LMFDB - Embedded newform 8002.2.a.e.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8002,2,Mod(1,8002)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8002.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8002, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment:select newform

sage:traces = [77,-77,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8002.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.69277 q^{3} +1.00000 q^{4} +0.930791 q^{5} +2.69277 q^{6} -2.75733 q^{7} -1.00000 q^{8} +4.25101 q^{9} -0.930791 q^{10} +1.87937 q^{11} -2.69277 q^{12} -1.88739 q^{13} +2.75733 q^{14} -2.50640 q^{15} +1.00000 q^{16} -3.64409 q^{17} -4.25101 q^{18} -3.78639 q^{19} +0.930791 q^{20} +7.42487 q^{21} -1.87937 q^{22} -1.94353 q^{23} +2.69277 q^{24} -4.13363 q^{25} +1.88739 q^{26} -3.36868 q^{27} -2.75733 q^{28} -10.1187 q^{29} +2.50640 q^{30} -9.02661 q^{31} -1.00000 q^{32} -5.06072 q^{33} +3.64409 q^{34} -2.56650 q^{35} +4.25101 q^{36} -0.0607967 q^{37} +3.78639 q^{38} +5.08231 q^{39} -0.930791 q^{40} -6.84750 q^{41} -7.42487 q^{42} +9.48945 q^{43} +1.87937 q^{44} +3.95680 q^{45} +1.94353 q^{46} +8.53876 q^{47} -2.69277 q^{48} +0.602891 q^{49} +4.13363 q^{50} +9.81271 q^{51} -1.88739 q^{52} -1.06001 q^{53} +3.36868 q^{54} +1.74930 q^{55} +2.75733 q^{56} +10.1959 q^{57} +10.1187 q^{58} -5.02007 q^{59} -2.50640 q^{60} -7.43945 q^{61} +9.02661 q^{62} -11.7215 q^{63} +1.00000 q^{64} -1.75677 q^{65} +5.06072 q^{66} +6.67092 q^{67} -3.64409 q^{68} +5.23348 q^{69} +2.56650 q^{70} -11.3277 q^{71} -4.25101 q^{72} +1.49471 q^{73} +0.0607967 q^{74} +11.1309 q^{75} -3.78639 q^{76} -5.18206 q^{77} -5.08231 q^{78} -5.20048 q^{79} +0.930791 q^{80} -3.68195 q^{81} +6.84750 q^{82} -1.01903 q^{83} +7.42487 q^{84} -3.39189 q^{85} -9.48945 q^{86} +27.2473 q^{87} -1.87937 q^{88} -6.26525 q^{89} -3.95680 q^{90} +5.20417 q^{91} -1.94353 q^{92} +24.3066 q^{93} -8.53876 q^{94} -3.52434 q^{95} +2.69277 q^{96} -3.11516 q^{97} -0.602891 q^{98} +7.98923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19}+ \cdots + 63 q^{99}+O(q^{100}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.69277	−1.55467	−0.777336	−	0.629086i	$-0.783429\pi$
$3$	−2.69277	−1.55467	−0.777336	+	0.629086i	$0.783429\pi$
$4$	1.00000	0.500000
$5$	0.930791	0.416262	0.208131	−	0.978101i	$-0.433262\pi$
$5$	0.930791	0.416262	0.208131	+	0.978101i	$0.433262\pi$
$6$	2.69277	1.09932
$7$	−2.75733	−1.04217	−0.521087	−	0.853503i	$-0.674473\pi$
$7$	−2.75733	−1.04217	−0.521087	+	0.853503i	$0.674473\pi$
$8$	−1.00000	−0.353553
$9$	4.25101	1.41700
$10$	−0.930791	−0.294342
$11$	1.87937	0.566652	0.283326	−	0.959024i	$-0.408562\pi$
$11$	1.87937	0.566652	0.283326	+	0.959024i	$0.408562\pi$
$12$	−2.69277	−0.777336
$13$	−1.88739	−0.523468	−0.261734	−	0.965140i	$-0.584294\pi$
$13$	−1.88739	−0.523468	−0.261734	+	0.965140i	$0.584294\pi$
$14$	2.75733	0.736929
$15$	−2.50640	−0.647151
$16$	1.00000	0.250000
$17$	−3.64409	−0.883823	−0.441911	−	0.897059i	$-0.645699\pi$
$17$	−3.64409	−0.883823	−0.441911	+	0.897059i	$0.645699\pi$
$18$	−4.25101	−1.00197
$19$	−3.78639	−0.868657	−0.434329	−	0.900754i	$-0.643014\pi$
$19$	−3.78639	−0.868657	−0.434329	+	0.900754i	$0.643014\pi$
$20$	0.930791	0.208131
$21$	7.42487	1.62024
$22$	−1.87937	−0.400684
$23$	−1.94353	−0.405254	−0.202627	−	0.979256i	$-0.564948\pi$
$23$	−1.94353	−0.405254	−0.202627	+	0.979256i	$0.564948\pi$
$24$	2.69277	0.549659
$25$	−4.13363	−0.826726
$26$	1.88739	0.370148
$27$	−3.36868	−0.648303
$28$	−2.75733	−0.521087
$29$	−10.1187	−1.87899	−0.939496	−	0.342561i	$-0.888706\pi$
$29$	−10.1187	−1.87899	−0.939496	+	0.342561i	$0.888706\pi$
$30$	2.50640	0.457605
$31$	−9.02661	−1.62123	−0.810614	−	0.585581i	$-0.800866\pi$
$31$	−9.02661	−1.62123	−0.810614	+	0.585581i	$0.800866\pi$
$32$	−1.00000	−0.176777
$33$	−5.06072	−0.880958
$34$	3.64409	0.624957
$35$	−2.56650	−0.433818
$36$	4.25101	0.708502
$37$	−0.0607967	−0.00999492	−0.00499746	−	0.999988i	$-0.501591\pi$
$37$	−0.0607967	−0.00999492	−0.00499746	+	0.999988i	$0.501591\pi$
$38$	3.78639	0.614233
$39$	5.08231	0.813821
$40$	−0.930791	−0.147171
$41$	−6.84750	−1.06940	−0.534700	−	0.845042i	$-0.679575\pi$
$41$	−6.84750	−1.06940	−0.534700	+	0.845042i	$0.679575\pi$
$42$	−7.42487	−1.14568
$43$	9.48945	1.44713	0.723564	−	0.690257i	$-0.242503\pi$
$43$	9.48945	1.44713	0.723564	+	0.690257i	$0.242503\pi$
$44$	1.87937	0.283326
$45$	3.95680	0.589845
$46$	1.94353	0.286558
$47$	8.53876	1.24551	0.622753	−	0.782419i	$-0.286014\pi$
$47$	8.53876	1.24551	0.622753	+	0.782419i	$0.286014\pi$
$48$	−2.69277	−0.388668
$49$	0.602891	0.0861272
$50$	4.13363	0.584583
$51$	9.81271	1.37405
$52$	−1.88739	−0.261734
$53$	−1.06001	−0.145603	−0.0728015	−	0.997346i	$-0.523194\pi$
$53$	−1.06001	−0.145603	−0.0728015	+	0.997346i	$0.523194\pi$
$54$	3.36868	0.458419
$55$	1.74930	0.235876
$56$	2.75733	0.368464
$57$	10.1959	1.35048
$58$	10.1187	1.32865
$59$	−5.02007	−0.653557	−0.326779	−	0.945101i	$-0.605963\pi$
$59$	−5.02007	−0.653557	−0.326779	+	0.945101i	$0.605963\pi$
$60$	−2.50640	−0.323575
$61$	−7.43945	−0.952524	−0.476262	−	0.879303i	$-0.658009\pi$
$61$	−7.43945	−0.952524	−0.476262	+	0.879303i	$0.658009\pi$
$62$	9.02661	1.14638
$63$	−11.7215	−1.47676
$64$	1.00000	0.125000
$65$	−1.75677	−0.217900
$66$	5.06072	0.622931
$67$	6.67092	0.814983	0.407492	−	0.913209i	$-0.366404\pi$
$67$	6.67092	0.814983	0.407492	+	0.913209i	$0.366404\pi$
$68$	−3.64409	−0.441911
$69$	5.23348	0.630037
$70$	2.56650	0.306755
$71$	−11.3277	−1.34435	−0.672175	−	0.740393i	$-0.734640\pi$
$71$	−11.3277	−1.34435	−0.672175	+	0.740393i	$0.734640\pi$
