LMFDB - Embedded newform 507.4.b.k.337.8

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$0.107680i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} -0.863573 q^{10} +8.40842i q^{11} +23.3998 q^{12} -3.64325 q^{14} -5.79219i q^{15} +59.2386 q^{16} +52.1271 q^{17} -4.02550i q^{18} -48.8304i q^{19} -15.0596i q^{20} -24.4361i q^{21} +3.76090 q^{22} -88.9229 q^{23} -21.2009i q^{24} +121.272 q^{25} +27.0000 q^{27} -63.5334i q^{28} +191.979 q^{29} -2.59072 q^{30} -115.257i q^{31} -83.0319i q^{32} +25.2252i q^{33} -23.3153i q^{34} -15.7265 q^{35} +70.1995 q^{36} -136.716i q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077i q^{41} -10.9297 q^{42} -202.048 q^{43} +65.5852i q^{44} -17.3766i q^{45} +39.7733i q^{46} +618.160i q^{47} +177.716 q^{48} +276.653 q^{49} -54.2425i q^{50} +156.381 q^{51} -453.170 q^{53} -12.0765i q^{54} +16.2344 q^{55} -57.5631 q^{56} -146.491i q^{57} -85.8679i q^{58} +500.044i q^{59} -45.1787i q^{60} +480.502 q^{61} -51.5522 q^{62} -73.3083i q^{63} +436.771 q^{64} +11.2827 q^{66} -886.769i q^{67} +406.589 q^{68} -266.769 q^{69} +7.03412i q^{70} -123.732i q^{71} -63.6027i q^{72} +673.168i q^{73} -61.1501 q^{74} +363.817 q^{75} -380.875i q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374i q^{80} +81.0000 q^{81} -195.048 q^{82} +939.418i q^{83} -190.600i q^{84} -100.643i q^{85} +90.3716i q^{86} +575.936 q^{87} +59.4220 q^{88} -754.979i q^{89} -7.77215 q^{90} -693.594 q^{92} -345.772i q^{93} +276.489 q^{94} -94.2783 q^{95} -249.096i q^{96} -1051.10i q^{97} -123.741i q^{98} +75.6757i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * 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 $

$2$	− 0.447278i	− 0.158137i	−0.996869	−	0.0790684i	$-0.974805\pi$
$2$	− 0.447278i	− 0.158137i	0.996869	−	0.0790684i	$-0.0251945\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.79994	0.974993
$5$	− 1.93073i	− 0.172690i	−0.996265	−	0.0863448i	$-0.972481\pi$
$5$	− 1.93073i	− 0.172690i	0.996265	−	0.0863448i	$-0.0275187\pi$
$6$	− 1.34183i	− 0.0913003i
$7$	− 8.14537i	− 0.439809i	−0.975521	−	0.219904i	$-0.929425\pi$
$7$	− 8.14537i	− 0.439809i	0.975521	−	0.219904i	$-0.0705745\pi$
$8$	− 7.06697i	− 0.312319i
$9$	9.00000	0.333333
$10$	−0.863573	−0.0273086
$11$	8.40842i	0.230476i	0.993338	+	0.115238i	$0.0367630\pi$
$11$	8.40842i	0.230476i	−0.993338	+	0.115238i	$0.963237\pi$
$12$	23.3998	0.562912
$13$	0	0
$13$	0	0
$14$	−3.64325	−0.0695499
$15$	− 5.79219i	− 0.0997024i
$16$	59.2386	0.925604
$17$	52.1271	0.743688	0.371844	−	0.928295i	$-0.378726\pi$
$17$	52.1271	0.743688	0.371844	+	0.928295i	$0.378726\pi$
$18$	− 4.02550i	− 0.0527122i
$19$	− 48.8304i	− 0.589604i	−0.955558	−	0.294802i	$-0.904746\pi$
$19$	− 48.8304i	− 0.589604i	0.955558	−	0.294802i	$-0.0952537\pi$
$20$	− 15.0596i	− 0.168371i
$21$	− 24.4361i	− 0.253924i
$22$	3.76090	0.0364467
$23$	−88.9229	−0.806162	−0.403081	−	0.915164i	$-0.632061\pi$
$23$	−88.9229	−0.806162	−0.403081	+	0.915164i	$0.632061\pi$
$24$	− 21.2009i	− 0.180317i
$25$	121.272	0.970178
$26$	0	0
$27$	27.0000	0.192450
$28$	− 63.5334i	− 0.428810i
$29$	191.979	1.22930	0.614648	−	0.788802i	$-0.289298\pi$
$29$	191.979	1.22930	0.614648	+	0.788802i	$0.289298\pi$
$30$	−2.59072	−0.0157666
$31$	− 115.257i	− 0.667769i	−0.942614	−	0.333885i	$-0.891640\pi$
$31$	− 115.257i	− 0.667769i	0.942614	−	0.333885i	$-0.108360\pi$
$32$	− 83.0319i	− 0.458691i
$33$	25.2252i	0.133065i
$34$	− 23.3153i	− 0.117604i
$35$	−15.7265	−0.0759504
$36$	70.1995	0.324998
$37$	− 136.716i	− 0.607459i	−0.952758	−	0.303729i	$-0.901768\pi$
$37$	− 136.716i	− 0.607459i	0.952758	−	0.303729i	$-0.0982319\pi$
$38$	−21.8408	−0.0932380
$39$	0	0
$40$	−13.6444	−0.0539342
$41$	− 436.077i	− 1.66107i	−0.556967	−	0.830535i	$-0.688035\pi$
$41$	− 436.077i	− 1.66107i	0.556967	−	0.830535i	$-0.311965\pi$
$42$	−10.9297	−0.0401547
$43$	−202.048	−0.716559	−0.358279	−	0.933614i	$-0.616636\pi$
$43$	−202.048	−0.716559	−0.358279	+	0.933614i	$0.616636\pi$
$44$	65.5852i	0.224712i
$45$	− 17.3766i	− 0.0575632i
$46$	39.7733i	0.127484i
$47$	618.160i	1.91847i	0.282617	+	0.959233i	$0.408798\pi$
$47$	618.160i	1.91847i	−0.282617	+	0.959233i	$0.591202\pi$
$48$	177.716	0.534398
$49$	276.653	0.806568
$50$	− 54.2425i	− 0.153421i
$51$	156.381	0.429368
$52$	0	0
$53$	−453.170	−1.17448	−0.587242	−	0.809411i	$-0.699786\pi$
$53$	−453.170	−1.17448	−0.587242	+	0.809411i	$0.699786\pi$
$54$	− 12.0765i	− 0.0304334i
$55$	16.2344	0.0398008
$56$	−57.5631	−0.137361
$57$	− 146.491i	− 0.340408i
$58$	− 85.8679i	− 0.194397i
$59$	500.044i	1.10339i	0.834045	+	0.551697i	$0.186019\pi$
$59$	500.044i	1.10339i	−0.834045	+	0.551697i	$0.813981\pi$
$60$	− 45.1787i	− 0.0972091i
$61$	480.502	1.00856	0.504279	−	0.863541i	$-0.331758\pi$
$61$	480.502	1.00856	0.504279	+	0.863541i	$0.331758\pi$
$62$	−51.5522	−0.105599
$63$	− 73.3083i	− 0.146603i
$64$	436.771	0.853068
$65$	0	0
$66$	11.2827	0.0210425
$67$	− 886.769i	− 1.61696i	−0.588526	−	0.808478i	$-0.700292\pi$
$67$	− 886.769i	− 1.61696i	0.588526	−	0.808478i	$-0.299708\pi$
$68$	406.589	0.725090
$69$	−266.769	−0.465438
$70$	7.03412i	0.0120105i
$71$	− 123.732i	− 0.206821i	−0.994639	−	0.103410i	$-0.967025\pi$
$71$	− 123.732i	− 0.206821i	0.994639	−	0.103410i	$-0.0329755\pi$
$72$	− 63.6027i	− 0.104106i
$73$	673.168i	1.07929i	0.841892	+	0.539646i	$0.181442\pi$
$73$	673.168i	1.07929i	−0.841892	+	0.539646i	$0.818558\pi$
$74$	−61.1501	−0.0960616
$75$	363.817	0.560133
$76$	− 380.875i	− 0.574859i
$77$	68.4897	0.101365
$78$	0	0
$79$	−681.298	−0.970279	−0.485139	−	0.874437i	$-0.661231\pi$
$79$	−681.298	−0.970279	−0.485139	+	0.874437i	$0.661231\pi$
$80$	− 114.374i	− 0.159842i
$81$	81.0000	0.111111
