LMFDB - Embedded newform 5184.2.d.g.2593.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2593,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.4
Root			$1.93649 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.g.2593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.87298i q^{5} +3.87298 q^{7} +O(q^{10})$	$q+3.87298i q^{5} +3.87298 q^{7} -3.00000i q^{11} +3.87298i q^{13} -2.00000i q^{19} -3.87298 q^{23} -10.0000 q^{25} +3.87298i q^{29} +3.87298 q^{31} +15.0000i q^{35} +7.74597i q^{37} -9.00000 q^{41} -7.00000i q^{43} -3.87298 q^{47} +8.00000 q^{49} +7.74597i q^{53} +11.6190 q^{55} +3.00000i q^{59} +11.6190i q^{61} -15.0000 q^{65} +11.0000i q^{67} -7.74597 q^{71} +8.00000 q^{73} -11.6190i q^{77} -3.87298 q^{79} +3.00000i q^{83} +12.0000 q^{89} +15.0000i q^{91} +7.74597 q^{95} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 40 q^{25} - 36 q^{41} + 32 q^{49} - 60 q^{65} + 32 q^{73} + 48 q^{89} + 4 q^{97}+O(q^{100}) $ 4 * q - 40 * q^25 - 36 * q^41 + 32 * q^49 - 60 * q^65 + 32 * q^73 + 48 * q^89 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	3.87298i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$5$	3.87298i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	3.87298	1.46385	0.731925	−	0.681385i	$-0.238622\pi$
$7$	3.87298	1.46385	0.731925	+	0.681385i	$0.238622\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	3.87298i	1.07417i	0.843527	+	0.537086i	$0.180475\pi$
$13$	3.87298i	1.07417i	−0.843527	+	0.537086i	$0.819525\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.87298	−0.807573	−0.403786	−	0.914853i	$-0.632306\pi$
$23$	−3.87298	−0.807573	−0.403786	+	0.914853i	$0.632306\pi$
$24$	0	0
$25$	−10.0000	−2.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.87298i	0.719195i	0.933108	+	0.359597i	$0.117086\pi$
$29$	3.87298i	0.719195i	−0.933108	+	0.359597i	$0.882914\pi$
$30$	0	0
$31$	3.87298	0.695608	0.347804	−	0.937567i	$-0.386927\pi$
$31$	3.87298	0.695608	0.347804	+	0.937567i	$0.386927\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.0000i	2.53546i
$36$	0	0
$37$	7.74597i	1.27343i	0.771100	+	0.636715i	$0.219707\pi$
$37$	7.74597i	1.27343i	−0.771100	+	0.636715i	$0.780293\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 7.00000i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$	− 7.00000i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.87298	−0.564933	−0.282466	−	0.959277i	$-0.591153\pi$
$47$	−3.87298	−0.564933	−0.282466	+	0.959277i	$0.591153\pi$
$48$	0	0
$49$	8.00000	1.14286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.74597i	1.06399i	0.846747	+	0.531995i	$0.178558\pi$
$53$	7.74597i	1.06399i	−0.846747	+	0.531995i	$0.821442\pi$
$54$	0	0
$55$	11.6190	1.56670
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.00000i	0.390567i	0.980747	+	0.195283i	$0.0625627\pi$
$59$	3.00000i	0.390567i	−0.980747	+	0.195283i	$0.937437\pi$
$60$	0	0
$61$	11.6190i	1.48765i	0.668372	+	0.743827i	$0.266991\pi$
$61$	11.6190i	1.48765i	−0.668372	+	0.743827i	$0.733009\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.0000	−1.86052
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.74597	−0.919277	−0.459639	−	0.888106i	$-0.652021\pi$
$71$	−7.74597	−0.919277	−0.459639	+	0.888106i	$0.652021\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.6190i	− 1.32410i
$78$	0	0
$79$	−3.87298	−0.435745	−0.217872	−	0.975977i	$-0.569912\pi$
$79$	−3.87298	−0.435745	−0.217872	+	0.975977i	$0.569912\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.00000i	0.329293i	0.986353	+	0.164646i	$0.0526483\pi$
$83$	3.00000i	0.329293i	−0.986353	+	0.164646i	$0.947352\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	15.0000i	1.57243i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.74597	0.794719
