LMFDB - Embedded newform 5577.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5577 = 3 \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5577.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:	$44.5325692073$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 21x^{4} + 13x^{3} - 33x^{2} - 7x + 7 $ x^7 - 3x^6 - 7x^5 + 21x^4 + 13x^3 - 33x^2 - 7x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.15754$ of defining polynomial
Character	$\chi$	$=$	5577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} +0.710210 q^{5} -2.15754 q^{6} +2.30964 q^{7} -1.41315 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} +0.710210 q^{5} -2.15754 q^{6} +2.30964 q^{7} -1.41315 q^{8} +1.00000 q^{9} -1.53231 q^{10} +1.00000 q^{11} +2.65498 q^{12} -4.98315 q^{14} +0.710210 q^{15} -2.26104 q^{16} -6.68027 q^{17} -2.15754 q^{18} -0.242517 q^{19} +1.88559 q^{20} +2.30964 q^{21} -2.15754 q^{22} +9.53531 q^{23} -1.41315 q^{24} -4.49560 q^{25} +1.00000 q^{27} +6.13206 q^{28} -2.95273 q^{29} -1.53231 q^{30} -4.02017 q^{31} +7.70458 q^{32} +1.00000 q^{33} +14.4130 q^{34} +1.64033 q^{35} +2.65498 q^{36} +3.42304 q^{37} +0.523240 q^{38} -1.00363 q^{40} +9.46022 q^{41} -4.98315 q^{42} -11.8791 q^{43} +2.65498 q^{44} +0.710210 q^{45} -20.5728 q^{46} -12.9178 q^{47} -2.26104 q^{48} -1.66555 q^{49} +9.69944 q^{50} -6.68027 q^{51} +6.78954 q^{53} -2.15754 q^{54} +0.710210 q^{55} -3.26386 q^{56} -0.242517 q^{57} +6.37063 q^{58} +7.81320 q^{59} +1.88559 q^{60} +0.910585 q^{61} +8.67368 q^{62} +2.30964 q^{63} -12.1009 q^{64} -2.15754 q^{66} +9.32216 q^{67} -17.7360 q^{68} +9.53531 q^{69} -3.53908 q^{70} +12.9704 q^{71} -1.41315 q^{72} +8.51665 q^{73} -7.38534 q^{74} -4.49560 q^{75} -0.643877 q^{76} +2.30964 q^{77} +1.82360 q^{79} -1.60581 q^{80} +1.00000 q^{81} -20.4108 q^{82} +4.64671 q^{83} +6.13206 q^{84} -4.74440 q^{85} +25.6297 q^{86} -2.95273 q^{87} -1.41315 q^{88} +14.7714 q^{89} -1.53231 q^{90} +25.3161 q^{92} -4.02017 q^{93} +27.8707 q^{94} -0.172238 q^{95} +7.70458 q^{96} +1.86542 q^{97} +3.59349 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 7 q^{3} + 9 q^{4} + 6 q^{5} + 3 q^{6} + 6 q^{7} + 15 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q + 3 * q^2 + 7 * q^3 + 9 * q^4 + 6 * q^5 + 3 * q^6 + 6 * q^7 + 15 * q^8 + 7 * q^9	$ 7 q + 3 q^{2} + 7 q^{3} + 9 q^{4} + 6 q^{5} + 3 q^{6} + 6 q^{7} + 15 q^{8} + 7 q^{9} + 7 q^{11} + 9 q^{12} + 8 q^{14} + 6 q^{15} + 17 q^{16} - 2 q^{17} + 3 q^{18} + 8 q^{19} - 2 q^{20} + 6 q^{21} + 3 q^{22} + 4 q^{23} + 15 q^{24} + 13 q^{25} + 7 q^{27} + 12 q^{28} - 12 q^{29} - 10 q^{31} + 33 q^{32} + 7 q^{33} + 28 q^{34} - 4 q^{35} + 9 q^{36} + 6 q^{37} + 16 q^{38} - 10 q^{40} + 2 q^{41} + 8 q^{42} - 16 q^{43} + 9 q^{44} + 6 q^{45} - 26 q^{46} + 18 q^{47} + 17 q^{48} + 23 q^{49} + 39 q^{50} - 2 q^{51} + 10 q^{53} + 3 q^{54} + 6 q^{55} + 16 q^{56} + 8 q^{57} + 10 q^{58} + 2 q^{59} - 2 q^{60} - 10 q^{61} - 36 q^{62} + 6 q^{63} + 29 q^{64} + 3 q^{66} + 8 q^{67} - 10 q^{68} + 4 q^{69} - 20 q^{70} + 36 q^{71} + 15 q^{72} + 20 q^{73} + 13 q^{75} + 10 q^{76} + 6 q^{77} + 6 q^{79} - 20 q^{80} + 7 q^{81} - 10 q^{82} + 30 q^{83} + 12 q^{84} - 40 q^{85} + 6 q^{86} - 12 q^{87} + 15 q^{88} + 34 q^{89} - 12 q^{92} - 10 q^{93} + 32 q^{94} + 18 q^{95} + 33 q^{96} + 16 q^{97} + q^{98} + 7 q^{99}+O(q^{100}) $ 7 * q + 3 * q^2 + 7 * q^3 + 9 * q^4 + 6 * q^5 + 3 * q^6 + 6 * q^7 + 15 * q^8 + 7 * q^9 + 7 * q^11 + 9 * q^12 + 8 * q^14 + 6 * q^15 + 17 * q^16 - 2 * q^17 + 3 * q^18 + 8 * q^19 - 2 * q^20 + 6 * q^21 + 3 * q^22 + 4 * q^23 + 15 * q^24 + 13 * q^25 + 7 * q^27 + 12 * q^28 - 12 * q^29 - 10 * q^31 + 33 * q^32 + 7 * q^33 + 28 * q^34 - 4 * q^35 + 9 * q^36 + 6 * q^37 + 16 * q^38 - 10 * q^40 + 2 * q^41 + 8 * q^42 - 16 * q^43 + 9 * q^44 + 6 * q^45 - 26 * q^46 + 18 * q^47 + 17 * q^48 + 23 * q^49 + 39 * q^50 - 2 * q^51 + 10 * q^53 + 3 * q^54 + 6 * q^55 + 16 * q^56 + 8 * q^57 + 10 * q^58 + 2 * q^59 - 2 * q^60 - 10 * q^61 - 36 * q^62 + 6 * q^63 + 29 * q^64 + 3 * q^66 + 8 * q^67 - 10 * q^68 + 4 * q^69 - 20 * q^70 + 36 * q^71 + 15 * q^72 + 20 * q^73 + 13 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 - 20 * q^80 + 7 * q^81 - 10 * q^82 + 30 * q^83 + 12 * q^84 - 40 * q^85 + 6 * q^86 - 12 * q^87 + 15 * q^88 + 34 * q^89 - 12 * q^92 - 10 * q^93 + 32 * q^94 + 18 * q^95 + 33 * q^96 + 16 * q^97 + q^98 + 7 * 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 $

$2$	−2.15754	−1.52561	−0.762806	−	0.646628i	$-0.776179\pi$
$2$	−2.15754	−1.52561	−0.762806	+	0.646628i	$0.776179\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.65498	1.32749
$5$	0.710210	0.317616	0.158808	−	0.987310i	$-0.449235\pi$
$5$	0.710210	0.317616	0.158808	+	0.987310i	$0.449235\pi$
$6$	−2.15754	−0.880812
$7$	2.30964	0.872963	0.436482	−	0.899713i	$-0.356224\pi$
$7$	2.30964	0.872963	0.436482	+	0.899713i	$0.356224\pi$
$8$	−1.41315	−0.499622
$9$	1.00000	0.333333
$10$	−1.53231	−0.484558
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.65498	0.766427
$13$	0	0
$13$	0	0
$14$	−4.98315	−1.33180
$15$	0.710210	0.183375
$16$	−2.26104	−0.565261
$17$	−6.68027	−1.62020	−0.810102	−	0.586289i	$-0.800588\pi$
$17$	−6.68027	−1.62020	−0.810102	+	0.586289i	$0.800588\pi$
$18$	−2.15754	−0.508537
$19$	−0.242517	−0.0556372	−0.0278186	−	0.999613i	$-0.508856\pi$
$19$	−0.242517	−0.0556372	−0.0278186	+	0.999613i	$0.508856\pi$
$20$	1.88559	0.421631
$21$	2.30964	0.504005
$22$	−2.15754	−0.459989
$23$	9.53531	1.98825	0.994125	−	0.108242i	$-0.0345220\pi$
$23$	9.53531	1.98825	0.994125	+	0.108242i	$0.0345220\pi$
$24$	−1.41315	−0.288457
$25$	−4.49560	−0.899120
$26$	0	0
$27$	1.00000	0.192450
$28$	6.13206	1.15885
$29$	−2.95273	−0.548308	−0.274154	−	0.961686i	$-0.588398\pi$
$29$	−2.95273	−0.548308	−0.274154	+	0.961686i	$0.588398\pi$
$30$	−1.53231	−0.279760
$31$	−4.02017	−0.722044	−0.361022	−	0.932557i	$-0.617572\pi$
$31$	−4.02017	−0.722044	−0.361022	+	0.932557i	$0.617572\pi$
$32$	7.70458	1.36199
$33$	1.00000	0.174078
$34$	14.4130	2.47180
$35$	1.64033	0.277267
$36$	2.65498	0.442497