$82$	−195.048	−0.262676
$83$	939.418i	1.24234i	0.783675	+	0.621172i	$0.213343\pi$
$83$	939.418i	1.24234i	−0.783675	+	0.621172i	$0.786657\pi$
$84$	− 190.600i	− 0.247574i
$85$	− 100.643i	− 0.128427i
$86$	90.3716i	0.113314i
$87$	575.936	0.709734
$88$	59.4220	0.0719819
$89$	− 754.979i	− 0.899187i	−0.893233	−	0.449593i	$-0.851569\pi$
$89$	− 754.979i	− 0.899187i	0.893233	−	0.449593i	$-0.148431\pi$
$90$	−7.77215	−0.00910286
$91$	0	0
$92$	−693.594	−0.786002
$93$	− 345.772i	− 0.385537i
$94$	276.489	0.303380
$95$	−94.2783	−0.101818
$96$	− 249.096i	− 0.264825i
$97$	− 1051.10i	− 1.10024i	−0.835087	−	0.550118i	$-0.814583\pi$
$97$	− 1051.10i	− 1.10024i	0.835087	−	0.550118i	$-0.185417\pi$
$98$	− 123.741i	− 0.127548i
$99$	75.6757i	0.0768252i
$100$	945.917	0.945917
$101$	−599.873	−0.590986	−0.295493	−	0.955345i	$-0.595484\pi$
$101$	−599.873	−0.590986	−0.295493	+	0.955345i	$0.595484\pi$
$102$	− 69.9460i	− 0.0678989i
$103$	293.312	0.280591	0.140295	−	0.990110i	$-0.455195\pi$
$103$	293.312	0.280591	0.140295	+	0.990110i	$0.455195\pi$
$104$	0	0
$105$	−47.1795	−0.0438500
$106$	202.693i	0.185729i
$107$	1533.69	1.38568	0.692839	−	0.721092i	$-0.256360\pi$
$107$	1533.69	1.38568	0.692839	+	0.721092i	$0.256360\pi$
$108$	210.598	0.187637
$109$	− 590.718i	− 0.519088i	−0.965731	−	0.259544i	$-0.916428\pi$
$109$	− 590.718i	− 0.519088i	0.965731	−	0.259544i	$-0.0835722\pi$
$110$	− 7.26128i	− 0.00629396i
$111$	− 410.148i	− 0.350717i
$112$	− 482.521i	− 0.407089i
$113$	−653.785	−0.544274	−0.272137	−	0.962259i	$-0.587730\pi$
$113$	−653.785	−0.544274	−0.272137	+	0.962259i	$0.587730\pi$
$114$	−65.5224	−0.0538310
$115$	171.686i	0.139216i
$116$	1497.42	1.19855
$117$	0	0
$118$	223.659	0.174487
$119$	− 424.595i	− 0.327080i
$120$	−40.9332	−0.0311389
$121$	1260.30	0.946881
$122$	− 214.918i	− 0.159490i
$123$	− 1308.23i	− 0.959019i
$124$	− 899.002i	− 0.651070i
$125$	− 475.485i	− 0.340229i
$126$	−32.7892	−0.0231833
$127$	141.203	0.0986595	0.0493297	−	0.998783i	$-0.484291\pi$
$127$	141.203	0.0986595	0.0493297	+	0.998783i	$0.484291\pi$
$128$	− 859.613i	− 0.593592i
$129$	−606.144	−0.413705
$130$	0	0
$131$	−731.910	−0.488147	−0.244074	−	0.969757i	$-0.578484\pi$
$131$	−731.910	−0.488147	−0.244074	+	0.969757i	$0.578484\pi$
$132$	196.755i	0.129738i
$133$	−397.742	−0.259313
$134$	−396.633	−0.255700
$135$	− 52.1297i	− 0.0332341i
$136$	− 368.381i	− 0.232268i
$137$	1762.27i	1.09898i	0.835499	+	0.549492i	$0.185179\pi$
$137$	1762.27i	1.09898i	−0.835499	+	0.549492i	$0.814821\pi$
$138$	119.320i	0.0736028i
$139$	−664.776	−0.405652	−0.202826	−	0.979215i	$-0.565013\pi$
$139$	−664.776	−0.405652	−0.202826	+	0.979215i	$0.565013\pi$
$140$	−122.666	−0.0740511
$141$	1854.48i	1.10763i
$142$	−55.3425	−0.0327059
$143$	0	0
$144$	533.148	0.308535
$145$	− 370.659i	− 0.212286i
$146$	301.094	0.170676
$147$	829.959	0.465672
$148$	− 1066.38i	− 0.592268i
$149$	3300.71i	1.81480i	0.420270	+	0.907399i	$0.361936\pi$
$149$	3300.71i	1.81480i	−0.420270	+	0.907399i	$0.638064\pi$
$150$	− 162.727i	− 0.0885776i
$151$	1464.15i	0.789079i	0.918879	+	0.394540i	$0.129096\pi$
$151$	1464.15i	0.789079i	−0.918879	+	0.394540i	$0.870904\pi$
$152$	−345.083	−0.184144
$153$	469.144	0.247896
$154$	− 30.6339i	− 0.0160296i
$155$	−222.531	−0.115317
$156$	0	0
$157$	1535.32	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$157$	1535.32	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$158$	304.730i	0.153437i
$159$	−1359.51	−0.678089
$160$	−160.312	−0.0792111
$161$	724.310i	0.354557i
$162$	− 36.2295i	− 0.0175707i
$163$	793.554i	0.381325i	0.981656	+	0.190663i	$0.0610636\pi$
$163$	793.554i	0.381325i	−0.981656	+	0.190663i	$0.938936\pi$
$164$	− 3401.38i	− 1.61953i
$165$	48.7031	0.0229790
$166$	420.181	0.196460
$167$	− 1336.87i	− 0.619460i	−0.950825	−	0.309730i	$-0.899761\pi$
$167$	− 1336.87i	− 0.619460i	0.950825	−	0.309730i	$-0.100239\pi$
$168$	−172.689	−0.0793052
$169$	0	0
$170$	−45.0156	−0.0203090
$171$	− 439.474i	− 0.196535i
$172$	−1575.96	−0.698640
$173$	−568.460	−0.249822	−0.124911	−	0.992168i	$-0.539865\pi$
$173$	−568.460	−0.249822	−0.124911	+	0.992168i	$0.539865\pi$
$174$	− 257.604i	− 0.112235i
$175$	− 987.808i	− 0.426693i
$176$	498.103i	0.213329i
$177$	1500.13i	0.637044i
$178$	−337.686	−0.142194
$179$	−1546.00	−0.645552	−0.322776	−	0.946475i	$-0.604616\pi$
$179$	−1546.00	−0.645552	−0.322776	+	0.946475i	$0.604616\pi$
$180$	− 135.536i	− 0.0561237i
$181$	−3408.96	−1.39992	−0.699960	−	0.714182i	$-0.746799\pi$
$181$	−3408.96	−1.39992	−0.699960	+	0.714182i	$0.746799\pi$
$182$	0	0
$183$	1441.51	0.582291
$184$	628.416i	0.251780i
$185$	−263.962	−0.104902
$186$	−154.656	−0.0609675
$187$	438.307i	0.171402i
$188$	4821.61i	1.87049i
$189$	− 219.925i	− 0.0846412i
$190$	42.1686i	0.0161012i
$191$	−3464.71	−1.31255	−0.656277	−	0.754520i	$-0.727870\pi$
$191$	−3464.71	−1.31255	−0.656277	+	0.754520i	$0.727870\pi$
$192$	1310.31	0.492519
$193$	4652.24i	1.73510i	0.497346	+	0.867552i	$0.334308\pi$
$193$	4652.24i	1.73510i	−0.497346	+	0.867552i	$0.665692\pi$
$194$	−470.134	−0.173988
$195$	0	0
$196$	2157.88	0.786398
$197$	2870.98i	1.03832i	0.854677	+	0.519160i	$0.173755\pi$
$197$	2870.98i	1.03832i	−0.854677	+	0.519160i	$0.826245\pi$
$198$	33.8481	0.0121489
$199$	−25.0330	−0.00891730	−0.00445865	−	0.999990i	$-0.501419\pi$
$199$	−25.0330	−0.00891730	−0.00445865	+	0.999990i	$0.501419\pi$
$200$	− 857.028i	− 0.303005i
$201$	− 2660.31i	− 0.933550i
$202$	268.310i	0.0934565i
$203$	− 1563.74i	− 0.540655i
$204$	1219.77	0.418631
$205$	−841.947	−0.286849
$206$	− 131.192i	− 0.0443717i
$207$	−800.307	−0.268721
$208$	0	0
$209$	410.587	0.135889
$210$	21.1024i	0.00693429i
$211$	−3605.91	−1.17650	−0.588249	−	0.808679i	$-0.700183\pi$
