LMFDB - Embedded newform 3450.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +9.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} +1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +6.00000 q^{51} -1.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} -6.00000 q^{59} -4.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +3.00000 q^{77} +1.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} -15.0000 q^{83} -1.00000 q^{84} +1.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} -18.0000 q^{89} +1.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -4.00000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	3.00000	0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	1.00000	0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	6.00000	0.840168
$52$	−1.00000	−0.138675
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	9.00000	1.18176
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	3.00000	0.341882
$78$	1.00000	0.113228
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	1.00000	0.107833
$87$	−9.00000	−0.964901
$88$	3.00000	0.319801
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−1.00000	−0.104257
$93$	−4.00000	−0.414781
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	6.00000	0.606092
$99$	−3.00000	−0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−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
$109$	−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
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−9.00000	−0.835629
$117$	−1.00000	−0.0924500
$118$	6.00000	0.552345
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	9.00000	0.811503
$124$	−4.00000	−0.359211
$125$	0	0
$126$	1.00000	0.0890871
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−3.00000	−0.261116
$133$	1.00000	0.0867110
$134$	4.00000	0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.00000	0.0851257
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	6.00000	0.503509
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	7.00000	0.579324
$147$	−6.00000	−0.494872
$148$	−4.00000	−0.328798
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	1.00000	0.0811107
$153$	6.00000	0.485071
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	1.00000	0.0795557
$159$	12.0000	0.951662
$160$	0	0
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	15.0000	1.16423
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−1.00000	−0.0762493
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−3.00000	−0.226134
$177$	−6.00000	−0.450988
$178$	18.0000	1.34916
$179$	−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
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−1.00000	−0.0741249
$183$	−4.00000	−0.295689
$184$	1.00000	0.0737210
$185$	0	0
$186$	4.00000	0.293294
$187$	−18.0000	−1.31629
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	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
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	3.00000	0.213201
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	−6.00000	−0.422159
$203$	9.00000	0.631676
$204$	6.00000	0.420084
$205$	0	0
$206$	7.00000	0.487713
$207$	−1.00000	−0.0695048
$208$	−1.00000	−0.0693375
$209$	3.00000	0.207514
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	12.0000	0.824163
$213$	−6.00000	−0.411113
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	4.00000	0.271538
$218$	4.00000	0.270914
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−6.00000	−0.403604
$222$	4.00000	0.268462
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−18.0000	−1.19734
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−1.00000	−0.0662266
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	9.00000	0.590879
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−1.00000	−0.0649570
$238$	6.00000	0.388922
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−9.00000	−0.573819
$247$	1.00000	0.0636285
$248$	4.00000	0.254000
$249$	−15.0000	−0.950586
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−1.00000	−0.0629941
$253$	3.00000	0.188608
$254$	−14.0000	−0.878438
$255$	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	$0.310261\pi$
$258$	1.00000	0.0622573
$259$	4.00000	0.248548
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	−18.0000	−1.10158
$268$	−4.00000	−0.244339
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	1.00000	0.0605228
$274$	12.0000	0.724947
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	10.0000	0.599760
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−6.00000	−0.357295
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−3.00000	−0.177394
$287$	−9.00000	−0.531253
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−4.00000	−0.234484
$292$	−7.00000	−0.409644
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	4.00000	0.232495
$297$	−3.00000	−0.174078
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	1.00000	0.0576390
$302$	10.0000	0.575435
$303$	6.00000	0.344691
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	3.00000	0.170941
$309$	−7.00000	−0.398216
