LMFDB - Embedded newform 5184.2.c.j.5183.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(5183,5184)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("5184.5183");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.c (of order $2$, degree $1$, not 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:	$8$
Coefficient field:	8.0.170772624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.2
Root			$0.335728 + 1.37379i$ of defining polynomial
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.j.5183.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434i q^{5} +1.27582i q^{7} +O(q^{10})$	$q-2.52434i q^{5} +1.27582i q^{7} +0.505408 q^{11} +2.37228 q^{13} +0.792287i q^{17} -4.70285i q^{19} -3.22060 q^{23} -1.37228 q^{25} -2.52434i q^{29} +8.12989i q^{31} +3.22060 q^{35} +6.74456 q^{37} -6.78073i q^{41} -7.72946i q^{43} -1.19897 q^{47} +5.37228 q^{49} -1.87953i q^{53} -1.27582i q^{55} -12.3770 q^{59} +2.37228 q^{61} -5.98844i q^{65} +7.72946i q^{67} +11.8716 q^{71} +3.37228 q^{73} +0.644810i q^{77} -9.88067i q^{79} -7.64018 q^{83} +2.00000 q^{85} -11.9769i q^{89} +3.02661i q^{91} -11.8716 q^{95} +10.4891 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{13} + 12 q^{25} + 8 q^{37} + 20 q^{49} - 4 q^{61} + 4 q^{73} + 16 q^{85} - 8 q^{97}+O(q^{100}) $ 8 * q - 4 * q^13 + 12 * q^25 + 8 * q^37 + 20 * q^49 - 4 * q^61 + 4 * q^73 + 16 * q^85 - 8 * 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
$5$	− 2.52434i	− 1.12892i	−0.825461	−	0.564459i	$-0.809085\pi$
$5$	− 2.52434i	− 1.12892i	0.825461	−	0.564459i	$-0.190915\pi$
$6$	0	0
$7$	1.27582i	0.482215i	0.970498	+	0.241107i	$0.0775106\pi$
$7$	1.27582i	0.482215i	−0.970498	+	0.241107i	$0.922489\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.505408	0.152386	0.0761931	−	0.997093i	$-0.475723\pi$
$11$	0.505408	0.152386	0.0761931	+	0.997093i	$0.475723\pi$
$12$	0	0
$13$	2.37228	0.657952	0.328976	−	0.944338i	$-0.393296\pi$
$13$	2.37228	0.657952	0.328976	+	0.944338i	$0.393296\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.792287i	0.192158i	0.995374	+	0.0960789i	$0.0306301\pi$
$17$	0.792287i	0.192158i	−0.995374	+	0.0960789i	$0.969370\pi$
$18$	0	0
$19$	− 4.70285i	− 1.07891i	−0.842015	−	0.539454i	$-0.818631\pi$
$19$	− 4.70285i	− 1.07891i	0.842015	−	0.539454i	$-0.181369\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.22060	−0.671542	−0.335771	−	0.941944i	$-0.608997\pi$
$23$	−3.22060	−0.671542	−0.335771	+	0.941944i	$0.608997\pi$
$24$	0	0
$25$	−1.37228	−0.274456
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.52434i	− 0.468758i	−0.972145	−	0.234379i	$-0.924694\pi$
$29$	− 2.52434i	− 0.468758i	0.972145	−	0.234379i	$-0.0753056\pi$
$30$	0	0
$31$	8.12989i	1.46017i	0.683356	+	0.730086i	$0.260520\pi$
$31$	8.12989i	1.46017i	−0.683356	+	0.730086i	$0.739480\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.22060	0.544381
$36$	0	0
$37$	6.74456	1.10880	0.554400	−	0.832251i	$-0.312948\pi$
$37$	6.74456	1.10880	0.554400	+	0.832251i	$0.312948\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.78073i	− 1.05897i	−0.848319	−	0.529486i	$-0.822385\pi$
$41$	− 6.78073i	− 1.05897i	0.848319	−	0.529486i	$-0.177615\pi$
$42$	0	0
$43$	− 7.72946i	− 1.17873i	−0.807866	−	0.589366i	$-0.799378\pi$
$43$	− 7.72946i	− 1.17873i	0.807866	−	0.589366i	$-0.200622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.19897	−0.174888	−0.0874439	−	0.996169i	$-0.527870\pi$
$47$	−1.19897	−0.174888	−0.0874439	+	0.996169i	$0.527870\pi$
$48$	0	0
$49$	5.37228	0.767469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.87953i	− 0.258173i	−0.991633	−	0.129086i	$-0.958796\pi$
$53$	− 1.87953i	− 0.258173i	0.991633	−	0.129086i	$-0.0412045\pi$
$54$	0	0
$55$	− 1.27582i	− 0.172032i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.3770	−1.61135	−0.805674	−	0.592359i	$-0.798197\pi$
$59$	−12.3770	−1.61135	−0.805674	+	0.592359i	$0.798197\pi$
$60$	0	0
$61$	2.37228	0.303739	0.151870	−	0.988401i	$-0.451471\pi$
$61$	2.37228	0.303739	0.151870	+	0.988401i	$0.451471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.98844i	− 0.742774i
$66$	0	0
$67$	7.72946i	0.944304i	0.881517	+	0.472152i	$0.156523\pi$
$67$	7.72946i	0.944304i	−0.881517	+	0.472152i	$0.843477\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.8716	1.40890	0.704450	−	0.709754i	$-0.251194\pi$
$71$	11.8716	1.40890	0.704450	+	0.709754i	$0.251194\pi$
$72$	0	0
$73$	3.37228	0.394696	0.197348	−	0.980334i	$-0.436767\pi$
$73$	3.37228	0.394696	0.197348	+	0.980334i	$0.436767\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.644810i	0.0734829i
$78$	0	0
$79$	− 9.88067i	− 1.11166i	−0.831295	−	0.555831i	$-0.812400\pi$
$79$	− 9.88067i	− 1.11166i	0.831295	−	0.555831i	$-0.187600\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.64018	−0.838618	−0.419309	−	0.907844i	$-0.637728\pi$
