LMFDB - Embedded newform 7200.2.f.bh.6049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(6049,7200)]

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, 0, 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("7200.6049");

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.f (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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.6049
Dual form			7200.2.f.bh.6049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{7} +O(q^{10})$	$q-2.82843i q^{7} +5.65685 q^{11} +2.00000i q^{13} -2.00000i q^{17} +2.82843i q^{23} +6.00000 q^{29} +5.65685 q^{31} -10.0000i q^{37} -2.00000 q^{41} +8.48528i q^{43} +2.82843i q^{47} -1.00000 q^{49} +6.00000i q^{53} +11.3137 q^{59} -2.00000 q^{61} -2.82843i q^{67} -5.65685 q^{71} +6.00000i q^{73} -16.0000i q^{77} -11.3137 q^{79} +2.82843i q^{83} +10.0000 q^{89} +5.65685 q^{91} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{29} - 8 q^{41} - 4 q^{49} - 8 q^{61} + 40 q^{89}+O(q^{100}) $ 4 * q + 24 * q^29 - 8 * q^41 - 4 * q^49 - 8 * q^61 + 40 * q^89

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
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.48528i	1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$	8.48528i	1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843i	0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$	2.82843i	0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.82843i	− 0.345547i	−0.984962	−	0.172774i	$-0.944727\pi$
$67$	− 2.82843i	− 0.345547i	0.984962	−	0.172774i	$-0.0552729\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.82843i	0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$	2.82843i	0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 14.1421i	− 1.39347i	−0.717331	−	0.696733i	$-0.754636\pi$
$103$	− 14.1421i	− 1.39347i	0.717331	−	0.696733i	$-0.245364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.82843i	− 0.250982i	−0.992095	−	0.125491i	$-0.959949\pi$
$127$	− 2.82843i	− 0.250982i	0.992095	−	0.125491i	$-0.0400507\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.3137i	0.946100i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−16.9706	−1.38104	−0.690522	−	0.723311i	$-0.742619\pi$
$151$	−16.9706	−1.38104	−0.690522	+	0.723311i	$0.742619\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 14.1421i	− 1.10770i	−0.832617	−	0.553849i	$-0.813159\pi$
$163$	− 14.1421i	− 1.10770i	0.832617	−	0.553849i	$-0.186841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.1421i	1.09435i	0.837018	+	0.547176i	$0.184297\pi$
$167$	14.1421i	1.09435i	−0.837018	+	0.547176i	$0.815703\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.3137i	− 0.827340i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	22.6274	1.60402	0.802008	−	0.597314i	$-0.203765\pi$
$199$	22.6274	1.60402	0.802008	+	0.597314i	$0.203765\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 16.9706i	− 1.19110i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 16.0000i	− 1.08615i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	8.48528i	0.568216i	0.958792	+	0.284108i	$0.0916975\pi$
$223$	8.48528i	0.568216i	−0.958792	+	0.284108i	$0.908302\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.7990i	− 1.31411i	−0.753845	−	0.657053i	$-0.771803\pi$
$227$	− 19.7990i	− 1.31411i	0.753845	−	0.657053i	$-0.228197\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	−28.2843	−1.75750
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 19.7990i	− 1.22086i	−0.792071	−	0.610429i	$-0.790997\pi$
$263$	− 19.7990i	− 1.22086i	0.792071	−	0.610429i	$-0.209003\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	8.48528i	0.504398i	0.967675	+	0.252199i	$0.0811537\pi$
$283$	8.48528i	0.504398i	−0.967675	+	0.252199i	$0.918846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685i	0.333914i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.65685	−0.327144
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.82843i	− 0.161427i	−0.996737	−	0.0807134i	$-0.974280\pi$
$307$	− 2.82843i	− 0.161427i	0.996737	−	0.0807134i	$-0.0257199\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	33.9411	1.90034
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	5.65685	0.310929	0.155464	−	0.987841i	$-0.450313\pi$