$211$	−3605.91	−1.17650	−0.588249	+	0.808679i	$0.700183\pi$
$212$	−3534.70	−1.14511
$213$	− 371.195i	− 0.119408i
$214$	− 685.987i	− 0.219127i
$215$	390.100i	0.123742i
$216$	− 190.808i	− 0.0601058i
$217$	−938.815	−0.293691
$218$	−264.215	−0.0820868
$219$	2019.50i	0.623130i
$220$	126.627	0.0388054
$221$	0	0
$222$	−183.450	−0.0554612
$223$	4304.37i	1.29256i	0.763098	+	0.646282i	$0.223677\pi$
$223$	4304.37i	1.29256i	−0.763098	+	0.646282i	$0.776323\pi$
$224$	−676.326	−0.201736
$225$	1091.45	0.323393
$226$	292.424i	0.0860697i
$227$	5475.19i	1.60089i	0.599409	+	0.800443i	$0.295402\pi$
$227$	5475.19i	1.60089i	−0.599409	+	0.800443i	$0.704598\pi$
$228$	− 1142.62i	− 0.331895i
$229$	378.108i	0.109110i	0.998511	+	0.0545548i	$0.0173740\pi$
$229$	378.108i	0.109110i	−0.998511	+	0.0545548i	$0.982626\pi$
$230$	76.7914	0.0220151
$231$	205.469	0.0585232
$232$	− 1356.71i	− 0.383932i
$233$	−2547.96	−0.716405	−0.358202	−	0.933644i	$-0.616610\pi$
$233$	−2547.96	−0.716405	−0.358202	+	0.933644i	$0.616610\pi$
$234$	0	0
$235$	1193.50	0.331299
$236$	3900.32i	1.07580i
$237$	−2043.89	−0.560191
$238$	−189.912	−0.0517234
$239$	6313.91i	1.70884i	0.519583	+	0.854420i	$0.326087\pi$
$239$	6313.91i	1.70884i	−0.519583	+	0.854420i	$0.673913\pi$
$240$	− 343.121i	− 0.0922849i
$241$	− 6763.73i	− 1.80784i	−0.427699	−	0.903921i	$-0.640676\pi$
$241$	− 6763.73i	− 1.80784i	0.427699	−	0.903921i	$-0.359324\pi$
$242$	− 563.704i	− 0.149737i
$243$	243.000	0.0641500
$244$	3747.89	0.983336
$245$	− 534.142i	− 0.139286i
$246$	−585.144	−0.151656
$247$	0	0
$248$	−814.521	−0.208557
$249$	2818.25i	0.717267i
$250$	−212.674	−0.0538027
$251$	3416.00	0.859028	0.429514	−	0.903060i	$-0.358685\pi$
$251$	3416.00	0.859028	0.429514	+	0.903060i	$0.358685\pi$
$252$	− 571.801i	− 0.142937i
$253$	− 747.701i	− 0.185801i
$254$	− 63.1571i	− 0.0156017i
$255$	− 301.930i	− 0.0741474i
$256$	3109.68	0.759199
$257$	−7002.36	−1.69959	−0.849796	−	0.527112i	$-0.823275\pi$
$257$	−7002.36	−1.69959	−0.849796	+	0.527112i	$0.823275\pi$
$258$	271.115i	0.0654220i
$259$	−1113.60	−0.267166
$260$	0	0
$261$	1727.81	0.409765
$262$	327.367i	0.0771940i
$263$	3369.76	0.790071	0.395035	−	0.918666i	$-0.370732\pi$
$263$	3369.76	0.790071	0.395035	+	0.918666i	$0.370732\pi$
$264$	178.266	0.0415588
$265$	874.948i	0.202821i
$266$	177.901i	0.0410069i
$267$	− 2264.94i	− 0.519146i
$268$	− 6916.75i	− 1.57652i
$269$	−7801.70	−1.76832	−0.884160	−	0.467185i	$-0.845268\pi$
$269$	−7801.70	−1.76832	−0.884160	+	0.467185i	$0.845268\pi$
$270$	−23.3165	−0.00525554
$271$	4410.05i	0.988528i	0.869312	+	0.494264i	$0.164562\pi$
$271$	4410.05i	0.988528i	−0.869312	+	0.494264i	$0.835438\pi$
$272$	3087.94	0.688360
$273$	0	0
$274$	788.225	0.173790
$275$	1019.71i	0.223603i
$276$	−2080.78	−0.453798
$277$	4636.44	1.00569	0.502846	−	0.864376i	$-0.332286\pi$
$277$	4636.44	1.00569	0.502846	+	0.864376i	$0.332286\pi$
$278$	297.340i	0.0641485i
$279$	− 1037.32i	− 0.222590i
$280$	111.139i	0.0237207i
$281$	1090.50i	0.231508i	0.993278	+	0.115754i	$0.0369285\pi$
$281$	1090.50i	0.231508i	−0.993278	+	0.115754i	$0.963072\pi$
$282$	829.468	0.175156
$283$	−1140.52	−0.239565	−0.119782	−	0.992800i	$-0.538220\pi$
$283$	−1140.52	−0.239565	−0.119782	+	0.992800i	$0.538220\pi$
$284$	− 965.101i	− 0.201649i
$285$	−282.835	−0.0587849
$286$	0	0
$287$	−3552.01	−0.730553
$288$	− 747.287i	− 0.152897i
$289$	−2195.76	−0.446929
$290$	−165.788	−0.0335703
$291$	− 3153.30i	− 0.635222i
$292$	5250.67i	1.05230i
$293$	− 335.079i	− 0.0668106i	−0.999442	−	0.0334053i	$-0.989365\pi$
$293$	− 335.079i	− 0.0668106i	0.999442	−	0.0334053i	$-0.0106352\pi$
$294$	− 371.223i	− 0.0736399i
$295$	965.449	0.190545
$296$	−966.168	−0.189721
$297$	227.027i	0.0443551i
$298$	1476.34	0.286986
$299$	0	0
$300$	2837.75	0.546125
$301$	1645.76i	0.315149i
$302$	654.883	0.124782
$303$	−1799.62	−0.341206
$304$	− 2892.65i	− 0.545739i
$305$	− 927.719i	− 0.174167i
$306$	− 209.838i	− 0.0392014i
$307$	2540.04i	0.472207i	0.971728	+	0.236104i	$0.0758705\pi$
$307$	2540.04i	0.472207i	−0.971728	+	0.236104i	$0.924130\pi$
$308$	534.215	0.0988303
$309$	879.935	0.161999
$310$	99.5332i	0.0182358i
$311$	−2376.36	−0.433283	−0.216642	−	0.976251i	$-0.569510\pi$
$311$	−2376.36	−0.433283	−0.216642	+	0.976251i	$0.569510\pi$
$312$	0	0
$313$	1315.78	0.237612	0.118806	−	0.992917i	$-0.462093\pi$
$313$	1315.78	0.237612	0.118806	+	0.992917i	$0.462093\pi$
$314$	− 686.713i	− 0.123419i
$315$	−141.538	−0.0253168
$316$	−5314.09	−0.946015
$317$	6043.13i	1.07071i	0.844626	+	0.535357i	$0.179823\pi$
$317$	6043.13i	1.07071i	−0.844626	+	0.535357i	$0.820177\pi$
$318$	608.079i	0.107231i
$319$	1614.24i	0.283323i
$320$	− 843.286i	− 0.147316i
$321$	4601.08	0.800022
$322$	323.968	0.0560685
$323$	− 2545.39i	− 0.438481i
$324$	631.795	0.108333
$325$	0	0
$326$	354.940	0.0603015
$327$	− 1772.16i	− 0.299695i
$328$	−3081.75	−0.518784
$329$	5035.14	0.843758
$330$	− 21.7838i	− 0.00363382i
$331$	− 8168.18i	− 1.35639i	−0.734884	−	0.678193i	$-0.762763\pi$
$331$	− 8168.18i	− 1.35639i	0.734884	−	0.678193i	$-0.237237\pi$
$332$	7327.40i	1.21128i
$333$	− 1230.44i	− 0.202486i
$334$	−597.951	−0.0979594
$335$	−1712.11	−0.279232
$336$	− 1447.56i	− 0.235033i
$337$	−3076.32	−0.497263	−0.248631	−	0.968598i	$-0.579981\pi$
$337$	−3076.32	−0.497263	−0.248631	+	0.968598i	$0.579981\pi$
$338$	0	0
$339$	−1961.36	−0.314237
$340$	− 785.012i	− 0.125215i
$341$	969.133	0.153905
$342$	−196.567	−0.0310793
$343$	− 5047.30i	− 0.794544i
$344$	1427.87i	0.223795i
$345$	515.058i	0.0803762i
$346$	254.260i	0.0395061i
$347$	−8754.07	−1.35430	−0.677152	−	0.735844i	$-0.736786\pi$
$347$	−8754.07	−1.35430	−0.677152	+	0.735844i	$0.736786\pi$
