LMFDB - Embedded newform 400.8.c.u.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-4292] 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:	$124.954010194$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 69x^{4} + 1164x^{2} + 16 $ x^6 + 69x^4 + 1164x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-5.41512i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.u.49.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-21.9866i q^{3} +1068.54i q^{7} +1703.59 q^{9} +671.263 q^{11} +1410.86i q^{13} -24141.5i q^{17} -6799.10 q^{19} +23493.7 q^{21} -41092.9i q^{23} -85540.9i q^{27} -174244. q^{29} +176924. q^{31} -14758.8i q^{33} -69922.1i q^{37} +31020.0 q^{39} -36103.9 q^{41} -4631.67i q^{43} +944966. i q^{47} -318245. q^{49} -530790. q^{51} +188685. i q^{53} +149489. i q^{57} +1.18358e6 q^{59} +1.80080e6 q^{61} +1.82036e6i q^{63} +1.24107e6i q^{67} -903493. q^{69} +1.07553e6 q^{71} -3.54474e6i q^{73} +717274. i q^{77} +7.07853e6 q^{79} +1.84500e6 q^{81} +7.53745e6i q^{83} +3.83103e6i q^{87} -6.14628e6 q^{89} -1.50756e6 q^{91} -3.88995e6i q^{93} -1.43256e7i q^{97} +1.14356e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4292 q^{9} + 5318 q^{11} + 111498 q^{19} - 123124 q^{21} - 131056 q^{29} + 289196 q^{31} + 1600008 q^{39} + 599486 q^{41} - 4720974 q^{49} + 3724630 q^{51} + 1966736 q^{59} + 11420236 q^{61} - 10781396 q^{69}+ \cdots + 106633020 q^{99}+O(q^{100}) $ 6 * q - 4292 * q^9 + 5318 * q^11 + 111498 * q^19 - 123124 * q^21 - 131056 * q^29 + 289196 * q^31 + 1600008 * q^39 + 599486 * q^41 - 4720974 * q^49 + 3724630 * q^51 + 1966736 * q^59 + 11420236 * q^61 - 10781396 * q^69 - 3867912 * q^71 + 11181356 * q^79 + 43424534 * q^81 - 44667418 * q^89 - 2063888 * q^91 + 106633020 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 21.9866i	− 0.470147i	−0.971978	−	0.235074i	$-0.924467\pi$
$3$	− 21.9866i	− 0.470147i	0.971978	−	0.235074i	$-0.0755331\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1068.54i	1.17747i	0.808326	+	0.588735i	$0.200374\pi$
$7$	1068.54i	1.17747i	−0.808326	+	0.588735i	$0.799626\pi$
$8$	0	0
$9$	1703.59	0.778962
$10$	0	0
$11$	671.263	0.152061	0.0760306	−	0.997105i	$-0.475775\pi$
$11$	671.263	0.152061	0.0760306	+	0.997105i	$0.475775\pi$
$12$	0	0
$13$	1410.86i	0.178107i	0.996027	+	0.0890536i	$0.0283842\pi$
$13$	1410.86i	0.178107i	−0.996027	+	0.0890536i	$0.971616\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 24141.5i	− 1.19177i	−0.803069	−	0.595886i	$-0.796801\pi$
$17$	− 24141.5i	− 1.19177i	0.803069	−	0.595886i	$-0.203199\pi$
$18$	0	0
$19$	−6799.10	−0.227412	−0.113706	−	0.993514i	$-0.536272\pi$
$19$	−6799.10	−0.227412	−0.113706	+	0.993514i	$0.536272\pi$
$20$	0	0
$21$	23493.7	0.553584
$22$	0	0
$23$	− 41092.9i	− 0.704238i	−0.935955	−	0.352119i	$-0.885461\pi$
$23$	− 41092.9i	− 0.704238i	0.935955	−	0.352119i	$-0.114539\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 85540.9i	− 0.836374i
$28$	0	0
$29$	−174244.	−1.32668	−0.663338	−	0.748320i	$-0.730861\pi$
$29$	−174244.	−1.32668	−0.663338	+	0.748320i	$0.730861\pi$
$30$	0	0
$31$	176924.	1.06664	0.533322	−	0.845912i	$-0.320943\pi$
$31$	176924.	1.06664	0.533322	+	0.845912i	$0.320943\pi$
$32$	0	0
$33$	− 14758.8i	− 0.0714911i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 69922.1i	− 0.226939i	−0.993542	−	0.113469i	$-0.963804\pi$
$37$	− 69922.1i	− 0.226939i	0.993542	−	0.113469i	$-0.0361963\pi$
$38$	0	0
$39$	31020.0	0.0837366
$40$	0	0
$41$	−36103.9	−0.0818107	−0.0409054	−	0.999163i	$-0.513024\pi$
$41$	−36103.9	−0.0818107	−0.0409054	+	0.999163i	$0.513024\pi$
$42$	0	0
$43$	− 4631.67i	− 0.00888378i	−0.999990	−	0.00444189i	$-0.998586\pi$
$43$	− 4631.67i	− 0.00888378i	0.999990	−	0.00444189i	$-0.00141390\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	944966.i	1.32762i	0.747901	+	0.663810i	$0.231062\pi$
$47$	944966.i	1.32762i	−0.747901	+	0.663810i	$0.768938\pi$
$48$	0	0
$49$	−318245.	−0.386433
$50$	0	0
$51$	−530790.	−0.560308
$52$	0	0
$53$	188685.i	0.174090i	0.996204	+	0.0870448i	$0.0277423\pi$
$53$	188685.i	0.174090i	−0.996204	+	0.0870448i	$0.972258\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	149489.i	0.106917i
$58$	0	0
$59$	1.18358e6	0.750267	0.375133	−	0.926971i	$-0.377597\pi$
$59$	1.18358e6	0.750267	0.375133	+	0.926971i	$0.377597\pi$
$60$	0	0
$61$	1.80080e6	1.01580	0.507902	−	0.861415i	$-0.330421\pi$
$61$	1.80080e6	1.01580	0.507902	+	0.861415i	$0.330421\pi$
$62$	0	0
$63$	1.82036e6i	0.917203i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.24107e6i	0.504119i	0.967712	+	0.252060i	$0.0811079\pi$
$67$	1.24107e6i	0.504119i	−0.967712	+	0.252060i	$0.918892\pi$
$68$	0	0
$69$	−903493.	−0.331095
$70$	0	0
$71$	1.07553e6	0.356632	0.178316	−	0.983973i	$-0.442935\pi$
$71$	1.07553e6	0.356632	0.178316	+	0.983973i	$0.442935\pi$
$72$	0	0
$73$	− 3.54474e6i	− 1.06649i	−0.845962	−	0.533243i	$-0.820973\pi$