$331$	5.65685	0.310929	0.155464	+	0.987841i	$0.450313\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.48528i	− 0.455514i	−0.973718	−	0.227757i	$-0.926861\pi$
$347$	− 8.48528i	− 0.455514i	0.973718	−	0.227757i	$-0.0731391\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.6274	1.19423	0.597115	−	0.802156i	$-0.296314\pi$
$359$	22.6274	1.19423	0.597115	+	0.802156i	$0.296314\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.48528i	0.442928i	0.975169	+	0.221464i	$0.0710835\pi$
$367$	8.48528i	0.442928i	−0.975169	+	0.221464i	$0.928916\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.9706	0.881068
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	22.6274	1.16229	0.581146	−	0.813799i	$-0.302604\pi$
$379$	22.6274	1.16229	0.581146	+	0.813799i	$0.302604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.7696i	1.87884i	0.342773	+	0.939418i	$0.388634\pi$
$383$	36.7696i	1.87884i	−0.342773	+	0.939418i	$0.611366\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	11.3137i	0.563576i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 56.5685i	− 2.80400i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 32.0000i	− 1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.3137	−0.552711	−0.276355	−	0.961056i	$-0.589127\pi$
$419$	−11.3137	−0.552711	−0.276355	+	0.961056i	$0.589127\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.65685i	0.273754i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.65685	0.272481	0.136241	−	0.990676i	$-0.456498\pi$
$431$	5.65685	0.272481	0.136241	+	0.990676i	$0.456498\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.82843i	0.134383i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	2.82843i	0.134383i	−0.997740	+	0.0671913i	$0.978596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	8.48528i	0.394344i	0.980369	+	0.197172i	$0.0631758\pi$
$463$	8.48528i	0.394344i	−0.980369	+	0.197172i	$0.936824\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.1421i	0.654420i	0.944952	+	0.327210i	$0.106108\pi$
$467$	14.1421i	0.654420i	−0.944952	+	0.327210i	$0.893892\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.9411	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$479$	−33.9411	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.1127i	1.40985i	0.709281	+	0.704925i	$0.249020\pi$
$487$	31.1127i	1.40985i	−0.709281	+	0.704925i	$0.750980\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.5980	−1.78703	−0.893516	−	0.449032i	$-0.851769\pi$
$491$	−39.5980	−1.78703	−0.893516	+	0.449032i	$0.851769\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.0000i	0.717698i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 8.48528i	− 0.378340i	−0.981944	−	0.189170i	$-0.939420\pi$
$503$	− 8.48528i	− 0.378340i	0.981944	−	0.189170i	$-0.0605797\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	16.9706	0.750733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	8.48528i	0.371035i	0.982641	+	0.185518i	$0.0593962\pi$
$523$	8.48528i	0.371035i	−0.982641	+	0.185518i	$0.940604\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11.3137i	− 0.492833i
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.65685	−0.243658
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	42.4264i	1.81402i	0.421107	+	0.907011i	$0.361642\pi$
$547$	42.4264i	1.81402i	−0.421107	+	0.907011i	$0.638358\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000i	1.36078i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 19.7990i	− 0.834428i	−0.908808	−	0.417214i	$-0.863007\pi$
$563$	− 19.7990i	− 0.834428i	0.908808	−	0.417214i	$-0.136993\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	28.2843	1.18366	0.591830	−	0.806063i	$-0.298406\pi$
$571$	28.2843	1.18366	0.591830	+	0.806063i	$0.298406\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	33.9411i	1.40570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.4558i	1.05068i	0.850894	+	0.525338i	$0.176061\pi$
$587$	25.4558i	1.05068i	−0.850894	+	0.525338i	$0.823939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.3137	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.82843i	− 0.114802i	−0.998351	−	0.0574012i	$-0.981719\pi$
$607$	− 2.82843i	− 0.114802i	0.998351	−	0.0574012i	$-0.0182814\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.65685	−0.228852
