LMFDB - Embedded newform 4410.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -7.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +7.00000 q^{26} +8.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} -3.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +9.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{46} -3.00000 q^{47} -1.00000 q^{50} -7.00000 q^{52} +1.00000 q^{53} +1.00000 q^{55} -8.00000 q^{58} +12.0000 q^{59} +4.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -7.00000 q^{65} +12.0000 q^{67} -4.00000 q^{68} +14.0000 q^{71} +14.0000 q^{73} +3.00000 q^{74} -1.00000 q^{76} +4.00000 q^{79} +1.00000 q^{80} -9.00000 q^{82} +12.0000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -1.00000 q^{88} -2.00000 q^{89} -1.00000 q^{92} +3.00000 q^{94} -1.00000 q^{95} +16.0000 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−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
$25$	1.00000	0.200000
$26$	7.00000	1.37281
$27$	0	0
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−7.00000	−0.970725
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.00000	−0.868243
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	3.00000	0.309426
$95$	−1.00000	−0.102598
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	8.00000	0.742781
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−6.00000	−0.538816
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.00000	0.613941
$131$	−13.0000	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$131$	−13.0000	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	4.00000	0.342997
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−14.0000	−1.17485
$143$	−7.00000	−0.585369
$144$	0	0
$145$	8.00000	0.664364
$146$	−14.0000	−1.15865
$147$	0	0
$148$	−3.00000	−0.246598
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−15.0000	−1.19713	−0.598565	−	0.801074i	$-0.704262\pi$
$157$	−15.0000	−1.19713	−0.598565	+	0.801074i	$0.704262\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−5.00000	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$167$	−5.00000	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	4.00000	0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	2.00000	0.149906
$179$	−13.0000	−0.971666	−0.485833	−	0.874052i	$-0.661484\pi$
$179$	−13.0000	−0.971666	−0.485833	+	0.874052i	$0.661484\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	0	0
$184$	1.00000	0.0737210
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−3.00000	−0.218797
$189$	0	0
$190$	1.00000	0.0725476
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.00000	0.628587
$206$	16.0000	1.11477
$207$	0	0
$208$	−7.00000	−0.485363
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	10.0000	0.677285
$219$	0	0
$220$	1.00000	0.0674200
$221$	28.0000	1.88348
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	7.00000	0.445399
$248$	6.00000	0.381000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	0	0
$260$	−7.00000	−0.434122
$261$	0	0
$262$	13.0000	0.803143
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	1.00000	0.0614295
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−2.00000	−0.120824
$275$	1.00000	0.0603023
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	7.00000	0.413919
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−8.00000	−0.469776
$291$	0	0
$292$	14.0000	0.819288
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	3.00000	0.174371
$297$	0	0
$298$	−4.00000	−0.231714
$299$	7.00000	0.404820
$300$	0	0
$301$	0	0
$302$	2.00000	0.115087
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	4.00000	0.229039
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	15.0000	0.846499
$315$	0	0
$316$	4.00000	0.225018
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−7.00000	−0.388290
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−9.00000	−0.494685	−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000	−0.494685	−0.247342	+	0.968928i	$0.579557\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	5.00000	0.273588
$335$	12.0000	0.655630
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−36.0000	−1.95814
$339$	0	0
$340$	−4.00000	−0.216930
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	21.0000	1.12897
$347$	34.0000	1.82522	0.912608	−	0.408836i	$-0.134065\pi$
$347$	34.0000	1.82522	0.912608	+	0.408836i	$0.134065\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−8.00000	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	−2.00000	−0.106000
$357$	0	0
$358$	13.0000	0.687071
$359$	−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
$361$	−18.0000	−0.947368
