LMFDB - Embedded newform 108.3.c.b.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,3,Mod(53,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("108.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	108.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.94278685509$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	108.53
Dual form			108.3.c.b.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{5} -7.00000 q^{7} +O(q^{10})$	$q-9.00000i q^{5} -7.00000 q^{7} -9.00000i q^{11} +14.0000 q^{13} -18.0000i q^{17} +8.00000 q^{19} +36.0000i q^{23} -56.0000 q^{25} +18.0000i q^{29} +35.0000 q^{31} +63.0000i q^{35} +44.0000 q^{37} -36.0000i q^{41} -22.0000 q^{43} +54.0000i q^{47} -9.00000i q^{53} -81.0000 q^{55} -18.0000i q^{59} +20.0000 q^{61} -126.000i q^{65} +14.0000 q^{67} -126.000i q^{71} +89.0000 q^{73} +63.0000i q^{77} +110.000 q^{79} -27.0000i q^{83} -162.000 q^{85} -18.0000i q^{89} -98.0000 q^{91} -72.0000i q^{95} +11.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7}+O(q^{10}) $ 2 * q - 14 * q^7	$ 2 q - 14 q^{7} + 28 q^{13} + 16 q^{19} - 112 q^{25} + 70 q^{31} + 88 q^{37} - 44 q^{43} - 162 q^{55} + 40 q^{61} + 28 q^{67} + 178 q^{73} + 220 q^{79} - 324 q^{85} - 196 q^{91} + 22 q^{97}+O(q^{100}) $ 2 * q - 14 * q^7 + 28 * q^13 + 16 * q^19 - 112 * q^25 + 70 * q^31 + 88 * q^37 - 44 * q^43 - 162 * q^55 + 40 * q^61 + 28 * q^67 + 178 * q^73 + 220 * q^79 - 324 * q^85 - 196 * q^91 + 22 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$1$

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
$5$	− 9.00000i	− 1.80000i	−0.435890	−	0.900000i	$-0.643566\pi$
$5$	− 9.00000i	− 1.80000i	0.435890	−	0.900000i	$-0.356434\pi$
$6$	0	0
$7$	−7.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$7$	−7.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 9.00000i	− 0.818182i	−0.912494	−	0.409091i	$-0.865846\pi$
$11$	− 9.00000i	− 0.818182i	0.912494	−	0.409091i	$-0.134154\pi$
$12$	0	0
$13$	14.0000	1.07692	0.538462	−	0.842650i	$-0.319006\pi$
$13$	14.0000	1.07692	0.538462	+	0.842650i	$0.319006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.0000i	− 1.05882i	−0.848365	−	0.529412i	$-0.822413\pi$
$17$	− 18.0000i	− 1.05882i	0.848365	−	0.529412i	$-0.177587\pi$
$18$	0	0
$19$	8.00000	0.421053	0.210526	−	0.977588i	$-0.432482\pi$
$19$	8.00000	0.421053	0.210526	+	0.977588i	$0.432482\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.0000i	1.56522i	0.622514	+	0.782609i	$0.286111\pi$
$23$	36.0000i	1.56522i	−0.622514	+	0.782609i	$0.713889\pi$
$24$	0	0
$25$	−56.0000	−2.24000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	18.0000i	0.620690i	0.950624	+	0.310345i	$0.100445\pi$
$29$	18.0000i	0.620690i	−0.950624	+	0.310345i	$0.899555\pi$
$30$	0	0
$31$	35.0000	1.12903	0.564516	−	0.825422i	$-0.309063\pi$
$31$	35.0000	1.12903	0.564516	+	0.825422i	$0.309063\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	63.0000i	1.80000i
$36$	0	0
$37$	44.0000	1.18919	0.594595	−	0.804026i	$-0.297313\pi$
$37$	44.0000	1.18919	0.594595	+	0.804026i	$0.297313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 36.0000i	− 0.878049i	−0.898475	−	0.439024i	$-0.855324\pi$
$41$	− 36.0000i	− 0.878049i	0.898475	−	0.439024i	$-0.144676\pi$
$42$	0	0
$43$	−22.0000	−0.511628	−0.255814	−	0.966726i	$-0.582343\pi$
$43$	−22.0000	−0.511628	−0.255814	+	0.966726i	$0.582343\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	54.0000i	1.14894i	0.818527	+	0.574468i	$0.194791\pi$
$47$	54.0000i	1.14894i	−0.818527	+	0.574468i	$0.805209\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.00000i	− 0.169811i	−0.996389	−	0.0849057i	$-0.972941\pi$
$53$	− 9.00000i	− 0.169811i	0.996389	−	0.0849057i	$-0.0270589\pi$
$54$	0	0
$55$	−81.0000	−1.47273
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 18.0000i	− 0.305085i	−0.988297	−	0.152542i	$-0.951254\pi$
$59$	− 18.0000i	− 0.305085i	0.988297	−	0.152542i	$-0.0487461\pi$
$60$	0	0
$61$	20.0000	0.327869	0.163934	−	0.986471i	$-0.447581\pi$
$61$	20.0000	0.327869	0.163934	+	0.986471i	$0.447581\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 126.000i	− 1.93846i
$66$	0	0
$67$	14.0000	0.208955	0.104478	−	0.994527i	$-0.466683\pi$
$67$	14.0000	0.208955	0.104478	+	0.994527i	$0.466683\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 126.000i	− 1.77465i	−0.461147	−	0.887324i	$-0.652562\pi$
$71$	− 126.000i	− 1.77465i	0.461147	−	0.887324i	$-0.347438\pi$
$72$	0	0
$73$	89.0000	1.21918	0.609589	−	0.792718i	$-0.291334\pi$
$73$	89.0000	1.21918	0.609589	+	0.792718i	$0.291334\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	63.0000i	0.818182i
$78$	0	0
$79$	110.000	1.39241	0.696203	−	0.717845i	$-0.254872\pi$
