LMFDB - Embedded newform 1856.2.a.bc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.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:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.68772992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 17x^{2} - 8 $ x^6 - 9x^4 + 17x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28790$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95528 q^{3} +2.85952 q^{5} -2.57579 q^{7} +0.823126 q^{9} +O(q^{10})$	$q+1.95528 q^{3} +2.85952 q^{5} -2.57579 q^{7} +0.823126 q^{9} +1.95528 q^{11} +2.85952 q^{13} +5.59117 q^{15} +7.03640 q^{17} +4.97066 q^{19} -5.03640 q^{21} -8.60656 q^{23} +3.17687 q^{25} -4.25640 q^{27} -1.00000 q^{29} +4.53107 q^{31} +3.82313 q^{33} -7.36554 q^{35} -7.71905 q^{37} +5.59117 q^{39} +6.68265 q^{41} -3.19630 q^{43} +2.35375 q^{45} +3.29005 q^{47} -0.365299 q^{49} +13.7581 q^{51} +3.21327 q^{53} +5.59117 q^{55} +9.71905 q^{57} +13.7581 q^{59} +7.03640 q^{61} -2.12020 q^{63} +8.17687 q^{65} -7.82113 q^{67} -16.8282 q^{69} +6.03076 q^{71} +1.64625 q^{73} +6.21168 q^{75} -5.03640 q^{77} -0.620510 q^{79} -10.7918 q^{81} -16.4277 q^{83} +20.1207 q^{85} -1.95528 q^{87} +7.39015 q^{89} -7.36554 q^{91} +8.85952 q^{93} +14.2137 q^{95} -4.75544 q^{97} +1.60944 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 6 * q - 4 * q^5 + 10 * q^9	$ 6 q - 4 q^{5} + 10 q^{9} - 4 q^{13} + 16 q^{17} - 4 q^{21} + 14 q^{25} - 6 q^{29} + 28 q^{33} - 4 q^{37} + 24 q^{41} + 4 q^{45} + 30 q^{49} - 12 q^{53} + 16 q^{57} + 16 q^{61} + 44 q^{65} + 20 q^{69} + 20 q^{73} - 4 q^{77} + 30 q^{81} + 20 q^{85} + 8 q^{89} + 32 q^{93} + 40 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 10 * q^9 - 4 * q^13 + 16 * q^17 - 4 * q^21 + 14 * q^25 - 6 * q^29 + 28 * q^33 - 4 * q^37 + 24 * q^41 + 4 * q^45 + 30 * q^49 - 12 * q^53 + 16 * q^57 + 16 * q^61 + 44 * q^65 + 20 * q^69 + 20 * q^73 - 4 * q^77 + 30 * q^81 + 20 * q^85 + 8 * q^89 + 32 * q^93 + 40 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	1.95528	1.12888	0.564441	−	0.825473i	$-0.309092\pi$
$3$	1.95528	1.12888	0.564441	+	0.825473i	$0.309092\pi$
$4$	0	0
$5$	2.85952	1.27882	0.639409	−	0.768867i	$-0.279179\pi$
$5$	2.85952	1.27882	0.639409	+	0.768867i	$0.279179\pi$
$6$	0	0
$7$	−2.57579	−0.973558	−0.486779	−	0.873525i	$-0.661828\pi$
$7$	−2.57579	−0.973558	−0.486779	+	0.873525i	$0.661828\pi$
$8$	0	0
$9$	0.823126	0.274375
$10$	0	0
$11$	1.95528	0.589540	0.294770	−	0.955568i	$-0.404757\pi$
$11$	1.95528	0.589540	0.294770	+	0.955568i	$0.404757\pi$
$12$	0	0
$13$	2.85952	0.793089	0.396545	−	0.918015i	$-0.370209\pi$
$13$	2.85952	0.793089	0.396545	+	0.918015i	$0.370209\pi$
$14$	0	0
$15$	5.59117	1.44363
$16$	0	0
$17$	7.03640	1.70658	0.853289	−	0.521439i	$-0.174605\pi$
$17$	7.03640	1.70658	0.853289	+	0.521439i	$0.174605\pi$
$18$	0	0
$19$	4.97066	1.14035	0.570174	−	0.821524i	$-0.306876\pi$
$19$	4.97066	1.14035	0.570174	+	0.821524i	$0.306876\pi$
$20$	0	0
$21$	−5.03640	−1.09903
$22$	0	0
$23$	−8.60656	−1.79459	−0.897295	−	0.441431i	$-0.854471\pi$
$23$	−8.60656	−1.79459	−0.897295	+	0.441431i	$0.854471\pi$
$24$	0	0
$25$	3.17687	0.635375
$26$	0	0
$27$	−4.25640	−0.819145
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.53107	0.813805	0.406902	−	0.913472i	$-0.366609\pi$
$31$	4.53107	0.813805	0.406902	+	0.913472i	$0.366609\pi$
$32$	0	0
$33$	3.82313	0.665521
$34$	0	0
$35$	−7.36554	−1.24500
$36$	0	0
$37$	−7.71905	−1.26900	−0.634502	−	0.772921i	$-0.718795\pi$
$37$	−7.71905	−1.26900	−0.634502	+	0.772921i	$0.718795\pi$
$38$	0	0
$39$	5.59117	0.895304
$40$	0	0
$41$	6.68265	1.04365	0.521827	−	0.853051i	$-0.325251\pi$
$41$	6.68265	1.04365	0.521827	+	0.853051i	$0.325251\pi$
$42$	0	0
$43$	−3.19630	−0.487431	−0.243716	−	0.969847i	$-0.578366\pi$
$43$	−3.19630	−0.487431	−0.243716	+	0.969847i	$0.578366\pi$
$44$	0	0
$45$	2.35375	0.350876
$46$	0	0
$47$	3.29005	0.479904	0.239952	−	0.970785i	$-0.422868\pi$
$47$	3.29005	0.479904	0.239952	+	0.970785i	$0.422868\pi$
$48$	0	0
$49$	−0.365299	−0.0521855
$50$	0	0
$51$	13.7581	1.92652
$52$	0	0
$53$	3.21327	0.441377	0.220688	−	0.975344i	$-0.429170\pi$
$53$	3.21327	0.441377	0.220688	+	0.975344i	$0.429170\pi$
$54$	0	0
$55$	5.59117	0.753914
$56$	0	0
$57$	9.71905	1.28732
$58$	0	0
$59$	13.7581	1.79116	0.895579	−	0.444904i	$-0.146762\pi$
$59$	13.7581	1.79116	0.895579	+	0.444904i	$0.146762\pi$
$60$	0	0
$61$	7.03640	0.900918	0.450459	−	0.892797i	$-0.351260\pi$
$61$	7.03640	0.900918	0.450459	+	0.892797i	$0.351260\pi$
$62$	0	0
$63$	−2.12020	−0.267120
$64$	0	0
$65$	8.17687	1.01422
$66$	0	0
$67$	−7.82113	−0.955503	−0.477751	−	0.878495i	$-0.658548\pi$
$67$	−7.82113	−0.955503	−0.477751	+	0.878495i	$0.658548\pi$
