LMFDB - Embedded newform 2058.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2058,2,Mod(1,2058)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2058, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2058.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2058 = 2 \cdot 3 \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2058.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:	$16.4332127360$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	2058.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.198062 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.198062 q^{10} -1.55496 q^{11} +1.00000 q^{12} +0.890084 q^{13} -0.198062 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} +2.66487 q^{19} -0.198062 q^{20} +1.55496 q^{22} -7.52111 q^{23} -1.00000 q^{24} -4.96077 q^{25} -0.890084 q^{26} +1.00000 q^{27} +6.59179 q^{29} +0.198062 q^{30} -0.850855 q^{31} -1.00000 q^{32} -1.55496 q^{33} +6.85086 q^{34} +1.00000 q^{36} -8.29590 q^{37} -2.66487 q^{38} +0.890084 q^{39} +0.198062 q^{40} -7.87263 q^{41} +3.70171 q^{43} -1.55496 q^{44} -0.198062 q^{45} +7.52111 q^{46} -1.10992 q^{47} +1.00000 q^{48} +4.96077 q^{50} -6.85086 q^{51} +0.890084 q^{52} -2.93362 q^{53} -1.00000 q^{54} +0.307979 q^{55} +2.66487 q^{57} -6.59179 q^{58} +6.31767 q^{59} -0.198062 q^{60} +5.87800 q^{61} +0.850855 q^{62} +1.00000 q^{64} -0.176292 q^{65} +1.55496 q^{66} -10.0000 q^{67} -6.85086 q^{68} -7.52111 q^{69} -2.19806 q^{71} -1.00000 q^{72} -0.176292 q^{73} +8.29590 q^{74} -4.96077 q^{75} +2.66487 q^{76} -0.890084 q^{78} -7.70171 q^{79} -0.198062 q^{80} +1.00000 q^{81} +7.87263 q^{82} -3.50604 q^{83} +1.35690 q^{85} -3.70171 q^{86} +6.59179 q^{87} +1.55496 q^{88} -1.14914 q^{89} +0.198062 q^{90} -7.52111 q^{92} -0.850855 q^{93} +1.10992 q^{94} -0.527811 q^{95} -1.00000 q^{96} +13.9758 q^{97} -1.55496 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 5 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 5 q^{10} - 5 q^{11} + 3 q^{12} + 2 q^{13} - 5 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 9 q^{19} - 5 q^{20} + 5 q^{22} - 7 q^{23}+ \cdots - 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 5 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 5 * q^10 - 5 * q^11 + 3 * q^12 + 2 * q^13 - 5 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 9 * q^19 - 5 * q^20 + 5 * q^22 - 7 * q^23 - 3 * q^24 - 2 * q^25 - 2 * q^26 + 3 * q^27 - 8 * q^29 + 5 * q^30 + 11 * q^31 - 3 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 - 11 * q^37 - 9 * q^38 + 2 * q^39 + 5 * q^40 - 7 * q^41 - 16 * q^43 - 5 * q^44 - 5 * q^45 + 7 * q^46 - 4 * q^47 + 3 * q^48 + 2 * q^50 - 7 * q^51 + 2 * q^52 - 2 * q^53 - 3 * q^54 + 6 * q^55 + 9 * q^57 + 8 * q^58 + 2 * q^59 - 5 * q^60 - 2 * q^61 - 11 * q^62 + 3 * q^64 - 8 * q^65 + 5 * q^66 - 30 * q^67 - 7 * q^68 - 7 * q^69 - 11 * q^71 - 3 * q^72 - 8 * q^73 + 11 * q^74 - 2 * q^75 + 9 * q^76 - 2 * q^78 + 4 * q^79 - 5 * q^80 + 3 * q^81 + 7 * q^82 - 20 * q^83 + 16 * q^86 - 8 * q^87 + 5 * q^88 - 17 * q^89 + 5 * q^90 - 7 * q^92 + 11 * q^93 + 4 * q^94 - 8 * q^95 - 3 * q^96 + 4 * q^97 - 5 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−0.198062	−0.0885761	−0.0442881	−	0.999019i	$-0.514102\pi$
$5$	−0.198062	−0.0885761	−0.0442881	+	0.999019i	$0.514102\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.198062	0.0626328
$11$	−1.55496	−0.468838	−0.234419	−	0.972136i	$-0.575319\pi$
$11$	−1.55496	−0.468838	−0.234419	+	0.972136i	$0.575319\pi$
$12$	1.00000	0.288675
$13$	0.890084	0.246865	0.123432	−	0.992353i	$-0.460610\pi$
$13$	0.890084	0.246865	0.123432	+	0.992353i	$0.460610\pi$
$14$	0	0
$15$	−0.198062	−0.0511395
$16$	1.00000	0.250000
$17$	−6.85086	−1.66158	−0.830788	−	0.556589i	$-0.812110\pi$
$17$	−6.85086	−1.66158	−0.830788	+	0.556589i	$0.812110\pi$
$18$	−1.00000	−0.235702
$19$	2.66487	0.611364	0.305682	−	0.952134i	$-0.401116\pi$
$19$	2.66487	0.611364	0.305682	+	0.952134i	$0.401116\pi$
$20$	−0.198062	−0.0442881
$21$	0	0
$22$	1.55496	0.331518
$23$	−7.52111	−1.56826	−0.784130	−	0.620597i	$-0.786890\pi$
$23$	−7.52111	−1.56826	−0.784130	+	0.620597i	$0.786890\pi$
$24$	−1.00000	−0.204124
$25$	−4.96077	−0.992154
$26$	−0.890084	−0.174560
$27$	1.00000	0.192450
$28$	0	0
$29$	6.59179	1.22407	0.612033	−	0.790832i	$-0.290352\pi$
$29$	6.59179	1.22407	0.612033	+	0.790832i	$0.290352\pi$
$30$	0.198062	0.0361611
$31$	−0.850855	−0.152818	−0.0764090	−	0.997077i	$-0.524345\pi$
$31$	−0.850855	−0.152818	−0.0764090	+	0.997077i	$0.524345\pi$
$32$	−1.00000	−0.176777
$33$	−1.55496	−0.270683
$34$	6.85086	1.17491
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.29590	−1.36384	−0.681919	−	0.731428i	$-0.738854\pi$
$37$	−8.29590	−1.36384	−0.681919	+	0.731428i	$0.738854\pi$
$38$	−2.66487	−0.432300
$39$	0.890084	0.142527
$40$	0.198062	0.0313164
$41$	−7.87263	−1.22950	−0.614749	−	0.788723i	$-0.710743\pi$
$41$	−7.87263	−1.22950	−0.614749	+	0.788723i	$0.710743\pi$
$42$	0	0
$43$	3.70171	0.564506	0.282253	−	0.959340i	$-0.408918\pi$
$43$	3.70171	0.564506	0.282253	+	0.959340i	$0.408918\pi$
$44$	−1.55496	−0.234419
$45$	−0.198062	−0.0295254
$46$	7.52111	1.10893
$47$	−1.10992	−0.161898	−0.0809490	−	0.996718i	$-0.525795\pi$
$47$	−1.10992	−0.161898	−0.0809490	+	0.996718i	$0.525795\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	4.96077	0.701559
$51$	−6.85086	−0.959312
$52$	0.890084	0.123432
$53$	−2.93362	−0.402964	−0.201482	−	0.979492i	$-0.564576\pi$
$53$	−2.93362	−0.402964	−0.201482	+	0.979492i	$0.564576\pi$
$54$	−1.00000	−0.136083
$55$	0.307979	0.0415278
$56$	0	0
$57$	2.66487	0.352971
$58$	−6.59179	−0.865545
