LMFDB - Embedded newform 136.2.s.b.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,2,Mod(3,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 1]))

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

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

chi := DirichletCharacter("136.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, 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:	$1.08596546749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			107.1
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	136.107
Dual form			136.2.s.b.75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.541196 + 1.30656i) q^{2} +(0.783227 + 1.17218i) q^{3} +(-1.41421 + 1.41421i) q^{4} +(-1.10765 + 1.65772i) q^{6} +(-2.61313 - 1.08239i) q^{8} +(0.387484 - 0.935468i) q^{9} +O(q^{10})$	\(q+(0.541196 + 1.30656i) q^{2} +(0.783227 + 1.17218i) q^{3} +(-1.41421 + 1.41421i) q^{4} +(-1.10765 + 1.65772i) q^{6} +(-2.61313 - 1.08239i) q^{8} +(0.387484 - 0.935468i) q^{9} +(0.816229 + 0.545387i) q^{11} +(-2.76537 - 0.550066i) q^{12} -4.00000i q^{16} +(1.46508 - 3.85403i) q^{17} +1.43195 q^{18} +(1.75490 + 4.23671i) q^{19} +(-0.270842 + 1.36162i) q^{22} +(-0.777910 - 3.91082i) q^{24} +(-4.61940 - 1.91342i) q^{25} +(5.54808 - 1.10358i) q^{27} +(5.22625 - 2.16478i) q^{32} +1.38393i q^{33} +(5.82843 - 0.171573i) q^{34} +(0.774967 + 1.87094i) q^{36} +(-4.58579 + 4.58579i) q^{38} +(-2.19952 - 11.0577i) q^{41} +(-1.53553 + 3.70711i) q^{43} +(-1.92562 + 0.383029i) q^{44} +(4.68873 - 3.13291i) q^{48} +(-6.46716 + 2.67878i) q^{49} -7.07107i q^{50} +(5.66511 - 1.30125i) q^{51} +(4.44450 + 6.65166i) q^{54} +(-3.59171 + 5.37538i) q^{57} +(-14.1924 - 5.87868i) q^{59} +(5.65685 + 5.65685i) q^{64} +(-1.80819 + 0.748978i) q^{66} +9.68714i q^{67} +(3.37849 + 7.52235i) q^{68} +(-2.02509 + 2.02509i) q^{72} +(-2.53605 + 12.7496i) q^{73} +(-1.37516 - 6.91342i) q^{75} +(-8.47343 - 3.50981i) q^{76} +(3.49107 + 3.49107i) q^{81} +(13.2573 - 8.85822i) q^{82} +(-16.3640 + 6.77817i) q^{83} -5.67459 q^{86} +(-1.54259 - 2.30864i) q^{88} +(1.17525 - 1.17525i) q^{89} +(6.63087 + 4.43060i) q^{96} +(18.2381 + 3.62778i) q^{97} +(-7.00000 - 7.00000i) q^{98} +(0.826467 - 0.552228i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 8 * q^6 - 16 * q^9	$ 8 q + 8 q^{3} - 8 q^{6} - 16 q^{9} - 16 q^{12} + 24 q^{22} + 32 q^{24} - 16 q^{27} + 24 q^{34} - 32 q^{36} - 48 q^{38} + 32 q^{41} + 16 q^{43} + 16 q^{44} - 32 q^{48} + 40 q^{51} - 8 q^{54} - 40 q^{57} - 40 q^{59} + 56 q^{66} + 48 q^{72} - 48 q^{73} + 56 q^{81} - 80 q^{83} + 32 q^{96} - 56 q^{98} + 32 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 8 * q^6 - 16 * q^9 - 16 * q^12 + 24 * q^22 + 32 * q^24 - 16 * q^27 + 24 * q^34 - 32 * q^36 - 48 * q^38 + 32 * q^41 + 16 * q^43 + 16 * q^44 - 32 * q^48 + 40 * q^51 - 8 * q^54 - 40 * q^57 - 40 * q^59 + 56 * q^66 + 48 * q^72 - 48 * q^73 + 56 * q^81 - 80 * q^83 + 32 * q^96 - 56 * q^98 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{16}\right)$

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 $

$2$	0.541196	+	1.30656i	0.382683	+	0.923880i
$2$	0.541196	+	1.30656i	0.382683	+	0.923880i
$3$	0.783227	+	1.17218i	0.452196	+	0.676760i	0.985599	−	0.169102i	$-0.0540867\pi$
$3$	0.783227	+	1.17218i	0.452196	+	0.676760i	−0.533402	+	0.845862i	$0.679087\pi$
$4$	−1.41421	+	1.41421i	−0.707107	+	0.707107i
$5$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$5$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$6$	−1.10765	+	1.65772i	−0.452196	+	0.676760i
$7$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$7$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$8$	−2.61313	−	1.08239i	−0.923880	−	0.382683i
$9$	0.387484	−	0.935468i	0.129161	−	0.311823i
$10$		0			0
$11$	0.816229	+	0.545387i	0.246102	+	0.164440i	0.672504	−	0.740094i	$-0.265219\pi$
$11$	0.816229	+	0.545387i	0.246102	+	0.164440i	−0.426401	+	0.904534i	$0.640219\pi$
$12$	−2.76537	−	0.550066i	−0.798293	−	0.158790i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		−	4.00000i		−	1.00000i
$17$	1.46508	−	3.85403i	0.355333	−	0.934740i
$17$	1.46508	−	3.85403i	0.355333	−	0.934740i
$18$	1.43195			0.337514
$19$	1.75490	+	4.23671i	0.402603	+	0.971969i	0.987032	+	0.160524i	$0.0513185\pi$
$19$	1.75490	+	4.23671i	0.402603	+	0.971969i	−0.584429	+	0.811445i	$0.698682\pi$
$20$		0			0
$21$		0			0
$22$	−0.270842	+	1.36162i	−0.0577438	+	0.290297i
$23$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$23$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$24$	−0.777910	−	3.91082i	−0.158790	−	0.798293i
$25$	−4.61940	−	1.91342i	−0.923880	−	0.382683i
$26$		0			0
$27$	5.54808	−	1.10358i	1.06773	−	0.212384i
$28$		0			0
$29$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$29$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$30$		0			0
$31$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$31$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$32$	5.22625	−	2.16478i	0.923880	−	0.382683i
$33$			1.38393i			0.240911i
$34$	5.82843	−	0.171573i	0.999567	−	0.0294245i
$35$		0			0
$36$	0.774967	+	1.87094i	0.129161	+	0.311823i
$37$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$	−4.58579	+	4.58579i	−0.743913	+	0.743913i
$39$		0			0
$40$		0			0
$41$	−2.19952	−	11.0577i	−0.343508	−	1.72693i	−0.636908	−	0.770940i	$-0.719787\pi$
$41$	−2.19952	−	11.0577i	−0.343508	−	1.72693i	0.293400	−	0.955990i	$-0.405213\pi$
$42$		0			0
$43$	−1.53553	+	3.70711i	−0.234167	+	0.565328i	−0.996660	−	0.0816682i	$-0.973975\pi$
$43$	−1.53553	+	3.70711i	−0.234167	+	0.565328i	0.762493	+	0.646997i	$0.223975\pi$
