LMFDB - Embedded newform 6002.2.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6002,2,Mod(1,6002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6002 = 2 \cdot 3001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6002.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:	$47.9262112932$
Analytic rank:	$0$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.51885 q^{3} +1.00000 q^{4} -4.29358 q^{5} -2.51885 q^{6} +3.16827 q^{7} +1.00000 q^{8} +3.34462 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.51885 q^{3} +1.00000 q^{4} -4.29358 q^{5} -2.51885 q^{6} +3.16827 q^{7} +1.00000 q^{8} +3.34462 q^{9} -4.29358 q^{10} +4.97347 q^{11} -2.51885 q^{12} +5.94922 q^{13} +3.16827 q^{14} +10.8149 q^{15} +1.00000 q^{16} +4.67739 q^{17} +3.34462 q^{18} -0.937762 q^{19} -4.29358 q^{20} -7.98041 q^{21} +4.97347 q^{22} -3.78774 q^{23} -2.51885 q^{24} +13.4348 q^{25} +5.94922 q^{26} -0.868051 q^{27} +3.16827 q^{28} +7.94150 q^{29} +10.8149 q^{30} -7.51048 q^{31} +1.00000 q^{32} -12.5274 q^{33} +4.67739 q^{34} -13.6032 q^{35} +3.34462 q^{36} +10.8529 q^{37} -0.937762 q^{38} -14.9852 q^{39} -4.29358 q^{40} +5.74253 q^{41} -7.98041 q^{42} +0.831399 q^{43} +4.97347 q^{44} -14.3604 q^{45} -3.78774 q^{46} -4.23012 q^{47} -2.51885 q^{48} +3.03795 q^{49} +13.4348 q^{50} -11.7817 q^{51} +5.94922 q^{52} -7.21562 q^{53} -0.868051 q^{54} -21.3540 q^{55} +3.16827 q^{56} +2.36208 q^{57} +7.94150 q^{58} -4.53090 q^{59} +10.8149 q^{60} -8.12871 q^{61} -7.51048 q^{62} +10.5967 q^{63} +1.00000 q^{64} -25.5435 q^{65} -12.5274 q^{66} +9.98418 q^{67} +4.67739 q^{68} +9.54077 q^{69} -13.6032 q^{70} +2.67275 q^{71} +3.34462 q^{72} +0.161516 q^{73} +10.8529 q^{74} -33.8403 q^{75} -0.937762 q^{76} +15.7573 q^{77} -14.9852 q^{78} +12.8153 q^{79} -4.29358 q^{80} -7.84737 q^{81} +5.74253 q^{82} -12.7824 q^{83} -7.98041 q^{84} -20.0827 q^{85} +0.831399 q^{86} -20.0035 q^{87} +4.97347 q^{88} +8.40279 q^{89} -14.3604 q^{90} +18.8488 q^{91} -3.78774 q^{92} +18.9178 q^{93} -4.23012 q^{94} +4.02635 q^{95} -2.51885 q^{96} -3.62669 q^{97} +3.03795 q^{98} +16.6344 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9}+O(q^{10}) $ 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9} + 18 q^{10} + 28 q^{11} + 17 q^{12} + 47 q^{13} + 19 q^{14} + 14 q^{15} + 79 q^{16} + 36 q^{17} + 118 q^{18} + 29 q^{19} + 18 q^{20} + 45 q^{21} + 28 q^{22} + 23 q^{23} + 17 q^{24} + 161 q^{25} + 47 q^{26} + 50 q^{27} + 19 q^{28} + 53 q^{29} + 14 q^{30} + 29 q^{31} + 79 q^{32} + 34 q^{33} + 36 q^{34} + 33 q^{35} + 118 q^{36} + 89 q^{37} + 29 q^{38} - 7 q^{39} + 18 q^{40} + 58 q^{41} + 45 q^{42} + 88 q^{43} + 28 q^{44} + 45 q^{45} + 23 q^{46} + 3 q^{47} + 17 q^{48} + 162 q^{49} + 161 q^{50} + 29 q^{51} + 47 q^{52} + 88 q^{53} + 50 q^{54} + 37 q^{55} + 19 q^{56} + 54 q^{57} + 53 q^{58} + 37 q^{59} + 14 q^{60} + 55 q^{61} + 29 q^{62} + 21 q^{63} + 79 q^{64} + 55 q^{65} + 34 q^{66} + 107 q^{67} + 36 q^{68} + 39 q^{69} + 33 q^{70} - 5 q^{71} + 118 q^{72} + 71 q^{73} + 89 q^{74} + 37 q^{75} + 29 q^{76} + 61 q^{77} - 7 q^{78} + 29 q^{79} + 18 q^{80} + 215 q^{81} + 58 q^{82} + 42 q^{83} + 45 q^{84} + 84 q^{85} + 88 q^{86} + 15 q^{87} + 28 q^{88} + 72 q^{89} + 45 q^{90} + 70 q^{91} + 23 q^{92} + 97 q^{93} + 3 q^{94} - 18 q^{95} + 17 q^{96} + 93 q^{97} + 162 q^{98} + 49 q^{99}+O(q^{100}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9 + 18 * q^10 + 28 * q^11 + 17 * q^12 + 47 * q^13 + 19 * q^14 + 14 * q^15 + 79 * q^16 + 36 * q^17 + 118 * q^18 + 29 * q^19 + 18 * q^20 + 45 * q^21 + 