LMFDB - Embedded newform 7200.2.k.q.3601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(3601,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("7200.3601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.k (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.212336640000.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{4} + 16 $ x^8 + 2*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.1
Root			$-1.26979 + 0.622597i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.q.3601.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.44949 q^{7} +O(q^{10})$	$q-3.44949 q^{7} -5.54048i q^{11} -3.87298i q^{13} +6.22069 q^{17} -7.03526i q^{19} +1.14152 q^{23} +3.05009i q^{29} +1.44949 q^{31} +6.32456i q^{37} -3.93765 q^{41} +0.710706i q^{43} +10.1583 q^{47} +4.89898 q^{49} -8.03087i q^{53} -4.98078i q^{59} -10.1975i q^{61} -5.61385i q^{67} -13.5829 q^{71} +6.89898 q^{73} +19.1118i q^{77} -2.89898 q^{79} +6.10018i q^{83} -7.87530 q^{89} +13.3598i q^{91} +15.6969 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 8 q^{31} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^31 + 16 * q^73 + 16 * q^79 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.44949	−1.30378	−0.651892	−	0.758312i	$-0.726025\pi$
$7$	−3.44949	−1.30378	−0.651892	+	0.758312i	$0.726025\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.54048i	− 1.67052i	−0.549857	−	0.835259i	$-0.685318\pi$
$11$	− 5.54048i	− 1.67052i	0.549857	−	0.835259i	$-0.314682\pi$
$12$	0	0
$13$	− 3.87298i	− 1.07417i	−0.843527	−	0.537086i	$-0.819525\pi$
$13$	− 3.87298i	− 1.07417i	0.843527	−	0.537086i	$-0.180475\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.22069	1.50874	0.754369	−	0.656451i	$-0.227943\pi$
$17$	6.22069	1.50874	0.754369	+	0.656451i	$0.227943\pi$
$18$	0	0
$19$	− 7.03526i	− 1.61400i	−0.590552	−	0.807000i	$-0.701090\pi$
$19$	− 7.03526i	− 1.61400i	0.590552	−	0.807000i	$-0.298910\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.14152	0.238023	0.119011	−	0.992893i	$-0.462027\pi$
$23$	1.14152	0.238023	0.119011	+	0.992893i	$0.462027\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.05009i	0.566388i	0.959063	+	0.283194i	$0.0913940\pi$
$29$	3.05009i	0.566388i	−0.959063	+	0.283194i	$0.908606\pi$
$30$	0	0
$31$	1.44949	0.260336	0.130168	−	0.991492i	$-0.458448\pi$
$31$	1.44949	0.260336	0.130168	+	0.991492i	$0.458448\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.32456i	1.03975i	0.854242	+	0.519875i	$0.174022\pi$
$37$	6.32456i	1.03975i	−0.854242	+	0.519875i	$0.825978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.93765	−0.614958	−0.307479	−	0.951555i	$-0.599485\pi$
$41$	−3.93765	−0.614958	−0.307479	+	0.951555i	$0.599485\pi$
$42$	0	0
$43$	0.710706i	0.108382i	0.998531	+	0.0541908i	$0.0172579\pi$
$43$	0.710706i	0.108382i	−0.998531	+	0.0541908i	$0.982742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.1583	1.48175	0.740873	−	0.671645i	$-0.234412\pi$
$47$	10.1583	1.48175	0.740873	+	0.671645i	$0.234412\pi$
$48$	0	0
$49$	4.89898	0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.03087i	− 1.10313i	−0.834134	−	0.551563i	$-0.814032\pi$
$53$	− 8.03087i	− 1.10313i	0.834134	−	0.551563i	$-0.185968\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.98078i	− 0.648442i	−0.945981	−	0.324221i	$-0.894898\pi$
$59$	− 4.98078i	− 0.648442i	0.945981	−	0.324221i	$-0.105102\pi$
$60$	0	0
$61$	− 10.1975i	− 1.30566i	−0.757504	−	0.652831i	$-0.773581\pi$
$61$	− 10.1975i	− 1.30566i	0.757504	−	0.652831i	$-0.226419\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.61385i	− 0.685841i	−0.939364	−	0.342920i	$-0.888584\pi$
$67$	− 5.61385i	− 0.685841i	0.939364	−	0.342920i	$-0.111416\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.5829	−1.61199	−0.805996	−	0.591921i	$-0.798370\pi$
$71$	−13.5829	−1.61199	−0.805996	+	0.591921i	$0.798370\pi$
$72$	0	0
$73$	6.89898	0.807464	0.403732	−	0.914877i	$-0.367713\pi$
$73$	6.89898	0.807464	0.403732	+	0.914877i	$0.367713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.1118i	2.17800i
$78$	0	0
$79$	−2.89898	−0.326161	−0.163080	−	0.986613i	$-0.552143\pi$
$79$	−2.89898	−0.326161	−0.163080	+	0.986613i	$0.552143\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.10018i	0.669582i	0.942292	+	0.334791i	$0.108666\pi$
$83$	6.10018i	0.669582i	−0.942292	+	0.334791i	$0.891334\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.87530	−0.834781	−0.417390	−	0.908727i	$-0.637055\pi$