$68$	0	0
$69$	−16.8282	−2.02588
$70$	0	0
$71$	6.03076	0.715720	0.357860	−	0.933775i	$-0.383507\pi$
$71$	6.03076	0.715720	0.357860	+	0.933775i	$0.383507\pi$
$72$	0	0
$73$	1.64625	0.192679	0.0963396	−	0.995349i	$-0.469287\pi$
$73$	1.64625	0.192679	0.0963396	+	0.995349i	$0.469287\pi$
$74$	0	0
$75$	6.21168	0.717263
$76$	0	0
$77$	−5.03640	−0.573951
$78$	0	0
$79$	−0.620510	−0.0698128	−0.0349064	−	0.999391i	$-0.511113\pi$
$79$	−0.620510	−0.0698128	−0.0349064	+	0.999391i	$0.511113\pi$
$80$	0	0
$81$	−10.7918	−1.19909
$82$	0	0
$83$	−16.4277	−1.80317	−0.901586	−	0.432600i	$-0.857596\pi$
$83$	−16.4277	−1.80317	−0.901586	+	0.432600i	$0.857596\pi$
$84$	0	0
$85$	20.1207	2.18240
$86$	0	0
$87$	−1.95528	−0.209628
$88$	0	0
$89$	7.39015	0.783354	0.391677	−	0.920103i	$-0.371895\pi$
$89$	7.39015	0.783354	0.391677	+	0.920103i	$0.371895\pi$
$90$	0	0
$91$	−7.36554	−0.772118
$92$	0	0
$93$	8.85952	0.918690
$94$	0	0
$95$	14.2137	1.45830
$96$	0	0
$97$	−4.75544	−0.482842	−0.241421	−	0.970420i	$-0.577614\pi$
$97$	−4.75544	−0.482842	−0.241421	+	0.970420i	$0.577614\pi$
$98$	0	0
$99$	1.60944	0.161755
$100$	0	0
$101$	−7.71905	−0.768074	−0.384037	−	0.923318i	$-0.625466\pi$
$101$	−7.71905	−0.768074	−0.384037	+	0.923318i	$0.625466\pi$
$102$	0	0
$103$	−3.91056	−0.385319	−0.192660	−	0.981266i	$-0.561711\pi$
$103$	−3.91056	−0.385319	−0.192660	+	0.981266i	$0.561711\pi$
$104$	0	0
$105$	−14.4017	−1.40546
$106$	0	0
$107$	−7.72737	−0.747033	−0.373517	−	0.927623i	$-0.621848\pi$
$107$	−7.72737	−0.747033	−0.373517	+	0.927623i	$0.621848\pi$
$108$	0	0
$109$	20.2248	1.93719	0.968593	−	0.248650i	$-0.0799870\pi$
$109$	20.2248	1.93719	0.968593	+	0.248650i	$0.0799870\pi$
$110$	0	0
$111$	−15.0929	−1.43256
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−24.6106	−2.29495
$116$	0	0
$117$	2.35375	0.217604
$118$	0	0
$119$	−18.1243	−1.66145
$120$	0	0
$121$	−7.17687	−0.652443
$122$	0	0
$123$	13.0665	1.17816
$124$	0	0
$125$	−5.21327	−0.466289
$126$	0	0
$127$	−15.2738	−1.35533	−0.677667	−	0.735369i	$-0.737009\pi$
$127$	−15.2738	−1.35533	−0.677667	+	0.735369i	$0.737009\pi$
$128$	0	0
$129$	−6.24967	−0.550253
$130$	0	0
$131$	12.2424	1.06963	0.534814	−	0.844970i	$-0.320382\pi$
$131$	12.2424	1.06963	0.534814	+	0.844970i	$0.320382\pi$
$132$	0	0
$133$	−12.8034	−1.11019
$134$	0	0
$135$	−12.1713	−1.04754
$136$	0	0
$137$	11.7190	1.00123	0.500613	−	0.865671i	$-0.333108\pi$
$137$	11.7190	1.00123	0.500613	+	0.865671i	$0.333108\pi$
$138$	0	0
$139$	−16.4277	−1.39338	−0.696689	−	0.717373i	$-0.745344\pi$
$139$	−16.4277	−1.39338	−0.696689	+	0.717373i	$0.745344\pi$
$140$	0	0
$141$	6.43298	0.541755
$142$	0	0
$143$	5.59117	0.467557
$144$	0	0
$145$	−2.85952	−0.237470
$146$	0	0
$147$	−0.714262	−0.0589113
$148$	0	0
$149$	4.93232	0.404071	0.202036	−	0.979378i	$-0.435244\pi$
$149$	4.93232	0.404071	0.202036	+	0.979378i	$0.435244\pi$
$150$	0	0
$151$	−7.36554	−0.599399	−0.299699	−	0.954034i	$-0.596886\pi$
$151$	−7.36554	−0.599399	−0.299699	+	0.954034i	$0.596886\pi$
$152$	0	0
$153$	5.79184	0.468243
$154$	0	0
$155$	12.9567	1.04071
$156$	0	0
$157$	15.3901	1.22827	0.614134	−	0.789202i	$-0.289506\pi$
$157$	15.3901	1.22827	0.614134	+	0.789202i	$0.289506\pi$
$158$	0	0
$159$	6.28285	0.498262
$160$	0	0
$161$	22.1687	1.74714
$162$	0	0
$163$	0.714262	0.0559453	0.0279727	−	0.999609i	$-0.491095\pi$
$163$	0.714262	0.0559453	0.0279727	+	0.999609i	$0.491095\pi$
$164$	0	0
$165$	10.9323	0.851080
$166$	0	0
$167$	3.36122	0.260099	0.130050	−	0.991507i	$-0.458486\pi$
$167$	3.36122	0.260099	0.130050	+	0.991507i	$0.458486\pi$
$168$	0	0
$169$	−4.82313	−0.371010
$170$	0	0
$171$	4.09148	0.312883
$172$	0	0
$173$	−7.36530	−0.559973	−0.279987	−	0.960004i	$-0.590330\pi$
$173$	−7.36530	−0.559973	−0.279987	+	0.960004i	$0.590330\pi$
$174$	0	0
$175$	−8.18296	−0.618574
$176$	0	0
$177$	26.9010	2.02201
$178$	0	0
$179$	−13.8519	−1.03534	−0.517669	−	0.855581i	$-0.673200\pi$
$179$	−13.8519	−1.03534	−0.517669	+	0.855581i	$0.673200\pi$
$180$	0	0
$181$	−8.93232	−0.663934	−0.331967	−	0.943291i	$-0.607712\pi$
$181$	−8.93232	−0.663934	−0.331967	+	0.943291i	$0.607712\pi$
$182$	0	0
$183$	13.7581	1.01703
$184$	0	0
$185$	−22.0728	−1.62282
$186$	0	0
$187$	13.7581	1.00609
$188$	0	0
$189$	10.9636	0.797485
$190$	0	0
$191$	22.7331	1.64491	0.822455	−	0.568830i	$-0.192604\pi$
$191$	22.7331	1.64491	0.822455	+	0.568830i	$0.192604\pi$
$192$	0	0
$193$	−4.04795	−0.291378	−0.145689	−	0.989330i	$-0.546540\pi$
$193$	−4.04795	−0.291378	−0.145689	+	0.989330i	$0.546540\pi$
$194$	0	0
$195$	15.9881	1.14493
$196$	0	0
$197$	5.43809	0.387448	0.193724	−	0.981056i	$-0.437943\pi$
$197$	5.43809	0.387448	0.193724	+	0.981056i	$0.437943\pi$