$96$	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
$101$	− 3.87298i	− 0.385376i	−0.981260	−	0.192688i	$-0.938279\pi$
$101$	− 3.87298i	− 0.385376i	0.981260	−	0.192688i	$-0.0617206\pi$
$102$	0	0
$103$	11.6190	1.14485	0.572425	−	0.819957i	$-0.306003\pi$
$103$	11.6190	1.14485	0.572425	+	0.819957i	$0.306003\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	− 7.74597i	− 0.741929i	−0.928647	−	0.370965i	$-0.879027\pi$
$109$	− 7.74597i	− 0.741929i	0.928647	−	0.370965i	$-0.120973\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	− 15.0000i	− 1.39876i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 19.3649i	− 1.73205i
$126$	0	0
$127$	−15.4919	−1.37469	−0.687343	−	0.726333i	$-0.741223\pi$
$127$	−15.4919	−1.37469	−0.687343	+	0.726333i	$0.741223\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.00000i	0.262111i	0.991375	+	0.131056i	$0.0418366\pi$
$131$	3.00000i	0.262111i	−0.991375	+	0.131056i	$0.958163\pi$
$132$	0	0
$133$	− 7.74597i	− 0.671660i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	1.00000i	0.0848189i	0.999100	+	0.0424094i	$0.0135034\pi$
$139$	1.00000i	0.0848189i	−0.999100	+	0.0424094i	$0.986497\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.6190	0.971625
$144$	0	0
$145$	−15.0000	−1.24568
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.6190i	0.951861i	0.879483	+	0.475931i	$0.157889\pi$
$149$	11.6190i	0.951861i	−0.879483	+	0.475931i	$0.842111\pi$
$150$	0	0
$151$	11.6190	0.945537	0.472768	−	0.881187i	$-0.343255\pi$
$151$	11.6190	0.945537	0.472768	+	0.881187i	$0.343255\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.0000i	1.20483i
$156$	0	0
$157$	− 3.87298i	− 0.309098i	−0.987985	−	0.154549i	$-0.950608\pi$
$157$	− 3.87298i	− 0.309098i	0.987985	−	0.154549i	$-0.0493924\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.0000	−1.18217
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.87298	0.299700	0.149850	−	0.988709i	$-0.452121\pi$
$167$	3.87298	0.299700	0.149850	+	0.988709i	$0.452121\pi$
$168$	0	0
$169$	−2.00000	−0.153846
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 19.3649i	− 1.47229i	−0.676825	−	0.736144i	$-0.736645\pi$
$173$	− 19.3649i	− 1.47229i	0.676825	−	0.736144i	$-0.263355\pi$
$174$	0	0
$175$	−38.7298	−2.92770
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	0	0
$181$	− 7.74597i	− 0.575753i	−0.957668	−	0.287877i	$-0.907051\pi$
$181$	− 7.74597i	− 0.575753i	0.957668	−	0.287877i	$-0.0929493\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−30.0000	−2.20564
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.1109	1.96167	0.980837	−	0.194828i	$-0.0624150\pi$
$191$	27.1109	1.96167	0.980837	+	0.194828i	$0.0624150\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−7.74597	−0.549097	−0.274549	−	0.961573i	$-0.588528\pi$
$199$	−7.74597	−0.549097	−0.274549	+	0.961573i	$0.588528\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.0000i	1.05279i
$204$	0	0
$205$	− 34.8569i	− 2.43451i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	− 1.00000i	− 0.0688428i	−0.999407	−	0.0344214i	$-0.989041\pi$
$211$	− 1.00000i	− 0.0688428i	0.999407	−	0.0344214i	$-0.0109588\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.1109	1.84895
$216$	0	0
$217$	15.0000	1.01827
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−27.1109	−1.81548	−0.907740	−	0.419534i	$-0.862193\pi$
$223$	−27.1109	−1.81548	−0.907740	+	0.419534i	$0.862193\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.0000i	1.39382i	0.717159	+	0.696909i	$0.245442\pi$
$227$	21.0000i	1.39382i	−0.717159	+	0.696909i	$0.754558\pi$
$228$	0	0
$229$	− 3.87298i	− 0.255934i	−0.991778	−	0.127967i	$-0.959155\pi$
