LMFDB - Embedded newform 507.4.b.c.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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.c.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +27.0000 q^{10} -30.0000i q^{11} -3.00000 q^{12} +6.00000 q^{14} -27.0000i q^{15} -71.0000 q^{16} +111.000 q^{17} +27.0000i q^{18} -46.0000i q^{19} +9.00000i q^{20} -6.00000i q^{21} +90.0000 q^{22} +6.00000 q^{23} +63.0000i q^{24} +44.0000 q^{25} +27.0000 q^{27} +2.00000i q^{28} -105.000 q^{29} +81.0000 q^{30} -100.000i q^{31} -45.0000i q^{32} -90.0000i q^{33} +333.000i q^{34} -18.0000 q^{35} -9.00000 q^{36} -17.0000i q^{37} +138.000 q^{38} +189.000 q^{40} -231.000i q^{41} +18.0000 q^{42} +514.000 q^{43} +30.0000i q^{44} -81.0000i q^{45} +18.0000i q^{46} +162.000i q^{47} -213.000 q^{48} +339.000 q^{49} +132.000i q^{50} +333.000 q^{51} +639.000 q^{53} +81.0000i q^{54} -270.000 q^{55} +42.0000 q^{56} -138.000i q^{57} -315.000i q^{58} -600.000i q^{59} +27.0000i q^{60} +233.000 q^{61} +300.000 q^{62} -18.0000i q^{63} -433.000 q^{64} +270.000 q^{66} +926.000i q^{67} -111.000 q^{68} +18.0000 q^{69} -54.0000i q^{70} -930.000i q^{71} +189.000i q^{72} +253.000i q^{73} +51.0000 q^{74} +132.000 q^{75} +46.0000i q^{76} -60.0000 q^{77} -1324.00 q^{79} +639.000i q^{80} +81.0000 q^{81} +693.000 q^{82} +810.000i q^{83} +6.00000i q^{84} -999.000i q^{85} +1542.00i q^{86} -315.000 q^{87} +630.000 q^{88} -498.000i q^{89} +243.000 q^{90} -6.00000 q^{92} -300.000i q^{93} -486.000 q^{94} -414.000 q^{95} -135.000i q^{96} +1358.00i q^{97} +1017.00i q^{98} -270.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9	$ 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9} + 54 q^{10} - 6 q^{12} + 12 q^{14} - 142 q^{16} + 222 q^{17} + 180 q^{22} + 12 q^{23} + 88 q^{25} + 54 q^{27} - 210 q^{29} + 162 q^{30} - 36 q^{35} - 18 q^{36} + 276 q^{38} + 378 q^{40} + 36 q^{42} + 1028 q^{43} - 426 q^{48} + 678 q^{49} + 666 q^{51} + 1278 q^{53} - 540 q^{55} + 84 q^{56} + 466 q^{61} + 600 q^{62} - 866 q^{64} + 540 q^{66} - 222 q^{68} + 36 q^{69} + 102 q^{74} + 264 q^{75} - 120 q^{77} - 2648 q^{79} + 162 q^{81} + 1386 q^{82} - 630 q^{87} + 1260 q^{88} + 486 q^{90} - 12 q^{92} - 972 q^{94} - 828 q^{95}+O(q^{100}) $ 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9 + 54 * q^10 - 6 * q^12 + 12 * q^14 - 142 * q^16 + 222 * q^17 + 180 * q^22 + 12 * q^23 + 88 * q^25 + 54 * q^27 - 210 * q^29 + 162 * q^30 - 36 * q^35 - 18 * q^36 + 276 * q^38 + 378 * q^40 + 36 * q^42 + 1028 * q^43 - 426 * q^48 + 678 * q^49 + 666 * q^51 + 1278 * q^53 - 540 * q^55 + 84 * q^56 + 466 * q^61 + 600 * q^62 - 866 * q^64 + 540 * q^66 - 222 * q^68 + 36 * q^69 + 102 * q^74 + 264 * q^75 - 120 * q^77 - 2648 * q^79 + 162 * q^81 + 1386 * q^82 - 630 * q^87 + 1260 * q^88 + 486 * q^90 - 12 * q^92 - 972 * q^94 - 828 * 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 $

$2$	3.00000i	1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$	3.00000i	1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−1.00000	−0.125000
$5$	− 9.00000i	− 0.804984i	−0.915423	−	0.402492i	$-0.868144\pi$
$5$	− 9.00000i	− 0.804984i	0.915423	−	0.402492i	$-0.131856\pi$
$6$	9.00000i	0.612372i
$7$	− 2.00000i	− 0.107990i	−0.998541	−	0.0539949i	$-0.982805\pi$
$7$	− 2.00000i	− 0.107990i	0.998541	−	0.0539949i	$-0.0171955\pi$
$8$	21.0000i	0.928078i
$9$	9.00000	0.333333
$10$	27.0000	0.853815
$11$	− 30.0000i	− 0.822304i	−0.911567	−	0.411152i	$-0.865127\pi$
$11$	− 30.0000i	− 0.822304i	0.911567	−	0.411152i	$-0.134873\pi$
$12$	−3.00000	−0.0721688
$13$	0	0
$13$	0	0
$14$	6.00000	0.114541
$15$	− 27.0000i	− 0.464758i
$16$	−71.0000	−1.10938
$17$	111.000	1.58361	0.791807	−	0.610771i	$-0.209140\pi$
$17$	111.000	1.58361	0.791807	+	0.610771i	$0.209140\pi$
$18$	27.0000i	0.353553i
$19$	− 46.0000i	− 0.555428i	−0.960664	−	0.277714i	$-0.910423\pi$
$19$	− 46.0000i	− 0.555428i	0.960664	−	0.277714i	$-0.0895767\pi$
$20$	9.00000i	0.100623i
$21$	− 6.00000i	− 0.0623480i
$22$	90.0000	0.872185
$23$	6.00000	0.0543951	0.0271975	−	0.999630i	$-0.491342\pi$
$23$	6.00000	0.0543951	0.0271975	+	0.999630i	$0.491342\pi$
$24$	63.0000i	0.535826i
$25$	44.0000	0.352000
$26$	0	0
$27$	27.0000	0.192450
$28$	2.00000i	0.0134987i
$29$	−105.000	−0.672345	−0.336173	−	0.941800i	$-0.609133\pi$
$29$	−105.000	−0.672345	−0.336173	+	0.941800i	$0.609133\pi$
$30$	81.0000	0.492950
$31$	− 100.000i	− 0.579372i	−0.957122	−	0.289686i	$-0.906449\pi$
$31$	− 100.000i	− 0.579372i	0.957122	−	0.289686i	$-0.0935509\pi$
$32$	− 45.0000i	− 0.248592i
$33$	− 90.0000i	− 0.474757i
$34$	333.000i	1.67968i
$35$	−18.0000	−0.0869302
$36$	−9.00000	−0.0416667
$37$	− 17.0000i	− 0.0755347i	−0.999287	−	0.0377673i	$-0.987975\pi$
$37$	− 17.0000i	− 0.0755347i	0.999287	−	0.0377673i	$-0.0120246\pi$
$38$	138.000	0.589120
$39$	0	0
$40$	189.000	0.747088
$41$	− 231.000i	− 0.879906i	−0.898021	−	0.439953i	$-0.854995\pi$
$41$	− 231.000i	− 0.879906i	0.898021	−	0.439953i	$-0.145005\pi$
$42$	18.0000	0.0661300
$43$	514.000	1.82289	0.911445	−	0.411422i	$-0.134968\pi$
$43$	514.000	1.82289	0.911445	+	0.411422i	$0.134968\pi$
$44$	30.0000i	0.102788i
$45$	− 81.0000i	− 0.268328i
$46$	18.0000i	0.0576947i
$47$	162.000i	0.502769i	0.967887	+	0.251384i	$0.0808858\pi$
$47$	162.000i	0.502769i	−0.967887	+	0.251384i	$0.919114\pi$
$48$	−213.000	−0.640498
$49$	339.000	0.988338
$50$	132.000i	0.373352i
$51$	333.000	0.914301
$52$	0	0
$53$	639.000	1.65610	0.828051	−	0.560653i	$-0.189450\pi$
$53$	639.000	1.65610	0.828051	+	0.560653i	$0.189450\pi$
$54$	81.0000i	0.204124i
$55$	−270.000	−0.661942
$56$	42.0000	0.100223
$57$	− 138.000i	− 0.320676i
$58$	− 315.000i	− 0.713130i
$59$	− 600.000i	− 1.32396i	−0.749524	−	0.661978i	$-0.769717\pi$
$59$	− 600.000i	− 1.32396i	0.749524	−	0.661978i	$-0.230283\pi$
$60$	27.0000i	0.0580948i
$61$	233.000	0.489059	0.244529	−	0.969642i	$-0.421367\pi$
$61$	233.000	0.489059	0.244529	+	0.969642i	$0.421367\pi$
$62$	300.000	0.614517
$63$	− 18.0000i	− 0.0359966i
