LMFDB - Embedded newform 6035.2.a.a.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6035,2,Mod(1,6035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6035, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6035 = 5 \cdot 17 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6035.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:	$48.1897176198$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.890323 q^{2} -0.682987 q^{3} -1.20732 q^{4} +1.00000 q^{5} +0.608079 q^{6} +2.96744 q^{7} +2.85556 q^{8} -2.53353 q^{9} +O(q^{10})$	\(q-0.890323 q^{2} -0.682987 q^{3} -1.20732 q^{4} +1.00000 q^{5} +0.608079 q^{6} +2.96744 q^{7} +2.85556 q^{8} -2.53353 q^{9} -0.890323 q^{10} +3.71858 q^{11} +0.824587 q^{12} +5.77711 q^{13} -2.64198 q^{14} -0.682987 q^{15} -0.127719 q^{16} +1.00000 q^{17} +2.25566 q^{18} -4.79179 q^{19} -1.20732 q^{20} -2.02672 q^{21} -3.31074 q^{22} -8.42555 q^{23} -1.95031 q^{24} +1.00000 q^{25} -5.14350 q^{26} +3.77933 q^{27} -3.58266 q^{28} -0.00828316 q^{29} +0.608079 q^{30} +3.83066 q^{31} -5.59740 q^{32} -2.53974 q^{33} -0.890323 q^{34} +2.96744 q^{35} +3.05879 q^{36} -7.02127 q^{37} +4.26624 q^{38} -3.94569 q^{39} +2.85556 q^{40} -5.86192 q^{41} +1.80444 q^{42} -4.94908 q^{43} -4.48953 q^{44} -2.53353 q^{45} +7.50147 q^{46} -9.42907 q^{47} +0.0872305 q^{48} +1.80569 q^{49} -0.890323 q^{50} -0.682987 q^{51} -6.97485 q^{52} +2.37173 q^{53} -3.36482 q^{54} +3.71858 q^{55} +8.47368 q^{56} +3.27273 q^{57} +0.00737469 q^{58} +3.16394 q^{59} +0.824587 q^{60} -10.1659 q^{61} -3.41052 q^{62} -7.51809 q^{63} +5.23893 q^{64} +5.77711 q^{65} +2.26119 q^{66} -6.52705 q^{67} -1.20732 q^{68} +5.75454 q^{69} -2.64198 q^{70} -1.00000 q^{71} -7.23463 q^{72} -8.91081 q^{73} +6.25120 q^{74} -0.682987 q^{75} +5.78524 q^{76} +11.0347 q^{77} +3.51294 q^{78} -14.2740 q^{79} -0.127719 q^{80} +5.01936 q^{81} +5.21901 q^{82} -7.13670 q^{83} +2.44691 q^{84} +1.00000 q^{85} +4.40628 q^{86} +0.00565729 q^{87} +10.6186 q^{88} -16.6205 q^{89} +2.25566 q^{90} +17.1432 q^{91} +10.1724 q^{92} -2.61629 q^{93} +8.39492 q^{94} -4.79179 q^{95} +3.82295 q^{96} -14.0768 q^{97} -1.60764 q^{98} -9.42113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) $ 36 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 + 36 * q^5 - 10 * q^6 - 7 * q^7 - 9 * q^8 + 10 * q^9	\( 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9} - 3 q^{10} - 20 q^{11} - 8 q^{12} - 29 q^{13} - 12 q^{14} - 8 q^{15} + q^{16} + 36 q^{17} - 8 q^{18} - 19 q^{19} + 23 q^{20} - 19 q^{21} - 10 q^{22} - 10 q^{23} - 23 q^{24} + 36 q^{25} - 32 q^{26} - 23 q^{27} - 20 q^{28} - 52 q^{29} - 10 q^{30} - 15 q^{31} - 16 q^{32} - 19 q^{33} - 3 q^{34} - 7 q^{35} + 9 q^{36} - 52 q^{37} + 7 q^{38} - 10 q^{39} - 9 q^{40} - 51 q^{41} - 2 q^{42} - 13 q^{43} - 27 q^{44} + 10 q^{45} + 12 q^{46} - 24 q^{47} + 12 q^{48} - 15 q^{49} - 3 q^{50} - 8 q^{51} - 49 q^{52} - 13 q^{53} - 48 q^{54} - 20 q^{55} - 12 q^{56} - 20 q^{57} - 20 q^{58} - 14 q^{59} - 8 q^{60} - 75 q^{61} - 7 q^{62} + 16 q^{63} - 41 q^{64} - 29 q^{65} - q^{66} - 5 q^{67} + 23 q^{68} - 37 q^{69} - 12 q^{70} - 36 q^{71} - 23 q^{72} - 21 q^{73} + q^{74} - 8 q^{75} - 40 q^{76} - 31 q^{77} + 84 q^{78} - 49 q^{79} + q^{80} - 56 q^{81} - 51 q^{82} + 6 q^{83} + 10 q^{84} + 36 q^{85} - 41 q^{86} - 4 q^{87} - 21 q^{88} - 78 q^{89} - 8 q^{90} - 25 q^{91} - 24 q^{92} - 36 q^{93} + 6 q^{94} - 19 q^{95} - 71 q^{96} - 48 q^{97} + 51 q^{98} - 17 q^{99}+O(q^{100}) \) 36 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 + 36 * q^5 - 10 * q^6 - 7 * q^7 - 9 * q^8 + 10 * q^9 - 3 * q^10 - 20 * q^11 - 8 * q^12 - 29 * q^13 - 12 * q^14 - 8 * q^15 + q^16 + 36 * q^17 - 8 * q^18 - 19 * q^19 + 23 * q^20 - 19 * q^21 - 10 * q^22 - 10 * q^23 - 23 * q^24 + 36 * q^25 - 32 * q^26 - 23 * q^27 - 20 * q^28 - 52 * q^29 - 10 * q^30 - 15 * q^31 - 16 * q^32 - 19 * q^33 - 3 * q^34 - 7 * q^35 + 9 * q^36 - 52 * q^37 + 7 * q^38 - 10 * q^39 - 9 * q^40 - 51 * q^41 - 2 * q^42 - 13 * q^43 - 27 * q^44 + 10 * q^45 + 12 * q^46 - 24 * q^47 + 12 * q^48 - 15 * q^49 - 3 * q^50 - 8 * q^51 - 49 * q^52 - 13 * q^53 - 48 * q^54 - 20 * q^55 - 12 * q^56 - 20 * q^57 - 20 * q^58 - 14 * q^59 - 8 * q^60 - 75 * q^61 - 7 * q^62 + 16 * q^63 - 41 * q^64 - 29 * q^65 - q^66 - 5 * q^67 + 23 * q^68 - 37 * q^69 - 12 * q^70 - 36 * q^71 - 23 * q^72 - 21 * q^73 + q^74 - 8 * q^75 - 40 * q^76 - 31 * q^77 + 84 * q^78 - 49 * q^79 + q^80 - 56 * q^81 - 51 * q^82 + 6 * q^83 + 10 * q^84 + 36 * q^85 - 41 * q^86 - 4 * q^87 - 21 * q^88 - 78 * q^89 - 8 * q^90 - 25 * q^91 - 24 * q^92 - 36 * q^93 + 6 * q^94 - 19 * q^95 - 71 * q^96 - 48 * q^97 + 51 * q^98 - 17 * 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 $

$2$	−0.890323	−0.629554	−0.314777	−	0.949166i	$-0.601930\pi$
$2$	−0.890323	−0.629554	−0.314777	+	0.949166i	$0.601930\pi$
$3$	−0.682987	−0.394323	−0.197161	−	0.980371i	$-0.563172\pi$
$3$	−0.682987	−0.394323	−0.197161	+	0.980371i	$0.563172\pi$
$4$	−1.20732	−0.603662
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.608079	0.248247
$7$	2.96744	1.12159	0.560793	−	0.827956i	$-0.310496\pi$
$7$	2.96744	1.12159	0.560793	+	0.827956i	$0.310496\pi$
$8$	2.85556	1.00959
$9$	−2.53353	−0.844510
$10$	−0.890323	−0.281545
$11$	3.71858	1.12119	0.560597	−	0.828089i	$-0.310572\pi$
$11$	3.71858	1.12119	0.560597	+	0.828089i	$0.310572\pi$
$12$	0.824587	0.238038
$13$	5.77711	1.60228	0.801141	−	0.598475i	$-0.204227\pi$
$13$	5.77711	1.60228	0.801141	+	0.598475i	$0.204227\pi$
$14$	−2.64198	−0.706099
$15$	−0.682987	−0.176346
$16$	−0.127719	−0.0319298
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.25566	0.531664
$19$	−4.79179	−1.09931	−0.549656	−	0.835391i	$-0.685241\pi$
$19$	−4.79179	−1.09931	−0.549656	+	0.835391i	$0.685241\pi$
$20$	−1.20732	−0.269966
$21$	−2.02672	−0.442267
$22$	−3.31074	−0.705852
$23$	−8.42555	−1.75685	−0.878425	−	0.477881i	$-0.841405\pi$
$23$	−8.42555	−1.75685	−0.878425	+	0.477881i	$0.841405\pi$
$24$	−1.95031	−0.398105
$25$	1.00000	0.200000
$26$	−5.14350	−1.00872
$27$	3.77933	0.727332
$28$	−3.58266	−0.677059
$29$	−0.00828316	−0.00153814	−0.000769072	−	1.00000i	$-0.500245\pi$
$29$	−0.00828316	−0.00153814	−0.000769072		1.00000i	$0.500245\pi$
$30$	0.608079	0.111020
