LMFDB - Embedded newform 1183.2.c.d.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1183.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.44630255912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1183.337
Dual form			1183.2.c.d.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.41421 q^{3} -4.41421i q^{5} +2.00000i q^{6} +1.00000i q^{7} -2.82843i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -1.41421 q^{3} -4.41421i q^{5} +2.00000i q^{6} +1.00000i q^{7} -2.82843i q^{8} -1.00000 q^{9} -6.24264 q^{10} -4.24264i q^{11} +1.41421 q^{14} +6.24264i q^{15} -4.00000 q^{16} +1.41421 q^{17} +1.41421i q^{18} -1.24264i q^{19} -1.41421i q^{21} -6.00000 q^{22} +0.171573 q^{23} +4.00000i q^{24} -14.4853 q^{25} +5.65685 q^{27} +5.82843 q^{29} +8.82843 q^{30} +5.24264i q^{31} +6.00000i q^{33} -2.00000i q^{34} +4.41421 q^{35} -6.24264i q^{37} -1.75736 q^{38} -12.4853 q^{40} -3.17157i q^{41} -2.00000 q^{42} +5.00000 q^{43} +4.41421i q^{45} -0.242641i q^{46} +4.41421i q^{47} +5.65685 q^{48} -1.00000 q^{49} +20.4853i q^{50} -2.00000 q^{51} -5.82843 q^{53} -8.00000i q^{54} -18.7279 q^{55} +2.82843 q^{56} +1.75736i q^{57} -8.24264i q^{58} +11.6569i q^{59} +6.00000 q^{61} +7.41421 q^{62} -1.00000i q^{63} -8.00000 q^{64} +8.48528 q^{66} -2.48528i q^{67} -0.242641 q^{69} -6.24264i q^{70} -1.07107i q^{71} +2.82843i q^{72} -0.757359i q^{73} -8.82843 q^{74} +20.4853 q^{75} +4.24264 q^{77} -1.48528 q^{79} +17.6569i q^{80} -5.00000 q^{81} -4.48528 q^{82} -4.75736i q^{83} -6.24264i q^{85} -7.07107i q^{86} -8.24264 q^{87} -12.0000 q^{88} +4.41421i q^{89} +6.24264 q^{90} -7.41421i q^{93} +6.24264 q^{94} -5.48528 q^{95} +13.7279i q^{97} +1.41421i q^{98} +4.24264i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^10 - 16 * q^16 - 24 * q^22 + 12 * q^23 - 24 * q^25 + 12 * q^29 + 24 * q^30 + 12 * q^35 - 24 * q^38 - 16 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^49 - 8 * q^51 - 12 * q^53 - 24 * q^55 + 24 * q^61 + 24 * q^62 - 32 * q^64 + 16 * q^69 - 24 * q^74 + 48 * q^75 + 28 * q^79 - 20 * q^81 + 16 * q^82 - 16 * q^87 - 48 * q^88 + 8 * q^90 + 8 * q^94 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.41421i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	− 1.41421i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	− 4.41421i	− 1.97410i	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	− 4.41421i	− 1.97410i	0.160424	−	0.987048i	$-0.448714\pi$
$6$	2.00000i	0.816497i
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 2.82843i	− 1.00000i
$9$	−1.00000	−0.333333
$10$	−6.24264	−1.97410
$11$	− 4.24264i	− 1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$11$	− 4.24264i	− 1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.41421	0.377964
$15$	6.24264i	1.61184i
$16$	−4.00000	−1.00000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	1.41421i	0.333333i
$19$	− 1.24264i	− 0.285081i	−0.989789	−	0.142541i	$-0.954473\pi$
$19$	− 1.24264i	− 0.285081i	0.989789	−	0.142541i	$-0.0455272\pi$
$20$	0	0
$21$	− 1.41421i	− 0.308607i
$22$	−6.00000	−1.27920
$23$	0.171573	0.0357754	0.0178877	−	0.999840i	$-0.494306\pi$
$23$	0.171573	0.0357754	0.0178877	+	0.999840i	$0.494306\pi$
$24$	4.00000i	0.816497i
$25$	−14.4853	−2.89706
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	5.82843	1.08231	0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843	1.08231	0.541156	+	0.840922i	$0.317987\pi$
$30$	8.82843	1.61184
$31$	5.24264i	0.941606i	0.882238	+	0.470803i	$0.156036\pi$
$31$	5.24264i	0.941606i	−0.882238	+	0.470803i	$0.843964\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	− 2.00000i	− 0.342997i
$35$	4.41421	0.746138
$36$	0	0
$37$	− 6.24264i	− 1.02628i	−0.858304	−	0.513142i	$-0.828481\pi$
$37$	− 6.24264i	− 1.02628i	0.858304	−	0.513142i	$-0.171519\pi$
$38$	−1.75736	−0.285081
$39$	0	0
$40$	−12.4853	−1.97410
$41$	− 3.17157i	− 0.495316i	−0.968847	−	0.247658i	$-0.920339\pi$
$41$	− 3.17157i	− 0.495316i	0.968847	−	0.247658i	$-0.0796610\pi$
$42$	−2.00000	−0.308607
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	4.41421i	0.658032i
$46$	− 0.242641i	− 0.0357754i
$47$	4.41421i	0.643879i	0.946760	+	0.321940i	$0.104335\pi$
$47$	4.41421i	0.643879i	−0.946760	+	0.321940i	$0.895665\pi$
$48$	5.65685	0.816497
$49$	−1.00000	−0.142857
$50$	20.4853i	2.89706i
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−5.82843	−0.800596	−0.400298	−	0.916385i	$-0.631093\pi$
$53$	−5.82843	−0.800596	−0.400298	+	0.916385i	$0.631093\pi$
$54$	− 8.00000i	− 1.08866i
$55$	−18.7279	−2.52527
$56$	2.82843	0.377964
$57$	1.75736i	0.232768i
$58$	− 8.24264i	− 1.08231i
$59$	11.6569i	1.51759i	0.651328	+	0.758797i	$0.274212\pi$
$59$	11.6569i	1.51759i	−0.651328	+	0.758797i	$0.725788\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	7.41421	0.941606
$63$	− 1.00000i	− 0.125988i
$64$	−8.00000	−1.00000
$65$	0	0
$66$	8.48528	1.04447
$67$	− 2.48528i	− 0.303625i	−0.988409	−	0.151813i	$-0.951489\pi$
$67$	− 2.48528i	− 0.303625i	0.988409	−	0.151813i	$-0.0485111\pi$
$68$	0	0
$69$	−0.242641	−0.0292105
$70$	− 6.24264i	− 0.746138i
$71$	− 1.07107i	− 0.127112i	−0.997978	−	0.0635562i	$-0.979756\pi$
$71$	− 1.07107i	− 0.127112i	0.997978	−	0.0635562i	$-0.0202442\pi$
$72$	2.82843i	0.333333i
$73$	− 0.757359i	− 0.0886422i	−0.999017	−	0.0443211i	$-0.985888\pi$
$73$	− 0.757359i	− 0.0886422i	0.999017	−	0.0443211i	$-0.0141125\pi$
