LMFDB - Embedded newform 400.8.c.l.49.1

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 = [2,0,0,0,0,0,0,0,4302] 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:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.l.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.00000i q^{3} +706.000i q^{7} +2151.00 q^{9} +3840.00 q^{11} +4054.00i q^{13} +858.000i q^{17} +21044.0 q^{19} +4236.00 q^{21} +85338.0i q^{23} -26028.0i q^{27} +83106.0 q^{29} +145564. q^{31} -23040.0i q^{33} -498886. i q^{37} +24324.0 q^{39} -689514. q^{41} +867890. i q^{43} -235638. i q^{47} +325107. q^{49} +5148.00 q^{51} -1.83544e6i q^{53} -126264. i q^{57} +629508. q^{59} -2.66796e6 q^{61} +1.51861e6i q^{63} +3.37331e6i q^{67} +512028. q^{69} +2.60005e6 q^{71} +1.62849e6i q^{73} +2.71104e6i q^{77} -4.24353e6 q^{79} +4.54807e6 q^{81} +1.25138e6i q^{83} -498636. i q^{87} -6.29947e6 q^{89} -2.86212e6 q^{91} -873384. i q^{93} +3.97651e6i q^{97} +8.25984e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4302 q^{9} + 7680 q^{11} + 42088 q^{19} + 8472 q^{21} + 166212 q^{29} + 291128 q^{31} + 48648 q^{39} - 1379028 q^{41} + 650214 q^{49} + 10296 q^{51} + 1259016 q^{59} - 5335916 q^{61} + 1024056 q^{69}+ \cdots + 16519680 q^{99}+O(q^{100}) $ 2 * q + 4302 * q^9 + 7680 * q^11 + 42088 * q^19 + 8472 * q^21 + 166212 * q^29 + 291128 * q^31 + 48648 * q^39 - 1379028 * q^41 + 650214 * q^49 + 10296 * q^51 + 1259016 * q^59 - 5335916 * q^61 + 1024056 * q^69 + 5200104 * q^71 - 8487056 * q^79 + 9096138 * q^81 - 12598932 * q^89 - 5724248 * q^91 + 16519680 * 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
$3$	− 6.00000i	− 0.128300i	−0.997940	−	0.0641500i	$-0.979566\pi$
$3$	− 6.00000i	− 0.128300i	0.997940	−	0.0641500i	$-0.0204336\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	706.000i	0.777968i	0.921245	+	0.388984i	$0.127174\pi$
$7$	706.000i	0.777968i	−0.921245	+	0.388984i	$0.872826\pi$
$8$	0	0
$9$	2151.00	0.983539
$10$	0	0
$11$	3840.00	0.869875	0.434937	−	0.900461i	$-0.356770\pi$
$11$	3840.00	0.869875	0.434937	+	0.900461i	$0.356770\pi$
$12$	0	0
$13$	4054.00i	0.511778i	0.966706	+	0.255889i	$0.0823682\pi$
$13$	4054.00i	0.511778i	−0.966706	+	0.255889i	$0.917632\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	858.000i	0.0423561i	0.999776	+	0.0211781i	$0.00674169\pi$
$17$	858.000i	0.0423561i	−0.999776	+	0.0211781i	$0.993258\pi$
$18$	0	0
$19$	21044.0	0.703867	0.351934	−	0.936025i	$-0.385524\pi$
$19$	21044.0	0.703867	0.351934	+	0.936025i	$0.385524\pi$
$20$	0	0
$21$	4236.00	0.0998133
$22$	0	0
$23$	85338.0i	1.46250i	0.682111	+	0.731249i	$0.261062\pi$
$23$	85338.0i	1.46250i	−0.682111	+	0.731249i	$0.738938\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 26028.0i	− 0.254488i
$28$	0	0
$29$	83106.0	0.632761	0.316380	−	0.948632i	$-0.397532\pi$
$29$	83106.0	0.632761	0.316380	+	0.948632i	$0.397532\pi$
$30$	0	0
$31$	145564.	0.877583	0.438791	−	0.898589i	$-0.355407\pi$
$31$	145564.	0.877583	0.438791	+	0.898589i	$0.355407\pi$
$32$	0	0
$33$	− 23040.0i	− 0.111605i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 498886.i	− 1.61918i	−0.586995	−	0.809590i	$-0.699689\pi$
$37$	− 498886.i	− 1.61918i	0.586995	−	0.809590i	$-0.300311\pi$
$38$	0	0
$39$	24324.0	0.0656612
$40$	0	0
$41$	−689514.	−1.56243	−0.781213	−	0.624264i	$-0.785399\pi$
$41$	−689514.	−1.56243	−0.781213	+	0.624264i	$0.785399\pi$
$42$	0	0
$43$	867890.i	1.66466i	0.554282	+	0.832329i	$0.312993\pi$
$43$	867890.i	1.66466i	−0.554282	+	0.832329i	$0.687007\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 235638.i	− 0.331057i	−0.986205	−	0.165529i	$-0.947067\pi$
$47$	− 235638.i	− 0.331057i	0.986205	−	0.165529i	$-0.0529330\pi$
$48$	0	0
$49$	325107.	0.394766
$50$	0	0
$51$	5148.00	0.00543429
$52$	0	0
$53$	− 1.83544e6i	− 1.69346i	−0.532022	−	0.846730i	$-0.678568\pi$
$53$	− 1.83544e6i	− 1.69346i	0.532022	−	0.846730i	$-0.321432\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 126264.i	− 0.0903062i
$58$	0	0
$59$	629508.	0.399043	0.199521	−	0.979893i	$-0.436061\pi$
$59$	629508.	0.399043	0.199521	+	0.979893i	$0.436061\pi$
$60$	0	0
$61$	−2.66796e6	−1.50496	−0.752479	−	0.658616i	$-0.771142\pi$
$61$	−2.66796e6	−1.50496	−0.752479	+	0.658616i	$0.771142\pi$
$62$	0	0
$63$	1.51861e6i	0.765162i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.37331e6i	1.37023i	0.728434	+	0.685116i	$0.240248\pi$
$67$	3.37331e6i	1.37023i	−0.728434	+	0.685116i	$0.759752\pi$
$68$	0	0
$69$	512028.	0.187638
$70$	0	0
$71$	2.60005e6	0.862140	0.431070	−	0.902318i	$-0.358136\pi$
$71$	2.60005e6	0.862140	0.431070	+	0.902318i	$0.358136\pi$
$72$	0	0
$73$	1.62849e6i	0.489955i	0.969529	+	0.244977i	$0.0787806\pi$
$73$	1.62849e6i	0.489955i	−0.969529	+	0.244977i	$0.921219\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.71104e6i	0.676735i
$78$	0	0
$79$	−4.24353e6	−0.968350	−0.484175	−	0.874971i	$-0.660880\pi$
