LMFDB - Embedded newform 169.3.d.b.70.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,3,Mod(70,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("169.70");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	169.d (of order $4$, degree $2$, 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:	$4.60491646769$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121 $ x^4 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			70.2
Root			$2.34521 - 2.34521i$ of defining polynomial
Character	$\chi$	$=$	169.70
Dual form			169.3.d.b.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.34521 - 2.34521i) q^{2} -3.00000 q^{3} -7.00000i q^{4} +(-2.34521 + 2.34521i) q^{5} +(-7.03562 + 7.03562i) q^{6} +(-7.03562 - 7.03562i) q^{7} +(-7.03562 - 7.03562i) q^{8} +O(q^{10})$	\(q+(2.34521 - 2.34521i) q^{2} -3.00000 q^{3} -7.00000i q^{4} +(-2.34521 + 2.34521i) q^{5} +(-7.03562 + 7.03562i) q^{6} +(-7.03562 - 7.03562i) q^{7} +(-7.03562 - 7.03562i) q^{8} +11.0000i q^{10} +(-4.69042 - 4.69042i) q^{11} +21.0000i q^{12} -33.0000 q^{14} +(7.03562 - 7.03562i) q^{15} -5.00000 q^{16} -3.00000i q^{17} +(14.0712 - 14.0712i) q^{19} +(16.4165 + 16.4165i) q^{20} +(21.1069 + 21.1069i) q^{21} -22.0000 q^{22} -12.0000i q^{23} +(21.1069 + 21.1069i) q^{24} +14.0000i q^{25} +27.0000 q^{27} +(-49.2494 + 49.2494i) q^{28} -42.0000 q^{29} -33.0000i q^{30} +(28.1425 - 28.1425i) q^{31} +(16.4165 - 16.4165i) q^{32} +(14.0712 + 14.0712i) q^{33} +(-7.03562 - 7.03562i) q^{34} +33.0000 q^{35} +(-35.1781 - 35.1781i) q^{37} -66.0000i q^{38} +33.0000 q^{40} +(32.8329 - 32.8329i) q^{41} +99.0000 q^{42} +49.0000i q^{43} +(-32.8329 + 32.8329i) q^{44} +(-28.1425 - 28.1425i) q^{46} +(2.34521 + 2.34521i) q^{47} +15.0000 q^{48} +50.0000i q^{49} +(32.8329 + 32.8329i) q^{50} +9.00000i q^{51} -24.0000 q^{53} +(63.3206 - 63.3206i) q^{54} +22.0000 q^{55} +99.0000i q^{56} +(-42.2137 + 42.2137i) q^{57} +(-98.4987 + 98.4987i) q^{58} +(-37.5233 - 37.5233i) q^{59} +(-49.2494 - 49.2494i) q^{60} +30.0000 q^{61} -132.000i q^{62} -97.0000i q^{64} +66.0000 q^{66} +(-28.1425 + 28.1425i) q^{67} -21.0000 q^{68} +36.0000i q^{69} +(77.3919 - 77.3919i) q^{70} +(-16.4165 + 16.4165i) q^{71} +(-28.1425 - 28.1425i) q^{73} -165.000 q^{74} -42.0000i q^{75} +(-98.4987 - 98.4987i) q^{76} +66.0000i q^{77} +54.0000 q^{79} +(11.7260 - 11.7260i) q^{80} -81.0000 q^{81} -154.000i q^{82} +(-32.8329 + 32.8329i) q^{83} +(147.748 - 147.748i) q^{84} +(7.03562 + 7.03562i) q^{85} +(114.915 + 114.915i) q^{86} +126.000 q^{87} +66.0000i q^{88} +(117.260 + 117.260i) q^{89} -84.0000 q^{92} +(-84.4275 + 84.4275i) q^{93} +11.0000 q^{94} +66.0000i q^{95} +(-49.2494 + 49.2494i) q^{96} +(14.0712 - 14.0712i) q^{97} +(117.260 + 117.260i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3}+O(q^{10}) $ 4 * q - 12 * q^3	$ 4 q - 12 q^{3} - 132 q^{14} - 20 q^{16} - 88 q^{22} + 108 q^{27} - 168 q^{29} + 132 q^{35} + 132 q^{40} + 396 q^{42} + 60 q^{48} - 96 q^{53} + 88 q^{55} + 120 q^{61} + 264 q^{66} - 84 q^{68} - 660 q^{74} + 216 q^{79} - 324 q^{81} + 504 q^{87} - 336 q^{92} + 44 q^{94}+O(q^{100}) $ 4 * q - 12 * q^3 - 132 * q^14 - 20 * q^16 - 88 * q^22 + 108 * q^27 - 168 * q^29 + 132 * q^35 + 132 * q^40 + 396 * q^42 + 60 * q^48 - 96 * q^53 + 88 * q^55 + 120 * q^61 + 264 * q^66 - 84 * q^68 - 660 * q^74 + 216 * q^79 - 324 * q^81 + 504 * q^87 - 336 * q^92 + 44 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.34521	−	2.34521i	1.17260	−	1.17260i	0.191017	−	0.981587i	$-0.438821\pi$
$2$	2.34521	−	2.34521i	1.17260	−	1.17260i	0.981587	−	0.191017i	$-0.0611786\pi$
$3$	−3.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−3.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$4$		−	7.00000i		−	1.75000i
$5$	−2.34521	+	2.34521i	−0.469042	+	0.469042i	−0.901604	−	0.432562i	$-0.857610\pi$
$5$	−2.34521	+	2.34521i	−0.469042	+	0.469042i	0.432562	+	0.901604i	$0.357610\pi$
$6$	−7.03562	+	7.03562i	−1.17260	+	1.17260i
$7$	−7.03562	−	7.03562i	−1.00509	−	1.00509i	−0.999987	−	0.00510211i	$-0.998376\pi$
$7$	−7.03562	−	7.03562i	−1.00509	−	1.00509i	−0.00510211	−	0.999987i	$-0.501624\pi$
$8$	−7.03562	−	7.03562i	−0.879453	−	0.879453i
$9$		0			0
$10$			11.0000i			1.10000i
$11$	−4.69042	−	4.69042i	−0.426401	−	0.426401i	0.460999	−	0.887401i	$-0.347491\pi$
$11$	−4.69042	−	4.69042i	−0.426401	−	0.426401i	−0.887401	+	0.460999i	$0.847491\pi$
$12$			21.0000i			1.75000i
$13$		0			0
$13$		0			0
$14$	−33.0000			−2.35714
$15$	7.03562	−	7.03562i	0.469042	−	0.469042i
$16$	−5.00000			−0.312500
$17$		−	3.00000i		−	0.176471i	−0.996100	−	0.0882353i	$-0.971877\pi$
$17$		−	3.00000i		−	0.176471i	0.996100	−	0.0882353i	$-0.0281227\pi$
$18$		0			0
$19$	14.0712	−	14.0712i	0.740592	−	0.740592i	−0.232100	−	0.972692i	$-0.574560\pi$
$19$	14.0712	−	14.0712i	0.740592	−	0.740592i	0.972692	+	0.232100i	$0.0745597\pi$
$20$	16.4165	+	16.4165i	0.820823	+	0.820823i
$21$	21.1069	+	21.1069i	1.00509	+	1.00509i
$22$	−22.0000			−1.00000
$23$		−	12.0000i		−	0.521739i	−0.965374	−	0.260870i	$-0.915991\pi$
$23$		−	12.0000i		−	0.521739i	0.965374	−	0.260870i	$-0.0840093\pi$
$24$	21.1069	+	21.1069i	0.879453	+	0.879453i
$25$			14.0000i			0.560000i
$26$		0			0
$27$	27.0000			1.00000
$28$	−49.2494	+	49.2494i	−1.75891	+	1.75891i
$29$	−42.0000			−1.44828			−0.724138	−	0.689655i	$-0.757762\pi$
$29$	−42.0000			−1.44828			−0.724138	+	0.689655i	$0.757762\pi$
$30$		−	33.0000i		−	1.10000i
$31$	28.1425	−	28.1425i	0.907822	−	0.907822i	−0.0882738	−	0.996096i	$-0.528135\pi$
$31$	28.1425	−	28.1425i	0.907822	−	0.907822i	0.996096	+	0.0882738i	$0.0281351\pi$
$32$	16.4165	−	16.4165i	0.513014	−	0.513014i
$33$	14.0712	+	14.0712i	0.426401	+	0.426401i
$34$	−7.03562	−	7.03562i	−0.206930	−	0.206930i
$35$	33.0000			0.942857
$36$		0			0
$37$	−35.1781	−	35.1781i	−0.950760	−	0.950760i	0.0480834	−	0.998843i	$-0.484689\pi$
$37$	−35.1781	−	35.1781i	−0.950760	−	0.950760i	−0.998843	+	0.0480834i	$0.984689\pi$
$38$		−	66.0000i		−	1.73684i
$39$		0			0
$40$	33.0000			0.825000
