LMFDB - Embedded newform 125.2.b.a.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:	4.0.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 11 $ x^4 + 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.2
Root			$-2.14896i$ of defining polynomial
Character	$\chi$	$=$	125.124
Dual form			125.2.b.a.124.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32813i q^{2} +2.14896i q^{3} +0.236068 q^{4} +2.85410 q^{6} +3.47709i q^{7} -2.96979i q^{8} -1.61803 q^{9} +O(q^{10})$	\(q-1.32813i q^{2} +2.14896i q^{3} +0.236068 q^{4} +2.85410 q^{6} +3.47709i q^{7} -2.96979i q^{8} -1.61803 q^{9} +2.00000 q^{11} +0.507301i q^{12} -2.65626i q^{13} +4.61803 q^{14} -3.47214 q^{16} -4.29792i q^{17} +2.14896i q^{18} -7.23607 q^{19} -7.47214 q^{21} -2.65626i q^{22} -0.820830i q^{23} +6.38197 q^{24} -3.52786 q^{26} +2.96979i q^{27} +0.820830i q^{28} -0.854102 q^{29} +2.00000 q^{31} -1.32813i q^{32} +4.29792i q^{33} -5.70820 q^{34} -0.381966 q^{36} +1.64166i q^{37} +9.61045i q^{38} +5.70820 q^{39} -6.09017 q^{41} +9.92398i q^{42} -3.79062i q^{43} +0.472136 q^{44} -1.09017 q^{46} +0.507301i q^{47} -7.46149i q^{48} -5.09017 q^{49} +9.23607 q^{51} -0.627058i q^{52} -8.59584i q^{53} +3.94427 q^{54} +10.3262 q^{56} -15.5500i q^{57} +1.13436i q^{58} +4.47214 q^{59} +5.09017 q^{61} -2.65626i q^{62} -5.62605i q^{63} -8.70820 q^{64} +5.70820 q^{66} -4.29792i q^{67} -1.01460i q^{68} +1.76393 q^{69} +8.18034 q^{71} +4.80522i q^{72} +16.5646i q^{73} +2.18034 q^{74} -1.70820 q^{76} +6.95418i q^{77} -7.58124i q^{78} -2.76393 q^{79} -11.2361 q^{81} +8.08854i q^{82} +5.11875i q^{83} -1.76393 q^{84} -5.03444 q^{86} -1.83543i q^{87} -5.93958i q^{88} +8.61803 q^{89} +9.23607 q^{91} -0.193772i q^{92} +4.29792i q^{93} +0.673762 q^{94} +2.85410 q^{96} +11.2521i q^{97} +6.76041i q^{98} -3.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 2 * q^6 - 2 * q^9	$ 4 q - 8 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 14 q^{14} + 4 q^{16} - 20 q^{19} - 12 q^{21} + 30 q^{24} - 32 q^{26} + 10 q^{29} + 8 q^{31} + 4 q^{34} - 6 q^{36} - 4 q^{39} - 2 q^{41} - 16 q^{44} + 18 q^{46} + 2 q^{49} + 28 q^{51} - 20 q^{54} + 10 q^{56} - 2 q^{61} - 8 q^{64} - 4 q^{66} + 16 q^{69} - 12 q^{71} - 36 q^{74} + 20 q^{76} - 20 q^{79} - 36 q^{81} - 16 q^{84} + 38 q^{86} + 30 q^{89} + 28 q^{91} + 34 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 14 * q^14 + 4 * q^16 - 20 * q^19 - 12 * q^21 + 30 * q^24 - 32 * q^26 + 10 * q^29 + 8 * q^31 + 4 * q^34 - 6 * q^36 - 4 * q^39 - 2 * q^41 - 16 * q^44 + 18 * q^46 + 2 * q^49 + 28 * q^51 - 20 * q^54 + 10 * q^56 - 2 * q^61 - 8 * q^64 - 4 * q^66 + 16 * q^69 - 12 * q^71 - 36 * q^74 + 20 * q^76 - 20 * q^79 - 36 * q^81 - 16 * q^84 + 38 * q^86 + 30 * q^89 + 28 * q^91 + 34 * q^94 - 2 * q^96 - 4 * 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 $

$2$	− 1.32813i	− 0.939130i	−0.882898	−	0.469565i	$-0.844411\pi$
$2$	− 1.32813i	− 0.939130i	0.882898	−	0.469565i	$-0.155589\pi$
$3$	2.14896i	1.24070i	0.784324	+	0.620352i	$0.213010\pi$
$3$	2.14896i	1.24070i	−0.784324	+	0.620352i	$0.786990\pi$
$4$	0.236068	0.118034
$5$	0	0
$5$	0	0
$6$	2.85410	1.16518
$7$	3.47709i	1.31422i	0.753796	+	0.657109i	$0.228221\pi$
$7$	3.47709i	1.31422i	−0.753796	+	0.657109i	$0.771779\pi$
$8$	− 2.96979i	− 1.04998i
$9$	−1.61803	−0.539345
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0.507301i	0.146445i
$13$	− 2.65626i	− 0.736715i	−0.929684	−	0.368357i	$-0.879920\pi$
$13$	− 2.65626i	− 0.736715i	0.929684	−	0.368357i	$-0.120080\pi$
$14$	4.61803	1.23422
$15$	0	0
$16$	−3.47214	−0.868034
$17$	− 4.29792i	− 1.04240i	−0.853435	−	0.521200i	$-0.825485\pi$
$17$	− 4.29792i	− 1.04240i	0.853435	−	0.521200i	$-0.174515\pi$
$18$	2.14896i	0.506515i
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	0	0
$21$	−7.47214	−1.63055
$22$	− 2.65626i	− 0.566317i
$23$	− 0.820830i	− 0.171155i	−0.996332	−	0.0855775i	$-0.972726\pi$
$23$	− 0.820830i	− 0.171155i	0.996332	−	0.0855775i	$-0.0272735\pi$
$24$	6.38197	1.30271
$25$	0	0
$26$	−3.52786	−0.691871
$27$	2.96979i	0.571537i
$28$	0.820830i	0.155122i
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 1.32813i	− 0.234783i
$33$	4.29792i	0.748172i
$34$	−5.70820	−0.978949
$35$	0	0
$36$	−0.381966	−0.0636610
$37$	1.64166i	0.269887i	0.990853	+	0.134944i	$0.0430853\pi$
$37$	1.64166i	0.269887i	−0.990853	+	0.134944i	$0.956915\pi$
$38$	9.61045i	1.55902i
$39$	5.70820	0.914044
$40$	0	0
$41$	−6.09017	−0.951125	−0.475562	−	0.879682i	$-0.657755\pi$
$41$	−6.09017	−0.951125	−0.475562	+	0.879682i	$0.657755\pi$
$42$	9.92398i	1.53130i
$43$	− 3.79062i	− 0.578064i	−0.957319	−	0.289032i	$-0.906667\pi$
$43$	− 3.79062i	− 0.578064i	0.957319	−	0.289032i	$-0.0933335\pi$
$44$	0.472136	0.0711772
$45$	0	0
$46$	−1.09017	−0.160737
$47$	0.507301i	0.0739974i	0.999315	+	0.0369987i	$0.0117797\pi$
$47$	0.507301i	0.0739974i	−0.999315	+	0.0369987i	$0.988220\pi$
$48$	− 7.46149i	− 1.07697i
$49$	−5.09017	−0.727167
$50$	0	0
$51$	9.23607	1.29331
$52$	− 0.627058i	− 0.0869574i
$53$	− 8.59584i	− 1.18073i	−0.807136	−	0.590365i	$-0.798984\pi$
$53$	− 8.59584i	− 1.18073i	0.807136	−	0.590365i	$-0.201016\pi$
$54$	3.94427	0.536747
$55$	0	0
$56$	10.3262	1.37990
$57$	− 15.5500i	− 2.05965i
$58$	1.13436i	0.148949i
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	5.09017	0.651729	0.325865	−	0.945416i	$-0.394345\pi$
$61$	5.09017	0.651729	0.325865	+	0.945416i	$0.394345\pi$
$62$	− 2.65626i	− 0.337346i
$63$	− 5.62605i	− 0.708816i
$64$	−8.70820	−1.08853
$65$	0	0
$66$	5.70820	0.702631
$67$	− 4.29792i	− 0.525075i	−0.964922	−	0.262537i	$-0.915441\pi$
$67$	− 4.29792i	− 0.525075i	0.964922	−	0.262537i	$-0.0845593\pi$
$68$	− 1.01460i	− 0.123039i
$69$	1.76393	0.212352
$70$	0	0
$71$	8.18034	0.970828	0.485414	−	0.874284i	$-0.338669\pi$
