LMFDB - Embedded newform 3040.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,4,0,-5,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.52616$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52616 q^{3} +1.00000 q^{5} -0.329157 q^{7} -0.670843 q^{9} +0.879127 q^{11} -1.35297 q^{13} -1.52616 q^{15} +5.60228 q^{17} -1.00000 q^{19} +0.502345 q^{21} -8.77078 q^{23} +1.00000 q^{25} +5.60228 q^{27} -2.54997 q^{29} +0.394001 q^{31} -1.34169 q^{33} -0.329157 q^{35} +9.34926 q^{37} +2.06484 q^{39} -3.05232 q^{41} -4.22081 q^{43} -0.670843 q^{45} -1.51487 q^{47} -6.89166 q^{49} -8.54997 q^{51} +0.669597 q^{53} +0.879127 q^{55} +1.52616 q^{57} -0.155969 q^{59} +9.95650 q^{61} +0.220813 q^{63} -1.35297 q^{65} -11.3095 q^{67} +13.3856 q^{69} -3.29406 q^{71} +0.445339 q^{73} -1.52616 q^{75} -0.289371 q^{77} -14.4416 q^{79} -6.53744 q^{81} +0.220813 q^{83} +5.60228 q^{85} +3.89166 q^{87} +4.04762 q^{89} +0.445339 q^{91} -0.601308 q^{93} -1.00000 q^{95} +5.19700 q^{97} -0.589756 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

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
$3$	−1.52616	−0.881128	−0.440564	−	0.897721i	$-0.645221\pi$
$3$	−1.52616	−0.881128	−0.440564	+	0.897721i	$0.645221\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.329157	−0.124410	−0.0622048	−	0.998063i	$-0.519813\pi$
$7$	−0.329157	−0.124410	−0.0622048	+	0.998063i	$0.519813\pi$
$8$	0	0
$9$	−0.670843	−0.223614
$10$	0	0
$11$	0.879127	0.265067	0.132533	−	0.991179i	$-0.457689\pi$
$11$	0.879127	0.265067	0.132533	+	0.991179i	$0.457689\pi$
$12$	0	0
$13$	−1.35297	−0.375246	−0.187623	−	0.982241i	$-0.560078\pi$
$13$	−1.35297	−0.375246	−0.187623	+	0.982241i	$0.560078\pi$
$14$	0	0
$15$	−1.52616	−0.394052
$16$	0	0
$17$	5.60228	1.35875	0.679377	−	0.733790i	$-0.262250\pi$
$17$	5.60228	1.35875	0.679377	+	0.733790i	$0.262250\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.502345	0.109621
$22$	0	0
$23$	−8.77078	−1.82883	−0.914417	−	0.404773i	$-0.867351\pi$
$23$	−8.77078	−1.82883	−0.914417	+	0.404773i	$0.867351\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.60228	1.07816
$28$	0	0
$29$	−2.54997	−0.473517	−0.236759	−	0.971568i	$-0.576085\pi$
$29$	−2.54997	−0.473517	−0.236759	+	0.971568i	$0.576085\pi$
$30$	0	0
$31$	0.394001	0.0707647	0.0353823	−	0.999374i	$-0.488735\pi$
$31$	0.394001	0.0707647	0.0353823	+	0.999374i	$0.488735\pi$
$32$	0	0
$33$	−1.34169	−0.233558
$34$	0	0
$35$	−0.329157	−0.0556377
$36$	0	0
$37$	9.34926	1.53701	0.768504	−	0.639845i	$-0.221001\pi$
$37$	9.34926	1.53701	0.768504	+	0.639845i	$0.221001\pi$
$38$	0	0
$39$	2.06484	0.330640
$40$	0	0
$41$	−3.05232	−0.476692	−0.238346	−	0.971180i	$-0.576605\pi$
$41$	−3.05232	−0.476692	−0.238346	+	0.971180i	$0.576605\pi$
$42$	0	0
$43$	−4.22081	−0.643668	−0.321834	−	0.946796i	$-0.604299\pi$
$43$	−4.22081	−0.643668	−0.321834	+	0.946796i	$0.604299\pi$
$44$	0	0
$45$	−0.670843	−0.100003
$46$	0	0
$47$	−1.51487	−0.220967	−0.110484	−	0.993878i	$-0.535240\pi$
$47$	−1.51487	−0.220967	−0.110484	+	0.993878i	$0.535240\pi$
$48$	0	0
$49$	−6.89166	−0.984522
$50$	0	0
$51$	−8.54997	−1.19724
$52$	0	0
$53$	0.669597	0.0919763	0.0459881	−	0.998942i	$-0.485356\pi$
$53$	0.669597	0.0919763	0.0459881	+	0.998942i	$0.485356\pi$
$54$	0	0
$55$	0.879127	0.118541
$56$	0	0
$57$	1.52616	0.202145
$58$	0	0
$59$	−0.155969	−0.0203054	−0.0101527	−	0.999948i	$-0.503232\pi$
$59$	−0.155969	−0.0203054	−0.0101527	+	0.999948i	$0.503232\pi$
$60$	0	0
$61$	9.95650	1.27480	0.637400	−	0.770533i	$-0.280010\pi$
$61$	9.95650	1.27480	0.637400	+	0.770533i	$0.280010\pi$
$62$	0	0
$63$	0.220813	0.0278198
$64$	0	0
$65$	−1.35297	−0.167815
$66$	0	0
$67$	−11.3095	−1.38167	−0.690836	−	0.723012i	$-0.742757\pi$
$67$	−11.3095	−1.38167	−0.690836	+	0.723012i	$0.742757\pi$
$68$	0	0
$69$	13.3856	1.61144
$70$	0	0
$71$	−3.29406	−0.390933	−0.195467	−	0.980710i	$-0.562622\pi$
$71$	−3.29406	−0.390933	−0.195467	+	0.980710i	$0.562622\pi$
$72$	0	0
$73$	0.445339	0.0521230	0.0260615	−	0.999660i	$-0.491703\pi$
$73$	0.445339	0.0521230	0.0260615	+	0.999660i	$0.491703\pi$
$74$	0	0
$75$	−1.52616	−0.176226
$76$	0	0
$77$	−0.289371	−0.0329769
$78$	0	0
$79$	−14.4416	−1.62481	−0.812405	−	0.583094i	$-0.801842\pi$
$79$	−14.4416	−1.62481	−0.812405	+	0.583094i	$0.801842\pi$
$80$	0	0
$81$	−6.53744	−0.726382
$82$	0	0
$83$	0.220813	0.0242373	0.0121187	−	0.999927i	$-0.496142\pi$
$83$	0.220813	0.0242373	0.0121187	+	0.999927i	$0.496142\pi$
$84$	0	0
$85$	5.60228	0.607653
$86$	0	0