$229$	− 3.87298i	− 0.255934i	0.991778	−	0.127967i	$-0.0408452\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	− 15.0000i	− 0.978492i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.3649	−1.25261	−0.626306	−	0.779577i	$-0.715434\pi$
$239$	−19.3649	−1.25261	−0.626306	+	0.779577i	$0.715434\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	30.9839i	1.97949i
$246$	0	0
$247$	7.74597	0.492864
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.00000i	− 0.378717i	−0.981908	−	0.189358i	$-0.939359\pi$
$251$	− 6.00000i	− 0.378717i	0.981908	−	0.189358i	$-0.0606408\pi$
$252$	0	0
$253$	11.6190i	0.730477i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	0	0
$259$	30.0000i	1.86411i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.6190	−0.716455	−0.358228	−	0.933634i	$-0.616619\pi$
$263$	−11.6190	−0.716455	−0.358228	+	0.933634i	$0.616619\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.2379i	1.41684i	0.705791	+	0.708420i	$0.250592\pi$
$269$	23.2379i	1.41684i	−0.705791	+	0.708420i	$0.749408\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.0000i	1.80907i
$276$	0	0
$277$	19.3649i	1.16353i	0.813359	+	0.581763i	$0.197637\pi$
$277$	19.3649i	1.16353i	−0.813359	+	0.581763i	$0.802363\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	− 23.0000i	− 1.36721i	−0.729853	−	0.683604i	$-0.760412\pi$
$283$	− 23.0000i	− 1.36721i	0.729853	−	0.683604i	$-0.239588\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.8569	−2.05753
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 27.1109i	− 1.58383i	−0.610628	−	0.791917i	$-0.709083\pi$
$293$	− 27.1109i	− 1.58383i	0.610628	−	0.791917i	$-0.290917\pi$
$294$	0	0
$295$	−11.6190	−0.676481
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 15.0000i	− 0.867472i
$300$	0	0
$301$	− 27.1109i	− 1.56265i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−45.0000	−2.57669
$306$	0	0
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.1109	1.53732	0.768659	−	0.639659i	$-0.220925\pi$
$311$	27.1109	1.53732	0.768659	+	0.639659i	$0.220925\pi$
$312$	0	0
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.6190i	− 0.652585i	−0.945269	−	0.326293i	$-0.894201\pi$
$317$	− 11.6190i	− 0.652585i	0.945269	−	0.326293i	$-0.105799\pi$
$318$	0	0
$319$	11.6190	0.650536
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 38.7298i	− 2.14834i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−15.0000	−0.826977
$330$	0	0
$331$	19.0000i	1.04433i	0.852843	+	0.522167i	$0.174876\pi$
$331$	19.0000i	1.04433i	−0.852843	+	0.522167i	$0.825124\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−42.6028	−2.32764
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 11.6190i	− 0.629201i
$342$	0	0
$343$	3.87298	0.209121
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.0000i	− 0.805242i	−0.915367	−	0.402621i	$-0.868099\pi$
$347$	− 15.0000i	− 0.805242i	0.915367	−	0.402621i	$-0.131901\pi$
$348$	0	0
$349$	19.3649i	1.03658i	0.855205	+	0.518290i	$0.173431\pi$
$349$	19.3649i	1.03658i	−0.855205	+	0.518290i	$0.826569\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	− 30.0000i	− 1.59223i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.9839	−1.63527	−0.817633	−	0.575740i	$-0.804714\pi$
$359$	−30.9839	−1.63527	−0.817633	+	0.575740i	$0.804714\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	30.9839i	1.62177i
$366$	0	0
$367$	11.6190	0.606504	0.303252	−	0.952910i	$-0.401928\pi$
$367$	11.6190	0.606504	0.303252	+	0.952910i	$0.401928\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	30.0000i	1.55752i
$372$	0	0
$373$	3.87298i	0.200535i	0.994960	+	0.100268i	$0.0319699\pi$
$373$	3.87298i	0.200535i	−0.994960	+	0.100268i	$0.968030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.0000	−0.772539