$71$	8.18034	0.970828	0.485414	+	0.874284i	$0.338669\pi$
$72$	4.80522i	0.566301i
$73$	16.5646i	1.93874i	0.245598	+	0.969372i	$0.421016\pi$
$73$	16.5646i	1.93874i	−0.245598	+	0.969372i	$0.578984\pi$
$74$	2.18034	0.253459
$75$	0	0
$76$	−1.70820	−0.195944
$77$	6.95418i	0.792503i
$78$	− 7.58124i	− 0.858407i
$79$	−2.76393	−0.310967	−0.155483	−	0.987839i	$-0.549693\pi$
$79$	−2.76393	−0.310967	−0.155483	+	0.987839i	$0.549693\pi$
$80$	0	0
$81$	−11.2361	−1.24845
$82$	8.08854i	0.893230i
$83$	5.11875i	0.561856i	0.959729	+	0.280928i	$0.0906422\pi$
$83$	5.11875i	0.561856i	−0.959729	+	0.280928i	$0.909358\pi$
$84$	−1.76393	−0.192461
$85$	0	0
$86$	−5.03444	−0.542878
$87$	− 1.83543i	− 0.196779i
$88$	− 5.93958i	− 0.633162i
$89$	8.61803	0.913510	0.456755	−	0.889593i	$-0.349012\pi$
$89$	8.61803	0.913510	0.456755	+	0.889593i	$0.349012\pi$
$90$	0	0
$91$	9.23607	0.968203
$92$	− 0.193772i	− 0.0202021i
$93$	4.29792i	0.445674i
$94$	0.673762	0.0694933
$95$	0	0
$96$	2.85410	0.291296
$97$	11.2521i	1.14248i	0.820784	+	0.571239i	$0.193537\pi$
$97$	11.2521i	1.14248i	−0.820784	+	0.571239i	$0.806463\pi$
$98$	6.76041i	0.682905i
$99$	−3.23607	−0.325237
$100$	0	0
$101$	−11.0902	−1.10351	−0.551757	−	0.834005i	$-0.686042\pi$
$101$	−11.0902	−1.10351	−0.551757	+	0.834005i	$0.686042\pi$
$102$	− 12.2667i	− 1.21459i
$103$	12.8938i	1.27046i	0.772323	+	0.635230i	$0.219095\pi$
$103$	12.8938i	1.27046i	−0.772323	+	0.635230i	$0.780905\pi$
$104$	−7.88854	−0.773535
$105$	0	0
$106$	−11.4164	−1.10886
$107$	10.1177i	0.978120i	0.872250	+	0.489060i	$0.162660\pi$
$107$	10.1177i	0.978120i	−0.872250	+	0.489060i	$0.837340\pi$
$108$	0.701073i	0.0674607i
$109$	12.5623	1.20325	0.601625	−	0.798778i	$-0.294520\pi$
$109$	12.5623	1.20325	0.601625	+	0.798778i	$0.294520\pi$
$110$	0	0
$111$	−3.52786	−0.334850
$112$	− 12.0729i	− 1.14079i
$113$	− 2.65626i	− 0.249880i	−0.992164	−	0.124940i	$-0.960126\pi$
$113$	− 2.65626i	− 0.249880i	0.992164	−	0.124940i	$-0.0398739\pi$
$114$	−20.6525	−1.93428
$115$	0	0
$116$	−0.201626	−0.0187205
$117$	4.29792i	0.397343i
$118$	− 5.93958i	− 0.546783i
$119$	14.9443	1.36994
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 6.76041i	− 0.612059i
$123$	− 13.0875i	− 1.18006i
$124$	0.472136	0.0423991
$125$	0	0
$126$	−7.47214	−0.665671
$127$	− 18.7136i	− 1.66056i	−0.557344	−	0.830281i	$-0.688180\pi$
$127$	− 18.7136i	− 1.66056i	0.557344	−	0.830281i	$-0.311820\pi$
$128$	8.90937i	0.787485i
$129$	8.14590	0.717206
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	1.01460i	0.0883098i
$133$	− 25.1605i	− 2.18169i
$134$	−5.70820	−0.493114
$135$	0	0
$136$	−12.7639	−1.09450
$137$	14.9230i	1.27496i	0.770469	+	0.637478i	$0.220022\pi$
$137$	14.9230i	1.27496i	−0.770469	+	0.637478i	$0.779978\pi$
$138$	− 2.34273i	− 0.199427i
$139$	−8.29180	−0.703301	−0.351650	−	0.936131i	$-0.614379\pi$
$139$	−8.29180	−0.703301	−0.351650	+	0.936131i	$0.614379\pi$
$140$	0	0
$141$	−1.09017	−0.0918089
$142$	− 10.8646i	− 0.911734i
$143$	− 5.31252i	− 0.444256i
$144$	5.61803	0.468169
$145$	0	0
$146$	22.0000	1.82073
$147$	− 10.9386i	− 0.902199i
$148$	0.387543i	0.0318559i
$149$	−19.2705	−1.57870	−0.789351	−	0.613942i	$-0.789583\pi$
$149$	−19.2705	−1.57870	−0.789351	+	0.613942i	$0.789583\pi$
$150$	0	0
$151$	18.1803	1.47950	0.739748	−	0.672885i	$-0.234945\pi$
$151$	18.1803	1.47950	0.739748	+	0.672885i	$0.234945\pi$
$152$	21.4896i	1.74304i
$153$	6.95418i	0.562212i
$154$	9.23607	0.744264
$155$	0	0
$156$	1.34752	0.107888
$157$	7.58124i	0.605049i	0.953142	+	0.302525i	$0.0978294\pi$
$157$	7.58124i	0.605049i	−0.953142	+	0.302525i	$0.902171\pi$
$158$	3.67086i	0.292038i
$159$	18.4721	1.46494
$160$	0	0
$161$	2.85410	0.224935
$162$	14.9230i	1.17246i
$163$	14.7292i	1.15368i	0.816857	+	0.576840i	$0.195714\pi$
$163$	14.7292i	1.15368i	−0.816857	+	0.576840i	$0.804286\pi$
$164$	−1.43769	−0.112265
$165$	0	0
$166$	6.79837	0.527656
$167$	6.44688i	0.498875i	0.968391	+	0.249437i	$0.0802457\pi$
$167$	6.44688i	0.498875i	−0.968391	+	0.249437i	$0.919754\pi$
$168$	22.1907i	1.71205i
$169$	5.94427	0.457252
$170$	0	0
$171$	11.7082	0.895349
$172$	− 0.894844i	− 0.0682312i
$173$	− 18.2063i	− 1.38420i	−0.721802	−	0.692099i	$-0.756686\pi$
$173$	− 18.2063i	− 1.38420i	0.721802	−	0.692099i	$-0.243314\pi$
$174$	−2.43769	−0.184801
$175$	0	0
$176$	−6.94427	−0.523444
$177$	9.61045i	0.722365i
$178$	− 11.4459i	− 0.857905i
$179$	−18.9443	−1.41596	−0.707981	−	0.706232i	$-0.750394\pi$
$179$	−18.9443	−1.41596	−0.707981	+	0.706232i	$0.750394\pi$
$180$	0	0
$181$	0.0901699	0.00670228	0.00335114	−	0.999994i	$-0.498933\pi$
$181$	0.0901699	0.00670228	0.00335114	+	0.999994i	$0.498933\pi$
$182$	− 12.2667i	− 0.909269i
$183$	10.9386i	0.808603i
$184$	−2.43769	−0.179709
$185$	0	0
$186$	5.70820	0.418546
$187$	− 8.59584i	− 0.628590i
$188$	0.119757i	0.00873421i
$189$	−10.3262	−0.751123
$190$	0	0
$191$	8.18034	0.591909	0.295954	−	0.955202i	$-0.404362\pi$
$191$	8.18034	0.591909	0.295954	+	0.955202i	$0.404362\pi$
$192$	− 18.7136i	− 1.35054i
$193$	6.95418i	0.500573i	0.968172	+	0.250287i	$0.0805248\pi$
$193$	6.95418i	0.500573i	−0.968172	+	0.250287i	$0.919475\pi$
$194$	14.9443	1.07294
$195$	0	0
$196$	−1.20163	−0.0858304
$197$	− 23.5188i	− 1.67565i	−0.545942	−	0.837823i	$-0.683828\pi$
$197$	− 23.5188i	− 1.67565i	0.545942	−	0.837823i	$-0.316172\pi$
$198$	4.29792i	0.305440i
$199$	16.1803	1.14699	0.573497	−	0.819208i	$-0.305586\pi$
$199$	16.1803	1.14699	0.573497	+	0.819208i	$0.305586\pi$
$200$	0	0
$201$	9.23607	0.651462
$202$	14.7292i	1.03634i
$203$	− 2.96979i	− 0.208438i
$204$	2.18034	0.152654
$205$	0	0
$206$	17.1246	1.19313
$207$	1.32813i	0.0923115i
$208$	9.22290i	0.639493i
$209$	−14.4721	−1.00106
$210$	0	0
$211$	5.81966	0.400642	0.200321	−	0.979730i	$-0.435802\pi$
$211$	5.81966	0.400642	0.200321	+	0.979730i	$0.435802\pi$