$72$	−4.25101	−0.500986
$73$	1.49471	0.174943	0.0874714	−	0.996167i	$-0.472121\pi$
$73$	1.49471	0.174943	0.0874714	+	0.996167i	$0.472121\pi$
$74$	0.0607967	0.00706747
$75$	11.1309	1.28529
$76$	−3.78639	−0.434329
$77$	−5.18206	−0.590550
$78$	−5.08231	−0.575459
$79$	−5.20048	−0.585100	−0.292550	−	0.956250i	$-0.594504\pi$
$79$	−5.20048	−0.585100	−0.292550	+	0.956250i	$0.594504\pi$
$80$	0.930791	0.104066
$81$	−3.68195	−0.409105
$82$	6.84750	0.756179
$83$	−1.01903	−0.111853	−0.0559265	−	0.998435i	$-0.517811\pi$
$83$	−1.01903	−0.111853	−0.0559265	+	0.998435i	$0.517811\pi$
$84$	7.42487	0.810119
$85$	−3.39189	−0.367902
$86$	−9.48945	−1.02327
$87$	27.2473	2.92121
$88$	−1.87937	−0.200342
$89$	−6.26525	−0.664115	−0.332057	−	0.943259i	$-0.607743\pi$
$89$	−6.26525	−0.664115	−0.332057	+	0.943259i	$0.607743\pi$
$90$	−3.95680	−0.417083
$91$	5.20417	0.545545
$92$	−1.94353	−0.202627
$93$	24.3066	2.52048
$94$	−8.53876	−0.880705
$95$	−3.52434	−0.361589
$96$	2.69277	0.274830
$97$	−3.11516	−0.316297	−0.158148	−	0.987415i	$-0.550552\pi$
$97$	−3.11516	−0.316297	−0.158148	+	0.987415i	$0.550552\pi$
$98$	−0.602891	−0.0609012
$99$	7.98923	0.802948
$100$	−4.13363	−0.413363
$101$	9.71379	0.966558	0.483279	−	0.875466i	$-0.339446\pi$
$101$	9.71379	0.966558	0.483279	+	0.875466i	$0.339446\pi$
$102$	−9.81271	−0.971603
$103$	−7.53370	−0.742318	−0.371159	−	0.928569i	$-0.621039\pi$
$103$	−7.53370	−0.742318	−0.371159	+	0.928569i	$0.621039\pi$
$104$	1.88739	0.185074
$105$	6.91099	0.674444
$106$	1.06001	0.102957
$107$	1.13067	0.109306	0.0546529	−	0.998505i	$-0.482595\pi$
$107$	1.13067	0.109306	0.0546529	+	0.998505i	$0.482595\pi$
$108$	−3.36868	−0.324151
$109$	18.5114	1.77307	0.886537	−	0.462659i	$-0.153104\pi$
$109$	18.5114	1.77307	0.886537	+	0.462659i	$0.153104\pi$
$110$	−1.74930	−0.166789
$111$	0.163712	0.0155388
$112$	−2.75733	−0.260544
$113$	8.73109	0.821352	0.410676	−	0.911781i	$-0.365293\pi$
$113$	8.73109	0.821352	0.410676	+	0.911781i	$0.365293\pi$
$114$	−10.1959	−0.954931
$115$	−1.80902	−0.168692
$116$	−10.1187	−0.939496
$117$	−8.02332	−0.741756
$118$	5.02007	0.462135
$119$	10.0480	0.921097
$120$	2.50640	0.228802
$121$	−7.46796	−0.678905
$122$	7.43945	0.673536
$123$	18.4387	1.66256
$124$	−9.02661	−0.810614
$125$	−8.50150	−0.760397
$126$	11.7215	1.04423
$127$	−17.2044	−1.52664	−0.763321	−	0.646020i	$-0.776432\pi$
$127$	−17.2044	−1.52664	−0.763321	+	0.646020i	$0.776432\pi$
$128$	−1.00000	−0.0883883
$129$	−25.5529	−2.24981
$130$	1.75677	0.154079
$131$	−1.57465	−0.137578	−0.0687890	−	0.997631i	$-0.521914\pi$
$131$	−1.57465	−0.137578	−0.0687890	+	0.997631i	$0.521914\pi$
$132$	−5.06072	−0.440479
$133$	10.4403	0.905292
$134$	−6.67092	−0.576280
$135$	−3.13554	−0.269864
$136$	3.64409	0.312478
$137$	3.48450	0.297701	0.148851	−	0.988860i	$-0.452443\pi$
$137$	3.48450	0.297701	0.148851	+	0.988860i	$0.452443\pi$
$138$	−5.23348	−0.445504
$139$	7.78296	0.660142	0.330071	−	0.943956i	$-0.392927\pi$
$139$	7.78296	0.660142	0.330071	+	0.943956i	$0.392927\pi$
$140$	−2.56650	−0.216909
$141$	−22.9929	−1.93635
$142$	11.3277	0.950599
$143$	−3.54711	−0.296625
$144$	4.25101	0.354251
$145$	−9.41837	−0.782153
$146$	−1.49471	−0.123703
$147$	−1.62345	−0.133900
$148$	−0.0607967	−0.00499746
$149$	−4.30914	−0.353019	−0.176509	−	0.984299i	$-0.556481\pi$
$149$	−4.30914	−0.353019	−0.176509	+	0.984299i	$0.556481\pi$
$150$	−11.1309	−0.908835
$151$	−8.76686	−0.713437	−0.356718	−	0.934212i	$-0.616104\pi$
$151$	−8.76686	−0.713437	−0.356718	+	0.934212i	$0.616104\pi$
$152$	3.78639	0.307117
$153$	−15.4911	−1.25238
$154$	5.18206	0.417582
$155$	−8.40189	−0.674856
$156$	5.08231	0.406911
$157$	−19.0598	−1.52114	−0.760568	−	0.649258i	$-0.775080\pi$
$157$	−19.0598	−1.52114	−0.760568	+	0.649258i	$0.775080\pi$
$158$	5.20048	0.413728
$159$	2.85435	0.226365
$160$	−0.930791	−0.0735855
$161$	5.35897	0.422346
$162$	3.68195	0.289281
$163$	−18.1259	−1.41973	−0.709864	−	0.704339i	$-0.751244\pi$
$163$	−18.1259	−1.41973	−0.709864	+	0.704339i	$0.751244\pi$
$164$	−6.84750	−0.534700
$165$	−4.71047	−0.366710
$166$	1.01903	0.0790920
$167$	16.7561	1.29663	0.648313	−	0.761374i	$-0.275475\pi$
$167$	16.7561	1.29663	0.648313	+	0.761374i	$0.275475\pi$
$168$	−7.42487	−0.572841
$169$	−9.43775	−0.725981
$170$	3.39189	0.260146
$171$	−16.0960	−1.23089
$172$	9.48945	0.723564
$173$	−0.220686	−0.0167784	−0.00838921	−	0.999965i	$-0.502670\pi$
$173$	−0.220686	−0.0167784	−0.00838921	+	0.999965i	$0.502670\pi$
$174$	−27.2473	−2.06561
$175$	11.3978	0.861592
$176$	1.87937	0.141663
$177$	13.5179	1.01607
$178$	6.26525	0.469600
$179$	−16.7066	−1.24871	−0.624356	−	0.781140i	$-0.714639\pi$
$179$	−16.7066	−1.24871	−0.624356	+	0.781140i	$0.714639\pi$
$180$	3.95680	0.294922
$181$	5.62445	0.418062	0.209031	−	0.977909i	$-0.432969\pi$
$181$	5.62445	0.418062	0.209031	+	0.977909i	$0.432969\pi$
$182$	−5.20417	−0.385759
$183$	20.0327	1.48086
$184$	1.94353	0.143279
$185$	−0.0565890	−0.00416051
$186$	−24.3066	−1.78225
$187$	−6.84861	−0.500820
$188$	8.53876	0.622753
$189$	9.28858	0.675645
$190$	3.52434	0.255682
$191$	−11.8780	−0.859464	−0.429732	−	0.902956i	$-0.641392\pi$
$191$	−11.8780	−0.859464	−0.429732	+	0.902956i	$0.641392\pi$
$192$	−2.69277	−0.194334
$193$	−11.5145	−0.828835	−0.414418	−	0.910087i	$-0.636015\pi$
$193$	−11.5145	−0.828835	−0.414418	+	0.910087i	$0.636015\pi$
$194$	3.11516	0.223656
$195$	4.73057	0.338763
$196$	0.602891	0.0430636
$197$	−20.8984	−1.48895	−0.744473	−	0.667653i	$-0.767299\pi$
$197$	−20.8984	−1.48895	−0.744473	+	0.667653i	$0.767299\pi$
$198$	−7.98923	−0.567770