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.87298	0.197900	0.0989501	−	0.995092i	$-0.468452\pi$
$383$	3.87298	0.197900	0.0989501	+	0.995092i	$0.468452\pi$
$384$	0	0
$385$	45.0000	2.29341
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 3.87298i	− 0.196368i	−0.995168	−	0.0981840i	$-0.968697\pi$
$389$	− 3.87298i	− 0.196368i	0.995168	−	0.0981840i	$-0.0313034\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 15.0000i	− 0.754732i
$396$	0	0
$397$	7.74597i	0.388759i	0.980926	+	0.194379i	$0.0622693\pi$
$397$	7.74597i	0.388759i	−0.980926	+	0.194379i	$0.937731\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	15.0000i	0.747203i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.2379	1.15186
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.6190i	0.571731i
$414$	0	0
$415$	−11.6190	−0.570352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.0000i	1.61216i	0.591810	+	0.806078i	$0.298414\pi$
$419$	33.0000i	1.61216i	−0.591810	+	0.806078i	$0.701586\pi$
$420$	0	0
$421$	− 3.87298i	− 0.188758i	−0.995536	−	0.0943788i	$-0.969914\pi$
$421$	− 3.87298i	− 0.188758i	0.995536	−	0.0943788i	$-0.0300865\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	45.0000i	2.17770i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.4919	0.746220	0.373110	−	0.927787i	$-0.378291\pi$
$431$	15.4919	0.746220	0.373110	+	0.927787i	$0.378291\pi$
$432$	0	0
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.74597i	0.370540i
$438$	0	0
$439$	11.6190	0.554542	0.277271	−	0.960792i	$-0.410570\pi$
$439$	11.6190	0.554542	0.277271	+	0.960792i	$0.410570\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0000i	0.712672i	0.934358	+	0.356336i	$0.115974\pi$
$443$	15.0000i	0.712672i	−0.934358	+	0.356336i	$0.884026\pi$
$444$	0	0
$445$	46.4758i	2.20316i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	27.0000i	1.27138i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−58.0948	−2.72352
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.8569i	1.62345i	0.584043	+	0.811723i	$0.301470\pi$
$461$	34.8569i	1.62345i	−0.584043	+	0.811723i	$0.698530\pi$
$462$	0	0
$463$	19.3649	0.899964	0.449982	−	0.893038i	$-0.351430\pi$
$463$	19.3649	0.899964	0.449982	+	0.893038i	$0.351430\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	42.6028i	1.96722i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.0000	−0.965581
$474$	0	0
$475$	20.0000i	0.917663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.6190	−0.530883	−0.265442	−	0.964127i	$-0.585518\pi$
$479$	−11.6190	−0.530883	−0.265442	+	0.964127i	$0.585518\pi$
$480$	0	0
$481$	−30.0000	−1.36788
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.87298i	0.175863i
$486$	0	0
$487$	30.9839	1.40401	0.702007	−	0.712171i	$-0.252288\pi$
$487$	30.9839	1.40401	0.702007	+	0.712171i	$0.252288\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21.0000i	− 0.947717i	−0.880601	−	0.473858i	$-0.842861\pi$
$491$	− 21.0000i	− 0.947717i	0.880601	−	0.473858i	$-0.157139\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−30.0000	−1.34568
$498$	0	0
$499$	− 7.00000i	− 0.313363i	−0.987649	−	0.156682i	$-0.949920\pi$
$499$	− 7.00000i	− 0.313363i	0.987649	−	0.156682i	$-0.0500796\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.74597	0.345376	0.172688	−	0.984977i	$-0.444755\pi$
$503$	7.74597	0.345376	0.172688	+	0.984977i	$0.444755\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.87298i	− 0.171667i	−0.996309	−	0.0858335i	$-0.972645\pi$
$509$	− 3.87298i	− 0.171667i	0.996309	−	0.0858335i	$-0.0273553\pi$
$510$	0	0
$511$	30.9839	1.37065
$512$	0	0
$513$	0	0
$514$	0	0
$515$	45.0000i	1.98294i
$516$	0	0
$517$	11.6190i	0.511001i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−8.00000	−0.347826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 34.8569i	− 1.50982i
