LMFDB - Embedded newform 7942.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.17009 q^{3} +1.00000 q^{4} +4.34017 q^{5} +1.17009 q^{6} +4.24846 q^{7} -1.00000 q^{8} -1.63090 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.17009 q^{3} +1.00000 q^{4} +4.34017 q^{5} +1.17009 q^{6} +4.24846 q^{7} -1.00000 q^{8} -1.63090 q^{9} -4.34017 q^{10} -1.00000 q^{11} -1.17009 q^{12} +3.63090 q^{13} -4.24846 q^{14} -5.07838 q^{15} +1.00000 q^{16} +1.12064 q^{17} +1.63090 q^{18} +4.34017 q^{20} -4.97107 q^{21} +1.00000 q^{22} +8.49693 q^{23} +1.17009 q^{24} +13.8371 q^{25} -3.63090 q^{26} +5.41855 q^{27} +4.24846 q^{28} +6.41855 q^{29} +5.07838 q^{30} -8.04945 q^{31} -1.00000 q^{32} +1.17009 q^{33} -1.12064 q^{34} +18.4391 q^{35} -1.63090 q^{36} +3.80098 q^{37} -4.24846 q^{39} -4.34017 q^{40} +0.539189 q^{41} +4.97107 q^{42} -3.87936 q^{43} -1.00000 q^{44} -7.07838 q^{45} -8.49693 q^{46} -9.58864 q^{47} -1.17009 q^{48} +11.0494 q^{49} -13.8371 q^{50} -1.31124 q^{51} +3.63090 q^{52} -2.34017 q^{53} -5.41855 q^{54} -4.34017 q^{55} -4.24846 q^{56} -6.41855 q^{58} +1.07838 q^{59} -5.07838 q^{60} -0.127826 q^{61} +8.04945 q^{62} -6.92881 q^{63} +1.00000 q^{64} +15.7587 q^{65} -1.17009 q^{66} +4.98667 q^{67} +1.12064 q^{68} -9.94214 q^{69} -18.4391 q^{70} -1.69594 q^{71} +1.63090 q^{72} -0.780465 q^{73} -3.80098 q^{74} -16.1906 q^{75} -4.24846 q^{77} +4.24846 q^{78} +8.34736 q^{79} +4.34017 q^{80} -1.44748 q^{81} -0.539189 q^{82} -4.03612 q^{83} -4.97107 q^{84} +4.86376 q^{85} +3.87936 q^{86} -7.51026 q^{87} +1.00000 q^{88} +9.18342 q^{89} +7.07838 q^{90} +15.4257 q^{91} +8.49693 q^{92} +9.41855 q^{93} +9.58864 q^{94} +1.17009 q^{96} -17.0722 q^{97} -11.0494 q^{98} +1.63090 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 - 3 * q^8 - q^9	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 3 q^{8} - q^{9} - 2 q^{10} - 3 q^{11} + 2 q^{12} + 7 q^{13} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 16 q^{17} + q^{18} + 2 q^{20} + 3 q^{22} + 8 q^{23} - 2 q^{24} + 13 q^{25} - 7 q^{26} + 2 q^{27} + 4 q^{28} + 5 q^{29} + 12 q^{30} - 6 q^{31} - 3 q^{32} - 2 q^{33} - 16 q^{34} + 8 q^{35} - q^{36} + 2 q^{37} - 4 q^{39} - 2 q^{40} + q^{43} - 3 q^{44} - 18 q^{45} - 8 q^{46} - 9 q^{47} + 2 q^{48} + 15 q^{49} - 13 q^{50} + 22 q^{51} + 7 q^{52} + 4 q^{53} - 2 q^{54} - 2 q^{55} - 4 q^{56} - 5 q^{58} - 12 q^{60} + 21 q^{61} + 6 q^{62} + 10 q^{63} + 3 q^{64} + 22 q^{65} + 2 q^{66} + 14 q^{67} + 16 q^{68} - 8 q^{70} + 3 q^{71} + q^{72} - 26 q^{73} - 2 q^{74} - 10 q^{75} - 4 q^{77} + 4 q^{78} - 20 q^{79} + 2 q^{80} - 5 q^{81} + 7 q^{83} - 12 q^{85} - q^{86} - 6 q^{87} + 3 q^{88} + 23 q^{89} + 18 q^{90} - 2 q^{91} + 8 q^{92} + 14 q^{93} + 9 q^{94} - 2 q^{96} - 13 q^{97} - 15 q^{98} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 - 3 * q^8 - q^9 - 2 * q^10 - 3 * q^11 + 2 * q^12 + 7 * q^13 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 16 * q^17 + q^18 + 2 * q^20 + 3 * q^22 + 8 * q^23 - 2 * q^24 + 13 * q^25 - 7 * q^26 + 2 * q^27 + 4 * q^28 + 5 * q^29 + 12 * q^30 - 6 * q^31 - 3 * q^32 - 2 * q^33 - 16 * q^34 + 8 * q^35 - q^36 + 2 * q^37 - 4 * q^39 - 2 * q^40 + q^43 - 3 * q^44 - 18 * q^45 - 8 * q^46 - 9 * q^47 + 2 * q^48 + 15 * q^49 - 13 * q^50 + 22 * q^51 + 7 * q^52 + 4 * q^53 - 2 * q^54 - 2 * q^55 - 4 * q^56 - 5 * q^58 - 12 * q^60 + 21 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 + 22 * q^65 + 2 * q^66 + 14 * q^67 + 16 * q^68 - 8 * q^70 + 3 * q^71 + q^72 - 26 * q^73 - 2 * q^74 - 10 * q^75 - 4 * q^77 + 4 * q^78 - 20 * q^79 + 2 * q^80 - 5 * q^81 + 7 * q^83 - 12 * q^85 - q^86 - 6 * q^87 + 3 * q^88 + 23 * q^89 + 18 * q^90 - 2 * q^91 + 8 * q^92 + 14 * q^93 + 9 * q^94 - 2 * q^96 - 13 * q^97 - 15 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.17009	−0.675550	−0.337775	−	0.941227i	$-0.609674\pi$
$3$	−1.17009	−0.675550	−0.337775	+	0.941227i	$0.609674\pi$
$4$	1.00000	0.500000
$5$	4.34017	1.94098	0.970492	−	0.241133i	$-0.0775189\pi$
$5$	4.34017	1.94098	0.970492	+	0.241133i	$0.0775189\pi$
$6$	1.17009	0.477686
$7$	4.24846	1.60577	0.802884	−	0.596135i	$-0.203298\pi$
$7$	4.24846	1.60577	0.802884	+	0.596135i	$0.203298\pi$
$8$	−1.00000	−0.353553
$9$	−1.63090	−0.543633
$10$	−4.34017	−1.37248
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.17009	−0.337775
$13$	3.63090	1.00703	0.503515	−	0.863987i	$-0.332040\pi$
$13$	3.63090	1.00703	0.503515	+	0.863987i	$0.332040\pi$
$14$	−4.24846	−1.13545
$15$	−5.07838	−1.31123
$16$	1.00000	0.250000
$17$	1.12064	0.271795	0.135897	−	0.990723i	$-0.456608\pi$
$17$	1.12064	0.271795	0.135897	+	0.990723i	$0.456608\pi$
$18$	1.63090	0.384406
$19$	0	0
$19$	0	0
$20$	4.34017	0.970492
$21$	−4.97107	−1.08478
$22$	1.00000	0.213201
$23$	8.49693	1.77173	0.885866	−	0.463941i	$-0.153565\pi$
$23$	8.49693	1.77173	0.885866	+	0.463941i	$0.153565\pi$
$24$	1.17009	0.238843
$25$	13.8371	2.76742
$26$	−3.63090	−0.712078
$27$	5.41855	1.04280
$28$	4.24846	0.802884
$29$	6.41855	1.19189	0.595947	−	0.803023i	$-0.296777\pi$
$29$	6.41855	1.19189	0.595947	+	0.803023i	$0.296777\pi$
$30$	5.07838	0.927181
$31$	−8.04945	−1.44572	−0.722862	−	0.690993i	$-0.757174\pi$
$31$	−8.04945	−1.44572	−0.722862	+	0.690993i	$0.757174\pi$
$32$	−1.00000	−0.176777
$33$	1.17009	0.203686
$34$	−1.12064	−0.192188
$35$	18.4391	3.11677
$36$	−1.63090	−0.271816
$37$	3.80098	0.624878	0.312439	−	0.949938i	$-0.398854\pi$
$37$	3.80098	0.624878	0.312439	+	0.949938i	$0.398854\pi$
$38$	0	0
$39$	−4.24846	−0.680299
$40$	−4.34017	−0.686242
$41$	0.539189	0.0842072	0.0421036	−	0.999113i	$-0.486594\pi$
$41$	0.539189	0.0842072	0.0421036	+	0.999113i	$0.486594\pi$
$42$	4.97107	0.767053
$43$	−3.87936	−0.591597	−0.295799	−	0.955250i	$-0.595586\pi$
$43$	−3.87936	−0.591597	−0.295799	+	0.955250i	$0.595586\pi$
$44$	−1.00000	−0.150756
$45$	−7.07838	−1.05518
$46$	−8.49693	−1.25280
$47$	−9.58864	−1.39865	−0.699323	−	0.714806i	$-0.746515\pi$
$47$	−9.58864	−1.39865	−0.699323	+	0.714806i	$0.746515\pi$