$79$	−4.24353e6	−0.968350	−0.484175	+	0.874971i	$0.660880\pi$
$80$	0	0
$81$	4.54807e6	0.950888
$82$	0	0
$83$	1.25138e6i	0.240223i	0.992760	+	0.120112i	$0.0383253\pi$
$83$	1.25138e6i	0.240223i	−0.992760	+	0.120112i	$0.961675\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 498636.i	− 0.0811832i
$88$	0	0
$89$	−6.29947e6	−0.947194	−0.473597	−	0.880742i	$-0.657045\pi$
$89$	−6.29947e6	−0.947194	−0.473597	+	0.880742i	$0.657045\pi$
$90$	0	0
$91$	−2.86212e6	−0.398147
$92$	0	0
$93$	− 873384.i	− 0.112594i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.97651e6i	0.442386i	0.975230	+	0.221193i	$0.0709951\pi$
$97$	3.97651e6i	0.442386i	−0.975230	+	0.221193i	$0.929005\pi$
$98$	0	0
$99$	8.25984e6	0.855556
$100$	0	0
$101$	1.25053e7	1.20773	0.603865	−	0.797086i	$-0.293626\pi$
$101$	1.25053e7	1.20773	0.603865	+	0.797086i	$0.293626\pi$
$102$	0	0
$103$	− 2.17226e7i	− 1.95876i	−0.202015	−	0.979382i	$-0.564749\pi$
$103$	− 2.17226e7i	− 1.95876i	0.202015	−	0.979382i	$-0.435251\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.04592e6i	− 0.240367i	−0.992752	−	0.120184i	$-0.961652\pi$
$107$	− 3.04592e6i	− 0.240367i	0.992752	−	0.120184i	$-0.0383483\pi$
$108$	0	0
$109$	1.37261e7	1.01521	0.507603	−	0.861591i	$-0.330532\pi$
$109$	1.37261e7	1.01521	0.507603	+	0.861591i	$0.330532\pi$
$110$	0	0
$111$	−2.99332e6	−0.207741
$112$	0	0
$113$	2.03152e7i	1.32449i	0.749289	+	0.662243i	$0.230395\pi$
$113$	2.03152e7i	1.32449i	−0.749289	+	0.662243i	$0.769605\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.72015e6i	0.503354i
$118$	0	0
$119$	−605748.	−0.0329517
$120$	0	0
$121$	−4.74157e6	−0.243318
$122$	0	0
$123$	4.13708e6i	0.200459i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.44019e7i	0.623888i	0.950100	+	0.311944i	$0.100980\pi$
$127$	1.44019e7i	0.623888i	−0.950100	+	0.311944i	$0.899020\pi$
$128$	0	0
$129$	5.20734e6	0.213576
$130$	0	0
$131$	−3.51283e7	−1.36524	−0.682618	−	0.730775i	$-0.739159\pi$
$131$	−3.51283e7	−1.36524	−0.682618	+	0.730775i	$0.739159\pi$
$132$	0	0
$133$	1.48571e7i	0.547586i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.66669e7i	0.886036i	0.896513	+	0.443018i	$0.146092\pi$
$137$	2.66669e7i	0.886036i	−0.896513	+	0.443018i	$0.853908\pi$
$138$	0	0
$139$	5.37105e7	1.69632	0.848159	−	0.529742i	$-0.177711\pi$
$139$	5.37105e7	1.69632	0.848159	+	0.529742i	$0.177711\pi$
$140$	0	0
$141$	−1.41383e6	−0.0424746
$142$	0	0
$143$	1.55674e7i	0.445183i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.95064e6i	− 0.0506485i
$148$	0	0
$149$	3.57914e7	0.886393	0.443197	−	0.896424i	$-0.353844\pi$
$149$	3.57914e7	0.886393	0.443197	+	0.896424i	$0.353844\pi$
$150$	0	0
$151$	8.13922e6	0.192382	0.0961908	−	0.995363i	$-0.469334\pi$
$151$	8.13922e6	0.192382	0.0961908	+	0.995363i	$0.469334\pi$
$152$	0	0
$153$	1.84556e6i	0.0416589i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.10735e6i	0.105329i	0.998612	+	0.0526643i	$0.0167713\pi$
$157$	5.10735e6i	0.105329i	−0.998612	+	0.0526643i	$0.983229\pi$
$158$	0	0
$159$	−1.10127e7	−0.217271
$160$	0	0
$161$	−6.02486e7	−1.13778
$162$	0	0
$163$	8.85622e7i	1.60174i	0.598839	+	0.800870i	$0.295629\pi$
$163$	8.85622e7i	1.60174i	−0.598839	+	0.800870i	$0.704371\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.12170e8i	1.86367i	0.362885	+	0.931834i	$0.381792\pi$
$167$	1.12170e8i	1.86367i	−0.362885	+	0.931834i	$0.618208\pi$
$168$	0	0
$169$	4.63136e7	0.738083
$170$	0	0
$171$	4.52656e7	0.692281
$172$	0	0
$173$	− 1.03754e7i	− 0.152350i	−0.997094	−	0.0761749i	$-0.975729\pi$
$173$	− 1.03754e7i	− 0.152350i	0.997094	−	0.0761749i	$-0.0242707\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.77705e6i	− 0.0511972i
$178$	0	0
$179$	1.91537e7	0.249613	0.124806	−	0.992181i	$-0.460169\pi$
$179$	1.91537e7	0.249613	0.124806	+	0.992181i	$0.460169\pi$
$180$	0	0
$181$	−2.46040e7	−0.308411	−0.154206	−	0.988039i	$-0.549282\pi$
$181$	−2.46040e7	−0.308411	−0.154206	+	0.988039i	$0.549282\pi$
$182$	0	0
$183$	1.60077e7i	0.193086i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.29472e6i	0.0368445i
$188$	0	0
$189$	1.83758e7	0.197984
$190$	0	0
$191$	1.15013e8	1.19434	0.597171	−	0.802114i	$-0.296291\pi$
$191$	1.15013e8	1.19434	0.597171	+	0.802114i	$0.296291\pi$
$192$	0	0
$193$	− 1.31761e8i	− 1.31928i	−0.751582	−	0.659640i	$-0.770709\pi$
$193$	− 1.31761e8i	− 1.31928i	0.751582	−	0.659640i	$-0.229291\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.46842e8i	1.36841i	0.729288	+	0.684206i	$0.239851\pi$
$197$	1.46842e8i	1.36841i	−0.729288	+	0.684206i	$0.760149\pi$
$198$	0	0
$199$	1.27107e8	1.14336	0.571679	−	0.820477i	$-0.306292\pi$
$199$	1.27107e8	1.14336	0.571679	+	0.820477i	$0.306292\pi$
$200$	0	0
$201$	2.02398e7	0.175801
$202$	0	0
$203$	5.86728e7i	0.492267i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.83562e8i	1.43842i
$208$	0	0
$209$	8.08090e7	0.612276
$210$	0	0
$211$	−8.90501e7	−0.652598	−0.326299	−	0.945267i	$-0.605802\pi$