$73$	− 3.54474e6i	− 1.06649i	0.845962	−	0.533243i	$-0.179027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	717274.i	0.179047i
$78$	0	0
$79$	7.07853e6	1.61528	0.807641	−	0.589674i	$-0.200744\pi$
$79$	7.07853e6	1.61528	0.807641	+	0.589674i	$0.200744\pi$
$80$	0	0
$81$	1.84500e6	0.385743
$82$	0	0
$83$	7.53745e6i	1.44694i	0.690355	+	0.723471i	$0.257454\pi$
$83$	7.53745e6i	1.44694i	−0.690355	+	0.723471i	$0.742546\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.83103e6i	0.623733i
$88$	0	0
$89$	−6.14628e6	−0.924160	−0.462080	−	0.886838i	$-0.652897\pi$
$89$	−6.14628e6	−0.924160	−0.462080	+	0.886838i	$0.652897\pi$
$90$	0	0
$91$	−1.50756e6	−0.209716
$92$	0	0
$93$	− 3.88995e6i	− 0.501480i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.43256e7i	− 1.59371i	−0.604167	−	0.796857i	$-0.706494\pi$
$97$	− 1.43256e7i	− 1.59371i	0.604167	−	0.796857i	$-0.293506\pi$
$98$	0	0
$99$	1.14356e6	0.118450
$100$	0	0
$101$	1.87022e7	1.80620	0.903102	−	0.429426i	$-0.141284\pi$
$101$	1.87022e7	1.80620	0.903102	+	0.429426i	$0.141284\pi$
$102$	0	0
$103$	− 1.26676e7i	− 1.14226i	−0.820861	−	0.571128i	$-0.806506\pi$
$103$	− 1.26676e7i	− 1.14226i	0.820861	−	0.571128i	$-0.193494\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.14177e7i	0.901025i	0.892770	+	0.450513i	$0.148759\pi$
$107$	1.14177e7i	0.901025i	−0.892770	+	0.450513i	$0.851241\pi$
$108$	0	0
$109$	−2.30961e7	−1.70823	−0.854115	−	0.520085i	$-0.825900\pi$
$109$	−2.30961e7	−1.70823	−0.854115	+	0.520085i	$0.825900\pi$
$110$	0	0
$111$	−1.53735e6	−0.106695
$112$	0	0
$113$	− 2.70703e7i	− 1.76490i	−0.470410	−	0.882448i	$-0.655894\pi$
$113$	− 2.70703e7i	− 1.76490i	0.470410	−	0.882448i	$-0.344106\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.40352e6i	0.138739i
$118$	0	0
$119$	2.57963e7	1.40328
$120$	0	0
$121$	−1.90366e7	−0.976877
$122$	0	0
$123$	793802.i	0.0384631i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.57965e7i	0.684301i	0.939645	+	0.342151i	$0.111155\pi$
$127$	1.57965e7i	0.684301i	−0.939645	+	0.342151i	$0.888845\pi$
$128$	0	0
$129$	−101835.	−0.00417668
$130$	0	0
$131$	5.51632e6	0.214388	0.107194	−	0.994238i	$-0.465813\pi$
$131$	5.51632e6	0.214388	0.107194	+	0.994238i	$0.465813\pi$
$132$	0	0
$133$	− 7.26514e6i	− 0.267771i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.55243e6i	− 0.0848070i	−0.999101	−	0.0424035i	$-0.986498\pi$
$137$	− 2.55243e6i	− 0.0848070i	0.999101	−	0.0424035i	$-0.0135015\pi$
$138$	0	0
$139$	6.02032e7	1.90138	0.950688	−	0.310150i	$-0.100379\pi$
$139$	6.02032e7	1.90138	0.950688	+	0.310150i	$0.100379\pi$
$140$	0	0
$141$	2.07766e7	0.624177
$142$	0	0
$143$	947056.i	0.0270832i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.99712e6i	0.181681i
$148$	0	0
$149$	4.27550e7	1.05885	0.529425	−	0.848356i	$-0.322408\pi$
$149$	4.27550e7	1.05885	0.529425	+	0.848356i	$0.322408\pi$
$150$	0	0
$151$	−1.50072e6	−0.0354717	−0.0177358	−	0.999843i	$-0.505646\pi$
$151$	−1.50072e6	−0.0354717	−0.0177358	+	0.999843i	$0.505646\pi$
$152$	0	0
$153$	− 4.11272e7i	− 0.928345i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.41082e7i	0.290954i	0.989362	+	0.145477i	$0.0464716\pi$
$157$	1.41082e7i	0.290954i	−0.989362	+	0.145477i	$0.953528\pi$
$158$	0	0
$159$	4.14855e6	0.0818478
$160$	0	0
$161$	4.39096e7	0.829218
$162$	0	0
$163$	− 7.39232e7i	− 1.33698i	−0.743722	−	0.668489i	$-0.766942\pi$
$163$	− 7.39232e7i	− 1.33698i	0.743722	−	0.668489i	$-0.233058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.65021e6i	0.0772617i	0.999254	+	0.0386309i	$0.0122996\pi$
$167$	4.65021e6i	0.0772617i	−0.999254	+	0.0386309i	$0.987700\pi$
$168$	0	0
$169$	6.07580e7	0.968278
$170$	0	0
$171$	−1.15829e7	−0.177145
$172$	0	0
$173$	− 4.25336e7i	− 0.624556i	−0.949991	−	0.312278i	$-0.898908\pi$
$173$	− 4.25336e7i	− 0.624556i	0.949991	−	0.312278i	$-0.101092\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.60229e7i	− 0.352736i
$178$	0	0
$179$	−2.68052e7	−0.349329	−0.174664	−	0.984628i	$-0.555884\pi$
$179$	−2.68052e7	−0.349329	−0.174664	+	0.984628i	$0.555884\pi$
$180$	0	0
$181$	−7.80758e6	−0.0978682	−0.0489341	−	0.998802i	$-0.515582\pi$
$181$	−7.80758e6	−0.0978682	−0.0489341	+	0.998802i	$0.515582\pi$
$182$	0	0
$183$	− 3.95934e7i	− 0.477578i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.62053e7i	− 0.181222i
$188$	0	0
$189$	9.14042e7	0.984804
$190$	0	0
$191$	1.26812e8	1.31687	0.658437	−	0.752636i	$-0.271218\pi$
$191$	1.26812e8	1.31687	0.658437	+	0.752636i	$0.271218\pi$
$192$	0	0
$193$	− 1.03526e8i	− 1.03657i	−0.855207	−	0.518287i	$-0.826570\pi$
$193$	− 1.03526e8i	− 1.03657i	0.855207	−	0.518287i	$-0.173430\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.83506e8i	1.71009i	0.518552	+	0.855046i	$0.326471\pi$
$197$	1.83506e8i	1.71009i	−0.518552	+	0.855046i	$0.673529\pi$
$198$	0	0
$199$	−5.17541e7	−0.465542	−0.232771	−	0.972532i	$-0.574779\pi$
$199$	−5.17541e7	−0.465542	−0.232771	+	0.972532i	$0.574779\pi$