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	1.00000	0.0566139
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−12.0000	−0.672927
$319$	27.0000	1.51171
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−1.00000	−0.0557278
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	−4.00000	−0.221201
$328$	−9.00000	−0.496942
$329$	−6.00000	−0.330791
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−15.0000	−0.823232
$333$	−4.00000	−0.219199
$334$	−24.0000	−1.31322
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	12.0000	0.652714
$339$	18.0000	0.977626
$340$	0	0
$341$	12.0000	0.649836
$342$	1.00000	0.0540738
$343$	13.0000	0.701934
$344$	1.00000	0.0539164
$345$	0	0
$346$	−15.0000	−0.806405
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−9.00000	−0.482451
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	3.00000	0.159901
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−6.00000	−0.317554
$358$	18.0000	0.951330
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−14.0000	−0.735824
$363$	−2.00000	−0.104973
$364$	1.00000	0.0524142
$365$	0	0
$366$	4.00000	0.209083
$367$	−37.0000	−1.93138	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−37.0000	−1.93138	−0.965692	+	0.259690i	$0.916380\pi$
$368$	−1.00000	−0.0521286
$369$	9.00000	0.468521
$370$	0	0
$371$	−12.0000	−0.623009
$372$	−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	$0.583357\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−6.00000	−0.309426
$377$	9.00000	0.463524
$378$	1.00000	0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	−15.0000	−0.767467
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−1.00000	−0.0508329
$388$	−4.00000	−0.203069
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	6.00000	0.303046
$393$	0	0
$394$	9.00000	0.453413
$395$	0	0
$396$	−3.00000	−0.150756
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	13.0000	0.651631
$399$	1.00000	0.0500626
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	4.00000	0.199502
$403$	4.00000	0.199254
$404$	6.00000	0.298511
$405$	0	0
$406$	−9.00000	−0.446663
$407$	12.0000	0.594818
$408$	−6.00000	−0.297044
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−7.00000	−0.344865
$413$	6.00000	0.295241
$414$	1.00000	0.0491473
$415$	0	0
$416$	1.00000	0.0490290
$417$	−10.0000	−0.489702
$418$	−3.00000	−0.146735
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	22.0000	1.07094
$423$	6.00000	0.291730
$424$	−12.0000	−0.582772
$425$	0	0
$426$	6.00000	0.290701
$427$	4.00000	0.193574
$428$	−12.0000	−0.580042
$429$	3.00000	0.144841
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−4.00000	−0.191565
$437$	1.00000	0.0478365
$438$	7.00000	0.334473
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	6.00000	0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	10.0000	0.473514
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	18.0000	0.846649
$453$	−10.0000	−0.469841
$454$	24.0000	1.12638
$455$	0	0
$456$	1.00000	0.0468293
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	4.00000	0.186908
$459$	6.00000	0.280056
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	−3.00000	−0.139573
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−9.00000	−0.416917
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	−1.00000	−0.0462250
$469$	4.00000	0.184703
$470$	0	0
$471$	8.00000	0.368621
$472$	6.00000	0.276172
$473$	3.00000	0.137940
$474$	1.00000	0.0459315
$475$	0	0
$476$	−6.00000	−0.275010
$477$	12.0000	0.549442
$478$	−12.0000	−0.548867
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	10.0000	0.455488
$483$	1.00000	0.0455016
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	4.00000	0.181071
$489$	−16.0000	−0.723545
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	9.00000	0.405751
$493$	−54.0000	−2.43204
$494$	−1.00000	−0.0449921
$495$	0	0
$496$	−4.00000	−0.179605
$497$	6.00000	0.269137
$498$	15.0000	0.672166
$499$	26.0000	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$499$	26.0000	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	−12.0000	−0.535586
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−3.00000	−0.133366
$507$	−12.0000	−0.532939
$508$	14.0000	0.621150
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	−18.0000	−0.791639
$518$	−4.00000	−0.175750
$519$	15.0000	0.658427
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	9.00000	0.393919
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	0	0
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−24.0000	−1.04546
$528$	−3.00000	−0.130558
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	1.00000	0.0433555
$533$	−9.00000	−0.389833
$534$	18.0000	0.778936
$535$	0	0
$536$	4.00000	0.172774
$537$	−18.0000	−0.776757
$538$	9.00000	0.388018
$539$	18.0000	0.775315
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	−8.00000	−0.343629
$543$	14.0000	0.600798
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−12.0000	−0.512615
$549$	−4.00000	−0.170716
$550$	0	0
$551$	9.00000	0.383413
$552$	1.00000	0.0425628
$553$	1.00000	0.0425243
$554$	19.0000	0.807233
$555$	0	0
$556$	−10.0000	−0.424094