$64$	−433.000	−0.845703
$65$	0	0
$66$	270.000	0.503556
$67$	926.000i	1.68849i	0.535957	+	0.844246i	$0.319951\pi$
$67$	926.000i	1.68849i	−0.535957	+	0.844246i	$0.680049\pi$
$68$	−111.000	−0.197952
$69$	18.0000	0.0314050
$70$	− 54.0000i	− 0.0922033i
$71$	− 930.000i	− 1.55452i	−0.629182	−	0.777258i	$-0.716610\pi$
$71$	− 930.000i	− 1.55452i	0.629182	−	0.777258i	$-0.283390\pi$
$72$	189.000i	0.309359i
$73$	253.000i	0.405636i	0.979216	+	0.202818i	$0.0650099\pi$
$73$	253.000i	0.405636i	−0.979216	+	0.202818i	$0.934990\pi$
$74$	51.0000	0.0801166
$75$	132.000	0.203227
$76$	46.0000i	0.0694284i
$77$	−60.0000	−0.0888004
$78$	0	0
$79$	−1324.00	−1.88559	−0.942795	−	0.333373i	$-0.891813\pi$
$79$	−1324.00	−1.88559	−0.942795	+	0.333373i	$0.891813\pi$
$80$	639.000i	0.893030i
$81$	81.0000	0.111111
$82$	693.000	0.933281
$83$	810.000i	1.07119i	0.844474	+	0.535597i	$0.179913\pi$
$83$	810.000i	1.07119i	−0.844474	+	0.535597i	$0.820087\pi$
$84$	6.00000i	0.00779350i
$85$	− 999.000i	− 1.27479i
$86$	1542.00i	1.93347i
$87$	−315.000	−0.388179
$88$	630.000	0.763162
$89$	− 498.000i	− 0.593122i	−0.955014	−	0.296561i	$-0.904160\pi$
$89$	− 498.000i	− 0.593122i	0.955014	−	0.296561i	$-0.0958399\pi$
$90$	243.000	0.284605
$91$	0	0
$92$	−6.00000	−0.00679938
$93$	− 300.000i	− 0.334501i
$94$	−486.000	−0.533267
$95$	−414.000	−0.447111
$96$	− 135.000i	− 0.143525i
$97$	1358.00i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	1358.00i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	1017.00i	1.04829i
$99$	− 270.000i	− 0.274101i
$100$	−44.0000	−0.0440000
$101$	357.000	0.351711	0.175856	−	0.984416i	$-0.443731\pi$
$101$	357.000	0.351711	0.175856	+	0.984416i	$0.443731\pi$
$102$	999.000i	0.969762i
$103$	−1118.00	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$103$	−1118.00	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$104$	0	0
$105$	−54.0000	−0.0501891
$106$	1917.00i	1.75656i
$107$	714.000	0.645093	0.322547	−	0.946554i	$-0.395461\pi$
$107$	714.000	0.645093	0.322547	+	0.946554i	$0.395461\pi$
$108$	−27.0000	−0.0240563
$109$	2006.00i	1.76275i	0.472416	+	0.881376i	$0.343382\pi$
$109$	2006.00i	1.76275i	−0.472416	+	0.881376i	$0.656618\pi$
$110$	− 810.000i	− 0.702095i
$111$	− 51.0000i	− 0.0436100i
$112$	142.000i	0.119801i
$113$	−1119.00	−0.931563	−0.465782	−	0.884900i	$-0.654227\pi$
$113$	−1119.00	−0.931563	−0.465782	+	0.884900i	$0.654227\pi$
$114$	414.000	0.340129
$115$	− 54.0000i	− 0.0437872i
$116$	105.000	0.0840431
$117$	0	0
$118$	1800.00	1.40427
$119$	− 222.000i	− 0.171014i
$120$	567.000	0.431332
$121$	431.000	0.323817
$122$	699.000i	0.518725i
$123$	− 693.000i	− 0.508014i
$124$	100.000i	0.0724215i
$125$	− 1521.00i	− 1.08834i
$126$	54.0000	0.0381802
$127$	604.000	0.422018	0.211009	−	0.977484i	$-0.432325\pi$
$127$	604.000	0.422018	0.211009	+	0.977484i	$0.432325\pi$
$128$	− 1659.00i	− 1.14560i
$129$	1542.00	1.05245
$130$	0	0
$131$	−1584.00	−1.05645	−0.528224	−	0.849105i	$-0.677142\pi$
$131$	−1584.00	−1.05645	−0.528224	+	0.849105i	$0.677142\pi$
$132$	90.0000i	0.0593447i
$133$	−92.0000	−0.0599805
$134$	−2778.00	−1.79092
$135$	− 243.000i	− 0.154919i
$136$	2331.00i	1.46972i
$137$	− 717.000i	− 0.447135i	−0.974688	−	0.223567i	$-0.928230\pi$
$137$	− 717.000i	− 0.447135i	0.974688	−	0.223567i	$-0.0717703\pi$
$138$	54.0000i	0.0333100i
$139$	−820.000	−0.500370	−0.250185	−	0.968198i	$-0.580492\pi$
$139$	−820.000	−0.500370	−0.250185	+	0.968198i	$0.580492\pi$
$140$	18.0000	0.0108663
$141$	486.000i	0.290274i
$142$	2790.00	1.64881
$143$	0	0
$144$	−639.000	−0.369792
$145$	945.000i	0.541227i
$146$	−759.000	−0.430242
$147$	1017.00	0.570617
$148$	17.0000i	0.00944183i
$149$	− 1749.00i	− 0.961635i	−0.876821	−	0.480818i	$-0.840340\pi$
$149$	− 1749.00i	− 0.961635i	0.876821	−	0.480818i	$-0.159660\pi$
$150$	396.000i	0.215555i
$151$	370.000i	0.199405i	0.995017	+	0.0997026i	$0.0317891\pi$
$151$	370.000i	0.199405i	−0.995017	+	0.0997026i	$0.968211\pi$
$152$	966.000	0.515480
$153$	999.000	0.527872
$154$	− 180.000i	− 0.0941871i
$155$	−900.000	−0.466385
$156$	0	0
$157$	−2611.00	−1.32726	−0.663632	−	0.748059i	$-0.730986\pi$
$157$	−2611.00	−1.32726	−0.663632	+	0.748059i	$0.730986\pi$
$158$	− 3972.00i	− 1.99997i
$159$	1917.00	0.956151
$160$	−405.000	−0.200113
$161$	− 12.0000i	− 0.00587411i
$162$	243.000i	0.117851i
$163$	1636.00i	0.786144i	0.919508	+	0.393072i	$0.128588\pi$
$163$	1636.00i	0.786144i	−0.919508	+	0.393072i	$0.871412\pi$
$164$	231.000i	0.109988i
$165$	−810.000	−0.382172
$166$	−2430.00	−1.13617
$167$	− 264.000i	− 0.122329i	−0.998128	−	0.0611645i	$-0.980519\pi$
$167$	− 264.000i	− 0.122329i	0.998128	−	0.0611645i	$-0.0194814\pi$
$168$	126.000	0.0578638
$169$	0	0
$170$	2997.00	1.35211
$171$	− 414.000i	− 0.185143i
$172$	−514.000	−0.227861
$173$	−1410.00	−0.619655	−0.309827	−	0.950793i	$-0.600271\pi$
$173$	−1410.00	−0.619655	−0.309827	+	0.950793i	$0.600271\pi$
$174$	− 945.000i	− 0.411726i
$175$	− 88.0000i	− 0.0380124i
$176$	2130.00i	0.912243i
$177$	− 1800.00i	− 0.764386i
$178$	1494.00	0.629101
$179$	474.000	0.197924	0.0989621	−	0.995091i	$-0.468448\pi$
$179$	474.000	0.197924	0.0989621	+	0.995091i	$0.468448\pi$
$180$	81.0000i	0.0335410i
$181$	−2249.00	−0.923574	−0.461787	−	0.886991i	$-0.652792\pi$
$181$	−2249.00	−0.923574	−0.461787	+	0.886991i	$0.652792\pi$
$182$	0	0
$183$	699.000	0.282358
$184$	126.000i	0.0504828i
$185$	−153.000	−0.0608042
$186$	900.000	0.354791
$187$	− 3330.00i	− 1.30221i
$188$	− 162.000i	− 0.0628461i
$189$	− 54.0000i	− 0.0207827i
$190$	− 1242.00i	− 0.474232i
$191$	3444.00	1.30471	0.652354	−	0.757915i	$-0.273782\pi$
$191$	3444.00	1.30471	0.652354	+	0.757915i	$0.273782\pi$
$192$	−1299.00	−0.488267
$193$	4273.00i	1.59366i	0.604201	+	0.796832i	$0.293493\pi$
$193$	4273.00i	1.59366i	−0.604201	+	0.796832i	$0.706507\pi$
$194$	−4074.00	−1.50771
$195$	0	0
$196$	−339.000	−0.123542
$197$	− 1986.00i	− 0.718257i	−0.933288	−	0.359129i	$-0.883074\pi$
$197$	− 1986.00i	− 0.718257i	0.933288	−	0.359129i	$-0.116926\pi$
$198$	810.000	0.290728
