LMFDB - Embedded newform 2160.2.h.c.431.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(431,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2160.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.h (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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.2.h.c.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +3.46410i q^{7} -2.00000 q^{13} +3.00000i q^{17} +1.73205i q^{19} +5.19615 q^{23} -1.00000 q^{25} -6.00000i q^{29} +5.19615i q^{31} -3.46410 q^{35} -10.0000 q^{37} -10.3923 q^{47} -5.00000 q^{49} +9.00000i q^{53} +10.3923 q^{59} -13.0000 q^{61} -2.00000i q^{65} -3.46410i q^{67} -8.00000 q^{73} +1.73205i q^{79} +5.19615 q^{83} -3.00000 q^{85} -12.0000i q^{89} -6.92820i q^{91} -1.73205 q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{13} - 4 q^{25} - 40 q^{37} - 20 q^{49} - 52 q^{61} - 32 q^{73} - 12 q^{85} + 16 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 - 4 * q^25 - 40 * q^37 - 20 * q^49 - 52 * q^61 - 32 * q^73 - 12 * q^85 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.46410i	1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$	3.46410i	1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	1.73205i	0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$	1.73205i	0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.19615	1.08347	0.541736	−	0.840548i	$-0.317767\pi$
$23$	5.19615	1.08347	0.541736	+	0.840548i	$0.317767\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	5.19615i	0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$	5.19615i	0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.00000i	− 0.248069i
$66$	0	0
$67$	− 3.46410i	− 0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	− 3.46410i	− 0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.73205i	0.194871i	0.995242	+	0.0974355i	$0.0310640\pi$
$79$	1.73205i	0.194871i	−0.995242	+	0.0974355i	$0.968936\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.0000i	− 1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$	− 12.0000i	− 1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$	0	0
$91$	− 6.92820i	− 0.726273i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.73205	−0.177705
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	13.8564i	1.36531i	0.730740	+	0.682656i	$0.239175\pi$
$103$	13.8564i	1.36531i	−0.730740	+	0.682656i	$0.760825\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.3923	−1.00466	−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923	−1.00466	−0.502331	+	0.864675i	$0.667524\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	5.19615i	0.484544i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.3923	−0.952661
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3923	0.907980	0.453990	−	0.891007i	$-0.350000\pi$
$131$	10.3923	0.907980	0.453990	+	0.891007i	$0.350000\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.0000i	1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$	15.0000i	1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$	0	0
$139$	3.46410i	0.293821i	0.989150	+	0.146911i	$0.0469330\pi$
$139$	3.46410i	0.293821i	−0.989150	+	0.146911i	$0.953067\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 24.0000i	− 1.96616i	−0.183186	−	0.983078i	$-0.558641\pi$
$149$	− 24.0000i	− 1.96616i	0.183186	−	0.983078i	$-0.441359\pi$
$150$	0	0
$151$	17.3205i	1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$	17.3205i	1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.19615	−0.417365
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.0000i	1.41860i
$162$	0	0
$163$	3.46410i	0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$	3.46410i	0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.5885	1.20627	0.603136	−	0.797639i	$-0.293918\pi$
$167$	15.5885	1.20627	0.603136	+	0.797639i	$0.293918\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	0	0
$175$	− 3.46410i	− 0.261861i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 10.0000i	− 0.735215i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3923	0.751961	0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	24.2487i	1.71895i	0.511182	+	0.859473i	$0.329208\pi$
$199$	24.2487i	1.71895i	−0.511182	+	0.859473i	$0.670792\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.7846	1.45879
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 1.73205i	− 0.119239i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	− 1.73205i	− 0.119239i	0.998221	−	0.0596196i	$-0.0189888\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.00000i	− 0.403604i
$222$	0	0