$87$	3.89166	0.417229
$88$	0	0
$89$	4.04762	0.429047	0.214524	−	0.976719i	$-0.431180\pi$
$89$	4.04762	0.429047	0.214524	+	0.976719i	$0.431180\pi$
$90$	0	0
$91$	0.445339	0.0466842
$92$	0	0
$93$	−0.601308	−0.0623527
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	5.19700	0.527675	0.263838	−	0.964567i	$-0.415012\pi$
$97$	5.19700	0.527675	0.263838	+	0.964567i	$0.415012\pi$
$98$	0	0
$99$	−0.589756	−0.0592727
$100$	0	0
$101$	−7.21612	−0.718031	−0.359015	−	0.933332i	$-0.616887\pi$
$101$	−7.21612	−0.718031	−0.359015	+	0.933332i	$0.616887\pi$
$102$	0	0
$103$	0.0761273	0.00750105	0.00375052	−	0.999993i	$-0.498806\pi$
$103$	0.0761273	0.00750105	0.00375052	+	0.999993i	$0.498806\pi$
$104$	0	0
$105$	0.502345	0.0490239
$106$	0	0
$107$	8.62610	0.833916	0.416958	−	0.908926i	$-0.363096\pi$
$107$	8.62610	0.833916	0.416958	+	0.908926i	$0.363096\pi$
$108$	0	0
$109$	−10.3652	−0.992809	−0.496404	−	0.868091i	$-0.665347\pi$
$109$	−10.3652	−0.992809	−0.496404	+	0.868091i	$0.665347\pi$
$110$	0	0
$111$	−14.2684	−1.35430
$112$	0	0
$113$	8.60888	0.809855	0.404928	−	0.914349i	$-0.367297\pi$
$113$	8.60888	0.809855	0.404928	+	0.914349i	$0.367297\pi$
$114$	0	0
$115$	−8.77078	−0.817880
$116$	0	0
$117$	0.907630	0.0839104
$118$	0	0
$119$	−1.84403	−0.169042
$120$	0	0
$121$	−10.2271	−0.929740
$122$	0	0
$123$	4.65831	0.420026
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.47013	−0.751602	−0.375801	−	0.926700i	$-0.622632\pi$
$127$	−8.47013	−0.751602	−0.375801	+	0.926700i	$0.622632\pi$
$128$	0	0
$129$	6.44163	0.567153
$130$	0	0
$131$	−12.7310	−1.11231	−0.556156	−	0.831078i	$-0.687724\pi$
$131$	−12.7310	−1.11231	−0.556156	+	0.831078i	$0.687724\pi$
$132$	0	0
$133$	0.329157	0.0285415
$134$	0	0
$135$	5.60228	0.482168
$136$	0	0
$137$	−19.1438	−1.63557	−0.817785	−	0.575524i	$-0.804798\pi$
$137$	−19.1438	−1.63557	−0.817785	+	0.575524i	$0.804798\pi$
$138$	0	0
$139$	5.80425	0.492310	0.246155	−	0.969231i	$-0.420833\pi$
$139$	5.80425	0.492310	0.246155	+	0.969231i	$0.420833\pi$
$140$	0	0
$141$	2.31194	0.194700
$142$	0	0
$143$	−1.18943	−0.0994652
$144$	0	0
$145$	−2.54997	−0.211763
$146$	0	0
$147$	10.5178	0.867490
$148$	0	0
$149$	−17.8194	−1.45982	−0.729910	−	0.683543i	$-0.760438\pi$
$149$	−17.8194	−1.45982	−0.729910	+	0.683543i	$0.760438\pi$
$150$	0	0
$151$	−14.7059	−1.19675	−0.598376	−	0.801215i	$-0.704187\pi$
$151$	−14.7059	−1.19675	−0.598376	+	0.801215i	$0.704187\pi$
$152$	0	0
$153$	−3.75825	−0.303837
$154$	0	0
$155$	0.394001	0.0316469
$156$	0	0
$157$	−10.4416	−0.833332	−0.416666	−	0.909060i	$-0.636802\pi$
$157$	−10.4416	−0.833332	−0.416666	+	0.909060i	$0.636802\pi$
$158$	0	0
$159$	−1.02191	−0.0810428
$160$	0	0
$161$	2.88696	0.227525
$162$	0	0
$163$	18.4207	1.44282	0.721410	−	0.692508i	$-0.243494\pi$
$163$	18.4207	1.44282	0.721410	+	0.692508i	$0.243494\pi$
$164$	0	0
$165$	−1.34169	−0.104450
$166$	0	0
$167$	−9.15101	−0.708126	−0.354063	−	0.935222i	$-0.615200\pi$
$167$	−9.15101	−0.708126	−0.354063	+	0.935222i	$0.615200\pi$
$168$	0	0
$169$	−11.1695	−0.859190
$170$	0	0
$171$	0.670843	0.0513006
$172$	0	0
$173$	12.8757	0.978920	0.489460	−	0.872026i	$-0.337194\pi$
$173$	12.8757	0.978920	0.489460	+	0.872026i	$0.337194\pi$
$174$	0	0
$175$	−0.329157	−0.0248819
$176$	0	0
$177$	0.238033	0.0178916
$178$	0	0
$179$	21.0675	1.57466	0.787328	−	0.616535i	$-0.211464\pi$
$179$	21.0675	1.57466	0.787328	+	0.616535i	$0.211464\pi$
$180$	0	0
$181$	−2.78800	−0.207231	−0.103615	−	0.994617i	$-0.533041\pi$
$181$	−2.78800	−0.207231	−0.103615	+	0.994617i	$0.533041\pi$
$182$	0	0
$183$	−15.1952	−1.12326
$184$	0	0
$185$	9.34926	0.687371
$186$	0	0
$187$	4.92512	0.360160
$188$	0	0
$189$	−1.84403	−0.134134
$190$	0	0
$191$	0.0439103	0.00317724	0.00158862	−	0.999999i	$-0.499494\pi$
$191$	0.0439103	0.00317724	0.00158862	+	0.999999i	$0.499494\pi$
$192$	0	0
$193$	−16.4241	−1.18224	−0.591118	−	0.806585i	$-0.701313\pi$
$193$	−16.4241	−1.18224	−0.591118	+	0.806585i	$0.701313\pi$
$194$	0	0
$195$	2.06484	0.147867
$196$	0	0
$197$	21.6236	1.54062	0.770310	−	0.637670i	$-0.220102\pi$
$197$	21.6236	1.54062	0.770310	+	0.637670i	$0.220102\pi$
$198$	0	0
$199$	−4.39772	−0.311746	−0.155873	−	0.987777i	$-0.549819\pi$
$199$	−4.39772	−0.311746	−0.155873	+	0.987777i	$0.549819\pi$
$200$	0	0
$201$	17.2600	1.21743
$202$	0	0
$203$	0.839340	0.0589102
$204$	0	0
$205$	−3.05232	−0.213183
$206$	0	0
$207$	5.88382	0.408954
$208$	0	0
$209$	−0.879127	−0.0608105
$210$	0	0
$211$	−5.20828	−0.358553	−0.179277	−	0.983799i	$-0.557376\pi$