$198$	0	0
$199$	−24.9405	−1.76798	−0.883992	−	0.467502i	$-0.845154\pi$
$199$	−24.9405	−1.76798	−0.883992	+	0.467502i	$0.845154\pi$
$200$	0	0
$201$	−15.2925	−1.07865
$202$	0	0
$203$	2.57579	0.180785
$204$	0	0
$205$	19.1092	1.33464
$206$	0	0
$207$	−7.08428	−0.492391
$208$	0	0
$209$	9.71905	0.672281
$210$	0	0
$211$	−4.62483	−0.318386	−0.159193	−	0.987247i	$-0.550889\pi$
$211$	−4.62483	−0.318386	−0.159193	+	0.987247i	$0.550889\pi$
$212$	0	0
$213$	11.7918	0.807964
$214$	0	0
$215$	−9.13990	−0.623336
$216$	0	0
$217$	−11.6711	−0.792286
$218$	0	0
$219$	3.21889	0.217512
$220$	0	0
$221$	20.1207	1.35347
$222$	0	0
$223$	6.39260	0.428080	0.214040	−	0.976825i	$-0.431338\pi$
$223$	6.39260	0.428080	0.214040	+	0.976825i	$0.431338\pi$
$224$	0	0
$225$	2.61497	0.174331
$226$	0	0
$227$	−16.8833	−1.12058	−0.560291	−	0.828296i	$-0.689311\pi$
$227$	−16.8833	−1.12058	−0.560291	+	0.828296i	$0.689311\pi$
$228$	0	0
$229$	−6.12074	−0.404470	−0.202235	−	0.979337i	$-0.564820\pi$
$229$	−6.12074	−0.404470	−0.202235	+	0.979337i	$0.564820\pi$
$230$	0	0
$231$	−9.84757	−0.647923
$232$	0	0
$233$	−1.82313	−0.119437	−0.0597185	−	0.998215i	$-0.519020\pi$
$233$	−1.82313	−0.119437	−0.0597185	+	0.998215i	$0.519020\pi$
$234$	0	0
$235$	9.40798	0.613709
$236$	0	0
$237$	−1.21327	−0.0788104
$238$	0	0
$239$	−19.0972	−1.23530	−0.617648	−	0.786454i	$-0.711915\pi$
$239$	−19.0972	−1.23530	−0.617648	+	0.786454i	$0.711915\pi$
$240$	0	0
$241$	−2.17687	−0.140225	−0.0701124	−	0.997539i	$-0.522336\pi$
$241$	−2.17687	−0.140225	−0.0701124	+	0.997539i	$0.522336\pi$
$242$	0	0
$243$	−8.33188	−0.534491
$244$	0	0
$245$	−1.04458	−0.0667358
$246$	0	0
$247$	14.2137	0.904398
$248$	0	0
$249$	−32.1207	−2.03557
$250$	0	0
$251$	−2.28512	−0.144236	−0.0721178	−	0.997396i	$-0.522976\pi$
$251$	−2.28512	−0.144236	−0.0721178	+	0.997396i	$0.522976\pi$
$252$	0	0
$253$	−16.8282	−1.05798
$254$	0	0
$255$	39.3417	2.46367
$256$	0	0
$257$	−25.4068	−1.58483	−0.792417	−	0.609980i	$-0.791177\pi$
$257$	−25.4068	−1.58483	−0.792417	+	0.609980i	$0.791177\pi$
$258$	0	0
$259$	19.8827	1.23545
$260$	0	0
$261$	−0.823126	−0.0509502
$262$	0	0
$263$	14.8342	0.914718	0.457359	−	0.889282i	$-0.348795\pi$
$263$	14.8342	0.914718	0.457359	+	0.889282i	$0.348795\pi$
$264$	0	0
$265$	9.18842	0.564440
$266$	0	0
$267$	14.4498	0.884314
$268$	0	0
$269$	−31.8398	−1.94131	−0.970653	−	0.240484i	$-0.922694\pi$
$269$	−31.8398	−1.94131	−0.970653	+	0.240484i	$0.922694\pi$
$270$	0	0
$271$	16.2628	0.987892	0.493946	−	0.869492i	$-0.335554\pi$
$271$	16.2628	0.987892	0.493946	+	0.869492i	$0.335554\pi$
$272$	0	0
$273$	−14.4017	−0.871630
$274$	0	0
$275$	6.21168	0.374579
$276$	0	0
$277$	−6.70750	−0.403014	−0.201507	−	0.979487i	$-0.564584\pi$
$277$	−6.70750	−0.403014	−0.201507	+	0.979487i	$0.564584\pi$
$278$	0	0
$279$	3.72964	0.223288
$280$	0	0
$281$	−31.3340	−1.86923	−0.934615	−	0.355660i	$-0.884256\pi$
$281$	−31.3340	−1.86923	−0.934615	+	0.355660i	$0.884256\pi$
$282$	0	0
$283$	−18.5479	−1.10256	−0.551279	−	0.834321i	$-0.685860\pi$
$283$	−18.5479	−1.10256	−0.551279	+	0.834321i	$0.685860\pi$
$284$	0	0
$285$	27.7918	1.64625
$286$	0	0
$287$	−17.2131	−1.01606
$288$	0	0
$289$	32.5109	1.91241
$290$	0	0
$291$	−9.29823	−0.545072
$292$	0	0
$293$	−11.9272	−0.696795	−0.348397	−	0.937347i	$-0.613274\pi$
$293$	−11.9272	−0.696795	−0.348397	+	0.937347i	$0.613274\pi$
$294$	0	0
$295$	39.3417	2.29056
$296$	0	0
$297$	−8.32246	−0.482918
$298$	0	0
$299$	−24.6106	−1.42327
$300$	0	0
$301$	8.23301	0.474543
$302$	0	0
$303$	−15.0929	−0.867065
$304$	0	0
$305$	20.1207	1.15211
$306$	0	0
$307$	−11.0174	−0.628798	−0.314399	−	0.949291i	$-0.601803\pi$
$307$	−11.0174	−0.628798	−0.314399	+	0.949291i	$0.601803\pi$
$308$	0	0
$309$	−7.64625	−0.434980
$310$	0	0
$311$	33.0363	1.87332	0.936658	−	0.350246i	$-0.113902\pi$
$311$	33.0363	1.87332	0.936658	+	0.350246i	$0.113902\pi$
$312$	0	0
$313$	−13.2612	−0.749568	−0.374784	−	0.927112i	$-0.622283\pi$
$313$	−13.2612	−0.749568	−0.374784	+	0.927112i	$0.622283\pi$
$314$	0	0
$315$	−6.06276	−0.341598
$316$	0	0
$317$	−8.96360	−0.503446	−0.251723	−	0.967799i	$-0.580997\pi$
$317$	−8.96360	−0.503446	−0.251723	+	0.967799i	$0.580997\pi$
$318$	0	0
$319$	−1.95528	−0.109475
$320$	0	0
$321$	−15.1092	−0.843313
$322$	0	0
$323$	34.9756	1.94609
$324$	0	0
$325$	9.08435	0.503909
$326$	0	0
$327$	39.5452	2.18686
$328$	0	0
$329$	−8.47449	−0.467214
$330$	0	0
$331$	31.8113	1.74851	0.874253	−	0.485471i	$-0.161352\pi$
$331$	31.8113	1.74851	0.874253	+	0.485471i	$0.161352\pi$
$332$	0	0
$333$	−6.35375	−0.348183
$334$	0	0
$335$	−22.3647	−1.22191
$336$	0	0
$337$	10.4745	0.570582	0.285291	−	0.958441i	$-0.407910\pi$
$337$	10.4745	0.570582	0.285291	+	0.958441i	$0.407910\pi$
