LMFDB - Embedded newform 4640.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4640,2,Mod(1,4640)] 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("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4640 = 2^{5} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4640.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,12,0,0,0,16,0,0,0,4,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.0505865379$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{26})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 25 $ x^4 - 16*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.77425$ of defining polynomial
Character	$\chi$	$=$	4640.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} -5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} +7.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} -5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} -12.4900 q^{33} -2.44949 q^{35} +3.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} +12.6475 q^{43} -3.00000 q^{45} -7.74855 q^{47} -1.00000 q^{49} +17.7465 q^{51} +8.00000 q^{53} +5.09902 q^{55} +6.00000 q^{57} -12.4475 q^{59} +10.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} +9.99800 q^{67} -12.4900 q^{69} +7.54851 q^{71} -13.2450 q^{73} +2.44949 q^{75} -12.4900 q^{77} +4.69894 q^{79} -9.00000 q^{81} -2.84957 q^{83} -7.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} -1.24500 q^{97} -15.2971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 12 * q^9 + 16 * q^13 + 4 * q^17 + 24 * q^21 + 4 * q^25 + 4 * q^29 - 12 * q^37 - 32 * q^41 - 12 * q^45 - 4 * q^49 + 32 * q^53 + 24 * q^57 - 8 * q^61 - 16 * q^65 - 28 * q^73 - 36 * q^81 - 4 * q^85 - 24 * q^89 + 72 * q^93 + 20 * q^97

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
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−5.09902	−1.53741	−0.768706	−	0.639602i	$-0.779099\pi$
$11$	−5.09902	−1.53741	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.44949	−0.632456
$16$	0	0
$17$	7.24500	1.75717	0.878585	−	0.477586i	$-0.158488\pi$
$17$	7.24500	1.75717	0.878585	+	0.477586i	$0.158488\pi$
$18$	0	0
$19$	2.44949	0.561951	0.280976	−	0.959715i	$-0.409342\pi$
$19$	2.44949	0.561951	0.280976	+	0.959715i	$0.409342\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	−5.09902	−1.06322	−0.531610	−	0.846990i	$-0.678413\pi$
$23$	−5.09902	−1.06322	−0.531610	+	0.846990i	$0.678413\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.34847	1.31982	0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847	1.31982	0.659912	+	0.751343i	$0.270594\pi$
$32$	0	0
$33$	−12.4900	−2.17423
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	3.24500	0.533474	0.266737	−	0.963769i	$-0.414054\pi$
$37$	3.24500	0.533474	0.266737	+	0.963769i	$0.414054\pi$
$38$	0	0
$39$	9.79796	1.56893
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	12.6475	1.92873	0.964365	−	0.264575i	$-0.0852318\pi$
$43$	12.6475	1.92873	0.964365	+	0.264575i	$0.0852318\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−7.74855	−1.13024	−0.565121	−	0.825008i	$-0.691171\pi$
$47$	−7.74855	−1.13024	−0.565121	+	0.825008i	$0.691171\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	17.7465	2.48501
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	5.09902	0.687552
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−12.4475	−1.62053	−0.810263	−	0.586067i	$-0.800675\pi$
$59$	−12.4475	−1.62053	−0.810263	+	0.586067i	$0.800675\pi$
$60$	0	0
$61$	10.4900	1.34311	0.671553	−	0.740956i	$-0.265627\pi$
$61$	10.4900	1.34311	0.671553	+	0.740956i	$0.265627\pi$
$62$	0	0
$63$	7.34847	0.925820
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	9.99800	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$67$	9.99800	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$68$	0	0
$69$	−12.4900	−1.50362
$70$	0	0
$71$	7.54851	0.895843	0.447922	−	0.894073i	$-0.352164\pi$
$71$	7.54851	0.895843	0.447922	+	0.894073i	$0.352164\pi$
$72$	0	0
$73$	−13.2450	−1.55021	−0.775105	−	0.631833i	$-0.782303\pi$
$73$	−13.2450	−1.55021	−0.775105	+	0.631833i	$0.782303\pi$
$74$	0	0
$75$	2.44949	0.282843
$76$	0	0
$77$	−12.4900	−1.42337
$78$	0	0
$79$	4.69894	0.528672	0.264336	−	0.964431i	$-0.414847\pi$
