LMFDB - Embedded newform 676.4.d.d.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,4,Mod(337,676)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("676.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	676.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.8852911639$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 $ x^8 + 51x^6 - 224x^5 + 2520x^4 - 5712x^3 + 16675x^2 + 9072x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-0.287051 - 0.497187i$ of defining polynomial
Character	$\chi$	$=$	676.337
Dual form			676.4.d.d.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.78841 q^{3} +2.49640i q^{5} +17.9693i q^{7} +68.8129 q^{9} +O(q^{10})$	$q-9.78841 q^{3} +2.49640i q^{5} +17.9693i q^{7} +68.8129 q^{9} +57.2769i q^{11} -24.4358i q^{15} +17.5058 q^{17} -22.1225i q^{19} -175.891i q^{21} +131.193 q^{23} +118.768 q^{25} -409.282 q^{27} +74.2527 q^{29} -110.463i q^{31} -560.649i q^{33} -44.8587 q^{35} +241.452i q^{37} -426.704i q^{41} +208.296 q^{43} +171.785i q^{45} -158.624i q^{47} +20.1031 q^{49} -171.354 q^{51} +47.9103 q^{53} -142.986 q^{55} +216.544i q^{57} +441.432i q^{59} +869.398 q^{61} +1236.52i q^{63} +801.045i q^{67} -1284.17 q^{69} +893.624i q^{71} +413.180i q^{73} -1162.55 q^{75} -1029.23 q^{77} -246.527 q^{79} +2148.27 q^{81} +333.857i q^{83} +43.7016i q^{85} -726.816 q^{87} -379.176i q^{89} +1081.26i q^{93} +55.2266 q^{95} -941.263i q^{97} +3941.39i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 140 q^{9}+O(q^{10}) $ 8 * q + 140 * q^9	$ 8 q + 140 q^{9} - 176 q^{17} + 40 q^{23} - 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} - 1008 q^{43} - 1844 q^{49} - 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} - 3476 q^{69} - 2896 q^{75} - 3972 q^{77} - 3968 q^{79} + 8264 q^{81} + 3320 q^{87} - 4800 q^{95}+O(q^{100}) $ 8 * q + 140 * q^9 - 176 * q^17 + 40 * q^23 - 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 - 1008 * q^43 - 1844 * q^49 - 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 - 3476 * q^69 - 2896 * q^75 - 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 - 4800 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.78841	−1.88378	−0.941890	−	0.335922i	$-0.890952\pi$
$3$	−9.78841	−1.88378	−0.941890	+	0.335922i	$0.890952\pi$
$4$	0	0
$5$	2.49640i	0.223285i	0.993748	+	0.111643i	$0.0356112\pi$
$5$	2.49640i	0.223285i	−0.993748	+	0.111643i	$0.964389\pi$
$6$	0	0
$7$	17.9693i	0.970253i	0.874444	+	0.485126i	$0.161226\pi$
$7$	17.9693i	0.970253i	−0.874444	+	0.485126i	$0.838774\pi$
$8$	0	0
$9$	68.8129	2.54863
$10$	0	0
$11$	57.2769i	1.56997i	0.619517	+	0.784983i	$0.287328\pi$
$11$	57.2769i	1.56997i	−0.619517	+	0.784983i	$0.712672\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 24.4358i	− 0.420620i
$16$	0	0
$17$	17.5058	0.249752	0.124876	−	0.992172i	$-0.460147\pi$
$17$	17.5058	0.249752	0.124876	+	0.992172i	$0.460147\pi$
$18$	0	0
$19$	− 22.1225i	− 0.267118i	−0.991041	−	0.133559i	$-0.957359\pi$
$19$	− 22.1225i	− 0.267118i	0.991041	−	0.133559i	$-0.0426406\pi$
$20$	0	0
$21$	− 175.891i	− 1.82774i
$22$	0	0
$23$	131.193	1.18937	0.594686	−	0.803958i	$-0.297276\pi$
$23$	131.193	1.18937	0.594686	+	0.803958i	$0.297276\pi$
$24$	0	0
$25$	118.768	0.950144
$26$	0	0
$27$	−409.282	−2.91727
$28$	0	0
$29$	74.2527	0.475462	0.237731	−	0.971331i	$-0.423596\pi$
$29$	74.2527	0.475462	0.237731	+	0.971331i	$0.423596\pi$
$30$	0	0
$31$	− 110.463i	− 0.639994i	−0.947419	−	0.319997i	$-0.896318\pi$
$31$	− 110.463i	− 0.639994i	0.947419	−	0.319997i	$-0.103682\pi$
$32$	0	0
$33$	− 560.649i	− 2.95747i
$34$	0	0
$35$	−44.8587	−0.216643
$36$	0	0
$37$	241.452i	1.07282i	0.843956	+	0.536412i	$0.180221\pi$
$37$	241.452i	1.07282i	−0.843956	+	0.536412i	$0.819779\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 426.704i	− 1.62537i	−0.582706	−	0.812683i	$-0.698006\pi$
$41$	− 426.704i	− 1.62537i	0.582706	−	0.812683i	$-0.301994\pi$
$42$	0	0
$43$	208.296	0.738718	0.369359	−	0.929287i	$-0.379577\pi$
$43$	208.296	0.738718	0.369359	+	0.929287i	$0.379577\pi$
$44$	0	0
$45$	171.785i	0.569070i
$46$	0	0
$47$	− 158.624i	− 0.492292i	−0.969233	−	0.246146i	$-0.920836\pi$
$47$	− 158.624i	− 0.492292i	0.969233	−	0.246146i	$-0.0791642\pi$
$48$	0	0
$49$	20.1031	0.0586096
$50$	0	0
$51$	−171.354	−0.470478
$52$	0	0
$53$	47.9103	0.124170	0.0620848	−	0.998071i	$-0.480225\pi$
$53$	47.9103	0.124170	0.0620848	+	0.998071i	$0.480225\pi$
$54$	0	0
$55$	−142.986	−0.350550
$56$	0	0
$57$	216.544i	0.503191i
$58$	0	0
$59$	441.432i	0.974060i	0.873385	+	0.487030i	$0.161920\pi$
$59$	441.432i	0.974060i	−0.873385	+	0.487030i	$0.838080\pi$
$60$	0	0
$61$	869.398	1.82484	0.912419	−	0.409258i	$-0.134213\pi$
$61$	869.398	1.82484	0.912419	+	0.409258i	$0.134213\pi$
$62$	0	0
$63$	1236.52i	2.47281i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	801.045i	1.46065i	0.683102	+	0.730323i	$0.260630\pi$
$67$	801.045i	1.46065i	−0.683102	+	0.730323i	$0.739370\pi$
