LMFDB - Embedded newform 816.2.e.a.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(239,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, 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("816.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.e (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:	$6.51579280494$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.a.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -3.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} -1.73205 q^{11} -5.00000 q^{13} +5.19615i q^{15} +1.00000i q^{17} -1.73205i q^{19} -6.00000i q^{21} +5.19615 q^{23} -4.00000 q^{25} -5.19615 q^{27} +6.00000i q^{29} +6.92820i q^{31} +3.00000 q^{33} +10.3923 q^{35} +2.00000 q^{37} +8.66025 q^{39} +3.00000i q^{41} +12.1244i q^{43} -9.00000i q^{45} -6.92820 q^{47} -5.00000 q^{49} -1.73205i q^{51} +12.0000i q^{53} +5.19615i q^{55} +3.00000i q^{57} -10.3923 q^{59} +14.0000 q^{61} +10.3923i q^{63} +15.0000i q^{65} +3.46410i q^{67} -9.00000 q^{69} -3.46410 q^{71} +4.00000 q^{73} +6.92820 q^{75} -6.00000i q^{77} -10.3923i q^{79} +9.00000 q^{81} -6.92820 q^{83} +3.00000 q^{85} -10.3923i q^{87} -6.00000i q^{89} -17.3205i q^{91} -12.0000i q^{93} -5.19615 q^{95} -8.00000 q^{97} -5.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9} - 20 q^{13} - 16 q^{25} + 12 q^{33} + 8 q^{37} - 20 q^{49} + 56 q^{61} - 36 q^{69} + 16 q^{73} + 36 q^{81} + 12 q^{85} - 32 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 20 * q^13 - 16 * q^25 + 12 * q^33 + 8 * q^37 - 20 * q^49 + 56 * q^61 - 36 * q^69 + 16 * q^73 + 36 * q^81 + 12 * q^85 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.73205	−1.00000
$3$	−1.73205	−1.00000
$4$	0	0
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	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
$9$	3.00000	1.00000
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	5.19615i	1.34164i
$16$	0	0
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	0	0
$19$	− 1.73205i	− 0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$19$	− 1.73205i	− 0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$20$	0	0
$21$	− 6.00000i	− 1.30931i
$22$	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
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	6.92820i	1.24434i	0.782881	+	0.622171i	$0.213749\pi$
$31$	6.92820i	1.24434i	−0.782881	+	0.622171i	$0.786251\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	10.3923	1.75662
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	8.66025	1.38675
$40$	0	0
$41$	3.00000i	0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$41$	3.00000i	0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$42$	0	0
$43$	12.1244i	1.84895i	0.381246	+	0.924473i	$0.375495\pi$
$43$	12.1244i	1.84895i	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	− 9.00000i	− 1.34164i
$46$	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
$49$	−5.00000	−0.714286
$50$	0	0
$51$	− 1.73205i	− 0.242536i
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	5.19615i	0.700649i
$56$	0	0
$57$	3.00000i	0.397360i
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	10.3923i	1.30931i
$64$	0	0
$65$	15.0000i	1.86052i
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	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
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	6.92820	0.800000
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	− 10.3923i	− 1.16923i	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	− 10.3923i	− 1.16923i	0.811312	−	0.584613i	$-0.198754\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−6.92820	−0.760469	−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820	−0.760469	−0.380235	+	0.924890i	$0.624157\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	− 10.3923i	− 1.11417i