$338$	0	0
$339$	3.91056	0.212393
$340$	0	0
$341$	8.85952	0.479770
$342$	0	0
$343$	18.9715	1.02436
$344$	0	0
$345$	−48.1207	−2.59073
$346$	0	0
$347$	−31.9762	−1.71657	−0.858286	−	0.513172i	$-0.828470\pi$
$347$	−31.9762	−1.71657	−0.858286	+	0.513172i	$0.828470\pi$
$348$	0	0
$349$	−18.2976	−0.979449	−0.489724	−	0.871877i	$-0.662903\pi$
$349$	−18.2976	−0.979449	−0.489724	+	0.871877i	$0.662903\pi$
$350$	0	0
$351$	−12.1713	−0.649655
$352$	0	0
$353$	33.5837	1.78748	0.893740	−	0.448586i	$-0.148072\pi$
$353$	33.5837	1.78748	0.893740	+	0.448586i	$0.148072\pi$
$354$	0	0
$355$	17.2451	0.915275
$356$	0	0
$357$	−35.4381	−1.87558
$358$	0	0
$359$	−29.2355	−1.54299	−0.771495	−	0.636236i	$-0.780491\pi$
$359$	−29.2355	−1.54299	−0.771495	+	0.636236i	$0.780491\pi$
$360$	0	0
$361$	5.70750	0.300395
$362$	0	0
$363$	−14.0328	−0.736531
$364$	0	0
$365$	4.70750	0.246402
$366$	0	0
$367$	14.7245	0.768612	0.384306	−	0.923206i	$-0.374441\pi$
$367$	14.7245	0.768612	0.384306	+	0.923206i	$0.374441\pi$
$368$	0	0
$369$	5.50066	0.286353
$370$	0	0
$371$	−8.27672	−0.429706
$372$	0	0
$373$	11.2133	0.580601	0.290301	−	0.956936i	$-0.406245\pi$
$373$	11.2133	0.580601	0.290301	+	0.956936i	$0.406245\pi$
$374$	0	0
$375$	−10.1934	−0.526386
$376$	0	0
$377$	−2.85952	−0.147273
$378$	0	0
$379$	7.45270	0.382820	0.191410	−	0.981510i	$-0.438694\pi$
$379$	7.45270	0.382820	0.191410	+	0.981510i	$0.438694\pi$
$380$	0	0
$381$	−29.8646	−1.53001
$382$	0	0
$383$	15.1867	0.776002	0.388001	−	0.921659i	$-0.373166\pi$
$383$	15.1867	0.776002	0.388001	+	0.921659i	$0.373166\pi$
$384$	0	0
$385$	−14.4017	−0.733978
$386$	0	0
$387$	−2.63096	−0.133739
$388$	0	0
$389$	−14.3537	−0.727764	−0.363882	−	0.931445i	$-0.618549\pi$
$389$	−14.3537	−0.727764	−0.363882	+	0.931445i	$0.618549\pi$
$390$	0	0
$391$	−60.5591	−3.06261
$392$	0	0
$393$	23.9374	1.20748
$394$	0	0
$395$	−1.77436	−0.0892779
$396$	0	0
$397$	−26.2976	−1.31984	−0.659920	−	0.751336i	$-0.729410\pi$
$397$	−26.2976	−1.31984	−0.659920	+	0.751336i	$0.729410\pi$
$398$	0	0
$399$	−25.0342	−1.25328
$400$	0	0
$401$	−32.6993	−1.63293	−0.816463	−	0.577398i	$-0.804068\pi$
$401$	−32.6993	−1.63293	−0.816463	+	0.577398i	$0.804068\pi$
$402$	0	0
$403$	12.9567	0.645420
$404$	0	0
$405$	−30.8595	−1.53342
$406$	0	0
$407$	−15.0929	−0.748128
$408$	0	0
$409$	32.8762	1.62562	0.812811	−	0.582527i	$-0.197936\pi$
$409$	32.8762	1.62562	0.812811	+	0.582527i	$0.197936\pi$
$410$	0	0
$411$	22.9140	1.13027
$412$	0	0
$413$	−35.4381	−1.74379
$414$	0	0
$415$	−46.9753	−2.30593
$416$	0	0
$417$	−32.1207	−1.57296
$418$	0	0
$419$	14.7631	0.721223	0.360612	−	0.932716i	$-0.382568\pi$
$419$	14.7631	0.721223	0.360612	+	0.932716i	$0.382568\pi$
$420$	0	0
$421$	1.52551	0.0743488	0.0371744	−	0.999309i	$-0.488164\pi$
$421$	1.52551	0.0743488	0.0371744	+	0.999309i	$0.488164\pi$
$422$	0	0
$423$	2.70813	0.131674
$424$	0	0
$425$	22.3537	1.08432
$426$	0	0
$427$	−18.1243	−0.877096
$428$	0	0
$429$	10.9323	0.527817
$430$	0	0
$431$	25.8197	1.24369	0.621845	−	0.783141i	$-0.286383\pi$
$431$	25.8197	1.24369	0.621845	+	0.783141i	$0.286383\pi$
$432$	0	0
$433$	13.0843	0.628794	0.314397	−	0.949292i	$-0.398198\pi$
$433$	13.0843	0.628794	0.314397	+	0.949292i	$0.398198\pi$
$434$	0	0
$435$	−5.59117	−0.268076
$436$	0	0
$437$	−42.7803	−2.04646
$438$	0	0
$439$	31.3331	1.49545	0.747723	−	0.664010i	$-0.231147\pi$
$439$	31.3331	1.49545	0.747723	+	0.664010i	$0.231147\pi$
$440$	0	0
$441$	−0.300687	−0.0143184
$442$	0	0
$443$	15.8232	0.751782	0.375891	−	0.926664i	$-0.377337\pi$
$443$	15.8232	0.751782	0.375891	+	0.926664i	$0.377337\pi$
$444$	0	0
$445$	21.1323	1.00177
$446$	0	0
$447$	9.64407	0.456149
$448$	0	0
$449$	15.5109	0.732004	0.366002	−	0.930614i	$-0.380726\pi$
$449$	15.5109	0.732004	0.366002	+	0.930614i	$0.380726\pi$
$450$	0	0
$451$	13.0665	0.615276
$452$	0	0
$453$	−14.4017	−0.676651
$454$	0	0
$455$	−21.0619	−0.987398
$456$	0	0
$457$	1.29250	0.0604608	0.0302304	−	0.999543i	$-0.490376\pi$
$457$	1.29250	0.0604608	0.0302304	+	0.999543i	$0.490376\pi$
$458$	0	0
$459$	−29.9497	−1.39793
$460$	0	0
$461$	−17.4381	−0.812173	−0.406086	−	0.913835i	$-0.633107\pi$
$461$	−17.4381	−0.812173	−0.406086	+	0.913835i	$0.633107\pi$
$462$	0	0
$463$	10.7268	0.498515	0.249257	−	0.968437i	$-0.419814\pi$
$463$	10.7268	0.498515	0.249257	+	0.968437i	$0.419814\pi$
$464$	0	0
$465$	25.3340	1.17484
$466$	0	0
$467$	−33.3821	−1.54474	−0.772370	−	0.635173i	$-0.780929\pi$
$467$	−33.3821	−1.54474	−0.772370	+	0.635173i	$0.780929\pi$
$468$	0	0
$469$	20.1456	0.930237
$470$	0	0
$471$	30.0921	1.38657
$472$	0	0
$473$	−6.24967	−0.287360
$474$	0	0
$475$	15.7912	0.724549
$476$	0	0
$477$	2.64493	0.121103