28 * q^22 + 23 * q^23 + 17 * q^24 + 161 * q^25 + 47 * q^26 + 50 * q^27 + 19 * q^28 + 53 * q^29 + 14 * q^30 + 29 * q^31 + 79 * q^32 + 34 * q^33 + 36 * q^34 + 33 * q^35 + 118 * q^36 + 89 * q^37 + 29 * q^38 - 7 * q^39 + 18 * q^40 + 58 * q^41 + 45 * q^42 + 88 * q^43 + 28 * q^44 + 45 * q^45 + 23 * q^46 + 3 * q^47 + 17 * q^48 + 162 * q^49 + 161 * q^50 + 29 * q^51 + 47 * q^52 + 88 * q^53 + 50 * q^54 + 37 * q^55 + 19 * q^56 + 54 * q^57 + 53 * q^58 + 37 * q^59 + 14 * q^60 + 55 * q^61 + 29 * q^62 + 21 * q^63 + 79 * q^64 + 55 * q^65 + 34 * q^66 + 107 * q^67 + 36 * q^68 + 39 * q^69 + 33 * q^70 - 5 * q^71 + 118 * q^72 + 71 * q^73 + 89 * q^74 + 37 * q^75 + 29 * q^76 + 61 * q^77 - 7 * q^78 + 29 * q^79 + 18 * q^80 + 215 * q^81 + 58 * q^82 + 42 * q^83 + 45 * q^84 + 84 * q^85 + 88 * q^86 + 15 * q^87 + 28 * q^88 + 72 * q^89 + 45 * q^90 + 70 * q^91 + 23 * q^92 + 97 * q^93 + 3 * q^94 - 18 * q^95 + 17 * q^96 + 93 * q^97 + 162 * q^98 + 49 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.51885	−1.45426	−0.727130	−	0.686499i	$-0.759146\pi$
$3$	−2.51885	−1.45426	−0.727130	+	0.686499i	$0.759146\pi$
$4$	1.00000	0.500000
$5$	−4.29358	−1.92015	−0.960073	−	0.279748i	$-0.909749\pi$
$5$	−4.29358	−1.92015	−0.960073	+	0.279748i	$0.909749\pi$
$6$	−2.51885	−1.02832
$7$	3.16827	1.19749	0.598747	−	0.800938i	$-0.295665\pi$
$7$	3.16827	1.19749	0.598747	+	0.800938i	$0.295665\pi$
$8$	1.00000	0.353553
$9$	3.34462	1.11487
$10$	−4.29358	−1.35775
$11$	4.97347	1.49956	0.749778	−	0.661689i	$-0.230160\pi$
$11$	4.97347	1.49956	0.749778	+	0.661689i	$0.230160\pi$
$12$	−2.51885	−0.727130
$13$	5.94922	1.65002	0.825009	−	0.565120i	$-0.191170\pi$
$13$	5.94922	1.65002	0.825009	+	0.565120i	$0.191170\pi$
$14$	3.16827	0.846756
$15$	10.8149	2.79239
$16$	1.00000	0.250000
$17$	4.67739	1.13443	0.567217	−	0.823568i	$-0.308020\pi$
$17$	4.67739	1.13443	0.567217	+	0.823568i	$0.308020\pi$
$18$	3.34462	0.788335
$19$	−0.937762	−0.215137	−0.107569	−	0.994198i	$-0.534307\pi$
$19$	−0.937762	−0.215137	−0.107569	+	0.994198i	$0.534307\pi$
$20$	−4.29358	−0.960073
$21$	−7.98041	−1.74147
$22$	4.97347	1.06035
$23$	−3.78774	−0.789799	−0.394899	−	0.918724i	$-0.629221\pi$
$23$	−3.78774	−0.789799	−0.394899	+	0.918724i	$0.629221\pi$
$24$	−2.51885	−0.514159
$25$	13.4348	2.68696
$26$	5.94922	1.16674
$27$	−0.868051	−0.167057
$28$	3.16827	0.598747
$29$	7.94150	1.47470	0.737349	−	0.675512i	$-0.236077\pi$
$29$	7.94150	1.47470	0.737349	+	0.675512i	$0.236077\pi$
$30$	10.8149	1.97452
$31$	−7.51048	−1.34892	−0.674461	−	0.738310i	$-0.735624\pi$
$31$	−7.51048	−1.34892	−0.674461	+	0.738310i	$0.735624\pi$
$32$	1.00000	0.176777
$33$	−12.5274	−2.18075
$34$	4.67739	0.802166
$35$	−13.6032	−2.29937
$36$	3.34462	0.557437
$37$	10.8529	1.78420	0.892102	−	0.451835i	$-0.149230\pi$
$37$	10.8529	1.78420	0.892102	+	0.451835i	$0.149230\pi$
$38$	−0.937762	−0.152125
$39$	−14.9852	−2.39956
$40$	−4.29358	−0.678874
$41$	5.74253	0.896833	0.448417	−	0.893825i	$-0.351988\pi$
$41$	5.74253	0.896833	0.448417	+	0.893825i	$0.351988\pi$
$42$	−7.98041	−1.23140
$43$	0.831399	0.126787	0.0633936	−	0.997989i	$-0.479808\pi$
$43$	0.831399	0.126787	0.0633936	+	0.997989i	$0.479808\pi$
$44$	4.97347	0.749778
$45$	−14.3604	−2.14072
$46$	−3.78774	−0.558472
$47$	−4.23012	−0.617027	−0.308514	−	0.951220i	$-0.599832\pi$
$47$	−4.23012	−0.617027	−0.308514	+	0.951220i	$0.599832\pi$
$48$	−2.51885	−0.363565
$49$	3.03795	0.433993
$50$	13.4348	1.89997
$51$	−11.7817	−1.64976
$52$	5.94922	0.825009
$53$	−7.21562	−0.991142	−0.495571	−	0.868567i	$-0.665041\pi$
