LMFDB - Embedded newform 164.2.o.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [164,2,Mod(7,164)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(164, base_ring=CyclotomicField(40))

chi = DirichletCharacter(H, H._module([20, 39]))

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

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

chi := DirichletCharacter("164.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	164.o (of order $40$, degree $16$, 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:	$1.30954659315$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{40}]$

Embedding invariants

Embedding label			11.1
Root			$0.453990 - 0.891007i$ of defining polynomial
Character	$\chi$	$=$	164.11
Dual form			164.2.o.a.15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39680 + 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(-0.935446 + 1.83592i) q^{5} +(-2.52015 + 1.28408i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})$	\(q+(-1.39680 + 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(-0.935446 + 1.83592i) q^{5} +(-2.52015 + 1.28408i) q^{8} +(2.12132 + 2.12132i) q^{9} +(0.900470 - 2.77136i) q^{10} +(1.02652 + 1.67513i) q^{13} +(3.23607 - 2.35114i) q^{16} +(0.617169 + 7.84187i) q^{17} +(-3.43237 - 2.49376i) q^{18} +(-0.644666 + 4.07026i) q^{20} +(0.443396 + 0.610282i) q^{25} +(-1.80444 - 2.11273i) q^{26} +(0.604768 - 7.68430i) q^{29} +(-4.00000 + 4.00000i) q^{32} +(-2.59693 - 10.8170i) q^{34} +(5.34604 + 2.72394i) q^{36} +(-3.58462 - 11.0323i) q^{37} -5.82797i q^{40} +(-1.69394 - 6.17499i) q^{41} +(-5.87895 + 1.91019i) q^{45} +(6.23705 - 3.17793i) q^{49} +(-0.754350 - 0.754350i) q^{50} +(2.98785 + 2.55187i) q^{52} +(0.619329 + 0.0487423i) q^{53} +(0.855270 + 10.8672i) q^{58} +(-0.541604 + 3.41955i) q^{61} +(4.70228 - 6.47214i) q^{64} +(-4.03566 + 0.317614i) q^{65} +(6.02047 + 14.5347i) q^{68} +(-8.06998 - 2.62210i) q^{72} +(-7.53112 + 7.53112i) q^{73} +(7.44770 + 14.6169i) q^{74} +(1.28933 + 8.14052i) q^{80} +9.00000i q^{81} +(3.73221 + 8.25049i) q^{82} +(-14.9743 - 6.20258i) q^{85} +(18.0947 - 4.34416i) q^{89} +(7.78913 - 3.96876i) q^{90} +(14.1224 - 12.0617i) q^{97} +(-8.00886 + 5.81878i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 8 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 8 * q^8	$ 16 q - 4 q^{2} + 8 q^{8} - 8 q^{13} + 16 q^{16} - 4 q^{17} - 20 q^{26} + 20 q^{29} - 64 q^{32} - 48 q^{34} - 20 q^{41} + 56 q^{50} + 96 q^{52} + 28 q^{53} + 12 q^{58} - 44 q^{61} - 112 q^{65} + 32 q^{68} - 36 q^{82} + 56 q^{85} + 32 q^{89} + 48 q^{90} + 36 q^{97}+O(q^{100}) $ 16 * q - 4 * q^2 + 8 * q^8 - 8 * q^13 + 16 * q^16 - 4 * q^17 - 20 * q^26 + 20 * q^29 - 64 * q^32 - 48 * q^34 - 20 * q^41 + 56 * q^50 + 96 * q^52 + 28 * q^53 + 12 * q^58 - 44 * q^61 - 112 * q^65 + 32 * q^68 - 36 * q^82 + 56 * q^85 + 32 * q^89 + 48 * q^90 + 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$-1$	$e\left(\frac{3}{40}\right)$

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 $

$2$	−1.39680	+	0.221232i	−0.987688	+	0.156434i
$2$	−1.39680	+	0.221232i	−0.987688	+	0.156434i
$3$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$3$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$4$	1.90211	−	0.618034i	0.951057	−	0.309017i
$5$	−0.935446	+	1.83592i	−0.418344	+	0.821047i	0.581627	+	0.813456i	$0.302416\pi$
$5$	−0.935446	+	1.83592i	−0.418344	+	0.821047i	−0.999971	+	0.00759122i	$0.997584\pi$
$6$		0			0
$7$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$7$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$8$	−2.52015	+	1.28408i	−0.891007	+	0.453990i
$9$	2.12132	+	2.12132i	0.707107	+	0.707107i
$10$	0.900470	−	2.77136i	0.284754	−	0.876382i
$11$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$11$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$12$		0			0
$13$	1.02652	+	1.67513i	0.284706	+	0.464599i	0.962739	−	0.270434i	$-0.0871670\pi$
$13$	1.02652	+	1.67513i	0.284706	+	0.464599i	−0.678032	+	0.735032i	$0.737167\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$	0.617169	+	7.84187i	0.149685	+	1.90193i	0.374020	+	0.927421i	$0.377979\pi$
$17$	0.617169	+	7.84187i	0.149685	+	1.90193i	−0.224335	+	0.974512i	$0.572021\pi$
$18$	−3.43237	−	2.49376i	−0.809017	−	0.587785i
$19$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$19$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$20$	−0.644666	+	4.07026i	−0.144152	+	0.910138i
$21$		0			0
$22$		0			0
$23$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$23$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$24$		0			0
$25$	0.443396	+	0.610282i	0.0886792	+	0.122056i
$26$	−1.80444	−	2.11273i	−0.353880	−	0.414341i
$27$		0			0
$28$		0			0
$29$	0.604768	−	7.68430i	0.112303	−	1.42694i	−0.640129	−	0.768268i	$-0.721119\pi$
$29$	0.604768	−	7.68430i	0.112303	−	1.42694i	0.752431	−	0.658671i	$-0.228881\pi$
$30$		0			0
$31$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$31$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	−2.59693	−	10.8170i	−0.445370	−	1.85510i
$35$		0			0
$36$	5.34604	+	2.72394i	0.891007	+	0.453990i
$37$	−3.58462	−	11.0323i	−0.589307	−	1.81370i	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	−3.58462	−	11.0323i	−0.589307	−	1.81370i	−0.00806896	−	0.999967i	$-0.502568\pi$
$38$		0			0
$39$		0			0
$40$		−	5.82797i		−	0.921482i
$41$	−1.69394	−	6.17499i	−0.264550	−	0.964372i
$41$	−1.69394	−	6.17499i	−0.264550	−	0.964372i
$42$		0			0
$43$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$43$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$44$		0			0