$199$	2386.00	0.849945	0.424973	−	0.905206i	$-0.360284\pi$
$199$	2386.00	0.849945	0.424973	+	0.905206i	$0.360284\pi$
$200$	924.000i	0.326683i
$201$	2778.00i	0.974851i
$202$	1071.00i	0.373046i
$203$	210.000i	0.0726065i
$204$	−333.000	−0.114288
$205$	−2079.00	−0.708311
$206$	− 3354.00i	− 1.13439i
$207$	54.0000	0.0181317
$208$	0	0
$209$	−1380.00	−0.456730
$210$	− 162.000i	− 0.0532336i
$211$	−1600.00	−0.522031	−0.261016	−	0.965335i	$-0.584057\pi$
$211$	−1600.00	−0.522031	−0.261016	+	0.965335i	$0.584057\pi$
$212$	−639.000	−0.207013
$213$	− 2790.00i	− 0.897501i
$214$	2142.00i	0.684225i
$215$	− 4626.00i	− 1.46740i
$216$	567.000i	0.178609i
$217$	−200.000	−0.0625663
$218$	−6018.00	−1.86968
$219$	759.000i	0.234194i
$220$	270.000	0.0827427
$221$	0	0
$222$	153.000	0.0462553
$223$	− 3832.00i	− 1.15072i	−0.817902	−	0.575358i	$-0.804863\pi$
$223$	− 3832.00i	− 1.15072i	0.817902	−	0.575358i	$-0.195137\pi$
$224$	−90.0000	−0.0268454
$225$	396.000	0.117333
$226$	− 3357.00i	− 0.988072i
$227$	− 1398.00i	− 0.408760i	−0.978892	−	0.204380i	$-0.934482\pi$
$227$	− 1398.00i	− 0.408760i	0.978892	−	0.204380i	$-0.0655178\pi$
$228$	138.000i	0.0400845i
$229$	− 4466.00i	− 1.28874i	−0.764714	−	0.644370i	$-0.777120\pi$
$229$	− 4466.00i	− 1.28874i	0.764714	−	0.644370i	$-0.222880\pi$
$230$	162.000	0.0464433
$231$	−180.000	−0.0512690
$232$	− 2205.00i	− 0.623989i
$233$	1638.00	0.460553	0.230277	−	0.973125i	$-0.426037\pi$
$233$	1638.00	0.460553	0.230277	+	0.973125i	$0.426037\pi$
$234$	0	0
$235$	1458.00	0.404721
$236$	600.000i	0.165494i
$237$	−3972.00	−1.08865
$238$	666.000	0.181388
$239$	− 594.000i	− 0.160764i	−0.996764	−	0.0803821i	$-0.974386\pi$
$239$	− 594.000i	− 0.160764i	0.996764	−	0.0803821i	$-0.0256141\pi$
$240$	1917.00i	0.515591i
$241$	− 2303.00i	− 0.615557i	−0.951458	−	0.307779i	$-0.900414\pi$
$241$	− 2303.00i	− 0.615557i	0.951458	−	0.307779i	$-0.0995856\pi$
$242$	1293.00i	0.343459i
$243$	243.000	0.0641500
$244$	−233.000	−0.0611324
$245$	− 3051.00i	− 0.795597i
$246$	2079.00	0.538830
$247$	0	0
$248$	2100.00	0.537702
$249$	2430.00i	0.618454i
$250$	4563.00	1.15436
$251$	−6324.00	−1.59031	−0.795154	−	0.606407i	$-0.792610\pi$
$251$	−6324.00	−1.59031	−0.795154	+	0.606407i	$0.792610\pi$
$252$	18.0000i	0.00449958i
$253$	− 180.000i	− 0.0447293i
$254$	1812.00i	0.447618i
$255$	− 2997.00i	− 0.735998i
$256$	1513.00	0.369385
$257$	−7833.00	−1.90120	−0.950601	−	0.310414i	$-0.899532\pi$
$257$	−7833.00	−1.90120	−0.950601	+	0.310414i	$0.899532\pi$
$258$	4626.00i	1.11629i
$259$	−34.0000	−0.00815698
$260$	0	0
$261$	−945.000	−0.224115
$262$	− 4752.00i	− 1.12053i
$263$	−3030.00	−0.710410	−0.355205	−	0.934788i	$-0.615589\pi$
$263$	−3030.00	−0.710410	−0.355205	+	0.934788i	$0.615589\pi$
$264$	1890.00	0.440612
$265$	− 5751.00i	− 1.33314i
$266$	− 276.000i	− 0.0636190i
$267$	− 1494.00i	− 0.342439i
$268$	− 926.000i	− 0.211061i
$269$	−534.000	−0.121036	−0.0605178	−	0.998167i	$-0.519275\pi$
$269$	−534.000	−0.121036	−0.0605178	+	0.998167i	$0.519275\pi$
$270$	729.000	0.164317
$271$	3688.00i	0.826679i	0.910577	+	0.413340i	$0.135638\pi$
$271$	3688.00i	0.826679i	−0.910577	+	0.413340i	$0.864362\pi$
$272$	−7881.00	−1.75682
$273$	0	0
$274$	2151.00	0.474258
$275$	− 1320.00i	− 0.289451i
$276$	−18.0000	−0.00392563
$277$	−1865.00	−0.404538	−0.202269	−	0.979330i	$-0.564832\pi$
$277$	−1865.00	−0.404538	−0.202269	+	0.979330i	$0.564832\pi$
$278$	− 2460.00i	− 0.530723i
$279$	− 900.000i	− 0.193124i
$280$	− 378.000i	− 0.0806779i
$281$	− 2997.00i	− 0.636249i	−0.948049	−	0.318125i	$-0.896947\pi$
$281$	− 2997.00i	− 0.636249i	0.948049	−	0.318125i	$-0.103053\pi$
$282$	−1458.00	−0.307882
$283$	4114.00	0.864141	0.432071	−	0.901840i	$-0.357783\pi$
$283$	4114.00	0.864141	0.432071	+	0.901840i	$0.357783\pi$
$284$	930.000i	0.194315i
$285$	−1242.00	−0.258139
$286$	0	0
$287$	−462.000	−0.0950209
$288$	− 405.000i	− 0.0828641i
$289$	7408.00	1.50784
$290$	−2835.00	−0.574058
$291$	4074.00i	0.820695i
$292$	− 253.000i	− 0.0507045i
$293$	4665.00i	0.930144i	0.885273	+	0.465072i	$0.153972\pi$
$293$	4665.00i	0.930144i	−0.885273	+	0.465072i	$0.846028\pi$
$294$	3051.00i	0.605231i
$295$	−5400.00	−1.06576
$296$	357.000	0.0701020
$297$	− 810.000i	− 0.158252i
$298$	5247.00	1.01997
$299$	0	0
$300$	−132.000	−0.0254034
$301$	− 1028.00i	− 0.196854i
$302$	−1110.00	−0.211501
$303$	1071.00	0.203061
$304$	3266.00i	0.616177i
$305$	− 2097.00i	− 0.393685i
$306$	2997.00i	0.559892i
$307$	− 1502.00i	− 0.279230i	−0.990206	−	0.139615i	$-0.955413\pi$
$307$	− 1502.00i	− 0.279230i	0.990206	−	0.139615i	$-0.0445865\pi$
$308$	60.0000	0.0111001
$309$	−3354.00	−0.617483
$310$	− 2700.00i	− 0.494676i
$311$	−2106.00	−0.383988	−0.191994	−	0.981396i	$-0.561495\pi$
$311$	−2106.00	−0.383988	−0.191994	+	0.981396i	$0.561495\pi$
$312$	0	0
$313$	−3898.00	−0.703923	−0.351962	−	0.936014i	$-0.614485\pi$
$313$	−3898.00	−0.703923	−0.351962	+	0.936014i	$0.614485\pi$
$314$	− 7833.00i	− 1.40778i
$315$	−162.000	−0.0289767
$316$	1324.00	0.235699
$317$	9351.00i	1.65680i	0.560140	+	0.828398i	$0.310747\pi$
$317$	9351.00i	1.65680i	−0.560140	+	0.828398i	$0.689253\pi$
$318$	5751.00i	1.01415i
$319$	3150.00i	0.552872i
$320$	3897.00i	0.680778i
$321$	2142.00	0.372445
$322$	36.0000	0.00623044
$323$	− 5106.00i	− 0.879583i
$324$	−81.0000	−0.0138889
$325$	0	0
$326$	−4908.00	−0.833831
$327$	6018.00i	1.01773i
$328$	4851.00	0.816621
$329$	324.000	0.0542939
$330$	− 2430.00i	− 0.405355i
$331$	− 9172.00i	− 1.52308i	−0.648119	−	0.761539i	$-0.724444\pi$
$331$	− 9172.00i	− 1.52308i	0.648119	−	0.761539i	$-0.275556\pi$
$332$	− 810.000i	− 0.133899i
$333$	− 153.000i	− 0.0251782i
$334$	792.000	0.129749
$335$	8334.00	1.35921
$336$	426.000i	0.0691673i
$337$	11089.0	1.79245	0.896226	−	0.443598i	$-0.146298\pi$
$337$	11089.0	1.79245	0.896226	+	0.443598i	$0.146298\pi$
$338$	0	0
$339$	−3357.00	−0.537838
$340$	999.000i	0.159348i
$341$	−3000.00	−0.476420
$342$	1242.00	0.196373
$343$	− 1364.00i	− 0.214720i
$344$	10794.0i	1.69178i
