LMFDB - Embedded newform 5184.2.c.d.5183.1

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.d.5183.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +1.73205i q^{7} +O(q^{10})$	$q-1.73205i q^{5} +1.73205i q^{7} +3.00000 q^{11} +5.00000 q^{13} -6.92820i q^{17} -3.46410i q^{19} -9.00000 q^{23} +2.00000 q^{25} +1.73205i q^{29} -5.19615i q^{31} +3.00000 q^{35} -2.00000 q^{37} -5.19615i q^{41} +5.19615i q^{43} -3.00000 q^{47} +4.00000 q^{49} -5.19615i q^{55} -3.00000 q^{59} +1.00000 q^{61} -8.66025i q^{65} -8.66025i q^{67} -12.0000 q^{71} -2.00000 q^{73} +5.19615i q^{77} +8.66025i q^{79} +15.0000 q^{83} -12.0000 q^{85} +6.92820i q^{89} +8.66025i q^{91} -6.00000 q^{95} -5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 6 q^{11} + 10 q^{13} - 18 q^{23} + 4 q^{25} + 6 q^{35} - 4 q^{37} - 6 q^{47} + 8 q^{49} - 6 q^{59} + 2 q^{61} - 24 q^{71} - 4 q^{73} + 30 q^{83} - 24 q^{85} - 12 q^{95} - 10 q^{97}+O(q^{100}) $ 2 * q + 6 * q^11 + 10 * q^13 - 18 * q^23 + 4 * q^25 + 6 * q^35 - 4 * q^37 - 6 * q^47 + 8 * q^49 - 6 * q^59 + 2 * q^61 - 24 * q^71 - 4 * q^73 + 30 * q^83 - 24 * q^85 - 12 * q^95 - 10 * 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
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	1.73205i	0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$	1.73205i	0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.92820i	− 1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	− 6.92820i	− 1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.73205i	0.321634i	0.986984	+	0.160817i	$0.0514129\pi$
$29$	1.73205i	0.321634i	−0.986984	+	0.160817i	$0.948587\pi$
$30$	0	0
$31$	− 5.19615i	− 0.933257i	−0.884454	−	0.466628i	$-0.845469\pi$
$31$	− 5.19615i	− 0.933257i	0.884454	−	0.466628i	$-0.154531\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.19615i	− 0.811503i	−0.913984	−	0.405751i	$-0.867010\pi$
$41$	− 5.19615i	− 0.811503i	0.913984	−	0.405751i	$-0.132990\pi$
$42$	0	0
$43$	5.19615i	0.792406i	0.918163	+	0.396203i	$0.129672\pi$
$43$	5.19615i	0.792406i	−0.918163	+	0.396203i	$0.870328\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 5.19615i	− 0.700649i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 8.66025i	− 1.07417i
$66$	0	0
$67$	− 8.66025i	− 1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$	− 8.66025i	− 1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.19615i	0.592157i
$78$	0	0
$79$	8.66025i	0.974355i	0.873303	+	0.487177i	$0.161973\pi$
$79$	8.66025i	0.974355i	−0.873303	+	0.487177i	$0.838027\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.92820i	0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$	6.92820i	0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$	0	0
$91$	8.66025i	0.907841i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.19615i	− 0.517036i	−0.966006	−	0.258518i	$-0.916766\pi$
$101$	− 5.19615i	− 0.517036i	0.966006	−	0.258518i	$-0.0832342\pi$
$102$	0	0
$103$	1.73205i	0.170664i	0.996353	+	0.0853320i	$0.0271951\pi$
$103$	1.73205i	0.170664i	−0.996353	+	0.0853320i	$0.972805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.1244i	− 1.14056i	−0.821449	−	0.570282i	$-0.806834\pi$
$113$	− 12.1244i	− 1.14056i	0.821449	−	0.570282i	$-0.193166\pi$
$114$	0	0
$115$	15.5885i	1.45363i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.66025i	− 0.739895i	−0.929053	−	0.369948i	$-0.879376\pi$
$137$	− 8.66025i	− 0.739895i	0.929053	−	0.369948i	$-0.120624\pi$
$138$	0	0
$139$	− 1.73205i	− 0.146911i	−0.997299	−	0.0734553i	$-0.976597\pi$