$45$	−5.87895	+	1.91019i	−0.876382	+	0.284754i
$46$		0			0
$47$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$47$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$48$		0			0
$49$	6.23705	−	3.17793i	0.891007	−	0.453990i
$50$	−0.754350	−	0.754350i	−0.106681	−	0.106681i
$51$		0			0
$52$	2.98785	+	2.55187i	0.414341	+	0.353880i
$53$	0.619329	+	0.0487423i	0.0850714	+	0.00669527i	0.120923	−	0.992662i	$-0.461414\pi$
$53$	0.619329	+	0.0487423i	0.0850714	+	0.00669527i	−0.0358519	+	0.999357i	$0.511414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.855270	+	10.8672i	0.112303	+	1.42694i
$59$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$59$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$60$		0			0
$61$	−0.541604	+	3.41955i	−0.0693453	+	0.437829i	0.928450	+	0.371458i	$0.121142\pi$
$61$	−0.541604	+	3.41955i	−0.0693453	+	0.437829i	−0.997795	+	0.0663709i	$0.978858\pi$
$62$		0			0
$63$		0			0
$64$	4.70228	−	6.47214i	0.587785	−	0.809017i
$65$	−4.03566	+	0.317614i	−0.500563	+	0.0393951i
$66$		0			0
$67$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$67$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$68$	6.02047	+	14.5347i	0.730089	+	1.76259i
$69$		0			0
$70$		0			0
$71$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$71$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$72$	−8.06998	−	2.62210i	−0.951057	−	0.309017i
$73$	−7.53112	+	7.53112i	−0.881451	+	0.881451i	−0.993682	−	0.112231i	$-0.964200\pi$
$73$	−7.53112	+	7.53112i	−0.881451	+	0.881451i	0.112231	+	0.993682i	$0.464200\pi$
$74$	7.44770	+	14.6169i	0.865777	+	1.69918i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$79$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$80$	1.28933	+	8.14052i	0.144152	+	0.910138i
$81$			9.00000i			1.00000i
$82$	3.73221	+	8.25049i	0.412154	+	0.911114i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−14.9743	−	6.20258i	−1.62420	−	0.672764i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	18.0947	−	4.34416i	1.91804	−	0.460480i	0.920468	−	0.390818i	$-0.127808\pi$
$89$	18.0947	−	4.34416i	1.91804	−	0.460480i	0.997571	−	0.0696627i	$-0.0221923\pi$
$90$	7.78913	−	3.96876i	0.821047	−	0.418344i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.1224	−	12.0617i	1.43391	−	1.22468i	0.500567	−	0.865698i	$-0.333125\pi$
$97$	14.1224	−	12.0617i	1.43391	−	1.22468i	0.933346	−	0.358979i	$-0.116875\pi$
$98$	−8.00886	+	5.81878i	−0.809017	+	0.587785i
$99$		0			0
$100$	1.22056	+	0.886792i	0.122056	+	0.0886792i
$101$	4.56909	−	7.45607i	0.454641	−	0.741907i	−0.540396	−	0.841411i	$-0.681726\pi$
$101$	4.56909	−	7.45607i	0.454641	−	0.741907i	0.995037	+	0.0995037i	$0.0317255\pi$
$102$		0			0
$103$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$103$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$104$	−4.73799	−	2.90345i	−0.464599	−	0.284706i
$105$		0			0
$106$	−0.875864	+	0.0689320i	−0.0850714	+	0.00669527i
$107$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$107$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$108$		0			0
$109$	4.01937	+	9.70362i	0.384986	+	0.929439i	0.990985	+	0.133973i	$0.0427736\pi$
$109$	4.01937	+	9.70362i	0.384986	+	0.929439i	−0.605999	+	0.795465i	$0.707226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.7719	−	6.09936i	−1.76591	−	0.573779i	−0.768126	−	0.640298i	$-0.778811\pi$
$113$	−18.7719	−	6.09936i	−1.76591	−	0.573779i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$		0			0
$116$	−3.59882	−	14.9902i	−0.334142	−	1.39180i
$117$	−1.37591	+	5.73108i	−0.127203	+	0.529839i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.72078	+	10.8646i	0.156434	+	0.987688i
$122$		−	4.89626i		−	0.443287i
$123$		0			0
$124$		0			0
$125$	−11.7108	+	1.85482i	−1.04745	+	0.165900i
$126$		0			0
$127$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$127$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$128$	−5.13632	+	10.0806i	−0.453990	+	0.891007i
$129$		0			0
$130$	5.56676	−	1.33646i	0.488237	−	0.117215i
$131$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$131$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−11.6249	−	18.9702i	−0.996830	−	1.62668i
$137$	18.8204	−	7.79568i	1.60794	−	0.666030i	0.615429	−	0.788192i	$-0.288983\pi$
$137$	18.8204	−	7.79568i	1.60794	−	0.666030i	0.992510	+	0.122162i	$0.0389827\pi$
$138$		0			0
$139$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$139$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.8523	+	1.87721i	0.987688	+	0.156434i
$145$	13.5420	+	8.29855i	1.12460	+	0.689157i
$146$	8.85336	−	12.1856i	0.732709	−	1.00849i
$147$		0			0
$148$	−13.6367	−	18.7693i	−1.12093	−	1.54283i
$149$	−15.7818	−	18.4782i	−1.29290	−	1.51379i	−0.719437	−	0.694558i	$-0.755600\pi$
$149$	−15.7818	−	18.4782i	−1.29290	−	1.51379i	−0.573462	−	0.819232i	$-0.694400\pi$
$150$		0			0
$151$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$151$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$152$		0			0
$153$	−15.3259	+	17.9443i	−1.23903	+	1.45071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.38976	−	9.95406i	−0.190724	−	0.794421i	−0.982902	−	0.184131i	$-0.941053\pi$
$157$	−2.38976	−	9.95406i	−0.190724	−	0.794421i	0.792178	−	0.610290i	$-0.208947\pi$
$158$		0			0
$159$		0			0
$160$	−3.60188	−	11.0855i	−0.284754	−	0.876382i
$161$		0			0
$162$	−1.99109	−	12.5712i	−0.156434	−	0.987688i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−7.03843	−	10.6986i	−0.549609	−	0.835422i
$165$		0			0
$166$		0			0
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$		0			0
