LMFDB - Embedded newform 507.4.b.g.337.2

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:	$6$
Coefficient field:	6.0.158155776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 $ x^6 - 2x^5 + 2x^4 + 24x^3 + 81x^2 + 54*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.42234 + 1.42234i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.g.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.73549i q^{2} +3.00000 q^{3} -5.95388 q^{4} -3.90776i q^{5} -11.2065i q^{6} -36.4129i q^{7} -7.64325i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.73549i q^{2} +3.00000 q^{3} -5.95388 q^{4} -3.90776i q^{5} -11.2065i q^{6} -36.4129i q^{7} -7.64325i q^{8} +9.00000 q^{9} -14.5974 q^{10} -19.1943i q^{11} -17.8616 q^{12} -136.020 q^{14} -11.7233i q^{15} -76.1823 q^{16} +83.8839 q^{17} -33.6194i q^{18} +46.8492i q^{19} +23.2664i q^{20} -109.239i q^{21} -71.7000 q^{22} -103.905 q^{23} -22.9298i q^{24} +109.729 q^{25} +27.0000 q^{27} +216.798i q^{28} +108.341 q^{29} -43.7922 q^{30} -147.532i q^{31} +223.432i q^{32} -57.5828i q^{33} -313.347i q^{34} -142.293 q^{35} -53.5849 q^{36} +160.012i q^{37} +175.005 q^{38} -29.8680 q^{40} +231.490i q^{41} -408.060 q^{42} +340.314 q^{43} +114.280i q^{44} -35.1699i q^{45} +388.135i q^{46} -119.653i q^{47} -228.547 q^{48} -982.902 q^{49} -409.893i q^{50} +251.652 q^{51} -732.879 q^{53} -100.858i q^{54} -75.0067 q^{55} -278.313 q^{56} +140.548i q^{57} -404.706i q^{58} +229.782i q^{59} +69.7991i q^{60} +108.943 q^{61} -551.104 q^{62} -327.716i q^{63} +225.170 q^{64} -215.100 q^{66} +10.3955i q^{67} -499.435 q^{68} -311.714 q^{69} +531.535i q^{70} -869.201i q^{71} -68.7893i q^{72} +1099.07i q^{73} +597.724 q^{74} +329.188 q^{75} -278.934i q^{76} -698.920 q^{77} +140.410 q^{79} +297.703i q^{80} +81.0000 q^{81} +864.729 q^{82} -159.474i q^{83} +650.395i q^{84} -327.799i q^{85} -1271.24i q^{86} +325.023 q^{87} -146.707 q^{88} -1067.93i q^{89} -131.377 q^{90} +618.636 q^{92} -442.596i q^{93} -446.964 q^{94} +183.075 q^{95} +670.297i q^{96} +858.881i q^{97} +3671.62i q^{98} -172.748i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) $ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9	$ 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) $ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 + 8 * q^10 - 60 * q^12 - 352 * q^14 - 220 * q^16 + 292 * q^17 - 112 * q^22 + 96 * q^23 - 290 * q^25 + 162 * q^27 - 4 * q^29 + 24 * q^30 + 160 * q^35 - 180 * q^36 - 624 * q^38 - 1032 * q^40 - 1056 * q^42 + 520 * q^43 - 660 * q^48 - 894 * q^49 + 876 * q^51 - 1356 * q^53 - 3104 * q^55 - 192 * q^56 + 460 * q^61 - 3904 * q^62 + 1500 * q^64 - 336 * q^66 - 920 * q^68 + 288 * q^69 + 3448 * q^74 - 870 * q^75 - 224 * q^77 - 48 * q^79 + 486 * q^81 - 1128 * q^82 - 12 * q^87 - 3216 * q^88 + 72 * q^90 - 2592 * q^92 - 3840 * q^94 + 8064 * 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 $

$2$	− 3.73549i	− 1.32069i	−0.750960	−	0.660347i	$-0.770409\pi$
$2$	− 3.73549i	− 1.32069i	0.750960	−	0.660347i	$-0.229591\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.95388	−0.744235
$5$	− 3.90776i	− 0.349521i	−0.984611	−	0.174761i	$-0.944085\pi$
$5$	− 3.90776i	− 0.349521i	0.984611	−	0.174761i	$-0.0559151\pi$
$6$	− 11.2065i	− 0.762504i
$7$	− 36.4129i	− 1.96611i	−0.183301	−	0.983057i	$-0.558678\pi$
$7$	− 36.4129i	− 1.96611i	0.183301	−	0.983057i	$-0.441322\pi$
$8$	− 7.64325i	− 0.337787i
$9$	9.00000	0.333333
$10$	−14.5974	−0.461611
$11$	− 19.1943i	− 0.526117i	−0.964780	−	0.263059i	$-0.915269\pi$
$11$	− 19.1943i	− 0.526117i	0.964780	−	0.263059i	$-0.0847313\pi$
$12$	−17.8616	−0.429684
$13$	0	0
$13$	0	0
$14$	−136.020	−2.59664
$15$	− 11.7233i	− 0.201796i
$16$	−76.1823	−1.19035
$17$	83.8839	1.19676	0.598378	−	0.801214i	$-0.295812\pi$
$17$	83.8839	1.19676	0.598378	+	0.801214i	$0.295812\pi$
$18$	− 33.6194i	− 0.440232i
$19$	46.8492i	0.565681i	0.959167	+	0.282840i	$0.0912767\pi$
$19$	46.8492i	0.565681i	−0.959167	+	0.282840i	$0.908723\pi$
$20$	23.2664i	0.260126i
$21$	− 109.239i	− 1.13514i
$22$	−71.7000	−0.694840
$23$	−103.905	−0.941983	−0.470991	−	0.882138i	$-0.656104\pi$
$23$	−103.905	−0.941983	−0.470991	+	0.882138i	$0.656104\pi$
$24$	− 22.9298i	− 0.195022i
$25$	109.729	0.877835
$26$	0	0
$27$	27.0000	0.192450
$28$	216.798i	1.46325i
$29$	108.341	0.693738	0.346869	−	0.937914i	$-0.387245\pi$
$29$	108.341	0.693738	0.346869	+	0.937914i	$0.387245\pi$
$30$	−43.7922	−0.266511
$31$	− 147.532i	− 0.854759i	−0.904072	−	0.427379i	$-0.859437\pi$
$31$	− 147.532i	− 0.854759i	0.904072	−	0.427379i	$-0.140563\pi$
$32$	223.432i	1.23430i
$33$	− 57.5828i	− 0.303754i
$34$	− 313.347i	− 1.58055i
$35$	−142.293	−0.687198
$36$	−53.5849	−0.248078
$37$	160.012i	0.710969i	0.934682	+	0.355484i	$0.115684\pi$
$37$	160.012i	0.710969i	−0.934682	+	0.355484i	$0.884316\pi$
$38$	175.005	0.747092
$39$	0	0
$40$	−29.8680	−0.118064
$41$	231.490i	0.881772i	0.897563	+	0.440886i	$0.145336\pi$
$41$	231.490i	0.881772i	−0.897563	+	0.440886i	$0.854664\pi$
$42$	−408.060	−1.49917
$43$	340.314	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$43$	340.314	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$44$	114.280i	0.391555i
$45$	− 35.1699i	− 0.116507i
$46$	388.135i	1.24407i
$47$	− 119.653i	− 0.371346i	−0.982612	−	0.185673i	$-0.940554\pi$
$47$	− 119.653i	− 0.371346i	0.982612	−	0.185673i	$-0.0594464\pi$
$48$	−228.547	−0.687248
$49$	−982.902	−2.86560
$50$	− 409.893i	− 1.15935i
$51$	251.652	0.690947
$52$	0	0
$53$	−732.879	−1.89941	−0.949705	−	0.313146i	$-0.898617\pi$
$53$	−732.879	−1.89941	−0.949705	+	0.313146i	$0.898617\pi$
$54$	− 100.858i	− 0.254168i
$55$	−75.0067	−0.183889
$56$	−278.313	−0.664128
$57$	140.548i	0.326596i
$58$	− 404.706i	− 0.916216i
$59$	229.782i	0.507035i	0.967331	+	0.253518i	$0.0815876\pi$
$59$	229.782i	0.507035i	−0.967331	+	0.253518i	$0.918412\pi$
$60$	69.7991i	0.150184i
$61$	108.943	0.228668	0.114334	−	0.993442i	$-0.463527\pi$
$61$	108.943	0.228668	0.114334	+	0.993442i	$0.463527\pi$