$199$	17.6607	1.25193	0.625966	−	0.779851i	$-0.284705\pi$
$199$	17.6607	1.25193	0.625966	+	0.779851i	$0.284705\pi$
$200$	4.13363	0.292292
$201$	−17.9633	−1.26703
$202$	−9.71379	−0.683460
$203$	27.9006	1.95824
$204$	9.81271	0.687027
$205$	−6.37358	−0.445150
$206$	7.53370	0.524898
$207$	−8.26197	−0.574247
$208$	−1.88739	−0.130867
$209$	−7.11604	−0.492227
$210$	−6.91099	−0.476904
$211$	22.4666	1.54666	0.773331	−	0.634003i	$-0.218589\pi$
$211$	22.4666	1.54666	0.773331	+	0.634003i	$0.218589\pi$
$212$	−1.06001	−0.0728015
$213$	30.5029	2.09002
$214$	−1.13067	−0.0772909
$215$	8.83269	0.602385
$216$	3.36868	0.229210
$217$	24.8894	1.68960
$218$	−18.5114	−1.25375
$219$	−4.02492	−0.271979
$220$	1.74930	0.117938
$221$	6.87783	0.462653
$222$	−0.163712	−0.0109876
$223$	21.9951	1.47290	0.736450	−	0.676492i	$-0.236501\pi$
$223$	21.9951	1.47290	0.736450	+	0.676492i	$0.236501\pi$
$224$	2.75733	0.184232
$225$	−17.5721	−1.17147
$226$	−8.73109	−0.580783
$227$	−26.4944	−1.75849	−0.879246	−	0.476368i	$-0.841953\pi$
$227$	−26.4944	−1.75849	−0.879246	+	0.476368i	$0.841953\pi$
$228$	10.1959	0.675238
$229$	0.892211	0.0589589	0.0294795	−	0.999565i	$-0.490615\pi$
$229$	0.892211	0.0589589	0.0294795	+	0.999565i	$0.490615\pi$
$230$	1.80902	0.119283
$231$	13.9541	0.918112
$232$	10.1187	0.664324
$233$	−10.8204	−0.708869	−0.354435	−	0.935081i	$-0.615327\pi$
$233$	−10.8204	−0.708869	−0.354435	+	0.935081i	$0.615327\pi$
$234$	8.02332	0.524501
$235$	7.94779	0.518457
$236$	−5.02007	−0.326779
$237$	14.0037	0.909638
$238$	−10.0480	−0.651314
$239$	−16.1783	−1.04649	−0.523243	−	0.852183i	$-0.675278\pi$
$239$	−16.1783	−1.04649	−0.523243	+	0.852183i	$0.675278\pi$
$240$	−2.50640	−0.161788
$241$	9.39784	0.605368	0.302684	−	0.953091i	$-0.402117\pi$
$241$	9.39784	0.605368	0.302684	+	0.953091i	$0.402117\pi$
$242$	7.46796	0.480058
$243$	20.0207	1.28433
$244$	−7.43945	−0.476262
$245$	0.561165	0.0358515
$246$	−18.4387	−1.17561
$247$	7.14640	0.454715
$248$	9.02661	0.573191
$249$	2.74401	0.173895
$250$	8.50150	0.537682
$251$	−7.38830	−0.466345	−0.233173	−	0.972435i	$-0.574911\pi$
$251$	−7.38830	−0.466345	−0.233173	+	0.972435i	$0.574911\pi$
$252$	−11.7215	−0.738382
$253$	−3.65262	−0.229638
$254$	17.2044	1.07950
$255$	9.13357	0.571967
$256$	1.00000	0.0625000
$257$	19.5737	1.22097	0.610487	−	0.792026i	$-0.290974\pi$
$257$	19.5737	1.22097	0.610487	+	0.792026i	$0.290974\pi$
$258$	25.5529	1.59085
$259$	0.167637	0.0104164
$260$	−1.75677	−0.108950
$261$	−43.0146	−2.66254
$262$	1.57465	0.0972824
$263$	3.04083	0.187505	0.0937527	−	0.995596i	$-0.470114\pi$
$263$	3.04083	0.187505	0.0937527	+	0.995596i	$0.470114\pi$
$264$	5.06072	0.311466
$265$	−0.986643	−0.0606090
$266$	−10.4403	−0.640138
$267$	16.8709	1.03248
$268$	6.67092	0.407492
$269$	−17.8887	−1.09069	−0.545347	−	0.838210i	$-0.683602\pi$
$269$	−17.8887	−1.09069	−0.545347	+	0.838210i	$0.683602\pi$
$270$	3.13554	0.190823
$271$	19.1775	1.16495	0.582476	−	0.812848i	$-0.302084\pi$
$271$	19.1775	1.16495	0.582476	+	0.812848i	$0.302084\pi$
$272$	−3.64409	−0.220956
$273$	−14.0136	−0.848144
$274$	−3.48450	−0.210507
$275$	−7.76863	−0.468466
$276$	5.23348	0.315019
$277$	4.39611	0.264137	0.132068	−	0.991241i	$-0.457838\pi$
$277$	4.39611	0.264137	0.132068	+	0.991241i	$0.457838\pi$
$278$	−7.78296	−0.466791
$279$	−38.3722	−2.29728
$280$	2.56650	0.153378
$281$	−17.0653	−1.01803	−0.509015	−	0.860757i	$-0.669990\pi$
$281$	−17.0653	−1.01803	−0.509015	+	0.860757i	$0.669990\pi$
$282$	22.9929	1.36921
$283$	8.83680	0.525293	0.262647	−	0.964892i	$-0.415405\pi$
$283$	8.83680	0.525293	0.262647	+	0.964892i	$0.415405\pi$
$284$	−11.3277	−0.672175
$285$	9.49022	0.562152
$286$	3.54711	0.209745
$287$	18.8808	1.11450
$288$	−4.25101	−0.250493
$289$	−3.72058	−0.218858
$290$	9.41837	0.553066
$291$	8.38841	0.491738
$292$	1.49471	0.0874714
$293$	16.1859	0.945592	0.472796	−	0.881172i	$-0.343245\pi$
$293$	16.1859	0.945592	0.472796	+	0.881172i	$0.343245\pi$
$294$	1.62345	0.0946813
$295$	−4.67263	−0.272051
$296$	0.0607967	0.00353374
$297$	−6.33101	−0.367362
$298$	4.30914	0.249622
$299$	3.66821	0.212138
$300$	11.1309	0.642643
$301$	−26.1656	−1.50816
$302$	8.76686	0.504476
$303$	−26.1570	−1.50268
$304$	−3.78639	−0.217164
$305$	−6.92457	−0.396500
$306$	15.4911	0.885566
$307$	28.5524	1.62957	0.814785	−	0.579763i	$-0.196855\pi$
$307$	28.5524	1.62957	0.814785	+	0.579763i	$0.196855\pi$
$308$	−5.18206	−0.295275
$309$	20.2865	1.15406
$310$	8.40189	0.477195
$311$	12.8152	0.726685	0.363342	−	0.931656i	$-0.381636\pi$
$311$	12.8152	0.726685	0.363342	+	0.931656i	$0.381636\pi$
$312$	−5.08231	−0.287729
$313$	−18.4176	−1.04103	−0.520513	−	0.853854i	$-0.674259\pi$
$313$	−18.4176	−1.04103	−0.520513	+	0.853854i	$0.674259\pi$
$314$	19.0598	1.07561
$315$	−10.9102	−0.614721
$316$	−5.20048	−0.292550
$317$	−32.1260	−1.80438	−0.902189	−	0.431340i	$-0.858041\pi$
$317$	−32.1260	−1.80438	−0.902189	+	0.431340i	$0.858041\pi$
$318$	−2.85435	−0.160064
$319$	−19.0168	−1.06473
$320$	0.930791	0.0520328
$321$	−3.04463	−0.169935
$322$	−5.35897	−0.298643
$323$	13.7980	0.767739
$324$	−3.68195	−0.204553
$325$	7.80178	0.432765
$326$	18.1259	1.00390
$327$	−49.8470	−2.75655
$328$	6.84750	0.378090
$329$	−23.5442	−1.29803
$330$	4.71047	0.259303
$331$	2.52325	0.138690	0.0693451	−	0.997593i	$-0.477909\pi$
$331$	2.52325	0.138690	0.0693451	+	0.997593i	$0.477909\pi$
$332$	−1.01903	−0.0559265
$333$	−0.258447	−0.0141628
$334$	−16.7561	−0.916853
$335$	6.20923	0.339247
$336$	7.42487	0.405060
$337$	−29.9672	−1.63242	−0.816208	−	0.577759i	$-0.803927\pi$