$169$	4.14955	−	8.14396i	0.319196	−	0.626458i
$170$	22.2884	+	5.35097i	1.70944	+	0.410401i
$171$		0			0
$172$		0			0
$173$	17.7160	+	17.7160i	1.34692	+	1.34692i	0.888986	+	0.457933i	$0.151410\pi$
$173$	17.7160	+	17.7160i	1.34692	+	1.34692i	0.457933	+	0.888986i	$0.348590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−24.3137	+	10.0711i	−1.82239	+	0.754858i
$179$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$179$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$180$	−10.0019	+	7.26678i	−0.745495	+	0.541634i
$181$	1.67515	+	21.2848i	0.124513	+	1.58209i	0.668965	+	0.743294i	$0.266738\pi$
$181$	1.67515	+	21.2848i	0.124513	+	1.58209i	−0.544452	+	0.838792i	$0.683262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	23.6076	+	3.73908i	1.73567	+	0.274903i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$191$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$192$		0			0
$193$	−1.92044	+	24.4015i	−0.138236	+	1.75646i	0.402644	+	0.915357i	$0.368091\pi$
$193$	−1.92044	+	24.4015i	−0.138236	+	1.75646i	−0.540880	+	0.841100i	$0.681909\pi$
$194$	−17.0578	+	19.9721i	−1.22468	+	1.43391i
$195$		0			0
$196$	9.89949	−	9.89949i	0.707107	−	0.707107i
$197$	−12.6973	−	24.9198i	−0.904643	−	1.77546i	−0.528651	−	0.848839i	$-0.677302\pi$
$197$	−12.6973	−	24.9198i	−0.904643	−	1.77546i	−0.375992	−	0.926623i	$-0.622698\pi$
$198$		0			0
$199$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$199$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$200$	−1.90107	−	0.968645i	−0.134426	−	0.0684936i
$201$		0			0
$202$	−4.73259	+	11.4255i	−0.332984	+	0.803895i
$203$		0			0
$204$		0			0
$205$	12.9214	+	2.66644i	0.902468	+	0.186232i
$206$		0			0
$207$		0			0
$208$	7.26038	+	3.00735i	0.503417	+	0.208522i
$209$		0			0
$210$		0			0
$211$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$211$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$212$	1.20816	−	0.290053i	0.0829767	−	0.0199209i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−7.76102	−	12.6648i	−0.525642	−	0.857771i
$219$		0			0
$220$		0			0
$221$	−12.5026	+	9.08370i	−0.841019	+	0.611036i
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	−0.354019	+	2.23519i	−0.0236013	+	0.149013i
$226$	27.5700	+	4.36666i	1.83393	+	0.290466i
$227$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$227$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$228$		0			0
$229$	−16.3911	+	1.29001i	−1.08316	+	0.0852463i	−0.607450	−	0.794358i	$-0.707808\pi$
$229$	−16.3911	+	1.29001i	−1.08316	+	0.0852463i	−0.475706	+	0.879604i	$0.657808\pi$
$230$		0			0
$231$		0			0
$232$	8.34315	+	20.1421i	0.547754	+	1.32240i
$233$	18.4405	−	11.3003i	1.20808	−	0.740310i	0.235269	−	0.971930i	$-0.424403\pi$
$233$	18.4405	−	11.3003i	1.20808	−	0.740310i	0.972806	+	0.231621i	$0.0744028\pi$
$234$	0.653978	−	8.30958i	0.0427519	−	0.543214i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$239$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$240$		0			0
$241$	−27.5162	−	14.0202i	−1.77247	−	0.903120i	−0.933151	−	0.359485i	$-0.882952\pi$
$241$	−27.5162	−	14.0202i	−1.77247	−	0.903120i	−0.839322	−	0.543635i	$-0.817048\pi$
$242$	−4.80718	−	14.7950i	−0.309017	−	0.951057i
$243$		0			0
$244$	1.08321	+	6.83911i	0.0693453	+	0.437829i
$245$			14.4235i			0.921482i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	15.9474	−	5.18162i	1.00860	−	0.327715i
$251$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$251$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	10.7169	+	9.15308i	0.668501	+	0.570953i	0.917591	−	0.397527i	$-0.130131\pi$
$257$	10.7169	+	9.15308i	0.668501	+	0.570953i	−0.249090	+	0.968480i	$0.580131\pi$
$258$		0			0
$259$		0			0
$260$	−7.47999	+	3.09831i	−0.463889	+	0.192149i
$261$	17.5838	−	15.0180i	1.08841	−	0.929588i
$262$		0			0
$263$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$263$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$264$		0			0
$265$	−0.668836	+	1.09144i	−0.0410863	+	0.0670467i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.9029	−	21.8884i	0.969615	−	1.33456i	0.0273737	−	0.999625i	$-0.491286\pi$
$269$	15.9029	−	21.8884i	0.969615	−	1.33456i	0.942241	−	0.334935i	$-0.108714\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$	20.4345	+	23.9258i	1.23903	+	1.45071i
$273$		0			0
$274$	−24.5638	+	15.0527i	−1.48395	+	0.909367i
$275$		0			0
$276$		0			0
$277$	12.7425	+	4.14028i	0.765621	+	0.248765i	0.665689	−	0.746229i	$-0.268138\pi$
$277$	12.7425	+	4.14028i	0.765621	+	0.248765i	0.0999315	+	0.994994i	$0.468138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.74115	+	7.25240i	−0.103868	+	0.432642i	−0.999936	−	0.0112742i	$-0.996411\pi$
$281$	−1.74115	+	7.25240i	−0.103868	+	0.432642i	0.896068	+	0.443916i	$0.146411\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.9706			−1.00000
$289$	−44.3233	+	7.02012i	−2.60725	+	0.412949i
$290$	−20.7514	−	8.59552i	−1.21856	−	0.504746i
$291$		0			0
$292$	−9.67055	+	18.9795i	−0.565926	+	1.11069i
$293$	−33.0551	−	7.93583i	−1.93110	−	0.463616i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−33.0551	−	7.93583i	−1.93110	−	0.463616i	−0.937948	−	0.346775i	$-0.887277\pi$
$294$		0			0
$295$		0			0
$296$	23.2001	+	23.2001i	1.34848	+	1.34848i
$297$		0			0
$298$	26.1321	+	22.3189i	1.51379	+	1.29290i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.77137	−	4.19315i	−0.330468	−	0.240099i
$306$	17.4374	−	28.4553i	0.996830	−	1.62668i
