LMFDB - Embedded newform 429.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(1,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, 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("429.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.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:	$3.42558224671$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	429.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.539189 q^{5} +2.17009 q^{6} +0.630898 q^{7} +1.53919 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.539189 q^{5} +2.17009 q^{6} +0.630898 q^{7} +1.53919 q^{8} +1.00000 q^{9} -1.17009 q^{10} +1.00000 q^{11} +2.70928 q^{12} +1.00000 q^{13} +1.36910 q^{14} -0.539189 q^{15} -2.07838 q^{16} +1.90829 q^{17} +2.17009 q^{18} -4.04945 q^{19} -1.46081 q^{20} +0.630898 q^{21} +2.17009 q^{22} +1.36910 q^{23} +1.53919 q^{24} -4.70928 q^{25} +2.17009 q^{26} +1.00000 q^{27} +1.70928 q^{28} -2.24846 q^{29} -1.17009 q^{30} -1.46081 q^{31} -7.58864 q^{32} +1.00000 q^{33} +4.14116 q^{34} -0.340173 q^{35} +2.70928 q^{36} -5.07838 q^{37} -8.78765 q^{38} +1.00000 q^{39} -0.829914 q^{40} -2.04945 q^{41} +1.36910 q^{42} +0.986669 q^{43} +2.70928 q^{44} -0.539189 q^{45} +2.97107 q^{46} +3.26180 q^{47} -2.07838 q^{48} -6.60197 q^{49} -10.2195 q^{50} +1.90829 q^{51} +2.70928 q^{52} +9.91548 q^{53} +2.17009 q^{54} -0.539189 q^{55} +0.971071 q^{56} -4.04945 q^{57} -4.87936 q^{58} -3.60197 q^{59} -1.46081 q^{60} +3.41855 q^{61} -3.17009 q^{62} +0.630898 q^{63} -12.3112 q^{64} -0.539189 q^{65} +2.17009 q^{66} +5.95774 q^{67} +5.17009 q^{68} +1.36910 q^{69} -0.738205 q^{70} -7.75872 q^{71} +1.53919 q^{72} +9.46800 q^{73} -11.0205 q^{74} -4.70928 q^{75} -10.9711 q^{76} +0.630898 q^{77} +2.17009 q^{78} +3.90829 q^{79} +1.12064 q^{80} +1.00000 q^{81} -4.44748 q^{82} +2.34017 q^{83} +1.70928 q^{84} -1.02893 q^{85} +2.14116 q^{86} -2.24846 q^{87} +1.53919 q^{88} +12.9783 q^{89} -1.17009 q^{90} +0.630898 q^{91} +3.70928 q^{92} -1.46081 q^{93} +7.07838 q^{94} +2.18342 q^{95} -7.58864 q^{96} -5.07838 q^{97} -14.3268 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + q^4 + q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{10} + 3 q^{11} + q^{12} + 3 q^{13} + 8 q^{14} - 3 q^{16} + 8 q^{17} + q^{18} + 6 q^{19} - 6 q^{20} - 2 q^{21} + q^{22} + 8 q^{23} + 3 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 2 q^{28} + 2 q^{29} + 2 q^{30} - 6 q^{31} - 3 q^{32} + 3 q^{33} - 8 q^{34} + 10 q^{35} + q^{36} - 12 q^{37} - 16 q^{38} + 3 q^{39} - 8 q^{40} + 12 q^{41} + 8 q^{42} + 2 q^{43} + q^{44} - 6 q^{46} + 2 q^{47} - 3 q^{48} - q^{49} - 7 q^{50} + 8 q^{51} + q^{52} - 2 q^{53} + q^{54} - 12 q^{56} + 6 q^{57} - 2 q^{58} + 8 q^{59} - 6 q^{60} - 4 q^{61} - 4 q^{62} - 2 q^{63} - 11 q^{64} + q^{66} + 2 q^{67} + 10 q^{68} + 8 q^{69} - 10 q^{70} + 2 q^{71} + 3 q^{72} - 4 q^{73} - 7 q^{75} - 18 q^{76} - 2 q^{77} + q^{78} + 14 q^{79} + 16 q^{80} + 3 q^{81} - 14 q^{82} - 4 q^{83} - 2 q^{84} - 18 q^{85} - 14 q^{86} + 2 q^{87} + 3 q^{88} - 10 q^{89} + 2 q^{90} - 2 q^{91} + 4 q^{92} - 6 q^{93} + 18 q^{94} + 2 q^{95} - 3 q^{96} - 12 q^{97} - 31 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + q^4 + q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^10 + 3 * q^11 + q^12 + 3 * q^13 + 8 * q^14 - 3 * q^16 + 8 * q^17 + q^18 + 6 * q^19 - 6 * q^20 - 2 * q^21 + q^22 + 8 * q^23 + 3 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 2 * q^28 + 2 * q^29 + 2 * q^30 - 6 * q^31 - 3 * q^32 + 3 * q^33 - 8 * q^34 + 10 * q^35 + q^36 - 12 * q^37 - 16 * q^38 + 3 * q^39 - 8 * q^40 + 12 * q^41 + 8 * q^42 + 2 * q^43 + q^44 - 6 * q^46 + 2 * q^47 - 3 * q^48 - q^49 - 7 * q^50 + 8 * q^51 + q^52 - 2 * q^53 + q^54 - 12 * q^56 + 6 * q^57 - 2 * q^58 + 8 * q^59 - 6 * q^60 - 4 * q^61 - 4 * q^62 - 2 * q^63 - 11 * q^64 + q^66 + 2 * q^67 + 10 * q^68 + 8 * q^69 - 10 * q^70 + 2 * q^71 + 3 * q^72 - 4 * q^73 - 7 * q^75 - 18 * q^76 - 2 * q^77 + q^78 + 14 * q^79 + 16 * q^80 + 3 * q^81 - 14 * q^82 - 4 * q^83 - 2 * q^84 - 18 * q^85 - 14 * q^86 + 2 * q^87 + 3 * q^88 - 10 * q^89 + 2 * q^90 - 2 * q^91 + 4 * q^92 - 6 * q^93 + 18 * q^94 + 2 * q^95 - 3 * q^96 - 12 * q^97 - 31 * q^98 + 3 * 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 $

$2$	2.17009	1.53448	0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.70928	1.35464
$5$	−0.539189	−0.241133	−0.120566	−	0.992705i	$-0.538471\pi$
$5$	−0.539189	−0.241133	−0.120566	+	0.992705i	$0.538471\pi$
$6$	2.17009	0.885934
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	1.53919	0.544185
$9$	1.00000	0.333333
$10$	−1.17009	−0.370014
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.70928	0.782100
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.36910	0.365908
$15$	−0.539189	−0.139218
$16$	−2.07838	−0.519594
$17$	1.90829	0.462829	0.231414	−	0.972855i	$-0.425665\pi$
$17$	1.90829	0.462829	0.231414	+	0.972855i	$0.425665\pi$
$18$	2.17009	0.511494
$19$	−4.04945	−0.929007	−0.464504	−	0.885571i	$-0.653767\pi$
$19$	−4.04945	−0.929007	−0.464504	+	0.885571i	$0.653767\pi$
$20$	−1.46081	−0.326647
$21$	0.630898	0.137673
$22$	2.17009	0.462664
$23$	1.36910	0.285478	0.142739	−	0.989760i	$-0.454409\pi$
$23$	1.36910	0.285478	0.142739	+	0.989760i	$0.454409\pi$
$24$	1.53919	0.314186
$25$	−4.70928	−0.941855
$26$	2.17009	0.425589
$27$	1.00000	0.192450
$28$	1.70928	0.323023
$29$	−2.24846	−0.417529	−0.208765	−	0.977966i	$-0.566944\pi$
$29$	−2.24846	−0.417529	−0.208765	+	0.977966i	$0.566944\pi$
$30$	−1.17009	−0.213628
$31$	−1.46081	−0.262369	−0.131185	−	0.991358i	$-0.541878\pi$
$31$	−1.46081	−0.262369	−0.131185	+	0.991358i	$0.541878\pi$
$32$	−7.58864	−1.34149
$33$	1.00000	0.174078
$34$	4.14116	0.710203
$35$	−0.340173	−0.0574997
$36$	2.70928	0.451546
$37$	−5.07838	−0.834880	−0.417440	−	0.908704i	$-0.637073\pi$
$37$	−5.07838	−0.834880	−0.417440	+	0.908704i	$0.637073\pi$
$38$	−8.78765	−1.42555
$39$	1.00000	0.160128
$40$	−0.829914	−0.131221
$41$	−2.04945	−0.320070	−0.160035	−	0.987111i	$-0.551161\pi$
$41$	−2.04945	−0.320070	−0.160035	+	0.987111i	$0.551161\pi$
$42$	1.36910	0.211257