$53$	−7.21562	−0.991142	−0.495571	+	0.868567i	$0.665041\pi$
$54$	−0.868051	−0.118127
$55$	−21.3540	−2.87937
$56$	3.16827	0.423378
$57$	2.36208	0.312866
$58$	7.94150	1.04277
$59$	−4.53090	−0.589872	−0.294936	−	0.955517i	$-0.595298\pi$
$59$	−4.53090	−0.589872	−0.294936	+	0.955517i	$0.595298\pi$
$60$	10.8149	1.39620
$61$	−8.12871	−1.04077	−0.520387	−	0.853930i	$-0.674212\pi$
$61$	−8.12871	−1.04077	−0.520387	+	0.853930i	$0.674212\pi$
$62$	−7.51048	−0.953832
$63$	10.5967	1.33506
$64$	1.00000	0.125000
$65$	−25.5435	−3.16828
$66$	−12.5274	−1.54202
$67$	9.98418	1.21976	0.609881	−	0.792493i	$-0.291217\pi$
$67$	9.98418	1.21976	0.609881	+	0.792493i	$0.291217\pi$
$68$	4.67739	0.567217
$69$	9.54077	1.14857
$70$	−13.6032	−1.62590
$71$	2.67275	0.317198	0.158599	−	0.987343i	$-0.449302\pi$
$71$	2.67275	0.317198	0.158599	+	0.987343i	$0.449302\pi$
$72$	3.34462	0.394167
$73$	0.161516	0.0189040	0.00945202	−	0.999955i	$-0.496991\pi$
$73$	0.161516	0.0189040	0.00945202	+	0.999955i	$0.496991\pi$
$74$	10.8529	1.26162
$75$	−33.8403	−3.90755
$76$	−0.937762	−0.107569
$77$	15.7573	1.79571
$78$	−14.9852	−1.69674
$79$	12.8153	1.44183	0.720917	−	0.693021i	$-0.243721\pi$
$79$	12.8153	1.44183	0.720917	+	0.693021i	$0.243721\pi$
$80$	−4.29358	−0.480037
$81$	−7.84737	−0.871930
$82$	5.74253	0.634157
$83$	−12.7824	−1.40305	−0.701525	−	0.712645i	$-0.747497\pi$
$83$	−12.7824	−1.40305	−0.701525	+	0.712645i	$0.747497\pi$
$84$	−7.98041	−0.870735
$85$	−20.0827	−2.17828
$86$	0.831399	0.0896521
$87$	−20.0035	−2.14460
$88$	4.97347	0.530173
$89$	8.40279	0.890694	0.445347	−	0.895358i	$-0.353080\pi$
$89$	8.40279	0.890694	0.445347	+	0.895358i	$0.353080\pi$
$90$	−14.3604	−1.51372
$91$	18.8488	1.97589
$92$	−3.78774	−0.394899
$93$	18.9178	1.96168
$94$	−4.23012	−0.436304
$95$	4.02635	0.413095
$96$	−2.51885	−0.257079
$97$	−3.62669	−0.368234	−0.184117	−	0.982904i	$-0.558943\pi$
$97$	−3.62669	−0.368234	−0.184117	+	0.982904i	$0.558943\pi$
$98$	3.03795	0.306879
$99$	16.6344	1.67182
$100$	13.4348	1.34348
$101$	−0.0192504	−0.00191549	−0.000957744	−	1.00000i	$-0.500305\pi$
$101$	−0.0192504	−0.00191549	−0.000957744		1.00000i	$0.500305\pi$
$102$	−11.7817	−1.16656
$103$	−1.21162	−0.119385	−0.0596923	−	0.998217i	$-0.519012\pi$
$103$	−1.21162	−0.119385	−0.0596923	+	0.998217i	$0.519012\pi$
$104$	5.94922	0.583369
$105$	34.2645	3.34388
$106$	−7.21562	−0.700843
$107$	−3.84170	−0.371391	−0.185696	−	0.982607i	$-0.559454\pi$
$107$	−3.84170	−0.371391	−0.185696	+	0.982607i	$0.559454\pi$
$108$	−0.868051	−0.0835283
$109$	−5.99282	−0.574008	−0.287004	−	0.957929i	$-0.592659\pi$
$109$	−5.99282	−0.574008	−0.287004	+	0.957929i	$0.592659\pi$
$110$	−21.3540	−2.03602
$111$	−27.3368	−2.59470
$112$	3.16827	0.299374
$113$	−9.07014	−0.853247	−0.426623	−	0.904429i	$-0.640297\pi$
$113$	−9.07014	−0.853247	−0.426623	+	0.904429i	$0.640297\pi$
$114$	2.36208	0.221229
$115$	16.2630	1.51653
$116$	7.94150	0.737349
$117$	19.8979	1.83956
$118$	−4.53090	−0.417103
$119$	14.8192	1.35848
$120$	10.8149	0.987260
$121$	13.7354	1.24867
$122$	−8.12871	−0.735939
$123$	−14.4646	−1.30423
$124$	−7.51048	−0.674461
$125$	−36.2156	−3.23922
$126$	10.5967	0.944027
$127$	8.76616	0.777871	0.388935	−	0.921265i	$-0.372843\pi$
$127$	8.76616	0.777871	0.388935	+	0.921265i	$0.372843\pi$
$128$	1.00000	0.0883883
$129$	−2.09417	−0.184382
$130$	−25.5435	−2.24031
$131$	18.9700	1.65742	0.828708	−	0.559682i	$-0.189077\pi$
$131$	18.9700	1.65742	0.828708	+	0.559682i	$0.189077\pi$
$132$	−12.5274	−1.09037
$133$	−2.97108	−0.257626
$134$	9.98418	0.862502
$135$	3.72705	0.320773