$200$	0	0
$201$	2.72868e7	0.237010
$202$	0	0
$203$	− 1.86187e8i	− 1.56212i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.00054e7i	− 0.548574i
$208$	0	0
$209$	−4.56399e6	−0.0345806
$210$	0	0
$211$	1.77935e8	1.30398	0.651991	−	0.758227i	$-0.273934\pi$
$211$	1.77935e8	1.30398	0.651991	+	0.758227i	$0.273934\pi$
$212$	0	0
$213$	− 2.36474e7i	− 0.167669i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.89051e8i	1.25594i
$218$	0	0
$219$	−7.79369e7	−0.501405
$220$	0	0
$221$	3.40602e7	0.212263
$222$	0	0
$223$	− 2.49149e8i	− 1.50450i	−0.658876	−	0.752251i	$-0.728968\pi$
$223$	− 2.49149e8i	− 1.50450i	0.658876	−	0.752251i	$-0.271032\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.45411e7i	0.479708i	0.970809	+	0.239854i	$0.0770996\pi$
$227$	8.45411e7i	0.479708i	−0.970809	+	0.239854i	$0.922900\pi$
$228$	0	0
$229$	3.00270e8	1.65230	0.826148	−	0.563453i	$-0.190528\pi$
$229$	3.00270e8	1.65230	0.826148	+	0.563453i	$0.190528\pi$
$230$	0	0
$231$	1.57704e7	0.0841786
$232$	0	0
$233$	1.30550e8i	0.676133i	0.941122	+	0.338067i	$0.109773\pi$
$233$	1.30550e8i	0.676133i	−0.941122	+	0.338067i	$0.890227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.55633e8i	− 0.759421i
$238$	0	0
$239$	−3.15757e7	−0.149610	−0.0748050	−	0.997198i	$-0.523833\pi$
$239$	−3.15757e7	−0.149610	−0.0748050	+	0.997198i	$0.523833\pi$
$240$	0	0
$241$	2.48149e8	1.14197	0.570984	−	0.820962i	$-0.306562\pi$
$241$	2.48149e8	1.14197	0.570984	+	0.820962i	$0.306562\pi$
$242$	0	0
$243$	− 2.27643e8i	− 1.01773i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9.59256e6i	− 0.0405038i
$248$	0	0
$249$	1.65723e8	0.680275
$250$	0	0
$251$	3.52963e8	1.40887	0.704436	−	0.709767i	$-0.251200\pi$
$251$	3.52963e8	1.40887	0.704436	+	0.709767i	$0.251200\pi$
$252$	0	0
$253$	− 2.75841e7i	− 0.107087i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.27654e8i	0.836582i	0.908313	+	0.418291i	$0.137371\pi$
$257$	2.27654e8i	0.836582i	−0.908313	+	0.418291i	$0.862629\pi$
$258$	0	0
$259$	7.47148e7	0.267213
$260$	0	0
$261$	−2.96840e8	−1.03343
$262$	0	0
$263$	6.29841e7i	0.213494i	0.994286	+	0.106747i	$0.0340435\pi$
$263$	6.29841e7i	0.213494i	−0.994286	+	0.106747i	$0.965956\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.35136e8i	0.434491i
$268$	0	0
$269$	4.21072e8	1.31894	0.659468	−	0.751733i	$-0.270782\pi$
$269$	4.21072e8	1.31894	0.659468	+	0.751733i	$0.270782\pi$
$270$	0	0
$271$	6.75064e7	0.206040	0.103020	−	0.994679i	$-0.467149\pi$
$271$	6.75064e7	0.206040	0.103020	+	0.994679i	$0.467149\pi$
$272$	0	0
$273$	3.31462e7i	0.0985972i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.43096e8i	0.969920i	0.874536	+	0.484960i	$0.161166\pi$
$277$	3.43096e8i	0.969920i	−0.874536	+	0.484960i	$0.838834\pi$
$278$	0	0
$279$	3.01405e8	0.830875
$280$	0	0
$281$	2.96200e8	0.796365	0.398183	−	0.917306i	$-0.369641\pi$
$281$	2.96200e8	0.796365	0.398183	+	0.917306i	$0.369641\pi$
$282$	0	0
$283$	2.09353e8i	0.549068i	0.961577	+	0.274534i	$0.0885236\pi$
$283$	2.09353e8i	0.549068i	−0.961577	+	0.274534i	$0.911476\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.85786e7i	− 0.0963296i
$288$	0	0
$289$	−1.72474e8	−0.420321
$290$	0	0
$291$	−3.14971e8	−0.749281
$292$	0	0
$293$	− 2.23696e8i	− 0.519542i	−0.965670	−	0.259771i	$-0.916353\pi$
$293$	− 2.23696e8i	− 0.519542i	0.965670	−	0.259771i	$-0.0836471\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.74204e7i	− 0.127180i
$298$	0	0
$299$	5.79762e7	0.125430
$300$	0	0
$301$	4.94914e6	0.0104604
$302$	0	0
$303$	− 4.11197e8i	− 0.849182i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.79876e8i	− 0.552054i	−0.961150	−	0.276027i	$-0.910982\pi$
$307$	− 2.79876e8i	− 0.552054i	0.961150	−	0.276027i	$-0.0890180\pi$
$308$	0	0
$309$	−2.78517e8	−0.537029
$310$	0	0
$311$	3.50620e8	0.660959	0.330480	−	0.943813i	$-0.392789\pi$
$311$	3.50620e8	0.660959	0.330480	+	0.943813i	$0.392789\pi$
$312$	0	0
$313$	− 8.36678e8i	− 1.54224i	−0.636688	−	0.771122i	$-0.719696\pi$
$313$	− 8.36678e8i	− 1.54224i	0.636688	−	0.771122i	$-0.280304\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.09028e8i	− 0.368550i	−0.982875	−	0.184275i	$-0.941006\pi$
$317$	− 2.09028e8i	− 0.368550i	0.982875	−	0.184275i	$-0.0589938\pi$
$318$	0	0
$319$	−1.16964e8	−0.201736
$320$	0	0
$321$	2.51037e8	0.423615
$322$	0	0
$323$	1.64141e8i	0.271024i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.07805e8i	0.803119i
$328$	0	0
$329$	−1.00974e9	−1.56323
$330$	0	0
$331$	−1.16334e9	−1.76322	−0.881611	−	0.471976i	$-0.843541\pi$
$331$	−1.16334e9	−1.76322	−0.881611	+	0.471976i	$0.843541\pi$
$332$	0	0
$333$	− 1.19118e8i	− 0.176776i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.10570e9i	− 1.57374i	−0.617119	−	0.786870i	$-0.711700\pi$
$337$	− 1.10570e9i	− 1.57374i	0.617119	−	0.786870i	$-0.288300\pi$
$338$	0	0
$339$	−5.95185e8	−0.829761
$340$	0	0
$341$	1.18762e8	0.162195
$342$	0	0