$89$	−7.87530	−0.834781	−0.417390	+	0.908727i	$0.637055\pi$
$90$	0	0
$91$	13.3598i	1.40049i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.6969	1.59378	0.796891	−	0.604123i	$-0.206476\pi$
$97$	15.6969	1.59378	0.796891	+	0.604123i	$0.206476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.1311i	1.40609i	0.711144	+	0.703046i	$0.248177\pi$
$101$	14.1311i	1.40609i	−0.711144	+	0.703046i	$0.751823\pi$
$102$	0	0
$103$	−6.89898	−0.679777	−0.339888	−	0.940466i	$-0.610389\pi$
$103$	−6.89898	−0.679777	−0.339888	+	0.940466i	$0.610389\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.559702i	− 0.0541085i	−0.999634	−	0.0270542i	$-0.991387\pi$
$107$	− 0.559702i	− 0.0541085i	0.999634	−	0.0270542i	$-0.00861268\pi$
$108$	0	0
$109$	16.5221i	1.58253i	0.611474	+	0.791265i	$0.290577\pi$
$109$	16.5221i	1.58253i	−0.611474	+	0.791265i	$0.709423\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.4414	−1.17039	−0.585193	−	0.810894i	$-0.698981\pi$
$113$	−12.4414	−1.17039	−0.585193	+	0.810894i	$0.698981\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.4582	−1.96707
$120$	0	0
$121$	−19.6969	−1.79063
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.79796	−0.337014	−0.168507	−	0.985700i	$-0.553895\pi$
$127$	−3.79796	−0.337014	−0.168507	+	0.985700i	$0.553895\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.5213i	− 0.919247i	−0.888114	−	0.459623i	$-0.847984\pi$
$131$	− 10.5213i	− 0.919247i	0.888114	−	0.459623i	$-0.152016\pi$
$132$	0	0
$133$	24.2681i	2.10431i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.1583	0.867885	0.433943	−	0.900940i	$-0.357122\pi$
$137$	10.1583	0.867885	0.433943	+	0.900940i	$0.357122\pi$
$138$	0	0
$139$	9.16738i	0.777567i	0.921329	+	0.388783i	$0.127105\pi$
$139$	9.16738i	0.777567i	−0.921329	+	0.388783i	$0.872895\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.4582	−1.79442
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 21.0425i	− 1.72387i	−0.507018	−	0.861935i	$-0.669252\pi$
$149$	− 21.0425i	− 1.72387i	0.507018	−	0.861935i	$-0.330748\pi$
$150$	0	0
$151$	12.3485	1.00490	0.502452	−	0.864605i	$-0.332431\pi$
$151$	12.3485	1.00490	0.502452	+	0.864605i	$0.332431\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.77613i	0.700411i	0.936673	+	0.350206i	$0.113888\pi$
$157$	8.77613i	0.700411i	−0.936673	+	0.350206i	$0.886112\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.93765	−0.310330
$162$	0	0
$163$	− 14.7812i	− 1.15776i	−0.815414	−	0.578878i	$-0.803491\pi$
$163$	− 14.7812i	− 1.15776i	0.815414	−	0.578878i	$-0.196509\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.42455	−0.265000	−0.132500	−	0.991183i	$-0.542300\pi$
$167$	−3.42455	−0.265000	−0.132500	+	0.991183i	$0.542300\pi$
$168$	0	0
$169$	−2.00000	−0.153846
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.93069i	− 0.146787i	−0.997303	−	0.0733937i	$-0.976617\pi$
$173$	− 1.93069i	− 0.146787i	0.997303	−	0.0733937i	$-0.0233830\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 21.0425i	− 1.57279i	−0.617723	−	0.786396i	$-0.711945\pi$
$179$	− 21.0425i	− 1.57279i	0.617723	−	0.786396i	$-0.288055\pi$
$180$	0	0
$181$	5.29439i	0.393529i	0.980451	+	0.196765i	$0.0630435\pi$
$181$	5.29439i	0.393529i	−0.980451	+	0.196765i	$0.936957\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 34.4656i	− 2.52037i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.5829	0.982823	0.491412	−	0.870927i	$-0.336481\pi$
$191$	13.5829	0.982823	0.491412	+	0.870927i	$0.336481\pi$
$192$	0	0
$193$	−11.8990	−0.856507	−0.428254	−	0.903659i	$-0.640871\pi$
$193$	−11.8990	−0.856507	−0.428254	+	0.903659i	$0.640871\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.2813i	− 1.65873i	−0.558710	−	0.829363i	$-0.688704\pi$
$197$	− 23.2813i	− 1.65873i	0.558710	−	0.829363i	$-0.311296\pi$
$198$	0	0
$199$	−21.2474	−1.50619	−0.753096	−	0.657911i	$-0.771440\pi$
$199$	−21.2474	−1.50619	−0.753096	+	0.657911i	$0.771440\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 10.5213i	− 0.738448i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−38.9787	−2.69622
$210$	0	0
$211$	− 2.13212i	− 0.146781i	−0.997303	−	0.0733905i	$-0.976618\pi$
$211$	− 2.13212i	− 0.146781i	0.997303	−	0.0733905i	$-0.0233819\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 24.0926i	− 1.62064i
$222$	0	0
$223$	−0.348469	−0.0233352	−0.0116676	−	0.999932i	$-0.503714\pi$