$48$	−1.17009	−0.168887
$49$	11.0494	1.57849
$50$	−13.8371	−1.95686
$51$	−1.31124	−0.183611
$52$	3.63090	0.503515
$53$	−2.34017	−0.321447	−0.160724	−	0.986999i	$-0.551383\pi$
$53$	−2.34017	−0.321447	−0.160724	+	0.986999i	$0.551383\pi$
$54$	−5.41855	−0.737371
$55$	−4.34017	−0.585229
$56$	−4.24846	−0.567725
$57$	0	0
$58$	−6.41855	−0.842797
$59$	1.07838	0.140393	0.0701964	−	0.997533i	$-0.477637\pi$
$59$	1.07838	0.140393	0.0701964	+	0.997533i	$0.477637\pi$
$60$	−5.07838	−0.655616
$61$	−0.127826	−0.0163665	−0.00818323	−	0.999967i	$-0.502605\pi$
$61$	−0.127826	−0.0163665	−0.00818323	+	0.999967i	$0.502605\pi$
$62$	8.04945	1.02228
$63$	−6.92881	−0.872948
$64$	1.00000	0.125000
$65$	15.7587	1.95463
$66$	−1.17009	−0.144028
$67$	4.98667	0.609219	0.304609	−	0.952477i	$-0.401474\pi$
$67$	4.98667	0.609219	0.304609	+	0.952477i	$0.401474\pi$
$68$	1.12064	0.135897
$69$	−9.94214	−1.19689
$70$	−18.4391	−2.20389
$71$	−1.69594	−0.201272	−0.100636	−	0.994923i	$-0.532088\pi$
$71$	−1.69594	−0.201272	−0.100636	+	0.994923i	$0.532088\pi$
$72$	1.63090	0.192203
$73$	−0.780465	−0.0913465	−0.0456733	−	0.998956i	$-0.514543\pi$
$73$	−0.780465	−0.0913465	−0.0456733	+	0.998956i	$0.514543\pi$
$74$	−3.80098	−0.441855
$75$	−16.1906	−1.86953
$76$	0	0
$77$	−4.24846	−0.484157
$78$	4.24846	0.481044
$79$	8.34736	0.939151	0.469576	−	0.882892i	$-0.344407\pi$
$79$	8.34736	0.939151	0.469576	+	0.882892i	$0.344407\pi$
$80$	4.34017	0.485246
$81$	−1.44748	−0.160831
$82$	−0.539189	−0.0595435
$83$	−4.03612	−0.443021	−0.221511	−	0.975158i	$-0.571099\pi$
$83$	−4.03612	−0.443021	−0.221511	+	0.975158i	$0.571099\pi$
$84$	−4.97107	−0.542388
$85$	4.86376	0.527549
$86$	3.87936	0.418322
$87$	−7.51026	−0.805184
$88$	1.00000	0.106600
$89$	9.18342	0.973440	0.486720	−	0.873558i	$-0.338193\pi$
$89$	9.18342	0.973440	0.486720	+	0.873558i	$0.338193\pi$
$90$	7.07838	0.746127
$91$	15.4257	1.61706
$92$	8.49693	0.885866
$93$	9.41855	0.976658
$94$	9.58864	0.988992
$95$	0	0
$96$	1.17009	0.119421
$97$	−17.0722	−1.73342	−0.866711	−	0.498810i	$-0.833771\pi$
$97$	−17.0722	−1.73342	−0.866711	+	0.498810i	$0.833771\pi$
$98$	−11.0494	−1.11616
$99$	1.63090	0.163911
$100$	13.8371	1.38371
$101$	8.52586	0.848355	0.424177	−	0.905579i	$-0.360563\pi$
$101$	8.52586	0.848355	0.424177	+	0.905579i	$0.360563\pi$
$102$	1.31124	0.129832
$103$	−0.568118	−0.0559784	−0.0279892	−	0.999608i	$-0.508910\pi$
$103$	−0.568118	−0.0559784	−0.0279892	+	0.999608i	$0.508910\pi$
$104$	−3.63090	−0.356039
$105$	−21.5753	−2.10553
$106$	2.34017	0.227298
$107$	−12.6937	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$107$	−12.6937	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$108$	5.41855	0.521400
$109$	1.10731	0.106061	0.0530304	−	0.998593i	$-0.483112\pi$
$109$	1.10731	0.106061	0.0530304	+	0.998593i	$0.483112\pi$
$110$	4.34017	0.413819
$111$	−4.44748	−0.422136
$112$	4.24846	0.401442
$113$	−6.44521	−0.606315	−0.303157	−	0.952941i	$-0.598041\pi$
$113$	−6.44521	−0.606315	−0.303157	+	0.952941i	$0.598041\pi$
$114$	0	0
$115$	36.8781	3.43890
$116$	6.41855	0.595947
$117$	−5.92162	−0.547454
$118$	−1.07838	−0.0992727
$119$	4.76099	0.436439
$120$	5.07838	0.463590
$121$	1.00000	0.0909091
$122$	0.127826	0.0115728
$123$	−0.630898	−0.0568861
$124$	−8.04945	−0.722862
$125$	38.3545	3.43054
$126$	6.92881	0.617267
$127$	18.6537	1.65525	0.827623	−	0.561284i	$-0.189692\pi$
$127$	18.6537	1.65525	0.827623	+	0.561284i	$0.189692\pi$
$128$	−1.00000	−0.0883883
$129$	4.53919	0.399653
$130$	−15.7587	−1.38213
$131$	−11.3474	−0.991424	−0.495712	−	0.868487i	$-0.665093\pi$
$131$	−11.3474	−0.991424	−0.495712	+	0.868487i	$0.665093\pi$
$132$	1.17009	0.101843
$133$	0	0
$134$	−4.98667	−0.430783
$135$	23.5174	2.02406
$136$	−1.12064	−0.0960939
$137$	8.36910	0.715021	0.357510	−	0.933909i	$-0.383626\pi$
$137$	8.36910	0.715021	0.357510	+	0.933909i	$0.383626\pi$
$138$	9.94214	0.846331
$139$	−11.2351	−0.952952	−0.476476	−	0.879188i	$-0.658086\pi$
$139$	−11.2351	−0.952952	−0.476476	+	0.879188i	$0.658086\pi$
$140$	18.4391	1.55839
$141$	11.2195	0.944855
$142$	1.69594	0.142320
$143$	−3.63090	−0.303631
$144$	−1.63090	−0.135908
$145$	27.8576	2.31345
$146$	0.780465	0.0645918
$147$	−12.9288	−1.06635
$148$	3.80098	0.312439
$149$	9.70928	0.795415	0.397707	−	0.917512i	$-0.369806\pi$
$149$	9.70928	0.795415	0.397707	+	0.917512i	$0.369806\pi$
$150$	16.1906	1.32196
$151$	9.69368	0.788860	0.394430	−	0.918926i	$-0.370942\pi$
$151$	9.69368	0.788860	0.394430	+	0.918926i	$0.370942\pi$
$152$	0	0
$153$	−1.82765	−0.147756
$154$	4.24846	0.342351
$155$	−34.9360	−2.80613
$156$	−4.24846	−0.340149
$157$	−17.8576	−1.42519	−0.712597	−	0.701574i	$-0.752481\pi$
$157$	−17.8576	−1.42519	−0.712597	+	0.701574i	$0.752481\pi$
$158$	−8.34736	−0.664080
$159$	2.73820	0.217154
$160$	−4.34017	−0.343121
$161$	36.0989	2.84499
$162$	1.44748	0.113725
$163$	0.0266620	0.00208833	0.00104416	−	0.999999i	$-0.499668\pi$
$163$	0.0266620	0.00208833	0.00104416	+	0.999999i	$0.499668\pi$
$164$	0.539189	0.0421036
$165$	5.07838	0.395351
$166$	4.03612	0.313263
$167$	12.1906	0.943337	0.471669	−	0.881776i	$-0.343652\pi$
$167$	12.1906	0.943337	0.471669	+	0.881776i	$0.343652\pi$
$168$	4.97107	0.383526
$169$	0.183417	0.0141090
$170$	−4.86376	−0.373034
$171$	0	0
$172$	−3.87936	−0.295799
$173$	−1.70928	−0.129954	−0.0649769	−	0.997887i	$-0.520697\pi$
$173$	−1.70928	−0.129954	−0.0649769	+	0.997887i	$0.520697\pi$
$174$	7.51026	0.569351
$175$	58.7864	4.44384
$176$	−1.00000	−0.0753778
$177$	−1.26180	−0.0948423
$178$	−9.18342	−0.688326
$179$	−1.90110	−0.142095	−0.0710476	−	0.997473i	$-0.522634\pi$
$179$	−1.90110	−0.142095	−0.0710476	+	0.997473i	$0.522634\pi$
$180$	−7.07838	−0.527591
$181$	−5.06278	−0.376313	−0.188157	−	0.982139i	$-0.560251\pi$
$181$	−5.06278	−0.376313	−0.188157	+	0.982139i	$0.560251\pi$
$182$	−15.4257	−1.14343
$183$	0.149568	0.0110564
$184$	−8.49693	−0.626402
$185$	16.4969	1.21288
$186$	−9.41855	−0.690602
$187$	−1.12064	−0.0819492