$343$	5.39934e8i	0.722456i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.36343e8i	0.946077i	0.881042	+	0.473039i	$0.156843\pi$
$347$	7.36343e8i	0.946077i	−0.881042	+	0.473039i	$0.843157\pi$
$348$	0	0
$349$	−2.15112e8	−0.270880	−0.135440	−	0.990786i	$-0.543245\pi$
$349$	−2.15112e8	−0.270880	−0.135440	+	0.990786i	$0.543245\pi$
$350$	0	0
$351$	1.20686e8	0.148964
$352$	0	0
$353$	5.45673e7i	0.0660269i	0.999455	+	0.0330134i	$0.0105104\pi$
$353$	5.45673e7i	0.0660269i	−0.999455	+	0.0330134i	$0.989490\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 5.67173e8i	− 0.659746i
$358$	0	0
$359$	1.36708e9	1.55942	0.779710	−	0.626141i	$-0.215367\pi$
$359$	1.36708e9	1.55942	0.779710	+	0.626141i	$0.215367\pi$
$360$	0	0
$361$	−8.47644e8	−0.948284
$362$	0	0
$363$	4.18550e8i	0.459276i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.97846e8i	− 1.05374i	−0.849947	−	0.526869i	$-0.823366\pi$
$367$	− 9.97846e8i	− 1.05374i	0.849947	−	0.526869i	$-0.176634\pi$
$368$	0	0
$369$	−6.15061e7	−0.0637274
$370$	0	0
$371$	−2.01619e8	−0.204985
$372$	0	0
$373$	7.42437e8i	0.740762i	0.928880	+	0.370381i	$0.120773\pi$
$373$	7.42437e8i	0.740762i	−0.928880	+	0.370381i	$0.879227\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.45833e8i	− 0.236290i
$378$	0	0
$379$	1.95656e9	1.84611	0.923053	−	0.384672i	$-0.125685\pi$
$379$	1.95656e9	1.84611	0.923053	+	0.384672i	$0.125685\pi$
$380$	0	0
$381$	3.47311e8	0.321722
$382$	0	0
$383$	1.19607e9i	1.08783i	0.839140	+	0.543916i	$0.183059\pi$
$383$	1.19607e9i	1.08783i	−0.839140	+	0.543916i	$0.816941\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.89046e6i	− 0.00692012i
$388$	0	0
$389$	−1.54299e9	−1.32905	−0.664523	−	0.747267i	$-0.731365\pi$
$389$	−1.54299e9	−1.32905	−0.664523	+	0.747267i	$0.731365\pi$
$390$	0	0
$391$	−9.92045e8	−0.839291
$392$	0	0
$393$	− 1.21285e8i	− 0.100794i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.18680e8i	0.335827i	0.985802	+	0.167913i	$0.0537029\pi$
$397$	4.18680e8i	0.335827i	−0.985802	+	0.167913i	$0.946297\pi$
$398$	0	0
$399$	−1.59736e8	−0.125892
$400$	0	0
$401$	−1.09037e9	−0.844442	−0.422221	−	0.906493i	$-0.638749\pi$
$401$	−1.09037e9	−0.844442	−0.422221	+	0.906493i	$0.638749\pi$
$402$	0	0
$403$	2.49614e8i	0.189977i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.69361e7i	− 0.0345085i
$408$	0	0
$409$	−9.97868e8	−0.721177	−0.360588	−	0.932725i	$-0.617424\pi$
$409$	−9.97868e8	−0.721177	−0.360588	+	0.932725i	$0.617424\pi$
$410$	0	0
$411$	−5.61193e7	−0.0398718
$412$	0	0
$413$	1.26471e9i	0.883416i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.32366e9i	− 0.893926i
$418$	0	0
$419$	−2.10530e9	−1.39819	−0.699095	−	0.715029i	$-0.746413\pi$
$419$	−2.10530e9	−1.39819	−0.699095	+	0.715029i	$0.746413\pi$
$420$	0	0
$421$	1.81656e8	0.118648	0.0593242	−	0.998239i	$-0.481105\pi$
$421$	1.81656e8	0.118648	0.0593242	+	0.998239i	$0.481105\pi$
$422$	0	0
$423$	1.60983e9i	1.03417i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.92423e9i	1.19608i
$428$	0	0
$429$	2.08226e7	0.0127331
$430$	0	0
$431$	−1.02927e9	−0.619241	−0.309620	−	0.950860i	$-0.600202\pi$
$431$	−1.02927e9	−0.619241	−0.309620	+	0.950860i	$0.600202\pi$
$432$	0	0
$433$	1.92403e9i	1.13895i	0.822009	+	0.569474i	$0.192853\pi$
$433$	1.92403e9i	1.13895i	−0.822009	+	0.569474i	$0.807147\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.79395e8i	0.160152i
$438$	0	0
$439$	−1.53719e8	−0.0867166	−0.0433583	−	0.999060i	$-0.513806\pi$
$439$	−1.53719e8	−0.0867166	−0.0433583	+	0.999060i	$0.513806\pi$
$440$	0	0
$441$	−5.42158e8	−0.301017
$442$	0	0
$443$	1.48910e9i	0.813786i	0.913476	+	0.406893i	$0.133388\pi$
$443$	1.48910e9i	0.813786i	−0.913476	+	0.406893i	$0.866612\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9.40037e8i	− 0.497816i
$448$	0	0
$449$	1.75334e9	0.914120	0.457060	−	0.889436i	$-0.348902\pi$
$449$	1.75334e9	0.914120	0.457060	+	0.889436i	$0.348902\pi$
$450$	0	0
$451$	−2.42352e7	−0.0124402
$452$	0	0
$453$	3.29958e7i	0.0166769i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.94898e8i	0.291565i	0.989317	+	0.145783i	$0.0465700\pi$
$457$	5.94898e8i	0.291565i	−0.989317	+	0.145783i	$0.953430\pi$
$458$	0	0
$459$	−2.06509e9	−0.996767
$460$	0	0
$461$	−6.49366e8	−0.308700	−0.154350	−	0.988016i	$-0.549328\pi$
$461$	−6.49366e8	−0.308700	−0.154350	+	0.988016i	$0.549328\pi$
$462$	0	0
$463$	− 1.41796e8i	− 0.0663942i	−0.999449	−	0.0331971i	$-0.989431\pi$
$463$	− 1.41796e8i	− 0.0663942i	0.999449	−	0.0331971i	$-0.0105689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.60598e8i	− 0.300143i	−0.988675	−	0.150072i	$-0.952050\pi$
$467$	− 6.60598e8i	− 0.300143i	0.988675	−	0.150072i	$-0.0479504\pi$
$468$	0	0
$469$	−1.32613e9	−0.593585
$470$	0	0
$471$	3.10192e8	0.136791
$472$	0	0
$473$	− 3.10907e6i	− 0.00135088i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.21442e8i	0.135609i
$478$	0	0
$479$	−2.90926e9	−1.20951	−0.604754	−	0.796413i	$-0.706728\pi$
