LMFDB - Embedded newform 735.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(1,735)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.86900454856$
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)

$f(q)$

$=$

$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} +3.00000 q^{13} +1.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -2.00000 q^{20} +12.0000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -2.00000 q^{30} -1.00000 q^{31} +8.00000 q^{32} +6.00000 q^{33} -8.00000 q^{34} +2.00000 q^{36} +7.00000 q^{37} +2.00000 q^{38} -3.00000 q^{39} +6.00000 q^{41} +1.00000 q^{43} -12.0000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -2.00000 q^{47} +4.00000 q^{48} -2.00000 q^{50} -4.00000 q^{51} +6.00000 q^{52} +4.00000 q^{53} +2.00000 q^{54} +6.00000 q^{55} +1.00000 q^{57} +16.0000 q^{58} +8.00000 q^{59} +2.00000 q^{60} +14.0000 q^{61} +2.00000 q^{62} -8.00000 q^{64} -3.00000 q^{65} -12.0000 q^{66} +7.00000 q^{67} +8.00000 q^{68} +4.00000 q^{69} +6.00000 q^{71} -1.00000 q^{73} -14.0000 q^{74} -1.00000 q^{75} -2.00000 q^{76} +6.00000 q^{78} -1.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -2.00000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +8.00000 q^{87} +12.0000 q^{89} +2.00000 q^{90} -8.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} +1.00000 q^{95} -8.00000 q^{96} +6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−2.00000	−0.577350
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	12.0000	2.55841
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−2.00000	−0.365148
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	8.00000	1.41421
$33$	6.00000	1.04447
$34$	−8.00000	−1.37199
$35$	0	0
$36$	2.00000	0.333333
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	2.00000	0.324443
$39$	−3.00000	−0.480384
$40$	0	0
$41$	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
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−12.0000	−1.80907
$45$	−1.00000	−0.149071
$46$	8.00000	1.17954
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	−2.00000	−0.282843
$51$	−4.00000	−0.560112
$52$	6.00000	0.832050
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	2.00000	0.272166
$55$	6.00000	0.809040
$56$	0	0
$57$	1.00000	0.132453
$58$	16.0000	2.10090
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	2.00000	0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−3.00000	−0.372104
$66$	−12.0000	−1.47710
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	8.00000	0.970143
$69$	4.00000	0.481543
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−14.0000	−1.62747
$75$	−1.00000	−0.115470
$76$	−2.00000	−0.229416
$77$	0	0
$78$	6.00000	0.679366
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−2.00000	−0.215666
$87$	8.00000	0.857690
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−8.00000	−0.834058
$93$	1.00000	0.103695
$94$	4.00000	0.412568
$95$	1.00000	0.102598
$96$	−8.00000	−0.816497
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	2.00000	0.200000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	8.00000	0.792118
$103$	19.0000	1.87213	0.936063	−	0.351833i	$-0.114441\pi$
$103$	19.0000	1.87213	0.936063	+	0.351833i	$0.114441\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.777029
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−2.00000	−0.192450
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	−12.0000	−1.14416
$111$	−7.00000	−0.664411
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.00000	−0.187317
$115$	4.00000	0.373002
$116$	−16.0000	−1.48556
$117$	3.00000	0.277350
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−28.0000	−2.53500
$123$	−6.00000	−0.541002
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	6.00000	0.526235
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	12.0000	1.04447
$133$	0	0
$134$	−14.0000	−1.20942
