LMFDB - Embedded newform 77.2.l.a.41.1

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			41.1
Root			$-0.373058 - 1.36412i$ of defining polynomial
Character	$\chi$	$=$	77.41
Dual form			77.2.l.a.62.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0784543 + 0.107983i) q^{2} +(0.612529 - 1.88517i) q^{4} +(2.51626 + 0.817582i) q^{7} +(0.505505 - 0.164249i) q^{8} +(-2.42705 + 1.76336i) q^{9} +O(q^{10})$	\(q+(0.0784543 + 0.107983i) q^{2} +(0.612529 - 1.88517i) q^{4} +(2.51626 + 0.817582i) q^{7} +(0.505505 - 0.164249i) q^{8} +(-2.42705 + 1.76336i) q^{9} +(-3.17317 + 0.964887i) q^{11} +(0.109126 + 0.335856i) q^{14} +(-3.14985 - 2.28850i) q^{16} +(-0.380825 - 0.123738i) q^{18} +(-0.353140 - 0.266949i) q^{22} -7.50465 q^{23} +(1.54508 + 4.75528i) q^{25} +(3.08256 - 4.24278i) q^{28} +(9.26082 + 3.00902i) q^{29} -1.58271i q^{32} +(1.83759 + 5.65551i) q^{36} +(2.53732 - 7.80906i) q^{37} -11.3343i q^{43} +(-0.124680 + 6.57298i) q^{44} +(-0.588772 - 0.810375i) q^{46} +(5.66312 + 4.11450i) q^{49} +(-0.392271 + 0.539915i) q^{50} +(1.51257 - 1.09894i) q^{53} +1.40627 q^{56} +(0.401627 + 1.23608i) q^{58} +(-7.54878 + 2.45275i) q^{63} +(-6.12879 + 4.45282i) q^{64} -6.09473 q^{67} +(-7.95588 - 5.78028i) q^{71} +(-0.937258 + 1.29003i) q^{72} +(1.04231 - 0.338667i) q^{74} +(-8.77339 - 0.166419i) q^{77} +(4.78485 + 6.58577i) q^{79} +(2.78115 - 8.55951i) q^{81} +(1.22392 - 0.889227i) q^{86} +(-1.44557 + 1.00894i) q^{88} +(-4.59682 + 14.1475i) q^{92} +0.934321i q^{98} +(6.00000 - 7.93725i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-1$	$e\left(\frac{3}{10}\right)$

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 $

$2$	0.0784543	+	0.107983i	0.0554756	+	0.0763556i	0.835853	−	0.548953i	$-0.184973\pi$
$2$	0.0784543	+	0.107983i	0.0554756	+	0.0763556i	−0.780378	+	0.625308i	$0.784973\pi$
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	0.612529	−	1.88517i	0.306264	−	0.942585i
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$		0			0
$7$	2.51626	+	0.817582i	0.951057	+	0.309017i
$7$	2.51626	+	0.817582i	0.951057	+	0.309017i
$8$	0.505505	−	0.164249i	0.178723	−	0.0580707i
$9$	−2.42705	+	1.76336i	−0.809017	+	0.587785i
$10$		0			0
$11$	−3.17317	+	0.964887i	−0.956746	+	0.290924i
$11$	−3.17317	+	0.964887i	−0.956746	+	0.290924i
$12$		0			0
$13$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$13$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$14$	0.109126	+	0.335856i	0.0291652	+	0.0897613i
$15$		0			0
$16$	−3.14985	−	2.28850i	−0.787462	−	0.572124i
$17$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$17$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$18$	−0.380825	−	0.123738i	−0.0897613	−	0.0291652i
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$		0			0
$21$		0			0
$22$	−0.353140	−	0.266949i	−0.0752897	−	0.0569137i
$23$	−7.50465			−1.56483			−0.782414	−	0.622758i	$-0.786012\pi$
$23$	−7.50465			−1.56483			−0.782414	+	0.622758i	$0.786012\pi$
$24$		0			0
$25$	1.54508	+	4.75528i	0.309017	+	0.951057i
$26$		0			0
$27$		0			0
$28$	3.08256	−	4.24278i	0.582549	−	0.801811i
$29$	9.26082	+	3.00902i	1.71969	+	0.558761i	0.991898	−	0.127036i	$-0.0405463\pi$
$29$	9.26082	+	3.00902i	1.71969	+	0.558761i	0.727793	+	0.685797i	$0.240546\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$		−	1.58271i		−	0.279787i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.83759	+	5.65551i	0.306264	+	0.942585i
$37$	2.53732	−	7.80906i	0.417133	−	1.28380i	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	2.53732	−	7.80906i	0.417133	−	1.28380i	0.910330	−	0.413884i	$-0.135828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$41$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$		−	11.3343i		−	1.72847i	−0.503088	−	0.864235i	$-0.667803\pi$
$43$		−	11.3343i		−	1.72847i	0.503088	−	0.864235i	$-0.332197\pi$
$44$	−0.124680	+	6.57298i	−0.0187962	+	0.990914i
$45$		0			0
$46$	−0.588772	−	0.810375i	−0.0868097	−	0.119483i
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$		0			0
$49$	5.66312	+	4.11450i	0.809017	+	0.587785i
$50$	−0.392271	+	0.539915i	−0.0554756	+	0.0763556i
$51$		0			0
$52$		0			0
$53$	1.51257	−	1.09894i	0.207767	−	0.150952i	−0.479036	−	0.877795i	$-0.659014\pi$
$53$	1.51257	−	1.09894i	0.207767	−	0.150952i	0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$		0			0
$56$	1.40627			0.187921
$57$		0			0
$58$	0.401627	+	1.23608i	0.0527363	+	0.162306i
$59$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$59$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$60$		0			0
$61$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$61$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$62$		0			0
$63$	−7.54878	+	2.45275i	−0.951057	+	0.309017i
$64$	−6.12879	+	4.45282i	−0.766098	+	0.556603i
$65$		0			0
$66$		0			0
$67$	−6.09473			−0.744590			−0.372295	−	0.928114i	$-0.621429\pi$