$37$	3.42304	0.562744	0.281372	−	0.959599i	$-0.409211\pi$
$37$	3.42304	0.562744	0.281372	+	0.959599i	$0.409211\pi$
$38$	0.523240	0.0848807
$39$	0	0
$40$	−1.00363	−0.158688
$41$	9.46022	1.47744	0.738719	−	0.674013i	$-0.235431\pi$
$41$	9.46022	1.47744	0.738719	+	0.674013i	$0.235431\pi$
$42$	−4.98315	−0.768916
$43$	−11.8791	−1.81155	−0.905776	−	0.423757i	$-0.860711\pi$
$43$	−11.8791	−1.81155	−0.905776	+	0.423757i	$0.860711\pi$
$44$	2.65498	0.400253
$45$	0.710210	0.105872
$46$	−20.5728	−3.03330
$47$	−12.9178	−1.88426	−0.942130	−	0.335249i	$-0.891180\pi$
$47$	−12.9178	−1.88426	−0.942130	+	0.335249i	$0.891180\pi$
$48$	−2.26104	−0.326353
$49$	−1.66555	−0.237935
$50$	9.69944	1.37171
$51$	−6.68027	−0.935425
$52$	0	0
$53$	6.78954	0.932615	0.466307	−	0.884623i	$-0.345584\pi$
$53$	6.78954	0.932615	0.466307	+	0.884623i	$0.345584\pi$
$54$	−2.15754	−0.293604
$55$	0.710210	0.0957647
$56$	−3.26386	−0.436152
$57$	−0.242517	−0.0321221
$58$	6.37063	0.836504
$59$	7.81320	1.01719	0.508596	−	0.861005i	$-0.330165\pi$
$59$	7.81320	1.01719	0.508596	+	0.861005i	$0.330165\pi$
$60$	1.88559	0.243429
$61$	0.910585	0.116588	0.0582942	−	0.998299i	$-0.481434\pi$
$61$	0.910585	0.116588	0.0582942	+	0.998299i	$0.481434\pi$
$62$	8.67368	1.10156
$63$	2.30964	0.290988
$64$	−12.1009	−1.51261
$65$	0	0
$66$	−2.15754	−0.265575
$67$	9.32216	1.13888	0.569442	−	0.822032i	$-0.307159\pi$
$67$	9.32216	1.13888	0.569442	+	0.822032i	$0.307159\pi$
$68$	−17.7360	−2.15080
$69$	9.53531	1.14792
$70$	−3.53908	−0.423001
$71$	12.9704	1.53930	0.769650	−	0.638466i	$-0.220431\pi$
$71$	12.9704	1.53930	0.769650	+	0.638466i	$0.220431\pi$
$72$	−1.41315	−0.166541
$73$	8.51665	0.996798	0.498399	−	0.866948i	$-0.333921\pi$
$73$	8.51665	0.996798	0.498399	+	0.866948i	$0.333921\pi$
$74$	−7.38534	−0.858529
$75$	−4.49560	−0.519107
$76$	−0.643877	−0.0738578
$77$	2.30964	0.263208
$78$	0	0
$79$	1.82360	0.205172	0.102586	−	0.994724i	$-0.467288\pi$
$79$	1.82360	0.205172	0.102586	+	0.994724i	$0.467288\pi$
$80$	−1.60581	−0.179536
$81$	1.00000	0.111111
$82$	−20.4108	−2.25400
$83$	4.64671	0.510042	0.255021	−	0.966935i	$-0.417918\pi$
$83$	4.64671	0.510042	0.255021	+	0.966935i	$0.417918\pi$
$84$	6.13206	0.669062
$85$	−4.74440	−0.514602
$86$	25.6297	2.76372
$87$	−2.95273	−0.316566
$88$	−1.41315	−0.150642
$89$	14.7714	1.56576	0.782881	−	0.622171i	$-0.213749\pi$
$89$	14.7714	1.56576	0.782881	+	0.622171i	$0.213749\pi$
$90$	−1.53231	−0.161519
$91$	0	0
$92$	25.3161	2.63938
$93$	−4.02017	−0.416872
$94$	27.8707	2.87465
$95$	−0.172238	−0.0176712
$96$	7.70458	0.786345
$97$	1.86542	0.189405	0.0947025	−	0.995506i	$-0.469810\pi$
$97$	1.86542	0.189405	0.0947025	+	0.995506i	$0.469810\pi$
$98$	3.59349	0.362997
$99$	1.00000	0.100504
$100$	−11.9357	−1.19357
$101$	−2.45086	−0.243870	−0.121935	−	0.992538i	$-0.538910\pi$
$101$	−2.45086	−0.243870	−0.121935	+	0.992538i	$0.538910\pi$
$102$	14.4130	1.42709
$103$	10.2811	1.01302	0.506512	−	0.862233i	$-0.330934\pi$
$103$	10.2811	1.01302	0.506512	+	0.862233i	$0.330934\pi$
$104$	0	0
$105$	1.64033	0.160080
$106$	−14.6487	−1.42281
$107$	19.6690	1.90148	0.950739	−	0.309993i	$-0.100327\pi$
$107$	19.6690	1.90148	0.950739	+	0.309993i	$0.100327\pi$
$108$	2.65498	0.255476
$109$	1.27506	0.122128	0.0610642	−	0.998134i	$-0.480551\pi$
$109$	1.27506	0.122128	0.0610642	+	0.998134i	$0.480551\pi$
$110$	−1.53231	−0.146100
$111$	3.42304	0.324900
$112$	−5.22220	−0.493452
$113$	−0.638739	−0.0600875	−0.0300438	−	0.999549i	$-0.509565\pi$
$113$	−0.638739	−0.0600875	−0.0300438	+	0.999549i	$0.509565\pi$
$114$	0.523240	0.0490059
$115$	6.77207	0.631499
$116$	−7.83943	−0.727873
$117$	0	0
$118$	−16.8573	−1.55184
$119$	−15.4290	−1.41438
$120$	−1.00363	−0.0916185
$121$	1.00000	0.0909091
$122$	−1.96462	−0.177869
$123$	9.46022	0.853000
$124$	−10.6735	−0.958506
$125$	−6.74387	−0.603190
$126$	−4.98315	−0.443934
$127$	10.3681	0.920021	0.460010	−	0.887914i	$-0.347846\pi$
$127$	10.3681	0.920021	0.460010	+	0.887914i	$0.347846\pi$
$128$	10.6989	0.945660
$129$	−11.8791	−1.04590
$130$	0	0
$131$	−5.68692	−0.496868	−0.248434	−	0.968649i	$-0.579916\pi$
$131$	−5.68692	−0.496868	−0.248434	+	0.968649i	$0.579916\pi$
$132$	2.65498	0.231086
$133$	−0.560127	−0.0485692
$134$	−20.1129	−1.73749
$135$	0.710210	0.0611251
$136$	9.44019	0.809490
$137$	4.86363	0.415528	0.207764	−	0.978179i	$-0.433381\pi$
$137$	4.86363	0.415528	0.207764	+	0.978179i	$0.433381\pi$
$138$	−20.5728	−1.75127
$139$	−4.41515	−0.374488	−0.187244	−	0.982313i	$-0.559956\pi$
$139$	−4.41515	−0.374488	−0.187244	+	0.982313i	$0.559956\pi$
$140$	4.35505	0.368069
$141$	−12.9178	−1.08788
$142$	−27.9841	−2.34837
$143$	0	0
$144$	−2.26104	−0.188420
$145$	−2.09706	−0.174151
$146$	−18.3750	−1.52073
$147$	−1.66555	−0.137372
$148$	9.08810	0.747037
$149$	9.50356	0.778562	0.389281	−	0.921119i	$-0.372724\pi$
$149$	9.50356	0.778562	0.389281	+	0.921119i	$0.372724\pi$
$150$	9.69944	0.791956
$151$	15.2126	1.23798	0.618991	−	0.785398i	$-0.287542\pi$
$151$	15.2126	1.23798	0.618991	+	0.785398i	$0.287542\pi$
$152$	0.342711	0.0277976
$153$	−6.68027	−0.540068
$154$	−4.98315	−0.401554
$155$	−2.85516	−0.229332
$156$	0	0
$157$	20.1949	1.61173	0.805866	−	0.592098i	$-0.201700\pi$
$157$	20.1949	1.61173	0.805866	+	0.592098i	$0.201700\pi$
$158$	−3.93450	−0.313012
$159$	6.78954	0.538445
$160$	5.47187	0.432589
$161$	22.0232	1.73567
$162$	−2.15754	−0.169512
$163$	−16.4974	−1.29217	−0.646087	−	0.763264i	$-0.723596\pi$
$163$	−16.4974	−1.29217	−0.646087	+	0.763264i	$0.723596\pi$
$164$	25.1167	1.96129
$165$	0.710210	0.0552898
$166$	−10.0255	−0.778126
$167$	−5.50507	−0.425995	−0.212998	−	0.977053i	$-0.568323\pi$
$167$	−5.50507	−0.425995	−0.212998	+	0.977053i	$0.568323\pi$
$168$	−3.26386	−0.251812
$169$	0	0
$170$	10.2362	0.785082
$171$	−0.242517	−0.0185457
$172$	−31.5389	−2.40482
$173$	−12.7126	−0.966524	−0.483262	−	0.875476i	$-0.660548\pi$
$173$	−12.7126	−0.966524	−0.483262	+	0.875476i	$0.660548\pi$
$174$	6.37063	0.482956
$175$	−10.3832	−0.784899
$176$	−2.26104	−0.170432
$177$	7.81320	0.587276
$178$	−31.8698	−2.38875
