LMFDB - Embedded newform 125.2.b.b.124.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [125,2,Mod(124,125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(125, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("125.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 125 = 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	125.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:	$0.998130025266$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	125.124
Dual form			125.2.b.b.124.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{2} -2.61803i q^{3} +1.61803 q^{4} -1.61803 q^{6} +3.00000i q^{7} -2.23607i q^{8} -3.85410 q^{9} +O(q^{10})$	\(q-0.618034i q^{2} -2.61803i q^{3} +1.61803 q^{4} -1.61803 q^{6} +3.00000i q^{7} -2.23607i q^{8} -3.85410 q^{9} -3.00000 q^{11} -4.23607i q^{12} +1.85410i q^{13} +1.85410 q^{14} +1.85410 q^{16} +0.236068i q^{17} +2.38197i q^{18} +1.38197 q^{19} +7.85410 q^{21} +1.85410i q^{22} +3.23607i q^{23} -5.85410 q^{24} +1.14590 q^{26} +2.23607i q^{27} +4.85410i q^{28} +6.70820 q^{29} -6.09017 q^{31} -5.61803i q^{32} +7.85410i q^{33} +0.145898 q^{34} -6.23607 q^{36} +9.70820i q^{37} -0.854102i q^{38} +4.85410 q^{39} -3.00000 q^{41} -4.85410i q^{42} -9.00000i q^{43} -4.85410 q^{44} +2.00000 q^{46} -7.32624i q^{47} -4.85410i q^{48} -2.00000 q^{49} +0.618034 q^{51} +3.00000i q^{52} +2.38197i q^{53} +1.38197 q^{54} +6.70820 q^{56} -3.61803i q^{57} -4.14590i q^{58} -10.8541 q^{59} +5.09017 q^{61} +3.76393i q^{62} -11.5623i q^{63} +0.236068 q^{64} +4.85410 q^{66} +7.14590i q^{67} +0.381966i q^{68} +8.47214 q^{69} -3.00000 q^{71} +8.61803i q^{72} -4.85410i q^{73} +6.00000 q^{74} +2.23607 q^{76} -9.00000i q^{77} -3.00000i q^{78} -9.47214 q^{79} -5.70820 q^{81} +1.85410i q^{82} -8.47214i q^{83} +12.7082 q^{84} -5.56231 q^{86} -17.5623i q^{87} +6.70820i q^{88} -13.4164 q^{89} -5.56231 q^{91} +5.23607i q^{92} +15.9443i q^{93} -4.52786 q^{94} -14.7082 q^{96} -1.14590i q^{97} +1.23607i q^{98} +11.5623 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	$ 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 6 q^{14} - 6 q^{16} + 10 q^{19} + 18 q^{21} - 10 q^{24} + 18 q^{26} - 2 q^{31} + 14 q^{34} - 16 q^{36} + 6 q^{39} - 12 q^{41} - 6 q^{44} + 8 q^{46} - 8 q^{49} - 2 q^{51} + 10 q^{54} - 30 q^{59} - 2 q^{61} - 8 q^{64} + 6 q^{66} + 16 q^{69} - 12 q^{71} + 24 q^{74} - 20 q^{79} + 4 q^{81} + 24 q^{84} + 18 q^{86} + 18 q^{91} - 36 q^{94} - 32 q^{96} + 6 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 - 6 * q^14 - 6 * q^16 + 10 * q^19 + 18 * q^21 - 10 * q^24 + 18 * q^26 - 2 * q^31 + 14 * q^34 - 16 * q^36 + 6 * q^39 - 12 * q^41 - 6 * q^44 + 8 * q^46 - 8 * q^49 - 2 * q^51 + 10 * q^54 - 30 * q^59 - 2 * q^61 - 8 * q^64 + 6 * q^66 + 16 * q^69 - 12 * q^71 + 24 * q^74 - 20 * q^79 + 4 * q^81 + 24 * q^84 + 18 * q^86 + 18 * q^91 - 36 * q^94 - 32 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/125\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 0.618034i	− 0.437016i	−0.975835	−	0.218508i	$-0.929881\pi$
$2$	− 0.618034i	− 0.437016i	0.975835	−	0.218508i	$-0.0701190\pi$
$3$	− 2.61803i	− 1.51152i	−0.654847	−	0.755761i	$-0.727267\pi$
$3$	− 2.61803i	− 1.51152i	0.654847	−	0.755761i	$-0.272733\pi$
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−3.85410	−1.28470
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 4.23607i	− 1.22285i
$13$	1.85410i	0.514235i	0.966380	+	0.257118i	$0.0827728\pi$
$13$	1.85410i	0.514235i	−0.966380	+	0.257118i	$0.917227\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	0.236068i	0.0572549i	0.999590	+	0.0286274i	$0.00911364\pi$
$17$	0.236068i	0.0572549i	−0.999590	+	0.0286274i	$0.990886\pi$
$18$	2.38197i	0.561435i
$19$	1.38197	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$19$	1.38197	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$20$	0	0
$21$	7.85410	1.71391
$22$	1.85410i	0.395296i
$23$	3.23607i	0.674767i	0.941367	+	0.337383i	$0.109542\pi$
$23$	3.23607i	0.674767i	−0.941367	+	0.337383i	$0.890458\pi$
$24$	−5.85410	−1.19496
$25$	0	0
$26$	1.14590	0.224729
$27$	2.23607i	0.430331i
$28$	4.85410i	0.917339i
$29$	6.70820	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$29$	6.70820	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$30$	0	0
$31$	−6.09017	−1.09383	−0.546913	−	0.837189i	$-0.684197\pi$
$31$	−6.09017	−1.09383	−0.546913	+	0.837189i	$0.684197\pi$
$32$	− 5.61803i	− 0.993137i
$33$	7.85410i	1.36722i
$34$	0.145898	0.0250213
$35$	0	0
$36$	−6.23607	−1.03934
$37$	9.70820i	1.59602i	0.602645	+	0.798009i	$0.294114\pi$
$37$	9.70820i	1.59602i	−0.602645	+	0.798009i	$0.705886\pi$
$38$	− 0.854102i	− 0.138554i
$39$	4.85410	0.777278
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	− 4.85410i	− 0.749004i
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	−4.85410	−0.731783
$45$	0	0
$46$	2.00000	0.294884
$47$	− 7.32624i	− 1.06864i	−0.845282	−	0.534321i	$-0.820567\pi$
$47$	− 7.32624i	− 1.06864i	0.845282	−	0.534321i	$-0.179433\pi$
$48$	− 4.85410i	− 0.700629i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0.618034	0.0865421
$52$	3.00000i	0.416025i
$53$	2.38197i	0.327188i	0.986528	+	0.163594i	$0.0523087\pi$
$53$	2.38197i	0.327188i	−0.986528	+	0.163594i	$0.947691\pi$
$54$	1.38197	0.188062
$55$	0	0
$56$	6.70820	0.896421
$57$	− 3.61803i	− 0.479220i
$58$	− 4.14590i	− 0.544383i
$59$	−10.8541	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$59$	−10.8541	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$60$	0	0
$61$	5.09017	0.651729	0.325865	−	0.945416i	$-0.394345\pi$
$61$	5.09017	0.651729	0.325865	+	0.945416i	$0.394345\pi$
$62$	3.76393i	0.478020i
$63$	− 11.5623i	− 1.45671i
$64$	0.236068	0.0295085
$65$	0	0
$66$	4.85410	0.597499
