LMFDB - Embedded newform 968.2.k.a.475.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(403,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 7]))

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

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

chi := DirichletCharacter("968.403");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.k (of order $10$, degree $4$, 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:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			475.1
Root			$0.831254 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	968.475
Dual form			968.2.k.a.699.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.34500 - 0.437016i) q^{2} +(-1.61803 + 1.17557i) q^{3} +(1.61803 + 1.17557i) q^{4} +(2.68999 - 0.874032i) q^{6} +(-1.66251 - 2.28825i) q^{8} +(0.309017 - 0.951057i) q^{9} -4.00000 q^{12} +(1.23607 + 3.80423i) q^{16} +(5.37999 - 1.74806i) q^{17} +(-0.831254 + 1.14412i) q^{18} +(-4.98752 - 6.86474i) q^{19} +(5.37999 + 1.74806i) q^{24} +(-4.04508 + 2.93893i) q^{25} +(-1.23607 - 3.80423i) q^{27} -5.65685i q^{32} -8.00000 q^{34} +(1.61803 - 1.17557i) q^{36} +(3.70820 + 11.4127i) q^{38} +(6.65003 + 9.15298i) q^{41} -8.48528i q^{43} +(-6.47214 - 4.70228i) q^{48} +(-2.16312 - 6.65740i) q^{49} +(6.72499 - 2.18508i) q^{50} +(-6.65003 + 9.15298i) q^{51} +5.65685i q^{54} +(16.1400 + 5.24419i) q^{57} +(4.85410 + 3.52671i) q^{59} +(-2.47214 + 7.60845i) q^{64} +14.0000 q^{67} +(10.7600 + 3.49613i) q^{68} +(-2.68999 + 0.874032i) q^{72} +(9.97505 - 13.7295i) q^{73} +(3.09017 - 9.51057i) q^{75} -16.9706i q^{76} +(8.89919 + 6.46564i) q^{81} +(-4.94427 - 15.2169i) q^{82} +(2.68999 - 0.874032i) q^{83} +(-3.70820 + 11.4127i) q^{86} +18.0000 q^{89} +(6.65003 + 9.15298i) q^{96} +(3.09017 - 9.51057i) q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 32 q^{12} - 8 q^{16} - 10 q^{25} + 8 q^{27} - 64 q^{34} + 4 q^{36} - 24 q^{38} - 16 q^{48} + 14 q^{49} + 12 q^{59} + 16 q^{64} + 112 q^{67} - 20 q^{75} + 22 q^{81} + 32 q^{82}+ \cdots - 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 - 32 * q^12 - 8 * q^16 - 10 * q^25 + 8 * q^27 - 64 * q^34 + 4 * q^36 - 24 * q^38 - 16 * q^48 + 14 * q^49 + 12 * q^59 + 16 * q^64 + 112 * q^67 - 20 * q^75 + 22 * q^81 + 32 * q^82 + 24 * q^86 + 144 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{10}\right)$

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 $

$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$3$	−1.61803	+	1.17557i	−0.934172	+	0.678716i	−0.947011	−	0.321202i	$-0.895913\pi$
$3$	−1.61803	+	1.17557i	−0.934172	+	0.678716i	0.0128385	+	0.999918i	$0.495913\pi$
$4$	1.61803	+	1.17557i	0.809017	+	0.587785i
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$	2.68999	−	0.874032i	1.09819	−	0.356822i
$7$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$7$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$8$	−1.66251	−	2.28825i	−0.587785	−	0.809017i
$9$	0.309017	−	0.951057i	0.103006	−	0.317019i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−4.00000			−1.15470
$13$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$13$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$14$		0			0
$15$		0			0
$16$	1.23607	+	3.80423i	0.309017	+	0.951057i
$17$	5.37999	−	1.74806i	1.30484	−	0.423968i	0.427576	−	0.903979i	$-0.359367\pi$
$17$	5.37999	−	1.74806i	1.30484	−	0.423968i	0.877262	+	0.480011i	$0.159367\pi$
$18$	−0.831254	+	1.14412i	−0.195928	+	0.269672i
$19$	−4.98752	−	6.86474i	−1.14422	−	1.57488i	−0.757709	−	0.652592i	$-0.773682\pi$
$19$	−4.98752	−	6.86474i	−1.14422	−	1.57488i	−0.386507	−	0.922287i	$-0.626318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	5.37999	+	1.74806i	1.09819	+	0.356822i
$25$	−4.04508	+	2.93893i	−0.809017	+	0.587785i
$26$		0			0
$27$	−1.23607	−	3.80423i	−0.237881	−	0.732124i
$28$		0			0
$29$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$29$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	−8.00000			−1.37199
$35$		0			0
$36$	1.61803	−	1.17557i	0.269672	−	0.195928i
$37$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$37$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$38$	3.70820	+	11.4127i	0.601550	+	1.85138i
$39$		0			0
$40$		0			0
$41$	6.65003	+	9.15298i	1.03856	+	1.42946i	0.898322	+	0.439338i	$0.144787\pi$
$41$	6.65003	+	9.15298i	1.03856	+	1.42946i	0.140238	+	0.990118i	$0.455213\pi$
$42$		0			0
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$	−6.47214	−	4.70228i	−0.934172	−	0.678716i
$49$	−2.16312	−	6.65740i	−0.309017	−	0.951057i
$50$	6.72499	−	2.18508i	0.951057	−	0.309017i
$51$	−6.65003	+	9.15298i	−0.931191	+	1.28167i
$52$		0			0
$53$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$53$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$54$			5.65685i			0.769800i
$55$		0			0
$56$		0			0
$57$	16.1400	+	5.24419i	2.13779	+	0.694610i
$58$		0			0
$59$	4.85410	+	3.52671i	0.631950	+	0.459139i	0.857075	−	0.515191i	$-0.172279\pi$