$59$	6.31767	0.822490	0.411245	−	0.911525i	$-0.365094\pi$
$59$	6.31767	0.822490	0.411245	+	0.911525i	$0.365094\pi$
$60$	−0.198062	−0.0255697
$61$	5.87800	0.752601	0.376301	−	0.926498i	$-0.377196\pi$
$61$	5.87800	0.752601	0.376301	+	0.926498i	$0.377196\pi$
$62$	0.850855	0.108059
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.176292	−0.0218663
$66$	1.55496	0.191402
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−6.85086	−0.830788
$69$	−7.52111	−0.905435
$70$	0	0
$71$	−2.19806	−0.260862	−0.130431	−	0.991457i	$-0.541636\pi$
$71$	−2.19806	−0.260862	−0.130431	+	0.991457i	$0.541636\pi$
$72$	−1.00000	−0.117851
$73$	−0.176292	−0.0206334	−0.0103167	−	0.999947i	$-0.503284\pi$
$73$	−0.176292	−0.0206334	−0.0103167	+	0.999947i	$0.503284\pi$
$74$	8.29590	0.964378
$75$	−4.96077	−0.572821
$76$	2.66487	0.305682
$77$	0	0
$78$	−0.890084	−0.100782
$79$	−7.70171	−0.866510	−0.433255	−	0.901271i	$-0.642635\pi$
$79$	−7.70171	−0.866510	−0.433255	+	0.901271i	$0.642635\pi$
$80$	−0.198062	−0.0221440
$81$	1.00000	0.111111
$82$	7.87263	0.869386
$83$	−3.50604	−0.384838	−0.192419	−	0.981313i	$-0.561633\pi$
$83$	−3.50604	−0.384838	−0.192419	+	0.981313i	$0.561633\pi$
$84$	0	0
$85$	1.35690	0.147176
$86$	−3.70171	−0.399166
$87$	6.59179	0.706714
$88$	1.55496	0.165759
$89$	−1.14914	−0.121809	−0.0609046	−	0.998144i	$-0.519399\pi$
$89$	−1.14914	−0.121809	−0.0609046	+	0.998144i	$0.519399\pi$
$90$	0.198062	0.0208776
$91$	0	0
$92$	−7.52111	−0.784130
$93$	−0.850855	−0.0882296
$94$	1.10992	0.114479
$95$	−0.527811	−0.0541523
$96$	−1.00000	−0.102062
$97$	13.9758	1.41903	0.709516	−	0.704690i	$-0.248914\pi$
$97$	13.9758	1.41903	0.709516	+	0.704690i	$0.248914\pi$
$98$	0	0
$99$	−1.55496	−0.156279
$100$	−4.96077	−0.496077
$101$	−14.7192	−1.46461	−0.732306	−	0.680976i	$-0.761556\pi$
$101$	−14.7192	−1.46461	−0.732306	+	0.680976i	$0.761556\pi$
$102$	6.85086	0.678336
$103$	−16.2717	−1.60330	−0.801651	−	0.597793i	$-0.796045\pi$
$103$	−16.2717	−1.60330	−0.801651	+	0.597793i	$0.796045\pi$
$104$	−0.890084	−0.0872799
$105$	0	0
$106$	2.93362	0.284939
$107$	−7.70410	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$107$	−7.70410	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$108$	1.00000	0.0962250
$109$	12.4644	1.19387	0.596937	−	0.802288i	$-0.296384\pi$
$109$	12.4644	1.19387	0.596937	+	0.802288i	$0.296384\pi$
$110$	−0.307979	−0.0293646
$111$	−8.29590	−0.787412
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−2.66487	−0.249588
$115$	1.48965	0.138910
$116$	6.59179	0.612033
$117$	0.890084	0.0822883
$118$	−6.31767	−0.581588
$119$	0	0
$120$	0.198062	0.0180805
$121$	−8.58211	−0.780191
$122$	−5.87800	−0.532169
$123$	−7.87263	−0.709851
$124$	−0.850855	−0.0764090
$125$	1.97285	0.176457
$126$	0	0
$127$	−4.17629	−0.370586	−0.185293	−	0.982683i	$-0.559323\pi$
$127$	−4.17629	−0.370586	−0.185293	+	0.982683i	$0.559323\pi$
$128$	−1.00000	−0.0883883
$129$	3.70171	0.325917
$130$	0.176292	0.0154618
$131$	22.3913	1.95634	0.978170	−	0.207805i	$-0.0666321\pi$
$131$	22.3913	1.95634	0.978170	+	0.207805i	$0.0666321\pi$
$132$	−1.55496	−0.135342
$133$	0	0
$134$	10.0000	0.863868
$135$	−0.198062	−0.0170465
$136$	6.85086	0.587456
$137$	−10.3720	−0.886136	−0.443068	−	0.896488i	$-0.646110\pi$
$137$	−10.3720	−0.886136	−0.443068	+	0.896488i	$0.646110\pi$
$138$	7.52111	0.640239
$139$	17.6256	1.49499	0.747494	−	0.664269i	$-0.231257\pi$
$139$	17.6256	1.49499	0.747494	+	0.664269i	$0.231257\pi$
$140$	0	0
$141$	−1.10992	−0.0934718
$142$	2.19806	0.184457
$143$	−1.38404	−0.115739
$144$	1.00000	0.0833333
$145$	−1.30559	−0.108423
$146$	0.176292	0.0145900
$147$	0	0
$148$	−8.29590	−0.681919
$149$	−1.42758	−0.116952	−0.0584761	−	0.998289i	$-0.518624\pi$
$149$	−1.42758	−0.116952	−0.0584761	+	0.998289i	$0.518624\pi$
$150$	4.96077	0.405045
$151$	0.944378	0.0768524	0.0384262	−	0.999261i	$-0.487766\pi$
$151$	0.944378	0.0768524	0.0384262	+	0.999261i	$0.487766\pi$
$152$	−2.66487	−0.216150
$153$	−6.85086	−0.553859
$154$	0	0
$155$	0.168522	0.0135360
$156$	0.890084	0.0712637
$157$	24.3720	1.94509	0.972547	−	0.232706i	$-0.0747580\pi$
$157$	24.3720	1.94509	0.972547	+	0.232706i	$0.0747580\pi$
$158$	7.70171	0.612715
$159$	−2.93362	−0.232652
$160$	0.198062	0.0156582
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−0.121998	−0.00955561	−0.00477780	−	0.999989i	$-0.501521\pi$
$163$	−0.121998	−0.00955561	−0.00477780	+	0.999989i	$0.501521\pi$
$164$	−7.87263	−0.614749
$165$	0.307979	0.0239761
$166$	3.50604	0.272122
$167$	−13.7995	−1.06784	−0.533920	−	0.845535i	$-0.679282\pi$
$167$	−13.7995	−1.06784	−0.533920	+	0.845535i	$0.679282\pi$
$168$	0	0
$169$	−12.2078	−0.939058
$170$	−1.35690	−0.104069
$171$	2.66487	0.203788
$172$	3.70171	0.282253
$173$	−1.56033	−0.118630	−0.0593150	−	0.998239i	$-0.518892\pi$
$173$	−1.56033	−0.118630	−0.0593150	+	0.998239i	$0.518892\pi$
$174$	−6.59179	−0.499723
$175$	0	0
$176$	−1.55496	−0.117209
$177$	6.31767	0.474865
$178$	1.14914	0.0861321
$179$	−11.0271	−0.824208	−0.412104	−	0.911137i	$-0.635206\pi$
$179$	−11.0271	−0.824208	−0.412104	+	0.911137i	$0.635206\pi$
$180$	−0.198062	−0.0147627
$181$	11.9215	0.886121	0.443061	−	0.896492i	$-0.353893\pi$
$181$	11.9215	0.886121	0.443061	+	0.896492i	$0.353893\pi$
$182$	0	0
$183$	5.87800	0.434514
$184$	7.52111	0.554463
$185$	1.64310	0.120803
$186$	0.850855	0.0623877
$187$	10.6528	0.779009
$188$	−1.10992	−0.0809490
$189$	0	0
$190$	0.527811	0.0382914
$191$	−25.1444	−1.81938	−0.909691	−	0.415286i	$-0.863682\pi$
$191$	−25.1444	−1.81938	−0.909691	+	0.415286i	$0.863682\pi$
$192$	1.00000	0.0721688
$193$	−18.3502	−1.32088	−0.660438	−	0.750881i	$-0.729629\pi$