$41$	32.8329	−	32.8329i	0.800803	−	0.800803i	−0.182418	−	0.983221i	$-0.558393\pi$
$41$	32.8329	−	32.8329i	0.800803	−	0.800803i	0.983221	+	0.182418i	$0.0583925\pi$
$42$	99.0000			2.35714
$43$			49.0000i			1.13953i	0.821806	+	0.569767i	$0.192967\pi$
$43$			49.0000i			1.13953i	−0.821806	+	0.569767i	$0.807033\pi$
$44$	−32.8329	+	32.8329i	−0.746203	+	0.746203i
$45$		0			0
$46$	−28.1425	−	28.1425i	−0.611793	−	0.611793i
$47$	2.34521	+	2.34521i	0.0498980	+	0.0498980i	0.731616	−	0.681717i	$-0.238767\pi$
$47$	2.34521	+	2.34521i	0.0498980	+	0.0498980i	−0.681717	+	0.731616i	$0.738767\pi$
$48$	15.0000			0.312500
$49$			50.0000i			1.02041i
$50$	32.8329	+	32.8329i	0.656658	+	0.656658i
$51$			9.00000i			0.176471i
$52$		0			0
$53$	−24.0000			−0.452830			−0.226415	−	0.974031i	$-0.572701\pi$
$53$	−24.0000			−0.452830			−0.226415	+	0.974031i	$0.572701\pi$
$54$	63.3206	−	63.3206i	1.17260	−	1.17260i
$55$	22.0000			0.400000
$56$			99.0000i			1.76786i
$57$	−42.2137	+	42.2137i	−0.740592	+	0.740592i
$58$	−98.4987	+	98.4987i	−1.69825	+	1.69825i
$59$	−37.5233	−	37.5233i	−0.635989	−	0.635989i	0.313575	−	0.949563i	$-0.398473\pi$
$59$	−37.5233	−	37.5233i	−0.635989	−	0.635989i	−0.949563	+	0.313575i	$0.898473\pi$
$60$	−49.2494	−	49.2494i	−0.820823	−	0.820823i
$61$	30.0000			0.491803			0.245902	−	0.969295i	$-0.420916\pi$
$61$	30.0000			0.491803			0.245902	+	0.969295i	$0.420916\pi$
$62$		−	132.000i		−	2.12903i
$63$		0			0
$64$		−	97.0000i		−	1.51562i
$65$		0			0
$66$	66.0000			1.00000
$67$	−28.1425	+	28.1425i	−0.420037	+	0.420037i	−0.885216	−	0.465179i	$-0.845990\pi$
$67$	−28.1425	+	28.1425i	−0.420037	+	0.420037i	0.465179	+	0.885216i	$0.345990\pi$
$68$	−21.0000			−0.308824
$69$			36.0000i			0.521739i
$70$	77.3919	−	77.3919i	1.10560	−	1.10560i
$71$	−16.4165	+	16.4165i	−0.231218	+	0.231218i	−0.813201	−	0.581983i	$-0.802277\pi$
$71$	−16.4165	+	16.4165i	−0.231218	+	0.231218i	0.581983	+	0.813201i	$0.302277\pi$
$72$		0			0
$73$	−28.1425	−	28.1425i	−0.385514	−	0.385514i	0.487570	−	0.873084i	$-0.337883\pi$
$73$	−28.1425	−	28.1425i	−0.385514	−	0.385514i	−0.873084	+	0.487570i	$0.837883\pi$
$74$	−165.000			−2.22973
$75$		−	42.0000i		−	0.560000i
$76$	−98.4987	−	98.4987i	−1.29604	−	1.29604i
$77$			66.0000i			0.857143i
$78$		0			0
$79$	54.0000			0.683544			0.341772	−	0.939783i	$-0.388973\pi$
$79$	54.0000			0.683544			0.341772	+	0.939783i	$0.388973\pi$
$80$	11.7260	−	11.7260i	0.146575	−	0.146575i
$81$	−81.0000			−1.00000
$82$		−	154.000i		−	1.87805i
$83$	−32.8329	+	32.8329i	−0.395577	+	0.395577i	−0.876670	−	0.481093i	$-0.840240\pi$
$83$	−32.8329	+	32.8329i	−0.395577	+	0.395577i	0.481093	+	0.876670i	$0.340240\pi$
$84$	147.748	−	147.748i	1.75891	−	1.75891i
$85$	7.03562	+	7.03562i	0.0827720	+	0.0827720i
$86$	114.915	+	114.915i	1.33622	+	1.33622i
$87$	126.000			1.44828
$88$			66.0000i			0.750000i
$89$	117.260	+	117.260i	1.31753	+	1.31753i	0.915723	+	0.401809i	$0.131618\pi$
$89$	117.260	+	117.260i	1.31753	+	1.31753i	0.401809	+	0.915723i	$0.368382\pi$
$90$		0			0
$91$		0			0
$92$	−84.0000			−0.913043
$93$	−84.4275	+	84.4275i	−0.907822	+	0.907822i
$94$	11.0000			0.117021
$95$			66.0000i			0.694737i
$96$	−49.2494	+	49.2494i	−0.513014	+	0.513014i
$97$	14.0712	−	14.0712i	0.145064	−	0.145064i	−0.630845	−	0.775909i	$-0.717292\pi$
$97$	14.0712	−	14.0712i	0.145064	−	0.145064i	0.775909	+	0.630845i	$0.217292\pi$
$98$	117.260	+	117.260i	1.19653	+	1.19653i
$99$		0			0
$100$	98.0000			0.980000
$101$			144.000i			1.42574i	0.701295	+	0.712871i	$0.252606\pi$
$101$			144.000i			1.42574i	−0.701295	+	0.712871i	$0.747394\pi$
$102$	21.1069	+	21.1069i	0.206930	+	0.206930i
$103$			26.0000i			0.252427i	0.992003	+	0.126214i	$0.0402825\pi$
$103$			26.0000i			0.252427i	−0.992003	+	0.126214i	$0.959718\pi$
$104$		0			0
$105$	−99.0000			−0.942857
$106$	−56.2850	+	56.2850i	−0.530990	+	0.530990i
$107$	114.000			1.06542			0.532710	−	0.846298i	$-0.321174\pi$
$107$	114.000			1.06542			0.532710	+	0.846298i	$0.321174\pi$
$108$		−	189.000i		−	1.75000i
$109$	119.606	−	119.606i	1.09730	−	1.09730i	0.102574	−	0.994725i	$-0.467292\pi$
$109$	119.606	−	119.606i	1.09730	−	1.09730i	0.994725	−	0.102574i	$-0.0327077\pi$
$110$	51.5946	−	51.5946i	0.469042	−	0.469042i
$111$	105.534	+	105.534i	0.950760	+	0.950760i
$112$	35.1781	+	35.1781i	0.314090	+	0.314090i
$113$	−126.000			−1.11504			−0.557522	−	0.830162i	$-0.688248\pi$
$113$	−126.000			−1.11504			−0.557522	+	0.830162i	$0.688248\pi$
$114$			198.000i			1.73684i
$115$	28.1425	+	28.1425i	0.244717	+	0.244717i
$116$			294.000i			2.53448i
$117$		0			0
$118$	−176.000			−1.49153
$119$	−21.1069	+	21.1069i	−0.177369	+	0.177369i
$120$	−99.0000			−0.825000
$121$		−	77.0000i		−	0.636364i
$122$	70.3562	−	70.3562i	0.576690	−	0.576690i
$123$	−98.4987	+	98.4987i	−0.800803	+	0.800803i
$124$	−196.997	−	196.997i	−1.58869	−	1.58869i
$125$	−91.4631	−	91.4631i	−0.731705	−	0.731705i
$126$		0			0
$127$		−	168.000i		−	1.32283i	−0.750018	−	0.661417i	$-0.769955\pi$
$127$		−	168.000i		−	1.32283i	0.750018	−	0.661417i	$-0.230045\pi$
$128$	−161.819	−	161.819i	−1.26421	−	1.26421i
$129$		−	147.000i		−	1.13953i
$130$		0			0
$131$	93.0000			0.709924			0.354962	−	0.934881i	$-0.384494\pi$
$131$	93.0000			0.709924			0.354962	+	0.934881i	$0.384494\pi$
$132$	98.4987	−	98.4987i	0.746203	−	0.746203i
$133$	−198.000			−1.48872
$134$			132.000i			0.985075i
$135$	−63.3206	+	63.3206i	−0.469042	+	0.469042i
$136$	−21.1069	+	21.1069i	−0.155198	+	0.155198i
$137$	−37.5233	−	37.5233i	−0.273893	−	0.273893i	0.556772	−	0.830665i	$-0.312040\pi$
$137$	−37.5233	−	37.5233i	−0.273893	−	0.273893i	−0.830665	+	0.556772i	$0.812040\pi$
$138$	84.4275	+	84.4275i	0.611793	+	0.611793i
$139$	−35.0000			−0.251799			−0.125899	−	0.992043i	$-0.540182\pi$
$139$	−35.0000			−0.251799			−0.125899	+	0.992043i	$0.540182\pi$
$140$		−	231.000i		−	1.65000i
$141$	−7.03562	−	7.03562i	−0.0498980	−	0.0498980i
$142$			77.0000i			0.542254i
$143$		0			0
$144$		0			0
$145$	98.4987	−	98.4987i	0.679302	−	0.679302i
$146$	−132.000			−0.904110
