LMFDB - Embedded newform 1856.2.a.ba.1.4

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:	$5$
Coefficient field:	5.5.230224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.757366$ 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)

$f(q)$	$=$	$q+1.26407 q^{3} +3.64073 q^{5} -1.51473 q^{7} -1.40213 q^{9} +O(q^{10})$	$q+1.26407 q^{3} +3.64073 q^{5} -1.51473 q^{7} -1.40213 q^{9} +1.84073 q^{11} +3.46406 q^{13} +4.60213 q^{15} +1.23860 q^{17} -4.85279 q^{19} -1.91472 q^{21} +8.04287 q^{23} +8.25493 q^{25} -5.56459 q^{27} +1.00000 q^{29} -5.45492 q^{31} +2.32680 q^{33} -5.51473 q^{35} +11.9576 q^{37} +4.37881 q^{39} +3.01340 q^{41} +1.84073 q^{43} -5.10479 q^{45} +11.3555 q^{47} -4.70559 q^{49} +1.56567 q^{51} +0.535937 q^{53} +6.70159 q^{55} -6.13426 q^{57} -5.01340 q^{59} -9.32433 q^{61} +2.12386 q^{63} +12.6117 q^{65} -2.67612 q^{67} +10.1667 q^{69} +8.03479 q^{71} +9.48039 q^{73} +10.4348 q^{75} -2.78821 q^{77} -16.4117 q^{79} -2.82762 q^{81} +2.71899 q^{83} +4.50940 q^{85} +1.26407 q^{87} -3.19619 q^{89} -5.24713 q^{91} -6.89539 q^{93} -17.6677 q^{95} +5.68953 q^{97} -2.58095 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * q^99

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
$3$	1.26407	0.729810	0.364905	−	0.931045i	$-0.381102\pi$
$3$	1.26407	0.729810	0.364905	+	0.931045i	$0.381102\pi$
$4$	0	0
$5$	3.64073	1.62818	0.814092	−	0.580735i	$-0.197235\pi$
$5$	3.64073	1.62818	0.814092	+	0.580735i	$0.197235\pi$
$6$	0	0
$7$	−1.51473	−0.572515	−0.286257	−	0.958153i	$-0.592411\pi$
$7$	−1.51473	−0.572515	−0.286257	+	0.958153i	$0.592411\pi$
$8$	0	0
$9$	−1.40213	−0.467378
$10$	0	0
$11$	1.84073	0.555000	0.277500	−	0.960726i	$-0.410494\pi$
$11$	1.84073	0.555000	0.277500	+	0.960726i	$0.410494\pi$
$12$	0	0
$13$	3.46406	0.960758	0.480379	−	0.877061i	$-0.340499\pi$
$13$	3.46406	0.960758	0.480379	+	0.877061i	$0.340499\pi$
$14$	0	0
$15$	4.60213	1.18826
$16$	0	0
$17$	1.23860	0.300404	0.150202	−	0.988655i	$-0.452008\pi$
$17$	1.23860	0.300404	0.150202	+	0.988655i	$0.452008\pi$
$18$	0	0
$19$	−4.85279	−1.11331	−0.556654	−	0.830745i	$-0.687915\pi$
$19$	−4.85279	−1.11331	−0.556654	+	0.830745i	$0.687915\pi$
$20$	0	0
$21$	−1.91472	−0.417827
$22$	0	0
$23$	8.04287	1.67705	0.838527	−	0.544860i	$-0.183417\pi$
$23$	8.04287	1.67705	0.838527	+	0.544860i	$0.183417\pi$
$24$	0	0
$25$	8.25493	1.65099
$26$	0	0
$27$	−5.56459	−1.07091
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.45492	−0.979733	−0.489867	−	0.871797i	$-0.662955\pi$
$31$	−5.45492	−0.979733	−0.489867	+	0.871797i	$0.662955\pi$
$32$	0	0
$33$	2.32680	0.405044
$34$	0	0
$35$	−5.51473	−0.932160
$36$	0	0
$37$	11.9576	1.96582	0.982908	−	0.184099i	$-0.0589367\pi$
$37$	11.9576	1.96582	0.982908	+	0.184099i	$0.0589367\pi$
$38$	0	0
$39$	4.37881	0.701170
$40$	0	0
$41$	3.01340	0.470614	0.235307	−	0.971921i	$-0.424390\pi$
$41$	3.01340	0.470614	0.235307	+	0.971921i	$0.424390\pi$
$42$	0	0
$43$	1.84073	0.280708	0.140354	−	0.990101i	$-0.455176\pi$
$43$	1.84073	0.280708	0.140354	+	0.990101i	$0.455176\pi$
$44$	0	0
$45$	−5.10479	−0.760978
$46$	0	0
$47$	11.3555	1.65636	0.828182	−	0.560459i	$-0.189375\pi$
$47$	11.3555	1.65636	0.828182	+	0.560459i	$0.189375\pi$
$48$	0	0
$49$	−4.70559	−0.672227
$50$	0	0
$51$	1.56567	0.219238
$52$	0	0
$53$	0.535937	0.0736167	0.0368083	−	0.999322i	$-0.488281\pi$
$53$	0.535937	0.0736167	0.0368083	+	0.999322i	$0.488281\pi$
$54$	0	0
$55$	6.70159	0.903643
$56$	0	0
$57$	−6.13426	−0.812502
$58$	0	0
$59$	−5.01340	−0.652689	−0.326345	−	0.945251i	$-0.605817\pi$
$59$	−5.01340	−0.652689	−0.326345	+	0.945251i	$0.605817\pi$
$60$	0	0
$61$	−9.32433	−1.19386	−0.596929	−	0.802294i	$-0.703613\pi$
$61$	−9.32433	−1.19386	−0.596929	+	0.802294i	$0.703613\pi$
$62$	0	0
$63$	2.12386	0.267581
$64$	0	0
$65$	12.6117	1.56429
$66$	0	0
$67$	−2.67612	−0.326941	−0.163470	−	0.986548i	$-0.552269\pi$
$67$	−2.67612	−0.326941	−0.163470	+	0.986548i	$0.552269\pi$
$68$	0	0
$69$	10.1667	1.22393
$70$	0	0
$71$	8.03479	0.953554	0.476777	−	0.879024i	$-0.341805\pi$
$71$	8.03479	0.953554	0.476777	+	0.879024i	$0.341805\pi$
$72$	0	0
$73$	9.48039	1.10960	0.554798	−	0.831985i	$-0.312796\pi$
$73$	9.48039	1.10960	0.554798	+	0.831985i	$0.312796\pi$
$74$	0	0
$75$	10.4348	1.20491
$76$	0	0
$77$	−2.78821	−0.317746
$78$	0	0
$79$	−16.4117	−1.84646	−0.923232	−	0.384243i	$-0.874462\pi$
$79$	−16.4117	−1.84646	−0.923232	+	0.384243i	$0.874462\pi$
$80$	0	0
$81$	−2.82762	−0.314180
$82$	0	0
$83$	2.71899	0.298448	0.149224	−	0.988803i	$-0.452322\pi$