$337$	−29.9672	−1.63242	−0.816208	+	0.577759i	$0.803927\pi$
$338$	9.43775	0.513346
$339$	−23.5108	−1.27693
$340$	−3.39189	−0.183951
$341$	−16.9644	−0.918672
$342$	16.0960	0.870371
$343$	17.6390	0.952415
$344$	−9.48945	−0.511637
$345$	4.87128	0.262261
$346$	0.220686	0.0118641
$347$	−5.24849	−0.281754	−0.140877	−	0.990027i	$-0.544992\pi$
$347$	−5.24849	−0.281754	−0.140877	+	0.990027i	$0.544992\pi$
$348$	27.2473	1.46061
$349$	16.6418	0.890816	0.445408	−	0.895328i	$-0.353059\pi$
$349$	16.6418	0.890816	0.445408	+	0.895328i	$0.353059\pi$
$350$	−11.3978	−0.609238
$351$	6.35802	0.339366
$352$	−1.87937	−0.100171
$353$	−1.92631	−0.102527	−0.0512636	−	0.998685i	$-0.516325\pi$
$353$	−1.92631	−0.102527	−0.0512636	+	0.998685i	$0.516325\pi$
$354$	−13.5179	−0.718468
$355$	−10.5437	−0.559602
$356$	−6.26525	−0.332057
$357$	−27.0569	−1.43200
$358$	16.7066	0.882973
$359$	−7.39255	−0.390164	−0.195082	−	0.980787i	$-0.562497\pi$
$359$	−7.39255	−0.390164	−0.195082	+	0.980787i	$0.562497\pi$
$360$	−3.95680	−0.208542
$361$	−4.66326	−0.245435
$362$	−5.62445	−0.295615
$363$	20.1095	1.05547
$364$	5.20417	0.272773
$365$	1.39126	0.0728221
$366$	−20.0327	−1.04713
$367$	35.2814	1.84168	0.920838	−	0.389945i	$-0.127506\pi$
$367$	35.2814	1.84168	0.920838	+	0.389945i	$0.127506\pi$
$368$	−1.94353	−0.101314
$369$	−29.1088	−1.51534
$370$	0.0565890	0.00294192
$371$	2.92279	0.151744
$372$	24.3066	1.26024
$373$	−11.5375	−0.597387	−0.298694	−	0.954349i	$-0.596551\pi$
$373$	−11.5375	−0.597387	−0.298694	+	0.954349i	$0.596551\pi$
$374$	6.84861	0.354133
$375$	22.8926	1.18217
$376$	−8.53876	−0.440353
$377$	19.0979	0.983593
$378$	−9.28858	−0.477753
$379$	−13.2385	−0.680015	−0.340007	−	0.940423i	$-0.610430\pi$
$379$	−13.2385	−0.680015	−0.340007	+	0.940423i	$0.610430\pi$
$380$	−3.52434	−0.180795
$381$	46.3274	2.37343
$382$	11.8780	0.607733
$383$	−6.63672	−0.339120	−0.169560	−	0.985520i	$-0.554235\pi$
$383$	−6.63672	−0.339120	−0.169560	+	0.985520i	$0.554235\pi$
$384$	2.69277	0.137415
$385$	−4.82341	−0.245824
$386$	11.5145	0.586075
$387$	40.3398	2.05058
$388$	−3.11516	−0.158148
$389$	−9.73799	−0.493736	−0.246868	−	0.969049i	$-0.579401\pi$
$389$	−9.73799	−0.493736	−0.246868	+	0.969049i	$0.579401\pi$
$390$	−4.73057	−0.239542
$391$	7.08241	0.358173
$392$	−0.602891	−0.0304506
$393$	4.24018	0.213889
$394$	20.8984	1.05284
$395$	−4.84056	−0.243555
$396$	7.98923	0.401474
$397$	19.8725	0.997373	0.498687	−	0.866782i	$-0.333816\pi$
$397$	19.8725	0.997373	0.498687	+	0.866782i	$0.333816\pi$
$398$	−17.6607	−0.885249
$399$	−28.1134	−1.40743
$400$	−4.13363	−0.206681
$401$	−37.2748	−1.86141	−0.930707	−	0.365765i	$-0.880807\pi$
$401$	−37.2748	−1.86141	−0.930707	+	0.365765i	$0.880807\pi$
$402$	17.9633	0.895926
$403$	17.0368	0.848661
$404$	9.71379	0.483279
$405$	−3.42712	−0.170295
$406$	−27.9006	−1.38468
$407$	−0.114260	−0.00566364
$408$	−9.81271	−0.485801
$409$	−40.2143	−1.98847	−0.994235	−	0.107221i	$-0.965805\pi$
$409$	−40.2143	−1.98847	−0.994235	+	0.107221i	$0.965805\pi$
$410$	6.37358	0.314769
$411$	−9.38297	−0.462828
$412$	−7.53370	−0.371159
$413$	13.8420	0.681121
$414$	8.26197	0.406054
$415$	−0.948503	−0.0465602
$416$	1.88739	0.0925370
$417$	−20.9577	−1.02630
$418$	7.11604	0.348057
$419$	21.5988	1.05517	0.527586	−	0.849502i	$-0.323097\pi$
$419$	21.5988	1.05517	0.527586	+	0.849502i	$0.323097\pi$
$420$	6.91099	0.337222
$421$	−23.6867	−1.15442	−0.577209	−	0.816596i	$-0.695858\pi$
$421$	−23.6867	−1.15442	−0.577209	+	0.816596i	$0.695858\pi$
$422$	−22.4666	−1.09365
$423$	36.2983	1.76489
$424$	1.06001	0.0514784
$425$	15.0633	0.730679
$426$	−30.5029	−1.47787
$427$	20.5131	0.992696
$428$	1.13067	0.0546529
$429$	9.55156	0.461154
$430$	−8.83269	−0.425950
$431$	37.8596	1.82363	0.911815	−	0.410601i	$-0.134681\pi$
$431$	37.8596	1.82363	0.911815	+	0.410601i	$0.134681\pi$
$432$	−3.36868	−0.162076
$433$	−36.1347	−1.73652	−0.868261	−	0.496107i	$-0.834762\pi$
$433$	−36.1347	−1.73652	−0.868261	+	0.496107i	$0.834762\pi$
$434$	−24.8894	−1.19473
$435$	25.3615	1.21599
$436$	18.5114	0.886537
$437$	7.35897	0.352027
$438$	4.02492	0.192318
$439$	6.00547	0.286625	0.143313	−	0.989677i	$-0.454225\pi$
$439$	6.00547	0.286625	0.143313	+	0.989677i	$0.454225\pi$
$440$	−1.74930	−0.0833947
$441$	2.56289	0.122043
$442$	−6.87783	−0.327145
$443$	0.145671	0.00692106	0.00346053	−	0.999994i	$-0.498898\pi$
$443$	0.145671	0.00692106	0.00346053	+	0.999994i	$0.498898\pi$
$444$	0.163712	0.00776941
$445$	−5.83163	−0.276446
$446$	−21.9951	−1.04150
$447$	11.6035	0.548828
$448$	−2.75733	−0.130272
$449$	38.5888	1.82112	0.910558	−	0.413380i	$-0.135652\pi$
$449$	38.5888	1.82112	0.910558	+	0.413380i	$0.135652\pi$
$450$	17.5721	0.828357
$451$	−12.8690	−0.605977
$452$	8.73109	0.410676
$453$	23.6071	1.10916
$454$	26.4944	1.24344
$455$	4.84399	0.227090
$456$	−10.1959	−0.477466
$457$	0.169863	0.00794587	0.00397293	−	0.999992i	$-0.498735\pi$
$457$	0.169863	0.00794587	0.00397293	+	0.999992i	$0.498735\pi$
$458$	−0.892211	−0.0416903
$459$	12.2758	0.572985
$460$	−1.80902	−0.0843460
$461$	28.2291	1.31476	0.657379	−	0.753560i	$-0.271665\pi$
$461$	28.2291	1.31476	0.657379	+	0.753560i	$0.271665\pi$
$462$	−13.9541	−0.649203
$463$	−27.8692	−1.29519	−0.647595	−	0.761985i	$-0.724225\pi$
$463$	−27.8692	−1.29519	−0.647595	+	0.761985i	$0.724225\pi$
$464$	−10.1187	−0.469748
$465$	22.6243	1.04918
$466$	10.8204	0.501246
$467$	23.9808	1.10970	0.554849	−	0.831951i	$-0.312776\pi$
$467$	23.9808	1.10970	0.554849	+	0.831951i	$0.312776\pi$
$468$	−8.02332	−0.370878
$469$	−18.3940	−0.849354
$470$	−7.94779	−0.366604
$471$	51.3236	2.36487
$472$	5.02007	0.231067
