LMFDB - Embedded newform 2160.2.h.d.431.1

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -1.73205i q^{7} -3.46410 q^{11} +1.00000 q^{13} +1.73205i q^{19} -3.46410 q^{23} -1.00000 q^{25} -6.00000i q^{29} -3.46410i q^{31} -1.73205 q^{35} -7.00000 q^{37} +6.00000i q^{41} -3.46410i q^{43} -6.92820 q^{47} +4.00000 q^{49} +6.00000i q^{53} +3.46410i q^{55} -13.8564 q^{59} -1.00000 q^{61} -1.00000i q^{65} +5.19615i q^{67} -3.46410 q^{71} +1.00000 q^{73} +6.00000i q^{77} +8.66025i q^{79} -10.3923 q^{83} -1.73205i q^{91} +1.73205 q^{95} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{13} - 4 q^{25} - 28 q^{37} + 16 q^{49} - 4 q^{61} + 4 q^{73} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^13 - 4 * q^25 - 28 * q^37 + 16 * q^49 - 4 * q^61 + 4 * q^73 + 4 * 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
$7$	− 1.73205i	− 0.654654i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	− 1.73205i	− 0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	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
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	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
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	3.46410i	0.467099i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.00000i	− 0.124035i
$66$	0	0
$67$	5.19615i	0.634811i	0.948290	+	0.317406i	$0.102812\pi$
$67$	5.19615i	0.634811i	−0.948290	+	0.317406i	$0.897188\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	8.66025i	0.974355i	0.873303	+	0.487177i	$0.161973\pi$
$79$	8.66025i	0.974355i	−0.873303	+	0.487177i	$0.838027\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3923	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 1.73205i	− 0.181568i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.73205	0.177705
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	− 8.66025i	− 0.853320i	−0.904412	−	0.426660i	$-0.859690\pi$
$103$	− 8.66025i	− 0.853320i	0.904412	−	0.426660i	$-0.140310\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.3205	−1.67444	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	−17.3205	−1.67444	−0.837218	+	0.546869i	$0.815820\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	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	$-0.847461\pi$
$127$	− 10.3923i	− 0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.8564	−1.21064	−0.605320	−	0.795982i	$-0.706955\pi$
$131$	−13.8564	−1.21064	−0.605320	+	0.795982i	$0.706955\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	22.5167i	1.90984i	0.296866	+	0.954919i	$0.404058\pi$
$139$	22.5167i	1.90984i	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000i	0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$	12.0000i	0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$	0	0
$151$	1.73205i	0.140952i	0.997513	+	0.0704761i	$0.0224519\pi$
$151$	1.73205i	0.140952i	−0.997513	+	0.0704761i	$0.977548\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.46410	−0.278243
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.00000i	0.472866i
$162$	0	0
$163$	− 15.5885i	− 1.22098i	−0.792023	−	0.610491i	$-0.790972\pi$
$163$	− 15.5885i	− 1.22098i	0.792023	−	0.610491i	$-0.209028\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	0	0
$175$	1.73205i	0.130931i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.46410	−0.258919	−0.129460	−	0.991585i	$-0.541324\pi$
$179$	−3.46410	−0.258919	−0.129460	+	0.991585i	$0.541324\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.00000i	0.514650i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	15.5885i	1.10504i	0.833501	+	0.552518i	$0.186333\pi$
$199$	15.5885i	1.10504i	−0.833501	+	0.552518i	$0.813667\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.3923	−0.729397
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.00000i	− 0.415029i
$210$	0	0
$211$	− 25.9808i	− 1.78859i	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	− 25.9808i	− 1.78859i	0.447478	−	0.894295i	$-0.352322\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 24.2487i	− 1.62381i	−0.583787	−	0.811907i	$-0.698430\pi$
