LMFDB - Embedded newform 2312.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(1,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,6,0,3,0,0,0,-6,0,-6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.857616$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.857616 q^{3} -0.406879 q^{5} -0.172637 q^{7} -2.26449 q^{9} +4.89805 q^{11} +0.998924 q^{13} +0.348946 q^{15} +1.97828 q^{19} +0.148056 q^{21} -2.31854 q^{23} -4.83445 q^{25} +4.51492 q^{27} -0.914234 q^{29} -4.87240 q^{31} -4.20065 q^{33} +0.0702421 q^{35} +8.21237 q^{37} -0.856693 q^{39} +8.26313 q^{41} -2.42752 q^{43} +0.921374 q^{45} -5.03377 q^{47} -6.97020 q^{49} +13.2813 q^{53} -1.99291 q^{55} -1.69661 q^{57} -9.59776 q^{59} +10.4503 q^{61} +0.390935 q^{63} -0.406441 q^{65} -7.29977 q^{67} +1.98842 q^{69} +4.45171 q^{71} +6.48377 q^{73} +4.14610 q^{75} -0.845584 q^{77} +15.5053 q^{79} +2.92142 q^{81} +13.3575 q^{83} +0.784061 q^{87} +15.3426 q^{89} -0.172451 q^{91} +4.17865 q^{93} -0.804921 q^{95} +7.06549 q^{97} -11.0916 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} + 3 q^{7} - 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} + 9 q^{23} + 6 q^{25} + 12 q^{27} + 27 q^{29} + 9 q^{31} + 3 q^{33} + 21 q^{35} + 15 q^{37} - 12 q^{39} + 21 q^{41} + 24 q^{45}+ \cdots - 42 q^{99}+O(q^{100}) $ 6 * q + 6 * q^5 + 3 * q^7 - 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 + 9 * q^23 + 6 * q^25 + 12 * q^27 + 27 * q^29 + 9 * q^31 + 3 * q^33 + 21 * q^35 + 15 * q^37 - 12 * q^39 + 21 * q^41 + 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 + 30 * q^57 + 18 * q^61 + 24 * q^63 - 9 * q^65 - 9 * q^67 - 3 * q^69 + 6 * q^71 + 27 * q^73 - 24 * q^75 - 45 * q^77 + 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 + 30 * q^91 - 30 * q^93 + 3 * q^95 + 51 * q^97 - 42 * q^99

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
$3$	−0.857616	−0.495145	−0.247572	−	0.968869i	$-0.579633\pi$
$3$	−0.857616	−0.495145	−0.247572	+	0.968869i	$0.579633\pi$
$4$	0	0
$5$	−0.406879	−0.181962	−0.0909808	−	0.995853i	$-0.529000\pi$
$5$	−0.406879	−0.181962	−0.0909808	+	0.995853i	$0.529000\pi$
$6$	0	0
$7$	−0.172637	−0.0652505	−0.0326253	−	0.999468i	$-0.510387\pi$
$7$	−0.172637	−0.0652505	−0.0326253	+	0.999468i	$0.510387\pi$
$8$	0	0
$9$	−2.26449	−0.754832
$10$	0	0
$11$	4.89805	1.47682	0.738410	−	0.674353i	$-0.235577\pi$
$11$	4.89805	1.47682	0.738410	+	0.674353i	$0.235577\pi$
$12$	0	0
$13$	0.998924	0.277052	0.138526	−	0.990359i	$-0.455764\pi$
$13$	0.998924	0.277052	0.138526	+	0.990359i	$0.455764\pi$
$14$	0	0
$15$	0.348946	0.0900974
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.97828	0.453849	0.226925	−	0.973912i	$-0.427133\pi$
$19$	1.97828	0.453849	0.226925	+	0.973912i	$0.427133\pi$
$20$	0	0
$21$	0.148056	0.0323085
$22$	0	0
$23$	−2.31854	−0.483449	−0.241724	−	0.970345i	$-0.577713\pi$
$23$	−2.31854	−0.483449	−0.241724	+	0.970345i	$0.577713\pi$
$24$	0	0
$25$	−4.83445	−0.966890
$26$	0	0
$27$	4.51492	0.868896
$28$	0	0
$29$	−0.914234	−0.169769	−0.0848845	−	0.996391i	$-0.527052\pi$
$29$	−0.914234	−0.169769	−0.0848845	+	0.996391i	$0.527052\pi$
$30$	0	0
$31$	−4.87240	−0.875108	−0.437554	−	0.899192i	$-0.644155\pi$
$31$	−4.87240	−0.875108	−0.437554	+	0.899192i	$0.644155\pi$
$32$	0	0
$33$	−4.20065	−0.731239
$34$	0	0
$35$	0.0702421	0.0118731
$36$	0	0
$37$	8.21237	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$37$	8.21237	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$38$	0	0
$39$	−0.856693	−0.137181
$40$	0	0
$41$	8.26313	1.29048	0.645242	−	0.763978i	$-0.276757\pi$
$41$	8.26313	1.29048	0.645242	+	0.763978i	$0.276757\pi$
$42$	0	0
$43$	−2.42752	−0.370193	−0.185097	−	0.982720i	$-0.559260\pi$
$43$	−2.42752	−0.370193	−0.185097	+	0.982720i	$0.559260\pi$
$44$	0	0
$45$	0.921374	0.137350
$46$	0	0
$47$	−5.03377	−0.734251	−0.367126	−	0.930171i	$-0.619658\pi$
$47$	−5.03377	−0.734251	−0.367126	+	0.930171i	$0.619658\pi$
$48$	0	0
$49$	−6.97020	−0.995742
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.2813	1.82432	0.912162	−	0.409830i	$-0.134412\pi$
$53$	13.2813	1.82432	0.912162	+	0.409830i	$0.134412\pi$
$54$	0	0
$55$	−1.99291	−0.268724
$56$	0	0
$57$	−1.69661	−0.224721
$58$	0	0
$59$	−9.59776	−1.24952	−0.624761	−	0.780816i	$-0.714804\pi$
$59$	−9.59776	−1.24952	−0.624761	+	0.780816i	$0.714804\pi$
$60$	0	0
$61$	10.4503	1.33802	0.669010	−	0.743254i	$-0.266719\pi$
$61$	10.4503	1.33802	0.669010	+	0.743254i	$0.266719\pi$
$62$	0	0
$63$	0.390935	0.0492531
$64$	0	0
$65$	−0.406441	−0.0504127
$66$	0	0
$67$	−7.29977	−0.891809	−0.445904	−	0.895081i	$-0.647118\pi$
$67$	−7.29977	−0.891809	−0.445904	+	0.895081i	$0.647118\pi$
$68$	0	0
$69$	1.98842	0.239377
$70$	0	0