$79$	4.69894	0.528672	0.264336	+	0.964431i	$0.414847\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−2.84957	−0.312781	−0.156390	−	0.987695i	$-0.549986\pi$
$83$	−2.84957	−0.312781	−0.156390	+	0.987695i	$0.549986\pi$
$84$	0	0
$85$	−7.24500	−0.785830
$86$	0	0
$87$	2.44949	0.262613
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	9.79796	1.02711
$92$	0	0
$93$	18.0000	1.86651
$94$	0	0
$95$	−2.44949	−0.251312
$96$	0	0
$97$	−1.24500	−0.126410	−0.0632052	−	0.998001i	$-0.520132\pi$
$97$	−1.24500	−0.126410	−0.0632052	+	0.998001i	$0.520132\pi$
$98$	0	0
$99$	−15.2971	−1.53741
$100$	0	0
$101$	12.4900	1.24280	0.621401	−	0.783493i	$-0.286564\pi$
$101$	12.4900	1.24280	0.621401	+	0.783493i	$0.286564\pi$
$102$	0	0
$103$	0.200040	0.0197105	0.00985526	−	0.999951i	$-0.496863\pi$
$103$	0.200040	0.0197105	0.00985526	+	0.999951i	$0.496863\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−7.34847	−0.710403	−0.355202	−	0.934790i	$-0.615588\pi$
$107$	−7.34847	−0.710403	−0.355202	+	0.934790i	$0.615588\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	7.94859	0.754447
$112$	0	0
$113$	19.7350	1.85651	0.928256	−	0.371942i	$-0.121308\pi$
$113$	19.7350	1.85651	0.928256	+	0.371942i	$0.121308\pi$
$114$	0	0
$115$	5.09902	0.475486
$116$	0	0
$117$	12.0000	1.10940
$118$	0	0
$119$	17.7465	1.62682
$120$	0	0
$121$	15.0000	1.36364
$122$	0	0
$123$	−19.5959	−1.76690
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.2474	−1.08679	−0.543393	−	0.839479i	$-0.682861\pi$
$127$	−12.2474	−1.08679	−0.543393	+	0.839479i	$0.682861\pi$
$128$	0	0
$129$	30.9800	2.72764
$130$	0	0
$131$	17.5465	1.53305	0.766523	−	0.642217i	$-0.221985\pi$
$131$	17.5465	1.53305	0.766523	+	0.642217i	$0.221985\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.24500	0.618982	0.309491	−	0.950902i	$-0.399841\pi$
$137$	7.24500	0.618982	0.309491	+	0.950902i	$0.399841\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−18.9800	−1.59840
$142$	0	0
$143$	−20.3961	−1.70561
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−2.44949	−0.202031
$148$	0	0
$149$	8.49000	0.695528	0.347764	−	0.937582i	$-0.386941\pi$
$149$	8.49000	0.695528	0.347764	+	0.937582i	$0.386941\pi$
$150$	0	0
$151$	−7.94859	−0.646847	−0.323424	−	0.946254i	$-0.604834\pi$
$151$	−7.94859	−0.646847	−0.323424	+	0.946254i	$0.604834\pi$
$152$	0	0
$153$	21.7350	1.75717
$154$	0	0
$155$	−7.34847	−0.590243
$156$	0	0
$157$	17.2450	1.37630	0.688150	−	0.725568i	$-0.258423\pi$
$157$	17.2450	1.37630	0.688150	+	0.725568i	$0.258423\pi$
$158$	0	0
$159$	19.5959	1.55406
$160$	0	0
$161$	−12.4900	−0.984350
$162$	0	0
$163$	−12.6475	−0.990631	−0.495315	−	0.868713i	$-0.664948\pi$
$163$	−12.6475	−0.990631	−0.495315	+	0.868713i	$0.664948\pi$
$164$	0	0
$165$	12.4900	0.972345
$166$	0	0
$167$	−14.8970	−1.15276	−0.576381	−	0.817181i	$-0.695536\pi$
$167$	−14.8970	−1.15276	−0.576381	+	0.817181i	$0.695536\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	7.34847	0.561951
$172$	0	0
$173$	−10.4900	−0.797540	−0.398770	−	0.917051i	$-0.630563\pi$
$173$	−10.4900	−0.797540	−0.398770	+	0.917051i	$0.630563\pi$
$174$	0	0
$175$	2.44949	0.185164
$176$	0	0
$177$	−30.4900	−2.29177
$178$	0	0
$179$	14.6969	1.09850	0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969	1.09850	0.549250	+	0.835658i	$0.314913\pi$
$180$	0	0
$181$	−14.4900	−1.07703	−0.538516	−	0.842615i	$-0.681015\pi$
$181$	−14.4900	−1.07703	−0.538516	+	0.842615i	$0.681015\pi$
$182$	0	0
$183$	25.6951	1.89944
$184$	0	0
$185$	−3.24500	−0.238577
$186$	0	0
$187$	−36.9424	−2.70149
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.49910	0.397901	0.198950	−	0.980010i	$-0.436247\pi$
$191$	5.49910	0.397901	0.198950	+	0.980010i	$0.436247\pi$
$192$	0	0
$193$	−17.7350	−1.27659	−0.638296	−	0.769791i	$-0.720361\pi$
$193$	−17.7350	−1.27659	−0.638296	+	0.769791i	$0.720361\pi$
$194$	0	0
$195$	−9.79796	−0.701646
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	22.6455	1.60530	0.802649	−	0.596451i	$-0.203423\pi$
$199$	22.6455	1.60530	0.802649	+	0.596451i	$0.203423\pi$
$200$	0	0
$201$	24.4900	1.72739
$202$	0	0
$203$	2.44949	0.171920
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	−15.2971	−1.06322
$208$	0	0
$209$	−12.4900	−0.863951
$210$	0	0
$211$	−5.49910	−0.378574	−0.189287	−	0.981922i	$-0.560618\pi$
$211$	−5.49910	−0.378574	−0.189287	+	0.981922i	$0.560618\pi$