$88$	0	0
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	− 17.3205i	− 1.81568i
$92$	0	0
$93$	− 12.0000i	− 1.24434i
$94$	0	0
$95$	−5.19615	−0.533114
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−5.19615	−0.522233
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	8.66025i	0.853320i	0.904412	+	0.426660i	$0.140310\pi$
$103$	8.66025i	0.853320i	−0.904412	+	0.426660i	$0.859690\pi$
$104$	0	0
$105$	−18.0000	−1.75662
$106$	0	0
$107$	1.73205	0.167444	0.0837218	−	0.996489i	$-0.473319\pi$
$107$	1.73205	0.167444	0.0837218	+	0.996489i	$0.473319\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−3.46410	−0.328798
$112$	0	0
$113$	3.00000i	0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$	3.00000i	0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$	0	0
$115$	− 15.5885i	− 1.45363i
$116$	0	0
$117$	−15.0000	−1.38675
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	− 5.19615i	− 0.468521i
$124$	0	0
$125$	− 3.00000i	− 0.268328i
$126$	0	0
$127$	19.0526i	1.69064i	0.534259	+	0.845321i	$0.320591\pi$
$127$	19.0526i	1.69064i	−0.534259	+	0.845321i	$0.679409\pi$
$128$	0	0
$129$	− 21.0000i	− 1.84895i
$130$	0	0
$131$	15.5885	1.36197	0.680985	−	0.732297i	$-0.261552\pi$
$131$	15.5885	1.36197	0.680985	+	0.732297i	$0.261552\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	15.5885i	1.34164i
$136$	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
$139$	− 17.3205i	− 1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	− 17.3205i	− 1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	8.66025	0.724207
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	8.66025	0.714286
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	3.46410i	0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$	3.46410i	0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	0	0
$155$	20.7846	1.66946
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	− 20.7846i	− 1.64833i
$160$	0	0
$161$	18.0000i	1.41860i
$162$	0	0
$163$	− 20.7846i	− 1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$	− 20.7846i	− 1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$	0	0
$165$	− 9.00000i	− 0.700649i
$166$	0	0
$167$	−5.19615	−0.402090	−0.201045	−	0.979582i	$-0.564434\pi$
$167$	−5.19615	−0.402090	−0.201045	+	0.979582i	$0.564434\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	− 5.19615i	− 0.397360i
$172$	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
$175$	− 13.8564i	− 1.04745i
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−24.2487	−1.79252
$184$	0	0
$185$	− 6.00000i	− 0.441129i
$186$	0	0
$187$	− 1.73205i	− 0.126660i
$188$	0	0
$189$	− 18.0000i	− 1.30931i
$190$	0	0
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	− 25.9808i	− 1.86052i
$196$	0	0
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	− 6.00000i	− 0.423207i
$202$	0	0
$203$	−20.7846	−1.45879
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	15.5885	1.08347
$208$	0	0
$209$	3.00000i	0.207514i
$210$	0	0
$211$	− 13.8564i	− 0.953914i	−0.878927	−	0.476957i	$-0.841740\pi$
$211$	− 13.8564i	− 0.953914i	0.878927	−	0.476957i	$-0.158260\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	36.3731	2.48062
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	−6.92820	−0.468165
$220$	0	0
$221$	− 5.00000i	− 0.336336i
$222$	0	0
$223$	− 15.5885i	− 1.04388i	−0.852982	−	0.521940i	$-0.825208\pi$
$223$	− 15.5885i	− 1.04388i	0.852982	−	0.521940i	$-0.174792\pi$
$224$	0	0
$225$	−12.0000	−0.800000
$226$	0	0
$227$	−8.66025	−0.574801	−0.287401	−	0.957810i	$-0.592791\pi$
$227$	−8.66025	−0.574801	−0.287401	+	0.957810i	$0.592791\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	10.3923i	0.683763i
$232$	0	0
$233$	− 3.00000i	− 0.196537i	−0.995160	−	0.0982683i	$-0.968670\pi$
$233$	− 3.00000i	− 0.196537i	0.995160	−	0.0982683i	$-0.0313303\pi$