$83$	2.71899	0.298448	0.149224	+	0.988803i	$0.452322\pi$
$84$	0	0
$85$	4.50940	0.489113
$86$	0	0
$87$	1.26407	0.135522
$88$	0	0
$89$	−3.19619	−0.338795	−0.169398	−	0.985548i	$-0.554182\pi$
$89$	−3.19619	−0.338795	−0.169398	+	0.985548i	$0.554182\pi$
$90$	0	0
$91$	−5.24713	−0.550048
$92$	0	0
$93$	−6.89539	−0.715019
$94$	0	0
$95$	−17.6677	−1.81267
$96$	0	0
$97$	5.68953	0.577684	0.288842	−	0.957377i	$-0.406730\pi$
$97$	5.68953	0.577684	0.288842	+	0.957377i	$0.406730\pi$
$98$	0	0
$99$	−2.58095	−0.259395
$100$	0	0
$101$	8.77746	0.873390	0.436695	−	0.899610i	$-0.356149\pi$
$101$	8.77746	0.873390	0.436695	+	0.899610i	$0.356149\pi$
$102$	0	0
$103$	−9.00533	−0.887322	−0.443661	−	0.896195i	$-0.646320\pi$
$103$	−9.00533	−0.887322	−0.443661	+	0.896195i	$0.646320\pi$
$104$	0	0
$105$	−6.97099	−0.680299
$106$	0	0
$107$	−12.6949	−1.22726	−0.613629	−	0.789594i	$-0.710291\pi$
$107$	−12.6949	−1.22726	−0.613629	+	0.789594i	$0.710291\pi$
$108$	0	0
$109$	−16.2744	−1.55881	−0.779405	−	0.626521i	$-0.784478\pi$
$109$	−16.2744	−1.55881	−0.779405	+	0.626521i	$0.784478\pi$
$110$	0	0
$111$	15.1152	1.43467
$112$	0	0
$113$	2.35334	0.221383	0.110692	−	0.993855i	$-0.464693\pi$
$113$	2.35334	0.221383	0.110692	+	0.993855i	$0.464693\pi$
$114$	0	0
$115$	29.2819	2.73055
$116$	0	0
$117$	−4.85708	−0.449037
$118$	0	0
$119$	−1.87614	−0.171986
$120$	0	0
$121$	−7.61172	−0.691975
$122$	0	0
$123$	3.80914	0.343459
$124$	0	0
$125$	11.8503	1.05992
$126$	0	0
$127$	−8.38626	−0.744160	−0.372080	−	0.928201i	$-0.621355\pi$
$127$	−8.38626	−0.744160	−0.372080	+	0.928201i	$0.621355\pi$
$128$	0	0
$129$	2.32680	0.204864
$130$	0	0
$131$	0.653865	0.0571284	0.0285642	−	0.999592i	$-0.490906\pi$
$131$	0.653865	0.0571284	0.0285642	+	0.999592i	$0.490906\pi$
$132$	0	0
$133$	7.35068	0.637385
$134$	0	0
$135$	−20.2592	−1.74363
$136$	0	0
$137$	8.31093	0.710050	0.355025	−	0.934857i	$-0.384472\pi$
$137$	8.31093	0.710050	0.355025	+	0.934857i	$0.384472\pi$
$138$	0	0
$139$	−14.6058	−1.23885	−0.619424	−	0.785057i	$-0.712634\pi$
$139$	−14.6058	−1.23885	−0.619424	+	0.785057i	$0.712634\pi$
$140$	0	0
$141$	14.3541	1.20883
$142$	0	0
$143$	6.37639	0.533221
$144$	0	0
$145$	3.64073	0.302346
$146$	0	0
$147$	−5.94818	−0.490598
$148$	0	0
$149$	7.36701	0.603529	0.301765	−	0.953382i	$-0.402424\pi$
$149$	7.36701	0.603529	0.301765	+	0.953382i	$0.402424\pi$
$150$	0	0
$151$	−13.0965	−1.06578	−0.532888	−	0.846186i	$-0.678893\pi$
$151$	−13.0965	−1.06578	−0.532888	+	0.846186i	$0.678893\pi$
$152$	0	0
$153$	−1.73668	−0.140402
$154$	0	0
$155$	−19.8599	−1.59519
$156$	0	0
$157$	−8.06967	−0.644030	−0.322015	−	0.946735i	$-0.604360\pi$
$157$	−8.06967	−0.644030	−0.322015	+	0.946735i	$0.604360\pi$
$158$	0	0
$159$	0.677461	0.0537261
$160$	0	0
$161$	−12.1828	−0.960138
$162$	0	0
$163$	5.96433	0.467162	0.233581	−	0.972337i	$-0.424956\pi$
$163$	5.96433	0.467162	0.233581	+	0.972337i	$0.424956\pi$
$164$	0	0
$165$	8.47127	0.659487
$166$	0	0
$167$	14.5281	1.12422	0.562110	−	0.827062i	$-0.309990\pi$
$167$	14.5281	1.12422	0.562110	+	0.827062i	$0.309990\pi$
$168$	0	0
$169$	−1.00027	−0.0769439
$170$	0	0
$171$	6.80427	0.520335
$172$	0	0
$173$	6.08573	0.462690	0.231345	−	0.972872i	$-0.425687\pi$
$173$	6.08573	0.462690	0.231345	+	0.972872i	$0.425687\pi$
$174$	0	0
$175$	−12.5040	−0.945214
$176$	0	0
$177$	−6.33728	−0.476339
$178$	0	0
$179$	−11.0804	−0.828188	−0.414094	−	0.910234i	$-0.635902\pi$
$179$	−11.0804	−0.828188	−0.414094	+	0.910234i	$0.635902\pi$
$180$	0	0
$181$	−7.37313	−0.548040	−0.274020	−	0.961724i	$-0.588353\pi$
$181$	−7.37313	−0.548040	−0.274020	+	0.961724i	$0.588353\pi$
$182$	0	0
$183$	−11.7866	−0.871289
$184$	0	0
$185$	43.5344	3.20071
$186$	0	0
$187$	2.27992	0.166724
$188$	0	0
$189$	8.42887	0.613110
$190$	0	0
$191$	−4.20347	−0.304153	−0.152076	−	0.988369i	$-0.548596\pi$
$191$	−4.20347	−0.304153	−0.152076	+	0.988369i	$0.548596\pi$
$192$	0	0
$193$	6.54687	0.471254	0.235627	−	0.971844i	$-0.424286\pi$
$193$	6.54687	0.471254	0.235627	+	0.971844i	$0.424286\pi$
$194$	0	0
$195$	15.9421	1.14164
$196$	0	0
$197$	2.50400	0.178403	0.0892013	−	0.996014i	$-0.471569\pi$
$197$	2.50400	0.178403	0.0892013	+	0.996014i	$0.471569\pi$
$198$	0	0
$199$	19.9071	1.41118	0.705589	−	0.708622i	$-0.250683\pi$
$199$	19.9071	1.41118	0.705589	+	0.708622i	$0.250683\pi$
$200$	0	0
$201$	−3.38280	−0.238604
$202$	0	0
$203$	−1.51473	−0.106313
$204$	0	0
$205$	10.9710	0.766247
$206$	0	0
$207$	−11.2772	−0.783818
$208$	0	0
$209$	−8.93267	−0.617886
$210$	0	0
