LMFDB - Embedded newform 507.2.b.g.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.g.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} -2.60388 q^{10} -5.18598i q^{11} -1.24698 q^{12} +6.20775 q^{14} -1.44504i q^{15} -4.93900 q^{16} +0.753020 q^{17} -1.80194i q^{18} -7.96077i q^{19} +1.80194i q^{20} +3.44504i q^{21} -9.34481 q^{22} +2.82908 q^{23} -1.35690i q^{24} +2.91185 q^{25} +1.00000 q^{27} -4.29590i q^{28} -3.91185 q^{29} -2.60388 q^{30} +4.89977i q^{31} +6.18598i q^{32} -5.18598i q^{33} -1.35690i q^{34} +4.97823 q^{35} -1.24698 q^{36} +6.24698i q^{37} -14.3448 q^{38} -1.96077 q^{40} -1.80194i q^{41} +6.20775 q^{42} +7.09783 q^{43} +6.46681i q^{44} -1.44504i q^{45} -5.09783i q^{46} +10.5526i q^{47} -4.93900 q^{48} -4.86831 q^{49} -5.24698i q^{50} +0.753020 q^{51} -3.08815 q^{53} -1.80194i q^{54} -7.49396 q^{55} +4.67456 q^{56} -7.96077i q^{57} +7.04892i q^{58} +1.87800i q^{59} +1.80194i q^{60} +3.34481 q^{61} +8.82908 q^{62} +3.44504i q^{63} +1.26875 q^{64} -9.34481 q^{66} +4.54288i q^{67} -0.939001 q^{68} +2.82908 q^{69} -8.97046i q^{70} +9.11960i q^{71} -1.35690i q^{72} +2.95108i q^{73} +11.2567 q^{74} +2.91185 q^{75} +9.92692i q^{76} +17.8659 q^{77} -9.43296 q^{79} +7.13706i q^{80} +1.00000 q^{81} -3.24698 q^{82} +6.46681i q^{83} -4.29590i q^{84} -1.08815i q^{85} -12.7899i q^{86} -3.91185 q^{87} -7.03684 q^{88} +1.15883i q^{89} -2.60388 q^{90} -3.52781 q^{92} +4.89977i q^{93} +19.0151 q^{94} -11.5036 q^{95} +6.18598i q^{96} -8.65817i q^{97} +8.77240i q^{98} -5.18598i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{14} - 10 q^{16} + 14 q^{17} - 10 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{27} - 16 q^{29} + 2 q^{30} + 36 q^{35} + 2 q^{36} - 40 q^{38} + 14 q^{40} + 2 q^{42} + 6 q^{43} - 10 q^{48} - 34 q^{49} + 14 q^{51} - 26 q^{53} - 26 q^{55} - 14 q^{56} - 26 q^{61} + 32 q^{62} - 8 q^{64} - 10 q^{66} + 14 q^{68} - 4 q^{69} + 14 q^{74} + 10 q^{75} + 30 q^{77} - 18 q^{79} + 6 q^{81} - 10 q^{82} - 16 q^{87} + 14 q^{88} + 2 q^{90} - 34 q^{92} + 64 q^{94} - 6 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^4 + 6 * q^9 + 2 * q^10 + 2 * q^12 + 2 * q^14 - 10 * q^16 + 14 * q^17 - 10 * q^22 - 4 * q^23 + 10 * q^25 + 6 * q^27 - 16 * q^29 + 2 * q^30 + 36 * q^35 + 2 * q^36 - 40 * q^38 + 14 * q^40 + 2 * q^42 + 6 * q^43 - 10 * q^48 - 34 * q^49 + 14 * q^51 - 26 * q^53 - 26 * q^55 - 14 * q^56 - 26 * q^61 + 32 * q^62 - 8 * q^64 - 10 * q^66 + 14 * q^68 - 4 * q^69 + 14 * q^74 + 10 * q^75 + 30 * q^77 - 18 * q^79 + 6 * q^81 - 10 * q^82 - 16 * q^87 + 14 * q^88 + 2 * q^90 - 34 * q^92 + 64 * q^94 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.80194i	− 1.27416i	−0.770797	−	0.637081i	$-0.780142\pi$
$2$	− 1.80194i	− 1.27416i	0.770797	−	0.637081i	$-0.219858\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.24698	−0.623490
$5$	− 1.44504i	− 0.646242i	−0.946358	−	0.323121i	$-0.895268\pi$
$5$	− 1.44504i	− 0.646242i	0.946358	−	0.323121i	$-0.104732\pi$
$6$	− 1.80194i	− 0.735638i
$7$	3.44504i	1.30210i	0.759033	+	0.651052i	$0.225672\pi$
$7$	3.44504i	1.30210i	−0.759033	+	0.651052i	$0.774328\pi$
$8$	− 1.35690i	− 0.479735i
$9$	1.00000	0.333333
$10$	−2.60388	−0.823418
$11$	− 5.18598i	− 1.56363i	−0.623509	−	0.781816i	$-0.714294\pi$
$11$	− 5.18598i	− 1.56363i	0.623509	−	0.781816i	$-0.285706\pi$
$12$	−1.24698	−0.359972
$13$	0	0
$13$	0	0
$14$	6.20775	1.65909
$15$	− 1.44504i	− 0.373108i
$16$	−4.93900	−1.23475
$17$	0.753020	0.182634	0.0913171	−	0.995822i	$-0.470892\pi$
$17$	0.753020	0.182634	0.0913171	+	0.995822i	$0.470892\pi$
$18$	− 1.80194i	− 0.424721i
$19$	− 7.96077i	− 1.82633i	−0.407594	−	0.913163i	$-0.633632\pi$
$19$	− 7.96077i	− 1.82633i	0.407594	−	0.913163i	$-0.366368\pi$
$20$	1.80194i	0.402926i
$21$	3.44504i	0.751770i
$22$	−9.34481	−1.99232
$23$	2.82908	0.589905	0.294952	−	0.955512i	$-0.404696\pi$
$23$	2.82908	0.589905	0.294952	+	0.955512i	$0.404696\pi$
$24$	− 1.35690i	− 0.276975i
$25$	2.91185	0.582371
$26$	0	0
$27$	1.00000	0.192450
$28$	− 4.29590i	− 0.811848i
$29$	−3.91185	−0.726413	−0.363207	−	0.931709i	$-0.618318\pi$
$29$	−3.91185	−0.726413	−0.363207	+	0.931709i	$0.618318\pi$
$30$	−2.60388	−0.475400
$31$	4.89977i	0.880025i	0.897992	+	0.440013i	$0.145026\pi$
$31$	4.89977i	0.880025i	−0.897992	+	0.440013i	$0.854974\pi$
$32$	6.18598i	1.09354i
$33$	− 5.18598i	− 0.902763i
$34$	− 1.35690i	− 0.232706i
$35$	4.97823	0.841474
$36$	−1.24698	−0.207830
$37$	6.24698i	1.02700i	0.858090	+	0.513499i	$0.171651\pi$
$37$	6.24698i	1.02700i	−0.858090	+	0.513499i	$0.828349\pi$
$38$	−14.3448	−2.32704
$39$	0	0
$40$	−1.96077	−0.310025
$41$	− 1.80194i	− 0.281415i	−0.990051	−	0.140708i	$-0.955062\pi$
$41$	− 1.80194i	− 0.281415i	0.990051	−	0.140708i	$-0.0449378\pi$
$42$	6.20775	0.957877
$43$	7.09783	1.08241	0.541205	−	0.840891i	$-0.317968\pi$
$43$	7.09783	1.08241	0.541205	+	0.840891i	$0.317968\pi$
$44$	6.46681i	0.974909i
$45$	− 1.44504i	− 0.215414i
$46$	− 5.09783i	− 0.751635i
$47$	10.5526i	1.53925i	0.638496	+	0.769625i	$0.279557\pi$
$47$	10.5526i	1.53925i	−0.638496	+	0.769625i	$0.720443\pi$
$48$	−4.93900	−0.712883
$49$	−4.86831	−0.695473
$50$	− 5.24698i	− 0.742035i
$51$	0.753020	0.105444
$52$	0	0
$53$	−3.08815	−0.424189	−0.212095	−	0.977249i	$-0.568029\pi$
$53$	−3.08815	−0.424189	−0.212095	+	0.977249i	$0.568029\pi$
$54$	− 1.80194i	− 0.245213i
$55$	−7.49396	−1.01049
$56$	4.67456	0.624665
$57$	− 7.96077i	− 1.05443i
$58$	7.04892i	0.925568i
$59$	1.87800i	0.244495i	0.992500	+	0.122248i	$0.0390102\pi$
$59$	1.87800i	0.244495i	−0.992500	+	0.122248i	$0.960990\pi$
$60$	1.80194i	0.232629i
$61$	3.34481	0.428260	0.214130	−	0.976805i	$-0.431308\pi$
$61$	3.34481	0.428260	0.214130	+	0.976805i	$0.431308\pi$
$62$	8.82908	1.12129
$63$	3.44504i	0.434034i
$64$	1.26875	0.158594
$65$	0	0
$66$	−9.34481	−1.15027
$67$	4.54288i	0.555001i	0.960726	+	0.277500i	$0.0895060\pi$
$67$	4.54288i	0.555001i	−0.960726	+	0.277500i	$0.910494\pi$
$68$	−0.939001	−0.113871
$69$	2.82908	0.340582