$136$	4.67739	0.401083
$137$	−10.9370	−0.934412	−0.467206	−	0.884149i	$-0.654739\pi$
$137$	−10.9370	−0.934412	−0.467206	+	0.884149i	$0.654739\pi$
$138$	9.54077	0.812164
$139$	3.11720	0.264397	0.132199	−	0.991223i	$-0.457796\pi$
$139$	3.11720	0.264397	0.132199	+	0.991223i	$0.457796\pi$
$140$	−13.6032	−1.14968
$141$	10.6551	0.897318
$142$	2.67275	0.224293
$143$	29.5883	2.47430
$144$	3.34462	0.278718
$145$	−34.0974	−2.83164
$146$	0.161516	0.0133672
$147$	−7.65215	−0.631139
$148$	10.8529	0.892102
$149$	4.61825	0.378342	0.189171	−	0.981944i	$-0.439420\pi$
$149$	4.61825	0.378342	0.189171	+	0.981944i	$0.439420\pi$
$150$	−33.8403	−2.76305
$151$	15.1252	1.23087	0.615435	−	0.788188i	$-0.288980\pi$
$151$	15.1252	1.23087	0.615435	+	0.788188i	$0.288980\pi$
$152$	−0.937762	−0.0760625
$153$	15.6441	1.26475
$154$	15.7573	1.26976
$155$	32.2468	2.59013
$156$	−14.9852	−1.19978
$157$	24.4915	1.95463	0.977316	−	0.211788i	$-0.0679287\pi$
$157$	24.4915	1.95463	0.977316	+	0.211788i	$0.0679287\pi$
$158$	12.8153	1.01953
$159$	18.1751	1.44138
$160$	−4.29358	−0.339437
$161$	−12.0006	−0.945780
$162$	−7.84737	−0.616548
$163$	−19.8103	−1.55166	−0.775831	−	0.630941i	$-0.782669\pi$
$163$	−19.8103	−1.55166	−0.775831	+	0.630941i	$0.782669\pi$
$164$	5.74253	0.448417
$165$	53.7875	4.18735
$166$	−12.7824	−0.992106
$167$	−12.8968	−0.997982	−0.498991	−	0.866607i	$-0.666296\pi$
$167$	−12.8968	−0.997982	−0.498991	+	0.866607i	$0.666296\pi$
$168$	−7.98041	−0.615702
$169$	22.3933	1.72256
$170$	−20.0827	−1.54028
$171$	−3.13646	−0.239851
$172$	0.831399	0.0633936
$173$	2.09580	0.159341	0.0796703	−	0.996821i	$-0.474613\pi$
$173$	2.09580	0.159341	0.0796703	+	0.996821i	$0.474613\pi$
$174$	−20.0035	−1.51646
$175$	42.5652	3.21762
$176$	4.97347	0.374889
$177$	11.4127	0.857828
$178$	8.40279	0.629816
$179$	−6.94468	−0.519070	−0.259535	−	0.965734i	$-0.583569\pi$
$179$	−6.94468	−0.519070	−0.259535	+	0.965734i	$0.583569\pi$
$180$	−14.3604	−1.07036
$181$	−12.1786	−0.905230	−0.452615	−	0.891706i	$-0.649509\pi$
$181$	−12.1786	−0.905230	−0.452615	+	0.891706i	$0.649509\pi$
$182$	18.8488	1.39716
$183$	20.4750	1.51356
$184$	−3.78774	−0.279236
$185$	−46.5977	−3.42593
$186$	18.9178	1.38712
$187$	23.2628	1.70115
$188$	−4.23012	−0.308514
$189$	−2.75022	−0.200049
$190$	4.02635	0.292102
$191$	−21.8928	−1.58411	−0.792053	−	0.610452i	$-0.790988\pi$
$191$	−21.8928	−1.58411	−0.792053	+	0.610452i	$0.790988\pi$
$192$	−2.51885	−0.181783
$193$	4.15746	0.299261	0.149630	−	0.988742i	$-0.452192\pi$
$193$	4.15746	0.299261	0.149630	+	0.988742i	$0.452192\pi$
$194$	−3.62669	−0.260381
$195$	64.3402	4.60750
$196$	3.03795	0.216997
$197$	13.1489	0.936823	0.468411	−	0.883510i	$-0.344827\pi$
$197$	13.1489	0.936823	0.468411	+	0.883510i	$0.344827\pi$
$198$	16.6344	1.18215
$199$	−8.89273	−0.630389	−0.315194	−	0.949027i	$-0.602070\pi$
$199$	−8.89273	−0.630389	−0.315194	+	0.949027i	$0.602070\pi$
$200$	13.4348	0.949985
$201$	−25.1487	−1.77385
$202$	−0.0192504	−0.00135445
$203$	25.1608	1.76594
$204$	−11.7817	−0.824881
$205$	−24.6560	−1.72205
$206$	−1.21162	−0.0844176
$207$	−12.6686	−0.880526
$208$	5.94922	0.412505
$209$	−4.66393	−0.322611
$210$	34.2645	2.36448
$211$	3.48571	0.239966	0.119983	−	0.992776i	$-0.461716\pi$
$211$	3.48571	0.239966	0.119983	+	0.992776i	$0.461716\pi$
$212$	−7.21562	−0.495571
$213$	−6.73228	−0.461288
$214$	−3.84170	−0.262613
$215$	−3.56968	−0.243450
$216$	−0.868051	−0.0590634
$217$	−23.7953	−1.61533
$218$	−5.99282	−0.405885
$219$	−0.406836	−0.0274914
$220$	−21.3540	−1.43968
$221$	27.8269	1.87184