$43$	0.986669	0.150466	0.0752328	−	0.997166i	$-0.476030\pi$
$43$	0.986669	0.150466	0.0752328	+	0.997166i	$0.476030\pi$
$44$	2.70928	0.408439
$45$	−0.539189	−0.0803775
$46$	2.97107	0.438060
$47$	3.26180	0.475782	0.237891	−	0.971292i	$-0.423544\pi$
$47$	3.26180	0.475782	0.237891	+	0.971292i	$0.423544\pi$
$48$	−2.07838	−0.299988
$49$	−6.60197	−0.943138
$50$	−10.2195	−1.44526
$51$	1.90829	0.267214
$52$	2.70928	0.375709
$53$	9.91548	1.36200	0.680998	−	0.732285i	$-0.261546\pi$
$53$	9.91548	1.36200	0.680998	+	0.732285i	$0.261546\pi$
$54$	2.17009	0.295311
$55$	−0.539189	−0.0727042
$56$	0.971071	0.129765
$57$	−4.04945	−0.536363
$58$	−4.87936	−0.640692
$59$	−3.60197	−0.468936	−0.234468	−	0.972124i	$-0.575335\pi$
$59$	−3.60197	−0.468936	−0.234468	+	0.972124i	$0.575335\pi$
$60$	−1.46081	−0.188590
$61$	3.41855	0.437701	0.218850	−	0.975758i	$-0.429769\pi$
$61$	3.41855	0.437701	0.218850	+	0.975758i	$0.429769\pi$
$62$	−3.17009	−0.402601
$63$	0.630898	0.0794856
$64$	−12.3112	−1.53891
$65$	−0.539189	−0.0668781
$66$	2.17009	0.267119
$67$	5.95774	0.727854	0.363927	−	0.931428i	$-0.381436\pi$
$67$	5.95774	0.727854	0.363927	+	0.931428i	$0.381436\pi$
$68$	5.17009	0.626965
$69$	1.36910	0.164821
$70$	−0.738205	−0.0882323
$71$	−7.75872	−0.920791	−0.460396	−	0.887714i	$-0.652292\pi$
$71$	−7.75872	−0.920791	−0.460396	+	0.887714i	$0.652292\pi$
$72$	1.53919	0.181395
$73$	9.46800	1.10815	0.554073	−	0.832468i	$-0.313073\pi$
$73$	9.46800	1.10815	0.554073	+	0.832468i	$0.313073\pi$
$74$	−11.0205	−1.28111
$75$	−4.70928	−0.543780
$76$	−10.9711	−1.25847
$77$	0.630898	0.0718975
$78$	2.17009	0.245714
$79$	3.90829	0.439717	0.219859	−	0.975532i	$-0.429440\pi$
$79$	3.90829	0.439717	0.219859	+	0.975532i	$0.429440\pi$
$80$	1.12064	0.125291
$81$	1.00000	0.111111
$82$	−4.44748	−0.491142
$83$	2.34017	0.256867	0.128434	−	0.991718i	$-0.459005\pi$
$83$	2.34017	0.256867	0.128434	+	0.991718i	$0.459005\pi$
$84$	1.70928	0.186497
$85$	−1.02893	−0.111603
$86$	2.14116	0.230887
$87$	−2.24846	−0.241061
$88$	1.53919	0.164078
$89$	12.9783	1.37569	0.687846	−	0.725856i	$-0.258556\pi$
$89$	12.9783	1.37569	0.687846	+	0.725856i	$0.258556\pi$
$90$	−1.17009	−0.123338
$91$	0.630898	0.0661360
$92$	3.70928	0.386719
$93$	−1.46081	−0.151479
$94$	7.07838	0.730079
$95$	2.18342	0.224014
$96$	−7.58864	−0.774512
$97$	−5.07838	−0.515631	−0.257816	−	0.966194i	$-0.583003\pi$
$97$	−5.07838	−0.515631	−0.257816	+	0.966194i	$0.583003\pi$
$98$	−14.3268	−1.44723
$99$	1.00000	0.100504
$100$	−12.7587	−1.27587
$101$	9.51026	0.946306	0.473153	−	0.880980i	$-0.343116\pi$
$101$	9.51026	0.946306	0.473153	+	0.880980i	$0.343116\pi$
$102$	4.14116	0.410036
$103$	−4.58145	−0.451424	−0.225712	−	0.974194i	$-0.572471\pi$
$103$	−4.58145	−0.451424	−0.225712	+	0.974194i	$0.572471\pi$
$104$	1.53919	0.150930
$105$	−0.340173	−0.0331975
$106$	21.5174	2.08996
$107$	11.6020	1.12160	0.560802	−	0.827950i	$-0.310493\pi$
$107$	11.6020	1.12160	0.560802	+	0.827950i	$0.310493\pi$
$108$	2.70928	0.260700
$109$	−13.1545	−1.25997	−0.629986	−	0.776607i	$-0.716939\pi$
$109$	−13.1545	−1.25997	−0.629986	+	0.776607i	$0.716939\pi$
$110$	−1.17009	−0.111563
$111$	−5.07838	−0.482018
$112$	−1.31124	−0.123901
$113$	6.18342	0.581687	0.290843	−	0.956771i	$-0.406064\pi$
$113$	6.18342	0.581687	0.290843	+	0.956771i	$0.406064\pi$
$114$	−8.78765	−0.823039
$115$	−0.738205	−0.0688379
$116$	−6.09171	−0.565601
$117$	1.00000	0.0924500
$118$	−7.81658	−0.719575
$119$	1.20394	0.110365
$120$	−0.829914	−0.0757604
$121$	1.00000	0.0909091
$122$	7.41855	0.671644
$123$	−2.04945	−0.184793
$124$	−3.95774	−0.355416
$125$	5.23513	0.468245
$126$	1.36910	0.121969
$127$	−20.4463	−1.81431	−0.907156	−	0.420795i	$-0.861751\pi$
$127$	−20.4463	−1.81431	−0.907156	+	0.420795i	$0.861751\pi$
$128$	−11.5392	−1.01993
$129$	0.986669	0.0868714
$130$	−1.17009	−0.102623
$131$	13.9421	1.21813	0.609065	−	0.793120i	$-0.291545\pi$
$131$	13.9421	1.21813	0.609065	+	0.793120i	$0.291545\pi$
$132$	2.70928	0.235812
$133$	−2.55479	−0.221528
$134$	12.9288	1.11688
$135$	−0.539189	−0.0464060
$136$	2.93722	0.251865
$137$	−2.90602	−0.248278	−0.124139	−	0.992265i	$-0.539617\pi$
$137$	−2.90602	−0.248278	−0.124139	+	0.992265i	$0.539617\pi$
$138$	2.97107	0.252914
$139$	18.6875	1.58506	0.792528	−	0.609836i	$-0.208765\pi$
$139$	18.6875	1.58506	0.792528	+	0.609836i	$0.208765\pi$
$140$	−0.921622	−0.0778913
$141$	3.26180	0.274693
$142$	−16.8371	−1.41294
$143$	1.00000	0.0836242
$144$	−2.07838	−0.173198
$145$	1.21235	0.100680
$146$	20.5464	1.70043
$147$	−6.60197	−0.544521
$148$	−13.7587	−1.13096
$149$	−15.9916	−1.31008	−0.655041	−	0.755593i	$-0.727349\pi$
$149$	−15.9916	−1.31008	−0.655041	+	0.755593i	$0.727349\pi$
$150$	−10.2195	−0.834422
$151$	10.7877	0.877887	0.438943	−	0.898515i	$-0.355353\pi$
$151$	10.7877	0.877887	0.438943	+	0.898515i	$0.355353\pi$
$152$	−6.23287	−0.505552
$153$	1.90829	0.154276
$154$	1.36910	0.110325
$155$	0.787653	0.0632658
$156$	2.70928	0.216916
$157$	−9.12783	−0.728480	−0.364240	−	0.931305i	$-0.618671\pi$
$157$	−9.12783	−0.728480	−0.364240	+	0.931305i	$0.618671\pi$
$158$	8.48133	0.674738
$159$	9.91548	0.786349
$160$	4.09171	0.323478
$161$	0.863763	0.0680741
$162$	2.17009	0.170498
$163$	−7.27739	−0.570009	−0.285005	−	0.958526i	$-0.591995\pi$
$163$	−7.27739	−0.570009	−0.285005	+	0.958526i	$0.591995\pi$
$164$	−5.55252	−0.433579
$165$	−0.539189	−0.0419758
$166$	5.07838	0.394159
$167$	16.0989	1.24577	0.622885	−	0.782313i	$-0.285960\pi$
$167$	16.0989	1.24577	0.622885	+	0.782313i	$0.285960\pi$
$168$	0.971071	0.0749197
$169$	1.00000	0.0769231
$170$	−2.23287	−0.171253
$171$	−4.04945	−0.309669
$172$	2.67316	0.203826
$173$	16.2485	1.23535	0.617674	−	0.786434i	$-0.288075\pi$
$173$	16.2485	1.23535	0.617674	+	0.786434i	$0.288075\pi$
$174$	−4.87936	−0.369903
$175$	−2.97107	−0.224592
$176$	−2.07838	−0.156664
$177$	−3.60197	−0.270741
$178$	28.1639	2.11098
$179$	18.5464	1.38622	0.693111	−	0.720831i	$-0.256240\pi$
$179$	18.5464	1.38622	0.693111	+	0.720831i	$0.256240\pi$
$180$	−1.46081	−0.108882
$181$	−10.8722	−0.808122	−0.404061	−	0.914732i	$-0.632402\pi$
$181$	−10.8722	−0.808122	−0.404061	+	0.914732i	$0.632402\pi$