$67$	−6.09473			−0.744590			−0.372295	+	0.928114i	$0.621429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.95588	−	5.78028i	−0.944189	−	0.685993i	0.00523645	−	0.999986i	$-0.498333\pi$
$71$	−7.95588	−	5.78028i	−0.944189	−	0.685993i	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−0.937258	+	1.29003i	−0.110457	+	0.152031i
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$	1.04231	−	0.338667i	0.121166	−	0.0393692i
$75$		0			0
$76$		0			0
$77$	−8.77339	−	0.166419i	−0.999820	−	0.0189652i
$78$		0			0
$79$	4.78485	+	6.58577i	0.538337	+	0.740957i	0.988372	−	0.152053i	$-0.0485886\pi$
$79$	4.78485	+	6.58577i	0.538337	+	0.740957i	−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	2.78115	−	8.55951i	0.309017	−	0.951057i
$82$		0			0
$83$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$83$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$84$		0			0
$85$		0			0
$86$	1.22392	−	0.889227i	0.131978	−	0.0958879i
$87$		0			0
$88$	−1.44557	+	1.00894i	−0.154098	+	0.107554i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−4.59682	+	14.1475i	−0.479251	+	1.47498i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$			0.934321i			0.0943807i
$99$	6.00000	−	7.93725i	0.603023	−	0.797724i
$100$	9.91092			0.991092
$101$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$101$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$102$		0			0
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$		0			0
$105$		0			0
$106$	0.237335	+	0.0771147i	0.0230520	+	0.00749004i
$107$	19.6453	−	6.38315i	1.89918	−	0.617082i	0.932447	−	0.361308i	$-0.117670\pi$
$107$	19.6453	−	6.38315i	1.89918	−	0.617082i	0.966736	−	0.255774i	$-0.0823304\pi$
$108$		0			0
$109$			20.3893i			1.95295i	0.215642	+	0.976473i	$0.430816\pi$
$109$			20.3893i			1.95295i	−0.215642	+	0.976473i	$0.569184\pi$
$110$		0			0
$111$		0			0
$112$	−6.05480	−	8.33371i	−0.572124	−	0.787462i
$113$	−4.34450	−	13.3710i	−0.408696	−	1.25784i	−0.917769	−	0.397114i	$-0.870012\pi$
$113$	−4.34450	−	13.3710i	−0.408696	−	1.25784i	0.509073	−	0.860724i	$-0.329988\pi$
$114$		0			0
$115$		0			0
$116$	11.3450	−	15.6151i	1.05336	−	1.44983i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.13799	−	6.12350i	0.830726	−	0.556682i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.857089	−	0.622712i	−0.0763556	−	0.0554756i
$127$	−11.8276	+	16.2794i	−1.04953	+	1.44456i	−0.160322	+	0.987065i	$0.551253\pi$
$127$	−11.8276	+	16.2794i	−1.04953	+	1.44456i	−0.889212	+	0.457495i	$0.848747\pi$
$128$	−3.97216	−	1.29063i	−0.351092	−	0.114077i
$129$		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
$134$	−0.478158	−	0.658128i	−0.0413066	−	0.0568536i
$135$		0			0
$136$		0			0
$137$	18.7856	+	13.6485i	1.60496	+	1.16607i	0.877051	+	0.480397i	$0.159507\pi$
$137$	18.7856	+	13.6485i	1.60496	+	1.16607i	0.727909	+	0.685674i	$0.240493\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$		−	1.31259i		−	0.110150i
$143$		0			0
$144$	11.6803			0.973356
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−13.1672	−	9.56655i	−1.08234	−	0.786366i
$149$	−6.22053	+	8.56183i	−0.509606	+	0.701413i	−0.983853	−	0.178979i	$-0.942720\pi$
$149$	−6.22053	+	8.56183i	−0.509606	+	0.701413i	0.474247	+	0.880392i	$0.342720\pi$
$150$		0			0
$151$	−20.1531	+	6.54813i	−1.64003	+	0.532879i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−20.1531	+	6.54813i	−1.64003	+	0.532879i	−0.663487	+	0.748187i	$0.730924\pi$
$152$		0			0
$153$		0			0
$154$	−0.670339	−	0.960433i	−0.0540175	−	0.0773939i
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$	−0.335760	+	1.03336i	−0.0267117	+	0.0822101i
$159$		0			0
$160$		0			0
$161$	−18.8837	−	6.13567i	−1.48824	−	0.483559i
$162$	1.14248	−	0.371213i	0.0897613	−	0.0291652i
$163$	7.21418	−	5.24141i	0.565058	−	0.410539i	−0.268249	−	0.963350i	$-0.586445\pi$
$163$	7.21418	−	5.24141i	0.565058	−	0.410539i	0.833307	+	0.552811i	$0.186445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$167$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$168$		0			0
$169$	−4.01722	+	12.3637i	−0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$	−21.3672	−	6.94261i	−1.62923	−	0.529369i
$173$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$173$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$	12.2031	+	4.22254i	0.919846	+	0.318286i
$177$		0			0
$178$		0			0
$179$	−7.39370	−	22.7555i	−0.552631	−	1.70082i	−0.702118	−	0.712060i	$-0.747762\pi$
$179$	−7.39370	−	22.7555i	−0.552631	−	1.70082i	0.149487	−	0.988764i	$-0.452238\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$	−3.79364	+	1.23263i	−0.279671	+	0.0908706i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.80563	+	20.9456i	−0.492438	+	1.51557i	0.328474	+	0.944513i	$0.393466\pi$
$191$	−6.80563	+	20.9456i	−0.492438	+	1.51557i	−0.820912	+	0.571055i	$0.806534\pi$
$192$		0			0
$193$	−3.84619	+	5.29383i	−0.276855	+	0.381058i	−0.924689	−	0.380724i	$-0.875675\pi$
$193$	−3.84619	+	5.29383i	−0.276855	+	0.381058i	0.647834	+	0.761781i	$0.275675\pi$
