LMFDB - Embedded newform 338.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,2,Mod(1,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.69894358832$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	338.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -1.00000 q^{12} -3.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -6.00000 q^{19} -3.00000 q^{20} +3.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{27} -3.00000 q^{28} +3.00000 q^{30} +1.00000 q^{32} -3.00000 q^{34} +9.00000 q^{35} -2.00000 q^{36} -3.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} +3.00000 q^{42} +1.00000 q^{43} +6.00000 q^{45} +6.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} -6.00000 q^{53} +5.00000 q^{54} -3.00000 q^{56} +6.00000 q^{57} +6.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} +6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +9.00000 q^{70} +15.0000 q^{71} -2.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} +10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} +3.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} +6.00000 q^{89} +6.00000 q^{90} +6.00000 q^{92} -3.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} -12.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−3.00000	−0.670820
$21$	3.00000	0.654654
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	3.00000	0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	9.00000	1.52128
$36$	−2.00000	−0.333333
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	3.00000	0.462910
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	6.00000	0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	3.00000	0.420084
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−3.00000	−0.400892
$57$	6.00000	0.794719
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	3.00000	0.387298
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−3.00000	−0.363803
$69$	−6.00000	−0.722315
$70$	9.00000	1.07571
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	−2.00000	−0.235702
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−3.00000	−0.348743
$75$	−4.00000	−0.461880
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	3.00000	0.327327
$85$	9.00000	0.976187
$86$	1.00000	0.107833
$87$	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	$0.396989\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	−3.00000	−0.309426
$95$	18.0000	1.84676
$96$	−1.00000	−0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	3.00000	0.297044
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−9.00000	−0.878310
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	5.00000	0.481125
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	−3.00000	−0.283473
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	6.00000	0.561951
$115$	−18.0000	−1.67851
$116$	0	0
$117$	0	0
$118$	6.00000	0.552345
$119$	9.00000	0.825029
$120$	3.00000	0.273861
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	6.00000	0.534522
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	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
$133$	18.0000	1.56080
$134$	−12.0000	−1.03664
$135$	−15.0000	−1.29099
$136$	−3.00000	−0.257248
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−6.00000	−0.510754
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	9.00000	0.760639
$141$	3.00000	0.252646
$142$	15.0000	1.25877
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−6.00000	−0.496564
$147$	−2.00000	−0.164957
$148$	−3.00000	−0.246598
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−4.00000	−0.326599
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	−6.00000	−0.486664
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	10.0000	0.795557
$159$	6.00000	0.475831
$160$	−3.00000	−0.237171
$161$	−18.0000	−1.41860
$162$	1.00000	0.0785674
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	9.00000	0.690268
$171$	12.0000	0.917663
$172$	1.00000	0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	−6.00000	−0.450988
$178$	6.00000	0.449719
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	6.00000	0.447214