$68$	0	0
$69$	−1284.17	−2.24052
$70$	0	0
$71$	893.624i	1.49371i	0.664985	+	0.746857i	$0.268438\pi$
$71$	893.624i	1.49371i	−0.664985	+	0.746857i	$0.731562\pi$
$72$	0	0
$73$	413.180i	0.662452i	0.943551	+	0.331226i	$0.107462\pi$
$73$	413.180i	0.662452i	−0.943551	+	0.331226i	$0.892538\pi$
$74$	0	0
$75$	−1162.55	−1.78986
$76$	0	0
$77$	−1029.23	−1.52326
$78$	0	0
$79$	−246.527	−0.351094	−0.175547	−	0.984471i	$-0.556169\pi$
$79$	−246.527	−0.351094	−0.175547	+	0.984471i	$0.556169\pi$
$80$	0	0
$81$	2148.27	2.94687
$82$	0	0
$83$	333.857i	0.441513i	0.975329	+	0.220756i	$0.0708526\pi$
$83$	333.857i	0.441513i	−0.975329	+	0.220756i	$0.929147\pi$
$84$	0	0
$85$	43.7016i	0.0557660i
$86$	0	0
$87$	−726.816	−0.895665
$88$	0	0
$89$	− 379.176i	− 0.451602i	−0.974174	−	0.225801i	$-0.927500\pi$
$89$	− 379.176i	− 0.451602i	0.974174	−	0.225801i	$-0.0724998\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1081.26i	1.20561i
$94$	0	0
$95$	55.2266	0.0596435
$96$	0	0
$97$	− 941.263i	− 0.985266i	−0.870237	−	0.492633i	$-0.836035\pi$
$97$	− 941.263i	− 0.985266i	0.870237	−	0.492633i	$-0.163965\pi$
$98$	0	0
$99$	3941.39i	4.00126i
$100$	0	0
$101$	−385.949	−0.380232	−0.190116	−	0.981762i	$-0.560886\pi$
$101$	−385.949	−0.380232	−0.190116	+	0.981762i	$0.560886\pi$
$102$	0	0
$103$	395.674	0.378514	0.189257	−	0.981928i	$-0.439392\pi$
$103$	395.674	0.378514	0.189257	+	0.981928i	$0.439392\pi$
$104$	0	0
$105$	439.095	0.408108
$106$	0	0
$107$	1228.14	1.10961	0.554807	−	0.831979i	$-0.312792\pi$
$107$	1228.14	1.10961	0.554807	+	0.831979i	$0.312792\pi$
$108$	0	0
$109$	542.176i	0.476431i	0.971212	+	0.238216i	$0.0765625\pi$
$109$	542.176i	0.476431i	−0.971212	+	0.238216i	$0.923438\pi$
$110$	0	0
$111$	− 2363.43i	− 2.02096i
$112$	0	0
$113$	−677.411	−0.563942	−0.281971	−	0.959423i	$-0.590988\pi$
$113$	−677.411	−0.563942	−0.281971	+	0.959423i	$0.590988\pi$
$114$	0	0
$115$	327.510i	0.265569i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	314.568i	0.242323i
$120$	0	0
$121$	−1949.64	−1.46479
$122$	0	0
$123$	4176.75i	3.06183i
$124$	0	0
$125$	608.543i	0.435438i
$126$	0	0
$127$	−1271.86	−0.888656	−0.444328	−	0.895864i	$-0.646558\pi$
$127$	−1271.86	−0.888656	−0.444328	+	0.895864i	$0.646558\pi$
$128$	0	0
$129$	−2038.89	−1.39158
$130$	0	0
$131$	1249.24	0.833179	0.416589	−	0.909095i	$-0.363225\pi$
$131$	1249.24	0.833179	0.416589	+	0.909095i	$0.363225\pi$
$132$	0	0
$133$	397.526	0.259172
$134$	0	0
$135$	− 1021.73i	− 0.651383i
$136$	0	0
$137$	1742.91i	1.08691i	0.839439	+	0.543454i	$0.182884\pi$
$137$	1742.91i	1.08691i	−0.839439	+	0.543454i	$0.817116\pi$
$138$	0	0
$139$	988.883	0.603424	0.301712	−	0.953399i	$-0.402442\pi$
$139$	988.883	0.603424	0.301712	+	0.953399i	$0.402442\pi$
$140$	0	0
$141$	1552.68i	0.927369i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	185.365i	0.106164i
$146$	0	0
$147$	−196.777	−0.110408
$148$	0	0
$149$	− 1696.58i	− 0.932815i	−0.884570	−	0.466408i	$-0.845548\pi$
$149$	− 1696.58i	− 0.932815i	0.884570	−	0.466408i	$-0.154452\pi$
$150$	0	0
$151$	− 2603.11i	− 1.40290i	−0.712719	−	0.701449i	$-0.752537\pi$
$151$	− 2603.11i	− 1.40290i	0.712719	−	0.701449i	$-0.247463\pi$
$152$	0	0
$153$	1204.63	0.636525
$154$	0	0
$155$	275.761	0.142901
$156$	0	0
$157$	−178.988	−0.0909859	−0.0454930	−	0.998965i	$-0.514486\pi$
$157$	−178.988	−0.0909859	−0.0454930	+	0.998965i	$0.514486\pi$
$158$	0	0
$159$	−468.966	−0.233908
$160$	0	0
$161$	2357.44i	1.15399i
$162$	0	0
$163$	2080.21i	0.999599i	0.866141	+	0.499799i	$0.166593\pi$
$163$	2080.21i	0.999599i	−0.866141	+	0.499799i	$0.833407\pi$
$164$	0	0
$165$	1399.61	0.660359
$166$	0	0
$167$	3039.90i	1.40859i	0.709907	+	0.704295i	$0.248737\pi$
$167$	3039.90i	1.40859i	−0.709907	+	0.704295i	$0.751263\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 1522.31i	− 0.680784i
$172$	0	0
$173$	−51.7355	−0.0227363	−0.0113681	−	0.999935i	$-0.503619\pi$
$173$	−51.7355	−0.0227363	−0.0113681	+	0.999935i	$0.503619\pi$
$174$	0	0
$175$	2134.18i	0.921880i
$176$	0	0
$177$	− 4320.92i	− 1.83492i
$178$	0	0
$179$	−628.894	−0.262602	−0.131301	−	0.991343i	$-0.541915\pi$
$179$	−628.894	−0.262602	−0.131301	+	0.991343i	$0.541915\pi$
$180$	0	0
$181$	−1661.09	−0.682145	−0.341072	−	0.940037i	$-0.610790\pi$
$181$	−1661.09	−0.682145	−0.341072	+	0.940037i	$0.610790\pi$
$182$	0	0
$183$	−8510.03	−3.43759
$184$	0	0
$185$	−602.762	−0.239546
$186$	0	0
$187$	1002.68i	0.392103i
$188$	0	0
$189$	− 7354.52i	− 2.83049i
$190$	0	0
$191$	1917.94	0.726581	0.363291	−	0.931676i	$-0.381653\pi$
$191$	1917.94	0.726581	0.363291	+	0.931676i	$0.381653\pi$
$192$	0	0
$193$	231.092i	0.0861883i	0.999071	+	0.0430942i	$0.0137215\pi$
$193$	231.092i	0.0861883i	−0.999071	+	0.0430942i	$0.986278\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1630.42i	− 0.589656i	−0.955550	−	0.294828i	$-0.904738\pi$
$197$	− 1630.42i	− 0.589656i	0.955550	−	0.294828i	$-0.0952624\pi$
$198$	0	0
$199$	3065.57	1.09202	0.546011	−	0.837778i	$-0.316146\pi$
