LMFDB - Embedded newform 507.4.b.i.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,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, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 $ x^10 + 70x^8 + 1645x^6 + 14700x^4 + 44100x^2 + 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$0.917374i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.i.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.917374i q^{2} +3.00000 q^{3} +7.15843 q^{4} -15.4704i q^{5} +2.75212i q^{6} +20.5833i q^{7} +13.9059i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.917374i q^{2} +3.00000 q^{3} +7.15843 q^{4} -15.4704i q^{5} +2.75212i q^{6} +20.5833i q^{7} +13.9059i q^{8} +9.00000 q^{9} +14.1922 q^{10} +65.8420i q^{11} +21.4753 q^{12} -18.8826 q^{14} -46.4112i q^{15} +44.5105 q^{16} -44.2956 q^{17} +8.25636i q^{18} +147.053i q^{19} -110.744i q^{20} +61.7500i q^{21} -60.4017 q^{22} -53.1586 q^{23} +41.7178i q^{24} -114.334 q^{25} +27.0000 q^{27} +147.344i q^{28} -38.6257 q^{29} +42.5765 q^{30} +88.3894i q^{31} +152.080i q^{32} +197.526i q^{33} -40.6357i q^{34} +318.433 q^{35} +64.4258 q^{36} -78.9587i q^{37} -134.903 q^{38} +215.131 q^{40} -354.966i q^{41} -56.6478 q^{42} +407.846 q^{43} +471.325i q^{44} -139.234i q^{45} -48.7663i q^{46} +67.9674i q^{47} +133.531 q^{48} -80.6738 q^{49} -104.887i q^{50} -132.887 q^{51} +226.572 q^{53} +24.7691i q^{54} +1018.60 q^{55} -286.231 q^{56} +441.160i q^{57} -35.4342i q^{58} -142.031i q^{59} -332.231i q^{60} +266.831 q^{61} -81.0862 q^{62} +185.250i q^{63} +216.569 q^{64} -181.205 q^{66} -411.187i q^{67} -317.087 q^{68} -159.476 q^{69} +292.122i q^{70} -91.5052i q^{71} +125.153i q^{72} -63.1328i q^{73} +72.4347 q^{74} -343.001 q^{75} +1052.67i q^{76} -1355.25 q^{77} -287.115 q^{79} -688.595i q^{80} +81.0000 q^{81} +325.637 q^{82} -373.812i q^{83} +442.033i q^{84} +685.272i q^{85} +374.147i q^{86} -115.877 q^{87} -915.595 q^{88} -119.403i q^{89} +127.729 q^{90} -380.532 q^{92} +265.168i q^{93} -62.3515 q^{94} +2274.97 q^{95} +456.241i q^{96} +554.650i q^{97} -74.0080i q^{98} +592.578i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	$ 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * 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 $

$2$	0.917374i	0.324341i	0.986763	+	0.162170i	$0.0518494\pi$
$2$	0.917374i	0.324341i	−0.986763	+	0.162170i	$0.948151\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.15843	0.894803
$5$	− 15.4704i	− 1.38372i	−0.722034	−	0.691858i	$-0.756792\pi$
$5$	− 15.4704i	− 1.38372i	0.722034	−	0.691858i	$-0.243208\pi$
$6$	2.75212i	0.187258i
$7$	20.5833i	1.11140i	0.831384	+	0.555698i	$0.187549\pi$
$7$	20.5833i	1.11140i	−0.831384	+	0.555698i	$0.812451\pi$
$8$	13.9059i	0.614562i
$9$	9.00000	0.333333
$10$	14.1922	0.448795
$11$	65.8420i	1.80474i	0.430964	+	0.902369i	$0.358173\pi$
$11$	65.8420i	1.80474i	−0.430964	+	0.902369i	$0.641827\pi$
$12$	21.4753	0.516615
$13$	0	0
$13$	0	0
$14$	−18.8826	−0.360471
$15$	− 46.4112i	− 0.798889i
$16$	44.5105	0.695476
$17$	−44.2956	−0.631957	−0.315979	−	0.948766i	$-0.602333\pi$
$17$	−44.2956	−0.631957	−0.315979	+	0.948766i	$0.602333\pi$
$18$	8.25636i	0.108114i
$19$	147.053i	1.77560i	0.460234	+	0.887798i	$0.347766\pi$
$19$	147.053i	1.77560i	−0.460234	+	0.887798i	$0.652234\pi$
$20$	− 110.744i	− 1.23815i
$21$	61.7500i	0.641665i
$22$	−60.4017	−0.585350
$23$	−53.1586	−0.481928	−0.240964	−	0.970534i	$-0.577464\pi$
$23$	−53.1586	−0.481928	−0.240964	+	0.970534i	$0.577464\pi$
$24$	41.7178i	0.354817i
$25$	−114.334	−0.914669
$26$	0	0
$27$	27.0000	0.192450
$28$	147.344i	0.994480i
$29$	−38.6257	−0.247331	−0.123666	−	0.992324i	$-0.539465\pi$
$29$	−38.6257	−0.247331	−0.123666	+	0.992324i	$0.539465\pi$
$30$	42.5765	0.259112
$31$	88.3894i	0.512104i	0.966663	+	0.256052i	$0.0824218\pi$
$31$	88.3894i	0.512104i	−0.966663	+	0.256052i	$0.917578\pi$
$32$	152.080i	0.840133i
$33$	197.526i	1.04197i
$34$	− 40.6357i	− 0.204969i
$35$	318.433	1.53786
$36$	64.4258	0.298268
$37$	− 78.9587i	− 0.350831i	−0.984495	−	0.175415i	$-0.943873\pi$
$37$	− 78.9587i	− 0.350831i	0.984495	−	0.175415i	$-0.0561268\pi$
$38$	−134.903	−0.575898
$39$	0	0
$40$	215.131	0.850379
$41$	− 354.966i	− 1.35211i	−0.736852	−	0.676054i	$-0.763689\pi$
$41$	− 354.966i	− 1.35211i	0.736852	−	0.676054i	$-0.236311\pi$
$42$	−56.6478	−0.208118
$43$	407.846	1.44642	0.723208	−	0.690630i	$-0.242667\pi$
$43$	407.846	1.44642	0.723208	+	0.690630i	$0.242667\pi$
$44$	471.325i	1.61489i
$45$	− 139.234i	− 0.461239i
$46$	− 48.7663i	− 0.156309i
$47$	67.9674i	0.210938i	0.994423	+	0.105469i	$0.0336343\pi$
$47$	67.9674i	0.210938i	−0.994423	+	0.105469i	$0.966366\pi$
$48$	133.531	0.401533
$49$	−80.6738	−0.235201
$50$	− 104.887i	− 0.296664i
$51$	−132.887	−0.364861
$52$	0	0
$53$	226.572	0.587209	0.293604	−	0.955927i	$-0.405145\pi$
$53$	226.572	0.587209	0.293604	+	0.955927i	$0.405145\pi$
$54$	24.7691i	0.0624194i
$55$	1018.60	2.49724
$56$	−286.231	−0.683021
$57$	441.160i	1.02514i
$58$	− 35.4342i	− 0.0802196i
$59$	− 142.031i	− 0.313404i	−0.987646	−	0.156702i	$-0.949914\pi$
$59$	− 142.031i	− 0.313404i	0.987646	−	0.156702i	$-0.0500862\pi$
$60$	− 332.231i	− 0.714848i
$61$	266.831	0.560069	0.280035	−	0.959990i	$-0.409654\pi$
$61$	266.831	0.560069	0.280035	+	0.959990i	$0.409654\pi$
$62$	−81.0862	−0.166096
$63$	185.250i	0.370465i
$64$	216.569	0.422987
$65$	0	0
$66$	−181.205	−0.337952
$67$	− 411.187i	− 0.749768i	−0.927072	−	0.374884i	$-0.877683\pi$
$67$	− 411.187i	− 0.749768i	0.927072	−	0.374884i	$-0.122317\pi$
$68$	−317.087	−0.565477
$69$	−159.476	−0.278241
$70$	292.122i	0.498789i
$71$	− 91.5052i	− 0.152953i	−0.997071	−	0.0764765i	$-0.975633\pi$
$71$	− 91.5052i	− 0.152953i	0.997071	−	0.0764765i	$-0.0243670\pi$
$72$	125.153i	0.204854i