$44$	−1.92562	+	0.383029i	−0.290297	+	0.0577438i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	4.68873	−	3.13291i	0.676760	−	0.452196i
$49$	−6.46716	+	2.67878i	−0.923880	+	0.382683i
$50$		−	7.07107i		−	1.00000i
$51$	5.66511	−	1.30125i	0.793275	−	0.182211i
$52$		0			0
$53$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$53$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$54$	4.44450	+	6.65166i	0.604819	+	0.905176i
$55$		0			0
$56$		0			0
$57$	−3.59171	+	5.37538i	−0.475734	+	0.711986i
$58$		0			0
$59$	−14.1924	−	5.87868i	−1.84769	−	0.765339i	−0.927117	−	0.374772i	$-0.877721\pi$
$59$	−14.1924	−	5.87868i	−1.84769	−	0.765339i	−0.920575	−	0.390567i	$-0.872279\pi$
$60$		0			0
$61$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$61$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	+	5.65685i	0.707107	+	0.707107i
$65$		0			0
$66$	−1.80819	+	0.748978i	−0.222573	+	0.0921928i
$67$			9.68714i			1.18347i	0.806132	+	0.591736i	$0.201557\pi$
$67$			9.68714i			1.18347i	−0.806132	+	0.591736i	$0.798443\pi$
$68$	3.37849	+	7.52235i	0.409702	+	0.912219i
$69$		0			0
$70$		0			0
$71$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$71$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$72$	−2.02509	+	2.02509i	−0.238659	+	0.238659i
$73$	−2.53605	+	12.7496i	−0.296822	+	1.49223i	0.488185	+	0.872740i	$0.337659\pi$
$73$	−2.53605	+	12.7496i	−0.296822	+	1.49223i	−0.785007	+	0.619486i	$0.787341\pi$
$74$		0			0
$75$	−1.37516	−	6.91342i	−0.158790	−	0.798293i
$76$	−8.47343	−	3.50981i	−0.971969	−	0.402603i
$77$		0			0
$78$		0			0
$79$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$79$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$80$		0			0
$81$	3.49107	+	3.49107i	0.387897	+	0.387897i
$82$	13.2573	−	8.85822i	1.46402	−	0.978227i
$83$	−16.3640	+	6.77817i	−1.79618	+	0.744001i	−0.808300	+	0.588771i	$0.799612\pi$
$83$	−16.3640	+	6.77817i	−1.79618	+	0.744001i	−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		0			0
$86$	−5.67459			−0.611907
$87$		0			0
$88$	−1.54259	−	2.30864i	−0.164440	−	0.246102i
$89$	1.17525	−	1.17525i	0.124576	−	0.124576i	−0.642070	−	0.766646i	$-0.721924\pi$
$89$	1.17525	−	1.17525i	0.124576	−	0.124576i	0.766646	+	0.642070i	$0.221924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	6.63087	+	4.43060i	0.676760	+	0.452196i
$97$	18.2381	+	3.62778i	1.85180	+	0.368345i	0.990246	−	0.139328i	$-0.0444942\pi$
$97$	18.2381	+	3.62778i	1.85180	+	0.368345i	0.861550	+	0.507673i	$0.169494\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$	0.826467	−	0.552228i	0.0830631	−	0.0555010i
$100$	9.23880	−	3.82683i	0.923880	−	0.382683i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	4.76610	+	6.69760i	0.471914	+	0.663161i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.01673	−	20.1935i	0.388312	−	1.95218i	0.0982914	−	0.995158i	$-0.468662\pi$
$107$	4.01673	−	20.1935i	0.388312	−	1.95218i	0.290021	−	0.957020i	$-0.406338\pi$
$108$	−6.28547	+	9.40687i	−0.604819	+	0.905176i
$109$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$109$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.1461	+	8.11574i	1.14261	+	0.763465i	0.974959	−	0.222383i	$-0.0713835\pi$
$113$	12.1461	+	8.11574i	1.14261	+	0.763465i	0.167646	+	0.985847i	$0.446383\pi$
$114$	−8.96709	−	1.78367i	−0.839845	−	0.167056i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	21.7248i		−	1.99993i
$119$		0			0
$120$		0			0
$121$	−3.84073	−	9.27235i	−0.349158	−	0.842941i
$122$		0			0
$123$	11.2390	−	11.2390i	1.01338	−	1.01338i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$127$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$128$	−4.32957	+	10.4525i	−0.382683	+	0.923880i
$129$	−5.54808	+	1.10358i	−0.488481	+	0.0971649i
$130$		0			0
$131$	6.97290	+	1.38700i	0.609225	+	0.121182i	0.490055	−	0.871692i	$-0.336977\pi$
$131$	6.97290	+	1.38700i	0.609225	+	0.121182i	0.119170	+	0.992874i	$0.461977\pi$
$132$	−1.95717	−	1.95717i	−0.170350	−	0.170350i
$133$		0			0
$134$	−12.6569	+	5.24264i	−1.09339	+	0.452895i
$135$		0			0
$136$	−8.00000	+	8.48528i	−0.685994	+	0.727607i
$137$	20.2426			1.72945			0.864723	−	0.502249i	$-0.167494\pi$
$137$	20.2426			1.72945			0.864723	+	0.502249i	$0.167494\pi$
$138$		0			0
$139$	−7.00809	−	10.4883i	−0.594418	−	0.889610i	0.405279	−	0.914193i	$-0.367174\pi$
$139$	−7.00809	−	10.4883i	−0.594418	−	0.889610i	−0.999698	+	0.0245830i	$0.992174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−3.74187	−	1.54993i	−0.311823	−	0.129161i
$145$		0			0
$146$	−18.0306	+	3.58652i	−1.49223	+	0.296822i
$147$	−8.20528	−	5.48259i	−0.676760	−	0.452196i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	8.28858	−	5.53825i	0.676760	−	0.452196i
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		−	12.9706i		−	1.05205i
$153$	−3.03763	−	2.86391i	−0.245578	−	0.231533i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−2.67195	+	6.45067i	−0.209928	+	0.506812i
$163$	24.1043	−	4.79464i	1.88799	−	0.375546i	0.891067	−	0.453872i	$-0.149958\pi$
$163$	24.1043	−	4.79464i	1.88799	−	0.375546i	0.996928	+	0.0783260i	$0.0249575\pi$
$164$	18.7486	+	12.5274i	1.46402	+	0.978227i
$165$		0			0
$166$	−17.7122	−	17.7122i	−1.37474	−	1.37474i
$167$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$167$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	4.64331			0.355083
$172$	−3.07107	−	7.41421i	−0.234167	−	0.565328i
$173$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$173$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$174$		0			0
$175$		0			0
$176$	2.18155	−	3.26492i	0.164440	−	0.246102i
$177$	−4.22498	−	21.2404i	−0.317569	−	1.59653i
$178$	2.17157	+	0.899495i	0.162766	+	0.0674200i