$59$	4.85410	+	3.52671i	0.631950	+	0.459139i	−0.225125	+	0.974330i	$0.572279\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	−2.47214	+	7.60845i	−0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	14.0000			1.71037			0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000			1.71037			0.855186	+	0.518321i	$0.173443\pi$
$68$	10.7600	+	3.49613i	1.30484	+	0.423968i
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$	−2.68999	+	0.874032i	−0.317019	+	0.103006i
$73$	9.97505	−	13.7295i	1.16749	−	1.60691i	0.489057	−	0.872252i	$-0.337341\pi$
$73$	9.97505	−	13.7295i	1.16749	−	1.60691i	0.678434	−	0.734662i	$-0.262659\pi$
$74$		0			0
$75$	3.09017	−	9.51057i	0.356822	−	1.09819i
$76$		−	16.9706i		−	1.94666i
$77$		0			0
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$		0			0
$81$	8.89919	+	6.46564i	0.988799	+	0.718404i
$82$	−4.94427	−	15.2169i	−0.546003	−	1.68043i
$83$	2.68999	−	0.874032i	0.295265	−	0.0959375i	−0.157639	−	0.987497i	$-0.550388\pi$
$83$	2.68999	−	0.874032i	0.295265	−	0.0959375i	0.452904	+	0.891559i	$0.350388\pi$
$84$		0			0
$85$		0			0
$86$	−3.70820	+	11.4127i	−0.399866	+	1.23066i
$87$		0			0
$88$		0			0
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			0.953998	+	0.299813i	$0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	6.65003	+	9.15298i	0.678716	+	0.934172i
$97$	3.09017	−	9.51057i	0.313759	−	0.965652i	−0.662503	−	0.749059i	$-0.730506\pi$
$97$	3.09017	−	9.51057i	0.313759	−	0.965652i	0.976262	−	0.216592i	$-0.0694942\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$	12.9443	−	9.40456i	1.28167	−	0.931191i
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.6376	−	16.0177i	−1.12504	−	1.54849i	−0.797154	−	0.603776i	$-0.793662\pi$
$107$	−11.6376	−	16.0177i	−1.12504	−	1.54849i	−0.327891	−	0.944716i	$-0.606338\pi$
$108$	2.47214	−	7.60845i	0.237881	−	0.732124i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.5623	−	10.5801i	1.36991	−	0.995295i	0.372162	−	0.928168i	$-0.378617\pi$
$113$	14.5623	−	10.5801i	1.36991	−	0.995295i	0.997744	−	0.0671276i	$-0.0213835\pi$
$114$	−19.4164	−	14.1068i	−1.81851	−	1.32123i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−4.98752	−	6.86474i	−0.459139	−	0.631950i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−21.5200	−	6.99226i	−1.94039	−	0.630471i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$127$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$128$	6.65003	−	9.15298i	0.587785	−	0.809017i
$129$	9.97505	+	13.7295i	0.878254	+	1.20881i
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$	−18.8300	−	6.11822i	−1.62666	−	0.528534i
$135$		0			0
$136$	−12.9443	−	9.40456i	−1.10996	−	0.806435i
$137$	1.85410	+	5.70634i	0.158407	+	0.487525i	0.998490	−	0.0549317i	$-0.0174941\pi$
$137$	1.85410	+	5.70634i	0.158407	+	0.487525i	−0.840083	+	0.542457i	$0.817494\pi$
$138$		0			0
$139$	−4.98752	+	6.86474i	−0.423036	+	0.582259i	−0.966337	−	0.257279i	$-0.917174\pi$
$139$	−4.98752	+	6.86474i	−0.423036	+	0.582259i	0.543301	+	0.839538i	$0.317174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.00000			0.333333
$145$		0			0
$146$	−19.4164	+	14.1068i	−1.60691	+	1.16749i
$147$	11.3262	+	8.22899i	0.934172	+	0.678716i
$148$		0			0
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$	−8.31254	+	11.4412i	−0.678716	+	0.934172i
$151$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$151$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$152$	−7.41641	+	22.8254i	−0.601550	+	1.85138i
$153$		−	5.65685i		−	0.457330i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$157$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−9.14379	−	12.5854i	−0.718404	−	0.988799i
$163$	−0.618034	+	1.90211i	−0.0484082	+	0.148985i	−0.972339	−	0.233575i	$-0.924958\pi$
$163$	−0.618034	+	1.90211i	−0.0484082	+	0.148985i	0.923931	+	0.382560i	$0.124958\pi$
$164$			22.6274i			1.76690i
$165$		0			0
$166$	−4.00000			−0.310460
$167$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$167$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$168$		0			0
$169$	10.5172	+	7.64121i	0.809017	+	0.587785i
$170$		0			0
$171$	−8.06998	+	2.62210i	−0.617127	+	0.200517i
$172$	9.97505	−	13.7295i	0.760590	−	1.04686i
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000			−0.901975
$178$	−24.2099	−	7.86629i	−1.81461	−	0.589603i
$179$	14.5623	−	10.5801i	1.08844	−	0.790796i	0.109303	−	0.994008i	$-0.465138\pi$
$179$	14.5623	−	10.5801i	1.08844	−	0.790796i	0.979135	+	0.203212i	$0.0651381\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$191$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$192$	−4.94427	−	15.2169i	−0.356822	−	1.09819i
$193$	−16.1400	+	5.24419i	−1.16178	+	0.377485i	−0.825569	−	0.564301i	$-0.809146\pi$
$193$	−16.1400	+	5.24419i	−1.16178	+	0.377485i	−0.336211	+	0.941787i	$0.609146\pi$
$194$	−8.31254	+	11.4412i	−0.596806	+	0.821432i
$195$		0			0