$212$	0	0
$213$	18.4900	1.26691
$214$	0	0
$215$	−12.6475	−0.862554
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	−32.4435	−2.19233
$220$	0	0
$221$	28.9800	1.94941
$222$	0	0
$223$	−0.200040	−0.0133957	−0.00669784	−	0.999978i	$-0.502132\pi$
$223$	−0.200040	−0.0133957	−0.00669784	+	0.999978i	$0.502132\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	0.200040	0.0132771	0.00663856	−	0.999978i	$-0.497887\pi$
$227$	0.200040	0.0132771	0.00663856	+	0.999978i	$0.497887\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−30.5941	−2.01295
$232$	0	0
$233$	4.49000	0.294149	0.147075	−	0.989125i	$-0.453014\pi$
$233$	4.49000	0.294149	0.147075	+	0.989125i	$0.453014\pi$
$234$	0	0
$235$	7.74855	0.505460
$236$	0	0
$237$	11.5100	0.747655
$238$	0	0
$239$	12.4475	0.805161	0.402581	−	0.915385i	$-0.368113\pi$
$239$	12.4475	0.805161	0.402581	+	0.915385i	$0.368113\pi$
$240$	0	0
$241$	−26.9800	−1.73793	−0.868967	−	0.494870i	$-0.835216\pi$
$241$	−26.9800	−1.73793	−0.868967	+	0.494870i	$0.835216\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	9.79796	0.623429
$248$	0	0
$249$	−6.97999	−0.442339
$250$	0	0
$251$	−19.7960	−1.24951	−0.624755	−	0.780821i	$-0.714801\pi$
$251$	−19.7960	−1.24951	−0.624755	+	0.780821i	$0.714801\pi$
$252$	0	0
$253$	26.0000	1.63461
$254$	0	0
$255$	−17.7465	−1.11133
$256$	0	0
$257$	14.4900	0.903861	0.451931	−	0.892053i	$-0.350735\pi$
$257$	14.4900	0.903861	0.451931	+	0.892053i	$0.350735\pi$
$258$	0	0
$259$	7.94859	0.493901
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	12.6475	0.779880	0.389940	−	0.920840i	$-0.372496\pi$
$263$	12.6475	0.779880	0.389940	+	0.920840i	$0.372496\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	−14.6969	−0.899438
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	15.2971	0.929230	0.464615	−	0.885513i	$-0.346193\pi$
$271$	15.2971	0.929230	0.464615	+	0.885513i	$0.346193\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	−5.09902	−0.307482
$276$	0	0
$277$	−28.9800	−1.74124	−0.870619	−	0.491957i	$-0.836282\pi$
$277$	−28.9800	−1.74124	−0.870619	+	0.491957i	$0.836282\pi$
$278$	0	0
$279$	22.0454	1.31982
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−5.49910	−0.326888	−0.163444	−	0.986553i	$-0.552260\pi$
$283$	−5.49910	−0.326888	−0.163444	+	0.986553i	$0.552260\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	−19.5959	−1.15671
$288$	0	0
$289$	35.4900	2.08765
$290$	0	0
$291$	−3.04961	−0.178771
$292$	0	0
$293$	−9.24500	−0.540099	−0.270049	−	0.962847i	$-0.587040\pi$
$293$	−9.24500	−0.540099	−0.270049	+	0.962847i	$0.587040\pi$
$294$	0	0
$295$	12.4475	0.724721
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.3961	−1.17954
$300$	0	0
$301$	30.9800	1.78566
$302$	0	0
$303$	30.5941	1.75759
$304$	0	0
$305$	−10.4900	−0.600655
$306$	0	0
$307$	−12.2474	−0.698999	−0.349499	−	0.936937i	$-0.613648\pi$
$307$	−12.2474	−0.698999	−0.349499	+	0.936937i	$0.613648\pi$
$308$	0	0
$309$	0.489996	0.0278749
$310$	0	0
$311$	−2.44949	−0.138898	−0.0694489	−	0.997586i	$-0.522124\pi$
$311$	−2.44949	−0.138898	−0.0694489	+	0.997586i	$0.522124\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	−7.34847	−0.414039
$316$	0	0
$317$	27.2450	1.53023	0.765116	−	0.643893i	$-0.222682\pi$
$317$	27.2450	1.53023	0.765116	+	0.643893i	$0.222682\pi$
$318$	0	0
$319$	−5.09902	−0.285490
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	17.7465	0.987444
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	39.1918	2.16731
$328$	0	0
$329$	−18.9800	−1.04640
$330$	0	0
$331$	−27.3445	−1.50299	−0.751494	−	0.659740i	$-0.770666\pi$
$331$	−27.3445	−1.50299	−0.751494	+	0.659740i	$0.770666\pi$
$332$	0	0
$333$	9.73499	0.533474
$334$	0	0
$335$	−9.99800	−0.546249
$336$	0	0
$337$	11.7350	0.639246	0.319623	−	0.947545i	$-0.396444\pi$
$337$	11.7350	0.639246	0.319623	+	0.947545i	$0.396444\pi$
$338$	0	0
$339$	48.3407	2.62550
$340$	0	0
$341$	−37.4700	−2.02911
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	12.4900	0.672439
$346$	0	0
$347$	−32.2434	−1.73092	−0.865459	−	0.500979i	$-0.832973\pi$
$347$	−32.2434	−1.73092	−0.865459	+	0.500979i	$0.832973\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.9800	−0.903754	−0.451877	−	0.892080i	$-0.649245\pi$
$353$	−16.9800	−0.903754	−0.451877	+	0.892080i	$0.649245\pi$
$354$	0	0