$79$	110.000	1.39241	0.696203	+	0.717845i	$0.254872\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 27.0000i	− 0.325301i	−0.986684	−	0.162651i	$-0.947996\pi$
$83$	− 27.0000i	− 0.325301i	0.986684	−	0.162651i	$-0.0520043\pi$
$84$	0	0
$85$	−162.000	−1.90588
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 18.0000i	− 0.202247i	−0.994874	−	0.101124i	$-0.967756\pi$
$89$	− 18.0000i	− 0.202247i	0.994874	−	0.101124i	$-0.0322438\pi$
$90$	0	0
$91$	−98.0000	−1.07692
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 72.0000i	− 0.757895i
$96$	0	0
$97$	11.0000	0.113402	0.0567010	−	0.998391i	$-0.481942\pi$
$97$	11.0000	0.113402	0.0567010	+	0.998391i	$0.481942\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	63.0000i	0.623762i	0.950121	+	0.311881i	$0.100959\pi$
$101$	63.0000i	0.623762i	−0.950121	+	0.311881i	$0.899041\pi$
$102$	0	0
$103$	−22.0000	−0.213592	−0.106796	−	0.994281i	$-0.534059\pi$
$103$	−22.0000	−0.213592	−0.106796	+	0.994281i	$0.534059\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	99.0000i	0.925234i	0.886558	+	0.462617i	$0.153089\pi$
$107$	99.0000i	0.925234i	−0.886558	+	0.462617i	$0.846911\pi$
$108$	0	0
$109$	−52.0000	−0.477064	−0.238532	−	0.971135i	$-0.576666\pi$
$109$	−52.0000	−0.477064	−0.238532	+	0.971135i	$0.576666\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	324.000	2.81739
$116$	0	0
$117$	0	0
$118$	0	0
$119$	126.000i	1.05882i
$120$	0	0
$121$	40.0000	0.330579
$122$	0	0
$123$	0	0
$124$	0	0
$125$	279.000i	2.23200i
$126$	0	0
$127$	−97.0000	−0.763780	−0.381890	−	0.924208i	$-0.624727\pi$
$127$	−97.0000	−0.763780	−0.381890	+	0.924208i	$0.624727\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 63.0000i	− 0.480916i	−0.970660	−	0.240458i	$-0.922702\pi$
$131$	− 63.0000i	− 0.480916i	0.970660	−	0.240458i	$-0.0772976\pi$
$132$	0	0
$133$	−56.0000	−0.421053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	126.000i	0.919708i	0.887995	+	0.459854i	$0.152098\pi$
$137$	126.000i	0.919708i	−0.887995	+	0.459854i	$0.847902\pi$
$138$	0	0
$139$	−220.000	−1.58273	−0.791367	−	0.611341i	$-0.790630\pi$
$139$	−220.000	−1.58273	−0.791367	+	0.611341i	$0.790630\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 126.000i	− 0.881119i
$144$	0	0
$145$	162.000	1.11724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	279.000i	1.87248i	0.351357	+	0.936242i	$0.385720\pi$
$149$	279.000i	1.87248i	−0.351357	+	0.936242i	$0.614280\pi$
$150$	0	0
$151$	−151.000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−151.000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 315.000i	− 2.03226i
$156$	0	0
$157$	44.0000	0.280255	0.140127	−	0.990133i	$-0.455249\pi$
$157$	44.0000	0.280255	0.140127	+	0.990133i	$0.455249\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 252.000i	− 1.56522i
$162$	0	0
$163$	−82.0000	−0.503067	−0.251534	−	0.967849i	$-0.580935\pi$
$163$	−82.0000	−0.503067	−0.251534	+	0.967849i	$0.580935\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	162.000i	0.970060i	0.874498	+	0.485030i	$0.161191\pi$
$167$	162.000i	0.970060i	−0.874498	+	0.485030i	$0.838809\pi$
$168$	0	0
$169$	27.0000	0.159763
$170$	0	0
$171$	0	0
$172$	0	0
$173$	189.000i	1.09249i	0.837627	+	0.546243i	$0.183942\pi$
$173$	189.000i	1.09249i	−0.837627	+	0.546243i	$0.816058\pi$
$174$	0	0
$175$	392.000	2.24000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 63.0000i	− 0.351955i	−0.984394	−	0.175978i	$-0.943691\pi$
$179$	− 63.0000i	− 0.351955i	0.984394	−	0.175978i	$-0.0563086\pi$
$180$	0	0
$181$	224.000	1.23757	0.618785	−	0.785561i	$-0.287625\pi$
$181$	224.000	1.23757	0.618785	+	0.785561i	$0.287625\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 396.000i	− 2.14054i
$186$	0	0
$187$	−162.000	−0.866310
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 342.000i	− 1.79058i	−0.445488	−	0.895288i	$-0.646970\pi$
$191$	− 342.000i	− 1.79058i	0.445488	−	0.895288i	$-0.353030\pi$
$192$	0	0
$193$	−193.000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−193.000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 45.0000i	− 0.228426i	−0.993456	−	0.114213i	$-0.963565\pi$
$197$	− 45.0000i	− 0.228426i	0.993456	−	0.114213i	$-0.0364347\pi$
$198$	0	0
$199$	−91.0000	−0.457286	−0.228643	−	0.973510i	$-0.573429\pi$
$199$	−91.0000	−0.457286	−0.228643	+	0.973510i	$0.573429\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 126.000i	− 0.620690i
$204$	0	0
$205$	−324.000	−1.58049
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 72.0000i	− 0.344498i
$210$	0	0
$211$	140.000	0.663507	0.331754	−	0.943366i	$-0.392360\pi$