$193$	−18.3502	−1.32088	−0.660438	+	0.750881i	$0.729629\pi$
$194$	−13.9758	−1.00341
$195$	−0.176292	−0.0126245
$196$	0	0
$197$	−22.2392	−1.58448	−0.792239	−	0.610211i	$-0.791085\pi$
$197$	−22.2392	−1.58448	−0.792239	+	0.610211i	$0.791085\pi$
$198$	1.55496	0.110506
$199$	6.25236	0.443218	0.221609	−	0.975136i	$-0.428869\pi$
$199$	6.25236	0.443218	0.221609	+	0.975136i	$0.428869\pi$
$200$	4.96077	0.350780
$201$	−10.0000	−0.705346
$202$	14.7192	1.03564
$203$	0	0
$204$	−6.85086	−0.479656
$205$	1.55927	0.108904
$206$	16.2717	1.13371
$207$	−7.52111	−0.522753
$208$	0.890084	0.0617162
$209$	−4.14377	−0.286630
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−2.93362	−0.201482
$213$	−2.19806	−0.150609
$214$	7.70410	0.526642
$215$	−0.733169	−0.0500017
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−12.4644	−0.844197
$219$	−0.176292	−0.0119127
$220$	0.307979	0.0207639
$221$	−6.09783	−0.410185
$222$	8.29590	0.556784
$223$	13.7942	0.923726	0.461863	−	0.886951i	$-0.347181\pi$
$223$	13.7942	0.923726	0.461863	+	0.886951i	$0.347181\pi$
$224$	0	0
$225$	−4.96077	−0.330718
$226$	0	0
$227$	−14.8853	−0.987972	−0.493986	−	0.869470i	$-0.664461\pi$
$227$	−14.8853	−0.987972	−0.493986	+	0.869470i	$0.664461\pi$
$228$	2.66487	0.176486
$229$	3.76941	0.249090	0.124545	−	0.992214i	$-0.460253\pi$
$229$	3.76941	0.249090	0.124545	+	0.992214i	$0.460253\pi$
$230$	−1.48965	−0.0982244
$231$	0	0
$232$	−6.59179	−0.432772
$233$	7.08575	0.464203	0.232102	−	0.972692i	$-0.425440\pi$
$233$	7.08575	0.464203	0.232102	+	0.972692i	$0.425440\pi$
$234$	−0.890084	−0.0581866
$235$	0.219833	0.0143403
$236$	6.31767	0.411245
$237$	−7.70171	−0.500280
$238$	0	0
$239$	3.06100	0.198000	0.0989998	−	0.995087i	$-0.468436\pi$
$239$	3.06100	0.198000	0.0989998	+	0.995087i	$0.468436\pi$
$240$	−0.198062	−0.0127849
$241$	−18.6896	−1.20390	−0.601952	−	0.798532i	$-0.705610\pi$
$241$	−18.6896	−1.20390	−0.601952	+	0.798532i	$0.705610\pi$
$242$	8.58211	0.551679
$243$	1.00000	0.0641500
$244$	5.87800	0.376301
$245$	0	0
$246$	7.87263	0.501940
$247$	2.37196	0.150924
$248$	0.850855	0.0540294
$249$	−3.50604	−0.222186
$250$	−1.97285	−0.124774
$251$	−13.4276	−0.847542	−0.423771	−	0.905769i	$-0.639294\pi$
$251$	−13.4276	−0.847542	−0.423771	+	0.905769i	$0.639294\pi$
$252$	0	0
$253$	11.6950	0.735259
$254$	4.17629	0.262044
$255$	1.35690	0.0849721
$256$	1.00000	0.0625000
$257$	16.2717	1.01500	0.507501	−	0.861651i	$-0.330569\pi$
$257$	16.2717	1.01500	0.507501	+	0.861651i	$0.330569\pi$
$258$	−3.70171	−0.230458
$259$	0	0
$260$	−0.176292	−0.0109332
$261$	6.59179	0.408022
$262$	−22.3913	−1.38334
$263$	−3.03385	−0.187075	−0.0935377	−	0.995616i	$-0.529818\pi$
$263$	−3.03385	−0.187075	−0.0935377	+	0.995616i	$0.529818\pi$
$264$	1.55496	0.0957011
$265$	0.581040	0.0356930
$266$	0	0
$267$	−1.14914	−0.0703265
$268$	−10.0000	−0.610847
$269$	6.36658	0.388178	0.194089	−	0.980984i	$-0.437825\pi$
$269$	6.36658	0.388178	0.194089	+	0.980984i	$0.437825\pi$
$270$	0.198062	0.0120537
$271$	−17.2597	−1.04845	−0.524225	−	0.851580i	$-0.675645\pi$
$271$	−17.2597	−1.04845	−0.524225	+	0.851580i	$0.675645\pi$
$272$	−6.85086	−0.415394
$273$	0	0
$274$	10.3720	0.626593
$275$	7.71379	0.465159
$276$	−7.52111	−0.452717
$277$	−11.8823	−0.713939	−0.356970	−	0.934116i	$-0.616190\pi$
$277$	−11.8823	−0.713939	−0.356970	+	0.934116i	$0.616190\pi$
$278$	−17.6256	−1.05712
$279$	−0.850855	−0.0509394
$280$	0	0
$281$	14.1414	0.843604	0.421802	−	0.906688i	$-0.361398\pi$
$281$	14.1414	0.843604	0.421802	+	0.906688i	$0.361398\pi$
$282$	1.10992	0.0660946
$283$	1.25773	0.0747645	0.0373822	−	0.999301i	$-0.488098\pi$
$283$	1.25773	0.0747645	0.0373822	+	0.999301i	$0.488098\pi$
$284$	−2.19806	−0.130431
$285$	−0.527811	−0.0312648
$286$	1.38404	0.0818402
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	29.9342	1.76084
$290$	1.30559	0.0766666
$291$	13.9758	0.819278
$292$	−0.176292	−0.0103167
$293$	21.3163	1.24531	0.622657	−	0.782495i	$-0.286053\pi$
$293$	21.3163	1.24531	0.622657	+	0.782495i	$0.286053\pi$
$294$	0	0
$295$	−1.25129	−0.0728530
$296$	8.29590	0.482189
$297$	−1.55496	−0.0902278
$298$	1.42758	0.0826977
$299$	−6.69441	−0.387148
$300$	−4.96077	−0.286410
$301$	0	0
$302$	−0.944378	−0.0543428
$303$	−14.7192	−0.845594
$304$	2.66487	0.152841
$305$	−1.16421	−0.0666625
$306$	6.85086	0.391637
$307$	14.6853	0.838135	0.419068	−	0.907955i	$-0.362357\pi$
$307$	14.6853	0.838135	0.419068	+	0.907955i	$0.362357\pi$
$308$	0	0
$309$	−16.2717	−0.925667
$310$	−0.168522	−0.00957142
$311$	−11.9022	−0.674910	−0.337455	−	0.941342i	$-0.609566\pi$
$311$	−11.9022	−0.674910	−0.337455	+	0.941342i	$0.609566\pi$
$312$	−0.890084	−0.0503911
$313$	−26.9202	−1.52162	−0.760810	−	0.648974i	$-0.775198\pi$
$313$	−26.9202	−1.52162	−0.760810	+	0.648974i	$0.775198\pi$
$314$	−24.3720	−1.37539
$315$	0	0
$316$	−7.70171	−0.433255
$317$	32.2500	1.81134	0.905669	−	0.423985i	$-0.139369\pi$
$317$	32.2500	1.81134	0.905669	+	0.423985i	$0.139369\pi$
$318$	2.93362	0.164509
$319$	−10.2500	−0.573888
$320$	−0.198062	−0.0110720
$321$	−7.70410	−0.430001
$322$	0	0
$323$	−18.2567	−1.01583
$324$	1.00000	0.0555556
$325$	−4.41550	−0.244928
$326$	0.121998	0.00675684
$327$	12.4644	0.689284
$328$	7.87263	0.434693
$329$	0	0
$330$	−0.307979	−0.0169537
$331$	−17.1836	−0.944495	−0.472248	−	0.881466i	$-0.656557\pi$
$331$	−17.1836	−0.944495	−0.472248	+	0.881466i	$0.656557\pi$
$332$	−3.50604	−0.192419
$333$	−8.29590	−0.454612
$334$	13.7995	0.755077
$335$	1.98062	0.108213
$336$	0	0
$337$	−21.2556	−1.15787	−0.578933	−	0.815375i	$-0.696531\pi$
$337$	−21.2556	−1.15787	−0.578933	+	0.815375i	$0.696531\pi$
$338$	12.2078	0.664014
$339$	0	0