$199$	3065.57	1.09202	0.546011	+	0.837778i	$0.316146\pi$
$200$	0	0
$201$	− 7840.96i	− 2.75154i
$202$	0	0
$203$	1334.27i	0.461318i
$204$	0	0
$205$	1065.23	0.362920
$206$	0	0
$207$	9027.75	3.03127
$208$	0	0
$209$	1267.11	0.419366
$210$	0	0
$211$	−3449.79	−1.12556	−0.562780	−	0.826606i	$-0.690268\pi$
$211$	−3449.79	−1.12556	−0.562780	+	0.826606i	$0.690268\pi$
$212$	0	0
$213$	− 8747.16i	− 2.81383i
$214$	0	0
$215$	519.991i	0.164945i
$216$	0	0
$217$	1984.95	0.620956
$218$	0	0
$219$	− 4044.37i	− 1.24791i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2524.48i	− 0.758080i	−0.925380	−	0.379040i	$-0.876254\pi$
$223$	− 2524.48i	− 0.758080i	0.925380	−	0.379040i	$-0.123746\pi$
$224$	0	0
$225$	8172.77	2.42156
$226$	0	0
$227$	− 1153.06i	− 0.337142i	−0.985689	−	0.168571i	$-0.946085\pi$
$227$	− 1153.06i	− 0.337142i	0.985689	−	0.168571i	$-0.0539153\pi$
$228$	0	0
$229$	− 3857.53i	− 1.11316i	−0.830795	−	0.556578i	$-0.812114\pi$
$229$	− 3857.53i	− 1.11316i	0.830795	−	0.556578i	$-0.187886\pi$
$230$	0	0
$231$	10074.5	2.86949
$232$	0	0
$233$	1018.55	0.286385	0.143192	−	0.989695i	$-0.454263\pi$
$233$	1018.55	0.286385	0.143192	+	0.989695i	$0.454263\pi$
$234$	0	0
$235$	395.990	0.109921
$236$	0	0
$237$	2413.11	0.661384
$238$	0	0
$239$	1263.24i	0.341893i	0.985280	+	0.170947i	$0.0546825\pi$
$239$	1263.24i	0.341893i	−0.985280	+	0.170947i	$0.945317\pi$
$240$	0	0
$241$	2067.94i	0.552729i	0.961053	+	0.276365i	$0.0891297\pi$
$241$	2067.94i	0.552729i	−0.961053	+	0.276365i	$0.910870\pi$
$242$	0	0
$243$	−9977.52	−2.63398
$244$	0	0
$245$	50.1855i	0.0130867i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 3267.93i	− 0.831713i
$250$	0	0
$251$	−5126.38	−1.28914	−0.644570	−	0.764545i	$-0.722964\pi$
$251$	−5126.38	−1.28914	−0.644570	+	0.764545i	$0.722964\pi$
$252$	0	0
$253$	7514.31i	1.86727i
$254$	0	0
$255$	− 427.769i	− 0.105051i
$256$	0	0
$257$	−3961.52	−0.961528	−0.480764	−	0.876850i	$-0.659641\pi$
$257$	−3961.52	−0.961528	−0.480764	+	0.876850i	$0.659641\pi$
$258$	0	0
$259$	−4338.73	−1.04091
$260$	0	0
$261$	5109.55	1.21177
$262$	0	0
$263$	−7104.63	−1.66574	−0.832871	−	0.553468i	$-0.813304\pi$
$263$	−7104.63	−1.66574	−0.832871	+	0.553468i	$0.813304\pi$
$264$	0	0
$265$	119.604i	0.0277252i
$266$	0	0
$267$	3711.53i	0.850718i
$268$	0	0
$269$	5613.88	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$269$	5613.88	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$270$	0	0
$271$	441.205i	0.0988978i	0.998777	+	0.0494489i	$0.0157465\pi$
$271$	441.205i	0.0988978i	−0.998777	+	0.0494489i	$0.984254\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6802.66i	1.49169i
$276$	0	0
$277$	−4676.87	−1.01446	−0.507231	−	0.861810i	$-0.669331\pi$
$277$	−4676.87	−1.01446	−0.507231	+	0.861810i	$0.669331\pi$
$278$	0	0
$279$	− 7601.31i	− 1.63110i
$280$	0	0
$281$	− 1253.34i	− 0.266079i	−0.991111	−	0.133040i	$-0.957526\pi$
$281$	− 1253.34i	− 0.266079i	0.991111	−	0.133040i	$-0.0424737\pi$
$282$	0	0
$283$	−4172.15	−0.876356	−0.438178	−	0.898888i	$-0.644376\pi$
$283$	−4172.15	−0.876356	−0.438178	+	0.898888i	$0.644376\pi$
$284$	0	0
$285$	−540.580	−0.112355
$286$	0	0
$287$	7667.59	1.57702
$288$	0	0
$289$	−4606.55	−0.937624
$290$	0	0
$291$	9213.47i	1.85602i
$292$	0	0
$293$	9779.07i	1.94983i	0.222585	+	0.974913i	$0.428550\pi$
$293$	9779.07i	1.94983i	−0.222585	+	0.974913i	$0.571450\pi$
$294$	0	0
$295$	−1101.99	−0.217493
$296$	0	0
$297$	− 23442.4i	− 4.58002i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3742.94i	0.716743i
$302$	0	0
$303$	3777.83	0.716273
$304$	0	0
$305$	2170.37i	0.407459i
$306$	0	0
$307$	− 1641.63i	− 0.305187i	−0.988289	−	0.152594i	$-0.951237\pi$
$307$	− 1641.63i	− 0.305187i	0.988289	−	0.152594i	$-0.0487626\pi$
$308$	0	0
$309$	−3873.02	−0.713037
$310$	0	0
$311$	−5900.48	−1.07584	−0.537919	−	0.842996i	$-0.680789\pi$
$311$	−5900.48	−1.07584	−0.537919	+	0.842996i	$0.680789\pi$
$312$	0	0
$313$	5268.10	0.951344	0.475672	−	0.879623i	$-0.342205\pi$
$313$	5268.10	0.951344	0.475672	+	0.879623i	$0.342205\pi$
$314$	0	0
$315$	−3086.86	−0.552142
$316$	0	0
$317$	20.4536i	0.00362395i	0.999998	+	0.00181197i	$0.000576769\pi$
$317$	20.4536i	0.00362395i	−0.999998	+	0.00181197i	$0.999423\pi$
$318$	0	0
$319$	4252.96i	0.746459i
$320$	0	0
$321$	−12021.5	−2.09027
$322$	0	0
$323$	− 387.272i	− 0.0667133i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5307.03i	− 0.897491i
$328$	0	0
$329$	2850.37	0.477647
$330$	0	0
$331$	− 6302.42i	− 1.04656i	−0.852160	−	0.523282i	$-0.824708\pi$
$331$	− 6302.42i	− 1.04656i	0.852160	−	0.523282i	$-0.175292\pi$
$332$	0	0
$333$	16615.0i	2.73423i
$334$	0	0
$335$	−1999.73	−0.326141
$336$	0	0
$337$	3060.14	0.494648	0.247324	−	0.968933i	$-0.420449\pi$
$337$	3060.14	0.494648	0.247324	+	0.968933i	$0.420449\pi$
$338$	0	0
$339$	6630.78	1.06234
$340$	0	0
$341$	6326.99	1.00477
$342$	0	0
$343$	6524.72i	1.02712i
$344$	0	0
$345$	− 3205.80i	− 0.500274i
