0.948198i
$756$	0	0
$757$	1.11681e13i	1.23608i	0.786147	+	0.618039i	$0.212073\pi$
$757$	1.11681e13i	1.23608i	−0.786147	+	0.618039i	$0.787927\pi$
$758$	0	0
$759$	2.24334e13	2.45362
$760$	0	0
$761$	−6.16538e12	−0.666391	−0.333195	−	0.942858i	$-0.608127\pi$
$761$	−6.16538e12	−0.666391	−0.333195	+	0.942858i	$0.608127\pi$
$762$	0	0
$763$	− 1.71581e12i	− 0.183277i
$764$	0	0
$765$	− 5.42429e13i	− 5.72620i
$766$	0	0
$767$	−8.85629e12	−0.924002
$768$	0	0
$769$	−4.00721e12	−0.413213	−0.206606	−	0.978424i	$-0.566242\pi$
$769$	−4.00721e12	−0.413213	−0.206606	+	0.978424i	$0.566242\pi$
$770$	0	0
$771$	− 2.61165e13i	− 2.66177i
$772$	0	0
$773$	− 1.57080e13i	− 1.58239i	−0.611567	−	0.791193i	$-0.709460\pi$
$773$	− 1.57080e13i	− 1.58239i	0.611567	−	0.791193i	$-0.290540\pi$
$774$	0	0
$775$	1.42407e12	0.141799
$776$	0	0
$777$	3.88781e12	0.382658
$778$	0	0
$779$	− 2.41083e12i	− 0.234557i
$780$	0	0
$781$	1.32268e13i	1.27211i
$782$	0	0
$783$	1.16256e13	1.10532
$784$	0	0
$785$	1.67757e13	1.57676
$786$	0	0
$787$	4.11748e12i	0.382600i	0.981532	+	0.191300i	$0.0612704\pi$
$787$	4.11748e12i	0.382600i	−0.981532	+	0.191300i	$0.938730\pi$
$788$	0	0
$789$	− 2.43463e13i	− 2.23659i
$790$	0	0
$791$	1.20068e12	0.109052
$792$	0	0
$793$	−7.27873e12	−0.653621
$794$	0	0
$795$	1.04111e13i	0.924371i
$796$	0	0
$797$	− 6.62590e12i	− 0.581677i	−0.956772	−	0.290839i	$-0.906066\pi$
$797$	− 6.62590e12i	− 0.581677i	0.956772	−	0.290839i	$-0.0939343\pi$
$798$	0	0
$799$	−2.03886e13	−1.76981
$800$	0	0
$801$	−2.74448e13	−2.35566
$802$	0	0
$803$	2.89985e11i	0.0246125i
$804$	0	0
$805$	− 1.28248e13i	− 1.07639i
$806$	0	0
$807$	−1.81320e13	−1.50493
$808$	0	0
$809$	−1.19920e12	−0.0984290	−0.0492145	−	0.998788i	$-0.515672\pi$
$809$	−1.19920e12	−0.0984290	−0.0492145	+	0.998788i	$0.515672\pi$
$810$	0	0
$811$	− 2.43378e12i	− 0.197555i	−0.995110	−	0.0987775i	$-0.968507\pi$
$811$	− 2.43378e12i	− 0.197555i	0.995110	−	0.0987775i	$-0.0314932\pi$
$812$	0	0
$813$	3.33166e13i	2.67457i
$814$	0	0
$815$	2.06191e13	1.63705
$816$	0	0
$817$	1.91070e12	0.150035
$818$	0	0
$819$	− 9.96192e12i	− 0.773688i
$820$	0	0
$821$	5.54529e12i	0.425971i	0.977055	+	0.212986i	$0.0683187\pi$
$821$	5.54529e12i	0.425971i	−0.977055	+	0.212986i	$0.931681\pi$
$822$	0	0
$823$	−1.48256e13	−1.12646	−0.563228	−	0.826302i	$-0.690441\pi$
$823$	−1.48256e13	−1.12646	−0.563228	+	0.826302i	$0.690441\pi$
$824$	0	0
$825$	2.28363e13	1.71626
$826$	0	0
$827$	− 1.41723e13i	− 1.05358i	−0.849997	−	0.526788i	$-0.823396\pi$
$827$	− 1.41723e13i	− 1.05358i	0.849997	−	0.526788i	$-0.176604\pi$
$828$	0	0
$829$	9.67424e11i	0.0711412i	0.999367	+	0.0355706i	$0.0113249\pi$
$829$	9.67424e11i	0.0711412i	−0.999367	+	0.0355706i	$0.988675\pi$
$830$	0	0