$479$	−2.90926e9	−1.20951	−0.604754	+	0.796413i	$0.706728\pi$
$480$	0	0
$481$	9.86501e7	0.0404194
$482$	0	0
$483$	− 9.65423e8i	− 0.389855i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.09182e9i	− 1.60533i	−0.596428	−	0.802667i	$-0.703414\pi$
$487$	− 4.09182e9i	− 1.60533i	0.596428	−	0.802667i	$-0.296586\pi$
$488$	0	0
$489$	−1.62532e9	−0.628576
$490$	0	0
$491$	−1.97082e9	−0.751386	−0.375693	−	0.926744i	$-0.622595\pi$
$491$	−1.97082e9	−0.751386	−0.375693	+	0.926744i	$0.622595\pi$
$492$	0	0
$493$	4.20651e9i	1.58110i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.14926e9i	0.419923i
$498$	0	0
$499$	−3.95387e9	−1.42453	−0.712263	−	0.701913i	$-0.752330\pi$
$499$	−3.95387e9	−1.42453	−0.712263	+	0.701913i	$0.752330\pi$
$500$	0	0
$501$	1.02242e8	0.0363244
$502$	0	0
$503$	3.13374e9i	1.09793i	0.835845	+	0.548965i	$0.184978\pi$
$503$	3.13374e9i	1.09793i	−0.835845	+	0.548965i	$0.815022\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.33586e9i	− 0.455233i
$508$	0	0
$509$	1.26250e9	0.424345	0.212172	−	0.977232i	$-0.431946\pi$
$509$	1.26250e9	0.424345	0.212172	+	0.977232i	$0.431946\pi$
$510$	0	0
$511$	3.78772e9	1.25575
$512$	0	0
$513$	5.81601e8i	0.190202i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.34321e8i	0.201879i
$518$	0	0
$519$	−9.35171e8	−0.293633
$520$	0	0
$521$	−3.03852e9	−0.941305	−0.470653	−	0.882319i	$-0.655981\pi$
$521$	−3.03852e9	−0.941305	−0.470653	+	0.882319i	$0.655981\pi$
$522$	0	0
$523$	− 2.00306e9i	− 0.612264i	−0.951989	−	0.306132i	$-0.900965\pi$
$523$	− 2.00306e9i	− 0.612264i	0.951989	−	0.306132i	$-0.0990349\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.27120e9i	− 1.27120i
$528$	0	0
$529$	1.71620e9	0.504049
$530$	0	0
$531$	2.01633e9	0.584429
$532$	0	0
$533$	− 5.09374e7i	− 0.0145711i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.89357e8i	0.164236i
$538$	0	0
$539$	−2.13626e8	−0.0587615
$540$	0	0
$541$	4.41041e8	0.119754	0.0598768	−	0.998206i	$-0.480929\pi$
$541$	4.41041e8	0.119754	0.0598768	+	0.998206i	$0.480929\pi$
$542$	0	0
$543$	1.71662e8i	0.0460125i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.66599e9i	0.957715i	0.877893	+	0.478857i	$0.158949\pi$
$547$	3.66599e9i	0.957715i	−0.877893	+	0.478857i	$0.841051\pi$
$548$	0	0
$549$	3.06782e9	0.791273
$550$	0	0
$551$	1.18470e9	0.301702
$552$	0	0
$553$	7.56373e9i	1.90195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.67236e9i	− 1.14563i	−0.819686	−	0.572813i	$-0.805852\pi$
$557$	− 4.67236e9i	− 1.14563i	0.819686	−	0.572813i	$-0.194148\pi$
$558$	0	0
$559$	6.53462e6	0.00158227
$560$	0	0
$561$	−3.56300e8	−0.0852012
$562$	0	0
$563$	− 5.21175e9i	− 1.23085i	−0.788196	−	0.615424i	$-0.788985\pi$
$563$	− 5.21175e9i	− 1.23085i	0.788196	−	0.615424i	$-0.211015\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.97146e9i	0.454200i
$568$	0	0
$569$	6.91998e9	1.57475	0.787375	−	0.616474i	$-0.211439\pi$
$569$	6.91998e9	1.57475	0.787375	+	0.616474i	$0.211439\pi$
$570$	0	0
$571$	−8.71208e9	−1.95837	−0.979186	−	0.202963i	$-0.934943\pi$
$571$	−8.71208e9	−1.95837	−0.979186	+	0.202963i	$0.934943\pi$
$572$	0	0
$573$	− 2.78817e9i	− 0.619125i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.53729e9i	1.41671i	0.705854	+	0.708357i	$0.250563\pi$
$577$	6.53729e9i	1.41671i	−0.705854	+	0.708357i	$0.749437\pi$
$578$	0	0
$579$	−2.27619e9	−0.487342
$580$	0	0
$581$	−8.05410e9	−1.70373
$582$	0	0
$583$	1.26658e8i	0.0264723i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.75278e9i	0.969871i	0.874550	+	0.484936i	$0.161157\pi$
$587$	4.75278e9i	0.969871i	−0.874550	+	0.484936i	$0.838843\pi$
$588$	0	0
$589$	−1.20292e9	−0.242568
$590$	0	0
$591$	4.03469e9	0.803995
$592$	0	0
$593$	1.81306e9i	0.357043i	0.983936	+	0.178521i	$0.0571314\pi$
$593$	1.81306e9i	0.357043i	−0.983936	+	0.178521i	$0.942869\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.13790e9i	0.218874i
$598$	0	0
$599$	8.63786e9	1.64215	0.821074	−	0.570822i	$-0.193376\pi$
$599$	8.63786e9	1.64215	0.821074	+	0.570822i	$0.193376\pi$
$600$	0	0
$601$	7.95803e9	1.49536	0.747678	−	0.664061i	$-0.231168\pi$
$601$	7.95803e9	1.49536	0.747678	+	0.664061i	$0.231168\pi$
$602$	0	0
$603$	2.11427e9i	0.392689i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.64426e9i	− 1.38731i	−0.720305	−	0.693657i	$-0.755998\pi$
$607$	− 7.64426e9i	− 1.38731i	0.720305	−	0.693657i	$-0.244002\pi$
$608$	0	0
$609$	−4.09363e9	−0.734426
$610$	0	0
$611$	−1.33321e9	−0.236459
$612$	0	0
$613$	− 5.36969e9i	− 0.941537i	−0.882257	−	0.470768i	$-0.843977\pi$
$613$	− 5.36969e9i	− 0.941537i	0.882257	−	0.470768i	$-0.156023\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.22651e9i	0.210220i	0.994461	+	0.105110i	$0.0335194\pi$
$617$	1.22651e9i	0.210220i	−0.994461	+	0.105110i	$0.966481\pi$
$618$	0	0
$619$	1.85402e8	0.0314194	0.0157097	−	0.999877i	$-0.494999\pi$
$619$	1.85402e8	0.0314194	0.0157097	+	0.999877i	$0.494999\pi$
$620$	0	0