$211$	19.7922	1.36255	0.681276	−	0.732027i	$-0.261426\pi$
$211$	19.7922	1.36255	0.681276	+	0.732027i	$0.261426\pi$
$212$	0	0
$213$	10.1565	0.695913
$214$	0	0
$215$	6.70159	0.457045
$216$	0	0
$217$	8.26275	0.560912
$218$	0	0
$219$	11.9839	0.809794
$220$	0	0
$221$	4.29058	0.288616
$222$	0	0
$223$	21.2738	1.42460	0.712301	−	0.701874i	$-0.247653\pi$
$223$	21.2738	1.42460	0.712301	+	0.701874i	$0.247653\pi$
$224$	0	0
$225$	−11.5745	−0.771634
$226$	0	0
$227$	−18.8815	−1.25321	−0.626604	−	0.779338i	$-0.715556\pi$
$227$	−18.8815	−1.25321	−0.626604	+	0.779338i	$0.715556\pi$
$228$	0	0
$229$	19.5495	1.29187	0.645934	−	0.763393i	$-0.276468\pi$
$229$	19.5495	1.29187	0.645934	+	0.763393i	$0.276468\pi$
$230$	0	0
$231$	−3.52448	−0.231894
$232$	0	0
$233$	3.25493	0.213237	0.106619	−	0.994300i	$-0.465998\pi$
$233$	3.25493	0.213237	0.106619	+	0.994300i	$0.465998\pi$
$234$	0	0
$235$	41.3422	2.69687
$236$	0	0
$237$	−20.7455	−1.34757
$238$	0	0
$239$	−22.2525	−1.43939	−0.719696	−	0.694290i	$-0.755719\pi$
$239$	−22.2525	−1.43939	−0.719696	+	0.694290i	$0.755719\pi$
$240$	0	0
$241$	0.326803	0.0210512	0.0105256	−	0.999945i	$-0.496650\pi$
$241$	0.326803	0.0210512	0.0105256	+	0.999945i	$0.496650\pi$
$242$	0	0
$243$	13.1195	0.841615
$244$	0	0
$245$	−17.1318	−1.09451
$246$	0	0
$247$	−16.8104	−1.06962
$248$	0	0
$249$	3.43699	0.217810
$250$	0	0
$251$	17.6458	1.11379	0.556896	−	0.830582i	$-0.311992\pi$
$251$	17.6458	1.11379	0.556896	+	0.830582i	$0.311992\pi$
$252$	0	0
$253$	14.8047	0.930765
$254$	0	0
$255$	5.70019	0.356960
$256$	0	0
$257$	−13.3035	−0.829847	−0.414923	−	0.909856i	$-0.636192\pi$
$257$	−13.3035	−0.829847	−0.414923	+	0.909856i	$0.636192\pi$
$258$	0	0
$259$	−18.1125	−1.12546
$260$	0	0
$261$	−1.40213	−0.0867899
$262$	0	0
$263$	0.770811	0.0475303	0.0237651	−	0.999718i	$-0.492435\pi$
$263$	0.770811	0.0475303	0.0237651	+	0.999718i	$0.492435\pi$
$264$	0	0
$265$	1.95120	0.119862
$266$	0	0
$267$	−4.04019	−0.247256
$268$	0	0
$269$	−11.1431	−0.679408	−0.339704	−	0.940532i	$-0.610327\pi$
$269$	−11.1431	−0.679408	−0.339704	+	0.940532i	$0.610327\pi$
$270$	0	0
$271$	2.40931	0.146355	0.0731777	−	0.997319i	$-0.476686\pi$
$271$	2.40931	0.146355	0.0731777	+	0.997319i	$0.476686\pi$
$272$	0	0
$273$	−6.63272	−0.401430
$274$	0	0
$275$	15.1951	0.916297
$276$	0	0
$277$	−26.6755	−1.60277	−0.801387	−	0.598147i	$-0.795904\pi$
$277$	−26.6755	−1.60277	−0.801387	+	0.598147i	$0.795904\pi$
$278$	0	0
$279$	7.64853	0.457906
$280$	0	0
$281$	25.8179	1.54016	0.770082	−	0.637945i	$-0.220215\pi$
$281$	25.8179	1.54016	0.770082	+	0.637945i	$0.220215\pi$
$282$	0	0
$283$	−14.5683	−0.865997	−0.432999	−	0.901395i	$-0.642545\pi$
$283$	−14.5683	−0.865997	−0.432999	+	0.901395i	$0.642545\pi$
$284$	0	0
$285$	−22.3332	−1.32290
$286$	0	0
$287$	−4.56450	−0.269434
$288$	0	0
$289$	−15.4659	−0.909757
$290$	0	0
$291$	7.19195	0.421599
$292$	0	0
$293$	−30.3542	−1.77331	−0.886657	−	0.462428i	$-0.846979\pi$
$293$	−30.3542	−1.77331	−0.886657	+	0.462428i	$0.846979\pi$
$294$	0	0
$295$	−18.2525	−1.06270
$296$	0	0
$297$	−10.2429	−0.594353
$298$	0	0
$299$	27.8610	1.61124
$300$	0	0
$301$	−2.78821	−0.160710
$302$	0	0
$303$	11.0953	0.637408
$304$	0	0
$305$	−33.9474	−1.94382
$306$	0	0
$307$	−30.7774	−1.75656	−0.878279	−	0.478149i	$-0.841308\pi$
$307$	−30.7774	−1.75656	−0.878279	+	0.478149i	$0.841308\pi$
$308$	0	0
$309$	−11.3833	−0.647576
$310$	0	0
$311$	−12.2035	−0.691996	−0.345998	−	0.938235i	$-0.612460\pi$
$311$	−12.2035	−0.691996	−0.345998	+	0.938235i	$0.612460\pi$
$312$	0	0
$313$	−7.83426	−0.442819	−0.221409	−	0.975181i	$-0.571066\pi$
$313$	−7.83426	−0.442819	−0.221409	+	0.975181i	$0.571066\pi$
$314$	0	0
$315$	7.73239	0.435671
$316$	0	0
$317$	−34.2100	−1.92143	−0.960714	−	0.277541i	$-0.910480\pi$
$317$	−34.2100	−1.92143	−0.960714	+	0.277541i	$0.910480\pi$
$318$	0	0
$319$	1.84073	0.103061
$320$	0	0
$321$	−16.0472	−0.895665
$322$	0	0
$323$	−6.01066	−0.334442
$324$	0	0
$325$	28.5956	1.58620
$326$	0	0
$327$	−20.5720	−1.13763
$328$	0	0
$329$	−17.2005	−0.948293
$330$	0	0
$331$	−27.6564	−1.52014	−0.760068	−	0.649844i	$-0.774834\pi$
$331$	−27.6564	−1.52014	−0.760068	+	0.649844i	$0.774834\pi$
$332$	0	0
$333$	−16.7661	−0.918779
$334$	0	0
$335$	−9.74305	−0.532320
$336$	0	0
$337$	12.8004	0.697284	0.348642	−	0.937256i	$-0.386643\pi$
$337$	12.8004	0.697284	0.348642	+	0.937256i	$0.386643\pi$
$338$	0	0
$339$	2.97478	0.161568
$340$	0	0
$341$	−10.0410	−0.543752
$342$	0	0
$343$	17.7308	0.957375
$344$	0	0
$345$	37.0143	1.99278
$346$	0	0