$74$	−8.82843	−1.02628
$75$	20.4853	2.36544
$76$	0	0
$77$	4.24264	0.483494
$78$	0	0
$79$	−1.48528	−0.167107	−0.0835536	−	0.996503i	$-0.526627\pi$
$79$	−1.48528	−0.167107	−0.0835536	+	0.996503i	$0.526627\pi$
$80$	17.6569i	1.97410i
$81$	−5.00000	−0.555556
$82$	−4.48528	−0.495316
$83$	− 4.75736i	− 0.522188i	−0.965313	−	0.261094i	$-0.915917\pi$
$83$	− 4.75736i	− 0.522188i	0.965313	−	0.261094i	$-0.0840833\pi$
$84$	0	0
$85$	− 6.24264i	− 0.677109i
$86$	− 7.07107i	− 0.762493i
$87$	−8.24264	−0.883704
$88$	−12.0000	−1.27920
$89$	4.41421i	0.467906i	0.972248	+	0.233953i	$0.0751661\pi$
$89$	4.41421i	0.467906i	−0.972248	+	0.233953i	$0.924834\pi$
$90$	6.24264	0.658032
$91$	0	0
$92$	0	0
$93$	− 7.41421i	− 0.768818i
$94$	6.24264	0.643879
$95$	−5.48528	−0.562778
$96$	0	0
$97$	13.7279i	1.39386i	0.717139	+	0.696930i	$0.245451\pi$
$97$	13.7279i	1.39386i	−0.717139	+	0.696930i	$0.754549\pi$
$98$	1.41421i	0.142857i
$99$	4.24264i	0.426401i
$100$	0	0
$101$	−1.75736	−0.174864	−0.0874319	−	0.996170i	$-0.527866\pi$
$101$	−1.75736	−0.174864	−0.0874319	+	0.996170i	$0.527866\pi$
$102$	2.82843i	0.280056i
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−6.24264	−0.609219
$106$	8.24264i	0.800596i
$107$	8.14214	0.787130	0.393565	−	0.919297i	$-0.371242\pi$
$107$	8.14214	0.787130	0.393565	+	0.919297i	$0.371242\pi$
$108$	0	0
$109$	8.72792i	0.835983i	0.908451	+	0.417992i	$0.137266\pi$
$109$	8.72792i	0.835983i	−0.908451	+	0.417992i	$0.862734\pi$
$110$	26.4853i	2.52527i
$111$	8.82843i	0.837957i
$112$	− 4.00000i	− 0.377964i
$113$	−20.3137	−1.91095	−0.955476	−	0.295067i	$-0.904658\pi$
$113$	−20.3137	−1.91095	−0.955476	+	0.295067i	$0.904658\pi$
$114$	2.48528	0.232768
$115$	− 0.757359i	− 0.0706241i
$116$	0	0
$117$	0	0
$118$	16.4853	1.51759
$119$	1.41421i	0.129641i
$120$	17.6569	1.61184
$121$	−7.00000	−0.636364
$122$	− 8.48528i	− 0.768221i
$123$	4.48528i	0.404424i
$124$	0	0
$125$	41.8701i	3.74497i
$126$	−1.41421	−0.125988
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	11.3137i	1.00000i
$129$	−7.07107	−0.622573
$130$	0	0
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	1.24264	0.107751
$134$	−3.51472	−0.303625
$135$	− 24.9706i	− 2.14912i
$136$	− 4.00000i	− 0.342997i
$137$	− 7.41421i	− 0.633439i	−0.948519	−	0.316720i	$-0.897419\pi$
$137$	− 7.41421i	− 0.633439i	0.948519	−	0.316720i	$-0.102581\pi$
$138$	0.343146i	0.0292105i
$139$	−2.24264	−0.190218	−0.0951092	−	0.995467i	$-0.530320\pi$
$139$	−2.24264	−0.190218	−0.0951092	+	0.995467i	$0.530320\pi$
$140$	0	0
$141$	− 6.24264i	− 0.525725i
$142$	−1.51472	−0.127112
$143$	0	0
$144$	4.00000	0.333333
$145$	− 25.7279i	− 2.13659i
$146$	−1.07107	−0.0886422
$147$	1.41421	0.116642
$148$	0	0
$149$	− 7.75736i	− 0.635508i	−0.948173	−	0.317754i	$-0.897071\pi$
$149$	− 7.75736i	− 0.635508i	0.948173	−	0.317754i	$-0.102929\pi$
$150$	− 28.9706i	− 2.36544i
$151$	− 18.2426i	− 1.48457i	−0.670087	−	0.742283i	$-0.733743\pi$
$151$	− 18.2426i	− 1.48457i	0.670087	−	0.742283i	$-0.266257\pi$
$152$	−3.51472	−0.285081
$153$	−1.41421	−0.114332
$154$	− 6.00000i	− 0.483494i
$155$	23.1421	1.85882
$156$	0	0
$157$	−12.2426	−0.977069	−0.488535	−	0.872545i	$-0.662468\pi$
$157$	−12.2426	−0.977069	−0.488535	+	0.872545i	$0.662468\pi$
$158$	2.10051i	0.167107i
$159$	8.24264	0.653684
$160$	0	0
$161$	0.171573i	0.0135218i
$162$	7.07107i	0.555556i
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	26.4853	2.06188
$166$	−6.72792	−0.522188
$167$	− 21.3848i	− 1.65480i	−0.561610	−	0.827402i	$-0.689818\pi$
$167$	− 21.3848i	− 1.65480i	0.561610	−	0.827402i	$-0.310182\pi$
$168$	−4.00000	−0.308607
$169$	0	0
$170$	−8.82843	−0.677109
$171$	1.24264i	0.0950271i
$172$	0	0
$173$	0.727922	0.0553429	0.0276714	−	0.999617i	$-0.491191\pi$
$173$	0.727922	0.0553429	0.0276714	+	0.999617i	$0.491191\pi$
$174$	11.6569i	0.883704i
$175$	− 14.4853i	− 1.09498i
$176$	16.9706i	1.27920i
$177$	− 16.4853i	− 1.23911i
$178$	6.24264	0.467906
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−6.72792	−0.500083	−0.250041	−	0.968235i	$-0.580444\pi$
$181$	−6.72792	−0.500083	−0.250041	+	0.968235i	$0.580444\pi$
$182$	0	0
$183$	−8.48528	−0.627250
$184$	− 0.485281i	− 0.0357754i
$185$	−27.5563	−2.02598
$186$	−10.4853	−0.768818
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	5.65685i	0.411476i
$190$	7.75736i	0.562778i
$191$	20.8284	1.50709	0.753546	−	0.657395i	$-0.228342\pi$
$191$	20.8284	1.50709	0.753546	+	0.657395i	$0.228342\pi$
$192$	11.3137	0.816497
$193$	2.48528i	0.178894i	0.995992	+	0.0894472i	$0.0285100\pi$
$193$	2.48528i	0.178894i	−0.995992	+	0.0894472i	$0.971490\pi$
$194$	19.4142	1.39386
$195$	0	0
$196$	0	0
$197$	− 23.6569i	− 1.68548i	−0.538320	−	0.842741i	$-0.680941\pi$
$197$	− 23.6569i	− 1.68548i	0.538320	−	0.842741i	$-0.319059\pi$
$198$	6.00000	0.426401
$199$	12.2426	0.867858	0.433929	−	0.900947i	$-0.357127\pi$
$199$	12.2426	0.867858	0.433929	+	0.900947i	$0.357127\pi$
$200$	40.9706i	2.89706i
$201$	3.51472i	0.247909i
$202$	2.48528i	0.174864i
$203$	5.82843i	0.409075i
$204$	0	0
$205$	−14.0000	−0.977802
$206$	− 11.3137i	− 0.788263i
$207$	−0.171573	−0.0119251
$208$	0	0
$209$	−5.27208	−0.364677
$210$	8.82843i	0.609219i
$211$	17.9706	1.23714	0.618572	−	0.785728i	$-0.287711\pi$
$211$	17.9706	1.23714	0.618572	+	0.785728i	$0.287711\pi$
$212$	0	0
$213$	1.51472i	0.103787i
$214$	− 11.5147i	− 0.787130i
$215$	− 22.0711i	− 1.50523i
$216$	− 16.0000i	− 1.08866i