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	4.00000	0.169334
$559$	1.00000	0.0422955
$560$	0	0
$561$	−18.0000	−0.759961
$562$	6.00000	0.253095
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	6.00000	0.251754
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	3.00000	0.125436
$573$	15.0000	0.626634
$574$	9.00000	0.375653
$575$	0	0
$576$	1.00000	0.0416667
$577$	−31.0000	−1.29055	−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000	−1.29055	−0.645273	+	0.763952i	$0.723257\pi$
$578$	−19.0000	−0.790296
$579$	2.00000	0.0831172
$580$	0	0
$581$	15.0000	0.622305
$582$	4.00000	0.165805
$583$	−36.0000	−1.49097
$584$	7.00000	0.289662
$585$	0	0
$586$	12.0000	0.495715
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	−6.00000	−0.247436
$589$	4.00000	0.164817
$590$	0	0
$591$	−9.00000	−0.370211
$592$	−4.00000	−0.164399
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	0	0
$597$	−13.0000	−0.532055
$598$	−1.00000	−0.0408930
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	−1.00000	−0.0407570
$603$	−4.00000	−0.162893
$604$	−10.0000	−0.406894
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	1.00000	0.0405554
$609$	9.00000	0.364698
$610$	0	0
$611$	−6.00000	−0.242734
$612$	6.00000	0.242536
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−3.00000	−0.120873
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	7.00000	0.281581
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−12.0000	−0.481156
$623$	18.0000	0.721155
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−8.00000	−0.319744
$627$	3.00000	0.119808
$628$	8.00000	0.319235
$629$	−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	$0.390121\pi$
$632$	1.00000	0.0397779
$633$	−22.0000	−0.874421
$634$	−3.00000	−0.119145
$635$	0	0
$636$	12.0000	0.475831
$637$	6.00000	0.237729
$638$	−27.0000	−1.06894
$639$	−6.00000	−0.237356
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	12.0000	0.473602
$643$	−49.0000	−1.93237	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000	−1.93237	−0.966186	+	0.257847i	$0.916987\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	6.00000	0.236067
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	0	0
$651$	4.00000	0.156772
$652$	−16.0000	−0.626608
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	9.00000	0.351391
$657$	−7.00000	−0.273096
$658$	6.00000	0.233904
$659$	−45.0000	−1.75295	−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000	−1.75295	−0.876476	+	0.481446i	$0.840112\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−8.00000	−0.310929
$663$	−6.00000	−0.233021
$664$	15.0000	0.582113
$665$	0	0
$666$	4.00000	0.154997
$667$	9.00000	0.348481
$668$	24.0000	0.928588
$669$	−10.0000	−0.386622
$670$	0	0
$671$	12.0000	0.463255
$672$	1.00000	0.0385758
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−12.0000	−0.461538
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−18.0000	−0.691286
$679$	4.00000	0.153506
$680$	0	0
$681$	−24.0000	−0.919682
$682$	−12.0000	−0.459504
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−4.00000	−0.152610
$688$	−1.00000	−0.0381246
$689$	−12.0000	−0.457164
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	15.0000	0.570214
$693$	3.00000	0.113961
$694$	−12.0000	−0.455514
$695$	0	0
$696$	9.00000	0.341144
$697$	54.0000	2.04540
$698$	−5.00000	−0.189253
$699$	9.00000	0.340411
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	1.00000	0.0377426
$703$	4.00000	0.150863
$704$	−3.00000	−0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	−6.00000	−0.225653
$708$	−6.00000	−0.225494
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	18.0000	0.674579
$713$	4.00000	0.149801
$714$	6.00000	0.224544
$715$	0	0
$716$	−18.0000	−0.672692
$717$	12.0000	0.448148
$718$	3.00000	0.111959
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	18.0000	0.669891
$723$	−10.0000	−0.371904
$724$	14.0000	0.520306
$725$	0	0
$726$	2.00000	0.0742270
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.00000	−0.221918
$732$	−4.00000	−0.147844
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	37.0000	1.36569
$735$	0	0
$736$	1.00000	0.0368605
$737$	12.0000	0.442026
$738$	−9.00000	−0.331295
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	12.0000	0.440534
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	10.0000	0.366126
$747$	−15.0000	−0.548821
$748$	−18.0000	−0.658145
$749$	12.0000	0.438470
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	6.00000	0.218797
$753$	12.0000	0.437304
$754$	−9.00000	−0.327761
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	−20.0000	−0.726433
$759$	3.00000	0.108893
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	−14.0000	−0.507166
$763$	4.00000	0.144810
$764$	15.0000	0.542681
$765$	0	0
$766$	27.0000	0.975550
$767$	6.00000	0.216647
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	2.00000	0.0719816
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	4.00000	0.143592
$777$	4.00000	0.143499
$778$	30.0000	1.07555
$779$	−9.00000	−0.322458
$780$	0	0
$781$	18.0000	0.644091
$782$	6.00000	0.214560
$783$	−9.00000	−0.321634
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	−9.00000	−0.320612