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−8.00000	−0.681005
$139$	−21.0000	−1.78120	−0.890598	−	0.454791i	$-0.849714\pi$
$139$	−21.0000	−1.78120	−0.890598	+	0.454791i	$0.849714\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	−12.0000	−1.00702
$143$	−18.0000	−1.50524
$144$	−4.00000	−0.333333
$145$	8.00000	0.664364
$146$	2.00000	0.165521
$147$	0	0
$148$	14.0000	1.15079
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	2.00000	0.163299
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	1.00000	0.0803219
$156$	−6.00000	−0.480384
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	2.00000	0.159111
$159$	−4.00000	−0.317221
$160$	−8.00000	−0.632456
$161$	0	0
$162$	−2.00000	−0.157135
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	12.0000	0.937043
$165$	−6.00000	−0.467099
$166$	4.00000	0.310460
$167$	10.0000	0.773823	0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000	0.773823	0.386912	+	0.922117i	$0.373542\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	8.00000	0.613572
$171$	−1.00000	−0.0764719
$172$	2.00000	0.152499
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	−16.0000	−1.21296
$175$	0	0
$176$	24.0000	1.80907
$177$	−8.00000	−0.601317
$178$	−24.0000	−1.79888
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−2.00000	−0.149071
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	−7.00000	−0.514650
$186$	−2.00000	−0.146647
$187$	−24.0000	−1.75505
$188$	−4.00000	−0.291730
$189$	0	0
$190$	−2.00000	−0.145095
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	8.00000	0.577350
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	−12.0000	−0.861550
$195$	3.00000	0.214834
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	12.0000	0.852803
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	−20.0000	−1.40720
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−6.00000	−0.419058
$206$	−38.0000	−2.64759
$207$	−4.00000	−0.278019
$208$	−12.0000	−0.832050
$209$	6.00000	0.415029
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	8.00000	0.549442
$213$	−6.00000	−0.411113
$214$	−24.0000	−1.64061
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	0	0
$218$	30.0000	2.03186
$219$	1.00000	0.0675737
$220$	12.0000	0.809040
$221$	12.0000	0.807207
$222$	14.0000	0.939618
$223$	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
$226$	12.0000	0.798228
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	2.00000	0.132453
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	2.00000	0.130466
$236$	16.0000	1.04151
$237$	1.00000	0.0649570
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	−4.00000	−0.258199
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−50.0000	−3.21412
$243$	−1.00000	−0.0641500
$244$	28.0000	1.79252
$245$	0	0
$246$	12.0000	0.765092
$247$	−3.00000	−0.190885
$248$	0	0
$249$	2.00000	0.126745
$250$	2.00000	0.126491
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−10.0000	−0.627456
$255$	4.00000	0.250490
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	−6.00000	−0.372104
$261$	−8.00000	−0.495188
$262$	4.00000	0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−12.0000	−0.734388
$268$	14.0000	0.855186
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	−2.00000	−0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−16.0000	−0.970143
$273$	0	0
$274$	−16.0000	−0.966595
$275$	−6.00000	−0.361814
$276$	8.00000	0.481543
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	42.0000	2.51899
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−4.00000	−0.238197
$283$	7.00000	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$283$	7.00000	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$284$	12.0000	0.712069
$285$	−1.00000	−0.0592349
$286$	36.0000	2.12872
$287$	0	0
$288$	8.00000	0.471405
$289$	−1.00000	−0.0588235
$290$	−16.0000	−0.939552
