LMFDB - Embedded newform 3960.2.d.j.3169.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.d (of order $2$, degree $1$, 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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 24x^{12} + 226x^{10} + 1052x^{8} + 2497x^{6} + 2788x^{4} + 1156x^{2} + 64 $ x^14 + 24x^12 + 226x^10 + 1052x^8 + 2497x^6 + 2788x^4 + 1156x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.8
Root			$-2.21557i$ of defining polynomial
Character	$\chi$	$=$	3960.3169
Dual form			3960.2.d.j.3169.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.101079 + 2.23378i) q^{5} -3.61424i q^{7} +1.00000 q^{11} -0.816900i q^{13} +1.81640i q^{17} -0.723148 q^{19} +4.63330i q^{23} +(-4.97957 + 0.451576i) q^{25} -0.539435 q^{29} +9.04600 q^{31} +(8.07343 - 0.365324i) q^{35} -9.34550i q^{37} +1.11112 q^{41} +5.63391i q^{43} +2.01967i q^{47} -6.06275 q^{49} +1.29901i q^{53} +(0.101079 + 2.23378i) q^{55} +7.11762 q^{59} +3.61313 q^{61} +(1.82478 - 0.0825713i) q^{65} +2.26541i q^{67} +11.8131 q^{71} -11.3652i q^{73} -3.61424i q^{77} +3.97230 q^{79} -2.50893i q^{83} +(-4.05744 + 0.183600i) q^{85} +6.19771 q^{89} -2.95247 q^{91} +(-0.0730949 - 1.61535i) q^{95} +0.0431733i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{11} - 2 q^{25} - 12 q^{35} + 32 q^{41} - 6 q^{49} + 24 q^{59} + 4 q^{61} + 4 q^{65} - 8 q^{71} - 32 q^{79} + 4 q^{85} - 48 q^{89} + 56 q^{91} - 8 q^{95}+O(q^{100}) $ 14 * q + 14 * q^11 - 2 * q^25 - 12 * q^35 + 32 * q^41 - 6 * q^49 + 24 * q^59 + 4 * q^61 + 4 * q^65 - 8 * q^71 - 32 * q^79 + 4 * q^85 - 48 * q^89 + 56 * q^91 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.101079	+	2.23378i	0.0452038	+	0.998978i
$5$	0.101079	+	2.23378i	0.0452038	+	0.998978i
$6$		0			0
$7$		−	3.61424i		−	1.36606i	−0.730393	−	0.683028i	$-0.760663\pi$
$7$		−	3.61424i		−	1.36606i	0.730393	−	0.683028i	$-0.239337\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		−	0.816900i		−	0.226567i	−0.993563	−	0.113284i	$-0.963863\pi$
$13$		−	0.816900i		−	0.226567i	0.993563	−	0.113284i	$-0.0361369\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.81640i			0.440542i	0.975439	+	0.220271i	$0.0706941\pi$
$17$			1.81640i			0.440542i	−0.975439	+	0.220271i	$0.929306\pi$
$18$		0			0
$19$	−0.723148			−0.165901			−0.0829507	−	0.996554i	$-0.526434\pi$
$19$	−0.723148			−0.165901			−0.0829507	+	0.996554i	$0.526434\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.63330i			0.966110i	0.875590	+	0.483055i	$0.160473\pi$
$23$			4.63330i			0.966110i	−0.875590	+	0.483055i	$0.839527\pi$
$24$		0			0
$25$	−4.97957	+	0.451576i	−0.995913	+	0.0903153i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.539435			−0.100171			−0.0500853	−	0.998745i	$-0.515949\pi$
$29$	−0.539435			−0.100171			−0.0500853	+	0.998745i	$0.515949\pi$
$30$		0			0
$31$	9.04600			1.62471			0.812355	−	0.583164i	$-0.198185\pi$
$31$	9.04600			1.62471			0.812355	+	0.583164i	$0.198185\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.07343	−	0.365324i	1.36466	−	0.0617510i
$36$		0			0
$37$		−	9.34550i		−	1.53639i	−0.640215	−	0.768196i	$-0.721155\pi$
$37$		−	9.34550i		−	1.53639i	0.640215	−	0.768196i	$-0.278845\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.11112			0.173527			0.0867636	−	0.996229i	$-0.472348\pi$
$41$	1.11112			0.173527			0.0867636	+	0.996229i	$0.472348\pi$
$42$		0			0
$43$			5.63391i			0.859164i	0.903028	+	0.429582i	$0.141339\pi$
$43$			5.63391i			0.859164i	−0.903028	+	0.429582i	$0.858661\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.01967i			0.294599i	0.989092	+	0.147300i	$0.0470581\pi$
$47$			2.01967i			0.294599i	−0.989092	+	0.147300i	$0.952942\pi$
$48$		0			0
$49$	−6.06275			−0.866107
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.29901i			0.178432i	0.996012	+	0.0892160i	$0.0284362\pi$
$53$			1.29901i			0.178432i	−0.996012	+	0.0892160i	$0.971564\pi$
$54$		0			0
$55$	0.101079	+	2.23378i	0.0136295	+	0.301203i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.11762			0.926635			0.463317	−	0.886192i	$-0.346659\pi$
$59$	7.11762			0.926635			0.463317	+	0.886192i	$0.346659\pi$
$60$		0			0
$61$	3.61313			0.462614			0.231307	−	0.972881i	$-0.425700\pi$
$61$	3.61313			0.462614			0.231307	+	0.972881i	$0.425700\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.82478	−	0.0825713i	0.226336	−	0.0102417i
$66$		0			0
$67$			2.26541i			0.276763i	0.990379	+	0.138382i	$0.0441901\pi$
$67$			2.26541i			0.276763i	−0.990379	+	0.138382i	$0.955810\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.8131			1.40195			0.700977	−	0.713184i	$-0.252748\pi$
$71$	11.8131			1.40195			0.700977	+	0.713184i	$0.252748\pi$
$72$		0			0
$73$		−	11.3652i		−	1.33019i	−0.746758	−	0.665096i	$-0.768391\pi$
$73$		−	11.3652i		−	1.33019i	0.746758	−	0.665096i	$-0.231609\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.61424i		−	0.411881i
$78$		0			0
$79$	3.97230			0.446919			0.223459	−	0.974713i	$-0.428265\pi$
$79$	3.97230			0.446919			0.223459	+	0.974713i	$0.428265\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.50893i		−	0.275391i	−0.990475	−	0.137696i	$-0.956030\pi$