$179$	−10.2283	+	24.6934i	−0.764501	+	1.84567i	−0.338330	+	0.941028i	$0.609862\pi$
$179$	−10.2283	+	24.6934i	−0.764501	+	1.84567i	−0.426172	+	0.904642i	$0.640138\pi$
$180$		0			0
$181$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$181$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3.29777	−	2.34674i	0.241157	−	0.171611i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	−2.20026	+	11.0615i	−0.158790	+	0.798293i
$193$	−13.8271	+	20.6937i	−0.995294	+	1.48956i	−0.127996	+	0.991775i	$0.540854\pi$
$193$	−13.8271	+	20.6937i	−0.995294	+	1.48956i	−0.867298	+	0.497788i	$0.834146\pi$
$194$	5.13045	+	25.7925i	0.368345	+	1.85180i
$195$		0			0
$196$	5.35757	−	12.9343i	0.382683	−	0.923880i
$197$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$197$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$198$	1.16880	+	0.780968i	0.0830631	+	0.0555010i
$199$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$199$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$200$	10.0000	+	10.0000i	0.707107	+	0.707107i
$201$	−11.3551	+	7.58723i	−0.800926	+	0.535162i
$202$		0			0
$203$		0			0
$204$	−6.17144	+	9.85192i	−0.432087	+	0.689772i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.878244	+	4.41523i	−0.0607494	+	0.305408i
$210$		0			0
$211$	5.64664	+	28.3876i	0.388731	+	1.95428i	0.278831	+	0.960340i	$0.410053\pi$
$211$	5.64664	+	28.3876i	0.388731	+	1.95428i	0.109900	+	0.993943i	$0.464947\pi$
$212$		0			0
$213$		0			0
$214$	28.5579	−	5.68052i	1.95218	−	0.388312i
$215$		0			0
$216$	−15.6923	−	3.12140i	−1.06773	−	0.212384i
$217$		0			0
$218$		0			0
$219$	−16.9311	+	7.01311i	−1.14410	+	0.473902i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$223$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$224$		0			0
$225$	−3.57988	+	3.57988i	−0.238659	+	0.238659i
$226$	−4.03032	+	20.2618i	−0.268093	+	1.34779i
$227$	−4.79279	+	7.17291i	−0.318108	+	0.476083i	−0.955721	−	0.294274i	$-0.904922\pi$
$227$	−4.79279	+	7.17291i	−0.318108	+	0.476083i	0.637613	+	0.770357i	$0.279922\pi$
$228$	−2.52248	−	12.6814i	−0.167056	−	0.839845i
$229$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$229$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	29.9406	+	5.95555i	1.96147	+	0.390161i	0.982683	+	0.185296i	$0.0593245\pi$
$233$	29.9406	+	5.95555i	1.96147	+	0.390161i	0.978790	+	0.204865i	$0.0656755\pi$
$234$		0			0
$235$		0			0
$236$	28.3848	−	11.7574i	1.84769	−	0.765339i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−17.2485	−	25.8143i	−1.11108	−	1.66284i	−0.564491	−	0.825439i	$-0.690928\pi$
$241$	−17.2485	−	25.8143i	−1.11108	−	1.66284i	−0.546585	−	0.837404i	$-0.684072\pi$
$242$	10.0363	−	10.0363i	0.645159	−	0.645159i
$243$	1.95287	−	9.81775i	0.125277	−	0.629809i
$244$		0			0
$245$		0			0
$246$	20.7669	+	8.60193i	1.32405	+	0.548439i
$247$		0			0
$248$		0			0
$249$	−20.7620	−	13.8727i	−1.31574	−	0.879146i
$250$		0			0
$251$	−4.49935	−	4.49935i	−0.283996	−	0.283996i	0.550704	−	0.834700i	$-0.314359\pi$
$251$	−4.49935	−	4.49935i	−0.283996	−	0.283996i	−0.834700	+	0.550704i	$0.814359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$	−8.94975	−	21.6066i	−0.558270	−	1.34778i	−0.911135	−	0.412108i	$-0.864792\pi$
$257$	−8.94975	−	21.6066i	−0.558270	−	1.34778i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	−4.44450	−	6.65166i	−0.276702	−	0.414114i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.96151	+	9.86116i	0.121182	+	0.609225i
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$	1.49796	−	3.61639i	0.0921928	−	0.222573i
$265$		0			0
$266$		0			0
$267$	2.29809	+	0.457118i	0.140641	+	0.0279752i
$268$	−13.6997	−	13.6997i	−0.836841	−	0.836841i
$269$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$269$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−15.4161	−	5.86030i	−0.934740	−	0.355333i
$273$		0			0
$274$	10.9552	+	26.4483i	0.661830	+	1.59780i
$275$	−2.72693	−	4.08114i	−0.164440	−	0.246102i
$276$		0			0
$277$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$277$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$278$	9.91093	−	14.8328i	0.594418	−	0.889610i
$279$		0			0
$280$		0			0
$281$	2.78290	−	6.71852i	0.166014	−	0.400794i	−0.818877	−	0.573969i	$-0.805403\pi$
$281$	2.78290	−	6.71852i	0.166014	−	0.400794i	0.984891	+	0.173176i	$0.0554029\pi$
$282$		0			0
$283$	−26.9686	−	18.0198i	−1.60312	−	1.07117i	−0.949234	−	0.314571i	$-0.898139\pi$
$283$	−26.9686	−	18.0198i	−1.60312	−	1.07117i	−0.653882	−	0.756596i	$-0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		−	5.72781i		−	0.337514i
$289$	−12.7071	−	11.2929i	−0.747477	−	0.664288i
$290$		0			0
$291$	10.0321	+	24.2197i	0.588094	+	1.41979i
$292$	−14.4441	−	21.6172i	−0.845278	−	1.26505i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	2.72269	−	13.6879i	0.158790	−	0.798293i
$295$		0			0
$296$		0			0
$297$	5.13038	+	2.12507i	0.297695	+	0.123309i
$298$		0			0
$299$		0			0
$300$	11.7218	+	7.83227i	0.676760	+	0.452196i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	16.9469	−	7.01962i	0.971969	−	0.402603i
$305$		0			0
$306$	2.09792	−	5.51879i	0.119930	−	0.315488i
$307$	−8.48528			−0.484281			−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528			−0.484281			−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$311$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$312$		0			0
$313$	−3.20522	−	16.1138i	−0.181170	−	0.910803i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	−3.20522	−	16.1138i	−0.181170	−	0.910803i	0.778063	−	0.628186i	$-0.216202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$317$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	26.8165	−	11.1077i	1.49675	−	0.619974i