$340$	1.35690	0.0735880
$341$	1.32304	0.0716469
$342$	−2.66487	−0.144100
$343$	0	0
$344$	−3.70171	−0.199583
$345$	1.48965	0.0801999
$346$	1.56033	0.0838841
$347$	7.39506	0.396988	0.198494	−	0.980102i	$-0.436395\pi$
$347$	7.39506	0.396988	0.198494	+	0.980102i	$0.436395\pi$
$348$	6.59179	0.353357
$349$	−10.9772	−0.587594	−0.293797	−	0.955868i	$-0.594919\pi$
$349$	−10.9772	−0.587594	−0.293797	+	0.955868i	$0.594919\pi$
$350$	0	0
$351$	0.890084	0.0475092
$352$	1.55496	0.0828795
$353$	5.64742	0.300582	0.150291	−	0.988642i	$-0.451979\pi$
$353$	5.64742	0.300582	0.150291	+	0.988642i	$0.451979\pi$
$354$	−6.31767	−0.335780
$355$	0.435353	0.0231061
$356$	−1.14914	−0.0609046
$357$	0	0
$358$	11.0271	0.582803
$359$	16.6571	0.879128	0.439564	−	0.898211i	$-0.355133\pi$
$359$	16.6571	0.879128	0.439564	+	0.898211i	$0.355133\pi$
$360$	0.198062	0.0104388
$361$	−11.8984	−0.626234
$362$	−11.9215	−0.626582
$363$	−8.58211	−0.450444
$364$	0	0
$365$	0.0349168	0.00182763
$366$	−5.87800	−0.307248
$367$	18.2000	0.950031	0.475016	−	0.879977i	$-0.342442\pi$
$367$	18.2000	0.950031	0.475016	+	0.879977i	$0.342442\pi$
$368$	−7.52111	−0.392065
$369$	−7.87263	−0.409833
$370$	−1.64310	−0.0854209
$371$	0	0
$372$	−0.850855	−0.0441148
$373$	21.8582	1.13177	0.565886	−	0.824483i	$-0.308534\pi$
$373$	21.8582	1.13177	0.565886	+	0.824483i	$0.308534\pi$
$374$	−10.6528	−0.550843
$375$	1.97285	0.101878
$376$	1.10992	0.0572396
$377$	5.86725	0.302179
$378$	0	0
$379$	−29.5555	−1.51817	−0.759083	−	0.650994i	$-0.774352\pi$
$379$	−29.5555	−1.51817	−0.759083	+	0.650994i	$0.774352\pi$
$380$	−0.527811	−0.0270761
$381$	−4.17629	−0.213958
$382$	25.1444	1.28650
$383$	30.5676	1.56193	0.780966	−	0.624573i	$-0.214727\pi$
$383$	30.5676	1.56193	0.780966	+	0.624573i	$0.214727\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.3502	0.934000
$387$	3.70171	0.188169
$388$	13.9758	0.709516
$389$	4.24400	0.215179	0.107590	−	0.994195i	$-0.465687\pi$
$389$	4.24400	0.215179	0.107590	+	0.994195i	$0.465687\pi$
$390$	0.176292	0.00892689
$391$	51.5260	2.60578
$392$	0	0
$393$	22.3913	1.12949
$394$	22.2392	1.12040
$395$	1.52542	0.0767521
$396$	−1.55496	−0.0781396
$397$	−2.79225	−0.140139	−0.0700695	−	0.997542i	$-0.522322\pi$
$397$	−2.79225	−0.140139	−0.0700695	+	0.997542i	$0.522322\pi$
$398$	−6.25236	−0.313402
$399$	0	0
$400$	−4.96077	−0.248039
$401$	33.8538	1.69058	0.845290	−	0.534308i	$-0.179428\pi$
$401$	33.8538	1.69058	0.845290	+	0.534308i	$0.179428\pi$
$402$	10.0000	0.498755
$403$	−0.757332	−0.0377254
$404$	−14.7192	−0.732306
$405$	−0.198062	−0.00984179
$406$	0	0
$407$	12.8998	0.639418
$408$	6.85086	0.339168
$409$	−29.3927	−1.45337	−0.726687	−	0.686969i	$-0.758941\pi$
$409$	−29.3927	−1.45337	−0.726687	+	0.686969i	$0.758941\pi$
$410$	−1.55927	−0.0770069
$411$	−10.3720	−0.511611
$412$	−16.2717	−0.801651
$413$	0	0
$414$	7.52111	0.369642
$415$	0.694414	0.0340875
$416$	−0.890084	−0.0436399
$417$	17.6256	0.863132
$418$	4.14377	0.202678
$419$	−22.8853	−1.11802	−0.559010	−	0.829161i	$-0.688819\pi$
$419$	−22.8853	−1.11802	−0.559010	+	0.829161i	$0.688819\pi$
$420$	0	0
$421$	37.3545	1.82055	0.910274	−	0.414007i	$-0.135871\pi$
$421$	37.3545	1.82055	0.910274	+	0.414007i	$0.135871\pi$
$422$	0	0
$423$	−1.10992	−0.0539660
$424$	2.93362	0.142469
$425$	33.9855	1.64854
$426$	2.19806	0.106496
$427$	0	0
$428$	−7.70410	−0.372392
$429$	−1.38404	−0.0668222
$430$	0.733169	0.0353566
$431$	3.96184	0.190835	0.0954175	−	0.995437i	$-0.469581\pi$
$431$	3.96184	0.190835	0.0954175	+	0.995437i	$0.469581\pi$
$432$	1.00000	0.0481125
$433$	32.6896	1.57096	0.785482	−	0.618885i	$-0.212415\pi$
$433$	32.6896	1.57096	0.785482	+	0.618885i	$0.212415\pi$
$434$	0	0
$435$	−1.30559	−0.0625980
$436$	12.4644	0.596937
$437$	−20.0428	−0.958777
$438$	0.176292	0.00842356
$439$	−31.4282	−1.49998	−0.749992	−	0.661446i	$-0.769943\pi$
$439$	−31.4282	−1.49998	−0.749992	+	0.661446i	$0.769943\pi$
$440$	−0.307979	−0.0146823
$441$	0	0
$442$	6.09783	0.290044
$443$	−9.55496	−0.453970	−0.226985	−	0.973898i	$-0.572887\pi$
$443$	−9.55496	−0.453970	−0.226985	+	0.973898i	$0.572887\pi$
$444$	−8.29590	−0.393706
$445$	0.227602	0.0107894
$446$	−13.7942	−0.653173
$447$	−1.42758	−0.0675224
$448$	0	0
$449$	19.7366	0.931429	0.465715	−	0.884935i	$-0.345797\pi$
$449$	19.7366	0.931429	0.465715	+	0.884935i	$0.345797\pi$
$450$	4.96077	0.233853
$451$	12.2416	0.576435
$452$	0	0
$453$	0.944378	0.0443707
$454$	14.8853	0.698602
$455$	0	0
$456$	−2.66487	−0.124794
$457$	28.0054	1.31004	0.655018	−	0.755613i	$-0.272661\pi$
$457$	28.0054	1.31004	0.655018	+	0.755613i	$0.272661\pi$
$458$	−3.76941	−0.176133
$459$	−6.85086	−0.319771
$460$	1.48965	0.0694552
$461$	−30.7482	−1.43209	−0.716044	−	0.698055i	$-0.754049\pi$
$461$	−30.7482	−1.43209	−0.716044	+	0.698055i	$0.754049\pi$
$462$	0	0
$463$	33.0944	1.53803	0.769013	−	0.639233i	$-0.220748\pi$
$463$	33.0944	1.53803	0.769013	+	0.639233i	$0.220748\pi$
$464$	6.59179	0.306016
$465$	0.168522	0.00781503
$466$	−7.08575	−0.328241
$467$	−16.8224	−0.778447	−0.389223	−	0.921143i	$-0.627257\pi$
$467$	−16.8224	−0.778447	−0.389223	+	0.921143i	$0.627257\pi$
$468$	0.890084	0.0411441
$469$	0	0
$470$	−0.219833	−0.0101401
$471$	24.3720	1.12300
$472$	−6.31767	−0.290794
$473$	−5.75600	−0.264661
$474$	7.70171	0.353751
$475$	−13.2198	−0.606568
$476$	0	0
$477$	−2.93362	−0.134321
$478$	−3.06100	−0.140007
$479$	20.1849	0.922272	0.461136	−	0.887329i	$-0.347442\pi$
$479$	20.1849	0.922272	0.461136	+	0.887329i	$0.347442\pi$
$480$	0.198062	0.00904026
$481$	−7.38404	−0.336683
$482$	18.6896	0.851289
$483$	0	0
$484$	−8.58211	−0.390096
$485$	−2.76809	−0.125692