$147$		−	150.000i		−	1.02041i
$148$	−246.247	+	246.247i	−1.66383	+	1.66383i
$149$	−46.9042	+	46.9042i	−0.314793	+	0.314793i	−0.846763	−	0.531970i	$-0.821452\pi$
$149$	−46.9042	+	46.9042i	−0.314793	+	0.314793i	0.531970	+	0.846763i	$0.321452\pi$
$150$	−98.4987	−	98.4987i	−0.656658	−	0.656658i
$151$	−119.606	−	119.606i	−0.792090	−	0.792090i	0.189744	−	0.981834i	$-0.439234\pi$
$151$	−119.606	−	119.606i	−0.792090	−	0.792090i	−0.981834	+	0.189744i	$0.939234\pi$
$152$	−198.000			−1.30263
$153$		0			0
$154$	154.784	+	154.784i	1.00509	+	1.00509i
$155$			132.000i			0.851613i
$156$		0			0
$157$	80.0000			0.509554			0.254777	−	0.967000i	$-0.417998\pi$
$157$	80.0000			0.509554			0.254777	+	0.967000i	$0.417998\pi$
$158$	126.641	−	126.641i	0.801527	−	0.801527i
$159$	72.0000			0.452830
$160$			77.0000i			0.481250i
$161$	−84.4275	+	84.4275i	−0.524394	+	0.524394i
$162$	−189.962	+	189.962i	−1.17260	+	1.17260i
$163$	84.4275	+	84.4275i	0.517960	+	0.517960i	0.916954	−	0.398994i	$-0.130640\pi$
$163$	84.4275	+	84.4275i	0.517960	+	0.517960i	−0.398994	+	0.916954i	$0.630640\pi$
$164$	−229.830	−	229.830i	−1.40140	−	1.40140i
$165$	−66.0000			−0.400000
$166$			154.000i			0.927711i
$167$	−65.6658	−	65.6658i	−0.393209	−	0.393209i	0.482621	−	0.875829i	$-0.339685\pi$
$167$	−65.6658	−	65.6658i	−0.393209	−	0.393209i	−0.875829	+	0.482621i	$0.839685\pi$
$168$		−	297.000i		−	1.76786i
$169$		0			0
$170$	33.0000			0.194118
$171$		0			0
$172$	343.000			1.99419
$173$			114.000i			0.658960i	0.944163	+	0.329480i	$0.106873\pi$
$173$			114.000i			0.658960i	−0.944163	+	0.329480i	$0.893127\pi$
$174$	295.496	−	295.496i	1.69825	−	1.69825i
$175$	98.4987	−	98.4987i	0.562850	−	0.562850i
$176$	23.4521	+	23.4521i	0.133250	+	0.133250i
$177$	112.570	+	112.570i	0.635989	+	0.635989i
$178$	550.000			3.08989
$179$		−	129.000i		−	0.720670i	−0.932823	−	0.360335i	$-0.882662\pi$
$179$		−	129.000i		−	0.720670i	0.932823	−	0.360335i	$-0.117338\pi$
$180$		0			0
$181$		−	182.000i		−	1.00552i	−0.864425	−	0.502762i	$-0.832317\pi$
$181$		−	182.000i		−	1.00552i	0.864425	−	0.502762i	$-0.167683\pi$
$182$		0			0
$183$	−90.0000			−0.491803
$184$	−84.4275	+	84.4275i	−0.458845	+	0.458845i
$185$	165.000			0.891892
$186$			396.000i			2.12903i
$187$	−14.0712	+	14.0712i	−0.0752473	+	0.0752473i
$188$	16.4165	−	16.4165i	0.0873216	−	0.0873216i
$189$	−189.962	−	189.962i	−1.00509	−	1.00509i
$190$	154.784	+	154.784i	0.814651	+	0.814651i
$191$	30.0000			0.157068			0.0785340	−	0.996911i	$-0.474976\pi$
$191$	30.0000			0.157068			0.0785340	+	0.996911i	$0.474976\pi$
$192$			291.000i			1.51562i
$193$	−126.641	−	126.641i	−0.656172	−	0.656172i	0.298300	−	0.954472i	$-0.403580\pi$
$193$	−126.641	−	126.641i	−0.656172	−	0.656172i	−0.954472	+	0.298300i	$0.903580\pi$
$194$		−	66.0000i		−	0.340206i
$195$		0			0
$196$	350.000			1.78571
$197$	−180.581	+	180.581i	−0.916655	+	0.916655i	−0.996784	−	0.0801296i	$-0.974467\pi$
$197$	−180.581	+	180.581i	−0.916655	+	0.916655i	0.0801296	+	0.996784i	$0.474467\pi$
$198$		0			0
$199$		−	276.000i		−	1.38693i	−0.720488	−	0.693467i	$-0.756082\pi$
$199$		−	276.000i		−	1.38693i	0.720488	−	0.693467i	$-0.243918\pi$
$200$	98.4987	−	98.4987i	0.492494	−	0.492494i
$201$	84.4275	−	84.4275i	0.420037	−	0.420037i
$202$	337.710	+	337.710i	1.67183	+	1.67183i
$203$	295.496	+	295.496i	1.45565	+	1.45565i
$204$	63.0000			0.308824
$205$			154.000i			0.751220i
$206$	60.9754	+	60.9754i	0.295997	+	0.295997i
$207$		0			0
$208$		0			0
$209$	−132.000			−0.631579
$210$	−232.176	+	232.176i	−1.10560	+	1.10560i
$211$	231.000			1.09479			0.547393	−	0.836875i	$-0.315620\pi$
$211$	231.000			1.09479			0.547393	+	0.836875i	$0.315620\pi$
$212$			168.000i			0.792453i
$213$	49.2494	−	49.2494i	0.231218	−	0.231218i
$214$	267.354	−	267.354i	1.24932	−	1.24932i
$215$	−114.915	−	114.915i	−0.534489	−	0.534489i
$216$	−189.962	−	189.962i	−0.879453	−	0.879453i
$217$	−396.000			−1.82488
$218$		−	561.000i		−	2.57339i
$219$	84.4275	+	84.4275i	0.385514	+	0.385514i
$220$		−	154.000i		−	0.700000i
$221$		0			0
$222$	495.000			2.22973
$223$	246.247	−	246.247i	1.10425	−	1.10425i	0.110353	−	0.993892i	$-0.464802\pi$
$223$	246.247	−	246.247i	1.10425	−	1.10425i	0.993892	−	0.110353i	$-0.0351983\pi$
$224$	−231.000			−1.03125
$225$		0			0
$226$	−295.496	+	295.496i	−1.30751	+	1.30751i
$227$	136.022	−	136.022i	0.599216	−	0.599216i	−0.340888	−	0.940104i	$-0.610728\pi$
$227$	136.022	−	136.022i	0.599216	−	0.599216i	0.940104	+	0.340888i	$0.110728\pi$
$228$	295.496	+	295.496i	1.29604	+	1.29604i
$229$	63.3206	+	63.3206i	0.276509	+	0.276509i	0.831714	−	0.555205i	$-0.187360\pi$
$229$	63.3206	+	63.3206i	0.276509	+	0.276509i	−0.555205	+	0.831714i	$0.687360\pi$
$230$	132.000			0.573913
$231$		−	198.000i		−	0.857143i
$232$	295.496	+	295.496i	1.27369	+	1.27369i
$233$		−	429.000i		−	1.84120i	−0.390505	−	0.920601i	$-0.627700\pi$
$233$		−	429.000i		−	1.84120i	0.390505	−	0.920601i	$-0.372300\pi$
$234$		0			0
$235$	−11.0000			−0.0468085
$236$	−262.663	+	262.663i	−1.11298	+	1.11298i
$237$	−162.000			−0.683544
$238$			99.0000i			0.415966i
$239$	180.581	−	180.581i	0.755569	−	0.755569i	−0.219944	−	0.975513i	$-0.570587\pi$
$239$	180.581	−	180.581i	0.755569	−	0.755569i	0.975513	+	0.219944i	$0.0705873\pi$
$240$	−35.1781	+	35.1781i	−0.146575	+	0.146575i
$241$	267.354	+	267.354i	1.10935	+	1.10935i	0.993236	+	0.116116i	$0.0370444\pi$
$241$	267.354	+	267.354i	1.10935	+	1.10935i	0.116116	+	0.993236i	$0.462956\pi$
$242$	−180.581	−	180.581i	−0.746203	−	0.746203i
$243$		0			0
$244$		−	210.000i		−	0.860656i
$245$	−117.260	−	117.260i	−0.478614	−	0.478614i
$246$			462.000i			1.87805i
$247$		0			0
$248$	−396.000			−1.59677
$249$	98.4987	−	98.4987i	0.395577	−	0.395577i
$250$	−429.000			−1.71600
$251$		−	42.0000i		−	0.167331i	−0.996494	−	0.0836653i	$-0.973337\pi$
$251$		−	42.0000i		−	0.167331i	0.996494	−	0.0836653i	$-0.0266627\pi$
$252$		0			0
$253$	−56.2850	+	56.2850i	−0.222470	+	0.222470i
$254$	−393.995	−	393.995i	−1.55116	−	1.55116i
$255$	−21.1069	−	21.1069i	−0.0827720	−	0.0827720i