$347$	32.2063	1.72892	0.864461	−	0.502699i	$-0.167660\pi$
$347$	32.2063	1.72892	0.864461	+	0.502699i	$0.167660\pi$
$348$	0	0
$349$	−34.2955	−1.83580	−0.917898	−	0.396816i	$-0.870115\pi$
$349$	−34.2955	−1.83580	−0.917898	+	0.396816i	$0.870115\pi$
$350$	0	0
$351$	−19.2761	−1.02888
$352$	0	0
$353$	−29.9887	−1.59614	−0.798068	−	0.602567i	$-0.794145\pi$
$353$	−29.9887	−1.59614	−0.798068	+	0.602567i	$0.794145\pi$
$354$	0	0
$355$	29.2525	1.55256
$356$	0	0
$357$	−2.37157	−0.125517
$358$	0	0
$359$	35.9425	1.89697	0.948487	−	0.316817i	$-0.102614\pi$
$359$	35.9425	1.89697	0.948487	+	0.316817i	$0.102614\pi$
$360$	0	0
$361$	4.54961	0.239453
$362$	0	0
$363$	−9.62173	−0.505010
$364$	0	0
$365$	34.5156	1.80663
$366$	0	0
$367$	32.3014	1.68612	0.843061	−	0.537819i	$-0.180751\pi$
$367$	32.3014	1.68612	0.843061	+	0.537819i	$0.180751\pi$
$368$	0	0
$369$	−4.22519	−0.219955
$370$	0	0
$371$	−0.811801	−0.0421466
$372$	0	0
$373$	16.6814	0.863730	0.431865	−	0.901938i	$-0.357856\pi$
$373$	16.6814	0.863730	0.431865	+	0.901938i	$0.357856\pi$
$374$	0	0
$375$	14.9796	0.773543
$376$	0	0
$377$	3.46406	0.178408
$378$	0	0
$379$	−20.2157	−1.03841	−0.519205	−	0.854649i	$-0.673772\pi$
$379$	−20.2157	−1.03841	−0.519205	+	0.854649i	$0.673772\pi$
$380$	0	0
$381$	−10.6008	−0.543095
$382$	0	0
$383$	27.8262	1.42185	0.710926	−	0.703267i	$-0.248276\pi$
$383$	27.8262	1.42185	0.710926	+	0.703267i	$0.248276\pi$
$384$	0	0
$385$	−10.1511	−0.517349
$386$	0	0
$387$	−2.58095	−0.131197
$388$	0	0
$389$	5.32973	0.270228	0.135114	−	0.990830i	$-0.456860\pi$
$389$	5.32973	0.270228	0.135114	+	0.990830i	$0.456860\pi$
$390$	0	0
$391$	9.96188	0.503794
$392$	0	0
$393$	0.826529	0.0416929
$394$	0	0
$395$	−59.7507	−3.00638
$396$	0	0
$397$	20.1238	1.00998	0.504992	−	0.863124i	$-0.331495\pi$
$397$	20.1238	1.00998	0.504992	+	0.863124i	$0.331495\pi$
$398$	0	0
$399$	9.29175	0.465170
$400$	0	0
$401$	−36.6489	−1.83016	−0.915080	−	0.403272i	$-0.867873\pi$
$401$	−36.6489	−1.83016	−0.915080	+	0.403272i	$0.867873\pi$
$402$	0	0
$403$	−18.8962	−0.941287
$404$	0	0
$405$	−10.2946	−0.511543
$406$	0	0
$407$	22.0107	1.09103
$408$	0	0
$409$	−24.4781	−1.21036	−0.605182	−	0.796087i	$-0.706900\pi$
$409$	−24.4781	−1.21036	−0.605182	+	0.796087i	$0.706900\pi$
$410$	0	0
$411$	10.5056	0.518202
$412$	0	0
$413$	7.59396	0.373674
$414$	0	0
$415$	9.89912	0.485929
$416$	0	0
$417$	−18.4627	−0.904123
$418$	0	0
$419$	−24.6407	−1.20378	−0.601888	−	0.798581i	$-0.705585\pi$
$419$	−24.6407	−1.20378	−0.601888	+	0.798581i	$0.705585\pi$
$420$	0	0
$421$	−7.16301	−0.349104	−0.174552	−	0.984648i	$-0.555848\pi$
$421$	−7.16301	−0.349104	−0.174552	+	0.984648i	$0.555848\pi$
$422$	0	0
$423$	−15.9219	−0.774148
$424$	0	0
$425$	10.2245	0.495963
$426$	0	0
$427$	14.1239	0.683501
$428$	0	0
$429$	8.06019	0.389150
$430$	0	0
$431$	14.6707	0.706664	0.353332	−	0.935498i	$-0.385049\pi$
$431$	14.6707	0.706664	0.353332	+	0.935498i	$0.385049\pi$
$432$	0	0
$433$	−22.8540	−1.09829	−0.549146	−	0.835727i	$-0.685047\pi$
$433$	−22.8540	−1.09829	−0.549146	+	0.835727i	$0.685047\pi$
$434$	0	0
$435$	4.60213	0.220655
$436$	0	0
$437$	−39.0304	−1.86708
$438$	0	0
$439$	−2.24023	−0.106920	−0.0534601	−	0.998570i	$-0.517025\pi$
$439$	−2.24023	−0.106920	−0.0534601	+	0.998570i	$0.517025\pi$
$440$	0	0
$441$	6.59787	0.314184
$442$	0	0
$443$	−6.57453	−0.312365	−0.156183	−	0.987728i	$-0.549919\pi$
$443$	−6.57453	−0.312365	−0.156183	+	0.987728i	$0.549919\pi$
$444$	0	0
$445$	−11.6365	−0.551621
$446$	0	0
$447$	9.31240	0.440461
$448$	0	0
$449$	35.7226	1.68586	0.842928	−	0.538027i	$-0.180830\pi$
$449$	35.7226	1.68586	0.842928	+	0.538027i	$0.180830\pi$
$450$	0	0
$451$	5.54685	0.261191
$452$	0	0
$453$	−16.5548	−0.777813
$454$	0	0
$455$	−19.1034	−0.895580
$456$	0	0
$457$	−34.9431	−1.63457	−0.817284	−	0.576235i	$-0.804521\pi$
$457$	−34.9431	−1.63457	−0.817284	+	0.576235i	$0.804521\pi$
$458$	0	0
$459$	−6.89229	−0.321705
$460$	0	0
$461$	13.7898	0.642253	0.321126	−	0.947036i	$-0.395939\pi$
$461$	13.7898	0.642253	0.321126	+	0.947036i	$0.395939\pi$
$462$	0	0
$463$	−14.6333	−0.680065	−0.340033	−	0.940414i	$-0.610438\pi$
$463$	−14.6333	−0.680065	−0.340033	+	0.940414i	$0.610438\pi$
$464$	0	0
$465$	−25.1043	−1.16418
$466$	0	0
$467$	−14.5248	−0.672129	−0.336065	−	0.941839i	$-0.609096\pi$
$467$	−14.5248	−0.672129	−0.336065	+	0.941839i	$0.609096\pi$
$468$	0	0
$469$	4.05361	0.187178
$470$	0	0
$471$	−10.2006	−0.470019
$472$	0	0
$473$	3.38828	0.155793
$474$	0	0
$475$	−40.0595	−1.83805
$476$	0	0
$477$	−0.751456	−0.0344068