$62$	−551.104	−1.12888
$63$	− 327.716i	− 0.655371i
$64$	225.170	0.439786
$65$	0	0
$66$	−215.100	−0.401166
$67$	10.3955i	0.0189555i	0.999955	+	0.00947774i	$0.00301690\pi$
$67$	10.3955i	0.0189555i	−0.999955	+	0.00947774i	$0.996983\pi$
$68$	−499.435	−0.890667
$69$	−311.714	−0.543854
$70$	531.535i	0.907579i
$71$	− 869.201i	− 1.45289i	−0.687224	−	0.726445i	$-0.741171\pi$
$71$	− 869.201i	− 1.45289i	0.687224	−	0.726445i	$-0.258829\pi$
$72$	− 68.7893i	− 0.112596i
$73$	1099.07i	1.76214i	0.472982	+	0.881072i	$0.343178\pi$
$73$	1099.07i	1.76214i	−0.472982	+	0.881072i	$0.656822\pi$
$74$	597.724	0.938973
$75$	329.188	0.506818
$76$	− 278.934i	− 0.421000i
$77$	−698.920	−1.03441
$78$	0	0
$79$	140.410	0.199967	0.0999835	−	0.994989i	$-0.468121\pi$
$79$	140.410	0.199967	0.0999835	+	0.994989i	$0.468121\pi$
$80$	297.703i	0.416052i
$81$	81.0000	0.111111
$82$	864.729	1.16455
$83$	− 159.474i	− 0.210898i	−0.994425	−	0.105449i	$-0.966372\pi$
$83$	− 159.474i	− 0.210898i	0.994425	−	0.105449i	$-0.0336280\pi$
$84$	650.395i	0.844808i
$85$	− 327.799i	− 0.418291i
$86$	− 1271.24i	− 1.59397i
$87$	325.023	0.400530
$88$	−146.707	−0.177716
$89$	− 1067.93i	− 1.27192i	−0.771723	−	0.635959i	$-0.780605\pi$
$89$	− 1067.93i	− 1.27192i	0.771723	−	0.635959i	$-0.219395\pi$
$90$	−131.377	−0.153870
$91$	0	0
$92$	618.636	0.701057
$93$	− 442.596i	− 0.493495i
$94$	−446.964	−0.490434
$95$	183.075	0.197717
$96$	670.297i	0.712624i
$97$	858.881i	0.899032i	0.893272	+	0.449516i	$0.148404\pi$
$97$	858.881i	0.899032i	−0.893272	+	0.449516i	$0.851596\pi$
$98$	3671.62i	3.78459i
$99$	− 172.748i	− 0.175372i
$100$	−653.316	−0.653316
$101$	1574.16	1.55084	0.775421	−	0.631444i	$-0.217538\pi$
$101$	1574.16	1.55084	0.775421	+	0.631444i	$0.217538\pi$
$102$	− 940.042i	− 0.912530i
$103$	129.724	0.124098	0.0620489	−	0.998073i	$-0.480237\pi$
$103$	129.724	0.124098	0.0620489	+	0.998073i	$0.480237\pi$
$104$	0	0
$105$	−426.879	−0.396754
$106$	2737.66i	2.50854i
$107$	−1957.43	−1.76853	−0.884263	−	0.466990i	$-0.845339\pi$
$107$	−1957.43	−1.76853	−0.884263	+	0.466990i	$0.845339\pi$
$108$	−160.755	−0.143228
$109$	1228.77i	1.07977i	0.841738	+	0.539886i	$0.181533\pi$
$109$	1228.77i	1.07977i	−0.841738	+	0.539886i	$0.818467\pi$
$110$	280.187i	0.242861i
$111$	480.037i	0.410478i
$112$	2774.02i	2.34036i
$113$	1629.50	1.35655	0.678275	−	0.734808i	$-0.262728\pi$
$113$	1629.50	1.35655	0.678275	+	0.734808i	$0.262728\pi$
$114$	525.014	0.431334
$115$	406.035i	0.329243i
$116$	−645.049	−0.516304
$117$	0	0
$118$	858.349	0.669639
$119$	− 3054.46i	− 2.35296i
$120$	−89.6041	−0.0681641
$121$	962.580	0.723201
$122$	− 406.956i	− 0.302000i
$123$	694.470i	0.509092i
$124$	878.388i	0.636142i
$125$	− 917.267i	− 0.656343i
$126$	−1224.18	−0.865546
$127$	−276.112	−0.192921	−0.0964607	−	0.995337i	$-0.530752\pi$
$127$	−276.112	−0.192921	−0.0964607	+	0.995337i	$0.530752\pi$
$128$	946.337i	0.653478i
$129$	1020.94	0.696813
$130$	0	0
$131$	−96.2240	−0.0641765	−0.0320883	−	0.999485i	$-0.510216\pi$
$131$	−96.2240	−0.0641765	−0.0320883	+	0.999485i	$0.510216\pi$
$132$	342.841i	0.226064i
$133$	1705.92	1.11219
$134$	38.8324	0.0250344
$135$	− 105.510i	− 0.0672653i
$136$	− 641.146i	− 0.404249i
$137$	− 2618.38i	− 1.63287i	−0.577438	−	0.816435i	$-0.695947\pi$
$137$	− 2618.38i	− 1.63287i	0.577438	−	0.816435i	$-0.304053\pi$
$138$	1164.40i	0.718265i
$139$	1963.34	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$139$	1963.34	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$140$	847.197	0.511437
$141$	− 358.960i	− 0.214396i
$142$	−3246.89	−1.91882
$143$	0	0
$144$	−685.641	−0.396783
$145$	− 423.370i	− 0.242476i
$146$	4105.57	2.32725
$147$	−2948.71	−1.65446
$148$	− 952.694i	− 0.529128i
$149$	− 301.111i	− 0.165557i	−0.996568	−	0.0827784i	$-0.973621\pi$
$149$	− 301.111i	− 0.165557i	0.996568	−	0.0827784i	$-0.0263794\pi$
$150$	− 1229.68i	− 0.669352i
$151$	− 342.973i	− 0.184839i	−0.995720	−	0.0924197i	$-0.970540\pi$
$151$	− 342.973i	− 0.184839i	0.995720	−	0.0924197i	$-0.0294602\pi$
$152$	358.080	0.191080
$153$	754.955	0.398918
$154$	2610.81i	1.36614i
$155$	−576.520	−0.298756
$156$	0	0
$157$	−1286.97	−0.654211	−0.327106	−	0.944988i	$-0.606073\pi$
$157$	−1286.97	−0.654211	−0.327106	+	0.944988i	$0.606073\pi$
$158$	− 524.501i	− 0.264095i
$159$	−2198.64	−1.09662
$160$	873.121	0.431414
$161$	3783.47i	1.85205i
$162$	− 302.575i	− 0.146744i
$163$	− 532.561i	− 0.255910i	−0.991780	−	0.127955i	$-0.959159\pi$
$163$	− 532.561i	− 0.255910i	0.991780	−	0.127955i	$-0.0408414\pi$
$164$	− 1378.26i	− 0.656246i
$165$	−225.020	−0.106168
$166$	−595.714	−0.278532
$167$	− 41.9542i	− 0.0194402i	−0.999953	−	0.00972011i	$-0.996906\pi$
$167$	− 41.9542i	− 0.0194402i	0.999953	−	0.00972011i	$-0.00309406\pi$
$168$	−834.940	−0.383435
$169$	0	0
$170$	−1224.49	−0.552435
$171$	421.643i	0.188560i
$172$	−2026.19	−0.898229
$173$	1066.50	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$173$	1066.50	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$174$	− 1214.12i	− 0.528977i
$175$	− 3995.57i	− 1.72592i
$176$	1462.26i	0.626263i
$177$	689.346i	0.292737i
$178$	−3989.25	−1.67982
$179$	−3174.61	−1.32559	−0.662797	−	0.748799i	$-0.730631\pi$
$179$	−3174.61	−1.32559	−0.662797	+	0.748799i	$0.730631\pi$
$180$	209.397i	0.0867086i
$181$	2725.43	1.11923	0.559613	−	0.828754i	$-0.310950\pi$
$181$	2725.43	1.11923	0.559613	+	0.828754i	$0.310950\pi$
$182$	0	0
$183$	326.829	0.132021
$184$	794.169i	0.318190i
$185$	625.290	0.248498
$186$	−1653.31	−0.651757
$187$	− 1610.09i	− 0.629634i
$188$	712.402i	0.276368i
$189$	− 983.149i	− 0.378379i
$190$	− 683.877i	− 0.261124i
$191$	784.888	0.297343	0.148672	−	0.988887i	$-0.452500\pi$
$191$	784.888	0.297343	0.148672	+	0.988887i	$0.452500\pi$
$192$	675.511	0.253910
$193$	1255.87i	0.468391i	0.972190	+	0.234195i	$0.0752455\pi$
$193$	1255.87i	0.468391i	−0.972190	+	0.234195i	$0.924754\pi$
$194$	3208.34	1.18735
$195$	0	0
$196$	5852.08	2.13268