$355$	−7.54851	−0.400633
$356$	0	0
$357$	43.4700	2.30068
$358$	0	0
$359$	−2.44949	−0.129279	−0.0646396	−	0.997909i	$-0.520590\pi$
$359$	−2.44949	−0.129279	−0.0646396	+	0.997909i	$0.520590\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	36.7423	1.92847
$364$	0	0
$365$	13.2450	0.693275
$366$	0	0
$367$	−32.6435	−1.70398	−0.851989	−	0.523560i	$-0.824604\pi$
$367$	−32.6435	−1.70398	−0.851989	+	0.523560i	$0.824604\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	19.5959	1.01737
$372$	0	0
$373$	28.4900	1.47516	0.737578	−	0.675262i	$-0.235969\pi$
$373$	28.4900	1.47516	0.737578	+	0.675262i	$0.235969\pi$
$374$	0	0
$375$	−2.44949	−0.126491
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	0.600120	0.0308261	0.0154130	−	0.999881i	$-0.495094\pi$
$379$	0.600120	0.0308261	0.0154130	+	0.999881i	$0.495094\pi$
$380$	0	0
$381$	−30.0000	−1.53695
$382$	0	0
$383$	−2.84957	−0.145606	−0.0728031	−	0.997346i	$-0.523194\pi$
$383$	−2.84957	−0.145606	−0.0728031	+	0.997346i	$0.523194\pi$
$384$	0	0
$385$	12.4900	0.636549
$386$	0	0
$387$	37.9426	1.92873
$388$	0	0
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−36.9424	−1.86826
$392$	0	0
$393$	42.9800	2.16805
$394$	0	0
$395$	−4.69894	−0.236429
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	14.6969	0.735767
$400$	0	0
$401$	−1.51000	−0.0754060	−0.0377030	−	0.999289i	$-0.512004\pi$
$401$	−1.51000	−0.0754060	−0.0377030	+	0.999289i	$0.512004\pi$
$402$	0	0
$403$	29.3939	1.46421
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−16.5463	−0.820170
$408$	0	0
$409$	26.4900	1.30985	0.654923	−	0.755696i	$-0.272701\pi$
$409$	26.4900	1.30985	0.654923	+	0.755696i	$0.272701\pi$
$410$	0	0
$411$	17.7465	0.875373
$412$	0	0
$413$	−30.4900	−1.50031
$414$	0	0
$415$	2.84957	0.139880
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.54851	−0.368769	−0.184384	−	0.982854i	$-0.559029\pi$
$419$	−7.54851	−0.368769	−0.184384	+	0.982854i	$0.559029\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	−23.2456	−1.13024
$424$	0	0
$425$	7.24500	0.351434
$426$	0	0
$427$	25.6951	1.24347
$428$	0	0
$429$	−49.9600	−2.41209
$430$	0	0
$431$	−34.6929	−1.67110	−0.835550	−	0.549414i	$-0.814851\pi$
$431$	−34.6929	−1.67110	−0.835550	+	0.549414i	$0.814851\pi$
$432$	0	0
$433$	−2.26501	−0.108849	−0.0544246	−	0.998518i	$-0.517332\pi$
$433$	−2.26501	−0.108849	−0.0544246	+	0.998518i	$0.517332\pi$
$434$	0	0
$435$	−2.44949	−0.117444
$436$	0	0
$437$	−12.4900	−0.597478
$438$	0	0
$439$	−18.1466	−0.866091	−0.433046	−	0.901372i	$-0.642561\pi$
$439$	−18.1466	−0.866091	−0.433046	+	0.901372i	$0.642561\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.6475	0.600902	0.300451	−	0.953797i	$-0.402863\pi$
$443$	12.6475	0.600902	0.300451	+	0.953797i	$0.402863\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	20.7962	0.983625
$448$	0	0
$449$	18.4900	0.872597	0.436298	−	0.899802i	$-0.356289\pi$
$449$	18.4900	0.872597	0.436298	+	0.899802i	$0.356289\pi$
$450$	0	0
$451$	40.7922	1.92083
$452$	0	0
$453$	−19.4700	−0.914780
$454$	0	0
$455$	−9.79796	−0.459335
$456$	0	0
$457$	−36.9800	−1.72985	−0.864926	−	0.501900i	$-0.832634\pi$
$457$	−36.9800	−1.72985	−0.864926	+	0.501900i	$0.832634\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.4900	−1.32691	−0.663456	−	0.748216i	$-0.730911\pi$
$461$	−28.4900	−1.32691	−0.663456	+	0.748216i	$0.730911\pi$
$462$	0	0
$463$	−29.5939	−1.37535	−0.687673	−	0.726021i	$-0.741368\pi$
$463$	−29.5939	−1.37535	−0.687673	+	0.726021i	$0.741368\pi$
$464$	0	0
$465$	−18.0000	−0.834730
$466$	0	0
$467$	2.44949	0.113349	0.0566744	−	0.998393i	$-0.481950\pi$
$467$	2.44949	0.113349	0.0566744	+	0.998393i	$0.481950\pi$
$468$	0	0
$469$	24.4900	1.13084
$470$	0	0
$471$	42.2414	1.94638
$472$	0	0
$473$	−64.4900	−2.96525
$474$	0	0
$475$	2.44949	0.112390
$476$	0	0
$477$	24.0000	1.09888
$478$	0	0
$479$	−25.4951	−1.16490	−0.582450	−	0.812866i	$-0.697906\pi$
$479$	−25.4951	−1.16490	−0.582450	+	0.812866i	$0.697906\pi$
$480$	0	0
$481$	12.9800	0.591837
$482$	0	0
$483$	−30.5941	−1.39208
$484$	0	0
$485$	1.24500	0.0565324
$486$	0	0
$487$	−12.2474	−0.554985	−0.277492	−	0.960728i	$-0.589503\pi$
$487$	−12.2474	−0.554985	−0.277492	+	0.960728i	$0.589503\pi$
$488$	0	0
$489$	−30.9800	−1.40096
$490$	0	0
$491$	−12.6475	−0.570775	−0.285387	−	0.958412i	$-0.592122\pi$