$83$		−	2.50893i		−	0.275391i	0.990475	−	0.137696i	$-0.0439695\pi$
$84$		0			0
$85$	−4.05744	+	0.183600i	−0.440091	+	0.0199142i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.19771			0.656956			0.328478	−	0.944512i	$-0.393464\pi$
$89$	6.19771			0.656956			0.328478	+	0.944512i	$0.393464\pi$
$90$		0			0
$91$	−2.95247			−0.309503
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0730949	−	1.61535i	−0.00749938	−	0.165732i
$96$		0			0
$97$			0.0431733i			0.00438359i	0.999998	+	0.00219179i	$0.000697670\pi$
$97$			0.0431733i			0.00438359i	−0.999998	+	0.00219179i	$0.999302\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.58142			−0.157357			−0.0786783	−	0.996900i	$-0.525070\pi$
$101$	−1.58142			−0.157357			−0.0786783	+	0.996900i	$0.525070\pi$
$102$		0			0
$103$			3.76535i			0.371011i	0.982643	+	0.185505i	$0.0593922\pi$
$103$			3.76535i			0.371011i	−0.982643	+	0.185505i	$0.940608\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.2756i		−	1.38007i	−0.723774	−	0.690037i	$-0.757594\pi$
$107$		−	14.2756i		−	1.38007i	0.723774	−	0.690037i	$-0.242406\pi$
$108$		0			0
$109$	0.939575			0.0899949			0.0449975	−	0.998987i	$-0.485672\pi$
$109$	0.939575			0.0899949			0.0449975	+	0.998987i	$0.485672\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.3723i		−	1.06982i	−0.844909	−	0.534909i	$-0.820346\pi$
$113$		−	11.3723i		−	1.06982i	0.844909	−	0.534909i	$-0.179654\pi$
$114$		0			0
$115$	−10.3498	+	0.468329i	−0.965122	+	0.0436719i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.56491			0.601804
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.51205	−	11.0776i	−0.135242	−	0.990813i
$126$		0			0
$127$			0.916571i			0.0813325i	0.999173	+	0.0406663i	$0.0129480\pi$
$127$			0.916571i			0.0813325i	−0.999173	+	0.0406663i	$0.987052\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.03198			0.701757			0.350879	−	0.936421i	$-0.385883\pi$
$131$	8.03198			0.701757			0.350879	+	0.936421i	$0.385883\pi$
$132$		0			0
$133$			2.61363i			0.226631i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.0807i			0.861252i	0.902530	+	0.430626i	$0.141707\pi$
$137$			10.0807i			0.861252i	−0.902530	+	0.430626i	$0.858293\pi$
$138$		0			0
$139$	−2.52945			−0.214545			−0.107273	−	0.994230i	$-0.534212\pi$
$139$	−2.52945			−0.214545			−0.107273	+	0.994230i	$0.534212\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	0.816900i		−	0.0683126i
$144$		0			0
$145$	−0.0545255	−	1.20498i	−0.00452810	−	0.100068i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.6623			1.11926			0.559629	−	0.828743i	$-0.310944\pi$
$149$	13.6623			1.11926			0.559629	+	0.828743i	$0.310944\pi$
$150$		0			0
$151$	10.8671			0.884351			0.442176	−	0.896928i	$-0.354207\pi$
$151$	10.8671			0.884351			0.442176	+	0.896928i	$0.354207\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.914359	+	20.2068i	0.0734431	+	1.62305i
$156$		0			0
$157$			14.8226i			1.18298i	0.806314	+	0.591488i	$0.201459\pi$
$157$			14.8226i			1.18298i	−0.806314	+	0.591488i	$0.798541\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.7459			1.31976
$162$		0			0
$163$			14.4699i			1.13337i	0.823935	+	0.566685i	$0.191774\pi$
$163$			14.4699i			1.13337i	−0.823935	+	0.566685i	$0.808226\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.9669i		−	1.15817i	−0.815266	−	0.579087i	$-0.803409\pi$
$167$		−	14.9669i		−	1.15817i	0.815266	−	0.579087i	$-0.196591\pi$
$168$		0			0
$169$	12.3327			0.948667
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.91447i			0.525698i	0.964837	+	0.262849i	$0.0846620\pi$
$173$			6.91447i			0.525698i	−0.964837	+	0.262849i	$0.915338\pi$
$174$		0			0
$175$	1.63211	+	17.9974i	0.123376	+	1.36047i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.4553			1.15519			0.577593	−	0.816325i	$-0.303992\pi$
$179$	15.4553			1.15519			0.577593	+	0.816325i	$0.303992\pi$
$180$		0			0
$181$	1.59528			0.118576			0.0592881	−	0.998241i	$-0.481117\pi$
$181$	1.59528			0.118576			0.0592881	+	0.998241i	$0.481117\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.8758	−	0.944633i	1.53482	−	0.0694508i
$186$		0			0
$187$			1.81640i			0.132828i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.39655			0.679910			0.339955	−	0.940442i	$-0.389588\pi$
$191$	9.39655			0.679910			0.339955	+	0.940442i	$0.389588\pi$
$192$		0			0
$193$		−	4.50410i		−	0.324212i	−0.986773	−	0.162106i	$-0.948171\pi$
$193$		−	4.50410i		−	0.324212i	0.986773	−	0.162106i	$-0.0518287\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.24472i			0.373671i	0.982391	+	0.186835i	$0.0598231\pi$
$197$			5.24472i			0.373671i	−0.982391	+	0.186835i	$0.940177\pi$
$198$		0			0
$199$	−8.78674			−0.622875			−0.311438	−	0.950267i	$-0.600811\pi$
$199$	−8.78674			−0.622875			−0.311438	+	0.950267i	$0.600811\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.94965i			0.136839i
$204$		0			0
$205$	0.112310	+	2.48199i	0.00784410	+	0.173350i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.723148			−0.0500212
$210$		0			0
