LMFDB - Embedded newform 368.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,2,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, 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("368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	368.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:	$2.93849479438$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} +7.00000 q^{13} -2.00000 q^{15} -4.00000 q^{17} +6.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +5.00000 q^{29} -3.00000 q^{31} +2.00000 q^{33} -8.00000 q^{35} +2.00000 q^{37} +7.00000 q^{39} -9.00000 q^{41} -8.00000 q^{43} +4.00000 q^{45} +1.00000 q^{47} +9.00000 q^{49} -4.00000 q^{51} -6.00000 q^{53} -4.00000 q^{55} +6.00000 q^{57} +8.00000 q^{59} -10.0000 q^{61} -8.00000 q^{63} -14.0000 q^{65} -2.00000 q^{67} +1.00000 q^{69} +13.0000 q^{71} -3.00000 q^{73} -1.00000 q^{75} +8.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} +8.00000 q^{85} +5.00000 q^{87} -4.00000 q^{89} +28.0000 q^{91} -3.00000 q^{93} -12.0000 q^{95} -8.00000 q^{97} -4.00000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	7.00000	1.12090
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−8.00000	−1.00791
$64$	0	0
$65$	−14.0000	−1.73649
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	5.00000	0.536056
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	28.0000	2.93520
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−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
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−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
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−14.0000	−1.29430
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	0	0
$143$	14.0000	1.17074
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	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
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	0	0
$195$	−14.0000	−1.00256
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	20.0000	1.40372
$204$	0	0
$205$	18.0000	1.25717
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	13.0000	0.890745
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	−28.0000	−1.88348
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	42.0000	2.67240
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	8.00000	0.500979
$256$	0	0
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	28.0000	1.69464
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	−36.0000	−2.12501
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	7.00000	0.404820
$300$	0	0
$301$	−32.0000	−1.84445
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	20.0000	1.14520
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	16.0000	0.901498
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−7.00000	−0.388290
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	−35.0000	−1.86816
$352$	0	0
$353$	33.0000	1.75641	0.878206	−	0.478282i	$-0.158740\pi$
$353$	33.0000	1.75641	0.878206	+	0.478282i	$0.158740\pi$
$354$	0	0
$355$	−26.0000	−1.37994
$356$	0	0
$357$	−16.0000	−0.846810
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	35.0000	1.80259
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−21.0000	−1.04608
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	20.0000	0.986527
$412$	0	0
$413$	32.0000	1.57462
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.0000	0.930434
$418$	0	0
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−40.0000	−1.93574
$428$	0	0
$429$	14.0000	0.675926
$430$	0	0
$431$	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
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	−10.0000	−0.479463
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	25.0000	1.18779	0.593893	−	0.804544i	$-0.297590\pi$
$443$	25.0000	1.18779	0.593893	+	0.804544i	$0.297590\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	−11.0000	−0.516825
$454$	0	0
$455$	−56.0000	−2.62532
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	1.00000	0.0465746	0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000	0.0465746	0.0232873	+	0.999729i	$0.492587\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	12.0000	0.549442
$478$	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
$481$	14.0000	0.638345
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	52.0000	2.33252
$498$	0	0
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	36.0000	1.59882
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	−30.0000	−1.32453
$514$	0	0
$515$	32.0000	1.41009
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−16.0000	−0.694341
$532$	0	0
$533$	−63.0000	−2.72883
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−5.00000	−0.215766
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.0000	1.66752	0.833760	−	0.552127i	$-0.186184\pi$
$547$	39.0000	1.66752	0.833760	+	0.552127i	$0.186184\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−56.0000	−2.36855
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	0	0
$579$	1.00000	0.0415586
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	28.0000	1.15766
$586$	0	0
$587$	17.0000	0.701665	0.350833	−	0.936438i	$-0.385899\pi$