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	6.00000	0.442326
$185$	9.00000	0.661693
$186$	0	0
$187$	0	0
$188$	−3.00000	−0.218797
$189$	−15.0000	−1.09109
$190$	18.0000	1.30586
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	4.00000	0.282843
$201$	12.0000	0.846415
$202$	−12.0000	−0.844317
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	−14.0000	−0.975426
$207$	−12.0000	−0.834058
$208$	0	0
$209$	0	0
$210$	−9.00000	−0.621059
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	−6.00000	−0.412082
$213$	−15.0000	−1.02778
$214$	−12.0000	−0.820303
$215$	−3.00000	−0.204598
$216$	5.00000	0.340207
$217$	0	0
$218$	9.00000	0.609557
$219$	6.00000	0.405442
$220$	0	0
$221$	0	0
$222$	3.00000	0.201347
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−3.00000	−0.200446
$225$	−8.00000	−0.533333
$226$	−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	$0.630377\pi$
$228$	6.00000	0.397360
$229$	−9.00000	−0.594737	−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000	−0.594737	−0.297368	+	0.954763i	$0.596109\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	6.00000	0.390567
$237$	−10.0000	−0.649570
$238$	9.00000	0.583383
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	3.00000	0.193649
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	−8.00000	−0.512148
$245$	−6.00000	−0.383326
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	3.00000	0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	6.00000	0.377964
$253$	0	0
$254$	2.00000	0.125491
$255$	−9.00000	−0.563602
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−1.00000	−0.0622573
$259$	9.00000	0.559233
$260$	0	0
$261$	0	0
$262$	−3.00000	−0.185341
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	18.0000	1.10365
$267$	−6.00000	−0.367194
$268$	−12.0000	−0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−15.0000	−0.912871
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	5.00000	0.299880
$279$	0	0
$280$	9.00000	0.537853
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	3.00000	0.178647
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	15.0000	0.890086
$285$	−18.0000	−1.06623
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	12.0000	0.703452
$292$	−6.00000	−0.351123
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−2.00000	−0.116642
$295$	−18.0000	−1.04800
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−4.00000	−0.230940
$301$	−3.00000	−0.172917
$302$	15.0000	0.863153
$303$	12.0000	0.689382
$304$	−6.00000	−0.344124
$305$	24.0000	1.37424
$306$	6.00000	0.342997
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−22.0000	−1.24153
$315$	−18.0000	−1.01419
$316$	10.0000	0.562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−3.00000	−0.167705
$321$	12.0000	0.669775
$322$	−18.0000	−1.00310
$323$	18.0000	1.00155
$324$	1.00000	0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	−9.00000	−0.497701
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	6.00000	0.329293
$333$	6.00000	0.328798
$334$	12.0000	0.656611
$335$	36.0000	1.96689
$336$	3.00000	0.163663
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	9.00000	0.488094
$341$	0	0
$342$	12.0000	0.648886
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	18.0000	0.969087
$346$	6.00000	0.322562
$347$	33.0000	1.77153	0.885766	−	0.464131i	$-0.153633\pi$
$347$	33.0000	1.77153	0.885766	+	0.464131i	$0.153633\pi$
$348$	0	0
$349$	21.0000	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$349$	21.0000	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$350$	−12.0000	−0.641427
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−6.00000	−0.318896
$355$	−45.0000	−2.38835
$356$	6.00000	0.317999
$357$	−9.00000	−0.476331
$358$	−15.0000	−0.792775
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	6.00000	0.316228
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	18.0000	0.942163
$366$	8.00000	0.418167
$367$	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
$370$	9.00000	0.467888
$371$	18.0000	0.934513
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−15.0000	−0.771517