$217$	−5.24264	−0.355894
$218$	12.3431	0.835983
$219$	1.07107i	0.0723761i
$220$	0	0
$221$	0	0
$222$	12.4853	0.837957
$223$	− 9.24264i	− 0.618933i	−0.950910	−	0.309466i	$-0.899850\pi$
$223$	− 9.24264i	− 0.618933i	0.950910	−	0.309466i	$-0.100150\pi$
$224$	0	0
$225$	14.4853	0.965685
$226$	28.7279i	1.91095i
$227$	− 21.1716i	− 1.40521i	−0.711582	−	0.702603i	$-0.752021\pi$
$227$	− 21.1716i	− 1.40521i	0.711582	−	0.702603i	$-0.247979\pi$
$228$	0	0
$229$	− 21.4558i	− 1.41784i	−0.705288	−	0.708921i	$-0.749182\pi$
$229$	− 21.4558i	− 1.41784i	0.705288	−	0.708921i	$-0.250818\pi$
$230$	−1.07107	−0.0706241
$231$	−6.00000	−0.394771
$232$	− 16.4853i	− 1.08231i
$233$	3.34315	0.219017	0.109508	−	0.993986i	$-0.465072\pi$
$233$	3.34315	0.219017	0.109508	+	0.993986i	$0.465072\pi$
$234$	0	0
$235$	19.4853	1.27108
$236$	0	0
$237$	2.10051	0.136442
$238$	2.00000	0.129641
$239$	− 20.4853i	− 1.32508i	−0.749025	−	0.662541i	$-0.769478\pi$
$239$	− 20.4853i	− 1.32508i	0.749025	−	0.662541i	$-0.230522\pi$
$240$	− 24.9706i	− 1.61184i
$241$	22.2132i	1.43088i	0.698675	+	0.715439i	$0.253773\pi$
$241$	22.2132i	1.43088i	−0.698675	+	0.715439i	$0.746227\pi$
$242$	9.89949i	0.636364i
$243$	−9.89949	−0.635053
$244$	0	0
$245$	4.41421i	0.282014i
$246$	6.34315	0.404424
$247$	0	0
$248$	14.8284	0.941606
$249$	6.72792i	0.426365i
$250$	59.2132	3.74497
$251$	−19.4142	−1.22541	−0.612707	−	0.790310i	$-0.709919\pi$
$251$	−19.4142	−1.22541	−0.612707	+	0.790310i	$0.709919\pi$
$252$	0	0
$253$	− 0.727922i	− 0.0457641i
$254$	2.82843i	0.177471i
$255$	8.82843i	0.552858i
$256$	0	0
$257$	16.5858	1.03459	0.517296	−	0.855806i	$-0.326938\pi$
$257$	16.5858	1.03459	0.517296	+	0.855806i	$0.326938\pi$
$258$	10.0000i	0.622573i
$259$	6.24264	0.387899
$260$	0	0
$261$	−5.82843	−0.360771
$262$	− 4.00000i	− 0.247121i
$263$	−31.9706	−1.97139	−0.985695	−	0.168541i	$-0.946095\pi$
$263$	−31.9706	−1.97139	−0.985695	+	0.168541i	$0.946095\pi$
$264$	16.9706	1.04447
$265$	25.7279i	1.58045i
$266$	− 1.75736i	− 0.107751i
$267$	− 6.24264i	− 0.382043i
$268$	0	0
$269$	14.8284	0.904105	0.452053	−	0.891991i	$-0.350692\pi$
$269$	14.8284	0.904105	0.452053	+	0.891991i	$0.350692\pi$
$270$	−35.3137	−2.14912
$271$	20.0000i	1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$	20.0000i	1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	−5.65685	−0.342997
$273$	0	0
$274$	−10.4853	−0.633439
$275$	61.4558i	3.70593i
$276$	0	0
$277$	−9.48528	−0.569915	−0.284958	−	0.958540i	$-0.591980\pi$
$277$	−9.48528	−0.569915	−0.284958	+	0.958540i	$0.591980\pi$
$278$	3.17157i	0.190218i
$279$	− 5.24264i	− 0.313869i
$280$	− 12.4853i	− 0.746138i
$281$	− 15.5563i	− 0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	− 15.5563i	− 0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$	−8.82843	−0.525725
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	7.75736	0.459506
$286$	0	0
$287$	3.17157	0.187212
$288$	0	0
$289$	−15.0000	−0.882353
$290$	−36.3848	−2.13659
$291$	− 19.4142i	− 1.13808i
$292$	0	0
$293$	− 21.3848i	− 1.24931i	−0.780900	−	0.624656i	$-0.785239\pi$
$293$	− 21.3848i	− 1.24931i	0.780900	−	0.624656i	$-0.214761\pi$
$294$	− 2.00000i	− 0.116642i
$295$	51.4558	2.99588
$296$	−17.6569	−1.02628
$297$	− 24.0000i	− 1.39262i
$298$	−10.9706	−0.635508
$299$	0	0
$300$	0	0
$301$	5.00000i	0.288195i
$302$	−25.7990	−1.48457
$303$	2.48528	0.142776
$304$	4.97056i	0.285081i
$305$	− 26.4853i	− 1.51654i
$306$	2.00000i	0.114332i
$307$	4.75736i	0.271517i	0.990742	+	0.135758i	$0.0433471\pi$
$307$	4.75736i	0.271517i	−0.990742	+	0.135758i	$0.956653\pi$
$308$	0	0
$309$	−11.3137	−0.643614
$310$	− 32.7279i	− 1.85882i
$311$	4.58579	0.260036	0.130018	−	0.991512i	$-0.458496\pi$
$311$	4.58579	0.260036	0.130018	+	0.991512i	$0.458496\pi$
$312$	0	0
$313$	−19.2132	−1.08599	−0.542997	−	0.839734i	$-0.682711\pi$
$313$	−19.2132	−1.08599	−0.542997	+	0.839734i	$0.682711\pi$
$314$	17.3137i	0.977069i
$315$	−4.41421	−0.248713
$316$	0	0
$317$	− 11.3137i	− 0.635441i	−0.948184	−	0.317721i	$-0.897083\pi$
$317$	− 11.3137i	− 0.635441i	0.948184	−	0.317721i	$-0.102917\pi$
$318$	− 11.6569i	− 0.653684i
$319$	− 24.7279i	− 1.38450i
$320$	35.3137i	1.97410i
$321$	−11.5147	−0.642689
$322$	0.242641	0.0135218
$323$	− 1.75736i	− 0.0977821i
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	− 12.3431i	− 0.682578i
$328$	−8.97056	−0.495316
$329$	−4.41421	−0.243363
$330$	− 37.4558i	− 2.06188i
$331$	18.0000i	0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$	18.0000i	0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$	0	0
$333$	6.24264i	0.342095i
$334$	−30.2426	−1.65480
$335$	−10.9706	−0.599386
$336$	5.65685i	0.308607i
$337$	33.0000	1.79762	0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000	1.79762	0.898812	+	0.438334i	$0.144431\pi$
$338$	0	0
$339$	28.7279	1.56029
$340$	0	0
$341$	22.2426	1.20451
$342$	1.75736	0.0950271
$343$	− 1.00000i	− 0.0539949i
$344$	− 14.1421i	− 0.762493i
$345$	1.07107i	0.0576644i
$346$	− 1.02944i	− 0.0553429i
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	− 0.272078i	− 0.0145640i	−0.999973	−	0.00728200i	$-0.997682\pi$
$349$	− 0.272078i	− 0.0145640i	0.999973	−	0.00728200i	$-0.00231795\pi$
$350$	−20.4853	−1.09498
$351$	0	0
$352$	0	0
$353$	− 8.48528i	− 0.451626i	−0.974171	−	0.225813i	$-0.927496\pi$
$353$	− 8.48528i	− 0.451626i	0.974171	−	0.225813i	$-0.0725038\pi$
$354$	−23.3137	−1.23911