$621$	−3.51512e9	−0.589006
$622$	0	0
$623$	− 6.56757e9i	− 1.08817i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.00347e8i	0.0162580i
$628$	0	0
$629$	−1.68802e9	−0.270459
$630$	0	0
$631$	1.00745e9	0.159633	0.0798165	−	0.996810i	$-0.474567\pi$
$631$	1.00745e9	0.159633	0.0798165	+	0.996810i	$0.474567\pi$
$632$	0	0
$633$	− 3.91218e9i	− 0.613063i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.48998e8i	− 0.0688266i
$638$	0	0
$639$	1.83227e9	0.277802
$640$	0	0
$641$	−1.57139e9	−0.235657	−0.117829	−	0.993034i	$-0.537593\pi$
$641$	−1.57139e9	−0.235657	−0.117829	+	0.993034i	$0.537593\pi$
$642$	0	0
$643$	1.49835e9i	0.222267i	0.993805	+	0.111133i	$0.0354481\pi$
$643$	1.49835e9i	0.222267i	−0.993805	+	0.111133i	$0.964552\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.34865e9i	0.195764i	0.995198	+	0.0978820i	$0.0312068\pi$
$647$	1.34865e9i	0.195764i	−0.995198	+	0.0978820i	$0.968793\pi$
$648$	0	0
$649$	7.94494e8	0.114086
$650$	0	0
$651$	4.15658e9	0.590477
$652$	0	0
$653$	− 1.26763e10i	− 1.78154i	−0.454457	−	0.890769i	$-0.650167\pi$
$653$	− 1.26763e10i	− 1.78154i	0.454457	−	0.890769i	$-0.349833\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.03879e9i	− 0.830751i
$658$	0	0
$659$	−6.26664e9	−0.852974	−0.426487	−	0.904494i	$-0.640249\pi$
$659$	−6.26664e9	−0.852974	−0.426487	+	0.904494i	$0.640249\pi$
$660$	0	0
$661$	1.21865e10	1.64124	0.820621	−	0.571473i	$-0.193628\pi$
$661$	1.21865e10	1.64124	0.820621	+	0.571473i	$0.193628\pi$
$662$	0	0
$663$	− 7.48869e8i	− 0.0997950i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.16019e9i	0.934295i
$668$	0	0
$669$	−5.47795e9	−0.707338
$670$	0	0
$671$	1.20881e9	0.154464
$672$	0	0
$673$	− 1.05615e10i	− 1.33559i	−0.744343	−	0.667797i	$-0.767237\pi$
$673$	− 1.05615e10i	− 1.33559i	0.744343	−	0.667797i	$-0.232763\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.77045e9i	− 0.962466i	−0.876593	−	0.481233i	$-0.840189\pi$
$677$	− 7.77045e9i	− 0.962466i	0.876593	−	0.481233i	$-0.159811\pi$
$678$	0	0
$679$	1.53075e10	1.87655
$680$	0	0
$681$	1.85877e9	0.225534
$682$	0	0
$683$	− 8.52102e8i	− 0.102334i	−0.998690	−	0.0511669i	$-0.983706\pi$
$683$	− 8.52102e8i	− 0.102334i	0.998690	−	0.0511669i	$-0.0162940\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.60192e9i	− 0.776822i
$688$	0	0
$689$	−2.66208e8	−0.0310066
$690$	0	0
$691$	−9.21838e9	−1.06287	−0.531437	−	0.847098i	$-0.678348\pi$
$691$	−9.21838e9	−1.06287	−0.531437	+	0.847098i	$0.678348\pi$
$692$	0	0
$693$	1.22194e9i	0.139471i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.71602e8i	0.0974997i
$698$	0	0
$699$	2.87036e9	0.317882
$700$	0	0
$701$	1.02862e10	1.12783	0.563915	−	0.825833i	$-0.309295\pi$
$701$	1.02862e10	1.12783	0.563915	+	0.825833i	$0.309295\pi$
$702$	0	0
$703$	4.75407e8i	0.0516086i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.99841e10i	2.12675i
$708$	0	0
$709$	7.03343e9	0.741149	0.370575	−	0.928803i	$-0.379161\pi$
$709$	7.03343e9	0.741149	0.370575	+	0.928803i	$0.379161\pi$
$710$	0	0
$711$	1.20589e10	1.25824
$712$	0	0
$713$	− 7.27030e9i	− 0.751171i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.94243e8i	0.0703387i
$718$	0	0
$719$	1.35138e10	1.35590	0.677948	−	0.735110i	$-0.262870\pi$
$719$	1.35138e10	1.35590	0.677948	+	0.735110i	$0.262870\pi$
$720$	0	0
$721$	1.35359e10	1.34497
$722$	0	0
$723$	− 5.45596e9i	− 0.536893i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 4.93058e9i	− 0.475913i	−0.971276	−	0.237956i	$-0.923522\pi$
$727$	− 4.93058e9i	− 0.475913i	0.971276	−	0.237956i	$-0.0764775\pi$
$728$	0	0
$729$	−9.70094e8	−0.0927401
$730$	0	0
$731$	−1.11815e8	−0.0105874
$732$	0	0
$733$	− 1.26842e8i	− 0.0118959i	−0.999982	−	0.00594796i	$-0.998107\pi$
$733$	− 1.26842e8i	− 0.0118959i	0.999982	−	0.00594796i	$-0.00189330\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.33082e8i	0.0766569i
$738$	0	0
$739$	8.65558e9	0.788934	0.394467	−	0.918910i	$-0.370929\pi$
$739$	8.65558e9	0.788934	0.394467	+	0.918910i	$0.370929\pi$
$740$	0	0
$741$	−2.10908e8	−0.0190427
$742$	0	0
$743$	1.09881e10i	0.982790i	0.870937	+	0.491395i	$0.163513\pi$
$743$	1.09881e10i	0.982790i	−0.870937	+	0.491395i	$0.836487\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.28407e10i	1.12711i
$748$	0	0
$749$	−1.22004e10	−1.06093
$750$	0	0
$751$	−4.76292e9	−0.410330	−0.205165	−	0.978727i	$-0.565773\pi$
$751$	−4.76292e9	−0.410330	−0.205165	+	0.978727i	$0.565773\pi$
$752$	0	0
$753$	− 7.76047e9i	− 0.662377i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.97900e9i	0.752302i	0.926558	+	0.376151i	$0.122753\pi$
$757$	8.97900e9i	0.752302i	−0.926558	+	0.376151i	$0.877247\pi$
$758$	0	0
$759$	−6.06482e8	−0.0503467
$760$	0	0
$761$	−9.14472e9	−0.752184	−0.376092	−	0.926582i	$-0.622732\pi$
$761$	−9.14472e9	−0.752184	−0.376092	+	0.926582i	$0.622732\pi$
$762$	0	0
$763$	− 2.46792e10i	− 2.01139i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.66986e9i	0.133628i
