LMFDB - Embedded newform 7440.2.a.bn.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,2,0,3,0,-2,0,2,0,3,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.74483 q^{7} +1.00000 q^{9} +5.48965 q^{11} -1.32331 q^{13} +1.00000 q^{15} +6.95558 q^{17} -5.06814 q^{19} -3.74483 q^{21} -2.11256 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.63227 q^{29} -1.00000 q^{31} -5.48965 q^{33} -3.74483 q^{35} +8.39145 q^{37} +1.32331 q^{39} +11.4897 q^{41} +0.421512 q^{43} -1.00000 q^{45} -1.46593 q^{47} +7.02372 q^{49} -6.95558 q^{51} +4.53407 q^{53} -5.48965 q^{55} +5.06814 q^{57} -4.78924 q^{59} +6.84302 q^{61} +3.74483 q^{63} +1.32331 q^{65} +4.39145 q^{67} +2.11256 q^{69} +7.34704 q^{71} -4.39145 q^{73} -1.00000 q^{75} +20.5578 q^{77} +14.0237 q^{79} +1.00000 q^{81} -5.04442 q^{83} -6.95558 q^{85} -3.63227 q^{87} -13.3470 q^{89} -4.95558 q^{91} +1.00000 q^{93} +5.06814 q^{95} -3.48965 q^{97} +5.48965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.74483	1.41541	0.707706	−	0.706507i	$-0.249730\pi$
$7$	3.74483	1.41541	0.707706	+	0.706507i	$0.249730\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.48965	1.65519	0.827596	−	0.561324i	$-0.189708\pi$
$11$	5.48965	1.65519	0.827596	+	0.561324i	$0.189708\pi$
$12$	0	0
$13$	−1.32331	−0.367021	−0.183511	−	0.983018i	$-0.558746\pi$
$13$	−1.32331	−0.367021	−0.183511	+	0.983018i	$0.558746\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	6.95558	1.68698	0.843488	−	0.537148i	$-0.180498\pi$
$17$	6.95558	1.68698	0.843488	+	0.537148i	$0.180498\pi$
$18$	0	0
$19$	−5.06814	−1.16271	−0.581356	−	0.813650i	$-0.697477\pi$
$19$	−5.06814	−1.16271	−0.581356	+	0.813650i	$0.697477\pi$
$20$	0	0
$21$	−3.74483	−0.817188
$22$	0	0
$23$	−2.11256	−0.440499	−0.220249	−	0.975444i	$-0.570687\pi$
$23$	−2.11256	−0.440499	−0.220249	+	0.975444i	$0.570687\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.63227	0.674495	0.337248	−	0.941416i	$-0.390504\pi$
$29$	3.63227	0.674495	0.337248	+	0.941416i	$0.390504\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−5.48965	−0.955626
$34$	0	0
$35$	−3.74483	−0.632991
$36$	0	0
$37$	8.39145	1.37955	0.689773	−	0.724025i	$-0.257710\pi$
$37$	8.39145	1.37955	0.689773	+	0.724025i	$0.257710\pi$
$38$	0	0
$39$	1.32331	0.211900
$40$	0	0
$41$	11.4897	1.79438	0.897191	−	0.441643i	$-0.145604\pi$
$41$	11.4897	1.79438	0.897191	+	0.441643i	$0.145604\pi$
$42$	0	0
$43$	0.421512	0.0642799	0.0321400	−	0.999483i	$-0.489768\pi$
$43$	0.421512	0.0642799	0.0321400	+	0.999483i	$0.489768\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.46593	−0.213828	−0.106914	−	0.994268i	$-0.534097\pi$
$47$	−1.46593	−0.213828	−0.106914	+	0.994268i	$0.534097\pi$
$48$	0	0
$49$	7.02372	1.00339
$50$	0	0
$51$	−6.95558	−0.973976
$52$	0	0
$53$	4.53407	0.622802	0.311401	−	0.950279i	$-0.399202\pi$
$53$	4.53407	0.622802	0.311401	+	0.950279i	$0.399202\pi$
$54$	0	0
$55$	−5.48965	−0.740225
$56$	0	0
$57$	5.06814	0.671292
$58$	0	0
$59$	−4.78924	−0.623506	−0.311753	−	0.950163i	$-0.600916\pi$
$59$	−4.78924	−0.623506	−0.311753	+	0.950163i	$0.600916\pi$
$60$	0	0
$61$	6.84302	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$61$	6.84302	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$62$	0	0
$63$	3.74483	0.471804
$64$	0	0
$65$	1.32331	0.164137
$66$	0	0
$67$	4.39145	0.536502	0.268251	−	0.963349i	$-0.413554\pi$
$67$	4.39145	0.536502	0.268251	+	0.963349i	$0.413554\pi$
$68$	0	0
$69$	2.11256	0.254322
$70$	0	0
$71$	7.34704	0.871933	0.435966	−	0.899963i	$-0.356407\pi$
$71$	7.34704	0.871933	0.435966	+	0.899963i	$0.356407\pi$
$72$	0	0
$73$	−4.39145	−0.513981	−0.256990	−	0.966414i	$-0.582731\pi$
$73$	−4.39145	−0.513981	−0.256990	+	0.966414i	$0.582731\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	20.5578	2.34278
$78$	0	0
$79$	14.0237	1.57779	0.788896	−	0.614527i	$-0.210653\pi$
$79$	14.0237	1.57779	0.788896	+	0.614527i	$0.210653\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.04442	−0.553697	−0.276848	−	0.960914i	$-0.589290\pi$
$83$	−5.04442	−0.553697	−0.276848	+	0.960914i	$0.589290\pi$
$84$	0	0
$85$	−6.95558	−0.754439
$86$	0	0
$87$	−3.63227	−0.389420
$88$	0	0
$89$	−13.3470	−1.41478	−0.707392	−	0.706822i	$-0.750128\pi$
$89$	−13.3470	−1.41478	−0.707392	+	0.706822i	$0.750128\pi$
$90$	0	0
$91$	−4.95558	−0.519486
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	5.06814	0.519980
$96$	0	0
$97$	−3.48965	−0.354320	−0.177160	−	0.984182i	$-0.556691\pi$