$346$	0	0
$347$	−203.618	−0.0315008	−0.0157504	−	0.999876i	$-0.505014\pi$
$347$	−203.618	−0.0315008	−0.0157504	+	0.999876i	$0.505014\pi$
$348$	0	0
$349$	1505.61i	0.230927i	0.993312	+	0.115464i	$0.0368354\pi$
$349$	1505.61i	0.230927i	−0.993312	+	0.115464i	$0.963165\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11133.4i	1.67867i	0.543612	+	0.839337i	$0.317056\pi$
$353$	11133.4i	1.67867i	−0.543612	+	0.839337i	$0.682944\pi$
$354$	0	0
$355$	−2230.85	−0.333524
$356$	0	0
$357$	− 3079.12i	− 0.456483i
$358$	0	0
$359$	4176.37i	0.613985i	0.951712	+	0.306992i	$0.0993226\pi$
$359$	4176.37i	0.613985i	−0.951712	+	0.306992i	$0.900677\pi$
$360$	0	0
$361$	6369.60	0.928648
$362$	0	0
$363$	19083.9	2.75935
$364$	0	0
$365$	−1031.46	−0.147916
$366$	0	0
$367$	4480.80	0.637319	0.318659	−	0.947869i	$-0.396767\pi$
$367$	4480.80	0.637319	0.318659	+	0.947869i	$0.396767\pi$
$368$	0	0
$369$	− 29362.8i	− 4.14245i
$370$	0	0
$371$	860.917i	0.120476i
$372$	0	0
$373$	−363.859	−0.0505092	−0.0252546	−	0.999681i	$-0.508040\pi$
$373$	−363.859	−0.0505092	−0.0252546	+	0.999681i	$0.508040\pi$
$374$	0	0
$375$	− 5956.67i	− 0.820270i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 9755.44i	− 1.32217i	−0.750310	−	0.661086i	$-0.770096\pi$
$379$	− 9755.44i	− 1.32217i	0.750310	−	0.661086i	$-0.229904\pi$
$380$	0	0
$381$	12449.5	1.67403
$382$	0	0
$383$	5252.34i	0.700737i	0.936612	+	0.350368i	$0.113944\pi$
$383$	5252.34i	0.700737i	−0.936612	+	0.350368i	$0.886056\pi$
$384$	0	0
$385$	− 2569.37i	− 0.340122i
$386$	0	0
$387$	14333.5	1.88272
$388$	0	0
$389$	−11266.0	−1.46841	−0.734205	−	0.678928i	$-0.762445\pi$
$389$	−11266.0	−1.46841	−0.734205	+	0.678928i	$0.762445\pi$
$390$	0	0
$391$	2296.64	0.297048
$392$	0	0
$393$	−12228.1	−1.56953
$394$	0	0
$395$	− 615.431i	− 0.0783941i
$396$	0	0
$397$	3001.18i	0.379407i	0.981841	+	0.189704i	$0.0607527\pi$
$397$	3001.18i	0.379407i	−0.981841	+	0.189704i	$0.939247\pi$
$398$	0	0
$399$	−3891.14	−0.488223
$400$	0	0
$401$	− 7998.63i	− 0.996091i	−0.867151	−	0.498045i	$-0.834051\pi$
$401$	− 7998.63i	− 0.996091i	0.867151	−	0.498045i	$-0.165949\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	5362.95i	0.657992i
$406$	0	0
$407$	−13829.6	−1.68430
$408$	0	0
$409$	− 10632.1i	− 1.28539i	−0.766123	−	0.642694i	$-0.777817\pi$
$409$	− 10632.1i	− 1.28539i	0.766123	−	0.642694i	$-0.222183\pi$
$410$	0	0
$411$	− 17060.3i	− 2.04750i
$412$	0	0
$413$	−7932.24	−0.945085
$414$	0	0
$415$	−833.442	−0.0985833
$416$	0	0
$417$	−9679.59	−1.13672
$418$	0	0
$419$	−329.538	−0.0384224	−0.0192112	−	0.999815i	$-0.506115\pi$
$419$	−329.538	−0.0384224	−0.0192112	+	0.999815i	$0.506115\pi$
$420$	0	0
$421$	10594.7i	1.22649i	0.789891	+	0.613247i	$0.210137\pi$
$421$	10594.7i	1.22649i	−0.789891	+	0.613247i	$0.789863\pi$
$422$	0	0
$423$	− 10915.4i	− 1.25467i
$424$	0	0
$425$	2079.13	0.237301
$426$	0	0
$427$	15622.5i	1.77055i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 12753.4i	− 1.42531i	−0.701514	−	0.712655i	$-0.747492\pi$
$431$	− 12753.4i	− 1.42531i	0.701514	−	0.712655i	$-0.252508\pi$
$432$	0	0
$433$	14009.8	1.55490	0.777448	−	0.628947i	$-0.216514\pi$
$433$	14009.8	1.55490	0.777448	+	0.628947i	$0.216514\pi$
$434$	0	0
$435$	− 1814.43i	− 0.199989i
$436$	0	0
$437$	− 2902.30i	− 0.317703i
$438$	0	0
$439$	−7296.19	−0.793230	−0.396615	−	0.917985i	$-0.629815\pi$
$439$	−7296.19	−0.793230	−0.396615	+	0.917985i	$0.629815\pi$
$440$	0	0
$441$	1383.35	0.149374
$442$	0	0
$443$	−6098.72	−0.654084	−0.327042	−	0.945010i	$-0.606052\pi$
$443$	−6098.72	−0.654084	−0.327042	+	0.945010i	$0.606052\pi$
$444$	0	0
$445$	946.575	0.100836
$446$	0	0
$447$	16606.8i	1.75722i
$448$	0	0
$449$	10846.2i	1.14001i	0.821640	+	0.570006i	$0.193059\pi$
$449$	10846.2i	1.14001i	−0.821640	+	0.570006i	$0.806941\pi$
$450$	0	0
$451$	24440.3	2.55177
$452$	0	0
$453$	25480.3i	2.64275i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5353.85i	0.548014i	0.961728	+	0.274007i	$0.0883492\pi$
$457$	5353.85i	0.548014i	−0.961728	+	0.274007i	$0.911651\pi$
$458$	0	0
$459$	−7164.82	−0.728595
$460$	0	0
$461$	15097.3i	1.52527i	0.646827	+	0.762637i	$0.276096\pi$
$461$	15097.3i	1.52527i	−0.646827	+	0.762637i	$0.723904\pi$
$462$	0	0
$463$	− 9543.47i	− 0.957932i	−0.877833	−	0.478966i	$-0.841012\pi$
$463$	− 9543.47i	− 0.957932i	0.877833	−	0.478966i	$-0.158988\pi$
$464$	0	0
$465$	−2699.26	−0.269194
$466$	0	0
$467$	−4319.08	−0.427972	−0.213986	−	0.976837i	$-0.568645\pi$
$467$	−4319.08	−0.427972	−0.213986	+	0.976837i	$0.568645\pi$
$468$	0	0
$469$	−14394.3	−1.41720
$470$	0	0
$471$	1752.01	0.171397
$472$	0	0
$473$	11930.6i	1.15976i
$474$	0	0
$475$	− 2627.44i	− 0.253800i
$476$	0	0
$477$	3296.85	0.316462
$478$	0	0
$479$	− 13244.2i	− 1.26335i	−0.775235	−	0.631673i	$-0.782368\pi$
$479$	− 13244.2i	− 1.26335i	0.775235	−	0.631673i	$-0.217632\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 23075.6i	− 2.17387i
$484$	0	0
$485$	2349.77	0.219995
$486$	0	0
