LMFDB - Embedded newform 77.2.l.a.62.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(6,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 9]))

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

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

chi := DirichletCharacter("77.6");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.l (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.614848095564$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.37515625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 $ x^8 - x^7 - x^6 + 3x^5 - x^4 + 6x^3 - 4x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			62.2
Root			$1.18208 - 0.776336i$ of defining polynomial
Character	$\chi$	$=$	77.62
Dual form			77.2.l.a.41.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.03958 - 1.43086i) q^{2} +(-0.348597 - 1.07287i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(1.46663 + 0.476537i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})$	\(q+(1.03958 - 1.43086i) q^{2} +(-0.348597 - 1.07287i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(1.46663 + 0.476537i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(-0.0629004 + 3.31603i) q^{11} +(-1.44601 + 4.45035i) q^{14} +(4.03181 - 2.92928i) q^{16} +(-5.04623 + 1.63962i) q^{18} +(4.67938 + 3.53728i) q^{22} +2.56038 q^{23} +(1.54508 - 4.75528i) q^{25} +(1.75432 + 2.41461i) q^{28} +(-7.02475 + 2.28248i) q^{29} -5.72996i q^{32} +(-1.04579 + 3.21861i) q^{36} +(-3.68322 - 11.3358i) q^{37} +2.77251i q^{43} +(3.57960 - 1.08847i) q^{44} +(2.66172 - 3.66355i) q^{46} +(5.66312 - 4.11450i) q^{49} +(-5.19790 - 7.15429i) q^{50} +(11.5776 + 8.41162i) q^{53} -4.08003 q^{56} +(-4.03688 + 12.4242i) q^{58} +(7.54878 + 2.45275i) q^{63} +(-0.135144 - 0.0981881i) q^{64} +12.5669 q^{67} +(-12.9884 + 9.43662i) q^{71} +(-2.71928 - 3.74277i) q^{72} +(-20.0489 - 6.51428i) q^{74} +(-2.55285 - 8.39541i) q^{77} +(-10.3127 + 14.1942i) q^{79} +(2.78115 + 8.55951i) q^{81} +(3.96707 + 2.88224i) q^{86} +(-1.67246 + 4.83341i) q^{88} +(-0.892541 - 2.74696i) q^{92} -12.3805i q^{98} +(6.00000 - 7.93725i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9}+O(q^{10}) $ 8 * q + 10 * q^4 - 10 * q^8 - 6 * q^9	$ 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9} - 4 q^{11} - 21 q^{14} + 8 q^{16} - 15 q^{18} - 14 q^{22} + 16 q^{23} - 10 q^{25} + 35 q^{28} + 30 q^{36} - 18 q^{37} + 25 q^{44} + 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} - 75 q^{72} - 14 q^{77} - 40 q^{79} - 18 q^{81} + 23 q^{86} - 8 q^{88} + 25 q^{92} + 48 q^{99}+O(q^{100}) $ 8 * q + 10 * q^4 - 10 * q^8 - 6 * q^9 - 4 * q^11 - 21 * q^14 + 8 * q^16 - 15 * q^18 - 14 * q^22 + 16 * q^23 - 10 * q^25 + 35 * q^28 + 30 * q^36 - 18 * q^37 + 25 * q^44 + 15 * q^46 + 14 * q^49 + 30 * q^53 - 42 * q^56 + 19 * q^58 - 34 * q^64 + 8 * q^67 - 48 * q^71 - 75 * q^72 - 14 * q^77 - 40 * q^79 - 18 * q^81 + 23 * q^86 - 8 * q^88 + 25 * q^92 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$

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 $

$2$	1.03958	−	1.43086i	0.735094	−	1.01177i	−0.263792	−	0.964580i	$-0.584973\pi$
$2$	1.03958	−	1.43086i	0.735094	−	1.01177i	0.998886	−	0.0471903i	$-0.0150267\pi$
$3$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$3$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$4$	−0.348597	−	1.07287i	−0.174298	−	0.536435i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$		0			0
$7$	−2.51626	+	0.817582i	−0.951057	+	0.309017i
$7$	−2.51626	+	0.817582i	−0.951057	+	0.309017i
$8$	1.46663	+	0.476537i	0.518532	+	0.168481i
$9$	−2.42705	−	1.76336i	−0.809017	−	0.587785i
$10$		0			0
$11$	−0.0629004	+	3.31603i	−0.0189652	+	0.999820i
$11$	−0.0629004	+	3.31603i	−0.0189652	+	0.999820i
$12$		0			0
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$	−1.44601	+	4.45035i	−0.386462	+	1.18941i
$15$		0			0
$16$	4.03181	−	2.92928i	1.00795	−	0.732321i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$	−5.04623	+	1.63962i	−1.18941	+	0.386462i
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$		0			0
$21$		0			0
$22$	4.67938	+	3.53728i	0.997647	+	0.754150i
$23$	2.56038			0.533877			0.266938	−	0.963714i	$-0.413988\pi$
$23$	2.56038			0.533877			0.266938	+	0.963714i	$0.413988\pi$
$24$		0			0
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$		0			0
$28$	1.75432	+	2.41461i	0.331535	+	0.456319i
$29$	−7.02475	+	2.28248i	−1.30446	+	0.423846i	−0.877132	−	0.480249i	$-0.840546\pi$
$29$	−7.02475	+	2.28248i	−1.30446	+	0.423846i	−0.427331	+	0.904095i	$0.640546\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$		−	5.72996i		−	1.01292i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.04579	+	3.21861i	−0.174298	+	0.536435i
$37$	−3.68322	−	11.3358i	−0.605517	−	1.86359i	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−3.68322	−	11.3358i	−0.605517	−	1.86359i	−0.112320	−	0.993672i	$-0.535828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$			2.77251i			0.422803i	0.977399	+	0.211402i	$0.0678028\pi$