$223$	− 10.3923i	− 0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	− 10.3923i	− 0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.5885	1.03464	0.517321	−	0.855791i	$-0.326929\pi$
$227$	15.5885	1.03464	0.517321	+	0.855791i	$0.326929\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	− 10.3923i	− 0.677919i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.00000i	− 0.319438i
$246$	0	0
$247$	− 3.46410i	− 0.220416i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.0000i	− 1.30994i	−0.755653	−	0.654972i	$-0.772680\pi$
$257$	− 21.0000i	− 1.30994i	0.755653	−	0.654972i	$-0.227320\pi$
$258$	0	0
$259$	− 34.6410i	− 2.15249i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.3923	−0.640817	−0.320408	−	0.947279i	$-0.603820\pi$
$263$	−10.3923	−0.640817	−0.320408	+	0.947279i	$0.603820\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.00000i	0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$	6.00000i	0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	0	0
$271$	− 8.66025i	− 0.526073i	−0.964786	−	0.263036i	$-0.915276\pi$
$271$	− 8.66025i	− 0.526073i	0.964786	−	0.263036i	$-0.0847240\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	0	0
$283$	− 20.7846i	− 1.23552i	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	− 20.7846i	− 1.23552i	0.786368	−	0.617758i	$-0.211959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	33.0000i	1.92788i	0.266119	+	0.963940i	$0.414259\pi$
$293$	33.0000i	1.92788i	−0.266119	+	0.963940i	$0.585741\pi$
$294$	0	0
$295$	10.3923i	0.605063i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.3923	−0.601003
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 13.0000i	− 0.744378i
$306$	0	0
$307$	− 10.3923i	− 0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$	− 10.3923i	− 0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.1769	1.76788	0.883940	−	0.467600i	$-0.154881\pi$
$311$	31.1769	1.76788	0.883940	+	0.467600i	$0.154881\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.0000i	− 1.17948i	−0.807594	−	0.589739i	$-0.799231\pi$
$317$	− 21.0000i	− 1.17948i	0.807594	−	0.589739i	$-0.200769\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.19615	−0.289122
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 36.0000i	− 1.98474i
$330$	0	0
$331$	− 17.3205i	− 0.952021i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	− 17.3205i	− 0.952021i	0.879440	−	0.476011i	$-0.157918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.46410	0.189264
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1769	1.67366	0.836832	−	0.547459i	$-0.184405\pi$
$347$	31.1769	1.67366	0.836832	+	0.547459i	$0.184405\pi$
$348$	0	0
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.1769	−1.64545	−0.822727	−	0.568436i	$-0.807549\pi$
$359$	−31.1769	−1.64545	−0.822727	+	0.568436i	$0.807549\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.00000i	− 0.418739i
$366$	0	0
$367$	24.2487i	1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$	24.2487i	1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−31.1769	−1.61862
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	− 19.0526i	− 0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$	− 19.0526i	− 0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.9808	1.32755	0.663777	−	0.747930i	$-0.268952\pi$
$383$	25.9808	1.32755	0.663777	+	0.747930i	$0.268952\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 36.0000i	− 1.82527i	−0.408773	−	0.912636i	$-0.634043\pi$
$389$	− 36.0000i	− 1.82527i	0.408773	−	0.912636i	$-0.365957\pi$
$390$	0	0
$391$	15.5885i	0.788342i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.73205	−0.0871489
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000i	0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$	6.00000i	0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$	0	0
$403$	− 10.3923i	− 0.517678i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	36.0000i	1.77144i
$414$	0	0
$415$	5.19615i	0.255069i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.7846	−1.01539	−0.507697	−	0.861536i	$-0.669503\pi$
$419$	−20.7846	−1.01539	−0.507697	+	0.861536i	$0.669503\pi$
$420$	0	0
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.00000i	− 0.145521i
$426$	0	0
$427$	− 45.0333i	− 2.17932i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.3923	0.500580	0.250290	−	0.968171i	$-0.419474\pi$
$431$	10.3923	0.500580	0.250290	+	0.968171i	$0.419474\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.00000i	0.430528i
$438$	0	0
$439$	− 5.19615i	− 0.247999i	−0.992282	−	0.123999i	$-0.960428\pi$