$182$	1.36910	0.101485
$183$	3.41855	0.252707
$184$	2.10731	0.155353
$185$	2.73820	0.201317
$186$	−3.17009	−0.232442
$187$	1.90829	0.139548
$188$	8.83710	0.644512
$189$	0.630898	0.0458910
$190$	4.73820	0.343746
$191$	−9.17727	−0.664044	−0.332022	−	0.943272i	$-0.607731\pi$
$191$	−9.17727	−0.664044	−0.332022	+	0.943272i	$0.607731\pi$
$192$	−12.3112	−0.888487
$193$	−17.1012	−1.23097	−0.615484	−	0.788149i	$-0.711040\pi$
$193$	−17.1012	−1.23097	−0.615484	+	0.788149i	$0.711040\pi$
$194$	−11.0205	−0.791227
$195$	−0.539189	−0.0386121
$196$	−17.8865	−1.27761
$197$	−3.34244	−0.238139	−0.119070	−	0.992886i	$-0.537991\pi$
$197$	−3.34244	−0.238139	−0.119070	+	0.992886i	$0.537991\pi$
$198$	2.17009	0.154221
$199$	−14.2557	−1.01056	−0.505278	−	0.862957i	$-0.668610\pi$
$199$	−14.2557	−1.01056	−0.505278	+	0.862957i	$0.668610\pi$
$200$	−7.24846	−0.512544
$201$	5.95774	0.420227
$202$	20.6381	1.45209
$203$	−1.41855	−0.0995627
$204$	5.17009	0.361978
$205$	1.10504	0.0771793
$206$	−9.94214	−0.692702
$207$	1.36910	0.0951592
$208$	−2.07838	−0.144110
$209$	−4.04945	−0.280106
$210$	−0.738205	−0.0509410
$211$	15.0133	1.03356	0.516780	−	0.856118i	$-0.327130\pi$
$211$	15.0133	1.03356	0.516780	+	0.856118i	$0.327130\pi$
$212$	26.8638	1.84501
$213$	−7.75872	−0.531619
$214$	25.1773	1.72108
$215$	−0.532001	−0.0362822
$216$	1.53919	0.104729
$217$	−0.921622	−0.0625638
$218$	−28.5464	−1.93340
$219$	9.46800	0.639788
$220$	−1.46081	−0.0984879
$221$	1.90829	0.128366
$222$	−11.0205	−0.739649
$223$	9.77432	0.654537	0.327269	−	0.944931i	$-0.393872\pi$
$223$	9.77432	0.654537	0.327269	+	0.944931i	$0.393872\pi$
$224$	−4.78765	−0.319889
$225$	−4.70928	−0.313952
$226$	13.4186	0.892589
$227$	−11.0700	−0.734740	−0.367370	−	0.930075i	$-0.619742\pi$
$227$	−11.0700	−0.734740	−0.367370	+	0.930075i	$0.619742\pi$
$228$	−10.9711	−0.726577
$229$	−17.0928	−1.12952	−0.564760	−	0.825255i	$-0.691031\pi$
$229$	−17.0928	−1.12952	−0.564760	+	0.825255i	$0.691031\pi$
$230$	−1.60197	−0.105631
$231$	0.630898	0.0415100
$232$	−3.46081	−0.227213
$233$	−8.00719	−0.524568	−0.262284	−	0.964991i	$-0.584476\pi$
$233$	−8.00719	−0.524568	−0.262284	+	0.964991i	$0.584476\pi$
$234$	2.17009	0.141863
$235$	−1.75872	−0.114726
$236$	−9.75872	−0.635239
$237$	3.90829	0.253871
$238$	2.61265	0.169353
$239$	−3.28458	−0.212462	−0.106231	−	0.994341i	$-0.533878\pi$
$239$	−3.28458	−0.212462	−0.106231	+	0.994341i	$0.533878\pi$
$240$	1.12064	0.0723369
$241$	13.5486	0.872745	0.436372	−	0.899766i	$-0.356263\pi$
$241$	13.5486	0.872745	0.436372	+	0.899766i	$0.356263\pi$
$242$	2.17009	0.139498
$243$	1.00000	0.0641500
$244$	9.26180	0.592926
$245$	3.55971	0.227421
$246$	−4.44748	−0.283561
$247$	−4.04945	−0.257660
$248$	−2.24846	−0.142778
$249$	2.34017	0.148302
$250$	11.3607	0.718513
$251$	4.18342	0.264055	0.132027	−	0.991246i	$-0.457851\pi$
$251$	4.18342	0.264055	0.132027	+	0.991246i	$0.457851\pi$
$252$	1.70928	0.107674
$253$	1.36910	0.0860747
$254$	−44.3701	−2.78403
$255$	−1.02893	−0.0644341
$256$	−0.418551	−0.0261594
$257$	8.52359	0.531687	0.265843	−	0.964016i	$-0.414350\pi$
$257$	8.52359	0.531687	0.265843	+	0.964016i	$0.414350\pi$
$258$	2.14116	0.133303
$259$	−3.20394	−0.199083
$260$	−1.46081	−0.0905957
$261$	−2.24846	−0.139176
$262$	30.2557	1.86920
$263$	22.7526	1.40298	0.701492	−	0.712677i	$-0.252518\pi$
$263$	22.7526	1.40298	0.701492	+	0.712677i	$0.252518\pi$
$264$	1.53919	0.0947305
$265$	−5.34632	−0.328422
$266$	−5.54411	−0.339931
$267$	12.9783	0.794257
$268$	16.1412	0.985978
$269$	−8.34017	−0.508509	−0.254255	−	0.967137i	$-0.581830\pi$
$269$	−8.34017	−0.508509	−0.254255	+	0.967137i	$0.581830\pi$
$270$	−1.17009	−0.0712092
$271$	26.7031	1.62210	0.811050	−	0.584977i	$-0.198896\pi$
$271$	26.7031	1.62210	0.811050	+	0.584977i	$0.198896\pi$
$272$	−3.96615	−0.240483
$273$	0.630898	0.0381837
$274$	−6.30632	−0.380979
$275$	−4.70928	−0.283980
$276$	3.70928	0.223272
$277$	7.62863	0.458360	0.229180	−	0.973384i	$-0.426396\pi$
$277$	7.62863	0.458360	0.229180	+	0.973384i	$0.426396\pi$
$278$	40.5536	2.43224
$279$	−1.46081	−0.0874565
$280$	−0.523590	−0.0312905
$281$	−16.0228	−0.955839	−0.477920	−	0.878404i	$-0.658609\pi$
$281$	−16.0228	−0.955839	−0.477920	+	0.878404i	$0.658609\pi$
$282$	7.07838	0.421511
$283$	16.2485	0.965871	0.482935	−	0.875656i	$-0.339571\pi$
$283$	16.2485	0.965871	0.482935	+	0.875656i	$0.339571\pi$
$284$	−21.0205	−1.24734
$285$	2.18342	0.129334
$286$	2.17009	0.128320
$287$	−1.29299	−0.0763229
$288$	−7.58864	−0.447165
$289$	−13.3584	−0.785790
$290$	2.63090	0.154492
$291$	−5.07838	−0.297700
$292$	25.6514	1.50114
$293$	−22.2329	−1.29886	−0.649429	−	0.760422i	$-0.724992\pi$
$293$	−22.2329	−1.29886	−0.649429	+	0.760422i	$0.724992\pi$
$294$	−14.3268	−0.835558
$295$	1.94214	0.113076
$296$	−7.81658	−0.454330
$297$	1.00000	0.0580259
$298$	−34.7031	−2.01030
$299$	1.36910	0.0791772
$300$	−12.7587	−0.736625
$301$	0.622487	0.0358796
$302$	23.4101	1.34710
$303$	9.51026	0.546350
$304$	8.41628	0.482707
$305$	−1.84324	−0.105544
$306$	4.14116	0.236734
$307$	−7.46800	−0.426221	−0.213111	−	0.977028i	$-0.568359\pi$
$307$	−7.46800	−0.426221	−0.213111	+	0.977028i	$0.568359\pi$
$308$	1.70928	0.0973950
$309$	−4.58145	−0.260630
$310$	1.70928	0.0970803
$311$	1.27021	0.0720268	0.0360134	−	0.999351i	$-0.488534\pi$
$311$	1.27021	0.0720268	0.0360134	+	0.999351i	$0.488534\pi$
$312$	1.53919	0.0871394
$313$	12.8287	0.725120	0.362560	−	0.931960i	$-0.381903\pi$
$313$	12.8287	0.725120	0.362560	+	0.931960i	$0.381903\pi$
$314$	−19.8082	−1.11784
$315$	−0.340173	−0.0191666
$316$	10.5886	0.595657
$317$	13.2918	0.746540	0.373270	−	0.927723i	$-0.378236\pi$
$317$	13.2918	0.746540	0.373270	+	0.927723i	$0.378236\pi$
$318$	21.5174	1.20664
$319$	−2.24846	−0.125890
$320$	6.63809	0.371080
$321$	11.6020	0.647559
$322$	1.87444	0.104459
$323$	−7.72753	−0.429971
$324$	2.70928	0.150515
$325$	−4.70928	−0.261224
$326$	−15.7926	−0.874670
$327$	−13.1545	−0.727445
$328$	−3.15449	−0.174177
$329$	2.05786	0.113453
$330$	−1.17009	−0.0644111
$331$	31.6586	1.74011	0.870057	−	0.492951i	$-0.164082\pi$