$789$	12.0000	0.427211
$790$	0	0
$791$	−18.0000	−0.640006
$792$	3.00000	0.106600
$793$	4.00000	0.142044
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−13.0000	−0.460773
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−1.00000	−0.0353996
$799$	36.0000	1.27359
$800$	0	0
$801$	−18.0000	−0.635999
$802$	−12.0000	−0.423735
$803$	21.0000	0.741074
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−9.00000	−0.316815
$808$	−6.00000	−0.211079
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	9.00000	0.315838
$813$	8.00000	0.280572
$814$	−12.0000	−0.420600
$815$	0	0
$816$	6.00000	0.210042
$817$	1.00000	0.0349856
$818$	7.00000	0.244749
$819$	1.00000	0.0349428
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	12.0000	0.418548
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−6.00000	−0.208767
$827$	27.0000	0.938882	0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000	0.938882	0.469441	+	0.882964i	$0.344455\pi$
$828$	−1.00000	−0.0347524
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	−1.00000	−0.0346688
$833$	−36.0000	−1.24733
$834$	10.0000	0.346272
$835$	0	0
$836$	3.00000	0.103757
$837$	−4.00000	−0.138260
$838$	9.00000	0.310900
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−14.0000	−0.482472
$843$	−6.00000	−0.206651
$844$	−22.0000	−0.757271
$845$	0	0
$846$	−6.00000	−0.206284
$847$	2.00000	0.0687208
$848$	12.0000	0.412082
$849$	−4.00000	−0.137280
$850$	0	0
$851$	4.00000	0.137118
$852$	−6.00000	−0.205557
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	−3.00000	−0.102418
$859$	38.0000	1.29654	0.648272	−	0.761409i	$-0.275492\pi$
$859$	38.0000	1.29654	0.648272	+	0.761409i	$0.275492\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	−24.0000	−0.817443
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	28.0000	0.951479
$867$	19.0000	0.645274
$868$	4.00000	0.135769
$869$	3.00000	0.101768
$870$	0	0
$871$	4.00000	0.135535
$872$	4.00000	0.135457
$873$	−4.00000	−0.135379
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	−7.00000	−0.236508
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−20.0000	−0.674967
$879$	−12.0000	−0.404750
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	6.00000	0.202031
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	24.0000	0.806296
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	4.00000	0.134231
$889$	−14.0000	−0.469545
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−10.0000	−0.334825
$893$	−6.00000	−0.200782
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	1.00000	0.0333890
$898$	18.0000	0.600668
$899$	36.0000	1.20067
$900$	0	0
$901$	72.0000	2.39867
$902$	27.0000	0.899002
$903$	1.00000	0.0332779
$904$	−18.0000	−0.598671
$905$	0	0
$906$	10.0000	0.332228
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	−24.0000	−0.796468
$909$	6.00000	0.199007
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	−1.00000	−0.0331133
$913$	45.0000	1.48928
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−4.00000	−0.132164
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	21.0000	0.691598
$923$	6.00000	0.197492
$924$	3.00000	0.0986928
$925$	0	0
$926$	4.00000	0.131448
$927$	−7.00000	−0.229910
$928$	9.00000	0.295439
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	9.00000	0.294805
$933$	12.0000	0.392862
$934$	21.0000	0.687141
$935$	0	0
$936$	1.00000	0.0326860
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	−4.00000	−0.130605
$939$	8.00000	0.261070
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	−8.00000	−0.260654
$943$	−9.00000	−0.293080
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−1.00000	−0.0324785
$949$	7.00000	0.227230
$950$	0	0
$951$	3.00000	0.0972817
$952$	6.00000	0.194461
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	12.0000	0.388108
$957$	27.0000	0.872786
$958$	−3.00000	−0.0969256
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	2.00000	0.0642824
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	1.00000	0.0320750
$973$	10.0000	0.320585
$974$	−14.0000	−0.448589
$975$	0	0
$976$	−4.00000	−0.128037
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	16.0000	0.511624
$979$	54.0000	1.72585
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−45.0000	−1.43528	−0.717639	−	0.696416i	$-0.754777\pi$
$983$	−45.0000	−1.43528	−0.717639	+	0.696416i	$0.754777\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	54.0000	1.71971
$987$	−6.00000	−0.190982
$988$	1.00000	0.0318142
$989$	1.00000	0.0317982
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	4.00000	0.127000
$993$	8.00000	0.253872
$994$	−6.00000	−0.190308
$995$	0	0
$996$	−15.0000	−0.475293
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	−26.0000	−0.823016
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.i.1.1	✓	1
5.2	odd	4		3450.2.d.c.2899.1		2
5.3	odd	4		3450.2.d.c.2899.2		2
5.4	even	2		3450.2.a.r.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.i.1.1	✓	1	1.1	even	1	trivial
3450.2.a.r.1.1	yes	1	5.4	even	2
3450.2.d.c.2899.1		2	5.2	odd	4
3450.2.d.c.2899.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more