$768$	0	0
$769$	−5.80655e9	−0.460443	−0.230222	−	0.973138i	$-0.573945\pi$
$769$	−5.80655e9	−0.460443	−0.230222	+	0.973138i	$0.573945\pi$
$770$	0	0
$771$	5.00533e9	0.393317
$772$	0	0
$773$	− 7.33269e9i	− 0.570999i	−0.958379	−	0.285499i	$-0.907841\pi$
$773$	− 7.33269e9i	− 0.570999i	0.958379	−	0.285499i	$-0.0921594\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.64273e9i	− 0.125629i
$778$	0	0
$779$	2.45474e8	0.0186048
$780$	0	0
$781$	7.21967e8	0.0542299
$782$	0	0
$783$	1.49050e10i	1.10960i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.71352e9i	0.637209i	0.947888	+	0.318604i	$0.103214\pi$
$787$	8.71352e9i	0.637209i	−0.947888	+	0.318604i	$0.896786\pi$
$788$	0	0
$789$	1.38481e9	0.100374
$790$	0	0
$791$	2.89259e10	2.07811
$792$	0	0
$793$	2.54067e9i	0.180922i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.78489e10i	1.24884i	0.781088	+	0.624421i	$0.214665\pi$
$797$	1.78489e10i	1.24884i	−0.781088	+	0.624421i	$0.785335\pi$
$798$	0	0
$799$	2.28129e10	1.58222
$800$	0	0
$801$	−1.04707e10	−0.719885
$802$	0	0
$803$	− 2.37946e9i	− 0.162171i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.25795e9i	− 0.620094i
$808$	0	0
$809$	2.08455e10	1.38418	0.692089	−	0.721812i	$-0.256690\pi$
$809$	2.08455e10	1.38418	0.692089	+	0.721812i	$0.256690\pi$
$810$	0	0
$811$	−2.05986e10	−1.35602	−0.678009	−	0.735054i	$-0.737157\pi$
$811$	−2.05986e10	−1.35602	−0.678009	+	0.735054i	$0.737157\pi$
$812$	0	0
$813$	− 1.48424e9i	− 0.0968693i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.14912e7i	0.00202028i
$818$	0	0
$819$	−2.56827e9	−0.163360
$820$	0	0
$821$	1.81452e10	1.14435	0.572177	−	0.820130i	$-0.306099\pi$
$821$	1.81452e10	1.14435	0.572177	+	0.820130i	$0.306099\pi$
$822$	0	0
$823$	− 9.49973e9i	− 0.594035i	−0.954872	−	0.297017i	$-0.904008\pi$
$823$	− 9.49973e9i	− 0.594035i	0.954872	−	0.297017i	$-0.0959919\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.21862e10i	1.36400i	0.731353	+	0.681999i	$0.238889\pi$
$827$	2.21862e10i	1.36400i	−0.731353	+	0.681999i	$0.761111\pi$
$828$	0	0
$829$	6.80171e9	0.414646	0.207323	−	0.978273i	$-0.433525\pi$
$829$	6.80171e9	0.414646	0.207323	+	0.978273i	$0.433525\pi$
$830$	0	0
$831$	7.54351e9	0.456005
$832$	0	0
$833$	7.68290e9i	0.460541i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.51342e10i	− 0.892114i
$838$	0	0
$839$	−1.73352e10	−1.01335	−0.506677	−	0.862136i	$-0.669126\pi$
$839$	−1.73352e10	−1.01335	−0.506677	+	0.862136i	$0.669126\pi$
$840$	0	0
$841$	1.31111e10	0.760068
$842$	0	0
$843$	− 6.51242e9i	− 0.374409i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.03414e10i	− 1.15024i
$848$	0	0
$849$	4.60296e9	0.258143
$850$	0	0
$851$	−2.87330e9	−0.159819
$852$	0	0
$853$	3.10458e10i	1.71270i	0.516398	+	0.856349i	$0.327273\pi$
$853$	3.10458e10i	1.71270i	−0.516398	+	0.856349i	$0.672727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.49340e9i	− 0.243861i	−0.992539	−	0.121931i	$-0.961091\pi$
$857$	− 4.49340e9i	− 0.243861i	0.992539	−	0.121931i	$-0.0389085\pi$
$858$	0	0
$859$	−2.55028e10	−1.37281	−0.686407	−	0.727218i	$-0.740813\pi$
$859$	−2.55028e10	−1.37281	−0.686407	+	0.727218i	$0.740813\pi$
$860$	0	0
$861$	−8.48212e8	−0.0452891
$862$	0	0
$863$	− 2.40822e8i	− 0.0127544i	−0.999980	−	0.00637718i	$-0.997970\pi$
$863$	− 2.40822e8i	− 0.0127544i	0.999980	−	0.00637718i	$-0.00202993\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.79212e9i	0.197613i
$868$	0	0
$869$	4.75156e9	0.245622
$870$	0	0
$871$	−1.75097e9	−0.0897872
$872$	0	0
$873$	− 2.44049e10i	− 1.24144i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.36881e10i	0.685244i	0.939473	+	0.342622i	$0.111315\pi$
$877$	1.36881e10i	0.685244i	−0.939473	+	0.342622i	$0.888685\pi$
$878$	0	0
$879$	−4.91831e9	−0.244261
$880$	0	0
$881$	−2.35657e10	−1.16109	−0.580543	−	0.814230i	$-0.697160\pi$
$881$	−2.35657e10	−1.16109	−0.580543	+	0.814230i	$0.697160\pi$
$882$	0	0
$883$	− 1.48379e10i	− 0.725287i	−0.931928	−	0.362644i	$-0.881874\pi$
$883$	− 1.48379e10i	− 0.725287i	0.931928	−	0.362644i	$-0.118126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.42165e10i	− 1.64628i	−0.567840	−	0.823139i	$-0.692221\pi$
$887$	− 3.42165e10i	− 1.64628i	0.567840	−	0.823139i	$-0.307779\pi$
$888$	0	0
$889$	−1.68792e10	−0.805744
$890$	0	0
$891$	1.23848e9	0.0586565
$892$	0	0
$893$	− 6.42492e9i	− 0.301917i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.27470e9i	− 0.0589705i
$898$	0	0
$899$	−3.08279e10	−1.41509
$900$	0	0
$901$	4.55515e9	0.207475
$902$	0	0
$903$	− 1.08815e8i	− 0.00491792i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.82264e10i	− 1.70113i	−0.525870	−	0.850565i	$-0.676260\pi$
$907$	− 3.82264e10i	− 1.70113i	0.525870	−	0.850565i	$-0.323740\pi$
$908$	0	0
$909$	3.18608e10	1.40696
$910$	0	0
$911$	1.33838e10	0.586496	0.293248	−	0.956036i	$-0.405264\pi$
$911$	1.33838e10	0.586496	0.293248	+	0.956036i	$0.405264\pi$
$912$	0	0