$491$	−12.6475	−0.570775	−0.285387	+	0.958412i	$0.592122\pi$
$492$	0	0
$493$	7.24500	0.326298
$494$	0	0
$495$	15.2971	0.687552
$496$	0	0
$497$	18.4900	0.829390
$498$	0	0
$499$	29.3939	1.31585	0.657925	−	0.753083i	$-0.271434\pi$
$499$	29.3939	1.31585	0.657925	+	0.753083i	$0.271434\pi$
$500$	0	0
$501$	−36.4900	−1.63025
$502$	0	0
$503$	−16.7463	−0.746683	−0.373341	−	0.927694i	$-0.621788\pi$
$503$	−16.7463	−0.746683	−0.373341	+	0.927694i	$0.621788\pi$
$504$	0	0
$505$	−12.4900	−0.555798
$506$	0	0
$507$	7.34847	0.326357
$508$	0	0
$509$	12.9800	0.575328	0.287664	−	0.957731i	$-0.407121\pi$
$509$	12.9800	0.575328	0.287664	+	0.957731i	$0.407121\pi$
$510$	0	0
$511$	−32.4435	−1.43522
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.200040	−0.00881482
$516$	0	0
$517$	39.5100	1.73765
$518$	0	0
$519$	−25.6951	−1.12789
$520$	0	0
$521$	24.9800	1.09439	0.547197	−	0.837004i	$-0.315695\pi$
$521$	24.9800	1.09439	0.547197	+	0.837004i	$0.315695\pi$
$522$	0	0
$523$	20.5961	0.900605	0.450303	−	0.892876i	$-0.351316\pi$
$523$	20.5961	0.900605	0.450303	+	0.892876i	$0.351316\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	53.2396	2.31916
$528$	0	0
$529$	3.00000	0.130435
$530$	0	0
$531$	−37.3425	−1.62053
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	7.34847	0.317702
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	5.09902	0.219630
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−35.4931	−1.52315
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−27.3445	−1.16916	−0.584582	−	0.811334i	$-0.698742\pi$
$547$	−27.3445	−1.16916	−0.584582	+	0.811334i	$0.698742\pi$
$548$	0	0
$549$	31.4700	1.34311
$550$	0	0
$551$	2.44949	0.104352
$552$	0	0
$553$	11.5100	0.489455
$554$	0	0
$555$	−7.94859	−0.337399
$556$	0	0
$557$	32.4900	1.37665	0.688323	−	0.725405i	$-0.258347\pi$
$557$	32.4900	1.37665	0.688323	+	0.725405i	$0.258347\pi$
$558$	0	0
$559$	50.5901	2.13973
$560$	0	0
$561$	−90.4900	−3.82049
$562$	0	0
$563$	42.0414	1.77183	0.885917	−	0.463844i	$-0.153530\pi$
$563$	42.0414	1.77183	0.885917	+	0.463844i	$0.153530\pi$
$564$	0	0
$565$	−19.7350	−0.830257
$566$	0	0
$567$	−22.0454	−0.925820
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	−35.4931	−1.48534	−0.742670	−	0.669658i	$-0.766441\pi$
$571$	−35.4931	−1.48534	−0.742670	+	0.669658i	$0.766441\pi$
$572$	0	0
$573$	13.4700	0.562717
$574$	0	0
$575$	−5.09902	−0.212644
$576$	0	0
$577$	−12.7550	−0.530998	−0.265499	−	0.964111i	$-0.585537\pi$
$577$	−12.7550	−0.530998	−0.265499	+	0.964111i	$0.585537\pi$
$578$	0	0
$579$	−43.4417	−1.80537
$580$	0	0
$581$	−6.97999	−0.289579
$582$	0	0
$583$	−40.7922	−1.68944
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−4.69894	−0.193946	−0.0969730	−	0.995287i	$-0.530916\pi$
$587$	−4.69894	−0.193946	−0.0969730	+	0.995287i	$0.530916\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	4.89898	0.201517
$592$	0	0
$593$	6.49000	0.266512	0.133256	−	0.991082i	$-0.457457\pi$
$593$	6.49000	0.266512	0.133256	+	0.991082i	$0.457457\pi$
$594$	0	0
$595$	−17.7465	−0.727538
$596$	0	0
$597$	55.4700	2.27024
$598$	0	0
$599$	25.4951	1.04170	0.520851	−	0.853648i	$-0.325615\pi$
$599$	25.4951	1.04170	0.520851	+	0.853648i	$0.325615\pi$
$600$	0	0
$601$	−16.9800	−0.692628	−0.346314	−	0.938119i	$-0.612567\pi$
$601$	−16.9800	−0.692628	−0.346314	+	0.938119i	$0.612567\pi$
$602$	0	0
$603$	29.9940	1.22145
$604$	0	0
$605$	−15.0000	−0.609837
$606$	0	0
$607$	−16.7463	−0.679713	−0.339857	−	0.940477i	$-0.610379\pi$
$607$	−16.7463	−0.679713	−0.339857	+	0.940477i	$0.610379\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−30.9942	−1.25389
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	19.5959	0.790184
$616$	0	0
$617$	−34.2250	−1.37785	−0.688923	−	0.724834i	$-0.741916\pi$
$617$	−34.2250	−1.37785	−0.688923	+	0.724834i	$0.741916\pi$
$618$	0	0
$619$	15.2971	0.614841	0.307420	−	0.951574i	$-0.400534\pi$
$619$	15.2971	0.614841	0.307420	+	0.951574i	$0.400534\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.6969	−0.588820
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−30.5941	−1.22181
$628$	0	0
$629$	23.5100	0.937405
$630$	0	0
$631$	−45.2911	−1.80301	−0.901504	−	0.432770i	$-0.857536\pi$
$631$	−45.2911	−1.80301	−0.901504	+	0.432770i	$0.857536\pi$