$139$	− 1.73205i	− 0.146911i	0.997299	−	0.0734553i	$-0.0234026\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.66025i	− 0.709476i	−0.934966	−	0.354738i	$-0.884570\pi$
$149$	− 8.66025i	− 0.709476i	0.934966	−	0.354738i	$-0.115430\pi$
$150$	0	0
$151$	8.66025i	0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$	8.66025i	0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.00000	−0.722897
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 15.5885i	− 1.22854i
$162$	0	0
$163$	− 17.3205i	− 1.35665i	−0.734763	−	0.678323i	$-0.762707\pi$
$163$	− 17.3205i	− 1.35665i	0.734763	−	0.678323i	$-0.237293\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 19.0526i	− 1.44854i	−0.689517	−	0.724270i	$-0.742177\pi$
$173$	− 19.0526i	− 1.44854i	0.689517	−	0.724270i	$-0.257823\pi$
$174$	0	0
$175$	3.46410i	0.261861i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.46410i	0.254686i
$186$	0	0
$187$	− 20.7846i	− 1.51992i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.92820i	0.493614i	0.969065	+	0.246807i	$0.0793814\pi$
$197$	6.92820i	0.493614i	−0.969065	+	0.246807i	$0.920619\pi$
$198$	0	0
$199$	− 24.2487i	− 1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$	− 24.2487i	− 1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 10.3923i	− 0.718851i
$210$	0	0
$211$	19.0526i	1.31163i	0.754921	+	0.655816i	$0.227675\pi$
$211$	19.0526i	1.31163i	−0.754921	+	0.655816i	$0.772325\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.00000	0.613795
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 34.6410i	− 2.33021i
$222$	0	0
$223$	− 25.9808i	− 1.73980i	−0.493228	−	0.869900i	$-0.664183\pi$
$223$	− 25.9808i	− 1.73980i	0.493228	−	0.869900i	$-0.335817\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	17.0000	1.12339	0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000	1.12339	0.561696	+	0.827344i	$0.310149\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.8564i	− 0.907763i	−0.891062	−	0.453882i	$-0.850039\pi$
$233$	− 13.8564i	− 0.907763i	0.891062	−	0.453882i	$-0.149961\pi$
$234$	0	0
$235$	5.19615i	0.338960i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.92820i	− 0.442627i
$246$	0	0
$247$	− 17.3205i	− 1.10208i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−27.0000	−1.69748
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.73205i	0.108042i	0.998540	+	0.0540212i	$0.0172039\pi$
$257$	1.73205i	0.108042i	−0.998540	+	0.0540212i	$0.982796\pi$
$258$	0	0
$259$	− 3.46410i	− 0.215249i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.7128i	− 1.68968i	−0.535019	−	0.844840i	$-0.679696\pi$
$269$	− 27.7128i	− 1.68968i	0.535019	−	0.844840i	$-0.320304\pi$
$270$	0	0
$271$	24.2487i	1.47300i	0.676435	+	0.736502i	$0.263524\pi$
$271$	24.2487i	1.47300i	−0.676435	+	0.736502i	$0.736476\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.0526i	1.13658i	0.822828	+	0.568290i	$0.192395\pi$
$281$	19.0526i	1.13658i	−0.822828	+	0.568290i	$0.807605\pi$
$282$	0	0
$283$	− 15.5885i	− 0.926638i	−0.886192	−	0.463319i	$-0.846658\pi$
$283$	− 15.5885i	− 0.926638i	0.886192	−	0.463319i	$-0.153342\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−31.0000	−1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.73205i	− 0.101187i	−0.998719	−	0.0505937i	$-0.983889\pi$
$293$	− 1.73205i	− 0.101187i	0.998719	−	0.0505937i	$-0.0161114\pi$
$294$	0	0
$295$	5.19615i	0.302532i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−45.0000	−2.60242