$223$	−0.348469	−0.0233352	−0.0116676	+	0.999932i	$0.503714\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.42108i	0.293437i	0.989178	+	0.146719i	$0.0468712\pi$
$227$	4.42108i	0.293437i	−0.989178	+	0.146719i	$0.953129\pi$
$228$	0	0
$229$	− 5.29439i	− 0.349863i	−0.984581	−	0.174932i	$-0.944030\pi$
$229$	− 5.29439i	− 0.349863i	0.984581	−	0.174932i	$-0.0559705\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.2543	−1.58895	−0.794477	−	0.607294i	$-0.792255\pi$
$233$	−24.2543	−1.58895	−0.794477	+	0.607294i	$0.792255\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.3335	−1.89743	−0.948713	−	0.316138i	$-0.897614\pi$
$239$	−29.3335	−1.89743	−0.948713	+	0.316138i	$0.897614\pi$
$240$	0	0
$241$	23.6969	1.52645	0.763227	−	0.646130i	$-0.223614\pi$
$241$	23.6969	1.52645	0.763227	+	0.646130i	$0.223614\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−27.2474	−1.73371
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.10018i	0.385040i	0.981293	+	0.192520i	$0.0616661\pi$
$251$	6.10018i	0.385040i	−0.981293	+	0.192520i	$0.938334\pi$
$252$	0	0
$253$	− 6.32456i	− 0.397621i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.28303	−0.142412	−0.0712059	−	0.997462i	$-0.522685\pi$
$257$	−2.28303	−0.142412	−0.0712059	+	0.997462i	$0.522685\pi$
$258$	0	0
$259$	− 21.8165i	− 1.35561i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.01682	−0.556001	−0.278001	−	0.960581i	$-0.589672\pi$
$263$	−9.01682	−0.556001	−0.278001	+	0.960581i	$0.589672\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.16950i	0.254219i	0.991889	+	0.127109i	$0.0405699\pi$
$269$	4.16950i	0.254219i	−0.991889	+	0.127109i	$0.959430\pi$
$270$	0	0
$271$	−5.10102	−0.309865	−0.154932	−	0.987925i	$-0.549516\pi$
$271$	−5.10102	−0.309865	−0.154932	+	0.987925i	$0.549516\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.2681i	1.45813i	0.684446	+	0.729063i	$0.260044\pi$
$277$	24.2681i	1.45813i	−0.684446	+	0.729063i	$0.739956\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.93765	0.234901	0.117450	−	0.993079i	$-0.462528\pi$
$281$	3.93765	0.234901	0.117450	+	0.993079i	$0.462528\pi$
$282$	0	0
$283$	0.710706i	0.0422471i	0.999777	+	0.0211235i	$0.00672433\pi$
$283$	0.710706i	0.0422471i	−0.999777	+	0.0211235i	$0.993276\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.5829	0.801773
$288$	0	0
$289$	21.6969	1.27629
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 16.0617i	− 0.938337i	−0.883109	−	0.469169i	$-0.844554\pi$
$293$	− 16.0617i	− 0.938337i	0.883109	−	0.469169i	$-0.155446\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.42108i	− 0.255677i
$300$	0	0
$301$	− 2.45157i	− 0.141306i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.7812i	0.843609i	0.906687	+	0.421805i	$0.138603\pi$
$307$	14.7812i	0.843609i	−0.906687	+	0.421805i	$0.861397\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.7506	−0.893135	−0.446568	−	0.894750i	$-0.647354\pi$
$311$	−15.7506	−0.893135	−0.446568	+	0.894750i	$0.647354\pi$
$312$	0	0
$313$	22.5959	1.27720	0.638598	−	0.769540i	$-0.279515\pi$
$313$	22.5959	1.27720	0.638598	+	0.769540i	$0.279515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0425i	1.18187i	0.806721	+	0.590933i	$0.201240\pi$
$317$	21.0425i	1.18187i	−0.806721	+	0.590933i	$0.798760\pi$
$318$	0	0
$319$	16.8990	0.946161
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 43.7642i	− 2.43510i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−35.0411	−1.93188
$330$	0	0
$331$	− 12.6491i	− 0.695258i	−0.937632	−	0.347629i	$-0.886987\pi$
$331$	− 12.6491i	− 0.695258i	0.937632	−	0.347629i	$-0.113013\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.7980	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$337$	18.7980	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.03087i	− 0.434896i
$342$	0	0
$343$	7.24745	0.391325
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0617i	0.862240i	0.902295	+	0.431120i	$0.141881\pi$
$347$	16.0617i	0.862240i	−0.902295	+	0.431120i	$0.858119\pi$
$348$	0	0
$349$	− 12.6491i	− 0.677091i	−0.940950	−	0.338546i	$-0.890065\pi$
$349$	− 12.6491i	− 0.677091i	0.940950	−	0.338546i	$-0.109935\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.628417	0.0334473	0.0167236	−	0.999860i	$-0.494676\pi$
$353$	0.628417	0.0334473	0.0167236	+	0.999860i	$0.494676\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.4582	−1.13252	−0.566260	−	0.824227i	$-0.691610\pi$