$331$	31.6586	1.74011	0.870057	+	0.492951i	$0.164082\pi$
$332$	6.34017	0.347962
$333$	−5.07838	−0.278293
$334$	34.9360	1.91161
$335$	−3.21235	−0.175509
$336$	−1.31124	−0.0715342
$337$	−1.71769	−0.0935683	−0.0467842	−	0.998905i	$-0.514897\pi$
$337$	−1.71769	−0.0935683	−0.0467842	+	0.998905i	$0.514897\pi$
$338$	2.17009	0.118037
$339$	6.18342	0.335837
$340$	−2.78765	−0.151182
$341$	−1.46081	−0.0791074
$342$	−8.78765	−0.475182
$343$	−8.58145	−0.463355
$344$	1.51867	0.0818812
$345$	−0.738205	−0.0397436
$346$	35.2606	1.89562
$347$	1.60197	0.0859982	0.0429991	−	0.999075i	$-0.486309\pi$
$347$	1.60197	0.0859982	0.0429991	+	0.999075i	$0.486309\pi$
$348$	−6.09171	−0.326550
$349$	16.4391	0.879963	0.439982	−	0.898007i	$-0.354985\pi$
$349$	16.4391	0.879963	0.439982	+	0.898007i	$0.354985\pi$
$350$	−6.44748	−0.344632
$351$	1.00000	0.0533761
$352$	−7.58864	−0.404476
$353$	−0.990545	−0.0527214	−0.0263607	−	0.999652i	$-0.508392\pi$
$353$	−0.990545	−0.0527214	−0.0263607	+	0.999652i	$0.508392\pi$
$354$	−7.81658	−0.415447
$355$	4.18342	0.222033
$356$	35.1617	1.86357
$357$	1.20394	0.0637191
$358$	40.2472	2.12713
$359$	−28.5464	−1.50662	−0.753310	−	0.657666i	$-0.771544\pi$
$359$	−28.5464	−1.50662	−0.753310	+	0.657666i	$0.771544\pi$
$360$	−0.829914	−0.0437403
$361$	−2.60197	−0.136946
$362$	−23.5936	−1.24005
$363$	1.00000	0.0524864
$364$	1.70928	0.0895904
$365$	−5.10504	−0.267210
$366$	7.41855	0.387774
$367$	−11.2351	−0.586469	−0.293235	−	0.956041i	$-0.594732\pi$
$367$	−11.2351	−0.586469	−0.293235	+	0.956041i	$0.594732\pi$
$368$	−2.84551	−0.148333
$369$	−2.04945	−0.106690
$370$	5.94214	0.308917
$371$	6.25565	0.324777
$372$	−3.95774	−0.205199
$373$	−19.0784	−0.987841	−0.493920	−	0.869507i	$-0.664437\pi$
$373$	−19.0784	−0.987841	−0.493920	+	0.869507i	$0.664437\pi$
$374$	4.14116	0.214134
$375$	5.23513	0.270341
$376$	5.02052	0.258913
$377$	−2.24846	−0.115802
$378$	1.36910	0.0704190
$379$	1.80098	0.0925103	0.0462552	−	0.998930i	$-0.485271\pi$
$379$	1.80098	0.0925103	0.0462552	+	0.998930i	$0.485271\pi$
$380$	5.91548	0.303458
$381$	−20.4463	−1.04749
$382$	−19.9155	−1.01896
$383$	−0.979481	−0.0500491	−0.0250246	−	0.999687i	$-0.507966\pi$
$383$	−0.979481	−0.0500491	−0.0250246	+	0.999687i	$0.507966\pi$
$384$	−11.5392	−0.588857
$385$	−0.340173	−0.0173368
$386$	−37.1110	−1.88890
$387$	0.986669	0.0501552
$388$	−13.7587	−0.698493
$389$	−30.2823	−1.53537	−0.767687	−	0.640825i	$-0.778592\pi$
$389$	−30.2823	−1.53537	−0.767687	+	0.640825i	$0.778592\pi$
$390$	−1.17009	−0.0592496
$391$	2.61265	0.132127
$392$	−10.1617	−0.513242
$393$	13.9421	0.703288
$394$	−7.25338	−0.365420
$395$	−2.10731	−0.106030
$396$	2.70928	0.136146
$397$	−31.7998	−1.59598	−0.797992	−	0.602668i	$-0.794104\pi$
$397$	−31.7998	−1.59598	−0.797992	+	0.602668i	$0.794104\pi$
$398$	−30.9360	−1.55068
$399$	−2.55479	−0.127899
$400$	9.78765	0.489383
$401$	−1.34963	−0.0673972	−0.0336986	−	0.999432i	$-0.510729\pi$
$401$	−1.34963	−0.0673972	−0.0336986	+	0.999432i	$0.510729\pi$
$402$	12.9288	0.644830
$403$	−1.46081	−0.0727682
$404$	25.7659	1.28190
$405$	−0.539189	−0.0267925
$406$	−3.07838	−0.152777
$407$	−5.07838	−0.251726
$408$	2.93722	0.145414
$409$	4.16063	0.205730	0.102865	−	0.994695i	$-0.467199\pi$
$409$	4.16063	0.205730	0.102865	+	0.994695i	$0.467199\pi$
$410$	2.39803	0.118430
$411$	−2.90602	−0.143344
$412$	−12.4124	−0.611515
$413$	−2.27247	−0.111821
$414$	2.97107	0.146020
$415$	−1.26180	−0.0619391
$416$	−7.58864	−0.372064
$417$	18.6875	0.915132
$418$	−8.78765	−0.429818
$419$	−33.8492	−1.65364	−0.826821	−	0.562465i	$-0.809853\pi$
$419$	−33.8492	−1.65364	−0.826821	+	0.562465i	$0.809853\pi$
$420$	−0.921622	−0.0449706
$421$	−23.2618	−1.13371	−0.566855	−	0.823817i	$-0.691840\pi$
$421$	−23.2618	−1.13371	−0.566855	+	0.823817i	$0.691840\pi$
$422$	32.5802	1.58598
$423$	3.26180	0.158594
$424$	15.2618	0.741178
$425$	−8.98667	−0.435917
$426$	−16.8371	−0.815760
$427$	2.15676	0.104373
$428$	31.4329	1.51937
$429$	1.00000	0.0482805
$430$	−1.15449	−0.0556744
$431$	−14.3090	−0.689239	−0.344620	−	0.938742i	$-0.611992\pi$
$431$	−14.3090	−0.689239	−0.344620	+	0.938742i	$0.611992\pi$
$432$	−2.07838	−0.0999960
$433$	−16.5236	−0.794073	−0.397037	−	0.917803i	$-0.629961\pi$
$433$	−16.5236	−0.794073	−0.397037	+	0.917803i	$0.629961\pi$
$434$	−2.00000	−0.0960031
$435$	1.21235	0.0581276
$436$	−35.6391	−1.70680
$437$	−5.54411	−0.265211
$438$	20.5464	0.981744
$439$	−13.7971	−0.658500	−0.329250	−	0.944243i	$-0.606796\pi$
$439$	−13.7971	−0.658500	−0.329250	+	0.944243i	$0.606796\pi$
$440$	−0.829914	−0.0395646
$441$	−6.60197	−0.314379
$442$	4.14116	0.196975
$443$	−24.5113	−1.16457	−0.582284	−	0.812986i	$-0.697841\pi$
$443$	−24.5113	−1.16457	−0.582284	+	0.812986i	$0.697841\pi$
$444$	−13.7587	−0.652960
$445$	−6.99773	−0.331724
$446$	21.2111	1.00438
$447$	−15.9916	−0.756376
$448$	−7.76713	−0.366963
$449$	−35.1617	−1.65938	−0.829691	−	0.558223i	$-0.811483\pi$
$449$	−35.1617	−1.65938	−0.829691	+	0.558223i	$0.811483\pi$
$450$	−10.2195	−0.481753
$451$	−2.04945	−0.0965048
$452$	16.7526	0.787975
$453$	10.7877	0.506848
$454$	−24.0228	−1.12745
$455$	−0.340173	−0.0159476
$456$	−6.23287	−0.291881
$457$	−36.2557	−1.69597	−0.847984	−	0.530022i	$-0.822184\pi$
$457$	−36.2557	−1.69597	−0.847984	+	0.530022i	$0.822184\pi$
$458$	−37.0928	−1.73323
$459$	1.90829	0.0890714
$460$	−2.00000	−0.0932505
$461$	23.6248	1.10031	0.550157	−	0.835061i	$-0.314568\pi$
$461$	23.6248	1.10031	0.550157	+	0.835061i	$0.314568\pi$
$462$	1.36910	0.0636964
$463$	−4.85270	−0.225524	−0.112762	−	0.993622i	$-0.535970\pi$
$463$	−4.85270	−0.225524	−0.112762	+	0.993622i	$0.535970\pi$
$464$	4.67316	0.216946
$465$	0.787653	0.0365265
$466$	−17.3763	−0.804941
$467$	15.8927	0.735426	0.367713	−	0.929939i	$-0.380141\pi$
$467$	15.8927	0.735426	0.367713	+	0.929939i	$0.380141\pi$
$468$	2.70928	0.125236
$469$	3.75872	0.173562
$470$	−3.81658	−0.176046
$471$	−9.12783	−0.420588
$472$	−5.54411	−0.255188
$473$	0.986669	0.0453671
$474$	8.48133	0.389560
$475$	19.0700	0.874990
$476$	3.26180	0.149504
$477$	9.91548	0.453999
$478$	−7.12783	−0.326019
$479$	−5.70928	−0.260863	−0.130432	−	0.991457i	$-0.541636\pi$