$211$	1.39770			0.0962218			0.0481109	−	0.998842i	$-0.484680\pi$
$211$	1.39770			0.0962218			0.0481109	+	0.998842i	$0.484680\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.5849	+	0.569469i	−0.858285	+	0.0388375i
$216$		0			0
$217$		−	32.6944i		−	2.21944i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.48382			0.0998123
$222$		0			0
$223$			28.6609i			1.91928i	0.281236	+	0.959639i	$0.409256\pi$
$223$			28.6609i			1.91928i	−0.281236	+	0.959639i	$0.590744\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			7.45142i			0.494568i	0.968943	+	0.247284i	$0.0795381\pi$
$227$			7.45142i			0.494568i	−0.968943	+	0.247284i	$0.920462\pi$
$228$		0			0
$229$	−16.1289			−1.06583			−0.532913	−	0.846170i	$-0.678903\pi$
$229$	−16.1289			−1.06583			−0.532913	+	0.846170i	$0.678903\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.5649i		−	1.01969i	−0.860266	−	0.509846i	$-0.829703\pi$
$233$		−	15.5649i		−	1.01969i	0.860266	−	0.509846i	$-0.170297\pi$
$234$		0			0
$235$	−4.51150	+	0.204146i	−0.294298	+	0.0133170i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.5692			1.45988			0.729939	−	0.683513i	$-0.239549\pi$
$239$	22.5692			1.45988			0.729939	+	0.683513i	$0.239549\pi$
$240$		0			0
$241$	21.1219			1.36058			0.680289	−	0.732944i	$-0.261854\pi$
$241$	21.1219			1.36058			0.680289	+	0.732944i	$0.261854\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.612816	−	13.5429i	−0.0391514	−	0.865222i
$246$		0			0
$247$			0.590739i			0.0375878i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.5841			−1.55173			−0.775866	−	0.630898i	$-0.782687\pi$
$251$	−24.5841			−1.55173			−0.775866	+	0.630898i	$0.782687\pi$
$252$		0			0
$253$			4.63330i			0.291293i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.0484i			1.25059i	0.780390	+	0.625293i	$0.215020\pi$
$257$			20.0484i			1.25059i	−0.780390	+	0.625293i	$0.784980\pi$
$258$		0			0
$259$	−33.7769			−2.09880
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.4128i			1.44369i	0.692053	+	0.721846i	$0.256706\pi$
$263$			23.4128i			1.44369i	−0.692053	+	0.721846i	$0.743294\pi$
$264$		0			0
$265$	−2.90169	+	0.131302i	−0.178250	+	0.00806582i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.2146			−1.04960			−0.524798	−	0.851227i	$-0.675859\pi$
$269$	−17.2146			−1.04960			−0.524798	+	0.851227i	$0.675859\pi$
$270$		0			0
$271$	6.86909			0.417268			0.208634	−	0.977994i	$-0.433098\pi$
$271$	6.86909			0.417268			0.208634	+	0.977994i	$0.433098\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.97957	+	0.451576i	−0.300279	+	0.0272311i
$276$		0			0
$277$		−	27.3132i		−	1.64109i	−0.571581	−	0.820546i	$-0.693670\pi$
$277$		−	27.3132i		−	1.64109i	0.571581	−	0.820546i	$-0.306330\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	27.1841			1.62167			0.810835	−	0.585275i	$-0.199013\pi$
$281$	27.1841			1.62167			0.810835	+	0.585275i	$0.199013\pi$
$282$		0			0
$283$		−	12.0909i		−	0.718728i	−0.933197	−	0.359364i	$-0.882994\pi$
$283$		−	12.0909i		−	0.718728i	0.933197	−	0.359364i	$-0.117006\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.01585i		−	0.237048i
$288$		0			0
$289$	13.7007			0.805923
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6.95136i			0.406103i	0.979168	+	0.203051i	$0.0650859\pi$
$293$			6.95136i			0.406103i	−0.979168	+	0.203051i	$0.934914\pi$
$294$		0			0
$295$	0.719441	+	15.8992i	0.0418875	+	0.925688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.78494			0.218889
$300$		0			0
$301$	20.3623			1.17366
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.365211	+	8.07095i	0.0209119	+	0.462141i
$306$		0			0
$307$			6.07977i			0.346991i	0.984835	+	0.173495i	$0.0555062\pi$
$307$			6.07977i			0.346991i	−0.984835	+	0.173495i	$0.944494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.8406			0.784829			0.392414	−	0.919789i	$-0.371640\pi$
$311$	13.8406			0.784829			0.392414	+	0.919789i	$0.371640\pi$
$312$		0			0
$313$		−	11.9924i		−	0.677851i	−0.940813	−	0.338926i	$-0.889936\pi$
$313$		−	11.9924i		−	0.677851i	0.940813	−	0.338926i	$-0.110064\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.1872i			1.52698i	0.645817	+	0.763492i	$0.276517\pi$
$317$			27.1872i			1.52698i	−0.645817	+	0.763492i	$0.723483\pi$
$318$		0			0
$319$	−0.539435			−0.0302026
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.31353i		−	0.0730865i
$324$		0			0
$325$	0.368893	+	4.06781i	0.0204625	+	0.225641i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.29958			0.402439
$330$		0			0
$331$	−19.2623			−1.05875			−0.529375	−	0.848388i	$-0.677573\pi$
$331$	−19.2623			−1.05875			−0.529375	+	0.848388i	$0.677573\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.06043	+	0.228985i	−0.276481	+	0.0125108i
$336$		0			0
$337$		−	34.6352i		−	1.88670i	−0.331799	−	0.943350i	$-0.607656\pi$
$337$		−	34.6352i		−	1.88670i	0.331799	−	0.943350i	$-0.392344\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.04600			0.489868
$342$		0			0
$343$		−	3.38745i		−	0.182905i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	13.7765i		−	0.739563i	−0.929119	−	0.369781i	$-0.879433\pi$