$67$	7.14590i	0.873010i	0.899702	+	0.436505i	$0.143784\pi$
$67$	7.14590i	0.873010i	−0.899702	+	0.436505i	$0.856216\pi$
$68$	0.381966i	0.0463202i
$69$	8.47214	1.01993
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	8.61803i	1.01565i
$73$	− 4.85410i	− 0.568130i	−0.958805	−	0.284065i	$-0.908317\pi$
$73$	− 4.85410i	− 0.568130i	0.958805	−	0.284065i	$-0.0916831\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	2.23607	0.256495
$77$	− 9.00000i	− 1.02565i
$78$	− 3.00000i	− 0.339683i
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	1.85410i	0.204751i
$83$	− 8.47214i	− 0.929938i	−0.885327	−	0.464969i	$-0.846066\pi$
$83$	− 8.47214i	− 0.929938i	0.885327	−	0.464969i	$-0.153934\pi$
$84$	12.7082	1.38658
$85$	0	0
$86$	−5.56231	−0.599799
$87$	− 17.5623i	− 1.88288i
$88$	6.70820i	0.715097i
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	−5.56231	−0.583088
$92$	5.23607i	0.545898i
$93$	15.9443i	1.65334i
$94$	−4.52786	−0.467014
$95$	0	0
$96$	−14.7082	−1.50115
$97$	− 1.14590i	− 0.116348i	−0.998306	−	0.0581742i	$-0.981472\pi$
$97$	− 1.14590i	− 0.116348i	0.998306	−	0.0581742i	$-0.0185279\pi$
$98$	1.23607i	0.124862i
$99$	11.5623	1.16206
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	− 0.381966i	− 0.0378203i
$103$	− 0.708204i	− 0.0697814i	−0.999391	−	0.0348907i	$-0.988892\pi$
$103$	− 0.708204i	− 0.0697814i	0.999391	−	0.0348907i	$-0.0111083\pi$
$104$	4.14590	0.406539
$105$	0	0
$106$	1.47214	0.142986
$107$	11.0902i	1.07213i	0.844178	+	0.536064i	$0.180089\pi$
$107$	11.0902i	1.07213i	−0.844178	+	0.536064i	$0.819911\pi$
$108$	3.61803i	0.348145i
$109$	10.3262	0.989074	0.494537	−	0.869157i	$-0.335338\pi$
$109$	10.3262	0.989074	0.494537	+	0.869157i	$0.335338\pi$
$110$	0	0
$111$	25.4164	2.41242
$112$	5.56231i	0.525589i
$113$	19.9443i	1.87620i	0.346366	+	0.938100i	$0.387416\pi$
$113$	19.9443i	1.87620i	−0.346366	+	0.938100i	$0.612584\pi$
$114$	−2.23607	−0.209427
$115$	0	0
$116$	10.8541	1.00778
$117$	− 7.14590i	− 0.660639i
$118$	6.70820i	0.617540i
$119$	−0.708204	−0.0649209
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 3.14590i	− 0.284816i
$123$	7.85410i	0.708181i
$124$	−9.85410	−0.884924
$125$	0	0
$126$	−7.14590	−0.636607
$127$	− 3.70820i	− 0.329050i	−0.986373	−	0.164525i	$-0.947391\pi$
$127$	− 3.70820i	− 0.329050i	0.986373	−	0.164525i	$-0.0526091\pi$
$128$	− 11.3820i	− 1.00603i
$129$	−23.5623	−2.07455
$130$	0	0
$131$	21.2705	1.85841	0.929207	−	0.369561i	$-0.120492\pi$
$131$	21.2705	1.85841	0.929207	+	0.369561i	$0.120492\pi$
$132$	12.7082i	1.10611i
$133$	4.14590i	0.359495i
$134$	4.41641	0.381520
$135$	0	0
$136$	0.527864	0.0452640
$137$	− 3.38197i	− 0.288941i	−0.989509	−	0.144470i	$-0.953852\pi$
$137$	− 3.38197i	− 0.288941i	0.989509	−	0.144470i	$-0.0461479\pi$
$138$	− 5.23607i	− 0.445724i
$139$	7.76393	0.658528	0.329264	−	0.944238i	$-0.393199\pi$
$139$	7.76393	0.658528	0.329264	+	0.944238i	$0.393199\pi$
$140$	0	0
$141$	−19.1803	−1.61528
$142$	1.85410i	0.155593i
$143$	− 5.56231i	− 0.465143i
$144$	−7.14590	−0.595492
$145$	0	0
$146$	−3.00000	−0.248282
$147$	5.23607i	0.431864i
$148$	15.7082i	1.29121i
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−9.90983	−0.806451	−0.403225	−	0.915101i	$-0.632111\pi$
$151$	−9.90983	−0.806451	−0.403225	+	0.915101i	$0.632111\pi$
$152$	− 3.09017i	− 0.250646i
$153$	− 0.909830i	− 0.0735554i
$154$	−5.56231	−0.448223
$155$	0	0
$156$	7.85410	0.628831
$157$	− 21.2705i	− 1.69757i	−0.528737	−	0.848786i	$-0.677334\pi$
$157$	− 21.2705i	− 1.69757i	0.528737	−	0.848786i	$-0.322666\pi$
$158$	5.85410i	0.465727i
$159$	6.23607	0.494552
$160$	0	0
$161$	−9.70820	−0.765114
$162$	3.52786i	0.277175i
$163$	− 13.1459i	− 1.02967i	−0.857290	−	0.514833i	$-0.827854\pi$
$163$	− 13.1459i	− 1.02967i	0.857290	−	0.514833i	$-0.172146\pi$
$164$	−4.85410	−0.379042
$165$	0	0
$166$	−5.23607	−0.406398
$167$	− 4.76393i	− 0.368644i	−0.982866	−	0.184322i	$-0.940991\pi$
$167$	− 4.76393i	− 0.368644i	0.982866	−	0.184322i	$-0.0590089\pi$
$168$	− 17.5623i	− 1.35496i
$169$	9.56231	0.735562
$170$	0	0
$171$	−5.32624	−0.407308
$172$	− 14.5623i	− 1.11037i
$173$	− 17.9443i	− 1.36428i	−0.731223	−	0.682139i	$-0.761050\pi$
$173$	− 17.9443i	− 1.36428i	0.731223	−	0.682139i	$-0.238950\pi$
$174$	−10.8541	−0.822847
$175$	0	0
$176$	−5.56231	−0.419275
$177$	28.4164i	2.13591i
$178$	8.29180i	0.621496i
$179$	6.70820	0.501395	0.250697	−	0.968066i	$-0.419340\pi$
$179$	6.70820	0.501395	0.250697	+	0.968066i	$0.419340\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	3.43769i	0.254819i
$183$	− 13.3262i	− 0.985104i
$184$	7.23607	0.533450
$185$	0	0
$186$	9.85410	0.722538
$187$	− 0.708204i	− 0.0517890i
$188$	− 11.8541i	− 0.864549i
$189$	−6.70820	−0.487950
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	− 0.618034i	− 0.0446028i
$193$	− 18.2705i	− 1.31514i	−0.753393	−	0.657570i	$-0.771584\pi$
$193$	− 18.2705i	− 1.31514i	0.753393	−	0.657570i	$-0.228416\pi$
$194$	−0.708204	−0.0508461
$195$	0	0
$196$	−3.23607	−0.231148
$197$	21.9443i	1.56346i	0.623614	+	0.781732i	$0.285664\pi$
$197$	21.9443i	1.56346i	−0.623614	+	0.781732i	$0.714336\pi$
$198$	− 7.14590i	− 0.507837i
$199$	6.18034	0.438113	0.219056	−	0.975712i	$-0.429702\pi$
$199$	6.18034	0.438113	0.219056	+	0.975712i	$0.429702\pi$
$200$	0	0
$201$	18.7082	1.31957
$202$	1.85410i	0.130454i
$203$	20.1246i	1.41247i
$204$	1.00000	0.0700140
$205$	0	0
$206$	−0.437694	−0.0304956
$207$	− 12.4721i	− 0.866873i
$208$	3.43769i	0.238361i
$209$	−4.14590	−0.286778
$210$	0	0
$211$	18.1803	1.25159	0.625793	−	0.779989i	$-0.284775\pi$
$211$	18.1803	1.25159	0.625793	+	0.779989i	$0.284775\pi$
$212$	3.85410i	0.264701i
$213$	7.85410i	0.538154i
$214$	6.85410	0.468537
$215$	0	0
$216$	5.00000	0.340207