$211$	−5.20828	−0.358553	−0.179277	+	0.983799i	$0.557376\pi$
$212$	0	0
$213$	5.02726	0.344462
$214$	0	0
$215$	−4.22081	−0.287857
$216$	0	0
$217$	−0.129688	−0.00880381
$218$	0	0
$219$	−0.679658	−0.0459270
$220$	0	0
$221$	−7.57972	−0.509867
$222$	0	0
$223$	−25.5475	−1.71079	−0.855394	−	0.517979i	$-0.826685\pi$
$223$	−25.5475	−1.71079	−0.855394	+	0.517979i	$0.826685\pi$
$224$	0	0
$225$	−0.670843	−0.0447229
$226$	0	0
$227$	−17.1247	−1.13661	−0.568304	−	0.822819i	$-0.692400\pi$
$227$	−17.1247	−1.13661	−0.568304	+	0.822819i	$0.692400\pi$
$228$	0	0
$229$	−7.98845	−0.527892	−0.263946	−	0.964538i	$-0.585024\pi$
$229$	−7.98845	−0.527892	−0.263946	+	0.964538i	$0.585024\pi$
$230$	0	0
$231$	0.441625	0.0290568
$232$	0	0
$233$	−25.5284	−1.67242	−0.836210	−	0.548410i	$-0.815234\pi$
$233$	−25.5284	−1.67242	−0.836210	+	0.548410i	$0.815234\pi$
$234$	0	0
$235$	−1.51487	−0.0988195
$236$	0	0
$237$	22.0402	1.43166
$238$	0	0
$239$	−15.6249	−1.01069	−0.505344	−	0.862918i	$-0.668634\pi$
$239$	−15.6249	−1.01069	−0.505344	+	0.862918i	$0.668634\pi$
$240$	0	0
$241$	−2.18474	−0.140731	−0.0703657	−	0.997521i	$-0.522417\pi$
$241$	−2.18474	−0.140731	−0.0703657	+	0.997521i	$0.522417\pi$
$242$	0	0
$243$	−6.82969	−0.438125
$244$	0	0
$245$	−6.89166	−0.440292
$246$	0	0
$247$	1.35297	0.0860874
$248$	0	0
$249$	−0.336995	−0.0213562
$250$	0	0
$251$	−14.5463	−0.918152	−0.459076	−	0.888397i	$-0.651819\pi$
$251$	−14.5463	−0.918152	−0.459076	+	0.888397i	$0.651819\pi$
$252$	0	0
$253$	−7.71063	−0.484763
$254$	0	0
$255$	−8.54997	−0.535420
$256$	0	0
$257$	10.6659	0.665320	0.332660	−	0.943047i	$-0.392054\pi$
$257$	10.6659	0.665320	0.332660	+	0.943047i	$0.392054\pi$
$258$	0	0
$259$	−3.07737	−0.191219
$260$	0	0
$261$	1.71063	0.105885
$262$	0	0
$263$	−0.936133	−0.0577244	−0.0288622	−	0.999583i	$-0.509188\pi$
$263$	−0.936133	−0.0577244	−0.0288622	+	0.999583i	$0.509188\pi$
$264$	0	0
$265$	0.669597	0.0411330
$266$	0	0
$267$	−6.17731	−0.378045
$268$	0	0
$269$	−19.4138	−1.18368	−0.591841	−	0.806055i	$-0.701599\pi$
$269$	−19.4138	−1.18368	−0.591841	+	0.806055i	$0.701599\pi$
$270$	0	0
$271$	−12.3517	−0.750314	−0.375157	−	0.926961i	$-0.622411\pi$
$271$	−12.3517	−0.750314	−0.375157	+	0.926961i	$0.622411\pi$
$272$	0	0
$273$	−0.679658	−0.0411348
$274$	0	0
$275$	0.879127	0.0530133
$276$	0	0
$277$	−11.5416	−0.693465	−0.346733	−	0.937964i	$-0.612709\pi$
$277$	−11.5416	−0.693465	−0.346733	+	0.937964i	$0.612709\pi$
$278$	0	0
$279$	−0.264313	−0.0158240
$280$	0	0
$281$	−8.41906	−0.502239	−0.251119	−	0.967956i	$-0.580799\pi$
$281$	−8.41906	−0.502239	−0.251119	+	0.967956i	$0.580799\pi$
$282$	0	0
$283$	30.7671	1.82891	0.914456	−	0.404685i	$-0.132619\pi$
$283$	30.7671	1.82891	0.914456	+	0.404685i	$0.132619\pi$
$284$	0	0
$285$	1.52616	0.0904018
$286$	0	0
$287$	1.00469	0.0593050
$288$	0	0
$289$	14.3856	0.846212
$290$	0	0
$291$	−7.93144	−0.464949
$292$	0	0
$293$	2.69466	0.157423	0.0787117	−	0.996897i	$-0.474919\pi$
$293$	2.69466	0.157423	0.0787117	+	0.996897i	$0.474919\pi$
$294$	0	0
$295$	−0.155969	−0.00908084
$296$	0	0
$297$	4.92512	0.285784
$298$	0	0
$299$	11.8666	0.686263
$300$	0	0
$301$	1.38931	0.0800785
$302$	0	0
$303$	11.0129	0.632677
$304$	0	0
$305$	9.95650	0.570108
$306$	0	0
$307$	29.8044	1.70103	0.850513	−	0.525954i	$-0.176292\pi$
$307$	29.8044	1.70103	0.850513	+	0.525954i	$0.176292\pi$
$308$	0	0
$309$	−0.116182	−0.00660938
$310$	0	0
$311$	−24.5159	−1.39017	−0.695083	−	0.718929i	$-0.744632\pi$
$311$	−24.5159	−1.39017	−0.695083	+	0.718929i	$0.744632\pi$
$312$	0	0
$313$	−23.0712	−1.30406	−0.652030	−	0.758193i	$-0.726082\pi$
$313$	−23.0712	−1.30406	−0.652030	+	0.758193i	$0.726082\pi$
$314$	0	0
$315$	0.220813	0.0124414
$316$	0	0
$317$	−2.01128	−0.112965	−0.0564825	−	0.998404i	$-0.517988\pi$
$317$	−2.01128	−0.112965	−0.0564825	+	0.998404i	$0.517988\pi$
$318$	0	0
$319$	−2.24175	−0.125514
$320$	0	0
$321$	−13.1648	−0.734787
$322$	0	0
$323$	−5.60228	−0.311719
$324$	0	0
$325$	−1.35297	−0.0750492
$326$	0	0
$327$	15.8190	0.874791
$328$	0	0
$329$	0.498632	0.0274904
$330$	0	0
$331$	12.4924	0.686646	0.343323	−	0.939217i	$-0.388447\pi$
$331$	12.4924	0.686646	0.343323	+	0.939217i	$0.388447\pi$
$332$	0	0
$333$	−6.27188	−0.343697
$334$	0	0
$335$	−11.3095	−0.617902
$336$	0	0
$337$	28.3691	1.54536	0.772681	−	0.634794i	$-0.218915\pi$
$337$	28.3691	1.54536	0.772681	+	0.634794i	$0.218915\pi$
$338$	0	0
$339$	−13.1385	−0.713586
$340$	0	0
$341$	0.346377	0.0187574
$342$	0	0
$343$	4.57254	0.246894
$344$	0	0
$345$	13.3856	0.720656