$97$	−3.48965	−0.354320	−0.177160	+	0.984182i	$0.556691\pi$
$98$	0	0
$99$	5.48965	0.551731
$100$	0	0
$101$	−13.4008	−1.33343	−0.666716	−	0.745312i	$-0.732300\pi$
$101$	−13.4008	−1.33343	−0.666716	+	0.745312i	$0.732300\pi$
$102$	0	0
$103$	−15.1456	−1.49234	−0.746172	−	0.665753i	$-0.768110\pi$
$103$	−15.1456	−1.49234	−0.746172	+	0.665753i	$0.768110\pi$
$104$	0	0
$105$	3.74483	0.365458
$106$	0	0
$107$	−12.9556	−1.25246	−0.626232	−	0.779637i	$-0.715404\pi$
$107$	−12.9556	−1.25246	−0.626232	+	0.779637i	$0.715404\pi$
$108$	0	0
$109$	−13.3771	−1.28129	−0.640647	−	0.767836i	$-0.721334\pi$
$109$	−13.3771	−1.28129	−0.640647	+	0.767836i	$0.721334\pi$
$110$	0	0
$111$	−8.39145	−0.796482
$112$	0	0
$113$	−1.26454	−0.118957	−0.0594787	−	0.998230i	$-0.518944\pi$
$113$	−1.26454	−0.118957	−0.0594787	+	0.998230i	$0.518944\pi$
$114$	0	0
$115$	2.11256	0.196997
$116$	0	0
$117$	−1.32331	−0.122340
$118$	0	0
$119$	26.0474	2.38777
$120$	0	0
$121$	19.1363	1.73966
$122$	0	0
$123$	−11.4897	−1.03599
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.3327	1.80423	0.902117	−	0.431492i	$-0.142013\pi$
$127$	20.3327	1.80423	0.902117	+	0.431492i	$0.142013\pi$
$128$	0	0
$129$	−0.421512	−0.0371120
$130$	0	0
$131$	−19.9937	−1.74685	−0.873427	−	0.486955i	$-0.838108\pi$
$131$	−19.9937	−1.74685	−0.873427	+	0.486955i	$0.838108\pi$
$132$	0	0
$133$	−18.9793	−1.64571
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	11.7986	1.00802	0.504011	−	0.863697i	$-0.331857\pi$
$137$	11.7986	1.00802	0.504011	+	0.863697i	$0.331857\pi$
$138$	0	0
$139$	−2.64663	−0.224484	−0.112242	−	0.993681i	$-0.535803\pi$
$139$	−2.64663	−0.224484	−0.112242	+	0.993681i	$0.535803\pi$
$140$	0	0
$141$	1.46593	0.123454
$142$	0	0
$143$	−7.26454	−0.607491
$144$	0	0
$145$	−3.63227	−0.301643
$146$	0	0
$147$	−7.02372	−0.579307
$148$	0	0
$149$	−0.510348	−0.0418093	−0.0209047	−	0.999781i	$-0.506655\pi$
$149$	−0.510348	−0.0418093	−0.0209047	+	0.999781i	$0.506655\pi$
$150$	0	0
$151$	1.24081	0.100976	0.0504880	−	0.998725i	$-0.483922\pi$
$151$	1.24081	0.100976	0.0504880	+	0.998725i	$0.483922\pi$
$152$	0	0
$153$	6.95558	0.562325
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−7.71477	−0.615706	−0.307853	−	0.951434i	$-0.599610\pi$
$157$	−7.71477	−0.615706	−0.307853	+	0.951434i	$0.599610\pi$
$158$	0	0
$159$	−4.53407	−0.359575
$160$	0	0
$161$	−7.91116	−0.623487
$162$	0	0
$163$	1.18703	0.0929756	0.0464878	−	0.998919i	$-0.485197\pi$
$163$	1.18703	0.0929756	0.0464878	+	0.998919i	$0.485197\pi$
$164$	0	0
$165$	5.48965	0.427369
$166$	0	0
$167$	−8.55779	−0.662222	−0.331111	−	0.943592i	$-0.607423\pi$
$167$	−8.55779	−0.662222	−0.331111	+	0.943592i	$0.607423\pi$
$168$	0	0
$169$	−11.2488	−0.865295
$170$	0	0
$171$	−5.06814	−0.387570
$172$	0	0
$173$	2.55779	0.194465	0.0972327	−	0.995262i	$-0.469001\pi$
$173$	2.55779	0.194465	0.0972327	+	0.995262i	$0.469001\pi$
$174$	0	0
$175$	3.74483	0.283082
$176$	0	0
$177$	4.78924	0.359982
$178$	0	0
$179$	−24.8905	−1.86040	−0.930200	−	0.367052i	$-0.880367\pi$
$179$	−24.8905	−1.86040	−0.930200	+	0.367052i	$0.880367\pi$
$180$	0	0
$181$	17.8223	1.32472	0.662362	−	0.749184i	$-0.269554\pi$
$181$	17.8223	1.32472	0.662362	+	0.749184i	$0.269554\pi$
$182$	0	0
$183$	−6.84302	−0.505851
$184$	0	0
$185$	−8.39145	−0.616952
$186$	0	0
$187$	38.1837	2.79227
$188$	0	0
$189$	−3.74483	−0.272396
$190$	0	0
$191$	10.9255	0.790543	0.395272	−	0.918564i	$-0.370650\pi$
$191$	10.9255	0.790543	0.395272	+	0.918564i	$0.370650\pi$
$192$	0	0
$193$	15.2645	1.09877	0.549383	−	0.835571i	$-0.314863\pi$
$193$	15.2645	1.09877	0.549383	+	0.835571i	$0.314863\pi$
$194$	0	0
$195$	−1.32331	−0.0947645
$196$	0	0
$197$	6.33768	0.451541	0.225770	−	0.974181i	$-0.427510\pi$
$197$	6.33768	0.451541	0.225770	+	0.974181i	$0.427510\pi$
$198$	0	0
$199$	3.55477	0.251991	0.125995	−	0.992031i	$-0.459788\pi$
$199$	3.55477	0.251991	0.125995	+	0.992031i	$0.459788\pi$
$200$	0	0
$201$	−4.39145	−0.309749
$202$	0	0
$203$	13.6022	0.954688
$204$	0	0
$205$	−11.4897	−0.802472
$206$	0	0
$207$	−2.11256	−0.146833
$208$	0	0
$209$	−27.8223	−1.92451
$210$	0	0
$211$	12.5578	0.864514	0.432257	−	0.901750i	$-0.357717\pi$
$211$	12.5578	0.864514	0.432257	+	0.901750i	$0.357717\pi$
$212$	0	0
$213$	−7.34704	−0.503411
$214$	0	0
$215$	−0.421512	−0.0287469
$216$	0	0
$217$	−3.74483	−0.254215
$218$	0	0
$219$	4.39145	0.296747
$220$	0	0
$221$	−9.20442	−0.619156
$222$	0	0