$487$	− 619.602i	− 0.0576526i	−0.999584	−	0.0288263i	$-0.990823\pi$
$487$	− 619.602i	− 0.0576526i	0.999584	−	0.0288263i	$-0.00917697\pi$
$488$	0	0
$489$	− 20361.9i	− 1.88302i
$490$	0	0
$491$	5462.14	0.502042	0.251021	−	0.967982i	$-0.419234\pi$
$491$	5462.14	0.502042	0.251021	+	0.967982i	$0.419234\pi$
$492$	0	0
$493$	1299.86	0.118748
$494$	0	0
$495$	−9839.30	−0.893421
$496$	0	0
$497$	−16057.8	−1.44928
$498$	0	0
$499$	− 16005.2i	− 1.43585i	−0.696119	−	0.717926i	$-0.745092\pi$
$499$	− 16005.2i	− 1.43585i	0.696119	−	0.717926i	$-0.254908\pi$
$500$	0	0
$501$	− 29755.8i	− 2.65347i
$502$	0	0
$503$	−10055.8	−0.891386	−0.445693	−	0.895186i	$-0.647043\pi$
$503$	−10055.8	−0.891386	−0.445693	+	0.895186i	$0.647043\pi$
$504$	0	0
$505$	− 963.485i	− 0.0849001i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14572.4i	1.26898i	0.772931	+	0.634490i	$0.218790\pi$
$509$	14572.4i	1.26898i	−0.772931	+	0.634490i	$0.781210\pi$
$510$	0	0
$511$	−7424.56	−0.642746
$512$	0	0
$513$	9054.32i	0.779255i
$514$	0	0
$515$	987.763i	0.0845166i
$516$	0	0
$517$	9085.49	0.772881
$518$	0	0
$519$	506.408	0.0428301
$520$	0	0
$521$	14409.9	1.21172	0.605862	−	0.795570i	$-0.292828\pi$
$521$	14409.9	1.21172	0.605862	+	0.795570i	$0.292828\pi$
$522$	0	0
$523$	−4248.32	−0.355193	−0.177597	−	0.984103i	$-0.556832\pi$
$523$	−4248.32	−0.355193	−0.177597	+	0.984103i	$0.556832\pi$
$524$	0	0
$525$	− 20890.2i	− 1.73662i
$526$	0	0
$527$	− 1933.75i	− 0.159840i
$528$	0	0
$529$	5044.52	0.414607
$530$	0	0
$531$	30376.2i	2.48252i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3065.93i	0.247761i
$536$	0	0
$537$	6155.87	0.494684
$538$	0	0
$539$	1151.44i	0.0920151i
$540$	0	0
$541$	14369.4i	1.14194i	0.820970	+	0.570971i	$0.193433\pi$
$541$	14369.4i	1.14194i	−0.820970	+	0.570971i	$0.806567\pi$
$542$	0	0
$543$	16259.5	1.28501
$544$	0	0
$545$	−1353.49	−0.106380
$546$	0	0
$547$	−18950.8	−1.48131	−0.740657	−	0.671884i	$-0.765485\pi$
$547$	−18950.8	−1.48131	−0.740657	+	0.671884i	$0.765485\pi$
$548$	0	0
$549$	59825.8	4.65083
$550$	0	0
$551$	− 1642.65i	− 0.127004i
$552$	0	0
$553$	− 4429.92i	− 0.340650i
$554$	0	0
$555$	5900.08	0.451251
$556$	0	0
$557$	11852.8i	0.901654i	0.892611	+	0.450827i	$0.148871\pi$
$557$	11852.8i	0.901654i	−0.892611	+	0.450827i	$0.851129\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 9814.63i	− 0.738635i
$562$	0	0
$563$	−11585.3	−0.867251	−0.433625	−	0.901093i	$-0.642766\pi$
$563$	−11585.3	−0.867251	−0.433625	+	0.901093i	$0.642766\pi$
$564$	0	0
$565$	− 1691.09i	− 0.125920i
$566$	0	0
$567$	38603.0i	2.85921i
$568$	0	0
$569$	−9242.34	−0.680948	−0.340474	−	0.940254i	$-0.610587\pi$
$569$	−9242.34	−0.680948	−0.340474	+	0.940254i	$0.610587\pi$
$570$	0	0
$571$	20877.1	1.53009	0.765043	−	0.643979i	$-0.222718\pi$
$571$	20877.1	1.53009	0.765043	+	0.643979i	$0.222718\pi$
$572$	0	0
$573$	−18773.5	−1.36872
$574$	0	0
$575$	15581.5	1.13007
$576$	0	0
$577$	− 19966.8i	− 1.44060i	−0.693661	−	0.720302i	$-0.744003\pi$
$577$	− 19966.8i	− 1.44060i	0.693661	−	0.720302i	$-0.255997\pi$
$578$	0	0
$579$	− 2262.02i	− 0.162360i
$580$	0	0
$581$	−5999.19	−0.428379
$582$	0	0
$583$	2744.15i	0.194942i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19411.9i	1.36493i	0.730917	+	0.682466i	$0.239093\pi$
$587$	19411.9i	1.36493i	−0.730917	+	0.682466i	$0.760907\pi$
$588$	0	0
$589$	−2443.72	−0.170954
$590$	0	0
$591$	15959.2i	1.11078i
$592$	0	0
$593$	19259.0i	1.33368i	0.745200	+	0.666841i	$0.232354\pi$
$593$	19259.0i	1.33368i	−0.745200	+	0.666841i	$0.767646\pi$
$594$	0	0
$595$	−785.289	−0.0541071
$596$	0	0
$597$	−30007.1	−2.05713
$598$	0	0
$599$	27804.6	1.89660	0.948301	−	0.317372i	$-0.102800\pi$
$599$	27804.6	1.89660	0.948301	+	0.317372i	$0.102800\pi$
$600$	0	0
$601$	−9927.10	−0.673769	−0.336884	−	0.941546i	$-0.609373\pi$
$601$	−9927.10	−0.673769	−0.336884	+	0.941546i	$0.609373\pi$
$602$	0	0
$603$	55122.3i	3.72264i
$604$	0	0
$605$	− 4867.09i	− 0.327066i
$606$	0	0
$607$	−14402.0	−0.963032	−0.481516	−	0.876437i	$-0.659914\pi$
$607$	−14402.0	−0.963032	−0.481516	+	0.876437i	$0.659914\pi$
$608$	0	0
$609$	− 13060.4i	− 0.869021i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 14778.2i	− 0.973713i	−0.873482	−	0.486856i	$-0.838144\pi$
$613$	− 14778.2i	− 0.973713i	0.873482	−	0.486856i	$-0.161856\pi$
$614$	0	0
$615$	−10426.9	−0.683661
$616$	0	0
$617$	− 17161.9i	− 1.11979i	−0.828562	−	0.559897i	$-0.810841\pi$
$617$	− 17161.9i	− 1.11979i	0.828562	−	0.559897i	$-0.189159\pi$
$618$	0	0
$619$	− 7592.09i	− 0.492975i	−0.969146	−	0.246488i	$-0.920724\pi$
$619$	− 7592.09i	− 0.492975i	0.969146	−	0.246488i	$-0.0792765\pi$
$620$	0	0
$621$	−53694.8	−3.46972
$622$	0	0
$623$	6813.53	0.438168
$624$	0	0
$625$	13326.8	0.852917
$626$	0	0
$627$	−12402.9	−0.789993
$628$	0	0
$629$	4226.82i	0.267940i
$630$	0	0
$631$	6757.72i	0.426340i	0.977015	+	0.213170i	$0.0683789\pi$
$631$	6757.72i	0.426340i	−0.977015	+	0.213170i	$0.931621\pi$