$831$	2.30857e13	1.67934
$832$	0	0
$833$	1.89101e13	1.36079
$834$	0	0
$835$	− 3.61186e13i	− 2.57124i
$836$	0	0
$837$	4.14741e12i	0.292087i
$838$	0	0
$839$	1.63583e13	1.13975	0.569875	−	0.821732i	$-0.306992\pi$
$839$	1.63583e13	1.13975	0.569875	+	0.821732i	$0.306992\pi$
$840$	0	0
$841$	1.09838e13	0.757131
$842$	0	0
$843$	− 1.89762e13i	− 1.29415i
$844$	0	0
$845$	− 1.03067e13i	− 0.695450i
$846$	0	0
$847$	−1.67961e12	−0.112133
$848$	0	0
$849$	−2.51812e13	−1.66338
$850$	0	0
$851$	− 1.05739e13i	− 0.691119i
$852$	0	0
$853$	8.51975e12i	0.551006i	0.961300	+	0.275503i	$0.0888444\pi$
$853$	8.51975e12i	0.551006i	−0.961300	+	0.275503i	$0.911156\pi$
$854$	0	0
$855$	1.47137e13	0.941619
$856$	0	0
$857$	−2.49683e12	−0.158116	−0.0790578	−	0.996870i	$-0.525191\pi$
$857$	−2.49683e12	−0.158116	−0.0790578	+	0.996870i	$0.525191\pi$
$858$	0	0
$859$	− 2.99678e13i	− 1.87796i	−0.343974	−	0.938979i	$-0.611773\pi$
$859$	− 2.99678e13i	− 1.87796i	0.343974	−	0.938979i	$-0.388227\pi$
$860$	0	0
$861$	− 1.12334e13i	− 0.696620i
$862$	0	0
$863$	−9.11663e12	−0.559482	−0.279741	−	0.960076i	$-0.590249\pi$
$863$	−9.11663e12	−0.559482	−0.279741	+	0.960076i	$0.590249\pi$
$864$	0	0
$865$	−8.80221e12	−0.534588
$866$	0	0
$867$	6.33700e13i	3.80889i
$868$	0	0
$869$	2.22252e13i	1.32208i
$870$	0	0
$871$	7.50984e12	0.442129
$872$	0	0
$873$	−4.93631e13	−2.87633
$874$	0	0
$875$	− 1.06514e12i	− 0.0614285i
$876$	0	0
$877$	1.62554e13i	0.927897i	0.885862	+	0.463948i	$0.153568\pi$
$877$	1.62554e13i	0.927897i	−0.885862	+	0.463948i	$0.846432\pi$
$878$	0	0
$879$	5.52233e13	3.12013
$880$	0	0
$881$	−3.34287e13	−1.86951	−0.934755	−	0.355294i	$-0.884381\pi$
$881$	−3.34287e13	−1.86951	−0.934755	+	0.355294i	$0.884381\pi$
$882$	0	0
$883$	1.62698e13i	0.900655i	0.892863	+	0.450328i	$0.148693\pi$
$883$	1.62698e13i	0.900655i	−0.892863	+	0.450328i	$0.851307\pi$
$884$	0	0
$885$	− 6.09503e13i	− 3.33988i
$886$	0	0
$887$	1.60289e13	0.869454	0.434727	−	0.900562i	$-0.356845\pi$
$887$	1.60289e13	0.869454	0.434727	+	0.900562i	$0.356845\pi$
$888$	0	0
$889$	1.03733e13	0.557006
$890$	0	0
$891$	2.95515e13i	1.57083i
$892$	0	0
$893$	− 5.53052e12i	− 0.291028i
$894$	0	0
$895$	−2.19502e13	−1.14350
$896$	0	0
$897$	−3.91623e13	−2.01977
$898$	0	0
$899$	− 1.25694e12i	− 0.0641795i
$900$	0	0
$901$	− 1.23946e13i	− 0.626570i
$902$	0	0
$903$	8.90301e12	0.445597
$904$	0	0
$905$	−4.33710e13	−2.14922
$906$	0	0
$907$	1.59734e13i	0.783726i	0.920024	+	0.391863i	$0.128169\pi$
$907$	1.59734e13i	0.783726i	−0.920024	+	0.391863i	$0.871831\pi$
$908$	0	0
$909$	1.25710e13i	0.610707i
$910$	0	0
$911$	−1.21013e13	−0.582104	−0.291052	−	0.956707i	$-0.594005\pi$
$911$	−1.21013e13	−0.582104	−0.291052	+	0.956707i	$0.594005\pi$
$912$	0	0