$43$			2.77251i			0.422803i	−0.977399	+	0.211402i	$0.932197\pi$
$44$	3.57960	−	1.08847i	0.539644	−	0.164093i
$45$		0			0
$46$	2.66172	−	3.66355i	0.392449	−	0.540160i
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$	5.66312	−	4.11450i	0.809017	−	0.587785i
$50$	−5.19790	−	7.15429i	−0.735094	−	1.01177i
$51$		0			0
$52$		0			0
$53$	11.5776	+	8.41162i	1.59031	+	1.15542i	0.903503	+	0.428581i	$0.140986\pi$
$53$	11.5776	+	8.41162i	1.59031	+	1.15542i	0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$		0			0
$56$	−4.08003			−0.545217
$57$		0			0
$58$	−4.03688	+	12.4242i	−0.530069	+	1.63138i
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$	7.54878	+	2.45275i	0.951057	+	0.309017i
$64$	−0.135144	−	0.0981881i	−0.0168930	−	0.0122735i
$65$		0			0
$66$		0			0
$67$	12.5669			1.53529			0.767644	−	0.640877i	$-0.221429\pi$
$67$	12.5669			1.53529			0.767644	+	0.640877i	$0.221429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.9884	+	9.43662i	−1.54144	+	1.11992i	−0.592014	+	0.805928i	$0.701667\pi$
$71$	−12.9884	+	9.43662i	−1.54144	+	1.11992i	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−2.71928	−	3.74277i	−0.320471	−	0.441090i
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$	−20.0489	−	6.51428i	−2.33064	−	0.757269i
$75$		0			0
$76$		0			0
$77$	−2.55285	−	8.39541i	−0.290924	−	0.956746i
$78$		0			0
$79$	−10.3127	+	14.1942i	−1.16027	+	1.59698i	−0.450035	+	0.893011i	$0.648589\pi$
$79$	−10.3127	+	14.1942i	−1.16027	+	1.59698i	−0.710235	+	0.703964i	$0.751411\pi$
$80$		0			0
$81$	2.78115	+	8.55951i	0.309017	+	0.951057i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$		0			0
$85$		0			0
$86$	3.96707	+	2.88224i	0.427780	+	0.310800i
$87$		0			0
$88$	−1.67246	+	4.83341i	−0.178285	+	0.515244i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−0.892541	−	2.74696i	−0.0930538	−	0.286390i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$97$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$98$		−	12.3805i		−	1.25062i
$99$	6.00000	−	7.93725i	0.603023	−	0.797724i
$100$	−5.64041			−0.564041
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$	24.0717	−	7.82136i	2.33805	−	0.759678i
$107$	16.5350	+	5.37256i	1.59850	+	0.519385i	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	16.5350	+	5.37256i	1.59850	+	0.519385i	0.631766	+	0.775159i	$0.282330\pi$
$108$		0			0
$109$		−	13.8487i		−	1.32646i	−0.748414	−	0.663232i	$-0.769184\pi$
$109$		−	13.8487i		−	1.32646i	0.748414	−	0.663232i	$-0.230816\pi$
$110$		0			0
$111$		0			0
$112$	−7.75016	+	10.6672i	−0.732321	+	1.00795i
$113$	3.34450	−	10.2933i	0.314624	−	0.968314i	−0.661285	−	0.750135i	$-0.729988\pi$
$113$	3.34450	−	10.2933i	0.314624	−	0.968314i	0.975909	−	0.218179i	$-0.0700116\pi$
$114$		0			0
$115$		0			0
$116$	4.89761	+	6.74098i	0.454732	+	0.625884i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.9921	−	0.417159i	−0.999281	−	0.0379235i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	11.3571	−	8.25141i	1.01177	−	0.735094i
$127$	−6.06090	−	8.34211i	−0.537818	−	0.740242i	0.450479	−	0.892787i	$-0.351253\pi$
$127$	−6.06090	−	8.34211i	−0.537818	−	0.740242i	−0.988297	+	0.152545i	$0.951253\pi$
$128$	10.6181	−	3.45001i	0.938512	−	0.304941i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	13.0643	−	17.9814i	1.12858	−	1.55336i
$135$		0			0
$136$		0			0
$137$	−13.7856	+	10.0158i	−1.17778	+	0.855708i	−0.991920	−	0.126868i	$-0.959507\pi$
$137$	−13.7856	+	10.0158i	−1.17778	+	0.855708i	−0.185861	+	0.982576i	$0.559507\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$			28.3947i			2.38283i
$143$		0			0
$144$	−14.9508			−1.24590
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−10.8779	+	7.90323i	−0.894154	+	0.649641i
$149$	6.22053	+	8.56183i	0.509606	+	0.701413i	0.983853	−	0.178979i	$-0.0572796\pi$
$149$	6.22053	+	8.56183i	0.509606	+	0.701413i	−0.474247	+	0.880392i	$0.657280\pi$
$150$		0			0
$151$	−23.2633	−	7.55872i	−1.89314	−	0.615120i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−23.2633	−	7.55872i	−1.89314	−	0.615120i	−0.916597	−	0.399811i	$-0.869076\pi$
$152$		0			0
$153$		0			0
$154$	−14.6665	−	5.07493i	−1.18186	−	0.408950i
$155$		0			0
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$	9.58905	+	29.5121i	0.762864	+	2.34785i
$159$		0			0
$160$		0			0
$161$	−6.44258	+	2.09332i	−0.507747	+	0.164977i
$162$	15.1387	+	4.91885i	1.18941	+	0.386462i
$163$	−17.2142	−	12.5068i	−1.34832	−	0.979611i	−0.999093	−	0.0425718i	$-0.986445\pi$
$163$	−17.2142	−	12.5068i	−1.34832	−	0.979611i	−0.349225	−	0.937039i	$-0.613555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−4.01722	−	12.3637i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	2.97454	−	0.966487i	0.226807	−	0.0736939i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$	9.45998	+	13.5539i	0.713073	+	1.02166i
$177$		0			0
$178$		0			0
$179$	8.15763	−	25.1066i	0.609730	−	1.87656i	0.149487	−	0.988764i	$-0.452238\pi$