$291$	−6.00000	−0.351726
$292$	−2.00000	−0.117041
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	6.00000	0.348155
$298$	8.00000	0.463428
$299$	−12.0000	−0.693978
$300$	−2.00000	−0.115470
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−10.0000	−0.574485
$304$	4.00000	0.229416
$305$	−14.0000	−0.801638
$306$	−8.00000	−0.457330
$307$	−3.00000	−0.171219	−0.0856095	−	0.996329i	$-0.527284\pi$
$307$	−3.00000	−0.171219	−0.0856095	+	0.996329i	$0.527284\pi$
$308$	0	0
$309$	−19.0000	−1.08087
$310$	−2.00000	−0.113592
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	−2.00000	−0.112509
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	8.00000	0.448618
$319$	48.0000	2.68748
$320$	8.00000	0.447214
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	2.00000	0.111111
$325$	3.00000	0.166410
$326$	24.0000	1.32924
$327$	15.0000	0.829502
$328$	0	0
$329$	0	0
$330$	12.0000	0.660578
$331$	−9.00000	−0.494685	−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000	−0.494685	−0.247342	+	0.968928i	$0.579557\pi$
$332$	−4.00000	−0.219529
$333$	7.00000	0.383598
$334$	−20.0000	−1.09435
$335$	−7.00000	−0.382451
$336$	0	0
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	8.00000	0.435143
$339$	6.00000	0.325875
$340$	−8.00000	−0.433861
$341$	6.00000	0.324918
$342$	2.00000	0.108148
$343$	0	0
$344$	0	0
$345$	−4.00000	−0.215353
$346$	48.0000	2.58050
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	16.0000	0.857690
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	−48.0000	−2.55841
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	16.0000	0.850390
$355$	−6.00000	−0.318447
$356$	24.0000	1.27200
$357$	0	0
$358$	−36.0000	−1.90266
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	26.0000	1.36653
$363$	−25.0000	−1.31216
$364$	0	0
$365$	1.00000	0.0523424
$366$	28.0000	1.46358
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	16.0000	0.834058
$369$	6.00000	0.312348
$370$	14.0000	0.727825
$371$	0	0
$372$	2.00000	0.103695
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	48.0000	2.48202
$375$	1.00000	0.0516398
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	2.00000	0.102598
$381$	−5.00000	−0.256158
$382$	−20.0000	−1.02329
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	0	0
$385$	0	0
$386$	18.0000	0.916176
$387$	1.00000	0.0508329
$388$	12.0000	0.609208
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−6.00000	−0.303822
$391$	−16.0000	−0.809155
$392$	0	0
$393$	2.00000	0.100887
$394$	−24.0000	−1.20910
$395$	1.00000	0.0503155
$396$	−12.0000	−0.603023
$397$	37.0000	1.85698	0.928488	−	0.371361i	$-0.121109\pi$
$397$	37.0000	1.85698	0.928488	+	0.371361i	$0.121109\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	14.0000	0.698257
$403$	−3.00000	−0.149441
$404$	20.0000	0.995037
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−42.0000	−2.08186
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	12.0000	0.592638
$411$	−8.00000	−0.394611
$412$	38.0000	1.87213
$413$	0	0
$414$	8.00000	0.393179
$415$	2.00000	0.0981761
$416$	24.0000	1.17670
$417$	21.0000	1.02837
$418$	−12.0000	−0.586939
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	40.0000	1.94717
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	4.00000	0.194029
$426$	12.0000	0.581402
$427$	0	0
$428$	24.0000	1.16008
$429$	18.0000	0.869048
$430$	2.00000	0.0964486
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	4.00000	0.192450
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	−30.0000	−1.43674
$437$	4.00000	0.191346
$438$	−2.00000	−0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−14.0000	−0.664411
$445$	−12.0000	−0.568855
$446$	−48.0000	−2.27287
$447$	4.00000	0.189194
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−2.00000	−0.0942809
$451$	−36.0000	−1.69517
$452$	−12.0000	−0.564433
$453$	−8.00000	−0.375873
