LMFDB - Embedded newform 1089.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

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:	yes
Analytic conductor:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} -4.00000 q^{10} +2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{19} -4.00000 q^{20} -4.00000 q^{23} +11.0000 q^{25} -2.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} +8.00000 q^{35} -6.00000 q^{37} -6.00000 q^{38} +12.0000 q^{40} -10.0000 q^{41} -6.00000 q^{43} +4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} -11.0000 q^{50} -2.00000 q^{52} +6.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +8.00000 q^{65} +8.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} +2.00000 q^{73} +6.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} -4.00000 q^{80} +10.0000 q^{82} +12.0000 q^{83} +8.00000 q^{85} +6.00000 q^{86} +4.00000 q^{91} +4.00000 q^{92} -8.00000 q^{94} +24.0000 q^{95} +2.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−4.00000	−1.26491
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	8.00000	1.35225
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	12.0000	1.89737
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−2.00000	−0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	8.00000	0.992278
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	6.00000	0.646997
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	24.0000	2.46235
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−11.0000	−1.10000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000	0.368230
$119$	4.00000	0.366679
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	0	0
$124$	−4.00000	−0.359211
$125$	24.0000	2.14663
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	−8.00000	−0.701646
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−24.0000	−1.99309
$146$	−2.00000	−0.165521
$147$	0	0
$148$	6.00000	0.493197
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	18.0000	1.45999
$153$	0	0
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−20.0000	−1.58114
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−8.00000	−0.613572
$171$	0	0
$172$	6.00000	0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	22.0000	1.66304
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−24.0000	−1.76452
$186$	0	0
$187$	0	0
$188$	−8.00000	−0.583460
$189$	0	0
$190$	−24.0000	−1.74114
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	33.0000	2.33345
$201$	0	0
$202$	14.0000	0.985037
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−40.0000	−2.79372
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−24.0000	−1.63679
$216$	0	0
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	12.0000	0.798228
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	−18.0000	−1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	32.0000	2.08745
$236$	4.00000	0.260378
$237$	0	0
$238$	−4.00000	−0.259281
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$244$	−6.00000	−0.384111
$245$	−12.0000	−0.766652
$246$	0	0
$247$	12.0000	0.763542
$248$	12.0000	0.762001
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	−10.0000	−0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	−8.00000	−0.496139
$261$	0	0
$262$	12.0000	0.741362
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	0	0
$268$	−8.00000	−0.488678
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	4.00000	0.241649
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	10.0000	0.599760
$279$	0	0
$280$	24.0000	1.43427
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	24.0000	1.40933
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	−18.0000	−1.04623
$297$	0	0
$298$	2.00000	0.115857
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−12.0000	−0.691669
$302$	14.0000	0.805609
$303$	0	0
$304$	−6.00000	−0.344124
$305$	24.0000	1.37424
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	0	0
$320$	28.0000	1.56525
$321$	0	0
$322$	8.00000	0.445823
$323$	12.0000	0.667698
$324$	0	0
$325$	22.0000	1.22034
$326$	20.0000	1.10770
$327$	0	0
$328$	−30.0000	−1.65647
$329$	16.0000	0.882109
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	32.0000	1.74835
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−8.00000	−0.433861
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−18.0000	−0.970495
$345$	0	0
$346$	6.00000	0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	−22.0000	−1.17595
$351$	0	0
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−24.0000	−1.26844
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−4.00000	−0.209657
$365$	8.00000	0.418739
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	24.0000	1.24770
$371$	0	0
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	0	0
$376$	24.0000	1.23771
$377$	−12.0000	−0.618031
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−24.0000	−1.23117
$381$	0	0
$382$	16.0000	0.818631
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−2.00000	−0.100759
$395$	40.0000	2.01262
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−11.0000	−0.550000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	14.0000	0.696526