$355$	−4.72792	−0.250932
$356$	0	0
$357$	− 2.00000i	− 0.105851i
$358$	− 12.7279i	− 0.672692i
$359$	− 8.10051i	− 0.427528i	−0.976885	−	0.213764i	$-0.931428\pi$
$359$	− 8.10051i	− 0.427528i	0.976885	−	0.213764i	$-0.0685724\pi$
$360$	12.4853	0.658032
$361$	17.4558	0.918729
$362$	9.51472i	0.500083i
$363$	9.89949	0.519589
$364$	0	0
$365$	−3.34315	−0.174988
$366$	12.0000i	0.627250i
$367$	1.75736	0.0917334	0.0458667	−	0.998948i	$-0.485395\pi$
$367$	1.75736	0.0917334	0.0458667	+	0.998948i	$0.485395\pi$
$368$	−0.686292	−0.0357754
$369$	3.17157i	0.165105i
$370$	38.9706i	2.02598i
$371$	− 5.82843i	− 0.302597i
$372$	0	0
$373$	−8.48528	−0.439351	−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528	−0.439351	−0.219676	+	0.975573i	$0.570500\pi$
$374$	−8.48528	−0.438763
$375$	− 59.2132i	− 3.05776i
$376$	12.4853	0.643879
$377$	0	0
$378$	8.00000	0.411476
$379$	− 32.2426i	− 1.65619i	−0.560585	−	0.828097i	$-0.689424\pi$
$379$	− 32.2426i	− 1.65619i	0.560585	−	0.828097i	$-0.310576\pi$
$380$	0	0
$381$	2.82843	0.144905
$382$	− 29.4558i	− 1.50709i
$383$	3.51472i	0.179594i	0.995960	+	0.0897969i	$0.0286218\pi$
$383$	3.51472i	0.179594i	−0.995960	+	0.0897969i	$0.971378\pi$
$384$	− 16.0000i	− 0.816497i
$385$	− 18.7279i	− 0.954463i
$386$	3.51472	0.178894
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−18.3431	−0.930034	−0.465017	−	0.885302i	$-0.653952\pi$
$389$	−18.3431	−0.930034	−0.465017	+	0.885302i	$0.653952\pi$
$390$	0	0
$391$	0.242641	0.0122709
$392$	2.82843i	0.142857i
$393$	−4.00000	−0.201773
$394$	−33.4558	−1.68548
$395$	6.55635i	0.329886i
$396$	0	0
$397$	24.2132i	1.21523i	0.794233	+	0.607613i	$0.207873\pi$
$397$	24.2132i	1.21523i	−0.794233	+	0.607613i	$0.792127\pi$
$398$	− 17.3137i	− 0.867858i
$399$	−1.75736	−0.0879780
$400$	57.9411	2.89706
$401$	17.6569i	0.881741i	0.897571	+	0.440871i	$0.145330\pi$
$401$	17.6569i	0.881741i	−0.897571	+	0.440871i	$0.854670\pi$
$402$	4.97056	0.247909
$403$	0	0
$404$	0	0
$405$	22.0711i	1.09672i
$406$	8.24264	0.409075
$407$	−26.4853	−1.31283
$408$	5.65685i	0.280056i
$409$	5.24264i	0.259232i	0.991564	+	0.129616i	$0.0413744\pi$
$409$	5.24264i	0.259232i	−0.991564	+	0.129616i	$0.958626\pi$
$410$	19.7990i	0.977802i
$411$	10.4853i	0.517201i
$412$	0	0
$413$	−11.6569	−0.573596
$414$	0.242641i	0.0119251i
$415$	−21.0000	−1.03085
$416$	0	0
$417$	3.17157	0.155313
$418$	7.45584i	0.364677i
$419$	32.8701	1.60581	0.802904	−	0.596109i	$-0.203287\pi$
$419$	32.8701	1.60581	0.802904	+	0.596109i	$0.203287\pi$
$420$	0	0
$421$	18.7279i	0.912743i	0.889789	+	0.456372i	$0.150851\pi$
$421$	18.7279i	0.912743i	−0.889789	+	0.456372i	$0.849149\pi$
$422$	− 25.4142i	− 1.23714i
$423$	− 4.41421i	− 0.214626i
$424$	16.4853i	0.800596i
$425$	−20.4853	−0.993682
$426$	2.14214	0.103787
$427$	6.00000i	0.290360i
$428$	0	0
$429$	0	0
$430$	−31.2132	−1.50523
$431$	23.6569i	1.13951i	0.821814	+	0.569755i	$0.192962\pi$
$431$	23.6569i	1.13951i	−0.821814	+	0.569755i	$0.807038\pi$
$432$	−22.6274	−1.08866
$433$	−8.97056	−0.431098	−0.215549	−	0.976493i	$-0.569154\pi$
$433$	−8.97056	−0.431098	−0.215549	+	0.976493i	$0.569154\pi$
$434$	7.41421i	0.355894i
$435$	36.3848i	1.74452i
$436$	0	0
$437$	− 0.213203i	− 0.0101989i
$438$	1.51472	0.0723761
$439$	17.5147	0.835932	0.417966	−	0.908463i	$-0.362743\pi$
$439$	17.5147	0.835932	0.417966	+	0.908463i	$0.362743\pi$
$440$	52.9706i	2.52527i
$441$	1.00000	0.0476190
$442$	0	0
$443$	−26.3137	−1.25020	−0.625101	−	0.780544i	$-0.714942\pi$
$443$	−26.3137	−1.25020	−0.625101	+	0.780544i	$0.714942\pi$
$444$	0	0
$445$	19.4853	0.923691
$446$	−13.0711	−0.618933
$447$	10.9706i	0.518890i
$448$	− 8.00000i	− 0.377964i
$449$	26.8284i	1.26611i	0.774106	+	0.633056i	$0.218200\pi$
$449$	26.8284i	1.26611i	−0.774106	+	0.633056i	$0.781800\pi$
$450$	− 20.4853i	− 0.965685i
$451$	−13.4558	−0.633611
$452$	0	0
$453$	25.7990i	1.21214i
$454$	−29.9411	−1.40521
$455$	0	0
$456$	4.97056	0.232768
$457$	− 35.2132i	− 1.64720i	−0.567168	−	0.823602i	$-0.691961\pi$
$457$	− 35.2132i	− 1.64720i	0.567168	−	0.823602i	$-0.308039\pi$
$458$	−30.3431	−1.41784
$459$	8.00000	0.373408
$460$	0	0
$461$	− 3.17157i	− 0.147715i	−0.997269	−	0.0738574i	$-0.976469\pi$
$461$	− 3.17157i	− 0.147715i	0.997269	−	0.0738574i	$-0.0235310\pi$
$462$	8.48528i	0.394771i
$463$	4.24264i	0.197172i	0.995129	+	0.0985861i	$0.0314320\pi$
$463$	4.24264i	0.197172i	−0.995129	+	0.0985861i	$0.968568\pi$
$464$	−23.3137	−1.08231
$465$	−32.7279	−1.51772
$466$	− 4.72792i	− 0.219017i
$467$	−15.8995	−0.735741	−0.367870	−	0.929877i	$-0.619913\pi$
$467$	−15.8995	−0.735741	−0.367870	+	0.929877i	$0.619913\pi$
$468$	0	0
$469$	2.48528	0.114760
$470$	− 27.5563i	− 1.27108i
$471$	17.3137	0.797774
$472$	32.9706	1.51759
$473$	− 21.2132i	− 0.975384i
$474$	− 2.97056i	− 0.136442i
$475$	18.0000i	0.825897i
$476$	0	0
$477$	5.82843	0.266865
$478$	−28.9706	−1.32508
$479$	36.2132i	1.65462i	0.561743	+	0.827312i	$0.310131\pi$
$479$	36.2132i	1.65462i	−0.561743	+	0.827312i	$0.689869\pi$
$480$	0	0
$481$	0	0
$482$	31.4142	1.43088
$483$	− 0.242641i	− 0.0110405i
$484$	0	0
$485$	60.5980	2.75161
$486$	14.0000i	0.635053i
$487$	− 39.4558i	− 1.78791i	−0.448152	−	0.893957i	$-0.647918\pi$
$487$	− 39.4558i	− 1.78791i	0.448152	−	0.893957i	$-0.352082\pi$
$488$	− 16.9706i	− 0.768221i
$489$	12.0000i	0.542659i
$490$	6.24264	0.282014
$491$	28.6274	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$491$	28.6274	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$492$	0	0