$71$	4.45171	0.528321	0.264161	−	0.964479i	$-0.414905\pi$
$71$	4.45171	0.528321	0.264161	+	0.964479i	$0.414905\pi$
$72$	0	0
$73$	6.48377	0.758868	0.379434	−	0.925219i	$-0.376119\pi$
$73$	6.48377	0.758868	0.379434	+	0.925219i	$0.376119\pi$
$74$	0	0
$75$	4.14610	0.478751
$76$	0	0
$77$	−0.845584	−0.0963632
$78$	0	0
$79$	15.5053	1.74448	0.872239	−	0.489080i	$-0.162668\pi$
$79$	15.5053	1.74448	0.872239	+	0.489080i	$0.162668\pi$
$80$	0	0
$81$	2.92142	0.324602
$82$	0	0
$83$	13.3575	1.46617	0.733086	−	0.680136i	$-0.238079\pi$
$83$	13.3575	1.46617	0.733086	+	0.680136i	$0.238079\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.784061	0.0840602
$88$	0	0
$89$	15.3426	1.62631	0.813156	−	0.582046i	$-0.197748\pi$
$89$	15.3426	1.62631	0.813156	+	0.582046i	$0.197748\pi$
$90$	0	0
$91$	−0.172451	−0.0180778
$92$	0	0
$93$	4.17865	0.433305
$94$	0	0
$95$	−0.804921	−0.0825832
$96$	0	0
$97$	7.06549	0.717392	0.358696	−	0.933454i	$-0.383221\pi$
$97$	7.06549	0.717392	0.358696	+	0.933454i	$0.383221\pi$
$98$	0	0
$99$	−11.0916	−1.11475
$100$	0	0
$101$	−11.1738	−1.11184	−0.555920	−	0.831236i	$-0.687634\pi$
$101$	−11.1738	−1.11184	−0.555920	+	0.831236i	$0.687634\pi$
$102$	0	0
$103$	16.9758	1.67268	0.836339	−	0.548212i	$-0.184691\pi$
$103$	16.9758	1.67268	0.836339	+	0.548212i	$0.184691\pi$
$104$	0	0
$105$	−0.0602408	−0.00587890
$106$	0	0
$107$	−11.5626	−1.11780	−0.558900	−	0.829235i	$-0.688777\pi$
$107$	−11.5626	−1.11780	−0.558900	+	0.829235i	$0.688777\pi$
$108$	0	0
$109$	7.73185	0.740577	0.370288	−	0.928917i	$-0.379259\pi$
$109$	7.73185	0.740577	0.370288	+	0.928917i	$0.379259\pi$
$110$	0	0
$111$	−7.04306	−0.668498
$112$	0	0
$113$	9.70547	0.913013	0.456507	−	0.889720i	$-0.349100\pi$
$113$	9.70547	0.913013	0.456507	+	0.889720i	$0.349100\pi$
$114$	0	0
$115$	0.943364	0.0879692
$116$	0	0
$117$	−2.26206	−0.209127
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.9909	1.18099
$122$	0	0
$123$	−7.08659	−0.638976
$124$	0	0
$125$	4.00143	0.357899
$126$	0	0
$127$	−8.04107	−0.713530	−0.356765	−	0.934194i	$-0.616120\pi$
$127$	−8.04107	−0.713530	−0.356765	+	0.934194i	$0.616120\pi$
$128$	0	0
$129$	2.08188	0.183299
$130$	0	0
$131$	0.951344	0.0831193	0.0415597	−	0.999136i	$-0.486767\pi$
$131$	0.951344	0.0831193	0.0415597	+	0.999136i	$0.486767\pi$
$132$	0	0
$133$	−0.341524	−0.0296139
$134$	0	0
$135$	−1.83702	−0.158106
$136$	0	0
$137$	5.84194	0.499110	0.249555	−	0.968361i	$-0.419716\pi$
$137$	5.84194	0.499110	0.249555	+	0.968361i	$0.419716\pi$
$138$	0	0
$139$	−13.2208	−1.12137	−0.560686	−	0.828028i	$-0.689463\pi$
$139$	−13.2208	−1.12137	−0.560686	+	0.828028i	$0.689463\pi$
$140$	0	0
$141$	4.31704	0.363561
$142$	0	0
$143$	4.89278	0.409155
$144$	0	0
$145$	0.371982	0.0308914
$146$	0	0
$147$	5.97775	0.493037
$148$	0	0
$149$	−1.24250	−0.101789	−0.0508947	−	0.998704i	$-0.516207\pi$
$149$	−1.24250	−0.101789	−0.0508947	+	0.998704i	$0.516207\pi$
$150$	0	0
$151$	16.9951	1.38304	0.691521	−	0.722356i	$-0.256941\pi$
$151$	16.9951	1.38304	0.691521	+	0.722356i	$0.256941\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.98247	0.159236
$156$	0	0
$157$	2.81037	0.224292	0.112146	−	0.993692i	$-0.464228\pi$
$157$	2.81037	0.224292	0.112146	+	0.993692i	$0.464228\pi$
$158$	0	0
$159$	−11.3902	−0.903304
$160$	0	0
$161$	0.400265	0.0315453
$162$	0	0
$163$	12.3002	0.963424	0.481712	−	0.876329i	$-0.340015\pi$
$163$	12.3002	0.963424	0.481712	+	0.876329i	$0.340015\pi$
$164$	0	0
$165$	1.70915	0.133058
$166$	0	0
$167$	−22.0092	−1.70312	−0.851561	−	0.524256i	$-0.824344\pi$
$167$	−22.0092	−1.70312	−0.851561	+	0.524256i	$0.824344\pi$
$168$	0	0
$169$	−12.0022	−0.923242
$170$	0	0
$171$	−4.47981	−0.342580
$172$	0	0
$173$	18.9695	1.44222	0.721110	−	0.692820i	$-0.243632\pi$
$173$	18.9695	1.44222	0.721110	+	0.692820i	$0.243632\pi$
$174$	0	0
$175$	0.834603	0.0630901
$176$	0	0
$177$	8.23120	0.618695
$178$	0	0
$179$	14.9918	1.12054	0.560269	−	0.828311i	$-0.310698\pi$
$179$	14.9918	1.12054	0.560269	+	0.828311i	$0.310698\pi$
$180$	0	0
$181$	−16.1876	−1.20322	−0.601608	−	0.798792i	$-0.705473\pi$
$181$	−16.1876	−1.20322	−0.601608	+	0.798792i	$0.705473\pi$
$182$	0	0
$183$	−8.96231	−0.662513
$184$	0	0
$185$	−3.34144	−0.245667
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.779440	−0.0566959
$190$	0	0
$191$	−10.8760	−0.786957	−0.393479	−	0.919334i	$-0.628728\pi$
$191$	−10.8760	−0.786957	−0.393479	+	0.919334i	$0.628728\pi$
$192$	0	0
$193$	12.3348	0.887877	0.443938	−	0.896057i	$-0.353581\pi$
$193$	12.3348	0.887877	0.443938	+	0.896057i	$0.353581\pi$
$194$	0	0
$195$	0.348570	0.0249616
$196$	0	0
$197$	2.05856	0.146666	0.0733331	−	0.997308i	$-0.476636\pi$
$197$	2.05856	0.146666	0.0733331	+	0.997308i	$0.476636\pi$