$70$	− 8.97046i	− 1.07218i
$71$	9.11960i	1.08230i	0.840927	+	0.541149i	$0.182011\pi$
$71$	9.11960i	1.08230i	−0.840927	+	0.541149i	$0.817989\pi$
$72$	− 1.35690i	− 0.159912i
$73$	2.95108i	0.345398i	0.984975	+	0.172699i	$0.0552488\pi$
$73$	2.95108i	0.345398i	−0.984975	+	0.172699i	$0.944751\pi$
$74$	11.2567	1.30856
$75$	2.91185	0.336232
$76$	9.92692i	1.13870i
$77$	17.8659	2.03601
$78$	0	0
$79$	−9.43296	−1.06129	−0.530645	−	0.847594i	$-0.678050\pi$
$79$	−9.43296	−1.06129	−0.530645	+	0.847594i	$0.678050\pi$
$80$	7.13706i	0.797948i
$81$	1.00000	0.111111
$82$	−3.24698	−0.358569
$83$	6.46681i	0.709825i	0.934900	+	0.354912i	$0.115489\pi$
$83$	6.46681i	0.709825i	−0.934900	+	0.354912i	$0.884511\pi$
$84$	− 4.29590i	− 0.468721i
$85$	− 1.08815i	− 0.118026i
$86$	− 12.7899i	− 1.37917i
$87$	−3.91185	−0.419395
$88$	−7.03684	−0.750129
$89$	1.15883i	0.122836i	0.998112	+	0.0614181i	$0.0195623\pi$
$89$	1.15883i	0.122836i	−0.998112	+	0.0614181i	$0.980438\pi$
$90$	−2.60388	−0.274473
$91$	0	0
$92$	−3.52781	−0.367800
$93$	4.89977i	0.508083i
$94$	19.0151	1.96125
$95$	−11.5036	−1.18025
$96$	6.18598i	0.631354i
$97$	− 8.65817i	− 0.879104i	−0.898217	−	0.439552i	$-0.855137\pi$
$97$	− 8.65817i	− 0.879104i	0.898217	−	0.439552i	$-0.144863\pi$
$98$	8.77240i	0.886146i
$99$	− 5.18598i	− 0.521211i
$100$	−3.63102	−0.363102
$101$	8.47650	0.843443	0.421722	−	0.906725i	$-0.361426\pi$
$101$	8.47650	0.843443	0.421722	+	0.906725i	$0.361426\pi$
$102$	− 1.35690i	− 0.134353i
$103$	−5.64742	−0.556456	−0.278228	−	0.960515i	$-0.589747\pi$
$103$	−5.64742	−0.556456	−0.278228	+	0.960515i	$0.589747\pi$
$104$	0	0
$105$	4.97823	0.485825
$106$	5.56465i	0.540486i
$107$	−6.73556	−0.651151	−0.325576	−	0.945516i	$-0.605558\pi$
$107$	−6.73556	−0.651151	−0.325576	+	0.945516i	$0.605558\pi$
$108$	−1.24698	−0.119991
$109$	2.07606i	0.198851i	0.995045	+	0.0994255i	$0.0317005\pi$
$109$	2.07606i	0.198851i	−0.995045	+	0.0994255i	$0.968300\pi$
$110$	13.5036i	1.28752i
$111$	6.24698i	0.592937i
$112$	− 17.0151i	− 1.60777i
$113$	6.16852	0.580286	0.290143	−	0.956983i	$-0.406297\pi$
$113$	6.16852	0.580286	0.290143	+	0.956983i	$0.406297\pi$
$114$	−14.3448	−1.34351
$115$	− 4.08815i	− 0.381222i
$116$	4.87800	0.452911
$117$	0	0
$118$	3.38404	0.311526
$119$	2.59419i	0.237809i
$120$	−1.96077	−0.178993
$121$	−15.8944	−1.44495
$122$	− 6.02715i	− 0.545672i
$123$	− 1.80194i	− 0.162475i
$124$	− 6.10992i	− 0.548687i
$125$	− 11.4330i	− 1.02260i
$126$	6.20775	0.553030
$127$	14.2620	1.26555	0.632776	−	0.774335i	$-0.281915\pi$
$127$	14.2620	1.26555	0.632776	+	0.774335i	$0.281915\pi$
$128$	10.0858i	0.891463i
$129$	7.09783	0.624929
$130$	0	0
$131$	22.6015	1.97470	0.987350	−	0.158554i	$-0.0506831\pi$
$131$	22.6015	1.97470	0.987350	+	0.158554i	$0.0506831\pi$
$132$	6.46681i	0.562864i
$133$	27.4252	2.37807
$134$	8.18598	0.707161
$135$	− 1.44504i	− 0.124369i
$136$	− 1.02177i	− 0.0876161i
$137$	− 13.6353i	− 1.16495i	−0.812850	−	0.582473i	$-0.802085\pi$
$137$	− 13.6353i	− 1.16495i	0.812850	−	0.582473i	$-0.197915\pi$
$138$	− 5.09783i	− 0.433957i
$139$	17.6015	1.49294	0.746469	−	0.665420i	$-0.231748\pi$
$139$	17.6015	1.49294	0.746469	+	0.665420i	$0.231748\pi$
$140$	−6.20775	−0.524651
$141$	10.5526i	0.888686i
$142$	16.4330	1.37902
$143$	0	0
$144$	−4.93900	−0.411583
$145$	5.65279i	0.469439i
$146$	5.31767	0.440093
$147$	−4.86831	−0.401532
$148$	− 7.78986i	− 0.640322i
$149$	− 12.7385i	− 1.04358i	−0.853073	−	0.521791i	$-0.825264\pi$
$149$	− 12.7385i	− 1.04358i	0.853073	−	0.521791i	$-0.174736\pi$
$150$	− 5.24698i	− 0.428414i
$151$	15.6407i	1.27282i	0.771350	+	0.636412i	$0.219582\pi$
$151$	15.6407i	1.27282i	−0.771350	+	0.636412i	$0.780418\pi$
$152$	−10.8019	−0.876153
$153$	0.753020	0.0608781
$154$	− 32.1933i	− 2.59421i
$155$	7.08038	0.568710
$156$	0	0
$157$	−0.823708	−0.0657391	−0.0328695	−	0.999460i	$-0.510465\pi$
$157$	−0.823708	−0.0657391	−0.0328695	+	0.999460i	$0.510465\pi$
$158$	16.9976i	1.35226i
$159$	−3.08815	−0.244906
$160$	8.93900	0.706690
$161$	9.74632i	0.768117i
$162$	− 1.80194i	− 0.141574i
$163$	6.26875i	0.491006i	0.969396	+	0.245503i	$0.0789532\pi$
$163$	6.26875i	0.491006i	−0.969396	+	0.245503i	$0.921047\pi$
$164$	2.24698i	0.175460i
$165$	−7.49396	−0.583404
$166$	11.6528	0.904432
$167$	− 7.45042i	− 0.576531i	−0.957551	−	0.288265i	$-0.906921\pi$
$167$	− 7.45042i	− 0.576531i	0.957551	−	0.288265i	$-0.0930785\pi$
$168$	4.67456	0.360650
$169$	0	0
$170$	−1.96077	−0.150384
$171$	− 7.96077i	− 0.608775i
$172$	−8.85086	−0.674871
$173$	2.00969	0.152794	0.0763969	−	0.997077i	$-0.475658\pi$
$173$	2.00969	0.152794	0.0763969	+	0.997077i	$0.475658\pi$
$174$	7.04892i	0.534377i
$175$	10.0315i	0.758307i
$176$	25.6136i	1.93070i
$177$	1.87800i	0.141159i
$178$	2.08815	0.156513
$179$	−20.0368	−1.49762	−0.748812	−	0.662783i	$-0.769375\pi$
$179$	−20.0368	−1.49762	−0.748812	+	0.662783i	$0.769375\pi$
$180$	1.80194i	0.134309i
$181$	−24.1226	−1.79302	−0.896509	−	0.443026i	$-0.853905\pi$
$181$	−24.1226	−1.79302	−0.896509	+	0.443026i	$0.853905\pi$
$182$	0	0
$183$	3.34481	0.247256
$184$	− 3.83877i	− 0.282998i
$185$	9.02715	0.663689
$186$	8.82908	0.647380
$187$	− 3.90515i	− 0.285573i
$188$	− 13.1588i	− 0.959707i
$189$	3.44504i	0.250590i
$190$	20.7289i	1.50383i
$191$	−7.08038	−0.512318	−0.256159	−	0.966635i	$-0.582457\pi$
$191$	−7.08038	−0.512318	−0.256159	+	0.966635i	$0.582457\pi$
$192$	1.26875	0.0915641
$193$	9.76809i	0.703122i	0.936165	+	0.351561i	$0.114349\pi$
$193$	9.76809i	0.703122i	−0.936165	+	0.351561i	$0.885651\pi$
$194$	−15.6015	−1.12012
$195$	0	0
$196$	6.07069	0.433621
$197$	− 23.4112i	− 1.66798i	−0.551781	−	0.833989i	$-0.686052\pi$
$197$	− 23.4112i	− 1.66798i	0.551781	−	0.833989i	$-0.313948\pi$
$198$	−9.34481	−0.664107
$199$	−4.02475	−0.285307	−0.142654	−	0.989773i	$-0.545563\pi$
$199$	−4.02475	−0.285307	−0.142654	+	0.989773i	$0.545563\pi$
$200$	− 3.95108i	− 0.279384i
$201$	4.54288i	0.320430i
$202$	− 15.2741i	− 1.07468i
$203$	− 13.4765i	− 0.945865i