$197$	− 2777.35i	− 1.00446i	−0.864734	−	0.502229i	$-0.832513\pi$
$197$	− 2777.35i	− 1.00446i	0.864734	−	0.502229i	$-0.167487\pi$
$198$	−645.300	−0.231613
$199$	−1490.43	−0.530924	−0.265462	−	0.964121i	$-0.585524\pi$
$199$	−1490.43	−0.530924	−0.265462	+	0.964121i	$0.585524\pi$
$200$	− 838.689i	− 0.296521i
$201$	31.1866i	0.0109440i
$202$	− 5880.27i	− 2.04819i
$203$	− 3945.01i	− 1.36397i
$204$	−1498.30	−0.514227
$205$	904.608	0.308198
$206$	− 484.582i	− 0.163895i
$207$	−935.141	−0.313994
$208$	0	0
$209$	899.236	0.297614
$210$	1594.60i	0.523991i
$211$	2305.63	0.752255	0.376127	−	0.926568i	$-0.377255\pi$
$211$	2305.63	0.752255	0.376127	+	0.926568i	$0.377255\pi$
$212$	4363.48	1.41361
$213$	− 2607.60i	− 0.838827i
$214$	7311.97i	2.33568i
$215$	− 1329.87i	− 0.421842i
$216$	− 206.368i	− 0.0650072i
$217$	−5372.07	−1.68055
$218$	4590.07	1.42605
$219$	3297.21i	1.01737i
$220$	446.581	0.136857
$221$	0	0
$222$	1793.17	0.542116
$223$	1241.98i	0.372956i	0.982459	+	0.186478i	$0.0597073\pi$
$223$	1241.98i	0.372956i	−0.982459	+	0.186478i	$0.940293\pi$
$224$	8135.83	2.42678
$225$	987.564	0.292612
$226$	− 6086.97i	− 1.79159i
$227$	− 1724.76i	− 0.504300i	−0.967688	−	0.252150i	$-0.918862\pi$
$227$	− 1724.76i	− 0.504300i	0.967688	−	0.252150i	$-0.0811376\pi$
$228$	− 836.803i	− 0.243064i
$229$	3273.72i	0.944688i	0.881414	+	0.472344i	$0.156592\pi$
$229$	3273.72i	0.944688i	−0.881414	+	0.472344i	$0.843408\pi$
$230$	1516.74	0.434829
$231$	−2096.76	−0.597215
$232$	− 828.076i	− 0.234336i
$233$	2129.52	0.598752	0.299376	−	0.954135i	$-0.403222\pi$
$233$	2129.52	0.598752	0.299376	+	0.954135i	$0.403222\pi$
$234$	0	0
$235$	−467.577	−0.129793
$236$	− 1368.10i	− 0.377354i
$237$	421.231	0.115451
$238$	−11409.9	−3.10754
$239$	− 5082.38i	− 1.37553i	−0.725933	−	0.687765i	$-0.758592\pi$
$239$	− 5082.38i	− 1.37553i	0.725933	−	0.687765i	$-0.241408\pi$
$240$	893.108i	0.240208i
$241$	− 4765.65i	− 1.27379i	−0.770953	−	0.636893i	$-0.780219\pi$
$241$	− 4765.65i	− 1.27379i	0.770953	−	0.636893i	$-0.219781\pi$
$242$	− 3595.71i	− 0.955127i
$243$	243.000	0.0641500
$244$	−648.634	−0.170183
$245$	3840.95i	1.00159i
$246$	2594.19	0.672355
$247$	0	0
$248$	−1127.62	−0.288727
$249$	− 478.422i	− 0.121762i
$250$	−3426.44	−0.866829
$251$	4339.96	1.09138	0.545689	−	0.837988i	$-0.316268\pi$
$251$	4339.96	1.09138	0.545689	+	0.837988i	$0.316268\pi$
$252$	1951.18i	0.487750i
$253$	1994.37i	0.495594i
$254$	1031.41i	0.254790i
$255$	− 983.396i	− 0.241500i
$256$	5336.40	1.30283
$257$	−4359.49	−1.05812	−0.529062	−	0.848583i	$-0.677456\pi$
$257$	−4359.49	−1.05812	−0.529062	+	0.848583i	$0.677456\pi$
$258$	− 3813.72i	− 0.920278i
$259$	5826.51	1.39785
$260$	0	0
$261$	975.068	0.231246
$262$	359.444i	0.0847576i
$263$	608.077	0.142569	0.0712844	−	0.997456i	$-0.477290\pi$
$263$	608.077	0.142569	0.0712844	+	0.997456i	$0.477290\pi$
$264$	−440.120	−0.102604
$265$	2863.92i	0.663884i
$266$	− 6372.43i	− 1.46887i
$267$	− 3203.80i	− 0.734342i
$268$	− 61.8938i	− 0.0141073i
$269$	3454.29	0.782942	0.391471	−	0.920190i	$-0.371966\pi$
$269$	3454.29	0.782942	0.391471	+	0.920190i	$0.371966\pi$
$270$	−394.130	−0.0888370
$271$	− 3703.72i	− 0.830204i	−0.909775	−	0.415102i	$-0.863746\pi$
$271$	− 3703.72i	− 0.830204i	0.909775	−	0.415102i	$-0.136254\pi$
$272$	−6390.47	−1.42456
$273$	0	0
$274$	−9780.92	−2.15652
$275$	− 2106.18i	− 0.461844i
$276$	1855.91	0.404755
$277$	3566.89	0.773696	0.386848	−	0.922144i	$-0.373564\pi$
$277$	3566.89	0.773696	0.386848	+	0.922144i	$0.373564\pi$
$278$	− 7334.04i	− 1.58225i
$279$	− 1327.79i	− 0.284920i
$280$	1087.58i	0.232127i
$281$	117.474i	0.0249392i	0.999922	+	0.0124696i	$0.00396929\pi$
$281$	117.474i	0.0249392i	−0.999922	+	0.0124696i	$0.996031\pi$
$282$	−1340.89	−0.283152
$283$	1737.62	0.364984	0.182492	−	0.983207i	$-0.441584\pi$
$283$	1737.62	0.364984	0.182492	+	0.983207i	$0.441584\pi$
$284$	5175.12i	1.08129i
$285$	549.226	0.114152
$286$	0	0
$287$	8429.23	1.73366
$288$	2010.89i	0.411434i
$289$	2123.51	0.432223
$290$	−1581.50	−0.320237
$291$	2576.64i	0.519057i
$292$	− 6543.74i	− 1.31145i
$293$	− 1904.05i	− 0.379643i	−0.981819	−	0.189822i	$-0.939209\pi$
$293$	− 1904.05i	− 0.379643i	0.981819	−	0.189822i	$-0.0607910\pi$
$294$	11014.9i	2.18503i
$295$	897.934	0.177219
$296$	1223.01	0.240156
$297$	− 518.245i	− 0.101251i
$298$	−1124.80	−0.218650
$299$	0	0
$300$	−1959.95	−0.377192
$301$	− 12391.8i	− 2.37293i
$302$	−1281.17	−0.244117
$303$	4722.49	0.895379
$304$	− 3569.08i	− 0.673358i
$305$	− 425.724i	− 0.0799242i
$306$	− 2820.13i	− 0.526850i
$307$	− 2862.39i	− 0.532134i	−0.963955	−	0.266067i	$-0.914276\pi$
$307$	− 2862.39i	− 0.532134i	0.963955	−	0.266067i	$-0.0857243\pi$
$308$	4161.29	0.769842
$309$	389.172	0.0716479
$310$	2153.58i	0.394566i
$311$	−4201.55	−0.766071	−0.383036	−	0.923734i	$-0.625121\pi$
$311$	−4201.55	−0.766071	−0.383036	+	0.923734i	$0.625121\pi$
$312$	0	0
$313$	3427.74	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$313$	3427.74	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$314$	4807.45i	0.864014i
$315$	−1280.64	−0.229066
$316$	−835.987	−0.148823
$317$	1676.09i	0.296966i	0.988915	+	0.148483i	$0.0474391\pi$
$317$	1676.09i	0.296966i	−0.988915	+	0.148483i	$0.952561\pi$
$318$	8212.99i	1.44831i
$319$	− 2079.52i	− 0.364987i
$320$	− 879.912i	− 0.153714i
$321$	−5872.30	−1.02106
$322$	14133.1	2.44599
$323$	3929.89i	0.676982i
$324$	−482.264	−0.0826928
$325$	0	0
$326$	−1989.37	−0.337979
$327$	3686.32i	0.623406i
$328$	1769.34	0.297851
$329$	−4356.93	−0.730108
$330$	840.560i	0.140216i
$331$	11156.6i	1.85264i	0.376740	+	0.926319i	$0.377045\pi$
$331$	11156.6i	1.85264i	−0.376740	+	0.926319i	$0.622955\pi$
$332$	949.490i	0.156958i
$333$	1440.11i	0.236990i
$334$	−156.720	−0.0256746
$335$	40.6233	0.00662534
$336$	8322.07i	1.35121i
$337$	−1636.44	−0.264517	−0.132259	−	0.991215i	$-0.542223\pi$
$337$	−1636.44	−0.264517	−0.132259	+	0.991215i	$0.542223\pi$
$338$	0	0
$339$	4888.49	0.783205
$340$	1951.67i	0.311307i