$346$	0	0
$347$	−14.4777	−0.777204	−0.388602	−	0.921406i	$-0.627042\pi$
$347$	−14.4777	−0.777204	−0.388602	+	0.921406i	$0.627042\pi$
$348$	0	0
$349$	−28.1198	−1.50522	−0.752608	−	0.658468i	$-0.771205\pi$
$349$	−28.1198	−1.50522	−0.752608	+	0.658468i	$0.771205\pi$
$350$	0	0
$351$	−7.57972	−0.404575
$352$	0	0
$353$	3.18572	0.169559	0.0847793	−	0.996400i	$-0.472981\pi$
$353$	3.18572	0.169559	0.0847793	+	0.996400i	$0.472981\pi$
$354$	0	0
$355$	−3.29406	−0.174831
$356$	0	0
$357$	2.81428	0.148948
$358$	0	0
$359$	3.77298	0.199130	0.0995652	−	0.995031i	$-0.468255\pi$
$359$	3.77298	0.199130	0.0995652	+	0.995031i	$0.468255\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	15.6082	0.819219
$364$	0	0
$365$	0.445339	0.0233101
$366$	0	0
$367$	34.3180	1.79139	0.895693	−	0.444673i	$-0.146680\pi$
$367$	34.3180	1.79139	0.895693	+	0.444673i	$0.146680\pi$
$368$	0	0
$369$	2.04762	0.106595
$370$	0	0
$371$	−0.220403	−0.0114427
$372$	0	0
$373$	−1.30534	−0.0675882	−0.0337941	−	0.999429i	$-0.510759\pi$
$373$	−1.30534	−0.0675882	−0.0337941	+	0.999429i	$0.510759\pi$
$374$	0	0
$375$	−1.52616	−0.0788104
$376$	0	0
$377$	3.45003	0.177686
$378$	0	0
$379$	19.0811	0.980130	0.490065	−	0.871686i	$-0.336973\pi$
$379$	19.0811	0.980130	0.490065	+	0.871686i	$0.336973\pi$
$380$	0	0
$381$	12.9268	0.662258
$382$	0	0
$383$	10.3330	0.527992	0.263996	−	0.964524i	$-0.414959\pi$
$383$	10.3330	0.527992	0.263996	+	0.964524i	$0.414959\pi$
$384$	0	0
$385$	−0.289371	−0.0147477
$386$	0	0
$387$	2.83150	0.143933
$388$	0	0
$389$	−23.1151	−1.17198	−0.585990	−	0.810318i	$-0.699294\pi$
$389$	−23.1151	−1.17198	−0.585990	+	0.810318i	$0.699294\pi$
$390$	0	0
$391$	−49.1364	−2.48494
$392$	0	0
$393$	19.4295	0.980089
$394$	0	0
$395$	−14.4416	−0.726637
$396$	0	0
$397$	−8.70594	−0.436939	−0.218469	−	0.975844i	$-0.570106\pi$
$397$	−8.70594	−0.436939	−0.218469	+	0.975844i	$0.570106\pi$
$398$	0	0
$399$	−0.502345	−0.0251487
$400$	0	0
$401$	9.07488	0.453178	0.226589	−	0.973990i	$-0.427243\pi$
$401$	9.07488	0.453178	0.226589	+	0.973990i	$0.427243\pi$
$402$	0	0
$403$	−0.533071	−0.0265542
$404$	0	0
$405$	−6.53744	−0.324848
$406$	0	0
$407$	8.21918	0.407410
$408$	0	0
$409$	23.2046	1.14739	0.573696	−	0.819068i	$-0.305509\pi$
$409$	23.2046	1.14739	0.573696	+	0.819068i	$0.305509\pi$
$410$	0	0
$411$	29.2165	1.44115
$412$	0	0
$413$	0.0513381	0.00252619
$414$	0	0
$415$	0.220813	0.0108393
$416$	0	0
$417$	−8.85819	−0.433788
$418$	0	0
$419$	18.0821	0.883367	0.441683	−	0.897171i	$-0.354381\pi$
$419$	18.0821	0.883367	0.441683	+	0.897171i	$0.354381\pi$
$420$	0	0
$421$	17.2009	0.838318	0.419159	−	0.907913i	$-0.362325\pi$
$421$	17.2009	0.838318	0.419159	+	0.907913i	$0.362325\pi$
$422$	0	0
$423$	1.01624	0.0494114
$424$	0	0
$425$	5.60228	0.271751
$426$	0	0
$427$	−3.27725	−0.158597
$428$	0	0
$429$	1.81526	0.0876416
$430$	0	0
$431$	−3.26900	−0.157462	−0.0787312	−	0.996896i	$-0.525087\pi$
$431$	−3.26900	−0.157462	−0.0787312	+	0.996896i	$0.525087\pi$
$432$	0	0
$433$	−8.41475	−0.404387	−0.202194	−	0.979346i	$-0.564807\pi$
$433$	−8.41475	−0.404387	−0.202194	+	0.979346i	$0.564807\pi$
$434$	0	0
$435$	3.89166	0.186591
$436$	0	0
$437$	8.77078	0.419563
$438$	0	0
$439$	29.1971	1.39350	0.696752	−	0.717312i	$-0.254628\pi$
$439$	29.1971	1.39350	0.696752	+	0.717312i	$0.254628\pi$
$440$	0	0
$441$	4.62322	0.220153
$442$	0	0
$443$	−30.3061	−1.43989	−0.719944	−	0.694032i	$-0.755833\pi$
$443$	−30.3061	−1.43989	−0.719944	+	0.694032i	$0.755833\pi$
$444$	0	0
$445$	4.04762	0.191876
$446$	0	0
$447$	27.1952	1.28629
$448$	0	0
$449$	1.83456	0.0865783	0.0432891	−	0.999063i	$-0.486216\pi$
$449$	1.83456	0.0865783	0.0432891	+	0.999063i	$0.486216\pi$
$450$	0	0
$451$	−2.68337	−0.126355
$452$	0	0
$453$	22.4436	1.05449
$454$	0	0
$455$	0.445339	0.0208778
$456$	0	0
$457$	24.9021	1.16487	0.582436	−	0.812877i	$-0.302100\pi$
$457$	24.9021	1.16487	0.582436	+	0.812877i	$0.302100\pi$
$458$	0	0
$459$	31.3856	1.46495
$460$	0	0
$461$	−8.63326	−0.402091	−0.201045	−	0.979582i	$-0.564434\pi$
$461$	−8.63326	−0.402091	−0.201045	+	0.979582i	$0.564434\pi$
$462$	0	0
$463$	−22.2109	−1.03223	−0.516114	−	0.856520i	$-0.672622\pi$
$463$	−22.2109	−1.03223	−0.516114	+	0.856520i	$0.672622\pi$
$464$	0	0
$465$	−0.601308	−0.0278850
$466$	0	0
$467$	−12.5898	−0.582584	−0.291292	−	0.956634i	$-0.594085\pi$
$467$	−12.5898	−0.582584	−0.291292	+	0.956634i	$0.594085\pi$
$468$	0	0
$469$	3.72259	0.171893
$470$	0	0
$471$	15.9356	0.734272
$472$	0	0
$473$	−3.71063	−0.170615