$211$	140.000	0.663507	0.331754	+	0.943366i	$0.392360\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	198.000i	0.920930i
$216$	0	0
$217$	−245.000	−1.12903
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 252.000i	− 1.14027i
$222$	0	0
$223$	14.0000	0.0627803	0.0313901	−	0.999507i	$-0.490007\pi$
$223$	14.0000	0.0627803	0.0313901	+	0.999507i	$0.490007\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	378.000i	1.66520i	0.553876	+	0.832599i	$0.313148\pi$
$227$	378.000i	1.66520i	−0.553876	+	0.832599i	$0.686852\pi$
$228$	0	0
$229$	350.000	1.52838	0.764192	−	0.644989i	$-0.223138\pi$
$229$	350.000	1.52838	0.764192	+	0.644989i	$0.223138\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	126.000i	0.540773i	0.962752	+	0.270386i	$0.0871514\pi$
$233$	126.000i	0.540773i	−0.962752	+	0.270386i	$0.912849\pi$
$234$	0	0
$235$	486.000	2.06809
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−322.000	−1.33610	−0.668050	−	0.744117i	$-0.732871\pi$
$241$	−322.000	−1.33610	−0.668050	+	0.744117i	$0.732871\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	112.000	0.453441
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 90.0000i	− 0.358566i	−0.983798	−	0.179283i	$-0.942622\pi$
$251$	− 90.0000i	− 0.358566i	0.983798	−	0.179283i	$-0.0573777\pi$
$252$	0	0
$253$	324.000	1.28063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	378.000i	1.47082i	0.677624	+	0.735409i	$0.263010\pi$
$257$	378.000i	1.47082i	−0.677624	+	0.735409i	$0.736990\pi$
$258$	0	0
$259$	−308.000	−1.18919
$260$	0	0
$261$	0	0
$262$	0	0
$263$	252.000i	0.958175i	0.877767	+	0.479087i	$0.159032\pi$
$263$	252.000i	0.958175i	−0.877767	+	0.479087i	$0.840968\pi$
$264$	0	0
$265$	−81.0000	−0.305660
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000i	0.0669145i	0.999440	+	0.0334572i	$0.0106518\pi$
$269$	18.0000i	0.0669145i	−0.999440	+	0.0334572i	$0.989348\pi$
$270$	0	0
$271$	−433.000	−1.59779	−0.798893	−	0.601473i	$-0.794581\pi$
$271$	−433.000	−1.59779	−0.798893	+	0.601473i	$0.794581\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	504.000i	1.83273i
$276$	0	0
$277$	284.000	1.02527	0.512635	−	0.858606i	$-0.328669\pi$
$277$	284.000	1.02527	0.512635	+	0.858606i	$0.328669\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 522.000i	− 1.85765i	−0.370517	−	0.928826i	$-0.620820\pi$
$281$	− 522.000i	− 1.85765i	0.370517	−	0.928826i	$-0.379180\pi$
$282$	0	0
$283$	−352.000	−1.24382	−0.621908	−	0.783090i	$-0.713642\pi$
$283$	−352.000	−1.24382	−0.621908	+	0.783090i	$0.713642\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	252.000i	0.878049i
$288$	0	0
$289$	−35.0000	−0.121107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	414.000i	1.41297i	0.707728	+	0.706485i	$0.249720\pi$
$293$	414.000i	1.41297i	−0.707728	+	0.706485i	$0.750280\pi$
$294$	0	0
$295$	−162.000	−0.549153
$296$	0	0
$297$	0	0
$298$	0	0
$299$	504.000i	1.68562i
$300$	0	0
$301$	154.000	0.511628
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 180.000i	− 0.590164i
$306$	0	0
$307$	290.000	0.944625	0.472313	−	0.881431i	$-0.343419\pi$
$307$	290.000	0.944625	0.472313	+	0.881431i	$0.343419\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 360.000i	− 1.15756i	−0.815485	−	0.578778i	$-0.803530\pi$
$311$	− 360.000i	− 1.15756i	0.815485	−	0.578778i	$-0.196470\pi$
$312$	0	0
$313$	497.000	1.58786	0.793930	−	0.608010i	$-0.208032\pi$
$313$	497.000	1.58786	0.793930	+	0.608010i	$0.208032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 99.0000i	− 0.312303i	−0.987733	−	0.156151i	$-0.950091\pi$
$317$	− 99.0000i	− 0.312303i	0.987733	−	0.156151i	$-0.0499088\pi$
$318$	0	0
$319$	162.000	0.507837
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 144.000i	− 0.445820i
$324$	0	0
$325$	−784.000	−2.41231
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 378.000i	− 1.14894i
$330$	0	0
$331$	506.000	1.52870	0.764350	−	0.644801i	$-0.223060\pi$
$331$	506.000	1.52870	0.764350	+	0.644801i	$0.223060\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 126.000i	− 0.376119i
$336$	0	0
$337$	230.000	0.682493	0.341246	−	0.939974i	$-0.389151\pi$
$337$	230.000	0.682493	0.341246	+	0.939974i	$0.389151\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 315.000i	− 0.923754i
$342$	0	0
$343$	343.000	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 189.000i	− 0.544669i	−0.962203	−	0.272334i	$-0.912204\pi$
$347$	− 189.000i	− 0.544669i	0.962203	−	0.272334i	$-0.0877957\pi$
$348$	0	0
$349$	56.0000	0.160458	0.0802292	−	0.996776i	$-0.474435\pi$
$349$	56.0000	0.160458	0.0802292	+	0.996776i	$0.474435\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 144.000i	− 0.407932i	−0.978978	−	0.203966i	$-0.934617\pi$