$632$	0	0
$633$	33767.9	2.12031
$634$	0	0
$635$	− 3175.08i	− 0.198424i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	61492.9i	3.80692i
$640$	0	0
$641$	−14252.7	−0.878235	−0.439117	−	0.898430i	$-0.644709\pi$
$641$	−14252.7	−0.878235	−0.439117	+	0.898430i	$0.644709\pi$
$642$	0	0
$643$	12887.1i	0.790383i	0.918599	+	0.395192i	$0.129322\pi$
$643$	12887.1i	0.790383i	−0.918599	+	0.395192i	$0.870678\pi$
$644$	0	0
$645$	− 5089.89i	− 0.310720i
$646$	0	0
$647$	−28352.4	−1.72280	−0.861398	−	0.507931i	$-0.830411\pi$
$647$	−28352.4	−1.72280	−0.861398	+	0.507931i	$0.830411\pi$
$648$	0	0
$649$	−25283.8	−1.52924
$650$	0	0
$651$	−19429.5	−1.16974
$652$	0	0
$653$	24016.2	1.43925	0.719623	−	0.694365i	$-0.244315\pi$
$653$	24016.2	1.43925	0.719623	+	0.694365i	$0.244315\pi$
$654$	0	0
$655$	3118.60i	0.186036i
$656$	0	0
$657$	28432.1i	1.68834i
$658$	0	0
$659$	−366.040	−0.0216371	−0.0108186	−	0.999941i	$-0.503444\pi$
$659$	−366.040	−0.0216371	−0.0108186	+	0.999941i	$0.503444\pi$
$660$	0	0
$661$	7308.10i	0.430033i	0.976610	+	0.215017i	$0.0689806\pi$
$661$	7308.10i	0.430033i	−0.976610	+	0.215017i	$0.931019\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	992.385i	0.0578692i
$666$	0	0
$667$	9741.42	0.565501
$668$	0	0
$669$	24710.6i	1.42806i
$670$	0	0
$671$	49796.4i	2.86493i
$672$	0	0
$673$	13894.0	0.795803	0.397901	−	0.917428i	$-0.369739\pi$
$673$	13894.0	0.795803	0.397901	+	0.917428i	$0.369739\pi$
$674$	0	0
$675$	−48609.6	−2.77183
$676$	0	0
$677$	23804.6	1.35138	0.675691	−	0.737185i	$-0.263845\pi$
$677$	23804.6	1.35138	0.675691	+	0.737185i	$0.263845\pi$
$678$	0	0
$679$	16913.9	0.955957
$680$	0	0
$681$	11286.6i	0.635102i
$682$	0	0
$683$	20866.6i	1.16902i	0.811388	+	0.584508i	$0.198712\pi$
$683$	20866.6i	1.16902i	−0.811388	+	0.584508i	$0.801288\pi$
$684$	0	0
$685$	−4350.99	−0.242690
$686$	0	0
$687$	37759.1i	2.09694i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 5142.24i	− 0.283097i	−0.989931	−	0.141549i	$-0.954792\pi$
$691$	− 5142.24i	− 0.283097i	0.989931	−	0.141549i	$-0.0452082\pi$
$692$	0	0
$693$	−70824.1	−3.88223
$694$	0	0
$695$	2468.65i	0.134736i
$696$	0	0
$697$	− 7469.81i	− 0.405939i
$698$	0	0
$699$	−9970.02	−0.539486
$700$	0	0
$701$	8584.99	0.462554	0.231277	−	0.972888i	$-0.425710\pi$
$701$	8584.99	0.462554	0.231277	+	0.972888i	$0.425710\pi$
$702$	0	0
$703$	5341.52	0.286571
$704$	0	0
$705$	−3876.11	−0.207068
$706$	0	0
$707$	− 6935.25i	− 0.368921i
$708$	0	0
$709$	− 21568.8i	− 1.14250i	−0.820775	−	0.571252i	$-0.806458\pi$
$709$	− 21568.8i	− 1.14250i	0.820775	−	0.571252i	$-0.193542\pi$
$710$	0	0
$711$	−16964.2	−0.894808
$712$	0	0
$713$	− 14492.0i	− 0.761191i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12365.1i	− 0.644051i
$718$	0	0
$719$	32635.3	1.69276	0.846379	−	0.532581i	$-0.178778\pi$
$719$	32635.3	1.69276	0.846379	+	0.532581i	$0.178778\pi$
$720$	0	0
$721$	7110.00i	0.367254i
$722$	0	0
$723$	− 20241.8i	− 1.04122i
$724$	0	0
$725$	8818.85	0.451757
$726$	0	0
$727$	6636.77	0.338575	0.169288	−	0.985567i	$-0.445853\pi$
$727$	6636.77	0.338575	0.169288	+	0.985567i	$0.445853\pi$
$728$	0	0
$729$	39660.8	2.01498
$730$	0	0
$731$	3646.40	0.184496
$732$	0	0
$733$	18221.6i	0.918188i	0.888388	+	0.459094i	$0.151826\pi$
$733$	18221.6i	0.918188i	−0.888388	+	0.459094i	$0.848174\pi$
$734$	0	0
$735$	− 491.236i	− 0.0246524i
$736$	0	0
$737$	−45881.4	−2.29316
$738$	0	0
$739$	− 25459.9i	− 1.26733i	−0.773607	−	0.633665i	$-0.781550\pi$
$739$	− 25459.9i	− 1.26733i	0.773607	−	0.633665i	$-0.218450\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 22559.2i	− 1.11388i	−0.830552	−	0.556942i	$-0.811975\pi$
$743$	− 22559.2i	− 1.11388i	0.830552	−	0.556942i	$-0.188025\pi$
$744$	0	0
$745$	4235.35	0.208284
$746$	0	0
$747$	22973.7i	1.12525i
$748$	0	0
$749$	22068.9i	1.07661i
$750$	0	0
$751$	21808.6	1.05966	0.529830	−	0.848104i	$-0.322256\pi$
$751$	21808.6	1.05966	0.529830	+	0.848104i	$0.322256\pi$
$752$	0	0
$753$	50179.1	2.42846
$754$	0	0
$755$	6498.40	0.313246
$756$	0	0
$757$	−12803.7	−0.614742	−0.307371	−	0.951590i	$-0.599449\pi$
$757$	−12803.7	−0.614742	−0.307371	+	0.951590i	$0.599449\pi$
$758$	0	0
$759$	− 73553.1i	− 3.51753i
$760$	0	0
$761$	− 4154.40i	− 0.197893i	−0.995093	−	0.0989467i	$-0.968453\pi$
$761$	− 4154.40i	− 0.197893i	0.995093	−	0.0989467i	$-0.0315473\pi$
$762$	0	0
$763$	−9742.53	−0.462259
$764$	0	0
$765$	3007.24i	0.142127i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 4122.77i	− 0.193330i	−0.995317	−	0.0966652i	$-0.969182\pi$
$769$	− 4122.77i	− 0.193330i	0.995317	−	0.0966652i	$-0.0308176\pi$
$770$	0	0
$771$	38776.9	1.81131
$772$	0	0
$773$	− 34102.7i	− 1.58679i	−0.608708	−	0.793394i	$-0.708312\pi$
$773$	− 34102.7i	− 1.58679i	0.608708	−	0.793394i	$-0.291688\pi$
$774$	0	0
$775$	− 13119.5i	− 0.608086i
$776$	0	0
$777$	42469.3	1.96085
$778$	0	0
$779$	−9439.74	−0.434164
$780$	0	0
$781$	−51184.0	−2.34508
$782$	0	0
$783$	−30390.3	−1.38705
$784$	0	0
