LMFDB - Embedded newform 8978.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8978,2,Mod(1,8978)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8978, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8978.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8978 = 2 \cdot 67^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8978.a (trivial)

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:	yes
Analytic conductor:	$71.6896909347$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.473.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 5x - 1 $ x^3 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 134)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.201640$ of defining polynomial
Character	$\chi$	$=$	8978.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.95934 q^{3} +1.00000 q^{4} -0.798360 q^{5} +2.95934 q^{6} -0.403279 q^{7} +1.00000 q^{8} +5.75770 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.95934 q^{3} +1.00000 q^{4} -0.798360 q^{5} +2.95934 q^{6} -0.403279 q^{7} +1.00000 q^{8} +5.75770 q^{9} -0.798360 q^{10} -3.16098 q^{11} +2.95934 q^{12} -6.75770 q^{13} -0.403279 q^{14} -2.36262 q^{15} +1.00000 q^{16} -0.798360 q^{17} +5.75770 q^{18} +2.00000 q^{19} -0.798360 q^{20} -1.19344 q^{21} -3.16098 q^{22} -6.95934 q^{23} +2.95934 q^{24} -4.36262 q^{25} -6.75770 q^{26} +8.16098 q^{27} -0.403279 q^{28} -2.36262 q^{30} -7.11212 q^{31} +1.00000 q^{32} -9.35442 q^{33} -0.798360 q^{34} +0.321962 q^{35} +5.75770 q^{36} +8.32196 q^{37} +2.00000 q^{38} -19.9983 q^{39} -0.798360 q^{40} -11.4341 q^{41} -1.19344 q^{42} -6.31376 q^{43} -3.16098 q^{44} -4.59672 q^{45} -6.95934 q^{46} -3.16098 q^{47} +2.95934 q^{48} -6.83737 q^{49} -4.36262 q^{50} -2.36262 q^{51} -6.75770 q^{52} +10.4747 q^{53} +8.16098 q^{54} +2.52360 q^{55} -0.403279 q^{56} +5.91868 q^{57} +1.20984 q^{59} -2.36262 q^{60} +2.47474 q^{61} -7.11212 q^{62} -2.32196 q^{63} +1.00000 q^{64} +5.39508 q^{65} -9.35442 q^{66} -0.798360 q^{68} -20.5951 q^{69} +0.321962 q^{70} -5.52360 q^{71} +5.75770 q^{72} -11.5643 q^{73} +8.32196 q^{74} -12.9105 q^{75} +2.00000 q^{76} +1.27476 q^{77} -19.9983 q^{78} +9.43409 q^{79} -0.798360 q^{80} +6.87802 q^{81} -11.4341 q^{82} +5.43409 q^{83} -1.19344 q^{84} +0.637379 q^{85} -6.31376 q^{86} -3.16098 q^{88} +3.44394 q^{89} -4.59672 q^{90} +2.72524 q^{91} -6.95934 q^{92} -21.0472 q^{93} -3.16098 q^{94} -1.59672 q^{95} +2.95934 q^{96} -6.72524 q^{97} -6.83737 q^{98} -18.2000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 - q^6 + 3 * q^8 + 8 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{8} + 8 q^{9} - 3 q^{10} + q^{11} - q^{12} - 11 q^{13} + 4 q^{15} + 3 q^{16} - 3 q^{17} + 8 q^{18} + 6 q^{19} - 3 q^{20} - 6 q^{21} + q^{22} - 11 q^{23} - q^{24} - 2 q^{25} - 11 q^{26} + 14 q^{27} + 4 q^{30} - 4 q^{31} + 3 q^{32} - 20 q^{33} - 3 q^{34} - 20 q^{35} + 8 q^{36} + 4 q^{37} + 6 q^{38} - 10 q^{39} - 3 q^{40} + 4 q^{41} - 6 q^{42} - q^{43} + q^{44} - 15 q^{45} - 11 q^{46} + q^{47} - q^{48} + 19 q^{49} - 2 q^{50} + 4 q^{51} - 11 q^{52} + 3 q^{53} + 14 q^{54} - 14 q^{55} - 2 q^{57} + 4 q^{60} - 21 q^{61} - 4 q^{62} + 14 q^{63} + 3 q^{64} + 18 q^{65} - 20 q^{66} - 3 q^{68} - 13 q^{69} - 20 q^{70} + 5 q^{71} + 8 q^{72} - 23 q^{73} + 4 q^{74} - 22 q^{75} + 6 q^{76} + 26 q^{77} - 10 q^{78} - 10 q^{79} - 3 q^{80} - 9 q^{81} + 4 q^{82} - 22 q^{83} - 6 q^{84} + 13 q^{85} - q^{86} + q^{88} + 19 q^{89} - 15 q^{90} - 14 q^{91} - 11 q^{92} - 20 q^{93} + q^{94} - 6 q^{95} - q^{96} + 2 q^{97} + 19 q^{98} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 - q^6 + 3 * q^8 + 8 * q^9 - 3 * q^10 + q^11 - q^12 - 11 * q^13 + 4 * q^15 + 3 * q^16 - 3 * q^17 + 8 * q^18 + 6 * q^19 - 3 * q^20 - 6 * q^21 + q^22 - 11 * q^23 - q^24 - 2 * q^25 - 11 * q^26 + 14 * q^27 + 4 * q^30 - 4 * q^31 + 3 * q^32 - 20 * q^33 - 3 * q^34 - 20 * q^35 + 8 * q^36 + 4 * q^37 + 6 * q^38 - 10 * q^39 - 3 * q^40 + 4 * q^41 - 6 * q^42 - q^43 + q^44 - 15 * q^45 - 11 * q^46 + q^47 - q^48 + 19 * q^49 - 2 * q^50 + 4 * q^51 - 11 * q^52 + 3 * q^53 + 14 * q^54 - 14 * q^55 - 2 * q^57 + 4 * q^60 - 21 * q^61 - 4 * q^62 + 14 * q^63 + 3 * q^64 + 18 * q^65 - 20 * q^66 - 3 * q^68 - 13 * q^69 - 20 * q^70 + 5 * q^71 + 8 * q^72 - 23 * q^73 + 4 * q^74 - 22 * q^75 + 6 * q^76 + 26 * q^77 - 10 * q^78 - 10 * q^79 - 3 * q^80 - 9 * q^81 + 4 * q^82 - 22 * q^83 - 6 * q^84 + 13 * q^85 - q^86 + q^88 + 19 * q^89 - 15 * q^90 - 14 * q^91 - 11 * q^92 - 20 * q^93 + q^94 - 6 * q^95 - q^96 + 2 * q^97 + 19 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.95934	1.70858	0.854288	−	0.519799i	$-0.173993\pi$
$3$	2.95934	1.70858	0.854288	+	0.519799i	$0.173993\pi$
$4$	1.00000	0.500000
$5$	−0.798360	−0.357038	−0.178519	−	0.983937i	$-0.557131\pi$
$5$	−0.798360	−0.357038	−0.178519	+	0.983937i	$0.557131\pi$
$6$	2.95934	1.20815
$7$	−0.403279	−0.152425	−0.0762126	−	0.997092i	$-0.524283\pi$
$7$	−0.403279	−0.152425	−0.0762126	+	0.997092i	$0.524283\pi$
$8$	1.00000	0.353553
$9$	5.75770	1.91923
$10$	−0.798360	−0.252464
$11$	−3.16098	−0.953072	−0.476536	−	0.879155i	$-0.658108\pi$
$11$	−3.16098	−0.953072	−0.476536	+	0.879155i	$0.658108\pi$
$12$	2.95934	0.854288
$13$	−6.75770	−1.87425	−0.937125	−	0.348995i	$-0.886523\pi$
$13$	−6.75770	−1.87425	−0.937125	+	0.348995i	$0.886523\pi$
$14$	−0.403279	−0.107781
$15$	−2.36262	−0.610026
$16$	1.00000	0.250000
$17$	−0.798360	−0.193631	−0.0968154	−	0.995302i	$-0.530866\pi$
$17$	−0.798360	−0.193631	−0.0968154	+	0.995302i	$0.530866\pi$
$18$	5.75770	1.35710
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−0.798360	−0.178519
$21$	−1.19344	−0.260430
$22$	−3.16098	−0.673923
$23$	−6.95934	−1.45112	−0.725562	−	0.688157i	$-0.758420\pi$
$23$	−6.95934	−1.45112	−0.725562	+	0.688157i	$0.758420\pi$
$24$	2.95934	0.604073
$25$	−4.36262	−0.872524
$26$	−6.75770	−1.32529
$27$	8.16098	1.57058
$28$	−0.403279	−0.0762126
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−2.36262	−0.431354
$31$	−7.11212	−1.27738	−0.638688	−	0.769466i	$-0.720522\pi$
$31$	−7.11212	−1.27738	−0.638688	+	0.769466i	$0.720522\pi$
$32$	1.00000	0.176777
$33$	−9.35442	−1.62840