$362$	−12.0000	−0.630706
$363$	0	0
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	3.00000	0.155963
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	3.00000	0.154713
$377$	−56.0000	−2.88415
$378$	0	0
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	−10.0000	−0.511645
$383$	13.0000	0.664269	0.332134	−	0.943232i	$-0.392231\pi$
$383$	13.0000	0.664269	0.332134	+	0.943232i	$0.392231\pi$
$384$	0	0
$385$	0	0
$386$	−26.0000	−1.32337
$387$	0	0
$388$	16.0000	0.812277
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	−3.00000	−0.151138
$395$	4.00000	0.201262
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	0	0
$403$	42.0000	2.09217
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−9.00000	−0.444478
$411$	0	0
$412$	−16.0000	−0.788263
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	7.00000	0.343203
$417$	0	0
$418$	1.00000	0.0489116
$419$	11.0000	0.537385	0.268693	−	0.963226i	$-0.413408\pi$
$419$	11.0000	0.537385	0.268693	+	0.963226i	$0.413408\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	15.0000	0.730189
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	18.0000	0.870063
$429$	0	0
$430$	4.00000	0.192897
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−40.0000	−1.92228	−0.961139	−	0.276066i	$-0.910969\pi$
$433$	−40.0000	−1.92228	−0.961139	+	0.276066i	$0.910969\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	1.00000	0.0478365
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−28.0000	−1.33182
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−4.00000	−0.189405
$447$	0	0
$448$	0	0
$449$	25.0000	1.17982	0.589911	−	0.807468i	$-0.299163\pi$
$449$	25.0000	1.17982	0.589911	+	0.807468i	$0.299163\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	6.00000	0.282216
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	33.0000	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$463$	33.0000	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	26.0000	1.20443
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	3.00000	0.138380
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	−6.00000	−0.274434
$479$	26.0000	1.18797	0.593985	−	0.804476i	$-0.297554\pi$
$479$	26.0000	1.18797	0.593985	+	0.804476i	$0.297554\pi$
$480$	0	0
$481$	21.0000	0.957518
$482$	7.00000	0.318841
$483$	0	0
$484$	−10.0000	−0.454545
$485$	16.0000	0.726523
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	−7.00000	−0.314945
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	3.00000	0.133897
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	0	0
$506$	1.00000	0.0444554
$507$	0	0
$508$	5.00000	0.221839
$509$	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
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.00000	0.352865
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−3.00000	−0.131940
$518$	0	0
$519$	0	0
$520$	7.00000	0.306970
$521$	−21.0000	−0.920027	−0.460013	−	0.887912i	$-0.652155\pi$
$521$	−21.0000	−0.920027	−0.460013	+	0.887912i	$0.652155\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−13.0000	−0.567908
$525$	0	0
$526$	−16.0000	−0.697633
$527$	24.0000	1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−1.00000	−0.0434372
$531$	0	0
$532$	0	0
$533$	−63.0000	−2.72883
$534$	0	0
$535$	18.0000	0.778208
$536$	−12.0000	−0.518321
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	4.00000	0.171499
$545$	−10.0000	−0.428353
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	4.00000	0.169638
$557$	−45.0000	−1.90671	−0.953356	−	0.301849i	$-0.902396\pi$
$557$	−45.0000	−1.90671	−0.953356	+	0.301849i	$0.902396\pi$
$558$	0	0
$559$	28.0000	1.18427
$560$	0	0
$561$	0	0
$562$	3.00000	0.126547
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	−14.0000	−0.587427
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−7.00000	−0.292685
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	8.00000	0.332182
$581$	0	0
$582$	0	0
$583$	1.00000	0.0414158
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−9.00000	−0.371787
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	−12.0000	−0.494032
$591$	0	0
$592$	−3.00000	−0.123299
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	−7.00000	−0.286251
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−4.00000	−0.161955
$611$	21.0000	0.849569
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	7.00000	0.281354	0.140677	−	0.990056i	$-0.455072\pi$
$619$	7.00000	0.281354	0.140677	+	0.990056i	$0.455072\pi$
$620$	−6.00000	−0.240966
$621$	0	0
$622$	−16.0000	−0.641542
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.0000	−0.959233
$627$	0	0