$345$	− 162.000i	− 0.0252805i
$346$	− 4230.00i	− 0.657243i
$347$	9762.00	1.51024	0.755118	−	0.655589i	$-0.227580\pi$
$347$	9762.00	1.51024	0.755118	+	0.655589i	$0.227580\pi$
$348$	315.000	0.0485223
$349$	8290.00i	1.27150i	0.771895	+	0.635750i	$0.219309\pi$
$349$	8290.00i	1.27150i	−0.771895	+	0.635750i	$0.780691\pi$
$350$	264.000	0.0403183
$351$	0	0
$352$	−1350.00	−0.204418
$353$	12405.0i	1.87040i	0.354119	+	0.935200i	$0.384781\pi$
$353$	12405.0i	1.87040i	−0.354119	+	0.935200i	$0.615219\pi$
$354$	5400.00	0.810754
$355$	−8370.00	−1.25136
$356$	498.000i	0.0741403i
$357$	− 666.000i	− 0.0987352i
$358$	1422.00i	0.209930i
$359$	1098.00i	0.161421i	0.996738	+	0.0807106i	$0.0257190\pi$
$359$	1098.00i	0.161421i	−0.996738	+	0.0807106i	$0.974281\pi$
$360$	1701.00	0.249029
$361$	4743.00	0.691500
$362$	− 6747.00i	− 0.979598i
$363$	1293.00	0.186956
$364$	0	0
$365$	2277.00	0.326530
$366$	2097.00i	0.299486i
$367$	−5734.00	−0.815565	−0.407783	−	0.913079i	$-0.633698\pi$
$367$	−5734.00	−0.815565	−0.407783	+	0.913079i	$0.633698\pi$
$368$	−426.000	−0.0603445
$369$	− 2079.00i	− 0.293302i
$370$	− 459.000i	− 0.0644926i
$371$	− 1278.00i	− 0.178842i
$372$	300.000i	0.0418126i
$373$	−8971.00	−1.24531	−0.622655	−	0.782496i	$-0.713946\pi$
$373$	−8971.00	−1.24531	−0.622655	+	0.782496i	$0.713946\pi$
$374$	9990.00	1.38120
$375$	− 4563.00i	− 0.628353i
$376$	−3402.00	−0.466608
$377$	0	0
$378$	162.000	0.0220433
$379$	7244.00i	0.981792i	0.871218	+	0.490896i	$0.163331\pi$
$379$	7244.00i	0.981792i	−0.871218	+	0.490896i	$0.836669\pi$
$380$	414.000	0.0558888
$381$	1812.00	0.243652
$382$	10332.0i	1.38385i
$383$	− 6312.00i	− 0.842110i	−0.907035	−	0.421055i	$-0.861660\pi$
$383$	− 6312.00i	− 0.842110i	0.907035	−	0.421055i	$-0.138340\pi$
$384$	− 4977.00i	− 0.661410i
$385$	540.000i	0.0714830i
$386$	−12819.0	−1.69034
$387$	4626.00	0.607630
$388$	− 1358.00i	− 0.177686i
$389$	−3627.00	−0.472741	−0.236370	−	0.971663i	$-0.575958\pi$
$389$	−3627.00	−0.472741	−0.236370	+	0.971663i	$0.575958\pi$
$390$	0	0
$391$	666.000	0.0861408
$392$	7119.00i	0.917255i
$393$	−4752.00	−0.609941
$394$	5958.00	0.761827
$395$	11916.0i	1.51787i
$396$	270.000i	0.0342627i
$397$	3898.00i	0.492783i	0.969170	+	0.246392i	$0.0792450\pi$
$397$	3898.00i	0.492783i	−0.969170	+	0.246392i	$0.920755\pi$
$398$	7158.00i	0.901503i
$399$	−276.000	−0.0346298
$400$	−3124.00	−0.390500
$401$	5703.00i	0.710210i	0.934826	+	0.355105i	$0.115555\pi$
$401$	5703.00i	0.710210i	−0.934826	+	0.355105i	$0.884445\pi$
$402$	−8334.00	−1.03399
$403$	0	0
$404$	−357.000	−0.0439639
$405$	− 729.000i	− 0.0894427i
$406$	−630.000	−0.0770108
$407$	−510.000	−0.0621124
$408$	6993.00i	0.848542i
$409$	6311.00i	0.762980i	0.924373	+	0.381490i	$0.124589\pi$
$409$	6311.00i	0.762980i	−0.924373	+	0.381490i	$0.875411\pi$
$410$	− 6237.00i	− 0.751277i
$411$	− 2151.00i	− 0.258153i
$412$	1118.00	0.133689
$413$	−1200.00	−0.142974
$414$	162.000i	0.0192316i
$415$	7290.00	0.862294
$416$	0	0
$417$	−2460.00	−0.288889
$418$	− 4140.00i	− 0.484435i
$419$	−2328.00	−0.271433	−0.135716	−	0.990748i	$-0.543334\pi$
$419$	−2328.00	−0.271433	−0.135716	+	0.990748i	$0.543334\pi$
$420$	54.0000	0.00627364
$421$	2045.00i	0.236739i	0.992970	+	0.118370i	$0.0377668\pi$
$421$	2045.00i	0.236739i	−0.992970	+	0.118370i	$0.962233\pi$
$422$	− 4800.00i	− 0.553697i
$423$	1458.00i	0.167590i
$424$	13419.0i	1.53699i
$425$	4884.00	0.557432
$426$	8370.00	0.951943
$427$	− 466.000i	− 0.0528134i
$428$	−714.000	−0.0806367
$429$	0	0
$430$	13878.0	1.55641
$431$	5034.00i	0.562597i	0.959620	+	0.281298i	$0.0907651\pi$
$431$	5034.00i	0.562597i	−0.959620	+	0.281298i	$0.909235\pi$
$432$	−1917.00	−0.213499
$433$	−4283.00	−0.475353	−0.237676	−	0.971344i	$-0.576386\pi$
$433$	−4283.00	−0.475353	−0.237676	+	0.971344i	$0.576386\pi$
$434$	− 600.000i	− 0.0663616i
$435$	2835.00i	0.312478i
$436$	− 2006.00i	− 0.220344i
$437$	− 276.000i	− 0.0302125i
$438$	−2277.00	−0.248400
$439$	1306.00	0.141986	0.0709931	−	0.997477i	$-0.477383\pi$
$439$	1306.00	0.141986	0.0709931	+	0.997477i	$0.477383\pi$
$440$	− 5670.00i	− 0.614333i
$441$	3051.00	0.329446
$442$	0	0
$443$	−5796.00	−0.621617	−0.310808	−	0.950473i	$-0.600600\pi$
$443$	−5796.00	−0.621617	−0.310808	+	0.950473i	$0.600600\pi$
$444$	51.0000i	0.00545125i
$445$	−4482.00	−0.477454
$446$	11496.0	1.22052
$447$	− 5247.00i	− 0.555200i
$448$	866.000i	0.0913274i
$449$	− 2706.00i	− 0.284419i	−0.989837	−	0.142209i	$-0.954579\pi$
$449$	− 2706.00i	− 0.284419i	0.989837	−	0.142209i	$-0.0454206\pi$
$450$	1188.00i	0.124451i
$451$	−6930.00	−0.723550
$452$	1119.00	0.116445
$453$	1110.00i	0.115127i
$454$	4194.00	0.433555
$455$	0	0
$456$	2898.00	0.297612
$457$	− 829.000i	− 0.0848555i	−0.999100	−	0.0424278i	$-0.986491\pi$
$457$	− 829.000i	− 0.0848555i	0.999100	−	0.0424278i	$-0.0135092\pi$
$458$	13398.0	1.36692
$459$	2997.00	0.304767
$460$	54.0000i	0.00547340i
$461$	− 5493.00i	− 0.554956i	−0.960732	−	0.277478i	$-0.910502\pi$
$461$	− 5493.00i	− 0.554956i	0.960732	−	0.277478i	$-0.0894985\pi$
$462$	− 540.000i	− 0.0543789i
$463$	15346.0i	1.54037i	0.637823	+	0.770183i	$0.279835\pi$
$463$	15346.0i	1.54037i	−0.637823	+	0.770183i	$0.720165\pi$
$464$	7455.00	0.745883
$465$	−2700.00	−0.269268
$466$	4914.00i	0.488491i
$467$	9594.00	0.950658	0.475329	−	0.879808i	$-0.342329\pi$
$467$	9594.00	0.950658	0.475329	+	0.879808i	$0.342329\pi$
$468$	0	0
$469$	1852.00	0.182340
$470$	4374.00i	0.429271i
$471$	−7833.00	−0.766296
$472$	12600.0	1.22873
$473$	− 15420.0i	− 1.49897i
$474$	− 11916.0i	− 1.15468i
$475$	− 2024.00i	− 0.195511i
$476$	222.000i	0.0213768i
$477$	5751.00	0.552034
$478$	1782.00	0.170516
$479$	12840.0i	1.22479i	0.790552	+	0.612395i	$0.209794\pi$
$479$	12840.0i	1.22479i	−0.790552	+	0.612395i	$0.790206\pi$
$480$	−1215.00	−0.115535
$481$	0	0
$482$	6909.00	0.652897
$483$	− 36.0000i	− 0.00339142i
$484$	−431.000	−0.0404771
$485$	12222.0	1.14427
$486$	729.000i	0.0680414i
$487$	− 14086.0i	− 1.31067i	−0.755337	−	0.655336i	$-0.772527\pi$