$341$	−2831.77	−0.449703
$342$	1575.04	0.249031
$343$	23300.7i	3.66799i
$344$	− 2601.10i	− 0.407681i
$345$	1218.10i	0.190088i
$346$	− 3983.88i	− 0.619002i
$347$	−2977.87	−0.460693	−0.230347	−	0.973109i	$-0.573986\pi$
$347$	−2977.87	−0.460693	−0.230347	+	0.973109i	$0.573986\pi$
$348$	−1935.15	−0.298088
$349$	− 9847.29i	− 1.51035i	−0.655521	−	0.755177i	$-0.727551\pi$
$349$	− 9847.29i	− 1.51035i	0.655521	−	0.755177i	$-0.272449\pi$
$350$	−14925.4	−2.27942
$351$	0	0
$352$	4288.62	0.649387
$353$	4687.34i	0.706747i	0.935482	+	0.353374i	$0.114966\pi$
$353$	4687.34i	0.706747i	−0.935482	+	0.353374i	$0.885034\pi$
$354$	2575.05	0.386616
$355$	−3396.63	−0.507816
$356$	6358.35i	0.946606i
$357$	− 9163.38i	− 1.35848i
$358$	11858.7i	1.75070i
$359$	2069.88i	0.304301i	0.988357	+	0.152151i	$0.0486199\pi$
$359$	2069.88i	0.304301i	−0.988357	+	0.152151i	$0.951380\pi$
$360$	−268.812	−0.0393546
$361$	4664.16	0.680005
$362$	− 10180.8i	− 1.47816i
$363$	2887.74	0.417540
$364$	0	0
$365$	4294.91	0.615906
$366$	− 1220.87i	− 0.174360i
$367$	−7299.16	−1.03818	−0.519092	−	0.854719i	$-0.673730\pi$
$367$	−7299.16	−1.03818	−0.519092	+	0.854719i	$0.673730\pi$
$368$	7915.70	1.12129
$369$	2083.41i	0.293924i
$370$	− 2335.76i	− 0.328191i
$371$	26686.3i	3.73446i
$372$	2635.16i	0.367277i
$373$	−8964.32	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$373$	−8964.32	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$374$	−6014.48	−0.831554
$375$	− 2751.80i	− 0.378940i
$376$	−914.541	−0.125436
$377$	0	0
$378$	−3672.54	−0.499723
$379$	− 4399.26i	− 0.596239i	−0.954529	−	0.298120i	$-0.903641\pi$
$379$	− 4399.26i	− 0.596239i	0.954529	−	0.298120i	$-0.0963594\pi$
$380$	−1090.01	−0.147148
$381$	−828.337	−0.111383
$382$	− 2931.94i	− 0.392700i
$383$	3529.74i	0.470917i	0.971884	+	0.235459i	$0.0756592\pi$
$383$	3529.74i	0.470917i	−0.971884	+	0.235459i	$0.924341\pi$
$384$	2839.01i	0.377286i
$385$	2731.21i	0.361547i
$386$	4691.28	0.618601
$387$	3062.82	0.402305
$388$	− 5113.68i	− 0.669092i
$389$	3034.77	0.395549	0.197775	−	0.980248i	$-0.436629\pi$
$389$	3034.77	0.395549	0.197775	+	0.980248i	$0.436629\pi$
$390$	0	0
$391$	−8715.93	−1.12732
$392$	7512.57i	0.967964i
$393$	−288.672	−0.0370523
$394$	−10374.8	−1.32658
$395$	− 548.690i	− 0.0698927i
$396$	1028.52i	0.130518i
$397$	3997.36i	0.505344i	0.967552	+	0.252672i	$0.0813094\pi$
$397$	3997.36i	0.505344i	−0.967552	+	0.252672i	$0.918691\pi$
$398$	5567.49i	0.701188i
$399$	5117.75	0.642125
$400$	−8359.44	−1.04493
$401$	9092.88i	1.13236i	0.824281	+	0.566181i	$0.191580\pi$
$401$	9092.88i	1.13236i	−0.824281	+	0.566181i	$0.808420\pi$
$402$	116.497	0.0144536
$403$	0	0
$404$	−9372.38	−1.15419
$405$	− 316.529i	− 0.0388357i
$406$	−14736.5	−1.80138
$407$	3071.32	0.374053
$408$	− 1923.44i	− 0.233393i
$409$	7143.54i	0.863631i	0.901962	+	0.431816i	$0.142127\pi$
$409$	7143.54i	0.863631i	−0.901962	+	0.431816i	$0.857873\pi$
$410$	− 3379.15i	− 0.407036i
$411$	− 7855.13i	− 0.942738i
$412$	−772.361	−0.0923579
$413$	8367.04	0.996889
$414$	3493.21i	0.414691i
$415$	−623.187	−0.0737134
$416$	0	0
$417$	5890.02	0.691692
$418$	− 3359.09i	− 0.393058i
$419$	8213.84	0.957691	0.478845	−	0.877899i	$-0.341055\pi$
$419$	8213.84	0.957691	0.478845	+	0.877899i	$0.341055\pi$
$420$	2541.59	0.295278
$421$	− 7997.40i	− 0.925818i	−0.886406	−	0.462909i	$-0.846806\pi$
$421$	− 7997.40i	− 0.925818i	0.886406	−	0.462909i	$-0.153194\pi$
$422$	− 8612.64i	− 0.993499i
$423$	− 1076.88i	− 0.123782i
$424$	5601.58i	0.641597i
$425$	9204.53	1.05055
$426$	−9740.68	−1.10783
$427$	− 3966.94i	− 0.449587i
$428$	11654.3	1.31620
$429$	0	0
$430$	−4967.70	−0.557125
$431$	− 13694.8i	− 1.53053i	−0.643718	−	0.765263i	$-0.722609\pi$
$431$	− 13694.8i	− 1.53053i	0.643718	−	0.765263i	$-0.277391\pi$
$432$	−2056.92	−0.229083
$433$	−6716.57	−0.745445	−0.372722	−	0.927943i	$-0.621576\pi$
$433$	−6716.57	−0.745445	−0.372722	+	0.927943i	$0.621576\pi$
$434$	20067.3i	2.21950i
$435$	− 1270.11i	− 0.139993i
$436$	− 7315.97i	− 0.803604i
$437$	− 4867.84i	− 0.532862i
$438$	12316.7	1.34364
$439$	5933.32	0.645061	0.322531	−	0.946559i	$-0.395466\pi$
$439$	5933.32	0.645061	0.322531	+	0.946559i	$0.395466\pi$
$440$	573.295i	0.0621154i
$441$	−8846.12	−0.955201
$442$	0	0
$443$	−6923.40	−0.742530	−0.371265	−	0.928527i	$-0.621076\pi$
$443$	−6923.40	−0.742530	−0.371265	+	0.928527i	$0.621076\pi$
$444$	− 2858.08i	− 0.305492i
$445$	−4173.23	−0.444562
$446$	4639.41	0.492561
$447$	− 903.333i	− 0.0955843i
$448$	− 8199.11i	− 0.864669i
$449$	− 8886.78i	− 0.934061i	−0.884241	−	0.467030i	$-0.845324\pi$
$449$	− 8886.78i	− 0.934061i	0.884241	−	0.467030i	$-0.154676\pi$
$450$	− 3689.04i	− 0.386451i
$451$	4443.28	0.463916
$452$	−9701.84	−1.00959
$453$	− 1028.92i	− 0.106717i
$454$	−6442.81	−0.666026
$455$	0	0
$456$	1074.24	0.110320
$457$	10965.0i	1.12237i	0.827691	+	0.561184i	$0.189654\pi$
$457$	10965.0i	1.12237i	−0.827691	+	0.561184i	$0.810346\pi$
$458$	12228.9	1.24764
$459$	2264.87	0.230316
$460$	− 2417.48i	− 0.245034i
$461$	10069.2i	1.01729i	0.860977	+	0.508644i	$0.169853\pi$
$461$	10069.2i	1.01729i	−0.860977	+	0.508644i	$0.830147\pi$
$462$	7832.42i	0.788739i
$463$	− 5599.72i	− 0.562076i	−0.959697	−	0.281038i	$-0.909321\pi$
$463$	− 5599.72i	− 0.562076i	0.959697	−	0.281038i	$-0.0906787\pi$
$464$	−8253.66	−0.825790
$465$	−1729.56	−0.172487
$466$	− 7954.78i	− 0.790769i
$467$	13247.8	1.31271	0.656355	−	0.754452i	$-0.272097\pi$
$467$	13247.8	1.31271	0.656355	+	0.754452i	$0.272097\pi$
$468$	0	0
$469$	378.532	0.0372686
$470$	1746.63i	0.171417i
$471$	−3860.90	−0.377709
$472$	1756.28	0.171270
$473$	− 6532.08i	− 0.634979i
$474$	− 1573.50i	− 0.152476i
$475$	5140.73i	0.496575i
$476$	18185.9i	1.75115i
$477$	−6595.92	−0.633137
$478$	−18985.2	−1.81666
$479$	− 16725.4i	− 1.59541i	−0.603045	−	0.797707i	$-0.706046\pi$
$479$	− 16725.4i	− 1.59541i	0.603045	−	0.797707i	$-0.293954\pi$
$480$	2619.36	0.249077
$481$	0	0
$482$	−17802.0	−1.68228
$483$	11350.4i	1.06928i
$484$	−5731.09	−0.538231
