LMFDB - Embedded newform 735.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,2,Mod(1,735)] 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("735.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
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	$\chi$	$=$	735.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} -6.00000 q^{39} +3.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -8.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -12.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{76} -6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} -5.00000 q^{96} +18.0000 q^{97} +O(q^{100})\)

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	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−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	$0.577979\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−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
$39$	−6.00000	−0.960769
$40$	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
$45$	−1.00000	−0.149071
$46$	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	$0.698301\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	−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	$0.583847\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−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
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	−8.00000	−0.917663
$77$	0	0
$78$	−6.00000	−0.679366
$79$	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
$81$	1.00000	0.111111
$82$	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
$87$	2.00000	0.214423
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−8.00000	−0.834058
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	−8.00000	−0.820783
$96$	−5.00000	−0.510310
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	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	$0.334245\pi$
$102$	2.00000	0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−18.0000	−1.76505
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−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
$111$	2.00000	0.189832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−8.00000	−0.749269
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	6.00000	0.554700
$118$	−4.00000	−0.368230
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	−6.00000	−0.541002
$124$	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
$129$	−4.00000	−0.352180
$130$	−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
$133$	0	0
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	6.00000	0.514496
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−8.00000	−0.681005
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	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	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	−1.00000	−0.0816497
$151$	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
$153$	−2.00000	−0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	6.00000	0.480384
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	−10.0000	−0.793052
$160$	−5.00000	−0.395285
$161$	0	0
$162$	1.00000	0.0785674
$163$	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
$171$	8.00000	0.611775
$172$	−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	$0.573247\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	6.00000	0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.00000	0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	−24.0000	−1.76930
$185$	2.00000	0.147043
$186$	4.00000	0.293294
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	−7.00000	−0.505181
$193$	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
$195$	6.00000	0.429669
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−3.00000	−0.212132
$201$	−4.00000	−0.282138
$202$	10.0000	0.703598
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	8.00000	0.556038
$208$	−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
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	3.00000	0.204124
$217$	0	0
$218$	−18.0000	−1.21911
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−12.0000	−0.807207
$222$	2.00000	0.134231
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	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	$0.457621\pi$
$228$	8.00000	0.529813
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	6.00000	0.392232
$235$	8.00000	0.521862
$236$	4.00000	0.260378
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	−1.00000	−0.0645497
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	48.0000	3.05417
$248$	12.0000	0.762001
$249$	−4.00000	−0.253490
$250$	−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
$255$	−2.00000	−0.125245
$256$	−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	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	6.00000	0.372104
$261$	−2.00000	−0.123797
$262$	−20.0000	−1.23560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−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	$0.640358\pi$
$270$	1.00000	0.0608581
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	8.00000	0.481543