$34$	−0.798360	−0.136918
$35$	0.321962	0.0544216
$36$	5.75770	0.959617
$37$	8.32196	1.36812	0.684061	−	0.729425i	$-0.260212\pi$
$37$	8.32196	1.36812	0.684061	+	0.729425i	$0.260212\pi$
$38$	2.00000	0.324443
$39$	−19.9983	−3.20230
$40$	−0.798360	−0.126232
$41$	−11.4341	−1.78570	−0.892852	−	0.450350i	$-0.851299\pi$
$41$	−11.4341	−1.78570	−0.892852	+	0.450350i	$0.851299\pi$
$42$	−1.19344	−0.184152
$43$	−6.31376	−0.962840	−0.481420	−	0.876490i	$-0.659879\pi$
$43$	−6.31376	−0.962840	−0.481420	+	0.876490i	$0.659879\pi$
$44$	−3.16098	−0.476536
$45$	−4.59672	−0.685239
$46$	−6.95934	−1.02610
$47$	−3.16098	−0.461076	−0.230538	−	0.973063i	$-0.574049\pi$
$47$	−3.16098	−0.461076	−0.230538	+	0.973063i	$0.574049\pi$
$48$	2.95934	0.427144
$49$	−6.83737	−0.976767
$50$	−4.36262	−0.616968
$51$	−2.36262	−0.330833
$52$	−6.75770	−0.937125
$53$	10.4747	1.43882	0.719408	−	0.694587i	$-0.244413\pi$
$53$	10.4747	1.43882	0.719408	+	0.694587i	$0.244413\pi$
$54$	8.16098	1.11057
$55$	2.52360	0.340282
$56$	−0.403279	−0.0538905
$57$	5.91868	0.783949
$58$	0	0
$59$	1.20984	0.157507	0.0787537	−	0.996894i	$-0.474906\pi$
$59$	1.20984	0.157507	0.0787537	+	0.996894i	$0.474906\pi$
$60$	−2.36262	−0.305013
$61$	2.47474	0.316859	0.158429	−	0.987370i	$-0.449357\pi$
$61$	2.47474	0.316859	0.158429	+	0.987370i	$0.449357\pi$
$62$	−7.11212	−0.903241
$63$	−2.32196	−0.292540
$64$	1.00000	0.125000
$65$	5.39508	0.669177
$66$	−9.35442	−1.15145
$67$	0	0
$67$	0	0
$68$	−0.798360	−0.0968154
$69$	−20.5951	−2.47935
$70$	0.321962	0.0384818
$71$	−5.52360	−0.655531	−0.327765	−	0.944759i	$-0.606296\pi$
$71$	−5.52360	−0.655531	−0.327765	+	0.944759i	$0.606296\pi$
$72$	5.75770	0.678552
$73$	−11.5643	−1.35349	−0.676747	−	0.736216i	$-0.736611\pi$
$73$	−11.5643	−1.35349	−0.676747	+	0.736216i	$0.736611\pi$
$74$	8.32196	0.967408
$75$	−12.9105	−1.49077
$76$	2.00000	0.229416
$77$	1.27476	0.145272
$78$	−19.9983	−2.26437
$79$	9.43409	1.06142	0.530709	−	0.847554i	$-0.321926\pi$
$79$	9.43409	1.06142	0.530709	+	0.847554i	$0.321926\pi$
$80$	−0.798360	−0.0892594
$81$	6.87802	0.764225
$82$	−11.4341	−1.26268
$83$	5.43409	0.596468	0.298234	−	0.954493i	$-0.403602\pi$
$83$	5.43409	0.596468	0.298234	+	0.954493i	$0.403602\pi$
$84$	−1.19344	−0.130215
$85$	0.637379	0.0691335
$86$	−6.31376	−0.680831
$87$	0	0
$88$	−3.16098	−0.336962
$89$	3.44394	0.365057	0.182528	−	0.983201i	$-0.441572\pi$
$89$	3.44394	0.365057	0.182528	+	0.983201i	$0.441572\pi$
$90$	−4.59672	−0.484537
$91$	2.72524	0.285683
$92$	−6.95934	−0.725562
$93$	−21.0472	−2.18249
$94$	−3.16098	−0.326030
$95$	−1.59672	−0.163820
$96$	2.95934	0.302037
$97$	−6.72524	−0.682845	−0.341422	−	0.939910i	$-0.610909\pi$
$97$	−6.72524	−0.682845	−0.341422	+	0.939910i	$0.610909\pi$
$98$	−6.83737	−0.690678
$99$	−18.2000	−1.82917
$100$	−4.36262	−0.436262
$101$	−3.44394	−0.342685	−0.171342	−	0.985212i	$-0.554810\pi$
$101$	−3.44394	−0.342685	−0.171342	+	0.985212i	$0.554810\pi$
$102$	−2.36262	−0.233934
$103$	16.8780	1.66304	0.831521	−	0.555494i	$-0.187471\pi$
$103$	16.8780	1.66304	0.831521	+	0.555494i	$0.187471\pi$
$104$	−6.75770	−0.662647
$105$	0.952796	0.0929834
$106$	10.4747	1.01740
$107$	14.6275	1.41410	0.707048	−	0.707165i	$-0.250026\pi$
$107$	14.6275	1.41410	0.707048	+	0.707165i	$0.250026\pi$
$108$	8.16098	0.785291
$109$	20.6357	1.97654	0.988272	−	0.152704i	$-0.0487980\pi$
$109$	20.6357	1.97654	0.988272	+	0.152704i	$0.0487980\pi$
$110$	2.52360	0.240616
$111$	24.6275	2.33754
$112$	−0.403279	−0.0381063
$113$	15.8374	1.48985	0.744927	−	0.667146i	$-0.232484\pi$
$113$	15.8374	1.48985	0.744927	+	0.667146i	$0.232484\pi$
$114$	5.91868	0.554335
$115$	5.55606	0.518105
$116$	0	0
$117$	−38.9088	−3.59712
$118$	1.20984	0.111375
$119$	0.321962	0.0295142
$120$	−2.36262	−0.215677
$121$	−1.00820	−0.0916544
$122$	2.47474	0.224053
$123$	−33.8374	−3.05101
$124$	−7.11212	−0.638688
$125$	7.47474	0.668562
$126$	−2.32196	−0.206857
$127$	−3.91048	−0.346999	−0.173500	−	0.984834i	$-0.555508\pi$
$127$	−3.91048	−0.346999	−0.173500	+	0.984834i	$0.555508\pi$
$128$	1.00000	0.0883883
$129$	−18.6846	−1.64509
$130$	5.39508	0.473180
$131$	−17.7560	−1.55135	−0.775677	−	0.631131i	$-0.782591\pi$
$131$	−17.7560	−1.55135	−0.775677	+	0.631131i	$0.782591\pi$
$132$	−9.35442	−0.814198
$133$	−0.806559	−0.0699375
$134$	0	0
$135$	−6.51540	−0.560757
$136$	−0.798360	−0.0684588
$137$	−4.40328	−0.376198	−0.188099	−	0.982150i	$-0.560233\pi$
$137$	−4.40328	−0.376198	−0.188099	+	0.982150i	$0.560233\pi$
$138$	−20.5951	−1.75317
$139$	16.6439	1.41172	0.705860	−	0.708352i	$-0.250561\pi$
$139$	16.6439	1.41172	0.705860	+	0.708352i	$0.250561\pi$
$140$	0.321962	0.0272108
$141$	−9.35442	−0.787784
$142$	−5.52360	−0.463530
$143$	21.3610	1.78629
$144$	5.75770	0.479808
$145$	0	0
$146$	−11.5643	−0.957065
$147$	−20.2341	−1.66888
$148$	8.32196	0.684061
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−12.9105	−1.05414
$151$	5.54786	0.451479	0.225739	−	0.974188i	$-0.427520\pi$
$151$	5.54786	0.451479	0.225739	+	0.974188i	$0.427520\pi$
$152$	2.00000	0.162221
$153$	−4.59672	−0.371623
$154$	1.27476	0.102723
$155$	5.67804	0.456071
$156$	−19.9983	−1.60115
$157$	−2.72524	−0.217498	−0.108749	−	0.994069i	$-0.534684\pi$
$157$	−2.72524	−0.217498	−0.108749	+	0.994069i	$0.534684\pi$
$158$	9.43409	0.750536
$159$	30.9983	2.45833
$160$	−0.798360	−0.0631159
$161$	2.80656	0.221188
$162$	6.87802	0.540389
$163$	−8.40328	−0.658196	−0.329098	−	0.944296i	$-0.606745\pi$
$163$	−8.40328	−0.658196	−0.329098	+	0.944296i	$0.606745\pi$
$164$	−11.4341	−0.892852
$165$	7.46820	0.581399
$166$	5.43409	0.421767
$167$	0.676385	0.0523402	0.0261701	−	0.999658i	$-0.491669\pi$
$167$	0.676385	0.0523402	0.0261701	+	0.999658i	$0.491669\pi$
$168$	−1.19344	−0.0920760
$169$	32.6665	2.51281
$170$	0.637379	0.0488848
$171$	11.5154	0.880605
$172$	−6.31376	−0.481420
$173$	7.53180	0.572632	0.286316	−	0.958135i	$-0.407569\pi$
$173$	7.53180	0.572632	0.286316	+	0.958135i	$0.407569\pi$
$174$	0	0
$175$	1.75935	0.132995
$176$	−3.16098	−0.238268
$177$	3.58032	0.269114
$178$	3.44394	0.258134
$179$	0.0714656	0.00534159	0.00267080	−	0.999996i	$-0.499150\pi$