$353$	− 144.000i	− 0.407932i	0.978978	−	0.203966i	$-0.0653832\pi$
$354$	0	0
$355$	−1134.00	−3.19437
$356$	0	0
$357$	0	0
$358$	0	0
$359$	594.000i	1.65460i	0.561763	+	0.827298i	$0.310123\pi$
$359$	594.000i	1.65460i	−0.561763	+	0.827298i	$0.689877\pi$
$360$	0	0
$361$	−297.000	−0.822715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 801.000i	− 2.19452i
$366$	0	0
$367$	−205.000	−0.558583	−0.279292	−	0.960206i	$-0.590100\pi$
$367$	−205.000	−0.558583	−0.279292	+	0.960206i	$0.590100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	63.0000i	0.169811i
$372$	0	0
$373$	−670.000	−1.79625	−0.898123	−	0.439744i	$-0.855069\pi$
$373$	−670.000	−1.79625	−0.898123	+	0.439744i	$0.855069\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	252.000i	0.668435i
$378$	0	0
$379$	512.000	1.35092	0.675462	−	0.737395i	$-0.263944\pi$
$379$	512.000	1.35092	0.675462	+	0.737395i	$0.263944\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 306.000i	− 0.798956i	−0.916743	−	0.399478i	$-0.869191\pi$
$383$	− 306.000i	− 0.798956i	0.916743	−	0.399478i	$-0.130809\pi$
$384$	0	0
$385$	567.000	1.47273
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 369.000i	− 0.948586i	−0.880367	−	0.474293i	$-0.842704\pi$
$389$	− 369.000i	− 0.948586i	0.880367	−	0.474293i	$-0.157296\pi$
$390$	0	0
$391$	648.000	1.65729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 990.000i	− 2.50633i
$396$	0	0
$397$	−286.000	−0.720403	−0.360202	−	0.932875i	$-0.617292\pi$
$397$	−286.000	−0.720403	−0.360202	+	0.932875i	$0.617292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	504.000i	1.25686i	0.777867	+	0.628429i	$0.216302\pi$
$401$	504.000i	1.25686i	−0.777867	+	0.628429i	$0.783698\pi$
$402$	0	0
$403$	490.000	1.21588
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 396.000i	− 0.972973i
$408$	0	0
$409$	77.0000	0.188264	0.0941320	−	0.995560i	$-0.469992\pi$
$409$	77.0000	0.188264	0.0941320	+	0.995560i	$0.469992\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	126.000i	0.305085i
$414$	0	0
$415$	−243.000	−0.585542
$416$	0	0
$417$	0	0
$418$	0	0
$419$	594.000i	1.41766i	0.705379	+	0.708831i	$0.250777\pi$
$419$	594.000i	1.41766i	−0.705379	+	0.708831i	$0.749223\pi$
$420$	0	0
$421$	308.000	0.731591	0.365796	−	0.930695i	$-0.380797\pi$
$421$	308.000	0.731591	0.365796	+	0.930695i	$0.380797\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1008.00i	2.37176i
$426$	0	0
$427$	−140.000	−0.327869
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 126.000i	− 0.292343i	−0.989259	−	0.146172i	$-0.953305\pi$
$431$	− 126.000i	− 0.292343i	0.989259	−	0.146172i	$-0.0466952\pi$
$432$	0	0
$433$	−133.000	−0.307159	−0.153580	−	0.988136i	$-0.549080\pi$
$433$	−133.000	−0.307159	−0.153580	+	0.988136i	$0.549080\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	288.000i	0.659039i
$438$	0	0
$439$	−175.000	−0.398633	−0.199317	−	0.979935i	$-0.563872\pi$
$439$	−175.000	−0.398633	−0.199317	+	0.979935i	$0.563872\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 594.000i	− 1.34086i	−0.741974	−	0.670429i	$-0.766110\pi$
$443$	− 594.000i	− 1.34086i	0.741974	−	0.670429i	$-0.233890\pi$
$444$	0	0
$445$	−162.000	−0.364045
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−324.000	−0.718404
$452$	0	0
$453$	0	0
$454$	0	0
$455$	882.000i	1.93846i
$456$	0	0
$457$	77.0000	0.168490	0.0842451	−	0.996445i	$-0.473152\pi$
$457$	77.0000	0.168490	0.0842451	+	0.996445i	$0.473152\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	477.000i	1.03471i	0.855772	+	0.517354i	$0.173083\pi$
$461$	477.000i	1.03471i	−0.855772	+	0.517354i	$0.826917\pi$
$462$	0	0
$463$	−517.000	−1.11663	−0.558315	−	0.829629i	$-0.688552\pi$
$463$	−517.000	−1.11663	−0.558315	+	0.829629i	$0.688552\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	585.000i	1.25268i	0.779551	+	0.626338i	$0.215447\pi$
$467$	585.000i	1.25268i	−0.779551	+	0.626338i	$0.784553\pi$
$468$	0	0
$469$	−98.0000	−0.208955
$470$	0	0
$471$	0	0
$472$	0	0
$473$	198.000i	0.418605i
$474$	0	0
$475$	−448.000	−0.943158
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 270.000i	− 0.563674i	−0.959462	−	0.281837i	$-0.909056\pi$
$479$	− 270.000i	− 0.563674i	0.959462	−	0.281837i	$-0.0909438\pi$
$480$	0	0
$481$	616.000	1.28067
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 99.0000i	− 0.204124i
$486$	0	0
$487$	350.000	0.718686	0.359343	−	0.933206i	$-0.383001\pi$
$487$	350.000	0.718686	0.359343	+	0.933206i	$0.383001\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 693.000i	− 1.41141i	−0.708508	−	0.705703i	$-0.750631\pi$
$491$	− 693.000i	− 1.41141i	0.708508	−	0.705703i	$-0.249369\pi$
$492$	0	0
$493$	324.000	0.657201
$494$	0	0
$495$	0	0
$496$	0	0