$31$	3.83066	0.688006	0.344003	−	0.938968i	$-0.388217\pi$
$31$	3.83066	0.688006	0.344003	+	0.938968i	$0.388217\pi$
$32$	−5.59740	−0.989490
$33$	−2.53974	−0.442112
$34$	−0.890323	−0.152689
$35$	2.96744	0.501588
$36$	3.05879	0.509799
$37$	−7.02127	−1.15429	−0.577145	−	0.816642i	$-0.695833\pi$
$37$	−7.02127	−1.15429	−0.577145	+	0.816642i	$0.695833\pi$
$38$	4.26624	0.692075
$39$	−3.94569	−0.631816
$40$	2.85556	0.451503
$41$	−5.86192	−0.915479	−0.457739	−	0.889086i	$-0.651341\pi$
$41$	−5.86192	−0.915479	−0.457739	+	0.889086i	$0.651341\pi$
$42$	1.80444	0.278431
$43$	−4.94908	−0.754727	−0.377364	−	0.926065i	$-0.623169\pi$
$43$	−4.94908	−0.754727	−0.377364	+	0.926065i	$0.623169\pi$
$44$	−4.48953	−0.676822
$45$	−2.53353	−0.377676
$46$	7.50147	1.10603
$47$	−9.42907	−1.37537	−0.687686	−	0.726009i	$-0.741373\pi$
$47$	−9.42907	−1.37537	−0.687686	+	0.726009i	$0.741373\pi$
$48$	0.0872305	0.0125906
$49$	1.80569	0.257955
$50$	−0.890323	−0.125911
$51$	−0.682987	−0.0956373
$52$	−6.97485	−0.967237
$53$	2.37173	0.325782	0.162891	−	0.986644i	$-0.447918\pi$
$53$	2.37173	0.325782	0.162891	+	0.986644i	$0.447918\pi$
$54$	−3.36482	−0.457895
$55$	3.71858	0.501413
$56$	8.47368	1.13234
$57$	3.27273	0.433483
$58$	0.00737469	0.000968344 0
$59$	3.16394	0.411909	0.205955	−	0.978562i	$-0.433970\pi$
$59$	3.16394	0.411909	0.205955	+	0.978562i	$0.433970\pi$
$60$	0.824587	0.106454
$61$	−10.1659	−1.30162	−0.650808	−	0.759243i	$-0.725570\pi$
$61$	−10.1659	−1.30162	−0.650808	+	0.759243i	$0.725570\pi$
$62$	−3.41052	−0.433137
$63$	−7.51809	−0.947190
$64$	5.23893	0.654867
$65$	5.77711	0.716563
$66$	2.26119	0.278333
$67$	−6.52705	−0.797406	−0.398703	−	0.917080i	$-0.630540\pi$
$67$	−6.52705	−0.797406	−0.398703	+	0.917080i	$0.630540\pi$
$68$	−1.20732	−0.146410
$69$	5.75454	0.692766
$70$	−2.64198	−0.315777
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	−7.23463	−0.852610
$73$	−8.91081	−1.04293	−0.521466	−	0.853272i	$-0.674615\pi$
$73$	−8.91081	−1.04293	−0.521466	+	0.853272i	$0.674615\pi$
$74$	6.25120	0.726687
$75$	−0.682987	−0.0788645
$76$	5.78524	0.663613
$77$	11.0347	1.25752
$78$	3.51294	0.397762
$79$	−14.2740	−1.60595	−0.802975	−	0.596013i	$-0.796751\pi$
$79$	−14.2740	−1.60595	−0.802975	+	0.596013i	$0.796751\pi$
$80$	−0.127719	−0.0142794
$81$	5.01936	0.557706
$82$	5.21901	0.576343
$83$	−7.13670	−0.783355	−0.391677	−	0.920103i	$-0.628105\pi$
$83$	−7.13670	−0.783355	−0.391677	+	0.920103i	$0.628105\pi$
$84$	2.44691	0.266980
$85$	1.00000	0.108465
$86$	4.40628	0.475141
$87$	0.00565729	0.000606525 0
$88$	10.6186	1.13195
$89$	−16.6205	−1.76177	−0.880887	−	0.473326i	$-0.843053\pi$
$89$	−16.6205	−1.76177	−0.880887	+	0.473326i	$0.843053\pi$
$90$	2.25566	0.237767
$91$	17.1432	1.79710
$92$	10.1724	1.06054
$93$	−2.61629	−0.271296
$94$	8.39492	0.865870
$95$	−4.79179	−0.491627
$96$	3.82295	0.390178
$97$	−14.0768	−1.42929	−0.714643	−	0.699489i	$-0.753411\pi$
$97$	−14.0768	−1.42929	−0.714643	+	0.699489i	$0.753411\pi$
$98$	−1.60764	−0.162397
$99$	−9.42113	−0.946859
$100$	−1.20732	−0.120732
$101$	9.27365	0.922763	0.461381	−	0.887202i	$-0.347354\pi$
$101$	9.27365	0.922763	0.461381	+	0.887202i	$0.347354\pi$
$102$	0.608079	0.0602088
$103$	5.12018	0.504507	0.252253	−	0.967661i	$-0.418828\pi$
$103$	5.12018	0.504507	0.252253	+	0.967661i	$0.418828\pi$
$104$	16.4969	1.61765
$105$	−2.02672	−0.197788
$106$	−2.11161	−0.205097
$107$	14.4455	1.39650	0.698248	−	0.715856i	$-0.253963\pi$
$107$	14.4455	1.39650	0.698248	+	0.715856i	$0.253963\pi$
$108$	−4.56287	−0.439063
$109$	2.19108	0.209867	0.104934	−	0.994479i	$-0.466537\pi$
$109$	2.19108	0.209867	0.104934	+	0.994479i	$0.466537\pi$
$110$	−3.31074	−0.315666
$111$	4.79544	0.455163
$112$	−0.378999	−0.0358120
$113$	2.95287	0.277782	0.138891	−	0.990308i	$-0.455646\pi$
$113$	2.95287	0.277782	0.138891	+	0.990308i	$0.455646\pi$
$114$	−2.91379	−0.272901
$115$	−8.42555	−0.785687
$116$	0.0100005	0.000928519 0
$117$	−14.6365	−1.35314
$118$	−2.81693	−0.259319
$119$	2.96744	0.272025
$120$	−1.95031	−0.178038
$121$	2.82783	0.257076
$122$	9.05098	0.819437
$123$	4.00362	0.360994
$124$	−4.62485	−0.415323
$125$	1.00000	0.0894427
$126$	6.69353	0.596307
$127$	3.93888	0.349519	0.174760	−	0.984611i	$-0.444085\pi$
$127$	3.93888	0.349519	0.174760	+	0.984611i	$0.444085\pi$
$128$	6.53045	0.577216
$129$	3.38015	0.297606
$130$	−5.14350	−0.451115
$131$	11.5843	1.01213	0.506063	−	0.862496i	$-0.331100\pi$
$131$	11.5843	1.01213	0.506063	+	0.862496i	$0.331100\pi$
$132$	3.06629	0.266886
$133$	−14.2193	−1.23297
$134$	5.81118	0.502010
$135$	3.77933	0.325273
$136$	2.85556	0.244862
$137$	−1.41706	−0.121068	−0.0605338	−	0.998166i	$-0.519280\pi$
$137$	−1.41706	−0.121068	−0.0605338	+	0.998166i	$0.519280\pi$
$138$	−5.12340	−0.436133
$139$	2.61305	0.221636	0.110818	−	0.993841i	$-0.464653\pi$
$139$	2.61305	0.221636	0.110818	+	0.993841i	$0.464653\pi$
$140$	−3.58266	−0.302790
$141$	6.43993	0.542340
$142$	0.890323	0.0747143
$143$	21.4826	1.79647
$144$	0.323580	0.0269650
$145$	−0.00828316	−0.000687879 0
$146$	7.93351	0.656582
$147$	−1.23326	−0.101718
$148$	8.47695	0.696801
$149$	10.5418	0.863620	0.431810	−	0.901965i	$-0.357875\pi$
$149$	10.5418	0.863620	0.431810	+	0.901965i	$0.357875\pi$
$150$	0.608079	0.0496495
$151$	19.6330	1.59771	0.798855	−	0.601524i	$-0.205439\pi$
$151$	19.6330	1.59771	0.798855	+	0.601524i	$0.205439\pi$
$152$	−13.6832	−1.10986
$153$	−2.53353	−0.204824
$154$	−9.82441	−0.791673
$155$	3.83066	0.307686
$156$	4.76373	0.381404
$157$	−6.01892	−0.480363	−0.240181	−	0.970728i	$-0.577207\pi$
$157$	−6.01892	−0.480363	−0.240181	+	0.970728i	$0.577207\pi$
$158$	12.7085	1.01103
$159$	−1.61986	−0.128463
$160$	−5.59740	−0.442513
$161$	−25.0023	−1.97046
$162$	−4.46885	−0.351106
$163$	11.5026	0.900952	0.450476	−	0.892789i	$-0.351254\pi$
$163$	11.5026	0.900952	0.450476	+	0.892789i	$0.351254\pi$
$164$	7.07724	0.552640
$165$	−2.53974	−0.197719
$166$	6.35397	0.493164
$167$	−21.4661	−1.66110	−0.830550	−	0.556944i	$-0.811974\pi$
$167$	−21.4661	−1.66110	−0.830550	+	0.556944i	$0.811974\pi$
$168$	−5.78741	−0.446509
$169$	20.3750	1.56731
$170$	−0.890323	−0.0682847
$171$	12.1401	0.928379
$172$	5.97514	0.455600
$173$	−15.2601	−1.16021	−0.580103	−	0.814543i	$-0.696988\pi$
$173$	−15.2601	−1.16021	−0.580103	+	0.814543i	$0.696988\pi$