$486$	−1.00000	−0.0453609
$487$	9.62325	0.436071	0.218036	−	0.975941i	$-0.430035\pi$
$487$	9.62325	0.436071	0.218036	+	0.975941i	$0.430035\pi$
$488$	−5.87800	−0.266085
$489$	−0.121998	−0.00551693
$490$	0	0
$491$	−1.13169	−0.0510723	−0.0255361	−	0.999674i	$-0.508129\pi$
$491$	−1.13169	−0.0510723	−0.0255361	+	0.999674i	$0.508129\pi$
$492$	−7.87263	−0.354925
$493$	−45.1594	−2.03388
$494$	−2.37196	−0.106720
$495$	0.307979	0.0138426
$496$	−0.850855	−0.0382045
$497$	0	0
$498$	3.50604	0.157109
$499$	−2.96376	−0.132676	−0.0663380	−	0.997797i	$-0.521132\pi$
$499$	−2.96376	−0.132676	−0.0663380	+	0.997797i	$0.521132\pi$
$500$	1.97285	0.0882287
$501$	−13.7995	−0.616518
$502$	13.4276	0.599302
$503$	−38.9396	−1.73623	−0.868115	−	0.496363i	$-0.834669\pi$
$503$	−38.9396	−1.73623	−0.868115	+	0.496363i	$0.834669\pi$
$504$	0	0
$505$	2.91531	0.129730
$506$	−11.6950	−0.519906
$507$	−12.2078	−0.542165
$508$	−4.17629	−0.185293
$509$	34.5483	1.53132	0.765662	−	0.643243i	$-0.222411\pi$
$509$	34.5483	1.53132	0.765662	+	0.643243i	$0.222411\pi$
$510$	−1.35690	−0.0600844
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.66487	0.117657
$514$	−16.2717	−0.717715
$515$	3.22282	0.142014
$516$	3.70171	0.162959
$517$	1.72587	0.0759038
$518$	0	0
$519$	−1.56033	−0.0684911
$520$	0.176292	0.00773092
$521$	31.5415	1.38186	0.690930	−	0.722922i	$-0.257201\pi$
$521$	31.5415	1.38186	0.690930	+	0.722922i	$0.257201\pi$
$522$	−6.59179	−0.288515
$523$	33.2277	1.45295	0.726473	−	0.687195i	$-0.241158\pi$
$523$	33.2277	1.45295	0.726473	+	0.687195i	$0.241158\pi$
$524$	22.3913	0.978170
$525$	0	0
$526$	3.03385	0.132282
$527$	5.82908	0.253919
$528$	−1.55496	−0.0676709
$529$	33.5670	1.45944
$530$	−0.581040	−0.0252388
$531$	6.31767	0.274163
$532$	0	0
$533$	−7.00730	−0.303520
$534$	1.14914	0.0497284
$535$	1.52589	0.0659701
$536$	10.0000	0.431934
$537$	−11.0271	−0.475857
$538$	−6.36658	−0.274483
$539$	0	0
$540$	−0.198062	−0.00852324
$541$	29.0358	1.24835	0.624173	−	0.781286i	$-0.285436\pi$
$541$	29.0358	1.24835	0.624173	+	0.781286i	$0.285436\pi$
$542$	17.2597	0.741366
$543$	11.9215	0.511602
$544$	6.85086	0.293728
$545$	−2.46873	−0.105749
$546$	0	0
$547$	−1.78017	−0.0761145	−0.0380572	−	0.999276i	$-0.512117\pi$
$547$	−1.78017	−0.0761145	−0.0380572	+	0.999276i	$0.512117\pi$
$548$	−10.3720	−0.443068
$549$	5.87800	0.250867
$550$	−7.71379	−0.328917
$551$	17.5663	0.748350
$552$	7.52111	0.320120
$553$	0	0
$554$	11.8823	0.504831
$555$	1.64310	0.0697459
$556$	17.6256	0.747494
$557$	30.8745	1.30820	0.654098	−	0.756410i	$-0.273048\pi$
$557$	30.8745	1.30820	0.654098	+	0.756410i	$0.273048\pi$
$558$	0.850855	0.0360196
$559$	3.29483	0.139357
$560$	0	0
$561$	10.6528	0.449761
$562$	−14.1414	−0.596518
$563$	44.9724	1.89536	0.947680	−	0.319222i	$-0.103422\pi$
$563$	44.9724	1.89536	0.947680	+	0.319222i	$0.103422\pi$
$564$	−1.10992	−0.0467359
$565$	0	0
$566$	−1.25773	−0.0528665
$567$	0	0
$568$	2.19806	0.0922286
$569$	6.53750	0.274066	0.137033	−	0.990566i	$-0.456243\pi$
$569$	6.53750	0.274066	0.137033	+	0.990566i	$0.456243\pi$
$570$	0.527811	0.0221076
$571$	37.7861	1.58130	0.790650	−	0.612268i	$-0.209743\pi$
$571$	37.7861	1.58130	0.790650	+	0.612268i	$0.209743\pi$
$572$	−1.38404	−0.0578697
$573$	−25.1444	−1.05042
$574$	0	0
$575$	37.3105	1.55595
$576$	1.00000	0.0416667
$577$	16.2064	0.674682	0.337341	−	0.941382i	$-0.390472\pi$
$577$	16.2064	0.674682	0.337341	+	0.941382i	$0.390472\pi$
$578$	−29.9342	−1.24510
$579$	−18.3502	−0.762608
$580$	−1.30559	−0.0542115
$581$	0	0
$582$	−13.9758	−0.579317
$583$	4.56166	0.188925
$584$	0.176292	0.00729501
$585$	−0.176292	−0.00728878
$586$	−21.3163	−0.880570
$587$	23.3163	0.962368	0.481184	−	0.876620i	$-0.340207\pi$
$587$	23.3163	0.962368	0.481184	+	0.876620i	$0.340207\pi$
$588$	0	0
$589$	−2.26742	−0.0934275
$590$	1.25129	0.0515149
$591$	−22.2392	−0.914799
$592$	−8.29590	−0.340959
$593$	−34.4432	−1.41441	−0.707207	−	0.707006i	$-0.750045\pi$
$593$	−34.4432	−1.41441	−0.707207	+	0.707006i	$0.750045\pi$
$594$	1.55496	0.0638007
$595$	0	0
$596$	−1.42758	−0.0584761
$597$	6.25236	0.255892
$598$	6.69441	0.273755
$599$	−5.07500	−0.207359	−0.103679	−	0.994611i	$-0.533062\pi$
$599$	−5.07500	−0.207359	−0.103679	+	0.994611i	$0.533062\pi$
$600$	4.96077	0.202523
$601$	−46.5870	−1.90032	−0.950162	−	0.311757i	$-0.899082\pi$
$601$	−46.5870	−1.90032	−0.950162	+	0.311757i	$0.899082\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0.944378	0.0384262
$605$	1.69979	0.0691063
$606$	14.7192	0.597925
$607$	13.5084	0.548290	0.274145	−	0.961688i	$-0.411605\pi$
$607$	13.5084	0.548290	0.274145	+	0.961688i	$0.411605\pi$
$608$	−2.66487	−0.108075
$609$	0	0
$610$	1.16421	0.0471375
$611$	−0.987918	−0.0399669
$612$	−6.85086	−0.276929
$613$	−29.8049	−1.20381	−0.601905	−	0.798568i	$-0.705591\pi$
$613$	−29.8049	−1.20381	−0.601905	+	0.798568i	$0.705591\pi$
$614$	−14.6853	−0.592651
$615$	1.55927	0.0628758
$616$	0	0
$617$	−20.3129	−0.817766	−0.408883	−	0.912587i	$-0.634082\pi$
$617$	−20.3129	−0.817766	−0.408883	+	0.912587i	$0.634082\pi$
$618$	16.2717	0.654545
$619$	2.07905	0.0835640	0.0417820	−	0.999127i	$-0.486697\pi$
$619$	2.07905	0.0835640	0.0417820	+	0.999127i	$0.486697\pi$
$620$	0.168522	0.00676802
$621$	−7.52111	−0.301812
$622$	11.9022	0.477233
$623$	0	0
$624$	0.890084	0.0356319
$625$	24.4131	0.976524
$626$	26.9202	1.07595
$627$	−4.14377	−0.165486
$628$	24.3720	0.972547
$629$	56.8340	2.26612
$630$	0	0
$631$	46.2344	1.84056	0.920282	−	0.391256i	$-0.127959\pi$
$631$	46.2344	1.84056	0.920282	+	0.391256i	$0.127959\pi$
$632$	7.70171	0.306358
$633$	0	0
$634$	−32.2500	−1.28081
$635$	0.827166	0.0328251