$179$	24.4279	1.82583	0.912913	−	0.408154i	$-0.133828\pi$
$179$	24.4279	1.82583	0.912913	+	0.408154i	$0.133828\pi$
$180$	1.88559	0.140544
$181$	0.931324	0.0692248	0.0346124	−	0.999401i	$-0.488980\pi$
$181$	0.931324	0.0692248	0.0346124	+	0.999401i	$0.488980\pi$
$182$	0	0
$183$	0.910585	0.0673124
$184$	−13.4748	−0.993374
$185$	2.43108	0.178736
$186$	8.67368	0.635985
$187$	−6.68027	−0.488510
$188$	−34.2966	−2.50134
$189$	2.30964	0.168002
$190$	0.371610	0.0269594
$191$	4.68209	0.338784	0.169392	−	0.985549i	$-0.445820\pi$
$191$	4.68209	0.338784	0.169392	+	0.985549i	$0.445820\pi$
$192$	−12.1009	−0.873304
$193$	−10.2671	−0.739039	−0.369520	−	0.929223i	$-0.620478\pi$
$193$	−10.2671	−0.739039	−0.369520	+	0.929223i	$0.620478\pi$
$194$	−4.02473	−0.288959
$195$	0	0
$196$	−4.42200	−0.315857
$197$	11.5842	0.825341	0.412671	−	0.910880i	$-0.364596\pi$
$197$	11.5842	0.825341	0.412671	+	0.910880i	$0.364596\pi$
$198$	−2.15754	−0.153330
$199$	17.2330	1.22162	0.610808	−	0.791779i	$-0.290845\pi$
$199$	17.2330	1.22162	0.610808	+	0.791779i	$0.290845\pi$
$200$	6.35294	0.449221
$201$	9.32216	0.657535
$202$	5.28783	0.372050
$203$	−6.81975	−0.478652
$204$	−17.7360	−1.24177
$205$	6.71875	0.469258
$206$	−22.1818	−1.54548
$207$	9.53531	0.662750
$208$	0	0
$209$	−0.242517	−0.0167752
$210$	−3.53908	−0.244220
$211$	−10.6905	−0.735967	−0.367983	−	0.929832i	$-0.619952\pi$
$211$	−10.6905	−0.735967	−0.367983	+	0.929832i	$0.619952\pi$
$212$	18.0261	1.23804
$213$	12.9704	0.888716
$214$	−42.4367	−2.90092
$215$	−8.43668	−0.575377
$216$	−1.41315	−0.0961523
$217$	−9.28516	−0.630318
$218$	−2.75099	−0.186320
$219$	8.51665	0.575502
$220$	1.88559	0.127127
$221$	0	0
$222$	−7.38534	−0.495672
$223$	−5.33992	−0.357587	−0.178794	−	0.983887i	$-0.557219\pi$
$223$	−5.33992	−0.357587	−0.178794	+	0.983887i	$0.557219\pi$
$224$	17.7948	1.18897
$225$	−4.49560	−0.299707
$226$	1.37811	0.0916702
$227$	−7.82431	−0.519318	−0.259659	−	0.965700i	$-0.583610\pi$
$227$	−7.82431	−0.519318	−0.259659	+	0.965700i	$0.583610\pi$
$228$	−0.643877	−0.0426418
$229$	−10.6805	−0.705784	−0.352892	−	0.935664i	$-0.614802\pi$
$229$	−10.6805	−0.705784	−0.352892	+	0.935664i	$0.614802\pi$
$230$	−14.6110	−0.963422
$231$	2.30964	0.151963
$232$	4.17263	0.273947
$233$	15.4509	1.01222	0.506112	−	0.862468i	$-0.331082\pi$
$233$	15.4509	1.01222	0.506112	+	0.862468i	$0.331082\pi$
$234$	0	0
$235$	−9.17437	−0.598470
$236$	20.7439	1.35031
$237$	1.82360	0.118456
$238$	33.2888	2.15779
$239$	−16.9365	−1.09553	−0.547765	−	0.836632i	$-0.684521\pi$
$239$	−16.9365	−1.09553	−0.547765	+	0.836632i	$0.684521\pi$
$240$	−1.60581	−0.103655
$241$	−0.0799646	−0.00515098	−0.00257549	−	0.999997i	$-0.500820\pi$
$241$	−0.0799646	−0.00515098	−0.00257549	+	0.999997i	$0.500820\pi$
$242$	−2.15754	−0.138692
$243$	1.00000	0.0641500
$244$	2.41758	0.154770
$245$	−1.18289	−0.0755720
$246$	−20.4108	−1.30135
$247$	0	0
$248$	5.68108	0.360749
$249$	4.64671	0.294473
$250$	14.5502	0.920234
$251$	−7.57174	−0.477924	−0.238962	−	0.971029i	$-0.576807\pi$
$251$	−7.57174	−0.477924	−0.238962	+	0.971029i	$0.576807\pi$
$252$	6.13206	0.386283
$253$	9.53531	0.599480
$254$	−22.3696	−1.40359
$255$	−4.74440	−0.297106
$256$	1.11835	0.0698970
$257$	23.0242	1.43621	0.718105	−	0.695935i	$-0.245010\pi$
$257$	23.0242	1.43621	0.718105	+	0.695935i	$0.245010\pi$
$258$	25.6297	1.59564
$259$	7.90600	0.491255
$260$	0	0
$261$	−2.95273	−0.182769
$262$	12.2698	0.758028
$263$	−5.95047	−0.366922	−0.183461	−	0.983027i	$-0.558730\pi$
$263$	−5.95047	−0.366922	−0.183461	+	0.983027i	$0.558730\pi$
$264$	−1.41315	−0.0869731
$265$	4.82200	0.296213
$266$	1.20850	0.0740977
$267$	14.7714	0.903994
$268$	24.7502	1.51186
$269$	−1.17571	−0.0716841	−0.0358420	−	0.999357i	$-0.511411\pi$
$269$	−1.17571	−0.0716841	−0.0358420	+	0.999357i	$0.511411\pi$
$270$	−1.53231	−0.0932532
$271$	−7.12706	−0.432938	−0.216469	−	0.976290i	$-0.569454\pi$
$271$	−7.12706	−0.432938	−0.216469	+	0.976290i	$0.569454\pi$
$272$	15.1044	0.915837
$273$	0	0
$274$	−10.4935	−0.633934
$275$	−4.49560	−0.271095
$276$	25.3161	1.52385
$277$	−11.9736	−0.719426	−0.359713	−	0.933063i	$-0.617125\pi$
$277$	−11.9736	−0.719426	−0.359713	+	0.933063i	$0.617125\pi$
$278$	9.52586	0.571323
$279$	−4.02017	−0.240681
$280$	−2.31803	−0.138529
$281$	18.7901	1.12092	0.560461	−	0.828181i	$-0.310624\pi$
$281$	18.7901	1.12092	0.560461	+	0.828181i	$0.310624\pi$
$282$	27.8707	1.65968
$283$	10.5237	0.625568	0.312784	−	0.949824i	$-0.398738\pi$
$283$	10.5237	0.625568	0.312784	+	0.949824i	$0.398738\pi$
$284$	34.4361	2.04341
$285$	−0.172238	−0.0102025
$286$	0	0
$287$	21.8497	1.28975
$288$	7.70458	0.453997
$289$	27.6260	1.62506
$290$	4.52448	0.265687
$291$	1.86542	0.109353
$292$	22.6115	1.32324
$293$	−14.0111	−0.818540	−0.409270	−	0.912413i	$-0.634217\pi$
$293$	−14.0111	−0.818540	−0.409270	+	0.912413i	$0.634217\pi$
$294$	3.59349	0.209576
$295$	5.54901	0.323076
$296$	−4.83725	−0.281159
$297$	1.00000	0.0580259
$298$	−20.5043	−1.18778
$299$	0	0
$300$	−11.9357	−0.689110
$301$	−27.4366	−1.58142
$302$	−32.8217	−1.88868
$303$	−2.45086	−0.140798
$304$	0.548341	0.0314495
$305$	0.646706	0.0370303
$306$	14.4130	0.823934
$307$	2.52126	0.143896	0.0719481	−	0.997408i	$-0.477078\pi$
$307$	2.52126	0.143896	0.0719481	+	0.997408i	$0.477078\pi$
$308$	6.13206	0.349406
$309$	10.2811	0.584870
$310$	6.16013	0.349872
$311$	−22.9380	−1.30069	−0.650347	−	0.759638i	$-0.725376\pi$
$311$	−22.9380	−1.30069	−0.650347	+	0.759638i	$0.725376\pi$
$312$	0	0
$313$	−21.6566	−1.22410	−0.612051	−	0.790818i	$-0.709655\pi$
$313$	−21.6566	−1.22410	−0.612051	+	0.790818i	$0.709655\pi$
$314$	−43.5714	−2.45888
$315$	1.64033	0.0924222
$316$	4.84163	0.272363
$317$	26.6050	1.49429	0.747143	−	0.664664i	$-0.231425\pi$
$317$	26.6050	1.49429	0.747143	+	0.664664i	$0.231425\pi$
$318$	−14.6487	−0.821458
$319$	−2.95273	−0.165321
$320$	−8.59415	−0.480428
$321$	19.6690	1.09782
$322$	−47.5159	−2.64796
$323$	1.62008	0.0901435
$324$	2.65498	0.147499
$325$	0	0
$326$	35.5937	1.97135
$327$	1.27506	0.0705108
$328$	−13.3687	−0.738161
$329$	−29.8356	−1.64489