$174$	−0.00503682	−0.000381840 0
$175$	2.96744	0.224317
$176$	−0.474934	−0.0357995
$177$	−2.16093	−0.162425
$178$	14.7977	1.10913
$179$	1.37016	0.102411	0.0512054	−	0.998688i	$-0.483694\pi$
$179$	1.37016	0.102411	0.0512054	+	0.998688i	$0.483694\pi$
$180$	3.05879	0.227989
$181$	−2.22948	−0.165716	−0.0828580	−	0.996561i	$-0.526405\pi$
$181$	−2.22948	−0.165716	−0.0828580	+	0.996561i	$0.526405\pi$
$182$	−15.2630	−1.13137
$183$	6.94321	0.513256
$184$	−24.0596	−1.77370
$185$	−7.02127	−0.516214
$186$	2.32934	0.170796
$187$	3.71858	0.271929
$188$	11.3839	0.830260
$189$	11.2149	0.815765
$190$	4.26624	0.309505
$191$	−12.2464	−0.886120	−0.443060	−	0.896492i	$-0.646107\pi$
$191$	−12.2464	−0.886120	−0.443060	+	0.896492i	$0.646107\pi$
$192$	−3.57812	−0.258229
$193$	2.27038	0.163425	0.0817127	−	0.996656i	$-0.473961\pi$
$193$	2.27038	0.163425	0.0817127	+	0.996656i	$0.473961\pi$
$194$	12.5329	0.899812
$195$	−3.94569	−0.282557
$196$	−2.18005	−0.155718
$197$	1.39301	0.0992477	0.0496239	−	0.998768i	$-0.484198\pi$
$197$	1.39301	0.0992477	0.0496239	+	0.998768i	$0.484198\pi$
$198$	8.38785	0.596099
$199$	−27.5530	−1.95318	−0.976590	−	0.215108i	$-0.930990\pi$
$199$	−27.5530	−1.95318	−0.976590	+	0.215108i	$0.930990\pi$
$200$	2.85556	0.201918
$201$	4.45789	0.314435
$202$	−8.25655	−0.580929
$203$	−0.0245798	−0.00172516
$204$	0.824587	0.0577326
$205$	−5.86192	−0.409415
$206$	−4.55862	−0.317614
$207$	21.3464	1.48368
$208$	−0.737848	−0.0511605
$209$	−17.8186	−1.23254
$210$	1.80444	0.124518
$211$	3.77772	0.260069	0.130035	−	0.991509i	$-0.458491\pi$
$211$	3.77772	0.260069	0.130035	+	0.991509i	$0.458491\pi$
$212$	−2.86345	−0.196662
$213$	0.682987	0.0467975
$214$	−12.8611	−0.879170
$215$	−4.94908	−0.337524
$216$	10.7921	0.734308
$217$	11.3672	0.771658
$218$	−1.95077	−0.132123
$219$	6.08597	0.411252
$220$	−4.48953	−0.302684
$221$	5.77711	0.388611
$222$	−4.26949	−0.286549
$223$	−14.6992	−0.984334	−0.492167	−	0.870501i	$-0.663795\pi$
$223$	−14.6992	−0.984334	−0.492167	+	0.870501i	$0.663795\pi$
$224$	−16.6099	−1.10980
$225$	−2.53353	−0.168902
$226$	−2.62901	−0.174879
$227$	−10.3963	−0.690024	−0.345012	−	0.938598i	$-0.612125\pi$
$227$	−10.3963	−0.690024	−0.345012	+	0.938598i	$0.612125\pi$
$228$	−3.95124	−0.261677
$229$	−17.3119	−1.14400	−0.572001	−	0.820253i	$-0.693833\pi$
$229$	−17.3119	−1.14400	−0.572001	+	0.820253i	$0.693833\pi$
$230$	7.50147	0.494632
$231$	−7.53652	−0.495867
$232$	−0.0236530	−0.00155290
$233$	−27.4401	−1.79766	−0.898831	−	0.438296i	$-0.855582\pi$
$233$	−27.4401	−1.79766	−0.898831	+	0.438296i	$0.855582\pi$
$234$	13.0312	0.851876
$235$	−9.42907	−0.615085
$236$	−3.81990	−0.248654
$237$	9.74895	0.633262
$238$	−2.64198	−0.171254
$239$	17.8848	1.15687	0.578435	−	0.815728i	$-0.303664\pi$
$239$	17.8848	1.15687	0.578435	+	0.815728i	$0.303664\pi$
$240$	0.0872305	0.00563071
$241$	22.4192	1.44415	0.722074	−	0.691816i	$-0.243189\pi$
$241$	22.4192	1.44415	0.722074	+	0.691816i	$0.243189\pi$
$242$	−2.51768	−0.161843
$243$	−14.7661	−0.947248
$244$	12.2736	0.785736
$245$	1.80569	0.115361
$246$	−3.56451	−0.227265
$247$	−27.6827	−1.76141
$248$	10.9387	0.694605
$249$	4.87427	0.308894
$250$	−0.890323	−0.0563090
$251$	9.76033	0.616066	0.308033	−	0.951376i	$-0.400329\pi$
$251$	9.76033	0.616066	0.308033	+	0.951376i	$0.400329\pi$
$252$	9.07677	0.571783
$253$	−31.3311	−1.96977
$254$	−3.50688	−0.220041
$255$	−0.682987	−0.0427703
$256$	−16.2921	−1.01826
$257$	22.9710	1.43289	0.716444	−	0.697644i	$-0.245768\pi$
$257$	22.9710	1.43289	0.716444	+	0.697644i	$0.245768\pi$
$258$	−3.00943	−0.187359
$259$	−20.8352	−1.29464
$260$	−6.97485	−0.432562
$261$	0.0209856	0.00129898
$262$	−10.3138	−0.637188
$263$	16.4018	1.01138	0.505690	−	0.862716i	$-0.331238\pi$
$263$	16.4018	1.01138	0.505690	+	0.862716i	$0.331238\pi$
$264$	−7.25237	−0.446353
$265$	2.37173	0.145694
$266$	12.6598	0.776222
$267$	11.3516	0.694708
$268$	7.88026	0.481364
$269$	−10.0822	−0.614722	−0.307361	−	0.951593i	$-0.599446\pi$
$269$	−10.0822	−0.614722	−0.307361	+	0.951593i	$0.599446\pi$
$270$	−3.36482	−0.204777
$271$	1.34822	0.0818984	0.0409492	−	0.999161i	$-0.486962\pi$
$271$	1.34822	0.0818984	0.0409492	+	0.999161i	$0.486962\pi$
$272$	−0.127719	−0.00774411
$273$	−11.7086	−0.708636
$274$	1.26164	0.0762186
$275$	3.71858	0.224239
$276$	−6.94760	−0.418196
$277$	12.2125	0.733780	0.366890	−	0.930264i	$-0.380423\pi$
$277$	12.2125	0.733780	0.366890	+	0.930264i	$0.380423\pi$
$278$	−2.32646	−0.139532
$279$	−9.70508	−0.581028
$280$	8.47368	0.506399
$281$	−16.5171	−0.985325	−0.492662	−	0.870221i	$-0.663976\pi$
$281$	−16.5171	−0.985325	−0.492662	+	0.870221i	$0.663976\pi$
$282$	−5.73362	−0.341432
$283$	12.2420	0.727712	0.363856	−	0.931455i	$-0.381460\pi$
$283$	12.2420	0.727712	0.363856	+	0.931455i	$0.381460\pi$
$284$	1.20732	0.0716415
$285$	3.27273	0.193860
$286$	−19.1265	−1.13097
$287$	−17.3949	−1.02679
$288$	14.1812	0.835634
$289$	1.00000	0.0588235
$290$	0.00737469	0.000433057 0
$291$	9.61429	0.563600
$292$	10.7582	0.629579
$293$	−4.16805	−0.243500	−0.121750	−	0.992561i	$-0.538851\pi$
$293$	−4.16805	−0.243500	−0.121750	+	0.992561i	$0.538851\pi$
$294$	1.09800	0.0640366
$295$	3.16394	0.184211
$296$	−20.0496	−1.16536
$297$	14.0537	0.815480
$298$	−9.38563	−0.543695
$299$	−48.6754	−2.81497
$300$	0.824587	0.0476075
$301$	−14.6861	−0.846491
$302$	−17.4797	−1.00584
$303$	−6.33378	−0.363866
$304$	0.612003	0.0351008
$305$	−10.1659	−0.582100
$306$	2.25566	0.128947
$307$	14.8936	0.850020	0.425010	−	0.905189i	$-0.360271\pi$
$307$	14.8936	0.850020	0.425010	+	0.905189i	$0.360271\pi$
$308$	−13.3224	−0.759114
$309$	−3.49702	−0.198938
$310$	−3.41052	−0.193705
$311$	−29.0084	−1.64492	−0.822459	−	0.568825i	$-0.807398\pi$
$311$	−29.0084	−1.64492	−0.822459	+	0.568825i	$0.807398\pi$
$312$	−11.2671	−0.637876
$313$	1.05351	0.0595481	0.0297740	−	0.999557i	$-0.490521\pi$
$313$	1.05351	0.0595481	0.0297740	+	0.999557i	$0.490521\pi$
$314$	5.35879	0.302414
$315$	−7.51809	−0.423596
$316$	17.2333	0.969451
$317$	15.0860	0.847315	0.423658	−	0.905822i	$-0.360746\pi$
$317$	15.0860	0.847315	0.423658	+	0.905822i	$0.360746\pi$
$318$	1.44220	0.0808746
$319$	−0.0308016	−0.00172456
$320$	5.23893	0.292865
$321$	−9.86607	−0.550670
$322$	22.2601	1.24051