$347$		−	13.7765i		−	0.739563i	0.929119	−	0.369781i	$-0.120567\pi$
$348$		0			0
$349$	25.0456			1.34066			0.670329	−	0.742064i	$-0.266153\pi$
$349$	25.0456			1.34066			0.670329	+	0.742064i	$0.266153\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.6377i			0.991985i	0.868327	+	0.495992i	$0.165196\pi$
$353$			18.6377i			0.991985i	−0.868327	+	0.495992i	$0.834804\pi$
$354$		0			0
$355$	1.19405	+	26.3878i	0.0633737	+	1.40052i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.81027			0.201098			0.100549	−	0.994932i	$-0.467940\pi$
$359$	3.81027			0.201098			0.100549	+	0.994932i	$0.467940\pi$
$360$		0			0
$361$	−18.4771			−0.972477
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.3873	−	1.14878i	1.32883	−	0.0601298i
$366$		0			0
$367$		−	17.8577i		−	0.932166i	−0.884741	−	0.466083i	$-0.845665\pi$
$367$		−	17.8577i		−	0.932166i	0.884741	−	0.466083i	$-0.154335\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.69492			0.243748
$372$		0			0
$373$			18.4880i			0.957274i	0.878013	+	0.478637i	$0.158869\pi$
$373$			18.4880i			0.957274i	−0.878013	+	0.478637i	$0.841131\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.440665i			0.0226954i
$378$		0			0
$379$	12.1809			0.625689			0.312845	−	0.949804i	$-0.398718\pi$
$379$	12.1809			0.625689			0.312845	+	0.949804i	$0.398718\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			9.81925i			0.501740i	0.968021	+	0.250870i	$0.0807167\pi$
$383$			9.81925i			0.501740i	−0.968021	+	0.250870i	$0.919283\pi$
$384$		0			0
$385$	8.07343	−	0.365324i	0.411460	−	0.0186186i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.9642			−1.31644			−0.658220	−	0.752826i	$-0.728690\pi$
$389$	−25.9642			−1.31644			−0.658220	+	0.752826i	$0.728690\pi$
$390$		0			0
$391$	−8.41593			−0.425612
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.401516	+	8.87326i	0.0202024	+	0.446462i
$396$		0			0
$397$		−	21.7476i		−	1.09148i	−0.837954	−	0.545740i	$-0.816249\pi$
$397$		−	21.7476i		−	1.09148i	0.837954	−	0.545740i	$-0.183751\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.2397			0.761034			0.380517	−	0.924774i	$-0.375746\pi$
$401$	15.2397			0.761034			0.380517	+	0.924774i	$0.375746\pi$
$402$		0			0
$403$		−	7.38967i		−	0.368106i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.34550i		−	0.463239i
$408$		0			0
$409$	−32.9324			−1.62840			−0.814201	−	0.580583i	$-0.802825\pi$
$409$	−32.9324			−1.62840			−0.814201	+	0.580583i	$0.802825\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	25.7248i		−	1.26583i
$414$		0			0
$415$	5.60441	−	0.253600i	0.275110	−	0.0124487i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.11582			0.152218			0.0761090	−	0.997100i	$-0.475750\pi$
$419$	3.11582			0.152218			0.0761090	+	0.997100i	$0.475750\pi$
$420$		0			0
$421$	−29.6279			−1.44398			−0.721988	−	0.691905i	$-0.756772\pi$
$421$	−29.6279			−1.44398			−0.721988	+	0.691905i	$0.756772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.820243	−	9.04489i	−0.0397877	−	0.438741i
$426$		0			0
$427$		−	13.0587i		−	0.631956i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.1805			−1.50191			−0.750955	−	0.660353i	$-0.770407\pi$
$431$	−31.1805			−1.50191			−0.750955	+	0.660353i	$0.770407\pi$
$432$		0			0
$433$		−	14.5723i		−	0.700299i	−0.936694	−	0.350149i	$-0.886131\pi$
$433$		−	14.5723i		−	0.700299i	0.936694	−	0.350149i	$-0.113869\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.35056i		−	0.160279i
$438$		0			0
$439$	36.3330			1.73408			0.867041	−	0.498238i	$-0.166019\pi$
$439$	36.3330			1.73408			0.867041	+	0.498238i	$0.166019\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	30.4100i		−	1.44482i	−0.691462	−	0.722412i	$-0.743033\pi$
$443$		−	30.4100i		−	1.44482i	0.691462	−	0.722412i	$-0.256967\pi$
$444$		0			0
$445$	0.626458	+	13.8443i	0.0296969	+	0.656285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.2169			0.576549			0.288275	−	0.957548i	$-0.406918\pi$
$449$	12.2169			0.576549			0.288275	+	0.957548i	$0.406918\pi$
$450$		0			0
$451$	1.11112			0.0523204
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.298433	−	6.59518i	−0.0139907	−	0.309187i
$456$		0			0
$457$		−	20.3685i		−	0.952798i	−0.879229	−	0.476399i	$-0.841942\pi$
$457$		−	20.3685i		−	0.952798i	0.879229	−	0.476399i	$-0.158058\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.6336			−1.05415			−0.527077	−	0.849818i	$-0.676712\pi$
$461$	−22.6336			−1.05415			−0.527077	+	0.849818i	$0.676712\pi$
$462$		0			0
$463$			15.0582i			0.699814i	0.936784	+	0.349907i	$0.113787\pi$
$463$			15.0582i			0.699814i	−0.936784	+	0.349907i	$0.886213\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			3.14202i			0.145395i	0.997354	+	0.0726976i	$0.0231608\pi$
$467$			3.14202i			0.145395i	−0.997354	+	0.0726976i	$0.976839\pi$
$468$		0			0
$469$	8.18773			0.378074
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.63391i			0.259048i
$474$		0			0
$475$	3.60096	−	0.326556i	0.165223	−	0.0149834i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.0401			−1.23549			−0.617747	−	0.786377i	$-0.711955\pi$
