LMFDB - Embedded newform 585.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, 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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$585 = 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	585.1

$q$ -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} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.00000 q^{28} -2.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} -2.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.00000 q^{53} -2.00000 q^{55} +12.0000 q^{56} -2.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{70} -6.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} +6.00000 q^{76} +8.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +16.0000 q^{83} -2.00000 q^{85} +10.0000 q^{86} +6.00000 q^{88} -2.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -4.00000 q^{94} -6.00000 q^{95} -2.00000 q^{97} +9.00000 q^{98} +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

)

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	4.00000	0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	12.0000	1.60357
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−4.00000	−0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	8.00000	0.911685
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.00000	0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	10.0000	1.07833
$87$	0	0
$88$	6.00000	0.639602
$89$	−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
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−4.00000	−0.412568
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	4.00000	0.377964
$113$	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
$115$	6.00000	0.559503
$116$	2.00000	0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	0	0
$124$	10.0000	0.898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	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
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−6.00000	−0.503509
$143$	2.00000	0.167248
$144$	0	0
$145$	−2.00000	−0.166091
$146$	−6.00000	−0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	18.0000	1.45999
$153$	0	0
$154$	8.00000	0.644658
$155$	−10.0000	−0.803219
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	5.00000	0.395285
$161$	−24.0000	−1.89146
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	16.0000	1.24184
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	2.00000	0.150756
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	−18.0000	−1.32698
$185$	−2.00000	−0.147043
$186$	0	0
$187$	4.00000	0.292509
$188$	4.00000	0.291730
$189$	0	0
$190$	−6.00000	−0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	18.0000	1.26648
$203$	8.00000	0.561490
$204$	0	0
$205$	6.00000	0.419058
$206$	2.00000	0.139347
$207$	0	0
$208$	1.00000	0.0693375
$209$	12.0000	0.830057
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−10.0000	−0.683586
$215$	10.0000	0.681994
$216$	0	0
$217$	40.0000	2.71538
$218$	10.0000	0.677285
$219$	0	0
$220$	2.00000	0.134840
$221$	2.00000	0.134535
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	14.0000	0.931266
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	6.00000	0.390567
$237$	0	0
$238$	8.00000	0.518563
$239$	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
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−2.00000	−0.128037
$245$	9.00000	0.574989
$246$	0	0
$247$	6.00000	0.381771
$248$	30.0000	1.90500
$249$	0	0
$250$	1.00000	0.0632456
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	−2.00000	−0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	1.00000	0.0620174
$261$	0	0
$262$	−20.0000	−1.23560
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	24.0000	1.47153
$267$	0	0
$268$	4.00000	0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	12.0000	0.717137
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	2.00000	0.118262
$287$	−24.0000	−1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	0	0
$292$	6.00000	0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	6.00000	0.348743
$297$	0	0
$298$	−18.0000	−1.04271
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−40.0000	−2.30556
$302$	10.0000	0.575435
$303$	0	0
$304$	6.00000	0.344124
$305$	2.00000	0.114520
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	−10.0000	−0.567962
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	12.0000	0.675053