$211$	−8.90501e7	−0.652598	−0.326299	+	0.945267i	$0.605802\pi$
$212$	0	0
$213$	− 1.56003e7i	− 0.110613i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.02768e8i	0.682731i
$218$	0	0
$219$	9.77096e6	0.0628612
$220$	0	0
$221$	−3.47833e6	−0.0216769
$222$	0	0
$223$	− 4.78436e7i	− 0.288906i	−0.989512	−	0.144453i	$-0.953858\pi$
$223$	− 4.78436e7i	− 0.288906i	0.989512	−	0.144453i	$-0.0461423\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.51060e7i	0.426171i	0.977034	+	0.213086i	$0.0683513\pi$
$227$	7.51060e7i	0.426171i	−0.977034	+	0.213086i	$0.931649\pi$
$228$	0	0
$229$	5.53497e7	0.304573	0.152286	−	0.988336i	$-0.451336\pi$
$229$	5.53497e7	0.304573	0.152286	+	0.988336i	$0.451336\pi$
$230$	0	0
$231$	1.62662e7	0.0868251
$232$	0	0
$233$	2.88910e8i	1.49629i	0.663533	+	0.748147i	$0.269056\pi$
$233$	2.88910e8i	1.49629i	−0.663533	+	0.748147i	$0.730944\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.54612e7i	0.124239i
$238$	0	0
$239$	−9.79382e7	−0.464044	−0.232022	−	0.972710i	$-0.574534\pi$
$239$	−9.79382e7	−0.464044	−0.232022	+	0.972710i	$0.574534\pi$
$240$	0	0
$241$	8.64399e7	0.397790	0.198895	−	0.980021i	$-0.436265\pi$
$241$	8.64399e7	0.397790	0.198895	+	0.980021i	$0.436265\pi$
$242$	0	0
$243$	− 8.42116e7i	− 0.376487i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.53124e7i	0.360224i
$248$	0	0
$249$	7.50827e6	0.0308207
$250$	0	0
$251$	−1.35898e8	−0.542443	−0.271221	−	0.962517i	$-0.587428\pi$
$251$	−1.35898e8	−0.542443	−0.271221	+	0.962517i	$0.587428\pi$
$252$	0	0
$253$	3.27698e8i	1.27219i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.52157e8i	1.66159i	0.556581	+	0.830793i	$0.312113\pi$
$257$	4.52157e8i	1.66159i	−0.556581	+	0.830793i	$0.687887\pi$
$258$	0	0
$259$	3.52214e8	1.25967
$260$	0	0
$261$	1.78761e8	0.622345
$262$	0	0
$263$	− 6.05973e7i	− 0.205404i	−0.994712	−	0.102702i	$-0.967251\pi$
$263$	− 6.05973e7i	− 0.205404i	0.994712	−	0.102702i	$-0.0327487\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.77968e7i	0.121525i
$268$	0	0
$269$	3.74181e7	0.117206	0.0586029	−	0.998281i	$-0.481335\pi$
$269$	3.74181e7	0.117206	0.0586029	+	0.998281i	$0.481335\pi$
$270$	0	0
$271$	−5.16074e8	−1.57514	−0.787571	−	0.616224i	$-0.788661\pi$
$271$	−5.16074e8	−1.57514	−0.787571	+	0.616224i	$0.788661\pi$
$272$	0	0
$273$	1.71727e7i	0.0510823i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.86991e7i	− 0.222480i	−0.993794	−	0.111240i	$-0.964518\pi$
$277$	− 7.86991e7i	− 0.222480i	0.993794	−	0.111240i	$-0.0354822\pi$
$278$	0	0
$279$	3.13108e8	0.863137
$280$	0	0
$281$	5.36047e7	0.144122	0.0720611	−	0.997400i	$-0.477042\pi$
$281$	5.36047e7	0.144122	0.0720611	+	0.997400i	$0.477042\pi$
$282$	0	0
$283$	1.17665e8i	0.308599i	0.988024	+	0.154299i	$0.0493121\pi$
$283$	1.17665e8i	0.308599i	−0.988024	+	0.154299i	$0.950688\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.86797e8i	− 1.21552i
$288$	0	0
$289$	4.09603e8	0.998206
$290$	0	0
$291$	2.38591e7	0.0567582
$292$	0	0
$293$	2.79303e8i	0.648693i	0.945938	+	0.324347i	$0.105144\pi$
$293$	2.79303e8i	0.648693i	−0.945938	+	0.324347i	$0.894856\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.99475e7i	− 0.221373i
$298$	0	0
$299$	−3.45960e8	−0.748475
$300$	0	0
$301$	−6.12730e8	−1.29505
$302$	0	0
$303$	− 7.50320e7i	− 0.154952i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.72857e8i	1.32721i	0.748085	+	0.663603i	$0.230973\pi$
$307$	6.72857e8i	1.32721i	−0.748085	+	0.663603i	$0.769027\pi$
$308$	0	0
$309$	−1.30336e8	−0.251310
$310$	0	0
$311$	−3.93153e7	−0.0741139	−0.0370570	−	0.999313i	$-0.511798\pi$
$311$	−3.93153e7	−0.0741139	−0.0370570	+	0.999313i	$0.511798\pi$
$312$	0	0
$313$	1.92264e8i	0.354399i	0.984175	+	0.177199i	$0.0567037\pi$
$313$	1.92264e8i	0.354399i	−0.984175	+	0.177199i	$0.943296\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.31614e7i	0.128995i	0.997918	+	0.0644977i	$0.0205445\pi$
$317$	7.31614e7i	0.128995i	−0.997918	+	0.0644977i	$0.979455\pi$
$318$	0	0
$319$	3.19127e8	0.550423
$320$	0	0
$321$	−1.82755e7	−0.0308391
$322$	0	0
$323$	1.80558e7i	0.0298131i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.23566e7i	− 0.130251i
$328$	0	0
$329$	1.66360e8	0.257552
$330$	0	0
$331$	9.16104e8	1.38850	0.694252	−	0.719732i	$-0.255735\pi$
$331$	9.16104e8	1.38850	0.694252	+	0.719732i	$0.255735\pi$
$332$	0	0
$333$	− 1.07310e9i	− 1.59253i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.56503e8i	− 1.07673i	−0.842712	−	0.538364i	$-0.819042\pi$
$337$	− 7.56503e8i	− 1.07673i	0.842712	−	0.538364i	$-0.180958\pi$
$338$	0	0
$339$	1.21891e8	0.169932
$340$	0	0
$341$	5.58966e8	0.763387
$342$	0	0
$343$	8.10947e8i	1.08508i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.87175e8i	− 0.497456i	−0.968573	−	0.248728i	$-0.919988\pi$
$347$	− 3.87175e8i	− 0.497456i	0.968573	−	0.248728i	$-0.0800125\pi$
$348$	0	0
$349$	1.99761e7	0.0251548	0.0125774	−	0.999921i	$-0.495996\pi$