$479$	−5.70928	−0.260863	−0.130432	+	0.991457i	$0.541636\pi$
$480$	4.09171	0.186760
$481$	−5.07838	−0.231554
$482$	29.4017	1.33921
$483$	0.863763	0.0393026
$484$	2.70928	0.123149
$485$	2.73820	0.124335
$486$	2.17009	0.0984371
$487$	−27.9265	−1.26547	−0.632736	−	0.774367i	$-0.718068\pi$
$487$	−27.9265	−1.26547	−0.632736	+	0.774367i	$0.718068\pi$
$488$	5.26180	0.238190
$489$	−7.27739	−0.329095
$490$	7.72487	0.348974
$491$	32.1711	1.45186	0.725931	−	0.687767i	$-0.241409\pi$
$491$	32.1711	1.45186	0.725931	+	0.687767i	$0.241409\pi$
$492$	−5.55252	−0.250327
$493$	−4.29072	−0.193245
$494$	−8.78765	−0.395375
$495$	−0.539189	−0.0242347
$496$	3.03612	0.136326
$497$	−4.89496	−0.219569
$498$	5.07838	0.227568
$499$	−16.2979	−0.729595	−0.364797	−	0.931087i	$-0.618862\pi$
$499$	−16.2979	−0.729595	−0.364797	+	0.931087i	$0.618862\pi$
$500$	14.1834	0.634302
$501$	16.0989	0.719246
$502$	9.07838	0.405188
$503$	−19.1194	−0.852493	−0.426246	−	0.904607i	$-0.640164\pi$
$503$	−19.1194	−0.852493	−0.426246	+	0.904607i	$0.640164\pi$
$504$	0.971071	0.0432549
$505$	−5.12783	−0.228185
$506$	2.97107	0.132080
$507$	1.00000	0.0444116
$508$	−55.3945	−2.45774
$509$	−23.5018	−1.04170	−0.520851	−	0.853648i	$-0.674385\pi$
$509$	−23.5018	−1.04170	−0.520851	+	0.853648i	$0.674385\pi$
$510$	−2.23287	−0.0988730
$511$	5.97334	0.264245
$512$	22.1701	0.979789
$513$	−4.04945	−0.178788
$514$	18.4969	0.815865
$515$	2.47027	0.108853
$516$	2.67316	0.117679
$517$	3.26180	0.143454
$518$	−6.95282	−0.305489
$519$	16.2485	0.713228
$520$	−0.829914	−0.0363941
$521$	−6.99386	−0.306406	−0.153203	−	0.988195i	$-0.548959\pi$
$521$	−6.99386	−0.306406	−0.153203	+	0.988195i	$0.548959\pi$
$522$	−4.87936	−0.213564
$523$	32.2050	1.40823	0.704113	−	0.710088i	$-0.251345\pi$
$523$	32.2050	1.40823	0.704113	+	0.710088i	$0.251345\pi$
$524$	37.7731	1.65013
$525$	−2.97107	−0.129668
$526$	49.3751	2.15286
$527$	−2.78765	−0.121432
$528$	−2.07838	−0.0904498
$529$	−21.1256	−0.918503
$530$	−11.6020	−0.503957
$531$	−3.60197	−0.156312
$532$	−6.92162	−0.300090
$533$	−2.04945	−0.0887715
$534$	28.1639	1.21877
$535$	−6.25565	−0.270455
$536$	9.17009	0.396087
$537$	18.5464	0.800335
$538$	−18.0989	−0.780299
$539$	−6.60197	−0.284367
$540$	−1.46081	−0.0628633
$541$	9.94668	0.427641	0.213821	−	0.976873i	$-0.431409\pi$
$541$	9.94668	0.427641	0.213821	+	0.976873i	$0.431409\pi$
$542$	57.9481	2.48908
$543$	−10.8722	−0.466570
$544$	−14.4813	−0.620882
$545$	7.09275	0.303820
$546$	1.36910	0.0585922
$547$	37.3412	1.59660	0.798298	−	0.602263i	$-0.205734\pi$
$547$	37.3412	1.59660	0.798298	+	0.602263i	$0.205734\pi$
$548$	−7.87322	−0.336327
$549$	3.41855	0.145900
$550$	−10.2195	−0.435762
$551$	9.10504	0.387888
$552$	2.10731	0.0896929
$553$	2.46573	0.104854
$554$	16.5548	0.703346
$555$	2.73820	0.116230
$556$	50.6297	2.14718
$557$	−2.31739	−0.0981908	−0.0490954	−	0.998794i	$-0.515634\pi$
$557$	−2.31739	−0.0981908	−0.0490954	+	0.998794i	$0.515634\pi$
$558$	−3.17009	−0.134200
$559$	0.986669	0.0417317
$560$	0.707008	0.0298765
$561$	1.90829	0.0805681
$562$	−34.7708	−1.46672
$563$	33.8888	1.42824	0.714122	−	0.700022i	$-0.246826\pi$
$563$	33.8888	1.42824	0.714122	+	0.700022i	$0.246826\pi$
$564$	8.83710	0.372109
$565$	−3.33403	−0.140264
$566$	35.2606	1.48211
$567$	0.630898	0.0264952
$568$	−11.9421	−0.501081
$569$	27.1389	1.13772	0.568861	−	0.822434i	$-0.307384\pi$
$569$	27.1389	1.13772	0.568861	+	0.822434i	$0.307384\pi$
$570$	4.73820	0.198462
$571$	4.64196	0.194260	0.0971300	−	0.995272i	$-0.469034\pi$
$571$	4.64196	0.194260	0.0971300	+	0.995272i	$0.469034\pi$
$572$	2.70928	0.113280
$573$	−9.17727	−0.383386
$574$	−2.80590	−0.117116
$575$	−6.44748	−0.268879
$576$	−12.3112	−0.512968
$577$	−33.4186	−1.39123	−0.695616	−	0.718414i	$-0.744869\pi$
$577$	−33.4186	−1.39123	−0.695616	+	0.718414i	$0.744869\pi$
$578$	−28.9889	−1.20578
$579$	−17.1012	−0.710700
$580$	3.28458	0.136385
$581$	1.47641	0.0612518
$582$	−11.0205	−0.456815
$583$	9.91548	0.410657
$584$	14.5730	0.603037
$585$	−0.539189	−0.0222927
$586$	−48.2472	−1.99308
$587$	0.595825	0.0245923	0.0122962	−	0.999924i	$-0.496086\pi$
$587$	0.595825	0.0245923	0.0122962	+	0.999924i	$0.496086\pi$
$588$	−17.8865	−0.737629
$589$	5.91548	0.243743
$590$	4.21461	0.173513
$591$	−3.34244	−0.137490
$592$	10.5548	0.433799
$593$	28.8554	1.18495	0.592474	−	0.805590i	$-0.298151\pi$
$593$	28.8554	1.18495	0.592474	+	0.805590i	$0.298151\pi$
$594$	2.17009	0.0890397
$595$	−0.649149	−0.0266125
$596$	−43.3256	−1.77469
$597$	−14.2557	−0.583445
$598$	2.97107	0.121496
$599$	45.0616	1.84117	0.920583	−	0.390548i	$-0.127714\pi$
$599$	45.0616	1.84117	0.920583	+	0.390548i	$0.127714\pi$
$600$	−7.24846	−0.295917
$601$	−13.1506	−0.536425	−0.268212	−	0.963360i	$-0.586433\pi$
$601$	−13.1506	−0.536425	−0.268212	+	0.963360i	$0.586433\pi$
$602$	1.35085	0.0550566
$603$	5.95774	0.242618
$604$	29.2267	1.18922
$605$	−0.539189	−0.0219211
$606$	20.6381	0.838365
$607$	22.4729	0.912148	0.456074	−	0.889942i	$-0.349255\pi$
$607$	22.4729	0.912148	0.456074	+	0.889942i	$0.349255\pi$
$608$	30.7298	1.24626
$609$	−1.41855	−0.0574826
$610$	−4.00000	−0.161955
$611$	3.26180	0.131958
$612$	5.17009	0.208988
$613$	−37.5136	−1.51516	−0.757579	−	0.652743i	$-0.773618\pi$
$613$	−37.5136	−1.51516	−0.757579	+	0.652743i	$0.773618\pi$
$614$	−16.2062	−0.654029
$615$	1.10504	0.0445595
$616$	0.971071	0.0391255
$617$	7.72875	0.311148	0.155574	−	0.987824i	$-0.450277\pi$
$617$	7.72875	0.311148	0.155574	+	0.987824i	$0.450277\pi$
$618$	−9.94214	−0.399932
$619$	−15.2918	−0.614628	−0.307314	−	0.951608i	$-0.599430\pi$
$619$	−15.2918	−0.614628	−0.307314	+	0.951608i	$0.599430\pi$
$620$	2.13397	0.0857023
$621$	1.36910	0.0549402
$622$	2.75646	0.110524
$623$	8.18795	0.328043
$624$	−2.07838	−0.0832017
$625$	20.7237	0.828946
$626$	27.8394	1.11268
$627$	−4.04945	−0.161719
$628$	−24.7298	−0.986826
$629$	−9.69102	−0.386406
$630$	−0.738205	−0.0294108
$631$	5.98440	0.238235	0.119118	−	0.992880i	$-0.461993\pi$
$631$	5.98440	0.238235	0.119118	+	0.992880i	$0.461993\pi$
$632$	6.01560	0.239288
$633$	15.0133	0.596726
