LMFDB - Embedded newform 1170.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.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:	$9.34249703649$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} +6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +4.00000 q^{35} -10.0000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} +6.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +12.0000 q^{53} -4.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} -10.0000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +14.0000 q^{67} -6.00000 q^{68} +4.00000 q^{70} -16.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +12.0000 q^{83} +6.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +12.0000 q^{94} +4.00000 q^{95} +8.00000 q^{97} +9.00000 q^{98} +O(q^{100})$

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−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
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	4.00000	0.676123
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	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
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	6.00000	0.625543
$93$	0	0
$94$	12.0000	1.23771
$95$	4.00000	0.410391
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	12.0000	1.16554
$107$	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
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−12.0000	−1.10469
$119$	24.0000	2.20008
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	14.0000	1.20942
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	−16.0000	−1.32417
$147$	0	0
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−24.0000	−1.89146
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.00000	0.460179
$171$	0	0
$172$	−4.00000	−0.304997
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	6.00000	0.442326
$185$	10.0000	0.735215
$186$	0	0
$187$	0	0
$188$	12.0000	0.875190
$189$	0	0
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	24.0000	1.68447
$204$	0	0
$205$	6.00000	0.419058
$206$	8.00000	0.557386
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	0	0
$217$	40.0000	2.71538
$218$	−4.00000	−0.270914
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−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
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−6.00000	−0.393919
$233$	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
$235$	−12.0000	−0.782794
$236$	−12.0000	−0.781133
$237$	0	0
$238$	24.0000	1.55569
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−10.0000	−0.640184
$245$	−9.00000	−0.574989
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−10.0000	−0.635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	40.0000	2.48548
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	6.00000	0.370681
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	16.0000	0.981023
$267$	0	0
$268$	14.0000	0.855186
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	4.00000	0.239046
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	0	0
$292$	−16.0000	−0.936329
$293$	−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
$295$	12.0000	0.698667
$296$	−10.0000	−0.581238
$297$	0	0
$298$	6.00000	0.347571
$299$	6.00000	0.346989
$300$	0	0
$301$	16.0000	0.922225
$302$	2.00000	0.115087
$303$	0	0
$304$	−4.00000	−0.229416
$305$	10.0000	0.572598
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	0	0
$310$	10.0000	0.567962
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−24.0000	−1.33747
$323$	24.0000	1.33540
$324$	0	0
$325$	1.00000	0.0554700
$326$	2.00000	0.110770
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−48.0000	−2.64633
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−14.0000	−0.764902
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−6.00000	−0.317110
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	0	0
$364$	−4.00000	−0.209657
$365$	16.0000	0.837478
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	10.0000	0.519875
$371$	−48.0000	−2.49204
$372$	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
$376$	12.0000	0.618853
$377$	−6.00000	−0.309016
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−16.0000	−0.814379
$387$	0	0
$388$	8.00000	0.406138
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	9.00000	0.454569
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	6.00000	0.298511
$405$	0	0
$406$	24.0000	1.19110
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	8.00000	0.394132
$413$	48.0000	2.36193
$414$	0	0
$415$	−12.0000	−0.589057
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	12.0000	0.582772
$425$	−6.00000	−0.291043
$426$	0	0
$427$	40.0000	1.93574
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	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
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	40.0000	1.92006
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	8.00000	0.378811
$447$	0	0
$448$	−4.00000	−0.188982
$449$	−42.0000	−1.98210	−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000	−1.98210	−0.991051	+	0.133482i	$0.957384\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−12.0000	−0.563188
$455$	4.00000	0.187523
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	−6.00000	−0.279751
$461$	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
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−56.0000	−2.58584
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	24.0000	1.10004
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−8.00000	−0.363261
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	−9.00000	−0.406579
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−10.0000	−0.449013
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−18.0000	−0.803379
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	64.0000	2.83119
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0000	−1.32324
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	40.0000	1.75750
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	6.00000	0.261612
$527$	60.0000	2.61364
$528$	0	0
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	0	0
$532$	16.0000	0.693688
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−12.0000	−0.518805
$536$	14.0000	0.604708
$537$	0	0
$538$	30.0000	1.29339
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	−6.00000	−0.257248
$545$	4.00000	0.171341
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	−32.0000	−1.36078
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	4.00000	0.169031
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	6.00000	0.250217