$256$	−371.000			−1.44922
$257$			261.000i			1.01556i	0.861486	+	0.507782i	$0.169534\pi$
$257$			261.000i			1.01556i	−0.861486	+	0.507782i	$0.830466\pi$
$258$	−344.746	−	344.746i	−1.33622	−	1.33622i
$259$			495.000i			1.91120i
$260$		0			0
$261$		0			0
$262$	218.104	−	218.104i	0.832459	−	0.832459i
$263$	270.000			1.02662			0.513308	−	0.858204i	$-0.328420\pi$
$263$	270.000			1.02662			0.513308	+	0.858204i	$0.328420\pi$
$264$		−	198.000i		−	0.750000i
$265$	56.2850	−	56.2850i	0.212396	−	0.212396i
$266$	−464.351	+	464.351i	−1.74568	+	1.74568i
$267$	−351.781	−	351.781i	−1.31753	−	1.31753i
$268$	196.997	+	196.997i	0.735065	+	0.735065i
$269$	186.000			0.691450			0.345725	−	0.938336i	$-0.387633\pi$
$269$	186.000			0.691450			0.345725	+	0.938336i	$0.387633\pi$
$270$			297.000i			1.10000i
$271$	−35.1781	−	35.1781i	−0.129809	−	0.129809i	0.639217	−	0.769026i	$-0.279258\pi$
$271$	−35.1781	−	35.1781i	−0.129809	−	0.129809i	−0.769026	+	0.639217i	$0.779258\pi$
$272$			15.0000i			0.0551471i
$273$		0			0
$274$	−176.000			−0.642336
$275$	65.6658	−	65.6658i	0.238785	−	0.238785i
$276$	252.000			0.913043
$277$			374.000i			1.35018i	0.737735	+	0.675090i	$0.235895\pi$
$277$			374.000i			1.35018i	−0.737735	+	0.675090i	$0.764105\pi$
$278$	−82.0823	+	82.0823i	−0.295260	+	0.295260i
$279$		0			0
$280$	−232.176	−	232.176i	−0.829199	−	0.829199i
$281$	32.8329	+	32.8329i	0.116843	+	0.116843i	0.763111	−	0.646268i	$-0.223671\pi$
$281$	32.8329	+	32.8329i	0.116843	+	0.116843i	−0.646268	+	0.763111i	$0.723671\pi$
$282$	−33.0000			−0.117021
$283$			66.0000i			0.233216i	0.993178	+	0.116608i	$0.0372020\pi$
$283$			66.0000i			0.233216i	−0.993178	+	0.116608i	$0.962798\pi$
$284$	114.915	+	114.915i	0.404631	+	0.404631i
$285$		−	198.000i		−	0.694737i
$286$		0			0
$287$	−462.000			−1.60976
$288$		0			0
$289$	280.000			0.968858
$290$		−	462.000i		−	1.59310i
$291$	−42.2137	+	42.2137i	−0.145064	+	0.145064i
$292$	−196.997	+	196.997i	−0.674649	+	0.674649i
$293$	114.915	+	114.915i	0.392202	+	0.392202i	0.875472	−	0.483270i	$-0.160551\pi$
$293$	114.915	+	114.915i	0.392202	+	0.392202i	−0.483270	+	0.875472i	$0.660551\pi$
$294$	−351.781	−	351.781i	−1.19653	−	1.19653i
$295$	176.000			0.596610
$296$			495.000i			1.67230i
$297$	−126.641	−	126.641i	−0.426401	−	0.426401i
$298$			220.000i			0.738255i
$299$		0			0
$300$	−294.000			−0.980000
$301$	344.746	−	344.746i	1.14533	−	1.14533i
$302$	−561.000			−1.85762
$303$		−	432.000i		−	1.42574i
$304$	−70.3562	+	70.3562i	−0.231435	+	0.231435i
$305$	−70.3562	+	70.3562i	−0.230676	+	0.230676i
$306$		0			0
$307$	−393.995	−	393.995i	−1.28337	−	1.28337i	−0.938737	−	0.344634i	$-0.888003\pi$
$307$	−393.995	−	393.995i	−1.28337	−	1.28337i	−0.344634	−	0.938737i	$-0.611997\pi$
$308$	462.000			1.50000
$309$		−	78.0000i		−	0.252427i
$310$	309.567	+	309.567i	0.998605	+	0.998605i
$311$			234.000i			0.752412i	0.926536	+	0.376206i	$0.122771\pi$
$311$			234.000i			0.752412i	−0.926536	+	0.376206i	$0.877229\pi$
$312$		0			0
$313$	93.0000			0.297125			0.148562	−	0.988903i	$-0.452535\pi$
$313$	93.0000			0.297125			0.148562	+	0.988903i	$0.452535\pi$
$314$	187.617	−	187.617i	0.597505	−	0.597505i
$315$		0			0
$316$		−	378.000i		−	1.19620i
$317$	−215.759	+	215.759i	−0.680628	+	0.680628i	−0.960142	−	0.279514i	$-0.909827\pi$
$317$	−215.759	+	215.759i	−0.680628	+	0.680628i	0.279514	+	0.960142i	$0.409827\pi$
$318$	168.855	−	168.855i	0.530990	−	0.530990i
$319$	196.997	+	196.997i	0.617547	+	0.617547i
$320$	227.485	+	227.485i	0.710891	+	0.710891i
$321$	−342.000			−1.06542
$322$			396.000i			1.22981i
$323$	−42.2137	−	42.2137i	−0.130693	−	0.130693i
$324$			567.000i			1.75000i
$325$		0			0
$326$	396.000			1.21472
$327$	−358.817	+	358.817i	−1.09730	+	1.09730i
$328$	−462.000			−1.40854
$329$		−	33.0000i		−	0.100304i
$330$	−154.784	+	154.784i	−0.469042	+	0.469042i
$331$	14.0712	−	14.0712i	0.0425113	−	0.0425113i	−0.685532	−	0.728043i	$-0.740430\pi$
$331$	14.0712	−	14.0712i	0.0425113	−	0.0425113i	0.728043	+	0.685532i	$0.240430\pi$
$332$	229.830	+	229.830i	0.692260	+	0.692260i
$333$		0			0
$334$	−308.000			−0.922156
$335$		−	132.000i		−	0.394030i
$336$	−105.534	−	105.534i	−0.314090	−	0.314090i
$337$		−	429.000i		−	1.27300i	−0.771278	−	0.636499i	$-0.780382\pi$
$337$		−	429.000i		−	1.27300i	0.771278	−	0.636499i	$-0.219618\pi$
$338$		0			0
$339$	378.000			1.11504
$340$	49.2494	−	49.2494i	0.144851	−	0.144851i
$341$	−264.000			−0.774194
$342$		0			0
$343$	7.03562	−	7.03562i	0.0205120	−	0.0205120i
$344$	344.746	−	344.746i	1.00217	−	1.00217i
$345$	−84.4275	−	84.4275i	−0.244717	−	0.244717i
$346$	267.354	+	267.354i	0.772699	+	0.772699i
$347$	−165.000			−0.475504			−0.237752	−	0.971326i	$-0.576411\pi$
$347$	−165.000			−0.475504			−0.237752	+	0.971326i	$0.576411\pi$
$348$		−	882.000i		−	2.53448i
$349$	147.748	+	147.748i	0.423347	+	0.423347i	0.886354	−	0.463007i	$-0.153230\pi$
$349$	147.748	+	147.748i	0.423347	+	0.423347i	−0.463007	+	0.886354i	$0.653230\pi$
$350$		−	462.000i		−	1.32000i
$351$		0			0
$352$	−154.000			−0.437500
$353$	−454.970	+	454.970i	−1.28887	+	1.28887i	−0.353393	+	0.935475i	$0.614972\pi$
$353$	−454.970	+	454.970i	−1.28887	+	1.28887i	−0.935475	+	0.353393i	$0.885028\pi$
$354$	528.000			1.49153
$355$		−	77.0000i		−	0.216901i
$356$	820.823	−	820.823i	2.30568	−	2.30568i
$357$	63.3206	−	63.3206i	0.177369	−	0.177369i
$358$	−302.532	−	302.532i	−0.845061	−	0.845061i
$359$	215.759	+	215.759i	0.601000	+	0.601000i	0.940578	−	0.339578i	$-0.110284\pi$
$359$	215.759	+	215.759i	0.601000	+	0.601000i	−0.339578	+	0.940578i	$0.610284\pi$
$360$		0			0
$361$		−	35.0000i		−	0.0969529i
$362$	−426.828	−	426.828i	−1.17908	−	1.17908i
$363$			231.000i			0.636364i
$364$		0			0
$365$	132.000			0.361644
$366$	−211.069	+	211.069i	−0.576690	+	0.576690i
$367$	−536.000			−1.46049			−0.730245	−	0.683185i	$-0.760594\pi$
$367$	−536.000			−1.46049			−0.730245	+	0.683185i	$0.760594\pi$
$368$			60.0000i			0.163043i
$369$		0			0
$370$	386.959	−	386.959i	1.04584	−	1.04584i