$179$	0.0714656	0.00534159	0.00267080	+	0.999996i	$0.499150\pi$
$180$	−4.59672	−0.342619
$181$	−14.7252	−1.09452	−0.547259	−	0.836963i	$-0.684329\pi$
$181$	−14.7252	−1.09452	−0.547259	+	0.836963i	$0.684329\pi$
$182$	2.72524	0.202008
$183$	7.32362	0.541377
$184$	−6.95934	−0.513049
$185$	−6.64392	−0.488471
$186$	−21.0472	−1.54326
$187$	2.52360	0.184544
$188$	−3.16098	−0.230538
$189$	−3.29116	−0.239396
$190$	−1.59672	−0.115838
$191$	−8.80656	−0.637220	−0.318610	−	0.947886i	$-0.603216\pi$
$191$	−8.80656	−0.637220	−0.318610	+	0.947886i	$0.603216\pi$
$192$	2.95934	0.213572
$193$	−6.16918	−0.444067	−0.222034	−	0.975039i	$-0.571269\pi$
$193$	−6.16918	−0.444067	−0.222034	+	0.975039i	$0.571269\pi$
$194$	−6.72524	−0.482844
$195$	15.9659	1.14334
$196$	−6.83737	−0.488383
$197$	−26.5951	−1.89482	−0.947410	−	0.320022i	$-0.896310\pi$
$197$	−26.5951	−1.89482	−0.947410	+	0.320022i	$0.896310\pi$
$198$	−18.2000	−1.29342
$199$	−3.91048	−0.277207	−0.138603	−	0.990348i	$-0.544261\pi$
$199$	−3.91048	−0.277207	−0.138603	+	0.990348i	$0.544261\pi$
$200$	−4.36262	−0.308484
$201$	0	0
$202$	−3.44394	−0.242315
$203$	0	0
$204$	−2.36262	−0.165417
$205$	9.12852	0.637564
$206$	16.8780	1.17595
$207$	−40.0698	−2.78504
$208$	−6.75770	−0.468562
$209$	−6.32196	−0.437299
$210$	0.952796	0.0657492
$211$	−11.0308	−0.759392	−0.379696	−	0.925111i	$-0.623971\pi$
$211$	−11.0308	−0.759392	−0.379696	+	0.925111i	$0.623971\pi$
$212$	10.4747	0.719408
$213$	−16.3462	−1.12002
$214$	14.6275	0.999917
$215$	5.04066	0.343770
$216$	8.16098	0.555284
$217$	2.86817	0.194704
$218$	20.6357	1.39763
$219$	−34.2226	−2.31255
$220$	2.52360	0.170141
$221$	5.39508	0.362912
$222$	24.6275	1.65289
$223$	1.52360	0.102028	0.0510140	−	0.998698i	$-0.483755\pi$
$223$	1.52360	0.102028	0.0510140	+	0.998698i	$0.483755\pi$
$224$	−0.403279	−0.0269452
$225$	−25.1187	−1.67458
$226$	15.8374	1.05349
$227$	−4.79016	−0.317934	−0.158967	−	0.987284i	$-0.550816\pi$
$227$	−4.79016	−0.317934	−0.158967	+	0.987284i	$0.550816\pi$
$228$	5.91868	0.391974
$229$	1.19344	0.0788648	0.0394324	−	0.999222i	$-0.487445\pi$
$229$	1.19344	0.0788648	0.0394324	+	0.999222i	$0.487445\pi$
$230$	5.55606	0.366356
$231$	3.77245	0.248209
$232$	0	0
$233$	28.1593	1.84478	0.922389	−	0.386261i	$-0.126234\pi$
$233$	28.1593	1.84478	0.922389	+	0.386261i	$0.126234\pi$
$234$	−38.9088	−2.54355
$235$	2.52360	0.164622
$236$	1.20984	0.0787537
$237$	27.9187	1.81351
$238$	0.321962	0.0208697
$239$	12.8879	0.833647	0.416824	−	0.908987i	$-0.363143\pi$
$239$	12.8879	0.833647	0.416824	+	0.908987i	$0.363143\pi$
$240$	−2.36262	−0.152507
$241$	−18.6600	−1.20200	−0.600998	−	0.799251i	$-0.705230\pi$
$241$	−18.6600	−1.20200	−0.600998	+	0.799251i	$0.705230\pi$
$242$	−1.00820	−0.0648094
$243$	−4.12852	−0.264845
$244$	2.47474	0.158429
$245$	5.45868	0.348742
$246$	−33.8374	−2.15739
$247$	−13.5154	−0.859965
$248$	−7.11212	−0.451620
$249$	16.0813	1.01911
$250$	7.47474	0.472744
$251$	−15.9269	−1.00530	−0.502648	−	0.864491i	$-0.667641\pi$
$251$	−15.9269	−1.00530	−0.502648	+	0.864491i	$0.667641\pi$
$252$	−2.32196	−0.146270
$253$	21.9983	1.38302
$254$	−3.91048	−0.245366
$255$	1.88622	0.118120
$256$	1.00000	0.0625000
$257$	−20.2731	−1.26460	−0.632301	−	0.774723i	$-0.717889\pi$
$257$	−20.2731	−1.26460	−0.632301	+	0.774723i	$0.717889\pi$
$258$	−18.6846	−1.16325
$259$	−3.35608	−0.208536
$260$	5.39508	0.334589
$261$	0	0
$262$	−17.7560	−1.09697
$263$	10.7967	0.665753	0.332877	−	0.942970i	$-0.391981\pi$
$263$	10.7967	0.665753	0.332877	+	0.942970i	$0.391981\pi$
$264$	−9.35442	−0.575725
$265$	−8.36262	−0.513712
$266$	−0.806559	−0.0494533
$267$	10.1918	0.623727
$268$	0	0
$269$	−2.80656	−0.171119	−0.0855595	−	0.996333i	$-0.527268\pi$
$269$	−2.80656	−0.171119	−0.0855595	+	0.996333i	$0.527268\pi$
$270$	−6.51540	−0.396515
$271$	−17.9023	−1.08749	−0.543743	−	0.839252i	$-0.682993\pi$
$271$	−17.9023	−1.08749	−0.543743	+	0.839252i	$0.682993\pi$
$272$	−0.798360	−0.0484077
$273$	8.06492	0.488111
$274$	−4.40328	−0.266012
$275$	13.7902	0.831578
$276$	−20.5951	−1.23968
$277$	13.7560	0.826521	0.413260	−	0.910613i	$-0.364390\pi$
$277$	13.7560	0.826521	0.413260	+	0.910613i	$0.364390\pi$
$278$	16.6439	0.998236
$279$	−40.9495	−2.45158
$280$	0.321962	0.0192409
$281$	−19.2098	−1.14596	−0.572981	−	0.819568i	$-0.694213\pi$
$281$	−19.2098	−1.14596	−0.572981	+	0.819568i	$0.694213\pi$
$282$	−9.35442	−0.557048
$283$	−5.98360	−0.355688	−0.177844	−	0.984059i	$-0.556912\pi$
$283$	−5.98360	−0.355688	−0.177844	+	0.984059i	$0.556912\pi$
$284$	−5.52360	−0.327765
$285$	−4.72524	−0.279899
$286$	21.3610	1.26310
$287$	4.61113	0.272186
$288$	5.75770	0.339276
$289$	−16.3626	−0.962507
$290$	0	0
$291$	−19.9023	−1.16669
$292$	−11.5643	−0.676747
$293$	10.7902	0.630368	0.315184	−	0.949031i	$-0.397934\pi$
$293$	10.7902	0.630368	0.315184	+	0.949031i	$0.397934\pi$
$294$	−20.2341	−1.18008
$295$	−0.965887	−0.0562361
$296$	8.32196	0.483704
$297$	−25.7967	−1.49688
$298$	6.00000	0.347571
$299$	47.0292	2.71977
$300$	−12.9105	−0.745387
$301$	2.54621	0.146761
$302$	5.54786	0.319244
$303$	−10.1918	−0.585503
$304$	2.00000	0.114708
$305$	−1.97574	−0.113130
$306$	−4.59672	−0.262777
$307$	9.03081	0.515415	0.257708	−	0.966223i	$-0.417033\pi$
$307$	9.03081	0.515415	0.257708	+	0.966223i	$0.417033\pi$
$308$	1.27476	0.0726361
$309$	49.9478	2.84143
$310$	5.67804	0.322491
$311$	−16.5462	−0.938250	−0.469125	−	0.883132i	$-0.655431\pi$
$311$	−16.5462	−0.938250	−0.469125	+	0.883132i	$0.655431\pi$
$312$	−19.9983	−1.13218
$313$	−4.48460	−0.253484	−0.126742	−	0.991936i	$-0.540452\pi$
$313$	−4.48460	−0.253484	−0.126742	+	0.991936i	$0.540452\pi$
$314$	−2.72524	−0.153794
$315$	1.85376	0.104448
$316$	9.43409	0.530709
$317$	19.2098	1.07893	0.539466	−	0.842007i	$-0.318626\pi$
$317$	19.2098	1.07893	0.539466	+	0.842007i	$0.318626\pi$
$318$	30.9983	1.73830
$319$	0	0
$320$	−0.798360	−0.0446297
$321$	43.2878	2.41609
$322$	2.80656	0.156403
$323$	−1.59672	−0.0888439
$324$	6.87802	0.382112
$325$	29.4813	1.63533
$326$	−8.40328	−0.465415
$327$	61.0682	3.37708
$328$	−11.4341	−0.631342
$329$	1.27476	0.0702797
$330$	7.46820	0.411111
$331$	6.33016	0.347937	0.173969	−	0.984751i	$-0.444341\pi$