$439$	− 5.19615i	− 0.247999i	0.992282	−	0.123999i	$-0.0395721\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.19615	0.246877	0.123438	−	0.992352i	$-0.460608\pi$
$443$	5.19615	0.246877	0.123438	+	0.992352i	$0.460608\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.0000i	− 0.849473i	−0.905317	−	0.424736i	$-0.860367\pi$
$449$	− 18.0000i	− 0.849473i	0.905317	−	0.424736i	$-0.139633\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.0000i	− 0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$	− 18.0000i	− 0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$	0	0
$463$	24.2487i	1.12693i	0.826139	+	0.563467i	$0.190533\pi$
$463$	24.2487i	1.12693i	−0.826139	+	0.563467i	$0.809467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5885	0.721348	0.360674	−	0.932692i	$-0.382547\pi$
$467$	15.5885	0.721348	0.360674	+	0.932692i	$0.382547\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 1.73205i	− 0.0794719i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.5692	−1.89935	−0.949673	−	0.313243i	$-0.898585\pi$
$479$	−41.5692	−1.89935	−0.949673	+	0.313243i	$0.898585\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.00000i	0.181631i
$486$	0	0
$487$	− 6.92820i	− 0.313947i	−0.987603	−	0.156973i	$-0.949826\pi$
$487$	− 6.92820i	− 0.313947i	0.987603	−	0.156973i	$-0.0501737\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.3923	0.468998	0.234499	−	0.972116i	$-0.424655\pi$
$491$	10.3923	0.468998	0.234499	+	0.972116i	$0.424655\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.3731i	1.62828i	0.580667	+	0.814141i	$0.302792\pi$
$499$	36.3731i	1.62828i	−0.580667	+	0.814141i	$0.697208\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.19615	0.231685	0.115842	−	0.993268i	$-0.463043\pi$
$503$	5.19615	0.231685	0.115842	+	0.993268i	$0.463043\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	− 27.7128i	− 1.22594i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.8564	−0.610586
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 3.46410i	− 0.151475i	−0.997128	−	0.0757373i	$-0.975869\pi$
$523$	− 3.46410i	− 0.151475i	0.997128	−	0.0757373i	$-0.0241310\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.5885	−0.679044
$528$	0	0
$529$	4.00000	0.173913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 10.3923i	− 0.449299i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 11.0000i	− 0.471188i
$546$	0	0
$547$	34.6410i	1.48114i	0.671978	+	0.740571i	$0.265445\pi$
$547$	34.6410i	1.48114i	−0.671978	+	0.740571i	$0.734555\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 6.00000i	− 0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$	− 6.00000i	− 0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$	0	0
$571$	− 29.4449i	− 1.23223i	−0.787657	−	0.616115i	$-0.788706\pi$
$571$	− 29.4449i	− 1.23223i	0.787657	−	0.616115i	$-0.211294\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.19615	−0.216695
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.0000i	0.746766i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.19615	−0.214468	−0.107234	−	0.994234i	$-0.534199\pi$
$587$	−5.19615	−0.214468	−0.107234	+	0.994234i	$0.534199\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.0000i	1.35515i	0.735455	+	0.677574i	$0.236969\pi$
$593$	33.0000i	1.35515i	−0.735455	+	0.677574i	$0.763031\pi$
$594$	0	0
$595$	− 10.3923i	− 0.426043i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.1769	1.27385	0.636927	−	0.770924i	$-0.280205\pi$
$599$	31.1769	1.27385	0.636927	+	0.770924i	$0.280205\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11.0000i	− 0.447214i
$606$	0	0
$607$	− 27.7128i	− 1.12483i	−0.826856	−	0.562414i	$-0.809873\pi$
$607$	− 27.7128i	− 1.12483i	0.826856	−	0.562414i	$-0.190127\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.7846	0.840855
$612$	0	0
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.0000i	0.603877i	0.953327	+	0.301939i	$0.0976338\pi$
$617$	15.0000i	0.603877i	−0.953327	+	0.301939i	$0.902366\pi$
$618$	0	0
$619$	− 45.0333i	− 1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$	− 45.0333i	− 1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	41.5692	1.66544
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 30.0000i	− 1.19618i
$630$	0	0
$631$	12.1244i	0.482663i	0.970443	+	0.241331i	$0.0775841\pi$
$631$	12.1244i	0.482663i	−0.970443	+	0.241331i	$0.922416\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3923	−0.412406
$636$	0	0
$637$	10.0000	0.396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000i	0.947943i	0.880540	+	0.473972i	$0.157180\pi$
$641$	24.0000i	0.947943i	−0.880540	+	0.473972i	$0.842820\pi$