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.73205i	− 0.0991769i
$306$	0	0
$307$	10.3923i	0.593120i	0.955014	+	0.296560i	$0.0958395\pi$
$307$	10.3923i	0.593120i	−0.955014	+	0.296560i	$0.904160\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.1244i	− 0.680972i	−0.940250	−	0.340486i	$-0.889408\pi$
$317$	− 12.1244i	− 0.680972i	0.940250	−	0.340486i	$-0.110592\pi$
$318$	0	0
$319$	5.19615i	0.290929i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	10.0000	0.554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.19615i	− 0.286473i
$330$	0	0
$331$	− 1.73205i	− 0.0952021i	−0.998866	−	0.0476011i	$-0.984842\pi$
$331$	− 1.73205i	− 0.0952021i	0.998866	−	0.0476011i	$-0.0151576\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.0000	−0.819538
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 15.5885i	− 0.844162i
$342$	0	0
$343$	19.0526i	1.02874i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	0	0
$349$	13.0000	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.19615i	0.276563i	0.990393	+	0.138282i	$0.0441579\pi$
$353$	5.19615i	0.276563i	−0.990393	+	0.138282i	$0.955842\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.46410i	0.181319i
$366$	0	0
$367$	− 19.0526i	− 0.994535i	−0.867597	−	0.497268i	$-0.834337\pi$
$367$	− 19.0526i	− 0.994535i	0.867597	−	0.497268i	$-0.165663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.66025i	0.446026i
$378$	0	0
$379$	17.3205i	0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$	17.3205i	0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.0000	1.37964	0.689818	−	0.723983i	$-0.257691\pi$
$383$	27.0000	1.37964	0.689818	+	0.723983i	$0.257691\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.3731i	1.84419i	0.386966	+	0.922094i	$0.373523\pi$
$389$	36.3731i	1.84419i	−0.386966	+	0.922094i	$0.626477\pi$
$390$	0	0
$391$	62.3538i	3.15337i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.0000	0.754732
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 19.0526i	− 0.951439i	−0.879597	−	0.475720i	$-0.842188\pi$
$401$	− 19.0526i	− 0.951439i	0.879597	−	0.475720i	$-0.157812\pi$
$402$	0	0
$403$	− 25.9808i	− 1.29419i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	31.0000	1.53285	0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000	1.53285	0.766426	+	0.642333i	$0.222033\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.19615i	− 0.255686i
$414$	0	0
$415$	− 25.9808i	− 1.27535i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 13.8564i	− 0.672134i
$426$	0	0
$427$	1.73205i	0.0838198i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.1769i	1.49139i
$438$	0	0
$439$	− 5.19615i	− 0.247999i	−0.992282	−	0.123999i	$-0.960428\pi$
$439$	− 5.19615i	− 0.247999i	0.992282	−	0.123999i	$-0.0395721\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.7846i	0.980886i	0.871473	+	0.490443i	$0.163165\pi$
$449$	20.7846i	0.980886i	−0.871473	+	0.490443i	$0.836835\pi$
$450$	0	0
$451$	− 15.5885i	− 0.734032i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.0000	0.703211
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 25.9808i	− 1.21004i	−0.796208	−	0.605022i	$-0.793164\pi$
$461$	− 25.9808i	− 1.21004i	0.796208	−	0.605022i	$-0.206836\pi$
$462$	0	0
$463$	− 19.0526i	− 0.885448i	−0.896658	−	0.442724i	$-0.854012\pi$
$463$	− 19.0526i	− 0.885448i	0.896658	−	0.442724i	$-0.145988\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	15.0000	0.692636
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.5885i	0.716758i