$371$	168.855	+	168.855i	0.455135	+	0.455135i
$372$	590.992	+	590.992i	1.58869	+	1.58869i
$373$	264.000			0.707775			0.353887	−	0.935288i	$-0.384860\pi$
$373$	264.000			0.707775			0.353887	+	0.935288i	$0.384860\pi$
$374$			66.0000i			0.176471i
$375$	274.389	+	274.389i	0.731705	+	0.731705i
$376$		−	33.0000i		−	0.0877660i
$377$		0			0
$378$	−891.000			−2.35714
$379$	−393.995	+	393.995i	−1.03956	+	1.03956i	−0.0403800	+	0.999184i	$0.512857\pi$
$379$	−393.995	+	393.995i	−1.03956	+	1.03956i	−0.999184	+	0.0403800i	$0.987143\pi$
$380$	462.000			1.21579
$381$			504.000i			1.32283i
$382$	70.3562	−	70.3562i	0.184179	−	0.184179i
$383$	−199.343	+	199.343i	−0.520477	+	0.520477i	−0.917715	−	0.397238i	$-0.869969\pi$
$383$	−199.343	+	199.343i	−0.520477	+	0.520477i	0.397238	+	0.917715i	$0.369969\pi$
$384$	485.458	+	485.458i	1.26421	+	1.26421i
$385$	−154.784	−	154.784i	−0.402036	−	0.402036i
$386$	−594.000			−1.53886
$387$		0			0
$388$	−98.4987	−	98.4987i	−0.253863	−	0.253863i
$389$			312.000i			0.802057i	0.916066	+	0.401028i	$0.131347\pi$
$389$			312.000i			0.802057i	−0.916066	+	0.401028i	$0.868653\pi$
$390$		0			0
$391$	−36.0000			−0.0920716
$392$	351.781	−	351.781i	0.897401	−	0.897401i
$393$	−279.000			−0.709924
$394$			847.000i			2.14975i
$395$	−126.641	+	126.641i	−0.320611	+	0.320611i
$396$		0			0
$397$	84.4275	+	84.4275i	0.212664	+	0.212664i	0.805398	−	0.592734i	$-0.201952\pi$
$397$	84.4275	+	84.4275i	0.212664	+	0.212664i	−0.592734	+	0.805398i	$0.701952\pi$
$398$	−647.277	−	647.277i	−1.62633	−	1.62633i
$399$	594.000			1.48872
$400$		−	70.0000i		−	0.175000i
$401$	544.088	+	544.088i	1.35683	+	1.35683i	0.877799	+	0.479030i	$0.159011\pi$
$401$	544.088	+	544.088i	1.35683	+	1.35683i	0.479030	+	0.877799i	$0.340989\pi$
$402$		−	396.000i		−	0.985075i
$403$		0			0
$404$	1008.00			2.49505
$405$	189.962	−	189.962i	0.469042	−	0.469042i
$406$	1386.00			3.41379
$407$			330.000i			0.810811i
$408$	63.3206	−	63.3206i	0.155198	−	0.155198i
$409$	−168.855	+	168.855i	−0.412848	+	0.412848i	−0.882730	−	0.469881i	$-0.844297\pi$
$409$	−168.855	+	168.855i	−0.412848	+	0.412848i	0.469881	+	0.882730i	$0.344297\pi$
$410$	361.162	+	361.162i	0.880883	+	0.880883i
$411$	112.570	+	112.570i	0.273893	+	0.273893i
$412$	182.000			0.441748
$413$			528.000i			1.27845i
$414$		0			0
$415$		−	154.000i		−	0.371084i
$416$		0			0
$417$	105.000			0.251799
$418$	−309.567	+	309.567i	−0.740592	+	0.740592i
$419$	387.000			0.923628			0.461814	−	0.886977i	$-0.347199\pi$
$419$	387.000			0.923628			0.461814	+	0.886977i	$0.347199\pi$
$420$			693.000i			1.65000i
$421$	−63.3206	+	63.3206i	−0.150405	+	0.150405i	−0.778299	−	0.627894i	$-0.783917\pi$
$421$	−63.3206	+	63.3206i	−0.150405	+	0.150405i	0.627894	+	0.778299i	$0.283917\pi$
$422$	541.743	−	541.743i	1.28375	−	1.28375i
$423$		0			0
$424$	168.855	+	168.855i	0.398243	+	0.398243i
$425$	42.0000			0.0988235
$426$		−	231.000i		−	0.542254i
$427$	−211.069	−	211.069i	−0.494306	−	0.494306i
$428$		−	798.000i		−	1.86449i
$429$		0			0
$430$	−539.000			−1.25349
$431$	307.222	−	307.222i	0.712813	−	0.712813i	−0.254310	−	0.967123i	$-0.581848\pi$
$431$	307.222	−	307.222i	0.712813	−	0.712813i	0.967123	+	0.254310i	$0.0818484\pi$
$432$	−135.000			−0.312500
$433$		−	315.000i		−	0.727483i	−0.931500	−	0.363741i	$-0.881499\pi$
$433$		−	315.000i		−	0.727483i	0.931500	−	0.363741i	$-0.118501\pi$
$434$	−928.702	+	928.702i	−2.13987	+	2.13987i
$435$	−295.496	+	295.496i	−0.679302	+	0.679302i
$436$	−837.239	−	837.239i	−1.92027	−	1.92027i
$437$	−168.855	−	168.855i	−0.386396	−	0.386396i
$438$	396.000			0.904110
$439$		−	688.000i		−	1.56720i	−0.621267	−	0.783599i	$-0.713382\pi$
$439$		−	688.000i		−	1.56720i	0.621267	−	0.783599i	$-0.286618\pi$
$440$	−154.784	−	154.784i	−0.351781	−	0.351781i
$441$		0			0
$442$		0			0
$443$	327.000			0.738149			0.369074	−	0.929400i	$-0.379675\pi$
$443$	327.000			0.738149			0.369074	+	0.929400i	$0.379675\pi$
$444$	738.740	−	738.740i	1.66383	−	1.66383i
$445$	−550.000			−1.23596
$446$		−	1155.00i		−	2.58969i
$447$	140.712	−	140.712i	0.314793	−	0.314793i
$448$	−682.455	+	682.455i	−1.52334	+	1.52334i
$449$	−342.400	−	342.400i	−0.762584	−	0.762584i	0.214205	−	0.976789i	$-0.431284\pi$
$449$	−342.400	−	342.400i	−0.762584	−	0.762584i	−0.976789	+	0.214205i	$0.931284\pi$
$450$		0			0
$451$	−308.000			−0.682927
$452$			882.000i			1.95133i
$453$	358.817	+	358.817i	0.792090	+	0.792090i
$454$		−	638.000i		−	1.40529i
$455$		0			0
$456$	594.000			1.30263
$457$	−211.069	+	211.069i	−0.461857	+	0.461857i	−0.899264	−	0.437407i	$-0.855897\pi$
$457$	−211.069	+	211.069i	−0.461857	+	0.461857i	0.437407	+	0.899264i	$0.355897\pi$
$458$	297.000			0.648472
$459$		−	81.0000i		−	0.176471i
$460$	196.997	−	196.997i	0.428255	−	0.428255i
$461$	166.510	−	166.510i	0.361193	−	0.361193i	−0.503059	−	0.864252i	$-0.667792\pi$
$461$	166.510	−	166.510i	0.361193	−	0.361193i	0.864252	+	0.503059i	$0.167792\pi$
$462$	−464.351	−	464.351i	−1.00509	−	1.00509i
$463$	−393.995	−	393.995i	−0.850961	−	0.850961i	0.139291	−	0.990252i	$-0.455518\pi$
$463$	−393.995	−	393.995i	−0.850961	−	0.850961i	−0.990252	+	0.139291i	$0.955518\pi$
$464$	210.000			0.452586
$465$		−	396.000i		−	0.851613i
$466$	−1006.09	−	1006.09i	−2.15900	−	2.15900i
$467$			390.000i			0.835118i	0.908650	+	0.417559i	$0.137114\pi$
$467$			390.000i			0.835118i	−0.908650	+	0.417559i	$0.862886\pi$
$468$		0			0
$469$	396.000			0.844350
$470$	−25.7973	+	25.7973i	−0.0548878	+	0.0548878i
$471$	−240.000			−0.509554
$472$			528.000i			1.11864i
$473$	229.830	−	229.830i	0.485899	−	0.485899i
$474$	−379.924	+	379.924i	−0.801527	+	0.801527i
$475$	196.997	+	196.997i	0.414731	+	0.414731i
$476$	147.748	+	147.748i	0.310395	+	0.310395i
$477$		0			0
$478$		−	847.000i		−	1.77197i
$479$	−340.055	−	340.055i	−0.709927	−	0.709927i	0.256592	−	0.966520i	$-0.417400\pi$
$479$	−340.055	−	340.055i	−0.709927	−	0.709927i	−0.966520	+	0.256592i	$0.917400\pi$
$480$		−	231.000i		−	0.481250i
$481$		0			0
$482$	1254.00			2.60166