$323$	−4.79179	−0.266622
$324$	−6.05999	−0.336666
$325$	5.77711	0.320457
$326$	−10.2410	−0.567197
$327$	−1.49648	−0.0827553
$328$	−16.7391	−0.924260
$329$	−27.9802	−1.54260
$330$	2.26119	0.124474
$331$	−31.7543	−1.74537	−0.872687	−	0.488280i	$-0.837624\pi$
$331$	−31.7543	−1.74537	−0.872687	+	0.488280i	$0.837624\pi$
$332$	8.61631	0.472882
$333$	17.7886	0.974809
$334$	19.1118	1.04575
$335$	−6.52705	−0.356611
$336$	0.258851	0.0141215
$337$	2.08905	0.113798	0.0568989	−	0.998380i	$-0.481879\pi$
$337$	2.08905	0.113798	0.0568989	+	0.998380i	$0.481879\pi$
$338$	−18.1404	−0.986705
$339$	−2.01677	−0.109536
$340$	−1.20732	−0.0654764
$341$	14.2446	0.771388
$342$	−10.8086	−0.584464
$343$	−15.4138	−0.832267
$344$	−14.1324	−0.761966
$345$	5.75454	0.309814
$346$	13.5864	0.730412
$347$	−10.9407	−0.587328	−0.293664	−	0.955909i	$-0.594875\pi$
$347$	−10.9407	−0.587328	−0.293664	+	0.955909i	$0.594875\pi$
$348$	−0.00683018	−0.000366136 0
$349$	30.4783	1.63146	0.815732	−	0.578430i	$-0.196335\pi$
$349$	30.4783	1.63146	0.815732	+	0.578430i	$0.196335\pi$
$350$	−2.64198	−0.141220
$351$	21.8336	1.16539
$352$	−20.8144	−1.10941
$353$	25.7528	1.37068	0.685341	−	0.728222i	$-0.259653\pi$
$353$	25.7528	1.37068	0.685341	+	0.728222i	$0.259653\pi$
$354$	1.92392	0.102255
$355$	−1.00000	−0.0530745
$356$	20.0664	1.06352
$357$	−2.02672	−0.107265
$358$	−1.21989	−0.0644731
$359$	−34.9341	−1.84375	−0.921876	−	0.387486i	$-0.873344\pi$
$359$	−34.9341	−1.84375	−0.921876	+	0.387486i	$0.873344\pi$
$360$	−7.23463	−0.381299
$361$	3.96121	0.208485
$362$	1.98496	0.104327
$363$	−1.93137	−0.101371
$364$	−20.6974	−1.08484
$365$	−8.91081	−0.466413
$366$	−6.18170	−0.323123
$367$	−8.44002	−0.440565	−0.220283	−	0.975436i	$-0.570698\pi$
$367$	−8.44002	−0.440565	−0.220283	+	0.975436i	$0.570698\pi$
$368$	1.07610	0.0560958
$369$	14.8514	0.773131
$370$	6.25120	0.324985
$371$	7.03796	0.365393
$372$	3.15871	0.163771
$373$	15.1244	0.783111	0.391556	−	0.920154i	$-0.371937\pi$
$373$	15.1244	0.783111	0.391556	+	0.920154i	$0.371937\pi$
$374$	−3.31074	−0.171194
$375$	−0.682987	−0.0352693
$376$	−26.9252	−1.38856
$377$	−0.0478527	−0.00246454
$378$	−9.98490	−0.513568
$379$	28.3648	1.45700	0.728501	−	0.685045i	$-0.240217\pi$
$379$	28.3648	1.45700	0.728501	+	0.685045i	$0.240217\pi$
$380$	5.78524	0.296777
$381$	−2.69020	−0.137823
$382$	10.9033	0.557860
$383$	23.5686	1.20430	0.602148	−	0.798384i	$-0.294312\pi$
$383$	23.5686	1.20430	0.602148	+	0.798384i	$0.294312\pi$
$384$	−4.46022	−0.227609
$385$	11.0347	0.562378
$386$	−2.02137	−0.102885
$387$	12.5386	0.637374
$388$	16.9953	0.862806
$389$	−24.8568	−1.26029	−0.630145	−	0.776477i	$-0.717005\pi$
$389$	−24.8568	−1.26029	−0.630145	+	0.776477i	$0.717005\pi$
$390$	3.51294	0.177885
$391$	−8.42555	−0.426099
$392$	5.15623	0.260429
$393$	−7.91193	−0.399104
$394$	−1.24023	−0.0624818
$395$	−14.2740	−0.718202
$396$	11.3744	0.571583
$397$	32.0695	1.60952	0.804760	−	0.593600i	$-0.202294\pi$
$397$	32.0695	1.60952	0.804760	+	0.593600i	$0.202294\pi$
$398$	24.5311	1.22963
$399$	9.71161	0.486189
$400$	−0.127719	−0.00638596
$401$	18.2067	0.909199	0.454600	−	0.890696i	$-0.349782\pi$
$401$	18.2067	0.909199	0.454600	+	0.890696i	$0.349782\pi$
$402$	−3.96896	−0.197954
$403$	22.1301	1.10238
$404$	−11.1963	−0.557037
$405$	5.01936	0.249414
$406$	0.0218839	0.00108608
$407$	−26.1092	−1.29418
$408$	−1.95031	−0.0965546
$409$	−29.9682	−1.48183	−0.740917	−	0.671597i	$-0.765609\pi$
$409$	−29.9682	−1.48183	−0.740917	+	0.671597i	$0.765609\pi$
$410$	5.21901	0.257748
$411$	0.967834	0.0477397
$412$	−6.18172	−0.304552
$413$	9.38878	0.461992
$414$	−19.0052	−0.934054
$415$	−7.13670	−0.350327
$416$	−32.3368	−1.58544
$417$	−1.78468	−0.0873960
$418$	15.8643	0.775951
$419$	25.9476	1.26762	0.633810	−	0.773488i	$-0.281490\pi$
$419$	25.9476	1.26762	0.633810	+	0.773488i	$0.281490\pi$
$420$	2.44691	0.119397
$421$	−17.7737	−0.866238	−0.433119	−	0.901337i	$-0.642587\pi$
$421$	−17.7737	−0.866238	−0.433119	+	0.901337i	$0.642587\pi$
$422$	−3.36340	−0.163728
$423$	23.8888	1.16151
$424$	6.77261	0.328907
$425$	1.00000	0.0485071
$426$	−0.608079	−0.0294615
$427$	−30.1668	−1.45987
$428$	−17.4404	−0.843012
$429$	−14.6724	−0.708389
$430$	4.40628	0.212490
$431$	−2.03912	−0.0982208	−0.0491104	−	0.998793i	$-0.515639\pi$
$431$	−2.03912	−0.0982208	−0.0491104	+	0.998793i	$0.515639\pi$
$432$	−0.482693	−0.0232236
$433$	−25.2728	−1.21453	−0.607266	−	0.794499i	$-0.707734\pi$
$433$	−25.2728	−1.21453	−0.607266	+	0.794499i	$0.707734\pi$
$434$	−10.1205	−0.485800
$435$	0.00565729	0.000271246 0
$436$	−2.64534	−0.126689
$437$	40.3735	1.93132
$438$	−5.41848	−0.258905
$439$	2.69440	0.128597	0.0642984	−	0.997931i	$-0.479519\pi$
$439$	2.69440	0.128597	0.0642984	+	0.997931i	$0.479519\pi$
$440$	10.6186	0.506222
$441$	−4.57476	−0.217845
$442$	−5.14350	−0.244651
$443$	12.3332	0.585969	0.292984	−	0.956117i	$-0.405352\pi$
$443$	12.3332	0.585969	0.292984	+	0.956117i	$0.405352\pi$
$444$	−5.78965	−0.274765
$445$	−16.6205	−0.787890
$446$	13.0871	0.619691
$447$	−7.19993	−0.340545
$448$	15.5462	0.734489
$449$	0.672029	0.0317150	0.0158575	−	0.999874i	$-0.494952\pi$
$449$	0.672029	0.0317150	0.0158575	+	0.999874i	$0.494952\pi$
$450$	2.25566	0.106333
$451$	−21.7980	−1.02643
$452$	−3.56507	−0.167687
$453$	−13.4091	−0.630013
$454$	9.25604	0.434407
$455$	17.1432	0.803686
$456$	9.34545	0.437641
$457$	−28.4691	−1.33173	−0.665865	−	0.746072i	$-0.731937\pi$
$457$	−28.4691	−1.33173	−0.665865	+	0.746072i	$0.731937\pi$
$458$	15.4132	0.720211
$459$	3.77933	0.176404
$460$	10.1724	0.474290
$461$	17.4075	0.810749	0.405375	−	0.914151i	$-0.367141\pi$
$461$	17.4075	0.810749	0.405375	+	0.914151i	$0.367141\pi$
$462$	6.70994	0.312175
$463$	−19.0927	−0.887312	−0.443656	−	0.896197i	$-0.646319\pi$
$463$	−19.0927	−0.887312	−0.443656	+	0.896197i	$0.646319\pi$
$464$	0.00105792	4.91126e−5 0
$465$	−2.61629	−0.121327
$466$	24.4306	1.13172
$467$	12.7378	0.589436	0.294718	−	0.955584i	$-0.404774\pi$
$467$	12.7378	0.589436	0.294718	+	0.955584i	$0.404774\pi$
$468$	17.6710	0.816841
$469$	−19.3686	−0.894359
$470$	8.39492	0.387229
$471$	4.11085	0.189418
$472$	9.03479	0.415860
$473$	−18.4035	−0.846195
$474$	−8.67972	−0.398673
$475$	−4.79179	−0.219862
$476$	−3.58266	−0.164211
$477$	−6.00885	−0.275126