$194$		0			0
$195$		0			0
$196$	11.2253	−	8.15569i	0.801811	−	0.582549i
$197$		−	21.4571i		−	1.52876i	−0.644767	−	0.764379i	$-0.723046\pi$
$197$		−	21.4571i		−	1.52876i	0.644767	−	0.764379i	$-0.276954\pi$
$198$	1.32781	+	0.0251868i	0.0943637	+	0.00178995i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.56210	+	2.15004i	0.110457	+	0.152031i
$201$		0			0
$202$		0			0
$203$	20.8425	+	15.1430i	1.46286	+	1.06283i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	18.2142	−	13.2334i	1.26597	−	0.919783i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	11.5138	+	15.8474i	0.792645	+	1.09098i	0.993774	+	0.111417i	$0.0355390\pi$
$211$	11.5138	+	15.8474i	0.792645	+	1.09098i	−0.201129	+	0.979565i	$0.564461\pi$
$212$	−1.14521	−	3.52458i	−0.0786530	−	0.242069i
$213$		0			0
$214$	2.23053	+	1.62057i	0.152476	+	0.110780i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.20170	+	1.59963i	−0.149118	+	0.108341i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$	1.29400	−	3.98251i	0.0864588	−	0.266093i
$225$	−12.1353	−	8.81678i	−0.809017	−	0.587785i
$226$	1.10300	−	1.51814i	0.0733703	−	0.100985i
$227$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$227$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$	5.17562			0.339796
$233$	−12.4411	−	17.1237i	−0.815042	−	1.12181i	−0.990526	−	0.137326i	$-0.956149\pi$
$233$	−12.4411	−	17.1237i	−0.815042	−	1.12181i	0.175484	−	0.984482i	$-0.443851\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29.3012	−	9.52055i	1.89534	−	0.615833i	0.921614	−	0.388108i	$-0.126871\pi$
$239$	29.3012	−	9.52055i	1.89534	−	0.615833i	0.973726	−	0.227725i	$-0.0731287\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	1.37815	+	0.506333i	0.0885907	+	0.0325483i
$243$		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	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$			15.7331i			0.991092i
$253$	23.8135	−	7.24115i	1.49714	−	0.455247i
$254$	−2.68582			−0.168524
$255$		0			0
$256$	4.50971	+	13.8795i	0.281857	+	0.867466i
$257$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$257$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$258$		0			0
$259$	12.7691	−	17.5752i	0.793433	−	1.09207i
$260$		0			0
$261$	−27.7825	+	9.02707i	−1.71969	+	0.558761i
$262$		0			0
$263$			14.5282i			0.895848i	0.894072	+	0.447924i	$0.147836\pi$
$263$			14.5282i			0.895848i	−0.894072	+	0.447924i	$0.852164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3.73320	+	11.4896i	−0.228041	+	0.701839i
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$			3.09931i			0.187236i
$275$	−9.49113	−	13.5985i	−0.572336	−	0.820019i
$276$		0			0
$277$	−11.3569	−	15.6315i	−0.682371	−	0.939204i	0.317588	−	0.948229i	$-0.397127\pi$
$277$	−11.3569	−	15.6315i	−0.682371	−	0.939204i	−0.999959	+	0.00902525i	$0.997127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.0478	+	26.2171i	−1.13630	+	1.56398i	−0.360782	+	0.932650i	$0.617490\pi$
$281$	−19.0478	+	26.2171i	−1.13630	+	1.56398i	−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$283$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$284$	−15.7700	+	11.4576i	−0.935778	+	0.679883i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.79088	+	3.84132i	0.164454	+	0.226352i
$289$	−5.25329	−	16.1680i	−0.309017	−	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$293$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$		0			0
$296$		−	4.36427i		−	0.253668i
$297$		0			0
$298$	−1.41256			−0.0818274
$299$		0			0
$300$		0			0
$301$	9.26675	−	28.5201i	0.534127	−	1.64387i
$302$	−2.28818	−	1.66246i	−0.131670	−	0.0956639i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−5.68768	+	16.4374i	−0.324086	+	0.936607i
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$313$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$314$		0			0
$315$		0			0
$316$	15.3462	−	4.98627i	0.863289	−	0.280500i
$317$	−27.2858	+	19.8243i	−1.53252	+	1.11344i	−0.577708	+	0.816243i	$0.696053\pi$
$317$	−27.2858	+	19.8243i	−1.53252	+	1.11344i	−0.954815	+	0.297200i	$0.903947\pi$
$318$		0			0
$319$	−32.2895	−	0.612486i	−1.80786	−	0.0342927i
$320$		0			0
$321$		0			0
$322$	−0.818955	−	2.52048i	−0.0456386	−	0.140461i
$323$		0			0
$324$	−14.4326	−	10.4859i	−0.801811	−	0.582549i
$325$		0			0
$326$	1.13197	+	0.367798i	0.0626938	+	0.0203705i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.09209			0.334852			0.167426	−	0.985885i	$-0.446455\pi$
$331$	6.09209			0.334852			0.167426	+	0.985885i	$0.446455\pi$
$332$		0			0
$333$	7.61195	+	23.4272i	0.417133	+	1.28380i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.9147	+	6.79560i	1.13930	+	0.370180i	0.817102	−	0.576493i	$-0.195579\pi$
$337$	20.9147	+	6.79560i	1.13930	+	0.370180i	0.322195	+	0.946673i	$0.395579\pi$
$338$	−1.65024	+	0.536196i	−0.0897613	+	0.0291652i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.8859	+	14.9832i	0.587785	+	0.809017i