$307$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$307$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$311$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$312$		0			0
$313$	7.93842	+	9.29470i	0.448706	+	0.525368i	0.937946	−	0.346782i	$-0.112726\pi$
$313$	7.93842	+	9.29470i	0.448706	+	0.525368i	−0.489240	+	0.872149i	$0.662726\pi$
$314$	5.54017	+	13.3752i	0.312650	+	0.754804i
$315$		0			0
$316$		0			0
$317$	15.6706	−	18.3479i	0.880146	−	1.03052i	−0.119145	−	0.992877i	$-0.538015\pi$
$317$	15.6706	−	18.3479i	0.880146	−	1.03052i	0.999291	−	0.0376418i	$-0.0119846\pi$
$318$		0			0
$319$		0			0
$320$	7.48357	+	14.6873i	0.418344	+	0.821047i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	5.56231	+	17.1190i	0.309017	+	0.951057i
$325$	−0.567148	+	1.36922i	−0.0314597	+	0.0759505i
$326$		0			0
$327$		0			0
$328$	12.1982	+	13.3867i	0.673531	+	0.739159i
$329$		0			0
$330$		0			0
$331$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$331$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$332$		0			0
$333$	15.7990	−	31.0072i	0.865777	−	1.69918i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.5972	−	23.5972i	−1.28542	−	1.28542i	−0.937539	−	0.347881i	$-0.886901\pi$
$337$	−23.5972	−	23.5972i	−1.28542	−	1.28542i	−0.347881	−	0.937539i	$-0.613099\pi$
$338$	−3.99440	+	12.2935i	−0.217267	+	0.668679i
$339$		0			0
$340$	−32.3163	−	2.54335i	−1.75260	−	0.137932i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−28.6650	−	20.8264i	−1.54104	−	1.11963i
$347$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$347$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$348$		0			0
$349$	28.8903	+	4.57578i	1.54646	+	0.244936i	0.870563	−	0.492057i	$-0.163755\pi$
$349$	28.8903	+	4.57578i	1.54646	+	0.244936i	0.675900	+	0.736993i	$0.263755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.16248	−	8.48193i	−0.327996	−	0.451448i	0.612892	−	0.790167i	$-0.290006\pi$
$353$	−6.16248	−	8.48193i	−0.327996	−	0.451448i	−0.940887	+	0.338719i	$0.890006\pi$
$354$		0			0
$355$		0			0
$356$	31.7334	−	19.4463i	1.68187	−	1.03065i
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$	12.3630	−	12.3630i	0.651587	−	0.651587i
$361$	−8.62582	−	16.9291i	−0.453990	−	0.891007i
$362$	−7.04872	−	29.3601i	−0.370473	−	1.54313i
$363$		0			0
$364$		0			0
$365$	−6.78155	−	20.8715i	−0.354963	−	1.09246i
$366$		0			0
$367$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$367$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$368$		0			0
$369$	9.50574	−	16.6925i	0.494849	−	0.868979i
$370$	−33.8024			−1.75730
$371$		0			0
$372$		0			0
$373$	−36.6768	+	11.9170i	−1.89905	+	0.617040i	−0.932005	+	0.362446i	$0.881942\pi$
$373$	−36.6768	+	11.9170i	−1.89905	+	0.617040i	−0.967048	+	0.254593i	$0.918058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.4930	−	6.87505i	0.694927	−	0.354083i
$378$		0			0
$379$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$379$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$		0			0
$385$		0			0
$386$	−2.71591	−	34.5089i	−0.138236	−	1.75646i
$387$		0			0
$388$	19.4079	−	31.6708i	0.985286	−	1.60784i
$389$	−5.74272	+	36.2581i	−0.291168	+	1.83836i	0.215852	+	0.976426i	$0.430747\pi$
$389$	−5.74272	+	36.2581i	−0.291168	+	1.83836i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$	−11.6376	+	16.0177i	−0.587785	+	0.809017i
$393$		0			0
$394$	23.2486	+	31.9990i	1.17125	+	1.61209i
$395$		0			0
$396$		0			0
$397$	8.20518	−	5.02814i	0.411806	−	0.252355i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	8.20518	−	5.02814i	0.411806	−	0.252355i	0.712938	+	0.701227i	$0.247364\pi$
$398$		0			0
$399$		0			0
$400$	2.86972	+	0.932428i	0.143486	+	0.0466214i
$401$	−17.7692	+	17.7692i	−0.887352	+	0.887352i	−0.994268	−	0.106916i	$-0.965902\pi$
$401$	−17.7692	+	17.7692i	−0.887352	+	0.887352i	0.106916	+	0.994268i	$0.465902\pi$
$402$		0			0
$403$		0			0
$404$	4.08282	−	17.0061i	0.203128	−	0.846088i
$405$	−16.5233	−	8.41902i	−0.821047	−	0.418344i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			12.8136i			0.633591i	0.948494	+	0.316795i	$0.102607\pi$
$409$			12.8136i			0.633591i	−0.948494	+	0.316795i	$0.897393\pi$
$410$	−18.6385	−	0.865866i	−0.920490	−	0.0427621i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−10.8066	−	2.59444i	−0.529839	−	0.127203i
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	30.5753	+	26.1138i	1.49015	+	1.27271i	0.874832	+	0.484427i	$0.160972\pi$
$421$	30.5753	+	26.1138i	1.49015	+	1.27271i	0.615317	+	0.788280i	$0.289028\pi$
$422$		0			0
$423$		0			0
$424$	−1.62339	+	0.672430i	−0.0788388	+	0.0326561i
$425$	−4.51210	+	3.85370i	−0.218869	+	0.186932i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$431$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$432$		0			0
$433$	24.4598	−	33.6660i	1.17546	−	1.61788i	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.4598	−	33.6660i	1.17546	−	1.61788i	0.598778	−	0.800915i	$-0.295653\pi$
$434$		0			0
$435$		0			0
$436$	13.6425	+	15.9733i	0.653356	+	0.764981i
$437$		0			0
$438$		0			0
$439$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$439$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$440$		0			0
$441$	19.9722	+	6.48936i	0.951057	+	0.309017i
$442$	15.4541	−	15.4541i	0.735077	−	0.735077i
$443$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$443$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$444$		0			0
$445$	−8.95114	+	37.2842i	−0.424325	+	1.76744i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.18425	−	7.47706i	−0.0558882	−	0.352864i	−0.999747	−	0.0225137i	$-0.992833\pi$
$449$	−1.18425	−	7.47706i	−0.0558882	−	0.352864i	0.943858	−	0.330350i	$-0.107167\pi$