$497$	882.000i	1.77465i
$498$	0	0
$499$	26.0000	0.0521042	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	26.0000	0.0521042	0.0260521	+	0.999661i	$0.491706\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	540.000i	1.07356i	0.843723	+	0.536779i	$0.180359\pi$
$503$	540.000i	1.07356i	−0.843723	+	0.536779i	$0.819641\pi$
$504$	0	0
$505$	567.000	1.12277
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 495.000i	− 0.972495i	−0.873821	−	0.486248i	$-0.838365\pi$
$509$	− 495.000i	− 0.972495i	0.873821	−	0.486248i	$-0.161635\pi$
$510$	0	0
$511$	−623.000	−1.21918
$512$	0	0
$513$	0	0
$514$	0	0
$515$	198.000i	0.384466i
$516$	0	0
$517$	486.000	0.940039
$518$	0	0
$519$	0	0
$520$	0	0
$521$	738.000i	1.41651i	0.705958	+	0.708253i	$0.250517\pi$
$521$	738.000i	1.41651i	−0.705958	+	0.708253i	$0.749483\pi$
$522$	0	0
$523$	854.000	1.63289	0.816444	−	0.577425i	$-0.195942\pi$
$523$	854.000	1.63289	0.816444	+	0.577425i	$0.195942\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 630.000i	− 1.19545i
$528$	0	0
$529$	−767.000	−1.44991
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 504.000i	− 0.945591i
$534$	0	0
$535$	891.000	1.66542
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−460.000	−0.850277	−0.425139	−	0.905128i	$-0.639775\pi$
$541$	−460.000	−0.850277	−0.425139	+	0.905128i	$0.639775\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	468.000i	0.858716i
$546$	0	0
$547$	−790.000	−1.44424	−0.722121	−	0.691767i	$-0.756832\pi$
$547$	−790.000	−1.44424	−0.722121	+	0.691767i	$0.756832\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	144.000i	0.261343i
$552$	0	0
$553$	−770.000	−1.39241
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 315.000i	− 0.565530i	−0.959189	−	0.282765i	$-0.908748\pi$
$557$	− 315.000i	− 0.565530i	0.959189	−	0.282765i	$-0.0912516\pi$
$558$	0	0
$559$	−308.000	−0.550984
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 783.000i	− 1.39076i	−0.718640	−	0.695382i	$-0.755235\pi$
$563$	− 783.000i	− 1.39076i	0.718640	−	0.695382i	$-0.244765\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 288.000i	− 0.506151i	−0.967447	−	0.253076i	$-0.918558\pi$
$569$	− 288.000i	− 0.506151i	0.967447	−	0.253076i	$-0.0814421\pi$
$570$	0	0
$571$	158.000	0.276708	0.138354	−	0.990383i	$-0.455819\pi$
$571$	158.000	0.276708	0.138354	+	0.990383i	$0.455819\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2016.00i	− 3.50609i
$576$	0	0
$577$	−658.000	−1.14038	−0.570191	−	0.821512i	$-0.693131\pi$
$577$	−658.000	−1.14038	−0.570191	+	0.821512i	$0.693131\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	189.000i	0.325301i
$582$	0	0
$583$	−81.0000	−0.138937
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 693.000i	− 1.18058i	−0.807192	−	0.590290i	$-0.799014\pi$
$587$	− 693.000i	− 1.18058i	0.807192	−	0.590290i	$-0.200986\pi$
$588$	0	0
$589$	280.000	0.475382
$590$	0	0
$591$	0	0
$592$	0	0
$593$	198.000i	0.333895i	0.985966	+	0.166948i	$0.0533911\pi$
$593$	198.000i	0.333895i	−0.985966	+	0.166948i	$0.946609\pi$
$594$	0	0
$595$	1134.00	1.90588
$596$	0	0
$597$	0	0
$598$	0	0
$599$	198.000i	0.330551i	0.986247	+	0.165275i	$0.0528513\pi$
$599$	198.000i	0.330551i	−0.986247	+	0.165275i	$0.947149\pi$
$600$	0	0
$601$	−931.000	−1.54908	−0.774542	−	0.632522i	$-0.782020\pi$
$601$	−931.000	−1.54908	−0.774542	+	0.632522i	$0.782020\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 360.000i	− 0.595041i
$606$	0	0
$607$	770.000	1.26853	0.634267	−	0.773114i	$-0.281302\pi$
$607$	770.000	1.26853	0.634267	+	0.773114i	$0.281302\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	756.000i	1.23732i
$612$	0	0
$613$	602.000	0.982055	0.491028	−	0.871144i	$-0.336621\pi$
$613$	602.000	0.982055	0.491028	+	0.871144i	$0.336621\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 144.000i	− 0.233387i	−0.993168	−	0.116694i	$-0.962770\pi$
$617$	− 144.000i	− 0.233387i	0.993168	−	0.116694i	$-0.0372296\pi$
$618$	0	0
$619$	758.000	1.22456	0.612278	−	0.790643i	$-0.290253\pi$
$619$	758.000	1.22456	0.612278	+	0.790643i	$0.290253\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	126.000i	0.202247i
$624$	0	0
$625$	1111.00	1.77760
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 792.000i	− 1.25914i
$630$	0	0
$631$	−631.000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−631.000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	873.000i	1.37480i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 684.000i	− 1.06708i	−0.845774	−	0.533541i	$-0.820861\pi$
$641$	− 684.000i	− 1.06708i	0.845774	−	0.533541i	$-0.179139\pi$
$642$	0	0
$643$	−1144.00	−1.77916	−0.889580	−	0.456779i	$-0.849003\pi$