$73$	− 63.1328i	− 0.101221i	−0.998718	−	0.0506105i	$-0.983883\pi$
$73$	− 63.1328i	− 0.101221i	0.998718	−	0.0506105i	$-0.0161167\pi$
$74$	72.4347	0.113789
$75$	−343.001	−0.528084
$76$	1052.67i	1.58881i
$77$	−1355.25	−2.00578
$78$	0	0
$79$	−287.115	−0.408899	−0.204449	−	0.978877i	$-0.565540\pi$
$79$	−287.115	−0.408899	−0.204449	+	0.978877i	$0.565540\pi$
$80$	− 688.595i	− 0.962341i
$81$	81.0000	0.111111
$82$	325.637	0.438543
$83$	− 373.812i	− 0.494352i	−0.968971	−	0.247176i	$-0.920497\pi$
$83$	− 373.812i	− 0.494352i	0.968971	−	0.247176i	$-0.0795026\pi$
$84$	442.033i	0.574164i
$85$	685.272i	0.874449i
$86$	374.147i	0.469132i
$87$	−115.877	−0.142797
$88$	−915.595	−1.10912
$89$	− 119.403i	− 0.142209i	−0.997469	−	0.0711047i	$-0.977348\pi$
$89$	− 119.403i	− 0.142209i	0.997469	−	0.0711047i	$-0.0226525\pi$
$90$	127.729	0.149598
$91$	0	0
$92$	−380.532	−0.431231
$93$	265.168i	0.295663i
$94$	−62.3515	−0.0684157
$95$	2274.97	2.45692
$96$	456.241i	0.485051i
$97$	554.650i	0.580579i	0.956939	+	0.290290i	$0.0937516\pi$
$97$	554.650i	0.580579i	−0.956939	+	0.290290i	$0.906248\pi$
$98$	− 74.0080i	− 0.0762851i
$99$	592.578i	0.601579i
$100$	−818.449	−0.818449
$101$	462.565	0.455712	0.227856	−	0.973695i	$-0.426828\pi$
$101$	462.565	0.455712	0.227856	+	0.973695i	$0.426828\pi$
$102$	− 121.907i	− 0.118339i
$103$	−1122.07	−1.07341	−0.536704	−	0.843771i	$-0.680331\pi$
$103$	−1122.07	−1.07341	−0.536704	+	0.843771i	$0.680331\pi$
$104$	0	0
$105$	955.298	0.887881
$106$	207.851i	0.190456i
$107$	602.756	0.544585	0.272293	−	0.962214i	$-0.412218\pi$
$107$	602.756	0.544585	0.272293	+	0.962214i	$0.412218\pi$
$108$	193.277	0.172205
$109$	1421.89i	1.24947i	0.780837	+	0.624735i	$0.214793\pi$
$109$	1421.89i	1.24947i	−0.780837	+	0.624735i	$0.785207\pi$
$110$	934.440i	0.809958i
$111$	− 236.876i	− 0.202552i
$112$	916.174i	0.772949i
$113$	−396.719	−0.330267	−0.165134	−	0.986271i	$-0.552806\pi$
$113$	−396.719	−0.330267	−0.165134	+	0.986271i	$0.552806\pi$
$114$	−404.708	−0.332495
$115$	822.386i	0.666851i
$116$	−276.499	−0.221313
$117$	0	0
$118$	130.295	0.101650
$119$	− 911.752i	− 0.702354i
$120$	645.392	0.490966
$121$	−3004.17	−2.25708
$122$	244.784i	0.181653i
$123$	− 1064.90i	− 0.780639i
$124$	632.729i	0.458232i
$125$	− 165.013i	− 0.118074i
$126$	−169.944	−0.120157
$127$	−437.868	−0.305941	−0.152970	−	0.988231i	$-0.548884\pi$
$127$	−437.868	−0.305941	−0.152970	+	0.988231i	$0.548884\pi$
$128$	1415.32i	0.977324i
$129$	1223.54	0.835089
$130$	0	0
$131$	1657.44	1.10543	0.552715	−	0.833370i	$-0.313592\pi$
$131$	1657.44	1.10543	0.552715	+	0.833370i	$0.313592\pi$
$132$	1413.98i	0.932354i
$133$	−3026.85	−1.97339
$134$	377.212	0.243180
$135$	− 417.701i	− 0.266296i
$136$	− 615.973i	− 0.388377i
$137$	− 1967.42i	− 1.22692i	−0.789725	−	0.613460i	$-0.789777\pi$
$137$	− 1967.42i	− 1.22692i	0.789725	−	0.613460i	$-0.210223\pi$
$138$	− 146.299i	− 0.0902449i
$139$	2825.13	1.72392	0.861960	−	0.506977i	$-0.169237\pi$
$139$	2825.13	1.72392	0.861960	+	0.506977i	$0.169237\pi$
$140$	2279.48	1.37608
$141$	203.902i	0.121785i
$142$	83.9445	0.0496089
$143$	0	0
$144$	400.594	0.231825
$145$	597.555i	0.342236i
$146$	57.9164	0.0328301
$147$	−242.021	−0.135793
$148$	− 565.220i	− 0.313924i
$149$	797.658i	0.438569i	0.975661	+	0.219284i	$0.0703722\pi$
$149$	797.658i	0.438569i	−0.975661	+	0.219284i	$0.929628\pi$
$150$	− 314.660i	− 0.171279i
$151$	161.987i	0.0873003i	0.999047	+	0.0436501i	$0.0138987\pi$
$151$	161.987i	0.0873003i	−0.999047	+	0.0436501i	$0.986101\pi$
$152$	−2044.91	−1.09121
$153$	−398.661	−0.210652
$154$	− 1243.27i	− 0.650555i
$155$	1367.42	0.708606
$156$	0	0
$157$	−342.000	−0.173851	−0.0869255	−	0.996215i	$-0.527704\pi$
$157$	−342.000	−0.173851	−0.0869255	+	0.996215i	$0.527704\pi$
$158$	− 263.392i	− 0.132622i
$159$	679.716	0.339025
$160$	2352.74	1.16250
$161$	− 1094.18i	− 0.535613i
$162$	74.3073i	0.0360378i
$163$	556.597i	0.267460i	0.991018	+	0.133730i	$0.0426956\pi$
$163$	556.597i	0.267460i	−0.991018	+	0.133730i	$0.957304\pi$
$164$	− 2541.00i	− 1.20987i
$165$	3055.81	1.44178
$166$	342.926	0.160339
$167$	− 3136.42i	− 1.45332i	−0.686999	−	0.726658i	$-0.741072\pi$
$167$	− 3136.42i	− 1.45332i	0.686999	−	0.726658i	$-0.258928\pi$
$168$	−858.692	−0.394342
$169$	0	0
$170$	−628.650	−0.283619
$171$	1323.48i	0.591865i
$172$	2919.53	1.29426
$173$	−3324.19	−1.46089	−0.730444	−	0.682972i	$-0.760687\pi$
$173$	−3324.19	−1.46089	−0.730444	+	0.682972i	$0.760687\pi$
$174$	− 106.303i	− 0.0463148i
$175$	− 2353.37i	− 1.01656i
$176$	2930.66i	1.25515i
$177$	− 426.092i	− 0.180944i
$178$	109.537	0.0461243
$179$	3313.25	1.38349	0.691743	−	0.722144i	$-0.256843\pi$
$179$	3313.25	1.38349	0.691743	+	0.722144i	$0.256843\pi$
$180$	− 996.694i	− 0.412718i
$181$	−76.0118	−0.0312150	−0.0156075	−	0.999878i	$-0.504968\pi$
$181$	−76.0118	−0.0312150	−0.0156075	+	0.999878i	$0.504968\pi$
$182$	0	0
$183$	800.494	0.323356
$184$	− 739.221i	− 0.296174i
$185$	−1221.52	−0.485450
$186$	−243.258	−0.0958956
$187$	− 2916.51i	− 1.14052i
$188$	486.540i	0.188748i
$189$	555.750i	0.213888i
$190$	2087.00i	0.796879i
$191$	−4073.39	−1.54314	−0.771572	−	0.636142i	$-0.780529\pi$
$191$	−4073.39	−1.54314	−0.771572	+	0.636142i	$0.780529\pi$
$192$	649.708	0.244211
$193$	− 867.581i	− 0.323574i	−0.986826	−	0.161787i	$-0.948274\pi$
$193$	− 867.581i	− 0.323574i	0.986826	−	0.161787i	$-0.0517258\pi$
$194$	−508.821	−0.188305
$195$	0	0
$196$	−577.497	−0.210458
$197$	− 1887.69i	− 0.682703i	−0.939936	−	0.341352i	$-0.889115\pi$
$197$	− 1887.69i	− 0.682703i	0.939936	−	0.341352i	$-0.110885\pi$
$198$	−543.616	−0.195117
$199$	−2786.22	−0.992511	−0.496256	−	0.868176i	$-0.665292\pi$
$199$	−2786.22	−0.992511	−0.496256	+	0.868176i	$0.665292\pi$
$200$	− 1589.92i	− 0.562120i
$201$	− 1233.56i	− 0.432879i
$202$	424.345i	0.147806i
$203$	− 795.045i	− 0.274883i