$478$	−15.9232	−0.728312
$479$	19.5405	0.892829	0.446415	−	0.894826i	$-0.352701\pi$
$479$	19.5405	0.892829	0.446415	+	0.894826i	$0.352701\pi$
$480$	3.82295	0.174493
$481$	−40.5627	−1.84950
$482$	−19.9603	−0.909169
$483$	17.0762	0.776996
$484$	−3.41411	−0.155187
$485$	−14.0768	−0.639196
$486$	13.1466	0.596344
$487$	15.1009	0.684289	0.342145	−	0.939647i	$-0.388847\pi$
$487$	15.1009	0.684289	0.342145	+	0.939647i	$0.388847\pi$
$488$	−29.0294	−1.31410
$489$	−7.85611	−0.355266
$490$	−1.60764	−0.0726259
$491$	−18.4992	−0.834856	−0.417428	−	0.908710i	$-0.637068\pi$
$491$	−18.4992	−0.834856	−0.417428	+	0.908710i	$0.637068\pi$
$492$	−4.83367	−0.217918
$493$	−0.00828316	−0.000373055 0
$494$	24.6465	1.10890
$495$	−9.42113	−0.423448
$496$	−0.489248	−0.0219679
$497$	−2.96744	−0.133108
$498$	−4.33968	−0.194466
$499$	−39.9154	−1.78686	−0.893429	−	0.449205i	$-0.851707\pi$
$499$	−39.9154	−1.78686	−0.893429	+	0.449205i	$0.851707\pi$
$500$	−1.20732	−0.0539932
$501$	14.6611	0.655009
$502$	−8.68985	−0.387847
$503$	27.1700	1.21145	0.605725	−	0.795674i	$-0.292883\pi$
$503$	27.1700	1.21145	0.605725	+	0.795674i	$0.292883\pi$
$504$	−21.4683	−0.956275
$505$	9.27365	0.412672
$506$	27.8948	1.24008
$507$	−13.9159	−0.618026
$508$	−4.75551	−0.210991
$509$	−8.08223	−0.358239	−0.179119	−	0.983827i	$-0.557325\pi$
$509$	−8.08223	−0.358239	−0.179119	+	0.983827i	$0.557325\pi$
$510$	0.608079	0.0269262
$511$	−26.4423	−1.16974
$512$	1.44431	0.0638302
$513$	−18.1097	−0.799564
$514$	−20.4516	−0.902080
$515$	5.12018	0.225622
$516$	−4.08094	−0.179653
$517$	−35.0627	−1.54206
$518$	18.5501	0.815042
$519$	10.4225	0.457495
$520$	16.4969	0.723435
$521$	−25.4835	−1.11645	−0.558227	−	0.829688i	$-0.688518\pi$
$521$	−25.4835	−1.11645	−0.558227	+	0.829688i	$0.688518\pi$
$522$	−0.0186840	−0.000817776 0
$523$	23.0397	1.00746	0.503729	−	0.863862i	$-0.331961\pi$
$523$	23.0397	1.00746	0.503729	+	0.863862i	$0.331961\pi$
$524$	−13.9860	−0.610982
$525$	−2.02672	−0.0884534
$526$	−14.6029	−0.636717
$527$	3.83066	0.166866
$528$	0.324374	0.0141165
$529$	47.9900	2.08652
$530$	−2.11161	−0.0917224
$531$	−8.01592	−0.347861
$532$	17.1673	0.744298
$533$	−33.8650	−1.46686
$534$	−10.1066	−0.437356
$535$	14.4455	0.624532
$536$	−18.6383	−0.805054
$537$	−0.935804	−0.0403829
$538$	8.97641	0.387000
$539$	6.71458	0.289218
$540$	−4.56287	−0.196355
$541$	9.78008	0.420479	0.210239	−	0.977650i	$-0.432576\pi$
$541$	9.78008	0.420479	0.210239	+	0.977650i	$0.432576\pi$
$542$	−1.20035	−0.0515594
$543$	1.52271	0.0653455
$544$	−5.59740	−0.239987
$545$	2.19108	0.0938554
$546$	10.4244	0.446125
$547$	−28.0104	−1.19764	−0.598819	−	0.800884i	$-0.704363\pi$
$547$	−28.0104	−1.19764	−0.598819	+	0.800884i	$0.704363\pi$
$548$	1.71085	0.0730840
$549$	25.7557	1.09923
$550$	−3.31074	−0.141170
$551$	0.0396911	0.00169090
$552$	16.4324	0.699410
$553$	−42.3572	−1.80121
$554$	−10.8731	−0.461954
$555$	4.79544	0.203555
$556$	−3.15480	−0.133793
$557$	−4.29251	−0.181879	−0.0909397	−	0.995856i	$-0.528987\pi$
$557$	−4.29251	−0.181879	−0.0909397	+	0.995856i	$0.528987\pi$
$558$	8.64066	0.365788
$559$	−28.5914	−1.20929
$560$	−0.378999	−0.0160156
$561$	−2.53974	−0.107228
$562$	14.7055	0.620315
$563$	−26.6740	−1.12418	−0.562088	−	0.827078i	$-0.690002\pi$
$563$	−26.6740	−1.12418	−0.562088	+	0.827078i	$0.690002\pi$
$564$	−7.77509	−0.327390
$565$	2.95287	0.124228
$566$	−10.8993	−0.458134
$567$	14.8946	0.625515
$568$	−2.85556	−0.119816
$569$	19.3746	0.812224	0.406112	−	0.913823i	$-0.366884\pi$
$569$	19.3746	0.812224	0.406112	+	0.913823i	$0.366884\pi$
$570$	−2.91379	−0.122045
$571$	−31.4549	−1.31635	−0.658173	−	0.752867i	$-0.728670\pi$
$571$	−31.4549	−1.31635	−0.658173	+	0.752867i	$0.728670\pi$
$572$	−25.9365	−1.08446
$573$	8.36415	0.349417
$574$	15.4871	0.646418
$575$	−8.42555	−0.351370
$576$	−13.2730	−0.553041
$577$	8.11328	0.337760	0.168880	−	0.985637i	$-0.445985\pi$
$577$	8.11328	0.337760	0.168880	+	0.985637i	$0.445985\pi$
$578$	−0.890323	−0.0370326
$579$	−1.55064	−0.0644423
$580$	0.0100005	0.000415246 0
$581$	−21.1777	−0.878600
$582$	−8.55983	−0.354816
$583$	8.81947	0.365265
$584$	−25.4453	−1.05294
$585$	−14.6365	−0.605144
$586$	3.71091	0.153296
$587$	32.0825	1.32419	0.662093	−	0.749421i	$-0.269668\pi$
$587$	32.0825	1.32419	0.662093	+	0.749421i	$0.269668\pi$
$588$	1.48894	0.0614030
$589$	−18.3557	−0.756333
$590$	−2.81693	−0.115971
$591$	−0.951407	−0.0391356
$592$	0.896751	0.0368562
$593$	−8.26554	−0.339425	−0.169713	−	0.985494i	$-0.554284\pi$
$593$	−8.26554	−0.339425	−0.169713	+	0.985494i	$0.554284\pi$
$594$	−12.5124	−0.513388
$595$	2.96744	0.121653
$596$	−12.7274	−0.521335
$597$	18.8183	0.770183
$598$	43.3368	1.77217
$599$	−28.4579	−1.16276	−0.581379	−	0.813633i	$-0.697487\pi$
$599$	−28.4579	−1.16276	−0.581379	+	0.813633i	$0.697487\pi$
$600$	−1.95031	−0.0796210
$601$	0.815002	0.0332446	0.0166223	−	0.999862i	$-0.494709\pi$
$601$	0.815002	0.0332446	0.0166223	+	0.999862i	$0.494709\pi$
$602$	13.0754	0.532912
$603$	16.5365	0.673417
$604$	−23.7034	−0.964477
$605$	2.82783	0.114968
$606$	5.63911	0.229073
$607$	−31.4559	−1.27675	−0.638377	−	0.769724i	$-0.720394\pi$
$607$	−31.4559	−1.27675	−0.638377	+	0.769724i	$0.720394\pi$
$608$	26.8215	1.08776
$609$	0.0167876	0.000680270 0
$610$	9.05098	0.366463
$611$	−54.4728	−2.20373
$612$	3.05879	0.123644
$613$	29.0874	1.17483	0.587414	−	0.809287i	$-0.300146\pi$
$613$	29.0874	1.17483	0.587414	+	0.809287i	$0.300146\pi$
$614$	−13.2601	−0.535133
$615$	4.00362	0.161441
$616$	31.5101	1.26958
$617$	−33.2965	−1.34047	−0.670233	−	0.742151i	$-0.733806\pi$
$617$	−33.2965	−1.34047	−0.670233	+	0.742151i	$0.733806\pi$
$618$	3.11348	0.125242
$619$	−15.1085	−0.607262	−0.303631	−	0.952790i	$-0.598199\pi$
$619$	−15.1085	−0.607262	−0.303631	+	0.952790i	$0.598199\pi$
$620$	−4.62485	−0.185738
$621$	−31.8429	−1.27781
$622$	25.8269	1.03556
$623$	−49.3204	−1.97598
$624$	0.503940	0.0201738
$625$	1.00000	0.0400000
$626$	−0.937967	−0.0374887
$627$	12.1699	0.486019
$628$	7.26679	0.289977
$629$	−7.02127	−0.279956
$630$	6.69353	0.266677
$631$	−28.5221	−1.13545	−0.567723	−	0.823220i	$-0.692175\pi$
$631$	−28.5221	−1.13545	−0.567723	+	0.823220i	$0.692175\pi$
$632$	−40.7602	−1.62135
$633$	−2.58014	−0.102551
$634$	−13.4314	−0.533430