$478$	0	0
$479$	−20.1733	−0.921743	−0.460871	−	0.887467i	$-0.652463\pi$
$479$	−20.1733	−0.921743	−0.460871	+	0.887467i	$0.652463\pi$
$480$	0	0
$481$	−22.0728	−1.00643
$482$	0	0
$483$	43.3460	1.97231
$484$	0	0
$485$	−13.5983	−0.617467
$486$	0	0
$487$	10.3969	0.471129	0.235565	−	0.971859i	$-0.424306\pi$
$487$	10.3969	0.471129	0.235565	+	0.971859i	$0.424306\pi$
$488$	0	0
$489$	1.39658	0.0631557
$490$	0	0
$491$	−13.8293	−0.624108	−0.312054	−	0.950064i	$-0.601017\pi$
$491$	−13.8293	−0.624108	−0.312054	+	0.950064i	$0.601017\pi$
$492$	0	0
$493$	−7.03640	−0.316903
$494$	0	0
$495$	4.60224	0.206855
$496$	0	0
$497$	−15.5340	−0.696795
$498$	0	0
$499$	−6.48635	−0.290369	−0.145185	−	0.989405i	$-0.546378\pi$
$499$	−6.48635	−0.290369	−0.145185	+	0.989405i	$0.546378\pi$
$500$	0	0
$501$	6.57213	0.293621
$502$	0	0
$503$	−19.0746	−0.850496	−0.425248	−	0.905077i	$-0.639813\pi$
$503$	−19.0746	−0.850496	−0.425248	+	0.905077i	$0.639813\pi$
$504$	0	0
$505$	−22.0728	−0.982226
$506$	0	0
$507$	−9.43057	−0.418826
$508$	0	0
$509$	13.9439	0.618051	0.309026	−	0.951054i	$-0.399997\pi$
$509$	13.9439	0.618051	0.309026	+	0.951054i	$0.399997\pi$
$510$	0	0
$511$	−4.24040	−0.187584
$512$	0	0
$513$	−21.1571	−0.934111
$514$	0	0
$515$	−11.1823	−0.492753
$516$	0	0
$517$	6.43298	0.282922
$518$	0	0
$519$	−14.4012	−0.632144
$520$	0	0
$521$	23.8959	1.04690	0.523450	−	0.852057i	$-0.324645\pi$
$521$	23.8959	1.04690	0.523450	+	0.852057i	$0.324645\pi$
$522$	0	0
$523$	22.9140	1.00196	0.500980	−	0.865459i	$-0.332973\pi$
$523$	22.9140	1.00196	0.500980	+	0.865459i	$0.332973\pi$
$524$	0	0
$525$	−16.0000	−0.698297
$526$	0	0
$527$	31.8824	1.38882
$528$	0	0
$529$	51.0728	2.22056
$530$	0	0
$531$	11.3247	0.491449
$532$	0	0
$533$	19.1092	0.827711
$534$	0	0
$535$	−22.0966	−0.955320
$536$	0	0
$537$	−27.0843	−1.16878
$538$	0	0
$539$	−0.714262	−0.0307654
$540$	0	0
$541$	21.1340	0.908623	0.454312	−	0.890843i	$-0.349885\pi$
$541$	21.1340	0.908623	0.454312	+	0.890843i	$0.349885\pi$
$542$	0	0
$543$	−17.4652	−0.749503
$544$	0	0
$545$	57.8334	2.47731
$546$	0	0
$547$	−22.4584	−0.960254	−0.480127	−	0.877199i	$-0.659409\pi$
$547$	−22.4584	−0.960254	−0.480127	+	0.877199i	$0.659409\pi$
$548$	0	0
$549$	5.79184	0.247190
$550$	0	0
$551$	−4.97066	−0.211757
$552$	0	0
$553$	1.59830	0.0679668
$554$	0	0
$555$	−43.1585	−1.83198
$556$	0	0
$557$	−32.0728	−1.35897	−0.679484	−	0.733690i	$-0.737796\pi$
$557$	−32.0728	−1.35897	−0.679484	+	0.733690i	$0.737796\pi$
$558$	0	0
$559$	−9.13990	−0.386576
$560$	0	0
$561$	26.9010	1.13576
$562$	0	0
$563$	38.5337	1.62400	0.812001	−	0.583656i	$-0.198378\pi$
$563$	38.5337	1.62400	0.812001	+	0.583656i	$0.198378\pi$
$564$	0	0
$565$	5.71905	0.240602
$566$	0	0
$567$	27.7975	1.16739
$568$	0	0
$569$	−46.2912	−1.94063	−0.970314	−	0.241850i	$-0.922246\pi$
$569$	−46.2912	−1.94063	−0.970314	+	0.241850i	$0.922246\pi$
$570$	0	0
$571$	15.0929	0.631619	0.315809	−	0.948823i	$-0.397724\pi$
$571$	15.0929	0.631619	0.315809	+	0.948823i	$0.397724\pi$
$572$	0	0
$573$	44.4496	1.85691
$574$	0	0
$575$	−27.3419	−1.14024
$576$	0	0
$577$	−19.7439	−0.821949	−0.410975	−	0.911647i	$-0.634811\pi$
$577$	−19.7439	−0.821949	−0.410975	+	0.911647i	$0.634811\pi$
$578$	0	0
$579$	−7.91488	−0.328931
$580$	0	0
$581$	42.3143	1.75549
$582$	0	0
$583$	6.28285	0.260209
$584$	0	0
$585$	6.73060	0.278276
$586$	0	0
$587$	13.9456	0.575598	0.287799	−	0.957691i	$-0.407076\pi$
$587$	13.9456	0.575598	0.287799	+	0.957691i	$0.407076\pi$
$588$	0	0
$589$	22.5224	0.928021
$590$	0	0
$591$	10.6330	0.437383
$592$	0	0
$593$	15.1884	0.623714	0.311857	−	0.950129i	$-0.399049\pi$
$593$	15.1884	0.623714	0.311857	+	0.950129i	$0.399049\pi$
$594$	0	0
$595$	−51.8268	−2.12469
$596$	0	0
$597$	−48.7657	−1.99585
$598$	0	0
$599$	−23.8964	−0.976380	−0.488190	−	0.872737i	$-0.662343\pi$
$599$	−23.8964	−0.976380	−0.488190	+	0.872737i	$0.662343\pi$
$600$	0	0
$601$	32.7554	1.33612	0.668061	−	0.744106i	$-0.267124\pi$
$601$	32.7554	1.33612	0.668061	+	0.744106i	$0.267124\pi$
$602$	0	0
$603$	−6.43777	−0.262166
$604$	0	0
$605$	−20.5224	−0.834356
$606$	0	0
$607$	−9.49515	−0.385396	−0.192698	−	0.981258i	$-0.561724\pi$
$607$	−9.49515	−0.385396	−0.192698	+	0.981258i	$0.561724\pi$
$608$	0	0
$609$	5.03640	0.204085
$610$	0	0
$611$	9.40798	0.380606
$612$	0	0
$613$	−13.8480	−0.559314	−0.279657	−	0.960100i	$-0.590221\pi$
$613$	−13.8480	−0.559314	−0.279657	+	0.960100i	$0.590221\pi$
$614$	0	0
$615$	37.3639	1.50666
$616$	0	0
$617$	31.8646	1.28282	0.641411	−	0.767197i	$-0.278349\pi$
$617$	31.8646	1.28282	0.641411	+	0.767197i	$0.278349\pi$
$618$	0	0
$619$	−42.4443	−1.70598	−0.852990	−	0.521928i	$-0.825213\pi$
$619$	−42.4443	−1.70598	−0.852990	+	0.521928i	$0.825213\pi$