$344$	−1.86165	−	5.72957i	−0.100373	−	0.308918i
$345$		0			0
$346$		0			0
$347$	16.2318	−	22.3412i	0.871371	−	1.19934i	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	16.2318	−	22.3412i	0.871371	−	1.19934i	0.978737	−	0.205120i	$-0.0657585\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$	−1.42848	+	1.03785i	−0.0763556	+	0.0554756i
$351$		0			0
$352$	1.52714	+	5.02221i	0.0813968	+	0.267685i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.87714	−	2.58366i	0.0992098	−	0.136551i
$359$	18.1220	+	5.88820i	0.956443	+	0.310767i	0.745331	−	0.666695i	$-0.232292\pi$
$359$	18.1220	+	5.88820i	0.956443	+	0.310767i	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	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$	23.6385	+	17.1744i	1.23224	+	0.895277i
$369$		0			0
$370$		0			0
$371$	4.70449	−	1.52858i	0.244245	−	0.0793599i
$372$		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
$379$	−31.4997	−	22.8859i	−1.61803	−	1.17557i	−0.809522	−	0.587090i	$-0.800274\pi$
$379$	−31.4997	−	22.8859i	−1.61803	−	1.17557i	−0.808511	−	0.588481i	$-0.799726\pi$
$380$		0			0
$381$		0			0
$382$	−2.79570	+	0.908377i	−0.143040	+	0.0464766i
$383$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$383$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$384$		0			0
$385$		0			0
$386$	−0.873393			−0.0444546
$387$	19.9865	+	27.5090i	1.01597	+	1.39836i
$388$		0			0
$389$	6.73894	−	20.7403i	0.341678	−	1.05158i	−0.621660	−	0.783287i	$-0.713542\pi$
$389$	6.73894	−	20.7403i	0.341678	−	1.05158i	0.963338	−	0.268290i	$-0.0864585\pi$
$390$		0			0
$391$		0			0
$392$	3.53854	+	1.14974i	0.178723	+	0.0580707i
$393$		0			0
$394$	2.31701	−	1.68340i	0.116729	−	0.0848087i
$395$		0			0
$396$	−11.2879	−	16.1728i	−0.567238	−	0.812714i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	6.01567	−	18.5143i	0.300784	−	0.925717i
$401$	7.78557	+	5.65655i	0.388793	+	0.282475i	0.764961	−	0.644077i	$-0.222758\pi$
$401$	7.78557	+	5.65655i	0.388793	+	0.282475i	−0.376168	+	0.926552i	$0.622758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$			3.43867i			0.170658i
$407$	−0.516471	+	27.2277i	−0.0256005	+	1.34963i
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	2.85796	+	0.928608i	0.140461	+	0.0456386i
$415$		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
$421$	11.8136	+	36.3585i	0.575759	+	1.77200i	0.633581	+	0.773676i	$0.281584\pi$
$421$	11.8136	+	36.3585i	0.575759	+	1.77200i	−0.0578225	+	0.998327i	$0.518416\pi$
$422$	−0.807944	+	2.48660i	−0.0393301	+	0.121046i
$423$		0			0
$424$	0.584110	−	0.803959i	0.0283669	−	0.0390437i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	40.9446i		−	1.97913i
$429$		0			0
$430$		0			0
$431$	−13.0829	−	18.0071i	−0.630182	−	0.867371i	0.367862	−	0.929880i	$-0.380090\pi$
$431$	−13.0829	−	18.0071i	−0.630182	−	0.867371i	−0.998044	+	0.0625092i	$0.980090\pi$
$432$		0			0
$433$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$433$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$434$		0			0
$435$		0			0
$436$	38.4374	+	12.4891i	1.84082	+	0.598117i
$437$		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
$443$	11.7279	+	36.0947i	0.557208	+	1.71491i	0.690039	+	0.723772i	$0.257593\pi$
$443$	11.7279	+	36.0947i	0.557208	+	1.71491i	−0.132831	+	0.991139i	$0.542407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−19.0622	+	6.19367i	−0.900603	+	0.292624i
$449$	−32.0711	+	23.3010i	−1.51353	+	1.09964i	−0.548950	+	0.835855i	$0.684972\pi$
$449$	−32.0711	+	23.3010i	−1.51353	+	1.09964i	−0.964580	+	0.263790i	$0.915028\pi$
$450$		−	2.00212i		−	0.0943807i
$451$		0			0
$452$	−27.8677			−1.31079
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0431	−	15.1995i	0.516575	−	0.711004i	−0.468436	−	0.883497i	$-0.655182\pi$
$457$	11.0431	−	15.1995i	0.516575	−	0.711004i	0.985011	+	0.172493i	$0.0551823\pi$
$458$		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
$463$	−41.6915			−1.93757			−0.968784	−	0.247907i	$-0.920257\pi$
$463$	−41.6915			−1.93757			−0.968784	+	0.247907i	$0.920257\pi$
$464$	−22.2840	−	30.6713i	−1.03451	−	1.42388i
$465$		0			0
$466$	0.873010	−	2.68685i	0.0404414	−	0.124466i
$467$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$467$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$468$		0			0
$469$	−15.3359	−	4.98295i	−0.708147	−	0.230091i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.9364	+	35.9658i	0.502854	+	1.65371i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.73325	+	5.33439i	−0.0793599	+	0.244245i
$478$	3.32687	+	2.41711i	0.152167	+	0.110556i
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−5.94655	−	20.9775i	−0.270298	−	0.953521i
$485$		0			0
$486$		0			0
$487$	0.727878	+	2.24018i	0.0329833	+	0.101512i	0.966193	−	0.257821i	$-0.0830043\pi$
$487$	0.727878	+	2.24018i	0.0329833	+	0.101512i	−0.933210	+	0.359333i	$0.883004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.03252	+	1.63516i	0.227114	+	0.0737939i	0.420363	−	0.907356i	$-0.361903\pi$