$635$	3.93888	0.156310
$636$	1.95570	0.0775485
$637$	10.4316	0.413317
$638$	0.0274234	0.00108570
$639$	2.53353	0.100225
$640$	6.53045	0.258139
$641$	21.7060	0.857333	0.428667	−	0.903463i	$-0.358983\pi$
$641$	21.7060	0.857333	0.428667	+	0.903463i	$0.358983\pi$
$642$	8.78399	0.346676
$643$	48.4571	1.91096	0.955482	−	0.295051i	$-0.0953365\pi$
$643$	48.4571	1.91096	0.955482	+	0.295051i	$0.0953365\pi$
$644$	30.1859	1.18949
$645$	3.38015	0.133093
$646$	4.26624	0.167853
$647$	11.2464	0.442141	0.221071	−	0.975258i	$-0.429045\pi$
$647$	11.2464	0.442141	0.221071	+	0.975258i	$0.429045\pi$
$648$	14.3330	0.563055
$649$	11.7653	0.461830
$650$	−5.14350	−0.201745
$651$	−7.76367	−0.304282
$652$	−13.8873	−0.543870
$653$	31.3892	1.22836	0.614178	−	0.789168i	$-0.289488\pi$
$653$	31.3892	1.22836	0.614178	+	0.789168i	$0.289488\pi$
$654$	1.33235	0.0520989
$655$	11.5843	0.452637
$656$	0.748680	0.0292310
$657$	22.5758	0.880766
$658$	24.9114	0.971148
$659$	48.5853	1.89261	0.946306	−	0.323271i	$-0.104783\pi$
$659$	48.5853	1.89261	0.946306	+	0.323271i	$0.104783\pi$
$660$	3.06629	0.119355
$661$	−20.3765	−0.792554	−0.396277	−	0.918131i	$-0.629698\pi$
$661$	−20.3765	−0.792554	−0.396277	+	0.918131i	$0.629698\pi$
$662$	28.2716	1.09881
$663$	−3.94569	−0.153238
$664$	−20.3792	−0.790868
$665$	−14.2193	−0.551402
$666$	−15.8376	−0.613695
$667$	0.0697902	0.00270229
$668$	25.9166	1.00274
$669$	10.0394	0.388145
$670$	5.81118	0.224506
$671$	−37.8029	−1.45936
$672$	11.3444	0.437618
$673$	17.7957	0.685974	0.342987	−	0.939340i	$-0.388561\pi$
$673$	17.7957	0.685974	0.342987	+	0.939340i	$0.388561\pi$
$674$	−1.85993	−0.0716418
$675$	3.77933	0.145466
$676$	−24.5993	−0.946125
$677$	−36.9687	−1.42082	−0.710411	−	0.703787i	$-0.751491\pi$
$677$	−36.9687	−1.42082	−0.710411	+	0.703787i	$0.751491\pi$
$678$	1.79558	0.0689587
$679$	−41.7721	−1.60307
$680$	2.85556	0.109506
$681$	7.10051	0.272092
$682$	−12.6823	−0.485630
$683$	−25.9025	−0.991131	−0.495566	−	0.868571i	$-0.665039\pi$
$683$	−25.9025	−0.991131	−0.495566	+	0.868571i	$0.665039\pi$
$684$	−14.6571	−0.560427
$685$	−1.41706	−0.0541431
$686$	13.7233	0.523957
$687$	11.8238	0.451106
$688$	0.632092	0.0240983
$689$	13.7018	0.521995
$690$	−5.12340	−0.195045
$691$	24.9696	0.949887	0.474943	−	0.880016i	$-0.342469\pi$
$691$	24.9696	0.949887	0.474943	+	0.880016i	$0.342469\pi$
$692$	18.4239	0.700372
$693$	−27.9566	−1.06198
$694$	9.74076	0.369754
$695$	2.61305	0.0991186
$696$	0.0161547	0.000612342 0
$697$	−5.86192	−0.222036
$698$	−27.1355	−1.02709
$699$	18.7412	0.708859
$700$	−3.58266	−0.135412
$701$	−33.9770	−1.28329	−0.641647	−	0.767000i	$-0.721749\pi$
$701$	−33.9770	−1.28329	−0.641647	+	0.767000i	$0.721749\pi$
$702$	−19.4390	−0.733676
$703$	33.6444	1.26892
$704$	19.4814	0.734233
$705$	6.43993	0.242542
$706$	−22.9283	−0.862918
$707$	27.5190	1.03496
$708$	2.60894	0.0980499
$709$	−11.2766	−0.423501	−0.211751	−	0.977324i	$-0.567916\pi$
$709$	−11.2766	−0.423501	−0.211751	+	0.977324i	$0.567916\pi$
$710$	0.890323	0.0334132
$711$	36.1636	1.35624
$712$	−47.4609	−1.77867
$713$	−32.2754	−1.20872
$714$	1.80444	0.0675294
$715$	21.4826	0.803405
$716$	−1.65423	−0.0618216
$717$	−12.2151	−0.456180
$718$	31.1026	1.16074
$719$	26.1636	0.975737	0.487869	−	0.872917i	$-0.337775\pi$
$719$	26.1636	0.975737	0.487869	+	0.872917i	$0.337775\pi$
$720$	0.323580	0.0120591
$721$	15.1938	0.565847
$722$	−3.52676	−0.131252
$723$	−15.3120	−0.569460
$724$	2.69171	0.100036
$725$	−0.00828316	−0.000307629 0
$726$	1.71954	0.0638183
$727$	27.2724	1.01148	0.505739	−	0.862687i	$-0.331220\pi$
$727$	27.2724	1.01148	0.505739	+	0.862687i	$0.331220\pi$
$728$	48.9534	1.81433
$729$	−4.97299	−0.184185
$730$	7.93351	0.293632
$731$	−4.94908	−0.183048
$732$	−8.38270	−0.309834
$733$	−21.8025	−0.805294	−0.402647	−	0.915355i	$-0.631910\pi$
$733$	−21.8025	−0.805294	−0.402647	+	0.915355i	$0.631910\pi$
$734$	7.51434	0.277360
$735$	−1.23326	−0.0454895
$736$	47.1612	1.73838
$737$	−24.2713	−0.894046
$738$	−13.2225	−0.486727
$739$	−22.9166	−0.843000	−0.421500	−	0.906828i	$-0.638496\pi$
$739$	−22.9166	−0.843000	−0.421500	+	0.906828i	$0.638496\pi$
$740$	8.47695	0.311619
$741$	18.9069	0.694563
$742$	−6.26606	−0.230034
$743$	47.4189	1.73963	0.869815	−	0.493379i	$-0.164238\pi$
$743$	47.4189	1.73963	0.869815	+	0.493379i	$0.164238\pi$
$744$	−7.47096	−0.273899
$745$	10.5418	0.386222
$746$	−13.4656	−0.493011
$747$	18.0810	0.661551
$748$	−4.48953	−0.164154
$749$	42.8660	1.56629
$750$	0.608079	0.0222039
$751$	−42.6833	−1.55753	−0.778767	−	0.627313i	$-0.784155\pi$
$751$	−42.6833	−1.55753	−0.778767	+	0.627313i	$0.784155\pi$
$752$	1.20427	0.0439153
$753$	−6.66618	−0.242929
$754$	0.0426044	0.00155156
$755$	19.6330	0.714518
$756$	−13.5400	−0.492447
$757$	2.82565	0.102700	0.0513500	−	0.998681i	$-0.483648\pi$
$757$	2.82565	0.102700	0.0513500	+	0.998681i	$0.483648\pi$
$758$	−25.2538	−0.917261
$759$	21.3987	0.776725
$760$	−13.6832	−0.496342
$761$	21.5575	0.781460	0.390730	−	0.920505i	$-0.372223\pi$
$761$	21.5575	0.781460	0.390730	+	0.920505i	$0.372223\pi$
$762$	2.39515	0.0867672
$763$	6.50188	0.235384
$764$	14.7854	0.534917
$765$	−2.53353	−0.0915999
$766$	−20.9836	−0.758169
$767$	18.2784	0.659995
$768$	11.1273	0.401521
$769$	2.03206	0.0732780	0.0366390	−	0.999329i	$-0.488335\pi$
$769$	2.03206	0.0732780	0.0366390	+	0.999329i	$0.488335\pi$
$770$	−9.82441	−0.354047
$771$	−15.6889	−0.565020
$772$	−2.74108	−0.0986537
$773$	−43.6536	−1.57011	−0.785055	−	0.619426i	$-0.787365\pi$
$773$	−43.6536	−1.57011	−0.785055	+	0.619426i	$0.787365\pi$
$774$	−11.1634	−0.401261
$775$	3.83066	0.137601
$776$	−40.1972	−1.44299
$777$	14.2302	0.510504
$778$	22.1306	0.793421
$779$	28.0891	1.00640
$780$	4.76373	0.170569
$781$	−3.71858	−0.133061
$782$	7.50147	0.268252
$783$	−0.0313048	−0.00111874
$784$	−0.230621	−0.00823645
$785$	−6.01892	−0.214825
$786$	7.04418	0.251258
$787$	38.7399	1.38093	0.690465	−	0.723366i	$-0.257406\pi$
$787$	38.7399	1.38093	0.690465	+	0.723366i	$0.257406\pi$
$788$	−1.68181	−0.0599121
$789$	−11.2022	−0.398810
$790$	12.7085	0.452147
$791$	8.76245	0.311557
$792$	−26.9026	−0.955941
$793$	−58.7298	−2.08556
$794$	−28.5522	−1.01328
$795$	−1.61986	−0.0574506
$796$	33.2654	1.17906
$797$	13.9754	0.495034	0.247517	−	0.968884i	$-0.420385\pi$