$474$	0	0
$475$	− 6.92820i	− 0.317888i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.66025i	0.393242i
$486$	0	0
$487$	− 3.46410i	− 0.156973i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	− 3.46410i	− 0.156973i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.0000	0.947717	0.473858	−	0.880601i	$-0.342861\pi$
$491$	21.0000	0.947717	0.473858	+	0.880601i	$0.342861\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.7846i	− 0.932317i
$498$	0	0
$499$	− 15.5885i	− 0.697835i	−0.937153	−	0.348918i	$-0.886549\pi$
$499$	− 15.5885i	− 0.697835i	0.937153	−	0.348918i	$-0.113451\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 36.3731i	− 1.61221i	−0.591774	−	0.806104i	$-0.701572\pi$
$509$	− 36.3731i	− 1.61221i	0.591774	−	0.806104i	$-0.298428\pi$
$510$	0	0
$511$	− 3.46410i	− 0.153243i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.00000	0.132196
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	17.3205i	0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$	17.3205i	0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.0000	−1.56818
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 25.9808i	− 1.12535i
$534$	0	0
$535$	− 20.7846i	− 0.898597i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 24.2487i	− 1.03870i
$546$	0	0
$547$	39.8372i	1.70331i	0.524099	+	0.851657i	$0.324402\pi$
$547$	39.8372i	1.70331i	−0.524099	+	0.851657i	$0.675598\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−15.0000	−0.637865
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.7128i	1.17423i	0.809504	+	0.587115i	$0.199736\pi$
$557$	27.7128i	1.17423i	−0.809504	+	0.587115i	$0.800264\pi$
$558$	0	0
$559$	25.9808i	1.09887i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	0	0
$565$	−21.0000	−0.883477
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.73205i	− 0.0726113i	−0.999341	−	0.0363057i	$-0.988441\pi$
$569$	− 1.73205i	− 0.0726113i	0.999341	−	0.0363057i	$-0.0115590\pi$
$570$	0	0
$571$	32.9090i	1.37720i	0.725143	+	0.688599i	$0.241774\pi$
$571$	32.9090i	1.37720i	−0.725143	+	0.688599i	$0.758226\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−18.0000	−0.750652
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25.9808i	1.07786i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 34.6410i	− 1.42254i	−0.702921	−	0.711268i	$-0.748121\pi$
$593$	− 34.6410i	− 1.42254i	0.702921	−	0.711268i	$-0.251879\pi$
$594$	0	0
$595$	− 20.7846i	− 0.852086i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.46410i	0.140836i
$606$	0	0
$607$	1.73205i	0.0703018i	0.999382	+	0.0351509i	$0.0111912\pi$
$607$	1.73205i	0.0703018i	−0.999382	+	0.0351509i	$0.988809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.0000	−0.606835
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.5167i	0.906487i	0.891387	+	0.453243i	$0.149733\pi$
$617$	22.5167i	0.906487i	−0.891387	+	0.453243i	$0.850267\pi$
$618$	0	0
$619$	− 29.4449i	− 1.18349i	−0.806126	−	0.591744i	$-0.798439\pi$
$619$	− 29.4449i	− 1.18349i	0.806126	−	0.591744i	$-0.201561\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.8564i	0.552491i
$630$	0	0
$631$	− 17.3205i	− 0.689519i	−0.938691	−	0.344759i	$-0.887961\pi$
$631$	− 17.3205i	− 0.689519i	0.938691	−	0.344759i	$-0.112039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	20.0000	0.792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 29.4449i	− 1.16300i	−0.813546	−	0.581501i	$-0.802466\pi$
$641$	− 29.4449i	− 1.16300i	0.813546	−	0.581501i	$-0.197534\pi$