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	26.0000	1.21490
$459$	−4.00000	−0.186704
$460$	8.00000	0.373002
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	3.00000	0.139422	0.0697109	−	0.997567i	$-0.477792\pi$
$463$	3.00000	0.139422	0.0697109	+	0.997567i	$0.477792\pi$
$464$	32.0000	1.48556
$465$	−1.00000	−0.0463739
$466$	−12.0000	−0.555889
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	−4.00000	−0.184506
$471$	10.0000	0.460776
$472$	0	0
$473$	−6.00000	−0.275880
$474$	−2.00000	−0.0918630
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	4.00000	0.183147
$478$	−28.0000	−1.28069
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	8.00000	0.365148
$481$	21.0000	0.957518
$482$	−36.0000	−1.63976
$483$	0	0
$484$	50.0000	2.27273
$485$	−6.00000	−0.272446
$486$	2.00000	0.0907218
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−12.0000	−0.541002
$493$	−32.0000	−1.44121
$494$	6.00000	0.269953
$495$	6.00000	0.269680
$496$	4.00000	0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	−2.00000	−0.0894427
$501$	−10.0000	−0.446767
$502$	24.0000	1.07117
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	−48.0000	−2.13386
$507$	4.00000	0.177646
$508$	10.0000	0.443678
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−8.00000	−0.354246
$511$	0	0
$512$	−32.0000	−1.41421
$513$	1.00000	0.0441511
$514$	−36.0000	−1.58789
$515$	−19.0000	−0.837240
$516$	−2.00000	−0.0880451
$517$	12.0000	0.527759
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	16.0000	0.700301
$523$	−11.0000	−0.480996	−0.240498	−	0.970650i	$-0.577311\pi$
$523$	−11.0000	−0.480996	−0.240498	+	0.970650i	$0.577311\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−8.00000	−0.348817
$527$	−4.00000	−0.174243
$528$	−24.0000	−1.04447
$529$	−7.00000	−0.304348
$530$	8.00000	0.347498
$531$	8.00000	0.347170
$532$	0	0
$533$	18.0000	0.779667
$534$	24.0000	1.03858
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−18.0000	−0.776757
$538$	−20.0000	−0.862261
$539$	0	0
$540$	2.00000	0.0860663
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	−48.0000	−2.06178
$543$	13.0000	0.557883
$544$	32.0000	1.37199
$545$	15.0000	0.642529
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	16.0000	0.683486
$549$	14.0000	0.597505
$550$	12.0000	0.511682
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	14.0000	0.594803
$555$	7.00000	0.297133
$556$	−42.0000	−1.78120
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	2.00000	0.0846668
$559$	3.00000	0.126886
$560$	0	0
$561$	24.0000	1.01328
$562$	24.0000	1.01238
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	4.00000	0.168430
$565$	6.00000	0.252422
$566$	−14.0000	−0.588464
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	2.00000	0.0837708
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	−36.0000	−1.50524
$573$	−10.0000	−0.417756
$574$	0	0
$575$	−4.00000	−0.166812
$576$	−8.00000	−0.333333
$577$	29.0000	1.20729	0.603643	−	0.797255i	$-0.293715\pi$
$577$	29.0000	1.20729	0.603643	+	0.797255i	$0.293715\pi$
$578$	2.00000	0.0831890
$579$	9.00000	0.374027
$580$	16.0000	0.664364
$581$	0	0
$582$	12.0000	0.497416
$583$	−24.0000	−0.993978
$584$	0	0
$585$	−3.00000	−0.124035
$586$	−32.0000	−1.32191
$587$	−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
$589$	1.00000	0.0412043
$590$	16.0000	0.658710
$591$	−12.0000	−0.493614
$592$	−28.0000	−1.15079
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	−8.00000	−0.327693
$597$	8.00000	0.327418
$598$	24.0000	0.981433
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	33.0000	1.34610	0.673049	−	0.739598i	$-0.264984\pi$
$601$	33.0000	1.34610	0.673049	+	0.739598i	$0.264984\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	16.0000	0.651031
$605$	−25.0000	−1.01639
$606$	20.0000	0.812444
$607$	−35.0000	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$607$	−35.0000	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	28.0000	1.13369