$642$	0	0
$643$	− 31.1769i	− 1.22950i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	− 31.1769i	− 1.22950i	0.788723	−	0.614749i	$-0.210743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.7654	−1.83854	−0.919268	−	0.393632i	$-0.871219\pi$
$647$	−46.7654	−1.83854	−0.919268	+	0.393632i	$0.871219\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.00000i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$653$	9.00000i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$654$	0	0
$655$	10.3923i	0.406061i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.3923	0.404827	0.202413	−	0.979300i	$-0.435122\pi$
$659$	10.3923	0.404827	0.202413	+	0.979300i	$0.435122\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6.00000i	− 0.232670i
$666$	0	0
$667$	− 31.1769i	− 1.20717i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	13.8564i	0.531760i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.19615	0.198825	0.0994126	−	0.995046i	$-0.468304\pi$
$683$	5.19615	0.198825	0.0994126	+	0.995046i	$0.468304\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 18.0000i	− 0.685745i
$690$	0	0
$691$	25.9808i	0.988355i	0.869361	+	0.494177i	$0.164531\pi$
$691$	25.9808i	0.988355i	−0.869361	+	0.494177i	$0.835469\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.46410	−0.131401
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.0000i	1.58632i	0.609015	+	0.793159i	$0.291565\pi$
$701$	42.0000i	1.58632i	−0.609015	+	0.793159i	$0.708435\pi$
$702$	0	0
$703$	− 17.3205i	− 0.653255i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.0000i	1.01116i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.3923	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	41.5692i	1.54172i	0.637006	+	0.770859i	$0.280172\pi$
$727$	41.5692i	1.54172i	−0.637006	+	0.770859i	$0.719828\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 5.19615i	− 0.191144i	−0.995423	−	0.0955718i	$-0.969532\pi$
$739$	− 5.19615i	− 0.191144i	0.995423	−	0.0955718i	$-0.0304679\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.3923	−0.381257	−0.190628	−	0.981662i	$-0.561053\pi$
$743$	−10.3923	−0.381257	−0.190628	+	0.981662i	$0.561053\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 36.0000i	− 1.31541i
$750$	0	0
$751$	12.1244i	0.442424i	0.975226	+	0.221212i	$0.0710013\pi$
$751$	12.1244i	0.442424i	−0.975226	+	0.221212i	$0.928999\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.3205	−0.630358
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 30.0000i	− 1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$	− 30.0000i	− 1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$	0	0
$763$	− 38.1051i	− 1.37950i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.7846	−0.750489
$768$	0	0
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 15.0000i	− 0.539513i	−0.962929	−	0.269756i	$-0.913057\pi$
$773$	− 15.0000i	− 0.539513i	0.962929	−	0.269756i	$-0.0869431\pi$
$774$	0	0
$775$	− 5.19615i	− 0.186651i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.00000i	− 0.142766i
$786$	0	0
$787$	6.92820i	0.246964i	0.992347	+	0.123482i	$0.0394061\pi$
$787$	6.92820i	0.246964i	−0.992347	+	0.123482i	$0.960594\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	26.0000	0.923287
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.00000i	0.106265i	0.998587	+	0.0531327i	$0.0169206\pi$
$797$	3.00000i	0.106265i	−0.998587	+	0.0531327i	$0.983079\pi$
$798$	0	0
$799$	− 31.1769i	− 1.10296i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−18.0000	−0.634417
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 48.0000i	− 1.68759i	−0.536666	−	0.843795i	$-0.680316\pi$
$809$	− 48.0000i	− 1.68759i	0.536666	−	0.843795i	$-0.319684\pi$
$810$	0	0
$811$	10.3923i	0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$	10.3923i	0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.46410	−0.121342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	48.4974i	1.69051i	0.534360	+	0.845257i	$0.320553\pi$
$823$	48.4974i	1.69051i	−0.534360	+	0.845257i	$0.679447\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.3731	1.26482	0.632408	−	0.774636i	$-0.282067\pi$
$827$	36.3731	1.26482	0.632408	+	0.774636i	$0.282067\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 15.0000i	− 0.519719i
$834$	0	0
$835$	15.5885i	0.539461i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.5692	1.43513	0.717564	−	0.696492i	$-0.245257\pi$
$839$	41.5692	1.43513	0.717564	+	0.696492i	$0.245257\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.00000i	− 0.309609i
$846$	0	0