$204$	−951.261	−0.326478
$205$	−5491.47	−1.87093
$206$	− 1029.36i	− 0.348150i
$207$	−478.428	−0.160643
$208$	0	0
$209$	−9682.28	−3.20448
$210$	876.365i	0.287976i
$211$	708.789	0.231256	0.115628	−	0.993293i	$-0.463112\pi$
$211$	708.789	0.231256	0.115628	+	0.993293i	$0.463112\pi$
$212$	1621.90	0.525436
$213$	− 274.516i	− 0.0883075i
$214$	552.953i	0.176631i
$215$	− 6309.54i	− 2.00143i
$216$	375.460i	0.118272i
$217$	−1819.35	−0.569150
$218$	−1304.40	−0.405254
$219$	− 189.398i	− 0.0584400i
$220$	7291.59	2.23454
$221$	0	0
$222$	217.304	0.0656959
$223$	5396.64i	1.62056i	0.586041	+	0.810281i	$0.300686\pi$
$223$	5396.64i	1.62056i	−0.586041	+	0.810281i	$0.699314\pi$
$224$	−3130.32	−0.933720
$225$	−1029.00	−0.304890
$226$	− 363.940i	− 0.107119i
$227$	3181.79i	0.930321i	0.885226	+	0.465160i	$0.154003\pi$
$227$	3181.79i	0.930321i	−0.885226	+	0.465160i	$0.845997\pi$
$228$	3158.01i	0.917299i
$229$	− 2034.00i	− 0.586945i	−0.955967	−	0.293473i	$-0.905189\pi$
$229$	− 2034.00i	− 0.586945i	0.955967	−	0.293473i	$-0.0948110\pi$
$230$	−754.435	−0.216287
$231$	−4065.75	−1.15804
$232$	− 537.126i	− 0.152000i
$233$	2794.22	0.785645	0.392823	−	0.919614i	$-0.371499\pi$
$233$	2794.22	0.785645	0.392823	+	0.919614i	$0.371499\pi$
$234$	0	0
$235$	1051.48	0.291878
$236$	− 1016.72i	− 0.280435i
$237$	−861.346	−0.236078
$238$	836.417	0.227802
$239$	− 5493.81i	− 1.48688i	−0.668800	−	0.743442i	$-0.733192\pi$
$239$	− 5493.81i	− 1.48688i	0.668800	−	0.743442i	$-0.266808\pi$
$240$	− 2065.79i	− 0.555608i
$241$	− 4061.27i	− 1.08552i	−0.839889	−	0.542758i	$-0.817380\pi$
$241$	− 4061.27i	− 1.08552i	0.839889	−	0.542758i	$-0.182620\pi$
$242$	− 2755.95i	− 0.732062i
$243$	243.000	0.0641500
$244$	1910.09	0.501152
$245$	1248.06i	0.325451i
$246$	976.910	0.253193
$247$	0	0
$248$	−1229.14	−0.314719
$249$	− 1121.44i	− 0.285414i
$250$	151.379	0.0382961
$251$	3570.88	0.897977	0.448988	−	0.893538i	$-0.351784\pi$
$251$	3570.88	0.897977	0.448988	+	0.893538i	$0.351784\pi$
$252$	1326.10i	0.331493i
$253$	− 3500.07i	− 0.869753i
$254$	− 401.688i	− 0.0992290i
$255$	2055.82i	0.504863i
$256$	434.179	0.106001
$257$	7518.26	1.82481	0.912405	−	0.409288i	$-0.134223\pi$
$257$	7518.26	1.82481	0.912405	+	0.409288i	$0.134223\pi$
$258$	1122.44i	0.270853i
$259$	1625.23	0.389912
$260$	0	0
$261$	−347.631	−0.0824437
$262$	1520.49i	0.358536i
$263$	−4660.18	−1.09262	−0.546310	−	0.837583i	$-0.683968\pi$
$263$	−4660.18	−1.09262	−0.546310	+	0.837583i	$0.683968\pi$
$264$	−2746.79	−0.640352
$265$	− 3505.16i	− 0.812530i
$266$	− 2776.75i	− 0.640050i
$267$	− 358.208i	− 0.0821047i
$268$	− 2943.45i	− 0.670895i
$269$	5347.44	1.21204	0.606021	−	0.795449i	$-0.292765\pi$
$269$	5347.44	1.21204	0.606021	+	0.795449i	$0.292765\pi$
$270$	383.188	0.0863707
$271$	− 2973.08i	− 0.666427i	−0.942851	−	0.333214i	$-0.891867\pi$
$271$	− 2973.08i	− 0.666427i	0.942851	−	0.333214i	$-0.108133\pi$
$272$	−1971.62	−0.439511
$273$	0	0
$274$	1804.86	0.397940
$275$	− 7527.96i	− 1.65074i
$276$	−1141.60	−0.248971
$277$	764.153	0.165753	0.0828764	−	0.996560i	$-0.473589\pi$
$277$	764.153	0.165753	0.0828764	+	0.996560i	$0.473589\pi$
$278$	2591.70i	0.559137i
$279$	795.505i	0.170701i
$280$	4428.11i	0.945107i
$281$	7040.34i	1.49463i	0.664469	+	0.747316i	$0.268658\pi$
$281$	7040.34i	1.49463i	−0.664469	+	0.747316i	$0.731342\pi$
$282$	−187.055	−0.0394998
$283$	9035.17	1.89783	0.948913	−	0.315537i	$-0.102185\pi$
$283$	9035.17	1.89783	0.948913	+	0.315537i	$0.102185\pi$
$284$	− 655.033i	− 0.136863i
$285$	6824.92	1.41850
$286$	0	0
$287$	7306.39	1.50273
$288$	1368.72i	0.280044i
$289$	−2950.90	−0.600630
$290$	−548.181	−0.111001
$291$	1663.95i	0.335197i
$292$	− 451.931i	− 0.0905729i
$293$	− 1785.23i	− 0.355954i	−0.984035	−	0.177977i	$-0.943045\pi$
$293$	− 1785.23i	− 0.355954i	0.984035	−	0.177977i	$-0.0569553\pi$
$294$	− 222.024i	− 0.0440432i
$295$	−2197.27	−0.433662
$296$	1098.00	0.215607
$297$	1777.73i	0.347322i
$298$	−731.751	−0.142246
$299$	0	0
$300$	−2455.35	−0.472532
$301$	8394.83i	1.60754i
$302$	−148.603	−0.0283150
$303$	1387.69	0.263105
$304$	6545.40i	1.23488i
$305$	− 4127.99i	− 0.774977i
$306$	− 365.721i	− 0.0683231i
$307$	− 5323.13i	− 0.989600i	−0.869007	−	0.494800i	$-0.835241\pi$
$307$	− 5323.13i	− 0.989600i	0.869007	−	0.494800i	$-0.164759\pi$
$308$	−9701.45	−1.79478
$309$	−3366.22	−0.619733
$310$	1254.44i	0.229830i
$311$	−6265.64	−1.14242	−0.571209	−	0.820805i	$-0.693525\pi$
$311$	−6265.64	−1.14242	−0.571209	+	0.820805i	$0.693525\pi$
$312$	0	0
$313$	7193.77	1.29909	0.649547	−	0.760322i	$-0.274959\pi$
$313$	7193.77	1.29909	0.649547	+	0.760322i	$0.274959\pi$
$314$	− 313.742i	− 0.0563869i
$315$	2865.89	0.512619
$316$	−2055.29	−0.365884
$317$	− 9576.87i	− 1.69682i	−0.529343	−	0.848408i	$-0.677561\pi$
$317$	− 9576.87i	− 1.69682i	0.529343	−	0.848408i	$-0.322439\pi$
$318$	623.554i	0.109960i
$319$	− 2543.19i	− 0.446368i
$320$	− 3350.41i	− 0.585293i
$321$	1808.27	0.314417
$322$	1003.77	0.173721
$323$	− 6513.81i	− 1.12210i
$324$	579.832	0.0994226
$325$	0	0
$326$	−510.608	−0.0867483
$327$	4265.67i	0.721382i
$328$	4936.14	0.830953
$329$	−1399.00	−0.234435
$330$	2803.32i	0.467629i
$331$	5773.13i	0.958670i	0.877632	+	0.479335i	$0.159122\pi$
$331$	5773.13i	0.958670i	−0.877632	+	0.479335i	$0.840878\pi$
$332$	− 2675.91i	− 0.442348i
$333$	− 710.629i	− 0.116944i
$334$	2877.27	0.471370
$335$	−6361.23	−1.03747
$336$	2748.52i	0.446262i
$337$	−1238.09	−0.200127	−0.100063	−	0.994981i	$-0.531905\pi$
$337$	−1238.09	−0.200127	−0.100063	+	0.994981i	$0.531905\pi$
$338$	0	0
$339$	−1190.16	−0.190680
$340$	4905.47i	0.782460i
$341$	−5819.74	−0.924213
$342$	−1214.12	−0.191966
$343$	5399.55i	0.849995i
$344$	5671.48i	0.888912i
$345$	2467.16i	0.385007i
$346$	− 3049.53i	− 0.473826i
$347$	5449.97	0.843140	0.421570	−	0.906796i	$-0.361479\pi$