$204$	−0.939001	−0.0657432
$205$	−2.60388	−0.181863
$206$	10.1763i	0.709016i
$207$	2.82908	0.196635
$208$	0	0
$209$	−41.2844	−2.85570
$210$	− 8.97046i	− 0.619021i
$211$	−3.91185	−0.269303	−0.134652	−	0.990893i	$-0.542992\pi$
$211$	−3.91185	−0.269303	−0.134652	+	0.990893i	$0.542992\pi$
$212$	3.85086	0.264478
$213$	9.11960i	0.624865i
$214$	12.1371i	0.829673i
$215$	− 10.2567i	− 0.699499i
$216$	− 1.35690i	− 0.0923251i
$217$	−16.8799	−1.14588
$218$	3.74094	0.253368
$219$	2.95108i	0.199416i
$220$	9.34481	0.630027
$221$	0	0
$222$	11.2567	0.755498
$223$	− 7.44935i	− 0.498846i	−0.968395	−	0.249423i	$-0.919759\pi$
$223$	− 7.44935i	− 0.498846i	0.968395	−	0.249423i	$-0.0802409\pi$
$224$	−21.3110	−1.42390
$225$	2.91185	0.194124
$226$	− 11.1153i	− 0.739378i
$227$	21.2500i	1.41041i	0.709004	+	0.705205i	$0.249145\pi$
$227$	21.2500i	1.41041i	−0.709004	+	0.705205i	$0.750855\pi$
$228$	9.92692i	0.657426i
$229$	9.29590i	0.614290i	0.951663	+	0.307145i	$0.0993737\pi$
$229$	9.29590i	0.614290i	−0.951663	+	0.307145i	$0.900626\pi$
$230$	−7.36658	−0.485738
$231$	17.8659	1.17549
$232$	5.30798i	0.348486i
$233$	−16.2107	−1.06200	−0.531000	−	0.847372i	$-0.678184\pi$
$233$	−16.2107	−1.06200	−0.531000	+	0.847372i	$0.678184\pi$
$234$	0	0
$235$	15.2489	0.994728
$236$	− 2.34183i	− 0.152440i
$237$	−9.43296	−0.612737
$238$	4.67456	0.303007
$239$	13.5090i	0.873826i	0.899504	+	0.436913i	$0.143928\pi$
$239$	13.5090i	0.873826i	−0.899504	+	0.436913i	$0.856072\pi$
$240$	7.13706i	0.460695i
$241$	6.26875i	0.403806i	0.979406	+	0.201903i	$0.0647125\pi$
$241$	6.26875i	0.403806i	−0.979406	+	0.201903i	$0.935287\pi$
$242$	28.6407i	1.84109i
$243$	1.00000	0.0641500
$244$	−4.17092	−0.267015
$245$	7.03492i	0.449444i
$246$	−3.24698	−0.207020
$247$	0	0
$248$	6.64848	0.422179
$249$	6.46681i	0.409818i
$250$	−20.6015	−1.30295
$251$	0.753020	0.0475302	0.0237651	−	0.999718i	$-0.492435\pi$
$251$	0.753020	0.0475302	0.0237651	+	0.999718i	$0.492435\pi$
$252$	− 4.29590i	− 0.270616i
$253$	− 14.6716i	− 0.922394i
$254$	− 25.6993i	− 1.61252i
$255$	− 1.08815i	− 0.0681423i
$256$	20.7114	1.29446
$257$	−19.7265	−1.23050	−0.615252	−	0.788331i	$-0.710946\pi$
$257$	−19.7265	−1.23050	−0.615252	+	0.788331i	$0.710946\pi$
$258$	− 12.7899i	− 0.796262i
$259$	−21.5211	−1.33726
$260$	0	0
$261$	−3.91185	−0.242138
$262$	− 40.7265i	− 2.51609i
$263$	−17.6093	−1.08583	−0.542917	−	0.839787i	$-0.682680\pi$
$263$	−17.6093	−1.08583	−0.542917	+	0.839787i	$0.682680\pi$
$264$	−7.03684	−0.433087
$265$	4.46250i	0.274129i
$266$	− 49.4185i	− 3.03004i
$267$	1.15883i	0.0709195i
$268$	− 5.66487i	− 0.346037i
$269$	−16.3870	−0.999135	−0.499567	−	0.866275i	$-0.666508\pi$
$269$	−16.3870	−0.999135	−0.499567	+	0.866275i	$0.666508\pi$
$270$	−2.60388	−0.158467
$271$	− 0.795233i	− 0.0483070i	−0.999708	−	0.0241535i	$-0.992311\pi$
$271$	− 0.795233i	− 0.0483070i	0.999708	−	0.0241535i	$-0.00768904\pi$
$272$	−3.71917	−0.225508
$273$	0	0
$274$	−24.5700	−1.48433
$275$	− 15.1008i	− 0.910614i
$276$	−3.52781	−0.212349
$277$	4.83340	0.290411	0.145205	−	0.989402i	$-0.453616\pi$
$277$	4.83340	0.290411	0.145205	+	0.989402i	$0.453616\pi$
$278$	− 31.7168i	− 1.90225i
$279$	4.89977i	0.293342i
$280$	− 6.75494i	− 0.403685i
$281$	18.7748i	1.12001i	0.828489	+	0.560005i	$0.189201\pi$
$281$	18.7748i	1.12001i	−0.828489	+	0.560005i	$0.810799\pi$
$282$	19.0151	1.13233
$283$	7.91723	0.470631	0.235315	−	0.971919i	$-0.424388\pi$
$283$	7.91723	0.470631	0.235315	+	0.971919i	$0.424388\pi$
$284$	− 11.3720i	− 0.674802i
$285$	−11.5036	−0.681417
$286$	0	0
$287$	6.20775	0.366432
$288$	6.18598i	0.364512i
$289$	−16.4330	−0.966645
$290$	10.1860	0.598141
$291$	− 8.65817i	− 0.507551i
$292$	− 3.67994i	− 0.215352i
$293$	6.57912i	0.384356i	0.981360	+	0.192178i	$0.0615552\pi$
$293$	6.57912i	0.384356i	−0.981360	+	0.192178i	$0.938445\pi$
$294$	8.77240i	0.511617i
$295$	2.71379	0.158003
$296$	8.47650	0.492687
$297$	− 5.18598i	− 0.300921i
$298$	−22.9541	−1.32969
$299$	0	0
$300$	−3.63102	−0.209637
$301$	24.4523i	1.40941i
$302$	28.1836	1.62178
$303$	8.47650	0.486962
$304$	39.3183i	2.25506i
$305$	− 4.83340i	− 0.276759i
$306$	− 1.35690i	− 0.0775686i
$307$	− 24.8649i	− 1.41911i	−0.704649	−	0.709556i	$-0.748895\pi$
$307$	− 24.8649i	− 1.41911i	0.704649	−	0.709556i	$-0.251105\pi$
$308$	−22.2784	−1.26943
$309$	−5.64742	−0.321270
$310$	− 12.7584i	− 0.724628i
$311$	17.0804	0.968539	0.484270	−	0.874919i	$-0.339085\pi$
$311$	17.0804	0.968539	0.484270	+	0.874919i	$0.339085\pi$
$312$	0	0
$313$	15.6974	0.887269	0.443635	−	0.896208i	$-0.353689\pi$
$313$	15.6974	0.887269	0.443635	+	0.896208i	$0.353689\pi$
$314$	1.48427i	0.0837622i
$315$	4.97823	0.280491
$316$	11.7627	0.661704
$317$	− 32.7821i	− 1.84123i	−0.390477	−	0.920613i	$-0.627690\pi$
$317$	− 32.7821i	− 1.84123i	0.390477	−	0.920613i	$-0.372310\pi$
$318$	5.56465i	0.312050i
$319$	20.2868i	1.13584i
$320$	− 1.83340i	− 0.102490i
$321$	−6.73556	−0.375942
$322$	17.5623	0.978706
$323$	− 5.99462i	− 0.333550i
$324$	−1.24698	−0.0692766
$325$	0	0
$326$	11.2959	0.625622
$327$	2.07606i	0.114807i
$328$	−2.44504	−0.135005
$329$	−36.3540	−2.00426
$330$	13.5036i	0.743351i
$331$	29.1618i	1.60288i	0.598076	+	0.801439i	$0.295932\pi$
$331$	29.1618i	1.60288i	−0.598076	+	0.801439i	$0.704068\pi$
$332$	− 8.06398i	− 0.442569i
$333$	6.24698i	0.342332i
$334$	−13.4252	−0.734594
$335$	6.56465	0.358665
$336$	− 17.0151i	− 0.928248i
$337$	33.2911	1.81348	0.906741	−	0.421688i	$-0.138562\pi$
$337$	33.2911	1.81348	0.906741	+	0.421688i	$0.138562\pi$
$338$	0	0
$339$	6.16852	0.335028
$340$	1.35690i	0.0735880i
$341$	25.4101	1.37604
$342$	−14.3448	−0.775679
$343$	7.34375i	0.396525i
$344$	− 9.63102i	− 0.519270i
$345$	− 4.08815i	− 0.220098i
$346$	− 3.62133i	− 0.194684i
$347$	−0.873690	−0.0469022	−0.0234511	−	0.999725i	$-0.507465\pi$
$347$	−0.873690	−0.0469022	−0.0234511	+	0.999725i	$0.507465\pi$
$348$	4.87800	0.261488
$349$	− 3.23191i	− 0.173000i	−0.996252	−	0.0865002i	$-0.972432\pi$