$487$	− 14086.0i	− 1.31067i	0.755337	−	0.655336i	$-0.227473\pi$
$488$	4893.00i	0.453885i
$489$	4908.00i	0.453880i
$490$	9153.00	0.843858
$491$	−11694.0	−1.07483	−0.537416	−	0.843317i	$-0.680600\pi$
$491$	−11694.0	−1.07483	−0.537416	+	0.843317i	$0.680600\pi$
$492$	693.000i	0.0635017i
$493$	−11655.0	−1.06474
$494$	0	0
$495$	−2430.00	−0.220647
$496$	7100.00i	0.642741i
$497$	−1860.00	−0.167872
$498$	−7290.00	−0.655969
$499$	− 3688.00i	− 0.330857i	−0.986222	−	0.165428i	$-0.947099\pi$
$499$	− 3688.00i	− 0.330857i	0.986222	−	0.165428i	$-0.0529007\pi$
$500$	1521.00i	0.136042i
$501$	− 792.000i	− 0.0706266i
$502$	− 18972.0i	− 1.68678i
$503$	−4746.00	−0.420703	−0.210352	−	0.977626i	$-0.567461\pi$
$503$	−4746.00	−0.420703	−0.210352	+	0.977626i	$0.567461\pi$
$504$	378.000	0.0334077
$505$	− 3213.00i	− 0.283122i
$506$	540.000	0.0474425
$507$	0	0
$508$	−604.000	−0.0527523
$509$	− 14505.0i	− 1.26311i	−0.775331	−	0.631555i	$-0.782417\pi$
$509$	− 14505.0i	− 1.26311i	0.775331	−	0.631555i	$-0.217583\pi$
$510$	8991.00	0.780643
$511$	506.000	0.0438045
$512$	− 8733.00i	− 0.753804i
$513$	− 1242.00i	− 0.106892i
$514$	− 23499.0i	− 2.01653i
$515$	10062.0i	0.860941i
$516$	−1542.00	−0.131556
$517$	4860.00	0.413429
$518$	− 102.000i	− 0.00865178i
$519$	−4230.00	−0.357758
$520$	0	0
$521$	5085.00	0.427597	0.213798	−	0.976878i	$-0.431416\pi$
$521$	5085.00	0.427597	0.213798	+	0.976878i	$0.431416\pi$
$522$	− 2835.00i	− 0.237710i
$523$	−10882.0	−0.909821	−0.454911	−	0.890537i	$-0.650329\pi$
$523$	−10882.0	−0.909821	−0.454911	+	0.890537i	$0.650329\pi$
$524$	1584.00	0.132056
$525$	− 264.000i	− 0.0219465i
$526$	− 9090.00i	− 0.753503i
$527$	− 11100.0i	− 0.917502i
$528$	6390.00i	0.526684i
$529$	−12131.0	−0.997041
$530$	17253.0	1.41400
$531$	− 5400.00i	− 0.441318i
$532$	92.0000	0.00749757
$533$	0	0
$534$	4482.00	0.363212
$535$	− 6426.00i	− 0.519290i
$536$	−19446.0	−1.56705
$537$	1422.00	0.114272
$538$	− 1602.00i	− 0.128378i
$539$	− 10170.0i	− 0.812714i
$540$	243.000i	0.0193649i
$541$	4699.00i	0.373430i	0.982414	+	0.186715i	$0.0597841\pi$
$541$	4699.00i	0.373430i	−0.982414	+	0.186715i	$0.940216\pi$
$542$	−11064.0	−0.876826
$543$	−6747.00	−0.533226
$544$	− 4995.00i	− 0.393674i
$545$	18054.0	1.41899
$546$	0	0
$547$	8270.00	0.646434	0.323217	−	0.946325i	$-0.395236\pi$
$547$	8270.00	0.646434	0.323217	+	0.946325i	$0.395236\pi$
$548$	717.000i	0.0558918i
$549$	2097.00	0.163020
$550$	3960.00	0.307009
$551$	4830.00i	0.373439i
$552$	378.000i	0.0291463i
$553$	2648.00i	0.203625i
$554$	− 5595.00i	− 0.429077i
$555$	−459.000	−0.0351053
$556$	820.000	0.0625463
$557$	22785.0i	1.73327i	0.498943	+	0.866635i	$0.333722\pi$
$557$	22785.0i	1.73327i	−0.498943	+	0.866635i	$0.666278\pi$
$558$	2700.00	0.204839
$559$	0	0
$560$	1278.00	0.0964381
$561$	− 9990.00i	− 0.751833i
$562$	8991.00	0.674844
$563$	11928.0	0.892905	0.446452	−	0.894807i	$-0.352687\pi$
$563$	11928.0	0.892905	0.446452	+	0.894807i	$0.352687\pi$
$564$	− 486.000i	− 0.0362842i
$565$	10071.0i	0.749894i
$566$	12342.0i	0.916560i
$567$	− 162.000i	− 0.0119989i
$568$	19530.0	1.44271
$569$	7962.00	0.586616	0.293308	−	0.956018i	$-0.405244\pi$
$569$	7962.00	0.586616	0.293308	+	0.956018i	$0.405244\pi$
$570$	− 3726.00i	− 0.273798i
$571$	−20618.0	−1.51110	−0.755549	−	0.655093i	$-0.772630\pi$
$571$	−20618.0	−1.51110	−0.755549	+	0.655093i	$0.772630\pi$
$572$	0	0
$573$	10332.0	0.753273
$574$	− 1386.00i	− 0.100785i
$575$	264.000	0.0191471
$576$	−3897.00	−0.281901
$577$	− 3493.00i	− 0.252020i	−0.992029	−	0.126010i	$-0.959783\pi$
$577$	− 3493.00i	− 0.252020i	0.992029	−	0.126010i	$-0.0402171\pi$
$578$	22224.0i	1.59930i
$579$	12819.0i	0.920103i
$580$	− 945.000i	− 0.0676534i
$581$	1620.00	0.115678
$582$	−12222.0	−0.870478
$583$	− 19170.0i	− 1.36182i
$584$	−5313.00	−0.376461
$585$	0	0
$586$	−13995.0	−0.986567
$587$	10416.0i	0.732392i	0.930538	+	0.366196i	$0.119340\pi$
$587$	10416.0i	0.732392i	−0.930538	+	0.366196i	$0.880660\pi$
$588$	−1017.00	−0.0713272
$589$	−4600.00	−0.321799
$590$	− 16200.0i	− 1.13041i
$591$	− 5958.00i	− 0.414686i
$592$	1207.00i	0.0837963i
$593$	− 2061.00i	− 0.142724i	−0.997450	−	0.0713618i	$-0.977266\pi$
$593$	− 2061.00i	− 0.142724i	0.997450	−	0.0713618i	$-0.0227345\pi$
$594$	2430.00	0.167852
$595$	−1998.00	−0.137664
$596$	1749.00i	0.120204i
$597$	7158.00	0.490716
$598$	0	0
$599$	12456.0	0.849647	0.424823	−	0.905276i	$-0.360336\pi$
$599$	12456.0	0.849647	0.424823	+	0.905276i	$0.360336\pi$
$600$	2772.00i	0.188611i
$601$	−781.000	−0.0530077	−0.0265039	−	0.999649i	$-0.508437\pi$
$601$	−781.000	−0.0530077	−0.0265039	+	0.999649i	$0.508437\pi$
$602$	3084.00	0.208795
$603$	8334.00i	0.562830i
$604$	− 370.000i	− 0.0249256i
$605$	− 3879.00i	− 0.260667i
$606$	3213.00i	0.215378i
$607$	19304.0	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$607$	19304.0	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$608$	−2070.00	−0.138075
$609$	630.000i	0.0419194i
$610$	6291.00	0.417566
$611$	0	0
$612$	−999.000	−0.0659840
$613$	12041.0i	0.793363i	0.917956	+	0.396681i	$0.129838\pi$
$613$	12041.0i	0.793363i	−0.917956	+	0.396681i	$0.870162\pi$
$614$	4506.00	0.296168
$615$	−6237.00	−0.408943
$616$	− 1260.00i	− 0.0824137i
$617$	9717.00i	0.634022i	0.948422	+	0.317011i	$0.102679\pi$
$617$	9717.00i	0.634022i	−0.948422	+	0.317011i	$0.897321\pi$
$618$	− 10062.0i	− 0.654940i
$619$	21040.0i	1.36619i	0.730332	+	0.683093i	$0.239366\pi$
$619$	21040.0i	1.36619i	−0.730332	+	0.683093i	$0.760634\pi$
$620$	900.000	0.0582982
$621$	162.000	0.0104683
$622$	− 6318.00i	− 0.407281i
$623$	−996.000	−0.0640512
$624$	0	0
$625$	−8189.00	−0.524096
$626$	− 11694.0i	− 0.746623i
$627$	−4140.00	−0.263693
$628$	2611.00	0.165908
$629$	− 1887.00i	− 0.119618i
$630$	− 486.000i	− 0.0307344i
$631$	5068.00i	0.319737i	0.987138	+	0.159868i	$0.0511070\pi$
$631$	5068.00i	0.319737i	−0.987138	+	0.159868i	$0.948893\pi$
$632$	− 27804.0i	− 1.74997i
$633$	−4800.00	−0.301395
$634$	−28053.0	−1.75730
$635$	− 5436.00i	− 0.339718i
$636$	−1917.00	−0.119519