$359$	−21.4582	−1.13252	−0.566260	+	0.824227i	$0.691610\pi$
$360$	0	0
$361$	−30.4949	−1.60499
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.4495	1.22405	0.612027	−	0.790837i	$-0.290354\pi$
$367$	23.4495	1.22405	0.612027	+	0.790837i	$0.290354\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	27.7024i	1.43824i
$372$	0	0
$373$	− 25.6895i	− 1.33015i	−0.746776	−	0.665075i	$-0.768399\pi$
$373$	− 25.6895i	− 1.33015i	0.746776	−	0.665075i	$-0.231601\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.8130	0.608398
$378$	0	0
$379$	− 13.3598i	− 0.686248i	−0.939290	−	0.343124i	$-0.888515\pi$
$379$	− 13.3598i	− 0.686248i	0.939290	−	0.343124i	$-0.111485\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.4414	−0.635724	−0.317862	−	0.948137i	$-0.602965\pi$
$383$	−12.4414	−0.635724	−0.317862	+	0.948137i	$0.602965\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 35.1736i	− 1.78337i	−0.452655	−	0.891686i	$-0.649523\pi$
$389$	− 35.1736i	− 1.78337i	0.452655	−	0.891686i	$-0.350477\pi$
$390$	0	0
$391$	7.10102	0.359114
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.45157i	− 0.123041i	−0.998106	−	0.0615204i	$-0.980405\pi$
$397$	− 2.45157i	− 0.123041i	0.998106	−	0.0615204i	$-0.0195949\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.1034	−1.55323	−0.776616	−	0.629975i	$-0.783065\pi$
$401$	−31.1034	−1.55323	−0.776616	+	0.629975i	$0.783065\pi$
$402$	0	0
$403$	− 5.61385i	− 0.279646i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.0411	1.73692
$408$	0	0
$409$	−7.89898	−0.390579	−0.195290	−	0.980746i	$-0.562565\pi$
$409$	−7.89898	−0.390579	−0.195290	+	0.980746i	$0.562565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.1811i	0.845429i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.8410i	1.16471i	0.812934	+	0.582355i	$0.197869\pi$
$419$	23.8410i	1.16471i	−0.812934	+	0.582355i	$0.802131\pi$
$420$	0	0
$421$	− 24.6593i	− 1.20182i	−0.799316	−	0.600911i	$-0.794805\pi$
$421$	− 24.6593i	− 1.20182i	0.799316	−	0.600911i	$-0.205195\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	35.1763i	1.70230i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.4582	−1.03360	−0.516802	−	0.856105i	$-0.672878\pi$
$431$	−21.4582	−1.03360	−0.516802	+	0.856105i	$0.672878\pi$
$432$	0	0
$433$	−12.5959	−0.605321	−0.302661	−	0.953098i	$-0.597875\pi$
$433$	−12.5959	−0.605321	−0.302661	+	0.953098i	$0.597875\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.03087i	− 0.384169i
$438$	0	0
$439$	17.0454	0.813533	0.406766	−	0.913532i	$-0.366656\pi$
$439$	17.0454	0.813533	0.406766	+	0.913532i	$0.366656\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.21959i	0.343013i	0.985183	+	0.171507i	$0.0548635\pi$
$443$	7.21959i	0.343013i	−0.985183	+	0.171507i	$0.945137\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.2905	0.910374	0.455187	−	0.890396i	$-0.349572\pi$
$449$	19.2905	0.910374	0.455187	+	0.890396i	$0.349572\pi$
$450$	0	0
$451$	21.8165i	1.02730i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.6969	−0.968162	−0.484081	−	0.875023i	$-0.660846\pi$
$457$	−20.6969	−0.968162	−0.484081	+	0.875023i	$0.660846\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 25.2120i	− 1.17424i	−0.809500	−	0.587120i	$-0.800262\pi$
$461$	− 25.2120i	− 1.17424i	0.809500	−	0.587120i	$-0.199738\pi$
$462$	0	0
$463$	6.89898	0.320623	0.160311	−	0.987066i	$-0.448750\pi$
$463$	6.89898	0.320623	0.160311	+	0.987066i	$0.448750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.6832i	1.51240i	0.654342	+	0.756199i	$0.272946\pi$
$467$	32.6832i	1.51240i	−0.654342	+	0.756199i	$0.727054\pi$
$468$	0	0
$469$	19.3649i	0.894189i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.93765	0.181053
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.87530	−0.359832	−0.179916	−	0.983682i	$-0.557583\pi$
$479$	−7.87530	−0.359832	−0.179916	+	0.983682i	$0.557583\pi$
$480$	0	0
$481$	24.4949	1.11687
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.04541	0.0473719	0.0236860	−	0.999719i	$-0.492460\pi$
$487$	1.04541	0.0473719	0.0236860	+	0.999719i	$0.492460\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.11940i	0.0505180i	0.999681	+	0.0252590i	$0.00804105\pi$
$491$	1.11940i	0.0505180i	−0.999681	+	0.0252590i	$0.991959\pi$
$492$	0	0
$493$	18.9737i	0.854531i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	46.8540	2.10169
$498$	0	0
$499$	19.6844i	0.881193i	0.897705	+	0.440597i	$0.145233\pi$