$188$	−9.58864	−0.699323
$189$	23.0205	1.67450
$190$	0	0
$191$	−12.8504	−0.929825	−0.464912	−	0.885357i	$-0.653914\pi$
$191$	−12.8504	−0.929825	−0.464912	+	0.885357i	$0.653914\pi$
$192$	−1.17009	−0.0844437
$193$	−25.9265	−1.86623	−0.933117	−	0.359574i	$-0.882922\pi$
$193$	−25.9265	−1.86623	−0.933117	+	0.359574i	$0.882922\pi$
$194$	17.0722	1.22572
$195$	−18.4391	−1.32045
$196$	11.0494	0.789246
$197$	−4.44521	−0.316708	−0.158354	−	0.987382i	$-0.550619\pi$
$197$	−4.44521	−0.316708	−0.158354	+	0.987382i	$0.550619\pi$
$198$	−1.63090	−0.115903
$199$	3.44029	0.243876	0.121938	−	0.992538i	$-0.461089\pi$
$199$	3.44029	0.243876	0.121938	+	0.992538i	$0.461089\pi$
$200$	−13.8371	−0.978431
$201$	−5.83483	−0.411557
$202$	−8.52586	−0.599877
$203$	27.2690	1.91391
$204$	−1.31124	−0.0918054
$205$	2.34017	0.163445
$206$	0.568118	0.0395827
$207$	−13.8576	−0.963171
$208$	3.63090	0.251757
$209$	0	0
$210$	21.5753	1.48884
$211$	−5.07838	−0.349610	−0.174805	−	0.984603i	$-0.555929\pi$
$211$	−5.07838	−0.349610	−0.174805	+	0.984603i	$0.555929\pi$
$212$	−2.34017	−0.160724
$213$	1.98440	0.135969
$214$	12.6937	0.867722
$215$	−16.8371	−1.14828
$216$	−5.41855	−0.368686
$217$	−34.1978	−2.32150
$218$	−1.10731	−0.0749963
$219$	0.913212	0.0617091
$220$	−4.34017	−0.292614
$221$	4.06892	0.273705
$222$	4.44748	0.298495
$223$	−11.9060	−0.797286	−0.398643	−	0.917106i	$-0.630519\pi$
$223$	−11.9060	−0.797286	−0.398643	+	0.917106i	$0.630519\pi$
$224$	−4.24846	−0.283862
$225$	−22.5669	−1.50446
$226$	6.44521	0.428729
$227$	−1.31124	−0.0870303	−0.0435151	−	0.999053i	$-0.513856\pi$
$227$	−1.31124	−0.0870303	−0.0435151	+	0.999053i	$0.513856\pi$
$228$	0	0
$229$	23.0361	1.52227	0.761135	−	0.648594i	$-0.224643\pi$
$229$	23.0361	1.52227	0.761135	+	0.648594i	$0.224643\pi$
$230$	−36.8781	−2.43167
$231$	4.97107	0.327072
$232$	−6.41855	−0.421398
$233$	−15.3763	−1.00733	−0.503667	−	0.863898i	$-0.668016\pi$
$233$	−15.3763	−1.00733	−0.503667	+	0.863898i	$0.668016\pi$
$234$	5.92162	0.387109
$235$	−41.6163	−2.71475
$236$	1.07838	0.0701964
$237$	−9.76713	−0.634444
$238$	−4.76099	−0.308609
$239$	10.5236	0.680714	0.340357	−	0.940296i	$-0.389452\pi$
$239$	10.5236	0.680714	0.340357	+	0.940296i	$0.389452\pi$
$240$	−5.07838	−0.327808
$241$	4.08452	0.263107	0.131554	−	0.991309i	$-0.458003\pi$
$241$	4.08452	0.263107	0.131554	+	0.991309i	$0.458003\pi$
$242$	−1.00000	−0.0642824
$243$	−14.5620	−0.934151
$244$	−0.127826	−0.00818323
$245$	47.9565	3.06383
$246$	0.630898	0.0402246
$247$	0	0
$248$	8.04945	0.511140
$249$	4.72261	0.299283
$250$	−38.3545	−2.42575
$251$	−6.74539	−0.425765	−0.212883	−	0.977078i	$-0.568285\pi$
$251$	−6.74539	−0.425765	−0.212883	+	0.977078i	$0.568285\pi$
$252$	−6.92881	−0.436474
$253$	−8.49693	−0.534197
$254$	−18.6537	−1.17044
$255$	−5.69102	−0.356386
$256$	1.00000	0.0625000
$257$	20.1256	1.25540	0.627699	−	0.778456i	$-0.283997\pi$
$257$	20.1256	1.25540	0.627699	+	0.778456i	$0.283997\pi$
$258$	−4.53919	−0.282598
$259$	16.1483	1.00341
$260$	15.7587	0.977315
$261$	−10.4680	−0.647953
$262$	11.3474	0.701042
$263$	2.48255	0.153081	0.0765404	−	0.997066i	$-0.475613\pi$
$263$	2.48255	0.153081	0.0765404	+	0.997066i	$0.475613\pi$
$264$	−1.17009	−0.0720138
$265$	−10.1568	−0.623925
$266$	0	0
$267$	−10.7454	−0.657607
$268$	4.98667	0.304609
$269$	30.4235	1.85495	0.927476	−	0.373883i	$-0.121974\pi$
$269$	30.4235	1.85495	0.927476	+	0.373883i	$0.121974\pi$
$270$	−23.5174	−1.43123
$271$	−19.6020	−1.19073	−0.595367	−	0.803454i	$-0.702993\pi$
$271$	−19.6020	−1.19073	−0.595367	+	0.803454i	$0.702993\pi$
$272$	1.12064	0.0679487
$273$	−18.0494	−1.09240
$274$	−8.36910	−0.505596
$275$	−13.8371	−0.834409
$276$	−9.94214	−0.598447
$277$	−32.6970	−1.96457	−0.982286	−	0.187389i	$-0.939998\pi$
$277$	−32.6970	−1.96457	−0.982286	+	0.187389i	$0.939998\pi$
$278$	11.2351	0.673839
$279$	13.1278	0.785942
$280$	−18.4391	−1.10195
$281$	−31.2339	−1.86326	−0.931629	−	0.363410i	$-0.881613\pi$
$281$	−31.2339	−1.86326	−0.931629	+	0.363410i	$0.881613\pi$
$282$	−11.2195	−0.668114
$283$	1.97334	0.117303	0.0586514	−	0.998279i	$-0.481320\pi$
$283$	1.97334	0.117303	0.0586514	+	0.998279i	$0.481320\pi$
$284$	−1.69594	−0.100636
$285$	0	0
$286$	3.63090	0.214699
$287$	2.29072	0.135217
$288$	1.63090	0.0961016
$289$	−15.7442	−0.926128
$290$	−27.8576	−1.63586
$291$	19.9760	1.17101
$292$	−0.780465	−0.0456733
$293$	−0.261795	−0.0152942	−0.00764712	−	0.999971i	$-0.502434\pi$
$293$	−0.261795	−0.0152942	−0.00764712	+	0.999971i	$0.502434\pi$
$294$	12.9288	0.754024
$295$	4.68035	0.272500
$296$	−3.80098	−0.220928
$297$	−5.41855	−0.314416
$298$	−9.70928	−0.562443
$299$	30.8515	1.78419
$300$	−16.1906	−0.934765
$301$	−16.4813	−0.949968
$302$	−9.69368	−0.557808
$303$	−9.97599	−0.573106
$304$	0	0
$305$	−0.554787	−0.0317670
$306$	1.82765	0.104480
$307$	−1.24846	−0.0712536	−0.0356268	−	0.999365i	$-0.511343\pi$
$307$	−1.24846	−0.0712536	−0.0356268	+	0.999365i	$0.511343\pi$
$308$	−4.24846	−0.242079
$309$	0.664748	0.0378162
$310$	34.9360	1.98423
$311$	−2.03999	−0.115677	−0.0578387	−	0.998326i	$-0.518421\pi$
$311$	−2.03999	−0.115677	−0.0578387	+	0.998326i	$0.518421\pi$
$312$	4.24846	0.240522
$313$	−8.06997	−0.456142	−0.228071	−	0.973645i	$-0.573242\pi$
$313$	−8.06997	−0.456142	−0.228071	+	0.973645i	$0.573242\pi$
$314$	17.8576	1.00776
$315$	−30.0722	−1.69438
$316$	8.34736	0.469576
$317$	11.9844	0.673111	0.336556	−	0.941664i	$-0.390738\pi$
$317$	11.9844	0.673111	0.336556	+	0.941664i	$0.390738\pi$
$318$	−2.73820	−0.153551
$319$	−6.41855	−0.359370
$320$	4.34017	0.242623
$321$	14.8527	0.828997
$322$	−36.0989	−2.01171
$323$	0	0
$324$	−1.44748	−0.0804156
$325$	50.2411	2.78687
$326$	−0.0266620	−0.00147667
$327$	−1.29565	−0.0716493
$328$	−0.539189	−0.0297717
$329$	−40.7370	−2.24590
$330$	−5.07838	−0.279555
$331$	−17.7926	−0.977968	−0.488984	−	0.872293i	$-0.662632\pi$
$331$	−17.7926	−0.977968	−0.488984	+	0.872293i	$0.662632\pi$
$332$	−4.03612	−0.221511
$333$	−6.19902	−0.339704