$223$	− 24.2487i	− 1.62381i	0.583787	−	0.811907i	$-0.301570\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	6.92820i	0.451946i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.46410	0.224074	0.112037	−	0.993704i	$-0.464262\pi$
$239$	3.46410	0.224074	0.112037	+	0.993704i	$0.464262\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.00000i	− 0.255551i
$246$	0	0
$247$	1.73205i	0.110208i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.46410	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	12.1244i	0.753371i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.7128	1.70885	0.854423	−	0.519579i	$-0.173911\pi$
$263$	27.7128	1.70885	0.854423	+	0.519579i	$0.173911\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.0000i	1.46331i	0.681677	+	0.731653i	$0.261251\pi$
$269$	24.0000i	1.46331i	−0.681677	+	0.731653i	$0.738749\pi$
$270$	0	0
$271$	8.66025i	0.526073i	0.964786	+	0.263036i	$0.0847240\pi$
$271$	8.66025i	0.526073i	−0.964786	+	0.263036i	$0.915276\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46410	0.208893
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.0000i	− 0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$	− 12.0000i	− 0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$	0	0
$283$	− 17.3205i	− 1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$	− 17.3205i	− 1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.3923	0.613438
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	13.8564i	0.806751i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.46410	−0.200334
$300$	0	0
$301$	−6.00000	−0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.00000i	0.0572598i
$306$	0	0
$307$	3.46410i	0.197707i	0.995102	+	0.0988534i	$0.0315175\pi$
$307$	3.46410i	0.197707i	−0.995102	+	0.0988534i	$0.968483\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.7128	1.57145	0.785725	−	0.618576i	$-0.212290\pi$
$311$	27.7128	1.57145	0.785725	+	0.618576i	$0.212290\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	20.7846i	1.16371i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0000i	0.661581i
$330$	0	0
$331$	− 19.0526i	− 1.04722i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	− 19.0526i	− 1.04722i	0.851957	−	0.523612i	$-0.175416\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.19615	0.283896
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.0000i	0.649836i
$342$	0	0
$343$	− 19.0526i	− 1.02874i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.3923	−0.557888	−0.278944	−	0.960307i	$-0.589984\pi$
$347$	−10.3923	−0.557888	−0.278944	+	0.960307i	$0.589984\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 12.0000i	− 0.638696i	−0.947638	−	0.319348i	$-0.896536\pi$
$353$	− 12.0000i	− 0.638696i	0.947638	−	0.319348i	$-0.103464\pi$
$354$	0	0
$355$	3.46410i	0.183855i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.2487	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.00000i	− 0.0523424i
$366$	0	0
$367$	− 15.5885i	− 0.813711i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	− 15.5885i	− 0.813711i	0.913493	−	0.406855i	$-0.133375\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	36.3731i	1.86836i	0.356803	+	0.934179i	$0.383867\pi$
$379$	36.3731i	1.86836i	−0.356803	+	0.934179i	$0.616133\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	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
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.66025	0.435745
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000i	0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$	18.0000i	0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$	0	0
$403$	− 3.46410i	− 0.172559i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.2487	1.20196
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	24.0000i	1.18096i
$414$	0	0
$415$	10.3923i	0.510138i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.73205i	0.0838198i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	38.1051i	1.81866i	0.416078	+	0.909329i	$0.363404\pi$
$439$	38.1051i	1.81866i	−0.416078	+	0.909329i	$0.636596\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8564	−0.658338	−0.329169	−	0.944271i	$-0.606769\pi$