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	18.0000	0.923381
$381$	−2.00000	−0.102463
$382$	12.0000	0.613973
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−2.00000	−0.101666
$388$	−12.0000	−0.609208
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	2.00000	0.101015
$393$	3.00000	0.151330
$394$	3.00000	0.151138
$395$	−30.0000	−1.50946
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	20.0000	1.00251
$399$	−18.0000	−0.901127
$400$	4.00000	0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	3.00000	0.148522
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	−14.0000	−0.689730
$413$	−18.0000	−0.885722
$414$	−12.0000	−0.589768
$415$	−18.0000	−0.883585
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	−9.00000	−0.439155
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	−23.0000	−1.11962
$423$	6.00000	0.291730
$424$	−6.00000	−0.291386
$425$	−12.0000	−0.582086
$426$	−15.0000	−0.726752
$427$	24.0000	1.16144
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−3.00000	−0.144673
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	5.00000	0.240563
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	9.00000	0.431022
$437$	−36.0000	−1.72211
$438$	6.00000	0.286691
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	3.00000	0.142374
$445$	−18.0000	−0.853282
$446$	−9.00000	−0.426162
$447$	6.00000	0.283790
$448$	−3.00000	−0.141737
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−15.0000	−0.704761
$454$	−12.0000	−0.563188
$455$	0	0
$456$	6.00000	0.280976
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−9.00000	−0.420542
$459$	−15.0000	−0.700140
$460$	−18.0000	−0.839254
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	21.0000	0.972806
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	9.00000	0.415139
$471$	22.0000	1.01371
$472$	6.00000	0.276172
$473$	0	0
$474$	−10.0000	−0.459315
$475$	−24.0000	−1.10120
$476$	9.00000	0.412514
$477$	12.0000	0.549442
$478$	9.00000	0.411650
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	3.00000	0.136931
$481$	0	0
$482$	30.0000	1.36646
$483$	18.0000	0.819028
$484$	−11.0000	−0.500000
$485$	36.0000	1.63468
$486$	−16.0000	−0.725775
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−8.00000	−0.362143
$489$	6.00000	0.271329
$490$	−6.00000	−0.271052
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−45.0000	−2.01853
$498$	−6.00000	−0.268866
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	3.00000	0.134164
$501$	−12.0000	−0.536120
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	6.00000	0.267261
$505$	36.0000	1.60198
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−9.00000	−0.398527
$511$	18.0000	0.796273
$512$	1.00000	0.0441942
$513$	−30.0000	−1.32453
$514$	−3.00000	−0.132324
$515$	42.0000	1.85074
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	9.00000	0.395437
$519$	−6.00000	−0.263371
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−3.00000	−0.131056
$525$	12.0000	0.523723
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	18.0000	0.781870
$531$	−12.0000	−0.520756
$532$	18.0000	0.780399
$533$	0	0
$534$	−6.00000	−0.259645
$535$	36.0000	1.55642
$536$	−12.0000	−0.518321
$537$	15.0000	0.647298
$538$	0	0
$539$	0	0
$540$	−15.0000	−0.645497
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	−15.0000	−0.644305
$543$	2.00000	0.0858282
$544$	−3.00000	−0.128624
$545$	−27.0000	−1.15655
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	−18.0000	−0.768922
$549$	16.0000	0.682863
$550$	0	0
$551$	0	0
$552$	−6.00000	−0.255377
$553$	−30.0000	−1.27573
$554$	−8.00000	−0.339887
$555$	−9.00000	−0.382029
$556$	5.00000	0.212047
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	0	0
$559$	0	0
$560$	9.00000	0.380319
$561$	0	0
$562$	30.0000	1.26547
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	3.00000	0.126323
$565$	18.0000	0.757266
$566$	−4.00000	−0.168133
$567$	−3.00000	−0.125988
$568$	15.0000	0.629386
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	−18.0000	−0.753937
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	24.0000	1.00087