$196$	4.32624	−	13.3148i	0.309017	−	0.951057i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	13.4500	+	4.37016i	0.951057	+	0.309017i
$201$	−22.6525	+	16.4580i	−1.59778	+	1.16086i
$202$		0			0
$203$		0			0
$204$	−21.5200	+	6.99226i	−1.50670	+	0.489556i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−24.2099	−	7.86629i	−1.66668	−	0.541538i	−0.684425	−	0.729083i	$-0.739947\pi$
$211$	−24.2099	−	7.86629i	−1.66668	−	0.541538i	−0.982256	+	0.187545i	$0.939947\pi$
$212$		0			0
$213$		0			0
$214$	8.65248	+	26.6296i	0.591471	+	1.82036i
$215$		0			0
$216$	−6.65003	+	9.15298i	−0.452477	+	0.622782i
$217$		0			0
$218$		0			0
$219$			33.9411i			2.29353i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$223$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	1.54508	+	4.75528i	0.103006	+	0.317019i
$226$	−24.2099	+	7.86629i	−1.61042	+	0.523258i
$227$	1.66251	−	2.28825i	0.110345	−	0.151876i	−0.750273	−	0.661128i	$-0.770078\pi$
$227$	1.66251	−	2.28825i	0.110345	−	0.151876i	0.860617	+	0.509252i	$0.170078\pi$
$228$	19.9501	+	27.4589i	1.32123	+	1.81851i
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.37999	+	1.74806i	0.352455	+	0.114519i	0.479893	−	0.877327i	$-0.340675\pi$
$233$	5.37999	+	1.74806i	0.352455	+	0.114519i	−0.127438	+	0.991847i	$0.540675\pi$
$234$		0			0
$235$		0			0
$236$	3.70820	+	11.4127i	0.241384	+	0.742902i
$237$		0			0
$238$		0			0
$239$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$239$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$240$		0			0
$241$		−	16.9706i		−	1.09317i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$		−	16.9706i		−	1.09317i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$	−10.0000			−0.641500
$244$		0			0
$245$		0			0
$246$	25.8885	+	18.8091i	1.65059	+	1.19923i
$247$		0			0
$248$		0			0
$249$	−3.32502	+	4.57649i	−0.210714	+	0.290023i
$250$		0			0
$251$	1.85410	−	5.70634i	0.117030	−	0.360181i	−0.875335	−	0.483517i	$-0.839359\pi$
$251$	1.85410	−	5.70634i	0.117030	−	0.360181i	0.992365	+	0.123336i	$0.0393592\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	−24.2705	−	17.6336i	−1.51395	−	1.09995i	−0.964385	−	0.264502i	$-0.914792\pi$
$257$	−24.2705	−	17.6336i	−1.51395	−	1.09995i	−0.549568	−	0.835449i	$-0.685208\pi$
$258$	−7.41641	−	22.8254i	−0.461725	−	1.42104i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	6.18034	−	19.0211i	0.381823	−	1.17513i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−29.1246	+	21.1603i	−1.78240	+	1.29499i
$268$	22.6525	+	16.4580i	1.38372	+	1.00533i
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$	13.3001	+	18.3060i	0.806435	+	1.10996i
$273$		0			0
$274$		−	8.48528i		−	0.512615i
$275$		0			0
$276$		0			0
$277$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$277$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$278$	9.70820	−	7.05342i	0.582259	−	0.423036i
$279$		0			0
$280$		0			0
$281$	−26.8999	+	8.74032i	−1.60472	+	0.521404i	−0.968268	−	0.249916i	$-0.919597\pi$
$281$	−26.8999	+	8.74032i	−1.60472	+	0.521404i	−0.636448	+	0.771319i	$0.719597\pi$
$282$		0			0
$283$	14.9626	+	20.5942i	0.889432	+	1.22420i	0.973718	+	0.227757i	$0.0731392\pi$
$283$	14.9626	+	20.5942i	0.889432	+	1.22420i	−0.0842855	+	0.996442i	$0.526861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−5.37999	−	1.74806i	−0.317019	−	0.103006i
$289$	12.1353	−	8.81678i	0.713839	−	0.518634i
$290$		0			0
$291$	6.18034	+	19.0211i	0.362298	+	1.11504i
$292$	32.2799	−	10.4884i	1.88904	−	0.613786i
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$	−11.6376	−	16.0177i	−0.678716	−	0.934172i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	16.1803	−	11.7557i	0.934172	−	0.678716i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	19.9501	−	27.4589i	1.14422	−	1.57488i
$305$		0			0
$306$	−2.47214	+	7.60845i	−0.141323	+	0.434946i
$307$		−	8.48528i		−	0.484281i	−0.970241	−	0.242140i	$-0.922151\pi$
$307$		−	8.48528i		−	0.484281i	0.970241	−	0.242140i	$-0.0778494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−3.09017	−	9.51057i	−0.174667	−	0.537569i	0.824951	−	0.565204i	$-0.191202\pi$
$313$	−3.09017	−	9.51057i	−0.174667	−	0.537569i	−0.999618	+	0.0276348i	$0.991202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	37.6599	+	12.2364i	2.10197	+	0.682972i
$322$		0			0
$323$	−38.8328	−	28.2137i	−2.16072	−	1.56985i
$324$	6.79837	+	20.9232i	0.377687	+	1.16240i
$325$		0			0
$326$	1.66251	−	2.28825i	0.0920778	−	0.126734i
$327$		0			0
$328$	9.88854	−	30.4338i	0.546003	−	1.68043i
$329$		0			0
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$	5.37999	+	1.74806i	0.295265	+	0.0959375i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	19.9501	−	27.4589i	1.08675	−	1.49578i	0.234886	−	0.972023i	$-0.424528\pi$
$337$	19.9501	−	27.4589i	1.08675	−	1.49578i	0.851865	−	0.523761i	$-0.175472\pi$
$338$	−10.8063	−	14.8736i	−0.587785	−	0.809017i
$339$	−11.1246	+	34.2380i	−0.604206	+	1.85955i
$340$		0			0
$341$		0			0
$342$	12.0000			0.648886
$343$		0			0