$499$	19.6844i	0.881193i	−0.897705	+	0.440597i	$0.854767\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.3073	−1.26216	−0.631080	−	0.775718i	$-0.717388\pi$
$503$	−28.3073	−1.26216	−0.631080	+	0.775718i	$0.717388\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 21.0425i	− 0.932693i	−0.884602	−	0.466347i	$-0.845570\pi$
$509$	− 21.0425i	− 0.932693i	0.884602	−	0.466347i	$-0.154430\pi$
$510$	0	0
$511$	−23.7980	−1.05276
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 56.2821i	− 2.47528i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.1034	−1.36267	−0.681333	−	0.731974i	$-0.738599\pi$
$521$	−31.1034	−1.36267	−0.681333	+	0.731974i	$0.738599\pi$
$522$	0	0
$523$	33.7549i	1.47600i	0.674802	+	0.737999i	$0.264229\pi$
$523$	33.7549i	1.47600i	−0.674802	+	0.737999i	$0.735771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.01682	0.392779
$528$	0	0
$529$	−21.6969	−0.943345
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.2505i	0.660571i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 27.1427i	− 1.16912i
$540$	0	0
$541$	8.77613i	0.377315i	0.982043	+	0.188658i	$0.0604136\pi$
$541$	8.77613i	0.377315i	−0.982043	+	0.188658i	$0.939586\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.16738i	− 0.391969i	−0.980607	−	0.195984i	$-0.937210\pi$
$547$	− 9.16738i	− 0.391969i	0.980607	−	0.195984i	$-0.0627902\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.4582	0.914150
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	2.75255	0.116421
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.7642i	1.84444i	0.386667	+	0.922220i	$0.373626\pi$
$563$	43.7642i	1.84444i	−0.386667	+	0.922220i	$0.626374\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.93765	0.165075	0.0825375	−	0.996588i	$-0.473698\pi$
$569$	3.93765	0.165075	0.0825375	+	0.996588i	$0.473698\pi$
$570$	0	0
$571$	8.45667i	0.353901i	0.984220	+	0.176950i	$0.0566232\pi$
$571$	8.45667i	0.353901i	−0.984220	+	0.176950i	$0.943377\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.6969	−0.653472	−0.326736	−	0.945116i	$-0.605949\pi$
$577$	−15.6969	−0.653472	−0.326736	+	0.945116i	$0.605949\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 21.0425i	− 0.872991i
$582$	0	0
$583$	−44.4949	−1.84279
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.6214i	0.686040i	0.939328	+	0.343020i	$0.111450\pi$
$587$	16.6214i	0.686040i	−0.939328	+	0.343020i	$0.888550\pi$
$588$	0	0
$589$	− 10.1975i	− 0.420182i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.5374	−1.08976	−0.544879	−	0.838514i	$-0.683425\pi$
$593$	−26.5374	−1.08976	−0.544879	+	0.838514i	$0.683425\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.2905	0.788187	0.394094	−	0.919070i	$-0.371059\pi$
$599$	19.2905	0.788187	0.394094	+	0.919070i	$0.371059\pi$
$600$	0	0
$601$	−0.101021	−0.00412071	−0.00206036	−	0.999998i	$-0.500656\pi$
$601$	−0.101021	−0.00412071	−0.00206036	+	0.999998i	$0.500656\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.6969	1.65184	0.825919	−	0.563789i	$-0.190657\pi$
$607$	40.6969	1.65184	0.825919	+	0.563789i	$0.190657\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 39.3431i	− 1.59165i
$612$	0	0
$613$	− 31.6228i	− 1.27723i	−0.769526	−	0.638616i	$-0.779507\pi$
$613$	− 31.6228i	− 1.27723i	0.769526	−	0.638616i	$-0.220493\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.0698	0.526170	0.263085	−	0.964773i	$-0.415260\pi$
$617$	13.0698	0.526170	0.263085	+	0.964773i	$0.415260\pi$
$618$	0	0
$619$	37.2366i	1.49667i	0.663323	+	0.748333i	$0.269146\pi$
$619$	37.2366i	1.49667i	−0.663323	+	0.748333i	$0.730854\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	27.1658	1.08837
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39.3431i	1.56871i
$630$	0	0
$631$	18.5505	0.738484	0.369242	−	0.929333i	$-0.379617\pi$
$631$	18.5505	0.738484	0.369242	+	0.929333i	$0.379617\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.9737i	− 0.751764i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	35.0411	1.38404	0.692020	−	0.721879i	$-0.256721\pi$
$641$	35.0411	1.38404	0.692020	+	0.721879i	$0.256721\pi$
$642$	0	0
$643$	34.4656i	1.35919i	0.733587	+	0.679595i	$0.237845\pi$
$643$	34.4656i	1.35919i	−0.733587	+	0.679595i	$0.762155\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.8827	0.978242	0.489121	−	0.872216i	$-0.337318\pi$
$647$	24.8827	0.978242	0.489121	+	0.872216i	$0.337318\pi$
$648$	0	0
$649$	−27.5959	−1.08323