$179$	8.15763	−	25.1066i	0.609730	−	1.87656i	0.460243	−	0.887793i	$-0.347762\pi$
$180$		0			0
$181$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$181$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$	3.75513	+	1.22012i	0.276832	+	0.0899482i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.80563	+	8.63483i	0.203008	+	0.624795i	0.999789	+	0.0205267i	$0.00653431\pi$
$191$	2.80563	+	8.63483i	0.203008	+	0.624795i	−0.796781	+	0.604268i	$0.793466\pi$
$192$		0			0
$193$	16.2839	+	22.4128i	1.17214	+	1.61331i	0.647834	+	0.761781i	$0.275675\pi$
$193$	16.2839	+	22.4128i	1.17214	+	1.61331i	0.524305	+	0.851530i	$0.324325\pi$
$194$		0			0
$195$		0			0
$196$	−6.38847	−	4.64149i	−0.456319	−	0.331535i
$197$			27.9978i			1.99476i	0.0723369	+	0.997380i	$0.476954\pi$
$197$			27.9978i			1.99476i	−0.0723369	+	0.997380i	$0.523046\pi$
$198$	−5.11961	−	16.8366i	−0.363835	−	1.19652i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	4.53214	−	6.23795i	0.320471	−	0.441090i
$201$		0			0
$202$		0			0
$203$	15.8100	−	11.4866i	1.10964	−	0.806203i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.21418	−	4.51486i	−0.431915	−	0.313805i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.90258	−	2.61867i	0.130979	−	0.180277i	−0.738490	−	0.674264i	$-0.764461\pi$
$211$	1.90258	−	2.61867i	0.130979	−	0.180277i	0.869469	+	0.493987i	$0.164461\pi$
$212$	4.98866	−	15.3535i	0.342623	−	1.05448i
$213$		0			0
$214$	24.8769	−	18.0741i	1.70055	−	1.23552i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−19.8155	−	14.3968i	−1.34208	−	0.975075i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$223$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$224$	4.68472	+	14.4181i	0.313011	+	0.963348i
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	−11.2514	−	15.4862i	−0.748432	−	1.03013i
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$		0			0
$231$		0			0
$232$	−11.3904			−0.747817
$233$	12.4411	−	17.1237i	0.815042	−	1.12181i	−0.175484	−	0.984482i	$-0.556149\pi$
$233$	12.4411	−	17.1237i	0.815042	−	1.12181i	0.990526	−	0.137326i	$-0.0438509\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.4127	−	3.70821i	−0.738226	−	0.239864i	−0.0843185	−	0.996439i	$-0.526871\pi$
$239$	−11.4127	−	3.70821i	−0.738226	−	0.239864i	−0.653907	+	0.756575i	$0.726871\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−12.0240	+	15.2945i	−0.772935	+	0.983165i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$		−	8.95388i		−	0.564041i
$253$	−0.161049	+	8.49030i	−0.0101251	+	0.533781i
$254$	−18.2372			−1.14430
$255$		0			0
$256$	6.20507	−	19.0972i	0.387817	−	1.19358i
$257$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$257$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$258$		0			0
$259$	18.5359	+	25.5124i	1.15176	+	1.58526i
$260$		0			0
$261$	21.0743	+	6.84744i	1.30446	+	0.423846i
$262$		0			0
$263$		−	23.0900i		−	1.42379i	−0.702284	−	0.711897i	$-0.747836\pi$
$263$		−	23.0900i		−	1.42379i	0.702284	−	0.711897i	$-0.252164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−4.38077	−	13.4826i	−0.267598	−	0.823582i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$			30.1374i			1.82067i
$275$	15.6715	+	5.42265i	0.945025	+	0.326998i
$276$		0			0
$277$	0.176583	−	0.243045i	0.0106098	−	0.0146032i	−0.803679	−	0.595063i	$-0.797127\pi$
$277$	0.176583	−	0.243045i	0.0106098	−	0.0146032i	0.814289	+	0.580460i	$0.197127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.08226	+	1.48960i	0.0645620	+	0.0888620i	0.840077	−	0.542467i	$-0.182510\pi$
$281$	1.08226	+	1.48960i	0.0645620	+	0.0888620i	−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	14.6520	+	10.6453i	0.869435	+	0.631682i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−10.1040	+	13.9069i	−0.595382	+	0.819473i
$289$	−5.25329	+	16.1680i	−0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$		−	18.3806i		−	1.06835i
$297$		0			0
$298$	18.7175			1.08428
$299$		0			0
$300$		0			0
$301$	−2.26675	−	6.97635i	−0.130653	−	0.402110i
$302$	−34.9995	+	25.4287i	−2.01400	+	1.46326i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−8.11727	+	5.66549i	−0.462525	+	0.322821i
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$313$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$	18.8235	+	6.11614i	1.05891	+	0.344060i
$317$	−17.2208	−	12.5116i	−0.967215	−	0.702723i	−0.0123997	−	0.999923i	$-0.503947\pi$
$317$	−17.2208	−	12.5116i	−0.967215	−	0.702723i	−0.954815	+	0.297200i	$0.903947\pi$
$318$		0			0
$319$	−7.12691	−	23.4378i	−0.399030	−	1.31227i
$320$		0			0
$321$		0			0
$322$	−3.70233	+	11.3946i	−0.206323	+	0.634996i
$323$		0			0
$324$	8.21374	−	5.96763i	0.456319	−	0.331535i
$325$		0			0
$326$	−35.7910	+	11.6292i	−1.98228	+	0.644082i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.1571			0.888076			0.444038	−	0.896008i	$-0.353545\pi$
$331$	16.1571			0.888076			0.444038	+	0.896008i	$0.353545\pi$
$332$		0			0