$473$	17.8342	0.820018
$474$	−14.0037	−0.643211
$475$	15.6515	0.718141
$476$	10.0480	0.460549
$477$	−4.50609	−0.206320
$478$	16.1783	0.739977
$479$	−30.1451	−1.37737	−0.688683	−	0.725063i	$-0.741811\pi$
$479$	−30.1451	−1.37737	−0.688683	+	0.725063i	$0.741811\pi$
$480$	2.50640	0.114401
$481$	0.114747	0.00523202
$482$	−9.39784	−0.428060
$483$	−14.4305	−0.656609
$484$	−7.46796	−0.339453
$485$	−2.89956	−0.131662
$486$	−20.0207	−0.908156
$487$	−12.5242	−0.567524	−0.283762	−	0.958895i	$-0.591583\pi$
$487$	−12.5242	−0.567524	−0.283762	+	0.958895i	$0.591583\pi$
$488$	7.43945	0.336768
$489$	48.8088	2.20721
$490$	−0.561165	−0.0253508
$491$	4.59365	0.207308	0.103654	−	0.994613i	$-0.466946\pi$
$491$	4.59365	0.207308	0.103654	+	0.994613i	$0.466946\pi$
$492$	18.4387	0.831282
$493$	36.8734	1.66070
$494$	−7.14640	−0.321532
$495$	7.43630	0.334237
$496$	−9.02661	−0.405307
$497$	31.2342	1.40105
$498$	−2.74401	−0.122962
$499$	13.7560	0.615802	0.307901	−	0.951418i	$-0.400373\pi$
$499$	13.7560	0.615802	0.307901	+	0.951418i	$0.400373\pi$
$500$	−8.50150	−0.380198
$501$	−45.1203	−2.01583
$502$	7.38830	0.329756
$503$	39.7287	1.77142	0.885708	−	0.464242i	$-0.153673\pi$
$503$	39.7287	1.77142	0.885708	+	0.464242i	$0.153673\pi$
$504$	11.7215	0.522115
$505$	9.04151	0.402342
$506$	3.65262	0.162379
$507$	25.4137	1.12866
$508$	−17.2044	−0.763321
$509$	10.8198	0.479577	0.239789	−	0.970825i	$-0.422922\pi$
$509$	10.8198	0.479577	0.239789	+	0.970825i	$0.422922\pi$
$510$	−9.13357	−0.404441
$511$	−4.12142	−0.182321
$512$	−1.00000	−0.0441942
$513$	12.7551	0.563153
$514$	−19.5737	−0.863359
$515$	−7.01230	−0.308999
$516$	−25.5529	−1.12490
$517$	16.0475	0.705769
$518$	−0.167637	−0.00736554
$519$	0.594256	0.0260849
$520$	1.75677	0.0770393
$521$	6.98607	0.306065	0.153033	−	0.988221i	$-0.451096\pi$
$521$	6.98607	0.306065	0.153033	+	0.988221i	$0.451096\pi$
$522$	43.0146	1.88270
$523$	−10.7300	−0.469192	−0.234596	−	0.972093i	$-0.575377\pi$
$523$	−10.7300	−0.469192	−0.234596	+	0.972093i	$0.575377\pi$
$524$	−1.57465	−0.0687890
$525$	−30.6916	−1.33949
$526$	−3.04083	−0.132586
$527$	32.8938	1.43288
$528$	−5.06072	−0.220240
$529$	−19.2227	−0.835769
$530$	0.986643	0.0428570
$531$	−21.3404	−0.926093
$532$	10.4403	0.452646
$533$	12.9239	0.559797
$534$	−16.8709	−0.730074
$535$	1.05242	0.0454999
$536$	−6.67092	−0.288140
$537$	44.9871	1.94134
$538$	17.8887	0.771238
$539$	1.13306	0.0488042
$540$	−3.13554	−0.134932
$541$	3.40665	0.146463	0.0732316	−	0.997315i	$-0.476669\pi$
$541$	3.40665	0.146463	0.0732316	+	0.997315i	$0.476669\pi$
$542$	−19.1775	−0.823745
$543$	−15.1454	−0.649949
$544$	3.64409	0.156239
$545$	17.2303	0.738063
$546$	14.0136	0.599728
$547$	−24.2792	−1.03810	−0.519052	−	0.854743i	$-0.673715\pi$
$547$	−24.2792	−1.03810	−0.519052	+	0.854743i	$0.673715\pi$
$548$	3.48450	0.148851
$549$	−31.6252	−1.34973
$550$	7.76863	0.331256
$551$	38.3133	1.63220
$552$	−5.23348	−0.222752
$553$	14.3395	0.609776
$554$	−4.39611	−0.186773
$555$	0.152381	0.00646822
$556$	7.78296	0.330071
$557$	−19.0532	−0.807308	−0.403654	−	0.914912i	$-0.632260\pi$
$557$	−19.0532	−0.807308	−0.403654	+	0.914912i	$0.632260\pi$
$558$	38.3722	1.62443
$559$	−17.9103	−0.757526
$560$	−2.56650	−0.108454
$561$	18.4417	0.778611
$562$	17.0653	0.719856
$563$	−37.7994	−1.59306	−0.796528	−	0.604602i	$-0.793332\pi$
$563$	−37.7994	−1.59306	−0.796528	+	0.604602i	$0.793332\pi$
$564$	−22.9929	−0.968176
$565$	8.12681	0.341898
$566$	−8.83680	−0.371438
$567$	10.1524	0.426359
$568$	11.3277	0.475299
$569$	2.29819	0.0963449	0.0481725	−	0.998839i	$-0.484660\pi$
$569$	2.29819	0.0963449	0.0481725	+	0.998839i	$0.484660\pi$
$570$	−9.49022	−0.397502
$571$	3.62670	0.151773	0.0758863	−	0.997116i	$-0.475821\pi$
$571$	3.62670	0.151773	0.0758863	+	0.997116i	$0.475821\pi$
$572$	−3.54711	−0.148312
$573$	31.9848	1.33618
$574$	−18.8808	−0.788071
$575$	8.03384	0.335034
$576$	4.25101	0.177125
$577$	−15.3947	−0.640892	−0.320446	−	0.947267i	$-0.603833\pi$
$577$	−15.3947	−0.640892	−0.320446	+	0.947267i	$0.603833\pi$
$578$	3.72058	0.154756
$579$	31.0060	1.28857
$580$	−9.41837	−0.391077
$581$	2.80981	0.116570
$582$	−8.38841	−0.347711
$583$	−1.99215	−0.0825062
$584$	−1.49471	−0.0618516
$585$	−7.46803	−0.308765
$586$	−16.1859	−0.668634
$587$	29.1237	1.20207	0.601033	−	0.799225i	$-0.294756\pi$
$587$	29.1237	1.20207	0.601033	+	0.799225i	$0.294756\pi$
$588$	−1.62345	−0.0669498
$589$	34.1783	1.40829
$590$	4.67263	0.192369
$591$	56.2745	2.31482
$592$	−0.0607967	−0.00249873
$593$	34.8150	1.42968	0.714840	−	0.699288i	$-0.246500\pi$
$593$	34.8150	1.42968	0.714840	+	0.699288i	$0.246500\pi$
$594$	6.33101	0.259764
$595$	9.35257	0.383418
$596$	−4.30914	−0.176509
$597$	−47.5561	−1.94634
$598$	−3.66821	−0.150004
$599$	18.8411	0.769827	0.384913	−	0.922953i	$-0.374231\pi$
$599$	18.8411	0.769827	0.384913	+	0.922953i	$0.374231\pi$
$600$	−11.1309	−0.454418
$601$	0.147052	0.00599838	0.00299919	−	0.999996i	$-0.499045\pi$
$601$	0.147052	0.00599838	0.00299919	+	0.999996i	$0.499045\pi$
$602$	26.1656	1.06643
$603$	28.3582	1.15483
$604$	−8.76686	−0.356718
$605$	−6.95110	−0.282603
$606$	26.1570	1.06256
$607$	25.3306	1.02814	0.514070	−	0.857748i	$-0.328137\pi$
$607$	25.3306	1.02814	0.514070	+	0.857748i	$0.328137\pi$
$608$	3.78639	0.153558
$609$	−75.1298	−3.04441
$610$	6.92457	0.280368
$611$	−16.1160	−0.651983
$612$	−15.4911	−0.626190
$613$	25.4159	1.02654	0.513269	−	0.858228i	$-0.328434\pi$
$613$	25.4159	1.02654	0.513269	+	0.858228i	$0.328434\pi$
$614$	−28.5524	−1.15228
$615$	17.1626	0.692063
$616$	5.18206	0.208791