$450$		−	3.20044i		−	0.150870i
$451$		0			0
$452$	−39.4759			−1.85679
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.55075	−	1.09254i	−0.212875	−	0.0511067i	0.125605	−	0.992080i	$-0.459913\pi$
$457$	−4.55075	−	1.09254i	−0.212875	−	0.0511067i	−0.338480	+	0.940974i	$0.609913\pi$
$458$	22.6098	−	5.42813i	1.05649	−	0.253640i
$459$		0			0
$460$		0			0
$461$	−13.2686	+	40.8365i	−0.617980	+	1.90195i	−0.296399	+	0.955064i	$0.595786\pi$
$461$	−13.2686	+	40.8365i	−0.617980	+	1.90195i	−0.321581	+	0.946882i	$0.604214\pi$
$462$		0			0
$463$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$463$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$464$	−16.1098	−	26.2888i	−0.747879	−	1.22043i
$465$		0			0
$466$	−23.2577	+	19.8640i	−1.07739	+	0.920180i
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$	0.924865	+	11.7515i	0.0427519	+	0.543214i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.21040	+	1.41719i	0.0554203	+	0.0648889i
$478$		0			0
$479$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$479$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$480$		0			0
$481$	14.8009	−	17.3296i	0.674863	−	0.790163i
$482$	41.5364	+	13.4960i	1.89193	+	0.614725i
$483$		0			0
$484$	9.98779	+	19.6021i	0.453990	+	0.891007i
$485$	8.93347	+	37.2106i	0.405648	+	1.68965i
$486$		0			0
$487$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$487$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$488$	−3.02606	−	9.31324i	−0.136983	−	0.421591i
$489$		0			0
$490$	−3.19093	−	20.1468i	−0.144152	−	0.910138i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	60.6325			2.73075
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$499$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$500$	−21.1290	+	10.7658i	−0.944918	+	0.481460i
$501$		0			0
$502$		0			0
$503$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$503$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$504$		0			0
$505$	9.41460	+	15.3632i	0.418944	+	0.683654i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.53997	+	44.9796i	0.156907	+	1.99369i	0.0905584	+	0.995891i	$0.471135\pi$
$509$	3.53997	+	44.9796i	0.156907	+	1.99369i	0.0663482	+	0.997797i	$0.478865\pi$
$510$		0			0
$511$		0			0
$512$	−3.53971	+	22.3488i	−0.156434	+	0.987688i
$513$		0			0
$514$	−16.9943	−	10.4141i	−0.749587	−	0.459348i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	9.76263	−	5.98254i	0.428119	−	0.262352i
$521$	−3.35074	+	42.5752i	−0.146799	+	1.86525i	0.278810	+	0.960346i	$0.410060\pi$
$521$	−3.35074	+	42.5752i	−0.146799	+	1.86525i	−0.425609	+	0.904907i	$0.639940\pi$
$522$	−21.2386	+	24.8672i	−0.929588	+	1.08841i
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.10739	−	21.8743i	−0.309017	−	0.951057i
$530$	0.692770	−	1.67250i	0.0300920	−	0.0726486i
$531$		0			0
$532$		0			0
$533$	8.60507	−	9.17636i	0.372727	−	0.397472i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−17.3707	+	34.0920i	−0.748906	+	1.46981i
$539$		0			0
$540$		0			0
$541$	−40.7497	+	20.7630i	−1.75196	+	0.892671i	−0.792964	+	0.609269i	$0.791463\pi$
$541$	−40.7497	+	20.7630i	−1.75196	+	0.892671i	−0.959001	+	0.283402i	$0.908537\pi$
$542$		0			0
$543$		0			0
$544$	−33.8362	−	28.8988i	−1.45071	−	1.23903i
$545$	−21.5750	−	1.69799i	−0.924169	−	0.0727337i
$546$		0			0
$547$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$547$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$548$	30.9806	−	26.4599i	1.32343	−	1.13031i
$549$	−8.40289	+	6.10505i	−0.358626	+	0.260557i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−18.7147	−	2.96411i	−0.795110	−	0.125933i
$555$		0			0
$556$		0			0
$557$	40.2860	−	3.17058i	1.70697	−	0.134342i	0.812942	−	0.582345i	$-0.197865\pi$
$557$	40.2860	−	3.17058i	1.70697	−	0.134342i	0.894032	+	0.448003i	$0.147865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.827578	−	10.5154i	0.0349093	−	0.443564i
$563$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$563$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$564$		0			0
$565$	28.7580	−	28.7580i	1.20986	−	1.20986i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.0189	−	5.61441i	−0.461936	−	0.235368i	0.207504	−	0.978234i	$-0.433466\pi$
$569$	−11.0189	−	5.61441i	−0.461936	−	0.235368i	−0.669440	+	0.742866i	$0.733466\pi$
$570$		0			0
$571$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$571$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.7045	−	3.75443i	0.987688	−	0.156434i
$577$	−8.55571	−	3.54389i	−0.356179	−	0.147534i	0.197417	−	0.980320i	$-0.436745\pi$
$577$	−8.55571	−	3.54389i	−0.356179	−	0.147534i	−0.553596	+	0.832785i	$0.686745\pi$
$578$	60.3578	−	19.6115i	2.51056	−	0.815729i
$579$		0			0
$580$	30.8872	+	7.41536i	1.28252	+	0.307906i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	9.30898	−	28.6501i	0.385208	−	1.18555i
$585$	−9.23470	−	7.88718i	−0.381808	−	0.326095i
$586$	47.9271	+	3.77194i	1.97985	+	0.155818i
$587$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$587$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−37.5386	−	27.2734i	−1.54283	−	1.12093i
$593$	−23.3049	+	38.0302i	−0.957019	+	1.56171i	−0.136079	+	0.990698i	$0.543450\pi$
$593$	−23.3049	+	38.0302i	−0.957019	+	1.56171i	−0.820940	+	0.571014i	$0.806550\pi$
$594$		0			0
$595$		0			0
$596$	−41.4390	−	25.3938i	−1.69741	−	1.04017i
$597$		0			0
$598$		0			0
$599$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$599$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$600$		0			0
$601$	−14.8395	−	35.8258i	−0.605317	−	1.46136i	−0.868040	−	0.496494i	$-0.834621\pi$