$797$	13.9754	0.495034	0.247517	+	0.968884i	$0.420385\pi$
$798$	−8.64648	−0.306082
$799$	−9.42907	−0.333577
$800$	−5.59740	−0.197898
$801$	42.1086	1.48784
$802$	−16.2099	−0.572390
$803$	−33.1356	−1.16933
$804$	−5.38212	−0.189813
$805$	−25.0023	−0.881216
$806$	−19.7030	−0.694008
$807$	6.88600	0.242399
$808$	26.4814	0.931613
$809$	21.2643	0.747613	0.373807	−	0.927507i	$-0.378052\pi$
$809$	21.2643	0.747613	0.373807	+	0.927507i	$0.378052\pi$
$810$	−4.46885	−0.157019
$811$	−25.5801	−0.898238	−0.449119	−	0.893472i	$-0.648262\pi$
$811$	−25.5801	−0.898238	−0.449119	+	0.893472i	$0.648262\pi$
$812$	0.0296757	0.00104141
$813$	−0.920815	−0.0322944
$814$	23.2456	0.814757
$815$	11.5026	0.402918
$816$	0.0872305	0.00305368
$817$	23.7149	0.829680
$818$	26.6814	0.932894
$819$	−43.4328	−1.51767
$820$	7.07724	0.247148
$821$	52.5774	1.83497	0.917483	−	0.397776i	$-0.130218\pi$
$821$	52.5774	1.83497	0.917483	+	0.397776i	$0.130218\pi$
$822$	−0.861685	−0.0300547
$823$	18.1455	0.632512	0.316256	−	0.948674i	$-0.397574\pi$
$823$	18.1455	0.632512	0.316256	+	0.948674i	$0.397574\pi$
$824$	14.6210	0.509346
$825$	−2.53974	−0.0884224
$826$	−8.35905	−0.290849
$827$	43.1188	1.49939	0.749694	−	0.661784i	$-0.230201\pi$
$827$	43.1188	1.49939	0.749694	+	0.661784i	$0.230201\pi$
$828$	−25.7720	−0.895639
$829$	2.11646	0.0735077	0.0367538	−	0.999324i	$-0.488298\pi$
$829$	2.11646	0.0735077	0.0367538	+	0.999324i	$0.488298\pi$
$830$	6.35397	0.220550
$831$	−8.34100	−0.289346
$832$	30.2659	1.04928
$833$	1.80569	0.0625633
$834$	1.58894	0.0550205
$835$	−21.4661	−0.742866
$836$	21.5129	0.744038
$837$	14.4773	0.500409
$838$	−23.1017	−0.798035
$839$	3.81073	0.131561	0.0657805	−	0.997834i	$-0.479046\pi$
$839$	3.81073	0.131561	0.0657805	+	0.997834i	$0.479046\pi$
$840$	−5.78741	−0.199685
$841$	−28.9999	−0.999998
$842$	15.8244	0.545344
$843$	11.2809	0.388536
$844$	−4.56094	−0.156994
$845$	20.3750	0.700922
$846$	−21.2688	−0.731236
$847$	8.39141	0.288332
$848$	−0.302916	−0.0104022
$849$	−8.36113	−0.286953
$850$	−0.890323	−0.0305378
$851$	59.1581	2.02791
$852$	−0.824587	−0.0282499
$853$	−32.3317	−1.10702	−0.553509	−	0.832843i	$-0.686712\pi$
$853$	−32.3317	−1.10702	−0.553509	+	0.832843i	$0.686712\pi$
$854$	26.8582	0.919069
$855$	12.1401	0.415184
$856$	41.2499	1.40989
$857$	−20.5494	−0.701955	−0.350978	−	0.936384i	$-0.614151\pi$
$857$	−20.5494	−0.701955	−0.350978	+	0.936384i	$0.614151\pi$
$858$	13.0632	0.445969
$859$	29.4183	1.00374	0.501870	−	0.864943i	$-0.332645\pi$
$859$	29.4183	1.00374	0.501870	+	0.864943i	$0.332645\pi$
$860$	5.97514	0.203751
$861$	11.8805	0.404886
$862$	1.81547	0.0618353
$863$	3.08601	0.105049	0.0525245	−	0.998620i	$-0.483273\pi$
$863$	3.08601	0.105049	0.0525245	+	0.998620i	$0.483273\pi$
$864$	−21.1544	−0.719688
$865$	−15.2601	−0.518860
$866$	22.5009	0.764613
$867$	−0.682987	−0.0231955
$868$	−13.7239	−0.465821
$869$	−53.0790	−1.80058
$870$	−0.00503682	−0.000170764 0
$871$	−37.7075	−1.27767
$872$	6.25674	0.211880
$873$	35.6641	1.20705
$874$	−35.9454	−1.21587
$875$	2.96744	0.100318
$876$	−7.34774	−0.248257
$877$	19.6057	0.662038	0.331019	−	0.943624i	$-0.392608\pi$
$877$	19.6057	0.662038	0.331019	+	0.943624i	$0.392608\pi$
$878$	−2.39889	−0.0809586
$879$	2.84672	0.0960175
$880$	−0.474934	−0.0160100
$881$	−18.7678	−0.632304	−0.316152	−	0.948709i	$-0.602391\pi$
$881$	−18.7678	−0.632304	−0.316152	+	0.948709i	$0.602391\pi$
$882$	4.07301	0.137145
$883$	29.3872	0.988960	0.494480	−	0.869189i	$-0.335359\pi$
$883$	29.3872	0.988960	0.494480	+	0.869189i	$0.335359\pi$
$884$	−6.97485	−0.234590
$885$	−2.16093	−0.0726387
$886$	−10.9805	−0.368899
$887$	−7.69546	−0.258388	−0.129194	−	0.991619i	$-0.541239\pi$
$887$	−7.69546	−0.258388	−0.129194	+	0.991619i	$0.541239\pi$
$888$	13.6936	0.459528
$889$	11.6884	0.392016
$890$	14.7977	0.496019
$891$	18.6649	0.625297
$892$	17.7467	0.594205
$893$	45.1821	1.51196
$894$	6.41026	0.214391
$895$	1.37016	0.0457995
$896$	19.3787	0.647397
$897$	33.2446	1.11001
$898$	−0.598323	−0.0199663
$899$	−0.0317299	−0.00105825
$900$	3.05879	0.101960
$901$	2.37173	0.0790138
$902$	19.4073	0.646192
$903$	10.0304	0.333791
$904$	8.43207	0.280447
$905$	−2.22948	−0.0741104
$906$	11.9384	0.396627
$907$	−27.2701	−0.905488	−0.452744	−	0.891641i	$-0.649555\pi$
$907$	−27.2701	−0.905488	−0.452744	+	0.891641i	$0.649555\pi$
$908$	12.5517	0.416542
$909$	−23.4951	−0.779282
$910$	−15.2630	−0.505964
$911$	15.4103	0.510565	0.255283	−	0.966867i	$-0.417832\pi$
$911$	15.4103	0.510565	0.255283	+	0.966867i	$0.417832\pi$
$912$	−0.417990	−0.0138410
$913$	−26.5384	−0.878292
$914$	25.3467	0.838396
$915$	6.94321	0.229535
$916$	20.9011	0.690591
$917$	34.3757	1.13519
$918$	−3.36482	−0.111056
$919$	−28.7520	−0.948442	−0.474221	−	0.880406i	$-0.657270\pi$
$919$	−28.7520	−0.948442	−0.474221	+	0.880406i	$0.657270\pi$
$920$	−24.0596	−0.793223
$921$	−10.1721	−0.335182
$922$	−15.4983	−0.510410
$923$	−5.77711	−0.190156
$924$	9.09903	0.299336
$925$	−7.02127	−0.230858
$926$	16.9987	0.558610
$927$	−12.9721	−0.426061
$928$	0.0463641	0.00152198
$929$	−20.3570	−0.667892	−0.333946	−	0.942592i	$-0.608380\pi$
$929$	−20.3570	−0.667892	−0.333946	+	0.942592i	$0.608380\pi$
$930$	2.32934	0.0763822
$931$	−8.65246	−0.283573
$932$	33.1291	1.08518
$933$	19.8124	0.648628
$934$	−11.3408	−0.371082
$935$	3.71858	0.121611
$936$	−41.7953	−1.36612
$937$	31.9253	1.04295	0.521476	−	0.853266i	$-0.325381\pi$
$937$	31.9253	1.04295	0.521476	+	0.853266i	$0.325381\pi$
$938$	17.2443	0.563047
$939$	−0.719536	−0.0234812
$940$	11.3839	0.371303
$941$	29.6774	0.967456	0.483728	−	0.875218i	$-0.339282\pi$
$941$	29.6774	0.967456	0.483728	+	0.875218i	$0.339282\pi$
$942$	−3.65998	−0.119249
$943$	49.3900	1.60836
$944$	−0.404095	−0.0131522
$945$	11.2149	0.364821
$946$	16.3851	0.532725
$947$	52.8494	1.71737	0.858687	−	0.512501i	$-0.171281\pi$
$947$	52.8494	1.71737	0.858687	+	0.512501i	$0.171281\pi$
$948$	−11.7701	−0.382277
$949$	−51.4788	−1.67107
$950$	4.26624	0.138415
$951$	−10.3036	−0.334116
$952$	8.47368	0.274634
$953$	17.9449	0.581291	0.290645	−	0.956831i	$-0.406130\pi$
$953$	17.9449	0.581291	0.290645	+	0.956831i	$0.406130\pi$
$954$	5.34982	0.173207
$955$	−12.2464	−0.396285
$956$	−21.5927	−0.698359
$957$	0.0210371	0.000680032 0