$485$	3356.30	0.314231
$486$	− 907.724i	− 0.0847226i
$487$	− 5305.86i	− 0.493699i	−0.969054	−	0.246850i	$-0.920605\pi$
$487$	− 5305.86i	− 0.493699i	0.969054	−	0.246850i	$-0.0793954\pi$
$488$	− 832.679i	− 0.0772410i
$489$	− 1597.68i	− 0.147750i
$490$	14347.8	1.32279
$491$	−16200.2	−1.48901	−0.744506	−	0.667616i	$-0.767315\pi$
$491$	−16200.2	−1.48901	−0.744506	+	0.667616i	$0.767315\pi$
$492$	− 4134.79i	− 0.378884i
$493$	9088.05	0.830234
$494$	0	0
$495$	−675.060	−0.0612963
$496$	11239.3i	1.01746i
$497$	−31650.2	−2.85655
$498$	−1787.14	−0.160811
$499$	4392.70i	0.394076i	0.980396	+	0.197038i	$0.0631323\pi$
$499$	4392.70i	0.394076i	−0.980396	+	0.197038i	$0.936868\pi$
$500$	5461.30i	0.488473i
$501$	− 125.863i	− 0.0112238i
$502$	− 16211.9i	− 1.44138i
$503$	−14955.2	−1.32568	−0.662841	−	0.748760i	$-0.730650\pi$
$503$	−14955.2	−1.32568	−0.662841	+	0.748760i	$0.730650\pi$
$504$	−2504.82	−0.221376
$505$	− 6151.46i	− 0.542052i
$506$	7449.96	0.654528
$507$	0	0
$508$	1643.94	0.143579
$509$	13403.4i	1.16719i	0.812047	+	0.583593i	$0.198353\pi$
$509$	13403.4i	1.16719i	−0.812047	+	0.583593i	$0.801647\pi$
$510$	−3673.46	−0.318948
$511$	40020.4	3.46458
$512$	− 12363.4i	− 1.06716i
$513$	1264.93i	0.108865i
$514$	16284.8i	1.39746i
$515$	− 506.930i	− 0.0433748i
$516$	−6078.57	−0.518593
$517$	−2296.66	−0.195371
$518$	− 21764.9i	− 1.84613i
$519$	3199.49	0.270601
$520$	0	0
$521$	19643.0	1.65178	0.825888	−	0.563834i	$-0.190674\pi$
$521$	19643.0	1.65178	0.825888	+	0.563834i	$0.190674\pi$
$522$	− 3642.35i	− 0.305405i
$523$	14657.4	1.22548	0.612738	−	0.790286i	$-0.290068\pi$
$523$	14657.4	1.22548	0.612738	+	0.790286i	$0.290068\pi$
$524$	572.906	0.0477624
$525$	− 11986.7i	− 0.996462i
$526$	− 2271.46i	− 0.188290i
$527$	− 12375.6i	− 1.02294i
$528$	4386.79i	0.361573i
$529$	−1370.83	−0.112668
$530$	10698.1	0.876788
$531$	2068.04i	0.169012i
$532$	−10156.8	−0.827733
$533$	0	0
$534$	−11967.8	−0.969842
$535$	7649.19i	0.618137i
$536$	79.4558	0.00640292
$537$	−9523.82	−0.765332
$538$	− 12903.4i	− 1.03403i
$539$	18866.1i	1.50764i
$540$	628.192i	0.0500612i
$541$	− 13921.3i	− 1.10633i	−0.833072	−	0.553164i	$-0.813420\pi$
$541$	− 13921.3i	− 1.10633i	0.833072	−	0.553164i	$-0.186580\pi$
$542$	−13835.2	−1.09645
$543$	8176.30	0.646185
$544$	18742.4i	1.47716i
$545$	4801.75	0.377403
$546$	0	0
$547$	−2324.11	−0.181667	−0.0908335	−	0.995866i	$-0.528953\pi$
$547$	−2324.11	−0.181667	−0.0908335	+	0.995866i	$0.528953\pi$
$548$	15589.5i	1.21524i
$549$	980.488	0.0762226
$550$	−7867.60	−0.609955
$551$	5075.68i	0.392434i
$552$	2382.51i	0.183707i
$553$	− 5112.75i	− 0.393158i
$554$	− 13324.1i	− 1.02182i
$555$	1875.87	0.143471
$556$	−11689.5	−0.891628
$557$	16962.8i	1.29037i	0.764027	+	0.645185i	$0.223220\pi$
$557$	16962.8i	1.29037i	−0.764027	+	0.645185i	$0.776780\pi$
$558$	−4959.94	−0.376292
$559$	0	0
$560$	10840.2	0.818006
$561$	− 4830.27i	− 0.363519i
$562$	438.823	0.0329370
$563$	389.000	0.0291197	0.0145599	−	0.999894i	$-0.495365\pi$
$563$	389.000	0.0291197	0.0145599	+	0.999894i	$0.495365\pi$
$564$	2137.21i	0.159561i
$565$	− 6367.69i	− 0.474143i
$566$	− 6490.85i	− 0.482033i
$567$	− 2949.45i	− 0.218457i
$568$	−6643.53	−0.490768
$569$	−2217.56	−0.163383	−0.0816914	−	0.996658i	$-0.526032\pi$
$569$	−2217.56	−0.163383	−0.0816914	+	0.996658i	$0.526032\pi$
$570$	− 2051.63i	− 0.150760i
$571$	17087.3	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$571$	17087.3	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$572$	0	0
$573$	2354.67	0.171671
$574$	− 31487.3i	− 2.28964i
$575$	−11401.4	−0.826906
$576$	2026.53	0.146595
$577$	− 3977.26i	− 0.286959i	−0.989653	−	0.143480i	$-0.954171\pi$
$577$	− 3977.26i	− 0.286959i	0.989653	−	0.143480i	$-0.0458291\pi$
$578$	− 7932.35i	− 0.570835i
$579$	3767.61i	0.270425i
$580$	2520.70i	0.180459i
$581$	−5806.92	−0.414650
$582$	9625.02	0.685516
$583$	14067.1i	0.999312i
$584$	8400.48	0.595230
$585$	0	0
$586$	−7112.54	−0.501393
$587$	− 16880.3i	− 1.18693i	−0.804861	−	0.593463i	$-0.797760\pi$
$587$	− 16880.3i	− 1.18693i	0.804861	−	0.593463i	$-0.202240\pi$
$588$	17556.2	1.23131
$589$	6911.75	0.483521
$590$	− 3354.22i	− 0.234053i
$591$	− 8332.06i	− 0.579924i
$592$	− 12190.1i	− 0.846301i
$593$	− 2423.25i	− 0.167810i	−0.996474	−	0.0839048i	$-0.973261\pi$
$593$	− 2423.25i	− 0.167810i	0.996474	−	0.0839048i	$-0.0267392\pi$
$594$	−1935.90	−0.133722
$595$	−11936.1	−0.822408
$596$	1792.78i	0.123213i
$597$	−4471.29	−0.306529
$598$	0	0
$599$	−3900.55	−0.266064	−0.133032	−	0.991112i	$-0.542471\pi$
$599$	−3900.55	−0.266064	−0.133032	+	0.991112i	$0.542471\pi$
$600$	− 2516.07i	− 0.171197i
$601$	28653.4	1.94476	0.972378	−	0.233413i	$-0.0749893\pi$
$601$	28653.4	1.94476	0.972378	+	0.233413i	$0.0749893\pi$
$602$	−46289.5	−3.13392
$603$	93.5599i	0.00631850i
$604$	2042.02i	0.137564i
$605$	− 3761.54i	− 0.252774i
$606$	− 17640.8i	− 1.18252i
$607$	214.736	0.0143589	0.00717946	−	0.999974i	$-0.497715\pi$
$607$	214.736	0.0143589	0.00717946	+	0.999974i	$0.497715\pi$
$608$	−10467.6	−0.698220
$609$	− 11835.0i	− 0.787487i
$610$	−1590.29	−0.105555
$611$	0	0
$612$	−4494.91	−0.296889
$613$	26438.5i	1.74199i	0.491290	+	0.870996i	$0.336525\pi$
$613$	26438.5i	1.74199i	−0.491290	+	0.870996i	$0.663475\pi$
$614$	−10692.4	−0.702787
$615$	2713.82	0.177938
$616$	5342.02i	0.349409i
$617$	6700.96i	0.437229i	0.975811	+	0.218615i	$0.0701538\pi$
$617$	6700.96i	0.437229i	−0.975811	+	0.218615i	$0.929846\pi$
$618$	− 1453.75i	− 0.0946250i
$619$	27319.1i	1.77391i	0.461860	+	0.886953i	$0.347182\pi$
$619$	27319.1i	1.77391i	−0.461860	+	0.886953i	$0.652818\pi$
$620$	3432.53	0.222345
$621$	−2805.42	−0.181285
$622$	15694.9i	1.01175i
$623$	−38886.6	−2.50074
$624$	0	0
$625$	10131.7	0.648429
$626$	− 12804.3i	− 0.817513i
$627$	2697.71	0.171828
$628$	7662.45	0.486887
$629$	13422.4i	0.850855i
$630$	4783.81i	0.302526i
$631$	7126.87i	0.449629i	0.974402	+	0.224815i	$0.0721776\pi$
$631$	7126.87i	0.449629i	−0.974402	+	0.224815i	$0.927822\pi$