$331$	6.33016	0.347937	0.173969	+	0.984751i	$0.444341\pi$
$332$	5.43409	0.298234
$333$	47.9154	2.62575
$334$	0.676385	0.0370101
$335$	0	0
$336$	−1.19344	−0.0651076
$337$	19.3364	1.05332	0.526660	−	0.850076i	$-0.323444\pi$
$337$	19.3364	1.05332	0.526660	+	0.850076i	$0.323444\pi$
$338$	32.6665	1.77683
$339$	46.8682	2.54553
$340$	0.637379	0.0345667
$341$	22.4813	1.21743
$342$	11.5154	0.622682
$343$	5.58032	0.301309
$344$	−6.31376	−0.340415
$345$	16.4423	0.885223
$346$	7.53180	0.404912
$347$	24.5544	1.31815	0.659075	−	0.752077i	$-0.270948\pi$
$347$	24.5544	1.31815	0.659075	+	0.752077i	$0.270948\pi$
$348$	0	0
$349$	−10.1429	−0.542939	−0.271469	−	0.962447i	$-0.587510\pi$
$349$	−10.1429	−0.542939	−0.271469	+	0.962447i	$0.587510\pi$
$350$	1.75935	0.0940415
$351$	−55.1495	−2.94366
$352$	−3.16098	−0.168481
$353$	3.83737	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$353$	3.83737	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$354$	3.58032	0.190292
$355$	4.40982	0.234049
$356$	3.44394	0.182528
$357$	0.952796	0.0504273
$358$	0.0714656	0.00377708
$359$	−5.61312	−0.296249	−0.148125	−	0.988969i	$-0.547324\pi$
$359$	−5.61312	−0.296249	−0.148125	+	0.988969i	$0.547324\pi$
$360$	−4.59672	−0.242268
$361$	−15.0000	−0.789474
$362$	−14.7252	−0.773942
$363$	−2.98360	−0.156599
$364$	2.72524	0.142841
$365$	9.23245	0.483248
$366$	7.32362	0.382812
$367$	−19.1121	−0.997645	−0.498822	−	0.866704i	$-0.666234\pi$
$367$	−19.1121	−0.997645	−0.498822	+	0.866704i	$0.666234\pi$
$368$	−6.95934	−0.362781
$369$	−65.8341	−3.42718
$370$	−6.64392	−0.345401
$371$	−4.22425	−0.219312
$372$	−21.0472	−1.09125
$373$	−13.7885	−0.713942	−0.356971	−	0.934115i	$-0.616190\pi$
$373$	−13.7885	−0.713942	−0.356971	+	0.934115i	$0.616190\pi$
$374$	2.52360	0.130492
$375$	22.1203	1.14229
$376$	−3.16098	−0.163015
$377$	0	0
$378$	−3.29116	−0.169279
$379$	6.51706	0.334759	0.167379	−	0.985893i	$-0.446470\pi$
$379$	6.51706	0.334759	0.167379	+	0.985893i	$0.446470\pi$
$380$	−1.59672	−0.0819100
$381$	−11.5725	−0.592875
$382$	−8.80656	−0.450583
$383$	20.8066	1.06317	0.531583	−	0.847006i	$-0.321597\pi$
$383$	20.8066	1.06317	0.531583	+	0.847006i	$0.321597\pi$
$384$	2.95934	0.151018
$385$	−1.01772	−0.0518676
$386$	−6.16918	−0.314003
$387$	−36.3528	−1.84792
$388$	−6.72524	−0.341422
$389$	11.0472	0.560115	0.280058	−	0.959983i	$-0.409646\pi$
$389$	11.0472	0.560115	0.280058	+	0.959983i	$0.409646\pi$
$390$	15.9659	0.808464
$391$	5.55606	0.280982
$392$	−6.83737	−0.345339
$393$	−52.5462	−2.65061
$394$	−26.5951	−1.33984
$395$	−7.53180	−0.378966
$396$	−18.2000	−0.914584
$397$	0.725242	0.0363988	0.0181994	−	0.999834i	$-0.494207\pi$
$397$	0.725242	0.0363988	0.0181994	+	0.999834i	$0.494207\pi$
$398$	−3.91048	−0.196015
$399$	−2.38688	−0.119494
$400$	−4.36262	−0.218131
$401$	−21.3364	−1.06549	−0.532744	−	0.846277i	$-0.678839\pi$
$401$	−21.3364	−1.06549	−0.532744	+	0.846277i	$0.678839\pi$
$402$	0	0
$403$	48.0616	2.39412
$404$	−3.44394	−0.171342
$405$	−5.49114	−0.272857
$406$	0	0
$407$	−26.3056	−1.30392
$408$	−2.36262	−0.116967
$409$	−17.5154	−0.866081	−0.433040	−	0.901374i	$-0.642559\pi$
$409$	−17.5154	−0.866081	−0.433040	+	0.901374i	$0.642559\pi$
$410$	9.12852	0.450826
$411$	−13.0308	−0.642762
$412$	16.8780	0.831521
$413$	−0.487903	−0.0240081
$414$	−40.0698	−1.96932
$415$	−4.33836	−0.212962
$416$	−6.75770	−0.331324
$417$	49.2551	2.41203
$418$	−6.32196	−0.309217
$419$	−18.7088	−0.913987	−0.456993	−	0.889470i	$-0.651074\pi$
$419$	−18.7088	−0.913987	−0.456993	+	0.889470i	$0.651074\pi$
$420$	0.952796	0.0464917
$421$	−1.58032	−0.0770203	−0.0385101	−	0.999258i	$-0.512261\pi$
$421$	−1.58032	−0.0770203	−0.0385101	+	0.999258i	$0.512261\pi$
$422$	−11.0308	−0.536971
$423$	−18.2000	−0.884914
$424$	10.4747	0.508699
$425$	3.48294	0.168948
$426$	−16.3462	−0.791977
$427$	−0.998014	−0.0482973
$428$	14.6275	0.707048
$429$	63.2144	3.05202
$430$	5.04066	0.243082
$431$	−31.0633	−1.49626	−0.748132	−	0.663549i	$-0.769049\pi$
$431$	−31.0633	−1.49626	−0.748132	+	0.663549i	$0.769049\pi$
$432$	8.16098	0.392645
$433$	3.75605	0.180504	0.0902521	−	0.995919i	$-0.471233\pi$
$433$	3.75605	0.180504	0.0902521	+	0.995919i	$0.471233\pi$
$434$	2.86817	0.137677
$435$	0	0
$436$	20.6357	0.988272
$437$	−13.9187	−0.665821
$438$	−34.2226	−1.63522
$439$	21.8472	1.04271	0.521355	−	0.853340i	$-0.325427\pi$
$439$	21.8472	1.04271	0.521355	+	0.853340i	$0.325427\pi$
$440$	2.52360	0.120308
$441$	−39.3675	−1.87464
$442$	5.39508	0.256618
$443$	−3.86983	−0.183861	−0.0919305	−	0.995765i	$-0.529304\pi$
$443$	−3.86983	−0.183861	−0.0919305	+	0.995765i	$0.529304\pi$
$444$	24.6275	1.16877
$445$	−2.74950	−0.130339
$446$	1.52360	0.0721446
$447$	17.7560	0.839832
$448$	−0.403279	−0.0190532
$449$	−9.60492	−0.453284	−0.226642	−	0.973978i	$-0.572775\pi$
$449$	−9.60492	−0.453284	−0.226642	+	0.973978i	$0.572775\pi$
$450$	−25.1187	−1.18411
$451$	36.1429	1.70190
$452$	15.8374	0.744927
$453$	16.4180	0.771386
$454$	−4.79016	−0.224813
$455$	−2.17572	−0.102000
$456$	5.91868	0.277168
$457$	−34.8764	−1.63145	−0.815724	−	0.578441i	$-0.803661\pi$
$457$	−34.8764	−1.63145	−0.815724	+	0.578441i	$0.803661\pi$
$458$	1.19344	0.0557658
$459$	−6.51540	−0.304113
$460$	5.55606	0.259053
$461$	4.40328	0.205081	0.102541	−	0.994729i	$-0.467303\pi$
$461$	4.40328	0.205081	0.102541	+	0.994729i	$0.467303\pi$
$462$	3.77245	0.175510
$463$	−4.30557	−0.200097	−0.100048	−	0.994983i	$-0.531900\pi$
$463$	−4.30557	−0.200097	−0.100048	+	0.994983i	$0.531900\pi$
$464$	0	0
$465$	16.8033	0.779232
$466$	28.1593	1.30446
$467$	−38.3187	−1.77318	−0.886588	−	0.462560i	$-0.846931\pi$
$467$	−38.3187	−1.77318	−0.886588	+	0.462560i	$0.846931\pi$
$468$	−38.9088	−1.79856
$469$	0	0
$470$	2.52360	0.116405
$471$	−8.06492	−0.371612
$472$	1.20984	0.0556873
$473$	19.9577	0.917655
$474$	27.9187	1.28235
$475$	−8.72524	−0.400342
$476$	0.321962	0.0147571
$477$	60.3105	2.76143
$478$	12.8879	0.589478
$479$	−34.1269	−1.55930	−0.779648	−	0.626218i	$-0.784602\pi$
$479$	−34.1269	−1.55930	−0.779648	+	0.626218i	$0.784602\pi$
$480$	−2.36262	−0.107838
$481$	−56.2373	−2.56420
$482$	−18.6600	−0.849939