$333$	−11.0496	+	34.0073i	−0.605517	+	1.86359i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	33.3558	−	10.8380i	1.81701	−	0.590381i	0.817102	−	0.576493i	$-0.195579\pi$
$337$	33.3558	−	10.8380i	1.81701	−	0.590381i	0.999904	−	0.0138879i	$-0.00442079\pi$
$338$	−21.8670	−	7.10501i	−1.18941	−	0.386462i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.8859	+	14.9832i	−0.587785	+	0.809017i
$344$	−1.32120	+	4.06624i	−0.0712345	+	0.219237i
$345$		0			0
$346$		0			0
$347$	−18.9958	−	26.1454i	−1.01975	−	1.40356i	−0.912381	−	0.409342i	$-0.865758\pi$
$347$	−18.9958	−	26.1454i	−1.01975	−	1.40356i	−0.107366	−	0.994220i	$-0.534242\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$	18.9285	+	13.7523i	1.01177	+	0.735094i
$351$		0			0
$352$	19.0007	+	0.360417i	1.01274	+	0.0192103i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−27.4435	−	37.7728i	−1.45043	−	1.99635i
$359$	−3.64987	+	1.18591i	−0.192633	+	0.0625901i	−0.403745	−	0.914872i	$-0.632292\pi$
$359$	−3.64987	+	1.18591i	−0.192633	+	0.0625901i	0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$367$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$368$	10.3230	−	7.50009i	0.538123	−	0.390969i
$369$		0			0
$370$		0			0
$371$	−36.0094	−	11.7002i	−1.86952	−	0.607443i
$372$		0			0
$373$		−	31.7490i		−	1.64390i	−0.569558	−	0.821951i	$-0.692886\pi$
$373$		−	31.7490i		−	1.64390i	0.569558	−	0.821951i	$-0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	25.4997	−	18.5267i	1.30983	−	0.951650i	0.309834	−	0.950791i	$-0.399726\pi$
$379$	25.4997	−	18.5267i	1.30983	−	0.951650i	1.00000		0.000859657i	$-0.000273637\pi$
$380$		0			0
$381$		0			0
$382$	15.2719	+	4.96214i	0.781378	+	0.253885i
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$		0			0
$385$		0			0
$386$	48.9980			2.49393
$387$	4.88892	−	6.72902i	0.248518	−	0.342055i
$388$		0			0
$389$	0.518410	+	1.59550i	0.0262844	+	0.0808951i	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	0.518410	+	1.59550i	0.0262844	+	0.0808951i	−0.937054	+	0.349185i	$0.886458\pi$
$390$		0			0
$391$		0			0
$392$	10.2664	−	3.33576i	0.518532	−	0.168481i
$393$		0			0
$394$	40.0609	+	29.1059i	2.01824	+	1.46634i
$395$		0			0
$396$	−10.6072	−	3.67032i	−0.533033	−	0.184441i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−7.70008	−	23.6984i	−0.385004	−	1.18492i
$401$	−24.7856	+	18.0078i	−1.23773	+	0.899265i	−0.997445	−	0.0714367i	$-0.977242\pi$
$401$	−24.7856	+	18.0078i	−1.23773	+	0.899265i	−0.240287	+	0.970702i	$0.577242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		−	34.5631i		−	1.71534i
$407$	37.8214	−	11.5006i	1.87474	−	0.570065i
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−12.9203	+	4.19805i	−0.634996	+	0.206323i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−6.84802	+	21.0760i	−0.333752	+	1.02718i	0.633581	+	0.773676i	$0.281584\pi$
$421$	−6.84802	+	21.0760i	−0.333752	+	1.02718i	−0.967333	+	0.253507i	$0.918416\pi$
$422$	−1.76907	−	5.44464i	−0.0861170	−	0.265041i
$423$		0			0
$424$	12.9716	+	17.8539i	0.629957	+	0.867062i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	19.6128i		−	0.948021i
$429$		0			0
$430$		0			0
$431$	−22.6942	+	31.2358i	−1.09314	+	1.50458i	−0.248963	+	0.968513i	$0.580090\pi$
$431$	−22.6942	+	31.2358i	−1.09314	+	1.50458i	−0.844177	+	0.536065i	$0.819910\pi$
$432$		0			0
$433$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$433$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$	−14.8578	+	4.82761i	−0.711562	+	0.231200i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$		0			0
$443$	−1.72788	+	5.31786i	−0.0820940	+	0.252659i	−0.983676	−	0.179949i	$-0.942407\pi$
$443$	−1.72788	+	5.31786i	−0.0820940	+	0.252659i	0.901582	+	0.432608i	$0.142407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.420335	+	0.136575i	0.0198590	+	0.00645256i
$449$	33.0711	+	24.0276i	1.56072	+	1.13393i	0.935413	+	0.353556i	$0.115028\pi$
$449$	33.0711	+	24.0276i	1.56072	+	1.13393i	0.625310	+	0.780376i	$0.284972\pi$
$450$			26.5296i			1.25062i
$451$		0			0
$452$	−12.2093			−0.574276
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.33490	−	5.96648i	−0.202778	−	0.279100i	0.695501	−	0.718525i	$-0.255182\pi$
$457$	−4.33490	−	5.96648i	−0.202778	−	0.279100i	−0.898279	+	0.439425i	$0.855182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−23.0299			−1.07029			−0.535145	−	0.844760i	$-0.679743\pi$
$463$	−23.0299			−1.07029			−0.535145	+	0.844760i	$0.679743\pi$
$464$	−21.6365	+	29.7800i	−1.00445	+	1.38250i
$465$		0			0
$466$	−11.5681	−	35.6028i	−0.535880	−	1.64927i
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$		0			0
$469$	−31.6215	+	10.2744i	−1.46015	+	0.474430i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.19371	−	0.174392i	−0.422727	−	0.00801854i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−13.2668	−	40.8309i	−0.607443	−	1.86952i
$478$	−17.1703	+	12.4750i	−0.785352	+	0.570592i