$83$	−7.64018	−0.838618	−0.419309	+	0.907844i	$0.637728\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 11.9769i	− 1.26955i	−0.772698	−	0.634773i	$-0.781093\pi$
$89$	− 11.9769i	− 1.26955i	0.772698	−	0.634773i	$-0.218907\pi$
$90$	0	0
$91$	3.02661i	0.317274i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.8716	−1.21800
$96$	0	0
$97$	10.4891	1.06501	0.532505	−	0.846427i	$-0.321251\pi$
$97$	10.4891	1.06501	0.532505	+	0.846427i	$0.321251\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.23472i	− 0.122859i	−0.998111	−	0.0614295i	$-0.980434\pi$
$101$	− 1.23472i	− 0.122859i	0.998111	−	0.0614295i	$-0.0195659\pi$
$102$	0	0
$103$	0.474964i	0.0467996i	0.999726	+	0.0233998i	$0.00744907\pi$
$103$	0.474964i	0.0467996i	−0.999726	+	0.0233998i	$0.992551\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.5652	1.21472	0.607360	−	0.794427i	$-0.292229\pi$
$107$	12.5652	1.21472	0.607360	+	0.794427i	$0.292229\pi$
$108$	0	0
$109$	−13.4891	−1.29202	−0.646012	−	0.763327i	$-0.723564\pi$
$109$	−13.4891	−1.29202	−0.646012	+	0.763327i	$0.723564\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.27806i	0.684662i	0.939579	+	0.342331i	$0.111216\pi$
$113$	7.27806i	0.684662i	−0.939579	+	0.342331i	$0.888784\pi$
$114$	0	0
$115$	8.12989i	0.758116i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.01082	−0.0926614
$120$	0	0
$121$	−10.7446	−0.976778
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.15759i	− 0.819080i
$126$	0	0
$127$	− 7.65492i	− 0.679265i	−0.940558	−	0.339632i	$-0.889697\pi$
$127$	− 7.65492i	− 0.679265i	0.940558	−	0.339632i	$-0.110303\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.64018	0.667525	0.333763	−	0.942657i	$-0.391682\pi$
$131$	7.64018	0.667525	0.333763	+	0.942657i	$0.391682\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.49111i	− 0.469137i	−0.972100	−	0.234568i	$-0.924632\pi$
$137$	− 5.49111i	− 0.469137i	0.972100	−	0.234568i	$-0.0753677\pi$
$138$	0	0
$139$	15.3844i	1.30489i	0.757838	+	0.652443i	$0.226256\pi$
$139$	15.3844i	1.30489i	−0.757838	+	0.652443i	$0.773744\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.19897	0.100263
$144$	0	0
$145$	−6.37228	−0.529189
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.9166i	− 1.05817i	−0.848568	−	0.529086i	$-0.822535\pi$
$149$	− 12.9166i	− 1.05817i	0.848568	−	0.529086i	$-0.177465\pi$
$150$	0	0
$151$	3.02661i	0.246302i	0.992388	+	0.123151i	$0.0392999\pi$
$151$	3.02661i	0.246302i	−0.992388	+	0.123151i	$0.960700\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.5226	1.64841
$156$	0	0
$157$	−11.8614	−0.946643	−0.473322	−	0.880890i	$-0.656945\pi$
$157$	−11.8614	−0.946643	−0.473322	+	0.880890i	$0.656945\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.10891i	− 0.323828i
$162$	0	0
$163$	− 1.75079i	− 0.137132i	−0.997647	−	0.0685660i	$-0.978158\pi$
$163$	− 1.75079i	− 0.137132i	0.997647	−	0.0685660i	$-0.0218424\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.4901	1.35343	0.676714	−	0.736246i	$-0.263403\pi$
$167$	17.4901	1.35343	0.676714	+	0.736246i	$0.263403\pi$
$168$	0	0
$169$	−7.37228	−0.567099
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 17.6704i	− 1.34345i	−0.740799	−	0.671726i	$-0.765553\pi$
$173$	− 17.6704i	− 1.34345i	0.740799	−	0.671726i	$-0.234447\pi$
$174$	0	0
$175$	− 1.75079i	− 0.132347i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.83915	−0.660669	−0.330334	−	0.943864i	$-0.607162\pi$
$179$	−8.83915	−0.660669	−0.330334	+	0.943864i	$0.607162\pi$
$180$	0	0
$181$	4.00000	0.297318	0.148659	−	0.988889i	$-0.452504\pi$
$181$	4.00000	0.297318	0.148659	+	0.988889i	$0.452504\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 17.0256i	− 1.25174i
$186$	0	0
$187$	0.400428i	0.0292822i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.0922	1.09203	0.546017	−	0.837774i	$-0.316144\pi$
$191$	15.0922	1.09203	0.546017	+	0.837774i	$0.316144\pi$
$192$	0	0
$193$	7.74456	0.557466	0.278733	−	0.960369i	$-0.410086\pi$
$193$	7.74456	0.557466	0.278733	+	0.960369i	$0.410086\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.9538i	− 1.70663i	−0.521392	−	0.853317i	$-0.674587\pi$
$197$	− 23.9538i	− 1.70663i	0.521392	−	0.853317i	$-0.325413\pi$
$198$	0	0
$199$	− 12.9073i	− 0.914973i	−0.889217	−	0.457486i	$-0.848750\pi$
$199$	− 12.9073i	− 0.914973i	0.889217	−	0.457486i	$-0.151250\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.22060	0.226042
$204$	0	0
$205$	−17.1168	−1.19549
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.37686i	− 0.164411i
$210$	0	0
$211$	− 17.5356i	− 1.20720i	−0.797287	−	0.603600i	$-0.793732\pi$
$211$	− 17.5356i	− 1.20720i	0.797287	−	0.603600i	$-0.206268\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.5118	−1.33069
$216$	0	0
$217$	−10.3723	−0.704116
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.87953i	0.126431i