$223$	−17.2044	−1.15209	−0.576047	−	0.817417i	$-0.695405\pi$
$223$	−17.2044	−1.15209	−0.576047	+	0.817417i	$0.695405\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−19.0919	−1.26717	−0.633586	−	0.773673i	$-0.718418\pi$
$227$	−19.0919	−1.26717	−0.633586	+	0.773673i	$0.718418\pi$
$228$	0	0
$229$	12.4690	0.823972	0.411986	−	0.911190i	$-0.364835\pi$
$229$	12.4690	0.823972	0.411986	+	0.911190i	$0.364835\pi$
$230$	0	0
$231$	−20.5578	−1.35260
$232$	0	0
$233$	−2.75919	−0.180760	−0.0903802	−	0.995907i	$-0.528808\pi$
$233$	−2.75919	−0.180760	−0.0903802	+	0.995907i	$0.528808\pi$
$234$	0	0
$235$	1.46593	0.0956267
$236$	0	0
$237$	−14.0237	−0.910939
$238$	0	0
$239$	6.61791	0.428077	0.214038	−	0.976825i	$-0.431338\pi$
$239$	6.61791	0.428077	0.214038	+	0.976825i	$0.431338\pi$
$240$	0	0
$241$	−5.26454	−0.339119	−0.169559	−	0.985520i	$-0.554234\pi$
$241$	−5.26454	−0.339119	−0.169559	+	0.985520i	$0.554234\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.02372	−0.448729
$246$	0	0
$247$	6.70674	0.426740
$248$	0	0
$249$	5.04442	0.319677
$250$	0	0
$251$	−20.0474	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$251$	−20.0474	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$252$	0	0
$253$	−11.5972	−0.729110
$254$	0	0
$255$	6.95558	0.435575
$256$	0	0
$257$	25.7385	1.60552	0.802761	−	0.596300i	$-0.203363\pi$
$257$	25.7385	1.60552	0.802761	+	0.596300i	$0.203363\pi$
$258$	0	0
$259$	31.4245	1.95263
$260$	0	0
$261$	3.63227	0.224832
$262$	0	0
$263$	−16.3327	−1.00712	−0.503558	−	0.863961i	$-0.667976\pi$
$263$	−16.3327	−1.00712	−0.503558	+	0.863961i	$0.667976\pi$
$264$	0	0
$265$	−4.53407	−0.278526
$266$	0	0
$267$	13.3470	0.816825
$268$	0	0
$269$	19.4546	1.18617	0.593084	−	0.805141i	$-0.297910\pi$
$269$	19.4546	1.18617	0.593084	+	0.805141i	$0.297910\pi$
$270$	0	0
$271$	28.7829	1.74844	0.874219	−	0.485533i	$-0.161374\pi$
$271$	28.7829	1.74844	0.874219	+	0.485533i	$0.161374\pi$
$272$	0	0
$273$	4.95558	0.299925
$274$	0	0
$275$	5.48965	0.331038
$276$	0	0
$277$	15.0381	0.903551	0.451775	−	0.892132i	$-0.350791\pi$
$277$	15.0381	0.903551	0.451775	+	0.892132i	$0.350791\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	18.7829	1.12049	0.560247	−	0.828325i	$-0.310706\pi$
$281$	18.7829	1.12049	0.560247	+	0.828325i	$0.310706\pi$
$282$	0	0
$283$	23.7923	1.41430	0.707152	−	0.707062i	$-0.249980\pi$
$283$	23.7923	1.41430	0.707152	+	0.707062i	$0.249980\pi$
$284$	0	0
$285$	−5.06814	−0.300211
$286$	0	0
$287$	43.0267	2.53979
$288$	0	0
$289$	31.3801	1.84589
$290$	0	0
$291$	3.48965	0.204567
$292$	0	0
$293$	18.3090	1.06962	0.534810	−	0.844972i	$-0.320383\pi$
$293$	18.3090	1.06962	0.534810	+	0.844972i	$0.320383\pi$
$294$	0	0
$295$	4.78924	0.278841
$296$	0	0
$297$	−5.48965	−0.318542
$298$	0	0
$299$	2.79558	0.161672
$300$	0	0
$301$	1.57849	0.0909825
$302$	0	0
$303$	13.4008	0.769857
$304$	0	0
$305$	−6.84302	−0.391830
$306$	0	0
$307$	−6.72413	−0.383766	−0.191883	−	0.981418i	$-0.561459\pi$
$307$	−6.72413	−0.383766	−0.191883	+	0.981418i	$0.561459\pi$
$308$	0	0
$309$	15.1456	0.861605
$310$	0	0
$311$	−24.8080	−1.40673	−0.703365	−	0.710828i	$-0.748320\pi$
$311$	−24.8080	−1.40673	−0.703365	+	0.710828i	$0.748320\pi$
$312$	0	0
$313$	29.5959	1.67286	0.836429	−	0.548075i	$-0.184639\pi$
$313$	29.5959	1.67286	0.836429	+	0.548075i	$0.184639\pi$
$314$	0	0
$315$	−3.74483	−0.210997
$316$	0	0
$317$	−13.7385	−0.771631	−0.385815	−	0.922576i	$-0.626080\pi$
$317$	−13.7385	−0.771631	−0.385815	+	0.922576i	$0.626080\pi$
$318$	0	0
$319$	19.9399	1.11642
$320$	0	0
$321$	12.9556	0.723110
$322$	0	0
$323$	−35.2519	−1.96147
$324$	0	0
$325$	−1.32331	−0.0734043
$326$	0	0
$327$	13.3771	0.739755
$328$	0	0
$329$	−5.48965	−0.302654
$330$	0	0
$331$	−0.908137	−0.0499157	−0.0249579	−	0.999689i	$-0.507945\pi$
$331$	−0.908137	−0.0499157	−0.0249579	+	0.999689i	$0.507945\pi$
$332$	0	0
$333$	8.39145	0.459849
$334$	0	0
$335$	−4.39145	−0.239931
$336$	0	0
$337$	13.2345	0.720928	0.360464	−	0.932773i	$-0.382618\pi$
$337$	13.2345	0.720928	0.360464	+	0.932773i	$0.382618\pi$
$338$	0	0
$339$	1.26454	0.0686801
$340$	0	0
$341$	−5.48965	−0.297281
$342$	0	0
$343$	0.0888361	0.00479670
$344$	0	0
$345$	−2.11256	−0.113736
$346$	0	0
$347$	−1.12825	−0.0605679	−0.0302839	−	0.999541i	$-0.509641\pi$
$347$	−1.12825	−0.0605679	−0.0302839	+	0.999541i	$0.509641\pi$
$348$	0	0
$349$	6.67035	0.357056	0.178528	−	0.983935i	$-0.442867\pi$
$349$	6.67035	0.357056	0.178528	+	0.983935i	$0.442867\pi$