$601$	−14.8395	−	35.8258i	−0.605317	−	1.46136i	0.262723	−	0.964871i	$-0.415379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−21.5561	−	7.00402i	−0.876382	−	0.284754i
$606$		0			0
$607$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$607$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$608$		0			0
$609$		0			0
$610$	8.98913	+	4.58019i	0.363959	+	0.185446i
$611$		0			0
$612$	−18.0614	+	43.6041i	−0.730089	+	1.76259i
$613$	−6.79873	−	42.9255i	−0.274598	−	1.73374i	−0.610651	−	0.791900i	$-0.709092\pi$
$613$	−6.79873	−	42.9255i	−0.274598	−	1.73374i	0.336053	−	0.941843i	$-0.390908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.0143	+	6.65442i	−1.69143	+	0.267897i	−0.926523	−	0.376239i	$-0.877217\pi$
$617$	−42.0143	+	6.65442i	−1.69143	+	0.267897i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.38405	−	19.6481i	0.255362	−	0.785923i
$626$	−13.1447	−	11.2266i	−0.525368	−	0.448706i
$627$		0			0
$628$	−10.6975	−	17.4568i	−0.426878	−	0.696602i
$629$	84.3016	−	34.9189i	3.36133	−	1.39231i
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	−17.8295	+	29.0952i	−0.708101	+	1.15552i
$635$		0			0
$636$		0			0
$637$	11.7259	+	7.18566i	0.464599	+	0.284706i
$638$		0			0
$639$		0			0
$640$	−13.7024	−	18.8597i	−0.541634	−	0.745495i
$641$	−24.9726	−	29.2391i	−0.986358	−	1.15488i	−0.987934	−	0.154878i	$-0.950502\pi$
$641$	−24.9726	−	29.2391i	−0.986358	−	1.15488i	0.00157563	−	0.999999i	$-0.499498\pi$
$642$		0			0
$643$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$643$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	−11.5567	−	22.6813i	−0.453990	−	0.891007i
$649$		0			0
$650$	0.489279	−	2.03800i	0.0191911	−	0.0799368i
$651$		0			0
$652$		0			0
$653$	16.0055	−	38.6407i	0.626344	−	1.51213i	−0.217789	−	0.975996i	$-0.569885\pi$
$653$	16.0055	−	38.6407i	0.626344	−	1.51213i	0.844134	−	0.536133i	$-0.180115\pi$
$654$		0			0
$655$		0			0
$656$	−20.0000	−	16.0000i	−0.780869	−	0.624695i
$657$	−31.9518			−1.24656
$658$		0			0
$659$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$659$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$660$		0			0
$661$	1.83329	−	3.59804i	0.0713068	−	0.139947i	−0.852601	−	0.522562i	$-0.824976\pi$
$661$	1.83329	−	3.59804i	0.0713068	−	0.139947i	0.923908	+	0.382615i	$0.124976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−15.2082	+	46.8061i	−0.589307	+	1.81370i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−3.96732	−	50.4096i	−0.152929	−	1.94315i	−0.299827	−	0.953994i	$-0.596929\pi$
$673$	−3.96732	−	50.4096i	−0.152929	−	1.94315i	0.146898	−	0.989152i	$-0.453071\pi$
$674$	38.1810	+	27.7401i	1.47068	+	1.06851i
$675$		0			0
$676$	2.85968	−	18.0553i	0.109988	−	0.694434i
$677$	−49.4565	−	7.83314i	−1.90077	−	0.301052i	−0.907796	−	0.419411i	$-0.862237\pi$
$677$	−49.4565	−	7.83314i	−1.90077	−	0.301052i	−0.992971	+	0.118359i	$0.962237\pi$
$678$		0			0
$679$		0			0
$680$	45.7022	−	3.59684i	1.75260	−	0.137932i
$681$		0			0
$682$		0			0
$683$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$683$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$684$		0			0
$685$	−3.29329	+	41.8452i	−0.125830	+	1.59882i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.554106	+	1.08750i	0.0211098	+	0.0414303i
$690$		0			0
$691$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$691$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$692$	44.6468	+	22.7487i	1.69722	+	0.864776i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	47.3781	−	17.0947i	1.79457	−	0.647508i
$698$	−41.3664			−1.56574
$699$		0			0
$700$		0			0
$701$	−1.65702	+	0.538398i	−0.0625847	+	0.0203350i	−0.340142	−	0.940374i	$-0.610475\pi$
$701$	−1.65702	+	0.538398i	−0.0625847	+	0.0203350i	0.277557	+	0.960709i	$0.410475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	10.4842	+	10.4842i	0.394580	+	0.394580i
$707$		0			0
$708$		0			0
$709$	−21.7739	−	1.71364i	−0.817735	−	0.0643572i	−0.337311	−	0.941393i	$-0.609517\pi$
$709$	−21.7739	−	1.71364i	−0.817735	−	0.0643572i	−0.480424	+	0.877036i	$0.659517\pi$
$710$		0			0
$711$		0			0
$712$	−40.0232	+	34.1830i	−1.49993	+	1.28106i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$719$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$720$	−14.5336	+	20.0037i	−0.541634	+	0.745495i
$721$		0			0
$722$	15.7938	+	21.7383i	0.587785	+	0.809017i
$723$		0			0
$724$	16.3410	+	39.4508i	0.607310	+	1.46618i
$725$	4.95774	−	3.03811i	0.184126	−	0.112833i
$726$		0			0
$727$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$727$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$728$		0			0
$729$	−19.0919	+	19.0919i	−0.707107	+	0.707107i
$730$	14.0899	+	27.6530i	0.521491	+	1.02348i
$731$		0			0
$732$		0			0
$733$	18.6679	+	9.51176i	0.689514	+	0.351325i	0.763386	−	0.645943i	$-0.223536\pi$
$733$	18.6679	+	9.51176i	0.689514	+	0.351325i	−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−9.58472	+	25.4191i	−0.352819	+	0.935692i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	47.2152	−	7.47816i	1.73567	−	0.274903i
$741$		0			0
$742$		0			0
$743$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$743$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$744$		0			0
$745$	48.6874	−	11.6888i	1.78377	−	0.428245i
$746$	48.5938	−	24.7598i	1.77915	−	0.906520i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$751$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$752$		0			0
$753$		0			0
$754$	−17.3261	+	12.5882i	−0.630980	+	0.458434i
$755$		0			0
$756$		0			0
