LMFDB - Embedded newform 315.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,2,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.51528766367$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} +1.00000 q^{20} -8.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} +8.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{52} -10.0000 q^{53} +3.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +6.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} +12.0000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +8.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} -6.00000 q^{91} +8.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -18.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.00000	1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	8.00000	1.17954
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	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	$0.421429\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	1.00000	0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	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
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	8.00000	0.834058
$93$	0	0
$94$	8.00000	0.825137
$95$	8.00000	0.820783
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	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
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−18.0000	−1.76505
$105$	0	0
$106$	10.0000	0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−2.00000	−0.185695
$117$	0	0
$118$	4.00000	0.368230
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	−6.00000	−0.526235
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	2.00000	0.165521
$147$	0	0
$148$	2.00000	0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−24.0000	−1.94666
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	5.00000	0.395285
$161$	−8.00000	−0.630488
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	−6.00000	−0.449719
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−24.0000	−1.76930
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−10.0000	−0.703598
$203$	2.00000	0.140372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	0	0
$208$	6.00000	0.416025
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.00000	0.271538
$218$	18.0000	1.21911
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	6.00000	0.399114
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	4.00000	0.260378
$237$	0	0
$238$	2.00000	0.129641
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	48.0000	3.05417
$248$	12.0000	0.762001
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−6.00000	−0.372104
$261$	0	0
$262$	20.0000	1.23560
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	8.00000	0.490511
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	0	0
$292$	2.00000	0.117041
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−6.00000	−0.348743
$297$	0	0
$298$	14.0000	0.810998
$299$	48.0000	2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	−8.00000	−0.460348
$303$	0	0
$304$	8.00000	0.458831
$305$	2.00000	0.114520
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	0	0
$320$	−7.00000	−0.391312
$321$	0	0
$322$	8.00000	0.445823
$323$	16.0000	0.890264
$324$	0	0
$325$	−6.00000	−0.332820
$326$	−12.0000	−0.664619
$327$	0	0
$328$	18.0000	0.993884
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−24.0000	−1.26844
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	0	0
$364$	6.00000	0.314485
$365$	2.00000	0.104685
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	−2.00000	−0.103975
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	−24.0000	−1.23771
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	4.00000	0.204658
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	−18.0000	−0.916176
$387$	0	0
$388$	18.0000	0.913812
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	3.00000	0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	30.0000	1.47087
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−30.0000	−1.45693
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	−12.0000	−0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	18.0000	0.862044
$437$	64.0000	3.06154
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	24.0000	1.13643
$447$	0	0
$448$	7.00000	0.330719
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−4.00000	−0.187729
$455$	6.00000	0.281284
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	−8.00000	−0.373002
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−12.0000	−0.552345
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	2.00000	0.0916698
$477$	0	0
$478$	−4.00000	−0.182956
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	6.00000	0.273293
$483$	0	0
$484$	11.0000	0.500000
$485$	18.0000	0.817338
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	1.00000	0.0451754
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	−48.0000	−2.15962
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.0000	0.538274
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	11.0000	0.486136
$513$	0	0
$514$	−6.00000	−0.264649
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	2.00000	0.0878750
$519$	0	0
$520$	18.0000	0.789352
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	16.0000	0.697633
$527$	−8.00000	−0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	−10.0000	−0.434372
$531$	0	0
$532$	8.00000	0.346844
$533$	−36.0000	−1.55933
$534$	0	0
$535$	−12.0000	−0.518805
$536$	12.0000	0.518321
$537$	0	0
$538$	14.0000	0.603583
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	10.0000	0.428746
$545$	18.0000	0.771035
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	8.00000	0.340195
$554$	−14.0000	−0.594803
$555$	0	0
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	1.00000	0.0422577
$561$	0	0
$562$	18.0000	0.759284
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	4.00000	0.168133
$567$	0	0
$568$	36.0000	1.51053
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	−8.00000	−0.333623
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	2.00000	0.0830455
$581$	4.00000	0.165948
$582$	0	0
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	14.0000	0.578335
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	−4.00000	−0.164677
$591$	0	0
$592$	2.00000	0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	14.0000	0.573462
$597$	0	0
$598$	−48.0000	−1.96287
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−8.00000	−0.325515
$605$	11.0000	0.447214
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	48.0000	1.94187