$347$	5449.97	0.843140	0.421570	+	0.906796i	$0.361479\pi$
$348$	−829.497	−0.127775
$349$	1374.03i	0.210746i	0.994433	+	0.105373i	$0.0336036\pi$
$349$	1374.03i	0.210746i	−0.994433	+	0.105373i	$0.966396\pi$
$350$	2158.92	0.329711
$351$	0	0
$352$	−10013.3	−1.51622
$353$	− 5970.44i	− 0.900211i	−0.892975	−	0.450106i	$-0.851386\pi$
$353$	− 5970.44i	− 0.900211i	0.892975	−	0.450106i	$-0.148614\pi$
$354$	390.886	0.0586874
$355$	−1415.62	−0.211644
$356$	− 854.734i	− 0.127249i
$357$	− 2735.26i	− 0.405504i
$358$	3039.49i	0.448720i
$359$	7813.71i	1.14872i	0.818602	+	0.574362i	$0.194750\pi$
$359$	7813.71i	1.14872i	−0.818602	+	0.574362i	$0.805250\pi$
$360$	1936.18	0.283460
$361$	−14765.6	−2.15274
$362$	− 69.7313i	− 0.0101243i
$363$	−9012.52	−1.30312
$364$	0	0
$365$	−976.690	−0.140061
$366$	734.352i	0.104878i
$367$	1688.39	0.240145	0.120073	−	0.992765i	$-0.461687\pi$
$367$	1688.39	0.240145	0.120073	+	0.992765i	$0.461687\pi$
$368$	−2366.11	−0.335169
$369$	− 3194.69i	− 0.450702i
$370$	− 1120.59i	− 0.157451i
$371$	4663.61i	0.652622i
$372$	1898.19i	0.264560i
$373$	1870.61	0.259670	0.129835	−	0.991536i	$-0.458555\pi$
$373$	1870.61	0.259670	0.129835	+	0.991536i	$0.458555\pi$
$374$	2675.53	0.369916
$375$	− 495.039i	− 0.0681699i
$376$	−945.151	−0.129634
$377$	0	0
$378$	−509.831	−0.0693726
$379$	− 11667.0i	− 1.58125i	−0.612298	−	0.790627i	$-0.709755\pi$
$379$	− 11667.0i	− 1.58125i	0.612298	−	0.790627i	$-0.290245\pi$
$380$	16285.2	2.19846
$381$	−1313.60	−0.176635
$382$	− 3736.82i	− 0.500504i
$383$	6676.60i	0.890753i	0.895343	+	0.445376i	$0.146930\pi$
$383$	6676.60i	0.890753i	−0.895343	+	0.445376i	$0.853070\pi$
$384$	4245.95i	0.564259i
$385$	20966.3i	2.77543i
$386$	795.896	0.104948
$387$	3670.61	0.482139
$388$	3970.42i	0.519504i
$389$	−14285.3	−1.86194	−0.930969	−	0.365099i	$-0.881035\pi$
$389$	−14285.3	−1.86194	−0.930969	+	0.365099i	$0.881035\pi$
$390$	0	0
$391$	2354.69	0.304558
$392$	− 1121.85i	− 0.144545i
$393$	4972.32	0.638220
$394$	1731.72	0.221428
$395$	4441.79i	0.565800i
$396$	4241.93i	0.538295i
$397$	3569.28i	0.451227i	0.974217	+	0.225614i	$0.0724387\pi$
$397$	3569.28i	0.451227i	−0.974217	+	0.225614i	$0.927561\pi$
$398$	− 2556.00i	− 0.321912i
$399$	−9080.54	−1.13934
$400$	−5089.04	−0.636130
$401$	499.348i	0.0621852i	0.999517	+	0.0310926i	$0.00989868\pi$
$401$	499.348i	0.0621852i	−0.999517	+	0.0310926i	$0.990101\pi$
$402$	1131.64	0.140400
$403$	0	0
$404$	3311.23	0.407772
$405$	− 1253.10i	− 0.153746i
$406$	729.354	0.0891557
$407$	5198.80	0.633158
$408$	− 1847.92i	− 0.224229i
$409$	− 10457.5i	− 1.26428i	−0.774855	−	0.632140i	$-0.782177\pi$
$409$	− 10457.5i	− 1.26428i	0.774855	−	0.632140i	$-0.217823\pi$
$410$	− 5037.73i	− 0.606819i
$411$	− 5902.26i	− 0.708363i
$412$	−8032.27	−0.960489
$413$	2923.47	0.348316
$414$	− 438.897i	− 0.0521029i
$415$	−5783.03	−0.684043
$416$	0	0
$417$	8475.40	0.995305
$418$	− 8882.27i	− 1.03934i
$419$	−1705.42	−0.198843	−0.0994215	−	0.995045i	$-0.531699\pi$
$419$	−1705.42	−0.198843	−0.0994215	+	0.995045i	$0.531699\pi$
$420$	6838.43	0.794479
$421$	8765.57i	1.01475i	0.861727	+	0.507373i	$0.169383\pi$
$421$	8765.57i	1.01475i	−0.861727	+	0.507373i	$0.830617\pi$
$422$	650.225i	0.0750058i
$423$	611.707i	0.0703126i
$424$	3150.70i	0.360876i
$425$	5064.48	0.578032
$426$	251.833	0.0286417
$427$	5492.28i	0.622459i
$428$	4314.79	0.487297
$429$	0	0
$430$	5788.21	0.649145
$431$	8080.37i	0.903057i	0.892257	+	0.451529i	$0.149121\pi$
$431$	8080.37i	0.903057i	−0.892257	+	0.451529i	$0.850879\pi$
$432$	1201.78	0.133844
$433$	4124.48	0.457760	0.228880	−	0.973455i	$-0.426494\pi$
$433$	4124.48	0.457760	0.228880	+	0.973455i	$0.426494\pi$
$434$	− 1669.02i	− 0.184598i
$435$	1792.66i	0.197590i
$436$	10178.5i	1.11803i
$437$	− 7817.14i	− 0.855709i
$438$	173.749	0.0189545
$439$	6115.52	0.664870	0.332435	−	0.943126i	$-0.392130\pi$
$439$	6115.52	0.664870	0.332435	+	0.943126i	$0.392130\pi$
$440$	14164.6i	1.53471i
$441$	−726.064	−0.0784002
$442$	0	0
$443$	−11058.8	−1.18605	−0.593025	−	0.805184i	$-0.702067\pi$
$443$	−11058.8	−1.18605	−0.593025	+	0.805184i	$0.702067\pi$
$444$	− 1695.66i	− 0.181244i
$445$	−1847.21	−0.196777
$446$	−4950.73	−0.525614
$447$	2392.97i	0.253208i
$448$	4457.72i	0.470106i
$449$	242.012i	0.0254371i	0.999919	+	0.0127185i	$0.00404854\pi$
$449$	242.012i	0.0254371i	−0.999919	+	0.0127185i	$0.995951\pi$
$450$	− 943.980i	− 0.0988881i
$451$	23371.7	2.44020
$452$	−2839.88	−0.295524
$453$	485.962i	0.0504028i
$454$	−2918.89	−0.301741
$455$	0	0
$456$	−6134.74	−0.630012
$457$	− 11052.2i	− 1.13129i	−0.824650	−	0.565644i	$-0.808628\pi$
$457$	− 11052.2i	− 1.13129i	0.824650	−	0.565644i	$-0.191372\pi$
$458$	1865.94	0.190370
$459$	−1195.98	−0.121620
$460$	5886.99i	0.596700i
$461$	273.763i	0.0276582i	0.999904	+	0.0138291i	$0.00440208\pi$
$461$	273.763i	0.0276582i	−0.999904	+	0.0138291i	$0.995598\pi$
$462$	− 3729.81i	− 0.375598i
$463$	− 11579.2i	− 1.16227i	−0.813808	−	0.581134i	$-0.802609\pi$
$463$	− 11579.2i	− 1.16227i	0.813808	−	0.581134i	$-0.197391\pi$
$464$	−1719.25	−0.172013
$465$	4102.26	0.409114
$466$	2563.34i	0.254817i
$467$	−902.915	−0.0894688	−0.0447344	−	0.998999i	$-0.514244\pi$
$467$	−902.915	−0.0894688	−0.0447344	+	0.998999i	$0.514244\pi$
$468$	0	0
$469$	8463.59	0.833289
$470$	964.604i	0.0946678i
$471$	−1026.00	−0.100373
$472$	1975.07	0.192606
$473$	26853.4i	2.61040i
$474$	− 790.176i	− 0.0765696i
$475$	− 16813.1i	− 1.62408i
$476$	− 6526.71i	− 0.628469i
$477$	2039.15	0.195736
$478$	5039.88	0.482257
$479$	12068.8i	1.15122i	0.817723	+	0.575611i	$0.195236\pi$
$479$	12068.8i	1.15122i	−0.817723	+	0.575611i	$0.804764\pi$
$480$	7058.23	0.671172
$481$	0	0
$482$	3725.70	0.352077
$483$	− 3282.55i	− 0.309236i
$484$	−21505.1	−2.01964
$485$	8580.66	0.803356