$348$	4492.27	0.691985
$349$	1064.97i	0.163343i	0.996659	+	0.0816715i	$0.0260258\pi$
$349$	1064.97i	0.163343i	−0.996659	+	0.0816715i	$0.973974\pi$
$350$	−441.825	−0.0674758
$351$	0	0
$352$	698.167	0.105717
$353$	− 5047.62i	− 0.761070i	−0.924767	−	0.380535i	$-0.875740\pi$
$353$	− 5047.62i	− 0.761070i	0.924767	−	0.380535i	$-0.124260\pi$
$354$	670.977	0.100740
$355$	−238.893	−0.0357158
$356$	− 5888.79i	− 0.876700i
$357$	− 1273.78i	− 0.188840i
$358$	691.494i	0.102085i
$359$	8152.39i	1.19851i	0.800557	+	0.599257i	$0.204537\pi$
$359$	8152.39i	1.19851i	−0.800557	+	0.599257i	$0.795463\pi$
$360$	−122.800	−0.0179781
$361$	4474.59	0.652367
$362$	1524.75i	0.221379i
$363$	3780.90	0.546682
$364$	0	0
$365$	1299.71	0.186383
$366$	− 644.754i	− 0.0920816i
$367$	11636.3	1.65507	0.827537	−	0.561412i	$-0.189742\pi$
$367$	11636.3	1.65507	0.827537	+	0.561412i	$0.189742\pi$
$368$	−5267.67	−0.746186
$369$	− 3924.70i	− 0.553690i
$370$	118.064i	0.0165888i
$371$	3691.24i	0.516548i
$372$	− 2697.00i	− 0.375896i
$373$	4720.59	0.655289	0.327644	−	0.944801i	$-0.393745\pi$
$373$	4720.59	0.655289	0.327644	+	0.944801i	$0.393745\pi$
$374$	196.045	0.0271049
$375$	− 1426.45i	− 0.196431i
$376$	4368.52	0.599173
$377$	0	0
$378$	−98.3677	−0.0133849
$379$	− 12933.7i	− 1.75293i	−0.481467	−	0.876464i	$-0.659896\pi$
$379$	− 12933.7i	− 1.75293i	0.481467	−	0.876464i	$-0.340104\pi$
$380$	−735.365	−0.0992722
$381$	423.609	0.0569611
$382$	1549.69i	0.207563i
$383$	− 554.945i	− 0.0740376i	−0.999315	−	0.0370188i	$-0.988214\pi$
$383$	− 554.945i	− 0.0740376i	0.999315	−	0.0370188i	$-0.0117861\pi$
$384$	− 2578.84i	− 0.342711i
$385$	− 132.235i	− 0.0175047i
$386$	2080.84	0.274384
$387$	−1818.43	−0.238853
$388$	− 8198.51i	− 1.07272i
$389$	−5091.35	−0.663604	−0.331802	−	0.943349i	$-0.607657\pi$
$389$	−5091.35	−0.663604	−0.331802	+	0.943349i	$0.607657\pi$
$390$	0	0
$391$	−4635.30	−0.599532
$392$	− 1955.10i	− 0.251907i
$393$	−2195.73	−0.281832
$394$	1284.13	0.164196
$395$	1315.40i	0.167557i
$396$	590.266i	0.0749040i
$397$	10254.6i	1.29638i	0.761480	+	0.648188i	$0.224473\pi$
$397$	10254.6i	1.29638i	−0.761480	+	0.648188i	$0.775527\pi$
$398$	11.1967i	0.00141015i
$399$	−1193.23	−0.149714
$400$	7184.00	0.898001
$401$	505.816i	0.0629906i	0.999504	+	0.0314953i	$0.0100269\pi$
$401$	505.816i	0.0629906i	−0.999504	+	0.0314953i	$0.989973\pi$
$402$	−1189.90	−0.147629
$403$	0	0
$404$	−4678.97	−0.576207
$405$	− 156.389i	− 0.0191877i
$406$	−699.426	−0.0854974
$407$	1149.57	0.140005
$408$	− 1105.14i	− 0.134100i
$409$	− 2219.21i	− 0.268295i	−0.990961	−	0.134148i	$-0.957170\pi$
$409$	− 2219.21i	− 0.268295i	0.990961	−	0.134148i	$-0.0428296\pi$
$410$	376.585i	0.0453614i
$411$	5286.81i	0.634499i
$412$	2287.82	0.273574
$413$	4073.04	0.485282
$414$	357.960i	0.0424946i
$415$	1813.76	0.214540
$416$	0	0
$417$	−1994.33	−0.234203
$418$	− 183.646i	− 0.0214891i
$419$	10249.3	1.19501	0.597506	−	0.801864i	$-0.296158\pi$
$419$	10249.3	1.19501	0.597506	+	0.801864i	$0.296158\pi$
$420$	−367.997	−0.0427534
$421$	− 97.9194i	− 0.0113356i	−0.999984	−	0.00566781i	$-0.998196\pi$
$421$	− 97.9194i	− 0.0113356i	0.999984	−	0.00566781i	$-0.00180413\pi$
$422$	1612.85i	0.186048i
$423$	5563.44i	0.639489i
$424$	3202.54i	0.366814i
$425$	6321.58	0.721510
$426$	−166.028	−0.0188828
$427$	− 3913.87i	− 0.443572i
$428$	11962.7	1.35103
$429$	0	0
$430$	174.483	0.0195682
$431$	− 11727.2i	− 1.31062i	−0.755360	−	0.655310i	$-0.772538\pi$
$431$	− 11727.2i	− 1.31062i	0.755360	−	0.655310i	$-0.227462\pi$
$432$	1599.44	0.178133
$433$	−6945.13	−0.770812	−0.385406	−	0.922747i	$-0.625939\pi$
$433$	−6945.13	−0.770812	−0.385406	+	0.922747i	$0.625939\pi$
$434$	419.911i	0.0464433i
$435$	− 1111.98i	− 0.122564i
$436$	− 4607.57i	− 0.506107i
$437$	4342.15i	0.475316i
$438$	903.281	0.0985398
$439$	−11446.3	−1.24442	−0.622210	−	0.782851i	$-0.713765\pi$
$439$	−11446.3	−1.24442	−0.622210	+	0.782851i	$0.713765\pi$
$440$	− 114.728i	− 0.0124305i
$441$	2489.88	0.268856
$442$	0	0
$443$	−10513.6	−1.12758	−0.563789	−	0.825919i	$-0.690657\pi$
$443$	−10513.6	−1.12758	−0.563789	+	0.825919i	$0.690657\pi$
$444$	− 3199.13i	− 0.341946i
$445$	−1457.66	−0.155280
$446$	1925.25	0.204402
$447$	9902.14i	1.04777i
$448$	− 3557.66i	− 0.375187i
$449$	− 9862.37i	− 1.03660i	−0.855199	−	0.518300i	$-0.826565\pi$
$449$	− 9862.37i	− 1.03660i	0.855199	−	0.518300i	$-0.173435\pi$
$450$	− 488.182i	− 0.0511403i
$451$	3666.72	0.382836
$452$	−5099.49	−0.530663
$453$	4392.45i	0.455575i
$454$	2448.93	0.253159
$455$	0	0
$456$	−1035.25	−0.106316
$457$	5990.48i	0.613179i	0.951842	+	0.306590i	$0.0991879\pi$
$457$	5990.48i	0.613179i	−0.951842	+	0.306590i	$0.900812\pi$
$458$	169.120	0.0172542
$459$	1407.43	0.143123
$460$	1339.14i	0.135734i
$461$	− 16579.5i	− 1.67502i	−0.546423	−	0.837510i	$-0.684011\pi$
$461$	− 16579.5i	− 1.67502i	0.546423	−	0.837510i	$-0.315989\pi$
$462$	− 91.9018i	− 0.00925467i
$463$	3606.65i	0.362020i	0.983481	+	0.181010i	$0.0579367\pi$
$463$	3606.65i	0.362020i	−0.983481	+	0.181010i	$0.942063\pi$
$464$	11372.6	1.13784
$465$	−667.593	−0.0665782
$466$	1139.65i	0.113290i
$467$	6789.61	0.672775	0.336387	−	0.941724i	$-0.390795\pi$
$467$	6789.61	0.672775	0.336387	+	0.941724i	$0.390795\pi$
$468$	0	0
$469$	−7223.06	−0.711152
$470$	− 533.826i	− 0.0523906i
$471$	4605.95	0.450596
$472$	3533.80	0.344611
$473$	− 1698.90i	− 0.165149i
$474$	914.189i	0.0885867i
$475$	− 5921.78i	− 0.572021i
$476$	− 3311.82i	− 0.318901i
$477$	−4078.53	−0.391495
$478$	2824.07	0.270230
$479$	− 12688.3i	− 1.21032i	−0.796103	−	0.605161i	$-0.793109\pi$
$479$	− 12688.3i	− 1.21032i	0.796103	−	0.605161i	$-0.206891\pi$
$480$	−480.936	−0.0457326
$481$	0	0
$482$	−3025.27	−0.285886