$334$	−12.1906	−0.667040
$335$	21.6430	1.18248
$336$	−4.97107	−0.271194
$337$	−2.21461	−0.120638	−0.0603189	−	0.998179i	$-0.519212\pi$
$337$	−2.21461	−0.120638	−0.0603189	+	0.998179i	$0.519212\pi$
$338$	−0.183417	−0.00997660
$339$	7.54146	0.409596
$340$	4.86376	0.263775
$341$	8.04945	0.435902
$342$	0	0
$343$	17.2039	0.928925
$344$	3.87936	0.209161
$345$	−43.1506	−2.32315
$346$	1.70928	0.0918912
$347$	15.5308	0.833736	0.416868	−	0.908967i	$-0.363128\pi$
$347$	15.5308	0.833736	0.416868	+	0.908967i	$0.363128\pi$
$348$	−7.51026	−0.402592
$349$	−7.81044	−0.418083	−0.209042	−	0.977907i	$-0.567034\pi$
$349$	−7.81044	−0.418083	−0.209042	+	0.977907i	$0.567034\pi$
$350$	−58.7864	−3.14227
$351$	19.6742	1.05013
$352$	1.00000	0.0533002
$353$	10.4186	0.554524	0.277262	−	0.960794i	$-0.410573\pi$
$353$	10.4186	0.554524	0.277262	+	0.960794i	$0.410573\pi$
$354$	1.26180	0.0670637
$355$	−7.36069	−0.390665
$356$	9.18342	0.486720
$357$	−5.57077	−0.294836
$358$	1.90110	0.100476
$359$	−12.1256	−0.639963	−0.319981	−	0.947424i	$-0.603677\pi$
$359$	−12.1256	−0.639963	−0.319981	+	0.947424i	$0.603677\pi$
$360$	7.07838	0.373063
$361$	0	0
$362$	5.06278	0.266094
$363$	−1.17009	−0.0614136
$364$	15.4257	0.808528
$365$	−3.38735	−0.177302
$366$	−0.149568	−0.00781802
$367$	31.2713	1.63235	0.816173	−	0.577808i	$-0.196092\pi$
$367$	31.2713	1.63235	0.816173	+	0.577808i	$0.196092\pi$
$368$	8.49693	0.442933
$369$	−0.879362	−0.0457777
$370$	−16.4969	−0.857634
$371$	−9.94214	−0.516170
$372$	9.41855	0.488329
$373$	−1.79380	−0.0928792	−0.0464396	−	0.998921i	$-0.514788\pi$
$373$	−1.79380	−0.0928792	−0.0464396	+	0.998921i	$0.514788\pi$
$374$	1.12064	0.0579468
$375$	−44.8781	−2.31750
$376$	9.58864	0.494496
$377$	23.3051	1.20027
$378$	−23.0205	−1.18405
$379$	−15.7659	−0.809840	−0.404920	−	0.914352i	$-0.632701\pi$
$379$	−15.7659	−0.809840	−0.404920	+	0.914352i	$0.632701\pi$
$380$	0	0
$381$	−21.8264	−1.11820
$382$	12.8504	0.657485
$383$	−12.4969	−0.638563	−0.319282	−	0.947660i	$-0.603442\pi$
$383$	−12.4969	−0.638563	−0.319282	+	0.947660i	$0.603442\pi$
$384$	1.17009	0.0597107
$385$	−18.4391	−0.939742
$386$	25.9265	1.31963
$387$	6.32684	0.321611
$388$	−17.0722	−0.866711
$389$	−20.3390	−1.03123	−0.515613	−	0.856822i	$-0.672436\pi$
$389$	−20.3390	−1.03123	−0.515613	+	0.856822i	$0.672436\pi$
$390$	18.4391	0.933699
$391$	9.52198	0.481547
$392$	−11.0494	−0.558081
$393$	13.2774	0.669756
$394$	4.44521	0.223947
$395$	36.2290	1.82288
$396$	1.63090	0.0819557
$397$	7.06278	0.354471	0.177235	−	0.984168i	$-0.443285\pi$
$397$	7.06278	0.354471	0.177235	+	0.984168i	$0.443285\pi$
$398$	−3.44029	−0.172446
$399$	0	0
$400$	13.8371	0.691855
$401$	30.2062	1.50843	0.754213	−	0.656630i	$-0.228019\pi$
$401$	30.2062	1.50843	0.754213	+	0.656630i	$0.228019\pi$
$402$	5.83483	0.291015
$403$	−29.2267	−1.45589
$404$	8.52586	0.424177
$405$	−6.28231	−0.312171
$406$	−27.2690	−1.35334
$407$	−3.80098	−0.188408
$408$	1.31124	0.0649162
$409$	36.0977	1.78492	0.892458	−	0.451131i	$-0.148979\pi$
$409$	36.0977	1.78492	0.892458	+	0.451131i	$0.148979\pi$
$410$	−2.34017	−0.115573
$411$	−9.79257	−0.483032
$412$	−0.568118	−0.0279892
$413$	4.58145	0.225438
$414$	13.8576	0.681065
$415$	−17.5174	−0.859898
$416$	−3.63090	−0.178019
$417$	13.1461	0.643766
$418$	0	0
$419$	12.8299	0.626782	0.313391	−	0.949624i	$-0.398535\pi$
$419$	12.8299	0.626782	0.313391	+	0.949624i	$0.398535\pi$
$420$	−21.5753	−1.05277
$421$	28.4235	1.38528	0.692638	−	0.721286i	$-0.256449\pi$
$421$	28.4235	1.38528	0.692638	+	0.721286i	$0.256449\pi$
$422$	5.07838	0.247212
$423$	15.6381	0.760350
$424$	2.34017	0.113649
$425$	15.5064	0.752170
$426$	−1.98440	−0.0961446
$427$	−0.543065	−0.0262807
$428$	−12.6937	−0.613572
$429$	4.24846	0.205118
$430$	16.8371	0.811957
$431$	−9.48360	−0.456809	−0.228404	−	0.973566i	$-0.573351\pi$
$431$	−9.48360	−0.456809	−0.228404	+	0.973566i	$0.573351\pi$
$432$	5.41855	0.260700
$433$	−11.7031	−0.562417	−0.281208	−	0.959647i	$-0.590735\pi$
$433$	−11.7031	−0.562417	−0.281208	+	0.959647i	$0.590735\pi$
$434$	34.1978	1.64155
$435$	−32.5958	−1.56285
$436$	1.10731	0.0530304
$437$	0	0
$438$	−0.913212	−0.0436349
$439$	−17.4596	−0.833301	−0.416650	−	0.909067i	$-0.636796\pi$
$439$	−17.4596	−0.833301	−0.416650	+	0.909067i	$0.636796\pi$
$440$	4.34017	0.206910
$441$	−18.0205	−0.858120
$442$	−4.06892	−0.193539
$443$	−22.3740	−1.06302	−0.531511	−	0.847051i	$-0.678376\pi$
$443$	−22.3740	−1.06302	−0.531511	+	0.847051i	$0.678376\pi$
$444$	−4.44748	−0.211068
$445$	39.8576	1.88943
$446$	11.9060	0.563767
$447$	−11.3607	−0.537342
$448$	4.24846	0.200721
$449$	7.06997	0.333652	0.166826	−	0.985986i	$-0.446648\pi$
$449$	7.06997	0.333652	0.166826	+	0.985986i	$0.446648\pi$
$450$	22.5669	1.06381
$451$	−0.539189	−0.0253894
$452$	−6.44521	−0.303157
$453$	−11.3424	−0.532914
$454$	1.31124	0.0615397
$455$	66.9504	3.13868
$456$	0	0
$457$	−7.77432	−0.363667	−0.181834	−	0.983329i	$-0.558203\pi$
$457$	−7.77432	−0.363667	−0.181834	+	0.983329i	$0.558203\pi$
$458$	−23.0361	−1.07641
$459$	6.07223	0.283428
$460$	36.8781	1.71945
$461$	1.02279	0.0476359	0.0238180	−	0.999716i	$-0.492418\pi$
$461$	1.02279	0.0476359	0.0238180	+	0.999716i	$0.492418\pi$
$462$	−4.97107	−0.231275
$463$	10.1122	0.469955	0.234978	−	0.972001i	$-0.424498\pi$
$463$	10.1122	0.469955	0.234978	+	0.972001i	$0.424498\pi$
$464$	6.41855	0.297974
$465$	40.8781	1.89568
$466$	15.3763	0.712293
$467$	−5.41136	−0.250408	−0.125204	−	0.992131i	$-0.539959\pi$
$467$	−5.41136	−0.250408	−0.125204	+	0.992131i	$0.539959\pi$
$468$	−5.92162	−0.273727
$469$	21.1857	0.978264
$470$	41.6163	1.91962
$471$	20.8950	0.962789
$472$	−1.07838	−0.0496364
$473$	3.87936	0.178373
$474$	9.76713	0.448619
$475$	0	0
$476$	4.76099	0.218220
$477$	3.81658	0.174749
$478$	−10.5236	−0.481338
$479$	7.26898	0.332128	0.166064	−	0.986115i	$-0.446894\pi$
$479$	7.26898	0.332128	0.166064	+	0.986115i	$0.446894\pi$
$480$	5.07838	0.231795
$481$	13.8010	0.629271
$482$	−4.08452	−0.186045