$636$	−2.93362	−0.116326
$637$	0	0
$638$	10.2500	0.405800
$639$	−2.19806	−0.0869540
$640$	0.198062	0.00782910
$641$	26.3720	1.04163	0.520815	−	0.853670i	$-0.325628\pi$
$641$	26.3720	1.04163	0.520815	+	0.853670i	$0.325628\pi$
$642$	7.70410	0.304057
$643$	15.5120	0.611734	0.305867	−	0.952074i	$-0.401054\pi$
$643$	15.5120	0.611734	0.305867	+	0.952074i	$0.401054\pi$
$644$	0	0
$645$	−0.733169	−0.0288685
$646$	18.2567	0.718299
$647$	27.0267	1.06253	0.531264	−	0.847206i	$-0.321717\pi$
$647$	27.0267	1.06253	0.531264	+	0.847206i	$0.321717\pi$
$648$	−1.00000	−0.0392837
$649$	−9.82371	−0.385614
$650$	4.41550	0.173190
$651$	0	0
$652$	−0.121998	−0.00477780
$653$	−15.2862	−0.598196	−0.299098	−	0.954222i	$-0.596686\pi$
$653$	−15.2862	−0.598196	−0.299098	+	0.954222i	$0.596686\pi$
$654$	−12.4644	−0.487397
$655$	−4.43488	−0.173285
$656$	−7.87263	−0.307374
$657$	−0.176292	−0.00687781
$658$	0	0
$659$	−4.43967	−0.172945	−0.0864724	−	0.996254i	$-0.527559\pi$
$659$	−4.43967	−0.172945	−0.0864724	+	0.996254i	$0.527559\pi$
$660$	0.307979	0.0119880
$661$	30.4892	1.18589	0.592946	−	0.805242i	$-0.297965\pi$
$661$	30.4892	1.18589	0.592946	+	0.805242i	$0.297965\pi$
$662$	17.1836	0.667859
$663$	−6.09783	−0.236820
$664$	3.50604	0.136061
$665$	0	0
$666$	8.29590	0.321459
$667$	−49.5776	−1.91965
$668$	−13.7995	−0.533920
$669$	13.7942	0.533313
$670$	−1.98062	−0.0765181
$671$	−9.14005	−0.352848
$672$	0	0
$673$	−41.0901	−1.58391	−0.791953	−	0.610582i	$-0.790935\pi$
$673$	−41.0901	−1.58391	−0.791953	+	0.610582i	$0.790935\pi$
$674$	21.2556	0.818735
$675$	−4.96077	−0.190940
$676$	−12.2078	−0.469529
$677$	31.3437	1.20464	0.602319	−	0.798255i	$-0.294243\pi$
$677$	31.3437	1.20464	0.602319	+	0.798255i	$0.294243\pi$
$678$	0	0
$679$	0	0
$680$	−1.35690	−0.0520346
$681$	−14.8853	−0.570406
$682$	−1.32304	−0.0506620
$683$	−40.8829	−1.56434	−0.782170	−	0.623065i	$-0.785887\pi$
$683$	−40.8829	−1.56434	−0.782170	+	0.623065i	$0.785887\pi$
$684$	2.66487	0.101894
$685$	2.05429	0.0784905
$686$	0	0
$687$	3.76941	0.143812
$688$	3.70171	0.141126
$689$	−2.61117	−0.0994777
$690$	−1.48965	−0.0567099
$691$	−17.6877	−0.672872	−0.336436	−	0.941706i	$-0.609222\pi$
$691$	−17.6877	−0.672872	−0.336436	+	0.941706i	$0.609222\pi$
$692$	−1.56033	−0.0593150
$693$	0	0
$694$	−7.39506	−0.280713
$695$	−3.49098	−0.132420
$696$	−6.59179	−0.249861
$697$	53.9342	2.04290
$698$	10.9772	0.415492
$699$	7.08575	0.268008
$700$	0	0
$701$	4.28754	0.161938	0.0809690	−	0.996717i	$-0.474199\pi$
$701$	4.28754	0.161938	0.0809690	+	0.996717i	$0.474199\pi$
$702$	−0.890084	−0.0335940
$703$	−22.1075	−0.833801
$704$	−1.55496	−0.0586047
$705$	0.219833	0.00827937
$706$	−5.64742	−0.212543
$707$	0	0
$708$	6.31767	0.237432
$709$	11.3556	0.426467	0.213234	−	0.977001i	$-0.431600\pi$
$709$	11.3556	0.426467	0.213234	+	0.977001i	$0.431600\pi$
$710$	−0.435353	−0.0163385
$711$	−7.70171	−0.288837
$712$	1.14914	0.0430660
$713$	6.39937	0.239658
$714$	0	0
$715$	0.274127	0.0102518
$716$	−11.0271	−0.412104
$717$	3.06100	0.114315
$718$	−16.6571	−0.621638
$719$	1.39480	0.0520171	0.0260086	−	0.999662i	$-0.491720\pi$
$719$	1.39480	0.0520171	0.0260086	+	0.999662i	$0.491720\pi$
$720$	−0.198062	−0.00738134
$721$	0	0
$722$	11.8984	0.442814
$723$	−18.6896	−0.695075
$724$	11.9215	0.443061
$725$	−32.7004	−1.21446
$726$	8.58211	0.318512
$727$	−31.4795	−1.16751	−0.583755	−	0.811930i	$-0.698417\pi$
$727$	−31.4795	−1.16751	−0.583755	+	0.811930i	$0.698417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−0.0349168	−0.00129233
$731$	−25.3599	−0.937969
$732$	5.87800	0.217257
$733$	4.67025	0.172500	0.0862498	−	0.996274i	$-0.472512\pi$
$733$	4.67025	0.172500	0.0862498	+	0.996274i	$0.472512\pi$
$734$	−18.2000	−0.671774
$735$	0	0
$736$	7.52111	0.277232
$737$	15.5496	0.572776
$738$	7.87263	0.289795
$739$	−35.5060	−1.30611	−0.653055	−	0.757310i	$-0.726513\pi$
$739$	−35.5060	−1.30611	−0.653055	+	0.757310i	$0.726513\pi$
$740$	1.64310	0.0604017
$741$	2.37196	0.0871362
$742$	0	0
$743$	−10.6987	−0.392498	−0.196249	−	0.980554i	$-0.562876\pi$
$743$	−10.6987	−0.392498	−0.196249	+	0.980554i	$0.562876\pi$
$744$	0.850855	0.0311939
$745$	0.282750	0.0103592
$746$	−21.8582	−0.800284
$747$	−3.50604	−0.128279
$748$	10.6528	0.389505
$749$	0	0
$750$	−1.97285	−0.0720384
$751$	−44.2646	−1.61524	−0.807618	−	0.589706i	$-0.799244\pi$
$751$	−44.2646	−1.61524	−0.807618	+	0.589706i	$0.799244\pi$
$752$	−1.10992	−0.0404745
$753$	−13.4276	−0.489328
$754$	−5.86725	−0.213673
$755$	−0.187046	−0.00680729
$756$	0	0
$757$	−18.9390	−0.688350	−0.344175	−	0.938906i	$-0.611841\pi$
$757$	−18.9390	−0.688350	−0.344175	+	0.938906i	$0.611841\pi$
$758$	29.5555	1.07351
$759$	11.6950	0.424502
$760$	0.527811	0.0191457
$761$	−30.0441	−1.08910	−0.544549	−	0.838729i	$-0.683299\pi$
$761$	−30.0441	−1.08910	−0.544549	+	0.838729i	$0.683299\pi$
$762$	4.17629	0.151291
$763$	0	0
$764$	−25.1444	−0.909691
$765$	1.35690	0.0490587
$766$	−30.5676	−1.10445
$767$	5.62325	0.203044
$768$	1.00000	0.0360844
$769$	19.6233	0.707633	0.353816	−	0.935315i	$-0.384884\pi$
$769$	19.6233	0.707633	0.353816	+	0.935315i	$0.384884\pi$
$770$	0	0
$771$	16.2717	0.586012
$772$	−18.3502	−0.660438
$773$	7.09219	0.255089	0.127544	−	0.991833i	$-0.459291\pi$
$773$	7.09219	0.255089	0.127544	+	0.991833i	$0.459291\pi$
$774$	−3.70171	−0.133055
$775$	4.22090	0.151619
$776$	−13.9758	−0.501703
$777$	0	0
$778$	−4.24400	−0.152155
$779$	−20.9796	−0.751671
$780$	−0.176292	−0.00631227
$781$	3.41789	0.122302
$782$	−51.5260	−1.84257
$783$	6.59179	0.235571
$784$	0	0
$785$	−4.82717	−0.172289
$786$	−22.3913	−0.798673
$787$	−38.5295	−1.37343	−0.686714	−	0.726928i	$-0.740947\pi$