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−0.449195	−0.0205672
$478$	0	0
$479$	−20.4909	−0.936252	−0.468126	−	0.883662i	$-0.655071\pi$
$479$	−20.4909	−0.936252	−0.468126	+	0.883662i	$0.655071\pi$
$480$	0	0
$481$	−12.6493	−0.576756
$482$	0	0
$483$	−4.40596	−0.200478
$484$	0	0
$485$	5.19700	0.235984
$486$	0	0
$487$	40.2509	1.82394	0.911972	−	0.410252i	$-0.134559\pi$
$487$	40.2509	1.82394	0.911972	+	0.410252i	$0.134559\pi$
$488$	0	0
$489$	−28.1129	−1.27131
$490$	0	0
$491$	−8.17482	−0.368925	−0.184462	−	0.982840i	$-0.559054\pi$
$491$	−8.17482	−0.368925	−0.184462	+	0.982840i	$0.559054\pi$
$492$	0	0
$493$	−14.2857	−0.643394
$494$	0	0
$495$	−0.589756	−0.0265076
$496$	0	0
$497$	1.08426	0.0486359
$498$	0	0
$499$	16.6624	0.745913	0.372957	−	0.927849i	$-0.378344\pi$
$499$	16.6624	0.745913	0.372957	+	0.927849i	$0.378344\pi$
$500$	0	0
$501$	13.9659	0.623950
$502$	0	0
$503$	16.3217	0.727750	0.363875	−	0.931448i	$-0.381454\pi$
$503$	16.3217	0.727750	0.363875	+	0.931448i	$0.381454\pi$
$504$	0	0
$505$	−7.21612	−0.321113
$506$	0	0
$507$	17.0464	0.757056
$508$	0	0
$509$	8.53438	0.378280	0.189140	−	0.981950i	$-0.439430\pi$
$509$	8.53438	0.378280	0.189140	+	0.981950i	$0.439430\pi$
$510$	0	0
$511$	−0.146587	−0.00648461
$512$	0	0
$513$	−5.60228	−0.247347
$514$	0	0
$515$	0.0761273	0.00335457
$516$	0	0
$517$	−1.33177	−0.0585710
$518$	0	0
$519$	−19.6503	−0.862553
$520$	0	0
$521$	2.03444	0.0891304	0.0445652	−	0.999006i	$-0.485810\pi$
$521$	2.03444	0.0891304	0.0445652	+	0.999006i	$0.485810\pi$
$522$	0	0
$523$	39.9353	1.74625	0.873124	−	0.487498i	$-0.162090\pi$
$523$	39.9353	1.74625	0.873124	+	0.487498i	$0.162090\pi$
$524$	0	0
$525$	0.502345	0.0219242
$526$	0	0
$527$	2.20731	0.0961518
$528$	0	0
$529$	53.9266	2.34464
$530$	0	0
$531$	0.104630	0.00454057
$532$	0	0
$533$	4.12969	0.178877
$534$	0	0
$535$	8.62610	0.372939
$536$	0	0
$537$	−32.1523	−1.38747
$538$	0	0
$539$	−6.05864	−0.260964
$540$	0	0
$541$	−7.51737	−0.323197	−0.161598	−	0.986857i	$-0.551665\pi$
$541$	−7.51737	−0.323197	−0.161598	+	0.986857i	$0.551665\pi$
$542$	0	0
$543$	4.25493	0.182597
$544$	0	0
$545$	−10.3652	−0.443998
$546$	0	0
$547$	10.2289	0.437357	0.218679	−	0.975797i	$-0.429825\pi$
$547$	10.2289	0.437357	0.218679	+	0.975797i	$0.429825\pi$
$548$	0	0
$549$	−6.67925	−0.285063
$550$	0	0
$551$	2.54997	0.108632
$552$	0	0
$553$	4.75356	0.202142
$554$	0	0
$555$	−14.2684	−0.605661
$556$	0	0
$557$	−16.7178	−0.708356	−0.354178	−	0.935178i	$-0.615239\pi$
$557$	−16.7178	−0.708356	−0.354178	+	0.935178i	$0.615239\pi$
$558$	0	0
$559$	5.71063	0.241534
$560$	0	0
$561$	−7.51651	−0.317347
$562$	0	0
$563$	−30.6926	−1.29354	−0.646769	−	0.762686i	$-0.723880\pi$
$563$	−30.6926	−1.29354	−0.646769	+	0.762686i	$0.723880\pi$
$564$	0	0
$565$	8.60888	0.362178
$566$	0	0
$567$	2.15184	0.0903690
$568$	0	0
$569$	−4.30451	−0.180454	−0.0902272	−	0.995921i	$-0.528759\pi$
$569$	−4.30451	−0.180454	−0.0902272	+	0.995921i	$0.528759\pi$
$570$	0	0
$571$	32.2930	1.35142	0.675709	−	0.737168i	$-0.263838\pi$
$571$	32.2930	1.35142	0.675709	+	0.737168i	$0.263838\pi$
$572$	0	0
$573$	−0.0670140	−0.00279955
$574$	0	0
$575$	−8.77078	−0.365767
$576$	0	0
$577$	9.70442	0.404000	0.202000	−	0.979385i	$-0.435256\pi$
$577$	9.70442	0.404000	0.202000	+	0.979385i	$0.435256\pi$
$578$	0	0
$579$	25.0658	1.04170
$580$	0	0
$581$	−0.0726820	−0.00301536
$582$	0	0
$583$	0.588661	0.0243798
$584$	0	0
$585$	0.907630	0.0375259
$586$	0	0
$587$	21.6096	0.891923	0.445962	−	0.895052i	$-0.352862\pi$
$587$	21.6096	0.891923	0.445962	+	0.895052i	$0.352862\pi$
$588$	0	0
$589$	−0.394001	−0.0162345
$590$	0	0
$591$	−33.0011	−1.35748
$592$	0	0
$593$	29.1971	1.19898	0.599491	−	0.800381i	$-0.295370\pi$
$593$	29.1971	1.19898	0.599491	+	0.800381i	$0.295370\pi$
$594$	0	0
$595$	−1.84403	−0.0755979
$596$	0	0
$597$	6.71161	0.274688
$598$	0	0
$599$	−11.0925	−0.453228	−0.226614	−	0.973985i	$-0.572766\pi$
$599$	−11.0925	−0.453228	−0.226614	+	0.973985i	$0.572766\pi$
$600$	0	0
$601$	28.2819	1.15364	0.576822	−	0.816870i	$-0.304293\pi$
$601$	28.2819	1.15364	0.576822	+	0.816870i	$0.304293\pi$
$602$	0	0
$603$	7.58688	0.308962
$604$	0	0
$605$	−10.2271	−0.415792
$606$	0	0
$607$	2.76693	0.112306	0.0561531	−	0.998422i	$-0.482117\pi$
$607$	2.76693	0.112306	0.0561531	+	0.998422i	$0.482117\pi$
$608$	0	0
$609$	−1.28097	−0.0519074
$610$	0	0
$611$	2.04958	0.0829171
$612$	0	0
$613$	27.6520	1.11685	0.558426	−	0.829554i	$-0.311406\pi$
$613$	27.6520	1.11685	0.558426	+	0.829554i	$0.311406\pi$