$349$	1.99761e7	0.0251548	0.0125774	+	0.999921i	$0.495996\pi$
$350$	0	0
$351$	1.05518e8	0.130242
$352$	0	0
$353$	− 1.36499e9i	− 1.65165i	−0.563923	−	0.825827i	$-0.690709\pi$
$353$	− 1.36499e9i	− 1.65165i	0.563923	−	0.825827i	$-0.309291\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.63449e6i	0.00422770i
$358$	0	0
$359$	−1.09937e9	−1.25404	−0.627022	−	0.779001i	$-0.715726\pi$
$359$	−1.09937e9	−1.25404	−0.627022	+	0.779001i	$0.715726\pi$
$360$	0	0
$361$	−4.51022e8	−0.504571
$362$	0	0
$363$	2.84494e7i	0.0312177i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.64118e8i	− 0.384513i	−0.981345	−	0.192256i	$-0.938419\pi$
$367$	− 3.64118e8i	− 0.384513i	0.981345	−	0.192256i	$-0.0615805\pi$
$368$	0	0
$369$	−1.48314e9	−1.53671
$370$	0	0
$371$	1.29582e9	1.31746
$372$	0	0
$373$	− 9.57949e7i	− 0.0955788i	−0.998857	−	0.0477894i	$-0.984782\pi$
$373$	− 9.57949e7i	− 0.0955788i	0.998857	−	0.0477894i	$-0.0152176\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.36912e8i	0.323833i
$378$	0	0
$379$	−6.84090e8	−0.645470	−0.322735	−	0.946489i	$-0.604602\pi$
$379$	−6.84090e8	−0.645470	−0.322735	+	0.946489i	$0.604602\pi$
$380$	0	0
$381$	8.64114e7	0.0800449
$382$	0	0
$383$	− 8.50349e8i	− 0.773395i	−0.922207	−	0.386698i	$-0.873616\pi$
$383$	− 8.50349e8i	− 0.773395i	0.922207	−	0.386698i	$-0.126384\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.86683e9i	1.63726i
$388$	0	0
$389$	1.28862e8	0.110994	0.0554972	−	0.998459i	$-0.482326\pi$
$389$	1.28862e8	0.110994	0.0554972	+	0.998459i	$0.482326\pi$
$390$	0	0
$391$	−7.32200e7	−0.0619457
$392$	0	0
$393$	2.10770e8i	0.175160i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.47383e9i	− 1.18217i	−0.806610	−	0.591084i	$-0.798700\pi$
$397$	− 1.47383e9i	− 1.18217i	0.806610	−	0.591084i	$-0.201300\pi$
$398$	0	0
$399$	8.91424e7	0.0702553
$400$	0	0
$401$	−1.00938e9	−0.781717	−0.390858	−	0.920451i	$-0.627822\pi$
$401$	−1.00938e9	−0.781717	−0.390858	+	0.920451i	$0.627822\pi$
$402$	0	0
$403$	5.90116e8i	0.449128i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.91572e9i	− 1.40848i
$408$	0	0
$409$	−8.79404e8	−0.635561	−0.317780	−	0.948164i	$-0.602937\pi$
$409$	−8.79404e8	−0.635561	−0.317780	+	0.948164i	$0.602937\pi$
$410$	0	0
$411$	1.60002e8	0.113678
$412$	0	0
$413$	4.44433e8i	0.310442i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.22263e8i	− 0.217638i
$418$	0	0
$419$	−2.38560e9	−1.58434	−0.792170	−	0.610300i	$-0.791049\pi$
$419$	−2.38560e9	−1.58434	−0.792170	+	0.610300i	$0.791049\pi$
$420$	0	0
$421$	1.41780e9	0.926034	0.463017	−	0.886349i	$-0.346767\pi$
$421$	1.41780e9	0.926034	0.463017	+	0.886349i	$0.346767\pi$
$422$	0	0
$423$	− 5.06857e8i	− 0.325608i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.88358e9i	− 1.17081i
$428$	0	0
$429$	9.34042e7	0.0571170
$430$	0	0
$431$	−2.47434e9	−1.48864	−0.744319	−	0.667825i	$-0.767226\pi$
$431$	−2.47434e9	−1.48864	−0.744319	+	0.667825i	$0.767226\pi$
$432$	0	0
$433$	1.42266e8i	0.0842160i	0.999113	+	0.0421080i	$0.0134074\pi$
$433$	1.42266e8i	0.0842160i	−0.999113	+	0.0421080i	$0.986593\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.79585e9i	1.02940i
$438$	0	0
$439$	2.80718e9	1.58359	0.791797	−	0.610785i	$-0.209146\pi$
$439$	2.80718e9	1.58359	0.791797	+	0.610785i	$0.209146\pi$
$440$	0	0
$441$	6.99305e8	0.388268
$442$	0	0
$443$	− 2.29630e9i	− 1.25492i	−0.778649	−	0.627460i	$-0.784095\pi$
$443$	− 2.29630e9i	− 1.25492i	0.778649	−	0.627460i	$-0.215905\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.14748e8i	− 0.113724i
$448$	0	0
$449$	−1.63361e8	−0.0851700	−0.0425850	−	0.999093i	$-0.513559\pi$
$449$	−1.63361e8	−0.0851700	−0.0425850	+	0.999093i	$0.513559\pi$
$450$	0	0
$451$	−2.64773e9	−1.35912
$452$	0	0
$453$	− 4.88353e7i	− 0.0246826i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.28894e9i	− 1.61194i	−0.591956	−	0.805970i	$-0.701644\pi$
$457$	− 3.28894e9i	− 1.61194i	0.591956	−	0.805970i	$-0.298356\pi$
$458$	0	0
$459$	2.23320e7	0.0107791
$460$	0	0
$461$	−9.55698e8	−0.454326	−0.227163	−	0.973857i	$-0.572945\pi$
$461$	−9.55698e8	−0.454326	−0.227163	+	0.973857i	$0.572945\pi$
$462$	0	0
$463$	− 1.26696e9i	− 0.593237i	−0.954996	−	0.296618i	$-0.904141\pi$
$463$	− 1.26696e9i	− 0.593237i	0.954996	−	0.296618i	$-0.0958589\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.93953e9i	− 1.78993i	−0.446140	−	0.894963i	$-0.647202\pi$
$467$	− 3.93953e9i	− 1.78993i	0.446140	−	0.894963i	$-0.352798\pi$
$468$	0	0
$469$	−2.38155e9	−1.06600
$470$	0	0
$471$	3.06441e7	0.0135137
$472$	0	0
$473$	3.33270e9i	1.44804i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.94804e9i	− 1.66558i
$478$	0	0
$479$	−3.23581e9	−1.34527	−0.672634	−	0.739975i	$-0.734837\pi$
$479$	−3.23581e9	−1.34527	−0.672634	+	0.739975i	$0.734837\pi$
$480$	0	0
$481$	2.02248e9	0.828662
$482$	0	0
$483$	3.61492e8i	0.145977i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.47239e9i	− 0.577658i	−0.957381	−	0.288829i	$-0.906734\pi$