$787$	−38.5295	−1.37343	−0.686714	+	0.726928i	$0.740947\pi$
$788$	−22.2392	−0.792239
$789$	−3.03385	−0.108008
$790$	−1.52542	−0.0542719
$791$	0	0
$792$	1.55496	0.0552530
$793$	5.23191	0.185791
$794$	2.79225	0.0990932
$795$	0.581040	0.0206074
$796$	6.25236	0.221609
$797$	1.19375	0.0422848	0.0211424	−	0.999776i	$-0.493270\pi$
$797$	1.19375	0.0422848	0.0211424	+	0.999776i	$0.493270\pi$
$798$	0	0
$799$	7.60388	0.269006
$800$	4.96077	0.175390
$801$	−1.14914	−0.0406030
$802$	−33.8538	−1.19542
$803$	0.274127	0.00967372
$804$	−10.0000	−0.352673
$805$	0	0
$806$	0.757332	0.0266759
$807$	6.36658	0.224114
$808$	14.7192	0.517819
$809$	36.9590	1.29941	0.649704	−	0.760187i	$-0.274893\pi$
$809$	36.9590	1.29941	0.649704	+	0.760187i	$0.274893\pi$
$810$	0.198062	0.00695920
$811$	−50.0103	−1.75610	−0.878049	−	0.478570i	$-0.841155\pi$
$811$	−50.0103	−1.75610	−0.878049	+	0.478570i	$0.841155\pi$
$812$	0	0
$813$	−17.2597	−0.605322
$814$	−12.8998	−0.452137
$815$	0.0241632	0.000846399 0
$816$	−6.85086	−0.239828
$817$	9.86459	0.345118
$818$	29.3927	1.02769
$819$	0	0
$820$	1.55927	0.0544521
$821$	47.4470	1.65591	0.827955	−	0.560794i	$-0.189504\pi$
$821$	47.4470	1.65591	0.827955	+	0.560794i	$0.189504\pi$
$822$	10.3720	0.361764
$823$	25.5555	0.890810	0.445405	−	0.895329i	$-0.353060\pi$
$823$	25.5555	0.890810	0.445405	+	0.895329i	$0.353060\pi$
$824$	16.2717	0.566853
$825$	7.71379	0.268560
$826$	0	0
$827$	−37.2922	−1.29678	−0.648388	−	0.761310i	$-0.724557\pi$
$827$	−37.2922	−1.29678	−0.648388	+	0.761310i	$0.724557\pi$
$828$	−7.52111	−0.261377
$829$	20.8552	0.724330	0.362165	−	0.932114i	$-0.382038\pi$
$829$	20.8552	0.724330	0.362165	+	0.932114i	$0.382038\pi$
$830$	−0.694414	−0.0241035
$831$	−11.8823	−0.412193
$832$	0.890084	0.0308581
$833$	0	0
$834$	−17.6256	−0.610326
$835$	2.73317	0.0945852
$836$	−4.14377	−0.143315
$837$	−0.850855	−0.0294099
$838$	22.8853	0.790559
$839$	−25.7453	−0.888825	−0.444412	−	0.895822i	$-0.646587\pi$
$839$	−25.7453	−0.888825	−0.444412	+	0.895822i	$0.646587\pi$
$840$	0	0
$841$	14.4517	0.498336
$842$	−37.3545	−1.28732
$843$	14.1414	0.487055
$844$	0	0
$845$	2.41789	0.0831781
$846$	1.10992	0.0381597
$847$	0	0
$848$	−2.93362	−0.100741
$849$	1.25773	0.0431653
$850$	−33.9855	−1.16569
$851$	62.3943	2.13885
$852$	−2.19806	−0.0753044
$853$	−32.9444	−1.12799	−0.563997	−	0.825777i	$-0.690737\pi$
$853$	−32.9444	−1.12799	−0.563997	+	0.825777i	$0.690737\pi$
$854$	0	0
$855$	−0.527811	−0.0180508
$856$	7.70410	0.263321
$857$	18.2476	0.623325	0.311663	−	0.950193i	$-0.399114\pi$
$857$	18.2476	0.623325	0.311663	+	0.950193i	$0.399114\pi$
$858$	1.38404	0.0472504
$859$	−55.4292	−1.89122	−0.945611	−	0.325301i	$-0.894534\pi$
$859$	−55.4292	−1.89122	−0.945611	+	0.325301i	$0.894534\pi$
$860$	−0.733169	−0.0250009
$861$	0	0
$862$	−3.96184	−0.134941
$863$	−37.0863	−1.26243	−0.631217	−	0.775606i	$-0.717444\pi$
$863$	−37.0863	−1.26243	−0.631217	+	0.775606i	$0.717444\pi$
$864$	−1.00000	−0.0340207
$865$	0.309043	0.0105078
$866$	−32.6896	−1.11084
$867$	29.9342	1.01662
$868$	0	0
$869$	11.9758	0.406252
$870$	1.30559	0.0442635
$871$	−8.90084	−0.301593
$872$	−12.4644	−0.422098
$873$	13.9758	0.473010
$874$	20.0428	0.677958
$875$	0	0
$876$	−0.176292	−0.00595635
$877$	20.2040	0.682242	0.341121	−	0.940019i	$-0.389193\pi$
$877$	20.2040	0.682242	0.341121	+	0.940019i	$0.389193\pi$
$878$	31.4282	1.06065
$879$	21.3163	0.718982
$880$	0.307979	0.0103820
$881$	28.7549	0.968779	0.484389	−	0.874853i	$-0.339042\pi$
$881$	28.7549	0.968779	0.484389	+	0.874853i	$0.339042\pi$
$882$	0	0
$883$	−38.1473	−1.28376	−0.641880	−	0.766805i	$-0.721845\pi$
$883$	−38.1473	−1.28376	−0.641880	+	0.766805i	$0.721845\pi$
$884$	−6.09783	−0.205092
$885$	−1.25129	−0.0420617
$886$	9.55496	0.321005
$887$	46.3866	1.55751	0.778754	−	0.627329i	$-0.215852\pi$
$887$	46.3866	1.55751	0.778754	+	0.627329i	$0.215852\pi$
$888$	8.29590	0.278392
$889$	0	0
$890$	−0.227602	−0.00762924
$891$	−1.55496	−0.0520931
$892$	13.7942	0.461863
$893$	−2.95779	−0.0989786
$894$	1.42758	0.0477455
$895$	2.18406	0.0730051
$896$	0	0
$897$	−6.69441	−0.223520
$898$	−19.7366	−0.658620
$899$	−5.60866	−0.187059
$900$	−4.96077	−0.165359
$901$	20.0978	0.669556
$902$	−12.2416	−0.407601
$903$	0	0
$904$	0	0
$905$	−2.36121	−0.0784892
$906$	−0.944378	−0.0313749
$907$	18.6305	0.618617	0.309309	−	0.950962i	$-0.399902\pi$
$907$	18.6305	0.618617	0.309309	+	0.950962i	$0.399902\pi$
$908$	−14.8853	−0.493986
$909$	−14.7192	−0.488204
$910$	0	0
$911$	7.96077	0.263752	0.131876	−	0.991266i	$-0.457900\pi$
$911$	7.96077	0.263752	0.131876	+	0.991266i	$0.457900\pi$
$912$	2.66487	0.0882428
$913$	5.45175	0.180426
$914$	−28.0054	−0.926336
$915$	−1.16421	−0.0384876
$916$	3.76941	0.124545
$917$	0	0
$918$	6.85086	0.226112
$919$	−48.7767	−1.60900	−0.804498	−	0.593956i	$-0.797565\pi$
$919$	−48.7767	−1.60900	−0.804498	+	0.593956i	$0.797565\pi$
$920$	−1.48965	−0.0491122
$921$	14.6853	0.483898
$922$	30.7482	1.01264
$923$	−1.95646	−0.0643976
$924$	0	0
$925$	41.1540	1.35314
$926$	−33.0944	−1.08755
$927$	−16.2717	−0.534434
$928$	−6.59179	−0.216386
$929$	9.55197	0.313390	0.156695	−	0.987647i	$-0.449916\pi$
$929$	9.55197	0.313390	0.156695	+	0.987647i	$0.449916\pi$
$930$	−0.168522	−0.00552606
$931$	0	0
$932$	7.08575	0.232102
$933$	−11.9022	−0.389659
$934$	16.8224	0.550445
$935$	−2.10992	−0.0690016
$936$	−0.890084	−0.0290933
$937$	−58.4892	−1.91076	−0.955379	−	0.295383i	$-0.904553\pi$
$937$	−58.4892	−1.91076	−0.955379	+	0.295383i	$0.904553\pi$
$938$	0	0
$939$	−26.9202	−0.878508
$940$	0.219833	0.00717015
$941$	6.68797	0.218022	0.109011	−	0.994041i	$-0.465232\pi$