$534$	0	0
$535$	−23.2379	−1.00466
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 24.0000i	− 1.03375i
$540$	0	0
$541$	30.9839i	1.33210i	0.745907	+	0.666050i	$0.232016\pi$
$541$	30.9839i	1.33210i	−0.745907	+	0.666050i	$0.767984\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	− 41.0000i	− 1.75303i	−0.481371	−	0.876517i	$-0.659861\pi$
$547$	− 41.0000i	− 1.75303i	0.481371	−	0.876517i	$-0.340139\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.74597	0.329989
$552$	0	0
$553$	−15.0000	−0.637865
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.2379i	− 0.984621i	−0.870420	−	0.492311i	$-0.836152\pi$
$557$	− 23.2379i	− 0.984621i	0.870420	−	0.492311i	$-0.163848\pi$
$558$	0	0
$559$	27.1109	1.14667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.0000i	1.39078i	0.718631	+	0.695392i	$0.244769\pi$
$563$	33.0000i	1.39078i	−0.718631	+	0.695392i	$0.755231\pi$
$564$	0	0
$565$	− 11.6190i	− 0.488813i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	− 41.0000i	− 1.71580i	−0.513820	−	0.857898i	$-0.671770\pi$
$571$	− 41.0000i	− 1.71580i	0.513820	−	0.857898i	$-0.328230\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.7298	1.61515
$576$	0	0
$577$	−44.0000	−1.83174	−0.915872	−	0.401470i	$-0.868499\pi$
$577$	−44.0000	−1.83174	−0.915872	+	0.401470i	$0.868499\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.6190i	0.482035i
$582$	0	0
$583$	23.2379	0.962415
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 45.0000i	− 1.85735i	−0.370896	−	0.928674i	$-0.620949\pi$
$587$	− 45.0000i	− 1.85735i	0.370896	−	0.928674i	$-0.379051\pi$
$588$	0	0
$589$	− 7.74597i	− 0.319167i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.3649	0.791229	0.395615	−	0.918417i	$-0.370532\pi$
$599$	19.3649	0.791229	0.395615	+	0.918417i	$0.370532\pi$
$600$	0	0
$601$	17.0000	0.693444	0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000	0.693444	0.346722	+	0.937968i	$0.387295\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.74597i	0.314918i
$606$	0	0
$607$	27.1109	1.10040	0.550198	−	0.835034i	$-0.314552\pi$
$607$	27.1109	1.10040	0.550198	+	0.835034i	$0.314552\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 15.0000i	− 0.606835i
$612$	0	0
$613$	23.2379i	0.938570i	0.883047	+	0.469285i	$0.155488\pi$
$613$	23.2379i	0.938570i	−0.883047	+	0.469285i	$0.844512\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.0000	1.57008	0.785040	−	0.619445i	$-0.212642\pi$
$617$	39.0000	1.57008	0.785040	+	0.619445i	$0.212642\pi$
$618$	0	0
$619$	− 17.0000i	− 0.683288i	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	− 17.0000i	− 0.683288i	0.939829	−	0.341644i	$-0.110984\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	46.4758	1.86201
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−15.4919	−0.616724	−0.308362	−	0.951269i	$-0.599781\pi$
$631$	−15.4919	−0.616724	−0.308362	+	0.951269i	$0.599781\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 60.0000i	− 2.38103i
$636$	0	0
$637$	30.9839i	1.22763i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.4919	0.609051	0.304525	−	0.952504i	$-0.401502\pi$
$647$	15.4919	0.609051	0.304525	+	0.952504i	$0.401502\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11.6190i	− 0.454685i	−0.973815	−	0.227342i	$-0.926996\pi$
$653$	− 11.6190i	− 0.454685i	0.973815	−	0.227342i	$-0.0730036\pi$
$654$	0	0
$655$	−11.6190	−0.453990
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 27.0000i	− 1.05177i	−0.850555	−	0.525885i	$-0.823734\pi$
$659$	− 27.0000i	− 1.05177i	0.850555	−	0.525885i	$-0.176266\pi$
$660$	0	0
$661$	11.6190i	0.451925i	0.974136	+	0.225962i	$0.0725526\pi$
$661$	11.6190i	0.451925i	−0.974136	+	0.225962i	$0.927447\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	30.0000	1.16335