$443$	−13.8564	−0.658338	−0.329169	+	0.944271i	$0.606769\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.0000i	− 1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$	− 24.0000i	− 1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$	0	0
$451$	− 20.7846i	− 0.978709i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.73205	−0.0811998
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 24.0000i	− 1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$	− 24.0000i	− 1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$	0	0
$463$	− 8.66025i	− 0.402476i	−0.979542	−	0.201238i	$-0.935504\pi$
$463$	− 8.66025i	− 0.402476i	0.979542	−	0.201238i	$-0.0644965\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.7128	1.28240	0.641198	−	0.767375i	$-0.278438\pi$
$467$	27.7128	1.28240	0.641198	+	0.767375i	$0.278438\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	− 1.73205i	− 0.0794719i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.3923	0.474837	0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923	0.474837	0.237418	+	0.971408i	$0.423699\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.00000i	− 0.0454077i
$486$	0	0
$487$	− 15.5885i	− 0.706380i	−0.935552	−	0.353190i	$-0.885097\pi$
$487$	− 15.5885i	− 0.706380i	0.935552	−	0.353190i	$-0.114903\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.3923	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000i	0.269137i
$498$	0	0
$499$	24.2487i	1.08552i	0.839887	+	0.542761i	$0.182621\pi$
$499$	24.2487i	1.08552i	−0.839887	+	0.542761i	$0.817379\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	− 1.73205i	− 0.0766214i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.66025	−0.381616
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.0000i	− 0.788594i	−0.918983	−	0.394297i	$-0.870988\pi$
$521$	− 18.0000i	− 0.788594i	0.918983	−	0.394297i	$-0.129012\pi$
$522$	0	0
$523$	− 29.4449i	− 1.28753i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	− 29.4449i	− 1.28753i	0.765222	−	0.643767i	$-0.222629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.00000i	0.259889i
$534$	0	0
$535$	17.3205i	0.748831i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13.8564	−0.596838
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 10.0000i	− 0.428353i
$546$	0	0
$547$	12.1244i	0.518400i	0.965824	+	0.259200i	$0.0834589\pi$
$547$	12.1244i	0.518400i	−0.965824	+	0.259200i	$0.916541\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	15.0000	0.637865
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	− 3.46410i	− 0.146516i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	− 19.0526i	− 0.797325i	−0.917098	−	0.398662i	$-0.869475\pi$
$571$	− 19.0526i	− 0.797325i	0.917098	−	0.398662i	$-0.130525\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.0000i	0.746766i
$582$	0	0
$583$	− 20.7846i	− 0.860811i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.3923	0.428936	0.214468	−	0.976731i	$-0.431198\pi$
$587$	10.3923	0.428936	0.214468	+	0.976731i	$0.431198\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	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
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.00000i	− 0.0406558i
$606$	0	0
$607$	− 8.66025i	− 0.351509i	−0.984434	−	0.175754i	$-0.943764\pi$
$607$	− 8.66025i	− 0.351509i	0.984434	−	0.175754i	$-0.0562365\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	22.5167i	0.905021i	0.891759	+	0.452510i	$0.149471\pi$
$619$	22.5167i	0.905021i	−0.891759	+	0.452510i	$0.850529\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 5.19615i	− 0.206856i	−0.994637	−	0.103428i	$-0.967019\pi$
$631$	− 5.19615i	− 0.206856i	0.994637	−	0.103428i	$-0.0329811\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3923	−0.412406
$636$	0	0
$637$	4.00000	0.158486
$638$	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
$647$	24.2487	0.953315	0.476658	−	0.879089i	$-0.341848\pi$
$647$	24.2487	0.953315	0.476658	+	0.879089i	$0.341848\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.0000i	− 1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$	− 36.0000i	− 1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	0	0
$655$	13.8564i	0.541415i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.00000i	− 0.116335i