$344$	−19.4164	+	14.1068i	−1.04686	+	0.760590i
$345$		0			0
$346$		0			0
$347$	34.9699	−	11.3624i	1.87728	−	0.609966i	0.888875	−	0.458149i	$-0.151488\pi$
$347$	34.9699	−	11.3624i	1.87728	−	0.609966i	0.988409	−	0.151817i	$-0.0485125\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$	16.1400	+	5.24419i	0.857829	+	0.278726i
$355$		0			0
$356$	29.1246	+	21.1603i	1.54360	+	1.12149i
$357$		0			0
$358$	−24.2099	+	7.86629i	−1.27954	+	0.415746i
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−16.3779	+	50.4060i	−0.861995	+	2.65295i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$		0			0
$369$	10.7600	−	3.49613i	0.560142	−	0.182001i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11.7426	−	36.1401i	−0.603179	−	1.85639i	−0.508853	−	0.860853i	$-0.669930\pi$
$379$	−11.7426	−	36.1401i	−0.603179	−	1.85639i	−0.0943260	−	0.995541i	$-0.530070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$			22.6274i			1.15470i
$385$		0			0
$386$	24.0000			1.22157
$387$	−8.06998	−	2.62210i	−0.410220	−	0.133289i
$388$	16.1803	−	11.7557i	0.821432	−	0.596806i
$389$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$389$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$390$		0			0
$391$		0			0
$392$	−11.6376	+	16.0177i	−0.587785	+	0.809017i
$393$	−16.6251	−	22.8825i	−0.838624	−	1.15427i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−16.1803	−	11.7557i	−0.809017	−	0.587785i
$401$	1.85410	+	5.70634i	0.0925894	+	0.284961i	0.986618	−	0.163049i	$-0.0521329\pi$
$401$	1.85410	+	5.70634i	0.0925894	+	0.284961i	−0.894029	+	0.448010i	$0.852133\pi$
$402$	37.6599	−	12.2364i	1.87831	−	0.610299i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	32.0000			1.58424
$409$	−32.2799	−	10.4884i	−1.59614	−	0.518617i	−0.629991	−	0.776603i	$-0.716941\pi$
$409$	−32.2799	−	10.4884i	−1.59614	−	0.518617i	−0.966149	+	0.257985i	$0.916941\pi$
$410$		0			0
$411$	−9.70820	−	7.05342i	−0.478870	−	0.347920i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	16.9706i		−	0.831052i
$418$		0			0
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	29.1246	+	21.1603i	1.41776	+	1.03007i
$423$		0			0
$424$		0			0
$425$	−16.6251	+	22.8825i	−0.806435	+	1.10996i
$426$		0			0
$427$		0			0
$428$		−	39.5980i		−	1.91404i
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$	12.9443	−	9.40456i	0.622782	−	0.452477i
$433$	−30.7426	−	22.3358i	−1.47740	−	1.07339i	−0.978384	−	0.206798i	$-0.933696\pi$
$433$	−30.7426	−	22.3358i	−1.47740	−	1.07339i	−0.499014	−	0.866594i	$-0.666304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	14.8328	−	45.6507i	0.708739	−	2.18128i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	33.9787	−	24.6870i	1.61438	−	1.17291i	0.767695	−	0.640816i	$-0.221404\pi$
$443$	33.9787	−	24.6870i	1.61438	−	1.17291i	0.846683	−	0.532098i	$-0.178596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.9787	−	39.9444i	0.612503	−	1.88509i	0.179303	−	0.983794i	$-0.442616\pi$
$449$	12.9787	−	39.9444i	0.612503	−	1.88509i	0.433200	−	0.901298i	$-0.357384\pi$
$450$		−	7.07107i		−	0.333333i
$451$		0			0
$452$	36.0000			1.69330
$453$		0			0
$454$	−3.23607	+	2.35114i	−0.151876	+	0.110345i
$455$		0			0
$456$	−14.8328	−	45.6507i	−0.694610	−	2.13779i
$457$	32.2799	−	10.4884i	1.50999	−	0.490626i	0.567078	−	0.823664i	$-0.308074\pi$
$457$	32.2799	−	10.4884i	1.50999	−	0.490626i	0.942913	+	0.333038i	$0.108074\pi$
$458$		0			0
$459$	−13.3001	−	18.3060i	−0.620794	−	0.854449i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−6.47214	−	4.70228i	−0.299816	−	0.217829i
$467$	9.27051	+	28.5317i	0.428988	+	1.32029i	0.899123	+	0.437695i	$0.144205\pi$
$467$	9.27051	+	28.5317i	0.428988	+	1.32029i	−0.470135	+	0.882594i	$0.655795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	16.9706i		−	0.781133i
$473$		0			0
$474$		0			0
$475$	40.3499	+	13.1105i	1.85138	+	0.601550i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		0			0
$481$		0			0
$482$	−7.41641	+	22.8254i	−0.337808	+	1.03967i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	13.4500	+	4.37016i	0.610103	+	0.198234i
$487$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$487$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$488$		0			0
$489$	−1.23607	−	3.80423i	−0.0558969	−	0.172033i
$490$		0			0
$491$	−8.31254	+	11.4412i	−0.375140	+	0.516335i	−0.954289	−	0.298886i	$-0.903385\pi$
$491$	−8.31254	+	11.4412i	−0.375140	+	0.516335i	0.579149	+	0.815222i	$0.303385\pi$
$492$	−26.6001	−	36.6119i	−1.19923	−	1.65059i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	6.47214	−	4.70228i	0.290023	−	0.210714i
$499$	11.3262	+	8.22899i	0.507032	+	0.368380i	0.811697	−	0.584079i	$-0.198544\pi$
$499$	11.3262	+	8.22899i	0.507032	+	0.368380i	−0.304664	+	0.952460i	$0.598544\pi$
$500$		0			0
$501$		0			0
$502$	−4.98752	+	6.86474i	−0.222604	+	0.306388i
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−26.0000			−1.15470
$508$		0			0
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	21.5200	−	6.99226i	0.951057	−	0.309017i