$493$	8.24264	0.371230
$494$	0	0
$495$	18.7279	0.841757
$496$	− 20.9706i	− 0.941606i
$497$	1.07107	0.0480440
$498$	9.51472	0.426365
$499$	38.7279i	1.73370i	0.498569	+	0.866850i	$0.333859\pi$
$499$	38.7279i	1.73370i	−0.498569	+	0.866850i	$0.666141\pi$
$500$	0	0
$501$	30.2426i	1.35114i
$502$	27.4558i	1.22541i
$503$	16.6274	0.741380	0.370690	−	0.928757i	$-0.379121\pi$
$503$	16.6274	0.741380	0.370690	+	0.928757i	$0.379121\pi$
$504$	−2.82843	−0.125988
$505$	7.75736i	0.345198i
$506$	−1.02944	−0.0457641
$507$	0	0
$508$	0	0
$509$	24.8995i	1.10365i	0.833960	+	0.551825i	$0.186069\pi$
$509$	24.8995i	1.10365i	−0.833960	+	0.551825i	$0.813931\pi$
$510$	12.4853	0.552858
$511$	0.757359	0.0335036
$512$	22.6274i	1.00000i
$513$	− 7.02944i	− 0.310357i
$514$	− 23.4558i	− 1.03459i
$515$	− 35.3137i	− 1.55611i
$516$	0	0
$517$	18.7279	0.823653
$518$	− 8.82843i	− 0.387899i
$519$	−1.02944	−0.0451873
$520$	0	0
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	8.24264i	0.360771i
$523$	−2.97056	−0.129894	−0.0649468	−	0.997889i	$-0.520688\pi$
$523$	−2.97056	−0.129894	−0.0649468	+	0.997889i	$0.520688\pi$
$524$	0	0
$525$	20.4853i	0.894051i
$526$	45.2132i	1.97139i
$527$	7.41421i	0.322968i
$528$	− 24.0000i	− 1.04447i
$529$	−22.9706	−0.998720
$530$	36.3848	1.58045
$531$	− 11.6569i	− 0.505864i
$532$	0	0
$533$	0	0
$534$	−8.82843	−0.382043
$535$	− 35.9411i	− 1.55387i
$536$	−7.02944	−0.303625
$537$	−12.7279	−0.549250
$538$	− 20.9706i	− 0.904105i
$539$	4.24264i	0.182743i
$540$	0	0
$541$	35.2132i	1.51393i	0.653453	+	0.756967i	$0.273320\pi$
$541$	35.2132i	1.51393i	−0.653453	+	0.756967i	$0.726680\pi$
$542$	28.2843	1.21491
$543$	9.51472	0.408316
$544$	0	0
$545$	38.5269	1.65031
$546$	0	0
$547$	18.5147	0.791632	0.395816	−	0.918330i	$-0.370462\pi$
$547$	18.5147	0.791632	0.395816	+	0.918330i	$0.370462\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	86.9117	3.70593
$551$	− 7.24264i	− 0.308547i
$552$	0.686292i	0.0292105i
$553$	− 1.48528i	− 0.0631606i
$554$	13.4142i	0.569915i
$555$	38.9706	1.65421
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	−7.41421	−0.313869
$559$	0	0
$560$	−17.6569	−0.746138
$561$	8.48528i	0.358249i
$562$	−22.0000	−0.928014
$563$	17.6569	0.744148	0.372074	−	0.928203i	$-0.378647\pi$
$563$	17.6569	0.744148	0.372074	+	0.928203i	$0.378647\pi$
$564$	0	0
$565$	89.6690i	3.77241i
$566$	12.0000i	0.504398i
$567$	− 5.00000i	− 0.209980i
$568$	−3.02944	−0.127112
$569$	23.1421	0.970169	0.485084	−	0.874467i	$-0.338789\pi$
$569$	23.1421	0.970169	0.485084	+	0.874467i	$0.338789\pi$
$570$	− 10.9706i	− 0.459506i
$571$	8.45584	0.353866	0.176933	−	0.984223i	$-0.443382\pi$
$571$	8.45584	0.353866	0.176933	+	0.984223i	$0.443382\pi$
$572$	0	0
$573$	−29.4558	−1.23054
$574$	− 4.48528i	− 0.187212i
$575$	−2.48528	−0.103643
$576$	8.00000	0.333333
$577$	− 26.9706i	− 1.12280i	−0.827545	−	0.561400i	$-0.810263\pi$
$577$	− 26.9706i	− 1.12280i	0.827545	−	0.561400i	$-0.189737\pi$
$578$	21.2132i	0.882353i
$579$	− 3.51472i	− 0.146067i
$580$	0	0
$581$	4.75736	0.197369
$582$	−27.4558	−1.13808
$583$	24.7279i	1.02413i
$584$	−2.14214	−0.0886422
$585$	0	0
$586$	−30.2426	−1.24931
$587$	− 24.5563i	− 1.01355i	−0.862079	−	0.506775i	$-0.830838\pi$
$587$	− 24.5563i	− 1.01355i	0.862079	−	0.506775i	$-0.169162\pi$
$588$	0	0
$589$	6.51472	0.268434
$590$	− 72.7696i	− 2.99588i
$591$	33.4558i	1.37619i
$592$	24.9706i	1.02628i
$593$	30.5563i	1.25480i	0.778698	+	0.627399i	$0.215881\pi$
$593$	30.5563i	1.25480i	−0.778698	+	0.627399i	$0.784119\pi$
$594$	−33.9411	−1.39262
$595$	6.24264	0.255923
$596$	0	0
$597$	−17.3137	−0.708603
$598$	0	0
$599$	10.7990	0.441235	0.220617	−	0.975360i	$-0.429193\pi$
$599$	10.7990	0.441235	0.220617	+	0.975360i	$0.429193\pi$
$600$	− 57.9411i	− 2.36544i
$601$	−5.02944	−0.205155	−0.102578	−	0.994725i	$-0.532709\pi$
$601$	−5.02944	−0.205155	−0.102578	+	0.994725i	$0.532709\pi$
$602$	7.07107	0.288195
$603$	2.48528i	0.101208i
$604$	0	0
$605$	30.8995i	1.25624i
$606$	− 3.51472i	− 0.142776i
$607$	−1.27208	−0.0516321	−0.0258160	−	0.999667i	$-0.508218\pi$
$607$	−1.27208	−0.0516321	−0.0258160	+	0.999667i	$0.508218\pi$
$608$	0	0
$609$	− 8.24264i	− 0.334009i
$610$	−37.4558	−1.51654
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	6.72792	0.271517
$615$	19.7990	0.798372
$616$	− 12.0000i	− 0.483494i
$617$	− 23.6569i	− 0.952389i	−0.879340	−	0.476195i	$-0.842016\pi$
$617$	− 23.6569i	− 0.952389i	0.879340	−	0.476195i	$-0.157984\pi$
$618$	16.0000i	0.643614i
$619$	− 32.9706i	− 1.32520i	−0.748974	−	0.662599i	$-0.769453\pi$
$619$	− 32.9706i	− 1.32520i	0.748974	−	0.662599i	$-0.230547\pi$
$620$	0	0
$621$	0.970563	0.0389473
$622$	− 6.48528i	− 0.260036i
$623$	−4.41421	−0.176852
$624$	0	0
$625$	112.397	4.49588
$626$	27.1716i	1.08599i
$627$	7.45584	0.297758
$628$	0	0
$629$	− 8.82843i	− 0.352012i
$630$	6.24264i	0.248713i
$631$	2.00000i	0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$	2.00000i	0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$	4.20101i	0.167107i
$633$	−25.4142	−1.01012
$634$	−16.0000	−0.635441
$635$	8.82843i	0.350345i
$636$	0	0
$637$	0	0
$638$	−34.9706	−1.38450
$639$	1.07107i	0.0423708i
$640$	49.9411	1.97410
$641$	−26.6569	−1.05288	−0.526441	−	0.850212i	$-0.676474\pi$
$641$	−26.6569	−1.05288	−0.526441	+	0.850212i	$0.676474\pi$
$642$	16.2843i	0.642689i