$350$	0	0
$351$	1.32331	0.0706333
$352$	0	0
$353$	−0.201395	−0.0107191	−0.00535957	−	0.999986i	$-0.501706\pi$
$353$	−0.201395	−0.0107191	−0.00535957	+	0.999986i	$0.501706\pi$
$354$	0	0
$355$	−7.34704	−0.389940
$356$	0	0
$357$	−26.0474	−1.37858
$358$	0	0
$359$	−14.2375	−0.751427	−0.375713	−	0.926736i	$-0.622602\pi$
$359$	−14.2375	−0.751427	−0.375713	+	0.926736i	$0.622602\pi$
$360$	0	0
$361$	6.68605	0.351897
$362$	0	0
$363$	−19.1363	−1.00439
$364$	0	0
$365$	4.39145	0.229859
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	11.4897	0.598127
$370$	0	0
$371$	16.9793	0.881522
$372$	0	0
$373$	24.8717	1.28781	0.643905	−	0.765105i	$-0.277313\pi$
$373$	24.8717	1.28781	0.643905	+	0.765105i	$0.277313\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−4.80663	−0.247554
$378$	0	0
$379$	31.2044	1.60286	0.801432	−	0.598086i	$-0.204072\pi$
$379$	31.2044	1.60286	0.801432	+	0.598086i	$0.204072\pi$
$380$	0	0
$381$	−20.3327	−1.04167
$382$	0	0
$383$	28.9142	1.47745	0.738723	−	0.674009i	$-0.235429\pi$
$383$	28.9142	1.47745	0.738723	+	0.674009i	$0.235429\pi$
$384$	0	0
$385$	−20.5578	−1.04772
$386$	0	0
$387$	0.421512	0.0214266
$388$	0	0
$389$	−33.1219	−1.67935	−0.839674	−	0.543091i	$-0.817254\pi$
$389$	−33.1219	−1.67935	−0.839674	+	0.543091i	$0.817254\pi$
$390$	0	0
$391$	−14.6941	−0.743111
$392$	0	0
$393$	19.9937	1.00855
$394$	0	0
$395$	−14.0237	−0.705610
$396$	0	0
$397$	13.5972	0.682424	0.341212	−	0.939986i	$-0.389163\pi$
$397$	13.5972	0.682424	0.341212	+	0.939986i	$0.389163\pi$
$398$	0	0
$399$	18.9793	0.950154
$400$	0	0
$401$	1.34704	0.0672678	0.0336339	−	0.999434i	$-0.489292\pi$
$401$	1.34704	0.0672678	0.0336339	+	0.999434i	$0.489292\pi$
$402$	0	0
$403$	1.32331	0.0659190
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	46.0662	2.28342
$408$	0	0
$409$	−36.8016	−1.81972	−0.909862	−	0.414911i	$-0.863813\pi$
$409$	−36.8016	−1.81972	−0.909862	+	0.414911i	$0.863813\pi$
$410$	0	0
$411$	−11.7986	−0.581982
$412$	0	0
$413$	−17.9349	−0.882518
$414$	0	0
$415$	5.04442	0.247621
$416$	0	0
$417$	2.64663	0.129606
$418$	0	0
$419$	−14.1012	−0.688890	−0.344445	−	0.938807i	$-0.611933\pi$
$419$	−14.1012	−0.688890	−0.344445	+	0.938807i	$0.611933\pi$
$420$	0	0
$421$	−29.0919	−1.41785	−0.708925	−	0.705284i	$-0.750820\pi$
$421$	−29.0919	−1.41785	−0.708925	+	0.705284i	$0.750820\pi$
$422$	0	0
$423$	−1.46593	−0.0712759
$424$	0	0
$425$	6.95558	0.337395
$426$	0	0
$427$	25.6259	1.24013
$428$	0	0
$429$	7.26454	0.350735
$430$	0	0
$431$	21.3658	1.02915	0.514576	−	0.857445i	$-0.327949\pi$
$431$	21.3658	1.02915	0.514576	+	0.857445i	$0.327949\pi$
$432$	0	0
$433$	−10.5277	−0.505931	−0.252965	−	0.967475i	$-0.581406\pi$
$433$	−10.5277	−0.505931	−0.252965	+	0.967475i	$0.581406\pi$
$434$	0	0
$435$	3.63227	0.174154
$436$	0	0
$437$	10.7067	0.512173
$438$	0	0
$439$	−24.0949	−1.14999	−0.574993	−	0.818158i	$-0.694995\pi$
$439$	−24.0949	−1.14999	−0.574993	+	0.818158i	$0.694995\pi$
$440$	0	0
$441$	7.02372	0.334463
$442$	0	0
$443$	33.8935	1.61033	0.805164	−	0.593052i	$-0.202077\pi$
$443$	33.8935	1.61033	0.805164	+	0.593052i	$0.202077\pi$
$444$	0	0
$445$	13.3470	0.632710
$446$	0	0
$447$	0.510348	0.0241386
$448$	0	0
$449$	−33.9048	−1.60007	−0.800034	−	0.599955i	$-0.795185\pi$
$449$	−33.9048	−1.60007	−0.800034	+	0.599955i	$0.795185\pi$
$450$	0	0
$451$	63.0742	2.97005
$452$	0	0
$453$	−1.24081	−0.0582985
$454$	0	0
$455$	4.95558	0.232321
$456$	0	0
$457$	2.36273	0.110524	0.0552620	−	0.998472i	$-0.482401\pi$
$457$	2.36273	0.110524	0.0552620	+	0.998472i	$0.482401\pi$
$458$	0	0
$459$	−6.95558	−0.324659
$460$	0	0
$461$	−1.10320	−0.0513810	−0.0256905	−	0.999670i	$-0.508178\pi$
$461$	−1.10320	−0.0513810	−0.0256905	+	0.999670i	$0.508178\pi$
$462$	0	0
$463$	16.6466	0.773634	0.386817	−	0.922156i	$-0.373574\pi$
$463$	16.6466	0.773634	0.386817	+	0.922156i	$0.373574\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	1.46593	0.0678351	0.0339176	−	0.999425i	$-0.489202\pi$
$467$	1.46593	0.0678351	0.0339176	+	0.999425i	$0.489202\pi$
$468$	0	0
$469$	16.4452	0.759370
$470$	0	0
$471$	7.71477	0.355478
$472$	0	0
$473$	2.31395	0.106396
$474$	0	0
$475$	−5.06814	−0.232542
$476$	0	0
$477$	4.53407	0.207601
$478$	0	0
$479$	−1.99367	−0.0910929	−0.0455464	−	0.998962i	$-0.514503\pi$
$479$	−1.99367	−0.0910929	−0.0455464	+	0.998962i	$0.514503\pi$
$480$	0	0
$481$	−11.1045	−0.506323
$482$	0	0
$483$	7.91116	0.359970
$484$	0	0
$485$	3.48965	0.158457