$330$	−1.53231	−0.0843507
$331$	−24.7670	−1.36132	−0.680659	−	0.732600i	$-0.738307\pi$
$331$	−24.7670	−1.36132	−0.680659	+	0.732600i	$0.738307\pi$
$332$	12.3369	0.677076
$333$	3.42304	0.187581
$334$	11.8774	0.649903
$335$	6.62069	0.361727
$336$	−5.22220	−0.284894
$337$	−28.2890	−1.54100	−0.770499	−	0.637441i	$-0.779993\pi$
$337$	−28.2890	−1.54100	−0.770499	+	0.637441i	$0.779993\pi$
$338$	0	0
$339$	−0.638739	−0.0346916
$340$	−12.5963	−0.683129
$341$	−4.02017	−0.217704
$342$	0.523240	0.0282936
$343$	−20.0143	−1.08067
$344$	16.7869	0.905091
$345$	6.77207	0.364596
$346$	27.4280	1.47454
$347$	−18.0814	−0.970661	−0.485331	−	0.874331i	$-0.661301\pi$
$347$	−18.0814	−0.970661	−0.485331	+	0.874331i	$0.661301\pi$
$348$	−7.83943	−0.420238
$349$	21.0326	1.12585	0.562924	−	0.826508i	$-0.309676\pi$
$349$	21.0326	1.12585	0.562924	+	0.826508i	$0.309676\pi$
$350$	22.4022	1.19745
$351$	0	0
$352$	7.70458	0.410655
$353$	2.20127	0.117162	0.0585810	−	0.998283i	$-0.481342\pi$
$353$	2.20127	0.117162	0.0585810	+	0.998283i	$0.481342\pi$
$354$	−16.8573	−0.895955
$355$	9.21169	0.488906
$356$	39.2177	2.07853
$357$	−15.4290	−0.816591
$358$	−52.7041	−2.78550
$359$	−7.76540	−0.409842	−0.204921	−	0.978778i	$-0.565694\pi$
$359$	−7.76540	−0.409842	−0.204921	+	0.978778i	$0.565694\pi$
$360$	−1.00363	−0.0528959
$361$	−18.9412	−0.996905
$362$	−2.00937	−0.105610
$363$	1.00000	0.0524864
$364$	0	0
$365$	6.04861	0.316599
$366$	−1.96462	−0.102692
$367$	−11.9604	−0.624326	−0.312163	−	0.950028i	$-0.601054\pi$
$367$	−11.9604	−0.624326	−0.312163	+	0.950028i	$0.601054\pi$
$368$	−21.5597	−1.12388
$369$	9.46022	0.492480
$370$	−5.24514	−0.272682
$371$	15.6814	0.814138
$372$	−10.6735	−0.553394
$373$	14.1243	0.731330	0.365665	−	0.930747i	$-0.380842\pi$
$373$	14.1243	0.731330	0.365665	+	0.930747i	$0.380842\pi$
$374$	14.4130	0.745276
$375$	−6.74387	−0.348252
$376$	18.2548	0.941418
$377$	0	0
$378$	−4.98315	−0.256305
$379$	−3.34432	−0.171786	−0.0858931	−	0.996304i	$-0.527374\pi$
$379$	−3.34432	−0.171786	−0.0858931	+	0.996304i	$0.527374\pi$
$380$	−0.457288	−0.0234584
$381$	10.3681	0.531174
$382$	−10.1018	−0.516853
$383$	−5.63621	−0.287997	−0.143998	−	0.989578i	$-0.545996\pi$
$383$	−5.63621	−0.287997	−0.143998	+	0.989578i	$0.545996\pi$
$384$	10.6989	0.545977
$385$	1.64033	0.0835991
$386$	22.1516	1.12749
$387$	−11.8791	−0.603850
$388$	4.95266	0.251433
$389$	14.3490	0.727523	0.363761	−	0.931492i	$-0.381492\pi$
$389$	14.3490	0.727523	0.363761	+	0.931492i	$0.381492\pi$
$390$	0	0
$391$	−63.6984	−3.22137
$392$	2.35366	0.118878
$393$	−5.68692	−0.286867
$394$	−24.9934	−1.25915
$395$	1.29514	0.0651657
$396$	2.65498	0.133418
$397$	−14.8769	−0.746648	−0.373324	−	0.927701i	$-0.621782\pi$
$397$	−14.8769	−0.746648	−0.373324	+	0.927701i	$0.621782\pi$
$398$	−37.1809	−1.86371
$399$	−0.560127	−0.0280414
$400$	10.1647	0.508237
$401$	19.6380	0.980675	0.490337	−	0.871533i	$-0.336874\pi$
$401$	19.6380	0.980675	0.490337	+	0.871533i	$0.336874\pi$
$402$	−20.1129	−1.00314
$403$	0	0
$404$	−6.50698	−0.323734
$405$	0.710210	0.0352906
$406$	14.7139	0.730237
$407$	3.42304	0.169674
$408$	9.44019	0.467359
$409$	23.1731	1.14584	0.572918	−	0.819613i	$-0.305811\pi$
$409$	23.1731	1.14584	0.572918	+	0.819613i	$0.305811\pi$
$410$	−14.4960	−0.715905
$411$	4.86363	0.239905
$412$	27.2961	1.34478
$413$	18.0457	0.887971
$414$	−20.5728	−1.01110
$415$	3.30014	0.161997
$416$	0	0
$417$	−4.41515	−0.216211
$418$	0.523240	0.0255925
$419$	−3.70294	−0.180900	−0.0904502	−	0.995901i	$-0.528831\pi$
$419$	−3.70294	−0.180900	−0.0904502	+	0.995901i	$0.528831\pi$
$420$	4.35505	0.212505
$421$	23.7239	1.15623	0.578116	−	0.815955i	$-0.303788\pi$
$421$	23.7239	1.15623	0.578116	+	0.815955i	$0.303788\pi$
$422$	23.0653	1.12280
$423$	−12.9178	−0.628086
$424$	−9.59460	−0.465955
$425$	30.0318	1.45676
$426$	−27.9841	−1.35583
$427$	2.10313	0.101777
$428$	52.2209	2.52419
$429$	0	0
$430$	18.2025	0.877802
$431$	−31.0695	−1.49656	−0.748282	−	0.663381i	$-0.769121\pi$
$431$	−31.0695	−1.49656	−0.748282	+	0.663381i	$0.769121\pi$
$432$	−2.26104	−0.108784
$433$	23.0541	1.10791	0.553954	−	0.832547i	$-0.313118\pi$
$433$	23.0541	1.10791	0.553954	+	0.832547i	$0.313118\pi$
$434$	20.0331	0.961620
$435$	−2.09706	−0.100546
$436$	3.38525	0.162124
$437$	−2.31247	−0.110621
$438$	−18.3750	−0.877992
$439$	2.39273	0.114199	0.0570994	−	0.998368i	$-0.481815\pi$
$439$	2.39273	0.114199	0.0570994	+	0.998368i	$0.481815\pi$
$440$	−1.00363	−0.0478462
$441$	−1.66555	−0.0793118
$442$	0	0
$443$	−17.1624	−0.815411	−0.407705	−	0.913113i	$-0.633671\pi$
$443$	−17.1624	−0.815411	−0.407705	+	0.913113i	$0.633671\pi$
$444$	9.08810	0.431302
$445$	10.4908	0.497311
$446$	11.5211	0.545539
$447$	9.50356	0.449503
$448$	−27.9487	−1.32045
$449$	29.0646	1.37164	0.685822	−	0.727769i	$-0.259443\pi$
$449$	29.0646	1.37164	0.685822	+	0.727769i	$0.259443\pi$
$450$	9.69944	0.457236
$451$	9.46022	0.445465
$452$	−1.69584	−0.0797656
$453$	15.2126	0.714749
$454$	16.8813	0.792277
$455$	0	0
$456$	0.342711	0.0160489
$457$	−31.9111	−1.49274	−0.746369	−	0.665532i	$-0.768205\pi$
$457$	−31.9111	−1.49274	−0.746369	+	0.665532i	$0.768205\pi$
$458$	23.0435	1.07675
$459$	−6.68027	−0.311808
$460$	17.9797	0.838308
$461$	−0.706685	−0.0329136	−0.0164568	−	0.999865i	$-0.505239\pi$
$461$	−0.706685	−0.0329136	−0.0164568	+	0.999865i	$0.505239\pi$
$462$	−4.98315	−0.231837
$463$	25.7813	1.19816	0.599079	−	0.800690i	$-0.295533\pi$
$463$	25.7813	1.19816	0.599079	+	0.800690i	$0.295533\pi$
$464$	6.67624	0.309937
$465$	−2.85516	−0.132405
$466$	−33.3360	−1.54426
$467$	37.3541	1.72854	0.864270	−	0.503028i	$-0.167781\pi$
$467$	37.3541	1.72854	0.864270	+	0.503028i	$0.167781\pi$
$468$	0	0
$469$	21.5309	0.994203
$470$	19.7941	0.913033
$471$	20.1949	0.930533
$472$	−11.0412	−0.508212
$473$	−11.8791	−0.546203
$474$	−3.93450	−0.180718
$475$	1.09026	0.0500245
$476$	−40.9638	−1.87757
$477$	6.78954	0.310872
$478$	36.5411	1.67135
$479$	1.74064	0.0795318	0.0397659	−	0.999209i	$-0.487339\pi$
$479$	1.74064	0.0795318	0.0397659	+	0.999209i	$0.487339\pi$