$234$	0	0
$235$	20.7846i	1.35584i
$236$	0	0
$237$	18.0000i	1.16923i
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	−15.5885	−1.00000
$244$	0	0
$245$	15.0000i	0.958315i
$246$	0	0
$247$	8.66025i	0.551039i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−24.2487	−1.53057	−0.765283	−	0.643695i	$-0.777401\pi$
$251$	−24.2487	−1.53057	−0.765283	+	0.643695i	$0.777401\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	−5.19615	−0.325396
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	6.92820i	0.430498i
$260$	0	0
$261$	18.0000i	1.11417i
$262$	0	0
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	10.3923i	0.635999i
$268$	0	0
$269$	− 3.00000i	− 0.182913i	−0.995809	−	0.0914566i	$-0.970848\pi$
$269$	− 3.00000i	− 0.182913i	0.995809	−	0.0914566i	$-0.0291523\pi$
$270$	0	0
$271$	− 19.0526i	− 1.15736i	−0.815555	−	0.578680i	$-0.803568\pi$
$271$	− 19.0526i	− 1.15736i	0.815555	−	0.578680i	$-0.196432\pi$
$272$	0	0
$273$	30.0000i	1.81568i
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	20.7846i	1.24434i
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	− 3.46410i	− 0.205919i	−0.994686	−	0.102960i	$-0.967169\pi$
$283$	− 3.46410i	− 0.205919i	0.994686	−	0.102960i	$-0.0328313\pi$
$284$	0	0
$285$	9.00000	0.533114
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	13.8564	0.812277
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	31.1769i	1.81519i
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	−25.9808	−1.50251
$300$	0	0
$301$	−42.0000	−2.42084
$302$	0	0
$303$	− 20.7846i	− 1.19404i
$304$	0	0
$305$	− 42.0000i	− 2.40491i
$306$	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
$309$	− 15.0000i	− 0.853320i
$310$	0	0
$311$	−3.46410	−0.196431	−0.0982156	−	0.995165i	$-0.531313\pi$
$311$	−3.46410	−0.196431	−0.0982156	+	0.995165i	$0.531313\pi$
$312$	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
$315$	31.1769	1.75662
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	− 10.3923i	− 0.581857i
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	1.73205	0.0963739
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	−6.92820	−0.383131
$328$	0	0
$329$	− 24.0000i	− 1.32316i
$330$	0	0
$331$	12.1244i	0.666415i	0.942854	+	0.333207i	$0.108131\pi$
$331$	12.1244i	0.666415i	−0.942854	+	0.333207i	$0.891869\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	10.3923	0.567792
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	− 5.19615i	− 0.282216i
$340$	0	0
$341$	− 12.0000i	− 0.649836i
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	27.0000i	1.45363i
$346$	0	0
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	0	0
$351$	25.9808	1.38675
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	10.3923i	0.551566i
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	17.3205	0.914141	0.457071	−	0.889430i	$-0.348899\pi$
$359$	17.3205	0.914141	0.457071	+	0.889430i	$0.348899\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	0	0
$363$	13.8564	0.727273
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	17.3205i	0.904123i	0.891987	+	0.452062i	$0.149311\pi$
$367$	17.3205i	0.904123i	−0.891987	+	0.452062i	$0.850689\pi$
$368$	0	0
$369$	9.00000i	0.468521i
$370$	0	0
$371$	−41.5692	−2.15817
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	5.19615i	0.268328i
$376$	0	0
$377$	− 30.0000i	− 1.54508i
$378$	0	0
$379$	24.2487i	1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$	24.2487i	1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$	0	0
$381$	− 33.0000i	− 1.69064i
$382$	0	0
$383$	34.6410	1.77007	0.885037	−	0.465521i	$-0.154133\pi$
$383$	34.6410	1.77007	0.885037	+	0.465521i	$0.154133\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	36.3731i	1.84895i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	5.19615i	0.262781i
$392$	0	0
$393$	−27.0000	−1.36197