$642$	0	0
$643$	5.19615i	0.204916i	0.994737	+	0.102458i	$0.0326708\pi$
$643$	5.19615i	0.204916i	−0.994737	+	0.102458i	$0.967329\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 15.5885i	− 0.610023i	−0.952349	−	0.305012i	$-0.901340\pi$
$653$	− 15.5885i	− 0.610023i	0.952349	−	0.305012i	$-0.0986604\pi$
$654$	0	0
$655$	5.19615i	0.203030i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	49.0000	1.90588	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000	1.90588	0.952940	+	0.303160i	$0.0980418\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 10.3923i	− 0.402996i
$666$	0	0
$667$	− 15.5885i	− 0.603587i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.3731i	1.39793i	0.715156	+	0.698965i	$0.246356\pi$
$677$	36.3731i	1.39793i	−0.715156	+	0.698965i	$0.753644\pi$
$678$	0	0
$679$	− 8.66025i	− 0.332350i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	5.19615i	0.197671i	0.995104	+	0.0988355i	$0.0315118\pi$
$691$	5.19615i	0.197671i	−0.995104	+	0.0988355i	$0.968488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.00000	−0.113796
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.8564i	0.523349i	0.965156	+	0.261675i	$0.0842747\pi$
$701$	13.8564i	0.523349i	−0.965156	+	0.261675i	$0.915725\pi$
$702$	0	0
$703$	6.92820i	0.261302i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.00000	0.338480
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	46.7654i	1.75138i
$714$	0	0
$715$	− 25.9808i	− 0.971625i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.46410i	0.128654i
$726$	0	0
$727$	36.3731i	1.34900i	0.738274	+	0.674501i	$0.235641\pi$
$727$	36.3731i	1.34900i	−0.738274	+	0.674501i	$0.764359\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 25.9808i	− 0.957014i
$738$	0	0
$739$	31.1769i	1.14686i	0.819254	+	0.573431i	$0.194388\pi$
$739$	31.1769i	1.14686i	−0.819254	+	0.573431i	$0.805612\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.7846i	0.759453i
$750$	0	0
$751$	− 12.1244i	− 0.442424i	−0.975226	−	0.221212i	$-0.928999\pi$
$751$	− 12.1244i	− 0.442424i	0.975226	−	0.221212i	$-0.0710013\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.0000	0.545906
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.5167i	0.816228i	0.912931	+	0.408114i	$0.133813\pi$
$761$	22.5167i	0.816228i	−0.912931	+	0.408114i	$0.866187\pi$
$762$	0	0
$763$	24.2487i	0.877862i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.0000	−0.541619
$768$	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
$773$	27.7128i	0.996761i	0.866959	+	0.498380i	$0.166072\pi$
$773$	27.7128i	0.996761i	−0.866959	+	0.498380i	$0.833928\pi$
$774$	0	0
$775$	− 10.3923i	− 0.373303i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.73205i	− 0.0618195i
$786$	0	0
$787$	− 29.4449i	− 1.04960i	−0.851227	−	0.524798i	$-0.824141\pi$
$787$	− 29.4449i	− 1.04960i	0.851227	−	0.524798i	$-0.175859\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21.0000	0.746674
$792$	0	0
$793$	5.00000	0.177555
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.9090i	1.16570i	0.812581	+	0.582848i	$0.198062\pi$
$797$	32.9090i	1.16570i	−0.812581	+	0.582848i	$0.801938\pi$
$798$	0	0
$799$	20.7846i	0.735307i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−27.0000	−0.951625
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.4974i	1.70508i	0.522663	+	0.852539i	$0.324939\pi$
$809$	48.4974i	1.70508i	−0.522663	+	0.852539i	$0.675061\pi$
$810$	0	0