$478$	0	0
$479$	9.18641	0.419738	0.209869	−	0.977730i	$-0.432696\pi$
$479$	9.18641	0.419738	0.209869	+	0.977730i	$0.432696\pi$
$480$	0	0
$481$	41.4218	1.88867
$482$	0	0
$483$	−15.3999	−0.700718
$484$	0	0
$485$	20.7140	0.940576
$486$	0	0
$487$	15.9431	0.722453	0.361226	−	0.932478i	$-0.382358\pi$
$487$	15.9431	0.722453	0.361226	+	0.932478i	$0.382358\pi$
$488$	0	0
$489$	7.53931	0.340939
$490$	0	0
$491$	−8.83807	−0.398856	−0.199428	−	0.979912i	$-0.563908\pi$
$491$	−8.83807	−0.398856	−0.199428	+	0.979912i	$0.563908\pi$
$492$	0	0
$493$	1.23860	0.0557836
$494$	0	0
$495$	−9.39653	−0.422343
$496$	0	0
$497$	−12.1706	−0.545924
$498$	0	0
$499$	14.8047	0.662750	0.331375	−	0.943499i	$-0.392487\pi$
$499$	14.8047	0.662750	0.331375	+	0.943499i	$0.392487\pi$
$500$	0	0
$501$	18.3645	0.820467
$502$	0	0
$503$	−2.86720	−0.127842	−0.0639211	−	0.997955i	$-0.520361\pi$
$503$	−2.86720	−0.127842	−0.0639211	+	0.997955i	$0.520361\pi$
$504$	0	0
$505$	31.9564	1.42204
$506$	0	0
$507$	−1.26441	−0.0561544
$508$	0	0
$509$	−0.589008	−0.0261073	−0.0130537	−	0.999915i	$-0.504155\pi$
$509$	−0.589008	−0.0261073	−0.0130537	+	0.999915i	$0.504155\pi$
$510$	0	0
$511$	−14.3603	−0.635260
$512$	0	0
$513$	27.0038	1.19225
$514$	0	0
$515$	−32.7860	−1.44472
$516$	0	0
$517$	20.9023	0.919282
$518$	0	0
$519$	7.69277	0.337675
$520$	0	0
$521$	−30.5147	−1.33687	−0.668436	−	0.743770i	$-0.733036\pi$
$521$	−30.5147	−1.33687	−0.668436	+	0.743770i	$0.733036\pi$
$522$	0	0
$523$	42.4582	1.85657	0.928284	−	0.371872i	$-0.121284\pi$
$523$	42.4582	1.85657	0.928284	+	0.371872i	$0.121284\pi$
$524$	0	0
$525$	−15.8059	−0.689826
$526$	0	0
$527$	−6.75646	−0.294316
$528$	0	0
$529$	41.6877	1.81251
$530$	0	0
$531$	7.02946	0.305053
$532$	0	0
$533$	10.4386	0.452147
$534$	0	0
$535$	−46.2186	−1.99820
$536$	0	0
$537$	−14.0064	−0.604420
$538$	0	0
$539$	−8.66170	−0.373086
$540$	0	0
$541$	−7.60425	−0.326932	−0.163466	−	0.986549i	$-0.552267\pi$
$541$	−7.60425	−0.326932	−0.163466	+	0.986549i	$0.552267\pi$
$542$	0	0
$543$	−9.32013	−0.399965
$544$	0	0
$545$	−59.2509	−2.53803
$546$	0	0
$547$	−19.2901	−0.824784	−0.412392	−	0.911006i	$-0.635307\pi$
$547$	−19.2901	−0.824784	−0.412392	+	0.911006i	$0.635307\pi$
$548$	0	0
$549$	13.0740	0.557983
$550$	0	0
$551$	−4.85279	−0.206736
$552$	0	0
$553$	24.8594	1.05713
$554$	0	0
$555$	55.0304	2.33591
$556$	0	0
$557$	−34.2685	−1.45200	−0.726002	−	0.687693i	$-0.758624\pi$
$557$	−34.2685	−1.45200	−0.726002	+	0.687693i	$0.758624\pi$
$558$	0	0
$559$	6.37639	0.269693
$560$	0	0
$561$	2.88197	0.121677
$562$	0	0
$563$	−5.52793	−0.232974	−0.116487	−	0.993192i	$-0.537163\pi$
$563$	−5.52793	−0.232974	−0.116487	+	0.993192i	$0.537163\pi$
$564$	0	0
$565$	8.56787	0.360453
$566$	0	0
$567$	4.28308	0.179873
$568$	0	0
$569$	0.760948	0.0319006	0.0159503	−	0.999873i	$-0.494923\pi$
$569$	0.760948	0.0319006	0.0159503	+	0.999873i	$0.494923\pi$
$570$	0	0
$571$	27.5817	1.15426	0.577130	−	0.816652i	$-0.304173\pi$
$571$	27.5817	1.15426	0.577130	+	0.816652i	$0.304173\pi$
$572$	0	0
$573$	−5.31347	−0.221974
$574$	0	0
$575$	66.3933	2.76879
$576$	0	0
$577$	36.1066	1.50314	0.751569	−	0.659655i	$-0.229298\pi$
$577$	36.1066	1.50314	0.751569	+	0.659655i	$0.229298\pi$
$578$	0	0
$579$	8.27568	0.343926
$580$	0	0
$581$	−4.11854	−0.170866
$582$	0	0
$583$	0.986515	0.0408573
$584$	0	0
$585$	−17.6833	−0.731116
$586$	0	0
$587$	−2.56833	−0.106006	−0.0530031	−	0.998594i	$-0.516879\pi$
$587$	−2.56833	−0.106006	−0.0530031	+	0.998594i	$0.516879\pi$
$588$	0	0
$589$	26.4716	1.09074
$590$	0	0
$591$	3.16523	0.130200
$592$	0	0
$593$	−4.75547	−0.195284	−0.0976419	−	0.995222i	$-0.531130\pi$
$593$	−4.75547	−0.195284	−0.0976419	+	0.995222i	$0.531130\pi$
$594$	0	0
$595$	−6.83053	−0.280025
$596$	0	0
$597$	25.1639	1.02989
$598$	0	0
$599$	−29.0701	−1.18777	−0.593887	−	0.804549i	$-0.702407\pi$
$599$	−29.0701	−1.18777	−0.593887	+	0.804549i	$0.702407\pi$
$600$	0	0
$601$	−5.22299	−0.213050	−0.106525	−	0.994310i	$-0.533972\pi$
$601$	−5.22299	−0.213050	−0.106525	+	0.994310i	$0.533972\pi$
$602$	0	0
$603$	3.75229	0.152805
$604$	0	0
$605$	−27.7122	−1.12666
$606$	0	0
$607$	5.30426	0.215293	0.107647	−	0.994189i	$-0.465668\pi$
$607$	5.30426	0.215293	0.107647	+	0.994189i	$0.465668\pi$
$608$	0	0
$609$	−1.91472	−0.0775885
$610$	0	0
$611$	39.3360	1.59137
$612$	0	0
$613$	36.0920	1.45774	0.728872	−	0.684650i	$-0.240045\pi$
$613$	36.0920	1.45774	0.728872	+	0.684650i	$0.240045\pi$
$614$	0	0
$615$	13.8681	0.559215
$616$	0	0
$617$	12.5586	0.505592	0.252796	−	0.967520i	$-0.418650\pi$