$617$	42.1789	1.69806	0.849029	−	0.528347i	$-0.177188\pi$
$617$	42.1789	1.69806	0.849029	+	0.528347i	$0.177188\pi$
$618$	−20.2865	−0.816044
$619$	12.3983	0.498329	0.249164	−	0.968461i	$-0.419844\pi$
$619$	12.3983	0.498329	0.249164	+	0.968461i	$0.419844\pi$
$620$	−8.40189	−0.337428
$621$	6.54714	0.262728
$622$	−12.8152	−0.513844
$623$	17.2754	0.692123
$624$	5.08231	0.203455
$625$	12.7550	0.510201
$626$	18.4176	0.736117
$627$	19.1619	0.765251
$628$	−19.0598	−0.760568
$629$	0.221549	0.00883373
$630$	10.9102	0.434673
$631$	7.26420	0.289183	0.144592	−	0.989491i	$-0.453813\pi$
$631$	7.26420	0.289183	0.144592	+	0.989491i	$0.453813\pi$
$632$	5.20048	0.206864
$633$	−60.4973	−2.40455
$634$	32.1260	1.27589
$635$	−16.0137	−0.635483
$636$	2.85435	0.113182
$637$	−1.13789	−0.0450849
$638$	19.0168	0.752881
$639$	−48.1541	−1.90495
$640$	−0.930791	−0.0367927
$641$	−3.21485	−0.126979	−0.0634894	−	0.997983i	$-0.520223\pi$
$641$	−3.21485	−0.126979	−0.0634894	+	0.997983i	$0.520223\pi$
$642$	3.04463	0.120162
$643$	15.8284	0.624210	0.312105	−	0.950048i	$-0.398966\pi$
$643$	15.8284	0.624210	0.312105	+	0.950048i	$0.398966\pi$
$644$	5.35897	0.211173
$645$	−23.7844	−0.936510
$646$	−13.7980	−0.542873
$647$	44.6033	1.75354	0.876768	−	0.480913i	$-0.159695\pi$
$647$	44.6033	1.75354	0.876768	+	0.480913i	$0.159695\pi$
$648$	3.68195	0.144641
$649$	−9.43458	−0.370340
$650$	−7.80178	−0.306011
$651$	−67.0214	−2.62678
$652$	−18.1259	−0.709864
$653$	0.0506975	0.00198395	0.000991974	−	1.00000i	$-0.499684\pi$
$653$	0.0506975	0.00198395	0.000991974		1.00000i	$0.499684\pi$
$654$	49.8470	1.94917
$655$	−1.46567	−0.0572685
$656$	−6.84750	−0.267350
$657$	6.35404	0.247895
$658$	23.5442	0.917849
$659$	16.7782	0.653586	0.326793	−	0.945096i	$-0.394032\pi$
$659$	16.7782	0.653586	0.326793	+	0.945096i	$0.394032\pi$
$660$	−4.71047	−0.183355
$661$	−43.9945	−1.71119	−0.855593	−	0.517649i	$-0.826807\pi$
$661$	−43.9945	−1.71119	−0.855593	+	0.517649i	$0.826807\pi$
$662$	−2.52325	−0.0980688
$663$	−18.5204	−0.719274
$664$	1.01903	0.0395460
$665$	9.71777	0.376839
$666$	0.258447	0.0100146
$667$	19.6660	0.761469
$668$	16.7561	0.648313
$669$	−59.2277	−2.28988
$670$	−6.20923	−0.239884
$671$	−13.9815	−0.539750
$672$	−7.42487	−0.286420
$673$	15.9081	0.613213	0.306606	−	0.951836i	$-0.400806\pi$
$673$	15.9081	0.613213	0.306606	+	0.951836i	$0.400806\pi$
$674$	29.9672	1.15429
$675$	13.9249	0.535969
$676$	−9.43775	−0.362990
$677$	19.1096	0.734442	0.367221	−	0.930134i	$-0.380309\pi$
$677$	19.1096	0.734442	0.367221	+	0.930134i	$0.380309\pi$
$678$	23.5108	0.902927
$679$	8.58954	0.329636
$680$	3.39189	0.130073
$681$	71.3432	2.73388
$682$	16.9644	0.649599
$683$	24.0946	0.921956	0.460978	−	0.887412i	$-0.347499\pi$
$683$	24.0946	0.921956	0.460978	+	0.887412i	$0.347499\pi$
$684$	−16.0960	−0.615445
$685$	3.24334	0.123922
$686$	−17.6390	−0.673459
$687$	−2.40252	−0.0916618
$688$	9.48945	0.361782
$689$	2.00065	0.0762185
$690$	−4.87128	−0.185446
$691$	20.8389	0.792749	0.396375	−	0.918089i	$-0.370268\pi$
$691$	20.8389	0.792749	0.396375	+	0.918089i	$0.370268\pi$
$692$	−0.220686	−0.00838921
$693$	−22.0290	−0.836812
$694$	5.24849	0.199230
$695$	7.24430	0.274792
$696$	−27.2473	−1.03281
$697$	24.9529	0.945159
$698$	−16.6418	−0.629902
$699$	29.1369	1.10206
$700$	11.3978	0.430796
$701$	16.3691	0.618253	0.309127	−	0.951021i	$-0.399963\pi$
$701$	16.3691	0.618253	0.309127	+	0.951021i	$0.399963\pi$
$702$	−6.35802	−0.239968
$703$	0.230200	0.00868216
$704$	1.87937	0.0708315
$705$	−21.4016	−0.806030
$706$	1.92631	0.0724976
$707$	−26.7842	−1.00732
$708$	13.5179	0.508033
$709$	−17.2049	−0.646144	−0.323072	−	0.946374i	$-0.604716\pi$
$709$	−17.2049	−0.646144	−0.323072	+	0.946374i	$0.604716\pi$
$710$	10.5437	0.395698
$711$	−22.1073	−0.829088
$712$	6.26525	0.234800
$713$	17.5435	0.657010
$714$	27.0569	1.01258
$715$	−3.30162	−0.123474
$716$	−16.7066	−0.624356
$717$	43.5644	1.62694
$718$	7.39255	0.275887
$719$	−35.4297	−1.32130	−0.660652	−	0.750692i	$-0.729720\pi$
$719$	−35.4297	−1.32130	−0.660652	+	0.750692i	$0.729720\pi$
$720$	3.95680	0.147461
$721$	20.7729	0.773625
$722$	4.66326	0.173548
$723$	−25.3062	−0.941148
$724$	5.62445	0.209031
$725$	41.8269	1.55341
$726$	−20.1095	−0.746333
$727$	−8.40604	−0.311763	−0.155881	−	0.987776i	$-0.549822\pi$
$727$	−8.40604	−0.311763	−0.155881	+	0.987776i	$0.549822\pi$
$728$	−5.20417	−0.192879
$729$	−42.8652	−1.58760
$730$	−1.39126	−0.0514930
$731$	−34.5805	−1.27900
$732$	20.0327	0.740431
$733$	−15.3711	−0.567745	−0.283873	−	0.958862i	$-0.591619\pi$
$733$	−15.3711	−0.567745	−0.283873	+	0.958862i	$0.591619\pi$
$734$	−35.2814	−1.30226
$735$	−1.51109	−0.0557373
$736$	1.94353	0.0716395
$737$	12.5372	0.461812
$738$	29.1088	1.07151
$739$	8.31178	0.305754	0.152877	−	0.988245i	$-0.451146\pi$
$739$	8.31178	0.305754	0.152877	+	0.988245i	$0.451146\pi$
$740$	−0.0565890	−0.00208025
$741$	−19.2436	−0.706932
$742$	−2.92279	−0.107299
$743$	−15.8483	−0.581416	−0.290708	−	0.956812i	$-0.593891\pi$
$743$	−15.8483	−0.581416	−0.290708	+	0.956812i	$0.593891\pi$
$744$	−24.3066	−0.891123
$745$	−4.01091	−0.146948
$746$	11.5375	0.422417
$747$	−4.33190	−0.158496
$748$	−6.84861	−0.250410
$749$	−3.11763	−0.113916
$750$	−22.8926	−0.835918
$751$	29.0199	1.05895	0.529475	−	0.848325i	$-0.322389\pi$
$751$	29.0199	1.05895	0.529475	+	0.848325i	$0.322389\pi$
$752$	8.53876	0.311376
$753$	19.8950	0.725013
$754$	−19.0979	−0.695505
$755$	−8.16011	−0.296977
$756$	9.28858	0.337822
$757$	44.4841	1.61680	0.808401	−	0.588632i	$-0.200333\pi$
$757$	44.4841	1.61680	0.808401	+	0.588632i	$0.200333\pi$