$643$	− 4.48528i	− 0.176882i	−0.996081	−	0.0884411i	$-0.971811\pi$
$643$	− 4.48528i	− 0.176882i	0.996081	−	0.0884411i	$-0.0281885\pi$
$644$	0	0
$645$	31.2132i	1.22902i
$646$	−2.48528	−0.0977821
$647$	50.5269	1.98642	0.993209	−	0.116344i	$-0.0371176\pi$
$647$	50.5269	1.98642	0.993209	+	0.116344i	$0.0371176\pi$
$648$	14.1421i	0.555556i
$649$	49.4558	1.94131
$650$	0	0
$651$	7.41421	0.290586
$652$	0	0
$653$	−5.31371	−0.207941	−0.103971	−	0.994580i	$-0.533155\pi$
$653$	−5.31371	−0.207941	−0.103971	+	0.994580i	$0.533155\pi$
$654$	−17.4558	−0.682578
$655$	− 12.4853i	− 0.487840i
$656$	12.6863i	0.495316i
$657$	0.757359i	0.0295474i
$658$	6.24264i	0.243363i
$659$	−26.6569	−1.03840	−0.519202	−	0.854652i	$-0.673771\pi$
$659$	−26.6569	−1.03840	−0.519202	+	0.854652i	$0.673771\pi$
$660$	0	0
$661$	− 19.2426i	− 0.748452i	−0.927338	−	0.374226i	$-0.877908\pi$
$661$	− 19.2426i	− 0.748452i	0.927338	−	0.374226i	$-0.122092\pi$
$662$	25.4558	0.989369
$663$	0	0
$664$	−13.4558	−0.522188
$665$	− 5.48528i	− 0.212710i
$666$	8.82843	0.342095
$667$	1.00000	0.0387202
$668$	0	0
$669$	13.0711i	0.505357i
$670$	15.5147i	0.599386i
$671$	− 25.4558i	− 0.982712i
$672$	0	0
$673$	−40.9411	−1.57816	−0.789082	−	0.614288i	$-0.789443\pi$
$673$	−40.9411	−1.57816	−0.789082	+	0.614288i	$0.789443\pi$
$674$	− 46.6690i	− 1.79762i
$675$	−81.9411	−3.15392
$676$	0	0
$677$	29.3553	1.12822	0.564109	−	0.825701i	$-0.309220\pi$
$677$	29.3553	1.12822	0.564109	+	0.825701i	$0.309220\pi$
$678$	− 40.6274i	− 1.56029i
$679$	−13.7279	−0.526829
$680$	−17.6569	−0.677109
$681$	29.9411i	1.14735i
$682$	− 31.4558i	− 1.20451i
$683$	− 26.8284i	− 1.02656i	−0.858221	−	0.513281i	$-0.828430\pi$
$683$	− 26.8284i	− 1.02656i	0.858221	−	0.513281i	$-0.171570\pi$
$684$	0	0
$685$	−32.7279	−1.25047
$686$	−1.41421	−0.0539949
$687$	30.3431i	1.15766i
$688$	−20.0000	−0.762493
$689$	0	0
$690$	1.51472	0.0576644
$691$	30.6985i	1.16783i	0.811816	+	0.583913i	$0.198479\pi$
$691$	30.6985i	1.16783i	−0.811816	+	0.583913i	$0.801521\pi$
$692$	0	0
$693$	−4.24264	−0.161165
$694$	− 8.00000i	− 0.303676i
$695$	9.89949i	0.375509i
$696$	23.3137i	0.883704i
$697$	− 4.48528i	− 0.169892i
$698$	−0.384776	−0.0145640
$699$	−4.72792	−0.178826
$700$	0	0
$701$	−28.7990	−1.08772	−0.543861	−	0.839175i	$-0.683038\pi$
$701$	−28.7990	−1.08772	−0.543861	+	0.839175i	$0.683038\pi$
$702$	0	0
$703$	−7.75736	−0.292574
$704$	33.9411i	1.27920i
$705$	−27.5563	−1.03783
$706$	−12.0000	−0.451626
$707$	− 1.75736i	− 0.0660923i
$708$	0	0
$709$	− 24.7279i	− 0.928677i	−0.885658	−	0.464338i	$-0.846292\pi$
$709$	− 24.7279i	− 0.928677i	0.885658	−	0.464338i	$-0.153708\pi$
$710$	6.68629i	0.250932i
$711$	1.48528	0.0557024
$712$	12.4853	0.467906
$713$	0.899495i	0.0336864i
$714$	−2.82843	−0.105851
$715$	0	0
$716$	0	0
$717$	28.9706i	1.08193i
$718$	−11.4558	−0.427528
$719$	10.2426	0.381986	0.190993	−	0.981591i	$-0.438829\pi$
$719$	10.2426	0.381986	0.190993	+	0.981591i	$0.438829\pi$
$720$	− 17.6569i	− 0.658032i
$721$	8.00000i	0.297936i
$722$	− 24.6863i	− 0.918729i
$723$	− 31.4142i	− 1.16831i
$724$	0	0
$725$	−84.4264	−3.13552
$726$	− 14.0000i	− 0.519589i
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	4.72792i	0.174988i
$731$	7.07107	0.261533
$732$	0	0
$733$	− 42.6985i	− 1.57710i	−0.614968	−	0.788552i	$-0.710831\pi$
$733$	− 42.6985i	− 1.57710i	0.614968	−	0.788552i	$-0.289169\pi$
$734$	− 2.48528i	− 0.0917334i
$735$	− 6.24264i	− 0.230263i
$736$	0	0
$737$	−10.5442	−0.388399
$738$	4.48528	0.165105
$739$	41.6985i	1.53390i	0.641705	+	0.766952i	$0.278227\pi$
$739$	41.6985i	1.53390i	−0.641705	+	0.766952i	$0.721773\pi$
$740$	0	0
$741$	0	0
$742$	−8.24264	−0.302597
$743$	18.3431i	0.672945i	0.941693	+	0.336472i	$0.109234\pi$
$743$	18.3431i	0.672945i	−0.941693	+	0.336472i	$0.890766\pi$
$744$	−20.9706	−0.768818
$745$	−34.2426	−1.25455
$746$	12.0000i	0.439351i
$747$	4.75736i	0.174063i
$748$	0	0
$749$	8.14214i	0.297507i
$750$	−83.7401	−3.05776
$751$	1.48528	0.0541987	0.0270993	−	0.999633i	$-0.491373\pi$
$751$	1.48528	0.0541987	0.0270993	+	0.999633i	$0.491373\pi$
$752$	− 17.6569i	− 0.643879i
$753$	27.4558	1.00055
$754$	0	0
$755$	−80.5269	−2.93067
$756$	0	0
$757$	4.51472	0.164090	0.0820451	−	0.996629i	$-0.473855\pi$
$757$	4.51472	0.164090	0.0820451	+	0.996629i	$0.473855\pi$
$758$	−45.5980	−1.65619
$759$	1.02944i	0.0373662i
$760$	15.5147i	0.562778i
$761$	43.2426i	1.56754i	0.621048	+	0.783772i	$0.286707\pi$
$761$	43.2426i	1.56754i	−0.621048	+	0.783772i	$0.713293\pi$
$762$	− 4.00000i	− 0.144905i
$763$	−8.72792	−0.315972
$764$	0	0
$765$	6.24264i	0.225703i
$766$	4.97056	0.179594
$767$	0	0
$768$	0	0
$769$	− 9.78680i	− 0.352921i	−0.984308	−	0.176460i	$-0.943535\pi$
$769$	− 9.78680i	− 0.352921i	0.984308	−	0.176460i	$-0.0564648\pi$
$770$	−26.4853	−0.954463
$771$	−23.4558	−0.844742
$772$	0	0
$773$	27.1716i	0.977294i	0.872482	+	0.488647i	$0.162509\pi$
$773$	27.1716i	0.977294i	−0.872482	+	0.488647i	$0.837491\pi$
$774$	7.07107i	0.254164i
$775$	− 75.9411i	− 2.72789i
$776$	38.8284	1.39386
$777$	−8.82843	−0.316718
$778$	25.9411i	0.930034i
$779$	−3.94113	−0.141205
$780$	0	0
$781$	−4.54416	−0.162603
$782$	− 0.343146i	− 0.0122709i
$783$	32.9706	1.17827
$784$	4.00000	0.142857
$785$	54.0416i	1.92883i
$786$	5.65685i	0.201773i
$787$	− 33.2426i	− 1.18497i	−0.805581	−	0.592486i	$-0.798147\pi$
$787$	− 33.2426i	− 1.18497i	0.805581	−	0.592486i	$-0.201853\pi$
$788$	0	0