$491$	5.03252	+	1.63516i	0.227114	+	0.0737939i	−0.193249	+	0.981150i	$0.561903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.2932	−	21.0493i	−0.685993	−	0.944189i
$498$		0			0
$499$	4.19437	−	12.9090i	0.187766	−	0.577884i	−0.812219	−	0.583352i	$-0.801741\pi$
$499$	4.19437	−	12.9090i	0.187766	−	0.577884i	0.999985	+	0.00546838i	$0.00174065\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$	−3.41309	+	2.47975i	−0.152031	+	0.110457i
$505$		0			0
$506$	2.65019	+	2.00336i	0.117815	+	0.0890601i
$507$		0			0
$508$	23.4446	+	32.2687i	1.04019	+	1.43169i
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$	−6.05480	+	8.33371i	−0.267587	+	0.368301i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	2.89961			0.127402
$519$		0			0
$520$		0			0
$521$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$521$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$522$	−3.15442	−	2.29182i	−0.138065	−	0.100310i
$523$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$523$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$524$		0			0
$525$		0			0
$526$	−1.56880	+	1.13980i	−0.0684030	+	0.0496977i
$527$		0			0
$528$		0			0
$529$	33.3198			1.44869
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−3.08092	+	1.00105i	−0.133075	+	0.0432388i
$537$		0			0
$538$		0			0
$539$	−21.9401	−	7.59172i	−0.945025	−	0.326998i
$540$		0			0
$541$	13.2398	+	18.2231i	0.569225	+	0.783470i	0.992463	−	0.122548i	$-0.0391066\pi$
$541$	13.2398	+	18.2231i	0.569225	+	0.783470i	−0.423238	+	0.906019i	$0.639107\pi$
$542$		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	$-0.710215\pi$
$547$	−15.0976	+	4.90549i	−0.645525	+	0.209744i	−0.0320849	+	0.999485i	$0.510215\pi$
$548$	37.2365	−	27.0539i	1.59066	−	1.15568i
$549$		0			0
$550$	0.723786	−	2.09174i	0.0308623	−	0.0891921i
$551$		0			0
$552$		0			0
$553$	6.65550	+	20.4835i	0.283021	+	0.871048i
$554$	0.796934	−	2.45271i	0.0338585	−	0.104206i
$555$		0			0
$556$		0			0
$557$	33.8576	+	11.0010i	1.43459	+	0.466127i	0.920207	−	0.391433i	$-0.128020\pi$
$557$	33.8576	+	11.0010i	1.43459	+	0.466127i	0.514384	+	0.857560i	$0.328020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−4.32538			−0.182455
$563$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$563$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	13.9962	−	19.2641i	0.587785	−	0.809017i
$568$	−4.97114	−	1.61522i	−0.208584	−	0.0677732i
$569$	40.2601	−	13.0813i	1.68779	−	0.548397i	0.701395	−	0.712773i	$-0.252561\pi$
$569$	40.2601	−	13.0813i	1.68779	−	0.548397i	0.986398	+	0.164375i	$0.0525608\pi$
$570$		0			0
$571$		−	18.5207i		−	0.775067i	−0.921856	−	0.387534i	$-0.873327\pi$
$571$		−	18.5207i		−	0.775067i	0.921856	−	0.387534i	$-0.126673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.5953	−	35.6868i	−0.483559	−	1.48824i
$576$	7.02297	−	21.6145i	0.292624	−	0.900603i
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	1.33372	−	1.83571i	0.0554756	−	0.0763556i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.73927	+	4.94659i	−0.154865	+	0.204867i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$587$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−25.8632	+	18.7907i	−1.06297	+	0.772293i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	12.3302	+	16.9711i	0.505067	+	0.695165i
$597$		0			0
$598$		0			0
$599$	−20.4997	−	14.8939i	−0.837597	−	0.608550i	0.0841014	−	0.996457i	$-0.473198\pi$
$599$	−20.4997	−	14.8939i	−0.837597	−	0.608550i	−0.921698	+	0.387907i	$0.873198\pi$
$600$		0			0
$601$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$601$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$602$	3.80671	−	1.23687i	0.155150	−	0.0504112i
$603$	14.7922	−	10.7472i	0.602386	−	0.437659i
$604$			42.0029i			1.70907i
$605$		0			0
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.18571	+	1.03510i	−0.128670	+	0.0418073i	−0.372644	−	0.927974i	$-0.621549\pi$
$613$	−3.18571	+	1.03510i	−0.128670	+	0.0418073i	0.243974	+	0.969782i	$0.421549\pi$
$614$		0			0
$615$		0			0
$616$	−4.46233	+	1.35689i	−0.179792	+	0.0546707i
$617$	45.9166			1.84853			0.924266	−	0.381749i	$-0.124678\pi$
$617$	45.9166			1.84853			0.924266	+	0.381749i	$0.124678\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$		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
$631$	−12.4683	+	38.3736i	−0.496357	+	1.52763i	0.318475	+	0.947931i	$0.396829\pi$
$631$	−12.4683	+	38.3736i	−0.496357	+	1.52763i	−0.814832	+	0.579698i	$0.803171\pi$
$632$	3.50047	+	2.54324i	0.139241	+	0.101165i
$633$		0			0
$634$	−4.28138	−	1.39110i	−0.170035	−	0.0552478i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−2.46711	−	3.53477i	−0.0976739	−	0.139943i
$639$	29.5020			1.16708
$640$		0			0
$641$	−15.3445	−	47.2255i	−0.606071	−	1.86530i	−0.489251	−	0.872143i	$-0.662730\pi$
$641$	−15.3445	−	47.2255i	−0.606071	−	1.86530i	−0.116820	−	0.993153i	$-0.537270\pi$