$222$	0	0
$223$	− 8.12989i	− 0.544418i	−0.962238	−	0.272209i	$-0.912246\pi$
$223$	− 8.12989i	− 0.544418i	0.962238	−	0.272209i	$-0.0877541\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.3986	−0.955671	−0.477835	−	0.878449i	$-0.658578\pi$
$227$	−14.3986	−0.955671	−0.477835	+	0.878449i	$0.658578\pi$
$228$	0	0
$229$	−3.62772	−0.239726	−0.119863	−	0.992790i	$-0.538246\pi$
$229$	−3.62772	−0.239726	−0.119863	+	0.992790i	$0.538246\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.84630i	− 0.317491i	−0.987320	−	0.158746i	$-0.949255\pi$
$233$	− 4.84630i	− 0.317491i	0.987320	−	0.158746i	$-0.0507450\pi$
$234$	0	0
$235$	3.02661i	0.197434i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0597	−0.780080	−0.390040	−	0.920798i	$-0.627539\pi$
$239$	−12.0597	−0.780080	−0.390040	+	0.920798i	$0.627539\pi$
$240$	0	0
$241$	−6.48913	−0.418001	−0.209001	−	0.977916i	$-0.567021\pi$
$241$	−6.48913	−0.418001	−0.209001	+	0.977916i	$0.567021\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 13.5615i	− 0.866409i
$246$	0	0
$247$	− 11.1565i	− 0.709871i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.1780	−0.705551	−0.352776	−	0.935708i	$-0.614762\pi$
$251$	−11.1780	−0.705551	−0.352776	+	0.935708i	$0.614762\pi$
$252$	0	0
$253$	−1.62772	−0.102334
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.9985i	− 0.935584i	−0.883839	−	0.467792i	$-0.845050\pi$
$257$	− 14.9985i	− 0.935584i	0.883839	−	0.467792i	$-0.154950\pi$
$258$	0	0
$259$	8.60485i	0.534680i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.9746	−1.72499	−0.862494	−	0.506067i	$-0.831099\pi$
$263$	−27.9746	−1.72499	−0.862494	+	0.506067i	$0.831099\pi$
$264$	0	0
$265$	−4.74456	−0.291456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.4843i	1.30992i	0.755663	+	0.654961i	$0.227315\pi$
$269$	21.4843i	1.30992i	−0.755663	+	0.654961i	$0.772685\pi$
$270$	0	0
$271$	29.9679i	1.82042i	0.414146	+	0.910211i	$0.364080\pi$
$271$	29.9679i	1.82042i	−0.414146	+	0.910211i	$0.635920\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.693562	−0.0418234
$276$	0	0
$277$	−6.37228	−0.382873	−0.191437	−	0.981505i	$-0.561315\pi$
$277$	−6.37228	−0.382873	−0.191437	+	0.981505i	$0.561315\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 23.0140i	− 1.37290i	−0.727177	−	0.686450i	$-0.759168\pi$
$281$	− 23.0140i	− 1.37290i	0.727177	−	0.686450i	$-0.240832\pi$
$282$	0	0
$283$	− 9.88067i	− 0.587345i	−0.955906	−	0.293673i	$-0.905122\pi$
$283$	− 9.88067i	− 0.587345i	0.955906	−	0.293673i	$-0.0948775\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.65099	0.510652
$288$	0	0
$289$	16.3723	0.963075
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 23.3089i	− 1.36172i	−0.732412	−	0.680862i	$-0.761605\pi$
$293$	− 23.3089i	− 1.36172i	0.732412	−	0.680862i	$-0.238395\pi$
$294$	0	0
$295$	31.2437i	1.81908i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.64018	−0.441843
$300$	0	0
$301$	9.86141	0.568402
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.98844i	− 0.342897i
$306$	0	0
$307$	1.20128i	0.0685609i	0.999412	+	0.0342805i	$0.0109140\pi$
$307$	1.20128i	0.0685609i	−0.999412	+	0.0342805i	$0.989086\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.1355	1.08507	0.542536	−	0.840032i	$-0.317464\pi$
$311$	19.1355	1.08507	0.542536	+	0.840032i	$0.317464\pi$
$312$	0	0
$313$	−18.4891	−1.04507	−0.522534	−	0.852619i	$-0.675013\pi$
$313$	−18.4891	−1.04507	−0.522534	+	0.852619i	$0.675013\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.5214i	1.82659i	0.407303	+	0.913293i	$0.366469\pi$
$317$	32.5214i	1.82659i	−0.407303	+	0.913293i	$0.633531\pi$
$318$	0	0
$319$	− 1.27582i	− 0.0714323i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.72601	0.207321
$324$	0	0
$325$	−3.25544	−0.180579
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.52967i	− 0.0843335i
$330$	0	0
$331$	− 27.7422i	− 1.52485i	−0.647078	−	0.762424i	$-0.724009\pi$
$331$	− 27.7422i	− 1.52485i	0.647078	−	0.762424i	$-0.275991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.5118	1.06604
$336$	0	0
$337$	−15.7446	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$337$	−15.7446	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.10891i	0.222510i
$342$	0	0
$343$	15.7848i	0.852300i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.94661	0.372914	0.186457	−	0.982463i	$-0.440300\pi$
$347$	6.94661	0.372914	0.186457	+	0.982463i	$0.440300\pi$
$348$	0	0
$349$	5.62772	0.301245	0.150622	−	0.988591i	$-0.451872\pi$
$349$	5.62772	0.301245	0.150622	+	0.988591i	$0.451872\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.36530i	0.445240i	0.974905	+	0.222620i	$0.0714609\pi$
$353$	8.36530i	0.445240i	−0.974905	+	0.222620i	$0.928539\pi$
$354$	0	0
$355$	− 29.9679i	− 1.59053i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.38712	−0.0732096	−0.0366048	−	0.999330i	$-0.511654\pi$