$394$	0	0
$395$	−31.1769	−1.56868
$396$	0	0
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	−10.3923	−0.520266
$400$	0	0
$401$	27.0000i	1.34832i	0.738587	+	0.674158i	$0.235493\pi$
$401$	27.0000i	1.34832i	−0.738587	+	0.674158i	$0.764507\pi$
$402$	0	0
$403$	− 34.6410i	− 1.72559i
$404$	0	0
$405$	− 27.0000i	− 1.34164i
$406$	0	0
$407$	−3.46410	−0.171709
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	31.1769i	1.53784i
$412$	0	0
$413$	− 36.0000i	− 1.77144i
$414$	0	0
$415$	20.7846i	1.02028i
$416$	0	0
$417$	30.0000i	1.46911i
$418$	0	0
$419$	24.2487	1.18463	0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487	1.18463	0.592314	+	0.805708i	$0.298215\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	0	0
$423$	−20.7846	−1.01058
$424$	0	0
$425$	− 4.00000i	− 0.194029i
$426$	0	0
$427$	48.4974i	2.34695i
$428$	0	0
$429$	−15.0000	−0.724207
$430$	0	0
$431$	3.46410	0.166860	0.0834300	−	0.996514i	$-0.473413\pi$
$431$	3.46410	0.166860	0.0834300	+	0.996514i	$0.473413\pi$
$432$	0	0
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	0	0
$435$	−31.1769	−1.49482
$436$	0	0
$437$	− 9.00000i	− 0.430528i
$438$	0	0
$439$	− 6.92820i	− 0.330665i	−0.986238	−	0.165333i	$-0.947130\pi$
$439$	− 6.92820i	− 0.330665i	0.986238	−	0.165333i	$-0.0528697\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	34.6410	1.64584	0.822922	−	0.568154i	$-0.192342\pi$
$443$	34.6410	1.64584	0.822922	+	0.568154i	$0.192342\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	10.3923i	0.491539i
$448$	0	0
$449$	− 30.0000i	− 1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$	− 30.0000i	− 1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$	0	0
$451$	− 5.19615i	− 0.244677i
$452$	0	0
$453$	− 6.00000i	− 0.281905i
$454$	0	0
$455$	−51.9615	−2.43599
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	− 5.19615i	− 0.242536i
$460$	0	0
$461$	18.0000i	0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$	18.0000i	0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	0	0
$463$	− 17.3205i	− 0.804952i	−0.915430	−	0.402476i	$-0.868150\pi$
$463$	− 17.3205i	− 0.804952i	0.915430	−	0.402476i	$-0.131850\pi$
$464$	0	0
$465$	−36.0000	−1.66946
$466$	0	0
$467$	−38.1051	−1.76329	−0.881647	−	0.471909i	$-0.843565\pi$
$467$	−38.1051	−1.76329	−0.881647	+	0.471909i	$0.843565\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	22.5167	1.03751
$472$	0	0
$473$	− 21.0000i	− 0.965581i
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	36.0000i	1.64833i
$478$	0	0
$479$	29.4449	1.34537	0.672685	−	0.739929i	$-0.265141\pi$
$479$	29.4449	1.34537	0.672685	+	0.739929i	$0.265141\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	− 31.1769i	− 1.41860i
$484$	0	0
$485$	24.0000i	1.08978i
$486$	0	0
$487$	24.2487i	1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$	24.2487i	1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$	0	0
$489$	36.0000i	1.62798i
$490$	0	0
$491$	13.8564	0.625331	0.312665	−	0.949863i	$-0.398778\pi$
$491$	13.8564	0.625331	0.312665	+	0.949863i	$0.398778\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	15.5885i	0.700649i
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	27.7128i	1.24060i	0.784366	+	0.620298i	$0.212988\pi$
$499$	27.7128i	1.24060i	−0.784366	+	0.620298i	$0.787012\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	−36.3731	−1.62179	−0.810897	−	0.585188i	$-0.801021\pi$
$503$	−36.3731	−1.62179	−0.810897	+	0.585188i	$0.801021\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	−20.7846	−0.923077
$508$	0	0
$509$	− 36.0000i	− 1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$	− 36.0000i	− 1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$	0	0
$511$	13.8564i	0.612971i
$512$	0	0
$513$	9.00000i	0.397360i
$514$	0	0
$515$	25.9808	1.14485