$483$	2172.93i	0.204703i
$484$	9830.26	0.923202
$485$	−2029.39	−0.189999
$486$	− 108.689i	− 0.0101445i
$487$	1217.28i	0.113265i	0.998395	+	0.0566325i	$0.0180363\pi$
$487$	1217.28i	0.113265i	−0.998395	+	0.0566325i	$0.981964\pi$
$488$	− 3395.69i	− 0.314991i
$489$	2380.66i	0.220158i
$490$	−238.910	−0.0220262
$491$	10496.3	0.964747	0.482374	−	0.875966i	$-0.339775\pi$
$491$	10496.3	0.964747	0.482374	+	0.875966i	$0.339775\pi$
$492$	− 10204.1i	− 0.935037i
$493$	10007.3	0.914211
$494$	0	0
$495$	146.109	0.0132669
$496$	− 6827.69i	− 0.618090i
$497$	−1007.84	−0.0909615
$498$	1260.54	0.113426
$499$	10516.5i	0.943450i	0.881746	+	0.471725i	$0.156368\pi$
$499$	10516.5i	0.943450i	−0.881746	+	0.471725i	$0.843632\pi$
$500$	− 3708.75i	− 0.331721i
$501$	− 4010.60i	− 0.357645i
$502$	− 1527.90i	− 0.135844i
$503$	13400.9	1.18790	0.593952	−	0.804500i	$-0.297567\pi$
$503$	13400.9	1.18790	0.593952	+	0.804500i	$0.297567\pi$
$504$	−518.068	−0.0457869
$505$	1158.19i	0.102057i
$506$	−334.430	−0.0293819
$507$	0	0
$508$	1101.38	0.0961923
$509$	− 4683.90i	− 0.407878i	−0.978984	−	0.203939i	$-0.934626\pi$
$509$	− 4683.90i	− 0.407878i	0.978984	−	0.203939i	$-0.0653745\pi$
$510$	−135.047	−0.0117254
$511$	5483.20	0.474682
$512$	− 8267.80i	− 0.713649i
$513$	− 1318.42i	− 0.113469i
$514$	3132.00i	0.268768i
$515$	− 566.305i	− 0.0484551i
$516$	−4727.89	−0.403360
$517$	−5197.75	−0.442160
$518$	498.090i	0.0422487i
$519$	−1705.38	−0.144235
$520$	0	0
$521$	15034.6	1.26426	0.632128	−	0.774864i	$-0.282182\pi$
$521$	15034.6	1.26426	0.632128	+	0.774864i	$0.282182\pi$
$522$	− 772.811i	− 0.0647989i
$523$	8009.33	0.669643	0.334822	−	0.942282i	$-0.391324\pi$
$523$	8009.33	0.669643	0.334822	+	0.942282i	$0.391324\pi$
$524$	−5708.86	−0.475940
$525$	− 2963.42i	− 0.246351i
$526$	− 1507.22i	− 0.124939i
$527$	− 6008.04i	− 0.496612i
$528$	1494.31i	0.123166i
$529$	−4259.71	−0.350103
$530$	391.345	0.0320735
$531$	4500.40i	0.367798i
$532$	−3102.36	−0.252828
$533$	0	0
$534$	−1013.06	−0.0820960
$535$	− 2961.14i	− 0.239292i
$536$	−6266.77	−0.505006
$537$	−4638.01	−0.372710
$538$	3489.53i	0.279636i
$539$	2326.21i	0.185894i
$540$	− 406.608i	− 0.0324030i
$541$	20833.2i	1.65561i	0.561013	+	0.827807i	$0.310412\pi$
$541$	20833.2i	1.65561i	−0.561013	+	0.827807i	$0.689588\pi$
$542$	1972.52	0.156323
$543$	−10226.9	−0.808245
$544$	− 4328.22i	− 0.341123i
$545$	−1140.52	−0.0896411
$546$	0	0
$547$	−14247.2	−1.11365	−0.556825	−	0.830630i	$-0.687981\pi$
$547$	−14247.2	−1.11365	−0.556825	+	0.830630i	$0.687981\pi$
$548$	13745.6i	1.07150i
$549$	4324.52	0.336186
$550$	456.093	0.0353598
$551$	− 9374.41i	− 0.724797i
$552$	1885.25i	0.145365i
$553$	5549.43i	0.426737i
$554$	− 2073.78i	− 0.159037i
$555$	−791.885	−0.0605651
$556$	−5185.22	−0.395508
$557$	2007.94i	0.152745i	0.997079	+	0.0763725i	$0.0243338\pi$
$557$	2007.94i	0.152745i	−0.997079	+	0.0763725i	$0.975666\pi$
$558$	−463.969	−0.0351996
$559$	0	0
$560$	−931.616	−0.0703000
$561$	1314.92i	0.0989589i
$562$	487.758	0.0366100
$563$	−10562.1	−0.790659	−0.395330	−	0.918539i	$-0.629370\pi$
$563$	−10562.1	−0.790659	−0.395330	+	0.918539i	$0.629370\pi$
$564$	14464.8i	1.07993i
$565$	1262.28i	0.0939905i
$566$	510.129i	0.0378840i
$567$	− 659.775i	− 0.0488676i
$568$	−874.409	−0.0645940
$569$	23207.2	1.70983	0.854916	−	0.518766i	$-0.173608\pi$
$569$	23207.2	1.70983	0.854916	+	0.518766i	$0.173608\pi$
$570$	126.506i	0.00929605i
$571$	−7987.78	−0.585426	−0.292713	−	0.956200i	$-0.594558\pi$
$571$	−7987.78	−0.585426	−0.292713	+	0.956200i	$0.594558\pi$
$572$	0	0
$573$	−10394.1	−0.757803
$574$	1588.74i	0.115527i
$575$	−10783.9	−0.782120
$576$	3930.94	0.284356
$577$	− 3224.41i	− 0.232641i	−0.993212	−	0.116321i	$-0.962890\pi$
$577$	− 3224.41i	− 0.232641i	0.993212	−	0.116321i	$-0.0371100\pi$
$578$	982.116i	0.0706759i
$579$	13956.7i	1.00176i
$580$	− 2891.12i	− 0.206978i
$581$	7651.90	0.546393
$582$	−1410.40	−0.100452
$583$	− 3810.44i	− 0.270690i
$584$	4757.26	0.337084
$585$	0	0
$586$	−149.873	−0.0105652
$587$	661.034i	0.0464800i	0.999730	+	0.0232400i	$0.00739819\pi$
$587$	661.034i	0.0464800i	−0.999730	+	0.0232400i	$0.992602\pi$
$588$	6473.63	0.454027
$589$	−5628.07	−0.393719
$590$	− 431.824i	− 0.0301321i
$591$	8612.94i	0.599474i
$592$	− 8098.87i	− 0.562266i
$593$	− 10172.5i	− 0.704443i	−0.935917	−	0.352221i	$-0.885426\pi$
$593$	− 10172.5i	− 0.704443i	0.935917	−	0.352221i	$-0.114574\pi$
$594$	101.544	0.00701417
$595$	−819.777	−0.0564834
$596$	25745.4i	1.76942i
$597$	−75.0990	−0.00514840
$598$	0	0
$599$	23462.2	1.60040	0.800200	−	0.599733i	$-0.204726\pi$
$599$	23462.2	1.60040	0.800200	+	0.599733i	$0.204726\pi$
$600$	− 2571.08i	− 0.174940i
$601$	−10482.6	−0.711474	−0.355737	−	0.934586i	$-0.615770\pi$
$601$	−10482.6	−0.711474	−0.355737	+	0.934586i	$0.615770\pi$
$602$	736.111	0.0498366
$603$	− 7980.92i	− 0.538986i
$604$	11420.3i	0.769347i
$605$	− 2433.29i	− 0.163516i
$606$	804.930i	0.0539572i
$607$	−24949.8	−1.66834	−0.834169	−	0.551508i	$-0.814053\pi$
$607$	−24949.8	−1.66834	−0.834169	+	0.551508i	$0.814053\pi$
$608$	−4054.48	−0.270446
$609$	− 4691.21i	− 0.312147i
$610$	−414.949	−0.0275423
$611$	0	0
$612$	3659.30	0.241697
$613$	− 8238.86i	− 0.542846i	−0.962460	−	0.271423i	$-0.912506\pi$
$613$	− 8238.86i	− 0.542846i	0.962460	−	0.271423i	$-0.0874942\pi$
$614$	1136.10	0.0746733
$615$	−2525.84	−0.165613
$616$	− 484.014i	− 0.0316583i
$617$	− 5248.85i	− 0.342481i	−0.985229	−	0.171241i	$-0.945222\pi$
$617$	− 5248.85i	− 0.342481i	0.985229	−	0.171241i	$-0.0547775\pi$
$618$	− 393.576i	− 0.0256180i
$619$	− 17820.9i	− 1.15716i	−0.815626	−	0.578580i	$-0.803607\pi$
$619$	− 17820.9i	− 1.15716i	0.815626	−	0.578580i	$-0.196393\pi$