$349$	− 3.23191i	− 0.173000i	0.996252	−	0.0865002i	$-0.0275683\pi$
$350$	18.0761	0.966206
$351$	0	0
$352$	32.0804	1.70989
$353$	− 8.14675i	− 0.433608i	−0.976215	−	0.216804i	$-0.930437\pi$
$353$	− 8.14675i	− 0.433608i	0.976215	−	0.216804i	$-0.0695632\pi$
$354$	3.38404	0.179860
$355$	13.1782	0.699427
$356$	− 1.44504i	− 0.0765871i
$357$	2.59419i	0.137299i
$358$	36.1051i	1.90822i
$359$	2.64071i	0.139371i	0.997569	+	0.0696857i	$0.0221996\pi$
$359$	2.64071i	0.139371i	−0.997569	+	0.0696857i	$0.977800\pi$
$360$	−1.96077	−0.103342
$361$	−44.3739	−2.33547
$362$	43.4674i	2.28460i
$363$	−15.8944	−0.834239
$364$	0	0
$365$	4.26444	0.223211
$366$	− 6.02715i	− 0.315044i
$367$	−2.90408	−0.151592	−0.0757960	−	0.997123i	$-0.524150\pi$
$367$	−2.90408	−0.151592	−0.0757960	+	0.997123i	$0.524150\pi$
$368$	−13.9729	−0.728385
$369$	− 1.80194i	− 0.0938051i
$370$	− 16.2664i	− 0.845648i
$371$	− 10.6388i	− 0.552339i
$372$	− 6.10992i	− 0.316784i
$373$	−8.39852	−0.434859	−0.217429	−	0.976076i	$-0.569767\pi$
$373$	−8.39852	−0.434859	−0.217429	+	0.976076i	$0.569767\pi$
$374$	−7.03684	−0.363866
$375$	− 11.4330i	− 0.590396i
$376$	14.3187	0.738432
$377$	0	0
$378$	6.20775	0.319292
$379$	− 15.7482i	− 0.808932i	−0.914553	−	0.404466i	$-0.867457\pi$
$379$	− 15.7482i	− 0.808932i	0.914553	−	0.404466i	$-0.132543\pi$
$380$	14.3448	0.735873
$381$	14.2620	0.730667
$382$	12.7584i	0.652776i
$383$	− 12.7385i	− 0.650909i	−0.945558	−	0.325455i	$-0.894483\pi$
$383$	− 12.7385i	− 0.650909i	0.945558	−	0.325455i	$-0.105517\pi$
$384$	10.0858i	0.514686i
$385$	− 25.8170i	− 1.31576i
$386$	17.6015	0.895892
$387$	7.09783	0.360803
$388$	10.7966i	0.548112i
$389$	0.310371	0.0157365	0.00786823	−	0.999969i	$-0.497495\pi$
$389$	0.310371	0.0157365	0.00786823	+	0.999969i	$0.497495\pi$
$390$	0	0
$391$	2.13036	0.107737
$392$	6.60579i	0.333643i
$393$	22.6015	1.14009
$394$	−42.1855	−2.12528
$395$	13.6310i	0.685851i
$396$	6.46681i	0.324970i
$397$	1.49098i	0.0748299i	0.999300	+	0.0374150i	$0.0119123\pi$
$397$	1.49098i	0.0748299i	−0.999300	+	0.0374150i	$0.988088\pi$
$398$	7.25236i	0.363528i
$399$	27.4252	1.37298
$400$	−14.3817	−0.719083
$401$	− 23.8334i	− 1.19018i	−0.803658	−	0.595092i	$-0.797116\pi$
$401$	− 23.8334i	− 1.19018i	0.803658	−	0.595092i	$-0.202884\pi$
$402$	8.18598	0.408280
$403$	0	0
$404$	−10.5700	−0.525878
$405$	− 1.44504i	− 0.0718047i
$406$	−24.2838	−1.20519
$407$	32.3967	1.60585
$408$	− 1.02177i	− 0.0505852i
$409$	− 4.26742i	− 0.211010i	−0.994419	−	0.105505i	$-0.966354\pi$
$409$	− 4.26742i	− 0.211010i	0.994419	−	0.105505i	$-0.0336460\pi$
$410$	4.69202i	0.231722i
$411$	− 13.6353i	− 0.672581i
$412$	7.04221	0.346945
$413$	−6.46980	−0.318358
$414$	− 5.09783i	− 0.250545i
$415$	9.34481	0.458719
$416$	0	0
$417$	17.6015	0.861948
$418$	74.3919i	3.63863i
$419$	29.6896	1.45043	0.725217	−	0.688521i	$-0.241740\pi$
$419$	29.6896	1.45043	0.725217	+	0.688521i	$0.241740\pi$
$420$	−6.20775	−0.302907
$421$	− 29.3991i	− 1.43282i	−0.697677	−	0.716412i	$-0.745783\pi$
$421$	− 29.3991i	− 1.43282i	0.697677	−	0.716412i	$-0.254217\pi$
$422$	7.04892i	0.343136i
$423$	10.5526i	0.513083i
$424$	4.19029i	0.203499i
$425$	2.19269	0.106361
$426$	16.4330	0.796180
$427$	11.5230i	0.557638i
$428$	8.39911	0.405986
$429$	0	0
$430$	−18.4819	−0.891275
$431$	33.0562i	1.59226i	0.605124	+	0.796131i	$0.293123\pi$
$431$	33.0562i	1.59226i	−0.605124	+	0.796131i	$0.706877\pi$
$432$	−4.93900	−0.237628
$433$	29.2664	1.40645	0.703226	−	0.710967i	$-0.251742\pi$
$433$	29.2664	1.40645	0.703226	+	0.710967i	$0.251742\pi$
$434$	30.4166i	1.46004i
$435$	5.65279i	0.271031i
$436$	− 2.58881i	− 0.123982i
$437$	− 22.5217i	− 1.07736i
$438$	5.31767	0.254088
$439$	2.13169	0.101740	0.0508699	−	0.998705i	$-0.483801\pi$
$439$	2.13169	0.101740	0.0508699	+	0.998705i	$0.483801\pi$
$440$	10.1685i	0.484765i
$441$	−4.86831	−0.231824
$442$	0	0
$443$	−22.9922	−1.09239	−0.546197	−	0.837657i	$-0.683925\pi$
$443$	−22.9922	−1.09239	−0.546197	+	0.837657i	$0.683925\pi$
$444$	− 7.78986i	− 0.369690i
$445$	1.67456	0.0793819
$446$	−13.4233	−0.635610
$447$	− 12.7385i	− 0.602513i
$448$	4.37090i	0.206505i
$449$	− 12.9379i	− 0.610579i	−0.952260	−	0.305289i	$-0.901247\pi$
$449$	− 12.9379i	− 0.610579i	0.952260	−	0.305289i	$-0.0987532\pi$
$450$	− 5.24698i	− 0.247345i
$451$	−9.34481	−0.440030
$452$	−7.69202	−0.361802
$453$	15.6407i	0.734865i
$454$	38.2911	1.79709
$455$	0	0
$456$	−10.8019	−0.505847
$457$	4.85325i	0.227025i	0.993537	+	0.113513i	$0.0362103\pi$
$457$	4.85325i	0.227025i	−0.993537	+	0.113513i	$0.963790\pi$
$458$	16.7506	0.782705
$459$	0.753020	0.0351480
$460$	5.09783i	0.237688i
$461$	− 18.8345i	− 0.877208i	−0.898680	−	0.438604i	$-0.855473\pi$
$461$	− 18.8345i	− 0.877208i	0.898680	−	0.438604i	$-0.144527\pi$
$462$	− 32.1933i	− 1.49777i
$463$	− 22.8767i	− 1.06317i	−0.847005	−	0.531585i	$-0.821597\pi$
$463$	− 22.8767i	− 1.06317i	0.847005	−	0.531585i	$-0.178403\pi$
$464$	19.3207	0.896939
$465$	7.08038	0.328345
$466$	29.2107i	1.35316i
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	−15.6504	−0.722668
$470$	− 27.4776i	− 1.26745i
$471$	−0.823708	−0.0379545
$472$	2.54825	0.117293
$473$	− 36.8092i	− 1.69249i
$474$	16.9976i	0.780726i
$475$	− 23.1806i	− 1.06360i
$476$	− 3.23490i	− 0.148271i
$477$	−3.08815	−0.141396
$478$	24.3424	1.11340
$479$	− 38.0901i	− 1.74038i	−0.492717	−	0.870190i	$-0.663996\pi$
$479$	− 38.0901i	− 1.74038i	0.492717	−	0.870190i	$-0.336004\pi$
$480$	8.93900	0.408008
$481$	0	0
$482$	11.2959	0.514514
$483$	9.74632i	0.443473i
$484$	19.8200	0.900909
$485$	−12.5114	−0.568114
$486$	− 1.80194i	− 0.0817376i
$487$	21.2500i	0.962928i	0.876466	+	0.481464i	$0.159895\pi$
$487$	21.2500i	0.962928i	−0.876466	+	0.481464i	$0.840105\pi$
$488$	− 4.53856i	− 0.205451i
$489$	6.26875i	0.283483i
$490$	12.6765	0.572665
$491$	6.35019	0.286580	0.143290	−	0.989681i	$-0.454232\pi$
$491$	6.35019	0.286580	0.143290	+	0.989681i	$0.454232\pi$
$492$	2.24698i	0.101302i
$493$	−2.94571	−0.132668
$494$	0	0