$483$	−42.2388	−1.92193
$484$	1.00000	0.0454545
$485$	−74.0965	−3.36455
$486$	14.5620	0.660545
$487$	9.32457	0.422537	0.211268	−	0.977428i	$-0.432241\pi$
$487$	9.32457	0.422537	0.211268	+	0.977428i	$0.432241\pi$
$488$	0.127826	0.00578641
$489$	−0.0311968	−0.00141077
$490$	−47.9565	−2.16645
$491$	−36.5958	−1.65155	−0.825773	−	0.564002i	$-0.809261\pi$
$491$	−36.5958	−1.65155	−0.825773	+	0.564002i	$0.809261\pi$
$492$	−0.630898	−0.0284431
$493$	7.19287	0.323951
$494$	0	0
$495$	7.07838	0.318149
$496$	−8.04945	−0.361431
$497$	−7.20516	−0.323196
$498$	−4.72261	−0.211625
$499$	−34.5380	−1.54613	−0.773066	−	0.634326i	$-0.781278\pi$
$499$	−34.5380	−1.54613	−0.773066	+	0.634326i	$0.781278\pi$
$500$	38.3545	1.71527
$501$	−14.2641	−0.637271
$502$	6.74539	0.301062
$503$	30.7070	1.36916	0.684579	−	0.728939i	$-0.259986\pi$
$503$	30.7070	1.36916	0.684579	+	0.728939i	$0.259986\pi$
$504$	6.92881	0.308634
$505$	37.0037	1.64664
$506$	8.49693	0.377735
$507$	−0.214614	−0.00953136
$508$	18.6537	0.827623
$509$	−32.9770	−1.46168	−0.730841	−	0.682548i	$-0.760872\pi$
$509$	−32.9770	−1.46168	−0.730841	+	0.682548i	$0.760872\pi$
$510$	5.69102	0.252003
$511$	−3.31578	−0.146681
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.1256	−0.887700
$515$	−2.46573	−0.108653
$516$	4.53919	0.199827
$517$	9.58864	0.421708
$518$	−16.1483	−0.709518
$519$	2.00000	0.0877903
$520$	−15.7587	−0.691066
$521$	−7.25953	−0.318046	−0.159023	−	0.987275i	$-0.550834\pi$
$521$	−7.25953	−0.318046	−0.159023	+	0.987275i	$0.550834\pi$
$522$	10.4680	0.458172
$523$	−14.4380	−0.631331	−0.315665	−	0.948871i	$-0.602228\pi$
$523$	−14.4380	−0.631331	−0.315665	+	0.948871i	$0.602228\pi$
$524$	−11.3474	−0.495712
$525$	−68.7852	−3.00203
$526$	−2.48255	−0.108244
$527$	−9.02052	−0.392940
$528$	1.17009	0.0509215
$529$	49.1978	2.13903
$530$	10.1568	0.441181
$531$	−1.75872	−0.0763221
$532$	0	0
$533$	1.95774	0.0847991
$534$	10.7454	0.464999
$535$	−55.0928	−2.38187
$536$	−4.98667	−0.215391
$537$	2.22446	0.0959923
$538$	−30.4235	−1.31165
$539$	−11.0494	−0.475933
$540$	23.5174	1.01203
$541$	10.5197	0.452278	0.226139	−	0.974095i	$-0.427390\pi$
$541$	10.5197	0.452278	0.226139	+	0.974095i	$0.427390\pi$
$542$	19.6020	0.841977
$543$	5.92389	0.254218
$544$	−1.12064	−0.0480470
$545$	4.80590	0.205862
$546$	18.0494	0.772445
$547$	18.0894	0.773449	0.386724	−	0.922195i	$-0.373606\pi$
$547$	18.0894	0.773449	0.386724	+	0.922195i	$0.373606\pi$
$548$	8.36910	0.357510
$549$	0.208471	0.00889734
$550$	13.8371	0.590016
$551$	0	0
$552$	9.94214	0.423166
$553$	35.4635	1.50806
$554$	32.6970	1.38916
$555$	−19.3028	−0.819360
$556$	−11.2351	−0.476476
$557$	22.2001	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$557$	22.2001	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$558$	−13.1278	−0.555745
$559$	−14.0856	−0.595756
$560$	18.4391	0.779193
$561$	1.31124	0.0553607
$562$	31.2339	1.31752
$563$	16.8638	0.710723	0.355361	−	0.934729i	$-0.384358\pi$
$563$	16.8638	0.710723	0.355361	+	0.934729i	$0.384358\pi$
$564$	11.2195	0.472428
$565$	−27.9733	−1.17685
$566$	−1.97334	−0.0829456
$567$	−6.14957	−0.258258
$568$	1.69594	0.0711602
$569$	2.26794	0.0950769	0.0475385	−	0.998869i	$-0.484862\pi$
$569$	2.26794	0.0950769	0.0475385	+	0.998869i	$0.484862\pi$
$570$	0	0
$571$	−38.4401	−1.60867	−0.804334	−	0.594177i	$-0.797478\pi$
$571$	−38.4401	−1.60867	−0.804334	+	0.594177i	$0.797478\pi$
$572$	−3.63090	−0.151815
$573$	15.0361	0.628143
$574$	−2.29072	−0.0956130
$575$	117.573	4.90313
$576$	−1.63090	−0.0679541
$577$	−9.34244	−0.388931	−0.194465	−	0.980909i	$-0.562297\pi$
$577$	−9.34244	−0.388931	−0.194465	+	0.980909i	$0.562297\pi$
$578$	15.7442	0.654871
$579$	30.3363	1.26073
$580$	27.8576	1.15672
$581$	−17.1473	−0.711390
$582$	−19.9760	−0.828031
$583$	2.34017	0.0969201
$584$	0.780465	0.0322959
$585$	−25.7009	−1.06260
$586$	0.261795	0.0108147
$587$	−37.9370	−1.56583	−0.782915	−	0.622129i	$-0.786268\pi$
$587$	−37.9370	−1.56583	−0.782915	+	0.622129i	$0.786268\pi$
$588$	−12.9288	−0.533175
$589$	0	0
$590$	−4.68035	−0.192687
$591$	5.20128	0.213952
$592$	3.80098	0.156219
$593$	39.9832	1.64191	0.820956	−	0.570991i	$-0.193441\pi$
$593$	39.9832	1.64191	0.820956	+	0.570991i	$0.193441\pi$
$594$	5.41855	0.222326
$595$	20.6635	0.847122
$596$	9.70928	0.397707
$597$	−4.02544	−0.164750
$598$	−30.8515	−1.26161
$599$	−26.4173	−1.07938	−0.539691	−	0.841863i	$-0.681459\pi$
$599$	−26.4173	−1.07938	−0.539691	+	0.841863i	$0.681459\pi$
$600$	16.1906	0.660979
$601$	−25.5031	−1.04029	−0.520146	−	0.854077i	$-0.674122\pi$
$601$	−25.5031	−1.04029	−0.520146	+	0.854077i	$0.674122\pi$
$602$	16.4813	0.671729
$603$	−8.13275	−0.331191
$604$	9.69368	0.394430
$605$	4.34017	0.176453
$606$	9.97599	0.405247
$607$	−36.6342	−1.48694	−0.743468	−	0.668771i	$-0.766821\pi$
$607$	−36.6342	−1.48694	−0.743468	+	0.668771i	$0.766821\pi$
$608$	0	0
$609$	−31.9071	−1.29294
$610$	0.554787	0.0224627
$611$	−34.8154	−1.40848
$612$	−1.82765	−0.0738782
$613$	−12.2123	−0.493252	−0.246626	−	0.969111i	$-0.579322\pi$
$613$	−12.2123	−0.493252	−0.246626	+	0.969111i	$0.579322\pi$
$614$	1.24846	0.0503839
$615$	−2.73820	−0.110415
$616$	4.24846	0.171176
$617$	−18.1422	−0.730378	−0.365189	−	0.930933i	$-0.618996\pi$
$617$	−18.1422	−0.730378	−0.365189	+	0.930933i	$0.618996\pi$
$618$	−0.664748	−0.0267401
$619$	17.2351	0.692738	0.346369	−	0.938098i	$-0.387414\pi$
$619$	17.2351	0.692738	0.346369	+	0.938098i	$0.387414\pi$
$620$	−34.9360	−1.40306
$621$	46.0410	1.84756
$622$	2.03999	0.0817963
$623$	39.0154	1.56312
$624$	−4.24846	−0.170075
$625$	97.2799	3.89119
$626$	8.06997	0.322541
$627$	0	0
$628$	−17.8576	−0.712597
$629$	4.25953	0.169838
$630$	30.0722	1.19811
$631$	26.1883	1.04254	0.521271	−	0.853391i	$-0.325458\pi$
$631$	26.1883	1.04254	0.521271	+	0.853391i	$0.325458\pi$
$632$	−8.34736	−0.332040
$633$	5.94214	0.236179
$634$	−11.9844	−0.475961
$635$	80.9602	3.21281
$636$	2.73820	0.108577
$637$	40.1194	1.58959
