LMFDB - Embedded newform 456.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.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:	$3.64117833217$
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	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -3.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{13} -3.00000 q^{15} -5.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{33} +9.00000 q^{35} +8.00000 q^{37} -2.00000 q^{39} -8.00000 q^{41} +13.0000 q^{43} -3.00000 q^{45} +13.0000 q^{47} +2.00000 q^{49} -5.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +1.00000 q^{57} +4.00000 q^{59} -13.0000 q^{61} -3.00000 q^{63} +6.00000 q^{65} +4.00000 q^{67} -4.00000 q^{69} -8.00000 q^{71} -3.00000 q^{73} +4.00000 q^{75} +3.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +15.0000 q^{85} -6.00000 q^{87} -6.00000 q^{89} +6.00000 q^{91} -2.00000 q^{93} -3.00000 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.00000	−0.654654
$22$	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
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	9.00000	1.52128
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	13.0000	1.98248	0.991241	−	0.132068i	$-0.0421616\pi$
$43$	13.0000	1.98248	0.991241	+	0.132068i	$0.0421616\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−5.00000	−0.700140
$52$	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
$55$	3.00000	0.404520
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	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
$75$	4.00000	0.461880
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	15.0000	1.62698
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	9.00000	0.878310
$106$	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
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	13.0000	1.14459
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	13.0000	1.09480
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−26.0000	−1.97674	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−26.0000	−1.97674	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−39.0000	−2.65978
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	−39.0000	−2.54408
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	15.0000	0.939336
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−15.0000	−0.901263	−0.450631	−	0.892710i	$-0.648801\pi$
$277$	−15.0000	−0.901263	−0.450631	+	0.892710i	$0.648801\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	33.0000	1.96165	0.980823	−	0.194900i	$-0.0624381\pi$
$283$	33.0000	1.96165	0.980823	+	0.194900i	$0.0624381\pi$
$284$	0	0
$285$	−3.00000	−0.177705
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−39.0000	−2.24792
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	39.0000	2.23313
$306$	0	0
$307$	−24.0000	−1.36975	−0.684876	−	0.728659i	$-0.740144\pi$
$307$	−24.0000	−1.36975	−0.684876	+	0.728659i	$0.740144\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	9.00000	0.507093
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−5.00000	−0.278207
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	−39.0000	−2.15014
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−7.00000	−0.375780	−0.187890	−	0.982190i	$-0.560165\pi$
$347$	−7.00000	−0.375780	−0.187890	+	0.982190i	$0.560165\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	15.0000	0.793884
$358$	0	0
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	18.0000	0.934513
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	0	0
$387$	13.0000	0.660827
$388$	0	0
$389$	13.0000	0.659126	0.329563	−	0.944134i	$-0.393099\pi$
$389$	13.0000	0.659126	0.329563	+	0.944134i	$0.393099\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−5.00000	−0.246632
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−15.0000	−0.734553
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	13.0000	0.632082
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	39.0000	1.88734
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	−5.00000	−0.236492
$448$	0	0
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	−18.0000	−0.843853
$456$	0	0
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	0	0
$463$	7.00000	0.325318	0.162659	−	0.986682i	$-0.447993\pi$
$463$	7.00000	0.325318	0.162659	+	0.986682i	$0.447993\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−13.0000	−0.597741
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	0	0
$501$	−14.0000	−0.625474
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−30.0000	−1.33498
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	18.0000	0.793175
$516$	0	0
$517$	−13.0000	−0.571739
$518$	0	0
$519$	−26.0000	−1.14127
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	0	0
$549$	−13.0000	−0.554826
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	−24.0000	−1.01874
$556$	0	0
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	0	0
$559$	−26.0000	−1.09968
$560$	0	0
$561$	5.00000	0.211100
$562$	0	0
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	−5.00000	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$577$	−5.00000	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	25.0000	1.03186	0.515930	−	0.856631i	$-0.327446\pi$
$587$	25.0000	1.03186	0.515930	+	0.856631i	$0.327446\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	22.0000	0.904959
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	−45.0000	−1.84482
$596$	0	0
$597$	−3.00000	−0.122782
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	30.0000	1.21967
$606$	0	0
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	−26.0000	−1.05185
$612$	0	0
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	0	0
$629$	−40.0000	−1.59490