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.10018i	− 0.238719i	−0.992851	−	0.119359i	$-0.961916\pi$
$653$	− 6.10018i	− 0.238719i	0.992851	−	0.119359i	$-0.0380840\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 22.7216i	− 0.885109i	−0.896742	−	0.442555i	$-0.854072\pi$
$659$	− 22.7216i	− 0.885109i	0.896742	−	0.442555i	$-0.145928\pi$
$660$	0	0
$661$	37.3084i	1.45113i	0.688154	+	0.725565i	$0.258421\pi$
$661$	37.3084i	1.45113i	−0.688154	+	0.725565i	$0.741579\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.48173i	0.134813i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−56.4993	−2.18113
$672$	0	0
$673$	−26.8990	−1.03688	−0.518440	−	0.855114i	$-0.673487\pi$
$673$	−26.8990	−1.03688	−0.518440	+	0.855114i	$0.673487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.1736i	1.35183i	0.736979	+	0.675915i	$0.236251\pi$
$677$	35.1736i	1.35183i	−0.736979	+	0.675915i	$0.763749\pi$
$678$	0	0
$679$	−54.1464	−2.07795
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33.2429i	− 1.27200i	−0.771687	−	0.636002i	$-0.780587\pi$
$683$	− 33.2429i	− 1.27200i	0.771687	−	0.636002i	$-0.219413\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−31.1034	−1.18495
$690$	0	0
$691$	− 37.9473i	− 1.44358i	−0.692110	−	0.721792i	$-0.743319\pi$
$691$	− 37.9473i	− 1.44358i	0.692110	−	0.721792i	$-0.256681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.4949	−0.927810
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.2738i	1.55889i	0.626472	+	0.779444i	$0.284498\pi$
$701$	41.2738i	1.55889i	−0.626472	+	0.779444i	$0.715502\pi$
$702$	0	0
$703$	44.4949	1.67816
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 48.7449i	− 1.83324i
$708$	0	0
$709$	− 1.03016i	− 0.0386885i	−0.999813	−	0.0193442i	$-0.993842\pi$
$709$	− 1.03016i	− 0.0386885i	0.999813	−	0.0193442i	$-0.00615785\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.65462	0.0619659
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.16772	0.0808422	0.0404211	−	0.999183i	$-0.487130\pi$
$719$	2.16772	0.0808422	0.0404211	+	0.999183i	$0.487130\pi$
$720$	0	0
$721$	23.7980	0.886282
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−43.4495	−1.61145	−0.805726	−	0.592288i	$-0.798225\pi$
$727$	−43.4495	−1.61145	−0.805726	+	0.592288i	$0.798225\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.42108i	0.163519i
$732$	0	0
$733$	30.9839i	1.14442i	0.820109	+	0.572208i	$0.193913\pi$
$733$	30.9839i	1.14442i	−0.820109	+	0.572208i	$0.806087\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.1034	−1.14571
$738$	0	0
$739$	30.9839i	1.13976i	0.821728	+	0.569880i	$0.193010\pi$
$739$	30.9839i	1.13976i	−0.821728	+	0.569880i	$0.806990\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.84910	0.251269	0.125635	−	0.992077i	$-0.459903\pi$
$743$	6.84910	0.251269	0.125635	+	0.992077i	$0.459903\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.93069i	0.0705458i
$750$	0	0
$751$	−35.7980	−1.30629	−0.653143	−	0.757235i	$-0.726550\pi$
$751$	−35.7980	−1.30629	−0.653143	+	0.757235i	$0.726550\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12.2579i	− 0.445519i	−0.974873	−	0.222760i	$-0.928493\pi$
$757$	− 12.2579i	− 0.445519i	0.974873	−	0.222760i	$-0.0715065\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.7506	−0.570959	−0.285480	−	0.958385i	$-0.592153\pi$
$761$	−15.7506	−0.570959	−0.285480	+	0.958385i	$0.592153\pi$
$762$	0	0
$763$	− 56.9928i	− 2.06328i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.2905	−0.696539
$768$	0	0
$769$	5.20204	0.187590	0.0937952	−	0.995592i	$-0.470100\pi$
$769$	5.20204	0.187590	0.0937952	+	0.995592i	$0.470100\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 9.15028i	− 0.329113i	−0.986368	−	0.164556i	$-0.947381\pi$
$773$	− 9.15028i	− 0.329113i	0.986368	−	0.164556i	$-0.0526192\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.7024i	0.992542i
$780$	0	0
$781$	75.2558i	2.69286i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.5170i	0.374890i	0.982275	+	0.187445i	$0.0600207\pi$
$787$	10.5170i	0.374890i	−0.982275	+	0.187445i	$0.939979\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	42.9164	1.52593
$792$	0	0
$793$	−39.4949	−1.40250
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.1427i	− 0.961444i	−0.876873	−	0.480722i	$-0.840375\pi$
$797$	− 27.1427i	− 0.961444i	0.876873	−	0.480722i	$-0.159625\pi$
$798$	0	0
$799$	63.1918	2.23557