$277$	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
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	−18.0000	−1.05518
$292$	−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
$300$	1.00000	0.0577350
$301$	0	0
$302$	8.00000	0.460348
$303$	−10.0000	−0.574485
$304$	−8.00000	−0.458831
$305$	−2.00000	−0.114520
$306$	−2.00000	−0.114332
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	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	$0.738219\pi$
$312$	18.0000	1.01905
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−10.0000	−0.560772
$319$	0	0
$320$	−7.00000	−0.391312
$321$	12.0000	0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	−1.00000	−0.0555556
$325$	6.00000	0.332820
$326$	12.0000	0.664619
$327$	18.0000	0.995402
$328$	−18.0000	−0.993884
$329$	0	0
$330$	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
$333$	−2.00000	−0.109599
$334$	−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
$339$	−6.00000	−0.325875
$340$	−2.00000	−0.108465
$341$	0	0
$342$	8.00000	0.432590
$343$	0	0
$344$	−12.0000	−0.646997
$345$	8.00000	0.430706
$346$	−6.00000	−0.322562
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−2.00000	−0.107211
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	4.00000	0.212598
$355$	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	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	3.00000	0.158114
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	−2.00000	−0.104685
$366$	−2.00000	−0.104542
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−8.00000	−0.417029
$369$	6.00000	0.312348
$370$	2.00000	0.103975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−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
$375$	1.00000	0.0516398
$376$	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
$381$	−8.00000	−0.409852
$382$	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	$0.195325\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	4.00000	0.203331
$388$	−18.0000	−0.913812
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	6.00000	0.303822
$391$	−16.0000	−0.809155
$392$	0	0
$393$	20.0000	1.00887
$394$	18.0000	0.906827
$395$	−8.00000	−0.402524
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−4.00000	−0.199502
$403$	−24.0000	−1.19553
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	−6.00000	−0.297044
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	−6.00000	−0.296319
$411$	10.0000	0.493264
$412$	8.00000	0.394132
$413$	0	0
$414$	8.00000	0.393179
$415$	−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
$423$	−8.00000	−0.388973
$424$	−30.0000	−1.45693
$425$	−2.00000	−0.0970143
$426$	12.0000	0.581402
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	18.0000	0.862044
$437$	64.0000	3.06154
$438$	−2.00000	−0.0955637
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−2.00000	−0.0949158
$445$	−6.00000	−0.284427
$446$	24.0000	1.13643
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	4.00000	0.187729
$455$	0	0
$456$	24.0000	1.12390
$457$	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
$459$	2.00000	0.0933520
$460$	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
$465$	−4.00000	−0.185496
$466$	−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	$0.724329\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	8.00000	0.369012
$471$	−14.0000	−0.645086
$472$	12.0000	0.552345
$473$	0	0
$474$	−8.00000	−0.367452
$475$	8.00000	0.367065
$476$	0	0
$477$	10.0000	0.457869
$478$	−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	$0.760973\pi$
$480$	5.00000	0.228218
$481$	−12.0000	−0.547153
$482$	6.00000	0.273293
$483$	0	0
$484$	11.0000	0.500000
$485$	−18.0000	−0.817338
$486$	−1.00000	−0.0453609
$487$	−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
$489$	−12.0000	−0.542659
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	4.00000	0.180151
$494$	48.0000	2.15962
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	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
$501$	8.00000	0.357414
$502$	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
$507$	−23.0000	−1.02147
$508$	−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	$0.485887\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−11.0000	−0.486136
$513$	−8.00000	−0.353209
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	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	$0.570297\pi$
$522$	−2.00000	−0.0875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	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
$531$	−4.00000	−0.173585
$532$	0	0
$533$	36.0000	1.55933
$534$	−6.00000	−0.259645
$535$	12.0000	0.518805
$536$	−12.0000	−0.518321
$537$	24.0000	1.03568
$538$	−14.0000	−0.603583
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	−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
$543$	−2.00000	−0.0858282
$544$	−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
$549$	2.00000	0.0853579
$550$	0	0
$551$	−16.0000	−0.681623
$552$	24.0000	1.02151
$553$	0	0
$554$	14.0000	0.594803
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−4.00000	−0.169334
$559$	24.0000	1.01509