$359$	−1.38712	−0.0732096	−0.0366048	+	0.999330i	$0.511654\pi$
$360$	0	0
$361$	−3.11684	−0.164044
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.51278i	− 0.445579i
$366$	0	0
$367$	− 6.37910i	− 0.332987i	−0.986043	−	0.166493i	$-0.946756\pi$
$367$	− 6.37910i	− 0.332987i	0.986043	−	0.166493i	$-0.0532444\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.39794	0.124495
$372$	0	0
$373$	19.8614	1.02838	0.514192	−	0.857675i	$-0.328092\pi$
$373$	19.8614	1.02838	0.514192	+	0.857675i	$0.328092\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.98844i	− 0.308420i
$378$	0	0
$379$	6.45364i	0.331501i	0.986168	+	0.165751i	$0.0530047\pi$
$379$	6.45364i	0.331501i	−0.986168	+	0.165751i	$0.946995\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.65099	0.442045	0.221023	−	0.975269i	$-0.429061\pi$
$383$	8.65099	0.442045	0.221023	+	0.975269i	$0.429061\pi$
$384$	0	0
$385$	1.62772	0.0829562
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 31.5268i	− 1.59847i	−0.601018	−	0.799235i	$-0.705238\pi$
$389$	− 31.5268i	− 1.59847i	0.601018	−	0.799235i	$-0.294762\pi$
$390$	0	0
$391$	− 2.55164i	− 0.129042i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.9422	−1.25498
$396$	0	0
$397$	18.7446	0.940763	0.470381	−	0.882463i	$-0.344116\pi$
$397$	18.7446	0.940763	0.470381	+	0.882463i	$0.344116\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.60625i	− 0.230025i	−0.993364	−	0.115012i	$-0.963309\pi$
$401$	− 4.60625i	− 0.230025i	0.993364	−	0.115012i	$-0.0366908\pi$
$402$	0	0
$403$	19.2864i	0.960723i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.40876	0.168966
$408$	0	0
$409$	13.7446	0.679625	0.339812	−	0.940493i	$-0.389636\pi$
$409$	13.7446	0.679625	0.339812	+	0.940493i	$0.389636\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 15.7908i	− 0.777016i
$414$	0	0
$415$	19.2864i	0.946731i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.9638	1.31727	0.658634	−	0.752464i	$-0.271135\pi$
$419$	26.9638	1.31727	0.658634	+	0.752464i	$0.271135\pi$
$420$	0	0
$421$	10.6060	0.516903	0.258452	−	0.966024i	$-0.416788\pi$
$421$	10.6060	0.516903	0.258452	+	0.966024i	$0.416788\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.08724i	− 0.0527389i
$426$	0	0
$427$	3.02661i	0.146468i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.1952	−1.50262	−0.751310	−	0.659949i	$-0.770578\pi$
$431$	−31.1952	−1.50262	−0.751310	+	0.659949i	$0.770578\pi$
$432$	0	0
$433$	−8.62772	−0.414622	−0.207311	−	0.978275i	$-0.566471\pi$
$433$	−8.62772	−0.414622	−0.207311	+	0.978275i	$0.566471\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.1460i	0.724533i
$438$	0	0
$439$	6.52818i	0.311573i	0.987791	+	0.155786i	$0.0497912\pi$
$439$	6.52818i	0.311573i	−0.987791	+	0.155786i	$0.950209\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.3770	0.588049	0.294025	−	0.955798i	$-0.405005\pi$
$443$	12.3770	0.588049	0.294025	+	0.955798i	$0.405005\pi$
$444$	0	0
$445$	−30.2337	−1.43321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.4024i	0.915657i	0.889041	+	0.457828i	$0.151373\pi$
$449$	19.4024i	0.915657i	−0.889041	+	0.457828i	$0.848627\pi$
$450$	0	0
$451$	− 3.42703i	− 0.161373i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.64018	0.358177
$456$	0	0
$457$	39.9783	1.87010	0.935052	−	0.354511i	$-0.115353\pi$
$457$	39.9783	1.87010	0.935052	+	0.354511i	$0.115353\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 0.349857i	− 0.0162944i	−0.999967	−	0.00814722i	$-0.997407\pi$
$461$	− 0.349857i	− 0.0162944i	0.999967	−	0.00814722i	$-0.00259337\pi$
$462$	0	0
$463$	− 4.77739i	− 0.222024i	−0.993819	−	0.111012i	$-0.964591\pi$
$463$	− 4.77739i	− 0.222024i	0.993819	−	0.111012i	$-0.0354092\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.11313	−0.236608	−0.118304	−	0.992977i	$-0.537746\pi$
$467$	−5.11313	−0.236608	−0.118304	+	0.992977i	$0.537746\pi$
$468$	0	0
$469$	−9.86141	−0.455357
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.90653i	− 0.179623i
$474$	0	0
$475$	6.45364i	0.296113i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.0706	−0.597209	−0.298605	−	0.954377i	$-0.596521\pi$
$479$	−13.0706	−0.597209	−0.298605	+	0.954377i	$0.596521\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 26.4781i	− 1.20231i
$486$	0	0
$487$	− 42.8752i	− 1.94286i	−0.237325	−	0.971430i	$-0.576271\pi$
$487$	− 42.8752i	− 1.94286i	0.237325	−	0.971430i	$-0.423729\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.7641	−0.621166	−0.310583	−	0.950546i	$-0.600524\pi$
$491$	−13.7641	−0.621166	−0.310583	+	0.950546i	$0.600524\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.1460i	0.679392i
$498$	0	0
$499$	− 3.42703i	− 0.153415i	−0.997054	−	0.0767076i	$-0.975559\pi$
$499$	− 3.42703i	− 0.153415i	0.997054	−	0.0767076i	$-0.0244408\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.3236	−0.861597	−0.430799	−	0.902448i	$-0.641768\pi$