$620$	0	0
$621$	36.6330	1.47003
$622$	0	0
$623$	−19.0355	−0.762640
$624$	0	0
$625$	−30.7918	−1.23167
$626$	0	0
$627$	19.0035	0.758926
$628$	0	0
$629$	−54.3143	−2.16565
$630$	0	0
$631$	−3.54872	−0.141272	−0.0706362	−	0.997502i	$-0.522503\pi$
$631$	−3.54872	−0.141272	−0.0706362	+	0.997502i	$0.522503\pi$
$632$	0	0
$633$	−9.04284	−0.359420
$634$	0	0
$635$	−43.6759	−1.73322
$636$	0	0
$637$	−1.04458	−0.0413878
$638$	0	0
$639$	4.96408	0.196376
$640$	0	0
$641$	−16.5473	−0.653578	−0.326789	−	0.945097i	$-0.605967\pi$
$641$	−16.5473	−0.653578	−0.326789	+	0.945097i	$0.605967\pi$
$642$	0	0
$643$	42.7029	1.68404	0.842020	−	0.539447i	$-0.181367\pi$
$643$	42.7029	1.68404	0.842020	+	0.539447i	$0.181367\pi$
$644$	0	0
$645$	−17.8711	−0.703673
$646$	0	0
$647$	9.15590	0.359955	0.179978	−	0.983671i	$-0.442397\pi$
$647$	9.15590	0.359955	0.179978	+	0.983671i	$0.442397\pi$
$648$	0	0
$649$	26.9010	1.05596
$650$	0	0
$651$	−22.8203	−0.894397
$652$	0	0
$653$	11.2446	0.440033	0.220017	−	0.975496i	$-0.429389\pi$
$653$	11.2446	0.440033	0.220017	+	0.975496i	$0.429389\pi$
$654$	0	0
$655$	35.0076	1.36786
$656$	0	0
$657$	1.35507	0.0528664
$658$	0	0
$659$	9.58890	0.373531	0.186765	−	0.982405i	$-0.440200\pi$
$659$	9.58890	0.373531	0.186765	+	0.982405i	$0.440200\pi$
$660$	0	0
$661$	35.3653	1.37555	0.687775	−	0.725924i	$-0.258587\pi$
$661$	35.3653	1.37555	0.687775	+	0.725924i	$0.258587\pi$
$662$	0	0
$663$	39.3417	1.52791
$664$	0	0
$665$	−36.6116	−1.41974
$666$	0	0
$667$	8.60656	0.333247
$668$	0	0
$669$	12.4993	0.483252
$670$	0	0
$671$	13.7581	0.531127
$672$	0	0
$673$	−41.7606	−1.60975	−0.804876	−	0.593444i	$-0.797768\pi$
$673$	−41.7606	−1.60975	−0.804876	+	0.593444i	$0.797768\pi$
$674$	0	0
$675$	−13.5221	−0.520464
$676$	0	0
$677$	−38.2415	−1.46974	−0.734870	−	0.678208i	$-0.762757\pi$
$677$	−38.2415	−1.46974	−0.734870	+	0.678208i	$0.762757\pi$
$678$	0	0
$679$	12.2490	0.470075
$680$	0	0
$681$	−33.0116	−1.26501
$682$	0	0
$683$	47.9483	1.83469	0.917345	−	0.398094i	$-0.130328\pi$
$683$	47.9483	1.83469	0.917345	+	0.398094i	$0.130328\pi$
$684$	0	0
$685$	33.5109	1.28039
$686$	0	0
$687$	−11.9678	−0.456599
$688$	0	0
$689$	9.18842	0.350051
$690$	0	0
$691$	−23.8252	−0.906354	−0.453177	−	0.891421i	$-0.649709\pi$
$691$	−23.8252	−0.906354	−0.453177	+	0.891421i	$0.649709\pi$
$692$	0	0
$693$	−4.14559	−0.157478
$694$	0	0
$695$	−46.9753	−1.78188
$696$	0	0
$697$	47.0218	1.78108
$698$	0	0
$699$	−3.56472	−0.134830
$700$	0	0
$701$	25.4942	0.962904	0.481452	−	0.876473i	$-0.340110\pi$
$701$	25.4942	0.962904	0.481452	+	0.876473i	$0.340110\pi$
$702$	0	0
$703$	−38.3688	−1.44711
$704$	0	0
$705$	18.3953	0.692806
$706$	0	0
$707$	19.8827	0.747764
$708$	0	0
$709$	−48.1623	−1.80877	−0.904386	−	0.426716i	$-0.859670\pi$
$709$	−48.1623	−1.80877	−0.904386	+	0.426716i	$0.859670\pi$
$710$	0	0
$711$	−0.510758	−0.0191549
$712$	0	0
$713$	−38.9969	−1.46045
$714$	0	0
$715$	15.9881	0.597921
$716$	0	0
$717$	−37.3405	−1.39450
$718$	0	0
$719$	−10.6330	−0.396544	−0.198272	−	0.980147i	$-0.563533\pi$
$719$	−10.6330	−0.396544	−0.198272	+	0.980147i	$0.563533\pi$
$720$	0	0
$721$	10.0728	0.375130
$722$	0	0
$723$	−4.25640	−0.158297
$724$	0	0
$725$	−3.17687	−0.117986
$726$	0	0
$727$	−33.3981	−1.23867	−0.619334	−	0.785128i	$-0.712597\pi$
$727$	−33.3981	−1.23867	−0.619334	+	0.785128i	$0.712597\pi$
$728$	0	0
$729$	16.0843	0.595717
$730$	0	0
$731$	−22.4904	−0.831839
$732$	0	0
$733$	9.64625	0.356292	0.178146	−	0.984004i	$-0.442990\pi$
$733$	9.64625	0.356292	0.178146	+	0.984004i	$0.442990\pi$
$734$	0	0
$735$	−2.04245	−0.0753369
$736$	0	0
$737$	−15.2925	−0.563307
$738$	0	0
$739$	−10.1062	−0.371764	−0.185882	−	0.982572i	$-0.559514\pi$
$739$	−10.1062	−0.371764	−0.185882	+	0.982572i	$0.559514\pi$
$740$	0	0
$741$	27.7918	1.02096
$742$	0	0
$743$	12.6043	0.462406	0.231203	−	0.972905i	$-0.425734\pi$
$743$	12.6043	0.462406	0.231203	+	0.972905i	$0.425734\pi$
$744$	0	0
$745$	14.1041	0.516733
$746$	0	0
$747$	−13.5221	−0.494746
$748$	0	0
$749$	19.9041	0.727280
$750$	0	0
$751$	40.3081	1.47086	0.735431	−	0.677600i	$-0.236980\pi$
$751$	40.3081	1.47086	0.735431	+	0.677600i	$0.236980\pi$
$752$	0	0
$753$	−4.46805	−0.162825
$754$	0	0
$755$	−21.0619	−0.766522
$756$	0	0
$757$	19.1820	0.697181	0.348591	−	0.937275i	$-0.386660\pi$
$757$	19.1820	0.697181	0.348591	+	0.937275i	$0.386660\pi$
$758$	0	0
$759$	−32.9039	−1.19434
$760$	0	0
$761$	−0.730598	−0.0264841	−0.0132421	−	0.999912i	$-0.504215\pi$
$761$	−0.730598	−0.0264841	−0.0132421	+	0.999912i	$0.504215\pi$
$762$	0	0
$763$	−52.0949	−1.88596
$764$	0	0
$765$	16.5619	0.598797
$766$	0	0
$767$	39.3417	1.42055
$768$	0	0