$480$	5.47187	0.249756
$481$	0	0
$482$	0.172527	0.00785839
$483$	22.0232	1.00209
$484$	2.65498	0.120681
$485$	1.32484	0.0601580
$486$	−2.15754	−0.0978680
$487$	5.69783	0.258193	0.129097	−	0.991632i	$-0.458792\pi$
$487$	5.69783	0.258193	0.129097	+	0.991632i	$0.458792\pi$
$488$	−1.28679	−0.0582502
$489$	−16.4974	−0.746037
$490$	2.55213	0.115293
$491$	−27.7388	−1.25183	−0.625917	−	0.779890i	$-0.715275\pi$
$491$	−27.7388	−1.25183	−0.625917	+	0.779890i	$0.715275\pi$
$492$	25.1167	1.13235
$493$	19.7250	0.888370
$494$	0	0
$495$	0.710210	0.0319216
$496$	9.08977	0.408143
$497$	29.9569	1.34375
$498$	−10.0255	−0.449251
$499$	8.27073	0.370249	0.185124	−	0.982715i	$-0.440731\pi$
$499$	8.27073	0.370249	0.185124	+	0.982715i	$0.440731\pi$
$500$	−17.9048	−0.800729
$501$	−5.50507	−0.245948
$502$	16.3363	0.729126
$503$	−12.7099	−0.566706	−0.283353	−	0.959016i	$-0.591447\pi$
$503$	−12.7099	−0.566706	−0.283353	+	0.959016i	$0.591447\pi$
$504$	−3.26386	−0.145384
$505$	−1.74062	−0.0774568
$506$	−20.5728	−0.914573
$507$	0	0
$508$	27.5271	1.22132
$509$	5.55978	0.246433	0.123216	−	0.992380i	$-0.460679\pi$
$509$	5.55978	0.246433	0.123216	+	0.992380i	$0.460679\pi$
$510$	10.2362	0.453268
$511$	19.6704	0.870168
$512$	−23.8107	−1.05230
$513$	−0.242517	−0.0107074
$514$	−49.6756	−2.19110
$515$	7.30172	0.321752
$516$	−31.5389	−1.38842
$517$	−12.9178	−0.568126
$518$	−17.0575	−0.749464
$519$	−12.7126	−0.558023
$520$	0	0
$521$	−4.41425	−0.193392	−0.0966958	−	0.995314i	$-0.530827\pi$
$521$	−4.41425	−0.193392	−0.0966958	+	0.995314i	$0.530827\pi$
$522$	6.37063	0.278835
$523$	15.5162	0.678474	0.339237	−	0.940701i	$-0.389831\pi$
$523$	15.5162	0.678474	0.339237	+	0.940701i	$0.389831\pi$
$524$	−15.0986	−0.659587
$525$	−10.3832	−0.453162
$526$	12.8384	0.559780
$527$	26.8558	1.16986
$528$	−2.26104	−0.0983992
$529$	67.9221	2.95314
$530$	−10.4037	−0.451906
$531$	7.81320	0.339064
$532$	−1.48713	−0.0644751
$533$	0	0
$534$	−31.8698	−1.37914
$535$	13.9691	0.603939
$536$	−13.1736	−0.569011
$537$	24.4279	1.05414
$538$	2.53663	0.109362
$539$	−1.66555	−0.0717402
$540$	1.88559	0.0811430
$541$	−27.9827	−1.20307	−0.601536	−	0.798846i	$-0.705444\pi$
$541$	−27.9827	−1.20307	−0.601536	+	0.798846i	$0.705444\pi$
$542$	15.3769	0.660495
$543$	0.931324	0.0399669
$544$	−51.4687	−2.20670
$545$	0.905559	0.0387899
$546$	0	0
$547$	−24.1470	−1.03245	−0.516226	−	0.856453i	$-0.672663\pi$
$547$	−24.1470	−1.03245	−0.516226	+	0.856453i	$0.672663\pi$
$548$	12.9128	0.551609
$549$	0.910585	0.0388628
$550$	9.69944	0.413586
$551$	0.716086	0.0305063
$552$	−13.4748	−0.573524
$553$	4.21188	0.179107
$554$	25.8336	1.09756
$555$	2.43108	0.103193
$556$	−11.7221	−0.497129
$557$	19.1326	0.810675	0.405337	−	0.914167i	$-0.367154\pi$
$557$	19.1326	0.810675	0.405337	+	0.914167i	$0.367154\pi$
$558$	8.67368	0.367186
$559$	0	0
$560$	−3.70886	−0.156728
$561$	−6.68027	−0.282041
$562$	−40.5404	−1.71009
$563$	−3.91199	−0.164871	−0.0824353	−	0.996596i	$-0.526270\pi$
$563$	−3.91199	−0.164871	−0.0824353	+	0.996596i	$0.526270\pi$
$564$	−34.2966	−1.44415
$565$	−0.453639	−0.0190847
$566$	−22.7053	−0.954373
$567$	2.30964	0.0969959
$568$	−18.3290	−0.769069
$569$	−15.8002	−0.662377	−0.331189	−	0.943565i	$-0.607450\pi$
$569$	−15.8002	−0.662377	−0.331189	+	0.943565i	$0.607450\pi$
$570$	0.371610	0.0155650
$571$	20.2458	0.847262	0.423631	−	0.905835i	$-0.360755\pi$
$571$	20.2458	0.847262	0.423631	+	0.905835i	$0.360755\pi$
$572$	0	0
$573$	4.68209	0.195597
$574$	−47.1417	−1.96766
$575$	−42.8669	−1.78768
$576$	−12.1009	−0.504202
$577$	−7.07073	−0.294358	−0.147179	−	0.989110i	$-0.547019\pi$
$577$	−7.07073	−0.294358	−0.147179	+	0.989110i	$0.547019\pi$
$578$	−59.6042	−2.47921
$579$	−10.2671	−0.426685
$580$	−5.56764	−0.231184
$581$	10.7322	0.445248
$582$	−4.02473	−0.166830
$583$	6.78954	0.281194
$584$	−12.0353	−0.498023
$585$	0	0
$586$	30.2296	1.24877
$587$	27.1767	1.12170	0.560851	−	0.827917i	$-0.310474\pi$
$587$	27.1767	1.12170	0.560851	+	0.827917i	$0.310474\pi$
$588$	−4.42200	−0.182360
$589$	0.974959	0.0401725
$590$	−11.9722	−0.492889
$591$	11.5842	0.476511
$592$	−7.73963	−0.318097
$593$	20.0431	0.823070	0.411535	−	0.911394i	$-0.364993\pi$
$593$	20.0431	0.823070	0.411535	+	0.911394i	$0.364993\pi$
$594$	−2.15754	−0.0885249
$595$	−10.9579	−0.449228
$596$	25.2318	1.03353
$597$	17.2330	0.705300
$598$	0	0
$599$	−24.5855	−1.00454	−0.502268	−	0.864712i	$-0.667501\pi$
$599$	−24.5855	−1.00454	−0.502268	+	0.864712i	$0.667501\pi$
$600$	6.35294	0.259358
$601$	−34.1098	−1.39137	−0.695683	−	0.718349i	$-0.744898\pi$
$601$	−34.1098	−1.39137	−0.695683	+	0.718349i	$0.744898\pi$
$602$	59.1955	2.41263
$603$	9.32216	0.379628
$604$	40.3891	1.64341
$605$	0.710210	0.0288741
$606$	5.28783	0.214803
$607$	20.9453	0.850143	0.425071	−	0.905160i	$-0.360249\pi$
$607$	20.9453	0.850143	0.425071	+	0.905160i	$0.360249\pi$
$608$	−1.86849	−0.0757773
$609$	−6.81975	−0.276350
$610$	−1.39530	−0.0564938
$611$	0	0
$612$	−17.7360	−0.716935
$613$	−22.2792	−0.899848	−0.449924	−	0.893067i	$-0.648549\pi$
$613$	−22.2792	−0.899848	−0.449924	+	0.893067i	$0.648549\pi$
$614$	−5.43973	−0.219530
$615$	6.71875	0.270926
$616$	−3.26386	−0.131505
$617$	27.4366	1.10456	0.552279	−	0.833660i	$-0.313758\pi$
$617$	27.4366	1.10456	0.552279	+	0.833660i	$0.313758\pi$
$618$	−22.1818	−0.892284
$619$	−47.9288	−1.92642	−0.963211	−	0.268747i	$-0.913390\pi$
$619$	−47.9288	−1.92642	−0.963211	+	0.268747i	$0.913390\pi$
$620$	−7.58040	−0.304436
$621$	9.53531	0.382639
$622$	49.4896	1.98435
$623$	34.1166	1.36685
$624$	0	0
$625$	17.6884	0.707538
$626$	46.7250	1.86751
$627$	−0.242517	−0.00968519
$628$	53.6171	2.13956
$629$	−22.8668	−0.911760
$630$	−3.53908	−0.141000
$631$	23.8277	0.948567	0.474284	−	0.880372i	$-0.342707\pi$
$631$	23.8277	0.948567	0.474284	+	0.880372i	$0.342707\pi$
$632$	−2.57702	−0.102508
$633$	−10.6905	−0.424911
$634$	−57.4013	−2.27970
$635$	7.36353	0.292213
$636$	18.0261	0.714781
$637$	0	0
$638$	6.37063	0.252216
$639$	12.9704	0.513100
$640$	7.59848	0.300356
$641$	−8.30355	−0.327970	−0.163985	−	0.986463i	$-0.552435\pi$