$198$	0	0
$199$	19.5758	1.38769	0.693846	−	0.720124i	$-0.255915\pi$
$199$	19.5758	1.38769	0.693846	+	0.720124i	$0.255915\pi$
$200$	0	0
$201$	6.26040	0.441575
$202$	0	0
$203$	0.157830	0.0110775
$204$	0	0
$205$	−3.36209	−0.234818
$206$	0	0
$207$	5.25032	0.364923
$208$	0	0
$209$	9.68974	0.670254
$210$	0	0
$211$	−7.19032	−0.495002	−0.247501	−	0.968888i	$-0.579609\pi$
$211$	−7.19032	−0.495002	−0.247501	+	0.968888i	$0.579609\pi$
$212$	0	0
$213$	−3.81786	−0.261596
$214$	0	0
$215$	0.987705	0.0673609
$216$	0	0
$217$	0.841154	0.0571013
$218$	0	0
$219$	−5.56058	−0.375749
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.728072	0.0487553	0.0243777	−	0.999703i	$-0.492240\pi$
$223$	0.728072	0.0487553	0.0243777	+	0.999703i	$0.492240\pi$
$224$	0	0
$225$	10.9476	0.729839
$226$	0	0
$227$	−2.29363	−0.152234	−0.0761169	−	0.997099i	$-0.524252\pi$
$227$	−2.29363	−0.152234	−0.0761169	+	0.997099i	$0.524252\pi$
$228$	0	0
$229$	−15.0576	−0.995033	−0.497517	−	0.867454i	$-0.665755\pi$
$229$	−15.0576	−0.995033	−0.497517	+	0.867454i	$0.665755\pi$
$230$	0	0
$231$	0.725186	0.0477137
$232$	0	0
$233$	−9.42684	−0.617573	−0.308786	−	0.951131i	$-0.599923\pi$
$233$	−9.42684	−0.617573	−0.308786	+	0.951131i	$0.599923\pi$
$234$	0	0
$235$	2.04813	0.133606
$236$	0	0
$237$	−13.2976	−0.863769
$238$	0	0
$239$	−5.86612	−0.379448	−0.189724	−	0.981837i	$-0.560759\pi$
$239$	−5.86612	−0.379448	−0.189724	+	0.981837i	$0.560759\pi$
$240$	0	0
$241$	10.1691	0.655051	0.327526	−	0.944842i	$-0.393785\pi$
$241$	10.1691	0.655051	0.327526	+	0.944842i	$0.393785\pi$
$242$	0	0
$243$	−16.0502	−1.02962
$244$	0	0
$245$	2.83602	0.181187
$246$	0	0
$247$	1.97615	0.125740
$248$	0	0
$249$	−11.4556	−0.725967
$250$	0	0
$251$	9.39157	0.592791	0.296395	−	0.955065i	$-0.404215\pi$
$251$	9.39157	0.592791	0.296395	+	0.955065i	$0.404215\pi$
$252$	0	0
$253$	−11.3563	−0.713967
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.375779	0.0234405	0.0117202	−	0.999931i	$-0.496269\pi$
$257$	0.375779	0.0234405	0.0117202	+	0.999931i	$0.496269\pi$
$258$	0	0
$259$	−1.41776	−0.0880951
$260$	0	0
$261$	2.07028	0.128147
$262$	0	0
$263$	22.7314	1.40168	0.700839	−	0.713319i	$-0.252809\pi$
$263$	22.7314	1.40168	0.700839	+	0.713319i	$0.252809\pi$
$264$	0	0
$265$	−5.40387	−0.331957
$266$	0	0
$267$	−13.1581	−0.805260
$268$	0	0
$269$	1.26787	0.0773037	0.0386518	−	0.999253i	$-0.487694\pi$
$269$	1.26787	0.0773037	0.0386518	+	0.999253i	$0.487694\pi$
$270$	0	0
$271$	31.2435	1.89790	0.948952	−	0.315419i	$-0.102145\pi$
$271$	31.2435	1.89790	0.948952	+	0.315419i	$0.102145\pi$
$272$	0	0
$273$	0.147897	0.00895111
$274$	0	0
$275$	−23.6794	−1.42792
$276$	0	0
$277$	19.5383	1.17394	0.586972	−	0.809608i	$-0.300320\pi$
$277$	19.5383	1.17394	0.586972	+	0.809608i	$0.300320\pi$
$278$	0	0
$279$	11.0335	0.660559
$280$	0	0
$281$	−17.9598	−1.07139	−0.535696	−	0.844411i	$-0.679951\pi$
$281$	−17.9598	−1.07139	−0.535696	+	0.844411i	$0.679951\pi$
$282$	0	0
$283$	7.58733	0.451020	0.225510	−	0.974241i	$-0.427595\pi$
$283$	7.58733	0.451020	0.225510	+	0.974241i	$0.427595\pi$
$284$	0	0
$285$	0.690314	0.0408906
$286$	0	0
$287$	−1.42652	−0.0842047
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−6.05948	−0.355213
$292$	0	0
$293$	13.4275	0.784443	0.392221	−	0.919871i	$-0.371707\pi$
$293$	13.4275	0.784443	0.392221	+	0.919871i	$0.371707\pi$
$294$	0	0
$295$	3.90512	0.227365
$296$	0	0
$297$	22.1143	1.28320
$298$	0	0
$299$	−2.31604	−0.133940
$300$	0	0
$301$	0.419079	0.0241553
$302$	0	0
$303$	9.58287	0.550521
$304$	0	0
$305$	−4.25199	−0.243468
$306$	0	0
$307$	19.7360	1.12639	0.563195	−	0.826324i	$-0.309572\pi$
$307$	19.7360	1.12639	0.563195	+	0.826324i	$0.309572\pi$
$308$	0	0
$309$	−14.5587	−0.828218
$310$	0	0
$311$	−1.08628	−0.0615971	−0.0307985	−	0.999526i	$-0.509805\pi$
$311$	−1.08628	−0.0615971	−0.0307985	+	0.999526i	$0.509805\pi$
$312$	0	0
$313$	−31.6016	−1.78623	−0.893115	−	0.449828i	$-0.851485\pi$
$313$	−31.6016	−1.78623	−0.893115	+	0.449828i	$0.851485\pi$
$314$	0	0
$315$	−0.159063	−0.00896218
$316$	0	0
$317$	1.55407	0.0872852	0.0436426	−	0.999047i	$-0.486104\pi$
$317$	1.55407	0.0872852	0.0436426	+	0.999047i	$0.486104\pi$
$318$	0	0
$319$	−4.47797	−0.250718
$320$	0	0
$321$	9.91629	0.553473
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.82925	−0.267878
$326$	0	0
$327$	−6.63096	−0.366693
$328$	0	0
$329$	0.869013	0.0479103
$330$	0	0
$331$	−18.8145	−1.03414	−0.517068	−	0.855944i	$-0.672977\pi$
$331$	−18.8145	−1.03414	−0.517068	+	0.855944i	$0.672977\pi$
$332$	0	0
$333$	−18.5969	−1.01910
$334$	0	0
$335$	2.97012	0.162275
$336$	0	0
$337$	30.0821	1.63868	0.819338	−	0.573311i	$-0.194341\pi$
$337$	30.0821	1.63868	0.819338	+	0.573311i	$0.194341\pi$