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	− 36.3731i	− 1.59660i
$520$	0	0
$521$	15.0000i	0.657162i	0.944476	+	0.328581i	$0.106570\pi$
$521$	15.0000i	0.657162i	−0.944476	+	0.328581i	$0.893430\pi$
$522$	0	0
$523$	31.1769i	1.36327i	0.731692	+	0.681636i	$0.238731\pi$
$523$	31.1769i	1.36327i	−0.731692	+	0.681636i	$0.761269\pi$
$524$	0	0
$525$	24.0000i	1.04745i
$526$	0	0
$527$	−6.92820	−0.301797
$528$	0	0
$529$	4.00000	0.173913
$530$	0	0
$531$	−31.1769	−1.35296
$532$	0	0
$533$	− 15.0000i	− 0.649722i
$534$	0	0
$535$	− 5.19615i	− 0.224649i
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	8.66025	0.373024
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	0	0
$543$	−17.3205	−0.743294
$544$	0	0
$545$	− 12.0000i	− 0.514024i
$546$	0	0
$547$	− 3.46410i	− 0.148114i	−0.997254	−	0.0740571i	$-0.976405\pi$
$547$	− 3.46410i	− 0.148114i	0.997254	−	0.0740571i	$-0.0235947\pi$
$548$	0	0
$549$	42.0000	1.79252
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	10.3923i	0.441129i
$556$	0	0
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	− 60.6218i	− 2.56403i
$560$	0	0
$561$	3.00000i	0.126660i
$562$	0	0
$563$	6.92820	0.291989	0.145994	−	0.989285i	$-0.453362\pi$
$563$	6.92820	0.291989	0.145994	+	0.989285i	$0.453362\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	0	0
$567$	31.1769i	1.30931i
$568$	0	0
$569$	24.0000i	1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$	24.0000i	1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$	0	0
$571$	13.8564i	0.579873i	0.957046	+	0.289936i	$0.0936341\pi$
$571$	13.8564i	0.579873i	−0.957046	+	0.289936i	$0.906366\pi$
$572$	0	0
$573$	30.0000	1.25327
$574$	0	0
$575$	−20.7846	−0.866778
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	0	0
$579$	27.7128	1.15171
$580$	0	0
$581$	− 24.0000i	− 0.995688i
$582$	0	0
$583$	− 20.7846i	− 0.860811i
$584$	0	0
$585$	45.0000i	1.86052i
$586$	0	0
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	− 25.9808i	− 1.06871i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	10.3923i	0.426043i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.46410	0.141539	0.0707697	−	0.997493i	$-0.477454\pi$
$599$	3.46410	0.141539	0.0707697	+	0.997493i	$0.477454\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	10.3923i	0.423207i
$604$	0	0
$605$	24.0000i	0.975739i
$606$	0	0
$607$	3.46410i	0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$	3.46410i	0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	34.6410	1.40143
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	0	0
$615$	−15.5885	−0.628587
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	− 31.1769i	− 1.25311i	−0.779379	−	0.626553i	$-0.784465\pi$
$619$	− 31.1769i	− 1.25311i	0.779379	−	0.626553i	$-0.215535\pi$
$620$	0	0
$621$	−27.0000	−1.08347
$622$	0	0
$623$	20.7846	0.832718
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	− 5.19615i	− 0.207514i
$628$	0	0
$629$	2.00000i	0.0797452i
$630$	0	0
$631$	− 8.66025i	− 0.344759i	−0.985031	−	0.172380i	$-0.944854\pi$
$631$	− 8.66025i	− 0.344759i	0.985031	−	0.172380i	$-0.0551456\pi$
$632$	0	0
$633$	24.0000i	0.953914i
$634$	0	0
$635$	57.1577	2.26823
$636$	0	0
$637$	25.0000	0.990536
$638$	0	0
$639$	−10.3923	−0.411113
$640$	0	0
$641$	21.0000i	0.829450i	0.909947	+	0.414725i	$0.136122\pi$
$641$	21.0000i	0.829450i	−0.909947	+	0.414725i	$0.863878\pi$
$642$	0	0
$643$	− 27.7128i	− 1.09289i	−0.837496	−	0.546443i	$-0.815981\pi$
$643$	− 27.7128i	− 1.09289i	0.837496	−	0.546443i	$-0.184019\pi$
$644$	0	0
$645$	−63.0000	−2.48062
$646$	0	0
$647$	17.3205	0.680939	0.340470	−	0.940255i	$-0.389414\pi$
$647$	17.3205	0.680939	0.340470	+	0.940255i	$0.389414\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	41.5692	1.62923
$652$	0	0
$653$	3.00000i	0.117399i	0.998276	+	0.0586995i	$0.0186954\pi$
$653$	3.00000i	0.117399i	−0.998276	+	0.0586995i	$0.981305\pi$