$322$		0			0
$323$	18.8995	−	0.556349i	1.05160	−	0.0309561i
$324$	−9.87425			−0.548569
$325$		0			0
$326$	19.3097	+	28.8989i	1.06946	+	1.60056i
$327$		0			0
$328$	−6.22119	+	31.2760i	−0.343508	+	1.72693i
$329$		0			0
$330$		0			0
$331$	31.1924	+	12.9203i	1.71449	+	0.710164i	0.999944	+	0.0105746i	$0.00336607\pi$
$331$	31.1924	+	12.9203i	1.71449	+	0.710164i	0.714545	+	0.699590i	$0.246634\pi$
$332$	13.5563	−	32.7279i	0.744001	−	1.79618i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.4078	+	16.9769i	−1.38405	+	0.924793i	−0.384052	+	0.923312i	$0.625472\pi$
$337$	−25.4078	+	16.9769i	−1.38405	+	0.924793i	−0.999999	+	0.00148149i	$0.999528\pi$
$338$	−16.9853	+	7.03555i	−0.923880	+	0.382683i
$339$			20.5939i			1.11851i
$340$		0			0
$341$		0			0
$342$	2.51294	+	6.06677i	0.135884	+	0.328054i
$343$		0			0
$344$	8.02509	−	8.02509i	0.432684	−	0.432684i
$345$		0			0
$346$		0			0
$347$	3.01205	+	15.1426i	0.161695	+	0.812897i	0.973450	+	0.228899i	$0.0735125\pi$
$347$	3.01205	+	15.1426i	0.161695	+	0.812897i	−0.811755	+	0.583998i	$0.801488\pi$
$348$		0			0
$349$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$349$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$		0			0
$352$	5.44646	+	1.08337i	0.290297	+	0.0577438i
$353$	22.9000	+	22.9000i	1.21884	+	1.21884i	0.968037	+	0.250808i	$0.0806962\pi$
$353$	22.9000	+	22.9000i	1.21884	+	1.21884i	0.250808	+	0.968037i	$0.419304\pi$
$354$	25.4654	−	17.0154i	1.35347	−	0.904360i
$355$		0			0
$356$			3.32410i			0.176177i
$357$		0			0
$358$	−37.7990			−1.99774
$359$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$359$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$360$		0			0
$361$	−1.43503	+	1.43503i	−0.0755278	+	0.0755278i
$362$		0			0
$363$	7.86072	−	11.7644i	0.412581	−	0.617471i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$367$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$368$		0			0
$369$	−11.1964	−	2.22711i	−0.582864	−	0.115939i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	4.85090	+	3.03870i	0.250834	+	0.157128i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	5.24435	−	26.3652i	0.269384	−	1.35429i	−0.574826	−	0.818275i	$-0.694930\pi$
$379$	5.24435	−	26.3652i	0.269384	−	1.35429i	0.844211	−	0.536011i	$-0.180070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$383$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$384$	−15.6433	+	3.11164i	−0.798293	+	0.158790i
$385$		0			0
$386$	−34.5207	−	6.86660i	−1.75706	−	0.349501i
$387$	2.87289	+	2.87289i	0.146037	+	0.146037i
$388$	−30.9230	+	20.6621i	−1.56988	+	1.04896i
$389$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$389$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$390$		0			0
$391$		0			0
$392$	19.7990			1.00000
$393$	3.83555	+	9.25984i	0.193478	+	0.467097i
$394$		0			0
$395$		0			0
$396$	−0.387833	+	1.94977i	−0.0194894	+	0.0979796i
$397$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$397$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$398$		0			0
$399$		0			0
$400$	−7.65367	+	18.4776i	−0.382683	+	0.923880i
$401$	−13.3484	+	2.65516i	−0.666586	+	0.132592i	−0.516774	−	0.856122i	$-0.672867\pi$
$401$	−13.3484	+	2.65516i	−0.666586	+	0.132592i	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−16.0585	−	10.7300i	−0.800926	−	0.535162i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−16.2121	−	2.73156i	−0.802619	−	0.135232i
$409$	7.33664			0.362774			0.181387	−	0.983412i	$-0.441941\pi$
$409$	7.33664			0.362774			0.181387	+	0.983412i	$0.441941\pi$
$410$		0			0
$411$	15.8546	+	23.7281i	0.782049	+	1.17042i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.80533	−	16.4295i	0.333258	−	0.804557i
$418$	−6.24408	+	1.24202i	−0.305408	+	0.0607494i
$419$	−27.0655	−	18.0846i	−1.32224	−	0.883490i	−0.324192	−	0.945991i	$-0.605092\pi$
$419$	−27.0655	−	18.0846i	−1.32224	−	0.883490i	−0.998045	+	0.0625011i	$0.980092\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	−34.0342	+	22.7410i	−1.65676	+	1.10701i
$423$		0			0
$424$		0			0
$425$	−14.1421	+	15.0000i	−0.685994	+	0.727607i
$426$		0			0
$427$		0			0
$428$	22.8774	+	34.2384i	1.10582	+	1.65498i
$429$		0			0
$430$		0			0
$431$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$431$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$432$	−4.41432	−	22.1923i	−0.212384	−	1.06773i
$433$	13.7381	+	5.69052i	0.660213	+	0.273469i	0.687528	−	0.726158i	$-0.258696\pi$
$433$	13.7381	+	5.69052i	0.660213	+	0.273469i	−0.0273152	+	0.999627i	$0.508696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−18.3261	−	18.3261i	−0.875657	−	0.875657i
$439$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$439$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$440$		0			0
$441$			7.08780i			0.337514i
$442$		0			0
$443$	18.6858			0.887791			0.443895	−	0.896079i	$-0.353596\pi$
$443$	18.6858			0.887791			0.443895	+	0.896079i	$0.353596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.516138	+	2.59480i	0.0243580	+	0.122456i	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	0.516138	+	2.59480i	0.0243580	+	0.122456i	−0.966693	+	0.255938i	$0.917616\pi$
$450$	−6.61476	−	2.73992i	−0.311823	−	0.129161i
$451$	4.23543	−	10.2252i	0.199439	−	0.481488i
$452$	−28.6545	+	5.69974i	−1.34779	+	0.268093i
$453$		0			0
$454$	−11.9657	−	2.38013i	−0.561578	−	0.111705i
$455$		0			0
$456$	15.2039	−	10.1589i	0.711986	−	0.475734i
$457$	39.1584	−	16.2200i	1.83175	−	0.758737i	0.865617	−	0.500706i	$-0.166926\pi$
$457$	39.1584	−	16.2200i	1.83175	−	0.758737i	0.966137	−	0.258031i	$-0.0830738\pi$
$458$		0			0
$459$	3.87512	−	22.9993i	0.180875	−	1.07351i