$576$	−2.00000	−0.0833333
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−8.00000	−0.332756
$579$	6.00000	0.249351
$580$	0	0
$581$	−18.0000	−0.746766
$582$	12.0000	0.497416
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	9.00000	0.371787
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	−18.0000	−0.741048
$591$	−3.00000	−0.123404
$592$	−3.00000	−0.123299
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	−27.0000	−1.10689
$596$	−6.00000	−0.245770
$597$	−20.0000	−0.818546
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−4.00000	−0.163299
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	−3.00000	−0.122271
$603$	24.0000	0.977356
$604$	15.0000	0.610341
$605$	33.0000	1.34164
$606$	12.0000	0.487467
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	24.0000	0.971732
$611$	0	0
$612$	6.00000	0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	14.0000	0.563163
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	18.0000	0.721734
$623$	−18.0000	−0.721155
$624$	0	0
$625$	−29.0000	−1.16000
$626$	19.0000	0.759393
$627$	0	0
$628$	−22.0000	−0.877896
$629$	9.00000	0.358854
$630$	−18.0000	−0.717137
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	10.0000	0.397779
$633$	23.0000	0.914168
$634$	18.0000	0.714871
$635$	−6.00000	−0.238103
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	−30.0000	−1.18678
$640$	−3.00000	−0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	−18.0000	−0.709299
$645$	3.00000	0.118125
$646$	18.0000	0.708201
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	−9.00000	−0.351928
$655$	9.00000	0.351659
$656$	0	0
$657$	12.0000	0.468165
$658$	9.00000	0.350857
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	30.0000	1.16598
$663$	0	0
$664$	6.00000	0.232845
$665$	−54.0000	−2.09403
$666$	6.00000	0.232495
$667$	0	0
$668$	12.0000	0.464294
$669$	9.00000	0.347960
$670$	36.0000	1.39080
$671$	0	0
$672$	3.00000	0.115728
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−13.0000	−0.500741
$675$	20.0000	0.769800
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	6.00000	0.230429
$679$	36.0000	1.38155
$680$	9.00000	0.345134
$681$	12.0000	0.459841
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	12.0000	0.458831
$685$	54.0000	2.06323
$686$	15.0000	0.572703
$687$	9.00000	0.343371
$688$	1.00000	0.0381246
$689$	0	0
$690$	18.0000	0.685248
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	33.0000	1.25266
$695$	−15.0000	−0.568982
$696$	0	0
$697$	0	0
$698$	21.0000	0.794862
$699$	−21.0000	−0.794293
$700$	−12.0000	−0.453557
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	−9.00000	−0.338960
$706$	6.00000	0.225813
$707$	36.0000	1.35392
$708$	−6.00000	−0.225494
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−45.0000	−1.68882
$711$	−20.0000	−0.750059
$712$	6.00000	0.224860
$713$	0	0
$714$	−9.00000	−0.336817
$715$	0	0
$716$	−15.0000	−0.560576
$717$	−9.00000	−0.336111
$718$	−24.0000	−0.895672
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	6.00000	0.223607
$721$	42.0000	1.56416
$722$	17.0000	0.632674
$723$	−30.0000	−1.11571
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	11.0000	0.408248
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	18.0000	0.666210
$731$	−3.00000	−0.110959
$732$	8.00000	0.295689
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	8.00000	0.295285
$735$	6.00000	0.221313
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	9.00000	0.330847
$741$	0	0
$742$	18.0000	0.660801
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	4.00000	0.146450
$747$	−12.0000	−0.439057
$748$	0	0
$749$	36.0000	1.31541
$750$	−3.00000	−0.109545
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−3.00000	−0.109399
$753$	12.0000	0.437304
$754$	0	0
$755$	−45.0000	−1.63772
$756$	−15.0000	−0.545545
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	18.0000	0.652929
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−2.00000	−0.0724524
$763$	−27.0000	−0.977466
$764$	12.0000	0.434145
$765$	−18.0000	−0.650791
$766$	−9.00000	−0.325183
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	−6.00000	−0.215945