$483$	8.30557	0.377916
$484$	−1.00820	−0.0458272
$485$	5.36917	0.243801
$486$	−4.12852	−0.187274
$487$	26.5462	1.20292	0.601462	−	0.798902i	$-0.294585\pi$
$487$	26.5462	1.20292	0.601462	+	0.798902i	$0.294585\pi$
$488$	2.47474	0.112026
$489$	−24.8682	−1.12458
$490$	5.45868	0.246598
$491$	25.1088	1.13315	0.566573	−	0.824012i	$-0.308269\pi$
$491$	25.1088	1.13315	0.566573	+	0.824012i	$0.308269\pi$
$492$	−33.8374	−1.52551
$493$	0	0
$494$	−13.5154	−0.608087
$495$	14.5301	0.653082
$496$	−7.11212	−0.319344
$497$	2.22755	0.0999195
$498$	16.0813	0.720621
$499$	−11.1220	−0.497888	−0.248944	−	0.968518i	$-0.580084\pi$
$499$	−11.1220	−0.497888	−0.248944	+	0.968518i	$0.580084\pi$
$500$	7.47474	0.334281
$501$	2.00165	0.0894273
$502$	−15.9269	−0.710851
$503$	25.3528	1.13042	0.565212	−	0.824946i	$-0.308794\pi$
$503$	25.3528	1.13042	0.565212	+	0.824946i	$0.308794\pi$
$504$	−2.32196	−0.103428
$505$	2.74950	0.122351
$506$	21.9983	0.977946
$507$	96.6714	4.29333
$508$	−3.91048	−0.173500
$509$	−6.07801	−0.269403	−0.134702	−	0.990886i	$-0.543008\pi$
$509$	−6.07801	−0.269403	−0.134702	+	0.990886i	$0.543008\pi$
$510$	1.88622	0.0835233
$511$	4.66363	0.206307
$512$	1.00000	0.0441942
$513$	16.3220	0.720632
$514$	−20.2731	−0.894208
$515$	−13.4747	−0.593768
$516$	−18.6846	−0.822543
$517$	9.99180	0.439439
$518$	−3.35608	−0.147457
$519$	22.2892	0.978386
$520$	5.39508	0.236590
$521$	−17.5482	−0.768800	−0.384400	−	0.923167i	$-0.625592\pi$
$521$	−17.5482	−0.768800	−0.384400	+	0.923167i	$0.625592\pi$
$522$	0	0
$523$	−27.9351	−1.22152	−0.610758	−	0.791817i	$-0.709135\pi$
$523$	−27.9351	−1.22152	−0.610758	+	0.791817i	$0.709135\pi$
$524$	−17.7560	−0.775677
$525$	5.20653	0.227232
$526$	10.7967	0.470759
$527$	5.67804	0.247339
$528$	−9.35442	−0.407099
$529$	25.4324	1.10576
$530$	−8.36262	−0.363249
$531$	6.96589	0.302294
$532$	−0.806559	−0.0349688
$533$	77.2681	3.34685
$534$	10.1918	0.441042
$535$	−11.6780	−0.504886
$536$	0	0
$537$	0.211491	0.00912652
$538$	−2.80656	−0.120999
$539$	21.6128	0.930929
$540$	−6.51540	−0.280378
$541$	−20.7544	−0.892301	−0.446151	−	0.894958i	$-0.647205\pi$
$541$	−20.7544	−0.892301	−0.446151	+	0.894958i	$0.647205\pi$
$542$	−17.9023	−0.768969
$543$	−43.5770	−1.87007
$544$	−0.798360	−0.0342294
$545$	−16.4747	−0.705701
$546$	8.06492	0.345147
$547$	23.7642	1.01609	0.508043	−	0.861332i	$-0.330369\pi$
$547$	23.7642	1.01609	0.508043	+	0.861332i	$0.330369\pi$
$548$	−4.40328	−0.188099
$549$	14.2488	0.608126
$550$	13.7902	0.588014
$551$	0	0
$552$	−20.5951	−0.876584
$553$	−3.80457	−0.161787
$554$	13.7560	0.584439
$555$	−19.6616	−0.834590
$556$	16.6439	0.705860
$557$	24.1429	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$557$	24.1429	1.02297	0.511484	+	0.859293i	$0.329096\pi$
$558$	−40.9495	−1.73353
$559$	42.6665	1.80460
$560$	0.321962	0.0136054
$561$	7.46820	0.315308
$562$	−19.2098	−0.810318
$563$	−7.77211	−0.327555	−0.163778	−	0.986497i	$-0.552368\pi$
$563$	−7.77211	−0.327555	−0.163778	+	0.986497i	$0.552368\pi$
$564$	−9.35442	−0.393892
$565$	−12.6439	−0.531934
$566$	−5.98360	−0.251510
$567$	−2.77377	−0.116487
$568$	−5.52360	−0.231765
$569$	4.86163	0.203810	0.101905	−	0.994794i	$-0.467506\pi$
$569$	4.86163	0.203810	0.101905	+	0.994794i	$0.467506\pi$
$570$	−4.72524	−0.197919
$571$	−36.0616	−1.50913	−0.754566	−	0.656224i	$-0.772152\pi$
$571$	−36.0616	−1.50913	−0.754566	+	0.656224i	$0.772152\pi$
$572$	21.3610	0.893147
$573$	−26.0616	−1.08874
$574$	4.61113	0.192465
$575$	30.3610	1.26614
$576$	5.75770	0.239904
$577$	−22.2406	−0.925890	−0.462945	−	0.886387i	$-0.653207\pi$
$577$	−22.2406	−0.925890	−0.462945	+	0.886387i	$0.653207\pi$
$578$	−16.3626	−0.680595
$579$	−18.2567	−0.758723
$580$	0	0
$581$	−2.19145	−0.0909169
$582$	−19.9023	−0.824976
$583$	−33.1105	−1.37130
$584$	−11.5643	−0.478533
$585$	31.0633	1.28431
$586$	10.7902	0.445737
$587$	9.52195	0.393013	0.196506	−	0.980503i	$-0.437040\pi$
$587$	9.52195	0.393013	0.196506	+	0.980503i	$0.437040\pi$
$588$	−20.2341	−0.834440
$589$	−14.2242	−0.586100
$590$	−0.965887	−0.0397649
$591$	−78.7039	−3.23745
$592$	8.32196	0.342031
$593$	6.70884	0.275499	0.137750	−	0.990467i	$-0.456013\pi$
$593$	6.70884	0.275499	0.137750	+	0.990467i	$0.456013\pi$
$594$	−25.7967	−1.05845
$595$	−0.257042	−0.0105377
$596$	6.00000	0.245770
$597$	−11.5725	−0.473629
$598$	47.0292	1.92317
$599$	26.1265	1.06750	0.533751	−	0.845642i	$-0.320782\pi$
$599$	26.1265	1.06750	0.533751	+	0.845642i	$0.320782\pi$
$600$	−12.9105	−0.527068
$601$	29.7010	1.21153	0.605764	−	0.795644i	$-0.292868\pi$
$601$	29.7010	1.21153	0.605764	+	0.795644i	$0.292868\pi$
$602$	2.54621	0.103776
$603$	0	0
$604$	5.54786	0.225739
$605$	0.804906	0.0327241
$606$	−10.1918	−0.414013
$607$	−13.4829	−0.547256	−0.273628	−	0.961836i	$-0.588224\pi$
$607$	−13.4829	−0.547256	−0.273628	+	0.961836i	$0.588224\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−1.97574	−0.0799953
$611$	21.3610	0.864172
$612$	−4.59672	−0.185811
$613$	−6.87148	−0.277536	−0.138768	−	0.990325i	$-0.544314\pi$
$613$	−6.87148	−0.277536	−0.138768	+	0.990325i	$0.544314\pi$
$614$	9.03081	0.364454
$615$	27.0144	1.08933
$616$	1.27476	0.0513615
$617$	−20.4197	−0.822065	−0.411033	−	0.911621i	$-0.634832\pi$
$617$	−20.4197	−0.822065	−0.411033	+	0.911621i	$0.634832\pi$
$618$	49.9478	2.00920
$619$	21.9187	0.880986	0.440493	−	0.897756i	$-0.354804\pi$
$619$	21.9187	0.880986	0.440493	+	0.897756i	$0.354804\pi$
$620$	5.67804	0.228035
$621$	−56.7951	−2.27911
$622$	−16.5462	−0.663443
$623$	−1.38887	−0.0556439
$624$	−19.9983	−0.800575
$625$	15.8456	0.633823
$626$	−4.48460	−0.179241
$627$	−18.7088	−0.747159
$628$	−2.72524	−0.108749
$629$	−6.64392	−0.264911
$630$	1.85376	0.0738557
$631$	19.5154	0.776896	0.388448	−	0.921471i	$-0.373011\pi$
$631$	19.5154	0.776896	0.388448	+	0.921471i	$0.373011\pi$
$632$	9.43409	0.375268
$633$	−32.6439	−1.29748
$634$	19.2098	0.762920
$635$	3.12198	0.123892
$636$	30.9983	1.22916
$637$	46.2049	1.83070
$638$	0	0
$639$	−31.8033	−1.25812
$640$	−0.798360	−0.0315580
$641$	−7.59672	−0.300052	−0.150026	−	0.988682i	$-0.547936\pi$
$641$	−7.59672	−0.300052	−0.150026	+	0.988682i	$0.547936\pi$