$642$		0			0
$643$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$643$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$644$	−23.1336	+	31.8406i	−0.911590	+	1.25470i
$645$		0			0
$646$		0			0
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$		−	4.78368i		−	0.187921i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−5.46205	−	16.8105i	−0.213910	−	0.658348i
$653$	−1.66431	+	5.12221i	−0.0651294	+	0.200448i	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−1.66431	+	5.12221i	−0.0651294	+	0.200448i	0.913196	+	0.407520i	$0.133606\pi$
$654$		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			$\pi$
$662$	0.477951	+	0.657843i	0.0185761	+	0.0255678i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.93255	+	2.65993i	−0.0748847	+	0.103070i
$667$	−69.4992	−	22.5817i	−2.69102	−	0.874366i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−30.4948	−	41.9725i	−1.17549	−	1.61792i	−0.597281	−	0.802032i	$-0.703752\pi$
$673$	−30.4948	−	41.9725i	−1.17549	−	1.61792i	−0.578208	−	0.815890i	$-0.696248\pi$
$674$	0.907039	+	2.79158i	0.0349378	+	0.107528i
$675$		0			0
$676$	20.8471	+	15.1463i	0.801811	+	0.582549i
$677$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$677$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.1014			−0.807423			−0.403711	−	0.914886i	$-0.632280\pi$
$683$	−21.1014			−0.807423			−0.403711	+	0.914886i	$0.632280\pi$
$684$		0			0
$685$		0			0
$686$	−0.763884	+	2.35099i	−0.0291652	+	0.0897613i
$687$		0			0
$688$	−25.9386	+	35.7014i	−0.988900	+	1.36110i
$689$		0			0
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$	21.5869	−	15.0667i	0.820019	−	0.572336i
$694$	3.68593			0.139916
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	24.9384	+	8.10299i	0.942585	+	0.306264i
$701$	−17.3604	+	5.64072i	−0.655691	+	0.213047i	−0.617922	−	0.786239i	$-0.712025\pi$
$701$	−17.3604	+	5.64072i	−0.655691	+	0.213047i	−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		0			0
$704$	15.1512	−	20.0431i	0.571032	−	0.755405i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−21.2355	−	15.4285i	−0.797517	−	0.579430i	0.112667	−	0.993633i	$-0.464061\pi$
$709$	−21.2355	−	15.4285i	−0.797517	−	0.579430i	−0.910185	+	0.414202i	$0.864061\pi$
$710$		0			0
$711$	−23.2261	−	7.54663i	−0.871048	−	0.283021i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−47.4268			−1.77242
$717$		0			0
$718$	0.785923	+	2.41882i	0.0293304	+	0.0902697i
$719$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$719$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$720$		0			0
$721$		0			0
$722$	2.41189	+	0.783671i	0.0897613	+	0.0291652i
$723$		0			0
$724$		0			0
$725$			48.6870i			1.80819i
$726$		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	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$		0			0
$735$		0			0
$736$			11.8777i			0.437818i
$737$	19.3396	−	5.88073i	0.712384	−	0.216620i
$738$		0			0
$739$	−10.4168	−	14.3375i	−0.383187	−	0.527412i	0.573238	−	0.819389i	$-0.305687\pi$
$739$	−10.4168	−	14.3375i	−0.383187	−	0.527412i	−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		0			0
$742$	0.534148	+	0.388081i	0.0196092	+	0.0142469i
$743$	31.4335	−	43.2645i	1.15318	−	1.58722i	0.419453	−	0.907777i	$-0.362222\pi$
$743$	31.4335	−	43.2645i	1.15318	−	1.58722i	0.733729	−	0.679442i	$-0.237778\pi$
$744$		0			0
$745$		0			0
$746$	3.42836	−	2.49085i	0.125521	−	0.0911964i
$747$		0			0
$748$		0			0
$749$	54.6514			1.99692
$750$		0			0
$751$	−3.19208	−	9.82420i	−0.116481	−	0.358490i	0.875772	−	0.482724i	$-0.160353\pi$
$751$	−3.19208	−	9.82420i	−0.116481	−	0.358490i	−0.992253	+	0.124234i	$0.960353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	30.3109	−	22.0222i	1.10167	−	0.800410i	0.120338	−	0.992733i	$-0.461602\pi$
$757$	30.3109	−	22.0222i	1.10167	−	0.800410i	0.981332	+	0.192323i	$0.0616021\pi$
$758$		−	5.19694i		−	0.188761i
$759$		0			0
$760$		0			0
$761$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$761$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$762$		0			0
$763$	−16.6700	+	51.3049i	−0.603493	+	1.85736i
$764$	35.3173	+	25.6595i	1.27774	+	0.928329i
$765$		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
$772$	7.62386	+	10.4933i	0.274389	+	0.377664i
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$	−1.40248	+	4.31640i	−0.0504112	+	0.155150i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	2.76830	−	0.899477i	0.0992485	−	0.0322478i
$779$		0			0
$780$		0			0
$781$	30.8227	+	10.6653i	1.10292	+	0.381634i
$782$		0			0
$783$		0			0
$784$	−8.42194	−	25.9201i	−0.300784	−	0.925717i
$785$		0			0
$786$		0			0
$787$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$787$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$788$	−40.4503	−	13.1431i	−1.44098	−	0.468204i
$789$		0			0
$790$		0			0
$791$		−	37.1969i		−	1.32257i