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	40.0000	1.97546
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−8.00000	−0.393654
$414$	0	0
$415$	48.0000	2.35623
$416$	−10.0000	−0.490290
$417$	0	0
$418$	0	0
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−6.00000	−0.292075
$423$	0	0
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	12.0000	0.580721
$428$	−12.0000	−0.580042
$429$	0	0
$430$	24.0000	1.15738
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	−4.00000	−0.190261
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	14.0000	0.661438
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	0	0
$454$	12.0000	0.563188
$455$	16.0000	0.750092
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	16.0000	0.746004
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−32.0000	−1.47605
$471$	0	0
$472$	−12.0000	−0.552345
$473$	0	0
$474$	0	0
$475$	66.0000	3.02829
$476$	−4.00000	−0.183340
$477$	0	0
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−26.0000	−1.18427
$483$	0	0
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	18.0000	0.814822
$489$	0	0
$490$	12.0000	0.542105
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	8.00000	0.357057
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	−56.0000	−2.49197
$506$	0	0
$507$	0	0
$508$	−10.0000	−0.443678
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000	0.486136
$513$	0	0
$514$	8.00000	0.352865
$515$	32.0000	1.41009
$516$	0	0
$517$	0	0
$518$	12.0000	0.527250
$519$	0	0
$520$	24.0000	1.05247
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	32.0000	1.39527
$527$	8.00000	0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−20.0000	−0.866296
$534$	0	0
$535$	48.0000	2.07522
$536$	24.0000	1.03664
$537$	0	0
$538$	−16.0000	−0.689809
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−10.0000	−0.428746
$545$	8.00000	0.342682
$546$	0	0
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	4.00000	0.170872
$549$	0	0
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	20.0000	0.850487
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	10.0000	0.424094
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	−8.00000	−0.338062
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	−48.0000	−2.01938
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	34.0000	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$571$	34.0000	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$572$	0	0
$573$	0	0
$574$	20.0000	0.834784
$575$	−44.0000	−1.83493
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	24.0000	0.996546
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	18.0000	0.743573
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	16.0000	0.658710
$591$	0	0
$592$	6.00000	0.246598
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	2.00000	0.0819232
$597$	0	0
$598$	8.00000	0.327144
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	14.0000	0.569652
$605$	0	0
$606$	0	0
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	−24.0000	−0.971732
$611$	16.0000	0.647291
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	34.0000	1.35891
$627$	0	0
$628$	10.0000	0.399043
$629$	−12.0000	−0.478471
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	30.0000	1.19334
$633$	0	0
$634$	−4.00000	−0.158860
$635$	40.0000	1.58735
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	12.0000	0.474342
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	0	0
$650$	−22.0000	−0.862911
$651$	0	0
$652$	20.0000	0.783260
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	10.0000	0.390434
$657$	0	0
$658$	−16.0000	−0.623745
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	36.0000	1.39707
$665$	48.0000	1.86136
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	−32.0000	−1.23627
$671$	0	0
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	24.0000	0.920358
$681$	0	0
$682$	0	0
$683$	32.0000	1.22445	0.612223	−	0.790685i	$-0.290275\pi$
$683$	32.0000	1.22445	0.612223	+	0.790685i	$0.290275\pi$
$684$	0	0
$685$	−16.0000	−0.611329
$686$	20.0000	0.763604
$687$	0	0
$688$	6.00000	0.228748
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−40.0000	−1.51729
$696$	0	0
$697$	−20.0000	−0.757554
$698$	18.0000	0.681310
$699$	0	0
$700$	−22.0000	−0.831522
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	0	0
$706$	24.0000	0.903252
$707$	−28.0000	−1.05305
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−8.00000	−0.298557
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−66.0000	−2.45118
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	12.0000	0.444750
$729$	0	0
$730$	−8.00000	−0.296093
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	24.0000	0.882258
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	34.0000	1.24483
$747$	0	0
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	12.0000	0.437014
$755$	−56.0000	−2.03805
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	72.0000	2.61171
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	16.0000	0.578860
$765$	0	0