$617$	12.5586	0.505592	0.252796	+	0.967520i	$0.418650\pi$
$618$	0	0
$619$	−22.5567	−0.906632	−0.453316	−	0.891350i	$-0.649759\pi$
$619$	−22.5567	−0.906632	−0.453316	+	0.891350i	$0.649759\pi$
$620$	0	0
$621$	−44.7553	−1.79597
$622$	0	0
$623$	4.84136	0.193965
$624$	0	0
$625$	1.86920	0.0747680
$626$	0	0
$627$	−11.2915	−0.450939
$628$	0	0
$629$	14.8106	0.590539
$630$	0	0
$631$	9.70826	0.386480	0.193240	−	0.981152i	$-0.438100\pi$
$631$	9.70826	0.386480	0.193240	+	0.981152i	$0.438100\pi$
$632$	0	0
$633$	25.0187	0.994403
$634$	0	0
$635$	−30.5321	−1.21163
$636$	0	0
$637$	−16.3005	−0.645847
$638$	0	0
$639$	−11.2659	−0.445670
$640$	0	0
$641$	−14.6230	−0.577573	−0.288786	−	0.957394i	$-0.593252\pi$
$641$	−14.6230	−0.577573	−0.288786	+	0.957394i	$0.593252\pi$
$642$	0	0
$643$	−37.1998	−1.46702	−0.733508	−	0.679681i	$-0.762118\pi$
$643$	−37.1998	−1.46702	−0.733508	+	0.679681i	$0.762118\pi$
$644$	0	0
$645$	8.47127	0.333556
$646$	0	0
$647$	45.4818	1.78807	0.894036	−	0.447995i	$-0.147862\pi$
$647$	45.4818	1.78807	0.894036	+	0.447995i	$0.147862\pi$
$648$	0	0
$649$	−9.22831	−0.362243
$650$	0	0
$651$	10.4447	0.409359
$652$	0	0
$653$	−28.5484	−1.11719	−0.558593	−	0.829442i	$-0.688659\pi$
$653$	−28.5484	−1.11719	−0.558593	+	0.829442i	$0.688659\pi$
$654$	0	0
$655$	2.38055	0.0930156
$656$	0	0
$657$	−13.2928	−0.518601
$658$	0	0
$659$	12.9306	0.503705	0.251853	−	0.967766i	$-0.418960\pi$
$659$	12.9306	0.503705	0.251853	+	0.967766i	$0.418960\pi$
$660$	0	0
$661$	−6.99891	−0.272226	−0.136113	−	0.990693i	$-0.543461\pi$
$661$	−6.99891	−0.272226	−0.136113	+	0.990693i	$0.543461\pi$
$662$	0	0
$663$	5.42358	0.210634
$664$	0	0
$665$	26.7619	1.03778
$666$	0	0
$667$	8.04287	0.311421
$668$	0	0
$669$	26.8916	1.03969
$670$	0	0
$671$	−17.1635	−0.662591
$672$	0	0
$673$	−3.87678	−0.149439	−0.0747195	−	0.997205i	$-0.523806\pi$
$673$	−3.87678	−0.149439	−0.0747195	+	0.997205i	$0.523806\pi$
$674$	0	0
$675$	−45.9353	−1.76805
$676$	0	0
$677$	−34.8046	−1.33765	−0.668825	−	0.743419i	$-0.733203\pi$
$677$	−34.8046	−1.33765	−0.668825	+	0.743419i	$0.733203\pi$
$678$	0	0
$679$	−8.61811	−0.330733
$680$	0	0
$681$	−23.8675	−0.914603
$682$	0	0
$683$	−22.3640	−0.855735	−0.427867	−	0.903842i	$-0.640735\pi$
$683$	−22.3640	−0.855735	−0.427867	+	0.903842i	$0.640735\pi$
$684$	0	0
$685$	30.2579	1.15609
$686$	0	0
$687$	24.7119	0.942818
$688$	0	0
$689$	1.85652	0.0707278
$690$	0	0
$691$	20.4079	0.776352	0.388176	−	0.921585i	$-0.373105\pi$
$691$	20.4079	0.776352	0.388176	+	0.921585i	$0.373105\pi$
$692$	0	0
$693$	3.90944	0.148507
$694$	0	0
$695$	−53.1758	−2.01707
$696$	0	0
$697$	3.73239	0.141374
$698$	0	0
$699$	4.11445	0.155623
$700$	0	0
$701$	−16.0533	−0.606323	−0.303162	−	0.952939i	$-0.598042\pi$
$701$	−16.0533	−0.606323	−0.303162	+	0.952939i	$0.598042\pi$
$702$	0	0
$703$	−58.0277	−2.18856
$704$	0	0
$705$	52.2593	1.96820
$706$	0	0
$707$	−13.2955	−0.500029
$708$	0	0
$709$	12.5478	0.471242	0.235621	−	0.971845i	$-0.424288\pi$
$709$	12.5478	0.471242	0.235621	+	0.971845i	$0.424288\pi$
$710$	0	0
$711$	23.0114	0.862997
$712$	0	0
$713$	−43.8732	−1.64307
$714$	0	0
$715$	23.2147	0.868182
$716$	0	0
$717$	−28.1286	−1.05048
$718$	0	0
$719$	−37.6717	−1.40492	−0.702459	−	0.711724i	$-0.747914\pi$
$719$	−37.6717	−1.40492	−0.702459	+	0.711724i	$0.747914\pi$
$720$	0	0
$721$	13.6407	0.508005
$722$	0	0
$723$	0.413101	0.0153634
$724$	0	0
$725$	8.25493	0.306580
$726$	0	0
$727$	32.0216	1.18762	0.593808	−	0.804607i	$-0.297624\pi$
$727$	32.0216	1.18762	0.593808	+	0.804607i	$0.297624\pi$
$728$	0	0
$729$	25.0668	0.928398
$730$	0	0
$731$	2.27992	0.0843259
$732$	0	0
$733$	26.8043	0.990038	0.495019	−	0.868882i	$-0.335161\pi$
$733$	26.8043	0.990038	0.495019	+	0.868882i	$0.335161\pi$
$734$	0	0
$735$	−21.6557	−0.798784
$736$	0	0
$737$	−4.92602	−0.181452
$738$	0	0
$739$	−10.0452	−0.369520	−0.184760	−	0.982784i	$-0.559151\pi$
$739$	−10.0452	−0.369520	−0.184760	+	0.982784i	$0.559151\pi$
$740$	0	0
$741$	−21.2495	−0.780618
$742$	0	0
$743$	−14.9975	−0.550203	−0.275102	−	0.961415i	$-0.588711\pi$
$743$	−14.9975	−0.550203	−0.275102	+	0.961415i	$0.588711\pi$
$744$	0	0
$745$	26.8213	0.982657
$746$	0	0
$747$	−3.81239	−0.139488
$748$	0	0
$749$	19.2293	0.702623
$750$	0	0
$751$	4.28921	0.156515	0.0782577	−	0.996933i	$-0.475064\pi$
$751$	4.28921	0.156515	0.0782577	+	0.996933i	$0.475064\pi$
$752$	0	0
$753$	22.3055	0.812856
$754$	0	0
$755$	−47.6807	−1.73528
$756$	0	0
$757$	9.72594	0.353495	0.176748	−	0.984256i	$-0.443442\pi$