$486$	0	0
$487$	−38.5164	−1.74534	−0.872672	−	0.488306i	$-0.837615\pi$
$487$	−38.5164	−1.74534	−0.872672	+	0.488306i	$0.837615\pi$
$488$	0	0
$489$	−1.18703	−0.0536795
$490$	0	0
$491$	−7.26454	−0.327844	−0.163922	−	0.986473i	$-0.552415\pi$
$491$	−7.26454	−0.327844	−0.163922	+	0.986473i	$0.552415\pi$
$492$	0	0
$493$	25.2645	1.13786
$494$	0	0
$495$	−5.48965	−0.246742
$496$	0	0
$497$	27.5134	1.23414
$498$	0	0
$499$	39.2519	1.75715	0.878577	−	0.477600i	$-0.158493\pi$
$499$	39.2519	1.75715	0.878577	+	0.477600i	$0.158493\pi$
$500$	0	0
$501$	8.55779	0.382334
$502$	0	0
$503$	5.69105	0.253751	0.126876	−	0.991919i	$-0.459505\pi$
$503$	5.69105	0.253751	0.126876	+	0.991919i	$0.459505\pi$
$504$	0	0
$505$	13.4008	0.596328
$506$	0	0
$507$	11.2488	0.499578
$508$	0	0
$509$	−28.3865	−1.25821	−0.629104	−	0.777321i	$-0.716578\pi$
$509$	−28.3865	−1.25821	−0.629104	+	0.777321i	$0.716578\pi$
$510$	0	0
$511$	−16.4452	−0.727494
$512$	0	0
$513$	5.06814	0.223764
$514$	0	0
$515$	15.1456	0.667397
$516$	0	0
$517$	−8.04744	−0.353926
$518$	0	0
$519$	−2.55779	−0.112275
$520$	0	0
$521$	12.7829	0.560029	0.280015	−	0.959996i	$-0.409661\pi$
$521$	12.7829	0.560029	0.280015	+	0.959996i	$0.409661\pi$
$522$	0	0
$523$	8.73546	0.381975	0.190988	−	0.981592i	$-0.438831\pi$
$523$	8.73546	0.381975	0.190988	+	0.981592i	$0.438831\pi$
$524$	0	0
$525$	−3.74483	−0.163438
$526$	0	0
$527$	−6.95558	−0.302990
$528$	0	0
$529$	−18.5371	−0.805961
$530$	0	0
$531$	−4.78924	−0.207835
$532$	0	0
$533$	−15.2044	−0.658577
$534$	0	0
$535$	12.9556	0.560119
$536$	0	0
$537$	24.8905	1.07410
$538$	0	0
$539$	38.5578	1.66080
$540$	0	0
$541$	4.65163	0.199989	0.0999946	−	0.994988i	$-0.468117\pi$
$541$	4.65163	0.199989	0.0999946	+	0.994988i	$0.468117\pi$
$542$	0	0
$543$	−17.8223	−0.764829
$544$	0	0
$545$	13.3771	0.573012
$546$	0	0
$547$	−38.9079	−1.66358	−0.831790	−	0.555091i	$-0.812684\pi$
$547$	−38.9079	−1.66358	−0.831790	+	0.555091i	$0.812684\pi$
$548$	0	0
$549$	6.84302	0.292053
$550$	0	0
$551$	−18.4088	−0.784243
$552$	0	0
$553$	52.5164	2.23322
$554$	0	0
$555$	8.39145	0.356197
$556$	0	0
$557$	19.6309	0.831789	0.415895	−	0.909413i	$-0.363469\pi$
$557$	19.6309	0.831789	0.415895	+	0.909413i	$0.363469\pi$
$558$	0	0
$559$	−0.557793	−0.0235921
$560$	0	0
$561$	−38.1837	−1.61212
$562$	0	0
$563$	16.7779	0.707105	0.353552	−	0.935415i	$-0.384974\pi$
$563$	16.7779	0.707105	0.353552	+	0.935415i	$0.384974\pi$
$564$	0	0
$565$	1.26454	0.0531994
$566$	0	0
$567$	3.74483	0.157268
$568$	0	0
$569$	−24.7004	−1.03549	−0.517747	−	0.855533i	$-0.673229\pi$
$569$	−24.7004	−1.03549	−0.517747	+	0.855533i	$0.673229\pi$
$570$	0	0
$571$	−9.80361	−0.410268	−0.205134	−	0.978734i	$-0.565763\pi$
$571$	−9.80361	−0.410268	−0.205134	+	0.978734i	$0.565763\pi$
$572$	0	0
$573$	−10.9255	−0.456420
$574$	0	0
$575$	−2.11256	−0.0880998
$576$	0	0
$577$	−2.70674	−0.112683	−0.0563416	−	0.998412i	$-0.517944\pi$
$577$	−2.70674	−0.112683	−0.0563416	+	0.998412i	$0.517944\pi$
$578$	0	0
$579$	−15.2645	−0.634372
$580$	0	0
$581$	−18.8905	−0.783709
$582$	0	0
$583$	24.8905	1.03086
$584$	0	0
$585$	1.32331	0.0547123
$586$	0	0
$587$	−4.33268	−0.178829	−0.0894143	−	0.995995i	$-0.528500\pi$
$587$	−4.33268	−0.178829	−0.0894143	+	0.995995i	$0.528500\pi$
$588$	0	0
$589$	5.06814	0.208829
$590$	0	0
$591$	−6.33768	−0.260697
$592$	0	0
$593$	15.1757	0.623191	0.311596	−	0.950215i	$-0.399137\pi$
$593$	15.1757	0.623191	0.311596	+	0.950215i	$0.399137\pi$
$594$	0	0
$595$	−26.0474	−1.06784
$596$	0	0
$597$	−3.55477	−0.145487
$598$	0	0
$599$	34.0411	1.39088	0.695441	−	0.718583i	$-0.255209\pi$
$599$	34.0411	1.39088	0.695441	+	0.718583i	$0.255209\pi$
$600$	0	0
$601$	11.1757	0.455866	0.227933	−	0.973677i	$-0.426803\pi$
$601$	11.1757	0.455866	0.227933	+	0.973677i	$0.426803\pi$
$602$	0	0
$603$	4.39145	0.178834
$604$	0	0
$605$	−19.1363	−0.778000
$606$	0	0
$607$	37.9699	1.54115	0.770576	−	0.637348i	$-0.219969\pi$
$607$	37.9699	1.54115	0.770576	+	0.637348i	$0.219969\pi$
$608$	0	0
$609$	−13.6022	−0.551189
$610$	0	0
$611$	1.93989	0.0784794
$612$	0	0
$613$	−44.7302	−1.80664	−0.903318	−	0.428972i	$-0.858876\pi$
$613$	−44.7302	−1.80664	−0.903318	+	0.428972i	$0.858876\pi$
$614$	0	0
$615$	11.4897	0.463307
$616$	0	0
$617$	14.1777	0.570772	0.285386	−	0.958413i	$-0.407878\pi$
$617$	14.1777	0.570772	0.285386	+	0.958413i	$0.407878\pi$
$618$	0	0
$619$	−11.5421	−0.463916	−0.231958	−	0.972726i	$-0.574513\pi$