$769$	−5.76699	−0.207963	−0.103982	−	0.994579i	$-0.533158\pi$
$769$	−5.76699	−0.207963	−0.103982	+	0.994579i	$0.533158\pi$
$770$	0	0
$771$	−49.6775	−1.78909
$772$	0	0
$773$	−42.2663	−1.52021	−0.760107	−	0.649798i	$-0.774854\pi$
$773$	−42.2663	−1.52021	−0.760107	+	0.649798i	$0.774854\pi$
$774$	0	0
$775$	14.3946	0.517071
$776$	0	0
$777$	38.8762	1.39468
$778$	0	0
$779$	33.2172	1.19013
$780$	0	0
$781$	11.7918	0.421945
$782$	0	0
$783$	4.25640	0.152111
$784$	0	0
$785$	44.0085	1.57073
$786$	0	0
$787$	−6.03076	−0.214974	−0.107487	−	0.994207i	$-0.534280\pi$
$787$	−6.03076	−0.214974	−0.107487	+	0.994207i	$0.534280\pi$
$788$	0	0
$789$	29.0051	1.03261
$790$	0	0
$791$	−5.15158	−0.183169
$792$	0	0
$793$	20.1207	0.714509
$794$	0	0
$795$	17.9660	0.637187
$796$	0	0
$797$	−8.96360	−0.317507	−0.158754	−	0.987318i	$-0.550748\pi$
$797$	−8.96360	−0.317507	−0.158754	+	0.987318i	$0.550748\pi$
$798$	0	0
$799$	23.1501	0.818992
$800$	0	0
$801$	6.08302	0.214933
$802$	0	0
$803$	3.21889	0.113592
$804$	0	0
$805$	63.3919	2.23427
$806$	0	0
$807$	−62.2558	−2.19151
$808$	0	0
$809$	−35.8149	−1.25919	−0.629593	−	0.776925i	$-0.716778\pi$
$809$	−35.8149	−1.25919	−0.629593	+	0.776925i	$0.716778\pi$
$810$	0	0
$811$	29.5427	1.03739	0.518693	−	0.854961i	$-0.326419\pi$
$811$	29.5427	1.03739	0.518693	+	0.854961i	$0.326419\pi$
$812$	0	0
$813$	31.7983	1.11521
$814$	0	0
$815$	2.04245	0.0715439
$816$	0	0
$817$	−15.8877	−0.555842
$818$	0	0
$819$	−6.06276	−0.211850
$820$	0	0
$821$	−7.16358	−0.250011	−0.125005	−	0.992156i	$-0.539895\pi$
$821$	−7.16358	−0.250011	−0.125005	+	0.992156i	$0.539895\pi$
$822$	0	0
$823$	31.2779	1.09028	0.545140	−	0.838345i	$-0.316477\pi$
$823$	31.2779	1.09028	0.545140	+	0.838345i	$0.316477\pi$
$824$	0	0
$825$	12.1456	0.422855
$826$	0	0
$827$	20.5969	0.716225	0.358112	−	0.933678i	$-0.383420\pi$
$827$	20.5969	0.716225	0.358112	+	0.933678i	$0.383420\pi$
$828$	0	0
$829$	35.1323	1.22019	0.610097	−	0.792326i	$-0.291130\pi$
$829$	35.1323	1.22019	0.610097	+	0.792326i	$0.291130\pi$
$830$	0	0
$831$	−13.1150	−0.454956
$832$	0	0
$833$	−2.57039	−0.0890586
$834$	0	0
$835$	9.61149	0.332619
$836$	0	0
$837$	−19.2861	−0.666624
$838$	0	0
$839$	55.8406	1.92783	0.963915	−	0.266209i	$-0.0857713\pi$
$839$	55.8406	1.92783	0.963915	+	0.266209i	$0.0857713\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−61.2668	−2.11014
$844$	0	0
$845$	−13.7918	−0.474454
$846$	0	0
$847$	18.4861	0.635191
$848$	0	0
$849$	−36.2663	−1.24466
$850$	0	0
$851$	66.4344	2.27734
$852$	0	0
$853$	16.6347	0.569561	0.284781	−	0.958593i	$-0.408079\pi$
$853$	16.6347	0.569561	0.284781	+	0.958593i	$0.408079\pi$
$854$	0	0
$855$	11.6997	0.400121
$856$	0	0
$857$	−38.6265	−1.31946	−0.659728	−	0.751504i	$-0.729329\pi$
$857$	−38.6265	−1.31946	−0.659728	+	0.751504i	$0.729329\pi$
$858$	0	0
$859$	6.05335	0.206538	0.103269	−	0.994653i	$-0.467070\pi$
$859$	6.05335	0.206538	0.103269	+	0.994653i	$0.467070\pi$
$860$	0	0
$861$	−33.6565	−1.14701
$862$	0	0
$863$	−6.94195	−0.236307	−0.118153	−	0.992995i	$-0.537697\pi$
$863$	−6.94195	−0.236307	−0.118153	+	0.992995i	$0.537697\pi$
$864$	0	0
$865$	−21.0612	−0.716104
$866$	0	0
$867$	63.5679	2.15888
$868$	0	0
$869$	−1.21327	−0.0411574
$870$	0	0
$871$	−22.3647	−0.757799
$872$	0	0
$873$	−3.91433	−0.132480
$874$	0	0
$875$	13.4283	0.453959
$876$	0	0
$877$	−2.05613	−0.0694306	−0.0347153	−	0.999397i	$-0.511052\pi$
$877$	−2.05613	−0.0694306	−0.0347153	+	0.999397i	$0.511052\pi$
$878$	0	0
$879$	−23.3210	−0.786599
$880$	0	0
$881$	−44.2184	−1.48976	−0.744878	−	0.667201i	$-0.767492\pi$
$881$	−44.2184	−1.48976	−0.744878	+	0.667201i	$0.767492\pi$
$882$	0	0
$883$	−24.3911	−0.820827	−0.410414	−	0.911899i	$-0.634616\pi$
$883$	−24.3911	−0.820827	−0.410414	+	0.911899i	$0.634616\pi$
$884$	0	0
$885$	76.9241	2.58578
$886$	0	0
$887$	13.2634	0.445341	0.222670	−	0.974894i	$-0.428523\pi$
$887$	13.2634	0.445341	0.222670	+	0.974894i	$0.428523\pi$
$888$	0	0
$889$	39.3422	1.31949
$890$	0	0
$891$	−21.1011	−0.706913
$892$	0	0
$893$	16.3537	0.547257
$894$	0	0
$895$	−39.6098	−1.32401
$896$	0	0
$897$	−48.1207	−1.60670
$898$	0	0
$899$	−4.53107	−0.151120
$900$	0	0
$901$	22.6099	0.753243
$902$	0	0
$903$	16.0978	0.535703
$904$	0	0
$905$	−25.5422	−0.849051
$906$	0	0
$907$	50.9731	1.69253	0.846267	−	0.532760i	$-0.178845\pi$
$907$	50.9731	1.69253	0.846267	+	0.532760i	$0.178845\pi$
$908$	0	0
$909$	−6.35375	−0.210741
$910$	0	0
$911$	−8.58397	−0.284400	−0.142200	−	0.989838i	$-0.545418\pi$
$911$	−8.58397	−0.284400	−0.142200	+	0.989838i	$0.545418\pi$
$912$	0	0
$913$	−32.1207	−1.06304
$914$	0	0
$915$	39.3417	1.30060
$916$	0	0
$917$	−31.5340	−1.04134
$918$	0	0
$919$	−11.9192	−0.393178	−0.196589	−	0.980486i	$-0.562986\pi$