$513$	−19.9501	+	27.4589i	−0.880818	+	1.21234i
$514$	24.9376	+	34.3237i	1.09995	+	1.51395i
$515$		0			0
$516$			33.9411i			1.49417i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.85410	+	3.52671i	0.212662	+	0.154508i	0.689017	−	0.724745i	$-0.258042\pi$
$521$	4.85410	+	3.52671i	0.212662	+	0.154508i	−0.476355	+	0.879253i	$0.658042\pi$
$522$		0			0
$523$	−24.2099	+	7.86629i	−1.05863	+	0.343969i	−0.786049	−	0.618165i	$-0.787876\pi$
$523$	−24.2099	+	7.86629i	−1.05863	+	0.343969i	−0.272578	+	0.962134i	$0.587876\pi$
$524$	−16.6251	+	22.8825i	−0.726270	+	0.999625i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	4.85410	−	3.52671i	0.210650	−	0.153046i
$532$		0			0
$533$		0			0
$534$	48.4199	−	15.7326i	2.09533	−	0.680815i
$535$		0			0
$536$	−23.2751	−	32.0354i	−1.00533	−	1.38372i
$537$	−11.1246	+	34.2380i	−0.480062	+	1.47748i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$		0			0
$544$	−9.88854	−	30.4338i	−0.423968	−	1.30484i
$545$		0			0
$546$		0			0
$547$	−4.98752	−	6.86474i	−0.213251	−	0.293515i	0.688969	−	0.724791i	$-0.258064\pi$
$547$	−4.98752	−	6.86474i	−0.213251	−	0.293515i	−0.902220	+	0.431276i	$0.858064\pi$
$548$	−3.70820	+	11.4127i	−0.158407	+	0.487525i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−16.1400	+	5.24419i	−0.684487	+	0.222403i
$557$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$557$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	40.0000			1.68730
$563$	34.9699	+	11.3624i	1.47381	+	0.478869i	0.932256	−	0.361799i	$-0.117837\pi$
$563$	34.9699	+	11.3624i	1.47381	+	0.478869i	0.541551	+	0.840668i	$0.317837\pi$
$564$		0			0
$565$		0			0
$566$	−11.1246	−	34.2380i	−0.467602	−	1.43913i
$567$		0			0
$568$		0			0
$569$	13.3001	+	18.3060i	0.557568	+	0.767426i	0.991015	−	0.133753i	$-0.0427029\pi$
$569$	13.3001	+	18.3060i	0.557568	+	0.767426i	−0.433447	+	0.901179i	$0.642703\pi$
$570$		0			0
$571$		−	42.4264i		−	1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$		−	42.4264i		−	1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	6.47214	+	4.70228i	0.269672	+	0.195928i
$577$	10.5066	+	32.3359i	0.437395	+	1.34616i	0.890613	+	0.454762i	$0.150276\pi$
$577$	10.5066	+	32.3359i	0.437395	+	1.34616i	−0.453218	+	0.891400i	$0.649724\pi$
$578$	−20.1750	+	6.55524i	−0.839168	+	0.272662i
$579$	19.9501	−	27.4589i	0.829098	−	1.14115i
$580$		0			0
$581$		0			0
$582$		−	28.2843i		−	1.17242i
$583$		0			0
$584$	−48.0000			−1.98625
$585$		0			0
$586$		0			0
$587$	4.85410	+	3.52671i	0.200350	+	0.145563i	0.683437	−	0.730010i	$-0.260484\pi$
$587$	4.85410	+	3.52671i	0.200350	+	0.145563i	−0.483087	+	0.875573i	$0.660484\pi$
$588$	8.65248	+	26.6296i	0.356822	+	1.09819i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$	−26.8999	+	8.74032i	−1.09819	+	0.356822i
$601$	9.97505	−	13.7295i	0.406891	−	0.560037i	−0.555566	−	0.831472i	$-0.687498\pi$
$601$	9.97505	−	13.7295i	0.406891	−	0.560037i	0.962457	+	0.271436i	$0.0874984\pi$
$602$		0			0
$603$	4.32624	−	13.3148i	0.176178	−	0.542220i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$	−38.8328	+	28.2137i	−1.57488	+	1.14422i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	6.65003	−	9.15298i	0.268812	−	0.369987i
$613$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$613$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$614$	−3.70820	+	11.4127i	−0.149651	+	0.460578i
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−21.0344	+	15.2824i	−0.845446	+	0.614252i	−0.923887	−	0.382667i	$-0.875006\pi$
$619$	−21.0344	+	15.2824i	−0.845446	+	0.614252i	0.0784409	+	0.996919i	$0.475006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	−	23.7764i	0.309017	−	0.951057i
$626$			14.1421i			0.565233i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$	48.4199	−	15.7326i	1.92452	−	0.625314i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	33.9787	−	24.6870i	1.34208	−	0.975077i	0.342714	−	0.939440i	$-0.388654\pi$
$641$	33.9787	−	24.6870i	1.34208	−	0.975077i	0.999365	−	0.0356372i	$-0.0113461\pi$
$642$	−45.3050	−	32.9160i	−1.78804	−	1.29909i
$643$	15.4508	+	47.5528i	0.609322	+	1.87530i	0.463786	+	0.885948i	$0.346491\pi$
$643$	15.4508	+	47.5528i	0.609322	+	1.87530i	0.145537	+	0.989353i	$0.453509\pi$
$644$		0			0
$645$		0			0
$646$	39.9002	+	54.9179i	1.56985	+	2.16072i
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$		−	31.1127i		−	1.22222i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−3.23607	+	2.35114i	−0.126734	+	0.0920778i
$653$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$653$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$654$		0			0
$655$		0			0
$656$	−26.6001	+	36.6119i	−1.03856	+	1.42946i
$657$	−9.97505	−	13.7295i	−0.389164	−	0.535638i
$658$		0			0
$659$		−	48.0833i		−	1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$		−	48.0833i		−	1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	34.9699	+	11.3624i	1.35914	+	0.441613i
