LMFDB - Embedded newform 7350.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1050)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -6.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} -3.00000 q^{41} -8.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} -3.00000 q^{51} +4.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +4.00000 q^{57} +6.00000 q^{58} +6.00000 q^{59} +2.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} +3.00000 q^{68} +3.00000 q^{69} -9.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -4.00000 q^{76} +4.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +6.00000 q^{83} +8.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} +3.00000 q^{89} -3.00000 q^{92} -5.00000 q^{93} +9.00000 q^{94} +1.00000 q^{96} -17.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	−3.00000	−0.420084
$52$	4.00000	0.554700
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	3.00000	0.363803
$69$	3.00000	0.361158
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	4.00000	0.452911
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	6.00000	0.643268
$88$	−6.00000	−0.639602
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	−5.00000	−0.518476
$94$	9.00000	0.928279
$95$	0	0
$96$	1.00000	0.102062
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	3.00000	0.297044
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	12.0000	1.16554
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	3.00000	0.270501
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	−3.00000	−0.255377
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	9.00000	0.755263
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	3.00000	0.242536
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	7.00000	0.556890
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−6.00000	−0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	6.00000	0.452267
$177$	−6.00000	−0.450988
$178$	−3.00000	−0.224860
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	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
$184$	3.00000	0.221163
$185$	0	0
$186$	5.00000	0.366618
$187$	18.0000	1.31629
$188$	−9.00000	−0.656392
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−1.00000	−0.0721688
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	17.0000	1.22053
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−6.00000	−0.426401
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−12.0000	−0.844317
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−13.0000	−0.905753
$207$	−3.00000	−0.208514
$208$	4.00000	0.277350
$209$	−24.0000	−1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−12.0000	−0.824163
$213$	9.00000	0.616670
$214$	6.00000	0.410152
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	4.00000	0.270914
$219$	14.0000	0.946032
$220$	0	0
$221$	12.0000	0.807207
$222$	−8.00000	−0.536925
$223$	−11.0000	−0.736614	−0.368307	−	0.929704i	$-0.620063\pi$
$223$	−11.0000	−0.736614	−0.368307	+	0.929704i	$0.620063\pi$
$224$	0	0
$225$	0	0
$226$	3.00000	0.199557
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	4.00000	0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	7.00000	0.454699
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−3.00000	−0.191273
$247$	−16.0000	−1.01806
$248$	−5.00000	−0.317500
$249$	−6.00000	−0.380235
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	18.0000	1.11204
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	−3.00000	−0.183597
$268$	−8.00000	−0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−15.0000	−0.906183
$275$	0	0
$276$	3.00000	0.180579
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−2.00000	−0.119952
$279$	5.00000	0.299342
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	−9.00000	−0.535942
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	17.0000	0.996558
$292$	−14.0000	−0.819288
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	−6.00000	−0.348155
$298$	6.00000	0.347571
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	−12.0000	−0.689382
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−3.00000	−0.171499
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	4.00000	0.226455
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−7.00000	−0.393781
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	−12.0000	−0.672927
$319$	−36.0000	−2.01561
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	4.00000	0.221201
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	6.00000	0.329293
$333$	−8.00000	−0.438397
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−3.00000	−0.163178
$339$	3.00000	0.162938
$340$	0	0
$341$	30.0000	1.62459
$342$	4.00000	0.216295
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	6.00000	0.321634
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−6.00000	−0.319801
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	18.0000	0.951330
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	−25.0000	−1.31216
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−3.00000	−0.156386
$369$	−3.00000	−0.156174
$370$	0	0
$371$	0	0
$372$	−5.00000	−0.259238
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	9.00000	0.464140
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−15.0000	−0.767467
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	5.00000	0.254493
$387$	−8.00000	−0.406663
$388$	−17.0000	−0.863044
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	0	0
$396$	6.00000	0.301511
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−5.00000	−0.250627
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−8.00000	−0.399004
$403$	20.0000	0.996271
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	−48.0000	−2.37927
$408$	3.00000	0.148522
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	13.0000	0.640464
$413$	0	0
$414$	3.00000	0.147442
$415$	0	0