$637$	0	0
$638$	−9450.00	−0.586409
$639$	− 8370.00i	− 0.518172i
$640$	−14931.0	−0.922187
$641$	−10185.0	−0.627587	−0.313794	−	0.949491i	$-0.601600\pi$
$641$	−10185.0	−0.627587	−0.313794	+	0.949491i	$0.601600\pi$
$642$	6426.00i	0.395037i
$643$	25928.0i	1.59020i	0.606476	+	0.795101i	$0.292582\pi$
$643$	25928.0i	1.59020i	−0.606476	+	0.795101i	$0.707418\pi$
$644$	12.0000i	0 0.000734264i
$645$	− 13878.0i	− 0.847203i
$646$	15318.0	0.932939
$647$	−23160.0	−1.40729	−0.703643	−	0.710554i	$-0.748444\pi$
$647$	−23160.0	−1.40729	−0.703643	+	0.710554i	$0.748444\pi$
$648$	1701.00i	0.103120i
$649$	−18000.0	−1.08869
$650$	0	0
$651$	−600.000	−0.0361227
$652$	− 1636.00i	− 0.0982680i
$653$	16626.0	0.996364	0.498182	−	0.867073i	$-0.334001\pi$
$653$	16626.0	0.996364	0.498182	+	0.867073i	$0.334001\pi$
$654$	−18054.0	−1.07946
$655$	14256.0i	0.850424i
$656$	16401.0i	0.976146i
$657$	2277.00i	0.135212i
$658$	972.000i	0.0575874i
$659$	−14808.0	−0.875323	−0.437661	−	0.899140i	$-0.644193\pi$
$659$	−14808.0	−0.875323	−0.437661	+	0.899140i	$0.644193\pi$
$660$	810.000	0.0477715
$661$	− 4853.00i	− 0.285567i	−0.989754	−	0.142784i	$-0.954395\pi$
$661$	− 4853.00i	− 0.285567i	0.989754	−	0.142784i	$-0.0456053\pi$
$662$	27516.0	1.61547
$663$	0	0
$664$	−17010.0	−0.994151
$665$	828.000i	0.0482834i
$666$	459.000	0.0267055
$667$	−630.000	−0.0365723
$668$	264.000i	0.0152911i
$669$	− 11496.0i	− 0.664366i
$670$	25002.0i	1.44166i
$671$	− 6990.00i	− 0.402155i
$672$	−270.000	−0.0154992
$673$	16165.0	0.925877	0.462938	−	0.886391i	$-0.346795\pi$
$673$	16165.0	0.925877	0.462938	+	0.886391i	$0.346795\pi$
$674$	33267.0i	1.90118i
$675$	1188.00	0.0677424
$676$	0	0
$677$	−25686.0	−1.45819	−0.729094	−	0.684414i	$-0.760058\pi$
$677$	−25686.0	−1.45819	−0.729094	+	0.684414i	$0.760058\pi$
$678$	− 10071.0i	− 0.570464i
$679$	2716.00	0.153506
$680$	20979.0	1.18310
$681$	− 4194.00i	− 0.235998i
$682$	− 9000.00i	− 0.505319i
$683$	19056.0i	1.06758i	0.845617	+	0.533790i	$0.179233\pi$
$683$	19056.0i	1.06758i	−0.845617	+	0.533790i	$0.820767\pi$
$684$	414.000i	0.0231428i
$685$	−6453.00	−0.359936
$686$	4092.00	0.227745
$687$	− 13398.0i	− 0.744055i
$688$	−36494.0	−2.02227
$689$	0	0
$690$	486.000	0.0268141
$691$	− 16390.0i	− 0.902323i	−0.892442	−	0.451161i	$-0.851010\pi$
$691$	− 16390.0i	− 0.902323i	0.892442	−	0.451161i	$-0.148990\pi$
$692$	1410.00	0.0774569
$693$	−540.000	−0.0296001
$694$	29286.0i	1.60185i
$695$	7380.00i	0.402790i
$696$	− 6615.00i	− 0.360260i
$697$	− 25641.0i	− 1.39343i
$698$	−24870.0	−1.34863
$699$	4914.00	0.265901
$700$	88.0000i	0.00475155i
$701$	27846.0	1.50033	0.750163	−	0.661253i	$-0.229975\pi$
$701$	27846.0	1.50033	0.750163	+	0.661253i	$0.229975\pi$
$702$	0	0
$703$	−782.000	−0.0419540
$704$	12990.0i	0.695425i
$705$	4374.00	0.233666
$706$	−37215.0	−1.98386
$707$	− 714.000i	− 0.0379812i
$708$	1800.00i	0.0955482i
$709$	12283.0i	0.650632i	0.945605	+	0.325316i	$0.105471\pi$
$709$	12283.0i	0.650632i	−0.945605	+	0.325316i	$0.894529\pi$
$710$	− 25110.0i	− 1.32727i
$711$	−11916.0	−0.628530
$712$	10458.0	0.550464
$713$	− 600.000i	− 0.0315150i
$714$	1998.00	0.104724
$715$	0	0
$716$	−474.000	−0.0247405
$717$	− 1782.00i	− 0.0928173i
$718$	−3294.00	−0.171213
$719$	−25512.0	−1.32328	−0.661639	−	0.749822i	$-0.730139\pi$
$719$	−25512.0	−1.32328	−0.661639	+	0.749822i	$0.730139\pi$
$720$	5751.00i	0.297677i
$721$	2236.00i	0.115497i
$722$	14229.0i	0.733447i
$723$	− 6909.00i	− 0.355392i
$724$	2249.00	0.115447
$725$	−4620.00	−0.236666
$726$	3879.00i	0.198296i
$727$	−6110.00	−0.311702	−0.155851	−	0.987781i	$-0.549812\pi$
$727$	−6110.00	−0.311702	−0.155851	+	0.987781i	$0.549812\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	6831.00i	0.346338i
$731$	57054.0	2.88676
$732$	−699.000	−0.0352948
$733$	− 27127.0i	− 1.36693i	−0.729984	−	0.683464i	$-0.760473\pi$
$733$	− 27127.0i	− 1.36693i	0.729984	−	0.683464i	$-0.239527\pi$
$734$	− 17202.0i	− 0.865037i
$735$	− 9153.00i	− 0.459338i
$736$	− 270.000i	− 0.0135222i
$737$	27780.0	1.38845
$738$	6237.00	0.311094
$739$	880.000i	0.0438042i	0.999760	+	0.0219021i	$0.00697222\pi$
$739$	880.000i	0.0438042i	−0.999760	+	0.0219021i	$0.993028\pi$
$740$	153.000	0.00760053
$741$	0	0
$742$	3834.00	0.189691
$743$	− 21876.0i	− 1.08015i	−0.841616	−	0.540076i	$-0.818396\pi$
$743$	− 21876.0i	− 1.08015i	0.841616	−	0.540076i	$-0.181604\pi$
$744$	6300.00	0.310442
$745$	−15741.0	−0.774102
$746$	− 26913.0i	− 1.32085i
$747$	7290.00i	0.357064i
$748$	3330.00i	0.162777i
$749$	− 1428.00i	− 0.0696635i
$750$	13689.0	0.666469
$751$	−11798.0	−0.573256	−0.286628	−	0.958042i	$-0.592534\pi$
$751$	−11798.0	−0.573256	−0.286628	+	0.958042i	$0.592534\pi$
$752$	− 11502.0i	− 0.557759i
$753$	−18972.0	−0.918165
$754$	0	0
$755$	3330.00	0.160518
$756$	54.0000i	0.00259783i
$757$	−8074.00	−0.387655	−0.193827	−	0.981036i	$-0.562090\pi$
$757$	−8074.00	−0.387655	−0.193827	+	0.981036i	$0.562090\pi$
$758$	−21732.0	−1.04135
$759$	− 540.000i	− 0.0258245i
$760$	− 8694.00i	− 0.414953i
$761$	− 19554.0i	− 0.931448i	−0.884930	−	0.465724i	$-0.845794\pi$
$761$	− 19554.0i	− 0.931448i	0.884930	−	0.465724i	$-0.154206\pi$
$762$	5436.00i	0.258432i
$763$	4012.00	0.190359
$764$	−3444.00	−0.163088
$765$	− 8991.00i	− 0.424928i
$766$	18936.0	0.893193
$767$	0	0
$768$	4539.00	0.213264
$769$	14030.0i	0.657913i	0.944345	+	0.328956i	$0.106697\pi$
$769$	14030.0i	0.657913i	−0.944345	+	0.328956i	$0.893303\pi$
$770$	−1620.00	−0.0758192
$771$	−23499.0	−1.09766
$772$	− 4273.00i	− 0.199208i
$773$	36042.0i	1.67703i	0.544882	+	0.838513i	$0.316574\pi$
$773$	36042.0i	1.67703i	−0.544882	+	0.838513i	$0.683426\pi$
$774$	13878.0i	0.644489i
$775$	− 4400.00i	− 0.203939i
$776$	−28518.0	−1.31925
$777$	−102.000	−0.00470943
$778$	− 10881.0i	− 0.501417i
$779$	−10626.0	−0.488724
$780$	0	0
$781$	−27900.0	−1.27828
$782$	1998.00i	0.0913662i
$783$	−2835.00	−0.129393
$784$	−24069.0	−1.09644
$785$	23499.0i	1.06843i
$786$	− 14256.0i	− 0.646940i
$787$	− 28628.0i	− 1.29667i	−0.761356	−	0.648334i	$-0.775466\pi$