$663$		0			0
$664$	−6.47214	−	4.70228i	−0.251168	−	0.182484i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−48.4199	−	15.7326i	−1.86645	−	0.606446i	−0.992784	−	0.119920i	$-0.961736\pi$
$673$	−48.4199	−	15.7326i	−1.86645	−	0.606446i	−0.873666	−	0.486526i	$-0.838264\pi$
$674$	−38.8328	+	28.2137i	−1.49578	+	1.08675i
$675$	16.1803	+	11.7557i	0.622782	+	0.452477i
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$677$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$678$	29.9251	−	41.1884i	1.14927	−	1.58183i
$679$		0			0
$680$		0			0
$681$			5.65685i			0.216771i
$682$		0			0
$683$	−42.0000			−1.60709			−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000			−1.60709			−0.803543	+	0.595247i	$0.797054\pi$
$684$	−16.1400	−	5.24419i	−0.617127	−	0.200517i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	32.2799	−	10.4884i	1.23066	−	0.399866i
$689$		0			0
$690$		0			0
$691$	−14.2148	+	43.7486i	−0.540756	+	1.66428i	0.190117	+	0.981761i	$0.439113\pi$
$691$	−14.2148	+	43.7486i	−0.540756	+	1.66428i	−0.730873	+	0.682514i	$0.760887\pi$
$692$		0			0
$693$		0			0
$694$	−52.0000			−1.97389
$695$		0			0
$696$		0			0
$697$	51.7771	+	37.6183i	1.96120	+	1.42489i
$698$		0			0
$699$	−10.7600	+	3.49613i	−0.406980	+	0.132236i
$700$		0			0
$701$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$701$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	40.3499	+	13.1105i	1.51859	+	0.493419i
$707$		0			0
$708$	−19.4164	−	14.1068i	−0.729713	−	0.530168i
$709$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$709$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$710$		0			0
$711$		0			0
$712$	−29.9251	−	41.1884i	−1.12149	−	1.54360i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	36.0000			1.34538
$717$		0			0
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$	44.0565	−	60.6385i	1.63961	−	2.25673i
$723$	19.9501	+	27.4589i	0.741952	+	1.02121i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−10.5172	+	7.64121i	−0.389527	+	0.283008i
$730$		0			0
$731$	−14.8328	−	45.6507i	−0.548612	−	1.68845i
$732$		0			0
$733$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−16.0000			−0.588968
$739$	40.3499	+	13.1105i	1.48430	+	0.482277i	0.935393	−	0.353610i	$-0.115046\pi$
$739$	40.3499	+	13.1105i	1.48430	+	0.482277i	0.548902	+	0.835886i	$0.315046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$743$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	2.82843i		−	0.103487i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$	3.70820	+	11.4127i	0.135134	+	0.415901i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$757$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$758$			53.7401i			1.95193i
$759$		0			0
$760$		0			0
$761$	10.7600	+	3.49613i	0.390049	+	0.126735i	0.497475	−	0.867479i	$-0.334261\pi$
$761$	10.7600	+	3.49613i	0.390049	+	0.126735i	−0.107426	+	0.994213i	$0.534261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	9.88854	−	30.4338i	0.356822	−	1.09819i
$769$			50.9117i			1.83592i	0.396670	+	0.917961i	$0.370166\pi$
$769$			50.9117i			1.83592i	−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	60.0000			2.16085
$772$	−32.2799	−	10.4884i	−1.16178	−	0.377485i
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$	9.70820	+	7.05342i	0.348954	+	0.253530i
$775$		0			0
$776$	−26.8999	+	8.74032i	−0.965652	+	0.313759i
$777$		0			0
$778$		0			0
$779$	29.6656	−	91.3014i	1.06288	−	3.27121i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.6525	−	16.4580i	0.809017	−	0.587785i
$785$		0			0
$786$	12.3607	+	38.0423i	0.440891	+	1.35692i
$787$	−24.2099	+	7.86629i	−0.862991	+	0.280403i	−0.706877	−	0.707336i	$-0.749897\pi$
$787$	−24.2099	+	7.86629i	−0.862991	+	0.280403i	−0.156114	+	0.987739i	$0.549897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$	16.6251	+	22.8825i	0.587785	+	0.809017i
$801$	5.56231	−	17.1190i	0.196534	−	0.604871i
$802$		−	8.48528i		−	0.299626i
$803$		0			0
$804$	−56.0000			−1.97497
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−53.7999	+	17.4806i	−1.89150	+	0.614587i	−0.913158	+	0.407605i	$0.866364\pi$
$809$	−53.7999	+	17.4806i	−1.89150	+	0.614587i	−0.978345	+	0.206981i	$0.933636\pi$
$810$		0			0
$811$	24.9376	+	34.3237i	0.875678	+	1.20527i	0.977599	+	0.210475i	$0.0675011\pi$
$811$	24.9376	+	34.3237i	0.875678	+	1.20527i	−0.101921	+	0.994792i	$0.532499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−43.0399	−	13.9845i	−1.50670	−	0.489556i
$817$	−58.2492	+	42.3205i	−2.03788	+	1.48061i
$818$	38.8328	+	28.2137i	1.35776	+	0.986469i
$819$		0			0
$820$		0			0
$821$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$821$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$822$	9.97505	+	13.7295i	0.347920	+	0.478870i
$823$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$823$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.8300	−	6.11822i	−0.654782	−	0.212752i	−0.0372604	−	0.999306i	$-0.511863\pi$