$503$	−19.3236	−0.861597	−0.430799	+	0.902448i	$0.641768\pi$
$504$	0	0
$505$	−3.11684	−0.138698
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.7962i	0.655829i	0.944707	+	0.327914i	$0.106346\pi$
$509$	14.7962i	0.655829i	−0.944707	+	0.327914i	$0.893654\pi$
$510$	0	0
$511$	4.30243i	0.190328i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.19897	0.0528329
$516$	0	0
$517$	−0.605969	−0.0266505
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 26.0357i	− 1.14064i	−0.821421	−	0.570322i	$-0.806819\pi$
$521$	− 26.0357i	− 1.14064i	0.821421	−	0.570322i	$-0.193181\pi$
$522$	0	0
$523$	9.40571i	0.411283i	0.978627	+	0.205641i	$0.0659281\pi$
$523$	9.40571i	0.411283i	−0.978627	+	0.205641i	$0.934072\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.44121	−0.280583
$528$	0	0
$529$	−12.6277	−0.549031
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 16.0858i	− 0.696753i
$534$	0	0
$535$	− 31.7187i	− 1.37132i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.71519	0.116952
$540$	0	0
$541$	8.97825	0.386005	0.193003	−	0.981198i	$-0.438177\pi$
$541$	8.97825	0.386005	0.193003	+	0.981198i	$0.438177\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	34.0511i	1.45859i
$546$	0	0
$547$	22.2385i	0.950848i	0.879757	+	0.475424i	$0.157705\pi$
$547$	22.2385i	0.950848i	−0.879757	+	0.475424i	$0.842295\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.8716	−0.505747
$552$	0	0
$553$	12.6060	0.536060
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.22316i	0.306055i	0.988222	+	0.153027i	$0.0489023\pi$
$557$	7.22316i	0.306055i	−0.988222	+	0.153027i	$0.951098\pi$
$558$	0	0
$559$	− 18.3365i	− 0.775549i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.7858	0.665290	0.332645	−	0.943052i	$-0.392059\pi$
$563$	15.7858	0.665290	0.332645	+	0.943052i	$0.392059\pi$
$564$	0	0
$565$	18.3723	0.772928
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 25.3909i	− 1.06444i	−0.846606	−	0.532220i	$-0.821358\pi$
$569$	− 25.3909i	− 1.06444i	0.846606	−	0.532220i	$-0.178642\pi$
$570$	0	0
$571$	4.22789i	0.176932i	0.996079	+	0.0884659i	$0.0281964\pi$
$571$	4.22789i	0.176932i	−0.996079	+	0.0884659i	$0.971804\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.41957	0.184309
$576$	0	0
$577$	31.8397	1.32550	0.662751	−	0.748840i	$-0.269389\pi$
$577$	31.8397	1.32550	0.662751	+	0.748840i	$0.269389\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.74749i	− 0.404394i
$582$	0	0
$583$	− 0.949929i	− 0.0393420i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.91416	−0.161555	−0.0807774	−	0.996732i	$-0.525740\pi$
$587$	−3.91416	−0.161555	−0.0807774	+	0.996732i	$0.525740\pi$
$588$	0	0
$589$	38.2337	1.57539
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.80773i	0.361690i	0.983512	+	0.180845i	$0.0578833\pi$
$593$	8.80773i	0.361690i	−0.983512	+	0.180845i	$0.942117\pi$
$594$	0	0
$595$	2.55164i	0.104607i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.188154	−0.00768776	−0.00384388	−	0.999993i	$-0.501224\pi$
$599$	−0.188154	−0.00768776	−0.00384388	+	0.999993i	$0.501224\pi$
$600$	0	0
$601$	15.9783	0.651766	0.325883	−	0.945410i	$-0.394338\pi$
$601$	15.9783	0.651766	0.325883	+	0.945410i	$0.394338\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	27.1229i	1.10270i
$606$	0	0
$607$	3.97653i	0.161403i	0.996738	+	0.0807013i	$0.0257160\pi$
$607$	3.97653i	0.161403i	−0.996738	+	0.0807013i	$0.974284\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.84429	−0.115068
$612$	0	0
$613$	−4.23369	−0.170997	−0.0854985	−	0.996338i	$-0.527248\pi$
$613$	−4.23369	−0.170997	−0.0854985	+	0.996338i	$0.527248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.90120i	0.197315i	0.995121	+	0.0986574i	$0.0314548\pi$
$617$	4.90120i	0.197315i	−0.995121	+	0.0986574i	$0.968545\pi$
$618$	0	0
$619$	9.33117i	0.375052i	0.982260	+	0.187526i	$0.0600468\pi$
$619$	9.33117i	0.375052i	−0.982260	+	0.187526i	$0.939953\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.2804	0.612194
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.34363i	0.213064i
$630$	0	0
$631$	42.8752i	1.70683i	0.521228	+	0.853417i	$0.325474\pi$
$631$	42.8752i	1.70683i	−0.521228	+	0.853417i	$0.674526\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.3236	−0.766834
$636$	0	0
$637$	12.7446	0.504958
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.9321i	0.826768i	0.910557	+	0.413384i	$0.135653\pi$
$641$	20.9321i	0.826768i	−0.910557	+	0.413384i	$0.864347\pi$
$642$	0	0
$643$	35.9466i	1.41760i	0.705412	+	0.708798i	$0.250762\pi$
$643$	35.9466i	1.41760i	−0.705412	+	0.708798i	$0.749238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.0993	1.81235	0.906174	−	0.422904i	$-0.138989\pi$
$647$	46.0993	1.81235	0.906174	+	0.422904i	$0.138989\pi$