$620$	−1735.73	−0.112433
$621$	−2400.92	−0.155146
$622$	1062.89i	0.0685180i
$623$	−6149.58	−0.395470
$624$	0	0
$625$	14241.0	0.911424
$626$	− 588.522i	− 0.0375752i
$627$	1231.76	0.0784557
$628$	11975.4	0.760938
$629$	− 7126.62i	− 0.451760i
$630$	63.3071i	0.00400351i
$631$	25445.0i	1.60531i	0.596446	+	0.802653i	$0.296579\pi$
$631$	25445.0i	1.60531i	−0.596446	+	0.802653i	$0.703421\pi$
$632$	4814.71i	0.303036i
$633$	−10817.7	−0.679252
$634$	2702.96	0.169319
$635$	− 272.625i	− 0.0170375i
$636$	−10604.1	−0.661132
$637$	0	0
$638$	722.013	0.0448037
$639$	− 1113.59i	− 0.0689402i
$640$	−1659.68	−0.102507
$641$	18701.0	1.15233	0.576167	−	0.817332i	$-0.304548\pi$
$641$	18701.0	1.15233	0.576167	+	0.817332i	$0.304548\pi$
$642$	− 2057.96i	− 0.126513i
$643$	28465.9i	1.74586i	0.487849	+	0.872928i	$0.337782\pi$
$643$	28465.9i	1.74586i	−0.487849	+	0.872928i	$0.662218\pi$
$644$	5649.58i	0.345690i
$645$	1170.30i	0.0714426i
$646$	−1138.50	−0.0693399
$647$	10924.6	0.663818	0.331909	−	0.943311i	$-0.392307\pi$
$647$	10924.6	0.663818	0.331909	+	0.943311i	$0.392307\pi$
$648$	− 572.425i	− 0.0347021i
$649$	−4204.58	−0.254305
$650$	0	0
$651$	−2816.44	−0.169562
$652$	6189.68i	0.371789i
$653$	−27110.9	−1.62470	−0.812351	−	0.583169i	$-0.801813\pi$
$653$	−27110.9	−1.62470	−0.812351	+	0.583169i	$0.801813\pi$
$654$	−792.646	−0.0473929
$655$	1413.12i	0.0842980i
$656$	− 25832.6i	− 1.53749i
$657$	6058.51i	0.359764i
$658$	− 2252.11i	− 0.133429i
$659$	1727.08	0.102090	0.0510452	−	0.998696i	$-0.483745\pi$
$659$	1727.08	0.102090	0.0510452	+	0.998696i	$0.483745\pi$
$660$	379.881	0.0224043
$661$	20651.1i	1.21518i	0.794249	+	0.607592i	$0.207864\pi$
$661$	20651.1i	1.21518i	−0.794249	+	0.607592i	$0.792136\pi$
$662$	−3653.45	−0.214494
$663$	0	0
$664$	6638.84	0.388007
$665$	767.932i	0.0447806i
$666$	−550.351	−0.0320205
$667$	−17071.3	−0.991010
$668$	− 10427.5i	− 0.603969i
$669$	12913.1i	0.746263i
$670$	765.790i	0.0441568i
$671$	4040.26i	0.232448i
$672$	−2028.98	−0.116472
$673$	−10858.0	−0.621907	−0.310954	−	0.950425i	$-0.600648\pi$
$673$	−10858.0	−0.621907	−0.310954	+	0.950425i	$0.600648\pi$
$674$	1375.97i	0.0786355i
$675$	3274.35	0.186711
$676$	0	0
$677$	29132.9	1.65387	0.826933	−	0.562301i	$-0.190084\pi$
$677$	29132.9	1.65387	0.826933	+	0.562301i	$0.190084\pi$
$678$	877.272i	0.0496924i
$679$	−8561.59	−0.483894
$680$	−711.244	−0.0401102
$681$	16425.6i	0.924272i
$682$	− 433.472i	− 0.0243380i
$683$	− 9443.65i	− 0.529065i	−0.964377	−	0.264532i	$-0.914782\pi$
$683$	− 9443.65i	− 0.529065i	0.964377	−	0.264532i	$-0.0852176\pi$
$684$	− 3427.87i	− 0.191620i
$685$	3402.46	0.189783
$686$	−2257.55	−0.125647
$687$	1134.33i	0.0629945i
$688$	−11969.0	−0.663249
$689$	0	0
$690$	230.374	0.0127104
$691$	23700.1i	1.30477i	0.757889	+	0.652384i	$0.226231\pi$
$691$	23700.1i	1.30477i	−0.757889	+	0.652384i	$0.773769\pi$
$692$	−4433.96	−0.243575
$693$	616.407	0.0337884
$694$	3915.51i	0.214165i
$695$	1283.50i	0.0700519i
$696$	− 4070.12i	− 0.221663i
$697$	− 22731.5i	− 1.23532i
$698$	476.340	0.0258305
$699$	−7643.88	−0.413616
$700$	− 7704.84i	− 0.416022i
$701$	−21643.1	−1.16612	−0.583059	−	0.812430i	$-0.698144\pi$
$701$	−21643.1	−1.16612	−0.583059	+	0.812430i	$0.698144\pi$
$702$	0	0
$703$	−6675.91	−0.358160
$704$	3672.55i	0.196611i
$705$	3580.50	0.191276
$706$	−2257.69	−0.120353
$707$	4886.18i	0.259921i
$708$	11700.9i	0.621114i
$709$	− 28965.3i	− 1.53429i	−0.641472	−	0.767146i	$-0.721676\pi$
$709$	− 28965.3i	− 1.53429i	0.641472	−	0.767146i	$-0.278324\pi$
$710$	106.851i	0.00564798i
$711$	−6131.68	−0.323426
$712$	−5335.41	−0.280833
$713$	10249.0i	0.538330i
$714$	−569.736	−0.0298625
$715$	0	0
$716$	−12058.7	−0.629408
$717$	18941.7i	0.986599i
$718$	3646.39	0.189529
$719$	21256.5	1.10255	0.551276	−	0.834323i	$-0.314141\pi$
$719$	21256.5	1.10255	0.551276	+	0.834323i	$0.314141\pi$
$720$	− 1029.36i	− 0.0532807i
$721$	− 2389.13i	− 0.123406i
$722$	− 2001.39i	− 0.103163i
$723$	− 20291.2i	− 1.04376i
$724$	−26589.7	−1.36491
$725$	23281.7	1.19264
$726$	− 1691.11i	− 0.0864505i
$727$	11868.9	0.605494	0.302747	−	0.953071i	$-0.402096\pi$
$727$	11868.9	0.605494	0.302747	+	0.953071i	$0.402096\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 581.330i	− 0.0294739i
$731$	−10532.2	−0.532896
$732$	11243.7	0.567729
$733$	− 27296.4i	− 1.37546i	−0.725965	−	0.687732i	$-0.758607\pi$
$733$	− 27296.4i	− 1.37546i	0.725965	−	0.687732i	$-0.241393\pi$
$734$	− 5204.68i	− 0.261728i
$735$	− 1602.43i	− 0.0804168i
$736$	7383.44i	0.369779i
$737$	7456.32	0.372669
$738$	−1755.43	−0.0875587
$739$	− 26424.6i	− 1.31535i	−0.753302	−	0.657674i	$-0.771540\pi$
$739$	− 26424.6i	− 1.31535i	0.753302	−	0.657674i	$-0.228460\pi$
$740$	−2058.89	−0.102279
$741$	0	0
$742$	1651.01	0.0816853
$743$	25974.8i	1.28253i	0.767318	+	0.641267i	$0.221591\pi$
$743$	25974.8i	1.28253i	−0.767318	+	0.641267i	$0.778409\pi$
$744$	−2443.56	−0.120410
$745$	6372.78	0.313397
$746$	− 2111.42i	− 0.103625i
$747$	8454.76i	0.414114i
$748$	3418.77i	0.167116i
$749$	− 12492.5i	− 0.609433i
$750$	−638.022	−0.0310630
$751$	−20234.8	−0.983195	−0.491598	−	0.870823i	$-0.663587\pi$
$751$	−20234.8	−0.983195	−0.491598	+	0.870823i	$0.663587\pi$
$752$	36619.0i	1.77574i
$753$	10248.0	0.495960
$754$	0	0
$755$	2826.88	0.136266
$756$	− 1715.40i	− 0.0825246i
$757$	−3363.48	−0.161490	−0.0807449	−	0.996735i	$-0.525730\pi$
$757$	−3363.48	−0.161490	−0.0807449	+	0.996735i	$0.525730\pi$
$758$	−5784.97	−0.277202
$759$	− 2243.10i	− 0.107272i
$760$	666.262i	0.0317998i
$761$	− 21311.3i	− 1.01516i	−0.861605	−	0.507579i	$-0.830541\pi$
$761$	− 21311.3i	− 1.01516i	0.861605	−	0.507579i	$-0.169459\pi$
$762$	− 189.471i	− 0.00900764i
$763$	−4811.62	−0.228299