$486$	222.922i	0.0208065i
$487$	11492.5i	1.06936i	0.845056	+	0.534678i	$0.179567\pi$
$487$	11492.5i	1.06936i	−0.845056	+	0.534678i	$0.820433\pi$
$488$	3710.54i	0.344197i
$489$	1669.79i	0.154418i
$490$	−1144.93	−0.105557
$491$	−15704.2	−1.44342	−0.721710	−	0.692196i	$-0.756643\pi$
$491$	−15704.2	−1.44342	−0.721710	+	0.692196i	$0.756643\pi$
$492$	− 7622.99i	− 0.698519i
$493$	1710.95	0.156303
$494$	0	0
$495$	9167.43	0.832415
$496$	3934.25i	0.356156i
$497$	1883.48	0.169991
$498$	1028.78	0.0925715
$499$	9019.80i	0.809181i	0.914498	+	0.404591i	$0.132586\pi$
$499$	9019.80i	0.809181i	−0.914498	+	0.404591i	$0.867414\pi$
$500$	− 1181.23i	− 0.105653i
$501$	− 9409.27i	− 0.839073i
$502$	3275.83i	0.291250i
$503$	−6033.84	−0.534862	−0.267431	−	0.963577i	$-0.586175\pi$
$503$	−6033.84	−0.534862	−0.267431	+	0.963577i	$0.586175\pi$
$504$	−2576.08	−0.227674
$505$	− 7156.06i	− 0.630576i
$506$	3210.87	0.282096
$507$	0	0
$508$	−3134.44	−0.273757
$509$	22451.5i	1.95510i	0.210699	+	0.977551i	$0.432426\pi$
$509$	22451.5i	1.95510i	−0.210699	+	0.977551i	$0.567574\pi$
$510$	−1885.95	−0.163748
$511$	1299.48	0.112497
$512$	11720.8i	1.01170i
$513$	3970.44i	0.341714i
$514$	6897.06i	0.591860i
$515$	17358.9i	1.48529i
$516$	8758.60	0.747240
$517$	−4475.11	−0.380687
$518$	1490.95i	0.126464i
$519$	−9972.58	−0.843445
$520$	0	0
$521$	15674.7	1.31808	0.659040	−	0.752108i	$-0.270963\pi$
$521$	15674.7	1.31808	0.659040	+	0.752108i	$0.270963\pi$
$522$	− 318.908i	− 0.0267399i
$523$	13510.8	1.12961	0.564804	−	0.825225i	$-0.308952\pi$
$523$	13510.8	1.12961	0.564804	+	0.825225i	$0.308952\pi$
$524$	11864.7	0.989142
$525$	− 7060.10i	− 0.586911i
$526$	− 4275.13i	− 0.354381i
$527$	− 3915.27i	− 0.323627i
$528$	8791.97i	0.724662i
$529$	−9341.16	−0.767746
$530$	3215.55	0.263537
$531$	− 1278.28i	− 0.104468i
$532$	−21667.4	−1.76580
$533$	0	0
$534$	328.610	0.0266299
$535$	− 9324.89i	− 0.753551i
$536$	5717.94	0.460778
$537$	9939.75	0.798756
$538$	4905.60i	0.393114i
$539$	− 5311.73i	− 0.424475i
$540$	− 2990.08i	− 0.238283i
$541$	− 12103.6i	− 0.961875i	−0.876755	−	0.480937i	$-0.840296\pi$
$541$	− 12103.6i	− 0.961875i	0.876755	−	0.480937i	$-0.159704\pi$
$542$	2727.43	0.216149
$543$	−228.036	−0.0180220
$544$	− 6736.49i	− 0.530928i
$545$	21997.2	1.72891
$546$	0	0
$547$	−15228.6	−1.19036	−0.595181	−	0.803592i	$-0.702920\pi$
$547$	−15228.6	−1.19036	−0.595181	+	0.803592i	$0.702920\pi$
$548$	− 14083.6i	− 1.09785i
$549$	2401.48	0.186690
$550$	6905.95	0.535401
$551$	− 5680.03i	− 0.439160i
$552$	− 2217.66i	− 0.170996i
$553$	− 5909.79i	− 0.454448i
$554$	701.014i	0.0537603i
$555$	−3664.57	−0.280275
$556$	20223.5	1.54257
$557$	− 23596.9i	− 1.79503i	−0.440982	−	0.897516i	$-0.645370\pi$
$557$	− 23596.9i	− 1.79503i	0.440982	−	0.897516i	$-0.354630\pi$
$558$	−729.775	−0.0553653
$559$	0	0
$560$	14173.6	1.06954
$561$	− 8749.54i	− 0.658478i
$562$	−6458.63	−0.484770
$563$	−7941.62	−0.594493	−0.297246	−	0.954801i	$-0.596068\pi$
$563$	−7941.62	−0.594493	−0.297246	+	0.954801i	$0.596068\pi$
$564$	1459.62i	0.108974i
$565$	6137.40i	0.456996i
$566$	8288.63i	0.615542i
$567$	1667.25i	0.123488i
$568$	1272.47	0.0939991
$569$	−2274.65	−0.167590	−0.0837948	−	0.996483i	$-0.526704\pi$
$569$	−2274.65	−0.167590	−0.0837948	+	0.996483i	$0.526704\pi$
$570$	6261.00i	0.460078i
$571$	4499.84	0.329794	0.164897	−	0.986311i	$-0.447271\pi$
$571$	4499.84	0.329794	0.164897	+	0.986311i	$0.447271\pi$
$572$	0	0
$573$	−12220.2	−0.890934
$574$	6702.69i	0.487395i
$575$	6077.82	0.440804
$576$	1949.12	0.140996
$577$	− 25253.3i	− 1.82202i	−0.412381	−	0.911011i	$-0.635303\pi$
$577$	− 25253.3i	− 1.82202i	0.412381	−	0.911011i	$-0.364697\pi$
$578$	− 2707.08i	− 0.194809i
$579$	− 2602.74i	− 0.186816i
$580$	4277.55i	0.306234i
$581$	7694.31	0.549421
$582$	−1526.46	−0.108718
$583$	14918.0i	1.05976i
$584$	877.921	0.0622066
$585$	0	0
$586$	1637.73	0.115450
$587$	− 11285.2i	− 0.793508i	−0.917925	−	0.396754i	$-0.870137\pi$
$587$	− 11285.2i	− 0.793508i	0.917925	−	0.396754i	$-0.129863\pi$
$588$	−1732.49	−0.121508
$589$	−12997.9	−0.909289
$590$	− 2015.72i	− 0.140654i
$591$	− 5663.08i	− 0.394159i
$592$	− 3514.49i	− 0.243994i
$593$	− 12824.5i	− 0.888090i	−0.896005	−	0.444045i	$-0.853543\pi$
$593$	− 12824.5i	− 0.888090i	0.896005	−	0.444045i	$-0.146457\pi$
$594$	−1630.85	−0.112651
$595$	−14105.2	−0.971859
$596$	5709.98i	0.392433i
$597$	−8358.65	−0.573027
$598$	0	0
$599$	−26180.3	−1.78581	−0.892905	−	0.450245i	$-0.851336\pi$
$599$	−26180.3	−1.78581	−0.892905	+	0.450245i	$0.851336\pi$
$600$	− 4769.75i	− 0.324540i
$601$	16012.6	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$601$	16012.6	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$602$	−7701.20	−0.521391
$603$	− 3700.68i	− 0.249923i
$604$	1159.57i	0.0781166i
$605$	46475.8i	3.12316i
$606$	1273.03i	0.0853357i
$607$	−9531.48	−0.637349	−0.318674	−	0.947864i	$-0.603238\pi$
$607$	−9531.48	−0.637349	−0.318674	+	0.947864i	$0.603238\pi$
$608$	−22363.9	−1.49174
$609$	− 2385.14i	− 0.158704i
$610$	3786.91	0.251356
$611$	0	0
$612$	−2853.78	−0.188492
$613$	10187.1i	0.671211i	0.942003	+	0.335605i	$0.108941\pi$
$613$	10187.1i	0.671211i	−0.942003	+	0.335605i	$0.891059\pi$
$614$	4883.30	0.320967
$615$	−16474.4	−1.08018
$616$	− 18846.0i	− 1.23267i
$617$	8012.62i	0.522813i	0.965229	+	0.261407i	$0.0841864\pi$
$617$	8012.62i	0.522813i	−0.965229	+	0.261407i	$0.915814\pi$
$618$	− 3088.08i	− 0.201004i
$619$	− 1886.59i	− 0.122501i	−0.998122	−	0.0612506i	$-0.980491\pi$
$619$	− 1886.59i	− 0.122501i	0.998122	−	0.0612506i	$-0.0195089\pi$
$620$	9788.58	0.634063
$621$	−1435.28	−0.0927470
$622$	− 5747.94i	− 0.370533i
$623$	2457.70	0.158051
$624$	0	0
$625$	−16844.5	−1.07805
$626$	6599.38i	0.421349i
$627$	−29046.8	−1.85011
$628$	−2448.18	−0.155562
$629$	3497.53i	0.221710i