$789$	45.2132	1.60963
$790$	9.27208	0.329886
$791$	− 20.3137i	− 0.722272i
$792$	12.0000	0.426401
$793$	0	0
$794$	34.2426	1.21523
$795$	− 36.3848i	− 1.29044i
$796$	0	0
$797$	35.6569	1.26303	0.631515	−	0.775363i	$-0.282433\pi$
$797$	35.6569	1.26303	0.631515	+	0.775363i	$0.282433\pi$
$798$	2.48528i	0.0879780i
$799$	6.24264i	0.220849i
$800$	0	0
$801$	− 4.41421i	− 0.155969i
$802$	24.9706	0.881741
$803$	−3.21320	−0.113391
$804$	0	0
$805$	0.757359	0.0266934
$806$	0	0
$807$	−20.9706	−0.738199
$808$	4.97056i	0.174864i
$809$	0.514719	0.0180965	0.00904827	−	0.999959i	$-0.497120\pi$
$809$	0.514719	0.0180965	0.00904827	+	0.999959i	$0.497120\pi$
$810$	31.2132	1.09672
$811$	− 45.9411i	− 1.61321i	−0.591090	−	0.806606i	$-0.701302\pi$
$811$	− 45.9411i	− 1.61321i	0.591090	−	0.806606i	$-0.298698\pi$
$812$	0	0
$813$	− 28.2843i	− 0.991973i
$814$	37.4558i	1.31283i
$815$	−37.4558	−1.31202
$816$	8.00000	0.280056
$817$	− 6.21320i	− 0.217372i
$818$	7.41421	0.259232
$819$	0	0
$820$	0	0
$821$	− 8.14214i	− 0.284162i	−0.989855	−	0.142081i	$-0.954621\pi$
$821$	− 8.14214i	− 0.284162i	0.989855	−	0.142081i	$-0.0453794\pi$
$822$	14.8284	0.517201
$823$	6.48528	0.226063	0.113031	−	0.993591i	$-0.463944\pi$
$823$	6.48528	0.226063	0.113031	+	0.993591i	$0.463944\pi$
$824$	− 22.6274i	− 0.788263i
$825$	− 86.9117i	− 3.02588i
$826$	16.4853i	0.573596i
$827$	1.45584i	0.0506247i	0.999680	+	0.0253123i	$0.00805803\pi$
$827$	1.45584i	0.0506247i	−0.999680	+	0.0253123i	$0.991942\pi$
$828$	0	0
$829$	−25.2721	−0.877736	−0.438868	−	0.898552i	$-0.644620\pi$
$829$	−25.2721	−0.877736	−0.438868	+	0.898552i	$0.644620\pi$
$830$	29.6985i	1.03085i
$831$	13.4142	0.465334
$832$	0	0
$833$	−1.41421	−0.0489996
$834$	− 4.48528i	− 0.155313i
$835$	−94.3970	−3.26674
$836$	0	0
$837$	29.6569i	1.02509i
$838$	− 46.4853i	− 1.60581i
$839$	− 38.8284i	− 1.34051i	−0.742133	−	0.670253i	$-0.766186\pi$
$839$	− 38.8284i	− 1.34051i	0.742133	−	0.670253i	$-0.233814\pi$
$840$	17.6569i	0.609219i
$841$	4.97056	0.171399
$842$	26.4853	0.912743
$843$	22.0000i	0.757720i
$844$	0	0
$845$	0	0
$846$	−6.24264	−0.214626
$847$	− 7.00000i	− 0.240523i
$848$	23.3137	0.800596
$849$	12.0000	0.411839
$850$	28.9706i	0.993682i
$851$	− 1.07107i	− 0.0367157i
$852$	0	0
$853$	14.2721i	0.488667i	0.969691	+	0.244333i	$0.0785691\pi$
$853$	14.2721i	0.488667i	−0.969691	+	0.244333i	$0.921431\pi$
$854$	8.48528	0.290360
$855$	5.48528	0.187593
$856$	− 23.0294i	− 0.787130i
$857$	−18.7696	−0.641156	−0.320578	−	0.947222i	$-0.603877\pi$
$857$	−18.7696	−0.641156	−0.320578	+	0.947222i	$0.603877\pi$
$858$	0	0
$859$	−2.97056	−0.101354	−0.0506771	−	0.998715i	$-0.516138\pi$
$859$	−2.97056	−0.101354	−0.0506771	+	0.998715i	$0.516138\pi$
$860$	0	0
$861$	−4.48528	−0.152858
$862$	33.4558	1.13951
$863$	28.2843i	0.962808i	0.876499	+	0.481404i	$0.159873\pi$
$863$	28.2843i	0.962808i	−0.876499	+	0.481404i	$0.840127\pi$
$864$	0	0
$865$	− 3.21320i	− 0.109252i
$866$	12.6863i	0.431098i
$867$	21.2132	0.720438
$868$	0	0
$869$	6.30152i	0.213764i
$870$	51.4558	1.74452
$871$	0	0
$872$	24.6863	0.835983
$873$	− 13.7279i	− 0.464620i
$874$	−0.301515	−0.0101989
$875$	−41.8701	−1.41547
$876$	0	0
$877$	1.75736i	0.0593418i	0.999560	+	0.0296709i	$0.00944593\pi$
$877$	1.75736i	0.0593418i	−0.999560	+	0.0296709i	$0.990554\pi$
$878$	− 24.7696i	− 0.835932i
$879$	30.2426i	1.02006i
$880$	74.9117	2.52527
$881$	49.1127	1.65465	0.827324	−	0.561724i	$-0.189862\pi$
$881$	49.1127	1.65465	0.827324	+	0.561724i	$0.189862\pi$
$882$	− 1.41421i	− 0.0476190i
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	−72.7696	−2.44612
$886$	37.2132i	1.25020i
$887$	−43.1127	−1.44758	−0.723791	−	0.690019i	$-0.757602\pi$
$887$	−43.1127	−1.44758	−0.723791	+	0.690019i	$0.757602\pi$
$888$	24.9706	0.837957
$889$	− 2.00000i	− 0.0670778i
$890$	− 27.5563i	− 0.923691i
$891$	21.2132i	0.710669i
$892$	0	0
$893$	5.48528	0.183558
$894$	15.5147	0.518890
$895$	− 39.7279i	− 1.32796i
$896$	−11.3137	−0.377964
$897$	0	0
$898$	37.9411	1.26611
$899$	30.5563i	1.01911i
$900$	0	0
$901$	−8.24264	−0.274602
$902$	19.0294i	0.633611i
$903$	− 7.07107i	− 0.235310i
$904$	57.4558i	1.91095i
$905$	29.6985i	0.987211i
$906$	36.4853	1.21214
$907$	31.9706	1.06157	0.530783	−	0.847508i	$-0.321898\pi$
$907$	31.9706	1.06157	0.530783	+	0.847508i	$0.321898\pi$
$908$	0	0
$909$	1.75736	0.0582879
$910$	0	0
$911$	−10.0294	−0.332290	−0.166145	−	0.986101i	$-0.553132\pi$
$911$	−10.0294	−0.332290	−0.166145	+	0.986101i	$0.553132\pi$
$912$	− 7.02944i	− 0.232768i
$913$	−20.1838	−0.667985
$914$	−49.7990	−1.64720
$915$	37.4558i	1.23825i
$916$	0	0
$917$	2.82843i	0.0934029i
$918$	− 11.3137i	− 0.373408i
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−2.14214	−0.0706241
$921$	− 6.72792i	− 0.221693i
$922$	−4.48528	−0.147715
$923$	0	0
$924$	0	0
$925$	90.4264i	2.97320i
$926$	6.00000	0.197172
$927$	−8.00000	−0.262754
$928$	0	0
$929$	− 17.1005i	− 0.561049i	−0.959847	−	0.280525i	$-0.909492\pi$
$929$	− 17.1005i	− 0.561049i	0.959847	−	0.280525i	$-0.0905085\pi$
$930$	46.2843i	1.51772i
$931$	1.24264i	0.0407259i
$932$	0	0
$933$	−6.48528	−0.212319
$934$	22.4853i	0.735741i
$935$	−26.4853	−0.866161
$936$	0	0
$937$	2.78680	0.0910407	0.0455203	−	0.998963i	$-0.485505\pi$
$937$	2.78680	0.0910407	0.0455203	+	0.998963i	$0.485505\pi$
$938$	− 3.51472i	− 0.114760i
$939$	27.1716	0.886711
$940$	0	0
$941$	− 25.9289i	− 0.845259i	−0.906303	−	0.422630i	$-0.861107\pi$