$479$	−27.0401			−1.23549			−0.617747	+	0.786377i	$0.711955\pi$
$480$		0			0
$481$	−7.63434			−0.348096
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.0964398	+	0.00436391i	−0.00437911	+	0.000198155i
$486$		0			0
$487$		−	22.5873i		−	1.02353i	−0.859126	−	0.511764i	$-0.828992\pi$
$487$		−	22.5873i		−	1.02353i	0.859126	−	0.511764i	$-0.171008\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.0875			−0.996797			−0.498399	−	0.866948i	$-0.666078\pi$
$491$	−22.0875			−0.996797			−0.498399	+	0.866948i	$0.666078\pi$
$492$		0			0
$493$		−	0.979831i		−	0.0441294i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	42.6953i		−	1.91515i
$498$		0			0
$499$	22.5568			1.00978			0.504891	−	0.863183i	$-0.331533\pi$
$499$	22.5568			1.00978			0.504891	+	0.863183i	$0.331533\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.76778i			0.301761i	0.988552	+	0.150880i	$0.0482108\pi$
$503$			6.76778i			0.301761i	−0.988552	+	0.150880i	$0.951789\pi$
$504$		0			0
$505$	−0.159848	−	3.53254i	−0.00711313	−	0.157196i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.1742			−0.539614			−0.269807	−	0.962914i	$-0.586960\pi$
$509$	−12.1742			−0.539614			−0.269807	+	0.962914i	$0.586960\pi$
$510$		0			0
$511$	−41.0764			−1.81711
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.41096	+	0.380597i	−0.370631	+	0.0167711i
$516$		0			0
$517$			2.01967i			0.0888250i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.5005			1.20482			0.602410	−	0.798187i	$-0.294207\pi$
$521$	27.5005			1.20482			0.602410	+	0.798187i	$0.294207\pi$
$522$		0			0
$523$			24.4783i			1.07036i	0.844738	+	0.535180i	$0.179756\pi$
$523$			24.4783i			1.07036i	−0.844738	+	0.535180i	$0.820244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.4312i			0.715752i
$528$		0			0
$529$	1.53253			0.0666317
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.907671i		−	0.0393156i
$534$		0			0
$535$	31.8886	−	1.44296i	1.37866	−	0.0623846i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.06275			−0.261141
$540$		0			0
$541$	−13.3690			−0.574778			−0.287389	−	0.957814i	$-0.592787\pi$
$541$	−13.3690			−0.574778			−0.287389	+	0.957814i	$0.592787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.0949711	+	2.09881i	0.00406812	+	0.0899029i
$546$		0			0
$547$			22.8525i			0.977103i	0.872535	+	0.488551i	$0.162474\pi$
$547$			22.8525i			0.977103i	−0.872535	+	0.488551i	$0.837526\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.390091			0.0166185
$552$		0			0
$553$		−	14.3569i		−	0.610516i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			12.5997i			0.533867i	0.963715	+	0.266933i	$0.0860103\pi$
$557$			12.5997i			0.533867i	−0.963715	+	0.266933i	$0.913990\pi$
$558$		0			0
$559$	4.60234			0.194658
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	13.1322i		−	0.553458i	−0.960948	−	0.276729i	$-0.910750\pi$
$563$		−	13.1322i		−	0.553458i	0.960948	−	0.276729i	$-0.0892504\pi$
$564$		0			0
$565$	25.4033	−	1.14950i	1.06873	−	0.0483599i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−43.3113			−1.81570			−0.907852	−	0.419290i	$-0.862279\pi$
$569$	−43.3113			−1.81570			−0.907852	+	0.419290i	$0.862279\pi$
$570$		0			0
$571$	31.3488			1.31191			0.655954	−	0.754801i	$-0.272266\pi$
$571$	31.3488			1.31191			0.655954	+	0.754801i	$0.272266\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.09229	−	23.0718i	−0.0872545	−	0.962162i
$576$		0			0
$577$		−	41.4085i		−	1.72386i	−0.507027	−	0.861930i	$-0.669256\pi$
$577$		−	41.4085i		−	1.72386i	0.507027	−	0.861930i	$-0.330744\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.06789			−0.376199
$582$		0			0
$583$			1.29901i			0.0537993i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			21.9686i			0.906742i	0.891322	+	0.453371i	$0.149779\pi$
$587$			21.9686i			0.906742i	−0.891322	+	0.453371i	$0.850221\pi$
$588$		0			0
$589$	−6.54159			−0.269542
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.5210i		−	0.637373i	−0.947860	−	0.318686i	$-0.896758\pi$
$593$		−	15.5210i		−	0.637373i	0.947860	−	0.318686i	$-0.103242\pi$
$594$		0			0
$595$	0.663574	+	14.6646i	0.0272039	+	0.601189i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.1776			0.661000			0.330500	−	0.943806i	$-0.392783\pi$
$599$	16.1776			0.661000			0.330500	+	0.943806i	$0.392783\pi$
$600$		0			0
$601$	−20.0214			−0.816688			−0.408344	−	0.912828i	$-0.633894\pi$
$601$	−20.0214			−0.816688			−0.408344	+	0.912828i	$0.633894\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.101079	+	2.23378i	0.00410944	+	0.0908162i
$606$		0			0
$607$			10.6411i			0.431911i	0.976403	+	0.215955i	$0.0692865\pi$
$607$			10.6411i			0.431911i	−0.976403	+	0.215955i	$0.930713\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.64987			0.0667465
$612$		0			0
$613$			20.8149i			0.840706i	0.907361	+	0.420353i	$0.138094\pi$
$613$			20.8149i			0.840706i	−0.907361	+	0.420353i	$0.861906\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.66579i			0.187838i	0.995580	+	0.0939188i	$0.0299394\pi$