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	45.2548	1.81895	0.909473	−	0.415764i	$-0.136486\pi$
$619$	45.2548	1.81895	0.909473	+	0.415764i	$0.136486\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 28.2843i	− 1.13319i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−16.9706	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	31.1127i	1.22697i	0.789708	+	0.613483i	$0.210232\pi$
$643$	31.1127i	1.22697i	−0.789708	+	0.613483i	$0.789768\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.1421i	0.555985i	0.960583	+	0.277992i	$0.0896690\pi$
$647$	14.1421i	0.555985i	−0.960583	+	0.277992i	$0.910331\pi$
$648$	0	0
$649$	64.0000	2.51222
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.9411	−1.32216	−0.661079	−	0.750316i	$-0.729901\pi$
$659$	−33.9411	−1.32216	−0.661079	+	0.750316i	$0.729901\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.9706i	0.657103i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	5.65685	0.217090
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.1421i	0.541134i	0.962701	+	0.270567i	$0.0872111\pi$
$683$	14.1421i	0.541134i	−0.962701	+	0.270567i	$0.912789\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.2843	−1.07598	−0.537992	−	0.842950i	$-0.680817\pi$
$691$	−28.2843	−1.07598	−0.537992	+	0.842950i	$0.680817\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.65685i	− 0.212748i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.3137	−0.421930	−0.210965	−	0.977494i	$-0.567661\pi$
$719$	−11.3137	−0.421930	−0.210965	+	0.977494i	$0.567661\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 25.4558i	− 0.944105i	−0.881570	−	0.472052i	$-0.843513\pi$
$727$	− 25.4558i	− 0.944105i	0.881570	−	0.472052i	$-0.156487\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.9706	0.627679
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	11.3137	0.416181	0.208091	−	0.978110i	$-0.433275\pi$
$739$	11.3137	0.416181	0.208091	+	0.978110i	$0.433275\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.1421i	0.518825i	0.965767	+	0.259412i	$0.0835289\pi$
$743$	14.1421i	0.518825i	−0.965767	+	0.259412i	$0.916471\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	−16.9706	−0.619265	−0.309632	−	0.950856i	$-0.600206\pi$
$751$	−16.9706	−0.619265	−0.309632	+	0.950856i	$0.600206\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	− 50.9117i	− 1.84313i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.6274i	0.817029i
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.48528i	0.302468i	0.988498	+	0.151234i	$0.0483246\pi$
$787$	8.48528i	0.302468i	−0.988498	+	0.151234i	$0.951675\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	0	0
$801$	0	0
$802$	0	0
$803$	33.9411i	1.19776i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 25.4558i	− 0.887335i	−0.896191	−	0.443667i	$-0.853677\pi$
$823$	− 25.4558i	− 0.887335i	0.896191	−	0.443667i	$-0.146323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31.1127i	− 1.08189i	−0.841057	−	0.540947i	$-0.818066\pi$
$827$	− 31.1127i	− 1.08189i	0.841057	−	0.540947i	$-0.181934\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.3137	0.390593	0.195296	−	0.980744i	$-0.437433\pi$
$839$	11.3137	0.390593	0.195296	+	0.980744i	$0.437433\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 59.3970i	− 2.04090i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.2843	0.969572
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−33.9411	−1.15806	−0.579028	−	0.815308i	$-0.696568\pi$
$859$	−33.9411	−1.15806	−0.579028	+	0.815308i	$0.696568\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 42.4264i	− 1.44421i	−0.691783	−	0.722106i	$-0.743174\pi$
$863$	− 42.4264i	− 1.44421i	0.691783	−	0.722106i	$-0.256826\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	5.65685	0.191675
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	− 2.82843i	− 0.0951842i	−0.998867	−	0.0475921i	$-0.984845\pi$
$883$	− 2.82843i	− 0.0951842i	0.998867	−	0.0475921i	$-0.0151548\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.82843i	0.0949693i	0.998872	+	0.0474846i	$0.0151205\pi$
$887$	2.82843i	0.0949693i	−0.998872	+	0.0474846i	$0.984879\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.9411	1.13200
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 25.4558i	− 0.845247i	−0.906305	−	0.422624i	$-0.861109\pi$