$632$	0	0
$633$	−13.4700	−0.535384
$634$	0	0
$635$	12.2474	0.486025
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	22.6455	0.895843
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	34.4929	1.36027	0.680134	−	0.733088i	$-0.261922\pi$
$643$	34.4929	1.36027	0.680134	+	0.733088i	$0.261922\pi$
$644$	0	0
$645$	−30.9800	−1.21984
$646$	0	0
$647$	−27.3445	−1.07502	−0.537511	−	0.843257i	$-0.680635\pi$
$647$	−27.3445	−1.07502	−0.537511	+	0.843257i	$0.680635\pi$
$648$	0	0
$649$	63.4700	2.49141
$650$	0	0
$651$	44.0908	1.72806
$652$	0	0
$653$	−13.2450	−0.518317	−0.259158	−	0.965835i	$-0.583445\pi$
$653$	−13.2450	−0.518317	−0.259158	+	0.965835i	$0.583445\pi$
$654$	0	0
$655$	−17.5465	−0.685599
$656$	0	0
$657$	−39.7350	−1.55021
$658$	0	0
$659$	−32.2434	−1.25603	−0.628013	−	0.778203i	$-0.716132\pi$
$659$	−32.2434	−1.25603	−0.628013	+	0.778203i	$0.716132\pi$
$660$	0	0
$661$	0.979992	0.0381173	0.0190586	−	0.999818i	$-0.493933\pi$
$661$	0.979992	0.0381173	0.0190586	+	0.999818i	$0.493933\pi$
$662$	0	0
$663$	70.9862	2.75688
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−5.09902	−0.197435
$668$	0	0
$669$	−0.489996	−0.0189443
$670$	0	0
$671$	−53.4887	−2.06491
$672$	0	0
$673$	50.9800	1.96513	0.982567	−	0.185908i	$-0.0595227\pi$
$673$	50.9800	1.96513	0.982567	+	0.185908i	$0.0595227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.24500	−0.201582	−0.100791	−	0.994908i	$-0.532137\pi$
$677$	−5.24500	−0.201582	−0.100791	+	0.994908i	$0.532137\pi$
$678$	0	0
$679$	−3.04961	−0.117033
$680$	0	0
$681$	0.489996	0.0187767
$682$	0	0
$683$	29.5939	1.13238	0.566190	−	0.824275i	$-0.308417\pi$
$683$	29.5939	1.13238	0.566190	+	0.824275i	$0.308417\pi$
$684$	0	0
$685$	−7.24500	−0.276817
$686$	0	0
$687$	34.2929	1.30835
$688$	0	0
$689$	32.0000	1.21910
$690$	0	0
$691$	−23.0456	−0.876696	−0.438348	−	0.898805i	$-0.644436\pi$
$691$	−23.0456	−0.876696	−0.438348	+	0.898805i	$0.644436\pi$
$692$	0	0
$693$	−37.4700	−1.42337
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−57.9600	−2.19539
$698$	0	0
$699$	10.9982	0.415990
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	7.94859	0.299787
$704$	0	0
$705$	18.9800	0.714828
$706$	0	0
$707$	30.5941	1.15061
$708$	0	0
$709$	24.0000	0.901339	0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000	0.901339	0.450669	+	0.892691i	$0.351185\pi$
$710$	0	0
$711$	14.0968	0.528672
$712$	0	0
$713$	−37.4700	−1.40326
$714$	0	0
$715$	20.3961	0.762770
$716$	0	0
$717$	30.4900	1.13867
$718$	0	0
$719$	−3.04961	−0.113731	−0.0568656	−	0.998382i	$-0.518111\pi$
$719$	−3.04961	−0.113731	−0.0568656	+	0.998382i	$0.518111\pi$
$720$	0	0
$721$	0.489996	0.0182484
$722$	0	0
$723$	−66.0872	−2.45781
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	16.7463	0.621088	0.310544	−	0.950559i	$-0.399489\pi$
$727$	16.7463	0.621088	0.310544	+	0.950559i	$0.399489\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	91.6313	3.38911
$732$	0	0
$733$	−24.7550	−0.914347	−0.457173	−	0.889378i	$-0.651138\pi$
$733$	−24.7550	−0.914347	−0.457173	+	0.889378i	$0.651138\pi$
$734$	0	0
$735$	2.44949	0.0903508
$736$	0	0
$737$	−50.9800	−1.87787
$738$	0	0
$739$	17.5465	0.645459	0.322729	−	0.946491i	$-0.395400\pi$
$739$	17.5465	0.645459	0.322729	+	0.946491i	$0.395400\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	−22.4455	−0.823445	−0.411723	−	0.911309i	$-0.635073\pi$
$743$	−22.4455	−0.823445	−0.411723	+	0.911309i	$0.635073\pi$
$744$	0	0
$745$	−8.49000	−0.311049
$746$	0	0
$747$	−8.54871	−0.312781
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	12.6475	0.461515	0.230757	−	0.973011i	$-0.425880\pi$
$751$	12.6475	0.461515	0.230757	+	0.973011i	$0.425880\pi$
$752$	0	0
$753$	−48.4900	−1.76707
$754$	0	0
$755$	7.94859	0.289279
$756$	0	0
$757$	−11.7350	−0.426516	−0.213258	−	0.976996i	$-0.568407\pi$
$757$	−11.7350	−0.426516	−0.213258	+	0.976996i	$0.568407\pi$
$758$	0	0
$759$	63.6867	2.31168
$760$	0	0
$761$	51.9600	1.88355	0.941774	−	0.336247i	$-0.109158\pi$
$761$	51.9600	1.88355	0.941774	+	0.336247i	$0.109158\pi$
$762$	0	0
$763$	39.1918	1.41884
$764$	0	0
$765$	−21.7350	−0.785830
$766$	0	0
$767$	−49.7900	−1.79781
$768$	0	0
$769$	20.4900	0.738888	0.369444	−	0.929253i	$-0.379548\pi$
$769$	20.4900	0.738888	0.369444	+	0.929253i	$0.379548\pi$
$770$	0	0