$758$	13.2385	0.480843
$759$	9.83567	0.357012
$760$	3.52434	0.127841
$761$	40.4438	1.46609	0.733043	−	0.680183i	$-0.238100\pi$
$761$	40.4438	1.46609	0.733043	+	0.680183i	$0.238100\pi$
$762$	−46.3274	−1.67827
$763$	−51.0422	−1.84785
$764$	−11.8780	−0.429732
$765$	−14.4189	−0.521318
$766$	6.63672	0.239794
$767$	9.47484	0.342117
$768$	−2.69277	−0.0971670
$769$	−0.856635	−0.0308910	−0.0154455	−	0.999881i	$-0.504917\pi$
$769$	−0.856635	−0.0308910	−0.0154455	+	0.999881i	$0.504917\pi$
$770$	4.82341	0.173824
$771$	−52.7075	−1.89821
$772$	−11.5145	−0.414418
$773$	35.7972	1.28754	0.643769	−	0.765220i	$-0.277370\pi$
$773$	35.7972	1.28754	0.643769	+	0.765220i	$0.277370\pi$
$774$	−40.3398	−1.44998
$775$	37.3127	1.34031
$776$	3.11516	0.111828
$777$	−0.451407	−0.0161942
$778$	9.73799	0.349124
$779$	25.9273	0.928941
$780$	4.73057	0.169382
$781$	−21.2890	−0.761779
$782$	−7.08241	−0.253266
$783$	34.0866	1.21816
$784$	0.602891	0.0215318
$785$	−17.7407	−0.633192
$786$	−4.24018	−0.151242
$787$	−48.6549	−1.73436	−0.867180	−	0.497995i	$-0.834070\pi$
$787$	−48.6549	−1.73436	−0.867180	+	0.497995i	$0.834070\pi$
$788$	−20.8984	−0.744473
$789$	−8.18825	−0.291509
$790$	4.84056	0.172219
$791$	−24.0745	−0.855992
$792$	−7.98923	−0.283885
$793$	14.0412	0.498616
$794$	−19.8725	−0.705249
$795$	2.65680	0.0942271
$796$	17.6607	0.625966
$797$	−2.97919	−0.105528	−0.0527642	−	0.998607i	$-0.516803\pi$
$797$	−2.97919	−0.105528	−0.0527642	+	0.998607i	$0.516803\pi$
$798$	28.1134	0.995205
$799$	−31.1160	−1.10081
$800$	4.13363	0.146146
$801$	−26.6336	−0.941053
$802$	37.2748	1.31622
$803$	2.80912	0.0991318
$804$	−17.9633	−0.633515
$805$	4.98807	0.175807
$806$	−17.0368	−0.600094
$807$	48.1702	1.69567
$808$	−9.71379	−0.341730
$809$	27.8000	0.977397	0.488699	−	0.872453i	$-0.337472\pi$
$809$	27.8000	0.977397	0.488699	+	0.872453i	$0.337472\pi$
$810$	3.42712	0.120417
$811$	18.5492	0.651349	0.325675	−	0.945482i	$-0.394409\pi$
$811$	18.5492	0.651349	0.325675	+	0.945482i	$0.394409\pi$
$812$	27.9006	0.979118
$813$	−51.6406	−1.81112
$814$	0.114260	0.00400480
$815$	−16.8714	−0.590979
$816$	9.81271	0.343513
$817$	−35.9308	−1.25706
$818$	40.2143	1.40606
$819$	22.1230	0.773039
$820$	−6.37358	−0.222575
$821$	5.69917	0.198902	0.0994512	−	0.995042i	$-0.468291\pi$
$821$	5.69917	0.198902	0.0994512	+	0.995042i	$0.468291\pi$
$822$	9.38297	0.327269
$823$	39.7512	1.38564	0.692819	−	0.721111i	$-0.256368\pi$
$823$	39.7512	1.38564	0.692819	+	0.721111i	$0.256368\pi$
$824$	7.53370	0.262449
$825$	20.9191	0.728311
$826$	−13.8420	−0.481625
$827$	−17.8431	−0.620466	−0.310233	−	0.950661i	$-0.600407\pi$
$827$	−17.8431	−0.620466	−0.310233	+	0.950661i	$0.600407\pi$
$828$	−8.26197	−0.287123
$829$	−36.0410	−1.25175	−0.625877	−	0.779922i	$-0.715259\pi$
$829$	−36.0410	−1.25175	−0.625877	+	0.779922i	$0.715259\pi$
$830$	0.948503	0.0329230
$831$	−11.8377	−0.410646
$832$	−1.88739	−0.0654335
$833$	−2.19699	−0.0761212
$834$	20.9577	0.725706
$835$	15.5964	0.539736
$836$	−7.11604	−0.246113
$837$	30.4078	1.05105
$838$	−21.5988	−0.746119
$839$	−44.7801	−1.54598	−0.772991	−	0.634417i	$-0.781240\pi$
$839$	−44.7801	−1.54598	−0.772991	+	0.634417i	$0.781240\pi$
$840$	−6.91099	−0.238452
$841$	73.3877	2.53061
$842$	23.6867	0.816297
$843$	45.9529	1.58270
$844$	22.4666	0.773331
$845$	−8.78457	−0.302198
$846$	−36.2983	−1.24796
$847$	20.5917	0.707538
$848$	−1.06001	−0.0364007
$849$	−23.7955	−0.816658
$850$	−15.0633	−0.516668
$851$	0.118160	0.00405048
$852$	30.5029	1.04501
$853$	−36.0059	−1.23282	−0.616410	−	0.787426i	$-0.711413\pi$
$853$	−36.0059	−1.23282	−0.616410	+	0.787426i	$0.711413\pi$
$854$	−20.5131	−0.701942
$855$	−14.9820	−0.512373
$856$	−1.13067	−0.0386455
$857$	31.3917	1.07232	0.536160	−	0.844116i	$-0.319874\pi$
$857$	31.3917	1.07232	0.536160	+	0.844116i	$0.319874\pi$
$858$	−9.55156	−0.326085
$859$	−50.0090	−1.70628	−0.853142	−	0.521679i	$-0.825306\pi$
$859$	−50.0090	−1.70628	−0.853142	+	0.521679i	$0.825306\pi$
$860$	8.83269	0.301192
$861$	−50.8417	−1.73268
$862$	−37.8596	−1.28950
$863$	36.8503	1.25440	0.627200	−	0.778858i	$-0.284201\pi$
$863$	36.8503	1.25440	0.627200	+	0.778858i	$0.284201\pi$
$864$	3.36868	0.114605
$865$	−0.205412	−0.00698423
$866$	36.1347	1.22791
$867$	10.0187	0.340252
$868$	24.8894	0.844801
$869$	−9.77364	−0.331548
$870$	−25.3615	−0.859835
$871$	−12.5906	−0.426618
$872$	−18.5114	−0.626876
$873$	−13.2426	−0.448194
$874$	−7.35897	−0.248921
$875$	23.4415	0.792466
$876$	−4.02492	−0.135989
$877$	2.94745	0.0995284	0.0497642	−	0.998761i	$-0.484153\pi$
$877$	2.94745	0.0995284	0.0497642	+	0.998761i	$0.484153\pi$
$878$	−6.00547	−0.202675
$879$	−43.5850	−1.47008
$880$	1.74930	0.0589690
$881$	12.1401	0.409009	0.204504	−	0.978866i	$-0.434442\pi$
$881$	12.1401	0.409009	0.204504	+	0.978866i	$0.434442\pi$
$882$	−2.56289	−0.0862971
$883$	9.43037	0.317357	0.158679	−	0.987330i	$-0.449277\pi$
$883$	9.43037	0.317357	0.158679	+	0.987330i	$0.449277\pi$
$884$	6.87783	0.231327
$885$	12.5823	0.422950
$886$	−0.145671	−0.00489393
$887$	−8.89817	−0.298771	−0.149386	−	0.988779i	$-0.547730\pi$
$887$	−8.89817	−0.298771	−0.149386	+	0.988779i	$0.547730\pi$
$888$	−0.163712	−0.00549380
$889$	47.4382	1.59103
$890$	5.83163	0.195477
$891$	−6.91975	−0.231820
$892$	21.9951	0.736450
$893$	−32.3311	−1.08192
$894$	−11.6035	−0.388080
$895$	−15.5504	−0.519792
$896$	2.75733	0.0921161
$897$	−9.87763	−0.329805
$898$	−38.5888	−1.28772
$899$	91.3374	3.04627
$900$	−17.5721	−0.585737
$901$	3.86276	0.128687
$902$	12.8690	0.428491
$903$	70.4579	2.34469
$904$	−8.73109	−0.290392
$905$	5.23519	0.174024
$906$	−23.6071	−0.784294