$217$	− 18.2705i	− 1.24028i
$218$	− 6.38197i	− 0.432241i
$219$	−12.7082	−0.858741
$220$	0	0
$221$	−0.437694	−0.0294425
$222$	− 15.7082i	− 1.05427i
$223$	− 4.85410i	− 0.325055i	−0.986704	−	0.162527i	$-0.948035\pi$
$223$	− 4.85410i	− 0.325055i	0.986704	−	0.162527i	$-0.0519646\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	12.3262	0.819929
$227$	4.38197i	0.290841i	0.989370	+	0.145421i	$0.0464535\pi$
$227$	4.38197i	0.290841i	−0.989370	+	0.145421i	$0.953546\pi$
$228$	− 5.85410i	− 0.387697i
$229$	3.61803	0.239086	0.119543	−	0.992829i	$-0.461857\pi$
$229$	3.61803	0.239086	0.119543	+	0.992829i	$0.461857\pi$
$230$	0	0
$231$	−23.5623	−1.55029
$232$	− 15.0000i	− 0.984798i
$233$	8.88854i	0.582308i	0.956676	+	0.291154i	$0.0940392\pi$
$233$	8.88854i	0.582308i	−0.956676	+	0.291154i	$0.905961\pi$
$234$	−4.41641	−0.288710
$235$	0	0
$236$	−17.5623	−1.14321
$237$	24.7984i	1.61083i
$238$	0.437694i	0.0283715i
$239$	−19.1459	−1.23845	−0.619223	−	0.785215i	$-0.712552\pi$
$239$	−19.1459	−1.23845	−0.619223	+	0.785215i	$0.712552\pi$
$240$	0	0
$241$	3.18034	0.204864	0.102432	−	0.994740i	$-0.467338\pi$
$241$	3.18034	0.204864	0.102432	+	0.994740i	$0.467338\pi$
$242$	1.23607i	0.0794575i
$243$	21.6525i	1.38901i
$244$	8.23607	0.527260
$245$	0	0
$246$	4.85410	0.309486
$247$	2.56231i	0.163036i
$248$	13.6180i	0.864746i
$249$	−22.1803	−1.40562
$250$	0	0
$251$	−27.2705	−1.72130	−0.860650	−	0.509198i	$-0.829942\pi$
$251$	−27.2705	−1.72130	−0.860650	+	0.509198i	$0.829942\pi$
$252$	− 18.7082i	− 1.17851i
$253$	− 9.70820i	− 0.610350i
$254$	−2.29180	−0.143800
$255$	0	0
$256$	−6.56231	−0.410144
$257$	15.3607i	0.958173i	0.877768	+	0.479086i	$0.159032\pi$
$257$	15.3607i	0.958173i	−0.877768	+	0.479086i	$0.840968\pi$
$258$	14.5623i	0.906610i
$259$	−29.1246	−1.80972
$260$	0	0
$261$	−25.8541	−1.60033
$262$	− 13.1459i	− 0.812156i
$263$	0.673762i	0.0415459i	0.999784	+	0.0207730i	$0.00661272\pi$
$263$	0.673762i	0.0415459i	−0.999784	+	0.0207730i	$0.993387\pi$
$264$	17.5623	1.08089
$265$	0	0
$266$	2.56231	0.157105
$267$	35.1246i	2.14959i
$268$	11.5623i	0.706280i
$269$	17.5623	1.07079	0.535396	−	0.844601i	$-0.320162\pi$
$269$	17.5623	1.07079	0.535396	+	0.844601i	$0.320162\pi$
$270$	0	0
$271$	8.90983	0.541234	0.270617	−	0.962687i	$-0.412772\pi$
$271$	8.90983	0.541234	0.270617	+	0.962687i	$0.412772\pi$
$272$	0.437694i	0.0265391i
$273$	14.5623i	0.881351i
$274$	−2.09017	−0.126272
$275$	0	0
$276$	13.7082	0.825137
$277$	− 18.7082i	− 1.12407i	−0.827114	−	0.562034i	$-0.810019\pi$
$277$	− 18.7082i	− 1.12407i	0.827114	−	0.562034i	$-0.189981\pi$
$278$	− 4.79837i	− 0.287787i
$279$	23.4721	1.40524
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	11.8541i	0.705901i
$283$	− 2.29180i	− 0.136233i	−0.997677	−	0.0681166i	$-0.978301\pi$
$283$	− 2.29180i	− 0.136233i	0.997677	−	0.0681166i	$-0.0216990\pi$
$284$	−4.85410	−0.288038
$285$	0	0
$286$	−3.43769	−0.203275
$287$	− 9.00000i	− 0.531253i
$288$	21.6525i	1.27588i
$289$	16.9443	0.996722
$290$	0	0
$291$	−3.00000	−0.175863
$292$	− 7.85410i	− 0.459627i
$293$	23.3607i	1.36475i	0.731004	+	0.682373i	$0.239052\pi$
$293$	23.3607i	1.36475i	−0.731004	+	0.682373i	$0.760948\pi$
$294$	3.23607	0.188731
$295$	0	0
$296$	21.7082	1.26176
$297$	− 6.70820i	− 0.389249i
$298$	− 9.27051i	− 0.537026i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	27.0000	1.55625
$302$	6.12461i	0.352432i
$303$	7.85410i	0.451206i
$304$	2.56231	0.146958
$305$	0	0
$306$	−0.562306	−0.0321449
$307$	27.2705i	1.55641i	0.628010	+	0.778205i	$0.283870\pi$
$307$	27.2705i	1.55641i	−0.628010	+	0.778205i	$0.716130\pi$
$308$	− 14.5623i	− 0.829764i
$309$	−1.85410	−0.105476
$310$	0	0
$311$	−12.2705	−0.695797	−0.347898	−	0.937532i	$-0.613105\pi$
$311$	−12.2705	−0.695797	−0.347898	+	0.937532i	$0.613105\pi$
$312$	− 10.8541i	− 0.614493i
$313$	− 7.41641i	− 0.419200i	−0.977787	−	0.209600i	$-0.932784\pi$
$313$	− 7.41641i	− 0.419200i	0.977787	−	0.209600i	$-0.0672162\pi$
$314$	−13.1459	−0.741866
$315$	0	0
$316$	−15.3262	−0.862168
$317$	9.05573i	0.508620i	0.967123	+	0.254310i	$0.0818484\pi$
$317$	9.05573i	0.508620i	−0.967123	+	0.254310i	$0.918152\pi$
$318$	− 3.85410i	− 0.216127i
$319$	−20.1246	−1.12676
$320$	0	0
$321$	29.0344	1.62054
$322$	6.00000i	0.334367i
$323$	0.326238i	0.0181524i
$324$	−9.23607	−0.513115
$325$	0	0
$326$	−8.12461	−0.449981
$327$	− 27.0344i	− 1.49501i
$328$	6.70820i	0.370399i
$329$	21.9787	1.21173
$330$	0	0
$331$	−9.90983	−0.544694	−0.272347	−	0.962199i	$-0.587800\pi$
$331$	−9.90983	−0.544694	−0.272347	+	0.962199i	$0.587800\pi$
$332$	− 13.7082i	− 0.752335i
$333$	− 37.4164i	− 2.05041i
$334$	−2.94427	−0.161103
$335$	0	0
$336$	14.5623	0.794439
$337$	− 23.8328i	− 1.29826i	−0.760679	−	0.649128i	$-0.775134\pi$
$337$	− 23.8328i	− 1.29826i	0.760679	−	0.649128i	$-0.224866\pi$
$338$	− 5.90983i	− 0.321452i
$339$	52.2148	2.83592
$340$	0	0
$341$	18.2705	0.989404
$342$	3.29180i	0.178000i
$343$	15.0000i	0.809924i
$344$	−20.1246	−1.08505
$345$	0	0
$346$	−11.0902	−0.596211
$347$	1.94427i	0.104374i	0.998637	+	0.0521870i	$0.0166192\pi$
$347$	1.94427i	0.104374i	−0.998637	+	0.0521870i	$0.983381\pi$
$348$	− 28.4164i	− 1.52328i
$349$	−27.3607	−1.46458	−0.732292	−	0.680991i	$-0.761549\pi$
$349$	−27.3607	−1.46458	−0.732292	+	0.680991i	$0.761549\pi$
$350$	0	0
$351$	−4.14590	−0.221292
$352$	16.8541i	0.898327i
$353$	− 31.8885i	− 1.69726i	−0.528990	−	0.848628i	$-0.677429\pi$
$353$	− 31.8885i	− 1.69726i	0.528990	−	0.848628i	$-0.322571\pi$
$354$	17.5623	0.933426
$355$	0	0
$356$	−21.7082	−1.15053
$357$	1.85410i	0.0981295i
$358$	− 4.14590i	− 0.219118i
$359$	4.14590	0.218812	0.109406	−	0.993997i	$-0.465105\pi$
$359$	4.14590	0.218812	0.109406	+	0.993997i	$0.465105\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	2.58359i	0.135791i