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	8.42242	+	42.3424i	0.390161	+	1.96147i
$467$	0.726905	+	0.301094i	0.0336372	+	0.0139330i	0.399439	−	0.916760i	$-0.369205\pi$
$467$	0.726905	+	0.301094i	0.0336372	+	0.0139330i	−0.365801	+	0.930693i	$0.619205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	30.7235	+	30.7235i	1.41416	+	1.41416i
$473$	−3.27515	+	2.18839i	−0.150592	+	0.100622i
$474$		0			0
$475$		−	22.9289i		−	1.05205i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$479$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$480$		0			0
$481$		0			0
$482$	24.3931	−	36.5069i	1.11108	−	1.66284i
$483$		0			0
$484$	18.5447	+	7.68147i	0.842941	+	0.349158i
$485$		0			0
$486$	13.8844	−	2.76178i	0.629809	−	0.125277i
$487$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$487$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$488$		0			0
$489$	24.4993	+	24.4993i	1.10790	+	1.10790i
$490$		0			0
$491$	−36.6766	+	15.1920i	−1.65519	+	0.685603i	−0.997695	−	0.0678537i	$-0.978385\pi$
$491$	−36.6766	+	15.1920i	−1.65519	+	0.685603i	−0.657497	+	0.753457i	$0.728385\pi$
$492$			31.7886i			1.43314i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	6.88926	−	34.6347i	0.308715	−	1.55202i
$499$	19.5624	−	29.2772i	0.875734	−	1.31063i	−0.0738993	−	0.997266i	$-0.523544\pi$
$499$	19.5624	−	29.2772i	0.875734	−	1.31063i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$		0			0
$502$	3.44365	−	8.31371i	0.153698	−	0.371059i
$503$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$503$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.2384	+	10.1820i	−0.676760	+	0.452196i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−8.65914	−	20.9050i	−0.382683	−	0.923880i
$513$	14.4119	+	21.5689i	0.636301	+	0.952292i
$514$	23.3868	−	23.3868i	1.03155	−	1.03155i
$515$		0			0
$516$	6.28547	−	9.40687i	0.276702	−	0.414114i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.0582	+	22.7570i	1.49212	+	0.997000i	0.991325	+	0.131432i	$0.0419576\pi$
$521$	34.0582	+	22.7570i	1.49212	+	0.997000i	0.500791	+	0.865568i	$0.333042\pi$
$522$		0			0
$523$	−26.9126	−	26.9126i	−1.17680	−	1.17680i	−0.980554	−	0.196250i	$-0.937124\pi$
$523$	−26.9126	−	26.9126i	−1.17680	−	1.17680i	−0.196250	−	0.980554i	$-0.562876\pi$
$524$	−11.8227	+	7.89966i	−0.516476	+	0.345098i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	5.53572			0.240911
$529$	−8.80172	−	21.2492i	−0.382683	−	0.923880i
$530$		0			0
$531$	−10.9986	+	10.9986i	−0.477300	+	0.477300i
$532$		0			0
$533$		0			0
$534$	0.646463	+	3.24999i	0.0279752	+	0.140641i
$535$		0			0
$536$	10.4853	−	25.3137i	0.452895	−	1.09339i
$537$	−36.9563	+	7.35106i	−1.59478	+	0.317221i
$538$		0			0
$539$	−6.73965	−	1.34060i	−0.290297	−	0.0577438i
$540$		0			0
$541$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$541$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$542$		0			0
$543$		0			0
$544$	−0.686292	−	23.3137i	−0.0294245	−	0.999567i
$545$		0			0
$546$		0			0
$547$	0.362185	+	0.542048i	0.0154859	+	0.0231763i	0.839132	−	0.543928i	$-0.183064\pi$
$547$	0.362185	+	0.542048i	0.0154859	+	0.0231763i	−0.823646	+	0.567104i	$0.808064\pi$
$548$	−28.6274	+	28.6274i	−1.22290	+	1.22290i
$549$		0			0
$550$	3.85647	−	5.77161i	0.164440	−	0.246102i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	24.7437	+	4.92183i	1.04937	+	0.208732i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.33371	+	2.02756i	0.225190	+	0.0856038i
$562$	10.2843			0.433816
$563$	−15.9914	−	38.6066i	−0.673956	−	1.62707i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	−15.9914	−	38.6066i	−0.673956	−	1.62707i	0.100870	−	0.994900i	$-0.467837\pi$
$564$		0			0
$565$		0			0
$566$	8.94875	−	44.9884i	0.376144	−	1.89100i
$567$		0			0
$568$		0			0
$569$	−43.8492	−	18.1630i	−1.83826	−	0.761431i	−0.957890	−	0.287135i	$-0.907297\pi$
$569$	−43.8492	−	18.1630i	−1.83826	−	0.761431i	−0.880366	−	0.474295i	$-0.842703\pi$
$570$		0			0
$571$	−22.6108	+	4.49756i	−0.946231	+	0.188217i	−0.644007	−	0.765020i	$-0.722729\pi$
$571$	−22.6108	+	4.49756i	−0.946231	+	0.188217i	−0.302224	+	0.953237i	$0.597729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	7.48375	−	3.09987i	0.311823	−	0.129161i
$577$			33.9411i			1.41299i	0.707719	+	0.706494i	$0.249724\pi$
$577$			33.9411i			1.41299i	−0.707719	+	0.706494i	$0.750276\pi$
$578$	7.87784	−	22.7143i	0.327675	−	0.944791i
$579$	−35.0865			−1.45815
$580$		0			0
$581$		0			0
$582$	−26.2152	+	26.2152i	−1.08666	+	1.08666i
$583$		0			0
$584$	20.4271	−	30.5713i	0.845278	−	1.26505i
$585$		0			0
$586$		0			0
$587$	−2.29610	+	5.54328i	−0.0947702	+	0.228796i	−0.964155	−	0.265341i	$-0.914516\pi$
$587$	−2.29610	+	5.54328i	−0.0947702	+	0.228796i	0.869385	+	0.494136i	$0.164516\pi$
$588$	19.3576	−	3.85046i	0.798293	−	0.158790i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.636039	−	0.263456i	0.0261190	−	0.0108188i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	0.636039	−	0.263456i	0.0261190	−	0.0108188i	0.395705	+	0.918378i	$0.370500\pi$
$594$			7.85325i			0.322223i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	−3.88955	+	19.5541i	−0.158790	+	0.798293i
$601$	23.2258	−	34.7599i	0.947400	−	1.41788i	0.0392547	−	0.999229i	$-0.487502\pi$
$601$	23.2258	−	34.7599i	0.947400	−	1.41788i	0.908145	−	0.418655i	$-0.137498\pi$
$602$		0			0
$603$	9.06201	+	3.75361i	0.369034	+	0.152859i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$607$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$608$	18.3431	+	18.3431i	0.743913	+	0.743913i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	8.34603	−	0.245684i	0.337368	−	0.00993120i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−4.59220	−	11.0866i	−0.185326	−	0.447417i
