LMFDB - Embedded newform 2175.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.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:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} +8.00000 q^{17} -2.00000 q^{18} -2.00000 q^{21} +6.00000 q^{22} -1.00000 q^{23} -8.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{29} -8.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} -16.0000 q^{34} +2.00000 q^{36} -7.00000 q^{37} +4.00000 q^{39} +7.00000 q^{41} +4.00000 q^{42} +9.00000 q^{43} -6.00000 q^{44} +2.00000 q^{46} -12.0000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +8.00000 q^{51} +8.00000 q^{52} +9.00000 q^{53} -2.00000 q^{54} -2.00000 q^{58} +10.0000 q^{59} +2.00000 q^{61} +16.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +6.00000 q^{66} +8.00000 q^{67} +16.0000 q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{73} +14.0000 q^{74} +6.00000 q^{77} -8.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -14.0000 q^{82} +9.00000 q^{83} -4.00000 q^{84} -18.0000 q^{86} +1.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} -2.00000 q^{92} -8.00000 q^{93} +24.0000 q^{94} +8.00000 q^{96} +13.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−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
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	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	$0.649385\pi$
$12$	2.00000	0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	−2.00000	−0.471405
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	6.00000	1.27920
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	−8.00000	−1.56893
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	−3.00000	−0.522233
$34$	−16.0000	−2.74398
$35$	0	0
$36$	2.00000	0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	4.00000	0.617213
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	2.00000	0.294884
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−4.00000	−0.577350
$49$	−3.00000	−0.428571
$50$	0	0
$51$	8.00000	1.12022
$52$	8.00000	1.10940
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−2.00000	−0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	16.0000	2.03200
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	6.00000	0.738549
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	16.0000	1.94029
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	6.00000	0.683763
$78$	−8.00000	−0.905822
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.0000	−1.54604
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−18.0000	−1.94099
$87$	1.00000	0.107211
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	−2.00000	−0.208514
$93$	−8.00000	−0.829561
$94$	24.0000	2.47541
$95$	0	0
$96$	8.00000	0.816497
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	6.00000	0.606092
$99$	−3.00000	−0.301511
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	−16.0000	−1.58424
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	−18.0000	−1.74831
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	2.00000	0.192450
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	8.00000	0.755929
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	−20.0000	−1.84115
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	7.00000	0.631169
$124$	−16.0000	−1.43684
$125$	0	0
$126$	4.00000	0.356348
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	0	0
$129$	9.00000	0.792406
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−16.0000	−1.38219
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	2.00000	0.170251
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	16.0000	1.34269
$143$	−12.0000	−1.00349
$144$	−4.00000	−0.333333
$145$	0	0
$146$	2.00000	0.165521
$147$	−3.00000	−0.247436
$148$	−14.0000	−1.15079
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	−12.0000	−0.966988
$155$	0	0
$156$	8.00000	0.640513
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	20.0000	1.59111
$159$	9.00000	0.713746
$160$	0	0
$161$	2.00000	0.157622
$162$	−2.00000	−0.157135
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	14.0000	1.09322
$165$	0	0
$166$	−18.0000	−1.39707
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	18.0000	1.37249
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	12.0000	0.904534
$177$	10.0000	0.751646
$178$	−20.0000	−1.49906
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	16.0000	1.18600
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	16.0000	1.17318
$187$	−24.0000	−1.75505