$632$	− 1073.19i	− 0.0675463i
$633$	6916.88	0.434315
$634$	6261.00	0.392202
$635$	1078.98i	0.0674301i
$636$	13090.4	0.816147
$637$	0	0
$638$	−7768.04	−0.482037
$639$	− 7822.81i	− 0.484297i
$640$	3698.06	0.228404
$641$	−23615.0	−1.45513	−0.727565	−	0.686039i	$-0.759348\pi$
$641$	−23615.0	−1.45513	−0.727565	+	0.686039i	$0.759348\pi$
$642$	21935.9i	1.34851i
$643$	− 8144.41i	− 0.499509i	−0.968309	−	0.249755i	$-0.919650\pi$
$643$	− 8144.41i	− 0.499509i	0.968309	−	0.249755i	$-0.0803499\pi$
$644$	− 22526.3i	− 1.37836i
$645$	− 3989.60i	− 0.243551i
$646$	14680.1	0.894086
$647$	−9682.00	−0.588313	−0.294157	−	0.955757i	$-0.595039\pi$
$647$	−9682.00	−0.588313	−0.294157	+	0.955757i	$0.595039\pi$
$648$	− 619.103i	− 0.0375319i
$649$	4410.50	0.266760
$650$	0	0
$651$	−16116.2	−0.970268
$652$	3170.80i	0.190457i
$653$	−18193.6	−1.09030	−0.545152	−	0.838337i	$-0.683528\pi$
$653$	−18193.6	−1.09030	−0.545152	+	0.838337i	$0.683528\pi$
$654$	13770.2	0.823329
$655$	376.020i	0.0224310i
$656$	− 17635.5i	− 1.04962i
$657$	9891.64i	0.587381i
$658$	16275.3i	0.964250i
$659$	9300.88	0.549789	0.274895	−	0.961474i	$-0.411357\pi$
$659$	9300.88	0.549789	0.274895	+	0.961474i	$0.411357\pi$
$660$	1339.74	0.0790143
$661$	5437.29i	0.319949i	0.987121	+	0.159974i	$0.0511412\pi$
$661$	5437.29i	0.319949i	−0.987121	+	0.159974i	$0.948859\pi$
$662$	41675.4	2.44677
$663$	0	0
$664$	−1218.90	−0.0712388
$665$	− 6666.32i	− 0.388735i
$666$	5379.51	0.312991
$667$	−11257.1	−0.653489
$668$	249.791i	0.0144681i
$669$	3725.94i	0.215326i
$670$	− 151.748i	− 0.00875005i
$671$	− 2091.08i	− 0.120306i
$672$	24407.5	1.40110
$673$	8682.75	0.497319	0.248659	−	0.968591i	$-0.420010\pi$
$673$	8682.75	0.497319	0.248659	+	0.968591i	$0.420010\pi$
$674$	6112.89i	0.349347i
$675$	2962.69	0.168939
$676$	0	0
$677$	13300.1	0.755041	0.377521	−	0.926001i	$-0.376777\pi$
$677$	13300.1	0.755041	0.377521	+	0.926001i	$0.376777\pi$
$678$	− 18260.9i	− 1.03437i
$679$	31274.4	1.76760
$680$	−2505.45	−0.141293
$681$	− 5174.27i	− 0.291158i
$682$	10578.0i	0.593921i
$683$	504.175i	0.0282455i	0.999900	+	0.0141228i	$0.00449557\pi$
$683$	504.175i	0.0282455i	−0.999900	+	0.0141228i	$0.995504\pi$
$684$	− 2510.41i	− 0.140333i
$685$	−10232.0	−0.570722
$686$	87039.6	4.84429
$687$	9821.16i	0.545416i
$688$	−25925.9	−1.43665
$689$	0	0
$690$	4550.21	0.251049
$691$	13443.8i	0.740124i	0.929007	+	0.370062i	$0.120664\pi$
$691$	13443.8i	0.740124i	−0.929007	+	0.370062i	$0.879336\pi$
$692$	−6349.79	−0.348819
$693$	−6290.28	−0.344802
$694$	11123.8i	0.608435i
$695$	− 7672.27i	− 0.418742i
$696$	− 2484.23i	− 0.135294i
$697$	19418.3i	1.05527i
$698$	−36784.4	−1.99472
$699$	6388.55	0.345690
$700$	23789.1i	1.28449i
$701$	−28735.6	−1.54826	−0.774128	−	0.633030i	$-0.781811\pi$
$701$	−28735.6	−1.54826	−0.774128	+	0.633030i	$0.781811\pi$
$702$	0	0
$703$	−7496.44	−0.402181
$704$	− 4321.98i	− 0.231379i
$705$	−1402.73	−0.0749361
$706$	17509.5	0.933398
$707$	− 57319.9i	− 3.04913i
$708$	− 4104.29i	− 0.217865i
$709$	− 17610.2i	− 0.932812i	−0.884571	−	0.466406i	$-0.845549\pi$
$709$	− 17610.2i	− 0.932812i	0.884571	−	0.466406i	$-0.154451\pi$
$710$	12688.1i	0.670670i
$711$	1263.69	0.0666557
$712$	−8162.48	−0.429638
$713$	15329.3i	0.805168i
$714$	−34229.7	−1.79414
$715$	0	0
$716$	18901.2	0.986553
$717$	− 15247.1i	− 0.794163i
$718$	7732.02	0.401889
$719$	9226.04	0.478544	0.239272	−	0.970953i	$-0.423091\pi$
$719$	9226.04	0.478544	0.239272	+	0.970953i	$0.423091\pi$
$720$	2679.32i	0.138684i
$721$	− 4723.63i	− 0.243990i
$722$	− 17422.9i	− 0.898079i
$723$	− 14296.9i	− 0.735420i
$724$	−16226.9	−0.832967
$725$	11888.2	0.608987
$726$	− 10787.1i	− 0.551443i
$727$	−33246.0	−1.69604	−0.848022	−	0.529961i	$-0.822207\pi$
$727$	−33246.0	−1.69604	−0.848022	+	0.529961i	$0.822207\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 16043.6i	− 0.813424i
$731$	28546.9	1.44438
$732$	−1945.90	−0.0982549
$733$	− 4423.26i	− 0.222888i	−0.993771	−	0.111444i	$-0.964452\pi$
$733$	− 4423.26i	− 0.222888i	0.993771	−	0.111444i	$-0.0355476\pi$
$734$	27266.0i	1.37112i
$735$	11522.8i	0.578267i
$736$	− 23215.6i	− 1.16269i
$737$	199.535	0.00997281
$738$	7782.56	0.388184
$739$	4529.56i	0.225470i	0.993625	+	0.112735i	$0.0359612\pi$
$739$	4529.56i	0.225470i	−0.993625	+	0.112735i	$0.964039\pi$
$740$	−3722.90	−0.184941
$741$	0	0
$742$	99686.4	4.93208
$743$	10851.5i	0.535803i	0.963446	+	0.267901i	$0.0863301\pi$
$743$	10851.5i	0.535803i	−0.963446	+	0.267901i	$0.913670\pi$
$744$	−3382.87	−0.166696
$745$	−1176.67	−0.0578656
$746$	33486.1i	1.64345i
$747$	− 1435.27i	− 0.0702994i
$748$	9586.29i	0.468595i
$749$	71275.9i	3.47712i
$750$	−10279.3	−0.500464
$751$	33022.6	1.60454	0.802272	−	0.596958i	$-0.203624\pi$
$751$	33022.6	1.60454	0.802272	+	0.596958i	$0.203624\pi$
$752$	9115.48i	0.442031i
$753$	13019.9	0.630107
$754$	0	0
$755$	−1340.26	−0.0646053
$756$	5853.55i	0.281603i
$757$	−3443.77	−0.165345	−0.0826724	−	0.996577i	$-0.526346\pi$
$757$	−3443.77	−0.165345	−0.0826724	+	0.996577i	$0.526346\pi$
$758$	−16433.4	−0.787450
$759$	5983.12i	0.286131i
$760$	− 1399.29i	− 0.0667864i
$761$	− 19562.6i	− 0.931858i	−0.884822	−	0.465929i	$-0.845720\pi$
$761$	− 19562.6i	− 0.931858i	0.884822	−	0.465929i	$-0.154280\pi$
$762$	3094.24i	0.147103i
$763$	44743.2	2.12295
$764$	−4673.13	−0.221293
$765$	− 2950.19i	− 0.139430i
$766$	13185.3	0.621938
$767$	0	0
$768$	16009.2	0.752190
$769$	− 17061.1i	− 0.800049i	−0.916504	−	0.400025i	$-0.869002\pi$
$769$	− 17061.1i	− 0.800049i	0.916504	−	0.400025i	$-0.130998\pi$
$770$	10202.4	0.477493
$771$	−13078.5	−0.610908
$772$	− 7477.29i	− 0.348593i
$773$	10798.4i	0.502448i	0.967929	+	0.251224i	$0.0808331\pi$
$773$	10798.4i	0.502448i	−0.967929	+	0.251224i	$0.919167\pi$
$774$	− 11441.1i	− 0.531322i
$775$	− 16188.6i	− 0.750337i
$776$	6564.64	0.303682
$777$	17479.5	0.807046
$778$	− 11336.3i	− 0.522400i
$779$	−10845.1	−0.498802
$780$	0	0
$781$	−16683.7	−0.764391
$782$	32558.2i	1.48885i
$783$	2925.20	0.133510