$648$	0	0
$649$	−6.25544	−0.245547
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.40387i	0.172337i	0.996281	+	0.0861683i	$0.0274623\pi$
$653$	4.40387i	0.172337i	−0.996281	+	0.0861683i	$0.972538\pi$
$654$	0	0
$655$	− 19.2864i	− 0.753581i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0597	−0.469781	−0.234891	−	0.972022i	$-0.575473\pi$
$659$	−12.0597	−0.469781	−0.234891	+	0.972022i	$0.575473\pi$
$660$	0	0
$661$	−15.6277	−0.607848	−0.303924	−	0.952696i	$-0.598297\pi$
$661$	−15.6277	−0.607848	−0.303924	+	0.952696i	$0.598297\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 15.1460i	− 0.587338i
$666$	0	0
$667$	8.12989i	0.314791i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.19897	0.0462857
$672$	0	0
$673$	29.8614	1.15107	0.575536	−	0.817776i	$-0.304793\pi$
$673$	29.8614	1.15107	0.575536	+	0.817776i	$0.304793\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.51900i	0.135246i	0.997711	+	0.0676232i	$0.0215416\pi$
$677$	3.51900i	0.135246i	−0.997711	+	0.0676232i	$0.978458\pi$
$678$	0	0
$679$	13.3822i	0.513563i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.0172	0.765936	0.382968	−	0.923762i	$-0.374902\pi$
$683$	20.0172	0.765936	0.382968	+	0.923762i	$0.374902\pi$
$684$	0	0
$685$	−13.8614	−0.529617
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.45877i	− 0.169866i
$690$	0	0
$691$	− 20.0872i	− 0.764155i	−0.924130	−	0.382077i	$-0.875209\pi$
$691$	− 20.0872i	− 0.764155i	0.924130	−	0.382077i	$-0.124791\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	38.8354	1.47311
$696$	0	0
$697$	5.37228	0.203490
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32.7615i	1.23738i	0.785634	+	0.618692i	$0.212337\pi$
$701$	32.7615i	1.23738i	−0.785634	+	0.618692i	$0.787663\pi$
$702$	0	0
$703$	− 31.7187i	− 1.19629i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.57528	0.0592444
$708$	0	0
$709$	−23.8614	−0.896134	−0.448067	−	0.894000i	$-0.647887\pi$
$709$	−23.8614	−0.896134	−0.448067	+	0.894000i	$0.647887\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 26.1831i	− 0.980566i
$714$	0	0
$715$	− 3.02661i	− 0.113189i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.2912	0.607558	0.303779	−	0.952743i	$-0.401752\pi$
$719$	16.2912	0.607558	0.303779	+	0.952743i	$0.401752\pi$
$720$	0	0
$721$	−0.605969	−0.0225675
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.46410i	0.128654i
$726$	0	0
$727$	− 38.0978i	− 1.41297i	−0.707728	−	0.706485i	$-0.750280\pi$
$727$	− 38.0978i	− 1.41297i	0.707728	−	0.706485i	$-0.249720\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.12395	0.226503
$732$	0	0
$733$	−0.372281	−0.0137505	−0.00687526	−	0.999976i	$-0.502188\pi$
$733$	−0.372281	−0.0137505	−0.00687526	+	0.999976i	$0.502188\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.90653i	0.143899i
$738$	0	0
$739$	− 6.45364i	− 0.237401i	−0.992930	−	0.118700i	$-0.962127\pi$
$739$	− 6.45364i	− 0.237401i	0.992930	−	0.118700i	$-0.0378728\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.1571	−0.776179	−0.388089	−	0.921622i	$-0.626865\pi$
$743$	−21.1571	−0.776179	−0.388089	+	0.921622i	$0.626865\pi$
$744$	0	0
$745$	−32.6060	−1.19459
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.0309i	0.585756i
$750$	0	0
$751$	21.6890i	0.791441i	0.918371	+	0.395721i	$0.129505\pi$
$751$	21.6890i	0.791441i	−0.918371	+	0.395721i	$0.870495\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.64018	0.278054
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17.3754i	− 0.629858i	−0.949115	−	0.314929i	$-0.898019\pi$
$761$	− 17.3754i	− 0.629858i	0.949115	−	0.314929i	$-0.101981\pi$
$762$	0	0
$763$	− 17.2097i	− 0.623033i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29.3617	−1.06019
$768$	0	0
$769$	8.09509	0.291917	0.145958	−	0.989291i	$-0.453373\pi$
$769$	8.09509	0.291917	0.145958	+	0.989291i	$0.453373\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.699713i	0.0251669i	0.999921	+	0.0125835i	$0.00400555\pi$
$773$	0.699713i	0.0251669i	−0.999921	+	0.0125835i	$0.995994\pi$
$774$	0	0
$775$	− 11.1565i	− 0.400753i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.8888	−1.14253
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	29.9422i	1.06868i
$786$	0	0
$787$	34.7453i	1.23854i	0.785180	+	0.619268i	$0.212571\pi$
$787$	34.7453i	1.23854i	−0.785180	+	0.619268i	$0.787429\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.28550	−0.330154
$792$	0	0
$793$	5.62772	0.199846
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.8989i	0.846541i	0.906003	+	0.423270i	$0.139118\pi$
$797$	23.8989i	0.846541i	−0.906003	+	0.423270i	$0.860882\pi$
$798$	0	0
$799$	− 0.949929i	− 0.0336061i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.70438	0.0601462
$804$	0	0
$805$	−10.3723	−0.365575
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.8381i	1.26000i	0.776595	+	0.630000i	$0.216945\pi$