$576$	0	0
$577$	−40.0000	−1.66522	−0.832611	−	0.553858i	$-0.813155\pi$
$577$	−40.0000	−1.66522	−0.832611	+	0.553858i	$0.813155\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	6.00000	0.249136
$581$	−48.0000	−1.99138
$582$	0	0
$583$	0	0
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	12.0000	0.494032
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	6.00000	0.245770
$597$	0	0
$598$	6.00000	0.245358
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	2.00000	0.0813788
$605$	11.0000	0.447214
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	10.0000	0.404888
$611$	12.0000	0.485468
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	12.0000	0.481156
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	2.00000	0.0798087
$629$	60.0000	2.39236
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−6.00000	−0.238290
$635$	16.0000	0.634941
$636$	0	0
$637$	9.00000	0.356593
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	24.0000	0.944267
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	2.00000	0.0783260
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−48.0000	−1.87123
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	12.0000	0.465690
$665$	−16.0000	−0.620453
$666$	0	0
$667$	−36.0000	−1.39393
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−14.0000	−0.540867
$671$	0	0
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	−12.0000	−0.455514
$695$	4.00000	0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	8.00000	0.302804
$699$	0	0
$700$	−4.00000	−0.151186
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	−24.0000	−0.902613
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	−60.0000	−2.24702
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	−3.00000	−0.111648
$723$	0	0
$724$	2.00000	0.0743294
$725$	−6.00000	−0.222834
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−4.00000	−0.148250
$729$	0	0
$730$	16.0000	0.592187
$731$	24.0000	0.887672
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	−48.0000	−1.76214
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−34.0000	−1.24483
$747$	0	0
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−6.00000	−0.218507
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	4.00000	0.145095
$761$	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
$763$	16.0000	0.579239
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	−16.0000	−0.575853
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	8.00000	0.287183
$777$	0	0
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	−36.0000	−1.28736
$783$	0	0
$784$	9.00000	0.321429
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−8.00000	−0.284627
$791$	24.0000	0.853342
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−10.0000	−0.354887
$795$	0	0
$796$	20.0000	0.708881
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	1.00000	0.0353553
$801$	0	0
$802$	−18.0000	−0.635602
$803$	0	0
$804$	0	0
$805$	24.0000	0.845889
$806$	−10.0000	−0.352235
$807$	0	0
$808$	6.00000	0.211079
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	16.0000	0.559769
$818$	−22.0000	−0.769212
$819$	0	0
$820$	6.00000	0.209529
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	48.0000	1.67013
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	1.00000	0.0346688
$833$	−54.0000	−1.87099
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	−18.0000	−0.621800
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−16.0000	−0.551396
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	44.0000	1.51186
$848$	12.0000	0.412082
$849$	0	0
$850$	−6.00000	−0.205798
$851$	−60.0000	−2.05677
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	24.0000	0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	40.0000	1.35769
$869$	0	0
$870$	0	0
$871$	14.0000	0.474372
$872$	−4.00000	−0.135457
$873$	0	0
$874$	−24.0000	−0.811812
$875$	4.00000	0.135225
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	0	0
$887$	54.0000	1.81314	0.906571	−	0.422053i	$-0.138690\pi$
$887$	54.0000	1.81314	0.906571	+	0.422053i	$0.138690\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	−6.00000	−0.201120
$891$	0	0
$892$	8.00000	0.267860
$893$	−48.0000	−1.60626
$894$	0	0
$895$	6.00000	0.200558
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−42.0000	−1.40156
$899$	60.0000	2.00111
$900$	0	0
$901$	−72.0000	−2.39867
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	4.00000	0.132599
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	8.00000	0.264616
$915$	0	0
$916$	−4.00000	−0.132164
$917$	−24.0000	−0.792550
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	32.0000	1.05159
$927$	0	0
$928$	−6.00000	−0.196960
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	6.00000	0.196537
$933$	0	0
$934$	−24.0000	−0.785304
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−56.0000	−1.82846
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−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
$943$	−36.0000	−1.17232
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	−4.00000	−0.129777
$951$	0	0
$952$	24.0000	0.777844
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	0	0
$958$	24.0000	0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	69.0000	2.22581
$962$	−10.0000	−0.322413
$963$	0	0
$964$	−10.0000	−0.322078
$965$	16.0000	0.515058
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−8.00000	−0.256865
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	20.0000	0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	0	0
$980$	−9.00000	−0.287494
$981$	0	0
$982$	18.0000	0.574403
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	36.0000	1.14647
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	0	0
$995$	−20.0000	−0.634043
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.h.1.1	yes	1
3.2	odd	2		1170.2.a.c.1.1	✓	1
4.3	odd	2		9360.2.a.ba.1.1		1
5.2	odd	4		5850.2.e.t.5149.2		2
5.3	odd	4		5850.2.e.t.5149.1		2
5.4	even	2		5850.2.a.z.1.1		1
12.11	even	2		9360.2.a.bz.1.1		1
15.2	even	4		5850.2.e.n.5149.1		2
15.8	even	4		5850.2.e.n.5149.2		2
15.14	odd	2		5850.2.a.ca.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.c.1.1	✓	1	3.2	odd	2
1170.2.a.h.1.1	yes	1	1.1	even	1	trivial
5850.2.a.z.1.1		1	5.4	even	2
5850.2.a.ca.1.1		1	15.14	odd	2
5850.2.e.n.5149.1		2	15.2	even	4
5850.2.e.n.5149.2		2	15.8	even	4
5850.2.e.t.5149.1		2	5.3	odd	4
5850.2.e.t.5149.2		2	5.2	odd	4
9360.2.a.ba.1.1		1	4.3	odd	2
9360.2.a.bz.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

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