$643$	−1144.00	−1.77916	−0.889580	+	0.456779i	$0.849003\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 756.000i	− 1.16847i	−0.811585	−	0.584235i	$-0.801395\pi$
$647$	− 756.000i	− 1.16847i	0.811585	−	0.584235i	$-0.198605\pi$
$648$	0	0
$649$	−162.000	−0.249615
$650$	0	0
$651$	0	0
$652$	0	0
$653$	441.000i	0.675345i	0.941264	+	0.337672i	$0.109640\pi$
$653$	441.000i	0.675345i	−0.941264	+	0.337672i	$0.890360\pi$
$654$	0	0
$655$	−567.000	−0.865649
$656$	0	0
$657$	0	0
$658$	0	0
$659$	441.000i	0.669196i	0.942361	+	0.334598i	$0.108600\pi$
$659$	441.000i	0.669196i	−0.942361	+	0.334598i	$0.891400\pi$
$660$	0	0
$661$	392.000	0.593041	0.296520	−	0.955027i	$-0.404174\pi$
$661$	392.000	0.593041	0.296520	+	0.955027i	$0.404174\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	504.000i	0.757895i
$666$	0	0
$667$	−648.000	−0.971514
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 180.000i	− 0.268256i
$672$	0	0
$673$	−889.000	−1.32095	−0.660475	−	0.750848i	$-0.729645\pi$
$673$	−889.000	−1.32095	−0.660475	+	0.750848i	$0.729645\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 882.000i	− 1.30281i	−0.758732	−	0.651403i	$-0.774181\pi$
$677$	− 882.000i	− 1.30281i	0.758732	−	0.651403i	$-0.225819\pi$
$678$	0	0
$679$	−77.0000	−0.113402
$680$	0	0
$681$	0	0
$682$	0	0
$683$	378.000i	0.553441i	0.960950	+	0.276720i	$0.0892476\pi$
$683$	378.000i	0.553441i	−0.960950	+	0.276720i	$0.910752\pi$
$684$	0	0
$685$	1134.00	1.65547
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 126.000i	− 0.182874i
$690$	0	0
$691$	140.000	0.202605	0.101302	−	0.994856i	$-0.467699\pi$
$691$	140.000	0.202605	0.101302	+	0.994856i	$0.467699\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1980.00i	2.84892i
$696$	0	0
$697$	−648.000	−0.929699
$698$	0	0
$699$	0	0
$700$	0	0
$701$	693.000i	0.988588i	0.869295	+	0.494294i	$0.164573\pi$
$701$	693.000i	0.988588i	−0.869295	+	0.494294i	$0.835427\pi$
$702$	0	0
$703$	352.000	0.500711
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 441.000i	− 0.623762i
$708$	0	0
$709$	128.000	0.180536	0.0902680	−	0.995918i	$-0.471228\pi$
$709$	128.000	0.180536	0.0902680	+	0.995918i	$0.471228\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1260.00i	1.76718i
$714$	0	0
$715$	−1134.00	−1.58601
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1188.00i	1.65229i	0.563454	+	0.826147i	$0.309472\pi$
$719$	1188.00i	1.65229i	−0.563454	+	0.826147i	$0.690528\pi$
$720$	0	0
$721$	154.000	0.213592
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1008.00i	− 1.39034i
$726$	0	0
$727$	725.000	0.997249	0.498624	−	0.866818i	$-0.333839\pi$
$727$	725.000	0.997249	0.498624	+	0.866818i	$0.333839\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	396.000i	0.541724i
$732$	0	0
$733$	224.000	0.305593	0.152797	−	0.988258i	$-0.451172\pi$
$733$	224.000	0.305593	0.152797	+	0.988258i	$0.451172\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 126.000i	− 0.170963i
$738$	0	0
$739$	−574.000	−0.776725	−0.388363	−	0.921507i	$-0.626959\pi$
$739$	−574.000	−0.776725	−0.388363	+	0.921507i	$0.626959\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	486.000i	0.654105i	0.945006	+	0.327052i	$0.106055\pi$
$743$	486.000i	0.654105i	−0.945006	+	0.327052i	$0.893945\pi$
$744$	0	0
$745$	2511.00	3.37047
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 693.000i	− 0.925234i
$750$	0	0
$751$	1355.00	1.80426	0.902130	−	0.431463i	$-0.142003\pi$
$751$	1355.00	1.80426	0.902130	+	0.431463i	$0.142003\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1359.00i	1.80000i
$756$	0	0
$757$	−190.000	−0.250991	−0.125495	−	0.992094i	$-0.540052\pi$
$757$	−190.000	−0.250991	−0.125495	+	0.992094i	$0.540052\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1080.00i	1.41919i	0.704612	+	0.709593i	$0.251121\pi$
$761$	1080.00i	1.41919i	−0.704612	+	0.709593i	$0.748879\pi$
$762$	0	0
$763$	364.000	0.477064
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 252.000i	− 0.328553i
$768$	0	0
$769$	−1309.00	−1.70221	−0.851105	−	0.524995i	$-0.824067\pi$
$769$	−1309.00	−1.70221	−0.851105	+	0.524995i	$0.824067\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 486.000i	− 0.628719i	−0.949304	−	0.314360i	$-0.898210\pi$
$773$	− 486.000i	− 0.628719i	0.949304	−	0.314360i	$-0.101790\pi$
$774$	0	0
$775$	−1960.00	−2.52903
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 288.000i	− 0.369705i
$780$	0	0
$781$	−1134.00	−1.45198
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 396.000i	− 0.504459i
$786$	0	0
$787$	1076.00	1.36722	0.683609	−	0.729849i	$-0.260410\pi$