$630$	2629.10i	0.166263i
$631$	− 14956.1i	− 0.943569i	−0.881714	−	0.471784i	$-0.843610\pi$
$631$	− 14956.1i	− 0.943569i	0.881714	−	0.471784i	$-0.156390\pi$
$632$	− 3992.61i	− 0.251293i
$633$	2126.37	0.133516
$634$	8785.57	0.550346
$635$	6773.99i	0.423335i
$636$	4865.70	0.303361
$637$	0	0
$638$	2333.06	0.144775
$639$	− 823.547i	− 0.0509844i
$640$	21895.5	1.35234
$641$	−23691.7	−1.45985	−0.729927	−	0.683525i	$-0.760446\pi$
$641$	−23691.7	−1.45985	−0.729927	+	0.683525i	$0.760446\pi$
$642$	1658.86i	0.101978i
$643$	13651.3i	0.837254i	0.908158	+	0.418627i	$0.137489\pi$
$643$	13651.3i	0.837254i	−0.908158	+	0.418627i	$0.862511\pi$
$644$	− 7832.62i	− 0.479268i
$645$	− 18928.6i	− 1.15553i
$646$	5975.60	0.363943
$647$	18132.3	1.10178	0.550892	−	0.834577i	$-0.314288\pi$
$647$	18132.3	1.10178	0.550892	+	0.834577i	$0.314288\pi$
$648$	1126.38i	0.0682846i
$649$	9351.59	0.565612
$650$	0	0
$651$	−5458.05	−0.328599
$652$	3984.36i	0.239324i
$653$	6363.47	0.381351	0.190675	−	0.981653i	$-0.438932\pi$
$653$	6363.47	0.381351	0.190675	+	0.981653i	$0.438932\pi$
$654$	−3913.21	−0.233973
$655$	− 25641.3i	− 1.52960i
$656$	− 15799.7i	− 0.940358i
$657$	− 568.195i	− 0.0337403i
$658$	− 1283.40i	− 0.0760369i
$659$	−1051.43	−0.0621518	−0.0310759	−	0.999517i	$-0.509893\pi$
$659$	−1051.43	−0.0621518	−0.0310759	+	0.999517i	$0.509893\pi$
$660$	21874.8	1.29011
$661$	− 8119.75i	− 0.477794i	−0.971045	−	0.238897i	$-0.923214\pi$
$661$	− 8119.75i	− 0.477794i	0.971045	−	0.238897i	$-0.0767858\pi$
$662$	−5296.12	−0.310936
$663$	0	0
$664$	5198.21	0.303810
$665$	46826.5i	2.73061i
$666$	651.912	0.0379296
$667$	2053.29	0.119196
$668$	− 22451.9i	− 1.30043i
$669$	16189.9i	0.935632i
$670$	− 5835.62i	− 0.336492i
$671$	17568.7i	1.01078i
$672$	−9390.96	−0.539083
$673$	190.264	0.0108977	0.00544885	−	0.999985i	$-0.498266\pi$
$673$	190.264	0.0108977	0.00544885	+	0.999985i	$0.498266\pi$
$674$	− 1135.79i	− 0.0649093i
$675$	−3087.01	−0.176028
$676$	0	0
$677$	4861.93	0.276010	0.138005	−	0.990431i	$-0.455931\pi$
$677$	4861.93	0.276010	0.138005	+	0.990431i	$0.455931\pi$
$678$	− 1091.82i	− 0.0618452i
$679$	−11416.5	−0.645253
$680$	−9529.35	−0.537403
$681$	9545.37i	0.537121i
$682$	− 5338.88i	− 0.299760i
$683$	14539.6i	0.814556i	0.913304	+	0.407278i	$0.133522\pi$
$683$	14539.6i	0.814556i	−0.913304	+	0.407278i	$0.866478\pi$
$684$	9474.02i	0.529603i
$685$	−30436.8	−1.69771
$686$	−4953.40	−0.275688
$687$	− 6102.00i	− 0.338873i
$688$	18153.4	1.00595
$689$	0	0
$690$	−2263.31	−0.124873
$691$	− 22106.5i	− 1.21703i	−0.793541	−	0.608517i	$-0.791765\pi$
$691$	− 22106.5i	− 1.21703i	0.793541	−	0.608517i	$-0.208235\pi$
$692$	−23796.0	−1.30721
$693$	−12197.2	−0.668593
$694$	4999.66i	0.273465i
$695$	− 43706.0i	− 2.38541i
$696$	− 1611.38i	− 0.0877574i
$697$	15723.4i	0.854474i
$698$	−1260.50	−0.0683534
$699$	8382.66	0.453593
$700$	− 16846.4i	− 0.909620i
$701$	−229.971	−0.0123907	−0.00619535	−	0.999981i	$-0.501972\pi$
$701$	−229.971	−0.0123907	−0.00619535	+	0.999981i	$0.501972\pi$
$702$	0	0
$703$	11611.1	0.622934
$704$	14259.4i	0.763380i
$705$	3154.45	0.168516
$706$	5477.13	0.291975
$707$	9521.12i	0.506476i
$708$	− 3050.15i	− 0.161909i
$709$	4802.22i	0.254374i	0.991879	+	0.127187i	$0.0405948\pi$
$709$	4802.22i	0.254374i	−0.991879	+	0.127187i	$0.959405\pi$
$710$	− 1298.66i	− 0.0686446i
$711$	−2584.04	−0.136300
$712$	1660.40	0.0873965
$713$	− 4698.66i	− 0.246797i
$714$	2509.25	0.131522
$715$	0	0
$716$	23717.6	1.23795
$717$	− 16481.4i	− 0.858453i
$718$	−7168.09	−0.372578
$719$	11016.9	0.571435	0.285718	−	0.958314i	$-0.407768\pi$
$719$	11016.9	0.571435	0.285718	+	0.958314i	$0.407768\pi$
$720$	− 6197.36i	− 0.320780i
$721$	− 23096.0i	− 1.19298i
$722$	− 13545.6i	− 0.698221i
$723$	− 12183.8i	− 0.626723i
$724$	−544.125	−0.0279313
$725$	4416.21	0.226226
$726$	− 8267.85i	− 0.422656i
$727$	13498.2	0.688612	0.344306	−	0.938857i	$-0.388114\pi$
$727$	13498.2	0.688612	0.344306	+	0.938857i	$0.388114\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 895.990i	− 0.0454275i
$731$	−18065.8	−0.914073
$732$	5730.27	0.289340
$733$	− 17014.7i	− 0.857368i	−0.903455	−	0.428684i	$-0.858977\pi$
$733$	− 17014.7i	− 0.857368i	0.903455	−	0.428684i	$-0.141023\pi$
$734$	1548.88i	0.0778888i
$735$	3744.17i	0.187899i
$736$	− 8084.38i	− 0.404883i
$737$	27073.4	1.35313
$738$	2930.73	0.146181
$739$	− 13392.6i	− 0.666651i	−0.942812	−	0.333326i	$-0.891829\pi$
$739$	− 13392.6i	− 0.666651i	0.942812	−	0.333326i	$-0.108171\pi$
$740$	−8744.19	−0.434382
$741$	0	0
$742$	−4278.27	−0.211672
$743$	11796.5i	0.582467i	0.956652	+	0.291233i	$0.0940656\pi$
$743$	11796.5i	0.582467i	−0.956652	+	0.291233i	$0.905934\pi$
$744$	−3687.42	−0.181703
$745$	12340.1	0.606854
$746$	1716.05i	0.0842214i
$747$	− 3364.31i	− 0.164784i
$748$	− 20877.6i	− 1.02054i
$749$	12406.7i	0.605250i
$750$	454.136	0.0221103
$751$	12450.5	0.604961	0.302480	−	0.953156i	$-0.402185\pi$
$751$	12450.5	0.604961	0.302480	+	0.953156i	$0.402185\pi$
$752$	3025.26i	0.146702i
$753$	10712.6	0.518447
$754$	0	0
$755$	2506.01	0.120799
$756$	3978.30i	0.191388i
$757$	−7029.84	−0.337522	−0.168761	−	0.985657i	$-0.553977\pi$
$757$	−7029.84	−0.337522	−0.168761	+	0.985657i	$0.553977\pi$
$758$	10703.0	0.512865
$759$	− 10500.2i	− 0.502152i
$760$	31635.6i	1.50993i
$761$	− 17740.6i	− 0.845067i	−0.906347	−	0.422533i	$-0.861141\pi$
$761$	− 17740.6i	− 0.845067i	0.906347	−	0.422533i	$-0.138859\pi$
$762$	− 1205.06i	− 0.0572899i
$763$	−29267.2	−1.38866
$764$	−29159.1	−1.38081
$765$	6167.45i	0.291483i
$766$	−6124.94	−0.288907
$767$	0	0
$768$	1302.54	0.0611995
$769$	31976.5i	1.49948i	0.661732	+	0.749741i	$0.269822\pi$
$769$	31976.5i	1.49948i	−0.661732	+	0.749741i	$0.730178\pi$
$770$	−19233.9	−0.900184
$771$	22554.8	1.05355
$772$	− 6210.51i	− 0.289535i