$587$	17.0000	0.701665	0.350833	+	0.936438i	$0.385899\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	32.0000	1.31187
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	0	0
$609$	20.0000	0.810441
$610$	0	0
$611$	7.00000	0.283190
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−14.0000	−0.555573
$636$	0	0
$637$	63.0000	2.49615
$638$	0	0
$639$	−26.0000	−1.02854
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	41.0000	1.61188	0.805938	−	0.592000i	$-0.201661\pi$
$647$	41.0000	1.61188	0.805938	+	0.592000i	$0.201661\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	15.0000	0.586995	0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000	0.586995	0.293498	+	0.955960i	$0.405181\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−28.0000	−1.08743
$664$	0	0
$665$	−48.0000	−1.86136
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−37.0000	−1.41577	−0.707883	−	0.706330i	$-0.750350\pi$
$683$	−37.0000	−1.41577	−0.707883	+	0.706330i	$0.750350\pi$
$684$	0	0
$685$	−40.0000	−1.52832
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	−42.0000	−1.60007
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	−16.0000	−0.607790
$694$	0	0
$695$	−38.0000	−1.44142
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	0	0
$707$	−56.0000	−2.10610
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−3.00000	−0.112351
$714$	0	0
$715$	−28.0000	−1.04714
$716$	0	0
$717$	−19.0000	−0.709568
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	0	0
$741$	42.0000	1.54291
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.0000	0.800662
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	5.00000	0.181250	0.0906249	−	0.995885i	$-0.471114\pi$
$761$	5.00000	0.181250	0.0906249	+	0.995885i	$0.471114\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−16.0000	−0.578481
$766$	0	0
$767$	56.0000	2.02204
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	0	0
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	−54.0000	−1.93475
$780$	0	0
$781$	26.0000	0.930353
$782$	0	0
$783$	−25.0000	−0.893427
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	0	0
$789$	−26.0000	−0.925625
$790$	0	0
$791$	−48.0000	−1.70668
$792$	0	0
$793$	−70.0000	−2.48577
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	−56.0000	−1.95680
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	17.0000	0.589723
$832$	0	0
$833$	−36.0000	−1.24733
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	15.0000	0.518476
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	−72.0000	−2.47688
$846$	0	0
$847$	−28.0000	−0.962091
$848$	0	0
$849$	22.0000	0.755038
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	43.0000	1.46885	0.734426	−	0.678689i	$-0.237451\pi$
$857$	43.0000	1.46885	0.734426	+	0.678689i	$0.237451\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−39.0000	−1.30949	−0.654746	−	0.755849i	$-0.727224\pi$
$887$	−39.0000	−1.30949	−0.654746	+	0.755849i	$0.727224\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	7.00000	0.233723
$898$	0	0
$899$	−15.0000	−0.500278
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	−32.0000	−1.06489
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	0	0
$909$	28.0000	0.928701
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.0000	0.661180
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	91.0000	2.99530
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	32.0000	1.05102
$928$	0	0
$929$	−41.0000	−1.34517	−0.672583	−	0.740022i	$-0.734815\pi$
$929$	−41.0000	−1.34517	−0.672583	+	0.740022i	$0.734815\pi$
$930$	0	0
$931$	54.0000	1.76978
$932$	0	0
$933$	−3.00000	−0.0982156
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	0	0
$945$	40.0000	1.30120
$946$	0	0
$947$	49.0000	1.59229	0.796143	−	0.605108i	$-0.206870\pi$
$947$	49.0000	1.59229	0.796143	+	0.605108i	$0.206870\pi$
$948$	0	0
$949$	−21.0000	−0.681689
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	80.0000	2.58333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	76.0000	2.43645
$974$	0	0
$975$	−7.00000	−0.224179
$976$	0	0
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	4.00000	0.127321
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	−28.0000	−0.887660
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.2.a.f.1.1		1
3.2	odd	2		3312.2.a.o.1.1		1
4.3	odd	2		184.2.a.b.1.1	✓	1
5.4	even	2		9200.2.a.l.1.1		1
8.3	odd	2		1472.2.a.k.1.1		1
8.5	even	2		1472.2.a.d.1.1		1
12.11	even	2		1656.2.a.g.1.1		1
20.3	even	4		4600.2.e.f.4049.1		2
20.7	even	4		4600.2.e.f.4049.2		2
20.19	odd	2		4600.2.a.k.1.1		1
23.22	odd	2		8464.2.a.m.1.1		1
28.27	even	2		9016.2.a.j.1.1		1
92.91	even	2		4232.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.b.1.1	✓	1	4.3	odd	2
368.2.a.f.1.1		1	1.1	even	1	trivial
1472.2.a.d.1.1		1	8.5	even	2
1472.2.a.k.1.1		1	8.3	odd	2
1656.2.a.g.1.1		1	12.11	even	2
3312.2.a.o.1.1		1	3.2	odd	2
4232.2.a.e.1.1		1	92.91	even	2
4600.2.a.k.1.1		1	20.19	odd	2
4600.2.e.f.4049.1		2	20.3	even	4
4600.2.e.f.4049.2		2	20.7	even	4
8464.2.a.m.1.1		1	23.22	odd	2
9016.2.a.j.1.1		1	28.27	even	2
9200.2.a.l.1.1		1	5.4	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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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