$766$	−20.0000	−0.722629
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	6.00000	0.215387
$777$	0	0
$778$	36.0000	1.29066
$779$	−60.0000	−2.14972
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	3.00000	0.107143
$785$	−40.0000	−1.42766
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	−40.0000	−1.42314
$791$	−24.0000	−0.853342
$792$	0	0
$793$	12.0000	0.426132
$794$	14.0000	0.496841
$795$	0	0
$796$	−12.0000	−0.425329
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	−55.0000	−1.94454
$801$	0	0
$802$	28.0000	0.988714
$803$	0	0
$804$	0	0
$805$	−32.0000	−1.12785
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−42.0000	−1.47755
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	0	0
$815$	−80.0000	−2.80228
$816$	0	0
$817$	−36.0000	−1.25948
$818$	6.00000	0.209785
$819$	0	0
$820$	40.0000	1.39686
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	8.00000	0.278356
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−48.0000	−1.66610
$831$	0	0
$832$	14.0000	0.485363
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−32.0000	−1.10542
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−34.0000	−1.17172
$843$	0	0
$844$	−6.00000	−0.206529
$845$	−36.0000	−1.23844
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−22.0000	−0.754594
$851$	24.0000	0.822709
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	36.0000	1.23045
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	24.0000	0.818393
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	2.00000	0.0679628
$867$	0	0
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	6.00000	0.203186
$873$	0	0
$874$	24.0000	0.811812
$875$	48.0000	1.62270
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−4.00000	−0.134383
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	48.0000	1.60626
$894$	0	0
$895$	96.0000	3.20893
$896$	6.00000	0.200446
$897$	0	0
$898$	−32.0000	−1.06785
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−36.0000	−1.19734
$905$	40.0000	1.32964
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	−16.0000	−0.530395
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	−24.0000	−0.792550
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	−48.0000	−1.58251
$921$	0	0
$922$	−6.00000	−0.197599
$923$	0	0
$924$	0	0
$925$	−66.0000	−2.17007
$926$	−16.0000	−0.525793
$927$	0	0
$928$	30.0000	0.984798
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−6.00000	−0.196537
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	−32.0000	−1.04372
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	40.0000	1.30258
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−66.0000	−2.14132
$951$	0	0
$952$	12.0000	0.388922
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	−64.0000	−2.07099
$956$	0	0
$957$	0	0
$958$	32.0000	1.03387
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	0	0
$964$	−26.0000	−0.837404
$965$	56.0000	1.80270
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	0	0
$969$	0	0
$970$	−8.00000	−0.256865
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	4.00000	0.128168
$975$	0	0
$976$	−6.00000	−0.192055
$977$	36.0000	1.15174	0.575871	−	0.817541i	$-0.304663\pi$
$977$	36.0000	1.15174	0.575871	+	0.817541i	$0.304663\pi$
$978$	0	0
$979$	0	0
$980$	12.0000	0.383326
$981$	0	0
$982$	28.0000	0.893516
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	12.0000	0.382158
$987$	0	0
$988$	−12.0000	−0.381771
$989$	24.0000	0.763156
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	0	0
$995$	48.0000	1.52170
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.d.1.1		1
3.2	odd	2		1089.2.a.h.1.1		1
11.10	odd	2		99.2.a.c.1.1	yes	1
33.32	even	2		99.2.a.a.1.1	✓	1
44.43	even	2		1584.2.a.r.1.1		1
55.32	even	4		2475.2.c.g.199.2		2
55.43	even	4		2475.2.c.g.199.1		2
55.54	odd	2		2475.2.a.c.1.1		1
77.76	even	2		4851.2.a.o.1.1		1
88.21	odd	2		6336.2.a.b.1.1		1
88.43	even	2		6336.2.a.f.1.1		1
99.32	even	6		891.2.e.j.298.1		2
99.43	odd	6		891.2.e.c.595.1		2
99.65	even	6		891.2.e.j.595.1		2
99.76	odd	6		891.2.e.c.298.1		2
132.131	odd	2		1584.2.a.b.1.1		1
165.32	odd	4		2475.2.c.b.199.1		2
165.98	odd	4		2475.2.c.b.199.2		2
165.164	even	2		2475.2.a.j.1.1		1
231.230	odd	2		4851.2.a.g.1.1		1
264.131	odd	2		6336.2.a.cm.1.1		1
264.197	even	2		6336.2.a.cl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.a.a.1.1	✓	1	33.32	even	2
99.2.a.c.1.1	yes	1	11.10	odd	2
891.2.e.c.298.1		2	99.76	odd	6
891.2.e.c.595.1		2	99.43	odd	6
891.2.e.j.298.1		2	99.32	even	6
891.2.e.j.595.1		2	99.65	even	6
1089.2.a.d.1.1		1	1.1	even	1	trivial
1089.2.a.h.1.1		1	3.2	odd	2
1584.2.a.b.1.1		1	132.131	odd	2
1584.2.a.r.1.1		1	44.43	even	2
2475.2.a.c.1.1		1	55.54	odd	2
2475.2.a.j.1.1		1	165.164	even	2
2475.2.c.b.199.1		2	165.32	odd	4
2475.2.c.b.199.2		2	165.98	odd	4
2475.2.c.g.199.1		2	55.43	even	4
2475.2.c.g.199.2		2	55.32	even	4
4851.2.a.g.1.1		1	231.230	odd	2
4851.2.a.o.1.1		1	77.76	even	2
6336.2.a.b.1.1		1	88.21	odd	2
6336.2.a.f.1.1		1	88.43	even	2
6336.2.a.cl.1.1		1	264.197	even	2
6336.2.a.cm.1.1		1	264.131	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more