$614$	0	0
$615$	4.65831	0.187841
$616$	0	0
$617$	4.35213	0.175210	0.0876051	−	0.996155i	$-0.472079\pi$
$617$	4.35213	0.175210	0.0876051	+	0.996155i	$0.472079\pi$
$618$	0	0
$619$	−21.4881	−0.863681	−0.431841	−	0.901950i	$-0.642136\pi$
$619$	−21.4881	−0.863681	−0.431841	+	0.901950i	$0.642136\pi$
$620$	0	0
$621$	−49.1364	−1.97178
$622$	0	0
$623$	−1.33230	−0.0533776
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.34169	0.0535818
$628$	0	0
$629$	52.3772	2.08842
$630$	0	0
$631$	34.6908	1.38102	0.690509	−	0.723324i	$-0.257387\pi$
$631$	34.6908	1.38102	0.690509	+	0.723324i	$0.257387\pi$
$632$	0	0
$633$	7.94866	0.315931
$634$	0	0
$635$	−8.47013	−0.336127
$636$	0	0
$637$	9.32420	0.369438
$638$	0	0
$639$	2.20980	0.0874183
$640$	0	0
$641$	−18.0058	−0.711185	−0.355592	−	0.934641i	$-0.615721\pi$
$641$	−18.0058	−0.711185	−0.355592	+	0.934641i	$0.615721\pi$
$642$	0	0
$643$	24.7425	0.975751	0.487875	−	0.872913i	$-0.337772\pi$
$643$	24.7425	0.975751	0.487875	+	0.872913i	$0.337772\pi$
$644$	0	0
$645$	6.44163	0.253639
$646$	0	0
$647$	38.7157	1.52207	0.761036	−	0.648709i	$-0.224691\pi$
$647$	38.7157	1.52207	0.761036	+	0.648709i	$0.224691\pi$
$648$	0	0
$649$	−0.137116	−0.00538228
$650$	0	0
$651$	0.197925	0.00775728
$652$	0	0
$653$	16.6641	0.652115	0.326058	−	0.945350i	$-0.394280\pi$
$653$	16.6641	0.652115	0.326058	+	0.945350i	$0.394280\pi$
$654$	0	0
$655$	−12.7310	−0.497441
$656$	0	0
$657$	−0.298753	−0.0116555
$658$	0	0
$659$	21.9676	0.855736	0.427868	−	0.903841i	$-0.359265\pi$
$659$	21.9676	0.855736	0.427868	+	0.903841i	$0.359265\pi$
$660$	0	0
$661$	27.3630	1.06430	0.532149	−	0.846651i	$-0.321385\pi$
$661$	27.3630	1.06430	0.532149	+	0.846651i	$0.321385\pi$
$662$	0	0
$663$	11.5678	0.449258
$664$	0	0
$665$	0.329157	0.0127642
$666$	0	0
$667$	22.3652	0.865985
$668$	0	0
$669$	38.9895	1.50742
$670$	0	0
$671$	8.75303	0.337907
$672$	0	0
$673$	14.5613	0.561295	0.280648	−	0.959811i	$-0.409451\pi$
$673$	14.5613	0.561295	0.280648	+	0.959811i	$0.409451\pi$
$674$	0	0
$675$	5.60228	0.215632
$676$	0	0
$677$	−38.7024	−1.48745	−0.743726	−	0.668484i	$-0.766943\pi$
$677$	−38.7024	−1.48745	−0.743726	+	0.668484i	$0.766943\pi$
$678$	0	0
$679$	−1.71063	−0.0656479
$680$	0	0
$681$	26.1350	1.00150
$682$	0	0
$683$	13.2688	0.507717	0.253859	−	0.967241i	$-0.418300\pi$
$683$	13.2688	0.507717	0.253859	+	0.967241i	$0.418300\pi$
$684$	0	0
$685$	−19.1438	−0.731449
$686$	0	0
$687$	12.1916	0.465140
$688$	0	0
$689$	−0.905944	−0.0345137
$690$	0	0
$691$	−33.3585	−1.26902	−0.634508	−	0.772916i	$-0.718797\pi$
$691$	−33.3585	−1.26902	−0.634508	+	0.772916i	$0.718797\pi$
$692$	0	0
$693$	0.194122	0.00737410
$694$	0	0
$695$	5.80425	0.220168
$696$	0	0
$697$	−17.0999	−0.647706
$698$	0	0
$699$	38.9603	1.47362
$700$	0	0
$701$	7.13601	0.269523	0.134762	−	0.990878i	$-0.456973\pi$
$701$	7.13601	0.269523	0.134762	+	0.990878i	$0.456973\pi$
$702$	0	0
$703$	−9.34926	−0.352614
$704$	0	0
$705$	2.31194	0.0870726
$706$	0	0
$707$	2.37524	0.0893300
$708$	0	0
$709$	−38.4986	−1.44585	−0.722923	−	0.690928i	$-0.757202\pi$
$709$	−38.4986	−1.44585	−0.722923	+	0.690928i	$0.757202\pi$
$710$	0	0
$711$	9.68806	0.363331
$712$	0	0
$713$	−3.45570	−0.129417
$714$	0	0
$715$	−1.18943	−0.0444822
$716$	0	0
$717$	23.8460	0.890545
$718$	0	0
$719$	33.8784	1.26345	0.631726	−	0.775192i	$-0.282347\pi$
$719$	33.8784	1.26345	0.631726	+	0.775192i	$0.282347\pi$
$720$	0	0
$721$	−0.0250578	−0.000933203 0
$722$	0	0
$723$	3.33426	0.124002
$724$	0	0
$725$	−2.54997	−0.0947035
$726$	0	0
$727$	35.1805	1.30477	0.652386	−	0.757887i	$-0.273768\pi$
$727$	35.1805	1.30477	0.652386	+	0.757887i	$0.273768\pi$
$728$	0	0
$729$	30.0355	1.11243
$730$	0	0
$731$	−23.6462	−0.874586
$732$	0	0
$733$	−4.33646	−0.160171	−0.0800854	−	0.996788i	$-0.525519\pi$
$733$	−4.33646	−0.160171	−0.0800854	+	0.996788i	$0.525519\pi$
$734$	0	0
$735$	10.5178	0.387953
$736$	0	0
$737$	−9.94246	−0.366235
$738$	0	0
$739$	6.80481	0.250319	0.125160	−	0.992137i	$-0.460056\pi$
$739$	6.80481	0.250319	0.125160	+	0.992137i	$0.460056\pi$
$740$	0	0
$741$	−2.06484	−0.0758539
$742$	0	0
$743$	49.0409	1.79914	0.899568	−	0.436780i	$-0.143881\pi$
$743$	49.0409	1.79914	0.899568	+	0.436780i	$0.143881\pi$
$744$	0	0
$745$	−17.8194	−0.652852
$746$	0	0
$747$	−0.148131	−0.00541982
$748$	0	0
$749$	−2.83934	−0.103747
$750$	0	0
$751$	−49.7208	−1.81434	−0.907169	−	0.420765i	$-0.861762\pi$
$751$	−49.7208	−1.81434	−0.907169	+	0.420765i	$0.861762\pi$
$752$	0	0
$753$	22.1999	0.809009
$754$	0	0