$188$	−24.0000	−1.75038
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−8.00000	−0.577350
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−26.0000	−1.86669
$195$	0	0
$196$	−6.00000	−0.428571
$197$	13.0000	0.926212	0.463106	−	0.886303i	$-0.346735\pi$
$197$	13.0000	0.926212	0.463106	+	0.886303i	$0.346735\pi$
$198$	6.00000	0.426401
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−34.0000	−2.39223
$203$	−2.00000	−0.140372
$204$	16.0000	1.12022
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−1.00000	−0.0695048
$208$	−16.0000	−1.10940
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	18.0000	1.23625
$213$	−8.00000	−0.548151
$214$	24.0000	1.64061
$215$	0	0
$216$	0	0
$217$	16.0000	1.08615
$218$	10.0000	0.677285
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	32.0000	2.15255
$222$	14.0000	0.939618
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−28.0000	−1.86253
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−8.00000	−0.522976
$235$	0	0
$236$	20.0000	1.30189
$237$	−10.0000	−0.649570
$238$	32.0000	2.07425
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	4.00000	0.257130
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−14.0000	−0.892607
$247$	0	0
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−4.00000	−0.251976
$253$	3.00000	0.188608
$254$	−6.00000	−0.376473
$255$	0	0
$256$	16.0000	1.00000
$257$	23.0000	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$257$	23.0000	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$258$	−18.0000	−1.12063
$259$	14.0000	0.869918
$260$	0	0
$261$	1.00000	0.0618984
$262$	−24.0000	−1.48272
$263$	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
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	16.0000	0.977356
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−32.0000	−1.94029
$273$	−8.00000	−0.484182
$274$	4.00000	0.241649
$275$	0	0
$276$	−2.00000	−0.120386
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	−10.0000	−0.599760
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	24.0000	1.42918
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	24.0000	1.41915
$287$	−14.0000	−0.826394
$288$	8.00000	0.471405
$289$	47.0000	2.76471
$290$	0	0
$291$	13.0000	0.762073
$292$	−2.00000	−0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−18.0000	−1.03750
$302$	6.00000	0.345261
$303$	17.0000	0.976624
$304$	0	0
$305$	0	0
$306$	−16.0000	−0.914659
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	12.0000	0.683763
$309$	4.00000	0.227552
$310$	0	0
$311$	17.0000	0.963982	0.481991	−	0.876176i	$-0.339914\pi$
$311$	17.0000	0.963982	0.481991	+	0.876176i	$0.339914\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−20.0000	−1.12509
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	−18.0000	−1.00939
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−4.00000	−0.222911
$323$	0	0
$324$	2.00000	0.111111
$325$	0	0
$326$	2.00000	0.110770
$327$	−5.00000	−0.276501
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	18.0000	0.987878
$333$	−7.00000	−0.383598
$334$	−16.0000	−0.875481
$335$	0	0
$336$	8.00000	0.436436
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−6.00000	−0.326357
$339$	14.0000	0.760376
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	−18.0000	−0.967686
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	2.00000	0.107211
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−24.0000	−1.27920
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	20.0000	1.06000
$357$	−16.0000	−0.846810
$358$	−20.0000	−1.05703
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−34.0000	−1.78700
$363$	−2.00000	−0.104973
$364$	−16.0000	−0.838628
$365$	0	0
$366$	−4.00000	−0.209083
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	4.00000	0.208514
$369$	7.00000	0.364405
$370$	0	0
$371$	−18.0000	−0.934513
$372$	−16.0000	−0.829561
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	48.0000	2.48202
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	4.00000	0.205738
$379$	−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
$381$	3.00000	0.153695
$382$	6.00000	0.306987
$383$	39.0000	1.99281	0.996403	−	0.0847358i	$-0.0270046\pi$
$383$	39.0000	1.99281	0.996403	+	0.0847358i	$0.0270046\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	9.00000	0.457496
$388$	26.0000	1.31995
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	12.0000	0.605320
$394$	−26.0000	−1.30986
$395$	0	0
$396$	−6.00000	−0.301511