$787$	− 28628.0i	− 1.29667i	0.761356	−	0.648334i	$-0.224534\pi$
$788$	1986.00i	0.0897821i
$789$	−9090.00	−0.410155
$790$	−35748.0	−1.60995
$791$	2238.00i	0.100599i
$792$	5670.00	0.254387
$793$	0	0
$794$	−11694.0	−0.522676
$795$	− 17253.0i	− 0.769687i
$796$	−2386.00	−0.106243
$797$	37434.0	1.66371	0.831857	−	0.554990i	$-0.187278\pi$
$797$	37434.0	1.66371	0.831857	+	0.554990i	$0.187278\pi$
$798$	− 828.000i	− 0.0367304i
$799$	17982.0i	0.796192i
$800$	− 1980.00i	− 0.0875045i
$801$	− 4482.00i	− 0.197707i
$802$	−17109.0	−0.753292
$803$	7590.00	0.333556
$804$	− 2778.00i	− 0.121856i
$805$	−108.000	−0.00472857
$806$	0	0
$807$	−1602.00	−0.0698799
$808$	7497.00i	0.326415i
$809$	37569.0	1.63270	0.816351	−	0.577556i	$-0.195994\pi$
$809$	37569.0	1.63270	0.816351	+	0.577556i	$0.195994\pi$
$810$	2187.00	0.0948683
$811$	5516.00i	0.238832i	0.992844	+	0.119416i	$0.0381023\pi$
$811$	5516.00i	0.238832i	−0.992844	+	0.119416i	$0.961898\pi$
$812$	− 210.000i	− 0.00907581i
$813$	11064.0i	0.477283i
$814$	− 1530.00i	− 0.0658802i
$815$	14724.0	0.632833
$816$	−23643.0	−1.01430
$817$	− 23644.0i	− 1.01248i
$818$	−18933.0	−0.809263
$819$	0	0
$820$	2079.00	0.0885388
$821$	8778.00i	0.373148i	0.982441	+	0.186574i	$0.0597384\pi$
$821$	8778.00i	0.373148i	−0.982441	+	0.186574i	$0.940262\pi$
$822$	6453.00	0.273813
$823$	3088.00	0.130791	0.0653955	−	0.997859i	$-0.479169\pi$
$823$	3088.00	0.130791	0.0653955	+	0.997859i	$0.479169\pi$
$824$	− 23478.0i	− 0.992591i
$825$	− 3960.00i	− 0.167115i
$826$	− 3600.00i	− 0.151647i
$827$	− 13176.0i	− 0.554020i	−0.960867	−	0.277010i	$-0.910657\pi$
$827$	− 13176.0i	− 0.554020i	0.960867	−	0.277010i	$-0.0893435\pi$
$828$	−54.0000	−0.00226646
$829$	2359.00	0.0988317	0.0494158	−	0.998778i	$-0.484264\pi$
$829$	2359.00	0.0988317	0.0494158	+	0.998778i	$0.484264\pi$
$830$	21870.0i	0.914601i
$831$	−5595.00	−0.233560
$832$	0	0
$833$	37629.0	1.56515
$834$	− 7380.00i	− 0.306413i
$835$	−2376.00	−0.0984729
$836$	1380.00	0.0570913
$837$	− 2700.00i	− 0.111500i
$838$	− 6984.00i	− 0.287898i
$839$	2676.00i	0.110114i	0.998483	+	0.0550571i	$0.0175341\pi$
$839$	2676.00i	0.110114i	−0.998483	+	0.0550571i	$0.982466\pi$
$840$	− 1134.00i	− 0.0465794i
$841$	−13364.0	−0.547952
$842$	−6135.00	−0.251100
$843$	− 8991.00i	− 0.367339i
$844$	1600.00	0.0652539
$845$	0	0
$846$	−4374.00	−0.177756
$847$	− 862.000i	− 0.0349689i
$848$	−45369.0	−1.83724
$849$	12342.0	0.498912
$850$	14652.0i	0.591246i
$851$	− 102.000i	− 0.00410871i
$852$	2790.00i	0.112188i
$853$	− 2477.00i	− 0.0994266i	−0.998764	−	0.0497133i	$-0.984169\pi$
$853$	− 2477.00i	− 0.0994266i	0.998764	−	0.0497133i	$-0.0158308\pi$
$854$	1398.00	0.0560171
$855$	−3726.00	−0.149037
$856$	14994.0i	0.598697i
$857$	17199.0	0.685539	0.342769	−	0.939420i	$-0.388635\pi$
$857$	17199.0	0.685539	0.342769	+	0.939420i	$0.388635\pi$
$858$	0	0
$859$	24338.0	0.966708	0.483354	−	0.875425i	$-0.339418\pi$
$859$	24338.0	0.966708	0.483354	+	0.875425i	$0.339418\pi$
$860$	4626.00i	0.183425i
$861$	−1386.00	−0.0548603
$862$	−15102.0	−0.596724
$863$	25146.0i	0.991865i	0.868361	+	0.495933i	$0.165174\pi$
$863$	25146.0i	0.991865i	−0.868361	+	0.495933i	$0.834826\pi$
$864$	− 1215.00i	− 0.0478416i
$865$	12690.0i	0.498813i
$866$	− 12849.0i	− 0.504188i
$867$	22224.0	0.870550
$868$	200.000	0.00782079
$869$	39720.0i	1.55053i
$870$	−8505.00	−0.331433
$871$	0	0
$872$	−42126.0	−1.63597
$873$	12222.0i	0.473828i
$874$	828.000	0.0320452
$875$	−3042.00	−0.117530
$876$	− 759.000i	− 0.0292742i
$877$	18089.0i	0.696490i	0.937403	+	0.348245i	$0.113222\pi$
$877$	18089.0i	0.696490i	−0.937403	+	0.348245i	$0.886778\pi$
$878$	3918.00i	0.150599i
$879$	13995.0i	0.537019i
$880$	19170.0	0.734342
$881$	15099.0	0.577410	0.288705	−	0.957418i	$-0.406775\pi$
$881$	15099.0	0.577410	0.288705	+	0.957418i	$0.406775\pi$
$882$	9153.00i	0.349430i
$883$	−33488.0	−1.27629	−0.638143	−	0.769918i	$-0.720297\pi$
$883$	−33488.0	−1.27629	−0.638143	+	0.769918i	$0.720297\pi$
$884$	0	0
$885$	−16200.0	−0.615319
$886$	− 17388.0i	− 0.659324i
$887$	−39768.0	−1.50539	−0.752694	−	0.658371i	$-0.771246\pi$
$887$	−39768.0	−1.50539	−0.752694	+	0.658371i	$0.771246\pi$
$888$	1071.00	0.0404734
$889$	− 1208.00i	− 0.0455737i
$890$	− 13446.0i	− 0.506417i
$891$	− 2430.00i	− 0.0913671i
$892$	3832.00i	0.143840i
$893$	7452.00	0.279252
$894$	15741.0	0.588879
$895$	− 4266.00i	− 0.159326i
$896$	−3318.00	−0.123713
$897$	0	0
$898$	8118.00	0.301672
$899$	10500.0i	0.389538i
$900$	−396.000	−0.0146667
$901$	70929.0	2.62263
$902$	− 20790.0i	− 0.767440i
$903$	− 3084.00i	− 0.113653i
$904$	− 23499.0i	− 0.864563i
$905$	20241.0i	0.743463i
$906$	−3330.00	−0.122110
$907$	−32156.0	−1.17720	−0.588601	−	0.808424i	$-0.700321\pi$
$907$	−32156.0	−1.17720	−0.588601	+	0.808424i	$0.700321\pi$
$908$	1398.00i	0.0510950i
$909$	3213.00	0.117237
$910$	0	0
$911$	11520.0	0.418962	0.209481	−	0.977813i	$-0.432823\pi$
$911$	11520.0	0.418962	0.209481	+	0.977813i	$0.432823\pi$
$912$	9798.00i	0.355750i
$913$	24300.0	0.880846
$914$	2487.00	0.0900029
$915$	− 6291.00i	− 0.227294i
$916$	4466.00i	0.161093i
$917$	3168.00i	0.114086i
$918$	8991.00i	0.323254i
$919$	4952.00	0.177749	0.0888745	−	0.996043i	$-0.471673\pi$
$919$	4952.00	0.177749	0.0888745	+	0.996043i	$0.471673\pi$
$920$	1134.00	0.0406379
$921$	− 4506.00i	− 0.161214i
$922$	16479.0	0.588619
$923$	0	0
$924$	180.000	0.00640862
$925$	− 748.000i	− 0.0265882i
$926$	−46038.0	−1.63380
$927$	−10062.0	−0.356504
$928$	4725.00i	0.167140i
$929$	8781.00i	0.310113i	0.987906	+	0.155057i	$0.0495560\pi$
$929$	8781.00i	0.310113i	−0.987906	+	0.155057i	$0.950444\pi$
$930$	− 8100.00i	− 0.285602i
$931$	− 15594.0i	− 0.548950i
$932$	−1638.00	−0.0575692
$933$	−6318.00	−0.221696
$934$	28782.0i	1.00833i
$935$	−29970.0	−1.04826
$936$	0	0
$937$	50039.0	1.74461	0.872307	−	0.488959i	$-0.162623\pi$
$937$	50039.0	1.74461	0.872307	+	0.488959i	$0.162623\pi$
$938$	5556.00i	0.193401i
$939$	−11694.0	−0.406410
$940$	−1458.00	−0.0505901