$416$	−4.00000	−0.196116
$417$	−2.00000	−0.0979404
$418$	24.0000	1.17388
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−8.00000	−0.389434
$423$	−9.00000	−0.437595
$424$	12.0000	0.582772
$425$	0	0
$426$	−9.00000	−0.436051
$427$	0	0
$428$	−6.00000	−0.290021
$429$	−24.0000	−1.15873
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	−1.00000	−0.0481125
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	12.0000	0.574038
$438$	−14.0000	−0.668946
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	11.0000	0.520865
$447$	6.00000	0.283790
$448$	0	0
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−3.00000	−0.141108
$453$	16.0000	0.751746
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	22.0000	1.02799
$459$	−3.00000	−0.140028
$460$	0	0
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	−6.00000	−0.276172
$473$	−48.0000	−2.20704
$474$	−7.00000	−0.321521
$475$	0	0
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−3.00000	−0.137217
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	22.0000	1.00207
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	25.0000	1.13286	0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000	1.13286	0.566429	+	0.824110i	$0.308325\pi$
$488$	−2.00000	−0.0905357
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	3.00000	0.135250
$493$	−18.0000	−0.810679
$494$	16.0000	0.719874
$495$	0	0
$496$	5.00000	0.224507
$497$	0	0
$498$	6.00000	0.268866
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	0	0
$501$	0	0
$502$	−24.0000	−1.07117
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	18.0000	0.800198
$507$	−3.00000	−0.133235
$508$	16.0000	0.709885
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−18.0000	−0.793946
$515$	0	0
$516$	8.00000	0.352180
$517$	−54.0000	−2.37492
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	6.00000	0.262613
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	−3.00000	−0.130806
$527$	15.0000	0.653410
$528$	−6.00000	−0.261116
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−12.0000	−0.519778
$534$	3.00000	0.129823
$535$	0	0
$536$	8.00000	0.345547
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	25.0000	1.07384
$543$	−2.00000	−0.0858282
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	15.0000	0.640768
$549$	2.00000	0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	−3.00000	−0.127688
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	2.00000	0.0848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−5.00000	−0.211667
$559$	−32.0000	−1.35346
$560$	0	0
$561$	−18.0000	−0.759961
$562$	−27.0000	−1.13893
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	−22.0000	−0.924729
$567$	0	0
$568$	9.00000	0.377632
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	24.0000	1.00349
$573$	−15.0000	−0.626634
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	8.00000	0.332756
$579$	5.00000	0.207793
$580$	0	0
$581$	0	0
$582$	−17.0000	−0.704673
$583$	−72.0000	−2.98194
$584$	14.0000	0.579324
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−5.00000	−0.204636
$598$	12.0000	0.490716
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−16.0000	−0.651031
$605$	0	0
$606$	12.0000	0.487467
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	−36.0000	−1.45640
$612$	3.00000	0.121268
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	13.0000	0.522937
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	−21.0000	−0.842023
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	17.0000	0.679457
$627$	24.0000	0.958468
$628$	−14.0000	−0.558661
$629$	−24.0000	−0.956943
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	7.00000	0.278445
$633$	−8.00000	−0.317971
$634$	12.0000	0.476581
$635$	0	0
$636$	12.0000	0.475831
$637$	0	0
$638$	36.0000	1.42525
$639$	−9.00000	−0.356034
$640$	0	0
$641$	−39.0000	−1.54041	−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000	−1.54041	−0.770204	+	0.637798i	$0.779845\pi$
$642$	−6.00000	−0.236801
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−1.00000	−0.0392837
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−3.00000	−0.117130
$657$	−14.0000	−0.546192
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−26.0000	−1.01052
$663$	−12.0000	−0.466041
$664$	−6.00000	−0.232845
$665$	0	0
$666$	8.00000	0.309994
$667$	18.0000	0.696963
$668$	0	0
$669$	11.0000	0.425285
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	3.00000	0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−3.00000	−0.115214
$679$	0	0
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−30.0000	−1.14876
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	22.0000	0.839352
$688$	−8.00000	−0.304997
$689$	−48.0000	−1.82865
$690$	0	0
$691$	−34.0000	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−9.00000	−0.340899
$698$	28.0000	1.05982
$699$	6.00000	0.226941
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	4.00000	0.150970
$703$	32.0000	1.20690
$704$	6.00000	0.226134
$705$	0	0
$706$	15.0000	0.564532
$707$	0	0
$708$	−6.00000	−0.225494
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	−3.00000	−0.112430
$713$	−15.0000	−0.561754
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−3.00000	−0.112037
$718$	36.0000	1.34351
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	22.0000	0.818189
$724$	2.00000	0.0743294
$725$	0	0
$726$	25.0000	0.927837
$727$	31.0000	1.14973	0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000	1.14973	0.574863	+	0.818250i	$0.305055\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	−2.00000	−0.0739221
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	3.00000	0.110581
$737$	−48.0000	−1.76810
$738$	3.00000	0.110432
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	8.00000	0.292901
$747$	6.00000	0.219529
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−9.00000	−0.328196
$753$	−24.0000	−0.874609
$754$	24.0000	0.874028