$785$	− 446.826i	− 0.0203158i
$786$	0	0
$787$	− 42573.1i	− 1.92829i	−0.265368	−	0.964147i	$-0.585493\pi$
$787$	− 42573.1i	− 1.92829i	0.265368	−	0.964147i	$-0.414507\pi$
$788$	0	0
$789$	69543.0	3.13789
$790$	0	0
$791$	− 12172.6i	− 0.547166i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 1170.73i	− 0.0522282i
$796$	0	0
$797$	6054.75	0.269097	0.134549	−	0.990907i	$-0.457042\pi$
$797$	6054.75	0.269097	0.134549	+	0.990907i	$0.457042\pi$
$798$	0	0
$799$	− 2776.85i	− 0.122951i
$800$	0	0
$801$	− 26092.2i	− 1.15096i
$802$	0	0
$803$	−23665.6	−1.04003
$804$	0	0
$805$	−5885.13	−0.257669
$806$	0	0
$807$	−54950.9	−2.39698
$808$	0	0
$809$	37902.7	1.64720	0.823601	−	0.567169i	$-0.191962\pi$
$809$	37902.7	1.64720	0.823601	+	0.567169i	$0.191962\pi$
$810$	0	0
$811$	− 2418.43i	− 0.104714i	−0.998628	−	0.0523568i	$-0.983327\pi$
$811$	− 2418.43i	− 0.104714i	0.998628	−	0.0523568i	$-0.0166733\pi$
$812$	0	0
$813$	− 4318.69i	− 0.186302i
$814$	0	0
$815$	−5193.04	−0.223196
$816$	0	0
$817$	− 4608.02i	− 0.197325i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40835.6i	1.73590i	0.496652	+	0.867950i	$0.334562\pi$
$821$	40835.6i	1.73590i	−0.496652	+	0.867950i	$0.665438\pi$
$822$	0	0
$823$	−1699.36	−0.0719755	−0.0359877	−	0.999352i	$-0.511458\pi$
$823$	−1699.36	−0.0719755	−0.0359877	+	0.999352i	$0.511458\pi$
$824$	0	0
$825$	− 66587.2i	− 2.81002i
$826$	0	0
$827$	28148.2i	1.18357i	0.806097	+	0.591784i	$0.201576\pi$
$827$	28148.2i	1.18357i	−0.806097	+	0.591784i	$0.798424\pi$
$828$	0	0
$829$	−1779.51	−0.0745535	−0.0372768	−	0.999305i	$-0.511868\pi$
$829$	−1779.51	−0.0745535	−0.0372768	+	0.999305i	$0.511868\pi$
$830$	0	0
$831$	45779.2	1.91102
$832$	0	0
$833$	351.922	0.0146379
$834$	0	0
$835$	−7588.82	−0.314517
$836$	0	0
$837$	45210.6i	1.86704i
$838$	0	0
$839$	− 27406.2i	− 1.12773i	−0.825867	−	0.563865i	$-0.809314\pi$
$839$	− 27406.2i	− 1.12773i	0.825867	−	0.563865i	$-0.190686\pi$
$840$	0	0
$841$	−18875.5	−0.773936
$842$	0	0
$843$	12268.2i	0.501234i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 35033.7i	− 1.42122i
$848$	0	0
$849$	40838.7	1.65086
$850$	0	0
$851$	31676.8i	1.27599i
$852$	0	0
$853$	− 49349.2i	− 1.98087i	−0.137964	−	0.990437i	$-0.544056\pi$
$853$	− 49349.2i	− 1.98087i	0.137964	−	0.990437i	$-0.455944\pi$
$854$	0	0
$855$	3800.30	0.152009
$856$	0	0
$857$	657.558	0.0262098	0.0131049	−	0.999914i	$-0.495828\pi$
$857$	657.558	0.0262098	0.0131049	+	0.999914i	$0.495828\pi$
$858$	0	0
$859$	2866.78	0.113869	0.0569343	−	0.998378i	$-0.481867\pi$
$859$	2866.78	0.113869	0.0569343	+	0.998378i	$0.481867\pi$
$860$	0	0
$861$	−75053.5	−2.97075
$862$	0	0
$863$	18625.8i	0.734683i	0.930086	+	0.367341i	$0.119732\pi$
$863$	18625.8i	0.734683i	−0.930086	+	0.367341i	$0.880268\pi$
$864$	0	0
$865$	− 129.153i	− 0.00507667i
$866$	0	0
$867$	45090.7	1.76628
$868$	0	0
$869$	− 14120.3i	− 0.551206i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 64771.1i	− 2.51107i
$874$	0	0
$875$	−10935.1	−0.422485
$876$	0	0
$877$	− 19727.9i	− 0.759596i	−0.925069	−	0.379798i	$-0.875994\pi$
$877$	− 19727.9i	− 0.759596i	0.925069	−	0.379798i	$-0.124006\pi$
$878$	0	0
$879$	− 95721.5i	− 3.67304i
$880$	0	0
$881$	50424.4	1.92831	0.964156	−	0.265337i	$-0.0854831\pi$
$881$	50424.4	1.92831	0.964156	+	0.265337i	$0.0854831\pi$
$882$	0	0
$883$	27255.8	1.03877	0.519384	−	0.854541i	$-0.326162\pi$
$883$	27255.8	1.03877	0.519384	+	0.854541i	$0.326162\pi$
$884$	0	0
$885$	10786.8	0.409709
$886$	0	0
$887$	11738.1	0.444336	0.222168	−	0.975008i	$-0.428687\pi$
$887$	11738.1	0.444336	0.222168	+	0.975008i	$0.428687\pi$
$888$	0	0
$889$	− 22854.5i	− 0.862221i
$890$	0	0
$891$	123046.i	4.62649i
$892$	0	0
$893$	−3509.16	−0.131500
$894$	0	0
$895$	− 1569.97i	− 0.0586351i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 8202.21i	− 0.304292i
$900$	0	0
$901$	838.710	0.0310117
$902$	0	0
$903$	− 36637.5i	− 1.35019i
$904$	0	0
$905$	− 4146.76i	− 0.152313i
$906$	0	0
$907$	−15434.7	−0.565051	−0.282526	−	0.959260i	$-0.591172\pi$
$907$	−15434.7	−0.565051	−0.282526	+	0.959260i	$0.591172\pi$
$908$	0	0
$909$	−26558.3	−0.969069
$910$	0	0
$911$	−24957.2	−0.907649	−0.453825	−	0.891091i	$-0.649941\pi$
$911$	−24957.2	−0.907649	−0.453825	+	0.891091i	$0.649941\pi$
$912$	0	0
$913$	−19122.3	−0.693160
$914$	0	0
$915$	− 21244.5i	− 0.767563i
$916$	0	0
$917$	22448.0i	0.808394i
$918$	0	0
$919$	16582.3	0.595213	0.297606	−	0.954689i	$-0.403812\pi$
$919$	16582.3	0.595213	0.297606	+	0.954689i	$0.403812\pi$
$920$	0	0
$921$	16068.9i	0.574906i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	28676.8i	1.01934i
$926$	0	0
$927$	27227.5	0.964691
$928$	0	0
$929$	5910.56i	0.208740i	0.994539	+	0.104370i	$0.0332826\pi$
$929$	5910.56i	0.208740i	−0.994539	+	0.104370i	$0.966717\pi$
$930$	0	0
$931$	− 444.730i	− 0.0156557i
$932$	0	0
$933$	57756.3	2.02664
$934$	0	0
$935$	−2503.09	−0.0875507
$936$	0	0
$937$	−7798.93	−0.271910	−0.135955	−	0.990715i	$-0.543410\pi$