$212$	− 2.02920i	− 0.139366i
$213$	17.5792i	1.20451i
$214$	13.4377	0.918582
$215$	0	0
$216$	8.81966	0.600102
$217$	6.95418i	0.472081i
$218$	− 16.6844i	− 1.13001i
$219$	−35.5967	−2.40541
$220$	0	0
$221$	−11.4164	−0.767951
$222$	4.68547i	0.314468i
$223$	− 7.46149i	− 0.499658i	−0.968290	−	0.249829i	$-0.919626\pi$
$223$	− 7.46149i	− 0.499658i	0.968290	−	0.249829i	$-0.0803744\pi$
$224$	4.61803	0.308555
$225$	0	0
$226$	−3.52786	−0.234670
$227$	− 12.0729i	− 0.801309i	−0.916229	−	0.400654i	$-0.868783\pi$
$227$	− 12.0729i	− 0.801309i	0.916229	−	0.400654i	$-0.131217\pi$
$228$	− 3.67086i	− 0.243109i
$229$	20.3262	1.34320	0.671598	−	0.740916i	$-0.265608\pi$
$229$	20.3262	1.34320	0.671598	+	0.740916i	$0.265608\pi$
$230$	0	0
$231$	−14.9443	−0.983261
$232$	2.53650i	0.166530i
$233$	1.01460i	0.0664688i	0.999448	+	0.0332344i	$0.0105808\pi$
$233$	1.01460i	0.0664688i	−0.999448	+	0.0332344i	$0.989419\pi$
$234$	5.70820	0.373157
$235$	0	0
$236$	1.05573	0.0687220
$237$	− 5.93958i	− 0.385817i
$238$	− 19.8480i	− 1.28655i
$239$	11.7082	0.757341	0.378670	−	0.925532i	$-0.376381\pi$
$239$	11.7082	0.757341	0.378670	+	0.925532i	$0.376381\pi$
$240$	0	0
$241$	−1.09017	−0.0702240	−0.0351120	−	0.999383i	$-0.511179\pi$
$241$	−1.09017	−0.0702240	−0.0351120	+	0.999383i	$0.511179\pi$
$242$	9.29692i	0.597628i
$243$	− 15.2365i	− 0.977422i
$244$	1.20163	0.0769262
$245$	0	0
$246$	−17.3820	−1.10823
$247$	19.2209i	1.22300i
$248$	− 5.93958i	− 0.377164i
$249$	−11.0000	−0.697097
$250$	0	0
$251$	−20.3607	−1.28515	−0.642577	−	0.766221i	$-0.722135\pi$
$251$	−20.3607	−1.28515	−0.642577	+	0.766221i	$0.722135\pi$
$252$	− 1.32813i	− 0.0836644i
$253$	− 1.64166i	− 0.103210i
$254$	−24.8541	−1.55949
$255$	0	0
$256$	−5.58359	−0.348975
$257$	− 13.9084i	− 0.867580i	−0.901014	−	0.433790i	$-0.857176\pi$
$257$	− 13.9084i	− 0.867580i	0.901014	−	0.433790i	$-0.142824\pi$
$258$	− 10.8188i	− 0.673550i
$259$	−5.70820	−0.354691
$260$	0	0
$261$	1.38197	0.0855415
$262$	18.8333i	1.16353i
$263$	− 0.119757i	− 0.00738456i	−0.999993	−	0.00369228i	$-0.998825\pi$
$263$	− 0.119757i	− 0.00738456i	0.999993	−	0.00369228i	$-0.00117529\pi$
$264$	12.7639	0.785566
$265$	0	0
$266$	−33.4164	−2.04889
$267$	18.5198i	1.13339i
$268$	− 1.01460i	− 0.0619767i
$269$	−1.05573	−0.0643689	−0.0321844	−	0.999482i	$-0.510246\pi$
$269$	−1.05573	−0.0643689	−0.0321844	+	0.999482i	$0.510246\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	14.9230i	0.904838i
$273$	19.8480i	1.20125i
$274$	19.8197	1.19735
$275$	0	0
$276$	0.416408	0.0250648
$277$	− 16.1771i	− 0.971987i	−0.873962	−	0.485993i	$-0.838458\pi$
$277$	− 16.1771i	− 0.971987i	0.873962	−	0.485993i	$-0.161542\pi$
$278$	11.0126i	0.660491i
$279$	−3.23607	−0.193738
$280$	0	0
$281$	31.2705	1.86544	0.932721	−	0.360599i	$-0.117428\pi$
$281$	31.2705	1.86544	0.932721	+	0.360599i	$0.117428\pi$
$282$	1.44789i	0.0862205i
$283$	1.01460i	0.0603118i	0.999545	+	0.0301559i	$0.00960038\pi$
$283$	1.01460i	0.0603118i	−0.999545	+	0.0301559i	$0.990400\pi$
$284$	1.93112	0.114591
$285$	0	0
$286$	−7.05573	−0.417214
$287$	− 21.1761i	− 1.24998i
$288$	2.14896i	0.126629i
$289$	−1.47214	−0.0865962
$290$	0	0
$291$	−24.1803	−1.41748
$292$	3.91038i	0.228838i
$293$	28.4438i	1.66170i	0.556493	+	0.830852i	$0.312147\pi$
$293$	28.4438i	1.66170i	−0.556493	+	0.830852i	$0.687853\pi$
$294$	−14.5279	−0.847282
$295$	0	0
$296$	4.87539	0.283376
$297$	5.93958i	0.344650i
$298$	25.5938i	1.48261i
$299$	−2.18034	−0.126092
$300$	0	0
$301$	13.1803	0.759702
$302$	− 24.1459i	− 1.38944i
$303$	− 23.8323i	− 1.36913i
$304$	25.1246	1.44100
$305$	0	0
$306$	9.23607	0.527991
$307$	− 9.80422i	− 0.559556i	−0.960065	−	0.279778i	$-0.909739\pi$
$307$	− 9.80422i	− 0.559556i	0.960065	−	0.279778i	$-0.0902609\pi$
$308$	1.64166i	0.0935423i
$309$	−27.7082	−1.57626
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	− 16.9522i	− 0.959728i
$313$	− 15.9376i	− 0.900845i	−0.892816	−	0.450422i	$-0.851273\pi$
$313$	− 15.9376i	− 0.900845i	0.892816	−	0.450422i	$-0.148727\pi$
$314$	10.0689	0.568220
$315$	0	0
$316$	−0.652476	−0.0367046
$317$	− 17.5792i	− 0.987348i	−0.869647	−	0.493674i	$-0.835654\pi$
$317$	− 17.5792i	− 0.987348i	0.869647	−	0.493674i	$-0.164346\pi$
$318$	− 24.5334i	− 1.37577i
$319$	−1.70820	−0.0956411
$320$	0	0
$321$	−21.7426	−1.21356
$322$	− 3.79062i	− 0.211243i
$323$	31.1001i	1.73045i
$324$	−2.65248	−0.147360
$325$	0	0
$326$	19.5623	1.08346
$327$	26.9959i	1.49288i
$328$	18.0865i	0.998662i
$329$	−1.76393	−0.0972487
$330$	0	0
$331$	5.81966	0.319877	0.159939	−	0.987127i	$-0.448870\pi$
$331$	5.81966	0.319877	0.159939	+	0.987127i	$0.448870\pi$
$332$	1.20837i	0.0663181i
$333$	− 2.65626i	− 0.145562i
$334$	8.56231	0.468509
$335$	0	0
$336$	25.9443	1.41538
$337$	1.64166i	0.0894269i	0.999000	+	0.0447135i	$0.0142375\pi$
$337$	1.64166i	0.0894269i	−0.999000	+	0.0447135i	$0.985763\pi$
$338$	− 7.89477i	− 0.429419i
$339$	5.70820	0.310027
$340$	0	0
$341$	4.00000	0.216612
$342$	− 15.5500i	− 0.840849i
$343$	6.64066i	0.358562i
$344$	−11.2574	−0.606956
$345$	0	0
$346$	−24.1803	−1.29994
$347$	28.6376i	1.53735i	0.639642	+	0.768673i	$0.279082\pi$
$347$	28.6376i	1.53735i	−0.639642	+	0.768673i	$0.720918\pi$
$348$	− 0.433287i	− 0.0232266i
$349$	9.27051	0.496239	0.248120	−	0.968729i	$-0.420187\pi$
$349$	9.27051	0.496239	0.248120	+	0.968729i	$0.420187\pi$
$350$	0	0
$351$	7.88854	0.421059
$352$	− 2.65626i	− 0.141579i
$353$	− 21.8772i	− 1.16440i	−0.813044	−	0.582202i	$-0.802191\pi$
$353$	− 21.8772i	− 1.16440i	0.813044	−	0.582202i	$-0.197809\pi$
$354$	12.7639	0.678395
$355$	0	0
$356$	2.03444	0.107825
$357$	32.1147i	1.69969i
$358$	25.1605i	1.32977i
$359$	−7.88854	−0.416341	−0.208171	−	0.978093i	$-0.566751\pi$
$359$	−7.88854	−0.416341	−0.208171	+	0.978093i	$0.566751\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	− 0.119757i	− 0.00629431i