$634$	28.8443	1.14555
$635$	11.0244	0.437490
$636$	26.8638	1.06522
$637$	−6.60197	−0.261580
$638$	−4.87936	−0.193176
$639$	−7.75872	−0.306930
$640$	6.22180	0.245938
$641$	−20.2023	−0.797944	−0.398972	−	0.916963i	$-0.630633\pi$
$641$	−20.2023	−0.797944	−0.398972	+	0.916963i	$0.630633\pi$
$642$	25.1773	0.993668
$643$	16.5659	0.653293	0.326647	−	0.945147i	$-0.394081\pi$
$643$	16.5659	0.653293	0.326647	+	0.945147i	$0.394081\pi$
$644$	2.34017	0.0922157
$645$	−0.532001	−0.0209475
$646$	−16.7694	−0.659783
$647$	0.183417	0.00721089	0.00360544	−	0.999994i	$-0.498852\pi$
$647$	0.183417	0.00721089	0.00360544	+	0.999994i	$0.498852\pi$
$648$	1.53919	0.0604650
$649$	−3.60197	−0.141390
$650$	−10.2195	−0.400843
$651$	−0.921622	−0.0361212
$652$	−19.7165	−0.772156
$653$	−43.5052	−1.70249	−0.851244	−	0.524770i	$-0.824151\pi$
$653$	−43.5052	−1.70249	−0.851244	+	0.524770i	$0.824151\pi$
$654$	−28.5464	−1.11625
$655$	−7.51745	−0.293731
$656$	4.25953	0.166307
$657$	9.46800	0.369382
$658$	4.46573	0.174092
$659$	7.57531	0.295092	0.147546	−	0.989055i	$-0.452863\pi$
$659$	7.57531	0.295092	0.147546	+	0.989055i	$0.452863\pi$
$660$	−1.46081	−0.0568620
$661$	47.0616	1.83048	0.915241	−	0.402906i	$-0.132000\pi$
$661$	47.0616	1.83048	0.915241	+	0.402906i	$0.132000\pi$
$662$	68.7019	2.67018
$663$	1.90829	0.0741119
$664$	3.60197	0.139783
$665$	1.37751	0.0534177
$666$	−11.0205	−0.427036
$667$	−3.07838	−0.119195
$668$	43.6163	1.68757
$669$	9.77432	0.377897
$670$	−6.97107	−0.269316
$671$	3.41855	0.131972
$672$	−4.78765	−0.184688
$673$	37.1605	1.43243	0.716215	−	0.697880i	$-0.245873\pi$
$673$	37.1605	1.43243	0.716215	+	0.697880i	$0.245873\pi$
$674$	−3.72753	−0.143579
$675$	−4.70928	−0.181260
$676$	2.70928	0.104203
$677$	32.6030	1.25304	0.626518	−	0.779407i	$-0.284480\pi$
$677$	32.6030	1.25304	0.626518	+	0.779407i	$0.284480\pi$
$678$	13.4186	0.515336
$679$	−3.20394	−0.122956
$680$	−1.58372	−0.0607328
$681$	−11.0700	−0.424202
$682$	−3.17009	−0.121389
$683$	−50.9914	−1.95113	−0.975566	−	0.219706i	$-0.929490\pi$
$683$	−50.9914	−1.95113	−0.975566	+	0.219706i	$0.929490\pi$
$684$	−10.9711	−0.419489
$685$	1.56690	0.0598680
$686$	−18.6225	−0.711010
$687$	−17.0928	−0.652129
$688$	−2.05067	−0.0781811
$689$	9.91548	0.377750
$690$	−1.60197	−0.0609859
$691$	15.9434	0.606514	0.303257	−	0.952909i	$-0.401926\pi$
$691$	15.9434	0.606514	0.303257	+	0.952909i	$0.401926\pi$
$692$	44.0216	1.67345
$693$	0.630898	0.0239658
$694$	3.47641	0.131963
$695$	−10.0761	−0.382209
$696$	−3.46081	−0.131182
$697$	−3.91094	−0.148138
$698$	35.6742	1.35029
$699$	−8.00719	−0.302860
$700$	−8.04945	−0.304241
$701$	4.88324	0.184437	0.0922187	−	0.995739i	$-0.470604\pi$
$701$	4.88324	0.184437	0.0922187	+	0.995739i	$0.470604\pi$
$702$	2.17009	0.0819046
$703$	20.5646	0.775610
$704$	−12.3112	−0.463997
$705$	−1.75872	−0.0662374
$706$	−2.14957	−0.0809000
$707$	6.00000	0.225653
$708$	−9.75872	−0.366755
$709$	−7.20394	−0.270550	−0.135275	−	0.990808i	$-0.543192\pi$
$709$	−7.20394	−0.270550	−0.135275	+	0.990808i	$0.543192\pi$
$710$	9.07838	0.340705
$711$	3.90829	0.146572
$712$	19.9760	0.748632
$713$	−2.00000	−0.0749006
$714$	2.61265	0.0977758
$715$	−0.539189	−0.0201645
$716$	50.2472	1.87783
$717$	−3.28458	−0.122665
$718$	−61.9481	−2.31188
$719$	−4.84939	−0.180852	−0.0904258	−	0.995903i	$-0.528823\pi$
$719$	−4.84939	−0.180852	−0.0904258	+	0.995903i	$0.528823\pi$
$720$	1.12064	0.0417637
$721$	−2.89043	−0.107645
$722$	−5.64650	−0.210141
$723$	13.5486	0.503880
$724$	−29.4557	−1.09471
$725$	10.5886	0.393252
$726$	2.17009	0.0805395
$727$	34.8515	1.29257	0.646285	−	0.763096i	$-0.276322\pi$
$727$	34.8515	1.29257	0.646285	+	0.763096i	$0.276322\pi$
$728$	0.971071	0.0359903
$729$	1.00000	0.0370370
$730$	−11.0784	−0.410029
$731$	1.88285	0.0696398
$732$	9.26180	0.342326
$733$	11.2579	0.415821	0.207910	−	0.978148i	$-0.433334\pi$
$733$	11.2579	0.415821	0.207910	+	0.978148i	$0.433334\pi$
$734$	−24.3812	−0.899927
$735$	3.55971	0.131302
$736$	−10.3896	−0.382967
$737$	5.95774	0.219456
$738$	−4.44748	−0.163714
$739$	16.1028	0.592350	0.296175	−	0.955134i	$-0.404289\pi$
$739$	16.1028	0.592350	0.296175	+	0.955134i	$0.404289\pi$
$740$	7.41855	0.272711
$741$	−4.04945	−0.148760
$742$	13.5753	0.498365
$743$	−21.1317	−0.775247	−0.387623	−	0.921818i	$-0.626704\pi$
$743$	−21.1317	−0.775247	−0.387623	+	0.921818i	$0.626704\pi$
$744$	−2.24846	−0.0824327
$745$	8.62249	0.315903
$746$	−41.4017	−1.51583
$747$	2.34017	0.0856225
$748$	5.17009	0.189037
$749$	7.31965	0.267454
$750$	11.3607	0.414834
$751$	51.3172	1.87259	0.936296	−	0.351213i	$-0.114231\pi$
$751$	51.3172	1.87259	0.936296	+	0.351213i	$0.114231\pi$
$752$	−6.77924	−0.247214
$753$	4.18342	0.152452
$754$	−4.87936	−0.177696
$755$	−5.81658	−0.211687
$756$	1.70928	0.0621657
$757$	48.3773	1.75830	0.879152	−	0.476541i	$-0.158110\pi$
$757$	48.3773	1.75830	0.879152	+	0.476541i	$0.158110\pi$
$758$	3.90829	0.141956
$759$	1.36910	0.0496953
$760$	3.36069	0.121905
$761$	34.4307	1.24811	0.624055	−	0.781380i	$-0.285484\pi$
$761$	34.4307	1.24811	0.624055	+	0.781380i	$0.285484\pi$
$762$	−44.3701	−1.60736
$763$	−8.29914	−0.300449
$764$	−24.8638	−0.899539
$765$	−1.02893	−0.0372010
$766$	−2.12556	−0.0767996
$767$	−3.60197	−0.130060
$768$	−0.418551	−0.0151031
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	−0.738205	−0.0266031
$771$	8.52359	0.306970
$772$	−46.3318	−1.66752
$773$	−38.7370	−1.39327	−0.696636	−	0.717425i	$-0.745321\pi$
$773$	−38.7370	−1.39327	−0.696636	+	0.717425i	$0.745321\pi$
$774$	2.14116	0.0769623
$775$	6.87936	0.247114
$776$	−7.81658	−0.280599
$777$	−3.20394	−0.114941
$778$	−65.7152	−2.35601
$779$	8.29914	0.297347
$780$	−1.46081	−0.0523054
$781$	−7.75872	−0.277629
$782$	5.66967	0.202747
$783$	−2.24846	−0.0803536
$784$	13.7214	0.490049
$785$	4.92162	0.175660
$786$	30.2557	1.07918
$787$	25.8927	0.922975	0.461487	−	0.887147i	$-0.347316\pi$
$787$	25.8927	0.922975	0.461487	+	0.887147i	$0.347316\pi$
$788$	−9.05559	−0.322592
$789$	22.7526	0.810013
$790$	−4.57304	−0.162701
$791$	3.90110	0.138707
$792$	1.53919	0.0546927
$793$	3.41855	0.121396
$794$	−69.0082	−2.44901