$611$	−6.00000	−0.242734
$612$	8.00000	0.323381
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	6.00000	0.242140
$615$	6.00000	0.241943
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	38.0000	1.52858
$619$	−3.00000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$619$	−3.00000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$620$	2.00000	0.0803219
$621$	4.00000	0.160514
$622$	12.0000	0.481156
$623$	0	0
$624$	12.0000	0.480384
$625$	1.00000	0.0400000
$626$	22.0000	0.879297
$627$	−6.00000	−0.239617
$628$	−20.0000	−0.798087
$629$	28.0000	1.11643
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	−40.0000	−1.58860
$635$	−5.00000	−0.198419
$636$	−8.00000	−0.317221
$637$	0	0
$638$	−96.0000	−3.80068
$639$	6.00000	0.237356
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	24.0000	0.947204
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	8.00000	0.314756
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−24.0000	−0.939913
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−30.0000	−1.17309
$655$	2.00000	0.0781465
$656$	−24.0000	−0.937043
$657$	−1.00000	−0.0390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−12.0000	−0.467099
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	18.0000	0.699590
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	−14.0000	−0.542489
$667$	32.0000	1.23904
$668$	20.0000	0.773823
$669$	−24.0000	−0.927894
$670$	14.0000	0.540867
$671$	−84.0000	−3.24278
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	−50.0000	−1.92593
$675$	−1.00000	−0.0384900
$676$	−8.00000	−0.307692
$677$	−16.0000	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$678$	−12.0000	−0.460857
$679$	0	0
$680$	0	0
$681$	−10.0000	−0.383201
$682$	−12.0000	−0.459504
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	−2.00000	−0.0764719
$685$	−8.00000	−0.305664
$686$	0	0
$687$	13.0000	0.495981
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	8.00000	0.304555
$691$	−27.0000	−1.02713	−0.513564	−	0.858051i	$-0.671675\pi$
$691$	−27.0000	−1.02713	−0.513564	+	0.858051i	$0.671675\pi$
$692$	−48.0000	−1.82469
$693$	0	0
$694$	−32.0000	−1.21470
$695$	21.0000	0.796575
$696$	0	0
$697$	24.0000	0.909065
$698$	4.00000	0.151402
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−44.0000	−1.66186	−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000	−1.66186	−0.830929	+	0.556379i	$0.812190\pi$
$702$	6.00000	0.226455
$703$	−7.00000	−0.264010
$704$	48.0000	1.80907
$705$	−2.00000	−0.0753244
$706$	−36.0000	−1.35488
$707$	0	0
$708$	−16.0000	−0.601317
$709$	−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
$711$	−1.00000	−0.0375029
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	18.0000	0.673162
$716$	36.0000	1.34538
$717$	−14.0000	−0.522840
$718$	48.0000	1.79134
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	36.0000	1.33978
$723$	−18.0000	−0.669427
$724$	−26.0000	−0.966282
$725$	−8.00000	−0.297113
$726$	50.0000	1.85567
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	4.00000	0.147945
$732$	−28.0000	−1.03491
$733$	43.0000	1.58824	0.794121	−	0.607760i	$-0.207932\pi$
$733$	43.0000	1.58824	0.794121	+	0.607760i	$0.207932\pi$
$734$	38.0000	1.40261
$735$	0	0
$736$	−32.0000	−1.17954
$737$	−42.0000	−1.54709
$738$	−12.0000	−0.441726
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−14.0000	−0.514650
$741$	3.00000	0.110208
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	−22.0000	−0.805477
$747$	−2.00000	−0.0731762
$748$	−48.0000	−1.75505
$749$	0	0
$750$	−2.00000	−0.0730297
$751$	29.0000	1.05823	0.529113	−	0.848552i	$-0.322525\pi$
$751$	29.0000	1.05823	0.529113	+	0.848552i	$0.322525\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	48.0000	1.74806
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−22.0000	−0.799076