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	7.00000	0.391312
$321$	0	0
$322$	−24.0000	−1.33747
$323$	12.0000	0.667698
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−18.0000	−0.993884
$329$	16.0000	0.882109
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	−16.0000	−0.878114
$333$	0	0
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	2.00000	0.108465
$341$	20.0000	1.08306
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−30.0000	−1.61749
$345$	0	0
$346$	6.00000	0.322562
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	2.00000	0.106000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−2.00000	−0.103975
$371$	8.00000	0.415339
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	12.0000	0.618853
$377$	2.00000	0.103005
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	−2.00000	−0.101797
$387$	0	0
$388$	2.00000	0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	−27.0000	−1.36371
$393$	0	0
$394$	6.00000	0.302276
$395$	−12.0000	−0.603786
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	−18.0000	−0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	4.00000	0.198273
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−2.00000	−0.0985329
$413$	24.0000	1.18096
$414$	0	0
$415$	16.0000	0.785409
$416$	−5.00000	−0.245145
$417$	0	0
$418$	12.0000	0.586939
$419$	40.0000	1.95413	0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000	1.95413	0.977064	+	0.212946i	$0.0683059\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−8.00000	−0.387147
$428$	10.0000	0.483368
$429$	0	0
$430$	10.0000	0.482243
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	40.0000	1.92006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−36.0000	−1.72211
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	2.00000	0.0951303
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	4.00000	0.189405
$447$	0	0
$448$	−28.0000	−1.32288
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−14.0000	−0.658505
$453$	0	0
$454$	−4.00000	−0.187729
$455$	4.00000	0.187523
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	−6.00000	−0.279751
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−10.0000	−0.463241
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−4.00000	−0.184506
$471$	0	0
$472$	18.0000	0.828517
$473$	−20.0000	−0.919601
$474$	0	0
$475$	−6.00000	−0.275299
$476$	−8.00000	−0.366679
$477$	0	0
$478$	6.00000	0.274434
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	10.0000	0.455488
$483$	0	0
$484$	7.00000	0.318182
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	9.00000	0.406579
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	6.00000	0.269953
$495$	0	0
$496$	10.0000	0.449013
$497$	24.0000	1.07655
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−24.0000	−1.07117
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	−12.0000	−0.533465
$507$	0	0
$508$	2.00000	0.0887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	2.00000	0.0881305
$516$	0	0
$517$	8.00000	0.351840
$518$	8.00000	0.351500
$519$	0	0
$520$	3.00000	0.131559
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−14.0000	−0.610429
$527$	20.0000	0.871214
$528$	0	0
$529$	13.0000	0.565217
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−24.0000	−1.04053
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−10.0000	−0.432338
$536$	12.0000	0.518321
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−10.0000	−0.428746
$545$	10.0000	0.428353
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	12.0000	0.511217
$552$	0	0
$553$	48.0000	2.04117
$554$	−14.0000	−0.594803
$555$	0	0
$556$	0	0
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	4.00000	0.169031
$561$	0	0
$562$	6.00000	0.253095
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	2.00000	0.0840663
$567$	0	0
$568$	18.0000	0.755263
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−24.0000	−1.00174
$575$	6.00000	0.250217
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	2.00000	0.0830455
$581$	−64.0000	−2.65517
$582$	0	0
$583$	4.00000	0.165663
$584$	18.0000	0.744845
$585$	0	0
$586$	−22.0000	−0.908812
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	−6.00000	−0.247016
$591$	0	0
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	18.0000	0.737309
$597$	0	0
$598$	−6.00000	−0.245358
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−40.0000	−1.63028
$603$	0	0
$604$	−10.0000	−0.406894
$605$	−7.00000	−0.284590
$606$	0	0