$483$	253.282	−	253.282i	0.524394	−	0.524394i
$484$	−539.000			−1.11364
$485$			66.0000i			0.136082i
$486$		0			0
$487$	−168.855	+	168.855i	−0.346725	+	0.346725i	−0.858888	−	0.512163i	$-0.828844\pi$
$487$	−168.855	+	168.855i	−0.346725	+	0.346725i	0.512163	+	0.858888i	$0.328844\pi$
$488$	−211.069	−	211.069i	−0.432518	−	0.432518i
$489$	−253.282	−	253.282i	−0.517960	−	0.517960i
$490$	−550.000			−1.12245
$491$			651.000i			1.32587i	0.748679	+	0.662933i	$0.230688\pi$
$491$			651.000i			1.32587i	−0.748679	+	0.662933i	$0.769312\pi$
$492$	689.491	+	689.491i	1.40140	+	1.40140i
$493$			126.000i			0.255578i
$494$		0			0
$495$		0			0
$496$	−140.712	+	140.712i	−0.283695	+	0.283695i
$497$	231.000			0.464789
$498$		−	462.000i		−	0.927711i
$499$	211.069	−	211.069i	0.422983	−	0.422983i	−0.463246	−	0.886230i	$-0.653315\pi$
$499$	211.069	−	211.069i	0.422983	−	0.422983i	0.886230	+	0.463246i	$0.153315\pi$
$500$	−640.242	+	640.242i	−1.28048	+	1.28048i
$501$	196.997	+	196.997i	0.393209	+	0.393209i
$502$	−98.4987	−	98.4987i	−0.196213	−	0.196213i
$503$	810.000			1.61034			0.805169	−	0.593045i	$-0.202075\pi$
$503$	810.000			1.61034			0.805169	+	0.593045i	$0.202075\pi$
$504$		0			0
$505$	−337.710	−	337.710i	−0.668733	−	0.668733i
$506$			264.000i			0.521739i
$507$		0			0
$508$	−1176.00			−2.31496
$509$	215.759	−	215.759i	0.423888	−	0.423888i	−0.462652	−	0.886540i	$-0.653102\pi$
$509$	215.759	−	215.759i	0.423888	−	0.423888i	0.886540	+	0.462652i	$0.153102\pi$
$510$	−99.0000			−0.194118
$511$			396.000i			0.774951i
$512$	−222.795	+	222.795i	−0.435146	+	0.435146i
$513$	379.924	−	379.924i	0.740592	−	0.740592i
$514$	612.099	+	612.099i	1.19085	+	1.19085i
$515$	−60.9754	−	60.9754i	−0.118399	−	0.118399i
$516$	−1029.00			−1.99419
$517$		−	22.0000i		−	0.0425532i
$518$	1160.88	+	1160.88i	2.24108	+	2.24108i
$519$		−	342.000i		−	0.658960i
$520$		0			0
$521$	−843.000			−1.61804			−0.809021	−	0.587780i	$-0.800002\pi$
$521$	−843.000			−1.61804			−0.809021	+	0.587780i	$0.800002\pi$
$522$		0			0
$523$	−250.000			−0.478011			−0.239006	−	0.971018i	$-0.576821\pi$
$523$	−250.000			−0.478011			−0.239006	+	0.971018i	$0.576821\pi$
$524$		−	651.000i		−	1.24237i
$525$	−295.496	+	295.496i	−0.562850	+	0.562850i
$526$	633.206	−	633.206i	1.20381	−	1.20381i
$527$	−84.4275	−	84.4275i	−0.160204	−	0.160204i
$528$	−70.3562	−	70.3562i	−0.133250	−	0.133250i
$529$	385.000			0.727788
$530$		−	264.000i		−	0.498113i
$531$		0			0
$532$			1386.00i			2.60526i
$533$		0			0
$534$	−1650.00			−3.08989
$535$	−267.354	+	267.354i	−0.499727	+	0.499727i
$536$	396.000			0.738806
$537$			387.000i			0.720670i
$538$	436.209	−	436.209i	0.810797	−	0.810797i
$539$	234.521	−	234.521i	0.435104	−	0.435104i
$540$	443.244	+	443.244i	0.820823	+	0.820823i
$541$	429.173	+	429.173i	0.793296	+	0.793296i	0.982029	−	0.188733i	$-0.0604380\pi$
$541$	429.173	+	429.173i	0.793296	+	0.793296i	−0.188733	+	0.982029i	$0.560438\pi$
$542$	−165.000			−0.304428
$543$			546.000i			1.00552i
$544$	−49.2494	−	49.2494i	−0.0905319	−	0.0905319i
$545$			561.000i			1.02936i
$546$		0			0
$547$	301.000			0.550274			0.275137	−	0.961405i	$-0.411277\pi$
$547$	301.000			0.550274			0.275137	+	0.961405i	$0.411277\pi$
$548$	−262.663	+	262.663i	−0.479313	+	0.479313i
$549$		0			0
$550$		−	308.000i		−	0.560000i
$551$	−590.992	+	590.992i	−1.07258	+	1.07258i
$552$	253.282	−	253.282i	0.458845	−	0.458845i
$553$	−379.924	−	379.924i	−0.687023	−	0.687023i
$554$	877.108	+	877.108i	1.58323	+	1.58323i
$555$	−495.000			−0.891892
$556$			245.000i			0.440647i
$557$	−340.055	−	340.055i	−0.610512	−	0.610512i	0.332568	−	0.943079i	$-0.392085\pi$
$557$	−340.055	−	340.055i	−0.610512	−	0.610512i	−0.943079	+	0.332568i	$0.892085\pi$
$558$		0			0
$559$		0			0
$560$	−165.000			−0.294643
$561$	42.2137	−	42.2137i	0.0752473	−	0.0752473i
$562$	154.000			0.274021
$563$			1011.00i			1.79574i	0.440264	+	0.897869i	$0.354885\pi$
$563$			1011.00i			1.79574i	−0.440264	+	0.897869i	$0.645115\pi$
$564$	−49.2494	+	49.2494i	−0.0873216	+	0.0873216i
$565$	295.496	−	295.496i	0.523002	−	0.523002i
$566$	154.784	+	154.784i	0.273469	+	0.273469i
$567$	569.886	+	569.886i	1.00509	+	1.00509i
$568$	231.000			0.406690
$569$			495.000i			0.869947i	0.900443	+	0.434974i	$0.143242\pi$
$569$			495.000i			0.869947i	−0.900443	+	0.434974i	$0.856758\pi$
$570$	−464.351	−	464.351i	−0.814651	−	0.814651i
$571$		−	39.0000i		−	0.0683012i	−0.999417	−	0.0341506i	$-0.989127\pi$
$571$		−	39.0000i		−	0.0683012i	0.999417	−	0.0341506i	$-0.0108726\pi$
$572$		0			0
$573$	−90.0000			−0.157068
$574$	−1083.49	+	1083.49i	−1.88761	+	1.88761i
$575$	168.000			0.292174
$576$		0			0
$577$	759.847	−	759.847i	1.31689	−	1.31689i	0.400671	−	0.916222i	$-0.368777\pi$
$577$	759.847	−	759.847i	1.31689	−	1.31689i	0.916222	−	0.400671i	$-0.131223\pi$
$578$	656.658	−	656.658i	1.13609	−	1.13609i
$579$	379.924	+	379.924i	0.656172	+	0.656172i
$580$	−689.491	−	689.491i	−1.18878	−	1.18878i
$581$	462.000			0.795181
$582$			198.000i			0.340206i
$583$	112.570	+	112.570i	0.193087	+	0.193087i
$584$			396.000i			0.678082i
$585$		0			0
$586$	539.000			0.919795
$587$	215.759	−	215.759i	0.367562	−	0.367562i	−0.499025	−	0.866588i	$-0.666308\pi$
$587$	215.759	−	215.759i	0.367562	−	0.367562i	0.866588	+	0.499025i	$0.166308\pi$
$588$	−1050.00			−1.78571
$589$		−	792.000i		−	1.34465i
$590$	412.757	−	412.757i	0.699587	−	0.699587i
$591$	541.743	−	541.743i	0.916655	−	0.916655i
$592$	175.891	+	175.891i	0.297112	+	0.297112i
$593$	276.735	+	276.735i	0.466669	+	0.466669i	0.900833	−	0.434165i	$-0.142956\pi$
$593$	276.735	+	276.735i	0.466669	+	0.466669i	−0.434165	+	0.900833i	$0.642956\pi$
$594$	−594.000			−1.00000
$595$		−	99.0000i		−	0.166387i
$596$	328.329	+	328.329i	0.550888	+	0.550888i
$597$			828.000i			1.38693i
$598$		0			0
$599$	−648.000			−1.08180			−0.540902	−	0.841086i	$-0.681917\pi$
$599$	−648.000			−1.08180			−0.540902	+	0.841086i	$0.681917\pi$
$600$	−295.496	+	295.496i	−0.492494	+	0.492494i
$601$	75.0000			0.124792			0.0623960	−	0.998051i	$-0.480126\pi$