$907$	8.66708	0.287786	0.143893	−	0.989593i	$-0.454038\pi$
$907$	8.66708	0.287786	0.143893	+	0.989593i	$0.454038\pi$
$908$	−26.4944	−0.879246
$909$	41.2934	1.36962
$910$	−4.84399	−0.160577
$911$	13.6303	0.451590	0.225795	−	0.974175i	$-0.427502\pi$
$911$	13.6303	0.451590	0.225795	+	0.974175i	$0.427502\pi$
$912$	10.1959	0.337619
$913$	−1.91514	−0.0633818
$914$	−0.169863	−0.00561858
$915$	18.6463	0.616427
$916$	0.892211	0.0294795
$917$	4.34184	0.143380
$918$	−12.2758	−0.405161
$919$	18.4237	0.607741	0.303870	−	0.952713i	$-0.401721\pi$
$919$	18.4237	0.607741	0.303870	+	0.952713i	$0.401721\pi$
$920$	1.80902	0.0596416
$921$	−76.8850	−2.53345
$922$	−28.2291	−0.929675
$923$	21.3798	0.703724
$924$	13.9541	0.459056
$925$	0.251311	0.00826306
$926$	27.8692	0.915838
$927$	−32.0258	−1.05187
$928$	10.1187	0.332162
$929$	−14.1475	−0.464165	−0.232083	−	0.972696i	$-0.574554\pi$
$929$	−14.1475	−0.464165	−0.232083	+	0.972696i	$0.574554\pi$
$930$	−22.6243	−0.741882
$931$	−2.28278	−0.0748151
$932$	−10.8204	−0.354435
$933$	−34.5085	−1.12976
$934$	−23.9808	−0.784675
$935$	−6.37462	−0.208472
$936$	8.02332	0.262250
$937$	−50.5113	−1.65013	−0.825067	−	0.565035i	$-0.808863\pi$
$937$	−50.5113	−1.65013	−0.825067	+	0.565035i	$0.808863\pi$
$938$	18.3940	0.600584
$939$	49.5944	1.61845
$940$	7.94779	0.259228
$941$	35.5256	1.15810	0.579052	−	0.815291i	$-0.303423\pi$
$941$	35.5256	1.15810	0.579052	+	0.815291i	$0.303423\pi$
$942$	−51.3236	−1.67221
$943$	13.3083	0.433379
$944$	−5.02007	−0.163389
$945$	8.64572	0.281245
$946$	−17.8342	−0.579841
$947$	−38.6413	−1.25567	−0.627837	−	0.778345i	$-0.716059\pi$
$947$	−38.6413	−1.25567	−0.627837	+	0.778345i	$0.716059\pi$
$948$	14.0037	0.454819
$949$	−2.82111	−0.0915771
$950$	−15.6515	−0.507803
$951$	86.5080	2.80522
$952$	−10.0480	−0.325657
$953$	7.13924	0.231263	0.115631	−	0.993292i	$-0.463111\pi$
$953$	7.13924	0.231263	0.115631	+	0.993292i	$0.463111\pi$
$954$	4.50609	0.145890
$955$	−11.0560	−0.357762
$956$	−16.1783	−0.523243
$957$	51.2078	1.65531
$958$	30.1451	0.973944
$959$	−9.60794	−0.310257
$960$	−2.50640	−0.0808939
$961$	50.4798	1.62838
$962$	−0.114747	−0.00369960
$963$	4.80648	0.154887
$964$	9.39784	0.302684
$965$	−10.7176	−0.345013
$966$	14.4305	0.464292
$967$	−25.9656	−0.834996	−0.417498	−	0.908678i	$-0.637093\pi$
$967$	−25.9656	−0.834996	−0.417498	+	0.908678i	$0.637093\pi$
$968$	7.46796	0.240029
$969$	−37.1547	−1.19358
$970$	2.89956	0.0930994
$971$	−10.2381	−0.328556	−0.164278	−	0.986414i	$-0.552529\pi$
$971$	−10.2381	−0.328556	−0.164278	+	0.986414i	$0.552529\pi$
$972$	20.0207	0.642164
$973$	−21.4602	−0.687983
$974$	12.5242	0.401300
$975$	−21.0084	−0.672807
$976$	−7.43945	−0.238131
$977$	2.39826	0.0767270	0.0383635	−	0.999264i	$-0.487786\pi$
$977$	2.39826	0.0767270	0.0383635	+	0.999264i	$0.487786\pi$
$978$	−48.8088	−1.56073
$979$	−11.7747	−0.376322
$980$	0.561165	0.0179258
$981$	78.6922	2.51245
$982$	−4.59365	−0.146589
$983$	46.0826	1.46981	0.734903	−	0.678172i	$-0.237228\pi$
$983$	46.0826	1.46981	0.734903	+	0.678172i	$0.237228\pi$
$984$	−18.4387	−0.587805
$985$	−19.4520	−0.619792
$986$	−36.8734	−1.17429
$987$	63.3991	2.01802
$988$	7.14640	0.227357
$989$	−18.4430	−0.586455
$990$	−7.43630	−0.236341
$991$	−15.6410	−0.496854	−0.248427	−	0.968651i	$-0.579914\pi$
$991$	−15.6410	−0.496854	−0.248427	+	0.968651i	$0.579914\pi$
$992$	9.02661	0.286595
$993$	−6.79453	−0.215618
$994$	−31.2342	−0.990689
$995$	16.4384	0.521132
$996$	2.74401	0.0869474
$997$	57.5113	1.82140	0.910701	−	0.413066i	$-0.135542\pi$
$997$	57.5113	1.82140	0.910701	+	0.413066i	$0.135542\pi$
$998$	−13.7560	−0.435438
$999$	0.204805	0.00647973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.6	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.6	✓	77	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.69277 q^{3} +1.00000 q^{4} +0.930791 q^{5} +2.69277 q^{6} -2.75733 q^{7} -1.00000 q^{8} +4.25101 q^{9} -0.930791 q^{10} +1.87937 q^{11} -2.69277 q^{12} -1.88739 q^{13} +2.75733 q^{14} -2.50640 q^{15} +1.00000 q^{16} -3.64409 q^{17} -4.25101 q^{18} -3.78639 q^{19} +0.930791 q^{20} +7.42487 q^{21} -1.87937 q^{22} -1.94353 q^{23} +2.69277 q^{24} -4.13363 q^{25} +1.88739 q^{26} -3.36868 q^{27} -2.75733 q^{28} -10.1187 q^{29} +2.50640 q^{30} -9.02661 q^{31} -1.00000 q^{32} -5.06072 q^{33} +3.64409 q^{34} -2.56650 q^{35} +4.25101 q^{36} -0.0607967 q^{37} +3.78639 q^{38} +5.08231 q^{39} -0.930791 q^{40} -6.84750 q^{41} -7.42487 q^{42} +9.48945 q^{43} +1.87937 q^{44} +3.95680 q^{45} +1.94353 q^{46} +8.53876 q^{47} -2.69277 q^{48} +0.602891 q^{49} +4.13363 q^{50} +9.81271 q^{51} -1.88739 q^{52} -1.06001 q^{53} +3.36868 q^{54} +1.74930 q^{55} +2.75733 q^{56} +10.1959 q^{57} +10.1187 q^{58} -5.02007 q^{59} -2.50640 q^{60} -7.43945 q^{61} +9.02661 q^{62} -11.7215 q^{63} +1.00000 q^{64} -1.75677 q^{65} +5.06072 q^{66} +6.67092 q^{67} -3.64409 q^{68} +5.23348 q^{69} +2.56650 q^{70} -11.3277 q^{71} -4.25101 q^{72} +1.49471 q^{73} +0.0607967 q^{74} +11.1309 q^{75} -3.78639 q^{76} -5.18206 q^{77} -5.08231 q^{78} -5.20048 q^{79} +0.930791 q^{80} -3.68195 q^{81} +6.84750 q^{82} -1.01903 q^{83} +7.42487 q^{84} -3.39189 q^{85} -9.48945 q^{86} +27.2473 q^{87} -1.87937 q^{88} -6.26525 q^{89} -3.95680 q^{90} +5.20417 q^{91} -1.94353 q^{92} +24.3066 q^{93} -8.53876 q^{94} -3.52434 q^{95} +2.69277 q^{96} -3.11516 q^{97} -0.602891 q^{98} +7.98923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19}+ \cdots + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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