$628$	−15.0000	−0.598565
$629$	12.0000	0.478471
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	10.0000	0.397151
$635$	5.00000	0.198419
$636$	0	0
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	23.0000	0.908445	0.454223	−	0.890888i	$-0.349917\pi$
$641$	23.0000	0.908445	0.454223	+	0.890888i	$0.349917\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	7.00000	0.274563
$651$	0	0
$652$	8.00000	0.313304
$653$	−29.0000	−1.13486	−0.567429	−	0.823422i	$-0.692062\pi$
$653$	−29.0000	−1.13486	−0.567429	+	0.823422i	$0.692062\pi$
$654$	0	0
$655$	−13.0000	−0.507952
$656$	9.00000	0.351391
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	9.00000	0.349795
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	−5.00000	−0.193456
$669$	0	0
$670$	−12.0000	−0.463600
$671$	4.00000	0.154418
$672$	0	0
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	0	0
$675$	0	0
$676$	36.0000	1.38462
$677$	1.00000	0.0384331	0.0192166	−	0.999815i	$-0.493883\pi$
$677$	1.00000	0.0384331	0.0192166	+	0.999815i	$0.493883\pi$
$678$	0	0
$679$	0	0
$680$	4.00000	0.153393
$681$	0	0
$682$	6.00000	0.229752
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−7.00000	−0.266679
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	−21.0000	−0.798300
$693$	0	0
$694$	−34.0000	−1.29062
$695$	4.00000	0.151729
$696$	0	0
$697$	−36.0000	−1.36360
$698$	−28.0000	−1.05982
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	3.00000	0.113147
$704$	1.00000	0.0376889
$705$	0	0
$706$	8.00000	0.301084
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	−14.0000	−0.525411
$711$	0	0
$712$	2.00000	0.0749532
$713$	6.00000	0.224702
$714$	0	0
$715$	−7.00000	−0.261785
$716$	−13.0000	−0.485833
$717$	0	0
$718$	36.0000	1.34351
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	0	0
$721$	0	0
$722$	18.0000	0.669891
$723$	0	0
$724$	12.0000	0.445976
$725$	8.00000	0.297113
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	0	0
$729$	0	0
$730$	−14.0000	−0.518163
$731$	16.0000	0.591781
$732$	0	0
$733$	−37.0000	−1.36663	−0.683313	−	0.730125i	$-0.739462\pi$
$733$	−37.0000	−1.36663	−0.683313	+	0.730125i	$0.739462\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	1.00000	0.0368605
$737$	12.0000	0.442026
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	−26.0000	−0.951928
$747$	0	0
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	56.0000	2.03940
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−1.00000	−0.0363216
$759$	0	0
$760$	1.00000	0.0362738
$761$	−17.0000	−0.616250	−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000	−0.616250	−0.308125	+	0.951346i	$0.599701\pi$
$762$	0	0
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	−13.0000	−0.469709
$767$	−84.0000	−3.03306
$768$	0	0
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	0	0
$771$	0	0
$772$	26.0000	0.935760
$773$	43.0000	1.54660	0.773301	−	0.634039i	$-0.218604\pi$
$773$	43.0000	1.54660	0.773301	+	0.634039i	$0.218604\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−14.0000	−0.501924
$779$	−9.00000	−0.322458
$780$	0	0
$781$	14.0000	0.500959
$782$	−4.00000	−0.143040
$783$	0	0
$784$	0	0
$785$	−15.0000	−0.535373
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	3.00000	0.106871
$789$	0	0
$790$	−4.00000	−0.142314
$791$	0	0
$792$	0	0
$793$	−28.0000	−0.994309
$794$	18.0000	0.638796
$795$	0	0
$796$	12.0000	0.425329
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−17.0000	−0.600291
$803$	14.0000	0.494049
$804$	0	0
$805$	0	0
$806$	−42.0000	−1.47939
$807$	0	0
$808$	0	0
$809$	−53.0000	−1.86338	−0.931690	−	0.363253i	$-0.881666\pi$
$809$	−53.0000	−1.86338	−0.931690	+	0.363253i	$0.881666\pi$
$810$	0	0
$811$	−37.0000	−1.29925	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000	−1.29925	−0.649623	+	0.760257i	$0.725073\pi$
$812$	0	0
$813$	0	0
$814$	3.00000	0.105150
$815$	8.00000	0.280228
$816$	0	0
$817$	4.00000	0.139942
$818$	10.0000	0.349642
$819$	0	0
$820$	9.00000	0.314294
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	0	0
$823$	48.0000	1.67317	0.836587	−	0.547833i	$-0.184547\pi$
$823$	48.0000	1.67317	0.836587	+	0.547833i	$0.184547\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	0	0
$829$	−12.0000	−0.416777	−0.208389	−	0.978046i	$-0.566822\pi$
$829$	−12.0000	−0.416777	−0.208389	+	0.978046i	$0.566822\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	−7.00000	−0.242681
$833$	0	0
$834$	0	0
$835$	−5.00000	−0.173032
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	−11.0000	−0.379989
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	14.0000	0.482472
$843$	0	0
$844$	−15.0000	−0.516321