$941$	6.68797	0.218022	0.109011	+	0.994041i	$0.465232\pi$
$942$	−24.3720	−0.794081
$943$	59.2109	1.92817
$944$	6.31767	0.205623
$945$	0	0
$946$	5.75600	0.187144
$947$	−20.6649	−0.671518	−0.335759	−	0.941948i	$-0.608993\pi$
$947$	−20.6649	−0.671518	−0.335759	+	0.941948i	$0.608993\pi$
$948$	−7.70171	−0.250140
$949$	−0.156915	−0.00509366
$950$	13.2198	0.428908
$951$	32.2500	1.04578
$952$	0	0
$953$	13.6668	0.442711	0.221355	−	0.975193i	$-0.428952\pi$
$953$	13.6668	0.442711	0.221355	+	0.975193i	$0.428952\pi$
$954$	2.93362	0.0949796
$955$	4.98015	0.161154
$956$	3.06100	0.0989998
$957$	−10.2500	−0.331334
$958$	−20.1849	−0.652145
$959$	0	0
$960$	−0.198062	−0.00639243
$961$	−30.2760	−0.976647
$962$	7.38404	0.238071
$963$	−7.70410	−0.248261
$964$	−18.6896	−0.601952
$965$	3.63448	0.116998
$966$	0	0
$967$	−56.3188	−1.81109	−0.905546	−	0.424248i	$-0.860538\pi$
$967$	−56.3188	−1.81109	−0.905546	+	0.424248i	$0.860538\pi$
$968$	8.58211	0.275839
$969$	−18.2567	−0.586489
$970$	2.76809	0.0888779
$971$	−43.8974	−1.40873	−0.704367	−	0.709836i	$-0.748769\pi$
$971$	−43.8974	−1.40873	−0.704367	+	0.709836i	$0.748769\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−9.62325	−0.308349
$975$	−4.41550	−0.141409
$976$	5.87800	0.188150
$977$	−26.7332	−0.855270	−0.427635	−	0.903952i	$-0.640653\pi$
$977$	−26.7332	−0.855270	−0.427635	+	0.903952i	$0.640653\pi$
$978$	0.121998	0.00390106
$979$	1.78687	0.0571087
$980$	0	0
$981$	12.4644	0.397958
$982$	1.13169	0.0361136
$983$	−47.9409	−1.52908	−0.764539	−	0.644578i	$-0.777033\pi$
$983$	−47.9409	−1.52908	−0.764539	+	0.644578i	$0.777033\pi$
$984$	7.87263	0.250970
$985$	4.40475	0.140347
$986$	45.1594	1.43817
$987$	0	0
$988$	2.37196	0.0754621
$989$	−27.8410	−0.885291
$990$	−0.307979	−0.00978820
$991$	23.0422	0.731960	0.365980	−	0.930623i	$-0.380734\pi$
$991$	23.0422	0.731960	0.365980	+	0.930623i	$0.380734\pi$
$992$	0.850855	0.0270147
$993$	−17.1836	−0.545305
$994$	0	0
$995$	−1.23836	−0.0392585
$996$	−3.50604	−0.111093
$997$	9.42280	0.298423	0.149211	−	0.988805i	$-0.452326\pi$
$997$	9.42280	0.298423	0.149211	+	0.988805i	$0.452326\pi$
$998$	2.96376	0.0938160
$999$	−8.29590	−0.262471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2058.2.a.e.1.3	yes	3
3.2	odd	2		6174.2.a.p.1.1		3
7.2	even	3		2058.2.e.h.361.1		6
7.3	odd	6		2058.2.e.i.667.3		6
7.4	even	3		2058.2.e.h.667.1		6
7.5	odd	6		2058.2.e.i.361.3		6
7.6	odd	2		2058.2.a.d.1.1	✓	3
21.20	even	2		6174.2.a.g.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2058.2.a.d.1.1	✓	3	7.6	odd	2
2058.2.a.e.1.3	yes	3	1.1	even	1	trivial
2058.2.e.h.361.1		6	7.2	even	3
2058.2.e.h.667.1		6	7.4	even	3
2058.2.e.i.361.3		6	7.5	odd	6
2058.2.e.i.667.3		6	7.3	odd	6
6174.2.a.g.1.3		3	21.20	even	2
6174.2.a.p.1.1		3	3.2	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:	\( N \)	\(=\)	\( 2058 = 2 \cdot 3 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2058.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.198062 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.198062 q^{10} -1.55496 q^{11} +1.00000 q^{12} +0.890084 q^{13} -0.198062 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} +2.66487 q^{19} -0.198062 q^{20} +1.55496 q^{22} -7.52111 q^{23} -1.00000 q^{24} -4.96077 q^{25} -0.890084 q^{26} +1.00000 q^{27} +6.59179 q^{29} +0.198062 q^{30} -0.850855 q^{31} -1.00000 q^{32} -1.55496 q^{33} +6.85086 q^{34} +1.00000 q^{36} -8.29590 q^{37} -2.66487 q^{38} +0.890084 q^{39} +0.198062 q^{40} -7.87263 q^{41} +3.70171 q^{43} -1.55496 q^{44} -0.198062 q^{45} +7.52111 q^{46} -1.10992 q^{47} +1.00000 q^{48} +4.96077 q^{50} -6.85086 q^{51} +0.890084 q^{52} -2.93362 q^{53} -1.00000 q^{54} +0.307979 q^{55} +2.66487 q^{57} -6.59179 q^{58} +6.31767 q^{59} -0.198062 q^{60} +5.87800 q^{61} +0.850855 q^{62} +1.00000 q^{64} -0.176292 q^{65} +1.55496 q^{66} -10.0000 q^{67} -6.85086 q^{68} -7.52111 q^{69} -2.19806 q^{71} -1.00000 q^{72} -0.176292 q^{73} +8.29590 q^{74} -4.96077 q^{75} +2.66487 q^{76} -0.890084 q^{78} -7.70171 q^{79} -0.198062 q^{80} +1.00000 q^{81} +7.87263 q^{82} -3.50604 q^{83} +1.35690 q^{85} -3.70171 q^{86} +6.59179 q^{87} +1.55496 q^{88} -1.14914 q^{89} +0.198062 q^{90} -7.52111 q^{92} -0.850855 q^{93} +1.10992 q^{94} -0.527811 q^{95} -1.00000 q^{96} +13.9758 q^{97} -1.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 5 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 5 q^{10} - 5 q^{11} + 3 q^{12} + 2 q^{13} - 5 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 9 q^{19} - 5 q^{20} + 5 q^{22} - 7 q^{23}+ \cdots - 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 5 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 5 * q^10 - 5 * q^11 + 3 * q^12 + 2 * q^13 - 5 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 9 * q^19 - 5 * q^20 + 5 * q^22 - 7 * q^23 - 3 * q^24 - 2 * q^25 - 2 * q^26 + 3 * q^27 - 8 * q^29 + 5 * q^30 + 11 * q^31 - 3 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 - 11 * q^37 - 9 * q^38 + 2 * q^39 + 5 * q^40 - 7 * q^41 - 16 * q^43 - 5 * q^44 - 5 * q^45 + 7 * q^46 - 4 * q^47 + 3 * q^48 + 2 * q^50 - 7 * q^51 + 2 * q^52 - 2 * q^53 - 3 * q^54 + 6 * q^55 + 9 * q^57 + 8 * q^58 + 2 * q^59 - 5 * q^60 - 2 * q^61 - 11 * q^62 + 3 * q^64 - 8 * q^65 + 5 * q^66 - 30 * q^67 - 7 * q^68 - 7 * q^69 - 11 * q^71 - 3 * q^72 - 8 * q^73 + 11 * q^74 - 2 * q^75 + 9 * q^76 - 2 * q^78 + 4 * q^79 - 5 * q^80 + 3 * q^81 + 7 * q^82 - 20 * q^83 + 16 * q^86 - 8 * q^87 + 5 * q^88 - 17 * q^89 + 5 * q^90 - 7 * q^92 + 11 * q^93 + 4 * q^94 - 8 * q^95 - 3 * q^96 + 4 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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