$913$	1.00983e13	0.480981
$914$	0	0
$915$	− 5.00932e13i	− 2.36257i
$916$	0	0
$917$	− 1.81061e13i	− 0.845596i
$918$	0	0
$919$	1.97502e13	0.913379	0.456689	−	0.889626i	$-0.349035\pi$
$919$	1.97502e13	0.913379	0.456689	+	0.889626i	$0.349035\pi$
$920$	0	0
$921$	−1.11188e13	−0.509201
$922$	0	0
$923$	− 2.30902e13i	− 1.04718i
$924$	0	0
$925$	− 1.07638e13i	− 0.483425i
$926$	0	0
$927$	−2.62181e13	−1.16612
$928$	0	0
$929$	7.13965e12	0.314490	0.157245	−	0.987560i	$-0.449739\pi$
$929$	7.13965e12	0.314490	0.157245	+	0.987560i	$0.449739\pi$
$930$	0	0
$931$	5.12949e12i	0.223769i
$932$	0	0
$933$	4.77838e13i	2.06449i
$934$	0	0
$935$	−5.21556e13	−2.23177
$936$	0	0
$937$	3.17448e13	1.34538	0.672689	−	0.739925i	$-0.265139\pi$
$937$	3.17448e13	1.34538	0.672689	+	0.739925i	$0.265139\pi$
$938$	0	0
$939$	5.94962e12i	0.249743i
$940$	0	0
$941$	− 4.59442e13i	− 1.91019i	−0.296296	−	0.955096i	$-0.595752\pi$
$941$	− 4.59442e13i	− 1.91019i	0.296296	−	0.955096i	$-0.404248\pi$
$942$	0	0
$943$	−3.05521e13	−1.25817
$944$	0	0
$945$	3.80215e13	1.55091
$946$	0	0
$947$	− 4.19613e12i	− 0.169541i	−0.996401	−	0.0847705i	$-0.972984\pi$
$947$	− 4.19613e12i	− 0.169541i	0.996401	−	0.0847705i	$-0.0270157\pi$
$948$	0	0
$949$	− 5.06232e11i	− 0.0202606i
$950$	0	0
$951$	−4.85164e13	−1.92343
$952$	0	0
$953$	−9.66942e11	−0.0379737	−0.0189868	−	0.999820i	$-0.506044\pi$
$953$	−9.66942e11	−0.0379737	−0.0189868	+	0.999820i	$0.506044\pi$
$954$	0	0
$955$	5.60715e13i	2.18136i
$956$	0	0
$957$	− 2.01563e13i	− 0.776795i
$958$	0	0
$959$	9.68296e12	0.369679
$960$	0	0
$961$	−2.59912e13	−0.983040
$962$	0	0
$963$	− 2.84836e13i	− 1.06727i
$964$	0	0
$965$	3.51493e13i	1.30480i
$966$	0	0
$967$	9.39038e12	0.345354	0.172677	−	0.984979i	$-0.444758\pi$
$967$	9.39038e12	0.345354	0.172677	+	0.984979i	$0.444758\pi$
$968$	0	0
$969$	−2.53192e13	−0.922558
$970$	0	0
$971$	2.23874e13i	0.808196i	0.914716	+	0.404098i	$0.132414\pi$
$971$	2.23874e13i	0.808196i	−0.914716	+	0.404098i	$0.867586\pi$
$972$	0	0
$973$	4.00828e12i	0.143367i
$974$	0	0
$975$	−3.98657e13	−1.41279
$976$	0	0
$977$	1.91366e13	0.671953	0.335977	−	0.941870i	$-0.390934\pi$
$977$	1.91366e13	0.671953	0.335977	+	0.941870i	$0.390934\pi$
$978$	0	0
$979$	2.63887e13i	0.918112i
$980$	0	0
$981$	− 2.49467e13i	− 0.860009i
$982$	0	0
$983$	−3.12487e13	−1.06744	−0.533718	−	0.845663i	$-0.679205\pi$
$983$	−3.12487e13	−1.06744	−0.533718	+	0.845663i	$0.679205\pi$
$984$	0	0
$985$	4.54719e13	1.53915
$986$	0	0
$987$	− 2.57697e13i	− 0.864336i
$988$	0	0
$989$	− 2.42141e13i	− 0.804793i
$990$	0	0
$991$	7.43111e12	0.244750	0.122375	−	0.992484i	$-0.460949\pi$
$991$	7.43111e12	0.244750	0.122375	+	0.992484i	$0.460949\pi$
$992$	0	0
$993$	1.07048e14	3.49386
$994$	0	0