$811$	− 31.1769i	− 1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$	− 31.1769i	− 1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−30.0000	−1.05085
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.4449i	1.02763i	0.857900	+	0.513816i	$0.171769\pi$
$821$	29.4449i	1.02763i	−0.857900	+	0.513816i	$0.828231\pi$
$822$	0	0
$823$	− 32.9090i	− 1.14713i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	− 32.9090i	− 1.14713i	0.819159	−	0.573567i	$-0.194441\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 27.7128i	− 0.960192i
$834$	0	0
$835$	− 5.19615i	− 0.179820i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.0000	1.55357	0.776786	−	0.629764i	$-0.216849\pi$
$839$	45.0000	1.55357	0.776786	+	0.629764i	$0.216849\pi$
$840$	0	0
$841$	26.0000	0.896552
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 20.7846i	− 0.715012i
$846$	0	0
$847$	− 3.46410i	− 0.119028i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.0000	0.617032
$852$	0	0
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.9090i	1.12415i	0.827087	+	0.562074i	$0.189997\pi$
$857$	32.9090i	1.12415i	−0.827087	+	0.562074i	$0.810003\pi$
$858$	0	0
$859$	− 22.5167i	− 0.768259i	−0.923279	−	0.384129i	$-0.874502\pi$
$859$	− 22.5167i	− 0.768259i	0.923279	−	0.384129i	$-0.125498\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−33.0000	−1.12203
$866$	0	0
$867$	0	0
$868$	0	0
$869$	25.9808i	0.881337i
$870$	0	0
$871$	− 43.3013i	− 1.46721i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21.0000	0.709930
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.7846i	0.700251i	0.936703	+	0.350126i	$0.113861\pi$
$881$	20.7846i	0.700251i	−0.936703	+	0.350126i	$0.886139\pi$
$882$	0	0
$883$	− 17.3205i	− 0.582882i	−0.956589	−	0.291441i	$-0.905865\pi$
$883$	− 17.3205i	− 0.582882i	0.956589	−	0.291441i	$-0.0941346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−57.0000	−1.91387	−0.956936	−	0.290298i	$-0.906246\pi$
$887$	−57.0000	−1.91387	−0.956936	+	0.290298i	$0.906246\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.3923i	0.347765i
$894$	0	0
$895$	20.7846i	0.694753i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.3205i	0.575753i
$906$	0	0
$907$	32.9090i	1.09272i	0.837549	+	0.546362i	$0.183988\pi$
$907$	32.9090i	1.09272i	−0.837549	+	0.546362i	$0.816012\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.0000	1.09334	0.546669	−	0.837349i	$-0.315895\pi$
$911$	33.0000	1.09334	0.546669	+	0.837349i	$0.315895\pi$
$912$	0	0
$913$	45.0000	1.48928
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.19615i	− 0.171592i
$918$	0	0
$919$	10.3923i	0.342811i	0.985201	+	0.171405i	$0.0548307\pi$
$919$	10.3923i	0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−60.0000	−1.97492
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.0526i	0.625094i	0.949902	+	0.312547i	$0.101182\pi$
$929$	19.0526i	0.625094i	−0.949902	+	0.312547i	$0.898818\pi$
$930$	0	0
$931$	− 13.8564i	− 0.454125i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−36.0000	−1.17733
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.6936i	1.75036i	0.483797	+	0.875180i	$0.339257\pi$
$941$	53.6936i	1.75036i	−0.483797	+	0.875180i	$0.660743\pi$
$942$	0	0
$943$	46.7654i	1.52289i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.4974i	1.57099i	0.618871	+	0.785493i	$0.287590\pi$
$953$	48.4974i	1.57099i	−0.618871	+	0.785493i	$0.712410\pi$
$954$	0	0
$955$	5.19615i	0.168144i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	4.00000	0.129032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.5167i	0.724837i