$222$	−27.3368	−1.83473
$223$	−20.0452	−1.34233	−0.671163	−	0.741310i	$-0.734205\pi$
$223$	−20.0452	−1.34233	−0.671163	+	0.741310i	$0.734205\pi$
$224$	3.16827	0.211689
$225$	44.9344	2.99563
$226$	−9.07014	−0.603336
$227$	−23.7164	−1.57411	−0.787056	−	0.616881i	$-0.788396\pi$
$227$	−23.7164	−1.57411	−0.787056	+	0.616881i	$0.788396\pi$
$228$	2.36208	0.156433
$229$	25.1050	1.65899	0.829493	−	0.558517i	$-0.188629\pi$
$229$	25.1050	1.65899	0.829493	+	0.558517i	$0.188629\pi$
$230$	16.2630	1.07235
$231$	−39.6903	−2.61143
$232$	7.94150	0.521385
$233$	18.4390	1.20798	0.603990	−	0.796992i	$-0.293577\pi$
$233$	18.4390	1.20798	0.603990	+	0.796992i	$0.293577\pi$
$234$	19.8979	1.30077
$235$	18.1624	1.18478
$236$	−4.53090	−0.294936
$237$	−32.2799	−2.09680
$238$	14.8192	0.960589
$239$	−21.9273	−1.41836	−0.709181	−	0.705026i	$-0.750935\pi$
$239$	−21.9273	−1.41836	−0.709181	+	0.705026i	$0.750935\pi$
$240$	10.8149	0.698098
$241$	15.1194	0.973925	0.486963	−	0.873423i	$-0.338105\pi$
$241$	15.1194	0.973925	0.486963	+	0.873423i	$0.338105\pi$
$242$	13.7354	0.882943
$243$	22.3705	1.43507
$244$	−8.12871	−0.520387
$245$	−13.0437	−0.833330
$246$	−14.4646	−0.922229
$247$	−5.57895	−0.354980
$248$	−7.51048	−0.476916
$249$	32.1970	2.04040
$250$	−36.2156	−2.29047
$251$	29.5872	1.86753	0.933763	−	0.357892i	$-0.116505\pi$
$251$	29.5872	1.86753	0.933763	+	0.357892i	$0.116505\pi$
$252$	10.5967	0.667528
$253$	−18.8382	−1.18435
$254$	8.76616	0.550038
$255$	50.5855	3.16779
$256$	1.00000	0.0625000
$257$	0.331417	0.0206732	0.0103366	−	0.999947i	$-0.496710\pi$
$257$	0.331417	0.0206732	0.0103366	+	0.999947i	$0.496710\pi$
$258$	−2.09417	−0.130377
$259$	34.3849	2.13657
$260$	−25.5435	−1.58414
$261$	26.5613	1.64410
$262$	18.9700	1.17197
$263$	12.3214	0.759768	0.379884	−	0.925034i	$-0.375964\pi$
$263$	12.3214	0.759768	0.379884	+	0.925034i	$0.375964\pi$
$264$	−12.5274	−0.771010
$265$	30.9808	1.90314
$266$	−2.97108	−0.182169
$267$	−21.1654	−1.29530
$268$	9.98418	0.609881
$269$	17.4852	1.06609	0.533044	−	0.846087i	$-0.321048\pi$
$269$	17.4852	1.06609	0.533044	+	0.846087i	$0.321048\pi$
$270$	3.72705	0.226821
$271$	−5.63118	−0.342070	−0.171035	−	0.985265i	$-0.554711\pi$
$271$	−5.63118	−0.342070	−0.171035	+	0.985265i	$0.554711\pi$
$272$	4.67739	0.283608
$273$	−47.4773	−2.87346
$274$	−10.9370	−0.660729
$275$	66.8176	4.02925
$276$	9.54077	0.574287
$277$	10.8981	0.654804	0.327402	−	0.944885i	$-0.393827\pi$
$277$	10.8981	0.654804	0.327402	+	0.944885i	$0.393827\pi$
$278$	3.11720	0.186957
$279$	−25.1197	−1.50388
$280$	−13.6032	−0.812948
$281$	−8.71707	−0.520017	−0.260009	−	0.965606i	$-0.583725\pi$
$281$	−8.71707	−0.520017	−0.260009	+	0.965606i	$0.583725\pi$
$282$	10.6551	0.634500
$283$	30.3538	1.80435	0.902173	−	0.431375i	$-0.141971\pi$
$283$	30.3538	1.80435	0.902173	+	0.431375i	$0.141971\pi$
$284$	2.67275	0.158599
$285$	−10.1418	−0.600748
$286$	29.5883	1.74959
$287$	18.1939	1.07395
$288$	3.34462	0.197084
$289$	4.87799	0.286940
$290$	−34.0974	−2.00227
$291$	9.13510	0.535509
$292$	0.161516	0.00945202
$293$	3.61928	0.211440	0.105720	−	0.994396i	$-0.466285\pi$
$293$	3.61928	0.211440	0.105720	+	0.994396i	$0.466285\pi$
$294$	−7.65215	−0.446283
$295$	19.4538	1.13264
$296$	10.8529	0.630811
$297$	−4.31723	−0.250511
$298$	4.61825	0.267528
$299$	−22.5341	−1.30318
$300$	−33.8403	−1.95377
$301$	2.63410	0.151827
$302$	15.1252	0.870357
$303$	0.0484890	0.00278562
$304$	−0.937762	−0.0537843
$305$	34.9013	1.99844
$306$	15.6441	0.894314
$307$	25.8314	1.47428	0.737139	−	0.675742i	$-0.236176\pi$