$619$	−11.5421	−0.463916	−0.231958	+	0.972726i	$0.574513\pi$
$620$	0	0
$621$	2.11256	0.0847740
$622$	0	0
$623$	−49.9823	−2.00250
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	27.8223	1.11112
$628$	0	0
$629$	58.3675	2.32726
$630$	0	0
$631$	−19.2519	−0.766405	−0.383202	−	0.923664i	$-0.625179\pi$
$631$	−19.2519	−0.766405	−0.383202	+	0.923664i	$0.625179\pi$
$632$	0	0
$633$	−12.5578	−0.499127
$634$	0	0
$635$	−20.3327	−0.806878
$636$	0	0
$637$	−9.29459	−0.368265
$638$	0	0
$639$	7.34704	0.290644
$640$	0	0
$641$	27.3945	1.08202	0.541008	−	0.841017i	$-0.318043\pi$
$641$	27.3945	1.08202	0.541008	+	0.841017i	$0.318043\pi$
$642$	0	0
$643$	11.2645	0.444230	0.222115	−	0.975020i	$-0.428704\pi$
$643$	11.2645	0.444230	0.222115	+	0.975020i	$0.428704\pi$
$644$	0	0
$645$	0.421512	0.0165970
$646$	0	0
$647$	−37.1630	−1.46103	−0.730515	−	0.682897i	$-0.760720\pi$
$647$	−37.1630	−1.46103	−0.730515	+	0.682897i	$0.760720\pi$
$648$	0	0
$649$	−26.2913	−1.03202
$650$	0	0
$651$	3.74483	0.146771
$652$	0	0
$653$	1.06314	0.0416039	0.0208020	−	0.999784i	$-0.493378\pi$
$653$	1.06314	0.0416039	0.0208020	+	0.999784i	$0.493378\pi$
$654$	0	0
$655$	19.9937	0.781217
$656$	0	0
$657$	−4.39145	−0.171327
$658$	0	0
$659$	14.3965	0.560806	0.280403	−	0.959882i	$-0.409532\pi$
$659$	14.3965	0.560806	0.280403	+	0.959882i	$0.409532\pi$
$660$	0	0
$661$	9.26454	0.360349	0.180174	−	0.983635i	$-0.442334\pi$
$661$	9.26454	0.360349	0.180174	+	0.983635i	$0.442334\pi$
$662$	0	0
$663$	9.20442	0.357470
$664$	0	0
$665$	18.9793	0.735986
$666$	0	0
$667$	−7.67338	−0.297114
$668$	0	0
$669$	17.2044	0.665161
$670$	0	0
$671$	37.5658	1.45021
$672$	0	0
$673$	−6.84169	−0.263728	−0.131864	−	0.991268i	$-0.542096\pi$
$673$	−6.84169	−0.263728	−0.131864	+	0.991268i	$0.542096\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−18.7178	−0.719383	−0.359692	−	0.933071i	$-0.617118\pi$
$677$	−18.7178	−0.719383	−0.359692	+	0.933071i	$0.617118\pi$
$678$	0	0
$679$	−13.0681	−0.501509
$680$	0	0
$681$	19.0919	0.731602
$682$	0	0
$683$	−35.5895	−1.36180	−0.680898	−	0.732378i	$-0.738410\pi$
$683$	−35.5895	−1.36180	−0.680898	+	0.732378i	$0.738410\pi$
$684$	0	0
$685$	−11.7986	−0.450802
$686$	0	0
$687$	−12.4690	−0.475720
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−7.21709	−0.274551	−0.137276	−	0.990533i	$-0.543835\pi$
$691$	−7.21709	−0.274551	−0.137276	+	0.990533i	$0.543835\pi$
$692$	0	0
$693$	20.5578	0.780926
$694$	0	0
$695$	2.64663	0.100392
$696$	0	0
$697$	79.9172	3.02708
$698$	0	0
$699$	2.75919	0.104362
$700$	0	0
$701$	−32.7228	−1.23592	−0.617961	−	0.786208i	$-0.712041\pi$
$701$	−32.7228	−1.23592	−0.617961	+	0.786208i	$0.712041\pi$
$702$	0	0
$703$	−42.5291	−1.60401
$704$	0	0
$705$	−1.46593	−0.0552101
$706$	0	0
$707$	−50.1837	−1.88735
$708$	0	0
$709$	−25.8223	−0.969778	−0.484889	−	0.874576i	$-0.661140\pi$
$709$	−25.8223	−0.969778	−0.484889	+	0.874576i	$0.661140\pi$
$710$	0	0
$711$	14.0237	0.525931
$712$	0	0
$713$	2.11256	0.0791159
$714$	0	0
$715$	7.26454	0.271678
$716$	0	0
$717$	−6.61791	−0.247150
$718$	0	0
$719$	40.4690	1.50924	0.754619	−	0.656164i	$-0.227822\pi$
$719$	40.4690	1.50924	0.754619	+	0.656164i	$0.227822\pi$
$720$	0	0
$721$	−56.7178	−2.11228
$722$	0	0
$723$	5.26454	0.195790
$724$	0	0
$725$	3.63227	0.134899
$726$	0	0
$727$	20.2138	0.749688	0.374844	−	0.927088i	$-0.377696\pi$
$727$	20.2138	0.749688	0.374844	+	0.927088i	$0.377696\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.93186	0.108439
$732$	0	0
$733$	16.3140	0.602570	0.301285	−	0.953534i	$-0.402585\pi$
$733$	16.3140	0.602570	0.301285	+	0.953534i	$0.402585\pi$
$734$	0	0
$735$	7.02372	0.259074
$736$	0	0
$737$	24.1076	0.888013
$738$	0	0
$739$	−23.7799	−0.874757	−0.437379	−	0.899277i	$-0.644093\pi$
$739$	−23.7799	−0.874757	−0.437379	+	0.899277i	$0.644093\pi$
$740$	0	0
$741$	−6.70674	−0.246378
$742$	0	0
$743$	45.7208	1.67733	0.838667	−	0.544644i	$-0.183335\pi$
$743$	45.7208	1.67733	0.838667	+	0.544644i	$0.183335\pi$
$744$	0	0
$745$	0.510348	0.0186977
$746$	0	0
$747$	−5.04442	−0.184566
$748$	0	0
$749$	−48.5164	−1.77275
$750$	0	0
$751$	2.81430	0.102695	0.0513477	−	0.998681i	$-0.483648\pi$
$751$	2.81430	0.102695	0.0513477	+	0.998681i	$0.483648\pi$
$752$	0	0
$753$	20.0474	0.730569
$754$	0	0
$755$	−1.24081	−0.0451578
$756$	0	0
$757$	−48.1724	−1.75086	−0.875428	−	0.483349i	$-0.839420\pi$
$757$	−48.1724	−1.75086	−0.875428	+	0.483349i	$0.839420\pi$
$758$	0	0
$759$	11.5972	0.420952
$760$	0	0