$615$		0			0
$616$		0			0
$617$	−0.574469	+	2.88805i	−0.0231272	+	0.116268i	−0.990624	−	0.136613i	$-0.956378\pi$
$617$	−0.574469	+	2.88805i	−0.0231272	+	0.116268i	0.967497	+	0.252882i	$0.0813783\pi$
$618$		0			0
$619$	7.12838	+	35.8368i	0.286514	+	1.44040i	0.809028	+	0.587770i	$0.199994\pi$
$619$	7.12838	+	35.8368i	0.286514	+	1.44040i	−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.6777	+	17.6777i	0.707107	+	0.707107i
$626$	19.3190	−	12.9085i	0.772142	−	0.515928i
$627$	−5.86332	+	2.42867i	−0.234158	+	0.0969916i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$631$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$632$		0			0
$633$	−28.8528	+	28.8528i	−1.14680	+	1.14680i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−45.9512	−	9.14027i	−1.81496	−	0.361019i	−0.833478	−	0.552552i	$-0.813654\pi$
$641$	−45.9512	−	9.14027i	−1.81496	−	0.361019i	−0.981486	+	0.191534i	$0.938654\pi$
$642$	29.0259	+	29.0259i	1.14556	+	1.14556i
$643$	28.9632	−	19.3526i	1.14220	−	0.763192i	0.167313	−	0.985904i	$-0.446491\pi$
$643$	28.9632	−	19.3526i	1.14220	−	0.763192i	0.974884	+	0.222712i	$0.0714908\pi$
$644$		0			0
$645$		0			0
$646$	10.9552	+	24.3923i	0.431028	+	0.959702i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−5.34391	−	12.9013i	−0.209928	−	0.506812i
$649$	−8.37808	−	12.5387i	−0.328869	−	0.492187i
$650$		0			0
$651$		0			0
$652$	−27.3080	+	40.8693i	−1.06946	+	1.60056i
$653$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$653$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$654$		0			0
$655$		0			0
$656$	−44.2310	+	8.79809i	−1.72693	+	0.343508i
$657$	10.9442	+	7.31265i	0.426972	+	0.285294i
$658$		0			0
$659$	−12.7279	−	12.7279i	−0.495809	−	0.495809i	0.414321	−	0.910131i	$-0.364019\pi$
$659$	−12.7279	−	12.7279i	−0.495809	−	0.495809i	−0.910131	+	0.414321i	$0.864019\pi$
$660$		0			0
$661$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$661$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$662$			47.7472i			1.85575i
$663$		0			0
$664$	50.0977			1.94417
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	35.8965	−	7.14025i	1.38371	−	0.275236i	0.553573	−	0.832801i	$-0.313264\pi$
$673$	35.8965	−	7.14025i	1.38371	−	0.275236i	0.830134	+	0.557564i	$0.188264\pi$
$674$	−35.9320	−	24.0090i	−1.38405	−	0.924793i
$675$	−27.7404	−	5.51791i	−1.06773	−	0.212384i
$676$	−18.3848	−	18.3848i	−0.707107	−	0.707107i
$677$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$677$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$678$	−26.9072	+	11.1453i	−1.03336	+	0.428033i
$679$		0			0
$680$		0			0
$681$	−12.1618			−0.466041
$682$		0			0
$683$	29.0047	+	43.4085i	1.10983	+	1.66098i	0.595247	+	0.803543i	$0.297054\pi$
$683$	29.0047	+	43.4085i	1.10983	+	1.66098i	0.514585	+	0.857439i	$0.327946\pi$
$684$	−6.56663	+	6.56663i	−0.251081	+	0.251081i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	14.8284	+	6.14214i	0.565328	+	0.234167i
$689$		0			0
$690$		0			0
$691$	20.0427	+	13.3921i	0.762458	+	0.509458i	0.874961	−	0.484193i	$-0.160887\pi$
$691$	20.0427	+	13.3921i	0.762458	+	0.509458i	−0.112503	+	0.993651i	$0.535887\pi$
$692$		0			0
$693$		0			0
$694$	−18.1546	+	12.1305i	−0.689141	+	0.460469i
$695$		0			0
$696$		0			0
$697$	−45.8393	−	7.72341i	−1.73629	−	0.292545i
$698$		0			0
$699$	16.4693	+	39.7604i	0.622926	+	1.50388i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$	1.53211	+	7.70246i	0.0577438	+	0.290297i
$705$		0			0
$706$	−17.5269	+	42.3137i	−0.659634	+	1.59250i
$707$		0			0
$708$	36.0135	+	24.0635i	1.35347	+	0.904360i
$709$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$709$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$710$		0			0
$711$		0			0
$712$	−4.34315	+	1.79899i	−0.162766	+	0.0674200i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−20.4567	−	49.3868i	−0.764501	−	1.84567i
$717$		0			0
$718$		0			0
$719$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$719$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$720$		0			0
$721$		0			0
$722$	−2.65159	−	1.09832i	−0.0986819	−	0.0408754i
$723$	16.7495	−	40.4369i	0.622921	−	1.50386i
$724$		0			0
$725$		0			0
$726$	19.6251	+	3.90368i	0.728357	+	0.144879i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	26.7217	−	11.0685i	0.989691	−	0.409944i
$730$		0			0
$731$	12.0376	+	11.3492i	0.445228	+	0.419765i
$732$		0			0
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.28324	+	7.90692i	−0.194610	+	0.291255i
$738$	−3.14961	−	15.8342i	−0.115939	−	0.582864i
$739$	39.1969	+	16.2359i	1.44188	+	0.597247i	0.960253	−	0.279129i	$-0.0900459\pi$
$739$	39.1969	+	16.2359i	1.44188	+	0.597247i	0.481627	+	0.876376i	$0.340046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$743$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			17.9344i			0.656185i
$748$	−1.34497	+	7.98255i	−0.0491769	+	0.291871i
$749$		0			0
$750$		0			0
$751$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$751$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$752$		0			0
$753$	1.75004	−	8.79807i	0.0637752	−	0.320619i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$	37.2860	−	7.41664i	1.35429	−	0.269384i
$759$		0			0
$760$		0			0
$761$	−32.6569	−	32.6569i	−1.18381	−	1.18381i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−32.6569	−	32.6569i	−1.18381	−	1.18381i	−0.205061	−	0.978749i	$-0.565739\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−12.5316	−	18.7549i	−0.452196	−	0.676760i
$769$	36.4558	−	36.4558i	1.31463	−	1.31463i	0.396670	−	0.917961i	$-0.370166\pi$