$941$	− 25.9289i	− 0.845259i	0.906303	−	0.422630i	$-0.138893\pi$
$942$	− 24.4853i	− 0.797774i
$943$	− 0.544156i	− 0.0177202i
$944$	− 46.6274i	− 1.51759i
$945$	24.9706	0.812292
$946$	−30.0000	−0.975384
$947$	20.8284i	0.676833i	0.940996	+	0.338416i	$0.109891\pi$
$947$	20.8284i	0.676833i	−0.940996	+	0.338416i	$0.890109\pi$
$948$	0	0
$949$	0	0
$950$	25.4558	0.825897
$951$	16.0000i	0.518836i
$952$	4.00000	0.129641
$953$	17.1421	0.555288	0.277644	−	0.960684i	$-0.410446\pi$
$953$	17.1421	0.555288	0.277644	+	0.960684i	$0.410446\pi$
$954$	− 8.24264i	− 0.266865i
$955$	− 91.9411i	− 2.97514i
$956$	0	0
$957$	34.9706i	1.13044i
$958$	51.2132	1.65462
$959$	7.41421	0.239417
$960$	− 49.9411i	− 1.61184i
$961$	3.51472	0.113378
$962$	0	0
$963$	−8.14214	−0.262377
$964$	0	0
$965$	10.9706	0.353155
$966$	−0.343146	−0.0110405
$967$	41.6985i	1.34093i	0.741940	+	0.670466i	$0.233906\pi$
$967$	41.6985i	1.34093i	−0.741940	+	0.670466i	$0.766094\pi$
$968$	19.7990i	0.636364i
$969$	2.48528i	0.0798387i
$970$	− 85.6985i	− 2.75161i
$971$	−7.45584	−0.239269	−0.119635	−	0.992818i	$-0.538172\pi$
$971$	−7.45584	−0.239269	−0.119635	+	0.992818i	$0.538172\pi$
$972$	0	0
$973$	− 2.24264i	− 0.0718958i
$974$	−55.7990	−1.78791
$975$	0	0
$976$	−24.0000	−0.768221
$977$	24.0416i	0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$977$	24.0416i	0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$978$	16.9706	0.542659
$979$	18.7279	0.598547
$980$	0	0
$981$	− 8.72792i	− 0.278661i
$982$	− 40.4853i	− 1.29194i
$983$	33.0416i	1.05386i	0.849907	+	0.526932i	$0.176658\pi$
$983$	33.0416i	1.05386i	−0.849907	+	0.526932i	$0.823342\pi$
$984$	12.6863	0.404424
$985$	−104.426	−3.32730
$986$	− 11.6569i	− 0.371230i
$987$	6.24264	0.198705
$988$	0	0
$989$	0.857864	0.0272785
$990$	− 26.4853i	− 0.841757i
$991$	34.9706	1.11088	0.555438	−	0.831558i	$-0.312551\pi$
$991$	34.9706	1.11088	0.555438	+	0.831558i	$0.312551\pi$
$992$	0	0
$993$	− 25.4558i	− 0.807817i
$994$	− 1.51472i	− 0.0480440i
$995$	− 54.0416i	− 1.71323i
$996$	0	0
$997$	−28.4853	−0.902138	−0.451069	−	0.892489i	$-0.648957\pi$
$997$	−28.4853	−0.902138	−0.451069	+	0.892489i	$0.648957\pi$
$998$	54.7696	1.73370
$999$	− 35.3137i	− 1.11728i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.2.c.d.337.1		4
13.5	odd	4		1183.2.a.d.1.1		2
13.8	odd	4		91.2.a.c.1.2	✓	2
13.12	even	2	inner	1183.2.c.d.337.3		4
39.8	even	4		819.2.a.h.1.1		2
52.47	even	4		1456.2.a.q.1.2		2
65.34	odd	4		2275.2.a.j.1.1		2
91.34	even	4		637.2.a.g.1.2		2
91.47	even	12		637.2.e.g.508.1		4
91.60	odd	12		637.2.e.f.79.1		4
91.73	even	12		637.2.e.g.79.1		4
91.83	even	4		8281.2.a.v.1.1		2
91.86	odd	12		637.2.e.f.508.1		4
104.21	odd	4		5824.2.a.bl.1.2		2
104.99	even	4		5824.2.a.bk.1.1		2
273.125	odd	4		5733.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.c.1.2	✓	2	13.8	odd	4
637.2.a.g.1.2		2	91.34	even	4
637.2.e.f.79.1		4	91.60	odd	12
637.2.e.f.508.1		4	91.86	odd	12
637.2.e.g.79.1		4	91.73	even	12
637.2.e.g.508.1		4	91.47	even	12
819.2.a.h.1.1		2	39.8	even	4
1183.2.a.d.1.1		2	13.5	odd	4
1183.2.c.d.337.1		4	1.1	even	1	trivial
1183.2.c.d.337.3		4	13.12	even	2	inner
1456.2.a.q.1.2		2	52.47	even	4
2275.2.a.j.1.1		2	65.34	odd	4
5733.2.a.s.1.1		2	273.125	odd	4
5824.2.a.bk.1.1		2	104.99	even	4
5824.2.a.bl.1.2		2	104.21	odd	4
8281.2.a.v.1.1		2	91.83	even	4

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:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.41421 q^{3} -4.41421i q^{5} +2.00000i q^{6} +1.00000i q^{7} -2.82843i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -1.41421 q^{3} -4.41421i q^{5} +2.00000i q^{6} +1.00000i q^{7} -2.82843i q^{8} -1.00000 q^{9} -6.24264 q^{10} -4.24264i q^{11} +1.41421 q^{14} +6.24264i q^{15} -4.00000 q^{16} +1.41421 q^{17} +1.41421i q^{18} -1.24264i q^{19} -1.41421i q^{21} -6.00000 q^{22} +0.171573 q^{23} +4.00000i q^{24} -14.4853 q^{25} +5.65685 q^{27} +5.82843 q^{29} +8.82843 q^{30} +5.24264i q^{31} +6.00000i q^{33} -2.00000i q^{34} +4.41421 q^{35} -6.24264i q^{37} -1.75736 q^{38} -12.4853 q^{40} -3.17157i q^{41} -2.00000 q^{42} +5.00000 q^{43} +4.41421i q^{45} -0.242641i q^{46} +4.41421i q^{47} +5.65685 q^{48} -1.00000 q^{49} +20.4853i q^{50} -2.00000 q^{51} -5.82843 q^{53} -8.00000i q^{54} -18.7279 q^{55} +2.82843 q^{56} +1.75736i q^{57} -8.24264i q^{58} +11.6569i q^{59} +6.00000 q^{61} +7.41421 q^{62} -1.00000i q^{63} -8.00000 q^{64} +8.48528 q^{66} -2.48528i q^{67} -0.242641 q^{69} -6.24264i q^{70} -1.07107i q^{71} +2.82843i q^{72} -0.757359i q^{73} -8.82843 q^{74} +20.4853 q^{75} +4.24264 q^{77} -1.48528 q^{79} +17.6569i q^{80} -5.00000 q^{81} -4.48528 q^{82} -4.75736i q^{83} -6.24264i q^{85} -7.07107i q^{86} -8.24264 q^{87} -12.0000 q^{88} +4.41421i q^{89} +6.24264 q^{90} -7.41421i q^{93} +6.24264 q^{94} -5.48528 q^{95} +13.7279i q^{97} +1.41421i q^{98} +4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^10 - 16 * q^16 - 24 * q^22 + 12 * q^23 - 24 * q^25 + 12 * q^29 + 24 * q^30 + 12 * q^35 - 24 * q^38 - 16 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^49 - 8 * q^51 - 12 * q^53 - 24 * q^55 + 24 * q^61 + 24 * q^62 - 32 * q^64 + 16 * q^69 - 24 * q^74 + 48 * q^75 + 28 * q^79 - 20 * q^81 + 16 * q^82 - 16 * q^87 - 48 * q^88 + 8 * q^90 + 8 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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