$795$	−5.34632	−0.189614
$796$	−38.6225	−1.36894
$797$	22.2967	0.789789	0.394895	−	0.918726i	$-0.370781\pi$
$797$	22.2967	0.789789	0.394895	+	0.918726i	$0.370781\pi$
$798$	−5.54411	−0.196259
$799$	6.22446	0.220205
$800$	35.7370	1.26349
$801$	12.9783	0.458564
$802$	−2.92881	−0.103420
$803$	9.46800	0.334118
$804$	16.1412	0.569255
$805$	−0.465732	−0.0164149
$806$	−3.17009	−0.111662
$807$	−8.34017	−0.293588
$808$	14.6381	0.514966
$809$	0.195140	0.00686077	0.00343038	−	0.999994i	$-0.498908\pi$
$809$	0.195140	0.00686077	0.00343038	+	0.999994i	$0.498908\pi$
$810$	−1.17009	−0.0411126
$811$	−45.3400	−1.59210	−0.796051	−	0.605229i	$-0.793081\pi$
$811$	−45.3400	−1.59210	−0.796051	+	0.605229i	$0.793081\pi$
$812$	−3.84324	−0.134871
$813$	26.7031	0.936520
$814$	−11.0205	−0.386269
$815$	3.92389	0.137448
$816$	−3.96615	−0.138843
$817$	−3.99547	−0.139784
$818$	9.02893	0.315689
$819$	0.630898	0.0220453
$820$	2.99386	0.104550
$821$	54.3605	1.89720	0.948598	−	0.316485i	$-0.102503\pi$
$821$	54.3605	1.89720	0.948598	+	0.316485i	$0.102503\pi$
$822$	−6.30632	−0.219958
$823$	−31.0205	−1.08131	−0.540654	−	0.841245i	$-0.681823\pi$
$823$	−31.0205	−1.08131	−0.540654	+	0.841245i	$0.681823\pi$
$824$	−7.05172	−0.245658
$825$	−4.70928	−0.163956
$826$	−4.93146	−0.171588
$827$	−50.0228	−1.73946	−0.869731	−	0.493525i	$-0.835708\pi$
$827$	−50.0228	−1.73946	−0.869731	+	0.493525i	$0.835708\pi$
$828$	3.70928	0.128906
$829$	38.9939	1.35431	0.677156	−	0.735839i	$-0.263212\pi$
$829$	38.9939	1.35431	0.677156	+	0.735839i	$0.263212\pi$
$830$	−2.73820	−0.0950445
$831$	7.62863	0.264634
$832$	−12.3112	−0.426816
$833$	−12.5985	−0.436511
$834$	40.5536	1.40426
$835$	−8.68035	−0.300396
$836$	−10.9711	−0.379442
$837$	−1.46081	−0.0504930
$838$	−73.4557	−2.53749
$839$	−34.2557	−1.18264	−0.591318	−	0.806438i	$-0.701392\pi$
$839$	−34.2557	−1.18264	−0.591318	+	0.806438i	$0.701392\pi$
$840$	−0.523590	−0.0180656
$841$	−23.9444	−0.825669
$842$	−50.4801	−1.73966
$843$	−16.0228	−0.551854
$844$	40.6752	1.40010
$845$	−0.539189	−0.0185487
$846$	7.07838	0.243360
$847$	0.630898	0.0216779
$848$	−20.6081	−0.707685
$849$	16.2485	0.557646
$850$	−19.5018	−0.668908
$851$	−6.95282	−0.238340
$852$	−21.0205	−0.720151
$853$	−34.6635	−1.18686	−0.593428	−	0.804887i	$-0.702226\pi$
$853$	−34.6635	−1.18686	−0.593428	+	0.804887i	$0.702226\pi$
$854$	4.68035	0.160158
$855$	2.18342	0.0746713
$856$	17.8576	0.610361
$857$	2.91443	0.0995552	0.0497776	−	0.998760i	$-0.484149\pi$
$857$	2.91443	0.0995552	0.0497776	+	0.998760i	$0.484149\pi$
$858$	2.17009	0.0740855
$859$	49.9109	1.70294	0.851470	−	0.524404i	$-0.175712\pi$
$859$	49.9109	1.70294	0.851470	+	0.524404i	$0.175712\pi$
$860$	−1.44134	−0.0491492
$861$	−1.29299	−0.0440651
$862$	−31.0517	−1.05763
$863$	19.0205	0.647466	0.323733	−	0.946148i	$-0.395062\pi$
$863$	19.0205	0.647466	0.323733	+	0.946148i	$0.395062\pi$
$864$	−7.58864	−0.258171
$865$	−8.76099	−0.297883
$866$	−35.8576	−1.21849
$867$	−13.3584	−0.453676
$868$	−2.49693	−0.0847513
$869$	3.90829	0.132580
$870$	2.63090	0.0891958
$871$	5.95774	0.201870
$872$	−20.2472	−0.685658
$873$	−5.07838	−0.171877
$874$	−12.0312	−0.406961
$875$	3.30283	0.111656
$876$	25.6514	0.866681
$877$	7.75872	0.261993	0.130997	−	0.991383i	$-0.458182\pi$
$877$	7.75872	0.261993	0.130997	+	0.991383i	$0.458182\pi$
$878$	−29.9409	−1.01046
$879$	−22.2329	−0.749896
$880$	1.12064	0.0377767
$881$	8.63931	0.291066	0.145533	−	0.989353i	$-0.453510\pi$
$881$	8.63931	0.291066	0.145533	+	0.989353i	$0.453510\pi$
$882$	−14.3268	−0.482410
$883$	25.4596	0.856783	0.428392	−	0.903593i	$-0.359080\pi$
$883$	25.4596	0.856783	0.428392	+	0.903593i	$0.359080\pi$
$884$	5.17009	0.173889
$885$	1.94214	0.0652844
$886$	−53.1917	−1.78701
$887$	−43.7198	−1.46797	−0.733983	−	0.679168i	$-0.762341\pi$
$887$	−43.7198	−1.46797	−0.733983	+	0.679168i	$0.762341\pi$
$888$	−7.81658	−0.262307
$889$	−12.8995	−0.432635
$890$	−15.1857	−0.509025
$891$	1.00000	0.0335013
$892$	26.4813	0.886661
$893$	−13.2085	−0.442005
$894$	−34.7031	−1.16065
$895$	−10.0000	−0.334263
$896$	−7.28005	−0.243209
$897$	1.36910	0.0457130
$898$	−76.3039	−2.54629
$899$	3.28458	0.109547
$900$	−12.7587	−0.425291
$901$	18.9216	0.630371
$902$	−4.44748	−0.148085
$903$	0.622487	0.0207151
$904$	9.51745	0.316546
$905$	5.86216	0.194865
$906$	23.4101	0.777750
$907$	−49.7875	−1.65317	−0.826583	−	0.562815i	$-0.809718\pi$
$907$	−49.7875	−1.65317	−0.826583	+	0.562815i	$0.809718\pi$
$908$	−29.9916	−0.995306
$909$	9.51026	0.315435
$910$	−0.738205	−0.0244712
$911$	−11.4497	−0.379347	−0.189674	−	0.981847i	$-0.560743\pi$
$911$	−11.4497	−0.379347	−0.189674	+	0.981847i	$0.560743\pi$
$912$	8.41628	0.278691
$913$	2.34017	0.0774484
$914$	−78.6779	−2.60243
$915$	−1.84324	−0.0609358
$916$	−46.3090	−1.53009
$917$	8.79606	0.290472
$918$	4.14116	0.136679
$919$	−9.41136	−0.310452	−0.155226	−	0.987879i	$-0.549611\pi$
$919$	−9.41136	−0.310452	−0.155226	+	0.987879i	$0.549611\pi$
$920$	−1.13624	−0.0374606
$921$	−7.46800	−0.246079
$922$	51.2678	1.68841
$923$	−7.75872	−0.255382
$924$	1.70928	0.0562310
$925$	23.9155	0.786336
$926$	−10.5308	−0.346063
$927$	−4.58145	−0.150475
$928$	17.0628	0.560113
$929$	30.5536	1.00243	0.501215	−	0.865323i	$-0.332887\pi$
$929$	30.5536	1.00243	0.501215	+	0.865323i	$0.332887\pi$
$930$	1.70928	0.0560493
$931$	26.7343	0.876182
$932$	−21.6937	−0.710600
$933$	1.27021	0.0415847
$934$	34.4885	1.12850
$935$	−1.02893	−0.0336496
$936$	1.53919	0.0503100
$937$	−44.7480	−1.46185	−0.730927	−	0.682455i	$-0.760912\pi$
$937$	−44.7480	−1.46185	−0.730927	+	0.682455i	$0.760912\pi$
$938$	8.15676	0.266328
$939$	12.8287	0.418649
$940$	−4.76487	−0.155413
$941$	−52.9153	−1.72499	−0.862495	−	0.506066i	$-0.831099\pi$
$941$	−52.9153	−1.72499	−0.862495	+	0.506066i	$0.831099\pi$
$942$	−19.8082	−0.645385
$943$	−2.80590	−0.0913728
$944$	7.48625	0.243657
$945$	−0.340173	−0.0110658
$946$	2.14116	0.0696150
$947$	52.6681	1.71148	0.855741	−	0.517404i	$-0.173102\pi$
$947$	52.6681	1.71148	0.855741	+	0.517404i	$0.173102\pi$
$948$	10.5886	0.343903
$949$	9.46800	0.307344
$950$	41.3835	1.34266
$951$	13.2918	0.431015