$617$			4.66579i			0.187838i	−0.995580	+	0.0939188i	$0.970061\pi$
$618$		0			0
$619$	−23.9917			−0.964309			−0.482155	−	0.876086i	$-0.660146\pi$
$619$	−23.9917			−0.964309			−0.482155	+	0.876086i	$0.660146\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	22.4000i		−	0.897438i
$624$		0			0
$625$	24.5922	−	4.49731i	0.983686	−	0.179892i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.9752			0.676844
$630$		0			0
$631$	−15.7945			−0.628768			−0.314384	−	0.949296i	$-0.601798\pi$
$631$	−15.7945			−0.628768			−0.314384	+	0.949296i	$0.601798\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.04742	+	0.0926459i	−0.0812494	+	0.00367654i
$636$		0			0
$637$			4.95266i			0.196232i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−47.9574			−1.89420			−0.947101	−	0.320935i	$-0.896003\pi$
$641$	−47.9574			−1.89420			−0.947101	+	0.320935i	$0.896003\pi$
$642$		0			0
$643$			36.5230i			1.44033i	0.693804	+	0.720164i	$0.255933\pi$
$643$			36.5230i			1.44033i	−0.693804	+	0.720164i	$0.744067\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	23.8327i		−	0.936962i	−0.883474	−	0.468481i	$-0.844801\pi$
$647$		−	23.8327i		−	0.936962i	0.883474	−	0.468481i	$-0.155199\pi$
$648$		0			0
$649$	7.11762			0.279391
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	17.5248i		−	0.685797i	−0.939372	−	0.342899i	$-0.888591\pi$
$653$		−	17.5248i		−	0.685797i	0.939372	−	0.342899i	$-0.111409\pi$
$654$		0			0
$655$	0.811863	+	17.9417i	0.0317221	+	0.701040i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.4640			0.680301			0.340150	−	0.940371i	$-0.389522\pi$
$659$	17.4640			0.680301			0.340150	+	0.940371i	$0.389522\pi$
$660$		0			0
$661$	−28.2391			−1.09837			−0.549186	−	0.835700i	$-0.685062\pi$
$661$	−28.2391			−1.09837			−0.549186	+	0.835700i	$0.685062\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.83828	+	0.264183i	−0.226399	+	0.0102446i
$666$		0			0
$667$		−	2.49937i		−	0.0967759i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.61313			0.139483
$672$		0			0
$673$		−	4.71350i		−	0.181692i	−0.995865	−	0.0908460i	$-0.971043\pi$
$673$		−	4.71350i		−	0.181692i	0.995865	−	0.0908460i	$-0.0289571\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	33.6515i		−	1.29333i	−0.762774	−	0.646665i	$-0.776163\pi$
$677$		−	33.6515i		−	1.29333i	0.762774	−	0.646665i	$-0.223837\pi$
$678$		0			0
$679$	0.156039			0.00598822
$680$		0			0
$681$		0			0
$682$		0			0
$683$			0.162226i			0.00620740i	0.999995	+	0.00310370i	$0.000987940\pi$
$683$			0.162226i			0.00620740i	−0.999995	+	0.00310370i	$0.999012\pi$
$684$		0			0
$685$	−22.5181	+	1.01895i	−0.860372	+	0.0389319i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.06116			0.0404269
$690$		0			0
$691$	14.8444			0.564706			0.282353	−	0.959311i	$-0.408885\pi$
$691$	14.8444			0.564706			0.282353	+	0.959311i	$0.408885\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.255674	−	5.65025i	−0.00969828	−	0.214326i
$696$		0			0
$697$			2.01823i			0.0764460i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.8929			−0.751344			−0.375672	−	0.926753i	$-0.622588\pi$
$701$	−19.8929			−0.751344			−0.375672	+	0.926753i	$0.622588\pi$
$702$		0			0
$703$			6.75818i			0.254889i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			5.71562i			0.214958i
$708$		0			0
$709$	−49.2654			−1.85020			−0.925100	−	0.379723i	$-0.876019\pi$
$709$	−49.2654			−1.85020			−0.925100	+	0.379723i	$0.876019\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			41.9128i			1.56965i
$714$		0			0
$715$	1.82478	−	0.0825713i	0.0682428	−	0.00308799i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.4245			−0.500651			−0.250326	−	0.968162i	$-0.580538\pi$
$719$	−13.4245			−0.500651			−0.250326	+	0.968162i	$0.580538\pi$
$720$		0			0
$721$	13.6089			0.506821
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.68615	−	0.243596i	0.0997613	−	0.00904694i
$726$		0			0
$727$		−	3.98768i		−	0.147895i	−0.997262	−	0.0739475i	$-0.976440\pi$
$727$		−	3.98768i		−	0.147895i	0.997262	−	0.0739475i	$-0.0235597\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.2334			−0.378497
$732$		0			0
$733$			23.2839i			0.860012i	0.902826	+	0.430006i	$0.141489\pi$
$733$			23.2839i			0.860012i	−0.902826	+	0.430006i	$0.858511\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2.26541i			0.0834473i
$738$		0			0
$739$	47.3733			1.74265			0.871327	−	0.490703i	$-0.163260\pi$
$739$	47.3733			1.74265			0.871327	+	0.490703i	$0.163260\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			42.4488i			1.55729i	0.627462	+	0.778647i	$0.284094\pi$
$743$			42.4488i			1.55729i	−0.627462	+	0.778647i	$0.715906\pi$
$744$		0			0
$745$	1.38097	+	30.5186i	0.0505948	+	1.11811i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−51.5955			−1.88526
$750$		0			0
$751$	−18.1889			−0.663724			−0.331862	−	0.943328i	$-0.607677\pi$
$751$	−18.1889			−0.663724			−0.331862	+	0.943328i	$0.607677\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.09843	+	24.2747i	0.0399761	+	0.883447i
$756$		0			0
$757$		−	1.11035i		−	0.0403563i	−0.999796	−	0.0201782i	$-0.993577\pi$