$755$	−14.7059	−0.535204
$756$	0	0
$757$	−29.0950	−1.05748	−0.528738	−	0.848785i	$-0.677335\pi$
$757$	−29.0950	−1.05748	−0.528738	+	0.848785i	$0.677335\pi$
$758$	0	0
$759$	11.7676	0.427138
$760$	0	0
$761$	−11.2444	−0.407608	−0.203804	−	0.979012i	$-0.565330\pi$
$761$	−11.2444	−0.407608	−0.203804	+	0.979012i	$0.565330\pi$
$762$	0	0
$763$	3.41179	0.123515
$764$	0	0
$765$	−3.75825	−0.135880
$766$	0	0
$767$	0.211021	0.00761951
$768$	0	0
$769$	−15.2518	−0.549993	−0.274997	−	0.961445i	$-0.588677\pi$
$769$	−15.2518	−0.549993	−0.274997	+	0.961445i	$0.588677\pi$
$770$	0	0
$771$	−16.2778	−0.586231
$772$	0	0
$773$	−53.9777	−1.94144	−0.970721	−	0.240211i	$-0.922784\pi$
$773$	−53.9777	−1.94144	−0.970721	+	0.240211i	$0.922784\pi$
$774$	0	0
$775$	0.394001	0.0141529
$776$	0	0
$777$	4.69656	0.168488
$778$	0	0
$779$	3.05232	0.109361
$780$	0	0
$781$	−2.89590	−0.103623
$782$	0	0
$783$	−14.2857	−0.510528
$784$	0	0
$785$	−10.4416	−0.372678
$786$	0	0
$787$	6.10436	0.217597	0.108798	−	0.994064i	$-0.465300\pi$
$787$	6.10436	0.217597	0.108798	+	0.994064i	$0.465300\pi$
$788$	0	0
$789$	1.42869	0.0508626
$790$	0	0
$791$	−2.83367	−0.100754
$792$	0	0
$793$	−13.4708	−0.478363
$794$	0	0
$795$	−1.02191	−0.0362434
$796$	0	0
$797$	−28.4179	−1.00661	−0.503307	−	0.864107i	$-0.667884\pi$
$797$	−28.4179	−1.00661	−0.503307	+	0.864107i	$0.667884\pi$
$798$	0	0
$799$	−8.48676	−0.300240
$800$	0	0
$801$	−2.71532	−0.0959411
$802$	0	0
$803$	0.391510	0.0138161
$804$	0	0
$805$	2.88696	0.101752
$806$	0	0
$807$	29.6286	1.04297
$808$	0	0
$809$	−6.94148	−0.244049	−0.122025	−	0.992527i	$-0.538939\pi$
$809$	−6.94148	−0.244049	−0.122025	+	0.992527i	$0.538939\pi$
$810$	0	0
$811$	−38.0172	−1.33496	−0.667482	−	0.744626i	$-0.732628\pi$
$811$	−38.0172	−1.33496	−0.667482	+	0.744626i	$0.732628\pi$
$812$	0	0
$813$	18.8507	0.661122
$814$	0	0
$815$	18.4207	0.645249
$816$	0	0
$817$	4.22081	0.147668
$818$	0	0
$819$	−0.298753	−0.0104393
$820$	0	0
$821$	5.92732	0.206865	0.103432	−	0.994636i	$-0.467017\pi$
$821$	5.92732	0.206865	0.103432	+	0.994636i	$0.467017\pi$
$822$	0	0
$823$	7.85255	0.273723	0.136861	−	0.990590i	$-0.456298\pi$
$823$	7.85255	0.273723	0.136861	+	0.990590i	$0.456298\pi$
$824$	0	0
$825$	−1.34169	−0.0467115
$826$	0	0
$827$	14.6762	0.510342	0.255171	−	0.966896i	$-0.417868\pi$
$827$	14.6762	0.510342	0.255171	+	0.966896i	$0.417868\pi$
$828$	0	0
$829$	20.2031	0.701681	0.350841	−	0.936435i	$-0.385896\pi$
$829$	20.2031	0.701681	0.350841	+	0.936435i	$0.385896\pi$
$830$	0	0
$831$	17.6142	0.611031
$832$	0	0
$833$	−38.6090	−1.33772
$834$	0	0
$835$	−9.15101	−0.316684
$836$	0	0
$837$	2.20731	0.0762957
$838$	0	0
$839$	8.95018	0.308994	0.154497	−	0.987993i	$-0.450624\pi$
$839$	8.95018	0.308994	0.154497	+	0.987993i	$0.450624\pi$
$840$	0	0
$841$	−22.4977	−0.775781
$842$	0	0
$843$	12.8488	0.442537
$844$	0	0
$845$	−11.1695	−0.384242
$846$	0	0
$847$	3.36633	0.115669
$848$	0	0
$849$	−46.9554	−1.61150
$850$	0	0
$851$	−82.0003	−2.81093
$852$	0	0
$853$	−34.7010	−1.18814	−0.594070	−	0.804413i	$-0.702480\pi$
$853$	−34.7010	−1.18814	−0.594070	+	0.804413i	$0.702480\pi$
$854$	0	0
$855$	0.670843	0.0229423
$856$	0	0
$857$	−2.59320	−0.0885821	−0.0442910	−	0.999019i	$-0.514103\pi$
$857$	−2.59320	−0.0885821	−0.0442910	+	0.999019i	$0.514103\pi$
$858$	0	0
$859$	5.44685	0.185844	0.0929221	−	0.995673i	$-0.470379\pi$
$859$	5.44685	0.185844	0.0929221	+	0.995673i	$0.470379\pi$
$860$	0	0
$861$	−1.53332	−0.0522553
$862$	0	0
$863$	−21.7523	−0.740457	−0.370229	−	0.928941i	$-0.620721\pi$
$863$	−21.7523	−0.740457	−0.370229	+	0.928941i	$0.620721\pi$
$864$	0	0
$865$	12.8757	0.437786
$866$	0	0
$867$	−21.9547	−0.745620
$868$	0	0
$869$	−12.6960	−0.430683
$870$	0	0
$871$	15.3014	0.518467
$872$	0	0
$873$	−3.48637	−0.117996
$874$	0	0
$875$	−0.329157	−0.0111275
$876$	0	0
$877$	33.2943	1.12427	0.562134	−	0.827046i	$-0.309980\pi$
$877$	33.2943	1.12427	0.562134	+	0.827046i	$0.309980\pi$
$878$	0	0
$879$	−4.11247	−0.138710
$880$	0	0
$881$	42.2767	1.42434	0.712169	−	0.702008i	$-0.247713\pi$
$881$	42.2767	1.42434	0.712169	+	0.702008i	$0.247713\pi$
$882$	0	0
$883$	19.8362	0.667541	0.333771	−	0.942654i	$-0.391679\pi$
$883$	19.8362	0.667541	0.333771	+	0.942654i	$0.391679\pi$
$884$	0	0
$885$	0.238033	0.00800138
$886$	0	0
$887$	−35.2314	−1.18295	−0.591477	−	0.806322i	$-0.701455\pi$
$887$	−35.2314	−1.18295	−0.591477	+	0.806322i	$0.701455\pi$
$888$	0	0
$889$	2.78800	0.0935066
$890$	0	0
$891$	−5.74724	−0.192540
$892$	0	0
$893$	1.51487	0.0506933
$894$	0	0