$307$	25.8314	1.47428	0.737139	+	0.675742i	$0.236176\pi$
$308$	15.7573	0.897855
$309$	3.05190	0.173616
$310$	32.2468	1.83150
$311$	−6.46948	−0.366850	−0.183425	−	0.983034i	$-0.558719\pi$
$311$	−6.46948	−0.366850	−0.183425	+	0.983034i	$0.558719\pi$
$312$	−14.9852	−0.848371
$313$	−24.4925	−1.38440	−0.692198	−	0.721708i	$-0.743358\pi$
$313$	−24.4925	−1.38440	−0.692198	+	0.721708i	$0.743358\pi$
$314$	24.4915	1.38213
$315$	−45.4977	−2.56350
$316$	12.8153	0.720917
$317$	−5.13868	−0.288617	−0.144309	−	0.989533i	$-0.546096\pi$
$317$	−5.13868	−0.288617	−0.144309	+	0.989533i	$0.546096\pi$
$318$	18.1751	1.01921
$319$	39.4968	2.21139
$320$	−4.29358	−0.240018
$321$	9.67668	0.540100
$322$	−12.0006	−0.668767
$323$	−4.38628	−0.244059
$324$	−7.84737	−0.435965
$325$	79.9268	4.43354
$326$	−19.8103	−1.09719
$327$	15.0950	0.834757
$328$	5.74253	0.317078
$329$	−13.4022	−0.738887
$330$	53.7875	2.96091
$331$	19.1733	1.05386	0.526929	−	0.849909i	$-0.323343\pi$
$331$	19.1733	1.05386	0.526929	+	0.849909i	$0.323343\pi$
$332$	−12.7824	−0.701525
$333$	36.2988	1.98916
$334$	−12.8968	−0.705680
$335$	−42.8679	−2.34212
$336$	−7.98041	−0.435367
$337$	−18.6397	−1.01537	−0.507684	−	0.861543i	$-0.669498\pi$
$337$	−18.6397	−1.01537	−0.507684	+	0.861543i	$0.669498\pi$
$338$	22.3933	1.21803
$339$	22.8463	1.24084
$340$	−20.0827	−1.08914
$341$	−37.3531	−2.02279
$342$	−3.13646	−0.169600
$343$	−12.5529	−0.677790
$344$	0.831399	0.0448260
$345$	−40.9640	−2.20543
$346$	2.09580	0.112671
$347$	6.23534	0.334730	0.167365	−	0.985895i	$-0.446474\pi$
$347$	6.23534	0.334730	0.167365	+	0.985895i	$0.446474\pi$
$348$	−20.0035	−1.07230
$349$	1.83984	0.0984843	0.0492422	−	0.998787i	$-0.484319\pi$
$349$	1.83984	0.0984843	0.0492422	+	0.998787i	$0.484319\pi$
$350$	42.5652	2.27520
$351$	−5.16423	−0.275646
$352$	4.97347	0.265087
$353$	24.1797	1.28695	0.643477	−	0.765465i	$-0.277491\pi$
$353$	24.1797	1.28695	0.643477	+	0.765465i	$0.277491\pi$
$354$	11.4127	0.606576
$355$	−11.4757	−0.609066
$356$	8.40279	0.445347
$357$	−37.3275	−1.97558
$358$	−6.94468	−0.367038
$359$	−15.9147	−0.839948	−0.419974	−	0.907536i	$-0.637961\pi$
$359$	−15.9147	−0.839948	−0.419974	+	0.907536i	$0.637961\pi$
$360$	−14.3604	−0.756859
$361$	−18.1206	−0.953716
$362$	−12.1786	−0.640094
$363$	−34.5974	−1.81589
$364$	18.8488	0.987944
$365$	−0.693483	−0.0362985
$366$	20.4750	1.07025
$367$	−3.07107	−0.160308	−0.0801542	−	0.996782i	$-0.525541\pi$
$367$	−3.07107	−0.160308	−0.0801542	+	0.996782i	$0.525541\pi$
$368$	−3.78774	−0.197450
$369$	19.2066	0.999856
$370$	−46.5977	−2.42250
$371$	−22.8611	−1.18689
$372$	18.9178	0.980842
$373$	−31.2257	−1.61680	−0.808402	−	0.588630i	$-0.799667\pi$
$373$	−31.2257	−1.61680	−0.808402	+	0.588630i	$0.799667\pi$
$374$	23.2628	1.20289
$375$	91.2217	4.71067
$376$	−4.23012	−0.218152
$377$	47.2457	2.43328
$378$	−2.75022	−0.141456
$379$	−27.3045	−1.40254	−0.701270	−	0.712896i	$-0.747383\pi$
$379$	−27.3045	−1.40254	−0.701270	+	0.712896i	$0.747383\pi$
$380$	4.02635	0.206548
$381$	−22.0807	−1.13123
$382$	−21.8928	−1.12013
$383$	−20.4956	−1.04728	−0.523638	−	0.851941i	$-0.675425\pi$
$383$	−20.4956	−1.04728	−0.523638	+	0.851941i	$0.675425\pi$
$384$	−2.51885	−0.128540
$385$	−67.6552	−3.44803
$386$	4.15746	0.211609
$387$	2.78072	0.141352
$388$	−3.62669	−0.184117
$389$	8.61229	0.436661	0.218330	−	0.975875i	$-0.429939\pi$
$389$	8.61229	0.436661	0.218330	+	0.975875i	$0.429939\pi$
$390$	64.3402	3.25799
$391$	−17.7168	−0.895975
$392$	3.03795	0.153440
$393$	−47.7826	−2.41031
$394$	13.1489	0.662434
$395$	−55.0235	−2.76853