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 38.2237i	− 1.34888i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.93765	−0.138440	−0.0692202	−	0.997601i	$-0.522051\pi$
$809$	−3.93765	−0.138440	−0.0692202	+	0.997601i	$0.522051\pi$
$810$	0	0
$811$	− 39.4405i	− 1.38494i	−0.721444	−	0.692472i	$-0.756521\pi$
$811$	− 39.4405i	− 1.38494i	0.721444	−	0.692472i	$-0.243479\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.0233i	0.908220i	0.890946	+	0.454110i	$0.150043\pi$
$821$	26.0233i	0.908220i	−0.890946	+	0.454110i	$0.849957\pi$
$822$	0	0
$823$	34.8434	1.21456	0.607282	−	0.794487i	$-0.292260\pi$
$823$	34.8434	1.21456	0.607282	+	0.794487i	$0.292260\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11.6407i	− 0.404786i	−0.979304	−	0.202393i	$-0.935128\pi$
$827$	− 11.6407i	− 0.404786i	0.979304	−	0.202393i	$-0.0648718\pi$
$828$	0	0
$829$	− 37.3084i	− 1.29578i	−0.761736	−	0.647888i	$-0.775653\pi$
$829$	− 37.3084i	− 1.29578i	0.761736	−	0.647888i	$-0.224347\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	30.4750	1.05590
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.70759	−0.197048	−0.0985239	−	0.995135i	$-0.531412\pi$
$839$	−5.70759	−0.197048	−0.0985239	+	0.995135i	$0.531412\pi$
$840$	0	0
$841$	19.6969	0.679205
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	67.9444	2.33460
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.21959i	0.247484i
$852$	0	0
$853$	− 5.29439i	− 0.181277i	−0.995884	−	0.0906383i	$-0.971109\pi$
$853$	− 5.29439i	− 0.181277i	0.995884	−	0.0906383i	$-0.0288907\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0336	−0.616017	−0.308009	−	0.951384i	$-0.599663\pi$
$857$	−18.0336	−0.616017	−0.308009	+	0.951384i	$0.599663\pi$
$858$	0	0
$859$	− 43.6330i	− 1.48874i	−0.667769	−	0.744369i	$-0.732750\pi$
$859$	− 43.6330i	− 1.48874i	0.667769	−	0.744369i	$-0.267250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.4414	0.423509	0.211755	−	0.977323i	$-0.432082\pi$
$863$	12.4414	0.423509	0.211755	+	0.977323i	$0.432082\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.0617i	0.544857i
$870$	0	0
$871$	−21.7423	−0.736711
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.7863i	0.701904i	0.936393	+	0.350952i	$0.114142\pi$
$877$	20.7863i	0.701904i	−0.936393	+	0.350952i	$0.885858\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.2905	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$881$	19.2905	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$882$	0	0
$883$	30.9121i	1.04027i	0.854083	+	0.520137i	$0.174119\pi$
$883$	30.9121i	1.04027i	−0.854083	+	0.520137i	$0.825881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.0747	−1.78207	−0.891037	−	0.453930i	$-0.850022\pi$
$887$	−53.0747	−1.78207	−0.891037	+	0.453930i	$0.850022\pi$
$888$	0	0
$889$	13.1010	0.439394
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 71.4666i	− 2.39154i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.42108i	0.147451i
$900$	0	0
$901$	− 49.9575i	− 1.66433i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	56.2821i	1.86882i	0.356204	+	0.934408i	$0.384071\pi$
$907$	56.2821i	1.86882i	−0.356204	+	0.934408i	$0.615929\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.6259	0.782761	0.391381	−	0.920229i	$-0.371998\pi$
$911$	23.6259	0.782761	0.391381	+	0.920229i	$0.371998\pi$
$912$	0	0
$913$	33.7980	1.11855
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.2930i	1.19850i
$918$	0	0
$919$	34.6413	1.14271	0.571356	−	0.820702i	$-0.306418\pi$
$919$	34.6413	1.14271	0.571356	+	0.820702i	$0.306418\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	52.6063i	1.73156i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.9787	1.27885	0.639425	−	0.768853i	$-0.279172\pi$
$929$	38.9787	1.27885	0.639425	+	0.768853i	$0.279172\pi$
$930$	0	0
$931$	− 34.4656i	− 1.12956i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.20204	0.0392690	0.0196345	−	0.999807i	$-0.493750\pi$
$937$	1.20204	0.0392690	0.0196345	+	0.999807i	$0.493750\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.4625i	1.31904i	0.751687	+	0.659520i	$0.229240\pi$
$941$	40.4625i	1.31904i	−0.751687	+	0.659520i	$0.770760\pi$
$942$	0	0
$943$	−4.49490	−0.146374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.42108i	0.143666i	0.997417	+	0.0718329i	$0.0228848\pi$
$947$	4.42108i	0.143666i	−0.997417	+	0.0718329i	$0.977115\pi$
$948$	0	0
$949$	− 26.7196i	− 0.867356i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.56607	0.147909	0.0739547	−	0.997262i	$-0.476438\pi$