$764$	−27024.5	−1.27973
$765$	− 905.790i	− 0.0428090i
$766$	−248.215	−0.0117081
$767$	0	0
$768$	9329.04	0.438324
$769$	34989.6i	1.64078i	0.571807	+	0.820388i	$0.306242\pi$
$769$	34989.6i	1.64078i	−0.571807	+	0.820388i	$0.693758\pi$
$770$	−59.1458	−0.00276814
$771$	−21007.1	−0.981260
$772$	36287.2i	1.69171i
$773$	− 20034.1i	− 0.932181i	−0.884737	−	0.466091i	$-0.845662\pi$
$773$	− 20034.1i	− 0.932181i	0.884737	−	0.466091i	$-0.154338\pi$
$774$	813.345i	0.0377714i
$775$	− 13977.5i	− 0.647855i
$776$	−7428.09	−0.343625
$777$	−3340.81	−0.154248
$778$	2277.25i	0.104940i
$779$	−21293.9	−0.979373
$780$	0	0
$781$	1040.39	0.0476671
$782$	2073.27i	0.0948081i
$783$	5183.43	0.236578
$784$	16388.5	0.746563
$785$	− 2964.28i	− 0.134777i
$786$	982.102i	0.0445680i
$787$	5591.98i	0.253282i	0.991949	+	0.126641i	$0.0404196\pi$
$787$	5591.98i	0.253282i	−0.991949	+	0.126641i	$0.959580\pi$
$788$	22393.5i	1.01235i
$789$	10109.3	0.456147
$790$	588.351	0.0264969
$791$	5325.32i	0.239376i
$792$	534.798	0.0239940
$793$	0	0
$794$	4586.64	0.205005
$795$	2624.84i	0.117099i
$796$	−195.256	−0.00869430
$797$	12738.6	0.566152	0.283076	−	0.959098i	$-0.408645\pi$
$797$	12738.6	0.566152	0.283076	+	0.959098i	$0.408645\pi$
$798$	533.704i	0.0236753i
$799$	32222.9i	1.42674i
$800$	− 10069.5i	− 0.445012i
$801$	− 6794.81i	− 0.299729i
$802$	226.240	0.00996113
$803$	−5660.28	−0.248751
$804$	− 20750.2i	− 0.910205i
$805$	1398.45	0.0612283
$806$	0	0
$807$	−23405.1	−1.02094
$808$	4239.28i	0.184576i
$809$	−8806.48	−0.382719	−0.191359	−	0.981520i	$-0.561290\pi$
$809$	−8806.48	−0.382719	−0.191359	+	0.981520i	$0.561290\pi$
$810$	−69.9494	−0.00303429
$811$	10565.8i	0.457478i	0.973488	+	0.228739i	$0.0734603\pi$
$811$	10565.8i	0.457478i	−0.973488	+	0.228739i	$0.926540\pi$
$812$	− 12197.1i	− 0.527134i
$813$	13230.1i	0.570727i
$814$	− 514.176i	− 0.0221399i
$815$	1532.14	0.0658509
$816$	9263.82	0.397425
$817$	9866.09i	0.422486i
$818$	−992.604	−0.0424273
$819$	0	0
$820$	−6567.14	−0.279676
$821$	− 422.966i	− 0.0179800i	−0.999960	−	0.00899002i	$-0.997138\pi$
$821$	− 422.966i	− 0.0179800i	0.999960	−	0.00899002i	$-0.00286165\pi$
$822$	2364.67	0.100338
$823$	−11148.8	−0.472202	−0.236101	−	0.971728i	$-0.575870\pi$
$823$	−11148.8	−0.472202	−0.236101	+	0.971728i	$0.575870\pi$
$824$	− 2072.83i	− 0.0876339i
$825$	3059.12i	0.129097i
$826$	− 1821.78i	− 0.0767409i
$827$	− 813.158i	− 0.0341914i	−0.999854	−	0.0170957i	$-0.994558\pi$
$827$	− 813.158i	− 0.0341914i	0.999854	−	0.0170957i	$-0.00544200\pi$
$828$	−6242.34	−0.262001
$829$	28320.8	1.18652	0.593258	−	0.805012i	$-0.297841\pi$
$829$	28320.8	1.18652	0.593258	+	0.805012i	$0.297841\pi$
$830$	− 811.255i	− 0.0339266i
$831$	13909.3	0.580637
$832$	0	0
$833$	14421.1	0.599835
$834$	892.020i	0.0370361i
$835$	−2581.13	−0.106974
$836$	3202.55	0.132491
$837$	− 3111.95i	− 0.128512i
$838$	− 4584.28i	− 0.188975i
$839$	− 22139.3i	− 0.911006i	−0.890234	−	0.455503i	$-0.849459\pi$
$839$	− 22139.3i	− 0.911006i	0.890234	−	0.455503i	$-0.150541\pi$
$840$	333.416i	0.0136952i
$841$	12466.8	0.511166
$842$	−43.7972	−0.00179258
$843$	3271.50i	0.133661i
$844$	−28125.9	−1.14708
$845$	0	0
$846$	2488.41	0.101127
$847$	− 10265.6i	− 0.416446i
$848$	−26845.2	−1.08711
$849$	−3421.56	−0.138313
$850$	− 2827.50i	− 0.114097i
$851$	12157.2i	0.489710i
$852$	− 2895.30i	− 0.116422i
$853$	− 14080.5i	− 0.565191i	−0.959239	−	0.282595i	$-0.908805\pi$
$853$	− 14080.5i	− 0.565191i	0.959239	−	0.282595i	$-0.0911953\pi$
$854$	−1750.59	−0.0701451
$855$	−848.505	−0.0339395
$856$	− 10838.6i	− 0.432774i
$857$	−10907.7	−0.434772	−0.217386	−	0.976086i	$-0.569753\pi$
$857$	−10907.7	−0.434772	−0.217386	+	0.976086i	$0.569753\pi$
$858$	0	0
$859$	7739.24	0.307403	0.153702	−	0.988117i	$-0.450881\pi$
$859$	7739.24	0.307403	0.153702	+	0.988117i	$0.450881\pi$
$860$	3042.76i	0.120648i
$861$	−10656.0	−0.421785
$862$	−5245.30	−0.207257
$863$	29072.5i	1.14674i	0.819295	+	0.573372i	$0.194365\pi$
$863$	29072.5i	1.14674i	−0.819295	+	0.573372i	$0.805635\pi$
$864$	− 2241.86i	− 0.0882751i
$865$	1097.54i	0.0431417i
$866$	3106.40i	0.121894i
$867$	−6587.28	−0.258034
$868$	−7322.70	−0.286346
$869$	− 5728.64i	− 0.223626i
$870$	−497.363	−0.0193818
$871$	0	0
$872$	−4174.59	−0.162121
$873$	− 9459.89i	− 0.366746i
$874$	1942.15	0.0751649
$875$	−3873.00	−0.149636
$876$	15752.0i	0.607547i
$877$	− 2391.96i	− 0.0920988i	−0.998939	−	0.0460494i	$-0.985337\pi$
$877$	− 2391.96i	− 0.0920988i	0.998939	−	0.0460494i	$-0.0146632\pi$
$878$	5119.66i	0.196788i
$879$	− 1005.24i	− 0.0385731i
$880$	961.702	0.0368397
$881$	11975.8	0.457974	0.228987	−	0.973430i	$-0.426459\pi$
$881$	11975.8	0.457974	0.228987	+	0.973430i	$0.426459\pi$
$882$	− 1113.67i	− 0.0425160i
$883$	44712.4	1.70407	0.852035	−	0.523485i	$-0.175368\pi$
$883$	44712.4	1.70407	0.852035	+	0.523485i	$0.175368\pi$
$884$	0	0
$885$	2896.35	0.110011
$886$	4702.51i	0.178311i
$887$	17023.3	0.644403	0.322202	−	0.946671i	$-0.395577\pi$
$887$	17023.3	0.644403	0.322202	+	0.946671i	$0.395577\pi$
$888$	−2898.51	−0.109535
$889$	− 1150.15i	− 0.0433913i
$890$	651.979i	0.0245555i
$891$	681.082i	0.0256084i
$892$	33573.8i	1.26024i
$893$	30185.0	1.13113
$894$	4429.01	0.165692
$895$	2984.91i	0.111480i
$896$	−7001.87	−0.261067
$897$	0	0
$898$	−4411.22	−0.163925
$899$	− 22127.0i	− 0.820886i
$900$	8513.25	0.315306
$901$	−23622.5	−0.873449
$902$	− 1640.04i	− 0.0605405i
$903$	4937.27i	0.181951i
$904$	4620.28i	0.169987i
$905$	6581.77i	0.241752i
$906$	1964.65	0.0720432
$907$	−2303.59	−0.0843324	−0.0421662	−	0.999111i	$-0.513426\pi$
$907$	−2303.59	−0.0843324	−0.0421662	+	0.999111i	$0.513426\pi$
$908$	42706.2i	1.56085i
$909$	−5398.85	−0.196995