$638$	6.41855	0.254113
$639$	2.76591	0.109418
$640$	−4.34017	−0.171560
$641$	38.1338	1.50619	0.753097	−	0.657909i	$-0.228559\pi$
$641$	38.1338	1.50619	0.753097	+	0.657909i	$0.228559\pi$
$642$	−14.8527	−0.586189
$643$	32.4052	1.27794	0.638969	−	0.769233i	$-0.279361\pi$
$643$	32.4052	1.27794	0.638969	+	0.769233i	$0.279361\pi$
$644$	36.0989	1.42250
$645$	19.7009	0.775721
$646$	0	0
$647$	−3.79484	−0.149191	−0.0745953	−	0.997214i	$-0.523766\pi$
$647$	−3.79484	−0.149191	−0.0745953	+	0.997214i	$0.523766\pi$
$648$	1.44748	0.0568624
$649$	−1.07838	−0.0423300
$650$	−50.2411	−1.97062
$651$	40.0144	1.56829
$652$	0.0266620	0.00104416
$653$	1.38858	0.0543392	0.0271696	−	0.999631i	$-0.491351\pi$
$653$	1.38858	0.0543392	0.0271696	+	0.999631i	$0.491351\pi$
$654$	1.29565	0.0506637
$655$	−49.2495	−1.92434
$656$	0.539189	0.0210518
$657$	1.27286	0.0496590
$658$	40.7370	1.58809
$659$	−29.0650	−1.13221	−0.566107	−	0.824332i	$-0.691551\pi$
$659$	−29.0650	−1.13221	−0.566107	+	0.824332i	$0.691551\pi$
$660$	5.07838	0.197676
$661$	27.9988	1.08903	0.544513	−	0.838752i	$-0.316714\pi$
$661$	27.9988	1.08903	0.544513	+	0.838752i	$0.316714\pi$
$662$	17.7926	0.691528
$663$	−4.76099	−0.184902
$664$	4.03612	0.156632
$665$	0	0
$666$	6.19902	0.240207
$667$	54.5380	2.11172
$668$	12.1906	0.471669
$669$	13.9311	0.538607
$670$	−21.6430	−0.836142
$671$	0.127826	0.00493467
$672$	4.97107	0.191763
$673$	31.7887	1.22536	0.612682	−	0.790329i	$-0.290091\pi$
$673$	31.7887	1.22536	0.612682	+	0.790329i	$0.290091\pi$
$674$	2.21461	0.0853038
$675$	74.9770	2.88587
$676$	0.183417	0.00705452
$677$	27.8143	1.06899	0.534495	−	0.845171i	$-0.320502\pi$
$677$	27.8143	1.06899	0.534495	+	0.845171i	$0.320502\pi$
$678$	−7.54146	−0.289628
$679$	−72.5308	−2.78348
$680$	−4.86376	−0.186517
$681$	1.53427	0.0587933
$682$	−8.04945	−0.308229
$683$	−28.3474	−1.08468	−0.542341	−	0.840159i	$-0.682462\pi$
$683$	−28.3474	−1.08468	−0.542341	+	0.840159i	$0.682462\pi$
$684$	0	0
$685$	36.3234	1.38784
$686$	−17.2039	−0.656849
$687$	−26.9542	−1.02837
$688$	−3.87936	−0.147899
$689$	−8.49693	−0.323707
$690$	43.1506	1.64272
$691$	29.9493	1.13933	0.569663	−	0.821878i	$-0.307074\pi$
$691$	29.9493	1.13933	0.569663	+	0.821878i	$0.307074\pi$
$692$	−1.70928	−0.0649769
$693$	6.92881	0.263204
$694$	−15.5308	−0.589540
$695$	−48.7624	−1.84966
$696$	7.51026	0.284676
$697$	0.604236	0.0228871
$698$	7.81044	0.295629
$699$	17.9916	0.680504
$700$	58.7864	2.22192
$701$	38.7009	1.46171	0.730856	−	0.682532i	$-0.239121\pi$
$701$	38.7009	1.46171	0.730856	+	0.682532i	$0.239121\pi$
$702$	−19.6742	−0.742555
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	48.6947	1.83395
$706$	−10.4186	−0.392107
$707$	36.2218	1.36226
$708$	−1.26180	−0.0474212
$709$	−1.63317	−0.0613348	−0.0306674	−	0.999530i	$-0.509763\pi$
$709$	−1.63317	−0.0613348	−0.0306674	+	0.999530i	$0.509763\pi$
$710$	7.36069	0.276242
$711$	−13.6137	−0.510553
$712$	−9.18342	−0.344163
$713$	−68.3956	−2.56143
$714$	5.57077	0.208481
$715$	−15.7587	−0.589343
$716$	−1.90110	−0.0710476
$717$	−12.3135	−0.459856
$718$	12.1256	0.452522
$719$	7.32457	0.273161	0.136580	−	0.990629i	$-0.456389\pi$
$719$	7.32457	0.273161	0.136580	+	0.990629i	$0.456389\pi$
$720$	−7.07838	−0.263796
$721$	−2.41363	−0.0898883
$722$	0	0
$723$	−4.77924	−0.177742
$724$	−5.06278	−0.188157
$725$	88.8141	3.29847
$726$	1.17009	0.0434260
$727$	−33.4957	−1.24229	−0.621143	−	0.783697i	$-0.713332\pi$
$727$	−33.4957	−1.24229	−0.621143	+	0.783697i	$0.713332\pi$
$728$	−15.4257	−0.571716
$729$	21.3812	0.791897
$730$	3.38735	0.125372
$731$	−4.34736	−0.160793
$732$	0.149568	0.00552818
$733$	−12.0351	−0.444526	−0.222263	−	0.974987i	$-0.571344\pi$
$733$	−12.0351	−0.444526	−0.222263	+	0.974987i	$0.571344\pi$
$734$	−31.2713	−1.15424
$735$	−56.1133	−2.06977
$736$	−8.49693	−0.313201
$737$	−4.98667	−0.183686
$738$	0.879362	0.0323698
$739$	−1.11837	−0.0411399	−0.0205700	−	0.999788i	$-0.506548\pi$
$739$	−1.11837	−0.0411399	−0.0205700	+	0.999788i	$0.506548\pi$
$740$	16.4969	0.606439
$741$	0	0
$742$	9.94214	0.364987
$743$	35.4934	1.30213	0.651064	−	0.759023i	$-0.274323\pi$
$743$	35.4934	1.30213	0.651064	+	0.759023i	$0.274323\pi$
$744$	−9.41855	−0.345301
$745$	42.1399	1.54389
$746$	1.79380	0.0656755
$747$	6.58249	0.240841
$748$	−1.12064	−0.0409746
$749$	−53.9286	−1.97051
$750$	44.8781	1.63872
$751$	−27.0833	−0.988284	−0.494142	−	0.869381i	$-0.664518\pi$
$751$	−27.0833	−0.988284	−0.494142	+	0.869381i	$0.664518\pi$
$752$	−9.58864	−0.349662
$753$	7.89269	0.287626
$754$	−23.3051	−0.848722
$755$	42.0722	1.53117
$756$	23.0205	0.837248
$757$	−22.4124	−0.814593	−0.407296	−	0.913296i	$-0.633528\pi$
$757$	−22.4124	−0.814593	−0.407296	+	0.913296i	$0.633528\pi$
$758$	15.7659	0.572644
$759$	9.94214	0.360877
$760$	0	0
$761$	4.55479	0.165111	0.0825555	−	0.996586i	$-0.473692\pi$
$761$	4.55479	0.165111	0.0825555	+	0.996586i	$0.473692\pi$
$762$	21.8264	0.790688
$763$	4.70435	0.170309
$764$	−12.8504	−0.464912
$765$	−7.93230	−0.286793
$766$	12.4969	0.451532
$767$	3.91548	0.141380
$768$	−1.17009	−0.0422219
$769$	−36.1133	−1.30228	−0.651139	−	0.758959i	$-0.725708\pi$
$769$	−36.1133	−1.30228	−0.651139	+	0.758959i	$0.725708\pi$
$770$	18.4391	0.664498
$771$	−23.5486	−0.848084
$772$	−25.9265	−0.933117
$773$	−10.6914	−0.384543	−0.192272	−	0.981342i	$-0.561585\pi$
$773$	−10.6914	−0.384543	−0.192272	+	0.981342i	$0.561585\pi$
$774$	−6.32684	−0.227414
$775$	−111.381	−4.00092
$776$	17.0722	0.612858
$777$	−18.8950	−0.677853
$778$	20.3390	0.729187
$779$	0	0
$780$	−18.4391	−0.660225
$781$	1.69594	0.0606857
$782$	−9.52198	−0.340505
$783$	34.7792	1.24291
$784$	11.0494	0.394623
$785$	−77.5052	−2.76628
$786$	−13.2774	−0.473589
$787$	35.2544	1.25668	0.628342	−	0.777937i	$-0.283734\pi$
$787$	35.2544	1.25668	0.628342	+	0.777937i	$0.283734\pi$
$788$	−4.44521	−0.158354
$789$	−2.90480	−0.103414
$790$	−36.2290	−1.28897
$791$	−27.3823	−0.973601
$792$	−1.63090	−0.0579514
$793$	−0.464123	−0.0164815