$338$	0	0
$339$	−8.32356	−0.452074
$340$	0	0
$341$	−23.8653	−1.29238
$342$	0	0
$343$	2.41177	0.130223
$344$	0	0
$345$	−0.809044	−0.0435575
$346$	0	0
$347$	−22.3365	−1.19909	−0.599543	−	0.800342i	$-0.704651\pi$
$347$	−22.3365	−1.19909	−0.599543	+	0.800342i	$0.704651\pi$
$348$	0	0
$349$	24.6602	1.32003	0.660015	−	0.751252i	$-0.270550\pi$
$349$	24.6602	1.32003	0.660015	+	0.751252i	$0.270550\pi$
$350$	0	0
$351$	4.51006	0.240729
$352$	0	0
$353$	17.6349	0.938611	0.469305	−	0.883036i	$-0.344504\pi$
$353$	17.6349	0.938611	0.469305	+	0.883036i	$0.344504\pi$
$354$	0	0
$355$	−1.81131	−0.0961342
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.9699	−1.74008	−0.870042	−	0.492977i	$-0.835909\pi$
$359$	−32.9699	−1.74008	−0.870042	+	0.492977i	$0.835909\pi$
$360$	0	0
$361$	−15.0864	−0.794021
$362$	0	0
$363$	−11.1412	−0.584763
$364$	0	0
$365$	−2.63811	−0.138085
$366$	0	0
$367$	−18.8844	−0.985756	−0.492878	−	0.870098i	$-0.664055\pi$
$367$	−18.8844	−0.985756	−0.492878	+	0.870098i	$0.664055\pi$
$368$	0	0
$369$	−18.7118	−0.974098
$370$	0	0
$371$	−2.29283	−0.119038
$372$	0	0
$373$	32.8087	1.69877	0.849385	−	0.527774i	$-0.176973\pi$
$373$	32.8087	1.69877	0.849385	+	0.527774i	$0.176973\pi$
$374$	0	0
$375$	−3.43169	−0.177212
$376$	0	0
$377$	−0.913249	−0.0470347
$378$	0	0
$379$	−24.2542	−1.24585	−0.622927	−	0.782280i	$-0.714056\pi$
$379$	−24.2542	−1.24585	−0.622927	+	0.782280i	$0.714056\pi$
$380$	0	0
$381$	6.89615	0.353301
$382$	0	0
$383$	−31.0607	−1.58713	−0.793564	−	0.608486i	$-0.791777\pi$
$383$	−31.0607	−1.58713	−0.793564	+	0.608486i	$0.791777\pi$
$384$	0	0
$385$	0.344050	0.0175344
$386$	0	0
$387$	5.49710	0.279433
$388$	0	0
$389$	−19.2057	−0.973766	−0.486883	−	0.873467i	$-0.661866\pi$
$389$	−19.2057	−0.973766	−0.486883	+	0.873467i	$0.661866\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−0.815888	−0.0411561
$394$	0	0
$395$	−6.30876	−0.317428
$396$	0	0
$397$	−7.82243	−0.392597	−0.196298	−	0.980544i	$-0.562892\pi$
$397$	−7.82243	−0.392597	−0.196298	+	0.980544i	$0.562892\pi$
$398$	0	0
$399$	0.292897	0.0146632
$400$	0	0
$401$	1.46302	0.0730595	0.0365298	−	0.999333i	$-0.488370\pi$
$401$	1.46302	0.0730595	0.0365298	+	0.999333i	$0.488370\pi$
$402$	0	0
$403$	−4.86715	−0.242450
$404$	0	0
$405$	−1.18866	−0.0590651
$406$	0	0
$407$	40.2247	1.99386
$408$	0	0
$409$	−5.87737	−0.290617	−0.145309	−	0.989386i	$-0.546418\pi$
$409$	−5.87737	−0.290617	−0.145309	+	0.989386i	$0.546418\pi$
$410$	0	0
$411$	−5.01014	−0.247132
$412$	0	0
$413$	1.65693	0.0815320
$414$	0	0
$415$	−5.43486	−0.266787
$416$	0	0
$417$	11.3384	0.555242
$418$	0	0
$419$	−36.8229	−1.79892	−0.899458	−	0.437007i	$-0.856038\pi$
$419$	−36.8229	−1.79892	−0.899458	+	0.437007i	$0.856038\pi$
$420$	0	0
$421$	−26.7509	−1.30376	−0.651879	−	0.758323i	$-0.726019\pi$
$421$	−26.7509	−1.30376	−0.651879	+	0.758323i	$0.726019\pi$
$422$	0	0
$423$	11.3989	0.554236
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.80410	−0.0873064
$428$	0	0
$429$	−4.19613	−0.202591
$430$	0	0
$431$	8.57254	0.412925	0.206462	−	0.978455i	$-0.433805\pi$
$431$	8.57254	0.412925	0.206462	+	0.978455i	$0.433805\pi$
$432$	0	0
$433$	7.49865	0.360362	0.180181	−	0.983633i	$-0.442332\pi$
$433$	7.49865	0.360362	0.180181	+	0.983633i	$0.442332\pi$
$434$	0	0
$435$	−0.319018	−0.0152957
$436$	0	0
$437$	−4.58673	−0.219413
$438$	0	0
$439$	21.8451	1.04261	0.521305	−	0.853370i	$-0.325445\pi$
$439$	21.8451	1.04261	0.521305	+	0.853370i	$0.325445\pi$
$440$	0	0
$441$	15.7840	0.751618
$442$	0	0
$443$	−21.9471	−1.04274	−0.521368	−	0.853332i	$-0.674578\pi$
$443$	−21.9471	−1.04274	−0.521368	+	0.853332i	$0.674578\pi$
$444$	0	0
$445$	−6.24257	−0.295926
$446$	0	0
$447$	1.06559	0.0504005
$448$	0	0
$449$	−10.9285	−0.515746	−0.257873	−	0.966179i	$-0.583022\pi$
$449$	−10.9285	−0.515746	−0.257873	+	0.966179i	$0.583022\pi$
$450$	0	0
$451$	40.4732	1.90581
$452$	0	0
$453$	−14.5753	−0.684806
$454$	0	0
$455$	0.0701665	0.00328946
$456$	0	0
$457$	−9.75317	−0.456234	−0.228117	−	0.973634i	$-0.573257\pi$
$457$	−9.75317	−0.456234	−0.228117	+	0.973634i	$0.573257\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.7974	−0.828910	−0.414455	−	0.910070i	$-0.636028\pi$
$461$	−17.7974	−0.828910	−0.414455	+	0.910070i	$0.636028\pi$
$462$	0	0
$463$	−24.9054	−1.15745	−0.578725	−	0.815523i	$-0.696450\pi$
$463$	−24.9054	−1.15745	−0.578725	+	0.815523i	$0.696450\pi$
$464$	0	0
$465$	−1.70020	−0.0788450
$466$	0	0
$467$	8.22223	0.380479	0.190240	−	0.981738i	$-0.439074\pi$
$467$	8.22223	0.380479	0.190240	+	0.981738i	$0.439074\pi$
$468$	0	0
$469$	1.26021	0.0581910
$470$	0	0
$471$	−2.41022	−0.111057
$472$	0	0
$473$	−11.8901	−0.546708
$474$	0	0
$475$	−9.56392	−0.438823