$641$	−8.30355	−0.327970	−0.163985	+	0.986463i	$0.552435\pi$
$642$	−42.4367	−1.67484
$643$	29.9185	1.17987	0.589934	−	0.807451i	$-0.299154\pi$
$643$	29.9185	1.17987	0.589934	+	0.807451i	$0.299154\pi$
$644$	58.4710	2.30408
$645$	−8.43668	−0.332194
$646$	−3.49538	−0.137524
$647$	5.87308	0.230895	0.115447	−	0.993314i	$-0.463170\pi$
$647$	5.87308	0.230895	0.115447	+	0.993314i	$0.463170\pi$
$648$	−1.41315	−0.0555136
$649$	7.81320	0.306695
$650$	0	0
$651$	−9.28516	−0.363914
$652$	−43.8002	−1.71535
$653$	−5.43262	−0.212595	−0.106297	−	0.994334i	$-0.533900\pi$
$653$	−5.43262	−0.212595	−0.106297	+	0.994334i	$0.533900\pi$
$654$	−2.75099	−0.107572
$655$	−4.03891	−0.157813
$656$	−21.3900	−0.835138
$657$	8.51665	0.332266
$658$	64.3715	2.50946
$659$	26.5954	1.03601	0.518004	−	0.855378i	$-0.326675\pi$
$659$	26.5954	1.03601	0.518004	+	0.855378i	$0.326675\pi$
$660$	1.88559	0.0733966
$661$	−27.7763	−1.08037	−0.540187	−	0.841545i	$-0.681646\pi$
$661$	−27.7763	−1.08037	−0.540187	+	0.841545i	$0.681646\pi$
$662$	53.4359	2.07684
$663$	0	0
$664$	−6.56647	−0.254828
$665$	−0.397808	−0.0154263
$666$	−7.38534	−0.286176
$667$	−28.1552	−1.09017
$668$	−14.6158	−0.565504
$669$	−5.33992	−0.206453
$670$	−14.2844	−0.551855
$671$	0.910585	0.0351527
$672$	17.7948	0.686450
$673$	3.00054	0.115663	0.0578313	−	0.998326i	$-0.481581\pi$
$673$	3.00054	0.115663	0.0578313	+	0.998326i	$0.481581\pi$
$674$	61.0346	2.35096
$675$	−4.49560	−0.173036
$676$	0	0
$677$	−25.1724	−0.967452	−0.483726	−	0.875219i	$-0.660717\pi$
$677$	−25.1724	−0.967452	−0.483726	+	0.875219i	$0.660717\pi$
$678$	1.37811	0.0529258
$679$	4.30846	0.165344
$680$	6.70452	0.257107
$681$	−7.82431	−0.299828
$682$	8.67368	0.332132
$683$	39.1652	1.49861	0.749307	−	0.662223i	$-0.230387\pi$
$683$	39.1652	1.49861	0.749307	+	0.662223i	$0.230387\pi$
$684$	−0.643877	−0.0246193
$685$	3.45420	0.131978
$686$	43.1817	1.64869
$687$	−10.6805	−0.407485
$688$	26.8592	1.02400
$689$	0	0
$690$	−14.6110	−0.556232
$691$	33.3472	1.26859	0.634294	−	0.773092i	$-0.281291\pi$
$691$	33.3472	1.26859	0.634294	+	0.773092i	$0.281291\pi$
$692$	−33.7518	−1.28305
$693$	2.30964	0.0877361
$694$	39.0114	1.48085
$695$	−3.13568	−0.118943
$696$	4.17263	0.158163
$697$	−63.1969	−2.39375
$698$	−45.3787	−1.71761
$699$	15.4509	0.584408
$700$	−27.5673	−1.04195
$701$	8.45704	0.319418	0.159709	−	0.987164i	$-0.448944\pi$
$701$	8.45704	0.319418	0.159709	+	0.987164i	$0.448944\pi$
$702$	0	0
$703$	−0.830144	−0.0313095
$704$	−12.1009	−0.456068
$705$	−9.17437	−0.345527
$706$	−4.74934	−0.178744
$707$	−5.66061	−0.212889
$708$	20.7439	0.779603
$709$	−7.03783	−0.264311	−0.132156	−	0.991229i	$-0.542190\pi$
$709$	−7.03783	−0.264311	−0.132156	+	0.991229i	$0.542190\pi$
$710$	−19.8746	−0.745880
$711$	1.82360	0.0683905
$712$	−20.8741	−0.782290
$713$	−38.3336	−1.43560
$714$	33.2888	1.24580
$715$	0	0
$716$	64.8555	2.42377
$717$	−16.9365	−0.632504
$718$	16.7542	0.625260
$719$	−5.52947	−0.206215	−0.103107	−	0.994670i	$-0.532879\pi$
$719$	−5.52947	−0.206215	−0.103107	+	0.994670i	$0.532879\pi$
$720$	−1.60581	−0.0598452
$721$	23.7456	0.884333
$722$	40.8664	1.52089
$723$	−0.0799646	−0.00297392
$724$	2.47265	0.0918952
$725$	13.2743	0.492995
$726$	−2.15754	−0.0800738
$727$	−38.7533	−1.43728	−0.718640	−	0.695383i	$-0.755235\pi$
$727$	−38.7533	−1.43728	−0.718640	+	0.695383i	$0.755235\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−13.0501	−0.483006
$731$	79.3558	2.93508
$732$	2.41758	0.0893565
$733$	9.60595	0.354804	0.177402	−	0.984138i	$-0.443231\pi$
$733$	9.60595	0.354804	0.177402	+	0.984138i	$0.443231\pi$
$734$	25.8050	0.952480
$735$	−1.18289	−0.0436315
$736$	73.4655	2.70798
$737$	9.32216	0.343386
$738$	−20.4108	−0.751332
$739$	21.6659	0.796995	0.398497	−	0.917169i	$-0.369532\pi$
$739$	21.6659	0.796995	0.398497	+	0.917169i	$0.369532\pi$
$740$	6.45446	0.237271
$741$	0	0
$742$	−33.8333	−1.24206
$743$	12.3149	0.451790	0.225895	−	0.974152i	$-0.427469\pi$
$743$	12.3149	0.451790	0.225895	+	0.974152i	$0.427469\pi$
$744$	5.68108	0.208279
$745$	6.74952	0.247283
$746$	−30.4738	−1.11572
$747$	4.64671	0.170014
$748$	−17.7360	−0.648492
$749$	45.4285	1.65992
$750$	14.5502	0.531297
$751$	2.50813	0.0915231	0.0457616	−	0.998952i	$-0.485429\pi$
$751$	2.50813	0.0915231	0.0457616	+	0.998952i	$0.485429\pi$
$752$	29.2078	1.06510
$753$	−7.57174	−0.275930
$754$	0	0
$755$	10.8041	0.393202
$756$	6.13206	0.223021
$757$	2.89889	0.105362	0.0526809	−	0.998611i	$-0.483223\pi$
$757$	2.89889	0.105362	0.0526809	+	0.998611i	$0.483223\pi$
$758$	7.21551	0.262079
$759$	9.53531	0.346110
$760$	0.243397	0.00882894
$761$	−33.6764	−1.22077	−0.610384	−	0.792105i	$-0.708985\pi$
$761$	−33.6764	−1.22077	−0.610384	+	0.792105i	$0.708985\pi$
$762$	−22.3696	−0.810365
$763$	2.94493	0.106614
$764$	12.4309	0.449733
$765$	−4.74440	−0.171534
$766$	12.1604	0.439371
$767$	0	0
$768$	1.11835	0.0403551
$769$	25.1224	0.905935	0.452968	−	0.891527i	$-0.350365\pi$
$769$	25.1224	0.905935	0.452968	+	0.891527i	$0.350365\pi$
$770$	−3.53908	−0.127540
$771$	23.0242	0.829196
$772$	−27.2588	−0.981067
$773$	31.6336	1.13778	0.568891	−	0.822413i	$-0.307373\pi$
$773$	31.6336	1.13778	0.568891	+	0.822413i	$0.307373\pi$
$774$	25.6297	0.921241
$775$	18.0731	0.649204
$776$	−2.63611	−0.0946310
$777$	7.90600	0.283626
$778$	−30.9585	−1.10992
$779$	−2.29426	−0.0822005
$780$	0	0
$781$	12.9704	0.464117
$782$	137.432	4.91456
$783$	−2.95273	−0.105522
$784$	3.76587	0.134495
$785$	14.3426	0.511911
$786$	12.2698	0.437648
$787$	23.2041	0.827137	0.413569	−	0.910473i	$-0.364282\pi$
$787$	23.2041	0.827137	0.413569	+	0.910473i	$0.364282\pi$
$788$	30.7559	1.09563
$789$	−5.95047	−0.211842
$790$	−2.79432	−0.0994175
$791$	−1.47526	−0.0524542
$792$	−1.41315	−0.0502139
$793$	0	0
$794$	32.0974	1.13909
$795$	4.82200	0.171019
$796$	45.7533	1.62168
$797$	16.5706	0.586960	0.293480	−	0.955965i	$-0.405187\pi$
$797$	16.5706	0.586960	0.293480	+	0.955965i	$0.405187\pi$
$798$	1.20850	0.0427803
$799$	86.2946	3.05288
$800$	−34.6367	−1.22459
$801$	14.7714	0.521921
$802$	−42.3698	−1.49613