$666$	0	0
$667$	20.7846i	0.804783i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.46410	0.133730
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	0	0
$679$	− 1.73205i	− 0.0664700i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.3205	−0.662751	−0.331375	−	0.943499i	$-0.607513\pi$
$683$	−17.3205	−0.662751	−0.331375	+	0.943499i	$0.607513\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.00000i	0.228582i
$690$	0	0
$691$	− 10.3923i	− 0.395342i	−0.980268	−	0.197671i	$-0.936662\pi$
$691$	− 10.3923i	− 0.395342i	0.980268	−	0.197671i	$-0.0633378\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.5167	0.854106
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0000i	1.13308i	0.824033	+	0.566542i	$0.191719\pi$
$701$	30.0000i	1.13308i	−0.824033	+	0.566542i	$0.808281\pi$
$702$	0	0
$703$	− 12.1244i	− 0.457279i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.7846	−0.781686
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.0000i	0.449404i
$714$	0	0
$715$	3.46410i	0.129550i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.3205	−0.645946	−0.322973	−	0.946408i	$-0.604682\pi$
$719$	−17.3205	−0.645946	−0.322973	+	0.946408i	$0.604682\pi$
$720$	0	0
$721$	−15.0000	−0.558629
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	− 24.2487i	− 0.899335i	−0.893196	−	0.449667i	$-0.851542\pi$
$727$	− 24.2487i	− 0.899335i	0.893196	−	0.449667i	$-0.148458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 18.0000i	− 0.663039i
$738$	0	0
$739$	31.1769i	1.14686i	0.819254	+	0.573431i	$0.194388\pi$
$739$	31.1769i	1.14686i	−0.819254	+	0.573431i	$0.805612\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.8564	−0.508342	−0.254171	−	0.967159i	$-0.581803\pi$
$743$	−13.8564	−0.508342	−0.254171	+	0.967159i	$0.581803\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	0	0
$748$	0	0
$749$	30.0000i	1.09618i
$750$	0	0
$751$	29.4449i	1.07446i	0.843436	+	0.537229i	$0.180529\pi$
$751$	29.4449i	1.07446i	−0.843436	+	0.537229i	$0.819471\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.73205	0.0630358
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.0000i	− 0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$	− 18.0000i	− 0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$	0	0
$763$	− 17.3205i	− 0.627044i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	3.46410i	0.124434i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.3923	−0.372343
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.00000i	− 0.0713831i
$786$	0	0
$787$	− 8.66025i	− 0.308705i	−0.988016	−	0.154352i	$-0.950671\pi$
$787$	− 8.66025i	− 0.308705i	0.988016	−	0.154352i	$-0.0493291\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.3923	0.369508
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.46410	−0.122245
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000i	0.421898i	0.977497	+	0.210949i	$0.0676553\pi$
$809$	12.0000i	0.421898i	−0.977497	+	0.210949i	$0.932345\pi$
$810$	0	0
$811$	− 38.1051i	− 1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$	− 38.1051i	− 1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.5885	−0.546040
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 15.5885i	− 0.543379i	−0.962385	−	0.271690i	$-0.912418\pi$
$823$	− 15.5885i	− 0.543379i	0.962385	−	0.271690i	$-0.0875824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.6410	−1.20459	−0.602293	−	0.798275i	$-0.705746\pi$
$827$	−34.6410	−1.20459	−0.602293	+	0.798275i	$0.705746\pi$
$828$	0	0
$829$	43.0000	1.49345	0.746726	−	0.665132i	$-0.231625\pi$
$829$	43.0000	1.49345	0.746726	+	0.665132i	$0.231625\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 13.8564i	− 0.479521i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.8564	0.478376	0.239188	−	0.970973i	$-0.423119\pi$
$839$	13.8564	0.478376	0.239188	+	0.970973i	$0.423119\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.0000i	0.412813i
$846$	0	0
$847$	− 1.73205i	− 0.0595140i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.2487	0.831235
$852$	0	0
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	43.3013i	1.47742i	0.674023	+	0.738710i	$0.264565\pi$