$642$	43.2878	1.70844
$643$	31.6911	1.24978	0.624888	−	0.780714i	$-0.285145\pi$
$643$	31.6911	1.24978	0.624888	+	0.780714i	$0.285145\pi$
$644$	2.80656	0.110594
$645$	14.9170	0.587357
$646$	−1.59672	−0.0628221
$647$	−33.7724	−1.32773	−0.663866	−	0.747852i	$-0.731085\pi$
$647$	−33.7724	−1.32773	−0.663866	+	0.747852i	$0.731085\pi$
$648$	6.87802	0.270194
$649$	−3.82428	−0.150116
$650$	29.4813	1.15635
$651$	8.48790	0.332667
$652$	−8.40328	−0.329098
$653$	11.2551	0.440444	0.220222	−	0.975450i	$-0.429322\pi$
$653$	11.2551	0.440444	0.220222	+	0.975450i	$0.429322\pi$
$654$	61.0682	2.38795
$655$	14.1757	0.553891
$656$	−11.4341	−0.446426
$657$	−66.5836	−2.59767
$658$	1.27476	0.0496953
$659$	−8.41968	−0.327984	−0.163992	−	0.986462i	$-0.552437\pi$
$659$	−8.41968	−0.327984	−0.163992	+	0.986462i	$0.552437\pi$
$660$	7.46820	0.290699
$661$	−45.5383	−1.77124	−0.885618	−	0.464414i	$-0.846265\pi$
$661$	−45.5383	−1.77124	−0.885618	+	0.464414i	$0.846265\pi$
$662$	6.33016	0.246029
$663$	15.9659	0.620064
$664$	5.43409	0.210883
$665$	0.643924	0.0249703
$666$	47.9154	1.85668
$667$	0	0
$668$	0.676385	0.0261701
$669$	4.50886	0.174323
$670$	0	0
$671$	−7.82262	−0.301989
$672$	−1.19344	−0.0460380
$673$	−43.5770	−1.67977	−0.839885	−	0.542764i	$-0.817378\pi$
$673$	−43.5770	−1.67977	−0.839885	+	0.542764i	$0.817378\pi$
$674$	19.3364	0.744809
$675$	−35.6033	−1.37037
$676$	32.6665	1.25641
$677$	15.2390	0.585682	0.292841	−	0.956161i	$-0.405399\pi$
$677$	15.2390	0.585682	0.292841	+	0.956161i	$0.405399\pi$
$678$	46.8682	1.79996
$679$	2.71215	0.104083
$680$	0.637379	0.0244424
$681$	−14.1757	−0.543215
$682$	22.4813	0.860853
$683$	−44.3590	−1.69735	−0.848675	−	0.528915i	$-0.822599\pi$
$683$	−44.3590	−1.69735	−0.848675	+	0.528915i	$0.822599\pi$
$684$	11.5154	0.440302
$685$	3.51540	0.134317
$686$	5.58032	0.213058
$687$	3.53180	0.134747
$688$	−6.31376	−0.240710
$689$	−70.7852	−2.69670
$690$	16.4423	0.625947
$691$	−17.5967	−0.669411	−0.334705	−	0.942323i	$-0.608637\pi$
$691$	−17.5967	−0.669411	−0.334705	+	0.942323i	$0.608637\pi$
$692$	7.53180	0.286316
$693$	7.33968	0.278811
$694$	24.5544	0.932073
$695$	−13.2878	−0.504037
$696$	0	0
$697$	9.12852	0.345767
$698$	−10.1429	−0.383916
$699$	83.3331	3.15195
$700$	1.75935	0.0664974
$701$	14.8066	0.559236	0.279618	−	0.960111i	$-0.409792\pi$
$701$	14.8066	0.559236	0.279618	+	0.960111i	$0.409792\pi$
$702$	−55.1495	−2.08148
$703$	16.6439	0.627738
$704$	−3.16098	−0.119134
$705$	7.46820	0.281269
$706$	3.83737	0.144421
$707$	1.38887	0.0522338
$708$	3.58032	0.134557
$709$	31.8990	1.19799	0.598996	−	0.800752i	$-0.295567\pi$
$709$	31.8990	1.19799	0.598996	+	0.800752i	$0.295567\pi$
$710$	4.40982	0.165498
$711$	54.3187	2.03711
$712$	3.44394	0.129067
$713$	49.4957	1.85363
$714$	0.952796	0.0356575
$715$	−17.0537	−0.637774
$716$	0.0714656	0.00267080
$717$	38.1396	1.42435
$718$	−5.61312	−0.209480
$719$	2.15113	0.0802236	0.0401118	−	0.999195i	$-0.487229\pi$
$719$	2.15113	0.0802236	0.0401118	+	0.999195i	$0.487229\pi$
$720$	−4.59672	−0.171310
$721$	−6.80656	−0.253489
$722$	−15.0000	−0.558242
$723$	−55.2213	−2.05370
$724$	−14.7252	−0.547259
$725$	0	0
$726$	−2.98360	−0.110732
$727$	−36.2373	−1.34397	−0.671984	−	0.740565i	$-0.734558\pi$
$727$	−36.2373	−1.34397	−0.671984	+	0.740565i	$0.734558\pi$
$728$	2.72524	0.101004
$729$	−32.8518	−1.21673
$730$	9.23245	0.341708
$731$	5.04066	0.186435
$732$	7.32362	0.270689
$733$	3.02261	0.111643	0.0558213	−	0.998441i	$-0.482222\pi$
$733$	3.02261	0.111643	0.0558213	+	0.998441i	$0.482222\pi$
$734$	−19.1121	−0.705441
$735$	16.1541	0.595853
$736$	−6.95934	−0.256525
$737$	0	0
$738$	−65.8341	−2.42339
$739$	−0.396734	−0.0145941	−0.00729705	−	0.999973i	$-0.502323\pi$
$739$	−0.396734	−0.0145941	−0.00729705	+	0.999973i	$0.502323\pi$
$740$	−6.64392	−0.244236
$741$	−39.9967	−1.46932
$742$	−4.22425	−0.155077
$743$	−30.2895	−1.11121	−0.555607	−	0.831445i	$-0.687514\pi$
$743$	−30.2895	−1.11121	−0.555607	+	0.831445i	$0.687514\pi$
$744$	−21.0472	−0.771628
$745$	−4.79016	−0.175498
$746$	−13.7885	−0.504833
$747$	31.2878	1.14476
$748$	2.52360	0.0922720
$749$	−5.89898	−0.215544
$750$	22.1203	0.807720
$751$	−1.52195	−0.0555367	−0.0277684	−	0.999614i	$-0.508840\pi$
$751$	−1.52195	−0.0555367	−0.0277684	+	0.999614i	$0.508840\pi$
$752$	−3.16098	−0.115269
$753$	−47.1331	−1.71762
$754$	0	0
$755$	−4.42919	−0.161195
$756$	−3.29116	−0.119698
$757$	−1.58852	−0.0577358	−0.0288679	−	0.999583i	$-0.509190\pi$
$757$	−1.58852	−0.0577358	−0.0288679	+	0.999583i	$0.509190\pi$
$758$	6.51706	0.236710
$759$	65.1006	2.36300
$760$	−1.59672	−0.0579191
$761$	−21.8374	−0.791604	−0.395802	−	0.918336i	$-0.629533\pi$
$761$	−21.8374	−0.791604	−0.395802	+	0.918336i	$0.629533\pi$
$762$	−11.5725	−0.419226
$763$	−8.32196	−0.301275
$764$	−8.80656	−0.318610
$765$	3.66984	0.132683
$766$	20.8066	0.751772
$767$	−8.17572	−0.295208
$768$	2.95934	0.106786
$769$	29.9967	1.08171	0.540854	−	0.841116i	$-0.318101\pi$
$769$	29.9967	1.08171	0.540854	+	0.841116i	$0.318101\pi$
$770$	−1.01772	−0.0366760
$771$	−59.9950	−2.16067
$772$	−6.16918	−0.222034
$773$	29.7560	1.07025	0.535125	−	0.844773i	$-0.320264\pi$
$773$	29.7560	1.07025	0.535125	+	0.844773i	$0.320264\pi$
$774$	−36.3528	−1.30667
$775$	31.0275	1.11454
$776$	−6.72524	−0.241422
$777$	−9.93177	−0.356300
$778$	11.0472	0.396061
$779$	−22.8682	−0.819337
$780$	15.9659	0.571670
$781$	17.4600	0.624768
$782$	5.55606	0.198684
$783$	0	0
$784$	−6.83737	−0.244192
$785$	2.17572	0.0776549
$786$	−52.5462	−1.87426
$787$	−1.00165	−0.0357051	−0.0178525	−	0.999841i	$-0.505683\pi$
$787$	−1.00165	−0.0357051	−0.0178525	+	0.999841i	$0.505683\pi$
$788$	−26.5951	−0.947410
$789$	31.9511	1.13749
$790$	−7.53180	−0.267969
$791$	−6.38688	−0.227091
$792$	−18.2000	−0.646708
$793$	−16.7236	−0.593872
$794$	0.725242	0.0257379
$795$	−24.7479	−0.877716
$796$	−3.91048	−0.138603
$797$	20.7055	0.733428	0.366714	−	0.930334i	$-0.380483\pi$
$797$	20.7055	0.733428	0.366714	+	0.930334i	$0.380483\pi$
$798$	−2.38688	−0.0844947
$799$	2.52360	0.0892786
$800$	−4.36262	−0.154242
$801$	19.8292	0.700629
$802$	−21.3364	−0.753414
$803$	36.5544	1.28998
$804$	0	0