$755$	0	0
$756$	0	0
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	10.0000	0.363216
$759$	18.0000	0.653359
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	15.0000	0.542681
$765$	0	0
$766$	21.0000	0.758761
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−5.00000	−0.179954
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	17.0000	0.610264
$777$	0	0
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−54.0000	−1.93227
$782$	9.00000	0.321839
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	−18.0000	−0.642039
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	0	0
$792$	−6.00000	−0.213201
$793$	8.00000	0.284088
$794$	2.00000	0.0709773
$795$	0	0
$796$	5.00000	0.177220
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	3.00000	0.106000
$802$	6.00000	0.211867
$803$	−84.0000	−2.96430
$804$	8.00000	0.282138
$805$	0	0
$806$	−20.0000	−0.704470
$807$	0	0
$808$	−12.0000	−0.422159
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	25.0000	0.876788
$814$	48.0000	1.68240
$815$	0	0
$816$	−3.00000	−0.105021
$817$	32.0000	1.11954
$818$	−29.0000	−1.01396
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	15.0000	0.523185
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	−3.00000	−0.104257
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	4.00000	0.138675
$833$	0	0
$834$	2.00000	0.0692543
$835$	0	0
$836$	−24.0000	−0.830057
$837$	−5.00000	−0.172825
$838$	−24.0000	−0.829066
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	4.00000	0.137849
$843$	−27.0000	−0.929929
$844$	8.00000	0.275371
$845$	0	0
$846$	9.00000	0.309426
$847$	0	0
$848$	−12.0000	−0.412082
$849$	−22.0000	−0.755038
$850$	0	0
$851$	24.0000	0.822709
$852$	9.00000	0.308335
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	0	0
$856$	6.00000	0.205076
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	24.0000	0.819346
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	−21.0000	−0.715263
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−7.00000	−0.237870
$867$	8.00000	0.271694
$868$	0	0
$869$	−42.0000	−1.42475
$870$	0	0
$871$	−32.0000	−1.08428
$872$	4.00000	0.135457
$873$	−17.0000	−0.575363
$874$	−12.0000	−0.405906
$875$	0	0
$876$	14.0000	0.473016
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	13.0000	0.438729
$879$	−18.0000	−0.607125
$880$	0	0
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	0	0
$891$	6.00000	0.201008
$892$	−11.0000	−0.368307
$893$	36.0000	1.20469
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	3.00000	0.100111
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−36.0000	−1.19933
$902$	18.0000	0.599334
$903$	0	0
$904$	3.00000	0.0997785
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	18.0000	0.597351
$909$	12.0000	0.398015
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	4.00000	0.132453
$913$	36.0000	1.19143
$914$	2.00000	0.0661541
$915$	0	0
$916$	−22.0000	−0.726900
$917$	0	0
$918$	3.00000	0.0990148
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−36.0000	−1.18560
$923$	−36.0000	−1.18495
$924$	0	0
$925$	0	0
$926$	−13.0000	−0.427207
$927$	13.0000	0.426976
$928$	6.00000	0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−21.0000	−0.687509
$934$	−30.0000	−0.981630
$935$	0	0
$936$	−4.00000	−0.130744
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	17.0000	0.554774
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	−14.0000	−0.456145
$943$	9.00000	0.293080
$944$	6.00000	0.195283
$945$	0	0
$946$	48.0000	1.56061
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	7.00000	0.227349
$949$	−56.0000	−1.81784
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	3.00000	0.0970269
$957$	36.0000	1.16371
$958$	−21.0000	−0.678479
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	32.0000	1.03172
$963$	−6.00000	−0.193347
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	31.0000	0.996893	0.498446	−	0.866921i	$-0.333904\pi$
$967$	31.0000	0.996893	0.498446	+	0.866921i	$0.333904\pi$
$968$	−25.0000	−0.803530
$969$	12.0000	0.385496
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−25.0000	−0.801052
$975$	0	0
$976$	2.00000	0.0640184
$977$	−57.0000	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$977$	−57.0000	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$978$	−8.00000	−0.255812
$979$	18.0000	0.575282
$980$	0	0
$981$	−4.00000	−0.127710
$982$	12.0000	0.382935
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−3.00000	−0.0956365
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	−16.0000	−0.509028
$989$	24.0000	0.763156
$990$	0	0
$991$	−43.0000	−1.36594	−0.682970	−	0.730446i	$-0.739312\pi$
$991$	−43.0000	−1.36594	−0.682970	+	0.730446i	$0.739312\pi$
$992$	−5.00000	−0.158750
$993$	−26.0000	−0.825085
$994$	0	0
$995$	0	0
$996$	−6.00000	−0.190117
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	−38.0000	−1.20287
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.s.1.1		1
5.4	even	2		7350.2.a.cy.1.1		1
7.2	even	3		1050.2.i.q.151.1	yes	2
7.4	even	3		1050.2.i.q.751.1	yes	2
7.6	odd	2		7350.2.a.bm.1.1		1
35.2	odd	12		1050.2.o.h.949.1		4
35.4	even	6		1050.2.i.c.751.1	yes	2
35.9	even	6		1050.2.i.c.151.1	✓	2
35.18	odd	12		1050.2.o.h.499.1		4
35.23	odd	12		1050.2.o.h.949.2		4
35.32	odd	12		1050.2.o.h.499.2		4
35.34	odd	2		7350.2.a.cg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.i.c.151.1	✓	2	35.9	even	6
1050.2.i.c.751.1	yes	2	35.4	even	6
1050.2.i.q.151.1	yes	2	7.2	even	3
1050.2.i.q.751.1	yes	2	7.4	even	3
1050.2.o.h.499.1		4	35.18	odd	12
1050.2.o.h.499.2		4	35.32	odd	12
1050.2.o.h.949.1		4	35.2	odd	12
1050.2.o.h.949.2		4	35.23	odd	12
7350.2.a.s.1.1		1	1.1	even	1	trivial
7350.2.a.bm.1.1		1	7.6	odd	2
7350.2.a.cg.1.1		1	35.34	odd	2
7350.2.a.cy.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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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