$476$	0	0
$477$	−30.0754	−1.37706
$478$	0	0
$479$	5.02177	0.229450	0.114725	−	0.993397i	$-0.463401\pi$
$479$	5.02177	0.229450	0.114725	+	0.993397i	$0.463401\pi$
$480$	0	0
$481$	8.20353	0.374049
$482$	0	0
$483$	−0.343274	−0.0156195
$484$	0	0
$485$	−2.87480	−0.130538
$486$	0	0
$487$	−19.7071	−0.893012	−0.446506	−	0.894781i	$-0.647332\pi$
$487$	−19.7071	−0.893012	−0.446506	+	0.894781i	$0.647332\pi$
$488$	0	0
$489$	−10.5488	−0.477035
$490$	0	0
$491$	−23.1304	−1.04386	−0.521930	−	0.852988i	$-0.674788\pi$
$491$	−23.1304	−1.04386	−0.521930	+	0.852988i	$0.674788\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.51294	0.202842
$496$	0	0
$497$	−0.768529	−0.0344732
$498$	0	0
$499$	0.923022	0.0413201	0.0206601	−	0.999787i	$-0.493423\pi$
$499$	0.923022	0.0413201	0.0206601	+	0.999787i	$0.493423\pi$
$500$	0	0
$501$	18.8754	0.843292
$502$	0	0
$503$	−33.5530	−1.49605	−0.748026	−	0.663669i	$-0.768998\pi$
$503$	−33.5530	−1.49605	−0.748026	+	0.663669i	$0.768998\pi$
$504$	0	0
$505$	4.54640	0.202312
$506$	0	0
$507$	10.2932	0.457139
$508$	0	0
$509$	−2.34164	−0.103791	−0.0518957	−	0.998653i	$-0.516526\pi$
$509$	−2.34164	−0.103791	−0.0518957	+	0.998653i	$0.516526\pi$
$510$	0	0
$511$	−1.11934	−0.0495165
$512$	0	0
$513$	8.93178	0.394348
$514$	0	0
$515$	−6.90710	−0.304363
$516$	0	0
$517$	−24.6557	−1.08436
$518$	0	0
$519$	−16.2685	−0.714108
$520$	0	0
$521$	−9.77606	−0.428297	−0.214148	−	0.976801i	$-0.568698\pi$
$521$	−9.77606	−0.428297	−0.214148	+	0.976801i	$0.568698\pi$
$522$	0	0
$523$	10.1328	0.443076	0.221538	−	0.975152i	$-0.428892\pi$
$523$	10.1328	0.443076	0.221538	+	0.975152i	$0.428892\pi$
$524$	0	0
$525$	−0.715769	−0.0312387
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.6244	−0.766277
$530$	0	0
$531$	21.7341	0.943179
$532$	0	0
$533$	8.25423	0.357530
$534$	0	0
$535$	4.70458	0.203397
$536$	0	0
$537$	−12.8572	−0.554828
$538$	0	0
$539$	−34.1404	−1.47053
$540$	0	0
$541$	39.6652	1.70534	0.852670	−	0.522450i	$-0.174982\pi$
$541$	39.6652	1.70534	0.852670	+	0.522450i	$0.174982\pi$
$542$	0	0
$543$	13.8828	0.595766
$544$	0	0
$545$	−3.14592	−0.134757
$546$	0	0
$547$	−15.5780	−0.666068	−0.333034	−	0.942915i	$-0.608072\pi$
$547$	−15.5780	−0.666068	−0.333034	+	0.942915i	$0.608072\pi$
$548$	0	0
$549$	−23.6646	−1.00998
$550$	0	0
$551$	−1.80861	−0.0770495
$552$	0	0
$553$	−2.67677	−0.113828
$554$	0	0
$555$	2.86567	0.121641
$556$	0	0
$557$	9.02229	0.382287	0.191143	−	0.981562i	$-0.438780\pi$
$557$	9.02229	0.382287	0.191143	+	0.981562i	$0.438780\pi$
$558$	0	0
$559$	−2.42490	−0.102563
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23.7152	−0.999479	−0.499739	−	0.866176i	$-0.666571\pi$
$563$	−23.7152	−0.999479	−0.499739	+	0.866176i	$0.666571\pi$
$564$	0	0
$565$	−3.94895	−0.166133
$566$	0	0
$567$	−0.504344	−0.0211805
$568$	0	0
$569$	2.33523	0.0978980	0.0489490	−	0.998801i	$-0.484413\pi$
$569$	2.33523	0.0978980	0.0489490	+	0.998801i	$0.484413\pi$
$570$	0	0
$571$	8.13480	0.340431	0.170215	−	0.985407i	$-0.445554\pi$
$571$	8.13480	0.340431	0.170215	+	0.985407i	$0.445554\pi$
$572$	0	0
$573$	9.32740	0.389658
$574$	0	0
$575$	11.2089	0.467442
$576$	0	0
$577$	17.0172	0.708435	0.354218	−	0.935163i	$-0.384747\pi$
$577$	17.0172	0.708435	0.354218	+	0.935163i	$0.384747\pi$
$578$	0	0
$579$	−10.5785	−0.439628
$580$	0	0
$581$	−2.30599	−0.0956684
$582$	0	0
$583$	65.0524	2.69420
$584$	0	0
$585$	0.920383	0.0380531
$586$	0	0
$587$	21.1414	0.872600	0.436300	−	0.899801i	$-0.356289\pi$
$587$	21.1414	0.872600	0.436300	+	0.899801i	$0.356289\pi$
$588$	0	0
$589$	−9.63899	−0.397168
$590$	0	0
$591$	−1.76545	−0.0726210
$592$	0	0
$593$	10.8412	0.445193	0.222597	−	0.974911i	$-0.428547\pi$
$593$	10.8412	0.445193	0.222597	+	0.974911i	$0.428547\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.7885	−0.687108
$598$	0	0
$599$	19.7689	0.807737	0.403868	−	0.914817i	$-0.367665\pi$
$599$	19.7689	0.807737	0.403868	+	0.914817i	$0.367665\pi$
$600$	0	0
$601$	21.6236	0.882044	0.441022	−	0.897496i	$-0.354616\pi$
$601$	21.6236	0.882044	0.441022	+	0.897496i	$0.354616\pi$
$602$	0	0
$603$	16.5303	0.673165
$604$	0	0
$605$	−5.28574	−0.214896
$606$	0	0
$607$	33.2656	1.35021	0.675105	−	0.737722i	$-0.264098\pi$
$607$	33.2656	1.35021	0.675105	+	0.737722i	$0.264098\pi$
$608$	0	0
$609$	−0.135358	−0.00548497
$610$	0	0
$611$	−5.02835	−0.203425
$612$	0	0
$613$	46.9354	1.89570	0.947852	−	0.318712i	$-0.103250\pi$
$613$	46.9354	1.89570	0.947852	+	0.318712i	$0.103250\pi$
$614$	0	0
$615$	2.88338	0.116269
$616$	0	0
$617$	−17.4750	−0.703519	−0.351759	−	0.936090i	$-0.614416\pi$
$617$	−17.4750	−0.703519	−0.351759	+	0.936090i	$0.614416\pi$
$618$	0	0
$619$	−38.2847	−1.53879	−0.769397	−	0.638771i	$-0.779443\pi$