$601$	75.0000			0.124792			0.0623960	+	0.998051i	$0.480126\pi$
$602$		−	1617.00i		−	2.68605i
$603$		0			0
$604$	−837.239	+	837.239i	−1.38616	+	1.38616i
$605$	180.581	+	180.581i	0.298481	+	0.298481i
$606$	−1013.13	−	1013.13i	−1.67183	−	1.67183i
$607$	−100.000			−0.164745			−0.0823723	−	0.996602i	$-0.526250\pi$
$607$	−100.000			−0.164745			−0.0823723	+	0.996602i	$0.526250\pi$
$608$		−	462.000i		−	0.759868i
$609$	−886.489	−	886.489i	−1.45565	−	1.45565i
$610$			330.000i			0.540984i
$611$		0			0
$612$		0			0
$613$	−28.1425	+	28.1425i	−0.0459095	+	0.0459095i	−0.729689	−	0.683779i	$-0.760335\pi$
$613$	−28.1425	+	28.1425i	−0.0459095	+	0.0459095i	0.683779	+	0.729689i	$0.260335\pi$
$614$	−1848.00			−3.00977
$615$		−	462.000i		−	0.751220i
$616$	464.351	−	464.351i	0.753817	−	0.753817i
$617$	−656.658	+	656.658i	−1.06428	+	1.06428i	−0.0664887	+	0.997787i	$0.521180\pi$
$617$	−656.658	+	656.658i	−1.06428	+	1.06428i	−0.997787	+	0.0664887i	$0.978820\pi$
$618$	−182.926	−	182.926i	−0.295997	−	0.295997i
$619$	520.636	+	520.636i	0.841092	+	0.841092i	0.989001	−	0.147909i	$-0.0472542\pi$
$619$	520.636	+	520.636i	0.841092	+	0.841092i	−0.147909	+	0.989001i	$0.547254\pi$
$620$	924.000			1.49032
$621$		−	324.000i		−	0.521739i
$622$	548.779	+	548.779i	0.882281	+	0.882281i
$623$		−	1650.00i		−	2.64848i
$624$		0			0
$625$	79.0000			0.126400
$626$	218.104	−	218.104i	0.348409	−	0.348409i
$627$	396.000			0.631579
$628$		−	560.000i		−	0.891720i
$629$	−105.534	+	105.534i	−0.167781	+	0.167781i
$630$		0			0
$631$	−738.740	−	738.740i	−1.17075	−	1.17075i	−0.982032	−	0.188713i	$-0.939568\pi$
$631$	−738.740	−	738.740i	−1.17075	−	1.17075i	−0.188713	−	0.982032i	$-0.560432\pi$
$632$	−379.924	−	379.924i	−0.601145	−	0.601145i
$633$	−693.000			−1.09479
$634$			1012.00i			1.59621i
$635$	393.995	+	393.995i	0.620464	+	0.620464i
$636$		−	504.000i		−	0.792453i
$637$		0			0
$638$	924.000			1.44828
$639$		0			0
$640$	759.000			1.18594
$641$		−	822.000i		−	1.28237i	−0.767386	−	0.641186i	$-0.778443\pi$
$641$		−	822.000i		−	1.28237i	0.767386	−	0.641186i	$-0.221557\pi$
$642$	−802.061	+	802.061i	−1.24932	+	1.24932i
$643$	196.997	−	196.997i	0.306372	−	0.306372i	−0.537128	−	0.843501i	$-0.680491\pi$
$643$	196.997	−	196.997i	0.306372	−	0.306372i	0.843501	+	0.537128i	$0.180491\pi$
$644$	590.992	+	590.992i	0.917690	+	0.917690i
$645$	344.746	+	344.746i	0.534489	+	0.534489i
$646$	−198.000			−0.306502
$647$		−	948.000i		−	1.46522i	−0.680646	−	0.732612i	$-0.738301\pi$
$647$		−	948.000i		−	1.46522i	0.680646	−	0.732612i	$-0.261699\pi$
$648$	569.886	+	569.886i	0.879453	+	0.879453i
$649$			352.000i			0.542373i
$650$		0			0
$651$	1188.00			1.82488
$652$	590.992	−	590.992i	0.906430	−	0.906430i
$653$	348.000			0.532925			0.266462	−	0.963845i	$-0.414145\pi$
$653$	348.000			0.532925			0.266462	+	0.963845i	$0.414145\pi$
$654$			1683.00i			2.57339i
$655$	−218.104	+	218.104i	−0.332984	+	0.332984i
$656$	−164.165	+	164.165i	−0.250251	+	0.250251i
$657$		0			0
$658$	−77.3919	−	77.3919i	−0.117617	−	0.117617i
$659$	−126.000			−0.191199			−0.0955994	−	0.995420i	$-0.530477\pi$
$659$	−126.000			−0.191199			−0.0955994	+	0.995420i	$0.530477\pi$
$660$			462.000i			0.700000i
$661$	422.137	+	422.137i	0.638635	+	0.638635i	0.950219	−	0.311584i	$-0.100860\pi$
$661$	422.137	+	422.137i	0.638635	+	0.638635i	−0.311584	+	0.950219i	$0.600860\pi$
$662$		−	66.0000i		−	0.0996979i
$663$		0			0
$664$	462.000			0.695783
$665$	464.351	−	464.351i	0.698272	−	0.698272i
$666$		0			0
$667$			504.000i			0.755622i
$668$	−459.661	+	459.661i	−0.688115	+	0.688115i
$669$	−738.740	+	738.740i	−1.10425	+	1.10425i
$670$	−309.567	−	309.567i	−0.462041	−	0.462041i
$671$	−140.712	−	140.712i	−0.209706	−	0.209706i
$672$	693.000			1.03125
$673$		−	77.0000i		−	0.114413i	−0.998362	−	0.0572065i	$-0.981781\pi$
$673$		−	77.0000i		−	0.114413i	0.998362	−	0.0572065i	$-0.0182194\pi$
$674$	−1006.09	−	1006.09i	−1.49272	−	1.49272i
$675$			378.000i			0.560000i
$676$		0			0
$677$	−726.000			−1.07238			−0.536189	−	0.844098i	$-0.680137\pi$
$677$	−726.000			−1.07238			−0.536189	+	0.844098i	$0.680137\pi$
$678$	886.489	−	886.489i	1.30751	−	1.30751i
$679$	−198.000			−0.291605
$680$		−	99.0000i		−	0.145588i
$681$	−408.066	+	408.066i	−0.599216	+	0.599216i
$682$	−619.135	+	619.135i	−0.907822	+	0.907822i
$683$	−586.302	−	586.302i	−0.858422	−	0.858422i	0.132731	−	0.991152i	$-0.457626\pi$
$683$	−586.302	−	586.302i	−0.858422	−	0.858422i	−0.991152	+	0.132731i	$0.957626\pi$
$684$		0			0
$685$	176.000			0.256934
$686$		−	33.0000i		−	0.0481050i
$687$	−189.962	−	189.962i	−0.276509	−	0.276509i
$688$		−	245.000i		−	0.356105i
$689$		0			0
$690$	−396.000			−0.573913
$691$	703.562	−	703.562i	1.01818	−	1.01818i	0.0183483	−	0.999832i	$-0.494159\pi$
$691$	703.562	−	703.562i	1.01818	−	1.01818i	0.999832	−	0.0183483i	$-0.00584078\pi$
$692$	798.000			1.15318
$693$		0			0
$694$	−386.959	+	386.959i	−0.557578	+	0.557578i
$695$	82.0823	−	82.0823i	0.118104	−	0.118104i
$696$	−886.489	−	886.489i	−1.27369	−	1.27369i
$697$	−98.4987	−	98.4987i	−0.141318	−	0.141318i
$698$	693.000			0.992837
$699$			1287.00i			1.84120i
$700$	−689.491	−	689.491i	−0.984987	−	0.984987i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−990.000			−1.40825
$704$	−454.970	+	454.970i	−0.646265	+	0.646265i
$705$	33.0000			0.0468085
$706$			2134.00i			3.02266i
$707$	1013.13	−	1013.13i	1.43300	−	1.43300i
$708$	787.990	−	787.990i	1.11298	−	1.11298i
$709$	84.4275	+	84.4275i	0.119080	+	0.119080i	0.764135	−	0.645056i	$-0.223166\pi$
$709$	84.4275	+	84.4275i	0.119080	+	0.119080i	−0.645056	+	0.764135i	$0.723166\pi$
$710$	−180.581	−	180.581i	−0.254339	−	0.254339i
$711$		0			0
$712$		−	1650.00i		−	2.31742i
$713$	−337.710	−	337.710i	−0.473646	−	0.473646i
$714$		−	297.000i		−	0.415966i
$715$		0			0
$716$	−903.000			−1.26117
$717$	−541.743	+	541.743i	−0.755569	+	0.755569i
$718$	1012.00			1.40947
$719$			348.000i			0.484006i	0.970276	+	0.242003i	$0.0778043\pi$