$761$	46.4212	1.68277	0.841384	−	0.540438i	$-0.181741\pi$
$761$	46.4212	1.68277	0.841384	+	0.540438i	$0.181741\pi$
$762$	0	0
$763$	−50.0949	−1.81356
$764$	0	0
$765$	−6.95558	−0.251480
$766$	0	0
$767$	6.33768	0.228840
$768$	0	0
$769$	−51.1580	−1.84481	−0.922403	−	0.386229i	$-0.873777\pi$
$769$	−51.1580	−1.84481	−0.922403	+	0.386229i	$0.873777\pi$
$770$	0	0
$771$	−25.7385	−0.926949
$772$	0	0
$773$	52.1661	1.87628	0.938141	−	0.346253i	$-0.112546\pi$
$773$	52.1661	1.87628	0.938141	+	0.346253i	$0.112546\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−31.4245	−1.12735
$778$	0	0
$779$	−58.2312	−2.08635
$780$	0	0
$781$	40.3327	1.44322
$782$	0	0
$783$	−3.63227	−0.129807
$784$	0	0
$785$	7.71477	0.275352
$786$	0	0
$787$	36.8016	1.31184	0.655918	−	0.754832i	$-0.272282\pi$
$787$	36.8016	1.31184	0.655918	+	0.754832i	$0.272282\pi$
$788$	0	0
$789$	16.3327	0.581459
$790$	0	0
$791$	−4.73546	−0.168374
$792$	0	0
$793$	−9.05547	−0.321569
$794$	0	0
$795$	4.53407	0.160807
$796$	0	0
$797$	24.8193	0.879145	0.439572	−	0.898207i	$-0.355130\pi$
$797$	24.8193	0.879145	0.439572	+	0.898207i	$0.355130\pi$
$798$	0	0
$799$	−10.1964	−0.360723
$800$	0	0
$801$	−13.3470	−0.471594
$802$	0	0
$803$	−24.1076	−0.850737
$804$	0	0
$805$	7.91116	0.278832
$806$	0	0
$807$	−19.4546	−0.684834
$808$	0	0
$809$	−29.9523	−1.05307	−0.526533	−	0.850155i	$-0.676508\pi$
$809$	−29.9523	−1.05307	−0.526533	+	0.850155i	$0.676508\pi$
$810$	0	0
$811$	−13.3534	−0.468900	−0.234450	−	0.972128i	$-0.575329\pi$
$811$	−13.3534	−0.468900	−0.234450	+	0.972128i	$0.575329\pi$
$812$	0	0
$813$	−28.7829	−1.00946
$814$	0	0
$815$	−1.18703	−0.0415800
$816$	0	0
$817$	−2.13628	−0.0747390
$818$	0	0
$819$	−4.95558	−0.173162
$820$	0	0
$821$	31.9937	1.11659	0.558293	−	0.829644i	$-0.311456\pi$
$821$	31.9937	1.11659	0.558293	+	0.829644i	$0.311456\pi$
$822$	0	0
$823$	9.48965	0.330788	0.165394	−	0.986228i	$-0.447110\pi$
$823$	9.48965	0.330788	0.165394	+	0.986228i	$0.447110\pi$
$824$	0	0
$825$	−5.48965	−0.191125
$826$	0	0
$827$	10.7779	0.374785	0.187392	−	0.982285i	$-0.439996\pi$
$827$	10.7779	0.374785	0.187392	+	0.982285i	$0.439996\pi$
$828$	0	0
$829$	14.1076	0.489976	0.244988	−	0.969526i	$-0.421216\pi$
$829$	14.1076	0.489976	0.244988	+	0.969526i	$0.421216\pi$
$830$	0	0
$831$	−15.0381	−0.521665
$832$	0	0
$833$	48.8541	1.69269
$834$	0	0
$835$	8.55779	0.296155
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	33.2295	1.14721	0.573605	−	0.819132i	$-0.305545\pi$
$839$	33.2295	1.14721	0.573605	+	0.819132i	$0.305545\pi$
$840$	0	0
$841$	−15.8066	−0.545056
$842$	0	0
$843$	−18.7829	−0.646918
$844$	0	0
$845$	11.2488	0.386972
$846$	0	0
$847$	71.6620	2.46234
$848$	0	0
$849$	−23.7923	−0.816549
$850$	0	0
$851$	−17.7274	−0.607689
$852$	0	0
$853$	−5.63860	−0.193062	−0.0965310	−	0.995330i	$-0.530775\pi$
$853$	−5.63860	−0.193062	−0.0965310	+	0.995330i	$0.530775\pi$
$854$	0	0
$855$	5.06814	0.173327
$856$	0	0
$857$	2.55779	0.0873725	0.0436863	−	0.999045i	$-0.486090\pi$
$857$	2.55779	0.0873725	0.0436863	+	0.999045i	$0.486090\pi$
$858$	0	0
$859$	46.8965	1.60009	0.800044	−	0.599941i	$-0.204809\pi$
$859$	46.8965	1.60009	0.800044	+	0.599941i	$0.204809\pi$
$860$	0	0
$861$	−43.0267	−1.46635
$862$	0	0
$863$	8.39279	0.285694	0.142847	−	0.989745i	$-0.454374\pi$
$863$	8.39279	0.285694	0.142847	+	0.989745i	$0.454374\pi$
$864$	0	0
$865$	−2.55779	−0.0869676
$866$	0	0
$867$	−31.3801	−1.06572
$868$	0	0
$869$	76.9854	2.61155
$870$	0	0
$871$	−5.81127	−0.196908
$872$	0	0
$873$	−3.48965	−0.118107
$874$	0	0
$875$	−3.74483	−0.126598
$876$	0	0
$877$	44.4877	1.50224	0.751121	−	0.660164i	$-0.229513\pi$
$877$	44.4877	1.50224	0.751121	+	0.660164i	$0.229513\pi$
$878$	0	0
$879$	−18.3090	−0.617546
$880$	0	0
$881$	−28.5515	−0.961923	−0.480962	−	0.876742i	$-0.659712\pi$
$881$	−28.5515	−0.961923	−0.480962	+	0.876742i	$0.659712\pi$
$882$	0	0
$883$	−33.4295	−1.12499	−0.562497	−	0.826799i	$-0.690159\pi$
$883$	−33.4295	−1.12499	−0.562497	+	0.826799i	$0.690159\pi$
$884$	0	0
$885$	−4.78924	−0.160989
$886$	0	0
$887$	12.0424	0.404346	0.202173	−	0.979350i	$-0.435200\pi$
$887$	12.0424	0.404346	0.202173	+	0.979350i	$0.435200\pi$
$888$	0	0
$889$	76.1423	2.55373
$890$	0	0
$891$	5.48965	0.183910
$892$	0	0
$893$	7.42954	0.248620
$894$	0	0
$895$	24.8905	0.831997
$896$	0	0
$897$	−2.79558	−0.0933417
$898$	0	0
$899$	−3.63227	−0.121143
$900$	0	0
$901$	31.5371	1.05065
$902$	0	0
$903$	−1.57849	−0.0525288