$757$		−	1.11035i		−	0.0403563i	0.999796	−	0.0201782i	$-0.00642335\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.8455			−1.33565			−0.667824	−	0.744319i	$-0.732774\pi$
$761$	−36.8455			−1.33565			−0.667824	+	0.744319i	$0.732774\pi$
$762$		0			0
$763$		−	3.39585i		−	0.122938i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	5.81438i		−	0.209945i
$768$		0			0
$769$	15.6879			0.565719			0.282859	−	0.959161i	$-0.408717\pi$
$769$	15.6879			0.565719			0.282859	+	0.959161i	$0.408717\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			25.4231i			0.914404i	0.889363	+	0.457202i	$0.151148\pi$
$773$			25.4231i			0.914404i	−0.889363	+	0.457202i	$0.848852\pi$
$774$		0			0
$775$	−45.0451	+	4.08496i	−1.61807	+	0.146736i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.803501			−0.0287884
$780$		0			0
$781$	11.8131			0.422705
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−33.1106	+	1.49826i	−1.18177	+	0.0534750i
$786$		0			0
$787$		−	36.7797i		−	1.31105i	−0.755172	−	0.655527i	$-0.772447\pi$
$787$		−	36.7797i		−	1.31105i	0.755172	−	0.655527i	$-0.227553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−41.1024			−1.46143
$792$		0			0
$793$		−	2.95157i		−	0.104813i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			26.9039i			0.952987i	0.879178	+	0.476493i	$0.158092\pi$
$797$			26.9039i			0.952987i	−0.879178	+	0.476493i	$0.841908\pi$
$798$		0			0
$799$	−3.66853			−0.129783
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	11.3652i		−	0.401068i
$804$		0			0
$805$	1.69265	+	37.4066i	0.0596582	+	1.31841i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−37.8033			−1.32909			−0.664547	−	0.747246i	$-0.731375\pi$
$809$	−37.8033			−1.32909			−0.664547	+	0.747246i	$0.731375\pi$
$810$		0			0
$811$	5.87234			0.206206			0.103103	−	0.994671i	$-0.467123\pi$
$811$	5.87234			0.206206			0.103103	+	0.994671i	$0.467123\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−32.3226	+	1.46260i	−1.13221	+	0.0512326i
$816$		0			0
$817$		−	4.07415i		−	0.142536i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−42.6585			−1.48879			−0.744396	−	0.667739i	$-0.767262\pi$
$821$	−42.6585			−1.48879			−0.744396	+	0.667739i	$0.767262\pi$
$822$		0			0
$823$		−	0.176308i		−	0.00614570i	−0.999995	−	0.00307285i	$-0.999022\pi$
$823$		−	0.176308i		−	0.00614570i	0.999995	−	0.00307285i	$-0.000978120\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			37.5050i			1.30418i	0.758143	+	0.652088i	$0.226107\pi$
$827$			37.5050i			1.30418i	−0.758143	+	0.652088i	$0.773893\pi$
$828$		0			0
$829$	48.4565			1.68296			0.841482	−	0.540285i	$-0.181683\pi$
$829$	48.4565			1.68296			0.841482	+	0.540285i	$0.181683\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.0124i		−	0.381556i
$834$		0			0
$835$	33.4328	−	1.51284i	1.15699	−	0.0523539i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.6085			1.02220			0.511100	−	0.859521i	$-0.329238\pi$
$839$	29.6085			1.02220			0.511100	+	0.859521i	$0.329238\pi$
$840$		0			0
$841$	−28.7090			−0.989966
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.24657	+	27.5485i	0.0428834	+	0.947698i
$846$		0			0
$847$		−	3.61424i		−	0.124187i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	43.3005			1.48432
$852$		0			0
$853$			15.3196i			0.524534i	0.964995	+	0.262267i	$0.0844701\pi$
$853$			15.3196i			0.524534i	−0.964995	+	0.262267i	$0.915530\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	15.7644i		−	0.538502i	−0.963070	−	0.269251i	$-0.913224\pi$
$857$		−	15.7644i		−	0.538502i	0.963070	−	0.269251i	$-0.0867762\pi$
$858$		0			0
$859$	−30.8230			−1.05167			−0.525834	−	0.850587i	$-0.676247\pi$
$859$	−30.8230			−1.05167			−0.525834	+	0.850587i	$0.676247\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			30.5651i			1.04045i	0.854030	+	0.520224i	$0.174152\pi$
$863$			30.5651i			1.04045i	−0.854030	+	0.520224i	$0.825848\pi$
$864$		0			0
$865$	−15.4454	+	0.698907i	−0.525160	+	0.0237635i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.97230			0.134751
$870$		0			0
$871$	1.85061			0.0627055
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−40.0372	+	5.46492i	−1.35350	+	0.184748i
$876$		0			0
$877$		−	29.9110i		−	1.01002i	−0.863112	−	0.505012i	$-0.831488\pi$
$877$		−	29.9110i		−	1.01002i	0.863112	−	0.505012i	$-0.168512\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.0435			1.14695			0.573477	−	0.819222i	$-0.305594\pi$
$881$	34.0435			1.14695			0.573477	+	0.819222i	$0.305594\pi$
$882$		0			0
$883$			8.32611i			0.280196i	0.990138	+	0.140098i	$0.0447417\pi$
$883$			8.32611i			0.280196i	−0.990138	+	0.140098i	$0.955258\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	14.9548i		−	0.502133i	−0.967970	−	0.251067i	$-0.919219\pi$
$887$		−	14.9548i		−	0.502133i	0.967970	−	0.251067i	$-0.0807813\pi$
$888$		0			0
$889$	3.31271			0.111105
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.46052i		−	0.0488744i
$894$		0			0
$895$	1.56221	+	34.5238i	0.0522188	+	1.15400i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.87973			−0.162748