$495$	−7.49396	−0.336828
$496$	− 24.2000i	− 1.08661i
$497$	−31.4174	−1.40926
$498$	11.6528	0.522174
$499$	4.65087i	0.208202i	0.994567	+	0.104101i	$0.0331965\pi$
$499$	4.65087i	0.208202i	−0.994567	+	0.104101i	$0.966804\pi$
$500$	14.2567i	0.637578i
$501$	− 7.45042i	− 0.332860i
$502$	− 1.35690i	− 0.0605612i
$503$	15.4752	0.690004	0.345002	−	0.938602i	$-0.387878\pi$
$503$	15.4752	0.690004	0.345002	+	0.938602i	$0.387878\pi$
$504$	4.67456	0.208222
$505$	− 12.2489i	− 0.545069i
$506$	−26.4373	−1.17528
$507$	0	0
$508$	−17.7845	−0.789059
$509$	20.5047i	0.908855i	0.890784	+	0.454428i	$0.150156\pi$
$509$	20.5047i	0.908855i	−0.890784	+	0.454428i	$0.849844\pi$
$510$	−1.96077	−0.0868244
$511$	−10.1666	−0.449744
$512$	− 17.1491i	− 0.757892i
$513$	− 7.96077i	− 0.351477i
$514$	35.5459i	1.56786i
$515$	8.16075i	0.359606i
$516$	−8.85086	−0.389637
$517$	54.7254	2.40682
$518$	38.7797i	1.70388i
$519$	2.00969	0.0882155
$520$	0	0
$521$	−42.0267	−1.84122	−0.920611	−	0.390481i	$-0.872309\pi$
$521$	−42.0267	−1.84122	−0.920611	+	0.390481i	$0.872309\pi$
$522$	7.04892i	0.308523i
$523$	−29.9885	−1.31131	−0.655653	−	0.755062i	$-0.727607\pi$
$523$	−29.9885	−1.31131	−0.655653	+	0.755062i	$0.727607\pi$
$524$	−28.1836	−1.23121
$525$	10.0315i	0.437809i
$526$	31.7308i	1.38353i
$527$	3.68963i	0.160723i
$528$	25.6136i	1.11469i
$529$	−14.9963	−0.652012
$530$	8.04115	0.349285
$531$	1.87800i	0.0814984i
$532$	−34.1987	−1.48270
$533$	0	0
$534$	2.08815	0.0903629
$535$	9.73317i	0.420802i
$536$	6.16421	0.266253
$537$	−20.0368	−0.864653
$538$	29.5284i	1.27306i
$539$	25.2470i	1.08746i
$540$	1.80194i	0.0775431i
$541$	36.3803i	1.56411i	0.623208	+	0.782056i	$0.285829\pi$
$541$	36.3803i	1.56411i	−0.623208	+	0.782056i	$0.714171\pi$
$542$	−1.43296	−0.0615509
$543$	−24.1226	−1.03520
$544$	4.65817i	0.199717i
$545$	3.00000	0.128506
$546$	0	0
$547$	−25.8159	−1.10381	−0.551905	−	0.833907i	$-0.686099\pi$
$547$	−25.8159	−1.10381	−0.551905	+	0.833907i	$0.686099\pi$
$548$	17.0030i	0.726331i
$549$	3.34481	0.142753
$550$	−27.2107	−1.16027
$551$	31.1414i	1.32667i
$552$	− 3.83877i	− 0.163389i
$553$	− 32.4969i	− 1.38191i
$554$	− 8.70948i	− 0.370030i
$555$	9.02715	0.383181
$556$	−21.9487	−0.930832
$557$	17.9903i	0.762274i	0.924519	+	0.381137i	$0.124467\pi$
$557$	17.9903i	0.762274i	−0.924519	+	0.381137i	$0.875533\pi$
$558$	8.82908	0.373765
$559$	0	0
$560$	−24.5875	−1.03901
$561$	− 3.90515i	− 0.164876i
$562$	33.8310	1.42707
$563$	39.1323	1.64923	0.824614	−	0.565695i	$-0.191392\pi$
$563$	39.1323	1.64923	0.824614	+	0.565695i	$0.191392\pi$
$564$	− 13.1588i	− 0.554087i
$565$	− 8.91377i	− 0.375005i
$566$	− 14.2664i	− 0.599660i
$567$	3.44504i	0.144678i
$568$	12.3744	0.519216
$569$	−30.6002	−1.28283	−0.641413	−	0.767196i	$-0.721651\pi$
$569$	−30.6002	−1.28283	−0.641413	+	0.767196i	$0.721651\pi$
$570$	20.7289i	0.868236i
$571$	2.96184	0.123949	0.0619745	−	0.998078i	$-0.480260\pi$
$571$	2.96184	0.123949	0.0619745	+	0.998078i	$0.480260\pi$
$572$	0	0
$573$	−7.08038	−0.295787
$574$	− 11.1860i	− 0.466894i
$575$	8.23788	0.343543
$576$	1.26875	0.0528646
$577$	0.819396i	0.0341119i	0.999855	+	0.0170560i	$0.00542934\pi$
$577$	0.819396i	0.0341119i	−0.999855	+	0.0170560i	$0.994571\pi$
$578$	29.6112i	1.23166i
$579$	9.76809i	0.405948i
$580$	− 7.04892i	− 0.292690i
$581$	−22.2784	−0.924265
$582$	−15.6015	−0.646702
$583$	16.0151i	0.663276i
$584$	4.00431	0.165700
$585$	0	0
$586$	11.8552	0.489732
$587$	31.7995i	1.31251i	0.754540	+	0.656254i	$0.227860\pi$
$587$	31.7995i	1.31251i	−0.754540	+	0.656254i	$0.772140\pi$
$588$	6.07069	0.250351
$589$	39.0060	1.60721
$590$	− 4.89008i	− 0.201322i
$591$	− 23.4112i	− 0.963008i
$592$	− 30.8538i	− 1.26808i
$593$	− 4.26337i	− 0.175076i	−0.996161	−	0.0875379i	$-0.972100\pi$
$593$	− 4.26337i	− 0.175076i	0.996161	−	0.0875379i	$-0.0278999\pi$
$594$	−9.34481	−0.383422
$595$	3.74871	0.153682
$596$	15.8847i	0.650663i
$597$	−4.02475	−0.164722
$598$	0	0
$599$	24.7278	1.01035	0.505175	−	0.863017i	$-0.331428\pi$
$599$	24.7278	1.01035	0.505175	+	0.863017i	$0.331428\pi$
$600$	− 3.95108i	− 0.161302i
$601$	6.82371	0.278345	0.139172	−	0.990268i	$-0.455556\pi$
$601$	6.82371	0.278345	0.139172	+	0.990268i	$0.455556\pi$
$602$	44.0616	1.79582
$603$	4.54288i	0.185000i
$604$	− 19.5036i	− 0.793592i
$605$	22.9681i	0.933785i
$606$	− 15.2741i	− 0.620469i
$607$	31.9963	1.29869	0.649344	−	0.760494i	$-0.275043\pi$
$607$	31.9963	1.29869	0.649344	+	0.760494i	$0.275043\pi$
$608$	49.2452	1.99716
$609$	− 13.4765i	− 0.546095i
$610$	−8.70948	−0.352637
$611$	0	0
$612$	−0.939001	−0.0379569
$613$	− 33.5875i	− 1.35659i	−0.734792	−	0.678293i	$-0.762720\pi$
$613$	− 33.5875i	− 1.35659i	0.734792	−	0.678293i	$-0.237280\pi$
$614$	−44.8049	−1.80818
$615$	−2.60388	−0.104998
$616$	− 24.2422i	− 0.976746i
$617$	26.5870i	1.07035i	0.844740	+	0.535176i	$0.179755\pi$
$617$	26.5870i	1.07035i	−0.844740	+	0.535176i	$0.820245\pi$
$618$	10.1763i	0.409350i
$619$	− 9.17928i	− 0.368946i	−0.982838	−	0.184473i	$-0.940942\pi$
$619$	− 9.17928i	− 0.368946i	0.982838	−	0.184473i	$-0.0590579\pi$
$620$	−8.82908	−0.354585
$621$	2.82908	0.113527
$622$	− 30.7778i	− 1.23408i
$623$	−3.99223	−0.159945
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	− 28.2857i	− 1.13053i
$627$	−41.2844	−1.64874
$628$	1.02715	0.0409876
$629$	4.70410i	0.187565i
$630$	− 8.97046i	− 0.357392i
$631$	17.0043i	0.676931i	0.940979	+	0.338465i	$0.109908\pi$
$631$	17.0043i	0.676931i	−0.940979	+	0.338465i	$0.890092\pi$
$632$	12.7995i	0.509139i
$633$	−3.91185	−0.155482
$634$	−59.0713	−2.34602
$635$	− 20.6093i	− 0.817853i
$636$	3.85086	0.152696
$637$	0	0
$638$	36.5555	1.44725
$639$	9.11960i	0.360766i
$640$	14.5743	0.576101
$641$	21.6649	0.855711	0.427856	−	0.903847i	$-0.359269\pi$
$641$	21.6649	0.855711	0.427856	+	0.903847i	$0.359269\pi$
$642$	12.1371i	0.479012i
$643$	− 9.35557i	− 0.368948i	−0.982837	−	0.184474i	$-0.940942\pi$
$643$	− 9.35557i	− 0.368948i	0.982837	−	0.184474i	$-0.0590581\pi$