$859$	43.3013i	1.47742i	−0.674023	+	0.738710i	$0.735435\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.5692	−1.41503	−0.707516	−	0.706697i	$-0.750184\pi$
$863$	−41.5692	−1.41503	−0.707516	+	0.706697i	$0.750184\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 30.0000i	− 1.01768i
$870$	0	0
$871$	5.19615i	0.176065i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.73205	0.0585540
$876$	0	0
$877$	5.00000	0.168838	0.0844190	−	0.996430i	$-0.473097\pi$
$877$	5.00000	0.168838	0.0844190	+	0.996430i	$0.473097\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 54.0000i	− 1.81931i	−0.415369	−	0.909653i	$-0.636347\pi$
$881$	− 54.0000i	− 1.81931i	0.415369	−	0.909653i	$-0.363653\pi$
$882$	0	0
$883$	39.8372i	1.34063i	0.742078	+	0.670314i	$0.233840\pi$
$883$	39.8372i	1.34063i	−0.742078	+	0.670314i	$0.766160\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.8564	−0.465253	−0.232626	−	0.972566i	$-0.574732\pi$
$887$	−13.8564	−0.465253	−0.232626	+	0.972566i	$0.574732\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	3.46410i	0.115792i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.7846	−0.693206
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.00000i	0.166206i
$906$	0	0
$907$	39.8372i	1.32277i	0.750046	+	0.661386i	$0.230031\pi$
$907$	39.8372i	1.32277i	−0.750046	+	0.661386i	$0.769969\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.4974	−1.60679	−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974	−1.60679	−0.803396	+	0.595446i	$0.796976\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	24.2487i	0.799891i	0.916539	+	0.399946i	$0.130971\pi$
$919$	24.2487i	0.799891i	−0.916539	+	0.399946i	$0.869029\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.46410	−0.114022
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000i	0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$	30.0000i	0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$	0	0
$931$	6.92820i	0.227063i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.0000	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$937$	−43.0000	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	0	0
$943$	− 20.7846i	− 0.676840i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	1.00000	0.0324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 60.0000i	− 1.94359i	−0.235826	−	0.971795i	$-0.575780\pi$
$953$	− 60.0000i	− 1.94359i	0.235826	−	0.971795i	$-0.424220\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.1769	−1.00676
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.00000i	0.225338i
$966$	0	0
$967$	− 43.3013i	− 1.39247i	−0.717812	−	0.696237i	$-0.754856\pi$
$967$	− 43.3013i	− 1.39247i	0.717812	−	0.696237i	$-0.245144\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−34.6410	−1.11168	−0.555842	−	0.831288i	$-0.687604\pi$
$971$	−34.6410	−1.11168	−0.555842	+	0.831288i	$0.687604\pi$
$972$	0	0
$973$	39.0000	1.25028
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.46410	0.110488	0.0552438	−	0.998473i	$-0.482406\pi$
$983$	3.46410	0.110488	0.0552438	+	0.998473i	$0.482406\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0000i	0.381578i
$990$	0	0
$991$	29.4449i	0.935347i	0.883901	+	0.467673i	$0.154908\pi$
$991$	29.4449i	0.935347i	−0.883901	+	0.467673i	$0.845092\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.5885	0.494187
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -1.73205i q^{7} -3.46410 q^{11} +1.00000 q^{13} +1.73205i q^{19} -3.46410 q^{23} -1.00000 q^{25} -6.00000i q^{29} -3.46410i q^{31} -1.73205 q^{35} -7.00000 q^{37} +6.00000i q^{41} -3.46410i q^{43} -6.92820 q^{47} +4.00000 q^{49} +6.00000i q^{53} +3.46410i q^{55} -13.8564 q^{59} -1.00000 q^{61} -1.00000i q^{65} +5.19615i q^{67} -3.46410 q^{71} +1.00000 q^{73} +6.00000i q^{77} +8.66025i q^{79} -10.3923 q^{83} -1.73205i q^{91} +1.73205 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{13} - 4 q^{25} - 28 q^{37} + 16 q^{49} - 4 q^{61} + 4 q^{73} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^13 - 4 * q^25 - 28 * q^37 + 16 * q^49 - 4 * q^61 + 4 * q^73 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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