$396$	16.6344	0.835908
$397$	−12.5662	−0.630679	−0.315339	−	0.948979i	$-0.602118\pi$
$397$	−12.5662	−0.630679	−0.315339	+	0.948979i	$0.602118\pi$
$398$	−8.89273	−0.445752
$399$	7.48373	0.374655
$400$	13.4348	0.671741
$401$	−16.0809	−0.803041	−0.401521	−	0.915850i	$-0.631518\pi$
$401$	−16.0809	−0.803041	−0.401521	+	0.915850i	$0.631518\pi$
$402$	−25.1487	−1.25430
$403$	−44.6815	−2.22575
$404$	−0.0192504	−0.000957744 0
$405$	33.6933	1.67423
$406$	25.1608	1.24871
$407$	53.9765	2.67551
$408$	−11.7817	−0.583279
$409$	11.5810	0.572643	0.286322	−	0.958134i	$-0.407567\pi$
$409$	11.5810	0.572643	0.286322	+	0.958134i	$0.407567\pi$
$410$	−24.6560	−1.21767
$411$	27.5487	1.35888
$412$	−1.21162	−0.0596923
$413$	−14.3551	−0.706369
$414$	−12.6686	−0.622626
$415$	54.8822	2.69406
$416$	5.94922	0.291685
$417$	−7.85176	−0.384502
$418$	−4.66393	−0.228120
$419$	19.5900	0.957034	0.478517	−	0.878078i	$-0.341175\pi$
$419$	19.5900	0.957034	0.478517	+	0.878078i	$0.341175\pi$
$420$	34.2645	1.67194
$421$	20.6745	1.00761	0.503807	−	0.863816i	$-0.331932\pi$
$421$	20.6745	1.00761	0.503807	+	0.863816i	$0.331932\pi$
$422$	3.48571	0.169682
$423$	−14.1482	−0.687907
$424$	−7.21562	−0.350422
$425$	62.8399	3.04818
$426$	−6.73228	−0.326180
$427$	−25.7540	−1.24632
$428$	−3.84170	−0.185696
$429$	−74.5285	−3.59827
$430$	−3.56968	−0.172145
$431$	27.3937	1.31951	0.659754	−	0.751482i	$-0.270660\pi$
$431$	27.3937	1.31951	0.659754	+	0.751482i	$0.270660\pi$
$432$	−0.868051	−0.0417641
$433$	−29.3484	−1.41039	−0.705197	−	0.709011i	$-0.749141\pi$
$433$	−29.3484	−1.41039	−0.705197	+	0.709011i	$0.749141\pi$
$434$	−23.7953	−1.14221
$435$	85.8864	4.11794
$436$	−5.99282	−0.287004
$437$	3.55200	0.169915
$438$	−0.406836	−0.0194394
$439$	−20.2339	−0.965711	−0.482855	−	0.875700i	$-0.660400\pi$
$439$	−20.2339	−0.965711	−0.482855	+	0.875700i	$0.660400\pi$
$440$	−21.3540	−1.01801
$441$	10.1608	0.483848
$442$	27.8269	1.32359
$443$	5.56166	0.264242	0.132121	−	0.991234i	$-0.457821\pi$
$443$	5.56166	0.264242	0.132121	+	0.991234i	$0.457821\pi$
$444$	−27.3368	−1.29735
$445$	−36.0781	−1.71026
$446$	−20.0452	−0.949167
$447$	−11.6327	−0.550208
$448$	3.16827	0.149687
$449$	−3.46701	−0.163619	−0.0818093	−	0.996648i	$-0.526070\pi$
$449$	−3.46701	−0.163619	−0.0818093	+	0.996648i	$0.526070\pi$
$450$	44.9344	2.11823
$451$	28.5603	1.34485
$452$	−9.07014	−0.426623
$453$	−38.0981	−1.79001
$454$	−23.7164	−1.11307
$455$	−80.9287	−3.79399
$456$	2.36208	0.110615
$457$	−1.45040	−0.0678471	−0.0339235	−	0.999424i	$-0.510800\pi$
$457$	−1.45040	−0.0678471	−0.0339235	+	0.999424i	$0.510800\pi$
$458$	25.1050	1.17308
$459$	−4.06022	−0.189515
$460$	16.2630	0.758265
$461$	−21.1407	−0.984618	−0.492309	−	0.870420i	$-0.663847\pi$
$461$	−21.1407	−0.984618	−0.492309	+	0.870420i	$0.663847\pi$
$462$	−39.6903	−1.84656
$463$	22.1072	1.02741	0.513705	−	0.857967i	$-0.328273\pi$
$463$	22.1072	1.02741	0.513705	+	0.857967i	$0.328273\pi$
$464$	7.94150	0.368675
$465$	−81.2251	−3.76672
$466$	18.4390	0.854171
$467$	6.15180	0.284672	0.142336	−	0.989818i	$-0.454539\pi$
$467$	6.15180	0.284672	0.142336	+	0.989818i	$0.454539\pi$
$468$	19.8979	0.919781
$469$	31.6326	1.46066
$470$	18.1624	0.837768
$471$	−61.6904	−2.84254
$472$	−4.53090	−0.208551
$473$	4.13494	0.190125
$474$	−32.2799	−1.48266
$475$	−12.5987	−0.578066
$476$	14.8192	0.679239
$477$	−24.1335	−1.10500
$478$	−21.9273	−1.00293
$479$	−34.9734	−1.59797	−0.798987	−	0.601349i	$-0.794630\pi$
$479$	−34.9734	−1.59797	−0.798987	+	0.601349i	$0.794630\pi$
$480$	10.8149	0.493630
$481$	64.5663	2.94397