$827$	−18.8300	−	6.11822i	−0.654782	−	0.212752i	−0.617521	+	0.786554i	$0.711863\pi$
$828$		0			0
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.2751	−	32.0354i	−0.806435	−	1.10996i
$834$	−7.41641	+	22.8254i	−0.256809	+	0.790377i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−24.2099	−	7.86629i	−0.836319	−	0.271736i
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$	33.2502	−	45.7649i	1.14520	−	1.57623i
$844$	−29.9251	−	41.1884i	−1.03007	−	1.41776i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−48.4199	−	15.7326i	−1.66177	−	0.539941i
$850$	32.3607	−	23.5114i	1.10996	−	0.806435i
$851$		0			0
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$		0			0
$855$		0			0
$856$	−17.3050	+	53.2592i	−0.591471	+	1.82036i
$857$			22.6274i			0.772938i	0.922302	+	0.386469i	$0.126305\pi$
$857$			22.6274i			0.772938i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	58.0000			1.97893			0.989467	−	0.144757i	$-0.0462401\pi$
$859$	58.0000			1.97893			0.989467	+	0.144757i	$0.0462401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$863$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$864$	−21.5200	+	6.99226i	−0.732124	+	0.237881i
$865$		0			0
$866$	31.5876	+	43.4767i	1.07339	+	1.47740i
$867$	−9.27051	+	28.5317i	−0.314843	+	0.968987i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−8.09017	−	5.87785i	−0.273811	−	0.198935i
$874$		0			0
$875$		0			0
$876$	−39.9002	+	54.9179i	−1.34810	+	1.85550i
$877$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$877$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$	9.41498	+	3.05911i	0.317019	+	0.103006i
$883$	−1.61803	+	1.17557i	−0.0544512	+	0.0395611i	−0.614678	−	0.788778i	$-0.710714\pi$
$883$	−1.61803	+	1.17557i	−0.0544512	+	0.0395611i	0.560227	+	0.828339i	$0.310714\pi$
$884$		0			0
$885$		0			0
$886$	−56.4899	+	18.3547i	−1.89781	+	0.616637i
$887$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$887$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−34.9127	+	48.0532i	−1.16505	+	1.60355i
$899$		0			0
$900$	−3.09017	+	9.51057i	−0.103006	+	0.317019i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−48.4199	−	15.7326i	−1.61042	−	0.523258i
$905$		0			0
$906$		0			0
$907$	−3.09017	−	9.51057i	−0.102607	−	0.315793i	0.886554	−	0.462625i	$-0.153092\pi$
$907$	−3.09017	−	9.51057i	−0.102607	−	0.315793i	−0.989161	+	0.146832i	$0.953092\pi$
$908$	5.37999	−	1.74806i	0.178541	−	0.0580115i
$909$		0			0
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$			67.8823i			2.24781i
$913$		0			0
$914$	−48.0000			−1.58770
$915$		0			0
$916$		0			0
$917$		0			0
$918$	9.88854	+	30.4338i	0.326371	+	1.00447i
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$	9.97505	+	13.7295i	0.328689	+	0.452402i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.6869	−	51.3571i	−0.547480	−	1.68497i	−0.715019	−	0.699105i	$-0.753582\pi$
$929$	−16.6869	−	51.3571i	−0.547480	−	1.68497i	0.167539	−	0.985865i	$-0.446418\pi$
$930$		0			0
$931$	−34.9127	+	48.0532i	−1.14422	+	1.57488i
$932$	6.65003	+	9.15298i	0.217829	+	0.299816i
$933$		0			0
$934$		−	42.4264i		−	1.38823i
$935$		0			0
$936$		0			0
$937$	−48.4199	−	15.7326i	−1.58181	−	0.513961i	−0.619287	−	0.785165i	$-0.712578\pi$
$937$	−48.4199	−	15.7326i	−1.58181	−	0.513961i	−0.962522	+	0.271204i	$0.912578\pi$
$938$		0			0
$939$	16.1803	+	11.7557i	0.528025	+	0.383633i
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	−7.41641	+	22.8254i	−0.241384	+	0.742902i
$945$		0			0
$946$		0			0
$947$	−30.0000			−0.974869			−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000			−0.974869			−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$	−48.5410	−	35.2671i	−1.57488	−	1.14422i
$951$		0			0
$952$		0			0
$953$	−26.6001	+	36.6119i	−0.861663	+	1.18598i	0.119508	+	0.992833i	$0.461868\pi$
$953$	−26.6001	+	36.6119i	−0.861663	+	1.18598i	−0.981171	+	0.193143i	$0.938132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$		0			0
$963$	−18.8300	+	6.11822i	−0.606787	+	0.197157i
$964$	19.9501	−	27.4589i	0.642549	−	0.884393i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	96.0000			3.08396
$970$		0			0
$971$	−43.6869	+	31.7404i	−1.40198	+	1.01860i	−0.407552	+	0.913182i	$0.633617\pi$
$971$	−43.6869	+	31.7404i	−1.40198	+	1.01860i	−0.994428	+	0.105416i	$0.966383\pi$
$972$	−16.1803	−	11.7557i	−0.518985	−	0.377064i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.85410	−	5.70634i	0.0593180	−	0.182562i	−0.917007	−	0.398871i	$-0.869402\pi$
$977$	1.85410	−	5.70634i	0.0593180	−	0.182562i	0.976325	+	0.216309i	$0.0694020\pi$
$978$			5.65685i			0.180886i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	16.1803	−	11.7557i	0.516335	−	0.375140i
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$	19.7771	+	60.8676i	0.630471	+	1.94039i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	42.0689	−	30.5648i	1.33502	−	0.969946i
$994$		0			0
$995$		0			0
$996$	−10.7600	+	3.49613i	−0.340943	+	0.110779i