$487$	− 1.47239e9i	− 0.577658i	0.957381	−	0.288829i	$-0.0932659\pi$
$488$	0	0
$489$	5.31373e8	0.205503
$490$	0	0
$491$	3.21107e9	1.22423	0.612116	−	0.790768i	$-0.290318\pi$
$491$	3.21107e9	1.22423	0.612116	+	0.790768i	$0.290318\pi$
$492$	0	0
$493$	7.13049e7i	0.0268013i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.83564e9i	0.670717i
$498$	0	0
$499$	−4.19098e8	−0.150995	−0.0754976	−	0.997146i	$-0.524055\pi$
$499$	−4.19098e8	−0.150995	−0.0754976	+	0.997146i	$0.524055\pi$
$500$	0	0
$501$	6.73019e8	0.239109
$502$	0	0
$503$	− 3.70706e9i	− 1.29880i	−0.760448	−	0.649399i	$-0.775021\pi$
$503$	− 3.70706e9i	− 1.29880i	0.760448	−	0.649399i	$-0.224979\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.77882e8i	− 0.0946961i
$508$	0	0
$509$	2.28893e9	0.769343	0.384672	−	0.923053i	$-0.374315\pi$
$509$	2.28893e9	0.769343	0.384672	+	0.923053i	$0.374315\pi$
$510$	0	0
$511$	−1.14972e9	−0.381169
$512$	0	0
$513$	− 5.47733e8i	− 0.179126i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.04850e8i	− 0.287978i
$518$	0	0
$519$	−6.22522e7	−0.0195465
$520$	0	0
$521$	1.90038e9	0.588719	0.294359	−	0.955695i	$-0.404894\pi$
$521$	1.90038e9	0.588719	0.294359	+	0.955695i	$0.404894\pi$
$522$	0	0
$523$	6.07040e9i	1.85550i	0.373200	+	0.927751i	$0.378261\pi$
$523$	6.07040e9i	1.85550i	−0.373200	+	0.927751i	$0.621739\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.24894e8i	0.0371710i
$528$	0	0
$529$	−3.87775e9	−1.13890
$530$	0	0
$531$	1.35407e9	0.392474
$532$	0	0
$533$	− 2.79529e9i	− 0.799616i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.14922e8i	− 0.0320253i
$538$	0	0
$539$	1.24841e9	0.343397
$540$	0	0
$541$	−4.10261e9	−1.11396	−0.556981	−	0.830526i	$-0.688040\pi$
$541$	−4.10261e9	−1.11396	−0.556981	+	0.830526i	$0.688040\pi$
$542$	0	0
$543$	1.47624e8i	0.0395692i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.53877e9i	0.663234i	0.943414	+	0.331617i	$0.107594\pi$
$547$	2.53877e9i	0.663234i	−0.943414	+	0.331617i	$0.892406\pi$
$548$	0	0
$549$	−5.73878e9	−1.48019
$550$	0	0
$551$	1.74888e9	0.445379
$552$	0	0
$553$	− 2.99593e9i	− 0.753345i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.19358e9i	0.783041i	0.920169	+	0.391520i	$0.128051\pi$
$557$	3.19358e9i	0.783041i	−0.920169	+	0.391520i	$0.871949\pi$
$558$	0	0
$559$	−3.51843e9	−0.851936
$560$	0	0
$561$	1.97683e7	0.00472715
$562$	0	0
$563$	3.52214e9i	0.831815i	0.909407	+	0.415908i	$0.136536\pi$
$563$	3.52214e9i	0.831815i	−0.909407	+	0.415908i	$0.863464\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.21094e9i	0.739760i
$568$	0	0
$569$	5.35081e9	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$569$	5.35081e9	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$570$	0	0
$571$	2.59117e9	0.582464	0.291232	−	0.956652i	$-0.405935\pi$
$571$	2.59117e9	0.582464	0.291232	+	0.956652i	$0.405935\pi$
$572$	0	0
$573$	− 6.90076e8i	− 0.153234i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.50238e8i	− 0.0759011i	−0.999280	−	0.0379505i	$-0.987917\pi$
$577$	− 3.50238e8i	− 0.0759011i	0.999280	−	0.0379505i	$-0.0120829\pi$
$578$	0	0
$579$	−7.90567e8	−0.169264
$580$	0	0
$581$	−8.83473e8	−0.186886
$582$	0	0
$583$	− 7.04810e9i	− 1.47310i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.42501e9i	0.494857i	0.968906	+	0.247429i	$0.0795856\pi$
$587$	2.42501e9i	0.494857i	−0.968906	+	0.247429i	$0.920414\pi$
$588$	0	0
$589$	3.06325e9	0.617702
$590$	0	0
$591$	8.81049e8	0.175567
$592$	0	0
$593$	− 7.57894e9i	− 1.49251i	−0.665661	−	0.746254i	$-0.731850\pi$
$593$	− 7.57894e9i	− 1.49251i	0.665661	−	0.746254i	$-0.268150\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7.62639e8i	− 0.146693i
$598$	0	0
$599$	−3.06431e9	−0.582557	−0.291279	−	0.956638i	$-0.594081\pi$
$599$	−3.06431e9	−0.582557	−0.291279	+	0.956638i	$0.594081\pi$
$600$	0	0
$601$	−8.51705e9	−1.60040	−0.800200	−	0.599733i	$-0.795273\pi$
$601$	−8.51705e9	−1.60040	−0.800200	+	0.599733i	$0.795273\pi$
$602$	0	0
$603$	7.25598e9i	1.34768i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.51105e9i	1.00017i	0.865976	+	0.500085i	$0.166698\pi$
$607$	5.51105e9i	1.00017i	−0.865976	+	0.500085i	$0.833302\pi$
$608$	0	0
$609$	3.52037e8	0.0631579
$610$	0	0
$611$	9.55276e8	0.169428
$612$	0	0
$613$	− 1.31574e9i	− 0.230706i	−0.993325	−	0.115353i	$-0.963200\pi$
$613$	− 1.31574e9i	− 0.230706i	0.993325	−	0.115353i	$-0.0368000\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.70804e8i	− 0.166392i	−0.996533	−	0.0831962i	$-0.973487\pi$
$617$	− 9.70804e8i	− 0.166392i	0.996533	−	0.0831962i	$-0.0265128\pi$
$618$	0	0
$619$	6.96530e9	1.18038	0.590191	−	0.807264i	$-0.299052\pi$
$619$	6.96530e9	1.18038	0.590191	+	0.807264i	$0.299052\pi$
$620$	0	0
$621$	2.22118e9	0.372188
$622$	0	0
$623$	− 4.44742e9i	− 0.736886i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.84854e8i	− 0.0785551i