$900$		0			0
$901$	−2.35951			−0.0786068
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.161249	+	3.56350i	0.00536010	+	0.118455i
$906$		0			0
$907$		−	43.1993i		−	1.43441i	−0.696863	−	0.717205i	$-0.745421\pi$
$907$		−	43.1993i		−	1.43441i	0.696863	−	0.717205i	$-0.254579\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.7918			−1.41776			−0.708879	−	0.705331i	$-0.750799\pi$
$911$	−42.7918			−1.41776			−0.708879	+	0.705331i	$0.750799\pi$
$912$		0			0
$913$		−	2.50893i		−	0.0830335i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	29.0295i		−	0.958639i
$918$		0			0
$919$	−20.7373			−0.684062			−0.342031	−	0.939689i	$-0.611115\pi$
$919$	−20.7373			−0.684062			−0.342031	+	0.939689i	$0.611115\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	9.65009i		−	0.317637i
$924$		0			0
$925$	4.22021	+	46.5365i	0.138760	+	1.53011i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−23.7519			−0.779275			−0.389638	−	0.920968i	$-0.627400\pi$
$929$	−23.7519			−0.779275			−0.389638	+	0.920968i	$0.627400\pi$
$930$		0			0
$931$	4.38426			0.143688
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.05744	+	0.183600i	−0.132693	+	0.00600435i
$936$		0			0
$937$		−	34.1030i		−	1.11410i	−0.830480	−	0.557048i	$-0.811934\pi$
$937$		−	34.1030i		−	1.11410i	0.830480	−	0.557048i	$-0.188066\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	54.1514			1.76529			0.882643	−	0.470044i	$-0.155762\pi$
$941$	54.1514			1.76529			0.882643	+	0.470044i	$0.155762\pi$
$942$		0			0
$943$			5.14814i			0.167646i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			40.1491i			1.30467i	0.757931	+	0.652335i	$0.226210\pi$
$947$			40.1491i			1.30467i	−0.757931	+	0.652335i	$0.773790\pi$
$948$		0			0
$949$	−9.28420			−0.301378
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.6388i			0.668557i	0.942474	+	0.334278i	$0.108493\pi$
$953$			20.6388i			0.668557i	−0.942474	+	0.334278i	$0.891507\pi$
$954$		0			0
$955$	0.949792	+	20.9898i	0.0307346	+	0.679215i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.4341			1.17652
$960$		0			0
$961$	50.8301			1.63968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.0612	−	0.455269i	0.323881	−	0.0146556i
$966$		0			0
$967$		−	44.9658i		−	1.44600i	−0.690847	−	0.723001i	$-0.742762\pi$
$967$		−	44.9658i		−	1.44600i	0.690847	−	0.723001i	$-0.257238\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.67273			−0.246230			−0.123115	−	0.992392i	$-0.539288\pi$
$971$	−7.67273			−0.246230			−0.123115	+	0.992392i	$0.539288\pi$
$972$		0			0
$973$			9.14206i			0.293081i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.0672i		−	0.642007i	−0.947078	−	0.321003i	$-0.895980\pi$
$977$		−	20.0672i		−	0.642007i	0.947078	−	0.321003i	$-0.104020\pi$
$978$		0			0
$979$	6.19771			0.198080
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	8.17776i		−	0.260830i	−0.991460	−	0.130415i	$-0.958369\pi$
$983$		−	8.17776i		−	0.260830i	0.991460	−	0.130415i	$-0.0416310\pi$
$984$		0			0
$985$	−11.7156	+	0.530130i	−0.373289	+	0.0168914i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.1036			−0.830046
$990$		0			0
$991$	−30.2960			−0.962385			−0.481193	−	0.876615i	$-0.659796\pi$
$991$	−30.2960			−0.962385			−0.481193	+	0.876615i	$0.659796\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.888154	−	19.6277i	−0.0281564	−	0.622239i
$996$		0			0
$997$			28.5847i			0.905286i	0.891692	+	0.452643i	$0.149519\pi$
$997$			28.5847i			0.905286i	−0.891692	+	0.452643i	$0.850481\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.d.j.3169.8	yes	14
3.2	odd	2		3960.2.d.i.3169.7	✓	14
5.4	even	2	inner	3960.2.d.j.3169.7	yes	14
15.14	odd	2		3960.2.d.i.3169.8	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.d.i.3169.7	✓	14	3.2	odd	2
3960.2.d.i.3169.8	yes	14	15.14	odd	2
3960.2.d.j.3169.7	yes	14	5.4	even	2	inner
3960.2.d.j.3169.8	yes	14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.101079 + 2.23378i) q^{5} -3.61424i q^{7} +1.00000 q^{11} -0.816900i q^{13} +1.81640i q^{17} -0.723148 q^{19} +4.63330i q^{23} +(-4.97957 + 0.451576i) q^{25} -0.539435 q^{29} +9.04600 q^{31} +(8.07343 - 0.365324i) q^{35} -9.34550i q^{37} +1.11112 q^{41} +5.63391i q^{43} +2.01967i q^{47} -6.06275 q^{49} +1.29901i q^{53} +(0.101079 + 2.23378i) q^{55} +7.11762 q^{59} +3.61313 q^{61} +(1.82478 - 0.0825713i) q^{65} +2.26541i q^{67} +11.8131 q^{71} -11.3652i q^{73} -3.61424i q^{77} +3.97230 q^{79} -2.50893i q^{83} +(-4.05744 + 0.183600i) q^{85} +6.19771 q^{89} -2.95247 q^{91} +(-0.0730949 - 1.61535i) q^{95} +0.0431733i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{11} - 2 q^{25} - 12 q^{35} + 32 q^{41} - 6 q^{49} + 24 q^{59} + 4 q^{61} + 4 q^{65} - 8 q^{71} - 32 q^{79} + 4 q^{85} - 48 q^{89} + 56 q^{91} - 8 q^{95}+O(q^{100}) \) 14 * q + 14 * q^11 - 2 * q^25 - 12 * q^35 + 32 * q^41 - 6 * q^49 + 24 * q^59 + 4 * q^61 + 4 * q^65 - 8 * q^71 - 32 * q^79 + 4 * q^85 - 48 * q^89 + 56 * q^91 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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