$805$	−2.24065	−0.0789724
$806$	48.0616	1.69290
$807$	−8.30557	−0.292370
$808$	−3.44394	−0.121157
$809$	−36.9659	−1.29965	−0.649826	−	0.760083i	$-0.725158\pi$
$809$	−36.9659	−1.29965	−0.649826	+	0.760083i	$0.725158\pi$
$810$	−5.49114	−0.192939
$811$	43.9088	1.54185	0.770924	−	0.636927i	$-0.219795\pi$
$811$	43.9088	1.54185	0.770924	+	0.636927i	$0.219795\pi$
$812$	0	0
$813$	−52.9790	−1.85805
$814$	−26.3056	−0.922010
$815$	6.70884	0.235001
$816$	−2.36262	−0.0827083
$817$	−12.6275	−0.441781
$818$	−17.5154	−0.612412
$819$	15.6911	0.548292
$820$	9.12852	0.318782
$821$	36.5298	1.27490	0.637450	−	0.770492i	$-0.279989\pi$
$821$	36.5298	1.27490	0.637450	+	0.770492i	$0.279989\pi$
$822$	−13.0308	−0.454502
$823$	13.6456	0.475655	0.237827	−	0.971307i	$-0.423565\pi$
$823$	13.6456	0.475655	0.237827	+	0.971307i	$0.423565\pi$
$824$	16.8780	0.587974
$825$	40.8098	1.42081
$826$	−0.487903	−0.0169763
$827$	11.4341	0.397602	0.198801	−	0.980040i	$-0.436295\pi$
$827$	11.4341	0.397602	0.198801	+	0.980040i	$0.436295\pi$
$828$	−40.0698	−1.39252
$829$	−2.48129	−0.0861788	−0.0430894	−	0.999071i	$-0.513720\pi$
$829$	−2.48129	−0.0861788	−0.0430894	+	0.999071i	$0.513720\pi$
$830$	−4.33836	−0.150587
$831$	40.7088	1.41217
$832$	−6.75770	−0.234281
$833$	5.45868	0.189132
$834$	49.2551	1.70556
$835$	−0.539999	−0.0186874
$836$	−6.32196	−0.218650
$837$	−58.0419	−2.00622
$838$	−18.7088	−0.646286
$839$	51.4078	1.77480	0.887398	−	0.461004i	$-0.152511\pi$
$839$	51.4078	1.77480	0.887398	+	0.461004i	$0.152511\pi$
$840$	0.952796	0.0328746
$841$	−29.0000	−1.00000
$842$	−1.58032	−0.0544616
$843$	−56.8485	−1.95797
$844$	−11.0308	−0.379696
$845$	−26.0797	−0.897168
$846$	−18.2000	−0.625728
$847$	0.406586	0.0139704
$848$	10.4747	0.359704
$849$	−17.7075	−0.607721
$850$	3.48294	0.119464
$851$	−57.9154	−1.98531
$852$	−16.3462	−0.560012
$853$	−24.0616	−0.823854	−0.411927	−	0.911217i	$-0.635144\pi$
$853$	−24.0616	−0.823854	−0.411927	+	0.911217i	$0.635144\pi$
$854$	−0.998014	−0.0341513
$855$	−9.19344	−0.314409
$856$	14.6275	0.499959
$857$	15.1285	0.516780	0.258390	−	0.966041i	$-0.416808\pi$
$857$	15.1285	0.516780	0.258390	+	0.966041i	$0.416808\pi$
$858$	63.2144	2.15810
$859$	39.2878	1.34048	0.670242	−	0.742143i	$-0.266190\pi$
$859$	39.2878	1.34048	0.670242	+	0.742143i	$0.266190\pi$
$860$	5.04066	0.171885
$861$	13.6459	0.465051
$862$	−31.0633	−1.05802
$863$	−39.3495	−1.33947	−0.669736	−	0.742600i	$-0.733593\pi$
$863$	−39.3495	−1.33947	−0.669736	+	0.742600i	$0.733593\pi$
$864$	8.16098	0.277642
$865$	−6.01309	−0.204451
$866$	3.75605	0.127636
$867$	−48.4226	−1.64452
$868$	2.86817	0.0973521
$869$	−29.8210	−1.01161
$870$	0	0
$871$	0	0
$872$	20.6357	0.698814
$873$	−38.7219	−1.31054
$874$	−13.9187	−0.470806
$875$	−3.01441	−0.101906
$876$	−34.2226	−1.15627
$877$	24.5462	0.828867	0.414433	−	0.910080i	$-0.363980\pi$
$877$	24.5462	0.828867	0.414433	+	0.910080i	$0.363980\pi$
$878$	21.8472	0.737308
$879$	31.9318	1.07703
$880$	2.52360	0.0850706
$881$	9.83082	0.331209	0.165604	−	0.986192i	$-0.447043\pi$
$881$	9.83082	0.331209	0.165604	+	0.986192i	$0.447043\pi$
$882$	−39.3675	−1.32557
$883$	−36.5138	−1.22879	−0.614393	−	0.789000i	$-0.710599\pi$
$883$	−36.5138	−1.22879	−0.614393	+	0.789000i	$0.710599\pi$
$884$	5.39508	0.181456
$885$	−2.85839	−0.0960837
$886$	−3.86983	−0.130009
$887$	−46.1347	−1.54905	−0.774526	−	0.632542i	$-0.782012\pi$
$887$	−46.1347	−1.54905	−0.774526	+	0.632542i	$0.782012\pi$
$888$	24.6275	0.826446
$889$	1.57702	0.0528915
$890$	−2.74950	−0.0921636
$891$	−21.7413	−0.728361
$892$	1.52360	0.0510140
$893$	−6.32196	−0.211556
$894$	17.7560	0.593851
$895$	−0.0570553	−0.00190715
$896$	−0.403279	−0.0134726
$897$	139.175	4.64693
$898$	−9.60492	−0.320520
$899$	0	0
$900$	−25.1187	−0.837289
$901$	−8.36262	−0.278599
$902$	36.1429	1.20343
$903$	7.53511	0.250753
$904$	15.8374	0.526743
$905$	11.7560	0.390784
$906$	16.4180	0.545452
$907$	−50.4813	−1.67620	−0.838102	−	0.545514i	$-0.816335\pi$
$907$	−50.4813	−1.67620	−0.838102	+	0.545514i	$0.816335\pi$
$908$	−4.79016	−0.158967
$909$	−19.8292	−0.657692
$910$	−2.17572	−0.0721246
$911$	41.3265	1.36921	0.684604	−	0.728915i	$-0.259975\pi$
$911$	41.3265	1.36921	0.684604	+	0.728915i	$0.259975\pi$
$912$	5.91868	0.195987
$913$	−17.1770	−0.568477
$914$	−34.8764	−1.15361
$915$	−5.84688	−0.193292
$916$	1.19344	0.0394324
$917$	7.16065	0.236465
$918$	−6.51540	−0.215040
$919$	20.4813	0.675615	0.337808	−	0.941215i	$-0.390315\pi$
$919$	20.4813	0.675615	0.337808	+	0.941215i	$0.390315\pi$
$920$	5.55606	0.183178
$921$	26.7252	0.880627
$922$	4.40328	0.145014
$923$	37.3269	1.22863
$924$	3.77245	0.124104
$925$	−36.3056	−1.19372
$926$	−4.30557	−0.141490
$927$	97.1786	3.19176
$928$	0	0
$929$	41.4341	1.35941	0.679704	−	0.733486i	$-0.262108\pi$
$929$	41.4341	1.35941	0.679704	+	0.733486i	$0.262108\pi$
$930$	16.8033	0.551000
$931$	−13.6747	−0.448171
$932$	28.1593	0.922389
$933$	−48.9659	−1.60307
$934$	−38.3187	−1.25382
$935$	−2.01474	−0.0658892
$936$	−38.9088	−1.27177
$937$	13.4505	0.439408	0.219704	−	0.975567i	$-0.429491\pi$
$937$	13.4505	0.439408	0.219704	+	0.975567i	$0.429491\pi$
$938$	0	0
$939$	−13.2715	−0.433097
$940$	2.52360	0.0823108
$941$	−7.41769	−0.241810	−0.120905	−	0.992664i	$-0.538580\pi$
$941$	−7.41769	−0.241810	−0.120905	+	0.992664i	$0.538580\pi$
$942$	−8.06492	−0.262769
$943$	79.5737	2.59128
$944$	1.20984	0.0393769
$945$	2.62753	0.0854735
$946$	19.9577	0.648880
$947$	−13.5967	−0.441834	−0.220917	−	0.975293i	$-0.570905\pi$
$947$	−13.5967	−0.441834	−0.220917	+	0.975293i	$0.570905\pi$
$948$	27.9187	0.906757
$949$	78.1478	2.53679
$950$	−8.72524	−0.283084
$951$	56.8485	1.84344
$952$	0.321962	0.0104349
$953$	6.79050	0.219966	0.109983	−	0.993933i	$-0.464920\pi$
$953$	6.79050	0.219966	0.109983	+	0.993933i	$0.464920\pi$
$954$	60.3105	1.95262
$955$	7.03081	0.227512
$956$	12.8879	0.416824
$957$	0	0
$958$	−34.1269	−1.10259
$959$	1.77575	0.0573420
$960$	−2.36262	−0.0762533
$961$	19.5823	0.631687
$962$	−56.2373	−1.81316
$963$	84.2209	2.71398
$964$	−18.6600	−0.600998
$965$	4.92523	0.158549
$966$	8.30557	0.267227
$967$	22.1511	0.712332	0.356166	−	0.934423i	$-0.384084\pi$