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	3.38425	+	11.9385i	0.153829	+	0.542659i
$485$		0			0
$486$		0			0
$487$	−12.7279	+	39.1724i	−0.576755	+	1.77507i	0.0533681	+	0.998575i	$0.483004\pi$
$487$	−12.7279	+	39.1724i	−0.576755	+	1.77507i	−0.630123	+	0.776495i	$0.716996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.03252	+	1.63516i	−0.227114	+	0.0737939i	−0.420363	−	0.907356i	$-0.638097\pi$
$491$	−5.03252	+	1.63516i	−0.227114	+	0.0737939i	0.193249	+	0.981150i	$0.438097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.9670	−	34.3641i	1.11992	−	1.54144i
$498$		0			0
$499$	13.8056	+	42.4894i	0.618025	+	1.90208i	0.314213	+	0.949352i	$0.398259\pi$
$499$	13.8056	+	42.4894i	0.618025	+	1.90208i	0.303812	+	0.952732i	$0.401741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$	9.90244	+	7.19455i	0.441090	+	0.320471i
$505$		0			0
$506$	11.9810	+	9.05678i	0.532620	+	0.402623i
$507$		0			0
$508$	−6.83719	+	9.41059i	−0.303351	+	0.417527i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$	−7.75016	−	10.6672i	−0.342512	−	0.471427i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	55.7742			2.45058
$519$		0			0
$520$		0			0
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$	31.7061	−	23.0358i	1.38774	−	1.00825i
$523$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$523$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$524$		0			0
$525$		0			0
$526$	−33.0386	−	24.0039i	−1.44055	−	1.04662i
$527$		0			0
$528$		0			0
$529$	−16.4444			−0.714976
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	18.4310	+	5.98858i	0.796096	+	0.258667i
$537$		0			0
$538$		0			0
$539$	13.2876	+	19.0379i	0.572336	+	0.820019i
$540$		0			0
$541$	24.7733	−	34.0976i	1.06509	−	1.46597i	0.190138	−	0.981757i	$-0.439107\pi$
$541$	24.7733	−	34.0976i	1.06509	−	1.46597i	0.874951	−	0.484211i	$-0.160893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.0976	+	4.90549i	0.645525	+	0.209744i	0.613440	−	0.789741i	$-0.289785\pi$
$547$	15.0976	+	4.90549i	0.645525	+	0.209744i	0.0320849	+	0.999485i	$0.489785\pi$
$548$	15.5513	+	11.2987i	0.664317	+	0.482655i
$549$		0			0
$550$	24.0508	−	16.7864i	1.02553	−	0.715773i
$551$		0			0
$552$		0			0
$553$	14.3445	−	44.1478i	0.609990	−	1.87736i
$554$	−0.164192	−	0.505329i	−0.00697583	−	0.0214694i
$555$		0			0
$556$		0			0
$557$	17.5720	−	5.70949i	0.744549	−	0.241919i	0.0879152	−	0.996128i	$-0.471980\pi$
$557$	17.5720	−	5.70949i	0.744549	−	0.241919i	0.656634	+	0.754209i	$0.271980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.25650			0.137367
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−13.9962	−	19.2641i	−0.587785	−	0.809017i
$568$	−23.5461	+	7.65059i	−0.987972	+	0.321011i
$569$	−40.2601	−	13.0813i	−1.68779	−	0.548397i	−0.701395	−	0.712773i	$-0.747439\pi$
$569$	−40.2601	−	13.0813i	−1.68779	−	0.548397i	−0.986398	+	0.164375i	$0.947439\pi$
$570$		0			0
$571$		−	10.9123i		−	0.456664i	−0.973583	−	0.228332i	$-0.926673\pi$
$571$		−	10.9123i		−	0.456664i	0.973583	−	0.228332i	$-0.0733271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.95601	−	12.1753i	0.164977	−	0.507747i
$576$	0.154862	+	0.476615i	0.00645256	+	0.0198590i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	17.6729	+	24.3246i	0.735094	+	1.01177i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−28.6214	+	37.8626i	−1.18538	+	1.56811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−48.0557	−	34.9145i	−1.97508	−	1.43498i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	7.01728	−	9.65845i	0.287439	−	0.395626i
$597$		0			0
$598$		0			0
$599$	36.4997	−	26.5186i	1.49134	−	1.08352i	0.517663	−	0.855584i	$-0.326802\pi$
$599$	36.4997	−	26.5186i	1.49134	−	1.08352i	0.973676	−	0.227937i	$-0.0731980\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$	−12.3386	−	4.00907i	−0.502885	−	0.163397i
$603$	−30.5004	−	22.1599i	−1.24207	−	0.902419i
$604$			27.5935i			1.12276i
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	45.6710	+	14.8394i	1.84463	+	0.599358i	0.997709	+	0.0676456i	$0.0215487\pi$
$613$	45.6710	+	14.8394i	1.84463	+	0.599358i	0.846925	+	0.531712i	$0.178451\pi$
$614$		0			0
$615$		0			0
$616$	0.256636	−	13.5295i	0.0103401	−	0.545119i
$617$	−3.84770			−0.154902			−0.0774512	−	0.996996i	$-0.524678\pi$
$617$	−3.84770			−0.154902			−0.0774512	+	0.996996i	$0.524678\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	15.5241	+	47.7782i	0.618003	+	1.90202i	0.318475	+	0.947931i	$0.396829\pi$
$631$	15.5241	+	47.7782i	0.618003	+	1.90202i	0.299528	+	0.954087i	$0.403171\pi$
$632$	−21.8890	+	15.9033i	−0.870698	+	0.632599i
$633$		0			0
$634$	−35.8047	+	11.6337i	−1.42199	+	0.462032i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−40.9452	−	14.1679i	−1.62104	−	0.560913i
$639$	48.1636			1.90532
$640$		0			0
$641$	−7.65550	+	23.5612i	−0.302374	+	0.930611i	0.678270	+	0.734813i	$0.262730\pi$
$641$	−7.65550	+	23.5612i	−0.302374	+	0.930611i	−0.980644	+	0.195799i	$0.937270\pi$
$642$		0			0