$363$	− 15.0427i	− 0.789538i
$364$	2.18034	0.114281
$365$	0	0
$366$	14.5279	0.759384
$367$	13.0875i	0.683164i	0.939852	+	0.341582i	$0.110963\pi$
$367$	13.0875i	0.683164i	−0.939852	+	0.341582i	$0.889037\pi$
$368$	2.85003i	0.148568i
$369$	9.85410	0.512984
$370$	0	0
$371$	29.8885	1.55174
$372$	1.01460i	0.0526047i
$373$	− 31.4876i	− 1.63037i	−0.579203	−	0.815183i	$-0.696636\pi$
$373$	− 31.4876i	− 1.63037i	0.579203	−	0.815183i	$-0.303364\pi$
$374$	−11.4164	−0.590328
$375$	0	0
$376$	1.50658	0.0776958
$377$	2.26872i	0.116845i
$378$	13.7146i	0.705403i
$379$	−28.9443	−1.48677	−0.743384	−	0.668865i	$-0.766780\pi$
$379$	−28.9443	−1.48677	−0.743384	+	0.668865i	$0.766780\pi$
$380$	0	0
$381$	40.2148	2.06027
$382$	− 10.8646i	− 0.555879i
$383$	− 1.52190i	− 0.0777656i	−0.999244	−	0.0388828i	$-0.987620\pi$
$383$	− 1.52190i	− 0.0777656i	0.999244	−	0.0388828i	$-0.0123799\pi$
$384$	−19.1459	−0.977035
$385$	0	0
$386$	9.23607	0.470103
$387$	6.13335i	0.311776i
$388$	2.65626i	0.134851i
$389$	23.6180	1.19748	0.598741	−	0.800943i	$-0.295668\pi$
$389$	23.6180	1.19748	0.598741	+	0.800943i	$0.295668\pi$
$390$	0	0
$391$	−3.52786	−0.178412
$392$	15.1167i	0.763511i
$393$	− 30.4730i	− 1.53716i
$394$	−31.2361	−1.57365
$395$	0	0
$396$	−0.763932	−0.0383890
$397$	− 31.7271i	− 1.59234i	−0.605074	−	0.796169i	$-0.706857\pi$
$397$	− 31.7271i	− 1.59234i	0.605074	−	0.796169i	$-0.293143\pi$
$398$	− 21.4896i	− 1.07718i
$399$	54.0689	2.70683
$400$	0	0
$401$	−37.2705	−1.86120	−0.930600	−	0.366037i	$-0.880714\pi$
$401$	−37.2705	−1.86120	−0.930600	+	0.366037i	$0.880714\pi$
$402$	− 12.2667i	− 0.611808i
$403$	− 5.31252i	− 0.264636i
$404$	−2.61803	−0.130252
$405$	0	0
$406$	−3.94427	−0.195751
$407$	3.28332i	0.162748i
$408$	− 27.4292i	− 1.35795i
$409$	13.7426	0.679530	0.339765	−	0.940510i	$-0.389652\pi$
$409$	13.7426	0.679530	0.339765	+	0.940510i	$0.389652\pi$
$410$	0	0
$411$	−32.0689	−1.58184
$412$	3.04381i	0.149958i
$413$	15.5500i	0.765167i
$414$	1.76393	0.0866925
$415$	0	0
$416$	−3.52786	−0.172968
$417$	− 17.8187i	− 0.872588i
$418$	19.2209i	0.940125i
$419$	−3.41641	−0.166902	−0.0834512	−	0.996512i	$-0.526594\pi$
$419$	−3.41641	−0.166902	−0.0834512	+	0.996512i	$0.526594\pi$
$420$	0	0
$421$	10.0902	0.491765	0.245882	−	0.969300i	$-0.420922\pi$
$421$	10.0902	0.491765	0.245882	+	0.969300i	$0.420922\pi$
$422$	− 7.72927i	− 0.376255i
$423$	− 0.820830i	− 0.0399101i
$424$	−25.5279	−1.23974
$425$	0	0
$426$	23.3475	1.13119
$427$	17.6990i	0.856514i
$428$	2.38848i	0.115451i
$429$	11.4164	0.551189
$430$	0	0
$431$	−14.1803	−0.683043	−0.341521	−	0.939874i	$-0.610942\pi$
$431$	−14.1803	−0.683043	−0.341521	+	0.939874i	$0.610942\pi$
$432$	− 10.3115i	− 0.496113i
$433$	14.2959i	0.687018i	0.939149	+	0.343509i	$0.111616\pi$
$433$	14.2959i	0.687018i	−0.939149	+	0.343509i	$0.888384\pi$
$434$	9.23607	0.443345
$435$	0	0
$436$	2.96556	0.142024
$437$	5.93958i	0.284129i
$438$	47.2771i	2.25899i
$439$	−10.6525	−0.508415	−0.254207	−	0.967150i	$-0.581815\pi$
$439$	−10.6525	−0.508415	−0.254207	+	0.967150i	$0.581815\pi$
$440$	0	0
$441$	8.23607	0.392194
$442$	15.1625i	0.721206i
$443$	24.3396i	1.15641i	0.815891	+	0.578206i	$0.196247\pi$
$443$	24.3396i	1.15641i	−0.815891	+	0.578206i	$0.803753\pi$
$444$	−0.832816	−0.0395237
$445$	0	0
$446$	−9.90983	−0.469244
$447$	− 41.4116i	− 1.95870i
$448$	− 30.2792i	− 1.43056i
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−12.1803	−0.573550
$452$	− 0.627058i	− 0.0294943i
$453$	39.0688i	1.83561i
$454$	−16.0344	−0.752534
$455$	0	0
$456$	−46.1803	−2.16259
$457$	− 13.9084i	− 0.650606i	−0.945610	−	0.325303i	$-0.894534\pi$
$457$	− 13.9084i	− 0.650606i	0.945610	−	0.325303i	$-0.105466\pi$
$458$	− 26.9959i	− 1.26144i
$459$	12.7639	0.595769
$460$	0	0
$461$	21.2705	0.990666	0.495333	−	0.868703i	$-0.335046\pi$
$461$	21.2705	0.990666	0.495333	+	0.868703i	$0.335046\pi$
$462$	19.8480i	0.923410i
$463$	22.9375i	1.06600i	0.846116	+	0.532998i	$0.178935\pi$
$463$	22.9375i	1.06600i	−0.846116	+	0.532998i	$0.821065\pi$
$464$	2.96556	0.137673
$465$	0	0
$466$	1.34752	0.0624229
$467$	− 21.6834i	− 1.00339i	−0.865045	−	0.501694i	$-0.832710\pi$
$467$	− 21.6834i	− 1.00339i	0.865045	−	0.501694i	$-0.167290\pi$
$468$	1.01460i	0.0469000i
$469$	14.9443	0.690062
$470$	0	0
$471$	−16.2918	−0.750686
$472$	− 13.2813i	− 0.611322i
$473$	− 7.58124i	− 0.348586i
$474$	−7.88854	−0.362333
$475$	0	0
$476$	3.52786	0.161699
$477$	13.9084i	0.636820i
$478$	− 15.5500i	− 0.711242i
$479$	29.5967	1.35231	0.676155	−	0.736759i	$-0.263645\pi$
$479$	29.5967	1.35231	0.676155	+	0.736759i	$0.263645\pi$
$480$	0	0
$481$	4.36068	0.198830
$482$	1.44789i	0.0659495i
$483$	6.13335i	0.279077i
$484$	−1.65248	−0.0751125
$485$	0	0
$486$	−20.2361	−0.917927
$487$	25.6678i	1.16312i	0.813504	+	0.581559i	$0.197557\pi$
$487$	25.6678i	1.16312i	−0.813504	+	0.581559i	$0.802443\pi$
$488$	− 15.1167i	− 0.684303i
$489$	−31.6525	−1.43137
$490$	0	0
$491$	−11.8197	−0.533414	−0.266707	−	0.963778i	$-0.585936\pi$
$491$	−11.8197	−0.533414	−0.266707	+	0.963778i	$0.585936\pi$
$492$	− 3.08955i	− 0.139288i
$493$	3.67086i	0.165327i
$494$	25.5279	1.14855
$495$	0	0
$496$	−6.94427	−0.311807
$497$	28.4438i	1.27588i
$498$	14.6094i	0.654665i
$499$	−16.1803	−0.724331	−0.362166	−	0.932114i	$-0.617963\pi$
$499$	−16.1803	−0.724331	−0.362166	+	0.932114i	$0.617963\pi$
$500$	0	0
$501$	−13.8541	−0.618956
$502$	27.0417i	1.20693i
$503$	8.78962i	0.391910i	0.980613	+	0.195955i	$0.0627806\pi$
$503$	8.78962i	0.391910i	−0.980613	+	0.195955i	$0.937219\pi$
$504$	−16.7082	−0.744243
$505$	0	0
$506$	−2.18034	−0.0969279
$507$	12.7740i	0.567314i
$508$	− 4.41768i	− 0.196003i
$509$	−15.5279	−0.688260	−0.344130	−	0.938922i	$-0.611826\pi$
$509$	−15.5279	−0.688260	−0.344130	+	0.938922i	$0.611826\pi$