$913$	5.05961e9i	0.220024i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.89443e9i	0.252435i
$918$	0	0
$919$	−4.40027e8	−0.0187014	−0.00935072	−	0.999956i	$-0.502976\pi$
$919$	−4.40027e8	−0.0187014	−0.00935072	+	0.999956i	$0.502976\pi$
$920$	0	0
$921$	−6.15353e9	−0.259547
$922$	0	0
$923$	1.51743e9i	0.0635187i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.15804e10i	− 0.889774i
$928$	0	0
$929$	4.86235e9	0.198972	0.0994859	−	0.995039i	$-0.468280\pi$
$929$	4.86235e9	0.198972	0.0994859	+	0.995039i	$0.468280\pi$
$930$	0	0
$931$	2.16378e9	0.0878797
$932$	0	0
$933$	− 7.70894e9i	− 0.310748i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6.59809e9i	− 0.262017i	−0.991381	−	0.131009i	$-0.958178\pi$
$937$	− 6.59809e9i	− 0.262017i	0.991381	−	0.131009i	$-0.0418216\pi$
$938$	0	0
$939$	−1.83957e10	−0.725082
$940$	0	0
$941$	−3.94166e10	−1.54211	−0.771055	−	0.636768i	$-0.780271\pi$
$941$	−3.94166e10	−1.54211	−0.771055	+	0.636768i	$0.780271\pi$
$942$	0	0
$943$	1.48361e9i	0.0576142i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.03861e10i	1.92790i	0.266076	+	0.963952i	$0.414273\pi$
$947$	5.03861e10i	1.92790i	−0.266076	+	0.963952i	$0.585727\pi$
$948$	0	0
$949$	5.00113e9	0.189949
$950$	0	0
$951$	−4.59582e9	−0.173273
$952$	0	0
$953$	2.07642e10i	0.777123i	0.921423	+	0.388562i	$0.127028\pi$
$953$	2.07642e10i	0.777123i	−0.921423	+	0.388562i	$0.872972\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.57163e9i	0.0948456i
$958$	0	0
$959$	2.72739e9	0.0998576
$960$	0	0
$961$	3.78932e9	0.137730
$962$	0	0
$963$	1.94511e10i	0.701864i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.90452e10i	− 1.38859i	−0.719689	−	0.694297i	$-0.755715\pi$
$967$	− 3.90452e10i	− 1.38859i	0.719689	−	0.694297i	$-0.244285\pi$
$968$	0	0
$969$	3.60890e9	0.127421
$970$	0	0
$971$	−2.96609e10	−1.03972	−0.519860	−	0.854251i	$-0.674016\pi$
$971$	−2.96609e10	−1.03972	−0.519860	+	0.854251i	$0.674016\pi$
$972$	0	0
$973$	6.43298e10i	2.23881i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.54794e10i	1.21715i	0.793495	+	0.608577i	$0.208259\pi$
$977$	3.54794e10i	1.21715i	−0.793495	+	0.608577i	$0.791741\pi$
$978$	0	0
$979$	−4.12577e9	−0.140529
$980$	0	0
$981$	−3.93463e10	−1.33064
$982$	0	0
$983$	2.82265e10i	0.947807i	0.880577	+	0.473903i	$0.157155\pi$
$983$	2.82265e10i	0.947807i	−0.880577	+	0.473903i	$0.842845\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.22007e10i	0.734949i
$988$	0	0
$989$	−1.90329e8	−0.00625629
$990$	0	0
$991$	−3.37277e10	−1.10085	−0.550426	−	0.834884i	$-0.685535\pi$
$991$	−3.37277e10	−1.10085	−0.550426	+	0.834884i	$0.685535\pi$
$992$	0	0
$993$	2.55778e10i	0.828974i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.90842e10i	0.609876i	0.952372	+	0.304938i	$0.0986358\pi$
$997$	1.90842e10i	0.609876i	−0.952372	+	0.304938i	$0.901364\pi$
$998$	0	0
$999$	−5.98119e9	−0.189805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.u.49.2		6
4.3	odd	2		200.8.c.j.49.5		6
5.2	odd	4		400.8.a.bh.1.2		3
5.3	odd	4		400.8.a.bi.1.2		3
5.4	even	2	inner	400.8.c.u.49.5		6
20.3	even	4		200.8.a.o.1.2	✓	3
20.7	even	4		200.8.a.p.1.2	yes	3
20.19	odd	2		200.8.c.j.49.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.8.a.o.1.2	✓	3	20.3	even	4
200.8.a.p.1.2	yes	3	20.7	even	4
200.8.c.j.49.2		6	20.19	odd	2
200.8.c.j.49.5		6	4.3	odd	2
400.8.a.bh.1.2		3	5.2	odd	4
400.8.a.bi.1.2		3	5.3	odd	4
400.8.c.u.49.2		6	1.1	even	1	trivial
400.8.c.u.49.5		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.9866i q^{3} +1068.54i q^{7} +1703.59 q^{9} +671.263 q^{11} +1410.86i q^{13} -24141.5i q^{17} -6799.10 q^{19} +23493.7 q^{21} -41092.9i q^{23} -85540.9i q^{27} -174244. q^{29} +176924. q^{31} -14758.8i q^{33} -69922.1i q^{37} +31020.0 q^{39} -36103.9 q^{41} -4631.67i q^{43} +944966. i q^{47} -318245. q^{49} -530790. q^{51} +188685. i q^{53} +149489. i q^{57} +1.18358e6 q^{59} +1.80080e6 q^{61} +1.82036e6i q^{63} +1.24107e6i q^{67} -903493. q^{69} +1.07553e6 q^{71} -3.54474e6i q^{73} +717274. i q^{77} +7.07853e6 q^{79} +1.84500e6 q^{81} +7.53745e6i q^{83} +3.83103e6i q^{87} -6.14628e6 q^{89} -1.50756e6 q^{91} -3.88995e6i q^{93} -1.43256e7i q^{97} +1.14356e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4292 q^{9} + 5318 q^{11} + 111498 q^{19} - 123124 q^{21} - 131056 q^{29} + 289196 q^{31} + 1600008 q^{39} + 599486 q^{41} - 4720974 q^{49} + 3724630 q^{51} + 1966736 q^{59} + 11420236 q^{61} - 10781396 q^{69}+ \cdots + 106633020 q^{99}+O(q^{100}) \) 6 * q - 4292 * q^9 + 5318 * q^11 + 111498 * q^19 - 123124 * q^21 - 131056 * q^29 + 289196 * q^31 + 1600008 * q^39 + 599486 * q^41 - 4720974 * q^49 + 3724630 * q^51 + 1966736 * q^59 + 11420236 * q^61 - 10781396 * q^69 - 3867912 * q^71 + 11181356 * q^79 + 43424534 * q^81 - 44667418 * q^89 - 2063888 * q^91 + 106633020 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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