$847$	− 38.1051i	− 1.30931i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−51.9615	−1.78122
$852$	0	0
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.0000i	1.74213i	0.491171	+	0.871063i	$0.336569\pi$
$857$	51.0000i	1.74213i	−0.491171	+	0.871063i	$0.663431\pi$
$858$	0	0
$859$	50.2295i	1.71381i	0.515476	+	0.856904i	$0.327615\pi$
$859$	50.2295i	1.71381i	−0.515476	+	0.856904i	$0.672385\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.19615	0.176879	0.0884395	−	0.996082i	$-0.471812\pi$
$863$	5.19615	0.176879	0.0884395	+	0.996082i	$0.471812\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	6.92820i	0.234753i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.46410	0.117108
$876$	0	0
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 18.0000i	− 0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$	− 18.0000i	− 0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$	0	0
$883$	45.0333i	1.51549i	0.652550	+	0.757746i	$0.273699\pi$
$883$	45.0333i	1.51549i	−0.652550	+	0.757746i	$0.726301\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.7654	−1.57023	−0.785114	−	0.619352i	$-0.787396\pi$
$887$	−46.7654	−1.57023	−0.785114	+	0.619352i	$0.787396\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	− 20.7846i	− 0.694753i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.1769	1.03981
$900$	0	0
$901$	−27.0000	−0.899500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.0000i	0.631581i
$906$	0	0
$907$	48.4974i	1.61033i	0.593051	+	0.805165i	$0.297923\pi$
$907$	48.4974i	1.61033i	−0.593051	+	0.805165i	$0.702077\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.9615	1.72156	0.860781	−	0.508975i	$-0.169976\pi$
$911$	51.9615	1.72156	0.860781	+	0.508975i	$0.169976\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.0000i	1.18882i
$918$	0	0
$919$	51.9615i	1.71405i	0.515273	+	0.857026i	$0.327691\pi$
$919$	51.9615i	1.71405i	−0.515273	+	0.857026i	$0.672309\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.0000i	0.393707i	0.980433	+	0.196854i	$0.0630724\pi$
$929$	12.0000i	0.393707i	−0.980433	+	0.196854i	$0.936928\pi$
$930$	0	0
$931$	− 8.66025i	− 0.283828i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.0000i	0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$	30.0000i	0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.5885	0.506557	0.253278	−	0.967393i	$-0.418491\pi$
$947$	15.5885	0.506557	0.253278	+	0.967393i	$0.418491\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	0	0
$955$	10.3923i	0.336287i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−51.9615	−1.67793
$960$	0	0
$961$	4.00000	0.129032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.00000i	0.0643823i
$966$	0	0
$967$	17.3205i	0.556990i	0.960438	+	0.278495i	$0.0898356\pi$
$967$	17.3205i	0.556990i	−0.960438	+	0.278495i	$0.910164\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−51.9615	−1.66752	−0.833762	−	0.552124i	$-0.813818\pi$
$971$	−51.9615	−1.66752	−0.833762	+	0.552124i	$0.813818\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15.5885	−0.497195	−0.248597	−	0.968607i	$-0.579970\pi$
$983$	−15.5885	−0.497195	−0.248597	+	0.968607i	$0.579970\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	19.0526i	0.605224i	0.953114	+	0.302612i	$0.0978587\pi$
$991$	19.0526i	0.605224i	−0.953114	+	0.302612i	$0.902141\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.2487	−0.768736
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.h.c.431.4	yes	4
3.2	odd	2	inner	2160.2.h.c.431.2	yes	4
4.3	odd	2	inner	2160.2.h.c.431.3	yes	4
12.11	even	2	inner	2160.2.h.c.431.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.2.h.c.431.1	✓	4	12.11	even	2	inner
2160.2.h.c.431.2	yes	4	3.2	odd	2	inner
2160.2.h.c.431.3	yes	4	4.3	odd	2	inner
2160.2.h.c.431.4	yes	4	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +3.46410i q^{7} -2.00000 q^{13} +3.00000i q^{17} +1.73205i q^{19} +5.19615 q^{23} -1.00000 q^{25} -6.00000i q^{29} +5.19615i q^{31} -3.46410 q^{35} -10.0000 q^{37} -10.3923 q^{47} -5.00000 q^{49} +9.00000i q^{53} +10.3923 q^{59} -13.0000 q^{61} -2.00000i q^{65} -3.46410i q^{67} -8.00000 q^{73} +1.73205i q^{79} +5.19615 q^{83} -3.00000 q^{85} -12.0000i q^{89} -6.92820i q^{91} -1.73205 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{13} - 4 q^{25} - 40 q^{37} - 20 q^{49} - 52 q^{61} - 32 q^{73} - 12 q^{85} + 16 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 - 4 * q^25 - 40 * q^37 - 20 * q^49 - 52 * q^61 - 32 * q^73 - 12 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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