$644$	− 12.1535i	− 0.478913i
$645$	− 10.2567i	− 0.403856i
$646$	−10.8019	−0.424997
$647$	0.702775	0.0276289	0.0138145	−	0.999905i	$-0.495603\pi$
$647$	0.702775	0.0276289	0.0138145	+	0.999905i	$0.495603\pi$
$648$	− 1.35690i	− 0.0533039i
$649$	9.73928	0.382300
$650$	0	0
$651$	−16.8799	−0.661576
$652$	− 7.81700i	− 0.306137i
$653$	−37.3411	−1.46127	−0.730635	−	0.682768i	$-0.760776\pi$
$653$	−37.3411	−1.46127	−0.730635	+	0.682768i	$0.760776\pi$
$654$	3.74094	0.146282
$655$	− 32.6601i	− 1.27614i
$656$	8.89977i	0.347478i
$657$	2.95108i	0.115133i
$658$	65.5077i	2.55376i
$659$	−0.735562	−0.0286534	−0.0143267	−	0.999897i	$-0.504560\pi$
$659$	−0.735562	−0.0286534	−0.0143267	+	0.999897i	$0.504560\pi$
$660$	9.34481	0.363746
$661$	13.8485i	0.538643i	0.963050	+	0.269321i	$0.0867994\pi$
$661$	13.8485i	0.538643i	−0.963050	+	0.269321i	$0.913201\pi$
$662$	52.5478	2.04233
$663$	0	0
$664$	8.77479	0.340528
$665$	− 39.6305i	− 1.53681i
$666$	11.2567	0.436187
$667$	−11.0670	−0.428515
$668$	9.29052i	0.359461i
$669$	− 7.44935i	− 0.288009i
$670$	− 11.8291i	− 0.456997i
$671$	− 17.3461i	− 0.669640i
$672$	−21.3110	−0.822088
$673$	6.35019	0.244782	0.122391	−	0.992482i	$-0.460944\pi$
$673$	6.35019	0.244782	0.122391	+	0.992482i	$0.460944\pi$
$674$	− 59.9885i	− 2.31067i
$675$	2.91185	0.112077
$676$	0	0
$677$	33.7241	1.29612	0.648061	−	0.761589i	$-0.275580\pi$
$677$	33.7241	1.29612	0.648061	+	0.761589i	$0.275580\pi$
$678$	− 11.1153i	− 0.426880i
$679$	29.8278	1.14468
$680$	−1.47650	−0.0566212
$681$	21.2500i	0.814300i
$682$	− 45.7875i	− 1.75329i
$683$	19.2687i	0.737298i	0.929569	+	0.368649i	$0.120180\pi$
$683$	19.2687i	0.737298i	−0.929569	+	0.368649i	$0.879820\pi$
$684$	9.92692i	0.379565i
$685$	−19.7036	−0.752837
$686$	13.2330	0.505237
$687$	9.29590i	0.354661i
$688$	−35.0562	−1.33651
$689$	0	0
$690$	−7.36658	−0.280441
$691$	39.4010i	1.49889i	0.662069	+	0.749443i	$0.269679\pi$
$691$	39.4010i	1.49889i	−0.662069	+	0.749443i	$0.730321\pi$
$692$	−2.50604	−0.0952654
$693$	17.8659	0.678670
$694$	1.57434i	0.0597610i
$695$	− 25.4349i	− 0.964800i
$696$	5.30798i	0.201198i
$697$	− 1.35690i	− 0.0513961i
$698$	−5.82371	−0.220431
$699$	−16.2107	−0.613146
$700$	− 12.5090i	− 0.472797i
$701$	−18.3985	−0.694902	−0.347451	−	0.937698i	$-0.612953\pi$
$701$	−18.3985	−0.694902	−0.347451	+	0.937698i	$0.612953\pi$
$702$	0	0
$703$	49.7308	1.87563
$704$	− 6.57971i	− 0.247982i
$705$	15.2489	0.574307
$706$	−14.6799	−0.552487
$707$	29.2019i	1.09825i
$708$	− 2.34183i	− 0.0880114i
$709$	− 38.4553i	− 1.44422i	−0.691778	−	0.722110i	$-0.743172\pi$
$709$	− 38.4553i	− 1.44422i	0.691778	−	0.722110i	$-0.256828\pi$
$710$	− 23.7463i	− 0.891183i
$711$	−9.43296	−0.353764
$712$	1.57242	0.0589288
$713$	13.8619i	0.519131i
$714$	4.67456	0.174941
$715$	0	0
$716$	24.9855	0.933753
$717$	13.5090i	0.504504i
$718$	4.75840	0.177582
$719$	−48.4999	−1.80874	−0.904371	−	0.426747i	$-0.859659\pi$
$719$	−48.4999	−1.80874	−0.904371	+	0.426747i	$0.859659\pi$
$720$	7.13706i	0.265983i
$721$	− 19.4556i	− 0.724564i
$722$	79.9590i	2.97576i
$723$	6.26875i	0.233137i
$724$	30.0804	1.11793
$725$	−11.3907	−0.423042
$726$	28.6407i	1.06296i
$727$	−19.0344	−0.705948	−0.352974	−	0.935633i	$-0.614830\pi$
$727$	−19.0344	−0.705948	−0.352974	+	0.935633i	$0.614830\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 7.68425i	− 0.284407i
$731$	5.34481	0.197685
$732$	−4.17092	−0.154161
$733$	− 24.6213i	− 0.909410i	−0.890642	−	0.454705i	$-0.849745\pi$
$733$	− 24.6213i	− 0.909410i	0.890642	−	0.454705i	$-0.150255\pi$
$734$	5.23298i	0.193153i
$735$	7.03492i	0.259487i
$736$	17.5007i	0.645083i
$737$	23.5593	0.867817
$738$	−3.24698	−0.119523
$739$	− 44.5115i	− 1.63738i	−0.574233	−	0.818692i	$-0.694700\pi$
$739$	− 44.5115i	− 1.63738i	0.574233	−	0.818692i	$-0.305300\pi$
$740$	−11.2567	−0.413803
$741$	0	0
$742$	−19.1704	−0.703769
$743$	− 10.4112i	− 0.381950i	−0.981595	−	0.190975i	$-0.938835\pi$
$743$	− 10.4112i	− 0.381950i	0.981595	−	0.190975i	$-0.0611649\pi$
$744$	6.64848	0.243745
$745$	−18.4077	−0.674407
$746$	15.1336i	0.554081i
$747$	6.46681i	0.236608i
$748$	4.86964i	0.178052i
$749$	− 23.2043i	− 0.847866i
$750$	−20.6015	−0.752260
$751$	−1.69979	−0.0620263	−0.0310131	−	0.999519i	$-0.509873\pi$
$751$	−1.69979	−0.0620263	−0.0310131	+	0.999519i	$0.509873\pi$
$752$	− 52.1191i	− 1.90059i
$753$	0.753020	0.0274416
$754$	0	0
$755$	22.6015	0.822552
$756$	− 4.29590i	− 0.156240i
$757$	27.4252	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$757$	27.4252	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$758$	−28.3773	−1.03071
$759$	− 14.6716i	− 0.532545i
$760$	15.6093i	0.566207i
$761$	− 5.02608i	− 0.182195i	−0.995842	−	0.0910977i	$-0.970962\pi$
$761$	− 5.02608i	− 0.182195i	0.995842	−	0.0910977i	$-0.0290375\pi$
$762$	− 25.6993i	− 0.930988i
$763$	−7.15213	−0.258924
$764$	8.82908	0.319425
$765$	− 1.08815i	− 0.0393420i
$766$	−22.9541	−0.829364
$767$	0	0
$768$	20.7114	0.747358
$769$	− 42.4456i	− 1.53063i	−0.643657	−	0.765314i	$-0.722584\pi$
$769$	− 42.4456i	− 1.53063i	0.643657	−	0.765314i	$-0.277416\pi$
$770$	−46.5206	−1.67649
$771$	−19.7265	−0.710431
$772$	− 12.1806i	− 0.438390i
$773$	− 26.3593i	− 0.948078i	−0.880504	−	0.474039i	$-0.842796\pi$
$773$	− 26.3593i	− 0.948078i	0.880504	−	0.474039i	$-0.157204\pi$
$774$	− 12.7899i	− 0.459722i
$775$	14.2674i	0.512501i
$776$	−11.7482	−0.421737
$777$	−21.5211	−0.772065
$778$	− 0.559270i	− 0.0200508i
$779$	−14.3448	−0.513956
$780$	0	0
$781$	47.2941	1.69232
$782$	− 3.83877i	− 0.137274i
$783$	−3.91185	−0.139798
$784$	24.0446	0.858736
$785$	1.19029i	0.0424834i
$786$	− 40.7265i	− 1.45266i
$787$	17.1424i	0.611062i	0.952182	+	0.305531i	$0.0988340\pi$
$787$	17.1424i	0.611062i	−0.952182	+	0.305531i	$0.901166\pi$
$788$	29.1933i	1.03997i
$789$	−17.6093	−0.626906
$790$	24.5623	0.873886
$791$	21.2508i	0.755592i
$792$	−7.03684	−0.250043
$793$	0	0
$794$	2.68664	0.0953455
$795$	4.46250i	0.158269i
$796$	5.01879	0.177886