$907$	− 25.4558i	− 0.845247i	0.906305	−	0.422624i	$-0.138891\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000i	0.528367i
$918$	0	0
$919$	−33.9411	−1.11961	−0.559807	−	0.828623i	$-0.689125\pi$
$919$	−33.9411	−1.11961	−0.559807	+	0.828623i	$0.689125\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 11.3137i	− 0.372395i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	− 5.65685i	− 0.184213i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.4264i	− 1.37867i	−0.724441	−	0.689336i	$-0.757902\pi$
$947$	− 42.4264i	− 1.37867i	0.724441	−	0.689336i	$-0.242098\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.9706	0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	42.4264i	1.36434i	0.731193	+	0.682171i	$0.238964\pi$
$967$	42.4264i	1.36434i	−0.731193	+	0.682171i	$0.761036\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28.2843	−0.907685	−0.453843	−	0.891082i	$-0.649947\pi$
$971$	−28.2843	−0.907685	−0.453843	+	0.891082i	$0.649947\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	56.5685	1.80794
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.82843i	0.0902128i	0.998982	+	0.0451064i	$0.0143627\pi$
$983$	2.82843i	0.0902128i	−0.998982	+	0.0451064i	$0.985637\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−16.9706	−0.539088	−0.269544	−	0.962988i	$-0.586873\pi$
$991$	−16.9706	−0.539088	−0.269544	+	0.962988i	$0.586873\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.f.bh.6049.2		4
3.2	odd	2		800.2.c.f.449.1		4
4.3	odd	2	inner	7200.2.f.bh.6049.3		4
5.2	odd	4		7200.2.a.cm.1.2		2
5.3	odd	4		1440.2.a.o.1.1		2
5.4	even	2	inner	7200.2.f.bh.6049.4		4
12.11	even	2		800.2.c.f.449.4		4
15.2	even	4		800.2.a.m.1.1		2
15.8	even	4		160.2.a.c.1.2	yes	2
15.14	odd	2		800.2.c.f.449.3		4
20.3	even	4		1440.2.a.o.1.2		2
20.7	even	4		7200.2.a.cm.1.1		2
20.19	odd	2	inner	7200.2.f.bh.6049.1		4
24.5	odd	2		1600.2.c.n.449.4		4
24.11	even	2		1600.2.c.n.449.1		4
40.3	even	4		2880.2.a.bk.1.2		2
40.13	odd	4		2880.2.a.bk.1.1		2
60.23	odd	4		160.2.a.c.1.1	✓	2
60.47	odd	4		800.2.a.m.1.2		2
60.59	even	2		800.2.c.f.449.2		4
105.83	odd	4		7840.2.a.bf.1.1		2
120.29	odd	2		1600.2.c.n.449.2		4
120.53	even	4		320.2.a.g.1.1		2
120.59	even	2		1600.2.c.n.449.3		4
120.77	even	4		1600.2.a.bc.1.2		2
120.83	odd	4		320.2.a.g.1.2		2
120.107	odd	4		1600.2.a.bc.1.1		2
240.53	even	4		1280.2.d.l.641.2		4
240.83	odd	4		1280.2.d.l.641.1		4
240.173	even	4		1280.2.d.l.641.3		4
240.203	odd	4		1280.2.d.l.641.4		4
420.83	even	4		7840.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.c.1.1	✓	2	60.23	odd	4
160.2.a.c.1.2	yes	2	15.8	even	4
320.2.a.g.1.1		2	120.53	even	4
320.2.a.g.1.2		2	120.83	odd	4
800.2.a.m.1.1		2	15.2	even	4
800.2.a.m.1.2		2	60.47	odd	4
800.2.c.f.449.1		4	3.2	odd	2
800.2.c.f.449.2		4	60.59	even	2
800.2.c.f.449.3		4	15.14	odd	2
800.2.c.f.449.4		4	12.11	even	2
1280.2.d.l.641.1		4	240.83	odd	4
1280.2.d.l.641.2		4	240.53	even	4
1280.2.d.l.641.3		4	240.173	even	4
1280.2.d.l.641.4		4	240.203	odd	4
1440.2.a.o.1.1		2	5.3	odd	4
1440.2.a.o.1.2		2	20.3	even	4
1600.2.a.bc.1.1		2	120.107	odd	4
1600.2.a.bc.1.2		2	120.77	even	4
1600.2.c.n.449.1		4	24.11	even	2
1600.2.c.n.449.2		4	120.29	odd	2
1600.2.c.n.449.3		4	120.59	even	2
1600.2.c.n.449.4		4	24.5	odd	2
2880.2.a.bk.1.1		2	40.13	odd	4
2880.2.a.bk.1.2		2	40.3	even	4
7200.2.a.cm.1.1		2	20.7	even	4
7200.2.a.cm.1.2		2	5.2	odd	4
7200.2.f.bh.6049.1		4	20.19	odd	2	inner
7200.2.f.bh.6049.2		4	1.1	even	1	trivial
7200.2.f.bh.6049.3		4	4.3	odd	2	inner
7200.2.f.bh.6049.4		4	5.4	even	2	inner
7840.2.a.bf.1.1		2	105.83	odd	4
7840.2.a.bf.1.2		2	420.83	even	4

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{7} +O(q^{10})\)	\(q-2.82843i q^{7} +5.65685 q^{11} +2.00000i q^{13} -2.00000i q^{17} +2.82843i q^{23} +6.00000 q^{29} +5.65685 q^{31} -10.0000i q^{37} -2.00000 q^{41} +8.48528i q^{43} +2.82843i q^{47} -1.00000 q^{49} +6.00000i q^{53} +11.3137 q^{59} -2.00000 q^{61} -2.82843i q^{67} -5.65685 q^{71} +6.00000i q^{73} -16.0000i q^{77} -11.3137 q^{79} +2.82843i q^{83} +10.0000 q^{89} +5.65685 q^{91} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{29} - 8 q^{41} - 4 q^{49} - 8 q^{61} + 40 q^{89}+O(q^{100}) \) 4 * q + 24 * q^29 - 8 * q^41 - 4 * q^49 - 8 * q^61 + 40 * q^89

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