$759$	−24.0000	−0.871145
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	10.0000	0.362262
$763$	0	0
$764$	20.0000	0.723575
$765$	−4.00000	−0.144620
$766$	56.0000	2.02336
$767$	24.0000	0.866590
$768$	−16.0000	−0.577350
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−18.0000	−0.647834
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−2.00000	−0.0718885
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−6.00000	−0.214972
$780$	6.00000	0.214834
$781$	−36.0000	−1.28818
$782$	32.0000	1.14432
$783$	8.00000	0.285897
$784$	0	0
$785$	10.0000	0.356915
$786$	−4.00000	−0.142675
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	24.0000	0.854965
$789$	−4.00000	−0.142404
$790$	−2.00000	−0.0711568
$791$	0	0
$792$	0	0
$793$	42.0000	1.49146
$794$	−74.0000	−2.62616
$795$	4.00000	0.141865
$796$	−16.0000	−0.567105
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	8.00000	0.282843
$801$	12.0000	0.423999
$802$	24.0000	0.847469
$803$	6.00000	0.211735
$804$	−14.0000	−0.493742
$805$	0	0
$806$	6.00000	0.211341
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	2.00000	0.0702728
$811$	48.0000	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$811$	48.0000	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	84.0000	2.94420
$815$	12.0000	0.420342
$816$	16.0000	0.560112
$817$	−1.00000	−0.0349856
$818$	10.0000	0.349642
$819$	0	0
$820$	−12.0000	−0.419058
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	16.0000	0.558064
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	−8.00000	−0.278019
$829$	−57.0000	−1.97969	−0.989846	−	0.142145i	$-0.954600\pi$
$829$	−57.0000	−1.97969	−0.989846	+	0.142145i	$0.954600\pi$
$830$	−4.00000	−0.138842
$831$	7.00000	0.242827
$832$	−24.0000	−0.832050
$833$	0	0
$834$	−42.0000	−1.45434
$835$	−10.0000	−0.346064
$836$	12.0000	0.415029
$837$	1.00000	0.0345651
$838$	12.0000	0.414533
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−2.00000	−0.0689246
$843$	12.0000	0.413302
$844$	−40.0000	−1.37686
$845$	4.00000	0.137604
$846$	4.00000	0.137523
$847$	0	0
$848$	−16.0000	−0.549442
$849$	−7.00000	−0.240239
$850$	−8.00000	−0.274398
$851$	−28.0000	−0.959828
$852$	−12.0000	−0.411113
$853$	9.00000	0.308154	0.154077	−	0.988059i	$-0.450760\pi$
$853$	9.00000	0.308154	0.154077	+	0.988059i	$0.450760\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	−36.0000	−1.22902
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	4.00000	0.136241
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−8.00000	−0.272166
$865$	24.0000	0.816024
$866$	−10.0000	−0.339814
$867$	1.00000	0.0339618
$868$	0	0
$869$	6.00000	0.203536
$870$	16.0000	0.542451
$871$	21.0000	0.711558
$872$	0	0
$873$	6.00000	0.203069
$874$	−8.00000	−0.270604
$875$	0	0
$876$	2.00000	0.0675737
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	32.0000	1.07995
$879$	−16.0000	−0.539667
$880$	−24.0000	−0.809040
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	7.00000	0.235569	0.117784	−	0.993039i	$-0.462421\pi$
$883$	7.00000	0.235569	0.117784	+	0.993039i	$0.462421\pi$
$884$	24.0000	0.807207
$885$	8.00000	0.268917
$886$	72.0000	2.41889
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	0	0
$889$	0	0
$890$	24.0000	0.804482
$891$	−6.00000	−0.201008
$892$	48.0000	1.60716
$893$	2.00000	0.0669274
$894$	−8.00000	−0.267560
$895$	−18.0000	−0.601674
$896$	0	0
$897$	12.0000	0.400668
$898$	−60.0000	−2.00223
$899$	8.00000	0.266815
$900$	2.00000	0.0666667
$901$	16.0000	0.533037
$902$	72.0000	2.39734
$903$	0	0
$904$	0	0
$905$	13.0000	0.432135
$906$	16.0000	0.531564
$907$	31.0000	1.02934	0.514669	−	0.857389i	$-0.327915\pi$
$907$	31.0000	1.02934	0.514669	+	0.857389i	$0.327915\pi$
$908$	20.0000	0.663723
$909$	10.0000	0.331679
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−4.00000	−0.132453
$913$	12.0000	0.397142
$914$	30.0000	0.992312