$628$	0	0
$629$	4.28044e8	0.0685822
$630$	0	0
$631$	−1.57127e9	−0.248971	−0.124485	−	0.992221i	$-0.539728\pi$
$631$	−1.57127e9	−0.248971	−0.124485	+	0.992221i	$0.539728\pi$
$632$	0	0
$633$	5.34300e8i	0.0837283i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.31798e9i	0.202033i
$638$	0	0
$639$	5.59271e9	0.847948
$640$	0	0
$641$	4.19744e9	0.629479	0.314740	−	0.949178i	$-0.398083\pi$
$641$	4.19744e9	0.629479	0.314740	+	0.949178i	$0.398083\pi$
$642$	0	0
$643$	5.02531e9i	0.745460i	0.927940	+	0.372730i	$0.121578\pi$
$643$	5.02531e9i	0.745460i	−0.927940	+	0.372730i	$0.878422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.28047e10i	− 1.85867i	−0.369232	−	0.929337i	$-0.620379\pi$
$647$	− 1.28047e10i	− 1.85867i	0.369232	−	0.929337i	$-0.379621\pi$
$648$	0	0
$649$	2.41731e9	0.347117
$650$	0	0
$651$	6.16609e8	0.0875944
$652$	0	0
$653$	1.11817e10i	1.57148i	0.618554	+	0.785742i	$0.287719\pi$
$653$	1.11817e10i	1.57148i	−0.618554	+	0.785742i	$0.712281\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.50289e9i	0.481890i
$658$	0	0
$659$	4.11341e9	0.559891	0.279945	−	0.960016i	$-0.409684\pi$
$659$	4.11341e9	0.559891	0.279945	+	0.960016i	$0.409684\pi$
$660$	0	0
$661$	4.55818e9	0.613884	0.306942	−	0.951728i	$-0.400694\pi$
$661$	4.55818e9	0.613884	0.306942	+	0.951728i	$0.400694\pi$
$662$	0	0
$663$	2.08700e7i	0.00278115i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.09210e9i	0.925410i
$668$	0	0
$669$	−2.87062e8	−0.0370667
$670$	0	0
$671$	−1.02450e10	−1.30913
$672$	0	0
$673$	7.02401e9i	0.888244i	0.895966	+	0.444122i	$0.146484\pi$
$673$	7.02401e9i	0.888244i	−0.895966	+	0.444122i	$0.853516\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.27731e9i	0.777523i	0.921338	+	0.388762i	$0.127097\pi$
$677$	6.27731e9i	0.777523i	−0.921338	+	0.388762i	$0.872903\pi$
$678$	0	0
$679$	−2.80742e9	−0.344162
$680$	0	0
$681$	4.50636e8	0.0546778
$682$	0	0
$683$	6.99148e9i	0.839647i	0.907606	+	0.419823i	$0.137908\pi$
$683$	6.99148e9i	0.839647i	−0.907606	+	0.419823i	$0.862092\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.32098e8i	− 0.0390767i
$688$	0	0
$689$	7.44088e9	0.866677
$690$	0	0
$691$	4.17342e9	0.481193	0.240596	−	0.970625i	$-0.422657\pi$
$691$	4.17342e9	0.481193	0.240596	+	0.970625i	$0.422657\pi$
$692$	0	0
$693$	5.83145e9i	0.665595i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.91603e8i	− 0.0661783i
$698$	0	0
$699$	1.73346e9	0.191975
$700$	0	0
$701$	−6.73860e9	−0.738851	−0.369425	−	0.929260i	$-0.620445\pi$
$701$	−6.73860e9	−0.738851	−0.369425	+	0.929260i	$0.620445\pi$
$702$	0	0
$703$	− 1.04986e10i	− 1.13969i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.82876e9i	0.939576i
$708$	0	0
$709$	7.10555e9	0.748749	0.374374	−	0.927278i	$-0.377858\pi$
$709$	7.10555e9	0.748749	0.374374	+	0.927278i	$0.377858\pi$
$710$	0	0
$711$	−9.12783e9	−0.952410
$712$	0	0
$713$	1.24221e10i	1.28346i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.87629e8i	0.0595369i
$718$	0	0
$719$	1.77811e10	1.78405	0.892024	−	0.451988i	$-0.149285\pi$
$719$	1.77811e10	1.78405	0.892024	+	0.451988i	$0.149285\pi$
$720$	0	0
$721$	1.53362e10	1.52386
$722$	0	0
$723$	− 5.18639e8i	− 0.0510365i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.27207e9i	0.315829i	0.987453	+	0.157915i	$0.0504771\pi$
$727$	3.27207e9i	0.315829i	−0.987453	+	0.157915i	$0.949523\pi$
$728$	0	0
$729$	9.44136e9	0.902585
$730$	0	0
$731$	−7.44650e8	−0.0705084
$732$	0	0
$733$	7.15509e9i	0.671044i	0.942032	+	0.335522i	$0.108913\pi$
$733$	7.15509e9i	0.671044i	−0.942032	+	0.335522i	$0.891087\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.29535e10i	1.19193i
$738$	0	0
$739$	1.70559e9	0.155460	0.0777299	−	0.996974i	$-0.475233\pi$
$739$	1.70559e9	0.155460	0.0777299	+	0.996974i	$0.475233\pi$
$740$	0	0
$741$	5.11874e8	0.0462168
$742$	0	0
$743$	− 9.59074e9i	− 0.857810i	−0.903350	−	0.428905i	$-0.858899\pi$
$743$	− 9.59074e9i	− 0.857810i	0.903350	−	0.428905i	$-0.141101\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.69171e9i	0.236269i
$748$	0	0
$749$	2.15042e9	0.186998
$750$	0	0
$751$	−6.33249e9	−0.545550	−0.272775	−	0.962078i	$-0.587941\pi$
$751$	−6.33249e9	−0.545550	−0.272775	+	0.962078i	$0.587941\pi$
$752$	0	0
$753$	8.15386e8i	0.0695954i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.95934e10i	1.64163i	0.571197	+	0.820813i	$0.306479\pi$
$757$	1.95934e10i	1.64163i	−0.571197	+	0.820813i	$0.693521\pi$
$758$	0	0
$759$	1.96619e9	0.163222
$760$	0	0
$761$	−1.16891e10	−0.961471	−0.480736	−	0.876866i	$-0.659630\pi$
$761$	−1.16891e10	−0.961471	−0.480736	+	0.876866i	$0.659630\pi$
$762$	0	0
$763$	9.69063e9i	0.789798i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.55203e9i	0.204221i
$768$	0	0
$769$	−2.10168e10	−1.66657	−0.833287	−	0.552841i	$-0.813544\pi$
$769$	−2.10168e10	−1.66657	−0.833287	+	0.552841i	$0.813544\pi$
$770$	0	0
$771$	2.71294e9	0.213182
$772$	0	0