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	50.0000	2.50627
$399$	0	0
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−16.0000	−0.798007
$403$	−32.0000	−1.59403
$404$	34.0000	1.69156
$405$	0	0
$406$	4.00000	0.198517
$407$	21.0000	1.04093
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	8.00000	0.394132
$413$	−20.0000	−0.984136
$414$	2.00000	0.0982946
$415$	0	0
$416$	32.0000	1.56893
$417$	5.00000	0.244851
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−24.0000	−1.16830
$423$	−12.0000	−0.583460
$424$	0	0
$425$	0	0
$426$	16.0000	0.775203
$427$	−4.00000	−0.193574
$428$	−24.0000	−1.16008
$429$	−12.0000	−0.579365
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−4.00000	−0.192450
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	2.00000	0.0955637
$439$	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
$441$	−3.00000	−0.142857
$442$	−64.0000	−3.04417
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	12.0000	0.568216
$447$	0	0
$448$	16.0000	0.755929
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	28.0000	1.31701
$453$	−3.00000	−0.140952
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	−40.0000	−1.86908
$459$	8.00000	0.373408
$460$	0	0
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	−12.0000	−0.558291
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	2.00000	0.0926482
$467$	−42.0000	−1.94353	−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000	−1.94353	−0.971764	+	0.235954i	$0.924178\pi$
$468$	8.00000	0.369800
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−27.0000	−1.24146
$474$	20.0000	0.918630
$475$	0	0
$476$	−32.0000	−1.46672
$477$	9.00000	0.412082
$478$	−60.0000	−2.74434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	6.00000	0.273293
$483$	2.00000	0.0910032
$484$	−4.00000	−0.181818
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	14.0000	0.631169
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	32.0000	1.43684
$497$	16.0000	0.717698
$498$	−18.0000	−0.806599
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	−24.0000	−1.07117
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	3.00000	0.133235
$508$	6.00000	0.266207
$509$	−40.0000	−1.77297	−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000	−1.77297	−0.886484	+	0.462758i	$0.846860\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−46.0000	−2.02897
$515$	0	0
$516$	18.0000	0.792406
$517$	36.0000	1.58328
$518$	−28.0000	−1.23025
$519$	9.00000	0.395056
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	−2.00000	−0.0875376
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	−8.00000	−0.348817
$527$	−64.0000	−2.78788
$528$	12.0000	0.522233
$529$	−22.0000	−0.956522
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	28.0000	1.21281
$534$	−20.0000	−0.865485
$535$	0	0
$536$	0	0
$537$	10.0000	0.431532
$538$	20.0000	0.862261
$539$	9.00000	0.387657
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−24.0000	−1.03089
$543$	17.0000	0.729540
$544$	64.0000	2.74398
$545$	0	0
$546$	16.0000	0.684737
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−4.00000	−0.170872
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	20.0000	0.850487
$554$	−56.0000	−2.37921
$555$	0	0
$556$	10.0000	0.424094
$557$	−47.0000	−1.99145	−0.995727	−	0.0923462i	$-0.970563\pi$
$557$	−47.0000	−1.99145	−0.995727	+	0.0923462i	$0.970563\pi$
$558$	16.0000	0.677334
$559$	36.0000	1.52264
$560$	0	0
$561$	−24.0000	−1.01328
$562$	36.0000	1.51857
$563$	−46.0000	−1.93867	−0.969334	−	0.245745i	$-0.920967\pi$
$563$	−46.0000	−1.93867	−0.969334	+	0.245745i	$0.920967\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	−28.0000	−1.17693
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	−24.0000	−1.00349
$573$	−3.00000	−0.125327
$574$	28.0000	1.16870
$575$	0	0
$576$	−8.00000	−0.333333
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−94.0000	−3.90988
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−18.0000	−0.746766
$582$	−26.0000	−1.07773
$583$	−27.0000	−1.11823
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	13.0000	0.534749
$592$	28.0000	1.15079
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	0	0
$597$	−25.0000	−1.02318
$598$	8.00000	0.327144
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	36.0000	1.46725
$603$	8.00000	0.325785
$604$	−6.00000	−0.244137
$605$	0	0
$606$	−34.0000	−1.38116
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−48.0000	−1.94187
$612$	16.0000	0.646762