$619$	−38.2847	−1.53879	−0.769397	+	0.638771i	$0.779443\pi$
$620$	0	0
$621$	−10.4680	−0.420067
$622$	0	0
$623$	−2.64869	−0.106118
$624$	0	0
$625$	22.5442	0.901766
$626$	0	0
$627$	−8.31008	−0.331873
$628$	0	0
$629$	0	0
$630$	0	0
$631$	15.7675	0.627692	0.313846	−	0.949474i	$-0.398382\pi$
$631$	15.7675	0.627692	0.313846	+	0.949474i	$0.398382\pi$
$632$	0	0
$633$	6.16653	0.245098
$634$	0	0
$635$	3.27174	0.129835
$636$	0	0
$637$	−6.96269	−0.275872
$638$	0	0
$639$	−10.0809	−0.398794
$640$	0	0
$641$	6.57685	0.259770	0.129885	−	0.991529i	$-0.458539\pi$
$641$	6.57685	0.259770	0.129885	+	0.991529i	$0.458539\pi$
$642$	0	0
$643$	23.6311	0.931918	0.465959	−	0.884806i	$-0.345709\pi$
$643$	23.6311	0.931918	0.465959	+	0.884806i	$0.345709\pi$
$644$	0	0
$645$	−0.847072	−0.0333534
$646$	0	0
$647$	−12.0348	−0.473139	−0.236569	−	0.971615i	$-0.576023\pi$
$647$	−12.0348	−0.473139	−0.236569	+	0.971615i	$0.576023\pi$
$648$	0	0
$649$	−47.0104	−1.84532
$650$	0	0
$651$	−0.721387	−0.0282734
$652$	0	0
$653$	11.9823	0.468902	0.234451	−	0.972128i	$-0.424671\pi$
$653$	11.9823	0.468902	0.234451	+	0.972128i	$0.424671\pi$
$654$	0	0
$655$	−0.387081	−0.0151245
$656$	0	0
$657$	−14.6825	−0.572817
$658$	0	0
$659$	30.4441	1.18593	0.592967	−	0.805227i	$-0.297957\pi$
$659$	30.4441	1.18593	0.592967	+	0.805227i	$0.297957\pi$
$660$	0	0
$661$	14.5940	0.567639	0.283819	−	0.958878i	$-0.408398\pi$
$661$	14.5940	0.567639	0.283819	+	0.958878i	$0.408398\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.138959	0.00538860
$666$	0	0
$667$	2.11969	0.0820746
$668$	0	0
$669$	−0.624406	−0.0241409
$670$	0	0
$671$	51.1860	1.97601
$672$	0	0
$673$	−25.7516	−0.992653	−0.496326	−	0.868136i	$-0.665318\pi$
$673$	−25.7516	−0.992653	−0.496326	+	0.868136i	$0.665318\pi$
$674$	0	0
$675$	−21.8271	−0.840127
$676$	0	0
$677$	−31.5471	−1.21245	−0.606226	−	0.795292i	$-0.707317\pi$
$677$	−31.5471	−1.21245	−0.606226	+	0.795292i	$0.707317\pi$
$678$	0	0
$679$	−1.21976	−0.0468102
$680$	0	0
$681$	1.96706	0.0753778
$682$	0	0
$683$	−25.5427	−0.977365	−0.488683	−	0.872462i	$-0.662522\pi$
$683$	−25.5427	−0.977365	−0.488683	+	0.872462i	$0.662522\pi$
$684$	0	0
$685$	−2.37696	−0.0908190
$686$	0	0
$687$	12.9136	0.492686
$688$	0	0
$689$	13.2670	0.505432
$690$	0	0
$691$	−16.6920	−0.634995	−0.317497	−	0.948259i	$-0.602842\pi$
$691$	−16.6920	−0.634995	−0.317497	+	0.948259i	$0.602842\pi$
$692$	0	0
$693$	1.91482	0.0727380
$694$	0	0
$695$	5.37925	0.204047
$696$	0	0
$697$	0	0
$698$	0	0
$699$	8.08461	0.305788
$700$	0	0
$701$	−25.3488	−0.957411	−0.478706	−	0.877975i	$-0.658894\pi$
$701$	−25.3488	−0.957411	−0.478706	+	0.877975i	$0.658894\pi$
$702$	0	0
$703$	16.2464	0.612745
$704$	0	0
$705$	−1.75651	−0.0661541
$706$	0	0
$707$	1.92901	0.0725481
$708$	0	0
$709$	−12.0083	−0.450983	−0.225491	−	0.974245i	$-0.572399\pi$
$709$	−12.0083	−0.450983	−0.225491	+	0.974245i	$0.572399\pi$
$710$	0	0
$711$	−35.1116	−1.31679
$712$	0	0
$713$	11.2968	0.423070
$714$	0	0
$715$	−1.99077	−0.0744505
$716$	0	0
$717$	5.03088	0.187882
$718$	0	0
$719$	−1.66572	−0.0621209	−0.0310604	−	0.999518i	$-0.509888\pi$
$719$	−1.66572	−0.0621209	−0.0310604	+	0.999518i	$0.509888\pi$
$720$	0	0
$721$	−2.93065	−0.109143
$722$	0	0
$723$	−8.72121	−0.324345
$724$	0	0
$725$	4.41982	0.164148
$726$	0	0
$727$	−44.4416	−1.64825	−0.824123	−	0.566411i	$-0.808332\pi$
$727$	−44.4416	−1.64825	−0.824123	+	0.566411i	$0.808332\pi$
$728$	0	0
$729$	5.00065	0.185209
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−22.6176	−0.835398	−0.417699	−	0.908585i	$-0.637163\pi$
$733$	−22.6176	−0.835398	−0.417699	+	0.908585i	$0.637163\pi$
$734$	0	0
$735$	−2.43222	−0.0897138
$736$	0	0
$737$	−35.7547	−1.31704
$738$	0	0
$739$	−32.5150	−1.19609	−0.598043	−	0.801464i	$-0.704055\pi$
$739$	−32.5150	−1.19609	−0.598043	+	0.801464i	$0.704055\pi$
$740$	0	0
$741$	−1.69478	−0.0622594
$742$	0	0
$743$	19.2272	0.705379	0.352690	−	0.935740i	$-0.385267\pi$
$743$	19.2272	0.705379	0.352690	+	0.935740i	$0.385267\pi$
$744$	0	0
$745$	0.505546	0.0185218
$746$	0	0
$747$	−30.2479	−1.10671
$748$	0	0
$749$	1.99613	0.0729371
$750$	0	0
$751$	50.3608	1.83769	0.918846	−	0.394617i	$-0.129123\pi$
$751$	50.3608	1.83769	0.918846	+	0.394617i	$0.129123\pi$
$752$	0	0
$753$	−8.05437	−0.293517
$754$	0	0
$755$	−6.91495	−0.251661
$756$	0	0
$757$	3.99667	0.145262	0.0726308	−	0.997359i	$-0.476861\pi$
$757$	3.99667	0.145262	0.0726308	+	0.997359i	$0.476861\pi$
$758$	0	0
$759$	9.73937	0.353517
$760$	0	0
$761$	−31.2422	−1.13253	−0.566265	−	0.824223i	$-0.691612\pi$
$761$	−31.2422	−1.13253	−0.566265	+	0.824223i	$0.691612\pi$
$762$	0	0
$763$	−1.33480	−0.0483230
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.58743	−0.346182