$363$	5.23607i	0.274822i
$364$	−9.00000	−0.471728
$365$	0	0
$366$	−8.23607	−0.430506
$367$	− 7.85410i	− 0.409981i	−0.978764	−	0.204990i	$-0.934284\pi$
$367$	− 7.85410i	− 0.409981i	0.978764	−	0.204990i	$-0.0657163\pi$
$368$	6.00000i	0.312772i
$369$	11.5623	0.601910
$370$	0	0
$371$	−7.14590	−0.370997
$372$	25.7984i	1.33758i
$373$	− 4.85410i	− 0.251336i	−0.992072	−	0.125668i	$-0.959893\pi$
$373$	− 4.85410i	− 0.251336i	0.992072	−	0.125668i	$-0.0401074\pi$
$374$	−0.437694	−0.0226326
$375$	0	0
$376$	−16.3820	−0.844835
$377$	12.4377i	0.640574i
$378$	4.14590i	0.213242i
$379$	−12.5623	−0.645282	−0.322641	−	0.946521i	$-0.604571\pi$
$379$	−12.5623	−0.645282	−0.322641	+	0.946521i	$0.604571\pi$
$380$	0	0
$381$	−9.70820	−0.497366
$382$	− 7.41641i	− 0.379456i
$383$	26.5279i	1.35551i	0.735288	+	0.677755i	$0.237047\pi$
$383$	26.5279i	1.35551i	−0.735288	+	0.677755i	$0.762953\pi$
$384$	−29.7984	−1.52064
$385$	0	0
$386$	−11.2918	−0.574737
$387$	34.6869i	1.76324i
$388$	− 1.85410i	− 0.0941278i
$389$	−28.4164	−1.44077	−0.720385	−	0.693575i	$-0.756035\pi$
$389$	−28.4164	−1.44077	−0.720385	+	0.693575i	$0.756035\pi$
$390$	0	0
$391$	−0.763932	−0.0386337
$392$	4.47214i	0.225877i
$393$	− 55.6869i	− 2.80903i
$394$	13.5623	0.683259
$395$	0	0
$396$	18.7082	0.940123
$397$	− 10.4164i	− 0.522785i	−0.965233	−	0.261392i	$-0.915818\pi$
$397$	− 10.4164i	− 0.522785i	0.965233	−	0.261392i	$-0.0841816\pi$
$398$	− 3.81966i	− 0.191462i
$399$	10.8541	0.543385
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	− 11.5623i	− 0.576675i
$403$	− 11.2918i	− 0.562484i
$404$	−4.85410	−0.241501
$405$	0	0
$406$	12.4377	0.617272
$407$	− 29.1246i	− 1.44365i
$408$	− 1.38197i	− 0.0684175i
$409$	27.2361	1.34674	0.673368	−	0.739307i	$-0.264847\pi$
$409$	27.2361	1.34674	0.673368	+	0.739307i	$0.264847\pi$
$410$	0	0
$411$	−8.85410	−0.436741
$412$	− 1.14590i	− 0.0564543i
$413$	− 32.5623i	− 1.60229i
$414$	−7.70820	−0.378838
$415$	0	0
$416$	10.4164	0.510706
$417$	− 20.3262i	− 0.995380i
$418$	2.56231i	0.125326i
$419$	−6.70820	−0.327717	−0.163859	−	0.986484i	$-0.552394\pi$
$419$	−6.70820	−0.327717	−0.163859	+	0.986484i	$0.552394\pi$
$420$	0	0
$421$	20.0902	0.979135	0.489567	−	0.871965i	$-0.337155\pi$
$421$	20.0902	0.979135	0.489567	+	0.871965i	$0.337155\pi$
$422$	− 11.2361i	− 0.546963i
$423$	28.2361i	1.37288i
$424$	5.32624	0.258665
$425$	0	0
$426$	4.85410	0.235182
$427$	15.2705i	0.738992i
$428$	17.9443i	0.867369i
$429$	−14.5623	−0.703075
$430$	0	0
$431$	21.2705	1.02456	0.512282	−	0.858817i	$-0.328800\pi$
$431$	21.2705	1.02456	0.512282	+	0.858817i	$0.328800\pi$
$432$	4.14590i	0.199470i
$433$	27.7082i	1.33157i	0.746143	+	0.665786i	$0.231903\pi$
$433$	27.7082i	1.33157i	−0.746143	+	0.665786i	$0.768097\pi$
$434$	−11.2918	−0.542024
$435$	0	0
$436$	16.7082	0.800178
$437$	4.47214i	0.213931i
$438$	7.85410i	0.375284i
$439$	10.1246	0.483221	0.241611	−	0.970373i	$-0.422324\pi$
$439$	10.1246	0.483221	0.241611	+	0.970373i	$0.422324\pi$
$440$	0	0
$441$	7.70820	0.367057
$442$	0.270510i	0.0128668i
$443$	− 15.1803i	− 0.721240i	−0.932713	−	0.360620i	$-0.882565\pi$
$443$	− 15.1803i	− 0.721240i	0.932713	−	0.360620i	$-0.117435\pi$
$444$	41.1246	1.95169
$445$	0	0
$446$	−3.00000	−0.142054
$447$	− 39.2705i	− 1.85743i
$448$	0.708204i	0.0334595i
$449$	5.72949	0.270391	0.135196	−	0.990819i	$-0.456834\pi$
$449$	5.72949	0.270391	0.135196	+	0.990819i	$0.456834\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	32.2705i	1.51788i
$453$	25.9443i	1.21897i
$454$	2.70820	0.127102
$455$	0	0
$456$	−8.09017	−0.378857
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	− 2.23607i	− 0.104485i
$459$	−0.527864	−0.0246386
$460$	0	0
$461$	2.72949	0.127125	0.0635625	−	0.997978i	$-0.479754\pi$
$461$	2.72949	0.127125	0.0635625	+	0.997978i	$0.479754\pi$
$462$	14.5623i	0.677500i
$463$	26.1246i	1.21411i	0.794658	+	0.607057i	$0.207650\pi$
$463$	26.1246i	1.21411i	−0.794658	+	0.607057i	$0.792350\pi$
$464$	12.4377	0.577405
$465$	0	0
$466$	5.49342	0.254478
$467$	− 34.7639i	− 1.60868i	−0.594167	−	0.804341i	$-0.702518\pi$
$467$	− 34.7639i	− 1.60868i	0.594167	−	0.804341i	$-0.297482\pi$
$468$	− 11.5623i	− 0.534468i
$469$	−21.4377	−0.989901
$470$	0	0
$471$	−55.6869	−2.56592
$472$	24.2705i	1.11714i
$473$	27.0000i	1.24146i
$474$	15.3262	0.703957
$475$	0	0
$476$	−1.14590	−0.0525222
$477$	− 9.18034i	− 0.420339i
$478$	11.8328i	0.541220i
$479$	12.4377	0.568293	0.284146	−	0.958781i	$-0.408290\pi$
$479$	12.4377	0.568293	0.284146	+	0.958781i	$0.408290\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	− 1.96556i	− 0.0895287i
$483$	25.4164i	1.15649i
$484$	−3.23607	−0.147094
$485$	0	0
$486$	13.3820	0.607018
$487$	9.70820i	0.439921i	0.975509	+	0.219960i	$0.0705928\pi$
$487$	9.70820i	0.439921i	−0.975509	+	0.219960i	$0.929407\pi$
$488$	− 11.3820i	− 0.515237i
$489$	−34.4164	−1.55636
$490$	0	0
$491$	2.72949	0.123180	0.0615901	−	0.998102i	$-0.480383\pi$
$491$	2.72949	0.123180	0.0615901	+	0.998102i	$0.480383\pi$
$492$	12.7082i	0.572930i
$493$	1.58359i	0.0713214i
$494$	1.58359	0.0712492
$495$	0	0
$496$	−11.2918	−0.507017
$497$	− 9.00000i	− 0.403705i
$498$	13.7082i	0.614279i
$499$	2.36068	0.105679	0.0528393	−	0.998603i	$-0.483173\pi$
$499$	2.36068	0.105679	0.0528393	+	0.998603i	$0.483173\pi$
$500$	0	0
$501$	−12.4721	−0.557214
$502$	16.8541i	0.752235i
$503$	1.20163i	0.0535779i	0.999641	+	0.0267889i	$0.00852820\pi$
$503$	1.20163i	0.0535779i	−0.999641	+	0.0267889i	$0.991472\pi$
$504$	−25.8541	−1.15163
$505$	0	0
$506$	−6.00000	−0.266733
$507$	− 25.0344i	− 1.11182i
$508$	− 6.00000i	− 0.266207i
$509$	13.4164	0.594672	0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164	0.594672	0.297336	+	0.954773i	$0.403902\pi$