$654$	0	0
$655$	− 46.7654i	− 1.82727i
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	−38.1051	−1.48436	−0.742182	−	0.670198i	$-0.766209\pi$
$659$	−38.1051	−1.48436	−0.742182	+	0.670198i	$0.766209\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	0	0
$663$	8.66025i	0.336336i
$664$	0	0
$665$	− 18.0000i	− 0.698010i
$666$	0	0
$667$	31.1769i	1.20717i
$668$	0	0
$669$	27.0000i	1.04388i
$670$	0	0
$671$	−24.2487	−0.936111
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	0	0
$675$	20.7846	0.800000
$676$	0	0
$677$	− 3.00000i	− 0.115299i	−0.998337	−	0.0576497i	$-0.981639\pi$
$677$	− 3.00000i	− 0.115299i	0.998337	−	0.0576497i	$-0.0183606\pi$
$678$	0	0
$679$	− 27.7128i	− 1.06352i
$680$	0	0
$681$	15.0000	0.574801
$682$	0	0
$683$	−25.9808	−0.994126	−0.497063	−	0.867714i	$-0.665588\pi$
$683$	−25.9808	−0.994126	−0.497063	+	0.867714i	$0.665588\pi$
$684$	0	0
$685$	−54.0000	−2.06323
$686$	0	0
$687$	45.0333	1.71813
$688$	0	0
$689$	− 60.0000i	− 2.28582i
$690$	0	0
$691$	34.6410i	1.31781i	0.752228	+	0.658903i	$0.228979\pi$
$691$	34.6410i	1.31781i	−0.752228	+	0.658903i	$0.771021\pi$
$692$	0	0
$693$	− 18.0000i	− 0.683763i
$694$	0	0
$695$	−51.9615	−1.97101
$696$	0	0
$697$	−3.00000	−0.113633
$698$	0	0
$699$	5.19615i	0.196537i
$700$	0	0
$701$	12.0000i	0.453234i	0.973984	+	0.226617i	$0.0727665\pi$
$701$	12.0000i	0.453234i	−0.973984	+	0.226617i	$0.927233\pi$
$702$	0	0
$703$	− 3.46410i	− 0.130651i
$704$	0	0
$705$	− 36.0000i	− 1.35584i
$706$	0	0
$707$	−41.5692	−1.56337
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	− 31.1769i	− 1.16923i
$712$	0	0
$713$	36.0000i	1.34821i
$714$	0	0
$715$	− 25.9808i	− 0.971625i
$716$	0	0
$717$	36.0000	1.34444
$718$	0	0
$719$	1.73205	0.0645946	0.0322973	−	0.999478i	$-0.489718\pi$
$719$	1.73205	0.0645946	0.0322973	+	0.999478i	$0.489718\pi$
$720$	0	0
$721$	−30.0000	−1.11726
$722$	0	0
$723$	3.46410	0.128831
$724$	0	0
$725$	− 24.0000i	− 0.891338i
$726$	0	0
$727$	3.46410i	0.128476i	0.997935	+	0.0642382i	$0.0204617\pi$
$727$	3.46410i	0.128476i	−0.997935	+	0.0642382i	$0.979538\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−12.1244	−0.448435
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	− 25.9808i	− 0.958315i
$736$	0	0
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	− 50.2295i	− 1.84772i	−0.382729	−	0.923861i	$-0.625016\pi$
$739$	− 50.2295i	− 1.84772i	0.382729	−	0.923861i	$-0.374984\pi$
$740$	0	0
$741$	− 15.0000i	− 0.551039i
$742$	0	0
$743$	3.46410	0.127086	0.0635428	−	0.997979i	$-0.479760\pi$
$743$	3.46410	0.127086	0.0635428	+	0.997979i	$0.479760\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−20.7846	−0.760469
$748$	0	0
$749$	6.00000i	0.219235i
$750$	0	0
$751$	13.8564i	0.505627i	0.967515	+	0.252814i	$0.0813560\pi$
$751$	13.8564i	0.505627i	−0.967515	+	0.252814i	$0.918644\pi$
$752$	0	0
$753$	42.0000	1.53057
$754$	0	0
$755$	10.3923	0.378215
$756$	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
$759$	15.5885	0.565825
$760$	0	0
$761$	− 24.0000i	− 0.869999i	−0.900431	−	0.435000i	$-0.856748\pi$
$761$	− 24.0000i	− 0.869999i	0.900431	−	0.435000i	$-0.143252\pi$
$762$	0	0
$763$	13.8564i	0.501636i
$764$	0	0
$765$	9.00000	0.325396
$766$	0	0
$767$	51.9615	1.87622
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	10.3923i	0.374270i
$772$	0	0
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	− 27.7128i	− 0.995474i
$776$	0	0
$777$	− 12.0000i	− 0.430498i
$778$	0	0
$779$	5.19615	0.186171
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	− 31.1769i	− 1.11417i
$784$	0	0
$785$	39.0000i	1.39197i
$786$	0	0
$787$	13.8564i	0.493928i	0.969025	+	0.246964i	$0.0794329\pi$
$787$	13.8564i	0.493928i	−0.969025	+	0.246964i	$0.920567\pi$
$788$	0	0
$789$	−42.0000	−1.49524
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−70.0000	−2.48577