$510$	0	0
$511$	−57.5967	−2.54793
$512$	25.2345i	1.11522i
$513$	− 21.4896i	− 0.948790i
$514$	−18.4721	−0.814771
$515$	0	0
$516$	1.92299	0.0846547
$517$	1.01460i	0.0446221i
$518$	7.58124i	0.333101i
$519$	39.1246	1.71738
$520$	0	0
$521$	23.9098	1.04751	0.523754	−	0.851869i	$-0.324531\pi$
$521$	23.9098	1.04751	0.523754	+	0.851869i	$0.324531\pi$
$522$	− 1.83543i	− 0.0803347i
$523$	− 15.6698i	− 0.685192i	−0.939483	−	0.342596i	$-0.888694\pi$
$523$	− 15.6698i	− 0.685192i	0.939483	−	0.342596i	$-0.111306\pi$
$524$	−3.34752	−0.146237
$525$	0	0
$526$	−0.159054	−0.00693507
$527$	− 8.59584i	− 0.374441i
$528$	− 14.9230i	− 0.649439i
$529$	22.3262	0.970706
$530$	0	0
$531$	−7.23607	−0.314019
$532$	− 5.93958i	− 0.257514i
$533$	16.1771i	0.700707i
$534$	24.5967	1.06441
$535$	0	0
$536$	−12.7639	−0.551318
$537$	− 40.7105i	− 1.75679i
$538$	1.40215i	0.0604508i
$539$	−10.1803	−0.438498
$540$	0	0
$541$	0.0901699	0.00387671	0.00193835	−	0.999998i	$-0.499383\pi$
$541$	0.0901699	0.00387671	0.00193835	+	0.999998i	$0.499383\pi$
$542$	10.6250i	0.456385i
$543$	0.193772i	0.00831554i
$544$	−5.70820	−0.244737
$545$	0	0
$546$	26.3607	1.12813
$547$	− 14.3417i	− 0.613205i	−0.951838	−	0.306602i	$-0.900808\pi$
$547$	− 14.3417i	− 0.613205i	0.951838	−	0.306602i	$-0.0991922\pi$
$548$	3.52284i	0.150488i
$549$	−8.23607	−0.351507
$550$	0	0
$551$	6.18034	0.263291
$552$	− 5.23851i	− 0.222966i
$553$	− 9.61045i	− 0.408678i
$554$	−21.4853	−0.912823
$555$	0	0
$556$	−1.95743	−0.0830134
$557$	34.1439i	1.44672i	0.690470	+	0.723361i	$0.257404\pi$
$557$	34.1439i	1.44672i	−0.690470	+	0.723361i	$0.742596\pi$
$558$	4.29792i	0.181946i
$559$	−10.0689	−0.425868
$560$	0	0
$561$	18.4721	0.779894
$562$	− 41.5313i	− 1.75189i
$563$	− 37.4272i	− 1.57737i	−0.614799	−	0.788684i	$-0.710763\pi$
$563$	− 37.4272i	− 1.57737i	0.614799	−	0.788684i	$-0.289237\pi$
$564$	−0.257354	−0.0108366
$565$	0	0
$566$	1.34752	0.0566407
$567$	− 39.0688i	− 1.64074i
$568$	− 24.2939i	− 1.01935i
$569$	23.2148	0.973214	0.486607	−	0.873621i	$-0.338234\pi$
$569$	23.2148	0.973214	0.486607	+	0.873621i	$0.338234\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 1.25412i	− 0.0524373i
$573$	17.5792i	0.734383i
$574$	−28.1246	−1.17390
$575$	0	0
$576$	14.0902	0.587090
$577$	− 21.2501i	− 0.884653i	−0.896854	−	0.442327i	$-0.854153\pi$
$577$	− 21.2501i	− 0.884653i	0.896854	−	0.442327i	$-0.145847\pi$
$578$	1.95519i	0.0813252i
$579$	−14.9443	−0.621063
$580$	0	0
$581$	−17.7984	−0.738401
$582$	32.1147i	1.33120i
$583$	− 17.1917i	− 0.712007i
$584$	49.1935	2.03564
$585$	0	0
$586$	37.7771	1.56056
$587$	− 11.6397i	− 0.480420i	−0.970721	−	0.240210i	$-0.922784\pi$
$587$	− 11.6397i	− 0.480420i	0.970721	−	0.240210i	$-0.0772163\pi$
$588$	− 2.58225i	− 0.106490i
$589$	−14.4721	−0.596314
$590$	0	0
$591$	50.5410	2.07898
$592$	− 5.70007i	− 0.234271i
$593$	− 36.0250i	− 1.47937i	−0.672953	−	0.739686i	$-0.734974\pi$
$593$	− 36.0250i	− 1.47937i	0.672953	−	0.739686i	$-0.265026\pi$
$594$	7.88854	0.323671
$595$	0	0
$596$	−4.54915	−0.186340
$597$	34.7709i	1.42308i
$598$	2.89578i	0.118417i
$599$	38.5410	1.57474	0.787372	−	0.616479i	$-0.211441\pi$
$599$	38.5410	1.57474	0.787372	+	0.616479i	$0.211441\pi$
$600$	0	0
$601$	6.27051	0.255779	0.127890	−	0.991788i	$-0.459180\pi$
$601$	6.27051	0.255779	0.127890	+	0.991788i	$0.459180\pi$
$602$	− 17.5052i	− 0.713459i
$603$	6.95418i	0.283196i
$604$	4.29180	0.174631
$605$	0	0
$606$	−31.6525	−1.28579
$607$	− 35.3980i	− 1.43676i	−0.695651	−	0.718380i	$-0.744884\pi$
$607$	− 35.3980i	− 1.43676i	0.695651	−	0.718380i	$-0.255116\pi$
$608$	9.61045i	0.389755i
$609$	6.38197	0.258610
$610$	0	0
$611$	1.34752	0.0545150
$612$	1.64166i	0.0663602i
$613$	10.6250i	0.429142i	0.976708	+	0.214571i	$0.0688353\pi$
$613$	10.6250i	0.429142i	−0.976708	+	0.214571i	$0.931165\pi$
$614$	−13.0213	−0.525496
$615$	0	0
$616$	20.6525	0.832112
$617$	38.6813i	1.55725i	0.627489	+	0.778625i	$0.284083\pi$
$617$	38.6813i	1.55725i	−0.627489	+	0.778625i	$0.715917\pi$
$618$	36.8001i	1.48032i
$619$	−39.5967	−1.59153	−0.795764	−	0.605607i	$-0.792930\pi$
$619$	−39.5967	−1.59153	−0.795764	+	0.605607i	$0.792930\pi$
$620$	0	0
$621$	2.43769	0.0978213
$622$	23.9064i	0.958558i
$623$	29.9657i	1.20055i
$624$	−19.8197	−0.793421
$625$	0	0
$626$	−21.1672	−0.846011
$627$	− 31.1001i	− 1.24202i
$628$	1.78969i	0.0714164i
$629$	7.05573	0.281330
$630$	0	0
$631$	−10.3607	−0.412452	−0.206226	−	0.978504i	$-0.566118\pi$
$631$	−10.3607	−0.412452	−0.206226	+	0.978504i	$0.566118\pi$
$632$	8.20830i	0.326509i
$633$	12.5062i	0.497078i
$634$	−23.3475	−0.927249
$635$	0	0
$636$	4.36068	0.172912
$637$	13.5208i	0.535715i
$638$	2.26872i	0.0898194i
$639$	−13.2361	−0.523611
$640$	0	0
$641$	26.2705	1.03762	0.518811	−	0.854889i	$-0.326375\pi$
$641$	26.2705	1.03762	0.518811	+	0.854889i	$0.326375\pi$
$642$	28.8771i	1.13969i
$643$	− 2.22298i	− 0.0876656i	−0.999039	−	0.0438328i	$-0.986043\pi$
$643$	− 2.22298i	− 0.0876656i	0.999039	−	0.0438328i	$-0.0139569\pi$
$644$	0.673762	0.0265499
$645$	0	0
$646$	41.3050	1.62512
$647$	19.4604i	0.765068i	0.923942	+	0.382534i	$0.124948\pi$
$647$	19.4604i	0.765068i	−0.923942	+	0.382534i	$0.875052\pi$
$648$	33.3688i	1.31085i
$649$	8.94427	0.351093
$650$	0	0
$651$	−14.9443	−0.585712
$652$	3.47709i	0.136173i
$653$	26.1751i	1.02431i	0.858893	+	0.512155i	$0.171153\pi$
$653$	26.1751i	1.02431i	−0.858893	+	0.512155i	$0.828847\pi$
$654$	35.8541	1.40201
$655$	0	0
$656$	21.1459	0.825609
$657$	− 26.8021i	− 1.04565i
$658$	2.34273i	0.0913292i
$659$	10.6525	0.414962	0.207481	−	0.978239i	$-0.433474\pi$
$659$	10.6525	0.414962	0.207481	+	0.978239i	$0.433474\pi$
$660$	0	0
$661$	−22.2705	−0.866222	−0.433111	−	0.901340i	$-0.642584\pi$
$661$	−22.2705	−0.866222	−0.433111	+	0.901340i	$0.642584\pi$