$803$	8.51665	0.300546
$804$	24.7502	0.872871
$805$	15.6411	0.551275
$806$	0	0
$807$	−1.17571	−0.0413868
$808$	3.46342	0.121843
$809$	8.97637	0.315592	0.157796	−	0.987472i	$-0.449561\pi$
$809$	8.97637	0.315592	0.157796	+	0.987472i	$0.449561\pi$
$810$	−1.53231	−0.0538398
$811$	2.09499	0.0735651	0.0367826	−	0.999323i	$-0.488289\pi$
$811$	2.09499	0.0735651	0.0367826	+	0.999323i	$0.488289\pi$
$812$	−18.1063	−0.635406
$813$	−7.12706	−0.249957
$814$	−7.38534	−0.258856
$815$	−11.7166	−0.410414
$816$	15.1044	0.528759
$817$	2.88089	0.100790
$818$	−49.9969	−1.74810
$819$	0	0
$820$	17.8381	0.622935
$821$	−32.8066	−1.14496	−0.572480	−	0.819919i	$-0.694018\pi$
$821$	−32.8066	−1.14496	−0.572480	+	0.819919i	$0.694018\pi$
$822$	−10.4935	−0.366002
$823$	−6.37447	−0.222200	−0.111100	−	0.993809i	$-0.535437\pi$
$823$	−6.37447	−0.222200	−0.111100	+	0.993809i	$0.535437\pi$
$824$	−14.5287	−0.506130
$825$	−4.49560	−0.156517
$826$	−38.9343	−1.35470
$827$	−14.7476	−0.512825	−0.256413	−	0.966567i	$-0.582541\pi$
$827$	−14.7476	−0.512825	−0.256413	+	0.966567i	$0.582541\pi$
$828$	25.3161	0.879794
$829$	−25.4837	−0.885085	−0.442542	−	0.896748i	$-0.645923\pi$
$829$	−25.4837	−0.885085	−0.442542	+	0.896748i	$0.645923\pi$
$830$	−7.12018	−0.247145
$831$	−11.9736	−0.415361
$832$	0	0
$833$	11.1263	0.385504
$834$	9.52586	0.329854
$835$	−3.90976	−0.135303
$836$	−0.643877	−0.0222690
$837$	−4.02017	−0.138957
$838$	7.98924	0.275984
$839$	−41.7953	−1.44293	−0.721467	−	0.692449i	$-0.756532\pi$
$839$	−41.7953	−1.44293	−0.721467	+	0.692449i	$0.756532\pi$
$840$	−2.31803	−0.0799795
$841$	−20.2814	−0.699359
$842$	−51.1852	−1.76396
$843$	18.7901	0.647165
$844$	−28.3831	−0.976988
$845$	0	0
$846$	27.8707	0.958216
$847$	2.30964	0.0793603
$848$	−15.3514	−0.527170
$849$	10.5237	0.361172
$850$	−64.7949	−2.22245
$851$	32.6397	1.11888
$852$	34.4361	1.17976
$853$	−25.2825	−0.865655	−0.432827	−	0.901477i	$-0.642484\pi$
$853$	−25.2825	−0.865655	−0.432827	+	0.901477i	$0.642484\pi$
$854$	−4.53758	−0.155273
$855$	−0.172238	−0.00589041
$856$	−27.7952	−0.950021
$857$	−26.8626	−0.917609	−0.458804	−	0.888537i	$-0.651722\pi$
$857$	−26.8626	−0.917609	−0.458804	+	0.888537i	$0.651722\pi$
$858$	0	0
$859$	6.11713	0.208714	0.104357	−	0.994540i	$-0.466722\pi$
$859$	6.11713	0.208714	0.104357	+	0.994540i	$0.466722\pi$
$860$	−22.3992	−0.763807
$861$	21.8497	0.744637
$862$	67.0336	2.28318
$863$	32.1517	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$863$	32.1517	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$864$	7.70458	0.262115
$865$	−9.02865	−0.306983
$866$	−49.7401	−1.69024
$867$	27.6260	0.938228
$868$	−24.6519	−0.836740
$869$	1.82360	0.0618616
$870$	4.52448	0.153394
$871$	0	0
$872$	−1.80184	−0.0610180
$873$	1.86542	0.0631350
$874$	4.98925	0.168764
$875$	−15.5759	−0.526563
$876$	22.6115	0.763973
$877$	−17.9062	−0.604648	−0.302324	−	0.953205i	$-0.597763\pi$
$877$	−17.9062	−0.604648	−0.302324	+	0.953205i	$0.597763\pi$
$878$	−5.16241	−0.174223
$879$	−14.0111	−0.472584
$880$	−1.60581	−0.0541320
$881$	40.5751	1.36701	0.683504	−	0.729947i	$-0.260455\pi$
$881$	40.5751	1.36701	0.683504	+	0.729947i	$0.260455\pi$
$882$	3.59349	0.120999
$883$	−24.6601	−0.829879	−0.414940	−	0.909849i	$-0.636197\pi$
$883$	−24.6601	−0.829879	−0.414940	+	0.909849i	$0.636197\pi$
$884$	0	0
$885$	5.54901	0.186528
$886$	37.0286	1.24400
$887$	27.2662	0.915510	0.457755	−	0.889078i	$-0.348654\pi$
$887$	27.2662	0.915510	0.457755	+	0.889078i	$0.348654\pi$
$888$	−4.83725	−0.162327
$889$	23.9466	0.803144
$890$	−22.6343	−0.758703
$891$	1.00000	0.0335013
$892$	−14.1774	−0.474693
$893$	3.13279	0.104835
$894$	−20.5043	−0.685767
$895$	17.3489	0.579911
$896$	24.7107	0.825526
$897$	0	0
$898$	−62.7081	−2.09260
$899$	11.8705	0.395902
$900$	−11.9357	−0.397858
$901$	−45.3560	−1.51103
$902$	−20.4108	−0.679606
$903$	−27.4366	−0.913032
$904$	0.902631	0.0300211
$905$	0.661436	0.0219869
$906$	−32.8217	−1.09043
$907$	0.121526	0.00403521	0.00201760	−	0.999998i	$-0.499358\pi$
$907$	0.121526	0.00403521	0.00201760	+	0.999998i	$0.499358\pi$
$908$	−20.7734	−0.689389
$909$	−2.45086	−0.0812898
$910$	0	0
$911$	−40.6736	−1.34758	−0.673788	−	0.738925i	$-0.735334\pi$
$911$	−40.6736	−1.34758	−0.673788	+	0.738925i	$0.735334\pi$
$912$	0.548341	0.0181574
$913$	4.64671	0.153784
$914$	68.8495	2.27734
$915$	0.646706	0.0213795
$916$	−28.3564	−0.936921
$917$	−13.1347	−0.433748
$918$	14.4130	0.475698
$919$	−26.0713	−0.860012	−0.430006	−	0.902826i	$-0.641489\pi$
$919$	−26.0713	−0.860012	−0.430006	+	0.902826i	$0.641489\pi$
$920$	−9.56992	−0.315511
$921$	2.52126	0.0830785
$922$	1.52470	0.0502134
$923$	0	0
$924$	6.13206	0.201730
$925$	−15.3886	−0.505975
$926$	−55.6242	−1.82792
$927$	10.2811	0.337675
$928$	−22.7495	−0.746790
$929$	−26.9849	−0.885346	−0.442673	−	0.896683i	$-0.645970\pi$
$929$	−26.9849	−0.885346	−0.442673	+	0.896683i	$0.645970\pi$
$930$	6.16013	0.201999
$931$	0.403923	0.0132381
$932$	41.0219	1.34372
$933$	−22.9380	−0.750956
$934$	−80.5929	−2.63708
$935$	−4.74440	−0.155158
$936$	0	0
$937$	31.9915	1.04512	0.522559	−	0.852603i	$-0.324977\pi$
$937$	31.9915	1.04512	0.522559	+	0.852603i	$0.324977\pi$
$938$	−46.4537	−1.51677
$939$	−21.6566	−0.706736
$940$	−24.3578	−0.794463
$941$	−31.7257	−1.03423	−0.517115	−	0.855916i	$-0.672994\pi$
$941$	−31.7257	−1.03423	−0.517115	+	0.855916i	$0.672994\pi$
$942$	−43.5714	−1.41963
$943$	90.2062	2.93752
$944$	−17.6660	−0.574979
$945$	1.64033	0.0533600
$946$	25.6297	0.833294
$947$	−46.5444	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$947$	−46.5444	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$948$	4.84163	0.157249
$949$	0	0
$950$	−2.35228	−0.0763180
$951$	26.6050	0.862726
$952$	21.8035	0.706655
$953$	37.1054	1.20196	0.600981	−	0.799263i	$-0.294777\pi$
$953$	37.1054	1.20196	0.600981	+	0.799263i	$0.294777\pi$
$954$	−14.6487	−0.474269
$955$	3.32527	0.107603
$956$	−44.9660	−1.45430
$957$	−2.95273	−0.0954481
$958$	−3.75550	−0.121335
$959$	11.2332	0.362740
$960$	−8.59415	−0.277375
$961$	−14.8382	−0.478653