$797$	−30.1629	−1.06842	−0.534212	−	0.845351i	$-0.679392\pi$
$797$	−30.1629	−1.06842	−0.534212	+	0.845351i	$0.679392\pi$
$798$	− 49.4185i	− 1.74940i
$799$	7.94630i	0.281120i
$800$	18.0127i	0.636844i
$801$	1.15883i	0.0409454i
$802$	−42.9463	−1.51649
$803$	15.3043	0.540076
$804$	− 5.66487i	− 0.199785i
$805$	14.0838	0.496390
$806$	0	0
$807$	−16.3870	−0.576851
$808$	− 11.5017i	− 0.404629i
$809$	−29.8504	−1.04948	−0.524742	−	0.851261i	$-0.675838\pi$
$809$	−29.8504	−1.04948	−0.524742	+	0.851261i	$0.675838\pi$
$810$	−2.60388	−0.0914909
$811$	− 47.7362i	− 1.67624i	−0.545484	−	0.838122i	$-0.683654\pi$
$811$	− 47.7362i	− 1.67624i	0.545484	−	0.838122i	$-0.316346\pi$
$812$	16.8049i	0.589737i
$813$	− 0.795233i	− 0.0278900i
$814$	− 58.3769i	− 2.04611i
$815$	9.05861	0.317309
$816$	−3.71917	−0.130197
$817$	− 56.5042i	− 1.97683i
$818$	−7.68963	−0.268862
$819$	0	0
$820$	3.24698	0.113389
$821$	17.9299i	0.625758i	0.949793	+	0.312879i	$0.101293\pi$
$821$	17.9299i	0.625758i	−0.949793	+	0.312879i	$0.898707\pi$
$822$	−24.5700	−0.856978
$823$	−54.3196	−1.89346	−0.946731	−	0.322026i	$-0.895636\pi$
$823$	−54.3196	−1.89346	−0.946731	+	0.322026i	$0.895636\pi$
$824$	7.66296i	0.266952i
$825$	− 15.1008i	− 0.525743i
$826$	11.6582i	0.405640i
$827$	49.1041i	1.70752i	0.520670	+	0.853758i	$0.325682\pi$
$827$	49.1041i	1.70752i	−0.520670	+	0.853758i	$0.674318\pi$
$828$	−3.52781	−0.122600
$829$	7.35796	0.255553	0.127776	−	0.991803i	$-0.459216\pi$
$829$	7.35796	0.255553	0.127776	+	0.991803i	$0.459216\pi$
$830$	− 16.8388i	− 0.584482i
$831$	4.83340	0.167669
$832$	0	0
$833$	−3.66594	−0.127017
$834$	− 31.7168i	− 1.09826i
$835$	−10.7662	−0.372579
$836$	51.4808	1.78050
$837$	4.89977i	0.169361i
$838$	− 53.4989i	− 1.84809i
$839$	− 36.5013i	− 1.26016i	−0.776529	−	0.630082i	$-0.783021\pi$
$839$	− 36.5013i	− 1.26016i	0.776529	−	0.630082i	$-0.216979\pi$
$840$	− 6.75494i	− 0.233068i
$841$	−13.6974	−0.472324
$842$	−52.9754	−1.82565
$843$	18.7748i	0.646638i
$844$	4.87800	0.167908
$845$	0	0
$846$	19.0151	0.653751
$847$	− 54.7569i	− 1.88147i
$848$	15.2524	0.523768
$849$	7.91723	0.271719
$850$	− 3.95108i	− 0.135521i
$851$	17.6732i	0.605831i
$852$	− 11.3720i	− 0.389597i
$853$	− 9.73855i	− 0.333441i	−0.986004	−	0.166721i	$-0.946682\pi$
$853$	− 9.73855i	− 0.333441i	0.986004	−	0.166721i	$-0.0533178\pi$
$854$	20.7638	0.710522
$855$	−11.5036	−0.393416
$856$	9.13946i	0.312380i
$857$	15.2030	0.519323	0.259662	−	0.965700i	$-0.416389\pi$
$857$	15.2030	0.519323	0.259662	+	0.965700i	$0.416389\pi$
$858$	0	0
$859$	−31.9885	−1.09143	−0.545717	−	0.837970i	$-0.683743\pi$
$859$	−31.9885	−1.09143	−0.545717	+	0.837970i	$0.683743\pi$
$860$	12.7899i	0.436130i
$861$	6.20775	0.211560
$862$	59.5652	2.02880
$863$	35.2905i	1.20130i	0.799511	+	0.600652i	$0.205092\pi$
$863$	35.2905i	1.20130i	−0.799511	+	0.600652i	$0.794908\pi$
$864$	6.18598i	0.210451i
$865$	− 2.90408i	− 0.0987418i
$866$	− 52.7362i	− 1.79205i
$867$	−16.4330	−0.558093
$868$	21.0489	0.714447
$869$	48.9191i	1.65947i
$870$	10.1860	0.345337
$871$	0	0
$872$	2.81700	0.0953958
$873$	− 8.65817i	− 0.293035i
$874$	−40.5827	−1.37273
$875$	39.3870	1.33152
$876$	− 3.67994i	− 0.124334i
$877$	15.3263i	0.517532i	0.965940	+	0.258766i	$0.0833159\pi$
$877$	15.3263i	0.517532i	−0.965940	+	0.258766i	$0.916684\pi$
$878$	− 3.84117i	− 0.129633i
$879$	6.57912i	0.221908i
$880$	37.0127	1.24770
$881$	36.4306	1.22738	0.613689	−	0.789548i	$-0.289685\pi$
$881$	36.4306	1.22738	0.613689	+	0.789548i	$0.289685\pi$
$882$	8.77240i	0.295382i
$883$	37.6819	1.26810	0.634048	−	0.773294i	$-0.281392\pi$
$883$	37.6819	1.26810	0.634048	+	0.773294i	$0.281392\pi$
$884$	0	0
$885$	2.71379	0.0912231
$886$	41.4306i	1.39189i
$887$	4.24890	0.142664	0.0713320	−	0.997453i	$-0.477275\pi$
$887$	4.24890	0.142664	0.0713320	+	0.997453i	$0.477275\pi$
$888$	8.47650	0.284453
$889$	49.1333i	1.64788i
$890$	− 3.01746i	− 0.101145i
$891$	− 5.18598i	− 0.173737i
$892$	9.28919i	0.311025i
$893$	84.0066	2.81117
$894$	−22.9541	−0.767699
$895$	28.9541i	0.967828i
$896$	−34.7458	−1.16078
$897$	0	0
$898$	−23.3134	−0.777977
$899$	− 19.1672i	− 0.639262i
$900$	−3.63102	−0.121034
$901$	−2.32544	−0.0774715
$902$	16.8388i	0.560670i
$903$	24.4523i	0.813723i
$904$	− 8.37004i	− 0.278383i
$905$	34.8582i	1.15872i
$906$	28.1836	0.936337
$907$	12.6183	0.418985	0.209493	−	0.977810i	$-0.432819\pi$
$907$	12.6183	0.418985	0.209493	+	0.977810i	$0.432819\pi$
$908$	− 26.4983i	− 0.879376i
$909$	8.47650	0.281148
$910$	0	0
$911$	6.77777	0.224558	0.112279	−	0.993677i	$-0.464185\pi$
$911$	6.77777	0.224558	0.112279	+	0.993677i	$0.464185\pi$
$912$	39.3183i	1.30196i
$913$	33.5368	1.10990
$914$	8.74525	0.289267
$915$	− 4.83340i	− 0.159787i
$916$	− 11.5918i	− 0.383004i
$917$	77.8631i	2.57126i
$918$	− 1.35690i	− 0.0447842i
$919$	−20.4674	−0.675157	−0.337579	−	0.941297i	$-0.609608\pi$
$919$	−20.4674	−0.675157	−0.337579	+	0.941297i	$0.609608\pi$
$920$	−5.54719	−0.182885
$921$	− 24.8649i	− 0.819325i
$922$	−33.9385	−1.11771
$923$	0	0
$924$	−22.2784	−0.732907
$925$	18.1903i	0.598093i
$926$	−41.2223	−1.35465
$927$	−5.64742	−0.185485
$928$	− 24.1987i	− 0.794360i
$929$	4.65220i	0.152634i	0.997084	+	0.0763169i	$0.0243161\pi$
$929$	4.65220i	0.152634i	−0.997084	+	0.0763169i	$0.975684\pi$
$930$	− 12.7584i	− 0.418364i
$931$	38.7555i	1.27016i
$932$	20.2145	0.662147
$933$	17.0804	0.559186
$934$	23.4252i	0.766496i
$935$	−5.64310	−0.184549
$936$	0	0
$937$	−41.8544	−1.36732	−0.683662	−	0.729798i	$-0.739614\pi$
$937$	−41.8544	−1.36732	−0.683662	+	0.729798i	$0.739614\pi$
$938$	28.2010i	0.920797i
$939$	15.6974	0.512265
$940$	−19.0151	−0.620203
$941$	− 30.3454i	− 0.989232i	−0.869112	−	0.494616i	$-0.835309\pi$
$941$	− 30.3454i	− 0.989232i	0.869112	−	0.494616i	$-0.164691\pi$
$942$	1.48427i	0.0483601i
$943$	− 5.09783i	− 0.166008i
$944$	− 9.27545i	− 0.301890i
$945$	4.97823	0.161942
$946$	−66.3279	−2.15651
$947$	12.0325i	0.391004i	0.980703	+	0.195502i	$0.0626337\pi$
$947$	12.0325i	0.391004i	−0.980703	+	0.195502i	$0.937366\pi$