$792$	1.72935	−	4.99782i	0.0614497	−	0.177590i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$797$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$		0			0
$800$	7.52624	−	2.44542i	0.266093	−	0.0864588i
$801$		0			0
$802$			1.28449i			0.0453569i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.5536	+	18.6550i	−0.476521	+	0.655874i	−0.977832	−	0.209393i	$-0.932851\pi$
$809$	−13.5536	+	18.6550i	−0.476521	+	0.655874i	0.501311	+	0.865267i	$0.332851\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$	41.3137	−	30.0161i	1.44983	−	1.05336i
$813$		0			0
$814$	−2.98065	+	2.08036i	−0.104472	+	0.0729165i
$815$		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	$-0.774580\pi$
$821$	−50.3252	−	16.3516i	−1.75636	−	0.570676i	−0.996813	+	0.0797750i	$0.974580\pi$
$822$		0			0
$823$	43.5906	−	31.6704i	1.51947	−	1.10396i	0.557725	−	0.830026i	$-0.311674\pi$
$823$	43.5906	−	31.6704i	1.51947	−	1.10396i	0.961748	−	0.273936i	$-0.0883256\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21.7719	+	29.9664i	0.757082	+	1.04203i	0.997451	+	0.0713526i	$0.0227315\pi$
$827$	21.7719	+	29.9664i	0.757082	+	1.04203i	−0.240369	+	0.970682i	$0.577268\pi$
$828$	−13.7904	−	42.4426i	−0.479251	−	1.47498i
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		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	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	53.2471	+	38.6863i	1.83611	+	1.33401i
$842$	−2.99927	+	4.12814i	−0.103362	+	0.142265i
$843$		0			0
$844$	36.9276	−	11.9985i	1.27110	−	0.413006i
$845$		0			0
$846$		0			0
$847$	28.0000	−	7.93725i	0.962091	−	0.272727i
$848$	−7.27929			−0.249972
$849$		0			0
$850$		0			0
$851$	−19.0417	+	58.6043i	−0.652741	+	2.00893i
$852$		0			0
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$	8.88238	−	6.45343i	0.303594	−	0.220574i
$857$		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
$862$	0.918050	−	2.82547i	0.0312689	−	0.0962358i
$863$	−6.47214	−	4.70228i	−0.220314	−	0.160068i	0.472154	−	0.881516i	$-0.343477\pi$
$863$	−6.47214	−	4.70228i	−0.220314	−	0.160068i	−0.692468	+	0.721449i	$0.743477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.5376	−	16.2809i	−0.730615	−	0.552293i
$870$		0			0
$871$		0			0
$872$	3.34892	+	10.3069i	0.113409	+	0.349036i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−52.3792	+	17.0190i	−1.76872	+	0.574692i	−0.998043	−	0.0625337i	$-0.980082\pi$
$877$	−52.3792	+	17.0190i	−1.76872	+	0.574692i	−0.770677	+	0.637226i	$0.780082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.64754	−	2.26764i	−0.0554756	−	0.0763556i
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	0.869072	−	0.494686i	$-0.164717\pi$
$883$	−3.70820	−	11.4127i	−0.124791	−	0.384067i	−0.993863	+	0.110619i	$0.964717\pi$
$884$		0			0
$885$		0			0
$886$	−2.97751	+	4.09820i	−0.100032	+	0.137682i
$887$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$887$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$	−43.0711	+	31.2930i	−1.44456	+	1.04953i
$890$		0			0
$891$	−0.566103	+	29.8443i	−0.0189652	+	0.999820i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−8.93978	−	6.49513i	−0.298657	−	0.216987i
$897$		0			0
$898$	−5.03224	−	1.63507i	−0.167928	−	0.0545631i
$899$		0			0
$900$	−24.0543	+	17.4765i	−0.801811	+	0.582549i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−4.39234	−	6.04553i	−0.146087	−	0.201071i
$905$		0			0
$906$		0			0
$907$	−36.7543	−	26.7035i	−1.22040	−	0.886676i	−0.224271	−	0.974527i	$-0.572000\pi$
$907$	−36.7543	−	26.7035i	−1.22040	−	0.886676i	−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	−0.352331	−	0.935875i	$-0.614611\pi$
$911$	12.9443	−	9.40456i	0.428863	−	0.311587i	0.781194	+	0.624288i	$0.214611\pi$
$912$		0			0
$913$		0			0
$914$	2.50767			0.0829464
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.03022	−	1.41798i	0.0339839	−	0.0467748i	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	1.03022	−	1.41798i	0.0339839	−	0.0467748i	0.825671	+	0.564152i	$0.190797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	41.0547			1.34987
$926$	−3.27088	−	4.50197i	−0.107488	−	0.147944i
$927$		0			0
$928$	4.76242	−	14.6572i	0.156334	−	0.481146i
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$		0			0
$932$	−39.9015	+	12.9648i	−1.30702	+	0.424676i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$	−0.665096	−	2.04695i	−0.0217161	−	0.0668354i
$939$		0			0
$940$		0			0
$941$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$941$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−3.02569	+	4.00261i	−0.0983736	+	0.130136i
$947$	−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
$953$	−58.6820	−	19.0669i	−1.90090	−	0.617639i	−0.961495	−	0.274822i	$-0.911381\pi$
$953$	−58.6820	−	19.0669i	−1.90090	−	0.617639i	−0.939402	−	0.342817i	$-0.888619\pi$
$954$	−0.712004	+	0.231344i	−0.0230520	+	0.00749004i
$955$		0			0
$956$		−	61.0694i		−	1.97513i
$957$		0			0
$958$		0			0
$959$	36.1106	+	49.7019i	1.16607	+	1.60496i