$962$	0	0
$963$	19.6690	0.633826
$964$	−0.212305	−0.00683787
$965$	−7.29177	−0.234730
$966$	−47.5159	−1.52880
$967$	33.1854	1.06717	0.533586	−	0.845746i	$-0.320844\pi$
$967$	33.1854	1.06717	0.533586	+	0.845746i	$0.320844\pi$
$968$	−1.41315	−0.0454202
$969$	1.62008	0.0520444
$970$	−2.85840	−0.0917777
$971$	23.0990	0.741283	0.370642	−	0.928776i	$-0.379138\pi$
$971$	23.0990	0.741283	0.370642	+	0.928776i	$0.379138\pi$
$972$	2.65498	0.0851585
$973$	−10.1974	−0.326914
$974$	−12.2933	−0.393903
$975$	0	0
$976$	−2.05887	−0.0659028
$977$	32.4063	1.03677	0.518385	−	0.855147i	$-0.326533\pi$
$977$	32.4063	1.03677	0.518385	+	0.855147i	$0.326533\pi$
$978$	35.5937	1.13816
$979$	14.7714	0.472095
$980$	−3.14055	−0.100321
$981$	1.27506	0.0407095
$982$	59.8475	1.90981
$983$	−1.17252	−0.0373976	−0.0186988	−	0.999825i	$-0.505952\pi$
$983$	−1.17252	−0.0373976	−0.0186988	+	0.999825i	$0.505952\pi$
$984$	−13.3687	−0.426178
$985$	8.22723	0.262141
$986$	−42.5575	−1.35531
$987$	−29.8356	−0.949677
$988$	0	0
$989$	−113.271	−3.60182
$990$	−1.53231	−0.0486999
$991$	−4.78159	−0.151892	−0.0759462	−	0.997112i	$-0.524198\pi$
$991$	−4.78159	−0.151892	−0.0759462	+	0.997112i	$0.524198\pi$
$992$	−30.9737	−0.983416
$993$	−24.7670	−0.785958
$994$	−64.6333	−2.05004
$995$	12.2391	0.388004
$996$	12.3369	0.390910
$997$	55.7577	1.76586	0.882932	−	0.469502i	$-0.155566\pi$
$997$	55.7577	1.76586	0.882932	+	0.469502i	$0.155566\pi$
$998$	−17.8444	−0.564856
$999$	3.42304	0.108300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.y.1.1		7
13.5	odd	4		429.2.b.b.298.12	yes	14
13.8	odd	4		429.2.b.b.298.3	✓	14
13.12	even	2		5577.2.a.x.1.7		7
39.5	even	4		1287.2.b.c.298.3		14
39.8	even	4		1287.2.b.c.298.12		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.b.b.298.3	✓	14	13.8	odd	4
429.2.b.b.298.12	yes	14	13.5	odd	4
1287.2.b.c.298.3		14	39.5	even	4
1287.2.b.c.298.12		14	39.8	even	4
5577.2.a.x.1.7		7	13.12	even	2
5577.2.a.y.1.1		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5577.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} +0.710210 q^{5} -2.15754 q^{6} +2.30964 q^{7} -1.41315 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.15754 q^{2} +1.00000 q^{3} +2.65498 q^{4} +0.710210 q^{5} -2.15754 q^{6} +2.30964 q^{7} -1.41315 q^{8} +1.00000 q^{9} -1.53231 q^{10} +1.00000 q^{11} +2.65498 q^{12} -4.98315 q^{14} +0.710210 q^{15} -2.26104 q^{16} -6.68027 q^{17} -2.15754 q^{18} -0.242517 q^{19} +1.88559 q^{20} +2.30964 q^{21} -2.15754 q^{22} +9.53531 q^{23} -1.41315 q^{24} -4.49560 q^{25} +1.00000 q^{27} +6.13206 q^{28} -2.95273 q^{29} -1.53231 q^{30} -4.02017 q^{31} +7.70458 q^{32} +1.00000 q^{33} +14.4130 q^{34} +1.64033 q^{35} +2.65498 q^{36} +3.42304 q^{37} +0.523240 q^{38} -1.00363 q^{40} +9.46022 q^{41} -4.98315 q^{42} -11.8791 q^{43} +2.65498 q^{44} +0.710210 q^{45} -20.5728 q^{46} -12.9178 q^{47} -2.26104 q^{48} -1.66555 q^{49} +9.69944 q^{50} -6.68027 q^{51} +6.78954 q^{53} -2.15754 q^{54} +0.710210 q^{55} -3.26386 q^{56} -0.242517 q^{57} +6.37063 q^{58} +7.81320 q^{59} +1.88559 q^{60} +0.910585 q^{61} +8.67368 q^{62} +2.30964 q^{63} -12.1009 q^{64} -2.15754 q^{66} +9.32216 q^{67} -17.7360 q^{68} +9.53531 q^{69} -3.53908 q^{70} +12.9704 q^{71} -1.41315 q^{72} +8.51665 q^{73} -7.38534 q^{74} -4.49560 q^{75} -0.643877 q^{76} +2.30964 q^{77} +1.82360 q^{79} -1.60581 q^{80} +1.00000 q^{81} -20.4108 q^{82} +4.64671 q^{83} +6.13206 q^{84} -4.74440 q^{85} +25.6297 q^{86} -2.95273 q^{87} -1.41315 q^{88} +14.7714 q^{89} -1.53231 q^{90} +25.3161 q^{92} -4.02017 q^{93} +27.8707 q^{94} -0.172238 q^{95} +7.70458 q^{96} +1.86542 q^{97} +3.59349 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 7 q^{3} + 9 q^{4} + 6 q^{5} + 3 q^{6} + 6 q^{7} + 15 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q + 3 * q^2 + 7 * q^3 + 9 * q^4 + 6 * q^5 + 3 * q^6 + 6 * q^7 + 15 * q^8 + 7 * q^9	\( 7 q + 3 q^{2} + 7 q^{3} + 9 q^{4} + 6 q^{5} + 3 q^{6} + 6 q^{7} + 15 q^{8} + 7 q^{9} + 7 q^{11} + 9 q^{12} + 8 q^{14} + 6 q^{15} + 17 q^{16} - 2 q^{17} + 3 q^{18} + 8 q^{19} - 2 q^{20} + 6 q^{21} + 3 q^{22} + 4 q^{23} + 15 q^{24} + 13 q^{25} + 7 q^{27} + 12 q^{28} - 12 q^{29} - 10 q^{31} + 33 q^{32} + 7 q^{33} + 28 q^{34} - 4 q^{35} + 9 q^{36} + 6 q^{37} + 16 q^{38} - 10 q^{40} + 2 q^{41} + 8 q^{42} - 16 q^{43} + 9 q^{44} + 6 q^{45} - 26 q^{46} + 18 q^{47} + 17 q^{48} + 23 q^{49} + 39 q^{50} - 2 q^{51} + 10 q^{53} + 3 q^{54} + 6 q^{55} + 16 q^{56} + 8 q^{57} + 10 q^{58} + 2 q^{59} - 2 q^{60} - 10 q^{61} - 36 q^{62} + 6 q^{63} + 29 q^{64} + 3 q^{66} + 8 q^{67} - 10 q^{68} + 4 q^{69} - 20 q^{70} + 36 q^{71} + 15 q^{72} + 20 q^{73} + 13 q^{75} + 10 q^{76} + 6 q^{77} + 6 q^{79} - 20 q^{80} + 7 q^{81} - 10 q^{82} + 30 q^{83} + 12 q^{84} - 40 q^{85} + 6 q^{86} - 12 q^{87} + 15 q^{88} + 34 q^{89} - 12 q^{92} - 10 q^{93} + 32 q^{94} + 18 q^{95} + 33 q^{96} + 16 q^{97} + q^{98} + 7 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 + 7 * q^3 + 9 * q^4 + 6 * q^5 + 3 * q^6 + 6 * q^7 + 15 * q^8 + 7 * q^9 + 7 * q^11 + 9 * q^12 + 8 * q^14 + 6 * q^15 + 17 * q^16 - 2 * q^17 + 3 * q^18 + 8 * q^19 - 2 * q^20 + 6 * q^21 + 3 * q^22 + 4 * q^23 + 15 * q^24 + 13 * q^25 + 7 * q^27 + 12 * q^28 - 12 * q^29 - 10 * q^31 + 33 * q^32 + 7 * q^33 + 28 * q^34 - 4 * q^35 + 9 * q^36 + 6 * q^37 + 16 * q^38 - 10 * q^40 + 2 * q^41 + 8 * q^42 - 16 * q^43 + 9 * q^44 + 6 * q^45 - 26 * q^46 + 18 * q^47 + 17 * q^48 + 23 * q^49 + 39 * q^50 - 2 * q^51 + 10 * q^53 + 3 * q^54 + 6 * q^55 + 16 * q^56 + 8 * q^57 + 10 * q^58 + 2 * q^59 - 2 * q^60 - 10 * q^61 - 36 * q^62 + 6 * q^63 + 29 * q^64 + 3 * q^66 + 8 * q^67 - 10 * q^68 + 4 * q^69 - 20 * q^70 + 36 * q^71 + 15 * q^72 + 20 * q^73 + 13 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 - 20 * q^80 + 7 * q^81 - 10 * q^82 + 30 * q^83 + 12 * q^84 - 40 * q^85 + 6 * q^86 - 12 * q^87 + 15 * q^88 + 34 * q^89 - 12 * q^92 - 10 * q^93 + 32 * q^94 + 18 * q^95 + 33 * q^96 + 16 * q^97 + q^98 + 7 * q^99

Introduction

L-functions

Modular forms

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