$794$	−7.06278	−0.250649
$795$	11.8843	0.421492
$796$	3.44029	0.121938
$797$	1.46081	0.0517446	0.0258723	−	0.999665i	$-0.491764\pi$
$797$	1.46081	0.0517446	0.0258723	+	0.999665i	$0.491764\pi$
$798$	0	0
$799$	−10.7454	−0.380145
$800$	−13.8371	−0.489215
$801$	−14.9772	−0.529194
$802$	−30.2062	−1.06662
$803$	0.780465	0.0275420
$804$	−5.83483	−0.205779
$805$	156.675	5.52208
$806$	29.2267	1.02947
$807$	−35.5981	−1.25311
$808$	−8.52586	−0.299939
$809$	−3.88550	−0.136607	−0.0683035	−	0.997665i	$-0.521759\pi$
$809$	−3.88550	−0.136607	−0.0683035	+	0.997665i	$0.521759\pi$
$810$	6.28231	0.220738
$811$	−24.7698	−0.869785	−0.434892	−	0.900482i	$-0.643214\pi$
$811$	−24.7698	−0.869785	−0.434892	+	0.900482i	$0.643214\pi$
$812$	27.2690	0.956954
$813$	22.9360	0.804401
$814$	3.80098	0.133224
$815$	0.115718	0.00405341
$816$	−1.31124	−0.0459027
$817$	0	0
$818$	−36.0977	−1.26213
$819$	−25.1578	−0.879085
$820$	2.34017	0.0817224
$821$	11.4368	0.399147	0.199574	−	0.979883i	$-0.436044\pi$
$821$	11.4368	0.399147	0.199574	+	0.979883i	$0.436044\pi$
$822$	9.79257	0.341555
$823$	23.2628	0.810892	0.405446	−	0.914119i	$-0.367116\pi$
$823$	23.2628	0.810892	0.405446	+	0.914119i	$0.367116\pi$
$824$	0.568118	0.0197913
$825$	16.1906	0.563685
$826$	−4.58145	−0.159409
$827$	19.9471	0.693627	0.346814	−	0.937934i	$-0.387264\pi$
$827$	19.9471	0.693627	0.346814	+	0.937934i	$0.387264\pi$
$828$	−13.8576	−0.481586
$829$	−8.93722	−0.310403	−0.155201	−	0.987883i	$-0.549603\pi$
$829$	−8.93722	−0.310403	−0.155201	+	0.987883i	$0.549603\pi$
$830$	17.5174	0.608039
$831$	38.2583	1.32717
$832$	3.63090	0.125879
$833$	12.3824	0.429026
$834$	−13.1461	−0.455211
$835$	52.9093	1.83100
$836$	0	0
$837$	−43.6163	−1.50760
$838$	−12.8299	−0.443202
$839$	−2.87322	−0.0991945	−0.0495973	−	0.998769i	$-0.515794\pi$
$839$	−2.87322	−0.0991945	−0.0495973	+	0.998769i	$0.515794\pi$
$840$	21.5753	0.744419
$841$	12.1978	0.420614
$842$	−28.4235	−0.979538
$843$	36.5464	1.25872
$844$	−5.07838	−0.174805
$845$	0.796064	0.0273854
$846$	−15.6381	−0.537648
$847$	4.24846	0.145979
$848$	−2.34017	−0.0803619
$849$	−2.30898	−0.0792439
$850$	−15.5064	−0.531865
$851$	32.2967	1.10712
$852$	1.98440	0.0679845
$853$	16.4619	0.563643	0.281822	−	0.959467i	$-0.409061\pi$
$853$	16.4619	0.563643	0.281822	+	0.959467i	$0.409061\pi$
$854$	0.543065	0.0185833
$855$	0	0
$856$	12.6937	0.433861
$857$	−38.9660	−1.33105	−0.665526	−	0.746375i	$-0.731793\pi$
$857$	−38.9660	−1.33105	−0.665526	+	0.746375i	$0.731793\pi$
$858$	−4.24846	−0.145040
$859$	32.3402	1.10343	0.551716	−	0.834032i	$-0.313973\pi$
$859$	32.3402	1.10343	0.551716	+	0.834032i	$0.313973\pi$
$860$	−16.8371	−0.574140
$861$	−2.68035	−0.0913459
$862$	9.48360	0.323013
$863$	−34.4980	−1.17432	−0.587162	−	0.809469i	$-0.699755\pi$
$863$	−34.4980	−1.17432	−0.587162	+	0.809469i	$0.699755\pi$
$864$	−5.41855	−0.184343
$865$	−7.41855	−0.252238
$866$	11.7031	0.397689
$867$	18.4220	0.625645
$868$	−34.1978	−1.16075
$869$	−8.34736	−0.283165
$870$	32.5958	1.10510
$871$	18.1061	0.613501
$872$	−1.10731	−0.0374982
$873$	27.8431	0.942345
$874$	0	0
$875$	162.948	5.50865
$876$	0.913212	0.0308546
$877$	9.68649	0.327089	0.163545	−	0.986536i	$-0.447707\pi$
$877$	9.68649	0.327089	0.163545	+	0.986536i	$0.447707\pi$
$878$	17.4596	0.589233
$879$	0.306323	0.0103320
$880$	−4.34017	−0.146307
$881$	15.1713	0.511134	0.255567	−	0.966791i	$-0.417738\pi$
$881$	15.1713	0.511134	0.255567	+	0.966791i	$0.417738\pi$
$882$	18.0205	0.606782
$883$	21.6814	0.729637	0.364818	−	0.931079i	$-0.381131\pi$
$883$	21.6814	0.729637	0.364818	+	0.931079i	$0.381131\pi$
$884$	4.06892	0.136853
$885$	−5.47641	−0.184087
$886$	22.3740	0.751670
$887$	−23.4524	−0.787455	−0.393727	−	0.919227i	$-0.628815\pi$
$887$	−23.4524	−0.787455	−0.393727	+	0.919227i	$0.628815\pi$
$888$	4.44748	0.149248
$889$	79.2495	2.65794
$890$	−39.8576	−1.33603
$891$	1.44748	0.0484924
$892$	−11.9060	−0.398643
$893$	0	0
$894$	11.3607	0.379958
$895$	−8.25112	−0.275804
$896$	−4.24846	−0.141931
$897$	−36.0989	−1.20531
$898$	−7.06997	−0.235928
$899$	−51.6658	−1.72315
$900$	−22.5669	−0.752230
$901$	−2.62249	−0.0873677
$902$	0.539189	0.0179530
$903$	19.2846	0.641751
$904$	6.44521	0.214365
$905$	−21.9733	−0.730418
$906$	11.3424	0.376827
$907$	−30.3356	−1.00728	−0.503639	−	0.863914i	$-0.668006\pi$
$907$	−30.3356	−1.00728	−0.503639	+	0.863914i	$0.668006\pi$
$908$	−1.31124	−0.0435151
$909$	−13.9048	−0.461193
$910$	−66.9504	−2.21938
$911$	52.4657	1.73827	0.869134	−	0.494577i	$-0.164677\pi$
$911$	52.4657	1.73827	0.869134	+	0.494577i	$0.164677\pi$
$912$	0	0
$913$	4.03612	0.133576
$914$	7.77432	0.257152
$915$	0.649149	0.0214602
$916$	23.0361	0.761135
$917$	−48.2089	−1.59200
$918$	−6.07223	−0.200414
$919$	39.5825	1.30571	0.652853	−	0.757485i	$-0.273572\pi$
$919$	39.5825	1.30571	0.652853	+	0.757485i	$0.273572\pi$
$920$	−36.8781	−1.21584
$921$	1.46081	0.0481354
$922$	−1.02279	−0.0336837
$923$	−6.15780	−0.202686
$924$	4.97107	0.163536
$925$	52.5946	1.72930
$926$	−10.1122	−0.332308
$927$	0.926543	0.0304317
$928$	−6.41855	−0.210699
$929$	29.4534	0.966336	0.483168	−	0.875528i	$-0.339486\pi$
$929$	29.4534	0.966336	0.483168	+	0.875528i	$0.339486\pi$
$930$	−40.8781	−1.34045
$931$	0	0
$932$	−15.3763	−0.503667
$933$	2.38697	0.0781458
$934$	5.41136	0.177065
$935$	−4.86376	−0.159062
$936$	5.92162	0.193554
$937$	21.3874	0.698694	0.349347	−	0.936993i	$-0.386403\pi$
$937$	21.3874	0.698694	0.349347	+	0.936993i	$0.386403\pi$
$938$	−21.1857	−0.691737
$939$	9.44256	0.308146
$940$	−41.6163	−1.35738
$941$	47.8492	1.55984	0.779920	−	0.625879i	$-0.215260\pi$
$941$	47.8492	1.55984	0.779920	+	0.625879i	$0.215260\pi$
$942$	−20.8950	−0.680795
$943$	4.58145	0.149193
$944$	1.07838	0.0350982
$945$	99.9130	3.25017
$946$	−3.87936	−0.126129
$947$	38.2290	1.24228	0.621138	−	0.783702i	$-0.286671\pi$
$947$	38.2290	1.24228	0.621138	+	0.783702i	$0.286671\pi$
$948$	−9.76713	−0.317222
$949$	−2.83379	−0.0919887
$950$	0	0
$951$	−14.0228	−0.454720