$948$	11.7627	0.382035
$949$	0	0
$950$	−41.7700	−1.35520
$951$	− 32.7821i	− 1.06303i
$952$	3.52004	0.114085
$953$	−22.9825	−0.744478	−0.372239	−	0.928137i	$-0.621410\pi$
$953$	−22.9825	−0.744478	−0.372239	+	0.928137i	$0.621410\pi$
$954$	5.56465i	0.180162i
$955$	10.2314i	0.331082i
$956$	− 16.8455i	− 0.544822i
$957$	20.2868i	0.655779i
$958$	−68.6359	−2.21753
$959$	46.9743	1.51688
$960$	− 1.83340i	− 0.0591726i
$961$	6.99223	0.225556
$962$	0	0
$963$	−6.73556	−0.217050
$964$	− 7.81700i	− 0.251769i
$965$	14.1153	0.454387
$966$	17.5623	0.565056
$967$	38.8883i	1.25056i	0.780399	+	0.625281i	$0.215016\pi$
$967$	38.8883i	1.25056i	−0.780399	+	0.625281i	$0.784984\pi$
$968$	21.5670i	0.693191i
$969$	− 5.99462i	− 0.192575i
$970$	22.5448i	0.723870i
$971$	−57.5133	−1.84569	−0.922845	−	0.385171i	$-0.874143\pi$
$971$	−57.5133	−1.84569	−0.922845	+	0.385171i	$0.874143\pi$
$972$	−1.24698	−0.0399969
$973$	60.6378i	1.94396i
$974$	38.2911	1.22693
$975$	0	0
$976$	−16.5200	−0.528794
$977$	− 16.3690i	− 0.523690i	−0.965110	−	0.261845i	$-0.915669\pi$
$977$	− 16.3690i	− 0.523690i	0.965110	−	0.261845i	$-0.0843309\pi$
$978$	11.2959	0.361203
$979$	6.00969	0.192070
$980$	− 8.77240i	− 0.280224i
$981$	2.07606i	0.0662836i
$982$	− 11.4426i	− 0.365150i
$983$	− 15.6963i	− 0.500635i	−0.968164	−	0.250318i	$-0.919465\pi$
$983$	− 15.6963i	− 0.500635i	0.968164	−	0.250318i	$-0.0805351\pi$
$984$	−2.44504	−0.0779451
$985$	−33.8301	−1.07792
$986$	5.30798i	0.169040i
$987$	−36.3540	−1.15716
$988$	0	0
$989$	20.0804	0.638519
$990$	13.5036i	0.429174i
$991$	−11.2644	−0.357827	−0.178913	−	0.983865i	$-0.557258\pi$
$991$	−11.2644	−0.357827	−0.178913	+	0.983865i	$0.557258\pi$
$992$	−30.3099	−0.962340
$993$	29.1618i	0.925422i
$994$	56.6122i	1.79563i
$995$	5.81594i	0.184378i
$996$	− 8.06398i	− 0.255517i
$997$	−7.70112	−0.243897	−0.121948	−	0.992536i	$-0.538914\pi$
$997$	−7.70112	−0.243897	−0.121948	+	0.992536i	$0.538914\pi$
$998$	8.38059	0.265283
$999$	6.24698i	0.197646i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.g.337.1		6
3.2	odd	2		1521.2.b.m.1351.6		6
13.2	odd	12		507.2.e.k.22.3		6
13.3	even	3		507.2.j.h.316.6		12
13.4	even	6		507.2.j.h.361.6		12
13.5	odd	4		507.2.a.j.1.1	✓	3
13.6	odd	12		507.2.e.k.484.3		6
13.7	odd	12		507.2.e.j.484.1		6
13.8	odd	4		507.2.a.k.1.3	yes	3
13.9	even	3		507.2.j.h.361.1		12
13.10	even	6		507.2.j.h.316.1		12
13.11	odd	12		507.2.e.j.22.1		6
13.12	even	2	inner	507.2.b.g.337.6		6
39.5	even	4		1521.2.a.q.1.3		3
39.8	even	4		1521.2.a.p.1.1		3
39.38	odd	2		1521.2.b.m.1351.1		6
52.31	even	4		8112.2.a.by.1.2		3
52.47	even	4		8112.2.a.cf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.1	✓	3	13.5	odd	4
507.2.a.k.1.3	yes	3	13.8	odd	4
507.2.b.g.337.1		6	1.1	even	1	trivial
507.2.b.g.337.6		6	13.12	even	2	inner
507.2.e.j.22.1		6	13.11	odd	12
507.2.e.j.484.1		6	13.7	odd	12
507.2.e.k.22.3		6	13.2	odd	12
507.2.e.k.484.3		6	13.6	odd	12
507.2.j.h.316.1		12	13.10	even	6
507.2.j.h.316.6		12	13.3	even	3
507.2.j.h.361.1		12	13.9	even	3
507.2.j.h.361.6		12	13.4	even	6
1521.2.a.p.1.1		3	39.8	even	4
1521.2.a.q.1.3		3	39.5	even	4
1521.2.b.m.1351.1		6	39.38	odd	2
1521.2.b.m.1351.6		6	3.2	odd	2
8112.2.a.by.1.2		3	52.31	even	4
8112.2.a.cf.1.2		3	52.47	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.80194i q^{2} +1.00000 q^{3} -1.24698 q^{4} -1.44504i q^{5} -1.80194i q^{6} +3.44504i q^{7} -1.35690i q^{8} +1.00000 q^{9} -2.60388 q^{10} -5.18598i q^{11} -1.24698 q^{12} +6.20775 q^{14} -1.44504i q^{15} -4.93900 q^{16} +0.753020 q^{17} -1.80194i q^{18} -7.96077i q^{19} +1.80194i q^{20} +3.44504i q^{21} -9.34481 q^{22} +2.82908 q^{23} -1.35690i q^{24} +2.91185 q^{25} +1.00000 q^{27} -4.29590i q^{28} -3.91185 q^{29} -2.60388 q^{30} +4.89977i q^{31} +6.18598i q^{32} -5.18598i q^{33} -1.35690i q^{34} +4.97823 q^{35} -1.24698 q^{36} +6.24698i q^{37} -14.3448 q^{38} -1.96077 q^{40} -1.80194i q^{41} +6.20775 q^{42} +7.09783 q^{43} +6.46681i q^{44} -1.44504i q^{45} -5.09783i q^{46} +10.5526i q^{47} -4.93900 q^{48} -4.86831 q^{49} -5.24698i q^{50} +0.753020 q^{51} -3.08815 q^{53} -1.80194i q^{54} -7.49396 q^{55} +4.67456 q^{56} -7.96077i q^{57} +7.04892i q^{58} +1.87800i q^{59} +1.80194i q^{60} +3.34481 q^{61} +8.82908 q^{62} +3.44504i q^{63} +1.26875 q^{64} -9.34481 q^{66} +4.54288i q^{67} -0.939001 q^{68} +2.82908 q^{69} -8.97046i q^{70} +9.11960i q^{71} -1.35690i q^{72} +2.95108i q^{73} +11.2567 q^{74} +2.91185 q^{75} +9.92692i q^{76} +17.8659 q^{77} -9.43296 q^{79} +7.13706i q^{80} +1.00000 q^{81} -3.24698 q^{82} +6.46681i q^{83} -4.29590i q^{84} -1.08815i q^{85} -12.7899i q^{86} -3.91185 q^{87} -7.03684 q^{88} +1.15883i q^{89} -2.60388 q^{90} -3.52781 q^{92} +4.89977i q^{93} +19.0151 q^{94} -11.5036 q^{95} +6.18598i q^{96} -8.65817i q^{97} +8.77240i q^{98} -5.18598i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{14} - 10 q^{16} + 14 q^{17} - 10 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{27} - 16 q^{29} + 2 q^{30} + 36 q^{35} + 2 q^{36} - 40 q^{38} + 14 q^{40} + 2 q^{42} + 6 q^{43} - 10 q^{48} - 34 q^{49} + 14 q^{51} - 26 q^{53} - 26 q^{55} - 14 q^{56} - 26 q^{61} + 32 q^{62} - 8 q^{64} - 10 q^{66} + 14 q^{68} - 4 q^{69} + 14 q^{74} + 10 q^{75} + 30 q^{77} - 18 q^{79} + 6 q^{81} - 10 q^{82} - 16 q^{87} + 14 q^{88} + 2 q^{90} - 34 q^{92} + 64 q^{94} - 6 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^4 + 6 * q^9 + 2 * q^10 + 2 * q^12 + 2 * q^14 - 10 * q^16 + 14 * q^17 - 10 * q^22 - 4 * q^23 + 10 * q^25 + 6 * q^27 - 16 * q^29 + 2 * q^30 + 36 * q^35 + 2 * q^36 - 40 * q^38 + 14 * q^40 + 2 * q^42 + 6 * q^43 - 10 * q^48 - 34 * q^49 + 14 * q^51 - 26 * q^53 - 26 * q^55 - 14 * q^56 - 26 * q^61 + 32 * q^62 - 8 * q^64 - 10 * q^66 + 14 * q^68 - 4 * q^69 + 14 * q^74 + 10 * q^75 + 30 * q^77 - 18 * q^79 + 6 * q^81 - 10 * q^82 - 16 * q^87 + 14 * q^88 + 2 * q^90 - 34 * q^92 + 64 * q^94 - 6 * q^95

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