$809$	35.8381i	1.26000i	−0.776595	+	0.630000i	$0.783055\pi$
$810$	0	0
$811$	− 1.20128i	− 0.0421828i	−0.999778	−	0.0210914i	$-0.993286\pi$
$811$	− 1.20128i	− 0.0421828i	0.999778	−	0.0210914i	$-0.00671410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.41957	−0.154811
$816$	0	0
$817$	−36.3505	−1.27174
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.2603i	0.637288i	0.947874	+	0.318644i	$0.103227\pi$
$821$	18.2603i	0.637288i	−0.947874	+	0.318644i	$0.896773\pi$
$822$	0	0
$823$	14.0340i	0.489195i	0.969625	+	0.244598i	$0.0786559\pi$
$823$	14.0340i	0.489195i	−0.969625	+	0.244598i	$0.921344\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.4864	1.65126	0.825632	−	0.564210i	$-0.190819\pi$
$827$	47.4864	1.65126	0.825632	+	0.564210i	$0.190819\pi$
$828$	0	0
$829$	48.2337	1.67523	0.837613	−	0.546265i	$-0.183951\pi$
$829$	48.2337	1.67523	0.837613	+	0.546265i	$0.183951\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.25639i	0.147475i
$834$	0	0
$835$	− 44.1510i	− 1.52791i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.6205	−1.47142	−0.735711	−	0.677296i	$-0.763152\pi$
$839$	−42.6205	−1.47142	−0.735711	+	0.677296i	$0.763152\pi$
$840$	0	0
$841$	22.6277	0.780266
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.6101i	0.640208i
$846$	0	0
$847$	− 13.7081i	− 0.471017i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.7216	−0.744605
$852$	0	0
$853$	−44.6060	−1.52728	−0.763640	−	0.645643i	$-0.776590\pi$
$853$	−44.6060	−1.52728	−0.763640	+	0.645643i	$0.776590\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.5701i	1.28337i	0.766968	+	0.641685i	$0.221764\pi$
$857$	37.5701i	1.28337i	−0.766968	+	0.641685i	$0.778236\pi$
$858$	0	0
$859$	1.82532i	0.0622791i	0.999515	+	0.0311396i	$0.00991364\pi$
$859$	1.82532i	0.0622791i	−0.999515	+	0.0311396i	$0.990086\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0344	1.36279	0.681393	−	0.731918i	$-0.261375\pi$
$863$	40.0344	1.36279	0.681393	+	0.731918i	$0.261375\pi$
$864$	0	0
$865$	−44.6060	−1.51665
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.99377i	− 0.169402i
$870$	0	0
$871$	18.3365i	0.621307i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.6834	0.394972
$876$	0	0
$877$	−21.6277	−0.730316	−0.365158	−	0.930946i	$-0.618985\pi$
$877$	−21.6277	−0.730316	−0.365158	+	0.930946i	$0.618985\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	52.9562i	1.78414i	0.451898	+	0.892070i	$0.350747\pi$
$881$	52.9562i	1.78414i	−0.451898	+	0.892070i	$0.649253\pi$
$882$	0	0
$883$	− 20.0127i	− 0.673481i	−0.941597	−	0.336741i	$-0.890675\pi$
$883$	− 20.0127i	− 0.673481i	0.941597	−	0.336741i	$-0.109325\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.5118	−0.655141	−0.327571	−	0.944827i	$-0.606230\pi$
$887$	−19.5118	−0.655141	−0.327571	+	0.944827i	$0.606230\pi$
$888$	0	0
$889$	9.76631	0.327552
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.63858i	0.188688i
$894$	0	0
$895$	22.3130i	0.745841i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.5226	0.684467
$900$	0	0
$901$	1.48913	0.0496100
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.0974i	− 0.335647i
$906$	0	0
$907$	29.8934i	0.992593i	0.868153	+	0.496297i	$0.165307\pi$
$907$	29.8934i	0.992593i	−0.868153	+	0.496297i	$0.834693\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.8029	1.18620	0.593102	−	0.805127i	$-0.297903\pi$
$911$	35.8029	1.18620	0.593102	+	0.805127i	$0.297903\pi$
$912$	0	0
$913$	−3.86141	−0.127794
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.74749i	0.321891i
$918$	0	0
$919$	36.9711i	1.21956i	0.792570	+	0.609781i	$0.208743\pi$
$919$	36.9711i	1.21956i	−0.792570	+	0.609781i	$0.791257\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.1628	0.926989
$924$	0	0
$925$	−9.25544	−0.304317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.5552i	0.608777i	0.952548	+	0.304389i	$0.0984521\pi$
$929$	18.5552i	0.608777i	−0.952548	+	0.304389i	$0.901548\pi$
$930$	0	0
$931$	− 25.2651i	− 0.828029i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.01082	0.0330572
$936$	0	0
$937$	45.7228	1.49370	0.746850	−	0.664993i	$-0.231565\pi$
$937$	45.7228	1.49370	0.746850	+	0.664993i	$0.231565\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.5065i	0.440301i	0.975466	+	0.220150i	$0.0706548\pi$
$941$	13.5065i	0.440301i	−0.975466	+	0.220150i	$0.929345\pi$
$942$	0	0
$943$	21.8380i	0.711144i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.57031	0.213506	0.106753	−	0.994286i	$-0.465955\pi$
$947$	6.57031	0.213506	0.106753	+	0.994286i	$0.465955\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.2997i	0.333641i	0.985987	+	0.166821i	$0.0533500\pi$
$953$	10.2997i	0.333641i	−0.985987	+	0.166821i	$0.946650\pi$
$954$	0	0
$955$	− 38.0978i	− 1.23282i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.00567	0.226225