$784$	74879.8	3.41107
$785$	5029.16i	0.228661i
$786$	1078.33i	0.0489348i
$787$	35607.0i	1.61277i	0.591390	+	0.806386i	$0.298580\pi$
$787$	35607.0i	1.61277i	−0.591390	+	0.806386i	$0.701420\pi$
$788$	16536.0i	0.747553i
$789$	1824.23	0.0823122
$790$	−2049.63	−0.0923069
$791$	− 59334.8i	− 2.66713i
$792$	−1320.36	−0.0592386
$793$	0	0
$794$	14932.1	0.667405
$795$	8591.76i	0.383293i
$796$	8873.85	0.395132
$797$	−22155.3	−0.984668	−0.492334	−	0.870406i	$-0.663856\pi$
$797$	−22155.3	−0.984668	−0.492334	+	0.870406i	$0.663856\pi$
$798$	− 19117.3i	− 0.848051i
$799$	− 10037.0i	− 0.444410i
$800$	24517.1i	1.08351i
$801$	− 9611.40i	− 0.423973i
$802$	33966.4	1.49550
$803$	21095.9	0.927094
$804$	− 185.681i	− 0.00814488i
$805$	14784.9	0.647329
$806$	0	0
$807$	10362.9	0.452032
$808$	− 12031.7i	− 0.523855i
$809$	−22524.6	−0.978889	−0.489445	−	0.872034i	$-0.662800\pi$
$809$	−22524.6	−0.978889	−0.489445	+	0.872034i	$0.662800\pi$
$810$	−1182.39	−0.0512901
$811$	4452.39i	0.192780i	0.995344	+	0.0963900i	$0.0307296\pi$
$811$	4452.39i	0.192780i	−0.995344	+	0.0963900i	$0.969270\pi$
$812$	23488.1i	1.01511i
$813$	− 11111.2i	− 0.479318i
$814$	− 11472.9i	− 0.494010i
$815$	−2081.12	−0.0894460
$816$	−19171.4	−0.822468
$817$	15943.4i	0.682729i
$818$	26684.6	1.14059
$819$	0	0
$820$	−5385.93	−0.229372
$821$	− 7097.27i	− 0.301701i	−0.988557	−	0.150851i	$-0.951799\pi$
$821$	− 7097.27i	− 0.301701i	0.988557	−	0.150851i	$-0.0482012\pi$
$822$	−29342.8	−1.24507
$823$	12193.8	0.516463	0.258231	−	0.966083i	$-0.416860\pi$
$823$	12193.8	0.516463	0.258231	+	0.966083i	$0.416860\pi$
$824$	− 991.512i	− 0.0419187i
$825$	− 6318.53i	− 0.266646i
$826$	− 31255.0i	− 1.31659i
$827$	− 7427.97i	− 0.312329i	−0.987731	−	0.156164i	$-0.950087\pi$
$827$	− 7427.97i	− 0.312329i	0.987731	−	0.156164i	$-0.0499130\pi$
$828$	5567.72	0.233686
$829$	−16966.2	−0.710810	−0.355405	−	0.934712i	$-0.615657\pi$
$829$	−16966.2	−0.710810	−0.355405	+	0.934712i	$0.615657\pi$
$830$	2327.91i	0.0973529i
$831$	10700.7	0.446693
$832$	0	0
$833$	−82449.7	−3.42943
$834$	− 22002.1i	− 0.913514i
$835$	−163.947	−0.00679477
$836$	−5353.94	−0.221495
$837$	− 3983.36i	− 0.164498i
$838$	− 30682.7i	− 1.26482i
$839$	− 12025.7i	− 0.494844i	−0.968908	−	0.247422i	$-0.920417\pi$
$839$	− 12025.7i	− 0.494844i	0.968908	−	0.247422i	$-0.0795834\pi$
$840$	3262.75i	0.134018i
$841$	−12651.3	−0.518728
$842$	−29874.2	−1.22272
$843$	352.422i	0.0143986i
$844$	−13727.4	−0.559855
$845$	0	0
$846$	−4022.68	−0.163478
$847$	− 35050.4i	− 1.42189i
$848$	55832.5	2.26096
$849$	5212.85	0.210724
$850$	− 34383.4i	− 1.38746i
$851$	− 16626.0i	− 0.669720i
$852$	15525.4i	0.624284i
$853$	22187.2i	0.890593i	0.895383	+	0.445297i	$0.146902\pi$
$853$	22187.2i	0.890593i	−0.895383	+	0.445297i	$0.853098\pi$
$854$	−14818.5	−0.593767
$855$	1647.68	0.0659058
$856$	14961.2i	0.597385i
$857$	−5746.19	−0.229038	−0.114519	−	0.993421i	$-0.536533\pi$
$857$	−5746.19	−0.229038	−0.114519	+	0.993421i	$0.536533\pi$
$858$	0	0
$859$	8305.66	0.329902	0.164951	−	0.986302i	$-0.447253\pi$
$859$	8305.66	0.329902	0.164951	+	0.986302i	$0.447253\pi$
$860$	7917.87i	0.313950i
$861$	25287.7	1.00093
$862$	−51156.9	−2.02136
$863$	38086.4i	1.50229i	0.660137	+	0.751146i	$0.270498\pi$
$863$	38086.4i	1.50229i	−0.660137	+	0.751146i	$0.729502\pi$
$864$	6032.67i	0.237541i
$865$	− 4167.61i	− 0.163819i
$866$	25089.7i	0.984505i
$867$	6370.53	0.249544
$868$	31984.7	1.25073
$869$	− 2695.07i	− 0.105206i
$870$	−4744.49	−0.184889
$871$	0	0
$872$	9391.82	0.364733
$873$	7729.93i	0.299677i
$874$	−18183.8	−0.703748
$875$	−33400.4	−1.29044
$876$	− 19631.2i	− 0.757166i
$877$	− 2098.53i	− 0.0808009i	−0.999184	−	0.0404005i	$-0.987137\pi$
$877$	− 2098.53i	− 0.0808009i	0.999184	−	0.0404005i	$-0.0128634\pi$
$878$	− 22163.8i	− 0.851929i
$879$	− 5712.14i	− 0.219187i
$880$	5714.18	0.218892
$881$	−14555.3	−0.556619	−0.278309	−	0.960491i	$-0.589774\pi$
$881$	−14555.3	−0.556619	−0.278309	+	0.960491i	$0.589774\pi$
$882$	33044.6i	1.26153i
$883$	−2122.88	−0.0809066	−0.0404533	−	0.999181i	$-0.512880\pi$
$883$	−2122.88	−0.0809066	−0.0404533	+	0.999181i	$0.512880\pi$
$884$	0	0
$885$	2693.80	0.102318
$886$	25862.3i	0.980656i
$887$	−12487.3	−0.472696	−0.236348	−	0.971668i	$-0.575951\pi$
$887$	−12487.3	−0.472696	−0.236348	+	0.971668i	$0.575951\pi$
$888$	3669.04	0.138654
$889$	10054.1i	0.379305i
$890$	15589.1i	0.587131i
$891$	− 1554.74i	− 0.0584575i
$892$	− 7394.61i	− 0.277567i
$893$	5605.66	0.210063
$894$	−3374.39	−0.126238
$895$	12405.6i	0.463323i
$896$	34458.9	1.28481
$897$	0	0
$898$	−33196.5	−1.23361
$899$	− 15983.7i	− 0.592978i
$900$	−5879.84	−0.217772
$901$	−61476.8	−2.27313
$902$	− 16597.8i	− 0.612691i
$903$	− 37175.5i	− 1.37001i
$904$	− 12454.7i	− 0.458226i
$905$	− 10650.3i	− 0.391193i
$906$	−3843.52	−0.140941
$907$	−29679.8	−1.08655	−0.543275	−	0.839555i	$-0.682816\pi$
$907$	−29679.8	−1.08655	−0.543275	+	0.839555i	$0.682816\pi$
$908$	10269.0i	0.375318i
$909$	14167.5	0.516948
$910$	0	0
$911$	24800.0	0.901934	0.450967	−	0.892541i	$-0.351079\pi$
$911$	24800.0	0.901934	0.450967	+	0.892541i	$0.351079\pi$
$912$	− 10707.2i	− 0.388763i
$913$	−3060.99	−0.110957
$914$	40959.7	1.48230
$915$	− 1277.17i	− 0.0461442i
$916$	− 19491.3i	− 0.703070i
$917$	3503.80i	0.126178i
$918$	− 8460.38i	− 0.304177i
$919$	−6597.90	−0.236828	−0.118414	−	0.992964i	$-0.537781\pi$
$919$	−6597.90	−0.236828	−0.118414	+	0.992964i	$0.537781\pi$
$920$	3103.43	0.111214
$921$	− 8587.17i	− 0.307228i
$922$	37613.4	1.34353
$923$	0	0
$924$	12483.9	0.444468
$925$	17558.0i	0.624113i
$926$	−20917.7	−0.742331
$927$	1167.51	0.0413659
$928$	24206.8i	0.856281i
$929$	15056.0i	0.531724i	0.964011	+	0.265862i	$0.0856565\pi$
$929$	15056.0i	0.531724i	−0.964011	+	0.265862i	$0.914344\pi$
$930$	6460.75i	0.227803i
$931$	− 46048.1i	− 1.62102i
$932$	−12678.9	−0.445612
$933$	−12604.7	−0.442291
$934$	− 49487.1i	− 1.73369i
$935$	−6291.85	−0.220070
$936$	0	0