$967$	22.1511	0.712332	0.356166	+	0.934423i	$0.384084\pi$
$968$	−1.00820	−0.0324047
$969$	−4.72524	−0.151797
$970$	5.36917	0.172394
$971$	35.3692	1.13505	0.567525	−	0.823356i	$-0.307901\pi$
$971$	35.3692	1.13505	0.567525	+	0.823356i	$0.307901\pi$
$972$	−4.12852	−0.132422
$973$	−6.71215	−0.215182
$974$	26.5462	0.850595
$975$	87.2452	2.79408
$976$	2.47474	0.0792147
$977$	54.2859	1.73676	0.868379	−	0.495901i	$-0.165162\pi$
$977$	54.2859	1.73676	0.868379	+	0.495901i	$0.165162\pi$
$978$	−24.8682	−0.795196
$979$	−10.8862	−0.347925
$980$	5.45868	0.174371
$981$	118.814	3.79345
$982$	25.1088	0.801255
$983$	−33.7724	−1.07717	−0.538587	−	0.842570i	$-0.681042\pi$
$983$	−33.7724	−1.07717	−0.538587	+	0.842570i	$0.681042\pi$
$984$	−33.8374	−1.07870
$985$	21.2324	0.676522
$986$	0	0
$987$	3.77245	0.120078
$988$	−13.5154	−0.429982
$989$	43.9396	1.39720
$990$	14.5301	0.461798
$991$	−44.4649	−1.41247	−0.706237	−	0.707976i	$-0.749609\pi$
$991$	−44.4649	−1.41247	−0.706237	+	0.707976i	$0.749609\pi$
$992$	−7.11212	−0.225810
$993$	18.7331	0.594477
$994$	2.22755	0.0706537
$995$	3.12198	0.0989733
$996$	16.0813	0.509556
$997$	26.5429	0.840622	0.420311	−	0.907380i	$-0.361921\pi$
$997$	26.5429	0.840622	0.420311	+	0.907380i	$0.361921\pi$
$998$	−11.1220	−0.352060
$999$	67.9154	2.14875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8978.2.a.i.1.3		3
67.66	odd	2		134.2.a.a.1.1	✓	3
201.200	even	2		1206.2.a.o.1.2		3
268.267	even	2		1072.2.a.j.1.3		3
335.133	even	4		3350.2.c.i.2949.4		6
335.267	even	4		3350.2.c.i.2949.3		6
335.334	odd	2		3350.2.a.m.1.3		3
469.468	even	2		6566.2.a.z.1.3		3
536.133	odd	2		4288.2.a.t.1.3		3
536.267	even	2		4288.2.a.u.1.1		3
804.803	odd	2		9648.2.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
134.2.a.a.1.1	✓	3	67.66	odd	2
1072.2.a.j.1.3		3	268.267	even	2
1206.2.a.o.1.2		3	201.200	even	2
3350.2.a.m.1.3		3	335.334	odd	2
3350.2.c.i.2949.3		6	335.267	even	4
3350.2.c.i.2949.4		6	335.133	even	4
4288.2.a.t.1.3		3	536.133	odd	2
4288.2.a.u.1.1		3	536.267	even	2
6566.2.a.z.1.3		3	469.468	even	2
8978.2.a.i.1.3		3	1.1	even	1	trivial
9648.2.a.bk.1.2		3	804.803	odd	2

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:	\( N \)	\(=\)	\( 8978 = 2 \cdot 67^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8978.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.95934 q^{3} +1.00000 q^{4} -0.798360 q^{5} +2.95934 q^{6} -0.403279 q^{7} +1.00000 q^{8} +5.75770 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.95934 q^{3} +1.00000 q^{4} -0.798360 q^{5} +2.95934 q^{6} -0.403279 q^{7} +1.00000 q^{8} +5.75770 q^{9} -0.798360 q^{10} -3.16098 q^{11} +2.95934 q^{12} -6.75770 q^{13} -0.403279 q^{14} -2.36262 q^{15} +1.00000 q^{16} -0.798360 q^{17} +5.75770 q^{18} +2.00000 q^{19} -0.798360 q^{20} -1.19344 q^{21} -3.16098 q^{22} -6.95934 q^{23} +2.95934 q^{24} -4.36262 q^{25} -6.75770 q^{26} +8.16098 q^{27} -0.403279 q^{28} -2.36262 q^{30} -7.11212 q^{31} +1.00000 q^{32} -9.35442 q^{33} -0.798360 q^{34} +0.321962 q^{35} +5.75770 q^{36} +8.32196 q^{37} +2.00000 q^{38} -19.9983 q^{39} -0.798360 q^{40} -11.4341 q^{41} -1.19344 q^{42} -6.31376 q^{43} -3.16098 q^{44} -4.59672 q^{45} -6.95934 q^{46} -3.16098 q^{47} +2.95934 q^{48} -6.83737 q^{49} -4.36262 q^{50} -2.36262 q^{51} -6.75770 q^{52} +10.4747 q^{53} +8.16098 q^{54} +2.52360 q^{55} -0.403279 q^{56} +5.91868 q^{57} +1.20984 q^{59} -2.36262 q^{60} +2.47474 q^{61} -7.11212 q^{62} -2.32196 q^{63} +1.00000 q^{64} +5.39508 q^{65} -9.35442 q^{66} -0.798360 q^{68} -20.5951 q^{69} +0.321962 q^{70} -5.52360 q^{71} +5.75770 q^{72} -11.5643 q^{73} +8.32196 q^{74} -12.9105 q^{75} +2.00000 q^{76} +1.27476 q^{77} -19.9983 q^{78} +9.43409 q^{79} -0.798360 q^{80} +6.87802 q^{81} -11.4341 q^{82} +5.43409 q^{83} -1.19344 q^{84} +0.637379 q^{85} -6.31376 q^{86} -3.16098 q^{88} +3.44394 q^{89} -4.59672 q^{90} +2.72524 q^{91} -6.95934 q^{92} -21.0472 q^{93} -3.16098 q^{94} -1.59672 q^{95} +2.95934 q^{96} -6.72524 q^{97} -6.83737 q^{98} -18.2000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 - q^6 + 3 * q^8 + 8 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} - q^{6} + 3 q^{8} + 8 q^{9} - 3 q^{10} + q^{11} - q^{12} - 11 q^{13} + 4 q^{15} + 3 q^{16} - 3 q^{17} + 8 q^{18} + 6 q^{19} - 3 q^{20} - 6 q^{21} + q^{22} - 11 q^{23} - q^{24} - 2 q^{25} - 11 q^{26} + 14 q^{27} + 4 q^{30} - 4 q^{31} + 3 q^{32} - 20 q^{33} - 3 q^{34} - 20 q^{35} + 8 q^{36} + 4 q^{37} + 6 q^{38} - 10 q^{39} - 3 q^{40} + 4 q^{41} - 6 q^{42} - q^{43} + q^{44} - 15 q^{45} - 11 q^{46} + q^{47} - q^{48} + 19 q^{49} - 2 q^{50} + 4 q^{51} - 11 q^{52} + 3 q^{53} + 14 q^{54} - 14 q^{55} - 2 q^{57} + 4 q^{60} - 21 q^{61} - 4 q^{62} + 14 q^{63} + 3 q^{64} + 18 q^{65} - 20 q^{66} - 3 q^{68} - 13 q^{69} - 20 q^{70} + 5 q^{71} + 8 q^{72} - 23 q^{73} + 4 q^{74} - 22 q^{75} + 6 q^{76} + 26 q^{77} - 10 q^{78} - 10 q^{79} - 3 q^{80} - 9 q^{81} + 4 q^{82} - 22 q^{83} - 6 q^{84} + 13 q^{85} - q^{86} + q^{88} + 19 q^{89} - 15 q^{90} - 14 q^{91} - 11 q^{92} - 20 q^{93} + q^{94} - 6 q^{95} - q^{96} + 2 q^{97} + 19 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 3 * q^5 - q^6 + 3 * q^8 + 8 * q^9 - 3 * q^10 + q^11 - q^12 - 11 * q^13 + 4 * q^15 + 3 * q^16 - 3 * q^17 + 8 * q^18 + 6 * q^19 - 3 * q^20 - 6 * q^21 + q^22 - 11 * q^23 - q^24 - 2 * q^25 - 11 * q^26 + 14 * q^27 + 4 * q^30 - 4 * q^31 + 3 * q^32 - 20 * q^33 - 3 * q^34 - 20 * q^35 + 8 * q^36 + 4 * q^37 + 6 * q^38 - 10 * q^39 - 3 * q^40 + 4 * q^41 - 6 * q^42 - q^43 + q^44 - 15 * q^45 - 11 * q^46 + q^47 - q^48 + 19 * q^49 - 2 * q^50 + 4 * q^51 - 11 * q^52 + 3 * q^53 + 14 * q^54 - 14 * q^55 - 2 * q^57 + 4 * q^60 - 21 * q^61 - 4 * q^62 + 14 * q^63 + 3 * q^64 + 18 * q^65 - 20 * q^66 - 3 * q^68 - 13 * q^69 - 20 * q^70 + 5 * q^71 + 8 * q^72 - 23 * q^73 + 4 * q^74 - 22 * q^75 + 6 * q^76 + 26 * q^77 - 10 * q^78 - 10 * q^79 - 3 * q^80 - 9 * q^81 + 4 * q^82 - 22 * q^83 - 6 * q^84 + 13 * q^85 - q^86 + q^88 + 19 * q^89 - 15 * q^90 - 14 * q^91 - 11 * q^92 - 20 * q^93 + q^94 - 6 * q^95 - q^96 + 2 * q^97 + 19 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more