$958$	−17.3974	−0.562084
$959$	−4.20504	−0.135788
$960$	−3.57812	−0.115483
$961$	−16.3261	−0.526647
$962$	36.1139	1.16436
$963$	−36.5980	−1.17935
$964$	−27.0673	−0.871778
$965$	2.27038	0.0730860
$966$	−15.2034	−0.489161
$967$	−55.1100	−1.77222	−0.886109	−	0.463476i	$-0.846602\pi$
$967$	−55.1100	−1.77222	−0.886109	+	0.463476i	$0.846602\pi$
$968$	8.07503	0.259541
$969$	3.27273	0.105135
$970$	12.5329	0.402408
$971$	52.2904	1.67808	0.839039	−	0.544071i	$-0.183118\pi$
$971$	52.2904	1.67808	0.839039	+	0.544071i	$0.183118\pi$
$972$	17.8275	0.571818
$973$	7.75406	0.248584
$974$	−13.4447	−0.430797
$975$	−3.94569	−0.126363
$976$	1.29839	0.0415603
$977$	6.27473	0.200746	0.100373	−	0.994950i	$-0.467996\pi$
$977$	6.27473	0.200746	0.100373	+	0.994950i	$0.467996\pi$
$978$	6.99448	0.223659
$979$	−61.8048	−1.97529
$980$	−2.18005	−0.0696391
$981$	−5.55115	−0.177235
$982$	16.4702	0.525587
$983$	−30.7995	−0.982352	−0.491176	−	0.871060i	$-0.663433\pi$
$983$	−30.7995	−0.982352	−0.491176	+	0.871060i	$0.663433\pi$
$984$	11.4326	0.364456
$985$	1.39301	0.0443849
$986$	0.00737469	0.000234858 0
$987$	19.1101	0.608281
$988$	33.4220	1.06329
$989$	41.6987	1.32594
$990$	8.38785	0.266583
$991$	48.3294	1.53523	0.767617	−	0.640909i	$-0.221443\pi$
$991$	48.3294	1.53523	0.767617	+	0.640909i	$0.221443\pi$
$992$	−21.4417	−0.680775
$993$	21.6878	0.688240
$994$	2.64198	0.0837985
$995$	−27.5530	−0.873489
$996$	−5.88483	−0.186468
$997$	−48.9637	−1.55070	−0.775348	−	0.631534i	$-0.782426\pi$
$997$	−48.9637	−1.55070	−0.775348	+	0.631534i	$0.782426\pi$
$998$	35.5376	1.12492
$999$	−26.5357	−0.839552

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6035.2.a.a.1.15	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6035.2.a.a.1.15	✓	36	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6035 = 5 \cdot 17 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6035.a (trivial)

\(f(q)\)	\(=\)	\(q-0.890323 q^{2} -0.682987 q^{3} -1.20732 q^{4} +1.00000 q^{5} +0.608079 q^{6} +2.96744 q^{7} +2.85556 q^{8} -2.53353 q^{9} +O(q^{10})\)	\(q-0.890323 q^{2} -0.682987 q^{3} -1.20732 q^{4} +1.00000 q^{5} +0.608079 q^{6} +2.96744 q^{7} +2.85556 q^{8} -2.53353 q^{9} -0.890323 q^{10} +3.71858 q^{11} +0.824587 q^{12} +5.77711 q^{13} -2.64198 q^{14} -0.682987 q^{15} -0.127719 q^{16} +1.00000 q^{17} +2.25566 q^{18} -4.79179 q^{19} -1.20732 q^{20} -2.02672 q^{21} -3.31074 q^{22} -8.42555 q^{23} -1.95031 q^{24} +1.00000 q^{25} -5.14350 q^{26} +3.77933 q^{27} -3.58266 q^{28} -0.00828316 q^{29} +0.608079 q^{30} +3.83066 q^{31} -5.59740 q^{32} -2.53974 q^{33} -0.890323 q^{34} +2.96744 q^{35} +3.05879 q^{36} -7.02127 q^{37} +4.26624 q^{38} -3.94569 q^{39} +2.85556 q^{40} -5.86192 q^{41} +1.80444 q^{42} -4.94908 q^{43} -4.48953 q^{44} -2.53353 q^{45} +7.50147 q^{46} -9.42907 q^{47} +0.0872305 q^{48} +1.80569 q^{49} -0.890323 q^{50} -0.682987 q^{51} -6.97485 q^{52} +2.37173 q^{53} -3.36482 q^{54} +3.71858 q^{55} +8.47368 q^{56} +3.27273 q^{57} +0.00737469 q^{58} +3.16394 q^{59} +0.824587 q^{60} -10.1659 q^{61} -3.41052 q^{62} -7.51809 q^{63} +5.23893 q^{64} +5.77711 q^{65} +2.26119 q^{66} -6.52705 q^{67} -1.20732 q^{68} +5.75454 q^{69} -2.64198 q^{70} -1.00000 q^{71} -7.23463 q^{72} -8.91081 q^{73} +6.25120 q^{74} -0.682987 q^{75} +5.78524 q^{76} +11.0347 q^{77} +3.51294 q^{78} -14.2740 q^{79} -0.127719 q^{80} +5.01936 q^{81} +5.21901 q^{82} -7.13670 q^{83} +2.44691 q^{84} +1.00000 q^{85} +4.40628 q^{86} +0.00565729 q^{87} +10.6186 q^{88} -16.6205 q^{89} +2.25566 q^{90} +17.1432 q^{91} +10.1724 q^{92} -2.61629 q^{93} +8.39492 q^{94} -4.79179 q^{95} +3.82295 q^{96} -14.0768 q^{97} -1.60764 q^{98} -9.42113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9}+O(q^{10}) \) 36 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 + 36 * q^5 - 10 * q^6 - 7 * q^7 - 9 * q^8 + 10 * q^9	\( 36 q - 3 q^{2} - 8 q^{3} + 23 q^{4} + 36 q^{5} - 10 q^{6} - 7 q^{7} - 9 q^{8} + 10 q^{9} - 3 q^{10} - 20 q^{11} - 8 q^{12} - 29 q^{13} - 12 q^{14} - 8 q^{15} + q^{16} + 36 q^{17} - 8 q^{18} - 19 q^{19} + 23 q^{20} - 19 q^{21} - 10 q^{22} - 10 q^{23} - 23 q^{24} + 36 q^{25} - 32 q^{26} - 23 q^{27} - 20 q^{28} - 52 q^{29} - 10 q^{30} - 15 q^{31} - 16 q^{32} - 19 q^{33} - 3 q^{34} - 7 q^{35} + 9 q^{36} - 52 q^{37} + 7 q^{38} - 10 q^{39} - 9 q^{40} - 51 q^{41} - 2 q^{42} - 13 q^{43} - 27 q^{44} + 10 q^{45} + 12 q^{46} - 24 q^{47} + 12 q^{48} - 15 q^{49} - 3 q^{50} - 8 q^{51} - 49 q^{52} - 13 q^{53} - 48 q^{54} - 20 q^{55} - 12 q^{56} - 20 q^{57} - 20 q^{58} - 14 q^{59} - 8 q^{60} - 75 q^{61} - 7 q^{62} + 16 q^{63} - 41 q^{64} - 29 q^{65} - q^{66} - 5 q^{67} + 23 q^{68} - 37 q^{69} - 12 q^{70} - 36 q^{71} - 23 q^{72} - 21 q^{73} + q^{74} - 8 q^{75} - 40 q^{76} - 31 q^{77} + 84 q^{78} - 49 q^{79} + q^{80} - 56 q^{81} - 51 q^{82} + 6 q^{83} + 10 q^{84} + 36 q^{85} - 41 q^{86} - 4 q^{87} - 21 q^{88} - 78 q^{89} - 8 q^{90} - 25 q^{91} - 24 q^{92} - 36 q^{93} + 6 q^{94} - 19 q^{95} - 71 q^{96} - 48 q^{97} + 51 q^{98} - 17 q^{99}+O(q^{100}) \) 36 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 + 36 * q^5 - 10 * q^6 - 7 * q^7 - 9 * q^8 + 10 * q^9 - 3 * q^10 - 20 * q^11 - 8 * q^12 - 29 * q^13 - 12 * q^14 - 8 * q^15 + q^16 + 36 * q^17 - 8 * q^18 - 19 * q^19 + 23 * q^20 - 19 * q^21 - 10 * q^22 - 10 * q^23 - 23 * q^24 + 36 * q^25 - 32 * q^26 - 23 * q^27 - 20 * q^28 - 52 * q^29 - 10 * q^30 - 15 * q^31 - 16 * q^32 - 19 * q^33 - 3 * q^34 - 7 * q^35 + 9 * q^36 - 52 * q^37 + 7 * q^38 - 10 * q^39 - 9 * q^40 - 51 * q^41 - 2 * q^42 - 13 * q^43 - 27 * q^44 + 10 * q^45 + 12 * q^46 - 24 * q^47 + 12 * q^48 - 15 * q^49 - 3 * q^50 - 8 * q^51 - 49 * q^52 - 13 * q^53 - 48 * q^54 - 20 * q^55 - 12 * q^56 - 20 * q^57 - 20 * q^58 - 14 * q^59 - 8 * q^60 - 75 * q^61 - 7 * q^62 + 16 * q^63 - 41 * q^64 - 29 * q^65 - q^66 - 5 * q^67 + 23 * q^68 - 37 * q^69 - 12 * q^70 - 36 * q^71 - 23 * q^72 - 21 * q^73 + q^74 - 8 * q^75 - 40 * q^76 - 31 * q^77 + 84 * q^78 - 49 * q^79 + q^80 - 56 * q^81 - 51 * q^82 + 6 * q^83 + 10 * q^84 + 36 * q^85 - 41 * q^86 - 4 * q^87 - 21 * q^88 - 78 * q^89 - 8 * q^90 - 25 * q^91 - 24 * q^92 - 36 * q^93 + 6 * q^94 - 19 * q^95 - 71 * q^96 - 48 * q^97 + 51 * q^98 - 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more