$937$	−7798.93	−0.271910	−0.135955	+	0.990715i	$0.543410\pi$
$938$	0	0
$939$	−51566.3	−1.79212
$940$	0	0
$941$	− 9738.22i	− 0.337361i	−0.985671	−	0.168681i	$-0.946049\pi$
$941$	− 9738.22i	− 0.337361i	0.985671	−	0.168681i	$-0.0539507\pi$
$942$	0	0
$943$	− 55980.5i	− 1.93316i
$944$	0	0
$945$	18359.9	0.632006
$946$	0	0
$947$	− 13774.3i	− 0.472656i	−0.971673	−	0.236328i	$-0.924056\pi$
$947$	− 13774.3i	− 0.472656i	0.971673	−	0.236328i	$-0.0759441\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 200.209i	− 0.00682672i
$952$	0	0
$953$	−30604.5	−1.04027	−0.520135	−	0.854084i	$-0.674118\pi$
$953$	−30604.5	−1.04027	−0.520135	+	0.854084i	$0.674118\pi$
$954$	0	0
$955$	4787.94i	0.162235i
$956$	0	0
$957$	− 41629.7i	− 1.40616i
$958$	0	0
$959$	−31318.8	−1.05458
$960$	0	0
$961$	17588.8	0.590408
$962$	0	0
$963$	84511.9	2.82799
$964$	0	0
$965$	−576.898	−0.0192446
$966$	0	0
$967$	− 48790.3i	− 1.62253i	−0.584677	−	0.811266i	$-0.698779\pi$
$967$	− 48790.3i	− 1.62253i	0.584677	−	0.811266i	$-0.301221\pi$
$968$	0	0
$969$	3790.78i	0.125673i
$970$	0	0
$971$	−33077.6	−1.09321	−0.546607	−	0.837389i	$-0.684081\pi$
$971$	−33077.6	−1.09321	−0.546607	+	0.837389i	$0.684081\pi$
$972$	0	0
$973$	17769.6i	0.585474i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13620.6i	0.446020i	0.974816	+	0.223010i	$0.0715883\pi$
$977$	13620.6i	0.446020i	−0.974816	+	0.223010i	$0.928412\pi$
$978$	0	0
$979$	21718.0	0.708999
$980$	0	0
$981$	37308.7i	1.21425i
$982$	0	0
$983$	23886.5i	0.775037i	0.921862	+	0.387519i	$0.126668\pi$
$983$	23886.5i	0.775037i	−0.921862	+	0.387519i	$0.873332\pi$
$984$	0	0
$985$	4070.17	0.131661
$986$	0	0
$987$	−27900.6	−0.899783
$988$	0	0
$989$	27326.9	0.878611
$990$	0	0
$991$	−3180.18	−0.101939	−0.0509695	−	0.998700i	$-0.516231\pi$
$991$	−3180.18	−0.101939	−0.0509695	+	0.998700i	$0.516231\pi$
$992$	0	0
$993$	61690.7i	1.97150i
$994$	0	0
$995$	7652.90i	0.243832i
$996$	0	0
$997$	−53027.4	−1.68445	−0.842223	−	0.539129i	$-0.818754\pi$
$997$	−53027.4	−1.68445	−0.842223	+	0.539129i	$0.818754\pi$
$998$	0	0
$999$	− 98822.0i	− 3.12972i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.4.d.d.337.2		8
13.2	odd	12		676.4.e.h.529.8		16
13.3	even	3		676.4.h.e.485.4		8
13.4	even	6		676.4.h.e.361.4		8
13.5	odd	4		676.4.a.g.1.2		8
13.6	odd	12		676.4.e.h.653.8		16
13.7	odd	12		676.4.e.h.653.7		16
13.8	odd	4		676.4.a.g.1.1		8
13.9	even	3		52.4.h.a.49.4	yes	8
13.10	even	6		52.4.h.a.17.4	✓	8
13.11	odd	12		676.4.e.h.529.7		16
13.12	even	2	inner	676.4.d.d.337.1		8
39.23	odd	6		468.4.t.g.433.2		8
39.35	odd	6		468.4.t.g.361.3		8
52.23	odd	6		208.4.w.c.17.1		8
52.35	odd	6		208.4.w.c.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.4	✓	8	13.10	even	6
52.4.h.a.49.4	yes	8	13.9	even	3
208.4.w.c.17.1		8	52.23	odd	6
208.4.w.c.49.1		8	52.35	odd	6
468.4.t.g.361.3		8	39.35	odd	6
468.4.t.g.433.2		8	39.23	odd	6
676.4.a.g.1.1		8	13.8	odd	4
676.4.a.g.1.2		8	13.5	odd	4
676.4.d.d.337.1		8	13.12	even	2	inner
676.4.d.d.337.2		8	1.1	even	1	trivial
676.4.e.h.529.7		16	13.11	odd	12
676.4.e.h.529.8		16	13.2	odd	12
676.4.e.h.653.7		16	13.7	odd	12
676.4.e.h.653.8		16	13.6	odd	12
676.4.h.e.361.4		8	13.4	even	6
676.4.h.e.485.4		8	13.3	even	3

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.78841 q^{3} +2.49640i q^{5} +17.9693i q^{7} +68.8129 q^{9} +O(q^{10})\)	\(q-9.78841 q^{3} +2.49640i q^{5} +17.9693i q^{7} +68.8129 q^{9} +57.2769i q^{11} -24.4358i q^{15} +17.5058 q^{17} -22.1225i q^{19} -175.891i q^{21} +131.193 q^{23} +118.768 q^{25} -409.282 q^{27} +74.2527 q^{29} -110.463i q^{31} -560.649i q^{33} -44.8587 q^{35} +241.452i q^{37} -426.704i q^{41} +208.296 q^{43} +171.785i q^{45} -158.624i q^{47} +20.1031 q^{49} -171.354 q^{51} +47.9103 q^{53} -142.986 q^{55} +216.544i q^{57} +441.432i q^{59} +869.398 q^{61} +1236.52i q^{63} +801.045i q^{67} -1284.17 q^{69} +893.624i q^{71} +413.180i q^{73} -1162.55 q^{75} -1029.23 q^{77} -246.527 q^{79} +2148.27 q^{81} +333.857i q^{83} +43.7016i q^{85} -726.816 q^{87} -379.176i q^{89} +1081.26i q^{93} +55.2266 q^{95} -941.263i q^{97} +3941.39i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 140 q^{9}+O(q^{10}) \) 8 * q + 140 * q^9	\( 8 q + 140 q^{9} - 176 q^{17} + 40 q^{23} - 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} - 1008 q^{43} - 1844 q^{49} - 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} - 3476 q^{69} - 2896 q^{75} - 3972 q^{77} - 3968 q^{79} + 8264 q^{81} + 3320 q^{87} - 4800 q^{95}+O(q^{100}) \) 8 * q + 140 * q^9 - 176 * q^17 + 40 * q^23 - 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 - 1008 * q^43 - 1844 * q^49 - 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 - 3476 * q^69 - 2896 * q^75 - 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 - 4800 * q^95

Introduction

L-functions

Modular forms

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Database

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