$794$	0	0
$795$	−62.3538	−2.21146
$796$	0	0
$797$	− 48.0000i	− 1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$	− 48.0000i	− 1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$	0	0
$799$	− 6.92820i	− 0.245102i
$800$	0	0
$801$	− 18.0000i	− 0.635999i
$802$	0	0
$803$	−6.92820	−0.244491
$804$	0	0
$805$	54.0000	1.90325
$806$	0	0
$807$	5.19615i	0.182913i
$808$	0	0
$809$	− 45.0000i	− 1.58212i	−0.611741	−	0.791058i	$-0.709531\pi$
$809$	− 45.0000i	− 1.58212i	0.611741	−	0.791058i	$-0.290469\pi$
$810$	0	0
$811$	− 27.7128i	− 0.973128i	−0.873645	−	0.486564i	$-0.838250\pi$
$811$	− 27.7128i	− 0.973128i	0.873645	−	0.486564i	$-0.161750\pi$
$812$	0	0
$813$	33.0000i	1.15736i
$814$	0	0
$815$	−62.3538	−2.18416
$816$	0	0
$817$	21.0000	0.734697
$818$	0	0
$819$	− 51.9615i	− 1.81568i
$820$	0	0
$821$	− 3.00000i	− 0.104701i	−0.998629	−	0.0523504i	$-0.983329\pi$
$821$	− 3.00000i	− 0.104701i	0.998629	−	0.0523504i	$-0.0166713\pi$
$822$	0	0
$823$	− 3.46410i	− 0.120751i	−0.998176	−	0.0603755i	$-0.980770\pi$
$823$	− 3.46410i	− 0.120751i	0.998176	−	0.0603755i	$-0.0192298\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	19.0526	0.662522	0.331261	−	0.943539i	$-0.392526\pi$
$827$	19.0526	0.662522	0.331261	+	0.943539i	$0.392526\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	13.8564	0.480673
$832$	0	0
$833$	− 5.00000i	− 0.173240i
$834$	0	0
$835$	15.5885i	0.539461i
$836$	0	0
$837$	− 36.0000i	− 1.24434i
$838$	0	0
$839$	−29.4449	−1.01655	−0.508275	−	0.861195i	$-0.669717\pi$
$839$	−29.4449	−1.01655	−0.508275	+	0.861195i	$0.669717\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	10.3923i	0.357930i
$844$	0	0
$845$	− 36.0000i	− 1.23844i
$846$	0	0
$847$	− 27.7128i	− 0.952224i
$848$	0	0
$849$	6.00000i	0.205919i
$850$	0	0
$851$	10.3923	0.356244
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−15.5885	−0.533114
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	− 17.3205i	− 0.590968i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	− 17.3205i	− 0.590968i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	−38.1051	−1.29711	−0.648557	−	0.761166i	$-0.724627\pi$
$863$	−38.1051	−1.29711	−0.648557	+	0.761166i	$0.724627\pi$
$864$	0	0
$865$	63.0000	2.14206
$866$	0	0
$867$	1.73205	0.0588235
$868$	0	0
$869$	18.0000i	0.610608i
$870$	0	0
$871$	− 17.3205i	− 0.586883i
$872$	0	0
$873$	−24.0000	−0.812277
$874$	0	0
$875$	10.3923	0.351324
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	− 51.9615i	− 1.75262i
$880$	0	0
$881$	30.0000i	1.01073i	0.862907	+	0.505363i	$0.168641\pi$
$881$	30.0000i	1.01073i	−0.862907	+	0.505363i	$0.831359\pi$
$882$	0	0
$883$	− 5.19615i	− 0.174864i	−0.996170	−	0.0874322i	$-0.972134\pi$
$883$	− 5.19615i	− 0.174864i	0.996170	−	0.0874322i	$-0.0278661\pi$
$884$	0	0
$885$	− 54.0000i	− 1.81519i
$886$	0	0
$887$	8.66025	0.290783	0.145391	−	0.989374i	$-0.453556\pi$
$887$	8.66025	0.290783	0.145391	+	0.989374i	$0.453556\pi$
$888$	0	0
$889$	−66.0000	−2.21357
$890$	0	0
$891$	−15.5885	−0.522233
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	− 41.5692i	− 1.38951i
$896$	0	0
$897$	45.0000	1.50251
$898$	0	0
$899$	−41.5692	−1.38641
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	72.7461	2.42084
$904$	0	0
$905$	− 30.0000i	− 0.997234i
$906$	0	0
$907$	− 3.46410i	− 0.115024i	−0.998345	−	0.0575118i	$-0.981683\pi$
$907$	− 3.46410i	− 0.115024i	0.998345	−	0.0575118i	$-0.0183167\pi$
$908$	0	0
$909$	36.0000i	1.19404i
$910$	0	0
$911$	32.9090	1.09032	0.545161	−	0.838331i	$-0.316468\pi$
$911$	32.9090	1.09032	0.545161	+	0.838331i	$0.316468\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	72.7461i	2.40491i
$916$	0	0
$917$	54.0000i	1.78324i
$918$	0	0
$919$	15.5885i	0.514216i	0.966383	+	0.257108i	$0.0827696\pi$
$919$	15.5885i	0.514216i	−0.966383	+	0.257108i	$0.917230\pi$
$920$	0	0