$643$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$643$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$644$	4.49173	+	6.18233i	0.176999	+	0.243618i
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$			13.8790i			0.545217i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−7.41740	+	22.8284i	−0.290488	+	0.894030i
$653$	−7.88484	−	24.2670i	−0.308558	−	0.949643i	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−7.88484	−	24.2670i	−0.308558	−	0.949643i	0.669768	−	0.742571i	$-0.266394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	16.7966	−	23.1186i	0.652819	−	0.898529i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	37.1727	+	51.1638i	1.44041	+	1.98256i
$667$	−17.9860	+	5.84402i	−0.696423	+	0.226281i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9.76533	−	13.4408i	0.376426	−	0.518106i	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	9.76533	−	13.4408i	0.376426	−	0.518106i	0.954633	+	0.297784i	$0.0962476\pi$
$674$	19.1684	−	58.9943i	0.738340	−	2.27238i
$675$		0			0
$676$	−11.8643	+	8.61991i	−0.456319	+	0.331535i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.0364			−0.422295			−0.211147	−	0.977454i	$-0.567720\pi$
$683$	−11.0364			−0.422295			−0.211147	+	0.977454i	$0.567720\pi$
$684$		0			0
$685$		0			0
$686$	10.1221	+	31.1525i	0.386462	+	1.18941i
$687$		0			0
$688$	8.12146	+	11.1782i	0.309628	+	0.426166i
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$	−8.60820	+	24.8777i	−0.326998	+	0.945025i
$694$	−57.1581			−2.16969
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	14.1927	−	4.61150i	0.536435	−	0.174298i
$701$	13.7423	+	4.46515i	0.519040	+	0.168646i	0.556810	−	0.830640i	$-0.312025\pi$
$701$	13.7423	+	4.46515i	0.519040	+	0.168646i	−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		0			0
$704$	0.334095	−	0.441966i	0.0125917	−	0.0166572i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	29.0896	−	21.1349i	1.09248	−	0.793736i	0.112667	−	0.993633i	$-0.464061\pi$
$709$	29.0896	−	21.1349i	1.09248	−	0.793736i	0.979817	+	0.199896i	$0.0640606\pi$
$710$		0			0
$711$	50.0589	−	16.2651i	1.87736	−	0.609990i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−29.7799			−1.11293
$717$		0			0
$718$	−2.09745	+	6.45530i	−0.0782762	+	0.240910i
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$		0			0
$722$	31.9594	−	10.3842i	1.18941	−	0.386462i
$723$		0			0
$724$		0			0
$725$			36.9313i			1.37159i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	8.34346	−	25.6785i	0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$		0			0
$736$		−	14.6709i		−	0.540777i
$737$	−0.790461	+	41.6721i	−0.0291170	+	1.53501i
$738$		0			0
$739$	−25.5143	+	35.1175i	−0.938560	+	1.29182i	0.0178655	+	0.999840i	$0.494313\pi$
$739$	−25.5143	+	35.1175i	−0.938560	+	1.29182i	−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		0			0
$742$	−54.1760	+	39.3612i	−1.98886	+	1.44499i
$743$	−3.79415	−	5.22220i	−0.139194	−	0.191584i	0.733729	−	0.679442i	$-0.237778\pi$
$743$	−3.79415	−	5.22220i	−0.139194	−	0.191584i	−0.872923	+	0.487858i	$0.837778\pi$
$744$		0			0
$745$		0			0
$746$	−45.4284	−	33.0056i	−1.66325	−	1.20842i
$747$		0			0
$748$		0			0
$749$	−45.9989			−1.68076
$750$		0			0
$751$	12.3593	−	38.0379i	0.450996	−	1.38802i	−0.424777	−	0.905298i	$-0.639647\pi$
$751$	12.3593	−	38.0379i	0.450996	−	1.38802i	0.875772	−	0.482724i	$-0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	40.3760	+	29.3349i	1.46749	+	1.06619i	0.981332	+	0.192323i	$0.0616021\pi$
$757$	40.3760	+	29.3349i	1.46749	+	1.06619i	0.486158	+	0.873871i	$0.338398\pi$
$758$		−	55.7465i		−	2.02480i
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$	11.3224	+	34.8469i	0.409900	+	1.26154i
$764$	8.28602	−	6.02015i	0.299778	−	0.217801i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	18.3696	−	25.2835i	0.661135	−	0.909974i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$	−4.54585	−	13.9907i	−0.163397	−	0.502885i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	2.82187	+	0.916880i	0.101169	+	0.0328717i
$779$		0			0
$780$		0			0
$781$	−30.4751	−	43.6635i	−1.09049	−	1.56240i
$782$		0			0
$783$		0			0
$784$	10.7801	−	33.1778i	0.385004	−	1.18492i
$785$		0			0
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$	30.0380	−	9.75994i	1.07006	−	0.347683i
$789$		0			0
$790$		0			0
$791$			28.6351i			1.01815i
$792$	12.5822	−	8.78180i	0.447088	−	0.312048i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−27.2476	−	8.85328i	−0.963348	−	0.313011i
$801$		0			0
$802$			54.1852i			1.91334i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28.9316	−	39.8210i	−1.01718	−	1.40003i	−0.914160	−	0.405353i	$-0.867149\pi$
$809$	−28.9316	−	39.8210i	−1.01718	−	1.40003i	−0.103022	−	0.994679i	$-0.532851\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$	−17.8350	−	12.9579i	−0.625884	−	0.454732i
$813$		0			0
$814$	22.8626	−	66.0729i	0.801334	−	2.31585i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	50.3252	−	16.3516i	1.75636	−	0.570676i	0.759548	−	0.650451i	$-0.225420\pi$