$960$		0			0
$961$	9.57953	−	29.4828i	0.309017	−	0.951057i
$962$		0			0
$963$	−36.4244	+	50.1339i	−1.17376	+	1.61554i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			23.3258i			0.750107i	0.927003	+	0.375053i	$0.122376\pi$
$967$			23.3258i			0.750107i	−0.927003	+	0.375053i	$0.877624\pi$
$968$	3.61352	−	4.59636i	0.116143	−	0.147733i
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$		0			0
$973$		0			0
$974$	−0.184796	+	0.254350i	−0.00592125	+	0.00814990i
$975$		0			0
$976$		0			0
$977$	−21.0711	+	15.3091i	−0.674126	+	0.489781i	−0.871404	−	0.490567i	$-0.836790\pi$
$977$	−21.0711	+	15.3091i	−0.674126	+	0.489781i	0.197278	+	0.980348i	$0.436790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−35.9537	−	49.4860i	−1.14791	−	1.57997i
$982$	0.218253	+	0.671712i	0.00696472	+	0.0214352i
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			85.0603i			2.70476i
$990$		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
$994$	1.07315	−	3.30281i	0.0340382	−	0.104759i
$995$		0			0
$996$		0			0
$997$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$997$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$998$	1.72301	−	0.559841i	0.0545411	−	0.0177215i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.l.a.41.1	✓	8
3.2	odd	2		693.2.bu.a.118.2		8
7.2	even	3		539.2.s.a.129.1		16
7.3	odd	6		539.2.s.a.19.2		16
7.4	even	3		539.2.s.a.19.2		16
7.5	odd	6		539.2.s.a.129.1		16
7.6	odd	2	CM	77.2.l.a.41.1	✓	8
11.2	odd	10		847.2.b.b.846.5		8
11.3	even	5		847.2.l.d.699.1		8
11.4	even	5		847.2.l.c.524.2		8
11.5	even	5		847.2.l.a.475.2		8
11.6	odd	10		847.2.l.d.475.1		8
11.7	odd	10	inner	77.2.l.a.62.1	yes	8
11.8	odd	10		847.2.l.a.699.2		8
11.9	even	5		847.2.b.b.846.4		8
11.10	odd	2		847.2.l.c.118.2		8
21.20	even	2		693.2.bu.a.118.2		8
33.29	even	10		693.2.bu.a.370.2		8
77.6	even	10		847.2.l.d.475.1		8
77.13	even	10		847.2.b.b.846.5		8
77.18	odd	30		539.2.s.a.117.1		16
77.20	odd	10		847.2.b.b.846.4		8
77.27	odd	10		847.2.l.a.475.2		8
77.40	even	30		539.2.s.a.227.2		16
77.41	even	10		847.2.l.a.699.2		8
77.48	odd	10		847.2.l.c.524.2		8
77.51	odd	30		539.2.s.a.227.2		16
77.62	even	10	inner	77.2.l.a.62.1	yes	8
77.69	odd	10		847.2.l.d.699.1		8
77.73	even	30		539.2.s.a.117.1		16
77.76	even	2		847.2.l.c.118.2		8
231.62	odd	10		693.2.bu.a.370.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.41.1	✓	8	1.1	even	1	trivial
77.2.l.a.41.1	✓	8	7.6	odd	2	CM
77.2.l.a.62.1	yes	8	11.7	odd	10	inner
77.2.l.a.62.1	yes	8	77.62	even	10	inner
539.2.s.a.19.2		16	7.3	odd	6
539.2.s.a.19.2		16	7.4	even	3
539.2.s.a.117.1		16	77.18	odd	30
539.2.s.a.117.1		16	77.73	even	30
539.2.s.a.129.1		16	7.2	even	3
539.2.s.a.129.1		16	7.5	odd	6
539.2.s.a.227.2		16	77.40	even	30
539.2.s.a.227.2		16	77.51	odd	30
693.2.bu.a.118.2		8	3.2	odd	2
693.2.bu.a.118.2		8	21.20	even	2
693.2.bu.a.370.2		8	33.29	even	10
693.2.bu.a.370.2		8	231.62	odd	10
847.2.b.b.846.4		8	11.9	even	5
847.2.b.b.846.4		8	77.20	odd	10
847.2.b.b.846.5		8	11.2	odd	10
847.2.b.b.846.5		8	77.13	even	10
847.2.l.a.475.2		8	11.5	even	5
847.2.l.a.475.2		8	77.27	odd	10
847.2.l.a.699.2		8	11.8	odd	10
847.2.l.a.699.2		8	77.41	even	10
847.2.l.c.118.2		8	11.10	odd	2
847.2.l.c.118.2		8	77.76	even	2
847.2.l.c.524.2		8	11.4	even	5
847.2.l.c.524.2		8	77.48	odd	10
847.2.l.d.475.1		8	11.6	odd	10
847.2.l.d.475.1		8	77.6	even	10
847.2.l.d.699.1		8	11.3	even	5
847.2.l.d.699.1		8	77.69	odd	10

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.0784543 + 0.107983i) q^{2} +(0.612529 - 1.88517i) q^{4} +(2.51626 + 0.817582i) q^{7} +(0.505505 - 0.164249i) q^{8} +(-2.42705 + 1.76336i) q^{9} +O(q^{10})\)	\(q+(0.0784543 + 0.107983i) q^{2} +(0.612529 - 1.88517i) q^{4} +(2.51626 + 0.817582i) q^{7} +(0.505505 - 0.164249i) q^{8} +(-2.42705 + 1.76336i) q^{9} +(-3.17317 + 0.964887i) q^{11} +(0.109126 + 0.335856i) q^{14} +(-3.14985 - 2.28850i) q^{16} +(-0.380825 - 0.123738i) q^{18} +(-0.353140 - 0.266949i) q^{22} -7.50465 q^{23} +(1.54508 + 4.75528i) q^{25} +(3.08256 - 4.24278i) q^{28} +(9.26082 + 3.00902i) q^{29} -1.58271i q^{32} +(1.83759 + 5.65551i) q^{36} +(2.53732 - 7.80906i) q^{37} -11.3343i q^{43} +(-0.124680 + 6.57298i) q^{44} +(-0.588772 - 0.810375i) q^{46} +(5.66312 + 4.11450i) q^{49} +(-0.392271 + 0.539915i) q^{50} +(1.51257 - 1.09894i) q^{53} +1.40627 q^{56} +(0.401627 + 1.23608i) q^{58} +(-7.54878 + 2.45275i) q^{63} +(-6.12879 + 4.45282i) q^{64} -6.09473 q^{67} +(-7.95588 - 5.78028i) q^{71} +(-0.937258 + 1.29003i) q^{72} +(1.04231 - 0.338667i) q^{74} +(-8.77339 - 0.166419i) q^{77} +(4.78485 + 6.58577i) q^{79} +(2.78115 - 8.55951i) q^{81} +(1.22392 - 0.889227i) q^{86} +(-1.44557 + 1.00894i) q^{88} +(-4.59682 + 14.1475i) q^{92} +0.934321i q^{98} +(6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9}+O(q^{10}) \) 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

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