$921$	18.0000i	0.593120i
$922$	0	0
$923$	17.3205	0.570111
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	25.9808i	0.853320i
$928$	0	0
$929$	15.0000i	0.492134i	0.969253	+	0.246067i	$0.0791383\pi$
$929$	15.0000i	0.492134i	−0.969253	+	0.246067i	$0.920862\pi$
$930$	0	0
$931$	8.66025i	0.283828i
$932$	0	0
$933$	6.00000	0.196431
$934$	0	0
$935$	−5.19615	−0.169932
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	6.92820	0.226093
$940$	0	0
$941$	− 6.00000i	− 0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$	− 6.00000i	− 0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$	0	0
$943$	15.5885i	0.507630i
$944$	0	0
$945$	−54.0000	−1.75662
$946$	0	0
$947$	31.1769	1.01311	0.506557	−	0.862207i	$-0.330918\pi$
$947$	31.1769	1.01311	0.506557	+	0.862207i	$0.330918\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	− 31.1769i	− 1.01098i
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	51.9615i	1.68144i
$956$	0	0
$957$	18.0000i	0.581857i
$958$	0	0
$959$	62.3538	2.01351
$960$	0	0
$961$	−17.0000	−0.548387
$962$	0	0
$963$	5.19615	0.167444
$964$	0	0
$965$	48.0000i	1.54517i
$966$	0	0
$967$	25.9808i	0.835485i	0.908565	+	0.417742i	$0.137179\pi$
$967$	25.9808i	0.835485i	−0.908565	+	0.417742i	$0.862821\pi$
$968$	0	0
$969$	−3.00000	−0.0963739
$970$	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
$973$	60.0000	1.92351
$974$	0	0
$975$	−34.6410	−1.10940
$976$	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
$979$	10.3923i	0.332140i
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	43.3013	1.38110	0.690548	−	0.723287i	$-0.257369\pi$
$983$	43.3013	1.38110	0.690548	+	0.723287i	$0.257369\pi$
$984$	0	0
$985$	45.0000	1.43382
$986$	0	0
$987$	41.5692i	1.32316i
$988$	0	0
$989$	63.0000i	2.00328i
$990$	0	0
$991$	− 20.7846i	− 0.660245i	−0.943938	−	0.330122i	$-0.892910\pi$
$991$	− 20.7846i	− 0.660245i	0.943938	−	0.330122i	$-0.107090\pi$
$992$	0	0
$993$	− 21.0000i	− 0.666415i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	0	0
$999$	−10.3923	−0.328798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.e.a.239.1	✓	4
3.2	odd	2	inner	816.2.e.a.239.4	yes	4
4.3	odd	2	inner	816.2.e.a.239.3	yes	4
12.11	even	2	inner	816.2.e.a.239.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.e.a.239.1	✓	4	1.1	even	1	trivial
816.2.e.a.239.2	yes	4	12.11	even	2	inner
816.2.e.a.239.3	yes	4	4.3	odd	2	inner
816.2.e.a.239.4	yes	4	3.2	odd	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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -3.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} -1.73205 q^{11} -5.00000 q^{13} +5.19615i q^{15} +1.00000i q^{17} -1.73205i q^{19} -6.00000i q^{21} +5.19615 q^{23} -4.00000 q^{25} -5.19615 q^{27} +6.00000i q^{29} +6.92820i q^{31} +3.00000 q^{33} +10.3923 q^{35} +2.00000 q^{37} +8.66025 q^{39} +3.00000i q^{41} +12.1244i q^{43} -9.00000i q^{45} -6.92820 q^{47} -5.00000 q^{49} -1.73205i q^{51} +12.0000i q^{53} +5.19615i q^{55} +3.00000i q^{57} -10.3923 q^{59} +14.0000 q^{61} +10.3923i q^{63} +15.0000i q^{65} +3.46410i q^{67} -9.00000 q^{69} -3.46410 q^{71} +4.00000 q^{73} +6.92820 q^{75} -6.00000i q^{77} -10.3923i q^{79} +9.00000 q^{81} -6.92820 q^{83} +3.00000 q^{85} -10.3923i q^{87} -6.00000i q^{89} -17.3205i q^{91} -12.0000i q^{93} -5.19615 q^{95} -8.00000 q^{97} -5.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9} - 20 q^{13} - 16 q^{25} + 12 q^{33} + 8 q^{37} - 20 q^{49} + 56 q^{61} - 36 q^{69} + 16 q^{73} + 36 q^{81} + 12 q^{85} - 32 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 20 * q^13 - 16 * q^25 + 12 * q^33 + 8 * q^37 - 20 * q^49 + 56 * q^61 - 36 * q^69 + 16 * q^73 + 36 * q^81 + 12 * q^85 - 32 * q^97

Introduction

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Database

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