$773$	− 2135.11i	− 0.0993463i	−0.998766	−	0.0496732i	$-0.984182\pi$
$773$	− 2135.11i	− 0.0993463i	0.998766	−	0.0496732i	$-0.0158180\pi$
$774$	3367.32i	0.156377i
$775$	− 10105.9i	− 0.468405i
$776$	−7712.93	−0.356802
$777$	4875.70	0.225116
$778$	− 13105.0i	− 0.603902i
$779$	52198.9	2.40080
$780$	0	0
$781$	6024.89	0.276040
$782$	2160.14i	0.0987804i
$783$	−1042.89	−0.0475989
$784$	−3590.83	−0.163576
$785$	5290.89i	0.240560i
$786$	4561.48i	0.207001i
$787$	11704.4i	0.530134i	0.964230	+	0.265067i	$0.0853940\pi$
$787$	11704.4i	0.530134i	−0.964230	+	0.265067i	$0.914606\pi$
$788$	− 13512.9i	− 0.610885i
$789$	−13980.6	−0.630825
$790$	−4074.78	−0.183512
$791$	− 8165.80i	− 0.367057i
$792$	−8240.36	−0.369708
$793$	0	0
$794$	−3274.37	−0.146351
$795$	− 10515.5i	− 0.469115i
$796$	−19944.9	−0.888102
$797$	149.733	0.00665474	0.00332737	−	0.999994i	$-0.498941\pi$
$797$	149.733	0.00665474	0.00332737	+	0.999994i	$0.498941\pi$
$798$	− 8330.25i	− 0.369533i
$799$	− 3010.66i	− 0.133304i
$800$	− 17387.9i	− 0.768443i
$801$	− 1074.62i	− 0.0474032i
$802$	−458.089	−0.0201692
$803$	4156.79	0.182677
$804$	− 8830.35i	− 0.387341i
$805$	−16927.4	−0.741135
$806$	0	0
$807$	16042.3	0.699772
$808$	6432.40i	0.280063i
$809$	23520.8	1.02219	0.511093	−	0.859526i	$-0.329241\pi$
$809$	23520.8	1.02219	0.511093	+	0.859526i	$0.329241\pi$
$810$	1149.56	0.0498661
$811$	29604.8i	1.28183i	0.767612	+	0.640915i	$0.221445\pi$
$811$	29604.8i	1.28183i	−0.767612	+	0.640915i	$0.778555\pi$
$812$	− 5691.27i	− 0.245966i
$813$	− 8919.24i	− 0.384762i
$814$	4769.25i	0.205359i
$815$	8610.79	0.370089
$816$	−5914.86	−0.253752
$817$	59975.0i	2.56825i
$818$	9593.44	0.410057
$819$	0	0
$820$	−39310.3	−1.67412
$821$	43034.7i	1.82938i	0.404154	+	0.914691i	$0.367566\pi$
$821$	43034.7i	1.82938i	−0.404154	+	0.914691i	$0.632434\pi$
$822$	5414.58	0.229751
$823$	−5584.95	−0.236548	−0.118274	−	0.992981i	$-0.537736\pi$
$823$	−5584.95	−0.236548	−0.118274	+	0.992981i	$0.537736\pi$
$824$	− 15603.5i	− 0.659675i
$825$	− 22583.9i	− 0.953054i
$826$	2681.91i	0.112973i
$827$	4788.13i	0.201330i	0.994920	+	0.100665i	$0.0320970\pi$
$827$	4788.13i	0.201330i	−0.994920	+	0.100665i	$0.967903\pi$
$828$	−3424.79	−0.143744
$829$	32392.2	1.35709	0.678546	−	0.734558i	$-0.262610\pi$
$829$	32392.2	1.35709	0.678546	+	0.734558i	$0.262610\pi$
$830$	− 5305.20i	− 0.221863i
$831$	2292.46	0.0956974
$832$	0	0
$833$	3573.50	0.148637
$834$	7775.11i	0.322818i
$835$	−48521.8	−2.01098
$836$	−69309.9	−2.86738
$837$	2386.51i	0.0985544i
$838$	− 1564.51i	− 0.0644929i
$839$	− 13598.9i	− 0.559577i	−0.960062	−	0.279788i	$-0.909736\pi$
$839$	− 13598.9i	− 0.559577i	0.960062	−	0.279788i	$-0.0902643\pi$
$840$	13284.3i	0.545658i
$841$	−22897.1	−0.938827
$842$	−8041.31	−0.329123
$843$	21121.0i	0.862926i
$844$	5073.82	0.206929
$845$	0	0
$846$	−561.164	−0.0228052
$847$	− 61835.9i	− 2.50851i
$848$	10084.8	0.408390
$849$	27105.5	1.09571
$850$	4646.02i	0.187479i
$851$	4197.34i	0.169075i
$852$	− 1965.10i	− 0.0790178i
$853$	− 41037.0i	− 1.64722i	−0.567155	−	0.823611i	$-0.691956\pi$
$853$	− 41037.0i	− 1.64722i	0.567155	−	0.823611i	$-0.308044\pi$
$854$	−5038.47	−0.201889
$855$	20474.8	0.818973
$856$	8381.89i	0.334681i
$857$	39959.3	1.59275	0.796374	−	0.604804i	$-0.206749\pi$
$857$	39959.3	1.59275	0.796374	+	0.604804i	$0.206749\pi$
$858$	0	0
$859$	−32570.5	−1.29371	−0.646853	−	0.762615i	$-0.723915\pi$
$859$	−32570.5	−1.29371	−0.646853	+	0.762615i	$0.723915\pi$
$860$	− 45166.4i	− 1.79088i
$861$	21919.2	0.867599
$862$	−7412.72	−0.292898
$863$	− 16951.8i	− 0.668652i	−0.942457	−	0.334326i	$-0.891491\pi$
$863$	− 16951.8i	− 0.668652i	0.942457	−	0.334326i	$-0.108509\pi$
$864$	4106.17i	0.161684i
$865$	51426.6i	2.02145i
$866$	3783.69i	0.148470i
$867$	−8852.69	−0.346774
$868$	−13023.7	−0.509277
$869$	− 18904.3i	− 0.737955i
$870$	−1644.54	−0.0640865
$871$	0	0
$872$	−19772.7	−0.767876
$873$	4991.85i	0.193526i
$874$	7171.24	0.277541
$875$	3396.52	0.131227
$876$	− 1355.79i	− 0.0522923i
$877$	− 45939.2i	− 1.76882i	−0.466709	−	0.884411i	$-0.654561\pi$
$877$	− 45939.2i	− 1.76882i	0.466709	−	0.884411i	$-0.345439\pi$
$878$	5610.22i	0.215644i
$879$	− 5355.70i	− 0.205510i
$880$	45338.5	1.73677
$881$	−37960.5	−1.45167	−0.725836	−	0.687868i	$-0.758547\pi$
$881$	−37960.5	−1.45167	−0.725836	+	0.687868i	$0.758547\pi$
$882$	− 666.072i	− 0.0254284i
$883$	−43172.9	−1.64539	−0.822697	−	0.568480i	$-0.807532\pi$
$883$	−43172.9	−1.64539	−0.822697	+	0.568480i	$0.807532\pi$
$884$	0	0
$885$	−6591.82	−0.250375
$886$	− 10145.1i	− 0.384684i
$887$	28706.3	1.08665	0.543327	−	0.839521i	$-0.317164\pi$
$887$	28706.3	1.08665	0.543327	+	0.839521i	$0.317164\pi$
$888$	3293.99	0.124481
$889$	− 9012.78i	− 0.340021i
$890$	− 1694.58i	− 0.0638229i
$891$	5333.20i	0.200526i
$892$	38631.4i	1.45008i
$893$	−9994.83	−0.374540
$894$	−2195.25	−0.0821255
$895$	− 51257.3i	− 1.91435i
$896$	−29131.9	−1.08619
$897$	0	0
$898$	−222.015	−0.00825027
$899$	− 3414.10i	− 0.126659i
$900$	−7366.04	−0.272816
$901$	−10036.2	−0.371091
$902$	21440.6i	0.791456i
$903$	25184.5i	0.928114i
$904$	− 5516.75i	− 0.202969i
$905$	1175.93i	0.0431927i
$906$	−445.809	−0.0163477
$907$	22356.1	0.818435	0.409218	−	0.912437i	$-0.365802\pi$
$907$	22356.1	0.818435	0.409218	+	0.912437i	$0.365802\pi$
$908$	22776.6i	0.832454i
$909$	4163.08	0.151904
$910$	0	0
$911$	6953.80	0.252897	0.126449	−	0.991973i	$-0.459642\pi$
$911$	6953.80	0.252897	0.126449	+	0.991973i	$0.459642\pi$
$912$	19636.2i	0.712960i
$913$	24612.6	0.892176
$914$	10139.0	0.366923
$915$	− 12384.0i	− 0.447433i
$916$	− 14560.2i	− 0.525200i
$917$	34115.7i	1.22857i
$918$	− 1097.16i	− 0.0394464i
$919$	40625.2	1.45822	0.729108	−	0.684399i	$-0.239935\pi$
$919$	40625.2	1.45822	0.729108	+	0.684399i	$0.239935\pi$
$920$	−11436.0	−0.409821
$921$	− 15969.4i	− 0.571346i