$666$	0	0
$667$	− 15.0000i	− 0.580802i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.8569	1.34563
$672$	0	0
$673$	−35.0000	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$673$	−35.0000	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19.3649i	− 0.744254i	−0.928182	−	0.372127i	$-0.878629\pi$
$677$	− 19.3649i	− 0.744254i	0.928182	−	0.372127i	$-0.121371\pi$
$678$	0	0
$679$	3.87298	0.148631
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	0	0
$685$	− 81.3327i	− 3.10756i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	1.00000i	0.0380418i	0.999819	+	0.0190209i	$0.00605490\pi$
$691$	1.00000i	0.0380418i	−0.999819	+	0.0190209i	$0.993945\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.87298	−0.146911
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.2379i	0.877683i	0.898564	+	0.438842i	$0.144611\pi$
$701$	23.2379i	0.877683i	−0.898564	+	0.438842i	$0.855389\pi$
$702$	0	0
$703$	15.4919	0.584289
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.0000i	− 0.564133i
$708$	0	0
$709$	3.87298i	0.145453i	0.997352	+	0.0727265i	$0.0231700\pi$
$709$	3.87298i	0.145453i	−0.997352	+	0.0727265i	$0.976830\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.0000	−0.561754
$714$	0	0
$715$	45.0000i	1.68290i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−46.4758	−1.73325	−0.866627	−	0.498956i	$-0.833717\pi$
$719$	−46.4758	−1.73325	−0.866627	+	0.498956i	$0.833717\pi$
$720$	0	0
$721$	45.0000	1.67589
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 38.7298i	− 1.43839i
$726$	0	0
$727$	−34.8569	−1.29277	−0.646385	−	0.763012i	$-0.723720\pi$
$727$	−34.8569	−1.29277	−0.646385	+	0.763012i	$0.723720\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	3.87298i	0.143052i	0.997439	+	0.0715260i	$0.0227869\pi$
$733$	3.87298i	0.143052i	−0.997439	+	0.0715260i	$0.977213\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.0000	1.21557
$738$	0	0
$739$	32.0000i	1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$739$	32.0000i	1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.3488	1.84712	0.923559	−	0.383457i	$-0.125266\pi$
$743$	50.3488	1.84712	0.923559	+	0.383457i	$0.125266\pi$
$744$	0	0
$745$	−45.0000	−1.64867
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.2379i	0.849094i
$750$	0	0
$751$	27.1109	0.989290	0.494645	−	0.869095i	$-0.335298\pi$
$751$	27.1109	0.989290	0.494645	+	0.869095i	$0.335298\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.0000i	1.63772i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	− 30.0000i	− 1.08607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.6190	−0.419536
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.7298i	− 1.39302i	−0.717549	−	0.696508i	$-0.754736\pi$
$773$	− 38.7298i	− 1.39302i	0.717549	−	0.696508i	$-0.245264\pi$
$774$	0	0
$775$	−38.7298	−1.39122
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000i	0.644917i
$780$	0	0
$781$	23.2379i	0.831517i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.0000	0.535373
$786$	0	0
$787$	25.0000i	0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$	25.0000i	0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.6190	−0.413122
$792$	0	0
$793$	−45.0000	−1.59800
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.3649i	− 0.685941i	−0.939346	−	0.342970i	$-0.888567\pi$
$797$	− 19.3649i	− 0.685941i	0.939346	−	0.342970i	$-0.111433\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	− 58.0948i	− 2.04757i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	− 34.0000i	− 1.19390i	−0.802278	−	0.596951i	$-0.796379\pi$
$811$	− 34.0000i	− 1.19390i	0.802278	−	0.596951i	$-0.203621\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.74597	0.271329