$757$	9.72594	0.353495	0.176748	+	0.984256i	$0.443442\pi$
$758$	0	0
$759$	18.7142	0.679281
$760$	0	0
$761$	3.42575	0.124183	0.0620917	−	0.998070i	$-0.480223\pi$
$761$	3.42575	0.124183	0.0620917	+	0.998070i	$0.480223\pi$
$762$	0	0
$763$	24.6514	0.892441
$764$	0	0
$765$	−6.32279	−0.228601
$766$	0	0
$767$	−17.3667	−0.627077
$768$	0	0
$769$	41.6738	1.50279	0.751397	−	0.659850i	$-0.229380\pi$
$769$	41.6738	1.50279	0.751397	+	0.659850i	$0.229380\pi$
$770$	0	0
$771$	−16.8165	−0.605630
$772$	0	0
$773$	−5.29167	−0.190328	−0.0951640	−	0.995462i	$-0.530338\pi$
$773$	−5.29167	−0.190328	−0.0951640	+	0.995462i	$0.530338\pi$
$774$	0	0
$775$	−45.0300	−1.61753
$776$	0	0
$777$	−22.8955	−0.821370
$778$	0	0
$779$	−14.6234	−0.523938
$780$	0	0
$781$	14.7899	0.529223
$782$	0	0
$783$	−5.56459	−0.198862
$784$	0	0
$785$	−29.3795	−1.04860
$786$	0	0
$787$	32.4739	1.15757	0.578784	−	0.815481i	$-0.303527\pi$
$787$	32.4739	1.15757	0.578784	+	0.815481i	$0.303527\pi$
$788$	0	0
$789$	0.974357	0.0346880
$790$	0	0
$791$	−3.56468	−0.126745
$792$	0	0
$793$	−32.3001	−1.14701
$794$	0	0
$795$	2.46645	0.0874761
$796$	0	0
$797$	1.72128	0.0609709	0.0304854	−	0.999535i	$-0.490295\pi$
$797$	1.72128	0.0609709	0.0304854	+	0.999535i	$0.490295\pi$
$798$	0	0
$799$	14.0648	0.497578
$800$	0	0
$801$	4.48148	0.158345
$802$	0	0
$803$	17.4508	0.615826
$804$	0	0
$805$	−44.3542	−1.56328
$806$	0	0
$807$	−14.0856	−0.495838
$808$	0	0
$809$	−16.2910	−0.572762	−0.286381	−	0.958116i	$-0.592452\pi$
$809$	−16.2910	−0.572762	−0.286381	+	0.958116i	$0.592452\pi$
$810$	0	0
$811$	5.94915	0.208903	0.104451	−	0.994530i	$-0.466691\pi$
$811$	5.94915	0.208903	0.104451	+	0.994530i	$0.466691\pi$
$812$	0	0
$813$	3.04554	0.106812
$814$	0	0
$815$	21.7145	0.760626
$816$	0	0
$817$	−8.93267	−0.312515
$818$	0	0
$819$	7.35717	0.257080
$820$	0	0
$821$	40.1196	1.40018	0.700092	−	0.714052i	$-0.253142\pi$
$821$	40.1196	1.40018	0.700092	+	0.714052i	$0.253142\pi$
$822$	0	0
$823$	43.4977	1.51623	0.758117	−	0.652118i	$-0.226119\pi$
$823$	43.4977	1.51623	0.758117	+	0.652118i	$0.226119\pi$
$824$	0	0
$825$	19.2076	0.668722
$826$	0	0
$827$	2.00179	0.0696091	0.0348045	−	0.999394i	$-0.488919\pi$
$827$	2.00179	0.0696091	0.0348045	+	0.999394i	$0.488919\pi$
$828$	0	0
$829$	40.4883	1.40622	0.703109	−	0.711083i	$-0.251795\pi$
$829$	40.4883	1.40622	0.703109	+	0.711083i	$0.251795\pi$
$830$	0	0
$831$	−33.7196	−1.16972
$832$	0	0
$833$	−5.82833	−0.201940
$834$	0	0
$835$	52.8930	1.83044
$836$	0	0
$837$	30.3544	1.04920
$838$	0	0
$839$	6.31534	0.218030	0.109015	−	0.994040i	$-0.465230\pi$
$839$	6.31534	0.218030	0.109015	+	0.994040i	$0.465230\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	32.6355	1.12403
$844$	0	0
$845$	−3.64172	−0.125279
$846$	0	0
$847$	11.5297	0.396166
$848$	0	0
$849$	−18.4153	−0.632013
$850$	0	0
$851$	96.1733	3.29678
$852$	0	0
$853$	0.502432	0.0172029	0.00860147	−	0.999963i	$-0.497262\pi$
$853$	0.502432	0.0172029	0.00860147	+	0.999963i	$0.497262\pi$
$854$	0	0
$855$	24.7725	0.847202
$856$	0	0
$857$	−12.4353	−0.424782	−0.212391	−	0.977185i	$-0.568125\pi$
$857$	−12.4353	−0.424782	−0.212391	+	0.977185i	$0.568125\pi$
$858$	0	0
$859$	16.0193	0.546570	0.273285	−	0.961933i	$-0.411890\pi$
$859$	16.0193	0.546570	0.273285	+	0.961933i	$0.411890\pi$
$860$	0	0
$861$	−5.76983	−0.196635
$862$	0	0
$863$	40.9575	1.39421	0.697104	−	0.716970i	$-0.254472\pi$
$863$	40.9575	1.39421	0.697104	+	0.716970i	$0.254472\pi$
$864$	0	0
$865$	22.1565	0.753344
$866$	0	0
$867$	−19.5499	−0.663950
$868$	0	0
$869$	−30.2095	−1.02479
$870$	0	0
$871$	−9.27026	−0.314111
$872$	0	0
$873$	−7.97748	−0.269997
$874$	0	0
$875$	−17.9501	−0.606823
$876$	0	0
$877$	−28.6865	−0.968673	−0.484336	−	0.874882i	$-0.660939\pi$
$877$	−28.6865	−0.968673	−0.484336	+	0.874882i	$0.660939\pi$
$878$	0	0
$879$	−38.3698	−1.29418
$880$	0	0
$881$	40.3075	1.35800	0.678998	−	0.734140i	$-0.262415\pi$
$881$	40.3075	1.35800	0.678998	+	0.734140i	$0.262415\pi$
$882$	0	0
$883$	50.2872	1.69230	0.846150	−	0.532944i	$-0.178914\pi$
$883$	50.2872	1.69230	0.846150	+	0.532944i	$0.178914\pi$
$884$	0	0
$885$	−23.0723	−0.775568
$886$	0	0
$887$	41.0714	1.37904	0.689522	−	0.724265i	$-0.257821\pi$
$887$	41.0714	1.37904	0.689522	+	0.724265i	$0.257821\pi$
$888$	0	0
$889$	12.7029	0.426043
$890$	0	0
$891$	−5.20487	−0.174370
$892$	0	0
$893$	−55.1057	−1.84404
$894$	0	0
$895$	−40.3408	−1.34844
$896$	0	0
$897$	35.2182	1.17590
$898$	0	0
$899$	−5.45492	−0.181932
$900$	0	0
$901$	0.663811	0.0221147
$902$	0	0
$903$	−3.52448	−0.117287
$904$	0	0