$662$	− 7.72927i	− 0.300407i
$663$	− 24.5334i	− 0.952799i
$664$	15.2016	0.589938
$665$	0	0
$666$	−3.52786	−0.136702
$667$	0.701073i	0.0271456i
$668$	1.52190i	0.0588842i
$669$	16.0344	0.619927
$670$	0	0
$671$	10.1803	0.393008
$672$	9.92398i	0.382826i
$673$	16.5646i	0.638520i	0.947667	+	0.319260i	$0.103434\pi$
$673$	16.5646i	0.638520i	−0.947667	+	0.319260i	$0.896566\pi$
$674$	2.18034	0.0839836
$675$	0	0
$676$	1.40325	0.0539712
$677$	5.31252i	0.204177i	0.994775	+	0.102088i	$0.0325525\pi$
$677$	5.31252i	0.204177i	−0.994775	+	0.102088i	$0.967448\pi$
$678$	− 7.58124i	− 0.291156i
$679$	−39.1246	−1.50146
$680$	0	0
$681$	25.9443	0.994187
$682$	− 5.31252i	− 0.203427i
$683$	− 36.2928i	− 1.38871i	−0.719634	−	0.694353i	$-0.755691\pi$
$683$	− 36.2928i	− 1.38871i	0.719634	−	0.694353i	$-0.244309\pi$
$684$	2.76393	0.105682
$685$	0	0
$686$	8.81966	0.336736
$687$	43.6803i	1.66651i
$688$	13.1616i	0.501779i
$689$	−22.8328	−0.869861
$690$	0	0
$691$	−14.1803	−0.539446	−0.269723	−	0.962938i	$-0.586932\pi$
$691$	−14.1803	−0.539446	−0.269723	+	0.962938i	$0.586932\pi$
$692$	− 4.29792i	− 0.163382i
$693$	− 11.2521i	− 0.427432i
$694$	38.0344	1.44377
$695$	0	0
$696$	−5.45085	−0.206614
$697$	26.1751i	0.991452i
$698$	− 12.3125i	− 0.466033i
$699$	−2.18034	−0.0824680
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	− 10.4770i	− 0.395430i
$703$	− 11.8792i	− 0.448031i
$704$	−17.4164	−0.656406
$705$	0	0
$706$	−29.0557	−1.09353
$707$	− 38.5615i	− 1.45026i
$708$	2.26872i	0.0852637i
$709$	−20.9787	−0.787872	−0.393936	−	0.919138i	$-0.628887\pi$
$709$	−20.9787	−0.787872	−0.393936	+	0.919138i	$0.628887\pi$
$710$	0	0
$711$	4.47214	0.167718
$712$	− 25.5938i	− 0.959167i
$713$	− 1.64166i	− 0.0614807i
$714$	42.6525	1.59623
$715$	0	0
$716$	−4.47214	−0.167132
$717$	25.1605i	0.939635i
$718$	10.4770i	0.390999i
$719$	−17.2361	−0.642797	−0.321398	−	0.946944i	$-0.604153\pi$
$719$	−17.2361	−0.642797	−0.321398	+	0.946944i	$0.604153\pi$
$720$	0	0
$721$	−44.8328	−1.66966
$722$	− 44.3074i	− 1.64895i
$723$	− 2.34273i	− 0.0871272i
$724$	0.0212862	0.000791097 0
$725$	0	0
$726$	−19.9787	−0.741480
$727$	37.5469i	1.39254i	0.717780	+	0.696269i	$0.245158\pi$
$727$	37.5469i	1.39254i	−0.717780	+	0.696269i	$0.754842\pi$
$728$	− 27.4292i	− 1.01659i
$729$	−0.965558	−0.0357614
$730$	0	0
$731$	−16.2918	−0.602574
$732$	2.58225i	0.0954426i
$733$	9.22290i	0.340656i	0.985387	+	0.170328i	$0.0544827\pi$
$733$	9.22290i	0.340656i	−0.985387	+	0.170328i	$0.945517\pi$
$734$	17.3820	0.641580
$735$	0	0
$736$	−1.09017	−0.0401842
$737$	− 8.59584i	− 0.316632i
$738$	− 13.0875i	− 0.481759i
$739$	−28.2918	−1.04073	−0.520365	−	0.853944i	$-0.674204\pi$
$739$	−28.2918	−1.04073	−0.520365	+	0.853944i	$0.674204\pi$
$740$	0	0
$741$	−41.3050	−1.51738
$742$	− 39.6959i	− 1.45728i
$743$	− 6.32713i	− 0.232120i	−0.993242	−	0.116060i	$-0.962974\pi$
$743$	− 6.32713i	− 0.232120i	0.993242	−	0.116060i	$-0.0370264\pi$
$744$	12.7639	0.467948
$745$	0	0
$746$	−41.8197	−1.53113
$747$	− 8.28232i	− 0.303034i
$748$	− 2.02920i	− 0.0741950i
$749$	−35.1803	−1.28546
$750$	0	0
$751$	−14.1803	−0.517448	−0.258724	−	0.965951i	$-0.583302\pi$
$751$	−14.1803	−0.517448	−0.258724	+	0.965951i	$0.583302\pi$
$752$	− 1.76142i	− 0.0642323i
$753$	− 43.7543i	− 1.59450i
$754$	3.01316	0.109733
$755$	0	0
$756$	−2.43769	−0.0886581
$757$	− 0.627058i	− 0.0227908i	−0.999935	−	0.0113954i	$-0.996373\pi$
$757$	− 0.627058i	− 0.0227908i	0.999935	−	0.0113954i	$-0.00362735\pi$
$758$	38.4418i	1.39627i
$759$	3.52786	0.128053
$760$	0	0
$761$	−37.2705	−1.35105	−0.675527	−	0.737335i	$-0.736084\pi$
$761$	−37.2705	−1.35105	−0.675527	+	0.737335i	$0.736084\pi$
$762$	− 53.4105i	− 1.93486i
$763$	43.6803i	1.58133i
$764$	1.93112	0.0698653
$765$	0	0
$766$	−2.02129	−0.0730320
$767$	− 11.8792i	− 0.428932i
$768$	− 11.9989i	− 0.432974i
$769$	8.49342	0.306281	0.153140	−	0.988204i	$-0.451061\pi$
$769$	8.49342	0.306281	0.153140	+	0.988204i	$0.451061\pi$
$770$	0	0
$771$	29.8885	1.07641
$772$	1.64166i	0.0590846i
$773$	3.28332i	0.118093i	0.998255	+	0.0590464i	$0.0188060\pi$
$773$	3.28332i	0.118093i	−0.998255	+	0.0590464i	$0.981194\pi$
$774$	8.14590	0.292798
$775$	0	0
$776$	33.4164	1.19958
$777$	− 12.2667i	− 0.440066i
$778$	− 31.3678i	− 1.12459i
$779$	44.0689	1.57893
$780$	0	0
$781$	16.3607	0.585431
$782$	4.68547i	0.167552i
$783$	− 2.53650i	− 0.0906473i
$784$	17.6738	0.631206
$785$	0	0
$786$	−40.4721	−1.44359
$787$	47.1574i	1.68098i	0.541828	+	0.840490i	$0.317733\pi$
$787$	47.1574i	1.68098i	−0.541828	+	0.840490i	$0.682267\pi$
$788$	− 5.55204i	− 0.197783i
$789$	0.257354	0.00916205
$790$	0	0
$791$	9.23607	0.328397
$792$	9.61045i	0.341492i
$793$	− 13.5208i	− 0.480139i
$794$	−42.1378	−1.49541
$795$	0	0
$796$	3.81966	0.135384
$797$	− 31.7271i	− 1.12383i	−0.827194	−	0.561916i	$-0.810064\pi$
$797$	− 31.7271i	− 1.12383i	0.827194	−	0.561916i	$-0.189936\pi$
$798$	− 71.8106i	− 2.54207i
$799$	2.18034	0.0771349
$800$	0	0
$801$	−13.9443	−0.492697
$802$	49.5001i	1.74791i
$803$	33.1293i	1.16911i
$804$	2.18034	0.0768947
$805$	0	0
$806$	−7.05573	−0.248527
$807$	− 2.26872i	− 0.0798627i
$808$	32.9355i	1.15867i
$809$	−54.7984	−1.92661	−0.963304	−	0.268412i	$-0.913501\pi$
$809$	−54.7984	−1.92661	−0.963304	+	0.268412i	$0.913501\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	− 0.701073i	− 0.0246028i
$813$	− 17.1917i	− 0.602939i
$814$	4.36068	0.152842
$815$	0	0
$816$	−32.0689	−1.12264
$817$	27.4292i	0.959626i
$818$	− 18.2520i	− 0.638167i
$819$	−14.9443	−0.522195
$820$	0	0
$821$	−27.2705	−0.951747	−0.475874	−	0.879514i	$-0.657868\pi$
$821$	−27.2705	−0.951747	−0.475874	+	0.879514i	$0.657868\pi$
$822$	42.5917i	1.48556i
$823$	− 18.2063i	− 0.634631i	−0.948320	−	0.317316i	$-0.897219\pi$
$823$	− 18.2063i	− 0.634631i	0.948320	−	0.317316i	$-0.102781\pi$