$768$	0	0
$769$	−27.7956	−1.00233	−0.501167	−	0.865351i	$-0.667096\pi$
$769$	−27.7956	−1.00233	−0.501167	+	0.865351i	$0.667096\pi$
$770$	0	0
$771$	−0.322274	−0.0116064
$772$	0	0
$773$	6.48801	0.233358	0.116679	−	0.993170i	$-0.462775\pi$
$773$	6.48801	0.233358	0.116679	+	0.993170i	$0.462775\pi$
$774$	0	0
$775$	23.5554	0.846134
$776$	0	0
$777$	1.21589	0.0436198
$778$	0	0
$779$	16.3468	0.585685
$780$	0	0
$781$	21.8047	0.780235
$782$	0	0
$783$	−4.12769	−0.147511
$784$	0	0
$785$	−1.14348	−0.0408126
$786$	0	0
$787$	11.0559	0.394099	0.197050	−	0.980394i	$-0.436864\pi$
$787$	11.0559	0.394099	0.197050	+	0.980394i	$0.436864\pi$
$788$	0	0
$789$	−19.4948	−0.694034
$790$	0	0
$791$	−1.67552	−0.0595746
$792$	0	0
$793$	10.4390	0.370700
$794$	0	0
$795$	4.63444	0.164367
$796$	0	0
$797$	40.5734	1.43718	0.718591	−	0.695432i	$-0.244787\pi$
$797$	40.5734	1.43718	0.718591	+	0.695432i	$0.244787\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−34.7432	−1.22759
$802$	0	0
$803$	31.7579	1.12071
$804$	0	0
$805$	−0.162859	−0.00574003
$806$	0	0
$807$	−1.08735	−0.0382765
$808$	0	0
$809$	23.8426	0.838261	0.419131	−	0.907926i	$-0.362335\pi$
$809$	23.8426	0.838261	0.419131	+	0.907926i	$0.362335\pi$
$810$	0	0
$811$	20.3117	0.713239	0.356619	−	0.934250i	$-0.383929\pi$
$811$	20.3117	0.713239	0.356619	+	0.934250i	$0.383929\pi$
$812$	0	0
$813$	−26.7949	−0.939738
$814$	0	0
$815$	−5.00468	−0.175306
$816$	0	0
$817$	−4.80232	−0.168012
$818$	0	0
$819$	0.390514	0.0136457
$820$	0	0
$821$	−29.0649	−1.01437	−0.507186	−	0.861837i	$-0.669314\pi$
$821$	−29.0649	−1.01437	−0.507186	+	0.861837i	$0.669314\pi$
$822$	0	0
$823$	26.5052	0.923914	0.461957	−	0.886902i	$-0.347147\pi$
$823$	26.5052	0.923914	0.461957	+	0.886902i	$0.347147\pi$
$824$	0	0
$825$	20.3078	0.707028
$826$	0	0
$827$	−32.9234	−1.14486	−0.572429	−	0.819954i	$-0.693999\pi$
$827$	−32.9234	−1.14486	−0.572429	+	0.819954i	$0.693999\pi$
$828$	0	0
$829$	39.3841	1.36787	0.683934	−	0.729544i	$-0.260268\pi$
$829$	39.3841	1.36787	0.683934	+	0.729544i	$0.260268\pi$
$830$	0	0
$831$	−16.7564	−0.581272
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.95506	0.309903
$836$	0	0
$837$	−21.9985	−0.760378
$838$	0	0
$839$	14.1181	0.487412	0.243706	−	0.969849i	$-0.421637\pi$
$839$	14.1181	0.487412	0.243706	+	0.969849i	$0.421637\pi$
$840$	0	0
$841$	−28.1642	−0.971179
$842$	0	0
$843$	15.4026	0.530494
$844$	0	0
$845$	4.88342	0.167995
$846$	0	0
$847$	−2.24271	−0.0770605
$848$	0	0
$849$	−6.50702	−0.223320
$850$	0	0
$851$	−19.0407	−0.652707
$852$	0	0
$853$	−17.3848	−0.595244	−0.297622	−	0.954684i	$-0.596194\pi$
$853$	−17.3848	−0.595244	−0.297622	+	0.954684i	$0.596194\pi$
$854$	0	0
$855$	1.82274	0.0623364
$856$	0	0
$857$	52.0006	1.77631	0.888153	−	0.459548i	$-0.151989\pi$
$857$	52.0006	1.77631	0.888153	+	0.459548i	$0.151989\pi$
$858$	0	0
$859$	37.0405	1.26380	0.631902	−	0.775048i	$-0.282275\pi$
$859$	37.0405	1.26380	0.631902	+	0.775048i	$0.282275\pi$
$860$	0	0
$861$	1.22340	0.0416935
$862$	0	0
$863$	−25.8555	−0.880131	−0.440066	−	0.897966i	$-0.645045\pi$
$863$	−25.8555	−0.880131	−0.440066	+	0.897966i	$0.645045\pi$
$864$	0	0
$865$	−7.71826	−0.262429
$866$	0	0
$867$	0	0
$868$	0	0
$869$	75.9456	2.57628
$870$	0	0
$871$	−7.29191	−0.247077
$872$	0	0
$873$	−15.9998	−0.541510
$874$	0	0
$875$	−0.690793	−0.0233531
$876$	0	0
$877$	29.9767	1.01224	0.506120	−	0.862463i	$-0.331079\pi$
$877$	29.9767	1.01224	0.506120	+	0.862463i	$0.331079\pi$
$878$	0	0
$879$	−11.5156	−0.388413
$880$	0	0
$881$	−34.8630	−1.17457	−0.587283	−	0.809382i	$-0.699802\pi$
$881$	−34.8630	−1.17457	−0.587283	+	0.809382i	$0.699802\pi$
$882$	0	0
$883$	−26.7792	−0.901193	−0.450596	−	0.892728i	$-0.648789\pi$
$883$	−26.7792	−0.901193	−0.450596	+	0.892728i	$0.648789\pi$
$884$	0	0
$885$	−3.34910	−0.112579
$886$	0	0
$887$	47.7971	1.60487	0.802435	−	0.596740i	$-0.203538\pi$
$887$	47.7971	1.60487	0.802435	+	0.596740i	$0.203538\pi$
$888$	0	0
$889$	1.38818	0.0465582
$890$	0	0
$891$	14.3093	0.479379
$892$	0	0
$893$	−9.95823	−0.333239
$894$	0	0
$895$	−6.09983	−0.203895
$896$	0	0
$897$	1.98628	0.0663198
$898$	0	0
$899$	4.45451	0.148566
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−0.359408	−0.0119604
$904$	0	0
$905$	6.58639	0.218939
$906$	0	0
$907$	36.8750	1.22441	0.612207	−	0.790698i	$-0.290282\pi$
$907$	36.8750	1.22441	0.612207	+	0.790698i	$0.290282\pi$
$908$	0	0
$909$	25.3031	0.839251
$910$	0	0
$911$	−55.1461	−1.82707	−0.913536	−	0.406758i	$-0.866659\pi$
$911$	−55.1461	−1.82707	−0.913536	+	0.406758i	$0.866659\pi$
$912$	0	0
$913$	65.4256	2.16527
$914$	0	0
$915$	3.64657	0.120552
$916$	0	0
$917$	−0.164237	−0.00542358
$918$	0	0
$919$	28.0520	0.925349	0.462674	−	0.886528i	$-0.346890\pi$