$771$	35.4931	1.27825
$772$	0	0
$773$	21.7350	0.781753	0.390877	−	0.920443i	$-0.372172\pi$
$773$	21.7350	0.781753	0.390877	+	0.920443i	$0.372172\pi$
$774$	0	0
$775$	7.34847	0.263965
$776$	0	0
$777$	19.4700	0.698482
$778$	0	0
$779$	−19.5959	−0.702097
$780$	0	0
$781$	−38.4900	−1.37728
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.2450	−0.615500
$786$	0	0
$787$	6.54831	0.233422	0.116711	−	0.993166i	$-0.462765\pi$
$787$	6.54831	0.233422	0.116711	+	0.993166i	$0.462765\pi$
$788$	0	0
$789$	30.9800	1.10292
$790$	0	0
$791$	48.3407	1.71880
$792$	0	0
$793$	41.9600	1.49004
$794$	0	0
$795$	−19.5959	−0.694996
$796$	0	0
$797$	−22.7550	−0.806024	−0.403012	−	0.915195i	$-0.632037\pi$
$797$	−22.7550	−0.806024	−0.403012	+	0.915195i	$0.632037\pi$
$798$	0	0
$799$	−56.1382	−1.98603
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	67.5365	2.38331
$804$	0	0
$805$	12.4900	0.440215
$806$	0	0
$807$	−58.7878	−2.06943
$808$	0	0
$809$	2.97999	0.104771	0.0523855	−	0.998627i	$-0.483318\pi$
$809$	2.97999	0.104771	0.0523855	+	0.998627i	$0.483318\pi$
$810$	0	0
$811$	−19.9960	−0.702154	−0.351077	−	0.936347i	$-0.614185\pi$
$811$	−19.9960	−0.702154	−0.351077	+	0.936347i	$0.614185\pi$
$812$	0	0
$813$	37.4700	1.31413
$814$	0	0
$815$	12.6475	0.443024
$816$	0	0
$817$	30.9800	1.08385
$818$	0	0
$819$	29.3939	1.02711
$820$	0	0
$821$	29.4700	1.02851	0.514255	−	0.857637i	$-0.328069\pi$
$821$	29.4700	1.02851	0.514255	+	0.857637i	$0.328069\pi$
$822$	0	0
$823$	−2.84957	−0.0993298	−0.0496649	−	0.998766i	$-0.515815\pi$
$823$	−2.84957	−0.0993298	−0.0496649	+	0.998766i	$0.515815\pi$
$824$	0	0
$825$	−12.4900	−0.434846
$826$	0	0
$827$	−17.9466	−0.624064	−0.312032	−	0.950072i	$-0.601010\pi$
$827$	−17.9466	−0.624064	−0.312032	+	0.950072i	$0.601010\pi$
$828$	0	0
$829$	−4.97999	−0.172962	−0.0864811	−	0.996253i	$-0.527562\pi$
$829$	−4.97999	−0.172962	−0.0864811	+	0.996253i	$0.527562\pi$
$830$	0	0
$831$	−70.9862	−2.46248
$832$	0	0
$833$	−7.24500	−0.251024
$834$	0	0
$835$	14.8970	0.515531
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.4929	1.19083	0.595414	−	0.803419i	$-0.296988\pi$
$839$	34.4929	1.19083	0.595414	+	0.803419i	$0.296988\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	29.3939	1.01238
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	36.7423	1.26248
$848$	0	0
$849$	−13.4700	−0.462289
$850$	0	0
$851$	−16.5463	−0.567200
$852$	0	0
$853$	−33.7350	−1.15506	−0.577532	−	0.816368i	$-0.695984\pi$
$853$	−33.7350	−1.15506	−0.577532	+	0.816368i	$0.695984\pi$
$854$	0	0
$855$	−7.34847	−0.251312
$856$	0	0
$857$	31.4700	1.07499	0.537497	−	0.843266i	$-0.319370\pi$
$857$	31.4700	1.07499	0.537497	+	0.843266i	$0.319370\pi$
$858$	0	0
$859$	−29.1938	−0.996081	−0.498040	−	0.867154i	$-0.665947\pi$
$859$	−29.1938	−0.996081	−0.498040	+	0.867154i	$0.665947\pi$
$860$	0	0
$861$	−48.0000	−1.63584
$862$	0	0
$863$	17.5465	0.597290	0.298645	−	0.954364i	$-0.403465\pi$
$863$	17.5465	0.597290	0.298645	+	0.954364i	$0.403465\pi$
$864$	0	0
$865$	10.4900	0.356671
$866$	0	0
$867$	86.9324	2.95238
$868$	0	0
$869$	−23.9600	−0.812787
$870$	0	0
$871$	39.9920	1.35508
$872$	0	0
$873$	−3.73499	−0.126410
$874$	0	0
$875$	−2.44949	−0.0828079
$876$	0	0
$877$	8.97999	0.303233	0.151616	−	0.988439i	$-0.451552\pi$
$877$	8.97999	0.303233	0.151616	+	0.988439i	$0.451552\pi$
$878$	0	0
$879$	−22.6455	−0.763815
$880$	0	0
$881$	−18.4900	−0.622944	−0.311472	−	0.950255i	$-0.600822\pi$
$881$	−18.4900	−0.622944	−0.311472	+	0.950255i	$0.600822\pi$
$882$	0	0
$883$	22.0454	0.741887	0.370944	−	0.928655i	$-0.379034\pi$
$883$	22.0454	0.741887	0.370944	+	0.928655i	$0.379034\pi$
$884$	0	0
$885$	30.4900	1.02491
$886$	0	0
$887$	13.0476	0.438096	0.219048	−	0.975714i	$-0.429705\pi$
$887$	13.0476	0.438096	0.219048	+	0.975714i	$0.429705\pi$
$888$	0	0
$889$	−30.0000	−1.00617
$890$	0	0
$891$	45.8912	1.53741
$892$	0	0
$893$	−18.9800	−0.635141
$894$	0	0
$895$	−14.6969	−0.491264
$896$	0	0
$897$	−49.9600	−1.66812
$898$	0	0
$899$	7.34847	0.245085
$900$	0	0
$901$	57.9600	1.93093
$902$	0	0
$903$	75.8852	2.52530
$904$	0	0
$905$	14.4900	0.481664
$906$	0	0
$907$	−22.4455	−0.745290	−0.372645	−	0.927974i	$-0.621549\pi$
$907$	−22.4455	−0.745290	−0.372645	+	0.927974i	$0.621549\pi$
$908$	0	0
$909$	37.4700	1.24280
$910$	0	0