$960$	0	0
$961$	−35.0951	−1.13210
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 19.5499i	− 0.629333i
$966$	0	0
$967$	46.8517i	1.50665i	0.657648	+	0.753325i	$0.271551\pi$
$967$	46.8517i	1.50665i	−0.657648	+	0.753325i	$0.728449\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.0019	−1.18745	−0.593724	−	0.804669i	$-0.702343\pi$
$971$	−37.0019	−1.18745	−0.593724	+	0.804669i	$0.702343\pi$
$972$	0	0
$973$	−19.6277	−0.629236
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.6882i	1.74963i	0.484455	+	0.874816i	$0.339018\pi$
$977$	54.6882i	1.74963i	−0.484455	+	0.874816i	$0.660982\pi$
$978$	0	0
$979$	− 6.05321i	− 0.193461i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.4158	1.09769	0.548847	−	0.835923i	$-0.315067\pi$
$983$	34.4158	1.09769	0.548847	+	0.835923i	$0.315067\pi$
$984$	0	0
$985$	−60.4674	−1.92665
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.8935i	0.791568i
$990$	0	0
$991$	7.65492i	0.243167i	0.992581	+	0.121583i	$0.0387972\pi$
$991$	7.65492i	0.243167i	−0.992581	+	0.121583i	$0.961203\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−32.5823	−1.03293
$996$	0	0
$997$	24.1386	0.764477	0.382238	−	0.924064i	$-0.375153\pi$
$997$	24.1386	0.764477	0.382238	+	0.924064i	$0.375153\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.j.5183.2		8
3.2	odd	2	inner	5184.2.c.j.5183.8		8
4.3	odd	2	inner	5184.2.c.j.5183.1		8
8.3	odd	2		324.2.b.b.323.8		8
8.5	even	2		324.2.b.b.323.2		8
9.2	odd	6		1728.2.s.f.1151.2		8
9.4	even	3		1728.2.s.f.575.1		8
9.5	odd	6		576.2.s.f.191.1		8
9.7	even	3		576.2.s.f.383.4		8
12.11	even	2	inner	5184.2.c.j.5183.7		8
24.5	odd	2		324.2.b.b.323.7		8
24.11	even	2		324.2.b.b.323.1		8
36.7	odd	6		576.2.s.f.383.1		8
36.11	even	6		1728.2.s.f.1151.1		8
36.23	even	6		576.2.s.f.191.4		8
36.31	odd	6		1728.2.s.f.575.2		8
72.5	odd	6		36.2.h.a.11.3	✓	8
72.11	even	6		108.2.h.a.71.2		8
72.13	even	6		108.2.h.a.35.2		8
72.29	odd	6		108.2.h.a.71.1		8
72.43	odd	6		36.2.h.a.23.3	yes	8
72.59	even	6		36.2.h.a.11.4	yes	8
72.61	even	6		36.2.h.a.23.4	yes	8
72.67	odd	6		108.2.h.a.35.1		8
360.43	even	12		900.2.o.a.599.1		16
360.59	even	6		900.2.r.c.551.1		8
360.77	even	12		900.2.o.a.299.1		16
360.133	odd	12		900.2.o.a.599.6		16
360.149	odd	6		900.2.r.c.551.2		8
360.187	even	12		900.2.o.a.599.8		16
360.203	odd	12		900.2.o.a.299.3		16
360.259	odd	6		900.2.r.c.851.2		8
360.277	odd	12		900.2.o.a.599.3		16
360.293	even	12		900.2.o.a.299.8		16
360.347	odd	12		900.2.o.a.299.6		16
360.349	even	6		900.2.r.c.851.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.h.a.11.3	✓	8	72.5	odd	6
36.2.h.a.11.4	yes	8	72.59	even	6
36.2.h.a.23.3	yes	8	72.43	odd	6
36.2.h.a.23.4	yes	8	72.61	even	6
108.2.h.a.35.1		8	72.67	odd	6
108.2.h.a.35.2		8	72.13	even	6
108.2.h.a.71.1		8	72.29	odd	6
108.2.h.a.71.2		8	72.11	even	6
324.2.b.b.323.1		8	24.11	even	2
324.2.b.b.323.2		8	8.5	even	2
324.2.b.b.323.7		8	24.5	odd	2
324.2.b.b.323.8		8	8.3	odd	2
576.2.s.f.191.1		8	9.5	odd	6
576.2.s.f.191.4		8	36.23	even	6
576.2.s.f.383.1		8	36.7	odd	6
576.2.s.f.383.4		8	9.7	even	3
900.2.o.a.299.1		16	360.77	even	12
900.2.o.a.299.3		16	360.203	odd	12
900.2.o.a.299.6		16	360.347	odd	12
900.2.o.a.299.8		16	360.293	even	12
900.2.o.a.599.1		16	360.43	even	12
900.2.o.a.599.3		16	360.277	odd	12
900.2.o.a.599.6		16	360.133	odd	12
900.2.o.a.599.8		16	360.187	even	12
900.2.r.c.551.1		8	360.59	even	6
900.2.r.c.551.2		8	360.149	odd	6
900.2.r.c.851.1		8	360.349	even	6
900.2.r.c.851.2		8	360.259	odd	6
1728.2.s.f.575.1		8	9.4	even	3
1728.2.s.f.575.2		8	36.31	odd	6
1728.2.s.f.1151.1		8	36.11	even	6
1728.2.s.f.1151.2		8	9.2	odd	6
5184.2.c.j.5183.1		8	4.3	odd	2	inner
5184.2.c.j.5183.2		8	1.1	even	1	trivial
5184.2.c.j.5183.7		8	12.11	even	2	inner
5184.2.c.j.5183.8		8	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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.52434i q^{5} +1.27582i q^{7} +O(q^{10})\)	\(q-2.52434i q^{5} +1.27582i q^{7} +0.505408 q^{11} +2.37228 q^{13} +0.792287i q^{17} -4.70285i q^{19} -3.22060 q^{23} -1.37228 q^{25} -2.52434i q^{29} +8.12989i q^{31} +3.22060 q^{35} +6.74456 q^{37} -6.78073i q^{41} -7.72946i q^{43} -1.19897 q^{47} +5.37228 q^{49} -1.87953i q^{53} -1.27582i q^{55} -12.3770 q^{59} +2.37228 q^{61} -5.98844i q^{65} +7.72946i q^{67} +11.8716 q^{71} +3.37228 q^{73} +0.644810i q^{77} -9.88067i q^{79} -7.64018 q^{83} +2.00000 q^{85} -11.9769i q^{89} +3.02661i q^{91} -11.8716 q^{95} +10.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{13} + 12 q^{25} + 8 q^{37} + 20 q^{49} - 4 q^{61} + 4 q^{73} + 16 q^{85} - 8 q^{97}+O(q^{100}) \) 8 * q - 4 * q^13 + 12 * q^25 + 8 * q^37 + 20 * q^49 - 4 * q^61 + 4 * q^73 + 16 * q^85 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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