$910$	0	0
$911$	−12897.0	−0.469042	−0.234521	−	0.972111i	$-0.575352\pi$
$911$	−12897.0	−0.469042	−0.234521	+	0.972111i	$0.575352\pi$
$912$	− 8677.95i	− 0.315083i
$913$	−7899.01	−0.286330
$914$	2679.41	0.0969662
$915$	− 2783.16i	− 0.100556i
$916$	2949.22i	0.106381i
$917$	5961.68i	0.214691i
$918$	− 629.514i	− 0.0226330i
$919$	16785.2	0.602496	0.301248	−	0.953546i	$-0.402597\pi$
$919$	16785.2	0.602496	0.301248	+	0.953546i	$0.402597\pi$
$920$	1213.30	0.0434797
$921$	7620.11i	0.272629i
$922$	−7415.64	−0.264882
$923$	0	0
$924$	1602.65	0.0570597
$925$	− 16579.9i	− 0.589344i
$926$	1613.18	0.0572487
$927$	2639.81	0.0935303
$928$	− 15940.4i	− 0.563866i
$929$	− 16348.0i	− 0.577352i	−0.957427	−	0.288676i	$-0.906785\pi$
$929$	− 16348.0i	− 0.577352i	0.957427	−	0.288676i	$-0.0932150\pi$
$930$	298.600i	0.0105285i
$931$	− 13509.1i	− 0.475556i
$932$	−19873.9	−0.698489
$933$	−7129.08	−0.250156
$934$	− 3036.85i	− 0.106390i
$935$	846.251	0.0295993
$936$	0	0
$937$	−26247.5	−0.915121	−0.457560	−	0.889179i	$-0.651277\pi$
$937$	−26247.5	−0.915121	−0.457560	+	0.889179i	$0.651277\pi$
$938$	3230.72i	0.112459i
$939$	3947.35	0.137185
$940$	9309.22	0.323014
$941$	− 43838.4i	− 1.51869i	−0.650686	−	0.759347i	$-0.725519\pi$
$941$	− 43838.4i	− 1.51869i	0.650686	−	0.759347i	$-0.274481\pi$
$942$	− 2060.14i	− 0.0712558i
$943$	38777.3i	1.33909i
$944$	29621.9i	1.02130i
$945$	−424.615	−0.0146167
$946$	−759.882	−0.0261162
$947$	− 45077.2i	− 1.54679i	−0.633924	−	0.773395i	$-0.718557\pi$
$947$	− 45077.2i	− 1.54679i	0.633924	−	0.773395i	$-0.281443\pi$
$948$	−15942.3	−0.546182
$949$	0	0
$950$	−2648.68	−0.0904575
$951$	18129.4i	0.618177i
$952$	−3000.60	−0.102153
$953$	−26479.1	−0.900046	−0.450023	−	0.893017i	$-0.648584\pi$
$953$	−26479.1	−0.900046	−0.450023	+	0.893017i	$0.648584\pi$
$954$	1824.24i	0.0619097i
$955$	6689.41i	0.226664i
$956$	49248.1i	1.66611i
$957$	4842.71i	0.163576i
$958$	−5675.21	−0.191396
$959$	14354.3	0.483343
$960$	− 2529.86i	− 0.0850529i
$961$	16506.7	0.554084
$962$	0	0
$963$	13803.2	0.461893
$964$	− 52756.7i	− 1.76263i
$965$	8982.20	0.299635
$966$	971.905	0.0323711
$967$	602.475i	0.0200355i	0.999950	+	0.0100177i	$0.00318880\pi$
$967$	602.475i	0.0200355i	−0.999950	+	0.0100177i	$0.996811\pi$
$968$	− 8906.49i	− 0.295729i
$969$	− 7636.17i	− 0.253157i
$970$	907.701i	0.0300459i
$971$	−11330.2	−0.374464	−0.187232	−	0.982316i	$-0.559952\pi$
$971$	−11330.2	−0.374464	−0.187232	+	0.982316i	$0.559952\pi$
$972$	1895.39	0.0625458
$973$	5414.85i	0.178409i
$974$	544.461	0.0179114
$975$	0	0
$976$	28464.3	0.933524
$977$	− 55633.6i	− 1.82178i	−0.412653	−	0.910888i	$-0.635398\pi$
$977$	− 55633.6i	− 1.82178i	0.412653	−	0.910888i	$-0.364602\pi$
$978$	1064.82	0.0348151
$979$	6348.18	0.207241
$980$	− 4166.27i	− 0.135803i
$981$	− 5316.47i	− 0.173029i
$982$	− 4694.76i	− 0.152562i
$983$	− 6884.90i	− 0.223392i	−0.993742	−	0.111696i	$-0.964372\pi$
$983$	− 6884.90i	− 0.223392i	0.993742	−	0.111696i	$-0.0356282\pi$
$984$	−9245.24	−0.299520
$985$	5543.08	0.179307
$986$	− 4476.05i	− 0.144570i
$987$	15105.4	0.487144
$988$	0	0
$989$	17966.7	0.577662
$990$	− 65.3515i	− 0.00209799i
$991$	52646.1	1.68755	0.843773	−	0.536700i	$-0.180329\pi$
$991$	52646.1	1.68755	0.843773	+	0.536700i	$0.180329\pi$
$992$	−9570.05	−0.306300
$993$	− 24504.5i	− 0.783109i
$994$	450.786i	0.0143844i
$995$	48.3319i	0.00153992i
$996$	21982.2i	0.699330i
$997$	26100.7	0.829106	0.414553	−	0.910025i	$-0.363938\pi$
$997$	26100.7	0.829106	0.414553	+	0.910025i	$0.363938\pi$
$998$	4703.78	0.149194
$999$	− 3691.33i	− 0.116906i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.8		18
13.5	odd	4		507.4.a.o.1.5	✓	9
13.8	odd	4		507.4.a.p.1.5	yes	9
13.12	even	2	inner	507.4.b.k.337.11		18
39.5	even	4		1521.4.a.bi.1.5		9
39.8	even	4		1521.4.a.bf.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.5	✓	9	13.5	odd	4
507.4.a.p.1.5	yes	9	13.8	odd	4
507.4.b.k.337.8		18	1.1	even	1	trivial
507.4.b.k.337.11		18	13.12	even	2	inner
1521.4.a.bf.1.5		9	39.8	even	4
1521.4.a.bi.1.5		9	39.5	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)

\(f(q)\)	\(=\)	\(q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} -0.863573 q^{10} +8.40842i q^{11} +23.3998 q^{12} -3.64325 q^{14} -5.79219i q^{15} +59.2386 q^{16} +52.1271 q^{17} -4.02550i q^{18} -48.8304i q^{19} -15.0596i q^{20} -24.4361i q^{21} +3.76090 q^{22} -88.9229 q^{23} -21.2009i q^{24} +121.272 q^{25} +27.0000 q^{27} -63.5334i q^{28} +191.979 q^{29} -2.59072 q^{30} -115.257i q^{31} -83.0319i q^{32} +25.2252i q^{33} -23.3153i q^{34} -15.7265 q^{35} +70.1995 q^{36} -136.716i q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077i q^{41} -10.9297 q^{42} -202.048 q^{43} +65.5852i q^{44} -17.3766i q^{45} +39.7733i q^{46} +618.160i q^{47} +177.716 q^{48} +276.653 q^{49} -54.2425i q^{50} +156.381 q^{51} -453.170 q^{53} -12.0765i q^{54} +16.2344 q^{55} -57.5631 q^{56} -146.491i q^{57} -85.8679i q^{58} +500.044i q^{59} -45.1787i q^{60} +480.502 q^{61} -51.5522 q^{62} -73.3083i q^{63} +436.771 q^{64} +11.2827 q^{66} -886.769i q^{67} +406.589 q^{68} -266.769 q^{69} +7.03412i q^{70} -123.732i q^{71} -63.6027i q^{72} +673.168i q^{73} -61.1501 q^{74} +363.817 q^{75} -380.875i q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374i q^{80} +81.0000 q^{81} -195.048 q^{82} +939.418i q^{83} -190.600i q^{84} -100.643i q^{85} +90.3716i q^{86} +575.936 q^{87} +59.4220 q^{88} -754.979i q^{89} -7.77215 q^{90} -693.594 q^{92} -345.772i q^{93} +276.489 q^{94} -94.2783 q^{95} -249.096i q^{96} -1051.10i q^{97} -123.741i q^{98} +75.6757i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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