$919$	−11.9192	−0.393178	−0.196589	+	0.980486i	$0.562986\pi$
$920$	0	0
$921$	−21.5422	−0.709839
$922$	0	0
$923$	17.2451	0.567630
$924$	0	0
$925$	−24.5224	−0.806293
$926$	0	0
$927$	−3.21889	−0.105722
$928$	0	0
$929$	−52.4599	−1.72115	−0.860576	−	0.509322i	$-0.829896\pi$
$929$	−52.4599	−1.72115	−0.860576	+	0.509322i	$0.829896\pi$
$930$	0	0
$931$	−1.81578	−0.0595097
$932$	0	0
$933$	64.5952	2.11475
$934$	0	0
$935$	39.3417	1.28661
$936$	0	0
$937$	5.19661	0.169766	0.0848829	−	0.996391i	$-0.472948\pi$
$937$	5.19661	0.169766	0.0848829	+	0.996391i	$0.472948\pi$
$938$	0	0
$939$	−25.9294	−0.846175
$940$	0	0
$941$	−11.7752	−0.383860	−0.191930	−	0.981409i	$-0.561475\pi$
$941$	−11.7752	−0.383860	−0.191930	+	0.981409i	$0.561475\pi$
$942$	0	0
$943$	−57.5146	−1.87293
$944$	0	0
$945$	31.3507	1.01984
$946$	0	0
$947$	7.29437	0.237035	0.118518	−	0.992952i	$-0.462186\pi$
$947$	7.29437	0.237035	0.118518	+	0.992952i	$0.462186\pi$
$948$	0	0
$949$	4.70750	0.152812
$950$	0	0
$951$	−17.5264	−0.568331
$952$	0	0
$953$	11.6878	0.378604	0.189302	−	0.981919i	$-0.439378\pi$
$953$	11.6878	0.378604	0.189302	+	0.981919i	$0.439378\pi$
$954$	0	0
$955$	65.0059	2.10354
$956$	0	0
$957$	−3.82313	−0.123584
$958$	0	0
$959$	−30.1858	−0.974751
$960$	0	0
$961$	−10.4694	−0.337722
$962$	0	0
$963$	−6.36060	−0.204968
$964$	0	0
$965$	−11.5752	−0.372619
$966$	0	0
$967$	−2.04903	−0.0658925	−0.0329462	−	0.999457i	$-0.510489\pi$
$967$	−2.04903	−0.0658925	−0.0329462	+	0.999457i	$0.510489\pi$
$968$	0	0
$969$	68.3871	2.19691
$970$	0	0
$971$	20.4254	0.655483	0.327741	−	0.944767i	$-0.393713\pi$
$971$	20.4254	0.655483	0.327741	+	0.944767i	$0.393713\pi$
$972$	0	0
$973$	42.3143	1.35653
$974$	0	0
$975$	17.7625	0.568854
$976$	0	0
$977$	10.2728	0.328655	0.164328	−	0.986406i	$-0.447455\pi$
$977$	10.2728	0.328655	0.164328	+	0.986406i	$0.447455\pi$
$978$	0	0
$979$	14.4498	0.461818
$980$	0	0
$981$	16.6476	0.531516
$982$	0	0
$983$	20.3157	0.647969	0.323984	−	0.946062i	$-0.394977\pi$
$983$	20.3157	0.647969	0.323984	+	0.946062i	$0.394977\pi$
$984$	0	0
$985$	15.5504	0.495476
$986$	0	0
$987$	−16.5700	−0.527429
$988$	0	0
$989$	27.5091	0.874740
$990$	0	0
$991$	52.0463	1.65331	0.826653	−	0.562712i	$-0.190242\pi$
$991$	52.0463	1.65331	0.826653	+	0.562712i	$0.190242\pi$
$992$	0	0
$993$	62.2000	1.97386
$994$	0	0
$995$	−71.3179	−2.26093
$996$	0	0
$997$	59.9605	1.89897	0.949485	−	0.313814i	$-0.101607\pi$
$997$	59.9605	1.89897	0.949485	+	0.313814i	$0.101607\pi$
$998$	0	0
$999$	32.8554	1.03950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.bc.1.5		6
4.3	odd	2	inner	1856.2.a.bc.1.2		6
8.3	odd	2		928.2.a.i.1.5	yes	6
8.5	even	2		928.2.a.i.1.2	✓	6
24.5	odd	2		8352.2.a.bi.1.5		6
24.11	even	2		8352.2.a.bi.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.i.1.2	✓	6	8.5	even	2
928.2.a.i.1.5	yes	6	8.3	odd	2
1856.2.a.bc.1.2		6	4.3	odd	2	inner
1856.2.a.bc.1.5		6	1.1	even	1	trivial
8352.2.a.bi.1.5		6	24.5	odd	2
8352.2.a.bi.1.6		6	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95528 q^{3} +2.85952 q^{5} -2.57579 q^{7} +0.823126 q^{9} +O(q^{10})\)	\(q+1.95528 q^{3} +2.85952 q^{5} -2.57579 q^{7} +0.823126 q^{9} +1.95528 q^{11} +2.85952 q^{13} +5.59117 q^{15} +7.03640 q^{17} +4.97066 q^{19} -5.03640 q^{21} -8.60656 q^{23} +3.17687 q^{25} -4.25640 q^{27} -1.00000 q^{29} +4.53107 q^{31} +3.82313 q^{33} -7.36554 q^{35} -7.71905 q^{37} +5.59117 q^{39} +6.68265 q^{41} -3.19630 q^{43} +2.35375 q^{45} +3.29005 q^{47} -0.365299 q^{49} +13.7581 q^{51} +3.21327 q^{53} +5.59117 q^{55} +9.71905 q^{57} +13.7581 q^{59} +7.03640 q^{61} -2.12020 q^{63} +8.17687 q^{65} -7.82113 q^{67} -16.8282 q^{69} +6.03076 q^{71} +1.64625 q^{73} +6.21168 q^{75} -5.03640 q^{77} -0.620510 q^{79} -10.7918 q^{81} -16.4277 q^{83} +20.1207 q^{85} -1.95528 q^{87} +7.39015 q^{89} -7.36554 q^{91} +8.85952 q^{93} +14.2137 q^{95} -4.75544 q^{97} +1.60944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 6 * q - 4 * q^5 + 10 * q^9	\( 6 q - 4 q^{5} + 10 q^{9} - 4 q^{13} + 16 q^{17} - 4 q^{21} + 14 q^{25} - 6 q^{29} + 28 q^{33} - 4 q^{37} + 24 q^{41} + 4 q^{45} + 30 q^{49} - 12 q^{53} + 16 q^{57} + 16 q^{61} + 44 q^{65} + 20 q^{69} + 20 q^{73} - 4 q^{77} + 30 q^{81} + 20 q^{85} + 8 q^{89} + 32 q^{93} + 40 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 10 * q^9 - 4 * q^13 + 16 * q^17 - 4 * q^21 + 14 * q^25 - 6 * q^29 + 28 * q^33 - 4 * q^37 + 24 * q^41 + 4 * q^45 + 30 * q^49 - 12 * q^53 + 16 * q^57 + 16 * q^61 + 44 * q^65 + 20 * q^69 + 20 * q^73 - 4 * q^77 + 30 * q^81 + 20 * q^85 + 8 * q^89 + 32 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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