$905$	−26.8436	−0.892311
$906$	0	0
$907$	26.7582	0.888492	0.444246	−	0.895905i	$-0.353472\pi$
$907$	26.7582	0.888492	0.444246	+	0.895905i	$0.353472\pi$
$908$	0	0
$909$	−12.3072	−0.408203
$910$	0	0
$911$	−35.7245	−1.18361	−0.591803	−	0.806082i	$-0.701584\pi$
$911$	−35.7245	−1.18361	−0.591803	+	0.806082i	$0.701584\pi$
$912$	0	0
$913$	5.00492	0.165639
$914$	0	0
$915$	−42.9118	−1.41862
$916$	0	0
$917$	−0.990429	−0.0327069
$918$	0	0
$919$	−52.9901	−1.74798	−0.873991	−	0.485942i	$-0.838477\pi$
$919$	−52.9901	−1.74798	−0.873991	+	0.485942i	$0.838477\pi$
$920$	0	0
$921$	−38.9047	−1.28195
$922$	0	0
$923$	27.8330	0.916135
$924$	0	0
$925$	98.7090	3.24553
$926$	0	0
$927$	12.6267	0.414715
$928$	0	0
$929$	−45.3477	−1.48781	−0.743904	−	0.668286i	$-0.767028\pi$
$929$	−45.3477	−1.48781	−0.743904	+	0.668286i	$0.767028\pi$
$930$	0	0
$931$	22.8353	0.748395
$932$	0	0
$933$	−15.4260	−0.505025
$934$	0	0
$935$	8.30058	0.271458
$936$	0	0
$937$	−7.30881	−0.238768	−0.119384	−	0.992848i	$-0.538092\pi$
$937$	−7.30881	−0.238768	−0.119384	+	0.992848i	$0.538092\pi$
$938$	0	0
$939$	−9.90303	−0.323173
$940$	0	0
$941$	43.3082	1.41181	0.705903	−	0.708308i	$-0.250541\pi$
$941$	43.3082	1.41181	0.705903	+	0.708308i	$0.250541\pi$
$942$	0	0
$943$	24.2364	0.789246
$944$	0	0
$945$	30.6872	0.998256
$946$	0	0
$947$	46.2761	1.50377	0.751885	−	0.659295i	$-0.229145\pi$
$947$	46.2761	1.50377	0.751885	+	0.659295i	$0.229145\pi$
$948$	0	0
$949$	32.8407	1.06605
$950$	0	0
$951$	−43.2438	−1.40228
$952$	0	0
$953$	16.9809	0.550066	0.275033	−	0.961435i	$-0.411311\pi$
$953$	16.9809	0.550066	0.275033	+	0.961435i	$0.411311\pi$
$954$	0	0
$955$	−15.3037	−0.495217
$956$	0	0
$957$	2.32680	0.0752149
$958$	0	0
$959$	−12.5888	−0.406514
$960$	0	0
$961$	−1.24381	−0.0401228
$962$	0	0
$963$	17.7999	0.573594
$964$	0	0
$965$	23.8354	0.767288
$966$	0	0
$967$	−43.8829	−1.41118	−0.705590	−	0.708620i	$-0.749318\pi$
$967$	−43.8829	−1.41118	−0.705590	+	0.708620i	$0.749318\pi$
$968$	0	0
$969$	−7.59788	−0.244079
$970$	0	0
$971$	20.5369	0.659061	0.329531	−	0.944145i	$-0.393110\pi$
$971$	20.5369	0.659061	0.329531	+	0.944145i	$0.393110\pi$
$972$	0	0
$973$	22.1239	0.709258
$974$	0	0
$975$	36.1467	1.15762
$976$	0	0
$977$	−9.55117	−0.305569	−0.152784	−	0.988260i	$-0.548824\pi$
$977$	−9.55117	−0.305569	−0.152784	+	0.988260i	$0.548824\pi$
$978$	0	0
$979$	−5.88331	−0.188031
$980$	0	0
$981$	22.8190	0.728553
$982$	0	0
$983$	28.3147	0.903098	0.451549	−	0.892246i	$-0.350872\pi$
$983$	28.3147	0.903098	0.451549	+	0.892246i	$0.350872\pi$
$984$	0	0
$985$	9.11640	0.290473
$986$	0	0
$987$	−21.7426	−0.692073
$988$	0	0
$989$	14.8047	0.470763
$990$	0	0
$991$	23.2632	0.738979	0.369490	−	0.929235i	$-0.379532\pi$
$991$	23.2632	0.738979	0.369490	+	0.929235i	$0.379532\pi$
$992$	0	0
$993$	−34.9596	−1.10941
$994$	0	0
$995$	72.4764	2.29766
$996$	0	0
$997$	−7.99414	−0.253177	−0.126589	−	0.991955i	$-0.540403\pi$
$997$	−7.99414	−0.253177	−0.126589	+	0.991955i	$0.540403\pi$
$998$	0	0
$999$	−66.5391	−2.10520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.ba.1.4		5
4.3	odd	2		1856.2.a.bb.1.2		5
8.3	odd	2		928.2.a.g.1.4	✓	5
8.5	even	2		928.2.a.h.1.2	yes	5
24.5	odd	2		8352.2.a.bh.1.5		5
24.11	even	2		8352.2.a.bg.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.g.1.4	✓	5	8.3	odd	2
928.2.a.h.1.2	yes	5	8.5	even	2
1856.2.a.ba.1.4		5	1.1	even	1	trivial
1856.2.a.bb.1.2		5	4.3	odd	2
8352.2.a.bg.1.5		5	24.11	even	2
8352.2.a.bh.1.5		5	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26407 q^{3} +3.64073 q^{5} -1.51473 q^{7} -1.40213 q^{9} +O(q^{10})\)	\(q+1.26407 q^{3} +3.64073 q^{5} -1.51473 q^{7} -1.40213 q^{9} +1.84073 q^{11} +3.46406 q^{13} +4.60213 q^{15} +1.23860 q^{17} -4.85279 q^{19} -1.91472 q^{21} +8.04287 q^{23} +8.25493 q^{25} -5.56459 q^{27} +1.00000 q^{29} -5.45492 q^{31} +2.32680 q^{33} -5.51473 q^{35} +11.9576 q^{37} +4.37881 q^{39} +3.01340 q^{41} +1.84073 q^{43} -5.10479 q^{45} +11.3555 q^{47} -4.70559 q^{49} +1.56567 q^{51} +0.535937 q^{53} +6.70159 q^{55} -6.13426 q^{57} -5.01340 q^{59} -9.32433 q^{61} +2.12386 q^{63} +12.6117 q^{65} -2.67612 q^{67} +10.1667 q^{69} +8.03479 q^{71} +9.48039 q^{73} +10.4348 q^{75} -2.78821 q^{77} -16.4117 q^{79} -2.82762 q^{81} +2.71899 q^{83} +4.50940 q^{85} +1.26407 q^{87} -3.19619 q^{89} -5.24713 q^{91} -6.89539 q^{93} -17.6677 q^{95} +5.68953 q^{97} -2.58095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * q^99

Introduction

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