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−8.00000	−0.321807
$619$	−30.0000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−34.0000	−1.36328
$623$	−20.0000	−0.801283
$624$	−16.0000	−0.640513
$625$	0	0
$626$	−8.00000	−0.319744
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−56.0000	−2.23287
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	24.0000	0.953162
$635$	0	0
$636$	18.0000	0.713746
$637$	−12.0000	−0.475457
$638$	6.00000	0.237542
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−43.0000	−1.69840	−0.849199	−	0.528073i	$-0.822915\pi$
$641$	−43.0000	−1.69840	−0.849199	+	0.528073i	$0.822915\pi$
$642$	24.0000	0.947204
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	0	0
$651$	16.0000	0.627089
$652$	−2.00000	−0.0783260
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−28.0000	−1.09322
$657$	−1.00000	−0.0390137
$658$	−48.0000	−1.87123
$659$	25.0000	0.973862	0.486931	−	0.873441i	$-0.338116\pi$
$659$	25.0000	0.973862	0.486931	+	0.873441i	$0.338116\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	−64.0000	−2.48743
$663$	32.0000	1.24278
$664$	0	0
$665$	0	0
$666$	14.0000	0.542489
$667$	−1.00000	−0.0387202
$668$	16.0000	0.619059
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−6.00000	−0.231627
$672$	−16.0000	−0.617213
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	44.0000	1.69482
$675$	0	0
$676$	6.00000	0.230769
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−28.0000	−1.07533
$679$	−26.0000	−0.997788
$680$	0	0
$681$	3.00000	0.114960
$682$	−48.0000	−1.83801
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	0	0
$685$	0	0
$686$	−40.0000	−1.52721
$687$	20.0000	0.763048
$688$	−36.0000	−1.37249
$689$	36.0000	1.37149
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	18.0000	0.684257
$693$	6.00000	0.227921
$694$	−6.00000	−0.227757
$695$	0	0
$696$	0	0
$697$	56.0000	2.12115
$698$	10.0000	0.378506
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−8.00000	−0.301941
$703$	0	0
$704$	24.0000	0.904534
$705$	0	0
$706$	52.0000	1.95705
$707$	−34.0000	−1.27870
$708$	20.0000	0.751646
$709$	−45.0000	−1.69001	−0.845005	−	0.534758i	$-0.820403\pi$
$709$	−45.0000	−1.69001	−0.845005	+	0.534758i	$0.820403\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	8.00000	0.299602
$714$	32.0000	1.19757
$715$	0	0
$716$	20.0000	0.747435
$717$	30.0000	1.12037
$718$	30.0000	1.11959
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	38.0000	1.41421
$723$	−3.00000	−0.111571
$724$	34.0000	1.26360
$725$	0	0
$726$	4.00000	0.148454
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	72.0000	2.66302
$732$	4.00000	0.147844
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−24.0000	−0.884051
$738$	−14.0000	−0.515347
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	36.0000	1.32160
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	0	0
$746$	−8.00000	−0.292901
$747$	9.00000	0.329293
$748$	−48.0000	−1.75505
$749$	24.0000	0.876941
$750$	0	0
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	48.0000	1.75038
$753$	12.0000	0.437304
$754$	−8.00000	−0.291343
$755$	0	0
$756$	−4.00000	−0.145479
$757$	3.00000	0.109037	0.0545184	−	0.998513i	$-0.482638\pi$
$757$	3.00000	0.109037	0.0545184	+	0.998513i	$0.482638\pi$
$758$	20.0000	0.726433
$759$	3.00000	0.108893
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−6.00000	−0.217357
$763$	10.0000	0.362024
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−78.0000	−2.81825
$767$	40.0000	1.44432
$768$	16.0000	0.577350
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	−12.0000	−0.431889
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−18.0000	−0.646997
$775$	0	0
$776$	0	0
$777$	14.0000	0.502247
$778$	50.0000	1.79259
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	16.0000	0.572159
$783$	1.00000	0.0357371
$784$	12.0000	0.428571
$785$	0	0
$786$	−24.0000	−0.856052
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	26.0000	0.926212
$789$	4.00000	0.142404
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	8.00000	0.284088
$794$	−16.0000	−0.567819
$795$	0	0
$796$	−50.0000	−1.77220
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	−96.0000	−3.39624
$800$	0	0
$801$	10.0000	0.353333
$802$	56.0000	1.97743
$803$	3.00000	0.105868
$804$	16.0000	0.564276
$805$	0	0
$806$	64.0000	2.25430
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	−4.00000	−0.140372
$813$	12.0000	0.420858
$814$	−42.0000	−1.47210
$815$	0	0
$816$	−32.0000	−1.12022
$817$	0	0