$937$	−35777.0	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$937$	−35777.0	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$938$	− 1414.00i	− 0.0492205i
$939$	10283.2	0.357381
$940$	2783.90	0.0965966
$941$	− 22973.6i	− 0.795873i	−0.917413	−	0.397937i	$-0.869726\pi$
$941$	− 22973.6i	− 0.795873i	0.917413	−	0.397937i	$-0.130274\pi$
$942$	14422.4i	0.498838i
$943$	− 24052.9i	− 0.830615i
$944$	− 17505.3i	− 0.603549i
$945$	−3841.92	−0.132251
$946$	−24400.5	−0.838614
$947$	51038.3i	1.75134i	0.482908	+	0.875671i	$0.339580\pi$
$947$	51038.3i	1.75134i	−0.482908	+	0.875671i	$0.660420\pi$
$948$	−2507.96	−0.0859227
$949$	0	0
$950$	19203.1	0.655823
$951$	5028.26i	0.171454i
$952$	−23346.0	−0.794799
$953$	22586.5	0.767733	0.383866	−	0.923389i	$-0.374592\pi$
$953$	22586.5	0.767733	0.383866	+	0.923389i	$0.374592\pi$
$954$	24639.0i	0.836180i
$955$	− 3067.16i	− 0.103928i
$956$	30259.9i	1.02372i
$957$	− 6238.57i	− 0.210726i
$958$	−62477.6	−2.10706
$959$	−95342.8	−3.21041
$960$	− 2639.74i	− 0.0887470i
$961$	8025.32	0.269387
$962$	0	0
$963$	−17616.9	−0.589509
$964$	28374.1i	0.947996i
$965$	4907.64	0.163712
$966$	42399.4	1.41219
$967$	− 36678.7i	− 1.21976i	−0.792494	−	0.609880i	$-0.791218\pi$
$967$	− 36678.7i	− 1.21976i	0.792494	−	0.609880i	$-0.208782\pi$
$968$	− 7357.24i	− 0.244288i
$969$	11789.7i	0.390855i
$970$	− 12537.4i	− 0.415003i
$971$	32635.2	1.07859	0.539296	−	0.842116i	$-0.318690\pi$
$971$	32635.2	1.07859	0.539296	+	0.842116i	$0.318690\pi$
$972$	−1446.79	−0.0477427
$973$	− 71491.0i	− 2.35549i
$974$	−19820.0	−0.652026
$975$	0	0
$976$	−8299.54	−0.272194
$977$	44432.6i	1.45499i	0.686112	+	0.727496i	$0.259316\pi$
$977$	44432.6i	1.45499i	−0.686112	+	0.727496i	$0.740684\pi$
$978$	−5968.12	−0.195133
$979$	−20498.2	−0.669178
$980$	− 22868.6i	− 0.745418i
$981$	11059.0i	0.359924i
$982$	60515.6i	1.96653i
$983$	− 484.485i	− 0.0157199i	−0.999969	−	0.00785996i	$-0.997498\pi$
$983$	− 484.485i	− 0.0157199i	0.999969	−	0.00785996i	$-0.00250193\pi$
$984$	5308.01	0.171965
$985$	−10853.2	−0.351079
$986$	− 33948.3i	− 1.09649i
$987$	−13070.8	−0.421528
$988$	0	0
$989$	−35360.2	−1.13689
$990$	2521.68i	0.0809538i
$991$	−48017.1	−1.53917	−0.769583	−	0.638546i	$-0.779536\pi$
$991$	−48017.1	−1.53917	−0.769583	+	0.638546i	$0.779536\pi$
$992$	32963.4	1.05503
$993$	33469.8i	1.06962i
$994$	118229.i	3.77263i
$995$	5824.25i	0.185569i
$996$	2848.47i	0.0906197i
$997$	−26561.9	−0.843755	−0.421877	−	0.906653i	$-0.638629\pi$
$997$	−26561.9	−0.843755	−0.421877	+	0.906653i	$0.638629\pi$
$998$	16408.9	0.520455
$999$	4320.33i	0.136826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.g.337.2		6
13.5	odd	4		39.4.a.c.1.1	✓	3
13.8	odd	4		507.4.a.h.1.3		3
13.12	even	2	inner	507.4.b.g.337.5		6
39.5	even	4		117.4.a.f.1.3		3
39.8	even	4		1521.4.a.u.1.1		3
52.31	even	4		624.4.a.t.1.2		3
65.44	odd	4		975.4.a.l.1.3		3
91.83	even	4		1911.4.a.k.1.1		3
104.5	odd	4		2496.4.a.bl.1.2		3
104.83	even	4		2496.4.a.bp.1.2		3
156.83	odd	4		1872.4.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.1	✓	3	13.5	odd	4
117.4.a.f.1.3		3	39.5	even	4
507.4.a.h.1.3		3	13.8	odd	4
507.4.b.g.337.2		6	1.1	even	1	trivial
507.4.b.g.337.5		6	13.12	even	2	inner
624.4.a.t.1.2		3	52.31	even	4
975.4.a.l.1.3		3	65.44	odd	4
1521.4.a.u.1.1		3	39.8	even	4
1872.4.a.bk.1.2		3	156.83	odd	4
1911.4.a.k.1.1		3	91.83	even	4
2496.4.a.bl.1.2		3	104.5	odd	4
2496.4.a.bp.1.2		3	104.83	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)

\(f(q)\)	\(=\)	\(q-3.73549i q^{2} +3.00000 q^{3} -5.95388 q^{4} -3.90776i q^{5} -11.2065i q^{6} -36.4129i q^{7} -7.64325i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.73549i q^{2} +3.00000 q^{3} -5.95388 q^{4} -3.90776i q^{5} -11.2065i q^{6} -36.4129i q^{7} -7.64325i q^{8} +9.00000 q^{9} -14.5974 q^{10} -19.1943i q^{11} -17.8616 q^{12} -136.020 q^{14} -11.7233i q^{15} -76.1823 q^{16} +83.8839 q^{17} -33.6194i q^{18} +46.8492i q^{19} +23.2664i q^{20} -109.239i q^{21} -71.7000 q^{22} -103.905 q^{23} -22.9298i q^{24} +109.729 q^{25} +27.0000 q^{27} +216.798i q^{28} +108.341 q^{29} -43.7922 q^{30} -147.532i q^{31} +223.432i q^{32} -57.5828i q^{33} -313.347i q^{34} -142.293 q^{35} -53.5849 q^{36} +160.012i q^{37} +175.005 q^{38} -29.8680 q^{40} +231.490i q^{41} -408.060 q^{42} +340.314 q^{43} +114.280i q^{44} -35.1699i q^{45} +388.135i q^{46} -119.653i q^{47} -228.547 q^{48} -982.902 q^{49} -409.893i q^{50} +251.652 q^{51} -732.879 q^{53} -100.858i q^{54} -75.0067 q^{55} -278.313 q^{56} +140.548i q^{57} -404.706i q^{58} +229.782i q^{59} +69.7991i q^{60} +108.943 q^{61} -551.104 q^{62} -327.716i q^{63} +225.170 q^{64} -215.100 q^{66} +10.3955i q^{67} -499.435 q^{68} -311.714 q^{69} +531.535i q^{70} -869.201i q^{71} -68.7893i q^{72} +1099.07i q^{73} +597.724 q^{74} +329.188 q^{75} -278.934i q^{76} -698.920 q^{77} +140.410 q^{79} +297.703i q^{80} +81.0000 q^{81} +864.729 q^{82} -159.474i q^{83} +650.395i q^{84} -327.799i q^{85} -1271.24i q^{86} +325.023 q^{87} -146.707 q^{88} -1067.93i q^{89} -131.377 q^{90} +618.636 q^{92} -442.596i q^{93} -446.964 q^{94} +183.075 q^{95} +670.297i q^{96} +858.881i q^{97} +3671.62i q^{98} -172.748i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) \) 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9	\( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) \) 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 + 8 * q^10 - 60 * q^12 - 352 * q^14 - 220 * q^16 + 292 * q^17 - 112 * q^22 + 96 * q^23 - 290 * q^25 + 162 * q^27 - 4 * q^29 + 24 * q^30 + 160 * q^35 - 180 * q^36 - 624 * q^38 - 1032 * q^40 - 1056 * q^42 + 520 * q^43 - 660 * q^48 - 894 * q^49 + 876 * q^51 - 1356 * q^53 - 3104 * q^55 - 192 * q^56 + 460 * q^61 - 3904 * q^62 + 1500 * q^64 - 336 * q^66 - 920 * q^68 + 288 * q^69 + 3448 * q^74 - 870 * q^75 - 224 * q^77 - 48 * q^79 + 486 * q^81 - 1128 * q^82 - 12 * q^87 - 3216 * q^88 + 72 * q^90 - 2592 * q^92 - 3840 * q^94 + 8064 * q^95

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