$915$	14.0000	0.462826
$916$	−26.0000	−0.859064
$917$	0	0
$918$	8.00000	0.264039
$919$	−9.00000	−0.296883	−0.148441	−	0.988921i	$-0.547426\pi$
$919$	−9.00000	−0.296883	−0.148441	+	0.988921i	$0.547426\pi$
$920$	0	0
$921$	3.00000	0.0988534
$922$	−16.0000	−0.526932
$923$	18.0000	0.592477
$924$	0	0
$925$	7.00000	0.230159
$926$	−6.00000	−0.197172
$927$	19.0000	0.624042
$928$	−64.0000	−2.10090
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	12.0000	0.393073
$933$	6.00000	0.196431
$934$	44.0000	1.43972
$935$	24.0000	0.784884
$936$	0	0
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	0	0
$939$	11.0000	0.358971
$940$	4.00000	0.130466
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−20.0000	−0.651635
$943$	−24.0000	−0.781548
$944$	−32.0000	−1.04151
$945$	0	0
$946$	12.0000	0.390154
$947$	26.0000	0.844886	0.422443	−	0.906389i	$-0.361173\pi$
$947$	26.0000	0.844886	0.422443	+	0.906389i	$0.361173\pi$
$948$	2.00000	0.0649570
$949$	−3.00000	−0.0973841
$950$	2.00000	0.0648886
$951$	−20.0000	−0.648544
$952$	0	0
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	−8.00000	−0.259010
$955$	−10.0000	−0.323592
$956$	28.0000	0.905585
$957$	−48.0000	−1.55162
$958$	−8.00000	−0.258468
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−30.0000	−0.967742
$962$	−42.0000	−1.35413
$963$	12.0000	0.386695
$964$	36.0000	1.15948
$965$	9.00000	0.289720
$966$	0	0
$967$	55.0000	1.76868	0.884340	−	0.466843i	$-0.154609\pi$
$967$	55.0000	1.76868	0.884340	+	0.466843i	$0.154609\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	12.0000	0.385297
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	26.0000	0.833094
$975$	−3.00000	−0.0960769
$976$	−56.0000	−1.79252
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	−24.0000	−0.767435
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−15.0000	−0.478913
$982$	24.0000	0.765871
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	64.0000	2.03818
$987$	0	0
$988$	−6.00000	−0.190885
$989$	−4.00000	−0.127193
$990$	−12.0000	−0.381385
$991$	−15.0000	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$991$	−15.0000	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$992$	−8.00000	−0.254000
$993$	9.00000	0.285606
$994$	0	0
$995$	8.00000	0.253617
$996$	4.00000	0.126745
$997$	25.0000	0.791758	0.395879	−	0.918303i	$-0.370440\pi$
$997$	25.0000	0.791758	0.395879	+	0.918303i	$0.370440\pi$
$998$	−58.0000	−1.83596
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.a.a.1.1		1
3.2	odd	2		2205.2.a.m.1.1		1
5.4	even	2		3675.2.a.p.1.1		1
7.2	even	3		735.2.i.f.361.1		2
7.3	odd	6		105.2.i.b.16.1	✓	2
7.4	even	3		735.2.i.f.226.1		2
7.5	odd	6		105.2.i.b.46.1	yes	2
7.6	odd	2		735.2.a.b.1.1		1
21.5	even	6		315.2.j.a.46.1		2
21.17	even	6		315.2.j.a.226.1		2
21.20	even	2		2205.2.a.k.1.1		1
28.3	even	6		1680.2.bg.l.961.1		2
28.19	even	6		1680.2.bg.l.1201.1		2
35.3	even	12		525.2.r.d.499.1		4
35.12	even	12		525.2.r.d.424.1		4
35.17	even	12		525.2.r.d.499.2		4
35.19	odd	6		525.2.i.a.151.1		2
35.24	odd	6		525.2.i.a.226.1		2
35.33	even	12		525.2.r.d.424.2		4
35.34	odd	2		3675.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.b.16.1	✓	2	7.3	odd	6
105.2.i.b.46.1	yes	2	7.5	odd	6
315.2.j.a.46.1		2	21.5	even	6
315.2.j.a.226.1		2	21.17	even	6
525.2.i.a.151.1		2	35.19	odd	6
525.2.i.a.226.1		2	35.24	odd	6
525.2.r.d.424.1		4	35.12	even	12
525.2.r.d.424.2		4	35.33	even	12
525.2.r.d.499.1		4	35.3	even	12
525.2.r.d.499.2		4	35.17	even	12
735.2.a.a.1.1		1	1.1	even	1	trivial
735.2.a.b.1.1		1	7.6	odd	2
735.2.i.f.226.1		2	7.4	even	3
735.2.i.f.361.1		2	7.2	even	3
1680.2.bg.l.961.1		2	28.3	even	6
1680.2.bg.l.1201.1		2	28.19	even	6
2205.2.a.k.1.1		1	21.20	even	2
2205.2.a.m.1.1		1	3.2	odd	2
3675.2.a.o.1.1		1	35.34	odd	2
3675.2.a.p.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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