$966$	0	0
$967$	8.66025i	0.278495i	0.990258	+	0.139247i	$0.0444684\pi$
$967$	8.66025i	0.278495i	−0.990258	+	0.139247i	$0.955532\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.5885i	0.498719i	0.968411	+	0.249359i	$0.0802201\pi$
$977$	15.5885i	0.498719i	−0.968411	+	0.249359i	$0.919780\pi$
$978$	0	0
$979$	20.7846i	0.664279i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15.0000	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 46.7654i	− 1.48705i
$990$	0	0
$991$	− 45.0333i	− 1.43053i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	− 45.0333i	− 1.43053i	0.698853	−	0.715265i	$-0.253694\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−42.0000	−1.33149
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.d.5183.1		2
3.2	odd	2		5184.2.c.b.5183.2		2
4.3	odd	2		5184.2.c.b.5183.1		2
8.3	odd	2		1296.2.c.c.1295.2		2
8.5	even	2		1296.2.c.a.1295.2		2
9.2	odd	6		576.2.s.c.383.1		2
9.4	even	3		576.2.s.b.191.1		2
9.5	odd	6		1728.2.s.c.575.1		2
9.7	even	3		1728.2.s.d.1151.1		2
12.11	even	2	inner	5184.2.c.d.5183.2		2
24.5	odd	2		1296.2.c.c.1295.1		2
24.11	even	2		1296.2.c.a.1295.1		2
36.7	odd	6		1728.2.s.c.1151.1		2
36.11	even	6		576.2.s.b.383.1		2
36.23	even	6		1728.2.s.d.575.1		2
36.31	odd	6		576.2.s.c.191.1		2
72.5	odd	6		432.2.s.a.143.1		2
72.11	even	6		144.2.s.b.95.1	yes	2
72.13	even	6		144.2.s.b.47.1	✓	2
72.29	odd	6		144.2.s.c.95.1	yes	2
72.43	odd	6		432.2.s.a.287.1		2
72.59	even	6		432.2.s.b.143.1		2
72.61	even	6		432.2.s.b.287.1		2
72.67	odd	6		144.2.s.c.47.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.b.47.1	✓	2	72.13	even	6
144.2.s.b.95.1	yes	2	72.11	even	6
144.2.s.c.47.1	yes	2	72.67	odd	6
144.2.s.c.95.1	yes	2	72.29	odd	6
432.2.s.a.143.1		2	72.5	odd	6
432.2.s.a.287.1		2	72.43	odd	6
432.2.s.b.143.1		2	72.59	even	6
432.2.s.b.287.1		2	72.61	even	6
576.2.s.b.191.1		2	9.4	even	3
576.2.s.b.383.1		2	36.11	even	6
576.2.s.c.191.1		2	36.31	odd	6
576.2.s.c.383.1		2	9.2	odd	6
1296.2.c.a.1295.1		2	24.11	even	2
1296.2.c.a.1295.2		2	8.5	even	2
1296.2.c.c.1295.1		2	24.5	odd	2
1296.2.c.c.1295.2		2	8.3	odd	2
1728.2.s.c.575.1		2	9.5	odd	6
1728.2.s.c.1151.1		2	36.7	odd	6
1728.2.s.d.575.1		2	36.23	even	6
1728.2.s.d.1151.1		2	9.7	even	3
5184.2.c.b.5183.1		2	4.3	odd	2
5184.2.c.b.5183.2		2	3.2	odd	2
5184.2.c.d.5183.1		2	1.1	even	1	trivial
5184.2.c.d.5183.2		2	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +1.73205i q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +1.73205i q^{7} +3.00000 q^{11} +5.00000 q^{13} -6.92820i q^{17} -3.46410i q^{19} -9.00000 q^{23} +2.00000 q^{25} +1.73205i q^{29} -5.19615i q^{31} +3.00000 q^{35} -2.00000 q^{37} -5.19615i q^{41} +5.19615i q^{43} -3.00000 q^{47} +4.00000 q^{49} -5.19615i q^{55} -3.00000 q^{59} +1.00000 q^{61} -8.66025i q^{65} -8.66025i q^{67} -12.0000 q^{71} -2.00000 q^{73} +5.19615i q^{77} +8.66025i q^{79} +15.0000 q^{83} -12.0000 q^{85} +6.92820i q^{89} +8.66025i q^{91} -6.00000 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 6 q^{11} + 10 q^{13} - 18 q^{23} + 4 q^{25} + 6 q^{35} - 4 q^{37} - 6 q^{47} + 8 q^{49} - 6 q^{59} + 2 q^{61} - 24 q^{71} - 4 q^{73} + 30 q^{83} - 24 q^{85} - 12 q^{95} - 10 q^{97}+O(q^{100}) \) 2 * q + 6 * q^11 + 10 * q^13 - 18 * q^23 + 4 * q^25 + 6 * q^35 - 4 * q^37 - 6 * q^47 + 8 * q^49 - 6 * q^59 + 2 * q^61 - 24 * q^71 - 4 * q^73 + 30 * q^83 - 24 * q^85 - 12 * q^95 - 10 * q^97

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