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−12.0000	−0.430775
$777$	−9.00000	−0.322873
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−18.0000	−0.643679
$783$	0	0
$784$	2.00000	0.0714286
$785$	66.0000	2.35564
$786$	3.00000	0.107006
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	3.00000	0.106871
$789$	−24.0000	−0.854423
$790$	−30.0000	−1.06735
$791$	18.0000	0.640006
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	−18.0000	−0.638394
$796$	20.0000	0.708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−18.0000	−0.637193
$799$	9.00000	0.318397
$800$	4.00000	0.141421
$801$	−12.0000	−0.423999
$802$	−30.0000	−1.05934
$803$	0	0
$804$	12.0000	0.423207
$805$	54.0000	1.90325
$806$	0	0
$807$	0	0
$808$	−12.0000	−0.422159
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	−3.00000	−0.105409
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	15.0000	0.526073
$814$	0	0
$815$	18.0000	0.630512
$816$	3.00000	0.105021
$817$	−6.00000	−0.209913
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	18.0000	0.627822
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−18.0000	−0.626300
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	−12.0000	−0.417029
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−18.0000	−0.624789
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	−5.00000	−0.173136
$835$	−36.0000	−1.24583
$836$	0	0
$837$	0	0
$838$	15.0000	0.518166
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	−9.00000	−0.310530
$841$	−29.0000	−1.00000
$842$	−15.0000	−0.516934
$843$	−30.0000	−1.03325
$844$	−23.0000	−0.791693
$845$	0	0
$846$	6.00000	0.206284
$847$	33.0000	1.13389
$848$	−6.00000	−0.206041
$849$	4.00000	0.137280
$850$	−12.0000	−0.411597
$851$	−18.0000	−0.617032
$852$	−15.0000	−0.513892
$853$	39.0000	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$854$	24.0000	0.821263
$855$	−36.0000	−1.23117
$856$	−12.0000	−0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	5.00000	0.170103
$865$	−18.0000	−0.612018
$866$	11.0000	0.373795
$867$	8.00000	0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	9.00000	0.304778
$873$	24.0000	0.812277
$874$	−36.0000	−1.21772
$875$	−9.00000	−0.304256
$876$	6.00000	0.202721
$877$	3.00000	0.101303	0.0506514	−	0.998716i	$-0.483870\pi$
$877$	3.00000	0.101303	0.0506514	+	0.998716i	$0.483870\pi$
$878$	10.0000	0.337484
$879$	−9.00000	−0.303562
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	−4.00000	−0.134687
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	−21.0000	−0.705509
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	3.00000	0.100673
$889$	−6.00000	−0.201234
$890$	−18.0000	−0.603361
$891$	0	0
$892$	−9.00000	−0.301342
$893$	18.0000	0.602347
$894$	6.00000	0.200670
$895$	45.0000	1.50418
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−24.0000	−0.800890
$899$	0	0
$900$	−8.00000	−0.266667
$901$	18.0000	0.599667
$902$	0	0
$903$	3.00000	0.0998337
$904$	−6.00000	−0.199557
$905$	6.00000	0.199447
$906$	−15.0000	−0.498342
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	−12.0000	−0.398234
$909$	24.0000	0.796030
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	6.00000	0.198680
$913$	0	0
$914$	18.0000	0.595387
$915$	−24.0000	−0.793416
$916$	−9.00000	−0.297368
$917$	9.00000	0.297206
$918$	−15.0000	−0.495074
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−18.0000	−0.593442
$921$	18.0000	0.593120
$922$	−15.0000	−0.493999
$923$	0	0
$924$	0	0
$925$	−12.0000	−0.394558
$926$	24.0000	0.788689
$927$	28.0000	0.919641
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	21.0000	0.687878
$933$	−18.0000	−0.589294
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	36.0000	1.17544
$939$	−19.0000	−0.620042
$940$	9.00000	0.293548
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	22.0000	0.716799
$943$	0	0
$944$	6.00000	0.195283
$945$	45.0000	1.46385
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−10.0000	−0.324785
$949$	0	0
$950$	−24.0000	−0.778663
$951$	−18.0000	−0.583690
$952$	9.00000	0.291692