$904$	0	0
$905$	−17.8223	−0.592434
$906$	0	0
$907$	−49.7509	−1.65195	−0.825975	−	0.563706i	$-0.809375\pi$
$907$	−49.7509	−1.65195	−0.825975	+	0.563706i	$0.809375\pi$
$908$	0	0
$909$	−13.4008	−0.444477
$910$	0	0
$911$	27.1757	0.900371	0.450186	−	0.892935i	$-0.351358\pi$
$911$	27.1757	0.900371	0.450186	+	0.892935i	$0.351358\pi$
$912$	0	0
$913$	−27.6921	−0.916475
$914$	0	0
$915$	6.84302	0.226223
$916$	0	0
$917$	−74.8728	−2.47252
$918$	0	0
$919$	−17.6860	−0.583409	−0.291704	−	0.956509i	$-0.594222\pi$
$919$	−17.6860	−0.583409	−0.291704	+	0.956509i	$0.594222\pi$
$920$	0	0
$921$	6.72413	0.221568
$922$	0	0
$923$	−9.72244	−0.320018
$924$	0	0
$925$	8.39145	0.275909
$926$	0	0
$927$	−15.1456	−0.497448
$928$	0	0
$929$	−38.5040	−1.26328	−0.631638	−	0.775264i	$-0.717617\pi$
$929$	−38.5040	−1.26328	−0.631638	+	0.775264i	$0.717617\pi$
$930$	0	0
$931$	−35.5972	−1.16665
$932$	0	0
$933$	24.8080	0.812176
$934$	0	0
$935$	−38.1837	−1.24874
$936$	0	0
$937$	4.96058	0.162055	0.0810276	−	0.996712i	$-0.474180\pi$
$937$	4.96058	0.162055	0.0810276	+	0.996712i	$0.474180\pi$
$938$	0	0
$939$	−29.5959	−0.965825
$940$	0	0
$941$	8.88413	0.289614	0.144807	−	0.989460i	$-0.453744\pi$
$941$	8.88413	0.289614	0.144807	+	0.989460i	$0.453744\pi$
$942$	0	0
$943$	−24.2726	−0.790423
$944$	0	0
$945$	3.74483	0.121819
$946$	0	0
$947$	6.06511	0.197090	0.0985449	−	0.995133i	$-0.468581\pi$
$947$	6.06511	0.197090	0.0985449	+	0.995133i	$0.468581\pi$
$948$	0	0
$949$	5.81127	0.188642
$950$	0	0
$951$	13.7385	0.445501
$952$	0	0
$953$	−28.6416	−0.927793	−0.463897	−	0.885889i	$-0.653549\pi$
$953$	−28.6416	−0.927793	−0.463897	+	0.885889i	$0.653549\pi$
$954$	0	0
$955$	−10.9255	−0.353542
$956$	0	0
$957$	−19.9399	−0.644565
$958$	0	0
$959$	44.1837	1.42677
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−12.9556	−0.417488
$964$	0	0
$965$	−15.2645	−0.491383
$966$	0	0
$967$	−36.5765	−1.17622	−0.588111	−	0.808780i	$-0.700128\pi$
$967$	−36.5765	−1.17622	−0.588111	+	0.808780i	$0.700128\pi$
$968$	0	0
$969$	35.2519	1.13245
$970$	0	0
$971$	6.16134	0.197727	0.0988634	−	0.995101i	$-0.468479\pi$
$971$	6.16134	0.197727	0.0988634	+	0.995101i	$0.468479\pi$
$972$	0	0
$973$	−9.91116	−0.317737
$974$	0	0
$975$	1.32331	0.0423800
$976$	0	0
$977$	−21.8223	−0.698158	−0.349079	−	0.937093i	$-0.613506\pi$
$977$	−21.8223	−0.698158	−0.349079	+	0.937093i	$0.613506\pi$
$978$	0	0
$979$	−73.2706	−2.34174
$980$	0	0
$981$	−13.3771	−0.427098
$982$	0	0
$983$	−9.06814	−0.289229	−0.144614	−	0.989488i	$-0.546194\pi$
$983$	−9.06814	−0.289229	−0.144614	+	0.989488i	$0.546194\pi$
$984$	0	0
$985$	−6.33768	−0.201935
$986$	0	0
$987$	5.48965	0.174738
$988$	0	0
$989$	−0.890468	−0.0283152
$990$	0	0
$991$	0.155004	0.00492385	0.00246192	−	0.999997i	$-0.499216\pi$
$991$	0.155004	0.00492385	0.00246192	+	0.999997i	$0.499216\pi$
$992$	0	0
$993$	0.908137	0.0288189
$994$	0	0
$995$	−3.55477	−0.112694
$996$	0	0
$997$	44.2438	1.40122	0.700608	−	0.713546i	$-0.252912\pi$
$997$	44.2438	1.40122	0.700608	+	0.713546i	$0.252912\pi$
$998$	0	0
$999$	−8.39145	−0.265494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bn.1.3		3
4.3	odd	2		1860.2.a.g.1.1	✓	3
12.11	even	2		5580.2.a.j.1.1		3
20.3	even	4		9300.2.g.q.3349.6		6
20.7	even	4		9300.2.g.q.3349.1		6
20.19	odd	2		9300.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.1	✓	3	4.3	odd	2
5580.2.a.j.1.1		3	12.11	even	2
7440.2.a.bn.1.3		3	1.1	even	1	trivial
9300.2.a.u.1.3		3	20.19	odd	2
9300.2.g.q.3349.1		6	20.7	even	4
9300.2.g.q.3349.6		6	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.74483 q^{7} +1.00000 q^{9} +5.48965 q^{11} -1.32331 q^{13} +1.00000 q^{15} +6.95558 q^{17} -5.06814 q^{19} -3.74483 q^{21} -2.11256 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.63227 q^{29} -1.00000 q^{31} -5.48965 q^{33} -3.74483 q^{35} +8.39145 q^{37} +1.32331 q^{39} +11.4897 q^{41} +0.421512 q^{43} -1.00000 q^{45} -1.46593 q^{47} +7.02372 q^{49} -6.95558 q^{51} +4.53407 q^{53} -5.48965 q^{55} +5.06814 q^{57} -4.78924 q^{59} +6.84302 q^{61} +3.74483 q^{63} +1.32331 q^{65} +4.39145 q^{67} +2.11256 q^{69} +7.34704 q^{71} -4.39145 q^{73} -1.00000 q^{75} +20.5578 q^{77} +14.0237 q^{79} +1.00000 q^{81} -5.04442 q^{83} -6.95558 q^{85} -3.63227 q^{87} -13.3470 q^{89} -4.95558 q^{91} +1.00000 q^{93} +5.06814 q^{95} -3.48965 q^{97} +5.48965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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