$560$	0	0
$561$	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	$0.526862\pi$
$564$	−8.00000	−0.336861
$565$	−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	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	8.00000	0.335083
$571$	−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
$573$	−4.00000	−0.167102
$574$	0	0
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−13.0000	−0.540729
$579$	−18.0000	−0.748054
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−18.0000	−0.746124
$583$	0	0
$584$	−6.00000	−0.248282
$585$	−6.00000	−0.248069
$586$	−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
$591$	−18.0000	−0.740421
$592$	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
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−4.00000	−0.163709
$598$	48.0000	1.96287
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	3.00000	0.122474
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	11.0000	0.447214
$606$	−10.0000	−0.406222
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	−48.0000	−1.94187
$612$	2.00000	0.0808452
$613$	−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
$615$	6.00000	0.241943
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	8.00000	0.321807
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	−4.00000	−0.160644
$621$	−8.00000	−0.321029
$622$	−24.0000	−0.962312
$623$	0	0
$624$	6.00000	0.240192
$625$	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
$633$	20.0000	0.794929
$634$	2.00000	0.0794301
$635$	−8.00000	−0.317470
$636$	10.0000	0.396526
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	12.0000	0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	−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	$0.656384\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	6.00000	0.235339
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	18.0000	0.703856
$655$	20.0000	0.781465
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	0	0
$659$	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	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−12.0000	−0.466393
$663$	12.0000	0.466041
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−16.0000	−0.619522
$668$	8.00000	0.309529
$669$	−24.0000	−0.927894
$670$	−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
$675$	−1.00000	−0.0384900
$676$	−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	$0.845134\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	−6.00000	−0.230089
$681$	−4.00000	−0.153280
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−8.00000	−0.305888
$685$	10.0000	0.382080
$686$	0	0
$687$	22.0000	0.839352
$688$	−4.00000	−0.152499
$689$	60.0000	2.28582
$690$	8.00000	0.304555
$691$	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
$696$	−6.00000	−0.227429
$697$	−12.0000	−0.454532
$698$	−14.0000	−0.529908
$699$	18.0000	0.680823
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	−6.00000	−0.226455
$703$	−16.0000	−0.603451
$704$	0	0
$705$	−8.00000	−0.301297
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−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
$711$	8.00000	0.300023
$712$	−18.0000	−0.674579
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	4.00000	0.149383
$718$	−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	$0.547662\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	45.0000	1.67473
$723$	−6.00000	−0.223142
$724$	−2.00000	−0.0743294
$725$	−2.00000	−0.0742781
$726$	11.0000	0.408248
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	−8.00000	−0.295891
$732$	2.00000	0.0739221
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	40.0000	1.47442
$737$	0	0
$738$	6.00000	0.220863
$739$	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
$741$	−48.0000	−1.76332
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−12.0000	−0.439941
$745$	−14.0000	−0.512920
$746$	−10.0000	−0.366126
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	1.00000	0.0365148
$751$	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
$753$	−12.0000	−0.437304
$754$	−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	$0.0657392\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−4.00000	−0.144715
$765$	2.00000	0.0723102
$766$	32.0000	1.15621
$767$	−24.0000	−0.866590
$768$	17.0000	0.613435
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−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	$0.681391\pi$
$774$	4.00000	0.143777
$775$	−4.00000	−0.143684
$776$	−54.0000	−1.93849
$777$	0	0
$778$	30.0000	1.07555
$779$	48.0000	1.71978
$780$	−6.00000	−0.214834
$781$	0	0
$782$	−16.0000	−0.572159
$783$	2.00000	0.0714742
$784$	0	0
$785$	−14.0000	−0.499681
$786$	20.0000	0.713376
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−18.0000	−0.641223
$789$	−16.0000	−0.569615
$790$	−8.00000	−0.284627
$791$	0	0
$792$	0	0
$793$	12.0000	0.426132
$794$	22.0000	0.780751
$795$	10.0000	0.354663
$796$	−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
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−24.0000	−0.845364
$807$	14.0000	0.492823
$808$	−30.0000	−1.05540