$510$	0	0
$511$	14.5623	0.644198
$512$	− 18.7082i	− 0.826794i
$513$	3.09017i	0.136434i
$514$	9.49342	0.418737
$515$	0	0
$516$	−38.1246	−1.67834
$517$	21.9787i	0.966623i
$518$	18.0000i	0.790875i
$519$	−46.9787	−2.06214
$520$	0	0
$521$	−27.2705	−1.19474	−0.597371	−	0.801965i	$-0.703788\pi$
$521$	−27.2705	−1.19474	−0.597371	+	0.801965i	$0.703788\pi$
$522$	15.9787i	0.699369i
$523$	− 10.5836i	− 0.462788i	−0.972860	−	0.231394i	$-0.925671\pi$
$523$	− 10.5836i	− 0.462788i	0.972860	−	0.231394i	$-0.0743287\pi$
$524$	34.4164	1.50349
$525$	0	0
$526$	0.416408	0.0181562
$527$	− 1.43769i	− 0.0626269i
$528$	14.5623i	0.633743i
$529$	12.5279	0.544690
$530$	0	0
$531$	41.8328	1.81539
$532$	6.70820i	0.290838i
$533$	− 5.56231i	− 0.240930i
$534$	21.7082	0.939406
$535$	0	0
$536$	15.9787	0.690175
$537$	− 17.5623i	− 0.757869i
$538$	− 10.8541i	− 0.467954i
$539$	6.00000	0.258438
$540$	0	0
$541$	−11.8197	−0.508167	−0.254083	−	0.967182i	$-0.581774\pi$
$541$	−11.8197	−0.508167	−0.254083	+	0.967182i	$0.581774\pi$
$542$	− 5.50658i	− 0.236528i
$543$	10.9443i	0.469664i
$544$	1.32624	0.0568620
$545$	0	0
$546$	9.00000	0.385164
$547$	28.8541i	1.23371i	0.787076	+	0.616856i	$0.211594\pi$
$547$	28.8541i	1.23371i	−0.787076	+	0.616856i	$0.788406\pi$
$548$	− 5.47214i	− 0.233758i
$549$	−19.6180	−0.837277
$550$	0	0
$551$	9.27051	0.394937
$552$	− 18.9443i	− 0.806322i
$553$	− 28.4164i	− 1.20839i
$554$	−11.5623	−0.491235
$555$	0	0
$556$	12.5623	0.532760
$557$	− 4.36068i	− 0.184768i	−0.995723	−	0.0923840i	$-0.970551\pi$
$557$	− 4.36068i	− 0.184768i	0.995723	−	0.0923840i	$-0.0294487\pi$
$558$	− 14.5066i	− 0.614112i
$559$	16.6869	0.705781
$560$	0	0
$561$	−1.85410	−0.0782802
$562$	− 7.41641i	− 0.312842i
$563$	24.9443i	1.05128i	0.850708	+	0.525638i	$0.176173\pi$
$563$	24.9443i	1.05128i	−0.850708	+	0.525638i	$0.823827\pi$
$564$	−31.0344	−1.30679
$565$	0	0
$566$	−1.41641	−0.0595361
$567$	− 17.1246i	− 0.719166i
$568$	6.70820i	0.281470i
$569$	26.8328	1.12489	0.562445	−	0.826835i	$-0.309861\pi$
$569$	26.8328	1.12489	0.562445	+	0.826835i	$0.309861\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	− 9.00000i	− 0.376309i
$573$	− 31.4164i	− 1.31244i
$574$	−5.56231	−0.232166
$575$	0	0
$576$	−0.909830	−0.0379096
$577$	− 9.43769i	− 0.392896i	−0.980514	−	0.196448i	$-0.937059\pi$
$577$	− 9.43769i	− 0.392896i	0.980514	−	0.196448i	$-0.0629407\pi$
$578$	− 10.4721i	− 0.435583i
$579$	−47.8328	−1.98786
$580$	0	0
$581$	25.4164	1.05445
$582$	1.85410i	0.0768550i
$583$	− 7.14590i	− 0.295953i
$584$	−10.8541	−0.449146
$585$	0	0
$586$	14.4377	0.596416
$587$	− 6.47214i	− 0.267134i	−0.991040	−	0.133567i	$-0.957357\pi$
$587$	− 6.47214i	− 0.267134i	0.991040	−	0.133567i	$-0.0426431\pi$
$588$	8.47214i	0.349385i
$589$	−8.41641	−0.346792
$590$	0	0
$591$	57.4508	2.36321
$592$	18.0000i	0.739795i
$593$	3.36068i	0.138007i	0.997616	+	0.0690033i	$0.0219819\pi$
$593$	3.36068i	0.138007i	−0.997616	+	0.0690033i	$0.978018\pi$
$594$	−4.14590	−0.170108
$595$	0	0
$596$	24.2705	0.994159
$597$	− 16.1803i	− 0.662217i
$598$	3.70820i	0.151640i
$599$	−9.27051	−0.378783	−0.189391	−	0.981902i	$-0.560651\pi$
$599$	−9.27051	−0.378783	−0.189391	+	0.981902i	$0.560651\pi$
$600$	0	0
$601$	−45.3607	−1.85030	−0.925150	−	0.379601i	$-0.876061\pi$
$601$	−45.3607	−1.85030	−0.925150	+	0.379601i	$0.876061\pi$
$602$	− 16.6869i	− 0.680108i
$603$	− 27.5410i	− 1.12156i
$604$	−16.0344	−0.652432
$605$	0	0
$606$	4.85410	0.197184
$607$	36.5410i	1.48315i	0.670868	+	0.741577i	$0.265922\pi$
$607$	36.5410i	1.48315i	−0.670868	+	0.741577i	$0.734078\pi$
$608$	− 7.76393i	− 0.314869i
$609$	52.6869	2.13498
$610$	0	0
$611$	13.5836	0.549533
$612$	− 1.47214i	− 0.0595076i
$613$	− 22.4164i	− 0.905390i	−0.891665	−	0.452695i	$-0.850463\pi$
$613$	− 22.4164i	− 0.905390i	0.891665	−	0.452695i	$-0.149537\pi$
$614$	16.8541	0.680176
$615$	0	0
$616$	−20.1246	−0.810844
$617$	30.2361i	1.21726i	0.793455	+	0.608629i	$0.208280\pi$
$617$	30.2361i	1.21726i	−0.793455	+	0.608629i	$0.791720\pi$
$618$	1.14590i	0.0460948i
$619$	29.4721	1.18459	0.592293	−	0.805723i	$-0.298223\pi$
$619$	29.4721	1.18459	0.592293	+	0.805723i	$0.298223\pi$
$620$	0	0
$621$	−7.23607	−0.290373
$622$	7.58359i	0.304074i
$623$	− 40.2492i	− 1.61255i
$624$	9.00000	0.360288
$625$	0	0
$626$	−4.58359	−0.183197
$627$	10.8541i	0.433471i
$628$	− 34.4164i	− 1.37336i
$629$	−2.29180	−0.0913799
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	21.1803i	0.842509i
$633$	− 47.5967i	− 1.89180i
$634$	5.59675	0.222275
$635$	0	0
$636$	10.0902	0.400101
$637$	− 3.70820i	− 0.146924i
$638$	12.4377i	0.492413i
$639$	11.5623	0.457398
$640$	0	0
$641$	−42.2705	−1.66958	−0.834792	−	0.550565i	$-0.814412\pi$
$641$	−42.2705	−1.66958	−0.834792	+	0.550565i	$0.814412\pi$
$642$	− 17.9443i	− 0.708204i
$643$	15.2705i	0.602210i	0.953591	+	0.301105i	$0.0973555\pi$
$643$	15.2705i	0.602210i	−0.953591	+	0.301105i	$0.902645\pi$
$644$	−15.7082	−0.618990
$645$	0	0
$646$	0.201626	0.00793287
$647$	15.7639i	0.619744i	0.950778	+	0.309872i	$0.100286\pi$
$647$	15.7639i	0.619744i	−0.950778	+	0.309872i	$0.899714\pi$
$648$	12.7639i	0.501415i
$649$	32.5623	1.27818
$650$	0	0
$651$	−47.8328	−1.87472
$652$	− 21.2705i	− 0.833017i
$653$	26.6525i	1.04299i	0.853254	+	0.521496i	$0.174626\pi$
$653$	26.6525i	1.04299i	−0.853254	+	0.521496i	$0.825374\pi$
$654$	−16.7082	−0.653342
$655$	0	0
$656$	−5.56231	−0.217172
$657$	18.7082i	0.729877i
$658$	− 13.5836i	− 0.529544i
$659$	19.1459	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$659$	19.1459	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$660$	0	0
$661$	−30.3607	−1.18089	−0.590447	−	0.807077i	$-0.701048\pi$
$661$	−30.3607	−1.18089	−0.590447	+	0.807077i	$0.701048\pi$