$997$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$997$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$998$	−11.6376	−	16.0177i	−0.368380	−	0.507032i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.k.a.475.1		8
8.3	odd	2	CM	968.2.k.a.475.1		8
11.2	odd	10	inner	968.2.k.a.723.1		8
11.3	even	5	inner	968.2.k.a.403.1		8
11.4	even	5		88.2.g.a.43.1	✓	2
11.5	even	5	inner	968.2.k.a.699.2		8
11.6	odd	10	inner	968.2.k.a.699.1		8
11.7	odd	10		88.2.g.a.43.2	yes	2
11.8	odd	10	inner	968.2.k.a.403.2		8
11.9	even	5	inner	968.2.k.a.723.2		8
11.10	odd	2	inner	968.2.k.a.475.2		8
33.26	odd	10		792.2.h.b.307.2		2
33.29	even	10		792.2.h.b.307.1		2
44.7	even	10		352.2.g.a.175.2		2
44.15	odd	10		352.2.g.a.175.1		2
88.3	odd	10	inner	968.2.k.a.403.1		8
88.19	even	10	inner	968.2.k.a.403.2		8
88.27	odd	10	inner	968.2.k.a.699.2		8
88.29	odd	10		352.2.g.a.175.2		2
88.35	even	10	inner	968.2.k.a.723.1		8
88.37	even	10		352.2.g.a.175.1		2
88.43	even	2	inner	968.2.k.a.475.2		8
88.51	even	10		88.2.g.a.43.2	yes	2
88.59	odd	10		88.2.g.a.43.1	✓	2
88.75	odd	10	inner	968.2.k.a.723.2		8
88.83	even	10	inner	968.2.k.a.699.1		8
132.59	even	10		3168.2.h.b.2287.2		2
132.95	odd	10		3168.2.h.b.2287.1		2
176.29	odd	20		2816.2.e.d.2815.3		4
176.37	even	20		2816.2.e.d.2815.1		4
176.51	even	20		2816.2.e.d.2815.2		4
176.59	odd	20		2816.2.e.d.2815.4		4
176.117	odd	20		2816.2.e.d.2815.2		4
176.125	even	20		2816.2.e.d.2815.4		4
176.139	even	20		2816.2.e.d.2815.3		4
176.147	odd	20		2816.2.e.d.2815.1		4
264.29	even	10		3168.2.h.b.2287.1		2
264.59	even	10		792.2.h.b.307.2		2
264.125	odd	10		3168.2.h.b.2287.2		2
264.227	odd	10		792.2.h.b.307.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.a.43.1	✓	2	11.4	even	5
88.2.g.a.43.1	✓	2	88.59	odd	10
88.2.g.a.43.2	yes	2	11.7	odd	10
88.2.g.a.43.2	yes	2	88.51	even	10
352.2.g.a.175.1		2	44.15	odd	10
352.2.g.a.175.1		2	88.37	even	10
352.2.g.a.175.2		2	44.7	even	10
352.2.g.a.175.2		2	88.29	odd	10
792.2.h.b.307.1		2	33.29	even	10
792.2.h.b.307.1		2	264.227	odd	10
792.2.h.b.307.2		2	33.26	odd	10
792.2.h.b.307.2		2	264.59	even	10
968.2.k.a.403.1		8	11.3	even	5	inner
968.2.k.a.403.1		8	88.3	odd	10	inner
968.2.k.a.403.2		8	11.8	odd	10	inner
968.2.k.a.403.2		8	88.19	even	10	inner
968.2.k.a.475.1		8	1.1	even	1	trivial
968.2.k.a.475.1		8	8.3	odd	2	CM
968.2.k.a.475.2		8	11.10	odd	2	inner
968.2.k.a.475.2		8	88.43	even	2	inner
968.2.k.a.699.1		8	11.6	odd	10	inner
968.2.k.a.699.1		8	88.83	even	10	inner
968.2.k.a.699.2		8	11.5	even	5	inner
968.2.k.a.699.2		8	88.27	odd	10	inner
968.2.k.a.723.1		8	11.2	odd	10	inner
968.2.k.a.723.1		8	88.35	even	10	inner
968.2.k.a.723.2		8	11.9	even	5	inner
968.2.k.a.723.2		8	88.75	odd	10	inner
2816.2.e.d.2815.1		4	176.37	even	20
2816.2.e.d.2815.1		4	176.147	odd	20
2816.2.e.d.2815.2		4	176.51	even	20
2816.2.e.d.2815.2		4	176.117	odd	20
2816.2.e.d.2815.3		4	176.29	odd	20
2816.2.e.d.2815.3		4	176.139	even	20
2816.2.e.d.2815.4		4	176.59	odd	20
2816.2.e.d.2815.4		4	176.125	even	20
3168.2.h.b.2287.1		2	132.95	odd	10
3168.2.h.b.2287.1		2	264.29	even	10
3168.2.h.b.2287.2		2	132.59	even	10
3168.2.h.b.2287.2		2	264.125	odd	10

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.k (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.34500 - 0.437016i) q^{2} +(-1.61803 + 1.17557i) q^{3} +(1.61803 + 1.17557i) q^{4} +(2.68999 - 0.874032i) q^{6} +(-1.66251 - 2.28825i) q^{8} +(0.309017 - 0.951057i) q^{9} -4.00000 q^{12} +(1.23607 + 3.80423i) q^{16} +(5.37999 - 1.74806i) q^{17} +(-0.831254 + 1.14412i) q^{18} +(-4.98752 - 6.86474i) q^{19} +(5.37999 + 1.74806i) q^{24} +(-4.04508 + 2.93893i) q^{25} +(-1.23607 - 3.80423i) q^{27} -5.65685i q^{32} -8.00000 q^{34} +(1.61803 - 1.17557i) q^{36} +(3.70820 + 11.4127i) q^{38} +(6.65003 + 9.15298i) q^{41} -8.48528i q^{43} +(-6.47214 - 4.70228i) q^{48} +(-2.16312 - 6.65740i) q^{49} +(6.72499 - 2.18508i) q^{50} +(-6.65003 + 9.15298i) q^{51} +5.65685i q^{54} +(16.1400 + 5.24419i) q^{57} +(4.85410 + 3.52671i) q^{59} +(-2.47214 + 7.60845i) q^{64} +14.0000 q^{67} +(10.7600 + 3.49613i) q^{68} +(-2.68999 + 0.874032i) q^{72} +(9.97505 - 13.7295i) q^{73} +(3.09017 - 9.51057i) q^{75} -16.9706i q^{76} +(8.89919 + 6.46564i) q^{81} +(-4.94427 - 15.2169i) q^{82} +(2.68999 - 0.874032i) q^{83} +(-3.70820 + 11.4127i) q^{86} +18.0000 q^{89} +(6.65003 + 9.15298i) q^{96} +(3.09017 - 9.51057i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 32 q^{12} - 8 q^{16} - 10 q^{25} + 8 q^{27} - 64 q^{34} + 4 q^{36} - 24 q^{38} - 16 q^{48} + 14 q^{49} + 12 q^{59} + 16 q^{64} + 112 q^{67} - 20 q^{75} + 22 q^{81} + 32 q^{82}+ \cdots - 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 - 32 * q^12 - 8 * q^16 - 10 * q^25 + 8 * q^27 - 64 * q^34 + 4 * q^36 - 24 * q^38 - 16 * q^48 + 14 * q^49 + 12 * q^59 + 16 * q^64 + 112 * q^67 - 20 * q^75 + 22 * q^81 + 32 * q^82 + 24 * q^86 + 144 * q^89 - 20 * q^97

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