$952$	1.85309	0.0600588
$953$	−14.0338	−0.454601	−0.227300	−	0.973825i	$-0.572990\pi$
$953$	−14.0338	−0.454601	−0.227300	+	0.973825i	$0.572990\pi$
$954$	21.5174	0.696653
$955$	4.94828	0.160123
$956$	−8.89884	−0.287809
$957$	−2.24846	−0.0726825
$958$	−12.3896	−0.400290
$959$	−1.83340	−0.0592037
$960$	6.63809	0.214243
$961$	−28.8660	−0.931162
$962$	−11.0205	−0.355316
$963$	11.6020	0.373868
$964$	36.7070	1.18225
$965$	9.22076	0.296827
$966$	1.87444	0.0603092
$967$	−18.3440	−0.589905	−0.294952	−	0.955512i	$-0.595304\pi$
$967$	−18.3440	−0.589905	−0.294952	+	0.955512i	$0.595304\pi$
$968$	1.53919	0.0494714
$969$	−7.72753	−0.248244
$970$	5.94214	0.190791
$971$	45.4968	1.46006	0.730030	−	0.683415i	$-0.239506\pi$
$971$	45.4968	1.46006	0.730030	+	0.683415i	$0.239506\pi$
$972$	2.70928	0.0869000
$973$	11.7899	0.377967
$974$	−60.6030	−1.94185
$975$	−4.70928	−0.150818
$976$	−7.10504	−0.227427
$977$	24.5802	0.786391	0.393196	−	0.919455i	$-0.371370\pi$
$977$	24.5802	0.786391	0.393196	+	0.919455i	$0.371370\pi$
$978$	−15.7926	−0.504991
$979$	12.9783	0.414787
$980$	9.64423	0.308074
$981$	−13.1545	−0.419990
$982$	69.8141	2.22786
$983$	−57.7198	−1.84097	−0.920487	−	0.390772i	$-0.872208\pi$
$983$	−57.7198	−1.84097	−0.920487	+	0.390772i	$0.872208\pi$
$984$	−3.15449	−0.100561
$985$	1.80221	0.0574231
$986$	−9.31124	−0.296530
$987$	2.05786	0.0655024
$988$	−10.9711	−0.349036
$989$	1.35085	0.0429546
$990$	−1.17009	−0.0371878
$991$	12.0221	0.381895	0.190948	−	0.981600i	$-0.438844\pi$
$991$	12.0221	0.381895	0.190948	+	0.981600i	$0.438844\pi$
$992$	11.0856	0.351967
$993$	31.6586	1.00466
$994$	−10.6225	−0.336925
$995$	7.68649	0.243678
$996$	6.34017	0.200896
$997$	−34.5958	−1.09566	−0.547830	−	0.836589i	$-0.684546\pi$
$997$	−34.5958	−1.09566	−0.547830	+	0.836589i	$0.684546\pi$
$998$	−35.3679	−1.11955
$999$	−5.07838	−0.160673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.a.f.1.3	✓	3
3.2	odd	2		1287.2.a.i.1.1		3
4.3	odd	2		6864.2.a.bp.1.2		3
11.10	odd	2		4719.2.a.t.1.1		3
13.12	even	2		5577.2.a.k.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.f.1.3	✓	3	1.1	even	1	trivial
1287.2.a.i.1.1		3	3.2	odd	2
4719.2.a.t.1.1		3	11.10	odd	2
5577.2.a.k.1.1		3	13.12	even	2
6864.2.a.bp.1.2		3	4.3	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.539189 q^{5} +2.17009 q^{6} +0.630898 q^{7} +1.53919 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.539189 q^{5} +2.17009 q^{6} +0.630898 q^{7} +1.53919 q^{8} +1.00000 q^{9} -1.17009 q^{10} +1.00000 q^{11} +2.70928 q^{12} +1.00000 q^{13} +1.36910 q^{14} -0.539189 q^{15} -2.07838 q^{16} +1.90829 q^{17} +2.17009 q^{18} -4.04945 q^{19} -1.46081 q^{20} +0.630898 q^{21} +2.17009 q^{22} +1.36910 q^{23} +1.53919 q^{24} -4.70928 q^{25} +2.17009 q^{26} +1.00000 q^{27} +1.70928 q^{28} -2.24846 q^{29} -1.17009 q^{30} -1.46081 q^{31} -7.58864 q^{32} +1.00000 q^{33} +4.14116 q^{34} -0.340173 q^{35} +2.70928 q^{36} -5.07838 q^{37} -8.78765 q^{38} +1.00000 q^{39} -0.829914 q^{40} -2.04945 q^{41} +1.36910 q^{42} +0.986669 q^{43} +2.70928 q^{44} -0.539189 q^{45} +2.97107 q^{46} +3.26180 q^{47} -2.07838 q^{48} -6.60197 q^{49} -10.2195 q^{50} +1.90829 q^{51} +2.70928 q^{52} +9.91548 q^{53} +2.17009 q^{54} -0.539189 q^{55} +0.971071 q^{56} -4.04945 q^{57} -4.87936 q^{58} -3.60197 q^{59} -1.46081 q^{60} +3.41855 q^{61} -3.17009 q^{62} +0.630898 q^{63} -12.3112 q^{64} -0.539189 q^{65} +2.17009 q^{66} +5.95774 q^{67} +5.17009 q^{68} +1.36910 q^{69} -0.738205 q^{70} -7.75872 q^{71} +1.53919 q^{72} +9.46800 q^{73} -11.0205 q^{74} -4.70928 q^{75} -10.9711 q^{76} +0.630898 q^{77} +2.17009 q^{78} +3.90829 q^{79} +1.12064 q^{80} +1.00000 q^{81} -4.44748 q^{82} +2.34017 q^{83} +1.70928 q^{84} -1.02893 q^{85} +2.14116 q^{86} -2.24846 q^{87} +1.53919 q^{88} +12.9783 q^{89} -1.17009 q^{90} +0.630898 q^{91} +3.70928 q^{92} -1.46081 q^{93} +7.07838 q^{94} +2.18342 q^{95} -7.58864 q^{96} -5.07838 q^{97} -14.3268 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + q^4 + q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9} + 2 q^{10} + 3 q^{11} + q^{12} + 3 q^{13} + 8 q^{14} - 3 q^{16} + 8 q^{17} + q^{18} + 6 q^{19} - 6 q^{20} - 2 q^{21} + q^{22} + 8 q^{23} + 3 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 2 q^{28} + 2 q^{29} + 2 q^{30} - 6 q^{31} - 3 q^{32} + 3 q^{33} - 8 q^{34} + 10 q^{35} + q^{36} - 12 q^{37} - 16 q^{38} + 3 q^{39} - 8 q^{40} + 12 q^{41} + 8 q^{42} + 2 q^{43} + q^{44} - 6 q^{46} + 2 q^{47} - 3 q^{48} - q^{49} - 7 q^{50} + 8 q^{51} + q^{52} - 2 q^{53} + q^{54} - 12 q^{56} + 6 q^{57} - 2 q^{58} + 8 q^{59} - 6 q^{60} - 4 q^{61} - 4 q^{62} - 2 q^{63} - 11 q^{64} + q^{66} + 2 q^{67} + 10 q^{68} + 8 q^{69} - 10 q^{70} + 2 q^{71} + 3 q^{72} - 4 q^{73} - 7 q^{75} - 18 q^{76} - 2 q^{77} + q^{78} + 14 q^{79} + 16 q^{80} + 3 q^{81} - 14 q^{82} - 4 q^{83} - 2 q^{84} - 18 q^{85} - 14 q^{86} + 2 q^{87} + 3 q^{88} - 10 q^{89} + 2 q^{90} - 2 q^{91} + 4 q^{92} - 6 q^{93} + 18 q^{94} + 2 q^{95} - 3 q^{96} - 12 q^{97} - 31 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + q^4 + q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9 + 2 * q^10 + 3 * q^11 + q^12 + 3 * q^13 + 8 * q^14 - 3 * q^16 + 8 * q^17 + q^18 + 6 * q^19 - 6 * q^20 - 2 * q^21 + q^22 + 8 * q^23 + 3 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 2 * q^28 + 2 * q^29 + 2 * q^30 - 6 * q^31 - 3 * q^32 + 3 * q^33 - 8 * q^34 + 10 * q^35 + q^36 - 12 * q^37 - 16 * q^38 + 3 * q^39 - 8 * q^40 + 12 * q^41 + 8 * q^42 + 2 * q^43 + q^44 - 6 * q^46 + 2 * q^47 - 3 * q^48 - q^49 - 7 * q^50 + 8 * q^51 + q^52 - 2 * q^53 + q^54 - 12 * q^56 + 6 * q^57 - 2 * q^58 + 8 * q^59 - 6 * q^60 - 4 * q^61 - 4 * q^62 - 2 * q^63 - 11 * q^64 + q^66 + 2 * q^67 + 10 * q^68 + 8 * q^69 - 10 * q^70 + 2 * q^71 + 3 * q^72 - 4 * q^73 - 7 * q^75 - 18 * q^76 - 2 * q^77 + q^78 + 14 * q^79 + 16 * q^80 + 3 * q^81 - 14 * q^82 - 4 * q^83 - 2 * q^84 - 18 * q^85 - 14 * q^86 + 2 * q^87 + 3 * q^88 - 10 * q^89 + 2 * q^90 - 2 * q^91 + 4 * q^92 - 6 * q^93 + 18 * q^94 + 2 * q^95 - 3 * q^96 - 12 * q^97 - 31 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more