$787$	1076.00	1.36722	0.683609	+	0.729849i	$0.260410\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	280.000	0.353090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 567.000i	− 0.711418i	−0.934597	−	0.355709i	$-0.884239\pi$
$797$	− 567.000i	− 0.711418i	0.934597	−	0.355709i	$-0.115761\pi$
$798$	0	0
$799$	972.000	1.21652
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 801.000i	− 0.997509i
$804$	0	0
$805$	−2268.00	−2.81739
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1314.00i	− 1.62423i	−0.583499	−	0.812114i	$-0.698317\pi$
$809$	− 1314.00i	− 1.62423i	0.583499	−	0.812114i	$-0.301683\pi$
$810$	0	0
$811$	56.0000	0.0690506	0.0345253	−	0.999404i	$-0.489008\pi$
$811$	56.0000	0.0690506	0.0345253	+	0.999404i	$0.489008\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	738.000i	0.905521i
$816$	0	0
$817$	−176.000	−0.215422
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1386.00i	1.68819i	0.536197	+	0.844093i	$0.319860\pi$
$821$	1386.00i	1.68819i	−0.536197	+	0.844093i	$0.680140\pi$
$822$	0	0
$823$	−1039.00	−1.26245	−0.631227	−	0.775598i	$-0.717448\pi$
$823$	−1039.00	−1.26245	−0.631227	+	0.775598i	$0.717448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 882.000i	− 1.06651i	−0.845956	−	0.533253i	$-0.820969\pi$
$827$	− 882.000i	− 1.06651i	0.845956	−	0.533253i	$-0.179031\pi$
$828$	0	0
$829$	−478.000	−0.576598	−0.288299	−	0.957540i	$-0.593090\pi$
$829$	−478.000	−0.576598	−0.288299	+	0.957540i	$0.593090\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1458.00	1.74611
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 162.000i	− 0.193087i	−0.995329	−	0.0965435i	$-0.969221\pi$
$839$	− 162.000i	− 0.193087i	0.995329	−	0.0965435i	$-0.0307787\pi$
$840$	0	0
$841$	517.000	0.614744
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 243.000i	− 0.287574i
$846$	0	0
$847$	−280.000	−0.330579
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1584.00i	1.86134i
$852$	0	0
$853$	686.000	0.804220	0.402110	−	0.915591i	$-0.368277\pi$
$853$	686.000	0.804220	0.402110	+	0.915591i	$0.368277\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1242.00i	− 1.44924i	−0.689148	−	0.724621i	$-0.742015\pi$
$857$	− 1242.00i	− 1.44924i	0.689148	−	0.724621i	$-0.257985\pi$
$858$	0	0
$859$	356.000	0.414435	0.207218	−	0.978295i	$-0.433559\pi$
$859$	356.000	0.414435	0.207218	+	0.978295i	$0.433559\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1476.00i	− 1.71031i	−0.518370	−	0.855156i	$-0.673461\pi$
$863$	− 1476.00i	− 1.71031i	0.518370	−	0.855156i	$-0.326539\pi$
$864$	0	0
$865$	1701.00	1.96647
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 990.000i	− 1.13924i
$870$	0	0
$871$	196.000	0.225029
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1953.00i	− 2.23200i
$876$	0	0
$877$	−310.000	−0.353478	−0.176739	−	0.984258i	$-0.556555\pi$
$877$	−310.000	−0.353478	−0.176739	+	0.984258i	$0.556555\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 630.000i	− 0.715096i	−0.933895	−	0.357548i	$-0.883613\pi$
$881$	− 630.000i	− 0.715096i	0.933895	−	0.357548i	$-0.116387\pi$
$882$	0	0
$883$	440.000	0.498301	0.249151	−	0.968465i	$-0.419849\pi$
$883$	440.000	0.498301	0.249151	+	0.968465i	$0.419849\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	450.000i	0.507328i	0.967292	+	0.253664i	$0.0816358\pi$
$887$	450.000i	0.507328i	−0.967292	+	0.253664i	$0.918364\pi$
$888$	0	0
$889$	679.000	0.763780
$890$	0	0
$891$	0	0
$892$	0	0
$893$	432.000i	0.483763i
$894$	0	0
$895$	−567.000	−0.633520
$896$	0	0
$897$	0	0
$898$	0	0
$899$	630.000i	0.700779i
$900$	0	0
$901$	−162.000	−0.179800
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2016.00i	− 2.22762i
$906$	0	0
$907$	−556.000	−0.613010	−0.306505	−	0.951869i	$-0.599160\pi$
$907$	−556.000	−0.613010	−0.306505	+	0.951869i	$0.599160\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	270.000i	0.296378i	0.988959	+	0.148189i	$0.0473443\pi$
$911$	270.000i	0.296378i	−0.988959	+	0.148189i	$0.952656\pi$
$912$	0	0
$913$	−243.000	−0.266156
$914$	0	0
$915$	0	0
$916$	0	0
$917$	441.000i	0.480916i
$918$	0	0
$919$	−187.000	−0.203482	−0.101741	−	0.994811i	$-0.532441\pi$
$919$	−187.000	−0.203482	−0.101741	+	0.994811i	$0.532441\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1764.00i	− 1.91116i
$924$	0	0
$925$	−2464.00	−2.66378
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 324.000i	− 0.348762i	−0.984678	−	0.174381i	$-0.944208\pi$
$929$	− 324.000i	− 0.348762i	0.984678	−	0.174381i	$-0.0557925\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1458.00i	1.55936i
$936$	0	0
$937$	−451.000	−0.481323	−0.240662	−	0.970609i	$-0.577364\pi$
$937$	−451.000	−0.481323	−0.240662	+	0.970609i	$0.577364\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1305.00i	1.38682i	0.720542	+	0.693411i	$0.243893\pi$