$818$	20.0000	0.699284
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	4.00000	0.139516
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	40.0000	1.39178
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	−2.00000	−0.0695048
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	−32.0000	−1.10940
$833$	−24.0000	−0.831551
$834$	−10.0000	−0.346272
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	56.0000	1.92989
$843$	−18.0000	−0.619953
$844$	24.0000	0.826114
$845$	0	0
$846$	24.0000	0.825137
$847$	4.00000	0.137442
$848$	−36.0000	−1.23625
$849$	14.0000	0.480479
$850$	0	0
$851$	7.00000	0.239957
$852$	−16.0000	−0.548151
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	0	0
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	24.0000	0.819346
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	0	0
$861$	−14.0000	−0.477119
$862$	−24.0000	−0.817443
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	−38.0000	−1.29129
$867$	47.0000	1.59620
$868$	32.0000	1.08615
$869$	30.0000	1.01768
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	13.0000	0.439983
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−40.0000	−1.34993
$879$	−6.00000	−0.202375
$880$	0	0
$881$	57.0000	1.92038	0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000	1.92038	0.960189	+	0.279350i	$0.0901189\pi$
$882$	6.00000	0.202031
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	64.0000	2.15255
$885$	0	0
$886$	12.0000	0.403148
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−12.0000	−0.401790
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.00000	−0.133556
$898$	−10.0000	−0.333704
$899$	−8.00000	−0.266815
$900$	0	0
$901$	72.0000	2.39867
$902$	42.0000	1.39845
$903$	−18.0000	−0.599002
$904$	0	0
$905$	0	0
$906$	6.00000	0.199337
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	6.00000	0.199117
$909$	17.0000	0.563854
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	−56.0000	−1.85232
$915$	0	0
$916$	40.0000	1.32164
$917$	−24.0000	−0.792550
$918$	−16.0000	−0.528079
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−17.0000	−0.560169
$922$	26.0000	0.856264
$923$	−32.0000	−1.05329
$924$	12.0000	0.394771
$925$	0	0
$926$	12.0000	0.394344
$927$	4.00000	0.131377
$928$	8.00000	0.262613
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	17.0000	0.556555
$934$	84.0000	2.74856
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	32.0000	1.04484
$939$	4.00000	0.130535
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	4.00000	0.130327
$943$	−7.00000	−0.227951
$944$	−40.0000	−1.30189
$945$	0	0
$946$	54.0000	1.75569
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−20.0000	−0.649570
$949$	−4.00000	−0.129845
$950$	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	$0.530982\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	60.0000	1.94054
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	33.0000	1.06452
$962$	56.0000	1.80551
$963$	−12.0000	−0.386695
$964$	−6.00000	−0.193247
$965$	0	0
$966$	−4.00000	−0.128698
$967$	3.00000	0.0964735	0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000	0.0964735	0.0482367	+	0.998836i	$0.484640\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	2.00000	0.0641500
$973$	−10.0000	−0.320585
$974$	44.0000	1.40985
$975$	0	0
$976$	−8.00000	−0.256074
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	2.00000	0.0639529
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−5.00000	−0.159638
$982$	56.0000	1.78703
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	0	0
$986$	−16.0000	−0.509544
$987$	24.0000	0.763928
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	−64.0000	−2.03200
$993$	32.0000	1.01549
$994$	−32.0000	−1.01498
$995$	0	0
$996$	18.0000	0.570352
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	−40.0000	−1.26618
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.a.1.1		1
3.2	odd	2		6525.2.a.l.1.1		1
5.2	odd	4		435.2.c.a.349.1	✓	2
5.3	odd	4		435.2.c.a.349.2	yes	2
5.4	even	2		2175.2.a.j.1.1		1
15.2	even	4		1305.2.c.a.784.2		2
15.8	even	4		1305.2.c.a.784.1		2
15.14	odd	2		6525.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.a.349.1	✓	2	5.2	odd	4
435.2.c.a.349.2	yes	2	5.3	odd	4
1305.2.c.a.784.1		2	15.8	even	4
1305.2.c.a.784.2		2	15.2	even	4
2175.2.a.a.1.1		1	1.1	even	1	trivial
2175.2.a.j.1.1		1	5.4	even	2
6525.2.a.b.1.1		1	15.14	odd	2
6525.2.a.l.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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