$662$	6.12461i	0.238040i
$663$	1.14590i	0.0445030i
$664$	−18.9443	−0.735180
$665$	0	0
$666$	−23.1246	−0.896061
$667$	21.7082i	0.840545i
$668$	− 7.70820i	− 0.298239i
$669$	−12.7082	−0.491328
$670$	0	0
$671$	−15.2705	−0.589511
$672$	− 44.1246i	− 1.70214i
$673$	28.6869i	1.10580i	0.833248	+	0.552900i	$0.186479\pi$
$673$	28.6869i	1.10580i	−0.833248	+	0.552900i	$0.813521\pi$
$674$	−14.7295	−0.567359
$675$	0	0
$676$	15.4721	0.595082
$677$	− 39.8885i	− 1.53304i	−0.642220	−	0.766521i	$-0.721986\pi$
$677$	− 39.8885i	− 1.53304i	0.642220	−	0.766521i	$-0.278014\pi$
$678$	− 32.2705i	− 1.23934i
$679$	3.43769	0.131927
$680$	0	0
$681$	11.4721	0.439613
$682$	− 11.2918i	− 0.432385i
$683$	15.0689i	0.576595i	0.957541	+	0.288297i	$0.0930892\pi$
$683$	15.0689i	0.576595i	−0.957541	+	0.288297i	$0.906911\pi$
$684$	−8.61803	−0.329519
$685$	0	0
$686$	9.27051	0.353950
$687$	− 9.47214i	− 0.361385i
$688$	− 16.6869i	− 0.636183i
$689$	−4.41641	−0.168252
$690$	0	0
$691$	48.1803	1.83287	0.916433	−	0.400188i	$-0.131055\pi$
$691$	48.1803	1.83287	0.916433	+	0.400188i	$0.131055\pi$
$692$	− 29.0344i	− 1.10372i
$693$	34.6869i	1.31765i
$694$	1.20163	0.0456131
$695$	0	0
$696$	−39.2705	−1.48854
$697$	− 0.708204i	− 0.0268251i
$698$	16.9098i	0.640047i
$699$	23.2705	0.880172
$700$	0	0
$701$	36.2705	1.36992	0.684959	−	0.728581i	$-0.259820\pi$
$701$	36.2705	1.36992	0.684959	+	0.728581i	$0.259820\pi$
$702$	2.56231i	0.0967080i
$703$	13.4164i	0.506009i
$704$	−0.708204	−0.0266914
$705$	0	0
$706$	−19.7082	−0.741728
$707$	− 9.00000i	− 0.338480i
$708$	45.9787i	1.72799i
$709$	−12.0344	−0.451963	−0.225981	−	0.974132i	$-0.572559\pi$
$709$	−12.0344	−0.451963	−0.225981	+	0.974132i	$0.572559\pi$
$710$	0	0
$711$	36.5066	1.36910
$712$	30.0000i	1.12430i
$713$	− 19.7082i	− 0.738078i
$714$	1.14590	0.0428842
$715$	0	0
$716$	10.8541	0.405637
$717$	50.1246i	1.87194i
$718$	− 2.56231i	− 0.0956244i
$719$	−27.4377	−1.02325	−0.511627	−	0.859208i	$-0.670957\pi$
$719$	−27.4377	−1.02325	−0.511627	+	0.859208i	$0.670957\pi$
$720$	0	0
$721$	2.12461	0.0791247
$722$	10.5623i	0.393088i
$723$	− 8.32624i	− 0.309656i
$724$	−6.76393	−0.251380
$725$	0	0
$726$	3.23607	0.120102
$727$	14.8328i	0.550119i	0.961427	+	0.275059i	$0.0886975\pi$
$727$	14.8328i	0.550119i	−0.961427	+	0.275059i	$0.911302\pi$
$728$	12.4377i	0.460972i
$729$	39.5623	1.46527
$730$	0	0
$731$	2.12461	0.0785816
$732$	− 21.5623i	− 0.796966i
$733$	3.43769i	0.126974i	0.997983	+	0.0634871i	$0.0202222\pi$
$733$	3.43769i	0.126974i	−0.997983	+	0.0634871i	$0.979778\pi$
$734$	−4.85410	−0.179168
$735$	0	0
$736$	18.1803	0.670136
$737$	− 21.4377i	− 0.789668i
$738$	− 7.14590i	− 0.263044i
$739$	−22.2361	−0.817967	−0.408983	−	0.912542i	$-0.634117\pi$
$739$	−22.2361	−0.817967	−0.408983	+	0.912542i	$0.634117\pi$
$740$	0	0
$741$	6.70820	0.246432
$742$	4.41641i	0.162131i
$743$	− 37.0902i	− 1.36071i	−0.732884	−	0.680353i	$-0.761826\pi$
$743$	− 37.0902i	− 1.36071i	0.732884	−	0.680353i	$-0.238174\pi$
$744$	35.6525	1.30708
$745$	0	0
$746$	−3.00000	−0.109838
$747$	32.6525i	1.19469i
$748$	− 1.14590i	− 0.0418982i
$749$	−33.2705	−1.21568
$750$	0	0
$751$	29.3607	1.07139	0.535693	−	0.844413i	$-0.320050\pi$
$751$	29.3607	1.07139	0.535693	+	0.844413i	$0.320050\pi$
$752$	− 13.5836i	− 0.495343i
$753$	71.3951i	2.60178i
$754$	7.68692	0.279941
$755$	0	0
$756$	−10.8541	−0.394760
$757$	− 27.0000i	− 0.981332i	−0.871348	−	0.490666i	$-0.836754\pi$
$757$	− 27.0000i	− 0.981332i	0.871348	−	0.490666i	$-0.163246\pi$
$758$	7.76393i	0.281999i
$759$	−25.4164	−0.922557
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	6.00000i	0.217357i
$763$	30.9787i	1.12150i
$764$	19.4164	0.702461
$765$	0	0
$766$	16.3951	0.592380
$767$	− 20.1246i	− 0.726658i
$768$	17.1803i	0.619942i
$769$	−22.8885	−0.825382	−0.412691	−	0.910871i	$-0.635411\pi$
$769$	−22.8885	−0.825382	−0.412691	+	0.910871i	$0.635411\pi$
$770$	0	0
$771$	40.2148	1.44830
$772$	− 29.5623i	− 1.06397i
$773$	− 16.7639i	− 0.602957i	−0.953473	−	0.301478i	$-0.902520\pi$
$773$	− 16.7639i	− 0.602957i	0.953473	−	0.301478i	$-0.0974801\pi$
$774$	21.4377	0.770562
$775$	0	0
$776$	−2.56231	−0.0919814
$777$	76.2492i	2.73543i
$778$	17.5623i	0.629639i
$779$	−4.14590	−0.148542
$780$	0	0
$781$	9.00000	0.322045
$782$	0.472136i	0.0168835i
$783$	15.0000i	0.536056i
$784$	−3.70820	−0.132436
$785$	0	0
$786$	−34.4164	−1.22759
$787$	24.7082i	0.880752i	0.897813	+	0.440376i	$0.145155\pi$
$787$	24.7082i	0.880752i	−0.897813	+	0.440376i	$0.854845\pi$
$788$	35.5066i	1.26487i
$789$	1.76393	0.0627976
$790$	0	0
$791$	−59.8328	−2.12741
$792$	− 25.8541i	− 0.918686i
$793$	9.43769i	0.335142i
$794$	−6.43769	−0.228465
$795$	0	0
$796$	10.0000	0.354441
$797$	16.2148i	0.574357i	0.957877	+	0.287179i	$0.0927173\pi$
$797$	16.2148i	0.574357i	−0.957877	+	0.287179i	$0.907283\pi$
$798$	− 6.70820i	− 0.237468i
$799$	1.72949	0.0611850
$800$	0	0
$801$	51.7082	1.82702
$802$	− 7.41641i	− 0.261882i
$803$	14.5623i	0.513893i
$804$	30.2705	1.06756
$805$	0	0
$806$	−6.97871	−0.245815
$807$	− 45.9787i	− 1.61853i
$808$	6.70820i	0.235994i
$809$	−35.1246	−1.23492	−0.617458	−	0.786604i	$-0.711837\pi$
$809$	−35.1246	−1.23492	−0.617458	+	0.786604i	$0.711837\pi$
$810$	0	0
$811$	−34.1803	−1.20023	−0.600117	−	0.799912i	$-0.704879\pi$
$811$	−34.1803	−1.20023	−0.600117	+	0.799912i	$0.704879\pi$
$812$	32.5623i	1.14271i
$813$	− 23.3262i	− 0.818087i
$814$	−18.0000	−0.630900
$815$	0	0
$816$	1.14590	0.0401145
$817$	− 12.4377i	− 0.435140i
$818$	− 16.8328i	− 0.588546i
$819$	21.4377	0.749094
$820$	0	0
$821$	−12.2705	−0.428244	−0.214122	−	0.976807i	$-0.568689\pi$
$821$	−12.2705	−0.428244	−0.214122	+	0.976807i	$0.568689\pi$
$822$	5.47214i	0.190863i
$823$	4.41641i	0.153946i	0.997033	+	0.0769732i	$0.0245256\pi$