$895$	21.0675	0.704207
$896$	0	0
$897$	−18.1103	−0.604685
$898$	0	0
$899$	−1.00469	−0.0335083
$900$	0	0
$901$	3.75127	0.124973
$902$	0	0
$903$	−2.12031	−0.0705594
$904$	0	0
$905$	−2.78800	−0.0926763
$906$	0	0
$907$	−9.71416	−0.322553	−0.161277	−	0.986909i	$-0.551561\pi$
$907$	−9.71416	−0.322553	−0.161277	+	0.986909i	$0.551561\pi$
$908$	0	0
$909$	4.84088	0.160562
$910$	0	0
$911$	−22.1616	−0.734248	−0.367124	−	0.930172i	$-0.619657\pi$
$911$	−22.1616	−0.734248	−0.367124	+	0.930172i	$0.619657\pi$
$912$	0	0
$913$	0.194122	0.00642451
$914$	0	0
$915$	−15.1952	−0.502337
$916$	0	0
$917$	4.19050	0.138382
$918$	0	0
$919$	−43.0888	−1.42137	−0.710684	−	0.703511i	$-0.751614\pi$
$919$	−43.0888	−1.42137	−0.710684	+	0.703511i	$0.751614\pi$
$920$	0	0
$921$	−45.4862	−1.49882
$922$	0	0
$923$	4.45676	0.146696
$924$	0	0
$925$	9.34926	0.307402
$926$	0	0
$927$	−0.0510695	−0.00167734
$928$	0	0
$929$	4.86440	0.159596	0.0797979	−	0.996811i	$-0.474573\pi$
$929$	4.86440	0.159596	0.0797979	+	0.996811i	$0.474573\pi$
$930$	0	0
$931$	6.89166	0.225865
$932$	0	0
$933$	37.4151	1.22491
$934$	0	0
$935$	4.92512	0.161069
$936$	0	0
$937$	−51.0062	−1.66630	−0.833150	−	0.553047i	$-0.813465\pi$
$937$	−51.0062	−1.66630	−0.833150	+	0.553047i	$0.813465\pi$
$938$	0	0
$939$	35.2102	1.14904
$940$	0	0
$941$	20.3390	0.663034	0.331517	−	0.943449i	$-0.392440\pi$
$941$	20.3390	0.663034	0.331517	+	0.943449i	$0.392440\pi$
$942$	0	0
$943$	26.7712	0.871790
$944$	0	0
$945$	−1.84403	−0.0599863
$946$	0	0
$947$	−59.0721	−1.91959	−0.959793	−	0.280709i	$-0.909430\pi$
$947$	−59.0721	−1.91959	−0.959793	+	0.280709i	$0.909430\pi$
$948$	0	0
$949$	−0.602530	−0.0195590
$950$	0	0
$951$	3.06954	0.0995365
$952$	0	0
$953$	−19.3592	−0.627105	−0.313553	−	0.949571i	$-0.601519\pi$
$953$	−19.3592	−0.627105	−0.313553	+	0.949571i	$0.601519\pi$
$954$	0	0
$955$	0.0439103	0.00142090
$956$	0	0
$957$	3.42126	0.110594
$958$	0	0
$959$	6.30133	0.203481
$960$	0	0
$961$	−30.8448	−0.994992
$962$	0	0
$963$	−5.78676	−0.186476
$964$	0	0
$965$	−16.4241	−0.528712
$966$	0	0
$967$	8.73875	0.281019	0.140510	−	0.990079i	$-0.455126\pi$
$967$	8.73875	0.281019	0.140510	+	0.990079i	$0.455126\pi$
$968$	0	0
$969$	8.54997	0.274665
$970$	0	0
$971$	−14.3238	−0.459673	−0.229836	−	0.973229i	$-0.573819\pi$
$971$	−14.3238	−0.459673	−0.229836	+	0.973229i	$0.573819\pi$
$972$	0	0
$973$	−1.91051	−0.0612481
$974$	0	0
$975$	2.06484	0.0661279
$976$	0	0
$977$	22.0252	0.704649	0.352324	−	0.935878i	$-0.385391\pi$
$977$	22.0252	0.704649	0.352324	+	0.935878i	$0.385391\pi$
$978$	0	0
$979$	3.55837	0.113726
$980$	0	0
$981$	6.95344	0.222006
$982$	0	0
$983$	6.31841	0.201526	0.100763	−	0.994910i	$-0.467872\pi$
$983$	6.31841	0.201526	0.100763	+	0.994910i	$0.467872\pi$
$984$	0	0
$985$	21.6236	0.688986
$986$	0	0
$987$	−0.760990	−0.0242226
$988$	0	0
$989$	37.0198	1.17716
$990$	0	0
$991$	17.3893	0.552390	0.276195	−	0.961102i	$-0.410926\pi$
$991$	17.3893	0.552390	0.276195	+	0.961102i	$0.410926\pi$
$992$	0	0
$993$	−19.0654	−0.605023
$994$	0	0
$995$	−4.39772	−0.139417
$996$	0	0
$997$	−35.1376	−1.11282	−0.556410	−	0.830908i	$-0.687822\pi$
$997$	−35.1376	−1.11282	−0.556410	+	0.830908i	$0.687822\pi$
$998$	0	0
$999$	52.3772	1.65714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.r.1.2	✓	4
4.3	odd	2		3040.2.a.t.1.3	yes	4
8.3	odd	2		6080.2.a.cd.1.2		4
8.5	even	2		6080.2.a.cf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.r.1.2	✓	4	1.1	even	1	trivial
3040.2.a.t.1.3	yes	4	4.3	odd	2
6080.2.a.cd.1.2		4	8.3	odd	2
6080.2.a.cf.1.3		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52616 q^{3} +1.00000 q^{5} -0.329157 q^{7} -0.670843 q^{9} +0.879127 q^{11} -1.35297 q^{13} -1.52616 q^{15} +5.60228 q^{17} -1.00000 q^{19} +0.502345 q^{21} -8.77078 q^{23} +1.00000 q^{25} +5.60228 q^{27} -2.54997 q^{29} +0.394001 q^{31} -1.34169 q^{33} -0.329157 q^{35} +9.34926 q^{37} +2.06484 q^{39} -3.05232 q^{41} -4.22081 q^{43} -0.670843 q^{45} -1.51487 q^{47} -6.89166 q^{49} -8.54997 q^{51} +0.669597 q^{53} +0.879127 q^{55} +1.52616 q^{57} -0.155969 q^{59} +9.95650 q^{61} +0.220813 q^{63} -1.35297 q^{65} -11.3095 q^{67} +13.3856 q^{69} -3.29406 q^{71} +0.445339 q^{73} -1.52616 q^{75} -0.289371 q^{77} -14.4416 q^{79} -6.53744 q^{81} +0.220813 q^{83} +5.60228 q^{85} +3.89166 q^{87} +4.04762 q^{89} +0.445339 q^{91} -0.601308 q^{93} -1.00000 q^{95} +5.19700 q^{97} -0.589756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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