$995$	6.20039e13i	2.00546i
$996$	0	0
$997$	− 2.49813e13i	− 0.800730i	−0.916356	−	0.400365i	$-0.868883\pi$
$997$	− 2.49813e13i	− 0.800730i	0.916356	−	0.400365i	$-0.131117\pi$
$998$	0	0
$999$	3.13482e13	0.995791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.10.b.l.129.4		4
4.3	odd	2		256.10.b.o.129.1		4
8.3	odd	2		256.10.b.o.129.4		4
8.5	even	2	inner	256.10.b.l.129.1		4
16.3	odd	4		64.10.a.m.1.2		2
16.5	even	4		32.10.a.e.1.2	yes	2
16.11	odd	4		32.10.a.b.1.1	✓	2
16.13	even	4		64.10.a.j.1.1		2
48.5	odd	4		288.10.a.e.1.1		2
48.11	even	4		288.10.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.10.a.b.1.1	✓	2	16.11	odd	4
32.10.a.e.1.2	yes	2	16.5	even	4
64.10.a.j.1.1		2	16.13	even	4
64.10.a.m.1.2		2	16.3	odd	4
256.10.b.l.129.1		4	8.5	even	2	inner
256.10.b.l.129.4		4	1.1	even	1	trivial
256.10.b.o.129.1		4	4.3	odd	2
256.10.b.o.129.4		4	8.3	odd	2
288.10.a.d.1.1		2	48.11	even	4
288.10.a.e.1.1		2	48.5	odd	4

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+252.730i q^{3} -2019.84i q^{5} -3039.30 q^{7} -44189.5 q^{9} +42489.0i q^{11} -74173.8i q^{13} +510474. q^{15} -607725. q^{17} -164849. i q^{19} -768123. i q^{21} -2.08911e6 q^{23} -2.12663e6 q^{25} -6.19353e6i q^{27} +1.87705e6i q^{29} -669635. q^{31} -1.07383e7 q^{33} +6.13890e6i q^{35} +5.06145e6i q^{37} +1.87459e7 q^{39} +1.46245e7 q^{41} +1.15906e7i q^{43} +8.92557e7i q^{45} +3.35490e7 q^{47} -3.11163e7 q^{49} -1.53590e8i q^{51} +2.03950e7i q^{53} +8.58211e7 q^{55} +4.16623e7 q^{57} -1.19399e8i q^{59} -9.81307e7i q^{61} +1.34305e8 q^{63} -1.49819e8 q^{65} +1.01247e8i q^{67} -5.27981e8i q^{69} +3.11299e8 q^{71} +6.82495e6 q^{73} -5.37464e8i q^{75} -1.29137e8i q^{77} +5.23080e8 q^{79} +6.95509e8 q^{81} -2.37668e8i q^{83} +1.22751e9i q^{85} -4.74388e8 q^{87} +6.21070e8 q^{89} +2.25436e8i q^{91} -1.69237e8i q^{93} -3.32969e8 q^{95} +1.11708e9 q^{97} -1.87757e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5568 q^{7} - 60788 q^{9} + 1115456 q^{15} - 944376 q^{17} - 7767360 q^{23} - 1105532 q^{25} + 9714432 q^{31} - 26938624 q^{33} + 41883584 q^{39} + 64718040 q^{41} - 41886080 q^{47} - 142809372 q^{49}+ \cdots + 3701486920 q^{97}+O(q^{100}) \) 4 * q - 5568 * q^7 - 60788 * q^9 + 1115456 * q^15 - 944376 * q^17 - 7767360 * q^23 - 1105532 * q^25 + 9714432 * q^31 - 26938624 * q^33 + 41883584 * q^39 + 64718040 * q^41 - 41886080 * q^47 - 142809372 * q^49 + 215481408 * q^55 + 228461312 * q^57 + 275654336 * q^63 - 334886448 * q^65 - 13610944 * q^71 + 46434840 * q^73 + 1453197696 * q^79 + 1539882020 * q^81 - 636858944 * q^87 + 142858200 * q^89 - 1830814144 * q^95 + 3701486920 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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