$911$	8.14863	0.269976	0.134988	−	0.990847i	$-0.456900\pi$
$911$	8.14863	0.269976	0.134988	+	0.990847i	$0.456900\pi$
$912$	0	0
$913$	14.5300	0.480873
$914$	0	0
$915$	−25.6951	−0.849455
$916$	0	0
$917$	42.9800	1.41932
$918$	0	0
$919$	−12.4475	−0.410605	−0.205302	−	0.978699i	$-0.565818\pi$
$919$	−12.4475	−0.410605	−0.205302	+	0.978699i	$0.565818\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	30.1940	0.993849
$924$	0	0
$925$	3.24500	0.106695
$926$	0	0
$927$	0.600120	0.0197105
$928$	0	0
$929$	−49.4700	−1.62306	−0.811529	−	0.584312i	$-0.801364\pi$
$929$	−49.4700	−1.62306	−0.811529	+	0.584312i	$0.801364\pi$
$930$	0	0
$931$	−2.44949	−0.0802788
$932$	0	0
$933$	−6.00000	−0.196431
$934$	0	0
$935$	36.9424	1.20815
$936$	0	0
$937$	−9.51000	−0.310678	−0.155339	−	0.987861i	$-0.549647\pi$
$937$	−9.51000	−0.310678	−0.155339	+	0.987861i	$0.549647\pi$
$938$	0	0
$939$	−68.5857	−2.23821
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	40.7922	1.32838
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.9466	−0.583186	−0.291593	−	0.956543i	$-0.594185\pi$
$947$	−17.9466	−0.583186	−0.291593	+	0.956543i	$0.594185\pi$
$948$	0	0
$949$	−52.9800	−1.71980
$950$	0	0
$951$	66.7363	2.16407
$952$	0	0
$953$	50.9800	1.65140	0.825702	−	0.564107i	$-0.190779\pi$
$953$	50.9800	1.65140	0.825702	+	0.564107i	$0.190779\pi$
$954$	0	0
$955$	−5.49910	−0.177947
$956$	0	0
$957$	−12.4900	−0.403744
$958$	0	0
$959$	17.7465	0.573066
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	−22.0454	−0.710403
$964$	0	0
$965$	17.7350	0.570910
$966$	0	0
$967$	−28.1446	−0.905070	−0.452535	−	0.891747i	$-0.649480\pi$
$967$	−28.1446	−0.905070	−0.452535	+	0.891747i	$0.649480\pi$
$968$	0	0
$969$	43.4700	1.39646
$970$	0	0
$971$	22.8456	0.733149	0.366575	−	0.930389i	$-0.380530\pi$
$971$	22.8456	0.733149	0.366575	+	0.930389i	$0.380530\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9.79796	0.313786
$976$	0	0
$977$	18.4900	0.591547	0.295774	−	0.955258i	$-0.404423\pi$
$977$	18.4900	0.591547	0.295774	+	0.955258i	$0.404423\pi$
$978$	0	0
$979$	30.5941	0.977792
$980$	0	0
$981$	48.0000	1.53252
$982$	0	0
$983$	16.7463	0.534126	0.267063	−	0.963679i	$-0.413947\pi$
$983$	16.7463	0.534126	0.267063	+	0.963679i	$0.413947\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−46.4913	−1.47983
$988$	0	0
$989$	−64.4900	−2.05066
$990$	0	0
$991$	−2.64953	−0.0841651	−0.0420825	−	0.999114i	$-0.513399\pi$
$991$	−2.64953	−0.0841651	−0.0420825	+	0.999114i	$0.513399\pi$
$992$	0	0
$993$	−66.9800	−2.12555
$994$	0	0
$995$	−22.6455	−0.717911
$996$	0	0
$997$	−38.2250	−1.21060	−0.605299	−	0.795998i	$-0.706946\pi$
$997$	−38.2250	−1.21060	−0.605299	+	0.795998i	$0.706946\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4640.2.a.o.1.3	yes	4
4.3	odd	2	inner	4640.2.a.o.1.2	✓	4
8.3	odd	2		9280.2.a.cd.1.3		4
8.5	even	2		9280.2.a.cd.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.o.1.2	✓	4	4.3	odd	2	inner
4640.2.a.o.1.3	yes	4	1.1	even	1	trivial
9280.2.a.cd.1.2		4	8.5	even	2
9280.2.a.cd.1.3		4	8.3	odd	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:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} -5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} +7.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} -5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} -12.4900 q^{33} -2.44949 q^{35} +3.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} +12.6475 q^{43} -3.00000 q^{45} -7.74855 q^{47} -1.00000 q^{49} +17.7465 q^{51} +8.00000 q^{53} +5.09902 q^{55} +6.00000 q^{57} -12.4475 q^{59} +10.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} +9.99800 q^{67} -12.4900 q^{69} +7.54851 q^{71} -13.2450 q^{73} +2.44949 q^{75} -12.4900 q^{77} +4.69894 q^{79} -9.00000 q^{81} -2.84957 q^{83} -7.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} -1.24500 q^{97} -15.2971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 12 * q^9 + 16 * q^13 + 4 * q^17 + 24 * q^21 + 4 * q^25 + 4 * q^29 - 12 * q^37 - 32 * q^41 - 12 * q^45 - 4 * q^49 + 32 * q^53 + 24 * q^57 - 8 * q^61 - 16 * q^65 - 28 * q^73 - 36 * q^81 - 4 * q^85 - 24 * q^89 + 72 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

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Database

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