$821$	50.3252	−	16.3516i	1.75636	−	0.570676i	0.996813	+	0.0797750i	$0.0254202\pi$
$822$		0			0
$823$	−1.70206	−	1.23662i	−0.0593301	−	0.0431058i	0.557725	−	0.830026i	$-0.311674\pi$
$823$	−1.70206	−	1.23662i	−0.0593301	−	0.0431058i	−0.617055	+	0.786920i	$0.711674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.7719	+	29.9664i	−0.757082	+	1.04203i	0.240369	+	0.970682i	$0.422732\pi$
$827$	−21.7719	+	29.9664i	−0.757082	+	1.04203i	−0.997451	+	0.0713526i	$0.977268\pi$
$828$	−2.67762	+	8.24087i	−0.0930538	+	0.286390i
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	20.6759	−	15.0219i	0.712963	−	0.517998i
$842$	23.0378	+	31.7088i	0.793934	+	1.09276i
$843$		0			0
$844$	−3.47273	−	1.12836i	−0.119536	−	0.0388397i
$845$		0			0
$846$		0			0
$847$	28.0000	−	7.93725i	0.962091	−	0.272727i
$848$	71.3188			2.44910
$849$		0			0
$850$		0			0
$851$	−9.43044	−	29.0239i	−0.323271	−	0.994927i
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	21.6906	+	15.7591i	0.741368	+	0.538636i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	21.1017	+	64.9443i	0.718726	+	2.21201i
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	−0.692468	−	0.721449i	$-0.743477\pi$
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	0.472154	+	0.881516i	$0.343477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−46.4198	−	35.0901i	−1.57468	−	1.19035i
$870$		0			0
$871$		0			0
$872$	6.59942	−	20.3109i	0.223484	−	0.687814i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−3.52249	−	1.14453i	−0.118946	−	0.0386480i	0.248939	−	0.968519i	$-0.419918\pi$
$877$	−3.52249	−	1.14453i	−0.118946	−	0.0386480i	−0.367885	+	0.929871i	$0.619918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−21.8312	+	30.0480i	−0.735094	+	1.01177i
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	−0.993863	−	0.110619i	$-0.964717\pi$
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	0.869072	+	0.494686i	$0.164717\pi$
$884$		0			0
$885$		0			0
$886$	5.81284	+	8.00069i	0.195286	+	0.268788i
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$	22.0711	+	16.0356i	0.740242	+	0.537818i
$890$		0			0
$891$	−28.5585	+	8.68399i	−0.956746	+	0.290924i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−23.8971	+	17.3623i	−0.798346	+	0.580032i
$897$		0			0
$898$	68.7602	−	22.3415i	2.29456	−	0.745547i
$899$		0			0
$900$	13.6896	+	9.94605i	0.456319	+	0.331535i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	9.81030	−	13.5027i	0.326286	−	0.449094i
$905$		0			0
$906$		0			0
$907$	−41.7868	+	30.3599i	−1.38751	+	1.00808i	−0.391373	+	0.920232i	$0.628000\pi$
$907$	−41.7868	+	30.3599i	−1.38751	+	1.00808i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.9443	+	9.40456i	0.428863	+	0.311587i	0.781194	−	0.624288i	$-0.214611\pi$
$911$	12.9443	+	9.40456i	0.428863	+	0.311587i	−0.352331	+	0.935875i	$0.614611\pi$
$912$		0			0
$913$		0			0
$914$	−13.0437			−0.431446
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−34.1974	−	47.0687i	−1.12807	−	1.55265i	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−34.1974	−	47.0687i	−1.12807	−	1.55265i	−0.336381	−	0.941726i	$-0.609203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−59.5957			−1.95949
$926$	−23.9414	+	32.9525i	−0.786763	+	1.08289i
$927$		0			0
$928$	13.0785	+	40.2516i	0.429324	+	1.32132i
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$	−22.7084	−	7.37840i	−0.743838	−	0.241688i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$	−18.1718	+	55.9270i	−0.593330	+	1.82608i
$939$		0			0
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−9.80713	+	12.9736i	−0.318857	+	0.421808i
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−46.2410	+	15.0246i	−1.49789	+	0.486694i	−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−46.2410	+	15.0246i	−1.49789	+	0.486694i	−0.558489	+	0.829512i	$0.688619\pi$
$954$	−72.2150	−	23.4641i	−2.33805	−	0.759678i
$955$		0			0
$956$			13.5370i			0.437818i
$957$		0			0
$958$		0			0
$959$	26.4993	−	36.4732i	0.855708	−	1.17778i
$960$		0			0
$961$	9.57953	+	29.4828i	0.309017	+	0.951057i
$962$		0			0
$963$	−30.6576	−	42.1966i	−0.987929	−	1.35977i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	52.7587i		−	1.69661i	−0.529511	−	0.848303i	$-0.677624\pi$
$967$		−	52.7587i		−	1.69661i	0.529511	−	0.848303i	$-0.322376\pi$
$968$	−15.9225	−	5.84996i	−0.511770	−	0.188025i
$969$		0			0
$970$		0			0
$971$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$971$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$972$		0			0
$973$		0			0
$974$	42.8185	+	58.9346i	1.37199	+	1.88839i
$975$		0			0
$976$		0			0
$977$	44.0711	+	32.0196i	1.40996	+	1.02440i	0.993328	+	0.115321i	$0.0367898\pi$
$977$	44.0711	+	32.0196i	1.40996	+	1.02440i	0.416632	+	0.909075i	$0.363210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−24.4202	+	33.6115i	−0.779676	+	1.07313i