$773$	− 2.12393e10i	− 1.65391i	−0.562267	−	0.826956i	$-0.690071\pi$
$773$	− 2.12393e10i	− 1.65391i	0.562267	−	0.826956i	$-0.309929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.11328e9i	− 0.161616i
$778$	0	0
$779$	−1.45101e10	−1.09974
$780$	0	0
$781$	9.98420e9	0.749954
$782$	0	0
$783$	− 2.16308e9i	− 0.161030i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.33601e9i	0.609602i	0.952416	+	0.304801i	$0.0985900\pi$
$787$	8.33601e9i	0.609602i	−0.952416	+	0.304801i	$0.901410\pi$
$788$	0	0
$789$	−3.63584e8	−0.0263533
$790$	0	0
$791$	−1.43426e10	−1.03041
$792$	0	0
$793$	− 1.08159e10i	− 0.770205i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.60775e10i	− 1.12490i	−0.826830	−	0.562451i	$-0.809858\pi$
$797$	− 1.60775e10i	− 1.12490i	0.826830	−	0.562451i	$-0.190142\pi$
$798$	0	0
$799$	2.02177e8	0.0140223
$800$	0	0
$801$	−1.35502e10	−0.931602
$802$	0	0
$803$	6.25342e9i	0.426200i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.24509e8i	− 0.0150375i
$808$	0	0
$809$	1.86150e9	0.123607	0.0618034	−	0.998088i	$-0.480315\pi$
$809$	1.86150e9	0.123607	0.0618034	+	0.998088i	$0.480315\pi$
$810$	0	0
$811$	2.58014e10	1.69852	0.849258	−	0.527979i	$-0.177050\pi$
$811$	2.58014e10	1.69852	0.849258	+	0.527979i	$0.177050\pi$
$812$	0	0
$813$	3.09644e9i	0.202091i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.82639e10i	1.17170i
$818$	0	0
$819$	−6.15643e9	−0.391593
$820$	0	0
$821$	−6.63649e9	−0.418540	−0.209270	−	0.977858i	$-0.567109\pi$
$821$	−6.63649e9	−0.418540	−0.209270	+	0.977858i	$0.567109\pi$
$822$	0	0
$823$	− 1.48106e8i	− 0.00926130i	−0.999989	−	0.00463065i	$-0.998526\pi$
$823$	− 1.48106e8i	− 0.00926130i	0.999989	−	0.00463065i	$-0.00147399\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.01466e10i	1.23861i	0.785152	+	0.619303i	$0.212585\pi$
$827$	2.01466e10i	1.23861i	−0.785152	+	0.619303i	$0.787415\pi$
$828$	0	0
$829$	1.98564e10	1.21049	0.605243	−	0.796041i	$-0.293076\pi$
$829$	1.98564e10	1.21049	0.605243	+	0.796041i	$0.293076\pi$
$830$	0	0
$831$	−4.72195e8	−0.0285442
$832$	0	0
$833$	2.78942e8i	0.0167208i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.78874e9i	− 0.223334i
$838$	0	0
$839$	1.71078e10	1.00006	0.500031	−	0.866007i	$-0.333322\pi$
$839$	1.71078e10	1.00006	0.500031	+	0.866007i	$0.333322\pi$
$840$	0	0
$841$	−1.03433e10	−0.599614
$842$	0	0
$843$	− 3.21628e8i	− 0.0184909i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.34755e9i	− 0.189293i
$848$	0	0
$849$	7.05990e8	0.0395933
$850$	0	0
$851$	4.25739e10	2.36805
$852$	0	0
$853$	− 2.00983e10i	− 1.10876i	−0.832264	−	0.554379i	$-0.812956\pi$
$853$	− 2.00983e10i	− 1.10876i	0.832264	−	0.554379i	$-0.187044\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.23744e10i	− 1.21428i	−0.794595	−	0.607140i	$-0.792317\pi$
$857$	− 2.23744e10i	− 1.21428i	0.794595	−	0.607140i	$-0.207683\pi$
$858$	0	0
$859$	−1.88899e10	−1.01684	−0.508421	−	0.861109i	$-0.669771\pi$
$859$	−1.88899e10	−1.01684	−0.508421	+	0.861109i	$0.669771\pi$
$860$	0	0
$861$	−2.92078e9	−0.155951
$862$	0	0
$863$	1.66361e10i	0.881075i	0.897734	+	0.440538i	$0.145212\pi$
$863$	1.66361e10i	0.881075i	−0.897734	+	0.440538i	$0.854788\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.45762e9i	− 0.128070i
$868$	0	0
$869$	−1.62951e10	−0.842343
$870$	0	0
$871$	−1.36754e10	−0.701255
$872$	0	0
$873$	8.55348e9i	0.435104i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.42290e10i	0.712323i	0.934424	+	0.356161i	$0.115915\pi$
$877$	1.42290e10i	0.712323i	−0.934424	+	0.356161i	$0.884085\pi$
$878$	0	0
$879$	1.67582e9	0.0832274
$880$	0	0
$881$	2.41869e10	1.19169	0.595847	−	0.803098i	$-0.296816\pi$
$881$	2.41869e10	1.19169	0.595847	+	0.803098i	$0.296816\pi$
$882$	0	0
$883$	3.23241e10i	1.58003i	0.613090	+	0.790013i	$0.289926\pi$
$883$	3.23241e10i	1.58003i	−0.613090	+	0.790013i	$0.710074\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.10944e9i	0.438287i	0.975693	+	0.219144i	$0.0703264\pi$
$887$	9.10944e9i	0.438287i	−0.975693	+	0.219144i	$0.929674\pi$
$888$	0	0
$889$	−1.01677e10	−0.485365
$890$	0	0
$891$	1.74646e10	0.827154
$892$	0	0
$893$	− 4.95877e9i	− 0.233020i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.07576e9i	0.0960293i
$898$	0	0
$899$	1.20972e10	0.555300
$900$	0	0
$901$	1.57481e9	0.0717284
$902$	0	0
$903$	3.67638e9i	0.166155i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 6.03785e9i	− 0.268693i	−0.990934	−	0.134347i	$-0.957106\pi$
$907$	− 6.03785e9i	− 0.268693i	0.990934	−	0.134347i	$-0.0428936\pi$
$908$	0	0
$909$	2.68990e10	1.18785
$910$	0	0
$911$	−3.61401e10	−1.58371	−0.791853	−	0.610711i	$-0.790884\pi$
$911$	−3.61401e10	−1.58371	−0.791853	+	0.610711i	$0.790884\pi$
$912$	0	0
$913$	4.80529e9i	0.208964i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.48006e10i	− 1.06211i
$918$	0	0
$919$	1.43783e10	0.611088	0.305544	−	0.952178i	$-0.401162\pi$
$919$	1.43783e10	0.611088	0.305544	+	0.952178i	$0.401162\pi$