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	12.0000	0.388514
$955$	−36.0000	−1.16493
$956$	9.00000	0.291081
$957$	0	0
$958$	−39.0000	−1.26003
$959$	54.0000	1.74375
$960$	3.00000	0.0968246
$961$	−31.0000	−1.00000
$962$	0	0
$963$	24.0000	0.773389
$964$	30.0000	0.966235
$965$	18.0000	0.579441
$966$	18.0000	0.579141
$967$	3.00000	0.0964735	0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000	0.0964735	0.0482367	+	0.998836i	$0.484640\pi$
$968$	−11.0000	−0.353553
$969$	−18.0000	−0.578243
$970$	36.0000	1.15589
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	−16.0000	−0.513200
$973$	−15.0000	−0.480878
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	6.00000	0.191859
$979$	0	0
$980$	−6.00000	−0.191663
$981$	−18.0000	−0.574696
$982$	−27.0000	−0.861605
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	0	0
$987$	−9.00000	−0.286473
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	−30.0000	−0.952021
$994$	−45.0000	−1.42731
$995$	−60.0000	−1.90213
$996$	−6.00000	−0.190117
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	−36.0000	−1.13956
$999$	−15.0000	−0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.a.d.1.1		1
3.2	odd	2		3042.2.a.g.1.1		1
4.3	odd	2		2704.2.a.j.1.1		1
5.4	even	2		8450.2.a.h.1.1		1
13.2	odd	12		338.2.e.c.147.2		4
13.3	even	3		338.2.c.b.191.1		2
13.4	even	6		338.2.c.f.315.1		2
13.5	odd	4		26.2.b.a.25.1	✓	2
13.6	odd	12		338.2.e.c.23.1		4
13.7	odd	12		338.2.e.c.23.2		4
13.8	odd	4		26.2.b.a.25.2	yes	2
13.9	even	3		338.2.c.b.315.1		2
13.10	even	6		338.2.c.f.191.1		2
13.11	odd	12		338.2.e.c.147.1		4
13.12	even	2		338.2.a.b.1.1		1
39.5	even	4		234.2.b.b.181.2		2
39.8	even	4		234.2.b.b.181.1		2
39.38	odd	2		3042.2.a.j.1.1		1
52.31	even	4		208.2.f.a.129.2		2
52.47	even	4		208.2.f.a.129.1		2
52.51	odd	2		2704.2.a.k.1.1		1
65.8	even	4		650.2.c.d.649.1		2
65.18	even	4		650.2.c.a.649.1		2
65.34	odd	4		650.2.d.b.51.1		2
65.44	odd	4		650.2.d.b.51.2		2
65.47	even	4		650.2.c.a.649.2		2
65.57	even	4		650.2.c.d.649.2		2
65.64	even	2		8450.2.a.u.1.1		1
91.5	even	12		1274.2.n.c.753.1		4
91.18	odd	12		1274.2.n.d.961.2		4
91.31	even	12		1274.2.n.c.961.2		4
91.34	even	4		1274.2.d.c.883.2		2
91.44	odd	12		1274.2.n.d.753.1		4
91.47	even	12		1274.2.n.c.753.2		4
91.60	odd	12		1274.2.n.d.961.1		4
91.73	even	12		1274.2.n.c.961.1		4
91.83	even	4		1274.2.d.c.883.1		2
91.86	odd	12		1274.2.n.d.753.2		4
104.5	odd	4		832.2.f.d.129.1		2
104.21	odd	4		832.2.f.d.129.2		2
104.83	even	4		832.2.f.b.129.1		2
104.99	even	4		832.2.f.b.129.2		2
156.47	odd	4		1872.2.c.f.1585.2		2
156.83	odd	4		1872.2.c.f.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.b.a.25.1	✓	2	13.5	odd	4
26.2.b.a.25.2	yes	2	13.8	odd	4
208.2.f.a.129.1		2	52.47	even	4
208.2.f.a.129.2		2	52.31	even	4
234.2.b.b.181.1		2	39.8	even	4
234.2.b.b.181.2		2	39.5	even	4
338.2.a.b.1.1		1	13.12	even	2
338.2.a.d.1.1		1	1.1	even	1	trivial
338.2.c.b.191.1		2	13.3	even	3
338.2.c.b.315.1		2	13.9	even	3
338.2.c.f.191.1		2	13.10	even	6
338.2.c.f.315.1		2	13.4	even	6
338.2.e.c.23.1		4	13.6	odd	12
338.2.e.c.23.2		4	13.7	odd	12
338.2.e.c.147.1		4	13.11	odd	12
338.2.e.c.147.2		4	13.2	odd	12
650.2.c.a.649.1		2	65.18	even	4
650.2.c.a.649.2		2	65.47	even	4
650.2.c.d.649.1		2	65.8	even	4
650.2.c.d.649.2		2	65.57	even	4
650.2.d.b.51.1		2	65.34	odd	4
650.2.d.b.51.2		2	65.44	odd	4
832.2.f.b.129.1		2	104.83	even	4
832.2.f.b.129.2		2	104.99	even	4
832.2.f.d.129.1		2	104.5	odd	4
832.2.f.d.129.2		2	104.21	odd	4
1274.2.d.c.883.1		2	91.83	even	4
1274.2.d.c.883.2		2	91.34	even	4
1274.2.n.c.753.1		4	91.5	even	12
1274.2.n.c.753.2		4	91.47	even	12
1274.2.n.c.961.1		4	91.73	even	12
1274.2.n.c.961.2		4	91.31	even	12
1274.2.n.d.753.1		4	91.44	odd	12
1274.2.n.d.753.2		4	91.86	odd	12
1274.2.n.d.961.1		4	91.60	odd	12
1274.2.n.d.961.2		4	91.18	odd	12
1872.2.c.f.1585.1		2	156.83	odd	4
1872.2.c.f.1585.2		2	156.47	odd	4
2704.2.a.j.1.1		1	4.3	odd	2
2704.2.a.k.1.1		1	52.51	odd	2
3042.2.a.g.1.1		1	3.2	odd	2
3042.2.a.j.1.1		1	39.38	odd	2
8450.2.a.h.1.1		1	5.4	even	2
8450.2.a.u.1.1		1	65.64	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more