$816$	0	0
$817$	−14.0000	−0.489798
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 42.6028i	− 1.48685i	−0.668820	−	0.743424i	$-0.733200\pi$
$821$	− 42.6028i	− 1.48685i	0.668820	−	0.743424i	$-0.266800\pi$
$822$	0	0
$823$	3.87298	0.135004	0.0675019	−	0.997719i	$-0.478497\pi$
$823$	3.87298	0.135004	0.0675019	+	0.997719i	$0.478497\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	7.74597i	0.269029i	0.990912	+	0.134514i	$0.0429474\pi$
$829$	7.74597i	0.269029i	−0.990912	+	0.134514i	$0.957053\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	15.0000i	0.519096i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.3488	−1.73823	−0.869117	−	0.494607i	$-0.835312\pi$
$839$	−50.3488	−1.73823	−0.869117	+	0.494607i	$0.835312\pi$
$840$	0	0
$841$	14.0000	0.482759
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 7.74597i	− 0.266469i
$846$	0	0
$847$	7.74597	0.266155
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 30.0000i	− 1.02839i
$852$	0	0
$853$	11.6190i	0.397825i	0.980017	+	0.198913i	$0.0637410\pi$
$853$	11.6190i	0.397825i	−0.980017	+	0.198913i	$0.936259\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	− 13.0000i	− 0.443554i	−0.975097	−	0.221777i	$-0.928814\pi$
$859$	− 13.0000i	− 0.443554i	0.975097	−	0.221777i	$-0.0711857\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.4919	0.527352	0.263676	−	0.964611i	$-0.415065\pi$
$863$	15.4919	0.527352	0.263676	+	0.964611i	$0.415065\pi$
$864$	0	0
$865$	75.0000	2.55008
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.6190i	0.394146i
$870$	0	0
$871$	−42.6028	−1.44354
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 75.0000i	− 2.53546i
$876$	0	0
$877$	− 27.1109i	− 0.915469i	−0.889089	−	0.457735i	$-0.848661\pi$
$877$	− 27.1109i	− 0.915469i	0.889089	−	0.457735i	$-0.151339\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	− 50.0000i	− 1.68263i	−0.540542	−	0.841317i	$-0.681781\pi$
$883$	− 50.0000i	− 1.68263i	0.540542	−	0.841317i	$-0.318219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.8569	−1.17038	−0.585189	−	0.810897i	$-0.698980\pi$
$887$	−34.8569	−1.17038	−0.585189	+	0.810897i	$0.698980\pi$
$888$	0	0
$889$	−60.0000	−2.01234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.74597i	0.259209i
$894$	0	0
$895$	−69.7137	−2.33027
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.0000i	0.500278i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.0000	0.997234
$906$	0	0
$907$	− 19.0000i	− 0.630885i	−0.948945	−	0.315442i	$-0.897847\pi$
$907$	− 19.0000i	− 0.630885i	0.948945	−	0.315442i	$-0.102153\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.3649	0.641588	0.320794	−	0.947149i	$-0.396050\pi$
$911$	19.3649	0.641588	0.320794	+	0.947149i	$0.396050\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.6190i	0.383692i
$918$	0	0
$919$	−46.4758	−1.53310	−0.766548	−	0.642188i	$-0.778027\pi$
$919$	−46.4758	−1.53310	−0.766548	+	0.642188i	$0.778027\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 30.0000i	− 0.987462i
$924$	0	0
$925$	− 77.4597i	− 2.54686i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	0	0
$931$	− 16.0000i	− 0.524379i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.87298i	0.126256i	0.998005	+	0.0631278i	$0.0201076\pi$
$941$	3.87298i	0.126256i	−0.998005	+	0.0631278i	$0.979892\pi$
$942$	0	0
$943$	34.8569	1.13510
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 51.0000i	− 1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$	− 51.0000i	− 1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$	0	0
$949$	30.9839i	1.00578i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	105.000i	3.39772i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−81.3327	−2.62637
$960$	0	0
$961$	−16.0000	−0.516129