$823$	4.41641i	0.153946i	−0.997033	+	0.0769732i	$0.975474\pi$
$824$	−1.58359	−0.0551670
$825$	0	0
$826$	−20.1246	−0.700225
$827$	29.3820i	1.02171i	0.859667	+	0.510856i	$0.170671\pi$
$827$	29.3820i	1.02171i	−0.859667	+	0.510856i	$0.829329\pi$
$828$	− 20.1803i	− 0.701315i
$829$	−48.7426	−1.69290	−0.846451	−	0.532467i	$-0.821265\pi$
$829$	−48.7426	−1.69290	−0.846451	+	0.532467i	$0.821265\pi$
$830$	0	0
$831$	−48.9787	−1.69905
$832$	0.437694i	0.0151743i
$833$	− 0.472136i	− 0.0163585i
$834$	−12.5623	−0.434997
$835$	0	0
$836$	−6.70820	−0.232008
$837$	− 13.6180i	− 0.470708i
$838$	4.14590i	0.143218i
$839$	44.3951	1.53269	0.766345	−	0.642429i	$-0.222073\pi$
$839$	44.3951	1.53269	0.766345	+	0.642429i	$0.222073\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	− 12.4164i	− 0.427898i
$843$	− 31.4164i	− 1.08204i
$844$	29.4164	1.01255
$845$	0	0
$846$	17.4508	0.599973
$847$	− 6.00000i	− 0.206162i
$848$	4.41641i	0.151660i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	−31.4164	−1.07694
$852$	12.7082i	0.435376i
$853$	47.8328i	1.63776i	0.573962	+	0.818882i	$0.305406\pi$
$853$	47.8328i	1.63776i	−0.573962	+	0.818882i	$0.694594\pi$
$854$	9.43769	0.322951
$855$	0	0
$856$	24.7984	0.847591
$857$	11.8197i	0.403752i	0.979411	+	0.201876i	$0.0647038\pi$
$857$	11.8197i	0.403752i	−0.979411	+	0.201876i	$0.935296\pi$
$858$	9.00000i	0.307255i
$859$	−15.1246	−0.516045	−0.258023	−	0.966139i	$-0.583071\pi$
$859$	−15.1246	−0.516045	−0.258023	+	0.966139i	$0.583071\pi$
$860$	0	0
$861$	−23.5623	−0.803001
$862$	− 13.1459i	− 0.447751i
$863$	− 11.2361i	− 0.382480i	−0.981543	−	0.191240i	$-0.938749\pi$
$863$	− 11.2361i	− 0.382480i	0.981543	−	0.191240i	$-0.0612509\pi$
$864$	12.5623	0.427378
$865$	0	0
$866$	17.1246	0.581918
$867$	− 44.3607i	− 1.50657i
$868$	− 29.5623i	− 1.00341i
$869$	28.4164	0.963961
$870$	0	0
$871$	−13.2492	−0.448933
$872$	− 23.0902i	− 0.781932i
$873$	4.41641i	0.149473i
$874$	2.76393	0.0934914
$875$	0	0
$876$	−20.5623	−0.694736
$877$	− 27.9787i	− 0.944774i	−0.881391	−	0.472387i	$-0.843392\pi$
$877$	− 27.9787i	− 0.944774i	0.881391	−	0.472387i	$-0.156608\pi$
$878$	− 6.25735i	− 0.211175i
$879$	61.1591	2.06284
$880$	0	0
$881$	−21.5410	−0.725735	−0.362868	−	0.931841i	$-0.618202\pi$
$881$	−21.5410	−0.725735	−0.362868	+	0.931841i	$0.618202\pi$
$882$	− 4.76393i	− 0.160410i
$883$	− 35.8328i	− 1.20587i	−0.797790	−	0.602935i	$-0.793998\pi$
$883$	− 35.8328i	− 1.20587i	0.797790	−	0.602935i	$-0.206002\pi$
$884$	−0.708204	−0.0238195
$885$	0	0
$886$	−9.38197	−0.315193
$887$	33.5279i	1.12576i	0.826540	+	0.562878i	$0.190306\pi$
$887$	33.5279i	1.12576i	−0.826540	+	0.562878i	$0.809694\pi$
$888$	− 56.8328i	− 1.90718i
$889$	11.1246	0.373108
$890$	0	0
$891$	17.1246	0.573696
$892$	− 7.85410i	− 0.262975i
$893$	− 10.1246i	− 0.338807i
$894$	−24.2705	−0.811727
$895$	0	0
$896$	34.1459	1.14073
$897$	15.7082i	0.524482i
$898$	− 3.54102i	− 0.118165i
$899$	−40.8541	−1.36256
$900$	0	0
$901$	−0.562306	−0.0187331
$902$	− 5.56231i	− 0.185205i
$903$	− 70.6869i	− 2.35231i
$904$	44.5967	1.48327
$905$	0	0
$906$	16.0344	0.532709
$907$	8.72949i	0.289858i	0.989442	+	0.144929i	$0.0462954\pi$
$907$	8.72949i	0.289858i	−0.989442	+	0.144929i	$0.953705\pi$
$908$	7.09017i	0.235296i
$909$	11.5623	0.383497
$910$	0	0
$911$	30.5410	1.01187	0.505935	−	0.862572i	$-0.331148\pi$
$911$	30.5410	1.01187	0.505935	+	0.862572i	$0.331148\pi$
$912$	− 6.70820i	− 0.222131i
$913$	25.4164i	0.841160i
$914$	11.1246	0.367969
$915$	0	0
$916$	5.85410	0.193425
$917$	63.8115i	2.10724i
$918$	0.326238i	0.0107675i
$919$	−34.0689	−1.12383	−0.561914	−	0.827195i	$-0.689935\pi$
$919$	−34.0689	−1.12383	−0.561914	+	0.827195i	$0.689935\pi$
$920$	0	0
$921$	71.3951	2.35255
$922$	− 1.68692i	− 0.0555557i
$923$	− 5.56231i	− 0.183086i
$924$	−38.1246	−1.25421
$925$	0	0
$926$	16.1459	0.530587
$927$	2.72949i	0.0896482i
$928$	− 37.6869i	− 1.23713i
$929$	−23.2918	−0.764179	−0.382090	−	0.924125i	$-0.624795\pi$
$929$	−23.2918	−0.764179	−0.382090	+	0.924125i	$0.624795\pi$
$930$	0	0
$931$	−2.76393	−0.0905842
$932$	14.3820i	0.471097i
$933$	32.1246i	1.05171i
$934$	−21.4853	−0.703020
$935$	0	0
$936$	−15.9787	−0.522281
$937$	− 29.5623i	− 0.965758i	−0.875687	−	0.482879i	$-0.839591\pi$
$937$	− 29.5623i	− 0.965758i	0.875687	−	0.482879i	$-0.160409\pi$
$938$	13.2492i	0.432602i
$939$	−19.4164	−0.633631
$940$	0	0
$941$	6.27051	0.204413	0.102206	−	0.994763i	$-0.467410\pi$
$941$	6.27051	0.204413	0.102206	+	0.994763i	$0.467410\pi$
$942$	34.4164i	1.12135i
$943$	− 9.70820i	− 0.316143i
$944$	−20.1246	−0.655000
$945$	0	0
$946$	16.6869	0.542538
$947$	− 26.5967i	− 0.864278i	−0.901807	−	0.432139i	$-0.857759\pi$
$947$	− 26.5967i	− 0.864278i	0.901807	−	0.432139i	$-0.142241\pi$
$948$	40.1246i	1.30319i
$949$	9.00000	0.292152
$950$	0	0
$951$	23.7082	0.768791
$952$	1.58359i	0.0513245i
$953$	− 56.1591i	− 1.81917i	−0.415518	−	0.909585i	$-0.636400\pi$
$953$	− 56.1591i	− 1.81917i	0.415518	−	0.909585i	$-0.363600\pi$
$954$	−5.67376	−0.183695
$955$	0	0
$956$	−30.9787	−1.00192
$957$	52.6869i	1.70313i
$958$	− 7.68692i	− 0.248353i
$959$	10.1459	0.327628
$960$	0	0
$961$	6.09017	0.196457
$962$	11.1246i	0.358672i
$963$	− 42.7426i	− 1.37736i
$964$	5.14590	0.165738
$965$	0	0
$966$	15.7082	0.505403
$967$	− 22.8541i	− 0.734938i	−0.930036	−	0.367469i	$-0.880224\pi$
$967$	− 22.8541i	− 0.734938i	0.930036	−	0.367469i	$-0.119776\pi$
$968$	4.47214i	0.143740i
$969$	0.854102	0.0274377
$970$	0	0
$971$	15.5410	0.498735	0.249368	−	0.968409i	$-0.419777\pi$
$971$	15.5410	0.498735	0.249368	+	0.968409i	$0.419777\pi$
$972$	35.0344i	1.12373i
$973$	23.2918i	0.746701i
$974$	6.00000	0.192252
$975$	0	0
$976$	9.43769	0.302093
$977$	− 25.6180i	− 0.819594i	−0.912177	−	0.409797i	$-0.865600\pi$