$757$	−24.2237	+	39.5294i	−0.880424	+	1.43672i	0.0190839	+	0.999818i	$0.493925\pi$
$757$	−24.2237	+	39.5294i	−0.880424	+	1.43672i	−0.899508	+	0.436904i	$0.856075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−26.7160	+	36.7715i	−0.968456	+	1.33296i	−0.0256326	+	0.999671i	$0.508160\pi$
$761$	−26.7160	+	36.7715i	−0.968456	+	1.33296i	−0.942823	+	0.333294i	$0.891840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−18.6077	−	44.9230i	−0.672764	−	1.62420i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−52.2788	−	16.9864i	−1.88522	−	0.612546i	−0.983700	−	0.179815i	$-0.942450\pi$
$769$	−52.2788	−	16.9864i	−1.88522	−	0.612546i	−0.901523	−	0.432731i	$-0.857550\pi$
$770$		0			0
$771$		0			0
$772$	11.4281	+	47.6013i	0.411305	+	1.71321i
$773$	−6.45230	+	26.8758i	−0.232073	+	0.966655i	0.727704	+	0.685892i	$0.240588\pi$
$773$	−6.45230	+	26.8758i	−0.232073	+	0.966655i	−0.959777	+	0.280763i	$0.909412\pi$
$774$		0			0
$775$		0			0
$776$	−20.1024	+	48.5315i	−0.721634	+	1.74218i
$777$		0			0
$778$		−	51.9159i		−	1.86128i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	12.7117	−	24.9482i	0.453990	−	0.891007i
$785$	20.5103	+	4.92409i	0.732045	+	0.175748i
$786$		0			0
$787$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$787$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$788$	−39.5529	−	39.5529i	−1.40901	−	1.40901i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6.28418	+	2.60299i	−0.223158	+	0.0924350i
$794$	−10.3486	+	8.83857i	−0.367259	+	0.313669i
$795$		0			0
$796$		0			0
$797$	10.3283	+	7.50394i	0.365847	+	0.265803i	0.755487	−	0.655164i	$-0.227401\pi$
$797$	10.3283	+	7.50394i	0.365847	+	0.265803i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$	−4.21471	−	0.667545i	−0.149013	−	0.0236013i
$801$	47.6001	+	29.1694i	1.68187	+	1.03065i
$802$	20.8889	−	28.7512i	0.737614	−	1.01524i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.94059	+	24.6575i	−0.0682696	+	0.867447i
$809$	23.4374	−	27.4417i	0.824017	−	0.964800i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	23.4374	−	27.4417i	0.824017	−	0.964800i	0.999807	+	0.0196280i	$0.00624818\pi$
$810$	24.9423	+	8.10423i	0.876382	+	0.284754i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−2.83477	−	17.8980i	−0.0991154	−	0.625790i
$819$		0			0
$820$	26.2259	−	2.91398i	0.915846	−	0.101761i
$821$	−2.24866			−0.0784787			−0.0392393	−	0.999230i	$-0.512493\pi$
$821$	−2.24866			−0.0784787			−0.0392393	+	0.999230i	$0.512493\pi$
$822$		0			0
$823$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$823$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$827$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$828$		0			0
$829$	−31.9448	−	31.9448i	−1.10949	−	1.10949i	−0.993218	−	0.116270i	$-0.962906\pi$
$829$	−31.9448	−	31.9448i	−1.10949	−	1.10949i	−0.116270	−	0.993218i	$-0.537094\pi$
$830$		0			0
$831$		0			0
$832$	15.6687	+	1.23315i	0.543214	+	0.0427519i
$833$	28.7702	+	46.9488i	0.996830	+	1.62668i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$839$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$840$		0			0
$841$	−30.0398	−	4.75783i	−1.03585	−	0.164063i
$842$	−48.4848	−	29.7115i	−1.67090	−	1.02393i
$843$		0			0
$844$		0			0
$845$	11.0699	+	15.2365i	0.380818	+	0.524150i
$846$		0			0
$847$		0			0
$848$	2.11879	−	1.29840i	0.0727596	−	0.0445872i
$849$		0			0
$850$	5.44995	−	6.38108i	0.186932	−	0.218869i
$851$		0			0
$852$		0			0
$853$	3.21020	+	6.30037i	0.109915	+	0.215720i	0.939411	−	0.342792i	$-0.111372\pi$
$853$	3.21020	+	6.30037i	0.109915	+	0.215720i	−0.829496	+	0.558512i	$0.811372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.362073	+	1.11435i	0.0123682	+	0.0380653i	0.957050	−	0.289922i	$-0.0936295\pi$
$857$	0.362073	+	1.11435i	0.0123682	+	0.0380653i	−0.944682	+	0.327988i	$0.893630\pi$
$858$		0			0
$859$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$859$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$863$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$864$		0			0
$865$	−49.0974	+	15.9527i	−1.66936	+	0.542408i
$866$	−26.7175	+	52.4360i	−0.907896	+	1.78185i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−22.5896	−	19.2934i	−0.764981	−	0.653356i
$873$	55.5448	+	4.37147i	1.87991	+	0.147952i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	43.6678	−	31.7265i	1.47456	−	1.07133i	0.495297	−	0.868723i	$-0.335059\pi$
$877$	43.6678	−	31.7265i	1.47456	−	1.07133i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.99109	−	12.5712i	0.0670814	−	0.423535i	−0.931178	−	0.364564i	$-0.881218\pi$
$881$	1.99109	−	12.5712i	0.0670814	−	0.423535i	0.998260	−	0.0589711i	$-0.0187820\pi$
$882$	−29.3328	−	4.64587i	−0.987688	−	0.156434i
$883$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$883$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$884$	−18.1674	+	25.0053i	−0.611036	+	0.841019i
$885$		0			0
$886$		0			0
$887$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$887$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$888$		0			0
$889$		0			0
$890$	4.25453	−	54.0589i	0.142612	−	1.81206i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	3.30832	+	10.1820i	0.110400	+	0.339777i
$899$		0			0
$900$	0.708038	+	4.47038i	0.0236013	+	0.149013i
$901$			4.88678i			0.162802i
$902$		0			0
$903$		0			0
$904$	55.1400	−	8.73332i	1.83393	−	0.290466i
$905$	−40.6441	−	16.8353i	−1.35106	−	0.559626i
$906$		0			0
$907$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$907$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$908$		0			0