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	2.00000	0.0809776
$611$	4.00000	0.161823
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−24.0000	−0.966988
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−2.00000	−0.0803868	−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	4.00000	0.160385
$623$	8.00000	0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	6.00000	0.239426
$629$	4.00000	0.159490
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	36.0000	1.43200
$633$	0	0
$634$	18.0000	0.714871
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	−9.00000	−0.356593
$638$	4.00000	0.158362
$639$	0	0
$640$	−3.00000	−0.118585
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	12.0000	0.472134
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	12.0000	0.469956
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	−6.00000	−0.234261
$657$	0	0
$658$	16.0000	0.623745
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−18.0000	−0.699590
$663$	0	0
$664$	−48.0000	−1.86276
$665$	24.0000	0.930680
$666$	0	0
$667$	−12.0000	−0.464642
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	6.00000	0.230089
$681$	0	0
$682$	20.0000	0.765840
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−10.0000	−0.381246
$689$	2.00000	0.0761939
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	22.0000	0.835109
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−30.0000	−1.13552
$699$	0	0
$700$	4.00000	0.151186
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−14.0000	−0.527645
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−72.0000	−2.70784
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	6.00000	0.224860
$713$	−60.0000	−2.24702
$714$	0	0
$715$	2.00000	0.0747958
$716$	12.0000	0.448461
$717$	0	0
$718$	10.0000	0.373197
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000	0.632674
$723$	0	0
$724$	22.0000	0.817624
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	−12.0000	−0.444750
$729$	0	0
$730$	−6.00000	−0.222070
$731$	−20.0000	−0.739727
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	30.0000	1.10581
$737$	8.00000	0.294684
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	8.00000	0.293689
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	34.0000	1.24483
$747$	0	0
$748$	−4.00000	−0.146254
$749$	40.0000	1.46157
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	2.00000	0.0728357
$755$	10.0000	0.363937
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	18.0000	0.652929
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	6.00000	0.216647
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	2.00000	0.0719816
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	6.00000	0.215387
$777$	0	0
$778$	10.0000	0.358517
$779$	−36.0000	−1.28983
$780$	0	0
$781$	12.0000	0.429394
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−9.00000	−0.321429
$785$	−6.00000	−0.214149
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−56.0000	−1.99113
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	6.00000	0.212932
$795$	0	0
$796$	16.0000	0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	5.00000	0.176777
$801$	0	0
$802$	−10.0000	−0.353112
$803$	12.0000	0.423471
$804$	0	0
$805$	−24.0000	−0.845889
$806$	10.0000	0.352235
$807$	0	0
$808$	−54.0000	−1.89971
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	4.00000	0.140200
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−60.0000	−2.09913
$818$	18.0000	0.629355
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	−46.0000	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	24.0000	0.835067
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	16.0000	0.555368
$831$	0	0
$832$	−7.00000	−0.242681
$833$	−18.0000	−0.623663
$834$	0	0
$835$	12.0000	0.415277
$836$	−12.0000	−0.415029
$837$	0	0
$838$	40.0000	1.38178
$839$	−38.0000	−1.31191	−0.655953	−	0.754802i	$-0.727733\pi$
$839$	−38.0000	−1.31191	−0.655953	+	0.754802i	$0.727733\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000	0.344623
$843$	0	0
$844$	−12.0000	−0.413057
$845$	1.00000	0.0344010
$846$	0	0
$847$	28.0000	0.962091
$848$	2.00000	0.0686803
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−12.0000	−0.411355
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	30.0000	1.02538
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	−14.0000	−0.476842
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	10.0000	0.339814