$922$	−251.143	−0.00897068
$923$	0	0
$924$	−29104.3	−1.03621
$925$	9027.64i	0.320894i
$926$	10622.4	0.376971
$927$	−10098.6	−0.357803
$928$	− 5874.20i	− 0.207791i
$929$	42813.8i	1.51203i	0.654555	+	0.756014i	$0.272856\pi$
$929$	42813.8i	1.51203i	−0.654555	+	0.756014i	$0.727144\pi$
$930$	3763.31i	0.132692i
$931$	− 11863.3i	− 0.417621i
$932$	20002.2	0.702998
$933$	−18796.9	−0.659575
$934$	− 828.311i	− 0.0290184i
$935$	−45119.7	−1.57815
$936$	0	0
$937$	43484.1	1.51608	0.758038	−	0.652210i	$-0.226158\pi$
$937$	43484.1	1.51608	0.758038	+	0.652210i	$0.226158\pi$
$938$	7764.28i	0.270269i
$939$	21581.3	0.750032
$940$	7526.97	0.261173
$941$	− 7108.58i	− 0.246263i	−0.992390	−	0.123131i	$-0.960706\pi$
$941$	− 7108.58i	− 0.246263i	0.992390	−	0.123131i	$-0.0392936\pi$
$942$	− 941.227i	− 0.0325550i
$943$	18869.5i	0.651618i
$944$	− 6321.85i	− 0.217965i
$945$	8597.68	0.295960
$946$	−24634.6	−0.846660
$947$	− 1938.99i	− 0.0665351i	−0.999446	−	0.0332676i	$-0.989409\pi$
$947$	− 1938.99i	− 0.0665351i	0.999446	−	0.0332676i	$-0.0105913\pi$
$948$	−6165.88	−0.211243
$949$	0	0
$950$	15423.9	0.526756
$951$	− 28730.6i	− 0.979657i
$952$	12678.8	0.431640
$953$	47806.7	1.62498	0.812492	−	0.582972i	$-0.198110\pi$
$953$	47806.7	1.62498	0.812492	+	0.582972i	$0.198110\pi$
$954$	1870.66i	0.0634852i
$955$	63017.1i	2.13527i
$956$	− 39327.1i	− 1.33047i
$957$	− 7629.58i	− 0.257711i
$958$	−11071.6	−0.373388
$959$	40496.1	1.36359
$960$	− 10051.2i	− 0.337919i
$961$	21978.3	0.737750
$962$	0	0
$963$	5424.81	0.181528
$964$	− 29072.3i	− 0.971323i
$965$	−13421.8	−0.447735
$966$	3011.32	0.100298
$967$	− 2832.71i	− 0.0942025i	−0.998890	−	0.0471013i	$-0.985002\pi$
$967$	− 2832.71i	− 0.0942025i	0.998890	−	0.0471013i	$-0.0149983\pi$
$968$	− 41775.8i	− 1.38711i
$969$	− 19541.4i	− 0.647845i
$970$	7871.68i	0.260561i
$971$	−41276.3	−1.36418	−0.682090	−	0.731268i	$-0.738929\pi$
$971$	−41276.3	−1.36418	−0.682090	+	0.731268i	$0.738929\pi$
$972$	1739.50	0.0574016
$973$	58150.7i	1.91596i
$974$	−10542.9	−0.346835
$975$	0	0
$976$	11876.8	0.389515
$977$	4278.89i	0.140117i	0.997543	+	0.0700583i	$0.0223185\pi$
$977$	4278.89i	0.140117i	−0.997543	+	0.0700583i	$0.977681\pi$
$978$	−1531.82	−0.0500842
$979$	7861.70	0.256651
$980$	8934.12i	0.291214i
$981$	12797.0i	0.416490i
$982$	− 14406.6i	− 0.468159i
$983$	− 22652.9i	− 0.735011i	−0.930021	−	0.367505i	$-0.880212\pi$
$983$	− 22652.9i	− 0.735011i	0.930021	−	0.367505i	$-0.119788\pi$
$984$	14808.4	0.479751
$985$	−29203.4	−0.944667
$986$	1569.58i	0.0506953i
$987$	−4196.99	−0.135351
$988$	0	0
$989$	−21680.5	−0.697068
$990$	8409.96i	0.269986i
$991$	22060.2	0.707131	0.353565	−	0.935410i	$-0.384969\pi$
$991$	22060.2	0.707131	0.353565	+	0.935410i	$0.384969\pi$
$992$	−13442.3	−0.430235
$993$	17319.4i	0.553488i
$994$	1727.86i	0.0551351i
$995$	43103.9i	1.37335i
$996$	− 8027.72i	− 0.255390i
$997$	35635.5	1.13198	0.565992	−	0.824411i	$-0.308493\pi$
$997$	35635.5	1.13198	0.565992	+	0.824411i	$0.308493\pi$
$998$	−8274.52	−0.262450
$999$	− 2131.89i	− 0.0675174i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.i.337.6		10
13.3	even	3		39.4.j.c.4.3	✓	10
13.4	even	6		39.4.j.c.10.3	yes	10
13.5	odd	4		507.4.a.r.1.6		10
13.8	odd	4		507.4.a.r.1.5		10
13.12	even	2	inner	507.4.b.i.337.5		10
39.5	even	4		1521.4.a.bk.1.5		10
39.8	even	4		1521.4.a.bk.1.6		10
39.17	odd	6		117.4.q.e.10.3		10
39.29	odd	6		117.4.q.e.82.3		10
52.3	odd	6		624.4.bv.h.433.2		10
52.43	odd	6		624.4.bv.h.49.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.3	✓	10	13.3	even	3
39.4.j.c.10.3	yes	10	13.4	even	6
117.4.q.e.10.3		10	39.17	odd	6
117.4.q.e.82.3		10	39.29	odd	6
507.4.a.r.1.5		10	13.8	odd	4
507.4.a.r.1.6		10	13.5	odd	4
507.4.b.i.337.5		10	13.12	even	2	inner
507.4.b.i.337.6		10	1.1	even	1	trivial
624.4.bv.h.49.4		10	52.43	odd	6
624.4.bv.h.433.2		10	52.3	odd	6
1521.4.a.bk.1.5		10	39.5	even	4
1521.4.a.bk.1.6		10	39.8	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.917374i q^{2} +3.00000 q^{3} +7.15843 q^{4} -15.4704i q^{5} +2.75212i q^{6} +20.5833i q^{7} +13.9059i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.917374i q^{2} +3.00000 q^{3} +7.15843 q^{4} -15.4704i q^{5} +2.75212i q^{6} +20.5833i q^{7} +13.9059i q^{8} +9.00000 q^{9} +14.1922 q^{10} +65.8420i q^{11} +21.4753 q^{12} -18.8826 q^{14} -46.4112i q^{15} +44.5105 q^{16} -44.2956 q^{17} +8.25636i q^{18} +147.053i q^{19} -110.744i q^{20} +61.7500i q^{21} -60.4017 q^{22} -53.1586 q^{23} +41.7178i q^{24} -114.334 q^{25} +27.0000 q^{27} +147.344i q^{28} -38.6257 q^{29} +42.5765 q^{30} +88.3894i q^{31} +152.080i q^{32} +197.526i q^{33} -40.6357i q^{34} +318.433 q^{35} +64.4258 q^{36} -78.9587i q^{37} -134.903 q^{38} +215.131 q^{40} -354.966i q^{41} -56.6478 q^{42} +407.846 q^{43} +471.325i q^{44} -139.234i q^{45} -48.7663i q^{46} +67.9674i q^{47} +133.531 q^{48} -80.6738 q^{49} -104.887i q^{50} -132.887 q^{51} +226.572 q^{53} +24.7691i q^{54} +1018.60 q^{55} -286.231 q^{56} +441.160i q^{57} -35.4342i q^{58} -142.031i q^{59} -332.231i q^{60} +266.831 q^{61} -81.0862 q^{62} +185.250i q^{63} +216.569 q^{64} -181.205 q^{66} -411.187i q^{67} -317.087 q^{68} -159.476 q^{69} +292.122i q^{70} -91.5052i q^{71} +125.153i q^{72} -63.1328i q^{73} +72.4347 q^{74} -343.001 q^{75} +1052.67i q^{76} -1355.25 q^{77} -287.115 q^{79} -688.595i q^{80} +81.0000 q^{81} +325.637 q^{82} -373.812i q^{83} +442.033i q^{84} +685.272i q^{85} +374.147i q^{86} -115.877 q^{87} -915.595 q^{88} -119.403i q^{89} +127.729 q^{90} -380.532 q^{92} +265.168i q^{93} -62.3515 q^{94} +2274.97 q^{95} +456.241i q^{96} +554.650i q^{97} -74.0080i q^{98} +592.578i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9	\( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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