$845$	36.0000	1.23844
$846$	0	0
$847$	0	0
$848$	1.00000	0.0343401
$849$	0	0
$850$	4.00000	0.137199
$851$	3.00000	0.102839
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	0	0
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−29.0000	−0.987171	−0.493586	−	0.869697i	$-0.664314\pi$
$863$	−29.0000	−0.987171	−0.493586	+	0.869697i	$0.664314\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	40.0000	1.35926
$867$	0	0
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−84.0000	−2.84623
$872$	10.0000	0.338643
$873$	0	0
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	0	0
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	16.0000	0.539974
$879$	0	0
$880$	1.00000	0.0337100
$881$	−25.0000	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$882$	0	0
$883$	−58.0000	−1.95186	−0.975928	−	0.218094i	$-0.930016\pi$
$883$	−58.0000	−1.95186	−0.975928	+	0.218094i	$0.930016\pi$
$884$	28.0000	0.941742
$885$	0	0
$886$	−36.0000	−1.20944
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	0	0
$890$	2.00000	0.0670402
$891$	0	0
$892$	4.00000	0.133930
$893$	3.00000	0.100391
$894$	0	0
$895$	−13.0000	−0.434542
$896$	0	0
$897$	0	0
$898$	−25.0000	−0.834261
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−4.00000	−0.133259
$902$	−9.00000	−0.299667
$903$	0	0
$904$	−6.00000	−0.199557
$905$	12.0000	0.398893
$906$	0	0
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	0	0
$911$	58.0000	1.92163	0.960813	−	0.277198i	$-0.0894057\pi$
$911$	58.0000	1.92163	0.960813	+	0.277198i	$0.0894057\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−10.0000	−0.330771
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	1.00000	0.0329690
$921$	0	0
$922$	28.0000	0.922131
$923$	−98.0000	−3.22571
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	−33.0000	−1.08445
$927$	0	0
$928$	−8.00000	−0.262613
$929$	31.0000	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$929$	31.0000	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$930$	0	0
$931$	0	0
$932$	−26.0000	−0.851658
$933$	0	0
$934$	−12.0000	−0.392652
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	0	0
$939$	0	0
$940$	−3.00000	−0.0978492
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	12.0000	0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	0	0
$949$	−98.0000	−3.18121
$950$	1.00000	0.0324443
$951$	0	0
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	6.00000	0.194054
$957$	0	0
$958$	−26.0000	−0.840022
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	−21.0000	−0.677067
$963$	0	0
$964$	−7.00000	−0.225455
$965$	26.0000	0.836970
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	−16.0000	−0.513729
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	0	0
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	4.00000	0.128037
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−33.0000	−1.05254	−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000	−1.05254	−0.526268	+	0.850319i	$0.676409\pi$
$984$	0	0
$985$	3.00000	0.0955879
$986$	32.0000	1.01909
$987$	0	0
$988$	7.00000	0.222700
$989$	4.00000	0.127193
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.q.1.1		1
3.2	odd	2		1470.2.a.k.1.1		1
7.3	odd	6		630.2.k.h.541.1		2
7.5	odd	6		630.2.k.h.361.1		2
7.6	odd	2		4410.2.a.g.1.1		1
15.14	odd	2		7350.2.a.ba.1.1		1
21.2	odd	6		1470.2.i.i.361.1		2
21.5	even	6		210.2.i.a.151.1	yes	2
21.11	odd	6		1470.2.i.i.961.1		2
21.17	even	6		210.2.i.a.121.1	✓	2
21.20	even	2		1470.2.a.r.1.1		1
84.47	odd	6		1680.2.bg.k.1201.1		2
84.59	odd	6		1680.2.bg.k.961.1		2
105.17	odd	12		1050.2.o.j.499.1		4
105.38	odd	12		1050.2.o.j.499.2		4
105.47	odd	12		1050.2.o.j.949.2		4
105.59	even	6		1050.2.i.s.751.1		2
105.68	odd	12		1050.2.o.j.949.1		4
105.89	even	6		1050.2.i.s.151.1		2
105.104	even	2		7350.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.a.121.1	✓	2	21.17	even	6
210.2.i.a.151.1	yes	2	21.5	even	6
630.2.k.h.361.1		2	7.5	odd	6
630.2.k.h.541.1		2	7.3	odd	6
1050.2.i.s.151.1		2	105.89	even	6
1050.2.i.s.751.1		2	105.59	even	6
1050.2.o.j.499.1		4	105.17	odd	12
1050.2.o.j.499.2		4	105.38	odd	12
1050.2.o.j.949.1		4	105.68	odd	12
1050.2.o.j.949.2		4	105.47	odd	12
1470.2.a.k.1.1		1	3.2	odd	2
1470.2.a.r.1.1		1	21.20	even	2
1470.2.i.i.361.1		2	21.2	odd	6
1470.2.i.i.961.1		2	21.11	odd	6
1680.2.bg.k.961.1		2	84.59	odd	6
1680.2.bg.k.1201.1		2	84.47	odd	6
4410.2.a.g.1.1		1	7.6	odd	2
4410.2.a.q.1.1		1	1.1	even	1	trivial
7350.2.a.j.1.1		1	105.104	even	2
7350.2.a.ba.1.1		1	15.14	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

Introduction

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