$952$	−4.76099	−0.154305
$953$	33.5318	1.08620	0.543101	−	0.839667i	$-0.317250\pi$
$953$	33.5318	1.08620	0.543101	+	0.839667i	$0.317250\pi$
$954$	−3.81658	−0.123566
$955$	−55.7731	−1.80478
$956$	10.5236	0.340357
$957$	7.51026	0.242772
$958$	−7.26898	−0.234850
$959$	35.5558	1.14816
$960$	−5.07838	−0.163904
$961$	33.7936	1.09012
$962$	−13.8010	−0.444962
$963$	20.7021	0.667115
$964$	4.08452	0.131554
$965$	−112.526	−3.62233
$966$	42.2388	1.35901
$967$	−33.5708	−1.07956	−0.539782	−	0.841805i	$-0.681493\pi$
$967$	−33.5708	−1.07956	−0.539782	+	0.841805i	$0.681493\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	74.0965	2.37909
$971$	27.3991	0.879278	0.439639	−	0.898175i	$-0.355106\pi$
$971$	27.3991	0.879278	0.439639	+	0.898175i	$0.355106\pi$
$972$	−14.5620	−0.467076
$973$	−47.7321	−1.53022
$974$	−9.32457	−0.298779
$975$	−58.7864	−1.88267
$976$	−0.127826	−0.00409161
$977$	58.8576	1.88302	0.941511	−	0.336982i	$-0.109406\pi$
$977$	58.8576	1.88302	0.941511	+	0.336982i	$0.109406\pi$
$978$	0.0311968	0.000997565 0
$979$	−9.18342	−0.293503
$980$	47.9565	1.53191
$981$	−1.80590	−0.0576581
$982$	36.5958	1.16782
$983$	12.4440	0.396902	0.198451	−	0.980111i	$-0.436409\pi$
$983$	12.4440	0.396902	0.198451	+	0.980111i	$0.436409\pi$
$984$	0.630898	0.0201123
$985$	−19.2930	−0.614726
$986$	−7.19287	−0.229068
$987$	47.6658	1.51722
$988$	0	0
$989$	−32.9627	−1.04815
$990$	−7.07838	−0.224966
$991$	2.69821	0.0857115	0.0428558	−	0.999081i	$-0.486354\pi$
$991$	2.69821	0.0857115	0.0428558	+	0.999081i	$0.486354\pi$
$992$	8.04945	0.255570
$993$	20.8188	0.660666
$994$	7.20516	0.228534
$995$	14.9315	0.473359
$996$	4.72261	0.149642
$997$	−49.4017	−1.56457	−0.782284	−	0.622922i	$-0.785946\pi$
$997$	−49.4017	−1.56457	−0.782284	+	0.622922i	$0.785946\pi$
$998$	34.5380	1.09328
$999$	20.5958	0.651623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.be.1.1		3
19.8	odd	6		418.2.e.h.45.1	✓	6
19.12	odd	6		418.2.e.h.353.1	yes	6
19.18	odd	2		7942.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.e.h.45.1	✓	6	19.8	odd	6
418.2.e.h.353.1	yes	6	19.12	odd	6
7942.2.a.be.1.1		3	1.1	even	1	trivial
7942.2.a.bh.1.3		3	19.18	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.17009 q^{3} +1.00000 q^{4} +4.34017 q^{5} +1.17009 q^{6} +4.24846 q^{7} -1.00000 q^{8} -1.63090 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.17009 q^{3} +1.00000 q^{4} +4.34017 q^{5} +1.17009 q^{6} +4.24846 q^{7} -1.00000 q^{8} -1.63090 q^{9} -4.34017 q^{10} -1.00000 q^{11} -1.17009 q^{12} +3.63090 q^{13} -4.24846 q^{14} -5.07838 q^{15} +1.00000 q^{16} +1.12064 q^{17} +1.63090 q^{18} +4.34017 q^{20} -4.97107 q^{21} +1.00000 q^{22} +8.49693 q^{23} +1.17009 q^{24} +13.8371 q^{25} -3.63090 q^{26} +5.41855 q^{27} +4.24846 q^{28} +6.41855 q^{29} +5.07838 q^{30} -8.04945 q^{31} -1.00000 q^{32} +1.17009 q^{33} -1.12064 q^{34} +18.4391 q^{35} -1.63090 q^{36} +3.80098 q^{37} -4.24846 q^{39} -4.34017 q^{40} +0.539189 q^{41} +4.97107 q^{42} -3.87936 q^{43} -1.00000 q^{44} -7.07838 q^{45} -8.49693 q^{46} -9.58864 q^{47} -1.17009 q^{48} +11.0494 q^{49} -13.8371 q^{50} -1.31124 q^{51} +3.63090 q^{52} -2.34017 q^{53} -5.41855 q^{54} -4.34017 q^{55} -4.24846 q^{56} -6.41855 q^{58} +1.07838 q^{59} -5.07838 q^{60} -0.127826 q^{61} +8.04945 q^{62} -6.92881 q^{63} +1.00000 q^{64} +15.7587 q^{65} -1.17009 q^{66} +4.98667 q^{67} +1.12064 q^{68} -9.94214 q^{69} -18.4391 q^{70} -1.69594 q^{71} +1.63090 q^{72} -0.780465 q^{73} -3.80098 q^{74} -16.1906 q^{75} -4.24846 q^{77} +4.24846 q^{78} +8.34736 q^{79} +4.34017 q^{80} -1.44748 q^{81} -0.539189 q^{82} -4.03612 q^{83} -4.97107 q^{84} +4.86376 q^{85} +3.87936 q^{86} -7.51026 q^{87} +1.00000 q^{88} +9.18342 q^{89} +7.07838 q^{90} +15.4257 q^{91} +8.49693 q^{92} +9.41855 q^{93} +9.58864 q^{94} +1.17009 q^{96} -17.0722 q^{97} -11.0494 q^{98} +1.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 - 3 * q^8 - q^9	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 3 q^{8} - q^{9} - 2 q^{10} - 3 q^{11} + 2 q^{12} + 7 q^{13} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 16 q^{17} + q^{18} + 2 q^{20} + 3 q^{22} + 8 q^{23} - 2 q^{24} + 13 q^{25} - 7 q^{26} + 2 q^{27} + 4 q^{28} + 5 q^{29} + 12 q^{30} - 6 q^{31} - 3 q^{32} - 2 q^{33} - 16 q^{34} + 8 q^{35} - q^{36} + 2 q^{37} - 4 q^{39} - 2 q^{40} + q^{43} - 3 q^{44} - 18 q^{45} - 8 q^{46} - 9 q^{47} + 2 q^{48} + 15 q^{49} - 13 q^{50} + 22 q^{51} + 7 q^{52} + 4 q^{53} - 2 q^{54} - 2 q^{55} - 4 q^{56} - 5 q^{58} - 12 q^{60} + 21 q^{61} + 6 q^{62} + 10 q^{63} + 3 q^{64} + 22 q^{65} + 2 q^{66} + 14 q^{67} + 16 q^{68} - 8 q^{70} + 3 q^{71} + q^{72} - 26 q^{73} - 2 q^{74} - 10 q^{75} - 4 q^{77} + 4 q^{78} - 20 q^{79} + 2 q^{80} - 5 q^{81} + 7 q^{83} - 12 q^{85} - q^{86} - 6 q^{87} + 3 q^{88} + 23 q^{89} + 18 q^{90} - 2 q^{91} + 8 q^{92} + 14 q^{93} + 9 q^{94} - 2 q^{96} - 13 q^{97} - 15 q^{98} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 - 3 * q^8 - q^9 - 2 * q^10 - 3 * q^11 + 2 * q^12 + 7 * q^13 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 16 * q^17 + q^18 + 2 * q^20 + 3 * q^22 + 8 * q^23 - 2 * q^24 + 13 * q^25 - 7 * q^26 + 2 * q^27 + 4 * q^28 + 5 * q^29 + 12 * q^30 - 6 * q^31 - 3 * q^32 - 2 * q^33 - 16 * q^34 + 8 * q^35 - q^36 + 2 * q^37 - 4 * q^39 - 2 * q^40 + q^43 - 3 * q^44 - 18 * q^45 - 8 * q^46 - 9 * q^47 + 2 * q^48 + 15 * q^49 - 13 * q^50 + 22 * q^51 + 7 * q^52 + 4 * q^53 - 2 * q^54 - 2 * q^55 - 4 * q^56 - 5 * q^58 - 12 * q^60 + 21 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 + 22 * q^65 + 2 * q^66 + 14 * q^67 + 16 * q^68 - 8 * q^70 + 3 * q^71 + q^72 - 26 * q^73 - 2 * q^74 - 10 * q^75 - 4 * q^77 + 4 * q^78 - 20 * q^79 + 2 * q^80 - 5 * q^81 + 7 * q^83 - 12 * q^85 - q^86 - 6 * q^87 + 3 * q^88 + 23 * q^89 + 18 * q^90 - 2 * q^91 + 8 * q^92 + 14 * q^93 + 9 * q^94 - 2 * q^96 - 13 * q^97 - 15 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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