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	0	0
$815$	−12.0000	−0.420342
$816$	−2.00000	−0.0700140
$817$	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	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	10.0000	0.348790
$823$	−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
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	−4.00000	−0.138842
$831$	−14.0000	−0.485655
$832$	42.0000	1.45609
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	4.00000	0.138260
$838$	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
$843$	−18.0000	−0.619953
$844$	20.0000	0.688428
$845$	−23.0000	−0.791224
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−10.0000	−0.343401
$849$	−4.00000	−0.137280
$850$	−2.00000	−0.0685994
$851$	−16.0000	−0.548473
$852$	−12.0000	−0.411113
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	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	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	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	$0.5\pi$
$864$	−5.00000	−0.170103
$865$	6.00000	0.204006
$866$	2.00000	0.0679628
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	24.0000	0.813209
$872$	54.0000	1.82867
$873$	18.0000	0.609208
$874$	64.0000	2.16483
$875$	0	0
$876$	2.00000	0.0675737
$877$	−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
$879$	14.0000	0.472208
$880$	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
$885$	−4.00000	−0.134459
$886$	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	$0.586566\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−24.0000	−0.803579
$893$	−64.0000	−2.14168
$894$	−14.0000	−0.468230
$895$	24.0000	0.802232
$896$	0	0
$897$	−48.0000	−1.60267
$898$	−30.0000	−1.00111
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	−20.0000	−0.666297
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−2.00000	−0.0664822
$906$	−8.00000	−0.265782
$907$	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
$909$	10.0000	0.331679
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	18.0000	0.595387
$915$	2.00000	0.0661180
$916$	22.0000	0.726900
$917$	0	0
$918$	2.00000	0.0660098
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	24.0000	0.791257
$921$	12.0000	0.395413
$922$	2.00000	0.0658665
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−24.0000	−0.788689
$927$	−8.00000	−0.262754
$928$	−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	$0.426238\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	18.0000	0.589610
$933$	24.0000	0.785725
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−18.0000	−0.588348
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	−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	$0.405216\pi$
$942$	−14.0000	−0.456145
$943$	48.0000	1.56310
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	8.00000	0.259828
$949$	12.0000	0.389536
$950$	8.00000	0.259554
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	10.0000	0.323762
$955$	−4.00000	−0.129437
$956$	4.00000	0.129369
$957$	0	0
$958$	−32.0000	−1.03387
$959$	0	0
$960$	7.00000	0.225924
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	−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
$969$	16.0000	0.513994
$970$	−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	$0.603990\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−6.00000	−0.192154
$976$	−2.00000	−0.0640184
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	18.0000	0.573819
$985$	−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
$993$	12.0000	0.380808
$994$	0	0
$995$	−4.00000	−0.126809
$996$	4.00000	0.126745
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	20.0000	0.633089
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.a.f.1.1		1
3.2	odd	2		2205.2.a.b.1.1		1
5.4	even	2		3675.2.a.f.1.1		1
7.2	even	3		735.2.i.b.361.1		2
7.3	odd	6		735.2.i.a.226.1		2
7.4	even	3		735.2.i.b.226.1		2
7.5	odd	6		735.2.i.a.361.1		2
7.6	odd	2		105.2.a.a.1.1	✓	1
21.20	even	2		315.2.a.a.1.1		1
28.27	even	2		1680.2.a.f.1.1		1
35.13	even	4		525.2.d.b.274.1		2
35.27	even	4		525.2.d.b.274.2		2
35.34	odd	2		525.2.a.a.1.1		1
56.13	odd	2		6720.2.a.p.1.1		1
56.27	even	2		6720.2.a.bk.1.1		1
84.83	odd	2		5040.2.a.d.1.1		1
105.62	odd	4		1575.2.d.b.1324.1		2
105.83	odd	4		1575.2.d.b.1324.2		2
105.104	even	2		1575.2.a.h.1.1		1
140.139	even	2		8400.2.a.co.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.a.1.1	✓	1	7.6	odd	2
315.2.a.a.1.1		1	21.20	even	2
525.2.a.a.1.1		1	35.34	odd	2
525.2.d.b.274.1		2	35.13	even	4
525.2.d.b.274.2		2	35.27	even	4
735.2.a.f.1.1		1	1.1	even	1	trivial
735.2.i.a.226.1		2	7.3	odd	6
735.2.i.a.361.1		2	7.5	odd	6
735.2.i.b.226.1		2	7.4	even	3
735.2.i.b.361.1		2	7.2	even	3
1575.2.a.h.1.1		1	105.104	even	2
1575.2.d.b.1324.1		2	105.62	odd	4
1575.2.d.b.1324.2		2	105.83	odd	4
1680.2.a.f.1.1		1	28.27	even	2
2205.2.a.b.1.1		1	3.2	odd	2
3675.2.a.f.1.1		1	5.4	even	2
5040.2.a.d.1.1		1	84.83	odd	2
6720.2.a.p.1.1		1	56.13	odd	2
6720.2.a.bk.1.1		1	56.27	even	2
8400.2.a.co.1.1		1	140.139	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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