$769$	36.4558	−	36.4558i	1.31463	−	1.31463i	0.917961	−	0.396670i	$-0.129834\pi$
$770$		0			0
$771$	18.3172	−	27.4136i	0.659678	−	0.987277i
$772$	−9.71084	−	48.8197i	−0.349501	−	1.75706i
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$	−2.19881	+	5.30840i	−0.0790347	+	0.190807i
$775$		0			0
$776$	−43.7317	−	29.2206i	−1.56988	−	1.04896i
$777$		0			0
$778$		0			0
$779$	42.9885	−	28.7240i	1.54022	−	1.02915i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.7151	+	25.8686i	0.382683	+	0.923880i
$785$		0			0
$786$	−10.0228	+	10.0228i	−0.357501	+	0.357501i
$787$	−10.5359	+	52.9678i	−0.375566	+	1.88810i	0.0781357	+	0.996943i	$0.475103\pi$
$787$	−10.5359	+	52.9678i	−0.375566	+	1.88810i	−0.453701	+	0.891154i	$0.649897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−2.75739	+	0.548479i	−0.0979796	+	0.0194894i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$	−28.2843			−1.00000
$801$	−0.644017	−	1.55479i	−0.0227552	−	0.0549360i
$802$	−10.6932	−	16.0035i	−0.377591	−	0.565105i
$803$	−9.02345	+	9.02345i	−0.318431	+	0.318431i
$804$	5.32856	−	26.7885i	0.187924	−	0.944757i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.05228	−	1.00496i	0.177629	−	0.0353326i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	5.05228	−	1.00496i	0.177629	−	0.0353326i	0.283103	+	0.959089i	$0.408636\pi$
$810$		0			0
$811$	−44.6717	−	8.88574i	−1.56863	−	0.312021i	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−44.6717	−	8.88574i	−1.56863	−	0.312021i	−0.901454	+	0.432876i	$0.857499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−5.20498	−	22.6605i	−0.182211	−	0.793275i
$817$	−18.4007			−0.643758
$818$	3.97056	+	9.58579i	0.138827	+	0.335159i
$819$		0			0
$820$		0			0
$821$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$821$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$822$	−22.4218	+	33.5566i	−0.782049	+	1.17042i
$823$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$823$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$824$		0			0
$825$	2.64804	−	6.39293i	0.0921928	−	0.222573i
$826$		0			0
$827$	47.2483	+	31.5703i	1.64298	+	1.09781i	0.907304	+	0.420476i	$0.138137\pi$
$827$	47.2483	+	31.5703i	1.64298	+	1.09781i	0.735679	+	0.677330i	$0.236863\pi$
$828$		0			0
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.849242	+	28.8492i	0.0294245	+	0.999567i
$834$	25.1492			0.870846
$835$		0			0
$836$	−5.00205	−	7.48610i	−0.173000	−	0.258912i
$837$		0			0
$838$	8.98091	−	45.1501i	0.310241	−	1.55968i
$839$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$839$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$840$		0			0
$841$	26.7925	+	11.0978i	0.923880	+	0.382683i
$842$		0			0
$843$	10.0550	−	2.00006i	0.346312	−	0.0688857i
$844$	−48.1317	−	32.1606i	−1.65676	−	1.10701i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	45.7257i		−	1.56930i
$850$	−27.2521	−	10.3596i	−0.934740	−	0.355333i
$851$		0			0
$852$		0			0
$853$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$853$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$854$		0			0
$855$		0			0
$856$	−32.3535	+	48.4204i	−1.10582	+	1.65498i
$857$	−9.52330	−	47.8768i	−0.325310	−	1.63544i	−0.704201	−	0.710001i	$-0.748695\pi$
$857$	−9.52330	−	47.8768i	−0.325310	−	1.63544i	0.378892	−	0.925441i	$-0.376305\pi$
$858$		0			0
$859$	−11.4939	+	27.7487i	−0.392167	+	0.946775i	0.597300	+	0.802018i	$0.296240\pi$
$859$	−11.4939	+	27.7487i	−0.392167	+	0.946775i	−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$	26.6066	−	17.7780i	0.905176	−	0.604819i
$865$		0			0
$866$			21.0294i			0.714609i
$867$	3.28478	−	23.7399i	0.111557	−	0.806251i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	10.4606	−	15.6554i	0.354039	−	0.529856i
$874$		0			0
$875$		0			0
$876$	14.0262	−	33.8623i	0.473902	−	1.14410i
$877$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$877$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.7933	−	20.5754i	1.03745	−	0.693203i	0.0845306	−	0.996421i	$-0.473061\pi$
$881$	30.7933	−	20.5754i	1.03745	−	0.693203i	0.952921	+	0.303218i	$0.0980609\pi$
$882$	−9.26066	+	3.83589i	−0.311823	+	0.129161i
$883$		−	54.1103i		−	1.82096i	−0.413558	−	0.910478i	$-0.635714\pi$
$883$		−	54.1103i		−	1.82096i	0.413558	−	0.910478i	$-0.364286\pi$
$884$		0			0
$885$		0			0
$886$	10.1127	+	24.4142i	0.339743	+	0.820212i
$887$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$887$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.945530	+	4.75350i	0.0316765	+	0.159248i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−3.11094	+	2.07866i	−0.103813	+	0.0693658i
$899$		0			0
$900$		−	10.1254i		−	0.337514i
$901$		0			0
$902$	15.6521			0.521159
$903$		0			0
$904$	−22.9548	−	34.3543i	−0.763465	−	1.14261i
$905$		0			0
$906$		0			0
$907$	−30.5244	+	45.6830i	−1.01355	+	1.51688i	−0.166022	+	0.986122i	$0.553092\pi$
$907$	−30.5244	+	45.6830i	−1.01355	+	1.51688i	−0.847524	+	0.530757i	$0.821908\pi$
$908$	−3.36601	−	16.9221i	−0.111705	−	0.561578i
$909$		0			0
$910$		0			0
$911$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$911$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$912$	21.5015	+	14.3669i	0.711986	+	0.475734i
$913$	−17.0535	−	3.39215i	−0.564387	−	0.112264i
$914$	42.3848	+	42.3848i	1.40196	+	1.40196i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	32.1472	−	7.38404i	1.06102	−	0.243710i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−6.64590	−	9.94630i	−0.218990	−	0.327742i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	39.4486	+	26.3587i	1.29427	+	0.864801i	0.995971	−	0.0896726i	$-0.0285821\pi$
$929$	39.4486	+	26.3587i	1.29427	+	0.864801i	0.298295	+	0.954474i	$0.403582\pi$