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.3649i	0.623379i
$966$	0	0
$967$	−58.0948	−1.86820	−0.934101	−	0.357010i	$-0.883796\pi$
$967$	−58.0948	−1.86820	−0.934101	+	0.357010i	$0.883796\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 24.0000i	− 0.770197i	−0.922876	−	0.385098i	$-0.874168\pi$
$971$	− 24.0000i	− 0.770197i	0.922876	−	0.385098i	$-0.125832\pi$
$972$	0	0
$973$	3.87298i	0.124162i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	0	0
$979$	− 36.0000i	− 1.15056i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.8569	1.11176	0.555880	−	0.831262i	$-0.312381\pi$
$983$	34.8569	1.11176	0.555880	+	0.831262i	$0.312381\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.1109i	0.862076i
$990$	0	0
$991$	46.4758	1.47635	0.738176	−	0.674608i	$-0.235687\pi$
$991$	46.4758	1.47635	0.738176	+	0.674608i	$0.235687\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 30.0000i	− 0.951064i
$996$	0	0
$997$	− 58.0948i	− 1.83988i	−0.392060	−	0.919940i	$-0.628237\pi$
$997$	− 58.0948i	− 1.83988i	0.392060	−	0.919940i	$-0.371763\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.g.2593.4		4
3.2	odd	2		5184.2.d.h.2593.2		4
4.3	odd	2	inner	5184.2.d.g.2593.3		4
8.3	odd	2	inner	5184.2.d.g.2593.1		4
8.5	even	2	inner	5184.2.d.g.2593.2		4
9.2	odd	6		576.2.r.d.481.4	yes	8
9.4	even	3		1728.2.r.d.289.3		8
9.5	odd	6		576.2.r.d.97.1	✓	8
9.7	even	3		1728.2.r.d.1441.1		8
12.11	even	2		5184.2.d.h.2593.1		4
24.5	odd	2		5184.2.d.h.2593.4		4
24.11	even	2		5184.2.d.h.2593.3		4
36.7	odd	6		1728.2.r.d.1441.2		8
36.11	even	6		576.2.r.d.481.2	yes	8
36.23	even	6		576.2.r.d.97.3	yes	8
36.31	odd	6		1728.2.r.d.289.4		8
72.5	odd	6		576.2.r.d.97.4	yes	8
72.11	even	6		576.2.r.d.481.3	yes	8
72.13	even	6		1728.2.r.d.289.1		8
72.29	odd	6		576.2.r.d.481.1	yes	8
72.43	odd	6		1728.2.r.d.1441.4		8
72.59	even	6		576.2.r.d.97.2	yes	8
72.61	even	6		1728.2.r.d.1441.3		8
72.67	odd	6		1728.2.r.d.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.d.97.1	✓	8	9.5	odd	6
576.2.r.d.97.2	yes	8	72.59	even	6
576.2.r.d.97.3	yes	8	36.23	even	6
576.2.r.d.97.4	yes	8	72.5	odd	6
576.2.r.d.481.1	yes	8	72.29	odd	6
576.2.r.d.481.2	yes	8	36.11	even	6
576.2.r.d.481.3	yes	8	72.11	even	6
576.2.r.d.481.4	yes	8	9.2	odd	6
1728.2.r.d.289.1		8	72.13	even	6
1728.2.r.d.289.2		8	72.67	odd	6
1728.2.r.d.289.3		8	9.4	even	3
1728.2.r.d.289.4		8	36.31	odd	6
1728.2.r.d.1441.1		8	9.7	even	3
1728.2.r.d.1441.2		8	36.7	odd	6
1728.2.r.d.1441.3		8	72.61	even	6
1728.2.r.d.1441.4		8	72.43	odd	6
5184.2.d.g.2593.1		4	8.3	odd	2	inner
5184.2.d.g.2593.2		4	8.5	even	2	inner
5184.2.d.g.2593.3		4	4.3	odd	2	inner
5184.2.d.g.2593.4		4	1.1	even	1	trivial
5184.2.d.h.2593.1		4	12.11	even	2
5184.2.d.h.2593.2		4	3.2	odd	2
5184.2.d.h.2593.3		4	24.11	even	2
5184.2.d.h.2593.4		4	24.5	odd	2

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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.87298i q^{5} +3.87298 q^{7} +O(q^{10})\)	\(q+3.87298i q^{5} +3.87298 q^{7} -3.00000i q^{11} +3.87298i q^{13} -2.00000i q^{19} -3.87298 q^{23} -10.0000 q^{25} +3.87298i q^{29} +3.87298 q^{31} +15.0000i q^{35} +7.74597i q^{37} -9.00000 q^{41} -7.00000i q^{43} -3.87298 q^{47} +8.00000 q^{49} +7.74597i q^{53} +11.6190 q^{55} +3.00000i q^{59} +11.6190i q^{61} -15.0000 q^{65} +11.0000i q^{67} -7.74597 q^{71} +8.00000 q^{73} -11.6190i q^{77} -3.87298 q^{79} +3.00000i q^{83} +12.0000 q^{89} +15.0000i q^{91} +7.74597 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 40 q^{25} - 36 q^{41} + 32 q^{49} - 60 q^{65} + 32 q^{73} + 48 q^{89} + 4 q^{97}+O(q^{100}) \) 4 * q - 40 * q^25 - 36 * q^41 + 32 * q^49 - 60 * q^65 + 32 * q^73 + 48 * q^89 + 4 * q^97

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