$953$	4.56607	0.147909	0.0739547	+	0.997262i	$0.476438\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−35.0411	−1.13154
$960$	0	0
$961$	−28.8990	−0.932225
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.29286	0.266680	0.133340	−	0.991070i	$-0.457430\pi$
$967$	8.29286	0.266680	0.133340	+	0.991070i	$0.457430\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 45.4433i	− 1.45834i	−0.684331	−	0.729172i	$-0.739906\pi$
$971$	− 45.4433i	− 1.45834i	0.684331	−	0.729172i	$-0.260094\pi$
$972$	0	0
$973$	− 31.6228i	− 1.01378i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.7842	−1.08085	−0.540427	−	0.841391i	$-0.681737\pi$
$977$	−33.7842	−1.08085	−0.540427	+	0.841391i	$0.681737\pi$
$978$	0	0
$979$	43.6330i	1.39452i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.5997	−0.720819	−0.360409	−	0.932794i	$-0.617363\pi$
$983$	−22.5997	−0.720819	−0.360409	+	0.932794i	$0.617363\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.811283i	0.0257973i
$990$	0	0
$991$	−4.75255	−0.150970	−0.0754849	−	0.997147i	$-0.524050\pi$
$991$	−4.75255	−0.150970	−0.0754849	+	0.997147i	$0.524050\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.6491i	0.400601i	0.979734	+	0.200301i	$0.0641919\pi$
$997$	12.6491i	0.400601i	−0.979734	+	0.200301i	$0.935808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.q.3601.1		8
3.2	odd	2	inner	7200.2.k.q.3601.3		8
4.3	odd	2		1800.2.k.s.901.2	yes	8
5.2	odd	4		7200.2.d.u.2449.1		16
5.3	odd	4		7200.2.d.u.2449.14		16
5.4	even	2		7200.2.k.t.3601.6		8
8.3	odd	2		1800.2.k.s.901.1	yes	8
8.5	even	2	inner	7200.2.k.q.3601.4		8
12.11	even	2		1800.2.k.s.901.7	yes	8
15.2	even	4		7200.2.d.u.2449.3		16
15.8	even	4		7200.2.d.u.2449.16		16
15.14	odd	2		7200.2.k.t.3601.8		8
20.3	even	4		1800.2.d.u.1549.11		16
20.7	even	4		1800.2.d.u.1549.6		16
20.19	odd	2		1800.2.k.r.901.7	yes	8
24.5	odd	2	inner	7200.2.k.q.3601.2		8
24.11	even	2		1800.2.k.s.901.8	yes	8
40.3	even	4		1800.2.d.u.1549.7		16
40.13	odd	4		7200.2.d.u.2449.15		16
40.19	odd	2		1800.2.k.r.901.8	yes	8
40.27	even	4		1800.2.d.u.1549.10		16
40.29	even	2		7200.2.k.t.3601.7		8
40.37	odd	4		7200.2.d.u.2449.4		16
60.23	odd	4		1800.2.d.u.1549.5		16
60.47	odd	4		1800.2.d.u.1549.12		16
60.59	even	2		1800.2.k.r.901.2	yes	8
120.29	odd	2		7200.2.k.t.3601.5		8
120.53	even	4		7200.2.d.u.2449.13		16
120.59	even	2		1800.2.k.r.901.1	✓	8
120.77	even	4		7200.2.d.u.2449.2		16
120.83	odd	4		1800.2.d.u.1549.9		16
120.107	odd	4		1800.2.d.u.1549.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.2.d.u.1549.5		16	60.23	odd	4
1800.2.d.u.1549.6		16	20.7	even	4
1800.2.d.u.1549.7		16	40.3	even	4
1800.2.d.u.1549.8		16	120.107	odd	4
1800.2.d.u.1549.9		16	120.83	odd	4
1800.2.d.u.1549.10		16	40.27	even	4
1800.2.d.u.1549.11		16	20.3	even	4
1800.2.d.u.1549.12		16	60.47	odd	4
1800.2.k.r.901.1	✓	8	120.59	even	2
1800.2.k.r.901.2	yes	8	60.59	even	2
1800.2.k.r.901.7	yes	8	20.19	odd	2
1800.2.k.r.901.8	yes	8	40.19	odd	2
1800.2.k.s.901.1	yes	8	8.3	odd	2
1800.2.k.s.901.2	yes	8	4.3	odd	2
1800.2.k.s.901.7	yes	8	12.11	even	2
1800.2.k.s.901.8	yes	8	24.11	even	2
7200.2.d.u.2449.1		16	5.2	odd	4
7200.2.d.u.2449.2		16	120.77	even	4
7200.2.d.u.2449.3		16	15.2	even	4
7200.2.d.u.2449.4		16	40.37	odd	4
7200.2.d.u.2449.13		16	120.53	even	4
7200.2.d.u.2449.14		16	5.3	odd	4
7200.2.d.u.2449.15		16	40.13	odd	4
7200.2.d.u.2449.16		16	15.8	even	4
7200.2.k.q.3601.1		8	1.1	even	1	trivial
7200.2.k.q.3601.2		8	24.5	odd	2	inner
7200.2.k.q.3601.3		8	3.2	odd	2	inner
7200.2.k.q.3601.4		8	8.5	even	2	inner
7200.2.k.t.3601.5		8	120.29	odd	2
7200.2.k.t.3601.6		8	5.4	even	2
7200.2.k.t.3601.7		8	40.29	even	2
7200.2.k.t.3601.8		8	15.14	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.44949 q^{7} +O(q^{10})\)	\(q-3.44949 q^{7} -5.54048i q^{11} -3.87298i q^{13} +6.22069 q^{17} -7.03526i q^{19} +1.14152 q^{23} +3.05009i q^{29} +1.44949 q^{31} +6.32456i q^{37} -3.93765 q^{41} +0.710706i q^{43} +10.1583 q^{47} +4.89898 q^{49} -8.03087i q^{53} -4.98078i q^{59} -10.1975i q^{61} -5.61385i q^{67} -13.5829 q^{71} +6.89898 q^{73} +19.1118i q^{77} -2.89898 q^{79} +6.10018i q^{83} -7.87530 q^{89} +13.3598i q^{91} +15.6969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 8 q^{31} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^31 + 16 * q^73 + 16 * q^79 + 8 * q^97

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