$920$	0	0
$921$	4.03714e9	0.170281
$922$	0	0
$923$	1.05406e10i	0.441225i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.67254e10i	− 1.92652i
$928$	0	0
$929$	9.95926e9	0.407542	0.203771	−	0.979019i	$-0.434680\pi$
$929$	9.95926e9	0.407542	0.203771	+	0.979019i	$0.434680\pi$
$930$	0	0
$931$	6.84155e9	0.277863
$932$	0	0
$933$	2.35892e8i	0.00950882i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8.59930e9i	− 0.341487i	−0.985315	−	0.170744i	$-0.945383\pi$
$937$	− 8.59930e9i	− 0.341487i	0.985315	−	0.170744i	$-0.0546170\pi$
$938$	0	0
$939$	1.15358e9	0.0454694
$940$	0	0
$941$	−3.62216e10	−1.41711	−0.708555	−	0.705656i	$-0.750653\pi$
$941$	−3.62216e10	−1.41711	−0.708555	+	0.705656i	$0.750653\pi$
$942$	0	0
$943$	− 5.88417e10i	− 2.28504i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.08322e9i	0.117972i	0.998259	+	0.0589861i	$0.0187868\pi$
$947$	3.08322e9i	0.117972i	−0.998259	+	0.0589861i	$0.981213\pi$
$948$	0	0
$949$	−6.60191e9	−0.250748
$950$	0	0
$951$	4.38968e8	0.0165501
$952$	0	0
$953$	− 5.78801e9i	− 0.216623i	−0.994117	−	0.108311i	$-0.965456\pi$
$953$	− 5.78801e9i	− 0.216623i	0.994117	−	0.108311i	$-0.0345443\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.91476e9i	− 0.0706192i
$958$	0	0
$959$	−1.88269e10	−0.689307
$960$	0	0
$961$	−6.32374e9	−0.229849
$962$	0	0
$963$	− 6.55177e9i	− 0.236410i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.82018e10i	− 0.647323i	−0.946173	−	0.323662i	$-0.895086\pi$
$967$	− 1.82018e10i	− 0.647323i	0.946173	−	0.323662i	$-0.104914\pi$
$968$	0	0
$969$	1.08335e8	0.00382502
$970$	0	0
$971$	−4.86220e10	−1.70438	−0.852188	−	0.523235i	$-0.824725\pi$
$971$	−4.86220e10	−1.70438	−0.852188	+	0.523235i	$0.824725\pi$
$972$	0	0
$973$	3.79196e10i	1.31968i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.78750e10i	− 1.29934i	−0.760218	−	0.649668i	$-0.774908\pi$
$977$	− 3.78750e10i	− 1.29934i	0.760218	−	0.649668i	$-0.225092\pi$
$978$	0	0
$979$	−2.41899e10	−0.823940
$980$	0	0
$981$	2.95248e10	0.998495
$982$	0	0
$983$	− 4.92346e10i	− 1.65323i	−0.562769	−	0.826614i	$-0.690264\pi$
$983$	− 4.92346e10i	− 1.65323i	0.562769	−	0.826614i	$-0.309736\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 9.98163e8i	− 0.0330439i
$988$	0	0
$989$	−7.40640e10	−2.43456
$990$	0	0
$991$	1.33475e10	0.435654	0.217827	−	0.975987i	$-0.430103\pi$
$991$	1.33475e10	0.435654	0.217827	+	0.975987i	$0.430103\pi$
$992$	0	0
$993$	− 5.49663e9i	− 0.178145i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.26641e10i	− 0.404709i	−0.979312	−	0.202354i	$-0.935141\pi$
$997$	− 1.26641e10i	− 0.404709i	0.979312	−	0.202354i	$-0.0648593\pi$
$998$	0	0
$999$	−1.29850e10	−0.412062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.l.49.1		2
4.3	odd	2		100.8.c.a.49.2		2
5.2	odd	4		400.8.a.j.1.1		1
5.3	odd	4		80.8.a.b.1.1		1
5.4	even	2	inner	400.8.c.l.49.2		2
20.3	even	4		20.8.a.a.1.1	✓	1
20.7	even	4		100.8.a.a.1.1		1
20.19	odd	2		100.8.c.a.49.1		2
40.3	even	4		320.8.a.e.1.1		1
40.13	odd	4		320.8.a.d.1.1		1
60.23	odd	4		180.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.a.a.1.1	✓	1	20.3	even	4
80.8.a.b.1.1		1	5.3	odd	4
100.8.a.a.1.1		1	20.7	even	4
100.8.c.a.49.1		2	20.19	odd	2
100.8.c.a.49.2		2	4.3	odd	2
180.8.a.c.1.1		1	60.23	odd	4
320.8.a.d.1.1		1	40.13	odd	4
320.8.a.e.1.1		1	40.3	even	4
400.8.a.j.1.1		1	5.2	odd	4
400.8.c.l.49.1		2	1.1	even	1	trivial
400.8.c.l.49.2		2	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)

\(f(q)\)	\(=\)	\(q-6.00000i q^{3} +706.000i q^{7} +2151.00 q^{9} +3840.00 q^{11} +4054.00i q^{13} +858.000i q^{17} +21044.0 q^{19} +4236.00 q^{21} +85338.0i q^{23} -26028.0i q^{27} +83106.0 q^{29} +145564. q^{31} -23040.0i q^{33} -498886. i q^{37} +24324.0 q^{39} -689514. q^{41} +867890. i q^{43} -235638. i q^{47} +325107. q^{49} +5148.00 q^{51} -1.83544e6i q^{53} -126264. i q^{57} +629508. q^{59} -2.66796e6 q^{61} +1.51861e6i q^{63} +3.37331e6i q^{67} +512028. q^{69} +2.60005e6 q^{71} +1.62849e6i q^{73} +2.71104e6i q^{77} -4.24353e6 q^{79} +4.54807e6 q^{81} +1.25138e6i q^{83} -498636. i q^{87} -6.29947e6 q^{89} -2.86212e6 q^{91} -873384. i q^{93} +3.97651e6i q^{97} +8.25984e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4302 q^{9} + 7680 q^{11} + 42088 q^{19} + 8472 q^{21} + 166212 q^{29} + 291128 q^{31} + 48648 q^{39} - 1379028 q^{41} + 650214 q^{49} + 10296 q^{51} + 1259016 q^{59} - 5335916 q^{61} + 1024056 q^{69}+ \cdots + 16519680 q^{99}+O(q^{100}) \) 2 * q + 4302 * q^9 + 7680 * q^11 + 42088 * q^19 + 8472 * q^21 + 166212 * q^29 + 291128 * q^31 + 48648 * q^39 - 1379028 * q^41 + 650214 * q^49 + 10296 * q^51 + 1259016 * q^59 - 5335916 * q^61 + 1024056 * q^69 + 5200104 * q^71 - 8487056 * q^79 + 9096138 * q^81 - 12598932 * q^89 - 5724248 * q^91 + 16519680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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