$941$	1305.00i	1.38682i	−0.720542	+	0.693411i	$0.756107\pi$
$942$	0	0
$943$	1296.00	1.37434
$944$	0	0
$945$	0	0
$946$	0	0
$947$	945.000i	0.997888i	0.866634	+	0.498944i	$0.166279\pi$
$947$	945.000i	0.997888i	−0.866634	+	0.498944i	$0.833721\pi$
$948$	0	0
$949$	1246.00	1.31296
$950$	0	0
$951$	0	0
$952$	0	0
$953$	864.000i	0.906611i	0.891355	+	0.453305i	$0.149755\pi$
$953$	864.000i	0.906611i	−0.891355	+	0.453305i	$0.850245\pi$
$954$	0	0
$955$	−3078.00	−3.22304
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 882.000i	− 0.919708i
$960$	0	0
$961$	264.000	0.274714
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1737.00i	1.80000i
$966$	0	0
$967$	329.000	0.340228	0.170114	−	0.985424i	$-0.445586\pi$
$967$	329.000	0.340228	0.170114	+	0.985424i	$0.445586\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	243.000i	0.250257i	0.992141	+	0.125129i	$0.0399344\pi$
$971$	243.000i	0.250257i	−0.992141	+	0.125129i	$0.960066\pi$
$972$	0	0
$973$	1540.00	1.58273
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 882.000i	− 0.902764i	−0.892331	−	0.451382i	$-0.850931\pi$
$977$	− 882.000i	− 0.902764i	0.892331	−	0.451382i	$-0.149069\pi$
$978$	0	0
$979$	−162.000	−0.165475
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 990.000i	− 1.00712i	−0.863960	−	0.503561i	$-0.832023\pi$
$983$	− 990.000i	− 1.00712i	0.863960	−	0.503561i	$-0.167977\pi$
$984$	0	0
$985$	−405.000	−0.411168
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 792.000i	− 0.800809i
$990$	0	0
$991$	143.000	0.144299	0.0721493	−	0.997394i	$-0.477014\pi$
$991$	143.000	0.144299	0.0721493	+	0.997394i	$0.477014\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	819.000i	0.823116i
$996$	0	0
$997$	698.000	0.700100	0.350050	−	0.936731i	$-0.386165\pi$
$997$	698.000	0.700100	0.350050	+	0.936731i	$0.386165\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.3.c.b.53.1	✓	2
3.2	odd	2	inner	108.3.c.b.53.2	yes	2
4.3	odd	2		432.3.e.g.161.1		2
5.2	odd	4		2700.3.b.f.1349.1		2
5.3	odd	4		2700.3.b.a.1349.2		2
5.4	even	2		2700.3.g.m.701.1		2
8.3	odd	2		1728.3.e.n.1025.2		2
8.5	even	2		1728.3.e.e.1025.2		2
9.2	odd	6		324.3.g.c.53.1		4
9.4	even	3		324.3.g.c.269.1		4
9.5	odd	6		324.3.g.c.269.2		4
9.7	even	3		324.3.g.c.53.2		4
12.11	even	2		432.3.e.g.161.2		2
15.2	even	4		2700.3.b.a.1349.1		2
15.8	even	4		2700.3.b.f.1349.2		2
15.14	odd	2		2700.3.g.m.701.2		2
24.5	odd	2		1728.3.e.e.1025.1		2
24.11	even	2		1728.3.e.n.1025.1		2
36.7	odd	6		1296.3.q.d.1025.2		4
36.11	even	6		1296.3.q.d.1025.1		4
36.23	even	6		1296.3.q.d.593.2		4
36.31	odd	6		1296.3.q.d.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.c.b.53.1	✓	2	1.1	even	1	trivial
108.3.c.b.53.2	yes	2	3.2	odd	2	inner
324.3.g.c.53.1		4	9.2	odd	6
324.3.g.c.53.2		4	9.7	even	3
324.3.g.c.269.1		4	9.4	even	3
324.3.g.c.269.2		4	9.5	odd	6
432.3.e.g.161.1		2	4.3	odd	2
432.3.e.g.161.2		2	12.11	even	2
1296.3.q.d.593.1		4	36.31	odd	6
1296.3.q.d.593.2		4	36.23	even	6
1296.3.q.d.1025.1		4	36.11	even	6
1296.3.q.d.1025.2		4	36.7	odd	6
1728.3.e.e.1025.1		2	24.5	odd	2
1728.3.e.e.1025.2		2	8.5	even	2
1728.3.e.n.1025.1		2	24.11	even	2
1728.3.e.n.1025.2		2	8.3	odd	2
2700.3.b.a.1349.1		2	15.2	even	4
2700.3.b.a.1349.2		2	5.3	odd	4
2700.3.b.f.1349.1		2	5.2	odd	4
2700.3.b.f.1349.2		2	15.8	even	4
2700.3.g.m.701.1		2	5.4	even	2
2700.3.g.m.701.2		2	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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	108.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{5} -7.00000 q^{7} +O(q^{10})\)	\(q-9.00000i q^{5} -7.00000 q^{7} -9.00000i q^{11} +14.0000 q^{13} -18.0000i q^{17} +8.00000 q^{19} +36.0000i q^{23} -56.0000 q^{25} +18.0000i q^{29} +35.0000 q^{31} +63.0000i q^{35} +44.0000 q^{37} -36.0000i q^{41} -22.0000 q^{43} +54.0000i q^{47} -9.00000i q^{53} -81.0000 q^{55} -18.0000i q^{59} +20.0000 q^{61} -126.000i q^{65} +14.0000 q^{67} -126.000i q^{71} +89.0000 q^{73} +63.0000i q^{77} +110.000 q^{79} -27.0000i q^{83} -162.000 q^{85} -18.0000i q^{89} -98.0000 q^{91} -72.0000i q^{95} +11.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7}+O(q^{10}) \) 2 * q - 14 * q^7	\( 2 q - 14 q^{7} + 28 q^{13} + 16 q^{19} - 112 q^{25} + 70 q^{31} + 88 q^{37} - 44 q^{43} - 162 q^{55} + 40 q^{61} + 28 q^{67} + 178 q^{73} + 220 q^{79} - 324 q^{85} - 196 q^{91} + 22 q^{97}+O(q^{100}) \) 2 * q - 14 * q^7 + 28 * q^13 + 16 * q^19 - 112 * q^25 + 70 * q^31 + 88 * q^37 - 44 * q^43 - 162 * q^55 + 40 * q^61 + 28 * q^67 + 178 * q^73 + 220 * q^79 - 324 * q^85 - 196 * q^91 + 22 * q^97

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