$941$	− 50670.0i	− 1.75536i	−0.479246	−	0.877681i	$-0.659090\pi$
$941$	− 50670.0i	− 1.75536i	0.479246	−	0.877681i	$-0.340910\pi$
$942$	− 23499.0i	− 0.812780i
$943$	− 1386.00i	− 0.0478625i
$944$	42600.0i	1.46876i
$945$	−486.000	−0.0167297
$946$	46260.0	1.58990
$947$	− 42384.0i	− 1.45438i	−0.686438	−	0.727188i	$-0.740827\pi$
$947$	− 42384.0i	− 1.45438i	0.686438	−	0.727188i	$-0.259173\pi$
$948$	3972.00	0.136081
$949$	0	0
$950$	6072.00	0.207370
$951$	28053.0i	0.956552i
$952$	4662.00	0.158715
$953$	50538.0	1.71782	0.858912	−	0.512123i	$-0.171141\pi$
$953$	50538.0	1.71782	0.858912	+	0.512123i	$0.171141\pi$
$954$	17253.0i	0.585520i
$955$	− 30996.0i	− 1.05027i
$956$	594.000i	0.0200955i
$957$	9450.00i	0.319201i
$958$	−38520.0	−1.29909
$959$	−1434.00	−0.0482860
$960$	11691.0i	0.393047i
$961$	19791.0	0.664328
$962$	0	0
$963$	6426.00	0.215031
$964$	2303.00i	0.0769446i
$965$	38457.0	1.28288
$966$	108.000	0.00359715
$967$	− 6886.00i	− 0.228996i	−0.993423	−	0.114498i	$-0.963474\pi$
$967$	− 6886.00i	− 0.228996i	0.993423	−	0.114498i	$-0.0365259\pi$
$968$	9051.00i	0.300527i
$969$	− 15318.0i	− 0.507828i
$970$	36666.0i	1.21368i
$971$	9060.00	0.299433	0.149716	−	0.988729i	$-0.452164\pi$
$971$	9060.00	0.299433	0.149716	+	0.988729i	$0.452164\pi$
$972$	−243.000	−0.00801875
$973$	1640.00i	0.0540349i
$974$	42258.0	1.39018
$975$	0	0
$976$	−16543.0	−0.542550
$977$	− 28311.0i	− 0.927072i	−0.886078	−	0.463536i	$-0.846581\pi$
$977$	− 28311.0i	− 0.927072i	0.886078	−	0.463536i	$-0.153419\pi$
$978$	−14724.0	−0.481413
$979$	−14940.0	−0.487727
$980$	3051.00i	0.0994496i
$981$	18054.0i	0.587584i
$982$	− 35082.0i	− 1.14003i
$983$	− 4284.00i	− 0.139001i	−0.997582	−	0.0695007i	$-0.977859\pi$
$983$	− 4284.00i	− 0.139001i	0.997582	−	0.0695007i	$-0.0221406\pi$
$984$	14553.0	0.471476
$985$	−17874.0	−0.578186
$986$	− 34965.0i	− 1.12932i
$987$	972.000	0.0313466
$988$	0	0
$989$	3084.00	0.0991562
$990$	− 7290.00i	− 0.234032i
$991$	−2458.00	−0.0787901	−0.0393950	−	0.999224i	$-0.512543\pi$
$991$	−2458.00	−0.0787901	−0.0393950	+	0.999224i	$0.512543\pi$
$992$	−4500.00	−0.144027
$993$	− 27516.0i	− 0.879349i
$994$	− 5580.00i	− 0.178055i
$995$	− 21474.0i	− 0.684193i
$996$	− 2430.00i	− 0.0773067i
$997$	24101.0	0.765583	0.382792	−	0.923835i	$-0.374963\pi$
$997$	24101.0	0.765583	0.382792	+	0.923835i	$0.374963\pi$
$998$	11064.0	0.350927
$999$	− 459.000i	− 0.0145367i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.c.337.2		2
13.2	odd	12		39.4.e.a.22.1	yes	2
13.5	odd	4		507.4.a.e.1.1		1
13.6	odd	12		39.4.e.a.16.1	✓	2
13.8	odd	4		507.4.a.a.1.1		1
13.12	even	2	inner	507.4.b.c.337.1		2
39.2	even	12		117.4.g.b.100.1		2
39.5	even	4		1521.4.a.c.1.1		1
39.8	even	4		1521.4.a.j.1.1		1
39.32	even	12		117.4.g.b.55.1		2
52.15	even	12		624.4.q.b.529.1		2
52.19	even	12		624.4.q.b.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.a.16.1	✓	2	13.6	odd	12
39.4.e.a.22.1	yes	2	13.2	odd	12
117.4.g.b.55.1		2	39.32	even	12
117.4.g.b.100.1		2	39.2	even	12
507.4.a.a.1.1		1	13.8	odd	4
507.4.a.e.1.1		1	13.5	odd	4
507.4.b.c.337.1		2	13.12	even	2	inner
507.4.b.c.337.2		2	1.1	even	1	trivial
624.4.q.b.289.1		2	52.19	even	12
624.4.q.b.529.1		2	52.15	even	12
1521.4.a.c.1.1		1	39.5	even	4
1521.4.a.j.1.1		1	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +27.0000 q^{10} -30.0000i q^{11} -3.00000 q^{12} +6.00000 q^{14} -27.0000i q^{15} -71.0000 q^{16} +111.000 q^{17} +27.0000i q^{18} -46.0000i q^{19} +9.00000i q^{20} -6.00000i q^{21} +90.0000 q^{22} +6.00000 q^{23} +63.0000i q^{24} +44.0000 q^{25} +27.0000 q^{27} +2.00000i q^{28} -105.000 q^{29} +81.0000 q^{30} -100.000i q^{31} -45.0000i q^{32} -90.0000i q^{33} +333.000i q^{34} -18.0000 q^{35} -9.00000 q^{36} -17.0000i q^{37} +138.000 q^{38} +189.000 q^{40} -231.000i q^{41} +18.0000 q^{42} +514.000 q^{43} +30.0000i q^{44} -81.0000i q^{45} +18.0000i q^{46} +162.000i q^{47} -213.000 q^{48} +339.000 q^{49} +132.000i q^{50} +333.000 q^{51} +639.000 q^{53} +81.0000i q^{54} -270.000 q^{55} +42.0000 q^{56} -138.000i q^{57} -315.000i q^{58} -600.000i q^{59} +27.0000i q^{60} +233.000 q^{61} +300.000 q^{62} -18.0000i q^{63} -433.000 q^{64} +270.000 q^{66} +926.000i q^{67} -111.000 q^{68} +18.0000 q^{69} -54.0000i q^{70} -930.000i q^{71} +189.000i q^{72} +253.000i q^{73} +51.0000 q^{74} +132.000 q^{75} +46.0000i q^{76} -60.0000 q^{77} -1324.00 q^{79} +639.000i q^{80} +81.0000 q^{81} +693.000 q^{82} +810.000i q^{83} +6.00000i q^{84} -999.000i q^{85} +1542.00i q^{86} -315.000 q^{87} +630.000 q^{88} -498.000i q^{89} +243.000 q^{90} -6.00000 q^{92} -300.000i q^{93} -486.000 q^{94} -414.000 q^{95} -135.000i q^{96} +1358.00i q^{97} +1017.00i q^{98} -270.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9	\( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9} + 54 q^{10} - 6 q^{12} + 12 q^{14} - 142 q^{16} + 222 q^{17} + 180 q^{22} + 12 q^{23} + 88 q^{25} + 54 q^{27} - 210 q^{29} + 162 q^{30} - 36 q^{35} - 18 q^{36} + 276 q^{38} + 378 q^{40} + 36 q^{42} + 1028 q^{43} - 426 q^{48} + 678 q^{49} + 666 q^{51} + 1278 q^{53} - 540 q^{55} + 84 q^{56} + 466 q^{61} + 600 q^{62} - 866 q^{64} + 540 q^{66} - 222 q^{68} + 36 q^{69} + 102 q^{74} + 264 q^{75} - 120 q^{77} - 2648 q^{79} + 162 q^{81} + 1386 q^{82} - 630 q^{87} + 1260 q^{88} + 486 q^{90} - 12 q^{92} - 972 q^{94} - 828 q^{95}+O(q^{100}) \) 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9 + 54 * q^10 - 6 * q^12 + 12 * q^14 - 142 * q^16 + 222 * q^17 + 180 * q^22 + 12 * q^23 + 88 * q^25 + 54 * q^27 - 210 * q^29 + 162 * q^30 - 36 * q^35 - 18 * q^36 + 276 * q^38 + 378 * q^40 + 36 * q^42 + 1028 * q^43 - 426 * q^48 + 678 * q^49 + 666 * q^51 + 1278 * q^53 - 540 * q^55 + 84 * q^56 + 466 * q^61 + 600 * q^62 - 866 * q^64 + 540 * q^66 - 222 * q^68 + 36 * q^69 + 102 * q^74 + 264 * q^75 - 120 * q^77 - 2648 * q^79 + 162 * q^81 + 1386 * q^82 - 630 * q^87 + 1260 * q^88 + 486 * q^90 - 12 * q^92 - 972 * q^94 - 828 * q^95

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