$977$	− 25.6180i	− 0.819594i	0.912177	−	0.409797i	$-0.134400\pi$
$978$	21.2705i	0.680156i
$979$	40.2492	1.28637
$980$	0	0
$981$	−39.7984	−1.27066
$982$	− 1.68692i	− 0.0538317i
$983$	− 47.7426i	− 1.52275i	−0.648309	−	0.761377i	$-0.724524\pi$
$983$	− 47.7426i	− 1.52275i	0.648309	−	0.761377i	$-0.275476\pi$
$984$	17.5623	0.559866
$985$	0	0
$986$	0.978714	0.0311686
$987$	− 57.5410i	− 1.83155i
$988$	4.14590i	0.131899i
$989$	29.1246	0.926109
$990$	0	0
$991$	−49.1803	−1.56226	−0.781132	−	0.624365i	$-0.785358\pi$
$991$	−49.1803	−1.56226	−0.781132	+	0.624365i	$0.785358\pi$
$992$	34.2148i	1.08632i
$993$	25.9443i	0.823317i
$994$	−5.56231	−0.176426
$995$	0	0
$996$	−35.8885	−1.13717
$997$	− 46.1459i	− 1.46146i	−0.682669	−	0.730728i	$-0.739181\pi$
$997$	− 46.1459i	− 1.46146i	0.682669	−	0.730728i	$-0.260819\pi$
$998$	− 1.45898i	− 0.0461832i
$999$	−21.7082	−0.686817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.b.b.124.2		4
3.2	odd	2		1125.2.b.f.874.3		4
4.3	odd	2		2000.2.c.e.1249.4		4
5.2	odd	4		125.2.a.a.1.2	✓	2
5.3	odd	4		125.2.a.b.1.1	yes	2
5.4	even	2	inner	125.2.b.b.124.3		4
15.2	even	4		1125.2.a.d.1.1		2
15.8	even	4		1125.2.a.c.1.2		2
15.14	odd	2		1125.2.b.f.874.2		4
20.3	even	4		2000.2.a.a.1.1		2
20.7	even	4		2000.2.a.l.1.2		2
20.19	odd	2		2000.2.c.e.1249.1		4
25.2	odd	20		625.2.d.j.501.1		4
25.3	odd	20		625.2.d.g.376.1		4
25.4	even	10		625.2.e.g.249.2		8
25.6	even	5		625.2.e.g.374.2		8
25.8	odd	20		625.2.d.g.251.1		4
25.9	even	10		625.2.e.d.499.1		8
25.11	even	5		625.2.e.d.124.1		8
25.12	odd	20		625.2.d.j.126.1		4
25.13	odd	20		625.2.d.a.126.1		4
25.14	even	10		625.2.e.d.124.2		8
25.16	even	5		625.2.e.d.499.2		8
25.17	odd	20		625.2.d.d.251.1		4
25.19	even	10		625.2.e.g.374.1		8
25.21	even	5		625.2.e.g.249.1		8
25.22	odd	20		625.2.d.d.376.1		4
25.23	odd	20		625.2.d.a.501.1		4
35.13	even	4		6125.2.a.g.1.1		2
35.27	even	4		6125.2.a.d.1.2		2
40.3	even	4		8000.2.a.u.1.2		2
40.13	odd	4		8000.2.a.d.1.1		2
40.27	even	4		8000.2.a.c.1.1		2
40.37	odd	4		8000.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.a.a.1.2	✓	2	5.2	odd	4
125.2.a.b.1.1	yes	2	5.3	odd	4
125.2.b.b.124.2		4	1.1	even	1	trivial
125.2.b.b.124.3		4	5.4	even	2	inner
625.2.d.a.126.1		4	25.13	odd	20
625.2.d.a.501.1		4	25.23	odd	20
625.2.d.d.251.1		4	25.17	odd	20
625.2.d.d.376.1		4	25.22	odd	20
625.2.d.g.251.1		4	25.8	odd	20
625.2.d.g.376.1		4	25.3	odd	20
625.2.d.j.126.1		4	25.12	odd	20
625.2.d.j.501.1		4	25.2	odd	20
625.2.e.d.124.1		8	25.11	even	5
625.2.e.d.124.2		8	25.14	even	10
625.2.e.d.499.1		8	25.9	even	10
625.2.e.d.499.2		8	25.16	even	5
625.2.e.g.249.1		8	25.21	even	5
625.2.e.g.249.2		8	25.4	even	10
625.2.e.g.374.1		8	25.19	even	10
625.2.e.g.374.2		8	25.6	even	5
1125.2.a.c.1.2		2	15.8	even	4
1125.2.a.d.1.1		2	15.2	even	4
1125.2.b.f.874.2		4	15.14	odd	2
1125.2.b.f.874.3		4	3.2	odd	2
2000.2.a.a.1.1		2	20.3	even	4
2000.2.a.l.1.2		2	20.7	even	4
2000.2.c.e.1249.1		4	20.19	odd	2
2000.2.c.e.1249.4		4	4.3	odd	2
6125.2.a.d.1.2		2	35.27	even	4
6125.2.a.g.1.1		2	35.13	even	4
8000.2.a.c.1.1		2	40.27	even	4
8000.2.a.d.1.1		2	40.13	odd	4
8000.2.a.u.1.2		2	40.3	even	4
8000.2.a.v.1.2		2	40.37	odd	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:	\( N \)	\(=\)	\( 125 = 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	125.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{2} -2.61803i q^{3} +1.61803 q^{4} -1.61803 q^{6} +3.00000i q^{7} -2.23607i q^{8} -3.85410 q^{9} +O(q^{10})\)	\(q-0.618034i q^{2} -2.61803i q^{3} +1.61803 q^{4} -1.61803 q^{6} +3.00000i q^{7} -2.23607i q^{8} -3.85410 q^{9} -3.00000 q^{11} -4.23607i q^{12} +1.85410i q^{13} +1.85410 q^{14} +1.85410 q^{16} +0.236068i q^{17} +2.38197i q^{18} +1.38197 q^{19} +7.85410 q^{21} +1.85410i q^{22} +3.23607i q^{23} -5.85410 q^{24} +1.14590 q^{26} +2.23607i q^{27} +4.85410i q^{28} +6.70820 q^{29} -6.09017 q^{31} -5.61803i q^{32} +7.85410i q^{33} +0.145898 q^{34} -6.23607 q^{36} +9.70820i q^{37} -0.854102i q^{38} +4.85410 q^{39} -3.00000 q^{41} -4.85410i q^{42} -9.00000i q^{43} -4.85410 q^{44} +2.00000 q^{46} -7.32624i q^{47} -4.85410i q^{48} -2.00000 q^{49} +0.618034 q^{51} +3.00000i q^{52} +2.38197i q^{53} +1.38197 q^{54} +6.70820 q^{56} -3.61803i q^{57} -4.14590i q^{58} -10.8541 q^{59} +5.09017 q^{61} +3.76393i q^{62} -11.5623i q^{63} +0.236068 q^{64} +4.85410 q^{66} +7.14590i q^{67} +0.381966i q^{68} +8.47214 q^{69} -3.00000 q^{71} +8.61803i q^{72} -4.85410i q^{73} +6.00000 q^{74} +2.23607 q^{76} -9.00000i q^{77} -3.00000i q^{78} -9.47214 q^{79} -5.70820 q^{81} +1.85410i q^{82} -8.47214i q^{83} +12.7082 q^{84} -5.56231 q^{86} -17.5623i q^{87} +6.70820i q^{88} -13.4164 q^{89} -5.56231 q^{91} +5.23607i q^{92} +15.9443i q^{93} -4.52786 q^{94} -14.7082 q^{96} -1.14590i q^{97} +1.23607i q^{98} +11.5623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 6 q^{14} - 6 q^{16} + 10 q^{19} + 18 q^{21} - 10 q^{24} + 18 q^{26} - 2 q^{31} + 14 q^{34} - 16 q^{36} + 6 q^{39} - 12 q^{41} - 6 q^{44} + 8 q^{46} - 8 q^{49} - 2 q^{51} + 10 q^{54} - 30 q^{59} - 2 q^{61} - 8 q^{64} + 6 q^{66} + 16 q^{69} - 12 q^{71} + 24 q^{74} - 20 q^{79} + 4 q^{81} + 24 q^{84} + 18 q^{86} + 18 q^{91} - 36 q^{94} - 32 q^{96} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 - 6 * q^14 - 6 * q^16 + 10 * q^19 + 18 * q^21 - 10 * q^24 + 18 * q^26 - 2 * q^31 + 14 * q^34 - 16 * q^36 + 6 * q^39 - 12 * q^41 - 6 * q^44 + 8 * q^46 - 8 * q^49 - 2 * q^51 + 10 * q^54 - 30 * q^59 - 2 * q^61 - 8 * q^64 + 6 * q^66 + 16 * q^69 - 12 * q^71 + 24 * q^74 - 20 * q^79 + 4 * q^81 + 24 * q^84 + 18 * q^86 + 18 * q^91 - 36 * q^94 - 32 * q^96 + 6 * q^99

Introduction

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