$867$	0	0
$868$	−40.0000	−1.35769
$869$	24.0000	0.814144
$870$	0	0
$871$	4.00000	0.135535
$872$	−30.0000	−1.01593
$873$	0	0
$874$	−36.0000	−1.21772
$875$	−4.00000	−0.135225
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	0	0
$879$	0	0
$880$	2.00000	0.0674200
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−14.0000	−0.470339
$887$	−58.0000	−1.94745	−0.973725	−	0.227728i	$-0.926870\pi$
$887$	−58.0000	−1.94745	−0.973725	+	0.227728i	$0.926870\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	−4.00000	−0.133930
$893$	24.0000	0.803129
$894$	0	0
$895$	−12.0000	−0.401116
$896$	12.0000	0.400892
$897$	0	0
$898$	6.00000	0.200223
$899$	20.0000	0.667037
$900$	0	0
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−42.0000	−1.39690
$905$	−22.0000	−0.731305
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	4.00000	0.132745
$909$	0	0
$910$	4.00000	0.132599
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	38.0000	1.25693
$915$	0	0
$916$	22.0000	0.726900
$917$	80.0000	2.64183
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	−10.0000	−0.329332
$923$	6.00000	0.197492
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−24.0000	−0.788689
$927$	0	0
$928$	−10.0000	−0.328266
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	10.0000	0.327561
$933$	0	0
$934$	10.0000	0.327210
$935$	4.00000	0.130814
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	4.00000	0.130466
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	6.00000	0.195283
$945$	0	0
$946$	−20.0000	−0.650256
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	−6.00000	−0.194666
$951$	0	0
$952$	−24.0000	−0.777844
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−30.0000	−0.969256
$959$	−8.00000	−0.258333
$960$	0	0
$961$	69.0000	2.22581
$962$	2.00000	0.0644826
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−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
$979$	4.00000	0.127841
$980$	−9.00000	−0.287494
$981$	0	0
$982$	24.0000	0.765871
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	4.00000	0.127386
$987$	0	0
$988$	−6.00000	−0.190885
$989$	60.0000	1.90789
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	−50.0000	−1.58750
$993$	0	0
$994$	24.0000	0.761234
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	10.0000	0.316544
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.h.1.1		1
3.2	odd	2		65.2.a.a.1.1	✓	1
4.3	odd	2		9360.2.a.ca.1.1		1
5.2	odd	4		2925.2.c.h.2224.2		2
5.3	odd	4		2925.2.c.h.2224.1		2
5.4	even	2		2925.2.a.f.1.1		1
12.11	even	2		1040.2.a.f.1.1		1
13.12	even	2		7605.2.a.f.1.1		1
15.2	even	4		325.2.b.b.274.1		2
15.8	even	4		325.2.b.b.274.2		2
15.14	odd	2		325.2.a.d.1.1		1
21.20	even	2		3185.2.a.e.1.1		1
24.5	odd	2		4160.2.a.q.1.1		1
24.11	even	2		4160.2.a.f.1.1		1
33.32	even	2		7865.2.a.c.1.1		1
39.2	even	12		845.2.m.b.316.1		4
39.5	even	4		845.2.c.a.506.2		2
39.8	even	4		845.2.c.a.506.1		2
39.11	even	12		845.2.m.b.316.2		4
39.17	odd	6		845.2.e.a.146.1		2
39.20	even	12		845.2.m.b.361.1		4
39.23	odd	6		845.2.e.a.191.1		2
39.29	odd	6		845.2.e.b.191.1		2
39.32	even	12		845.2.m.b.361.2		4
39.35	odd	6		845.2.e.b.146.1		2
39.38	odd	2		845.2.a.a.1.1		1
60.59	even	2		5200.2.a.d.1.1		1
195.194	odd	2		4225.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.a.1.1	✓	1	3.2	odd	2
325.2.a.d.1.1		1	15.14	odd	2
325.2.b.b.274.1		2	15.2	even	4
325.2.b.b.274.2		2	15.8	even	4
585.2.a.h.1.1		1	1.1	even	1	trivial
845.2.a.a.1.1		1	39.38	odd	2
845.2.c.a.506.1		2	39.8	even	4
845.2.c.a.506.2		2	39.5	even	4
845.2.e.a.146.1		2	39.17	odd	6
845.2.e.a.191.1		2	39.23	odd	6
845.2.e.b.146.1		2	39.35	odd	6
845.2.e.b.191.1		2	39.29	odd	6
845.2.m.b.316.1		4	39.2	even	12
845.2.m.b.316.2		4	39.11	even	12
845.2.m.b.361.1		4	39.20	even	12
845.2.m.b.361.2		4	39.32	even	12
1040.2.a.f.1.1		1	12.11	even	2
2925.2.a.f.1.1		1	5.4	even	2
2925.2.c.h.2224.1		2	5.3	odd	4
2925.2.c.h.2224.2		2	5.2	odd	4
3185.2.a.e.1.1		1	21.20	even	2
4160.2.a.f.1.1		1	24.11	even	2
4160.2.a.q.1.1		1	24.5	odd	2
4225.2.a.g.1.1		1	195.194	odd	2
5200.2.a.d.1.1		1	60.59	even	2
7605.2.a.f.1.1		1	13.12	even	2
7865.2.a.c.1.1		1	33.32	even	2
9360.2.a.ca.1.1		1	4.3	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Fields

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	585.2.a.h.1.1

Level	$585$
Weight	$2$
Character	585.1
Self dual	yes
Analytic conductor	$4.671$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$