$919$	28.0520	0.925349	0.462674	+	0.886528i	$0.346890\pi$
$920$	0	0
$921$	−16.9259	−0.557727
$922$	0	0
$923$	4.44692	0.146372
$924$	0	0
$925$	−39.7023	−1.30540
$926$	0	0
$927$	−38.4417	−1.26259
$928$	0	0
$929$	44.8919	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$929$	44.8919	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$930$	0	0
$931$	−13.7890	−0.451917
$932$	0	0
$933$	0.931608	0.0304995
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.2778	0.662446	0.331223	−	0.943552i	$-0.392539\pi$
$937$	20.2778	0.662446	0.331223	+	0.943552i	$0.392539\pi$
$938$	0	0
$939$	27.1021	0.884443
$940$	0	0
$941$	−32.1913	−1.04941	−0.524703	−	0.851285i	$-0.675824\pi$
$941$	−32.1913	−1.04941	−0.524703	+	0.851285i	$0.675824\pi$
$942$	0	0
$943$	−19.1584	−0.623883
$944$	0	0
$945$	0.317137	0.0103165
$946$	0	0
$947$	−48.2974	−1.56946	−0.784728	−	0.619841i	$-0.787197\pi$
$947$	−48.2974	−1.56946	−0.784728	+	0.619841i	$0.787197\pi$
$948$	0	0
$949$	6.47679	0.210245
$950$	0	0
$951$	−1.33279	−0.0432188
$952$	0	0
$953$	31.0668	1.00635	0.503176	−	0.864184i	$-0.332165\pi$
$953$	31.0668	1.00635	0.503176	+	0.864184i	$0.332165\pi$
$954$	0	0
$955$	4.42520	0.143196
$956$	0	0
$957$	3.84038	0.124142
$958$	0	0
$959$	−1.00853	−0.0325672
$960$	0	0
$961$	−7.25974	−0.234185
$962$	0	0
$963$	26.1835	0.843751
$964$	0	0
$965$	−5.01876	−0.161559
$966$	0	0
$967$	−29.5621	−0.950652	−0.475326	−	0.879810i	$-0.657670\pi$
$967$	−29.5621	−0.950652	−0.475326	+	0.879810i	$0.657670\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.5298	0.594650	0.297325	−	0.954776i	$-0.403906\pi$
$971$	18.5298	0.594650	0.297325	+	0.954776i	$0.403906\pi$
$972$	0	0
$973$	2.28239	0.0731701
$974$	0	0
$975$	4.14164	0.132639
$976$	0	0
$977$	−22.7303	−0.727208	−0.363604	−	0.931554i	$-0.618454\pi$
$977$	−22.7303	−0.727208	−0.363604	+	0.931554i	$0.618454\pi$
$978$	0	0
$979$	75.1489	2.40177
$980$	0	0
$981$	−17.5087	−0.559011
$982$	0	0
$983$	45.4406	1.44933	0.724666	−	0.689100i	$-0.241994\pi$
$983$	45.4406	1.44933	0.724666	+	0.689100i	$0.241994\pi$
$984$	0	0
$985$	−0.837584	−0.0266876
$986$	0	0
$987$	−0.745280	−0.0237225
$988$	0	0
$989$	5.62830	0.178969
$990$	0	0
$991$	51.9896	1.65150	0.825752	−	0.564034i	$-0.190751\pi$
$991$	51.9896	1.65150	0.825752	+	0.564034i	$0.190751\pi$
$992$	0	0
$993$	16.1356	0.512047
$994$	0	0
$995$	−7.96497	−0.252507
$996$	0	0
$997$	45.3841	1.43733	0.718664	−	0.695357i	$-0.244754\pi$
$997$	45.3841	1.43733	0.718664	+	0.695357i	$0.244754\pi$
$998$	0	0
$999$	37.0782	1.17310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.v.1.3	yes	6
4.3	odd	2		4624.2.a.bs.1.4		6
17.4	even	4		2312.2.b.o.577.8		12
17.13	even	4		2312.2.b.o.577.5		12
17.16	even	2		2312.2.a.u.1.4	✓	6
68.67	odd	2		4624.2.a.br.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.4	✓	6	17.16	even	2
2312.2.a.v.1.3	yes	6	1.1	even	1	trivial
2312.2.b.o.577.5		12	17.13	even	4
2312.2.b.o.577.8		12	17.4	even	4
4624.2.a.br.1.3		6	68.67	odd	2
4624.2.a.bs.1.4		6	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-0.857616 q^{3} -0.406879 q^{5} -0.172637 q^{7} -2.26449 q^{9} +4.89805 q^{11} +0.998924 q^{13} +0.348946 q^{15} +1.97828 q^{19} +0.148056 q^{21} -2.31854 q^{23} -4.83445 q^{25} +4.51492 q^{27} -0.914234 q^{29} -4.87240 q^{31} -4.20065 q^{33} +0.0702421 q^{35} +8.21237 q^{37} -0.856693 q^{39} +8.26313 q^{41} -2.42752 q^{43} +0.921374 q^{45} -5.03377 q^{47} -6.97020 q^{49} +13.2813 q^{53} -1.99291 q^{55} -1.69661 q^{57} -9.59776 q^{59} +10.4503 q^{61} +0.390935 q^{63} -0.406441 q^{65} -7.29977 q^{67} +1.98842 q^{69} +4.45171 q^{71} +6.48377 q^{73} +4.14610 q^{75} -0.845584 q^{77} +15.5053 q^{79} +2.92142 q^{81} +13.3575 q^{83} +0.784061 q^{87} +15.3426 q^{89} -0.172451 q^{91} +4.17865 q^{93} -0.804921 q^{95} +7.06549 q^{97} -11.0916 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} + 3 q^{7} - 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} + 9 q^{23} + 6 q^{25} + 12 q^{27} + 27 q^{29} + 9 q^{31} + 3 q^{33} + 21 q^{35} + 15 q^{37} - 12 q^{39} + 21 q^{41} + 24 q^{45}+ \cdots - 42 q^{99}+O(q^{100}) \) 6 * q + 6 * q^5 + 3 * q^7 - 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 + 9 * q^23 + 6 * q^25 + 12 * q^27 + 27 * q^29 + 9 * q^31 + 3 * q^33 + 21 * q^35 + 15 * q^37 - 12 * q^39 + 21 * q^41 + 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 + 30 * q^57 + 18 * q^61 + 24 * q^63 - 9 * q^65 - 9 * q^67 - 3 * q^69 + 6 * q^71 + 27 * q^73 - 24 * q^75 - 45 * q^77 + 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 + 30 * q^91 - 30 * q^93 + 3 * q^95 + 51 * q^97 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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