$824$	38.2918	1.33396
$825$	0	0
$826$	20.6525	0.718592
$827$	26.8021i	0.932002i	0.884784	+	0.466001i	$0.154306\pi$
$827$	26.8021i	0.932002i	−0.884784	+	0.466001i	$0.845694\pi$
$828$	0.313529i	0.0108959i
$829$	−10.8541	−0.376979	−0.188489	−	0.982075i	$-0.560359\pi$
$829$	−10.8541	−0.376979	−0.188489	+	0.982075i	$0.560359\pi$
$830$	0	0
$831$	34.7639	1.20595
$832$	23.1313i	0.801933i
$833$	21.8772i	0.757998i
$834$	−23.6656	−0.819474
$835$	0	0
$836$	−3.41641	−0.118159
$837$	5.93958i	0.205302i
$838$	4.53744i	0.156743i
$839$	31.7082	1.09469	0.547344	−	0.836907i	$-0.315639\pi$
$839$	31.7082	1.09469	0.547344	+	0.836907i	$0.315639\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	− 13.4011i	− 0.461831i
$843$	67.1991i	2.31446i
$844$	1.37384	0.0472894
$845$	0	0
$846$	−1.09017	−0.0374808
$847$	− 24.3396i	− 0.836320i
$848$	29.8459i	1.02491i
$849$	−2.18034	−0.0748291
$850$	0	0
$851$	1.34752	0.0461925
$852$	4.14989i	0.142173i
$853$	34.3834i	1.17726i	0.808401	+	0.588632i	$0.200333\pi$
$853$	34.3834i	1.17726i	−0.808401	+	0.588632i	$0.799667\pi$
$854$	23.5066	0.804379
$855$	0	0
$856$	30.0476	1.02701
$857$	− 0.627058i	− 0.0214199i	−0.999943	−	0.0107100i	$-0.996591\pi$
$857$	− 0.627058i	− 0.0214199i	0.999943	−	0.0107100i	$-0.00340915\pi$
$858$	− 15.1625i	− 0.517639i
$859$	26.8328	0.915524	0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328	0.915524	0.457762	+	0.889075i	$0.348651\pi$
$860$	0	0
$861$	45.5066	1.55086
$862$	18.8333i	0.641466i
$863$	49.5001i	1.68500i	0.538693	+	0.842502i	$0.318918\pi$
$863$	49.5001i	1.68500i	−0.538693	+	0.842502i	$0.681082\pi$
$864$	3.94427	0.134187
$865$	0	0
$866$	18.9868	0.645199
$867$	− 3.16356i	− 0.107440i
$868$	1.64166i	0.0557216i
$869$	−5.52786	−0.187520
$870$	0	0
$871$	−11.4164	−0.386830
$872$	− 37.3074i	− 1.26339i
$873$	− 18.2063i	− 0.616190i
$874$	7.88854	0.266834
$875$	0	0
$876$	−8.40325	−0.283920
$877$	− 29.4584i	− 0.994739i	−0.867539	−	0.497370i	$-0.834299\pi$
$877$	− 29.4584i	− 0.994739i	0.867539	−	0.497370i	$-0.165701\pi$
$878$	14.1479i	0.477468i
$879$	−61.1246	−2.06168
$880$	0	0
$881$	−13.4508	−0.453171	−0.226585	−	0.973991i	$-0.572756\pi$
$881$	−13.4508	−0.453171	−0.226585	+	0.973991i	$0.572756\pi$
$882$	− 10.9386i	− 0.368321i
$883$	− 37.8605i	− 1.27411i	−0.770820	−	0.637053i	$-0.780153\pi$
$883$	− 37.8605i	− 1.27411i	0.770820	−	0.637053i	$-0.219847\pi$
$884$	−2.69505	−0.0906443
$885$	0	0
$886$	32.3262	1.08602
$887$	− 10.6708i	− 0.358290i	−0.983823	−	0.179145i	$-0.942667\pi$
$887$	− 10.6708i	− 0.358290i	0.983823	−	0.179145i	$-0.0573331\pi$
$888$	10.4770i	0.351586i
$889$	65.0689	2.18234
$890$	0	0
$891$	−22.4721	−0.752845
$892$	− 1.76142i	− 0.0589766i
$893$	− 3.67086i	− 0.122841i
$894$	−55.0000	−1.83948
$895$	0	0
$896$	−30.9787	−1.03493
$897$	− 4.68547i	− 0.156443i
$898$	13.2813i	0.443203i
$899$	−1.70820	−0.0569718
$900$	0	0
$901$	−36.9443	−1.23079
$902$	16.1771i	0.538638i
$903$	28.3240i	0.942565i
$904$	−7.88854	−0.262369
$905$	0	0
$906$	51.8885	1.72388
$907$	11.6854i	0.388007i	0.981001	+	0.194004i	$0.0621473\pi$
$907$	11.6854i	0.388007i	−0.981001	+	0.194004i	$0.937853\pi$
$908$	− 2.85003i	− 0.0945817i
$909$	17.9443	0.595174
$910$	0	0
$911$	46.7214	1.54795	0.773974	−	0.633218i	$-0.218266\pi$
$911$	46.7214	1.54795	0.773974	+	0.633218i	$0.218266\pi$
$912$	53.9918i	1.78785i
$913$	10.2375i	0.338812i
$914$	−18.4721	−0.611004
$915$	0	0
$916$	4.79837	0.158543
$917$	− 49.3063i	− 1.62824i
$918$	− 16.9522i	− 0.559505i
$919$	12.7639	0.421043	0.210522	−	0.977589i	$-0.432484\pi$
$919$	12.7639	0.421043	0.210522	+	0.977589i	$0.432484\pi$
$920$	0	0
$921$	21.0689	0.694243
$922$	− 28.2500i	− 0.930365i
$923$	− 21.7291i	− 0.715223i
$924$	−3.52786	−0.116058
$925$	0	0
$926$	30.4640	1.00111
$927$	− 20.8626i	− 0.685216i
$928$	1.13436i	0.0372372i
$929$	17.9656	0.589431	0.294715	−	0.955585i	$-0.404775\pi$
$929$	17.9656	0.589431	0.294715	+	0.955585i	$0.404775\pi$
$930$	0	0
$931$	36.8328	1.20715
$932$	0.239515i	0.00784557i
$933$	− 38.6813i	− 1.26637i
$934$	−28.7984	−0.942312
$935$	0	0
$936$	12.7639	0.417202
$937$	− 6.56664i	− 0.214523i	−0.994231	−	0.107261i	$-0.965792\pi$
$937$	− 6.56664i	− 0.214523i	0.994231	−	0.107261i	$-0.0342082\pi$
$938$	− 19.8480i	− 0.648059i
$939$	34.2492	1.11768
$940$	0	0
$941$	54.3607	1.77211	0.886054	−	0.463583i	$-0.153436\pi$
$941$	54.3607	1.77211	0.886054	+	0.463583i	$0.153436\pi$
$942$	21.6376i	0.704992i
$943$	4.99899i	0.162790i
$944$	−15.5279	−0.505389
$945$	0	0
$946$	−10.0689	−0.327368
$947$	8.71560i	0.283219i	0.989923	+	0.141610i	$0.0452277\pi$
$947$	8.71560i	0.283219i	−0.989923	+	0.141610i	$0.954772\pi$
$948$	− 1.40215i	− 0.0455396i
$949$	44.0000	1.42830
$950$	0	0
$951$	37.7771	1.22501
$952$	− 44.3814i	− 1.43841i
$953$	− 8.59584i	− 0.278447i	−0.990261	−	0.139223i	$-0.955539\pi$
$953$	− 8.59584i	− 0.278447i	0.990261	−	0.139223i	$-0.0444606\pi$
$954$	18.4721	0.598057
$955$	0	0
$956$	2.76393	0.0893920
$957$	− 3.67086i	− 0.118662i
$958$	− 39.3084i	− 1.27000i
$959$	−51.8885	−1.67557
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 5.79155i	− 0.186727i
$963$	− 16.3709i	− 0.527544i
$964$	−0.257354	−0.00828882
$965$	0	0
$966$	8.14590	0.262090
$967$	6.44688i	0.207318i	0.994613	+	0.103659i	$0.0330550\pi$
$967$	6.44688i	0.207318i	−0.994613	+	0.103659i	$0.966945\pi$
$968$	20.7885i	0.668169i
$969$	−66.8328	−2.14698
$970$	0	0
$971$	−1.81966	−0.0583957	−0.0291978	−	0.999574i	$-0.509295\pi$
$971$	−1.81966	−0.0583957	−0.0291978	+	0.999574i	$0.509295\pi$
$972$	− 3.59685i	− 0.115369i
$973$	− 28.8313i	− 0.924290i
$974$	34.0902	1.09232
$975$	0	0
$976$	−17.6738	−0.565723
$977$	− 16.1771i	− 0.517551i	−0.965938	−	0.258775i	$-0.916681\pi$
$977$	− 16.1771i	− 0.517551i	0.965938	−	0.258775i	$-0.0833190\pi$
$978$	42.0386i	1.34425i