$982$	−2.89201	+	8.90071i	−0.0922879	+	0.284033i
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7.09868i			0.225725i
$990$		0			0
$991$	24.0000			0.762385			0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000			0.762385			0.381193	+	0.924496i	$0.375513\pi$
$992$		0			0
$993$		0			0
$994$	−23.2150	−	71.4484i	−0.736334	−	2.26620i
$995$		0			0
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	75.1483	+	24.4172i	2.37878	+	0.772912i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.l.a.62.2	yes	8
3.2	odd	2		693.2.bu.a.370.1		8
7.2	even	3		539.2.s.a.227.1		16
7.3	odd	6		539.2.s.a.117.2		16
7.4	even	3		539.2.s.a.117.2		16
7.5	odd	6		539.2.s.a.227.1		16
7.6	odd	2	CM	77.2.l.a.62.2	yes	8
11.2	odd	10		847.2.l.d.699.2		8
11.3	even	5		847.2.l.c.118.1		8
11.4	even	5		847.2.l.d.475.2		8
11.5	even	5		847.2.b.b.846.7		8
11.6	odd	10		847.2.b.b.846.2		8
11.7	odd	10		847.2.l.a.475.1		8
11.8	odd	10	inner	77.2.l.a.41.2	✓	8
11.9	even	5		847.2.l.a.699.1		8
11.10	odd	2		847.2.l.c.524.1		8
21.20	even	2		693.2.bu.a.370.1		8
33.8	even	10		693.2.bu.a.118.1		8
77.6	even	10		847.2.b.b.846.2		8
77.13	even	10		847.2.l.d.699.2		8
77.19	even	30		539.2.s.a.129.2		16
77.20	odd	10		847.2.l.a.699.1		8
77.27	odd	10		847.2.b.b.846.7		8
77.30	odd	30		539.2.s.a.129.2		16
77.41	even	10	inner	77.2.l.a.41.2	✓	8
77.48	odd	10		847.2.l.d.475.2		8
77.52	even	30		539.2.s.a.19.1		16
77.62	even	10		847.2.l.a.475.1		8
77.69	odd	10		847.2.l.c.118.1		8
77.74	odd	30		539.2.s.a.19.1		16
77.76	even	2		847.2.l.c.524.1		8
231.41	odd	10		693.2.bu.a.118.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.41.2	✓	8	11.8	odd	10	inner
77.2.l.a.41.2	✓	8	77.41	even	10	inner
77.2.l.a.62.2	yes	8	1.1	even	1	trivial
77.2.l.a.62.2	yes	8	7.6	odd	2	CM
539.2.s.a.19.1		16	77.52	even	30
539.2.s.a.19.1		16	77.74	odd	30
539.2.s.a.117.2		16	7.3	odd	6
539.2.s.a.117.2		16	7.4	even	3
539.2.s.a.129.2		16	77.19	even	30
539.2.s.a.129.2		16	77.30	odd	30
539.2.s.a.227.1		16	7.2	even	3
539.2.s.a.227.1		16	7.5	odd	6
693.2.bu.a.118.1		8	33.8	even	10
693.2.bu.a.118.1		8	231.41	odd	10
693.2.bu.a.370.1		8	3.2	odd	2
693.2.bu.a.370.1		8	21.20	even	2
847.2.b.b.846.2		8	11.6	odd	10
847.2.b.b.846.2		8	77.6	even	10
847.2.b.b.846.7		8	11.5	even	5
847.2.b.b.846.7		8	77.27	odd	10
847.2.l.a.475.1		8	11.7	odd	10
847.2.l.a.475.1		8	77.62	even	10
847.2.l.a.699.1		8	11.9	even	5
847.2.l.a.699.1		8	77.20	odd	10
847.2.l.c.118.1		8	11.3	even	5
847.2.l.c.118.1		8	77.69	odd	10
847.2.l.c.524.1		8	11.10	odd	2
847.2.l.c.524.1		8	77.76	even	2
847.2.l.d.475.2		8	11.4	even	5
847.2.l.d.475.2		8	77.48	odd	10
847.2.l.d.699.2		8	11.2	odd	10
847.2.l.d.699.2		8	77.13	even	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.l (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.03958 - 1.43086i) q^{2} +(-0.348597 - 1.07287i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(1.46663 + 0.476537i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)	\(q+(1.03958 - 1.43086i) q^{2} +(-0.348597 - 1.07287i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(1.46663 + 0.476537i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(-0.0629004 + 3.31603i) q^{11} +(-1.44601 + 4.45035i) q^{14} +(4.03181 - 2.92928i) q^{16} +(-5.04623 + 1.63962i) q^{18} +(4.67938 + 3.53728i) q^{22} +2.56038 q^{23} +(1.54508 - 4.75528i) q^{25} +(1.75432 + 2.41461i) q^{28} +(-7.02475 + 2.28248i) q^{29} -5.72996i q^{32} +(-1.04579 + 3.21861i) q^{36} +(-3.68322 - 11.3358i) q^{37} +2.77251i q^{43} +(3.57960 - 1.08847i) q^{44} +(2.66172 - 3.66355i) q^{46} +(5.66312 - 4.11450i) q^{49} +(-5.19790 - 7.15429i) q^{50} +(11.5776 + 8.41162i) q^{53} -4.08003 q^{56} +(-4.03688 + 12.4242i) q^{58} +(7.54878 + 2.45275i) q^{63} +(-0.135144 - 0.0981881i) q^{64} +12.5669 q^{67} +(-12.9884 + 9.43662i) q^{71} +(-2.71928 - 3.74277i) q^{72} +(-20.0489 - 6.51428i) q^{74} +(-2.55285 - 8.39541i) q^{77} +(-10.3127 + 14.1942i) q^{79} +(2.78115 + 8.55951i) q^{81} +(3.96707 + 2.88224i) q^{86} +(-1.67246 + 4.83341i) q^{88} +(-0.892541 - 2.74696i) q^{92} -12.3805i q^{98} +(6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9}+O(q^{10}) \) 8 * q + 10 * q^4 - 10 * q^8 - 6 * q^9	\( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9} - 4 q^{11} - 21 q^{14} + 8 q^{16} - 15 q^{18} - 14 q^{22} + 16 q^{23} - 10 q^{25} + 35 q^{28} + 30 q^{36} - 18 q^{37} + 25 q^{44} + 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} - 75 q^{72} - 14 q^{77} - 40 q^{79} - 18 q^{81} + 23 q^{86} - 8 q^{88} + 25 q^{92} + 48 q^{99}+O(q^{100}) \) 8 * q + 10 * q^4 - 10 * q^8 - 6 * q^9 - 4 * q^11 - 21 * q^14 + 8 * q^16 - 15 * q^18 - 14 * q^22 + 16 * q^23 - 10 * q^25 + 35 * q^28 + 30 * q^36 - 18 * q^37 + 25 * q^44 + 15 * q^46 + 14 * q^49 + 30 * q^53 - 42 * q^56 + 19 * q^58 - 34 * q^64 + 8 * q^67 - 48 * q^71 - 75 * q^72 - 14 * q^77 - 40 * q^79 - 18 * q^81 + 23 * q^86 - 8 * q^88 + 25 * q^92 + 48 * q^99

Introduction

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