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	29.0000	1.14365	0.571824	−	0.820376i	$-0.306236\pi$
$643$	29.0000	1.14365	0.571824	+	0.820376i	$0.306236\pi$
$644$	0	0
$645$	−39.0000	−1.53562
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	0	0
$657$	−3.00000	−0.117041
$658$	0	0
$659$	−50.0000	−1.94772	−0.973862	−	0.227142i	$-0.927062\pi$
$659$	−50.0000	−1.94772	−0.973862	+	0.227142i	$0.927062\pi$
$660$	0	0
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	0	0
$663$	10.0000	0.388368
$664$	0	0
$665$	9.00000	0.349005
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	13.0000	0.501859
$672$	0	0
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	0	0
$685$	15.0000	0.573121
$686$	0	0
$687$	−19.0000	−0.724895
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−21.0000	−0.798878	−0.399439	−	0.916760i	$-0.630795\pi$
$691$	−21.0000	−0.798878	−0.399439	+	0.916760i	$0.630795\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	45.0000	1.70695
$696$	0	0
$697$	40.0000	1.51511
$698$	0	0
$699$	−19.0000	−0.718646
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−39.0000	−1.46882
$706$	0	0
$707$	−30.0000	−1.12827
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	−9.00000	−0.336111
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	12.0000	0.446285
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−65.0000	−2.40411
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	9.00000	0.327978
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	11.0000	0.399802	0.199901	−	0.979816i	$-0.435938\pi$
$757$	11.0000	0.399802	0.199901	+	0.979816i	$0.435938\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	11.0000	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$761$	11.0000	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	15.0000	0.542326
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	−40.0000	−1.43870	−0.719350	−	0.694648i	$-0.755560\pi$
$773$	−40.0000	−1.43870	−0.719350	+	0.694648i	$0.755560\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−42.0000	−1.49904
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	5.00000	0.178005
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	26.0000	0.923287
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−65.0000	−2.29953
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	3.00000	0.105868
$804$	0	0
$805$	−36.0000	−1.26883
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	−28.0000	−0.982003
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	13.0000	0.454812
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	11.0000	0.383903	0.191951	−	0.981404i	$-0.438518\pi$
$821$	11.0000	0.383903	0.191951	+	0.981404i	$0.438518\pi$
$822$	0	0
$823$	37.0000	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$823$	37.0000	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	0	0
$831$	−15.0000	−0.520344
$832$	0	0
$833$	−10.0000	−0.346479
$834$	0	0
$835$	42.0000	1.45347
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−38.0000	−1.31191	−0.655953	−	0.754802i	$-0.727733\pi$
$839$	−38.0000	−1.31191	−0.655953	+	0.754802i	$0.727733\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	27.0000	0.928828
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	33.0000	1.13256
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	78.0000	2.65208
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	28.0000	0.944417
$880$	0	0
$881$	−1.00000	−0.0336909	−0.0168454	−	0.999858i	$-0.505362\pi$
$881$	−1.00000	−0.0336909	−0.0168454	+	0.999858i	$0.505362\pi$
$882$	0	0
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	13.0000	0.435028
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	−39.0000	−1.29784
$904$	0	0
$905$	42.0000	1.39613
$906$	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
$909$	10.0000	0.331679
$910$	0	0
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	39.0000	1.28930
$916$	0	0
$917$	45.0000	1.48603
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	9.00000	0.294647
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	−35.0000	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$937$	−35.0000	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−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
$943$	32.0000	1.04206
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	24.0000	0.778253
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	−33.0000	−1.06785
$956$	0	0
$957$	6.00000	0.193952
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	72.0000	2.31776
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	−5.00000	−0.160623
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	45.0000	1.44263
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	−66.0000	−2.10293
$986$	0	0
$987$	−39.0000	−1.24138
$988$	0	0
$989$	−52.0000	−1.65350
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	9.00000	0.285319
$996$	0	0
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.a.c.1.1	✓	1
3.2	odd	2		1368.2.a.i.1.1		1
4.3	odd	2		912.2.a.a.1.1		1
8.3	odd	2		3648.2.a.bj.1.1		1
8.5	even	2		3648.2.a.p.1.1		1
12.11	even	2		2736.2.a.u.1.1		1
19.18	odd	2		8664.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.c.1.1	✓	1	1.1	even	1	trivial
912.2.a.a.1.1		1	4.3	odd	2
1368.2.a.i.1.1		1	3.2	odd	2
2736.2.a.u.1.1		1	12.11	even	2
3648.2.a.p.1.1		1	8.5	even	2
3648.2.a.bj.1.1		1	8.3	odd	2
8664.2.a.a.1.1		1	19.18	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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

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L-functions

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Representations

Groups

Database

Properties

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