$909$	25.5092	−	6.12422i	0.846088	−	0.203128i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$	6.59820	+	0.519289i	0.218249	+	0.0171766i
$915$		0			0
$916$	−30.3805	+	12.5840i	−1.00380	+	0.415788i
$917$		0			0
$918$		0			0
$919$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$919$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$920$		0			0
$921$		0			0
$922$	9.49926	−	59.9760i	0.312842	−	1.97520i
$923$		0			0
$924$		0			0
$925$	5.14342	−	7.07931i	0.169115	−	0.232766i
$926$		0			0
$927$		0			0
$928$	28.3181	+	33.1563i	0.929588	+	1.08841i
$929$	13.8790	+	33.5069i	0.455355	+	1.09932i	0.970257	+	0.242076i	$0.0778283\pi$
$929$	13.8790	+	33.5069i	0.455355	+	1.09932i	−0.514902	+	0.857249i	$0.672172\pi$
$930$		0			0
$931$		0			0
$932$	28.0919	−	32.8914i	0.920180	−	1.07739i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−3.89166	−	16.2099i	−0.127203	−	0.529839i
$937$	3.90755	−	16.2761i	0.127654	−	0.531718i	−0.871492	−	0.490409i	$-0.836847\pi$
$937$	3.90755	−	16.2761i	0.127654	−	0.531718i	0.999146	−	0.0413087i	$-0.0131527\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.90683	+	43.6080i	0.225156	+	1.42158i	0.798366	+	0.602172i	$0.205698\pi$
$941$	6.90683	+	43.6080i	0.225156	+	1.42158i	−0.573210	+	0.819408i	$0.694302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$947$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$948$		0			0
$949$	−20.3465	−	4.88476i	−0.660475	−	0.158566i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.55524	+	20.1750i	−0.212345	+	0.653531i	0.786986	+	0.616970i	$0.211640\pi$
$953$	−6.55524	+	20.1750i	−0.212345	+	0.653531i	−0.999331	+	0.0365605i	$0.988360\pi$
$954$	−2.00422	−	1.71176i	−0.0648889	−	0.0554203i
$955$		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
$962$	−16.8401	+	27.4805i	−0.542946	+	0.886007i
$963$		0			0
$964$	−61.0038	−	9.66206i	−1.96480	−	0.311194i
$965$	−43.0026	−	26.3520i	−1.38430	−	0.848302i
$966$		0			0
$967$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$967$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$968$	−18.2876	−	25.1707i	−0.587785	−	0.809017i
$969$		0			0
$970$	−20.7105	−	49.9995i	−0.664973	−	1.60539i
$971$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$971$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	6.28719	+	12.3393i	0.201248	+	0.394971i
$977$	14.0866	+	58.6747i	0.450669	+	1.87717i	0.479421	+	0.877585i	$0.340847\pi$
$977$	14.0866	+	58.6747i	0.450669	+	1.87717i	−0.0287524	+	0.999587i	$0.509153\pi$
$978$		0			0
$979$		0			0
$980$	8.91420	+	27.4351i	0.284754	+	0.876382i
$981$	−12.0581	+	29.1109i	−0.384986	+	0.929439i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	57.6283			1.83619
$986$	−84.6916	+	13.4138i	−2.69713	+	0.427184i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$991$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	30.0361	+	49.0145i	0.951254	+	1.55231i	0.828589	+	0.559857i	$0.189144\pi$
$997$	30.0361	+	49.0145i	0.951254	+	1.55231i	0.122665	+	0.992448i	$0.460856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.a.11.1	✓	16
4.3	odd	2	CM	164.2.o.a.11.1	✓	16
41.15	odd	40	inner	164.2.o.a.15.1	yes	16
164.15	even	40	inner	164.2.o.a.15.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.a.11.1	✓	16	1.1	even	1	trivial
164.2.o.a.11.1	✓	16	4.3	odd	2	CM
164.2.o.a.15.1	yes	16	41.15	odd	40	inner
164.2.o.a.15.1	yes	16	164.15	even	40	inner

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:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.o (of order \(40\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(-0.935446 + 1.83592i) q^{5} +(-2.52015 + 1.28408i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)	\(q+(-1.39680 + 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(-0.935446 + 1.83592i) q^{5} +(-2.52015 + 1.28408i) q^{8} +(2.12132 + 2.12132i) q^{9} +(0.900470 - 2.77136i) q^{10} +(1.02652 + 1.67513i) q^{13} +(3.23607 - 2.35114i) q^{16} +(0.617169 + 7.84187i) q^{17} +(-3.43237 - 2.49376i) q^{18} +(-0.644666 + 4.07026i) q^{20} +(0.443396 + 0.610282i) q^{25} +(-1.80444 - 2.11273i) q^{26} +(0.604768 - 7.68430i) q^{29} +(-4.00000 + 4.00000i) q^{32} +(-2.59693 - 10.8170i) q^{34} +(5.34604 + 2.72394i) q^{36} +(-3.58462 - 11.0323i) q^{37} -5.82797i q^{40} +(-1.69394 - 6.17499i) q^{41} +(-5.87895 + 1.91019i) q^{45} +(6.23705 - 3.17793i) q^{49} +(-0.754350 - 0.754350i) q^{50} +(2.98785 + 2.55187i) q^{52} +(0.619329 + 0.0487423i) q^{53} +(0.855270 + 10.8672i) q^{58} +(-0.541604 + 3.41955i) q^{61} +(4.70228 - 6.47214i) q^{64} +(-4.03566 + 0.317614i) q^{65} +(6.02047 + 14.5347i) q^{68} +(-8.06998 - 2.62210i) q^{72} +(-7.53112 + 7.53112i) q^{73} +(7.44770 + 14.6169i) q^{74} +(1.28933 + 8.14052i) q^{80} +9.00000i q^{81} +(3.73221 + 8.25049i) q^{82} +(-14.9743 - 6.20258i) q^{85} +(18.0947 - 4.34416i) q^{89} +(7.78913 - 3.96876i) q^{90} +(14.1224 - 12.0617i) q^{97} +(-8.00886 + 5.81878i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 8 * q^8	\( 16 q - 4 q^{2} + 8 q^{8} - 8 q^{13} + 16 q^{16} - 4 q^{17} - 20 q^{26} + 20 q^{29} - 64 q^{32} - 48 q^{34} - 20 q^{41} + 56 q^{50} + 96 q^{52} + 28 q^{53} + 12 q^{58} - 44 q^{61} - 112 q^{65} + 32 q^{68} - 36 q^{82} + 56 q^{85} + 32 q^{89} + 48 q^{90} + 36 q^{97}+O(q^{100}) \) 16 * q - 4 * q^2 + 8 * q^8 - 8 * q^13 + 16 * q^16 - 4 * q^17 - 20 * q^26 + 20 * q^29 - 64 * q^32 - 48 * q^34 - 20 * q^41 + 56 * q^50 + 96 * q^52 + 28 * q^53 + 12 * q^58 - 44 * q^61 - 112 * q^65 + 32 * q^68 - 36 * q^82 + 56 * q^85 + 32 * q^89 + 48 * q^90 + 36 * q^97

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