$719$			348.000i			0.484006i	−0.970276	+	0.242003i	$0.922196\pi$
$720$		0			0
$721$	182.926	−	182.926i	0.253712	−	0.253712i
$722$	−82.0823	−	82.0823i	−0.113687	−	0.113687i
$723$	−802.061	−	802.061i	−1.10935	−	1.10935i
$724$	−1274.00			−1.75967
$725$		−	588.000i		−	0.811034i
$726$	541.743	+	541.743i	0.746203	+	0.746203i
$727$		−	702.000i		−	0.965612i	−0.875727	−	0.482806i	$-0.839618\pi$
$727$		−	702.000i		−	0.965612i	0.875727	−	0.482806i	$-0.160382\pi$
$728$		0			0
$729$	729.000			1.00000
$730$	309.567	−	309.567i	0.424065	−	0.424065i
$731$	147.000			0.201094
$732$			630.000i			0.860656i
$733$	−63.3206	+	63.3206i	−0.0863856	+	0.0863856i	−0.748979	−	0.662594i	$-0.769456\pi$
$733$	−63.3206	+	63.3206i	−0.0863856	+	0.0863856i	0.662594	+	0.748979i	$0.269456\pi$
$734$	−1257.03	+	1257.03i	−1.71258	+	1.71258i
$735$	351.781	+	351.781i	0.478614	+	0.478614i
$736$	−196.997	−	196.997i	−0.267660	−	0.267660i
$737$	264.000			0.358209
$738$		0			0
$739$	−126.641	−	126.641i	−0.171368	−	0.171368i	0.616212	−	0.787580i	$-0.288666\pi$
$739$	−126.641	−	126.641i	−0.171368	−	0.171368i	−0.787580	+	0.616212i	$0.788666\pi$
$740$		−	1155.00i		−	1.56081i
$741$		0			0
$742$	792.000			1.06739
$743$	2.34521	−	2.34521i	0.00315640	−	0.00315640i	−0.705527	−	0.708683i	$-0.749290\pi$
$743$	2.34521	−	2.34521i	0.00315640	−	0.00315640i	0.708683	+	0.705527i	$0.249290\pi$
$744$	1188.00			1.59677
$745$		−	220.000i		−	0.295302i
$746$	619.135	−	619.135i	0.829940	−	0.829940i
$747$		0			0
$748$	98.4987	+	98.4987i	0.131683	+	0.131683i
$749$	−802.061	−	802.061i	−1.07084	−	1.07084i
$750$	1287.00			1.71600
$751$			170.000i			0.226365i	0.993574	+	0.113182i	$0.0361045\pi$
$751$			170.000i			0.226365i	−0.993574	+	0.113182i	$0.963896\pi$
$752$	−11.7260	−	11.7260i	−0.0155931	−	0.0155931i
$753$			126.000i			0.167331i
$754$		0			0
$755$	561.000			0.743046
$756$	−1329.73	+	1329.73i	−1.75891	+	1.75891i
$757$	504.000			0.665786			0.332893	−	0.942965i	$-0.391975\pi$
$757$	504.000			0.665786			0.332893	+	0.942965i	$0.391975\pi$
$758$			1848.00i			2.43799i
$759$	168.855	−	168.855i	0.222470	−	0.222470i
$760$	464.351	−	464.351i	0.610988	−	0.610988i
$761$	572.231	+	572.231i	0.751946	+	0.751946i	0.974842	−	0.222896i	$-0.0715512\pi$
$761$	572.231	+	572.231i	0.751946	+	0.751946i	−0.222896	+	0.974842i	$0.571551\pi$
$762$	1181.98	+	1181.98i	1.55116	+	1.55116i
$763$	−1683.00			−2.20577
$764$		−	210.000i		−	0.274869i
$765$		0			0
$766$			935.000i			1.22063i
$767$		0			0
$768$	1113.00			1.44922
$769$	−393.995	+	393.995i	−0.512347	+	0.512347i	−0.915245	−	0.402898i	$-0.868003\pi$
$769$	−393.995	+	393.995i	−0.512347	+	0.512347i	0.402898	+	0.915245i	$0.368003\pi$
$770$	−726.000			−0.942857
$771$		−	783.000i		−	1.01556i
$772$	−886.489	+	886.489i	−1.14830	+	1.14830i
$773$	898.215	−	898.215i	1.16199	−	1.16199i	0.177945	−	0.984040i	$-0.443055\pi$
$773$	898.215	−	898.215i	1.16199	−	1.16199i	0.984040	−	0.177945i	$-0.0569449\pi$
$774$		0			0
$775$	393.995	+	393.995i	0.508381	+	0.508381i
$776$	−198.000			−0.255155

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	169.d (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.34521 - 2.34521i) q^{2} -3.00000 q^{3} -7.00000i q^{4} +(-2.34521 + 2.34521i) q^{5} +(-7.03562 + 7.03562i) q^{6} +(-7.03562 - 7.03562i) q^{7} +(-7.03562 - 7.03562i) q^{8} +O(q^{10})\)	\(q+(2.34521 - 2.34521i) q^{2} -3.00000 q^{3} -7.00000i q^{4} +(-2.34521 + 2.34521i) q^{5} +(-7.03562 + 7.03562i) q^{6} +(-7.03562 - 7.03562i) q^{7} +(-7.03562 - 7.03562i) q^{8} +11.0000i q^{10} +(-4.69042 - 4.69042i) q^{11} +21.0000i q^{12} -33.0000 q^{14} +(7.03562 - 7.03562i) q^{15} -5.00000 q^{16} -3.00000i q^{17} +(14.0712 - 14.0712i) q^{19} +(16.4165 + 16.4165i) q^{20} +(21.1069 + 21.1069i) q^{21} -22.0000 q^{22} -12.0000i q^{23} +(21.1069 + 21.1069i) q^{24} +14.0000i q^{25} +27.0000 q^{27} +(-49.2494 + 49.2494i) q^{28} -42.0000 q^{29} -33.0000i q^{30} +(28.1425 - 28.1425i) q^{31} +(16.4165 - 16.4165i) q^{32} +(14.0712 + 14.0712i) q^{33} +(-7.03562 - 7.03562i) q^{34} +33.0000 q^{35} +(-35.1781 - 35.1781i) q^{37} -66.0000i q^{38} +33.0000 q^{40} +(32.8329 - 32.8329i) q^{41} +99.0000 q^{42} +49.0000i q^{43} +(-32.8329 + 32.8329i) q^{44} +(-28.1425 - 28.1425i) q^{46} +(2.34521 + 2.34521i) q^{47} +15.0000 q^{48} +50.0000i q^{49} +(32.8329 + 32.8329i) q^{50} +9.00000i q^{51} -24.0000 q^{53} +(63.3206 - 63.3206i) q^{54} +22.0000 q^{55} +99.0000i q^{56} +(-42.2137 + 42.2137i) q^{57} +(-98.4987 + 98.4987i) q^{58} +(-37.5233 - 37.5233i) q^{59} +(-49.2494 - 49.2494i) q^{60} +30.0000 q^{61} -132.000i q^{62} -97.0000i q^{64} +66.0000 q^{66} +(-28.1425 + 28.1425i) q^{67} -21.0000 q^{68} +36.0000i q^{69} +(77.3919 - 77.3919i) q^{70} +(-16.4165 + 16.4165i) q^{71} +(-28.1425 - 28.1425i) q^{73} -165.000 q^{74} -42.0000i q^{75} +(-98.4987 - 98.4987i) q^{76} +66.0000i q^{77} +54.0000 q^{79} +(11.7260 - 11.7260i) q^{80} -81.0000 q^{81} -154.000i q^{82} +(-32.8329 + 32.8329i) q^{83} +(147.748 - 147.748i) q^{84} +(7.03562 + 7.03562i) q^{85} +(114.915 + 114.915i) q^{86} +126.000 q^{87} +66.0000i q^{88} +(117.260 + 117.260i) q^{89} -84.0000 q^{92} +(-84.4275 + 84.4275i) q^{93} +11.0000 q^{94} +66.0000i q^{95} +(-49.2494 + 49.2494i) q^{96} +(14.0712 - 14.0712i) q^{97} +(117.260 + 117.260i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3}+O(q^{10}) \) 4 * q - 12 * q^3	\( 4 q - 12 q^{3} - 132 q^{14} - 20 q^{16} - 88 q^{22} + 108 q^{27} - 168 q^{29} + 132 q^{35} + 132 q^{40} + 396 q^{42} + 60 q^{48} - 96 q^{53} + 88 q^{55} + 120 q^{61} + 264 q^{66} - 84 q^{68} - 660 q^{74} + 216 q^{79} - 324 q^{81} + 504 q^{87} - 336 q^{92} + 44 q^{94}+O(q^{100}) \) 4 * q - 12 * q^3 - 132 * q^14 - 20 * q^16 - 88 * q^22 + 108 * q^27 - 168 * q^29 + 132 * q^35 + 132 * q^40 + 396 * q^42 + 60 * q^48 - 96 * q^53 + 88 * q^55 + 120 * q^61 + 264 * q^66 - 84 * q^68 - 660 * q^74 + 216 * q^79 - 324 * q^81 + 504 * q^87 - 336 * q^92 + 44 * q^94

Introduction

L-functions

Modular forms

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