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	24.0000	0.962312
$623$	6.00000	0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	14.0000	0.558661
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	2.00000	0.0794301
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	−3.00000	−0.118585
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	6.00000	0.235339
$651$	0	0
$652$	−12.0000	−0.469956
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	0	0
$658$	8.00000	0.311872
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	12.0000	0.465690
$665$	8.00000	0.310227
$666$	0	0
$667$	−16.0000	−0.619522
$668$	8.00000	0.309529
$669$	0	0
$670$	4.00000	0.154533
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−23.0000	−0.884615
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	0	0
$679$	−18.0000	−0.690777
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	60.0000	2.28582
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−14.0000	−0.529908
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	10.0000	0.376089
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	18.0000	0.674579
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−36.0000	−1.34351
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−45.0000	−1.67473
$723$	0	0
$724$	2.00000	0.0743294
$725$	2.00000	0.0742781
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−18.0000	−0.667124
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	−8.00000	−0.295891
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	40.0000	1.47442
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	10.0000	0.367112
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	10.0000	0.366126
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	12.0000	0.437014
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	24.0000	0.870572
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	4.00000	0.144715
$765$	0	0
$766$	−32.0000	−1.15621
$767$	24.0000	0.866590
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−54.0000	−1.93849
$777$	0	0
$778$	30.0000	1.07555
$779$	−48.0000	−1.71978
$780$	0	0
$781$	0	0
$782$	−16.0000	−0.572159
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	14.0000	0.499681
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	8.00000	0.284627
$791$	−6.00000	−0.213335
$792$	0	0
$793$	12.0000	0.426132
$794$	22.0000	0.780751
$795$	0	0
$796$	4.00000	0.141776
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	−5.00000	−0.176777
$801$	0	0
$802$	18.0000	0.635602
$803$	0	0
$804$	0	0
$805$	8.00000	0.281963
$806$	24.0000	0.845364
$807$	0	0
$808$	30.0000	1.05540
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−32.0000	−1.11954
$818$	22.0000	0.769212
$819$	0	0
$820$	6.00000	0.209529
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	4.00000	0.139178
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	−42.0000	−1.45609
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−12.0000	−0.414533
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	0	0
$844$	20.0000	0.688428
$845$	−23.0000	−0.791224
$846$	0	0
$847$	−11.0000	−0.377964
$848$	10.0000	0.343401
$849$	0	0
$850$	2.00000	0.0685994
$851$	16.0000	0.548473
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	36.0000	1.23045
$857$	−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	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	28.0000	0.953684
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	2.00000	0.0679628
$867$	0	0
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−54.0000	−1.82867
$873$	0	0
$874$	−64.0000	−2.16483
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	6.00000	0.201120
$891$	0	0
$892$	24.0000	0.803579
$893$	64.0000	2.14168
$894$	0	0
$895$	−24.0000	−0.802232
$896$	3.00000	0.100223
$897$	0	0
$898$	−30.0000	−1.00111
$899$	8.00000	0.266815
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	2.00000	0.0664822
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−6.00000	−0.198898
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−22.0000	−0.726900
$917$	−20.0000	−0.660458
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	24.0000	0.791257
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	24.0000	0.788689
$927$	0	0
$928$	−10.0000	−0.328266
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	−18.0000	−0.589610
$933$	0	0
$934$	28.0000	0.916188
$935$	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	$0.510401\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−8.00000	−0.260931
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	8.00000	0.259554
$951$	0	0
$952$	−6.00000	−0.194461
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	−4.00000	−0.129369
$957$	0	0
$958$	32.0000	1.03387
$959$	10.0000	0.322917
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	0	0
$964$	6.00000	0.193247
$965$	−18.0000	−0.579441
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	−18.0000	−0.577945
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	0	0
$980$	1.00000	0.0319438
$981$	0	0
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	4.00000	0.127386
$987$	0	0
$988$	−48.0000	−1.52708
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	−12.0000	−0.380617
$995$	4.00000	0.126809
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.a.1.1		1
3.2	odd	2		105.2.a.a.1.1	✓	1
4.3	odd	2		5040.2.a.d.1.1		1
5.2	odd	4		1575.2.d.b.1324.1		2
5.3	odd	4		1575.2.d.b.1324.2		2
5.4	even	2		1575.2.a.h.1.1		1
7.6	odd	2		2205.2.a.b.1.1		1
12.11	even	2		1680.2.a.f.1.1		1
15.2	even	4		525.2.d.b.274.2		2
15.8	even	4		525.2.d.b.274.1		2
15.14	odd	2		525.2.a.a.1.1		1
21.2	odd	6		735.2.i.a.361.1		2
21.5	even	6		735.2.i.b.361.1		2
21.11	odd	6		735.2.i.a.226.1		2
21.17	even	6		735.2.i.b.226.1		2
21.20	even	2		735.2.a.f.1.1		1
24.5	odd	2		6720.2.a.p.1.1		1
24.11	even	2		6720.2.a.bk.1.1		1
60.59	even	2		8400.2.a.co.1.1		1
105.104	even	2		3675.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.a.1.1	✓	1	3.2	odd	2
315.2.a.a.1.1		1	1.1	even	1	trivial
525.2.a.a.1.1		1	15.14	odd	2
525.2.d.b.274.1		2	15.8	even	4
525.2.d.b.274.2		2	15.2	even	4
735.2.a.f.1.1		1	21.20	even	2
735.2.i.a.226.1		2	21.11	odd	6
735.2.i.a.361.1		2	21.2	odd	6
735.2.i.b.226.1		2	21.17	even	6
735.2.i.b.361.1		2	21.5	even	6
1575.2.a.h.1.1		1	5.4	even	2
1575.2.d.b.1324.1		2	5.2	odd	4
1575.2.d.b.1324.2		2	5.3	odd	4
1680.2.a.f.1.1		1	12.11	even	2
2205.2.a.b.1.1		1	7.6	odd	2
3675.2.a.f.1.1		1	105.104	even	2
5040.2.a.d.1.1		1	4.3	odd	2
6720.2.a.p.1.1		1	24.5	odd	2
6720.2.a.bk.1.1		1	24.11	even	2
8400.2.a.co.1.1		1	60.59	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more