$930$		0			0
$931$	−22.6985	−	22.6985i	−0.743913	−	0.743913i
$932$	−50.7648	+	33.9200i	−1.66286	+	1.11109i
$933$		0			0
$934$			1.11270i			0.0364086i
$935$		0			0
$936$		0			0
$937$	19.4831	+	47.0363i	0.636484	+	1.53661i	0.831333	+	0.555775i	$0.187578\pi$
$937$	19.4831	+	47.0363i	0.636484	+	1.53661i	−0.194849	+	0.980833i	$0.562422\pi$
$938$		0			0
$939$	16.3778	−	16.3778i	0.534470	−	0.534470i
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$		0			0
$944$	−23.5147	+	56.7696i	−0.765339	+	1.84769i
$945$		0			0
$946$	−4.63177	−	3.09485i	−0.150592	−	0.100622i
$947$	39.1409	+	7.78561i	1.27191	+	0.252998i	0.784474	−	0.620161i	$-0.212933\pi$
$947$	39.1409	+	7.78561i	1.27191	+	0.252998i	0.487435	+	0.873160i	$0.337933\pi$
$948$		0			0
$949$		0			0
$950$	29.9581	−	12.4090i	0.971969	−	0.402603i
$951$		0			0
$952$		0			0
$953$	−25.7373			−0.833713			−0.416857	−	0.908972i	$-0.636868\pi$
$953$	−25.7373			−0.833713			−0.416857	+	0.908972i	$0.636868\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.8632	−	28.6403i	0.382683	−	0.923880i
$962$		0			0
$963$	−17.3339	−	11.5822i	−0.558579	−	0.373230i
$964$	60.9000	+	12.1138i	1.96146	+	0.390158i
$965$		0			0
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$			28.3870i			0.912393i
$969$	15.4547	+	21.7179i	0.496478	+	0.697680i
$970$		0			0
$971$	18.9081	+	45.6482i	0.606790	+	1.46492i	0.866471	+	0.499227i	$0.166383\pi$
$971$	18.9081	+	45.6482i	0.606790	+	1.46492i	−0.259681	+	0.965694i	$0.583617\pi$
$972$	11.1226	+	16.6462i	0.356758	+	0.533926i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−55.2340	−	22.8787i	−1.76709	−	0.731954i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	−55.2340	−	22.8787i	−1.76709	−	0.731954i	−0.771709	−	0.635975i	$-0.780598\pi$
$978$	−18.7510	+	45.2689i	−0.599590	+	1.44754i
$979$	1.60023	−	0.318306i	0.0511437	−	0.0101731i
$980$		0			0
$981$		0			0
$982$	−39.6985	−	39.6985i	−1.26683	−	1.26683i
$983$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$983$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$984$	−41.5338	+	17.2039i	−1.32405	+	0.548439i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$991$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$992$		0			0
$993$	9.28577	+	46.6827i	0.294675	+	1.48143i
$994$		0			0
$995$		0			0
$996$	48.9808	−	9.74289i	1.55202	−	0.308715i
$997$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$997$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$998$	48.8396	+	9.71481i	1.54599	+	0.307517i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.2.s.b.107.1	yes	8
4.3	odd	2		544.2.cc.a.175.1		8
8.3	odd	2	CM	136.2.s.b.107.1	yes	8
8.5	even	2		544.2.cc.a.175.1		8
17.7	odd	16	inner	136.2.s.b.75.1	✓	8
68.7	even	16		544.2.cc.a.143.1		8
136.75	even	16	inner	136.2.s.b.75.1	✓	8
136.109	odd	16		544.2.cc.a.143.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.s.b.75.1	✓	8	17.7	odd	16	inner
136.2.s.b.75.1	✓	8	136.75	even	16	inner
136.2.s.b.107.1	yes	8	1.1	even	1	trivial
136.2.s.b.107.1	yes	8	8.3	odd	2	CM
544.2.cc.a.143.1		8	68.7	even	16
544.2.cc.a.143.1		8	136.109	odd	16
544.2.cc.a.175.1		8	4.3	odd	2
544.2.cc.a.175.1		8	8.5	even	2

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.541196 + 1.30656i) q^{2} +(0.783227 + 1.17218i) q^{3} +(-1.41421 + 1.41421i) q^{4} +(-1.10765 + 1.65772i) q^{6} +(-2.61313 - 1.08239i) q^{8} +(0.387484 - 0.935468i) q^{9} +O(q^{10})\)	\(q+(0.541196 + 1.30656i) q^{2} +(0.783227 + 1.17218i) q^{3} +(-1.41421 + 1.41421i) q^{4} +(-1.10765 + 1.65772i) q^{6} +(-2.61313 - 1.08239i) q^{8} +(0.387484 - 0.935468i) q^{9} +(0.816229 + 0.545387i) q^{11} +(-2.76537 - 0.550066i) q^{12} -4.00000i q^{16} +(1.46508 - 3.85403i) q^{17} +1.43195 q^{18} +(1.75490 + 4.23671i) q^{19} +(-0.270842 + 1.36162i) q^{22} +(-0.777910 - 3.91082i) q^{24} +(-4.61940 - 1.91342i) q^{25} +(5.54808 - 1.10358i) q^{27} +(5.22625 - 2.16478i) q^{32} +1.38393i q^{33} +(5.82843 - 0.171573i) q^{34} +(0.774967 + 1.87094i) q^{36} +(-4.58579 + 4.58579i) q^{38} +(-2.19952 - 11.0577i) q^{41} +(-1.53553 + 3.70711i) q^{43} +(-1.92562 + 0.383029i) q^{44} +(4.68873 - 3.13291i) q^{48} +(-6.46716 + 2.67878i) q^{49} -7.07107i q^{50} +(5.66511 - 1.30125i) q^{51} +(4.44450 + 6.65166i) q^{54} +(-3.59171 + 5.37538i) q^{57} +(-14.1924 - 5.87868i) q^{59} +(5.65685 + 5.65685i) q^{64} +(-1.80819 + 0.748978i) q^{66} +9.68714i q^{67} +(3.37849 + 7.52235i) q^{68} +(-2.02509 + 2.02509i) q^{72} +(-2.53605 + 12.7496i) q^{73} +(-1.37516 - 6.91342i) q^{75} +(-8.47343 - 3.50981i) q^{76} +(3.49107 + 3.49107i) q^{81} +(13.2573 - 8.85822i) q^{82} +(-16.3640 + 6.77817i) q^{83} -5.67459 q^{86} +(-1.54259 - 2.30864i) q^{88} +(1.17525 - 1.17525i) q^{89} +(6.63087 + 4.43060i) q^{96} +(18.2381 + 3.62778i) q^{97} +(-7.00000 - 7.00000i) q^{98} +(0.826467 - 0.552228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 8 * q^6 - 16 * q^9	\( 8 q + 8 q^{3} - 8 q^{6} - 16 q^{9} - 16 q^{12} + 24 q^{22} + 32 q^{24} - 16 q^{27} + 24 q^{34} - 32 q^{36} - 48 q^{38} + 32 q^{41} + 16 q^{43} + 16 q^{44} - 32 q^{48} + 40 q^{51} - 8 q^{54} - 40 q^{57} - 40 q^{59} + 56 q^{66} + 48 q^{72} - 48 q^{73} + 56 q^{81} - 80 q^{83} + 32 q^{96} - 56 q^{98} + 32 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 8 * q^6 - 16 * q^9 - 16 * q^12 + 24 * q^22 + 32 * q^24 - 16 * q^27 + 24 * q^34 - 32 * q^36 - 48 * q^38 + 32 * q^41 + 16 * q^43 + 16 * q^44 - 32 * q^48 + 40 * q^51 - 8 * q^54 - 40 * q^57 - 40 * q^59 + 56 * q^66 + 48 * q^72 - 48 * q^73 + 56 * q^81 - 80 * q^83 + 32 * q^96 - 56 * q^98 + 32 * q^99

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