$979$	17.2361	0.550867
$980$	0	0
$981$	−20.3262	−0.648967
$982$	15.6981i	0.500945i
$983$	− 41.9646i	− 1.33846i	−0.743054	−	0.669232i	$-0.766623\pi$
$983$	− 41.9646i	− 1.33846i	0.743054	−	0.669232i	$-0.233377\pi$
$984$	−38.8673	−1.23904
$985$	0	0
$986$	4.87539	0.155264
$987$	− 3.79062i	− 0.120657i
$988$	4.53744i	0.144355i
$989$	−3.11146	−0.0989386
$990$	0	0
$991$	−0.360680	−0.0114574	−0.00572869	−	0.999984i	$-0.501824\pi$
$991$	−0.360680	−0.0114574	−0.00572869	+	0.999984i	$0.501824\pi$
$992$	− 2.65626i	− 0.0843364i
$993$	12.5062i	0.396873i
$994$	37.7771	1.19822
$995$	0	0
$996$	−2.59675	−0.0822811
$997$	− 23.5188i	− 0.744848i	−0.928063	−	0.372424i	$-0.878527\pi$
$997$	− 23.5188i	− 0.744848i	0.928063	−	0.372424i	$-0.121473\pi$
$998$	21.4896i	0.680242i
$999$	−4.87539	−0.154250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.b.a.124.2		4
3.2	odd	2		1125.2.b.a.874.3		4
4.3	odd	2		2000.2.c.c.1249.1		4
5.2	odd	4		125.2.a.c.1.3	yes	4
5.3	odd	4		125.2.a.c.1.2	✓	4
5.4	even	2	inner	125.2.b.a.124.3		4
15.2	even	4		1125.2.a.k.1.2		4
15.8	even	4		1125.2.a.k.1.3		4
15.14	odd	2		1125.2.b.a.874.2		4
20.3	even	4		2000.2.a.o.1.4		4
20.7	even	4		2000.2.a.o.1.1		4
20.19	odd	2		2000.2.c.c.1249.4		4
25.2	odd	20		625.2.d.l.501.2		8
25.3	odd	20		625.2.d.k.376.2		8
25.4	even	10		625.2.e.h.249.2		8
25.6	even	5		625.2.e.h.374.2		8
25.8	odd	20		625.2.d.k.251.2		8
25.9	even	10		625.2.e.b.499.1		8
25.11	even	5		625.2.e.b.124.1		8
25.12	odd	20		625.2.d.l.126.2		8
25.13	odd	20		625.2.d.l.126.1		8
25.14	even	10		625.2.e.b.124.2		8
25.16	even	5		625.2.e.b.499.2		8
25.17	odd	20		625.2.d.k.251.1		8
25.19	even	10		625.2.e.h.374.1		8
25.21	even	5		625.2.e.h.249.1		8
25.22	odd	20		625.2.d.k.376.1		8
25.23	odd	20		625.2.d.l.501.1		8
35.13	even	4		6125.2.a.o.1.2		4
35.27	even	4		6125.2.a.o.1.3		4
40.3	even	4		8000.2.a.bk.1.1		4
40.13	odd	4		8000.2.a.bj.1.4		4
40.27	even	4		8000.2.a.bk.1.4		4
40.37	odd	4		8000.2.a.bj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.a.c.1.2	✓	4	5.3	odd	4
125.2.a.c.1.3	yes	4	5.2	odd	4
125.2.b.a.124.2		4	1.1	even	1	trivial
125.2.b.a.124.3		4	5.4	even	2	inner
625.2.d.k.251.1		8	25.17	odd	20
625.2.d.k.251.2		8	25.8	odd	20
625.2.d.k.376.1		8	25.22	odd	20
625.2.d.k.376.2		8	25.3	odd	20
625.2.d.l.126.1		8	25.13	odd	20
625.2.d.l.126.2		8	25.12	odd	20
625.2.d.l.501.1		8	25.23	odd	20
625.2.d.l.501.2		8	25.2	odd	20
625.2.e.b.124.1		8	25.11	even	5
625.2.e.b.124.2		8	25.14	even	10
625.2.e.b.499.1		8	25.9	even	10
625.2.e.b.499.2		8	25.16	even	5
625.2.e.h.249.1		8	25.21	even	5
625.2.e.h.249.2		8	25.4	even	10
625.2.e.h.374.1		8	25.19	even	10
625.2.e.h.374.2		8	25.6	even	5
1125.2.a.k.1.2		4	15.2	even	4
1125.2.a.k.1.3		4	15.8	even	4
1125.2.b.a.874.2		4	15.14	odd	2
1125.2.b.a.874.3		4	3.2	odd	2
2000.2.a.o.1.1		4	20.7	even	4
2000.2.a.o.1.4		4	20.3	even	4
2000.2.c.c.1249.1		4	4.3	odd	2
2000.2.c.c.1249.4		4	20.19	odd	2
6125.2.a.o.1.2		4	35.13	even	4
6125.2.a.o.1.3		4	35.27	even	4
8000.2.a.bj.1.1		4	40.37	odd	4
8000.2.a.bj.1.4		4	40.13	odd	4
8000.2.a.bk.1.1		4	40.3	even	4
8000.2.a.bk.1.4		4	40.27	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:	\( N \)	\(=\)	\( 125 = 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	125.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.32813i q^{2} +2.14896i q^{3} +0.236068 q^{4} +2.85410 q^{6} +3.47709i q^{7} -2.96979i q^{8} -1.61803 q^{9} +O(q^{10})\)	\(q-1.32813i q^{2} +2.14896i q^{3} +0.236068 q^{4} +2.85410 q^{6} +3.47709i q^{7} -2.96979i q^{8} -1.61803 q^{9} +2.00000 q^{11} +0.507301i q^{12} -2.65626i q^{13} +4.61803 q^{14} -3.47214 q^{16} -4.29792i q^{17} +2.14896i q^{18} -7.23607 q^{19} -7.47214 q^{21} -2.65626i q^{22} -0.820830i q^{23} +6.38197 q^{24} -3.52786 q^{26} +2.96979i q^{27} +0.820830i q^{28} -0.854102 q^{29} +2.00000 q^{31} -1.32813i q^{32} +4.29792i q^{33} -5.70820 q^{34} -0.381966 q^{36} +1.64166i q^{37} +9.61045i q^{38} +5.70820 q^{39} -6.09017 q^{41} +9.92398i q^{42} -3.79062i q^{43} +0.472136 q^{44} -1.09017 q^{46} +0.507301i q^{47} -7.46149i q^{48} -5.09017 q^{49} +9.23607 q^{51} -0.627058i q^{52} -8.59584i q^{53} +3.94427 q^{54} +10.3262 q^{56} -15.5500i q^{57} +1.13436i q^{58} +4.47214 q^{59} +5.09017 q^{61} -2.65626i q^{62} -5.62605i q^{63} -8.70820 q^{64} +5.70820 q^{66} -4.29792i q^{67} -1.01460i q^{68} +1.76393 q^{69} +8.18034 q^{71} +4.80522i q^{72} +16.5646i q^{73} +2.18034 q^{74} -1.70820 q^{76} +6.95418i q^{77} -7.58124i q^{78} -2.76393 q^{79} -11.2361 q^{81} +8.08854i q^{82} +5.11875i q^{83} -1.76393 q^{84} -5.03444 q^{86} -1.83543i q^{87} -5.93958i q^{88} +8.61803 q^{89} +9.23607 q^{91} -0.193772i q^{92} +4.29792i q^{93} +0.673762 q^{94} +2.85410 q^{96} +11.2521i q^{97} +6.76041i q^{98} -3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 2 * q^6 - 2 * q^9	\( 4 q - 8 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 14 q^{14} + 4 q^{16} - 20 q^{19} - 12 q^{21} + 30 q^{24} - 32 q^{26} + 10 q^{29} + 8 q^{31} + 4 q^{34} - 6 q^{36} - 4 q^{39} - 2 q^{41} - 16 q^{44} + 18 q^{46} + 2 q^{49} + 28 q^{51} - 20 q^{54} + 10 q^{56} - 2 q^{61} - 8 q^{64} - 4 q^{66} + 16 q^{69} - 12 q^{71} - 36 q^{74} + 20 q^{76} - 20 q^{79} - 36 q^{81} - 16 q^{84} + 38 q^{86} + 30 q^{89} + 28 q^{91} + 34 q^{94} - 2 q^{96} - 4 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 14 * q^14 + 4 * q^16 - 20 * q^19 - 12 * q^21 + 30 * q^24 - 32 * q^26 + 10 * q^29 + 8 * q^31 + 4 * q^34 - 6 * q^36 - 4 * q^39 - 2 * q^41 - 16 * q^44 + 18 * q^46 + 2 * q^49 + 28 * q^51 - 20 * q^54 + 10 * q^56 - 2 * q^61 - 8 * q^64 - 4 * q^66 + 16 * q^69 - 12 * q^71 - 36 * q^74 + 20 * q^76 - 20 * q^79 - 36 * q^81 - 16 * q^84 + 38 * q^86 + 30 * q^89 + 28 * q^91 + 34 * q^94 - 2 * q^96 - 4 * q^99

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