$482$	15.1194	0.688669
$483$	30.2278	1.37541
$484$	13.7354	0.624335
$485$	15.5715	0.707064
$486$	22.3705	1.01475
$487$	−11.0631	−0.501317	−0.250658	−	0.968076i	$-0.580647\pi$
$487$	−11.0631	−0.501317	−0.250658	+	0.968076i	$0.580647\pi$
$488$	−8.12871	−0.367970
$489$	49.8992	2.25652
$490$	−13.0437	−0.589254
$491$	−22.2560	−1.00440	−0.502200	−	0.864752i	$-0.667476\pi$
$491$	−22.2560	−1.00440	−0.502200	+	0.864752i	$0.667476\pi$
$492$	−14.4646	−0.652115
$493$	37.1455	1.67295
$494$	−5.57895	−0.251009
$495$	−71.4210	−3.21013
$496$	−7.51048	−0.337231
$497$	8.46802	0.379842
$498$	32.1970	1.44278
$499$	6.68878	0.299431	0.149715	−	0.988729i	$-0.452164\pi$
$499$	6.68878	0.299431	0.149715	+	0.988729i	$0.452164\pi$
$500$	−36.2156	−1.61961
$501$	32.4851	1.45133
$502$	29.5872	1.32054
$503$	−30.4597	−1.35813	−0.679066	−	0.734078i	$-0.737615\pi$
$503$	−30.4597	−1.35813	−0.679066	+	0.734078i	$0.737615\pi$
$504$	10.5967	0.472013
$505$	0.0826532	0.00367802
$506$	−18.8382	−0.837461
$507$	−56.4054	−2.50505
$508$	8.76616	0.388935
$509$	−22.9121	−1.01556	−0.507780	−	0.861487i	$-0.669534\pi$
$509$	−22.9121	−1.01556	−0.507780	+	0.861487i	$0.669534\pi$
$510$	50.5855	2.23996
$511$	0.511727	0.0226375
$512$	1.00000	0.0441942
$513$	0.814025	0.0359401
$514$	0.331417	0.0146182
$515$	5.20219	0.229236
$516$	−2.09417	−0.0921908
$517$	−21.0384	−0.925267
$518$	34.3849	1.51079
$519$	−5.27901	−0.231723
$520$	−25.5435	−1.12016
$521$	−42.8694	−1.87814	−0.939070	−	0.343725i	$-0.888311\pi$
$521$	−42.8694	−1.87814	−0.939070	+	0.343725i	$0.888311\pi$
$522$	26.5613	1.16256
$523$	−20.2699	−0.886340	−0.443170	−	0.896437i	$-0.646146\pi$
$523$	−20.2699	−0.886340	−0.443170	+	0.896437i	$0.646146\pi$
$524$	18.9700	0.828708
$525$	−107.215	−4.67926
$526$	12.3214	0.537237
$527$	−35.1295	−1.53026
$528$	−12.5274	−0.545187
$529$	−8.65301	−0.376218
$530$	30.9808	1.34572
$531$	−15.1541	−0.657633
$532$	−2.97108	−0.128813
$533$	34.1636	1.47979
$534$	−21.1654	−0.915917
$535$	16.4946	0.713126
$536$	9.98418	0.431251
$537$	17.4926	0.754863
$538$	17.4852	0.753839
$539$	15.1092	0.650797
$540$	3.72705	0.160387
$541$	−41.8645	−1.79989	−0.899947	−	0.436000i	$-0.856395\pi$
$541$	−41.8645	−1.79989	−0.899947	+	0.436000i	$0.856395\pi$
$542$	−5.63118	−0.241880
$543$	30.6762	1.31644
$544$	4.67739	0.200541
$545$	25.7306	1.10218
$546$	−47.4773	−2.03184
$547$	−11.6391	−0.497650	−0.248825	−	0.968548i	$-0.580044\pi$
$547$	−11.6391	−0.497650	−0.248825	+	0.968548i	$0.580044\pi$
$548$	−10.9370	−0.467206
$549$	−27.1875	−1.16033
$550$	66.8176	2.84911
$551$	−7.44723	−0.317263
$552$	9.54077	0.406082
$553$	40.6024	1.72659
$554$	10.8981	0.463016
$555$	117.373	4.98220
$556$	3.11720	0.132199
$557$	7.04015	0.298301	0.149150	−	0.988815i	$-0.452346\pi$
$557$	7.04015	0.298301	0.149150	+	0.988815i	$0.452346\pi$
$558$	−25.1197	−1.06340
$559$	4.94618	0.209201
$560$	−13.6032	−0.574841
$561$	−58.5957	−2.47391
$562$	−8.71707	−0.367708
$563$	34.4518	1.45197	0.725984	−	0.687711i	$-0.241385\pi$
$563$	34.4518	1.45197	0.725984	+	0.687711i	$0.241385\pi$
$564$	10.6551	0.448659
$565$	38.9433	1.63836
$566$	30.3538	1.27587
$567$	−24.8626	−1.04413
$568$	2.67275	0.112146
$569$	34.3820	1.44137	0.720684	−	0.693264i	$-0.243828\pi$
$569$	34.3820	1.44137	0.720684	+	0.693264i	$0.243828\pi$
$570$	−10.1418	−0.424793
$571$	−1.83265	−0.0766939	−0.0383469	−	0.999264i	$-0.512209\pi$
$571$	−1.83265	−0.0766939	−0.0383469	+	0.999264i	$0.512209\pi$
$572$	29.5883	1.23715
$573$	55.1447	2.30370
$574$	18.1939	0.759399
$575$	−50.8876	−2.12216
$576$	3.34462	0.139359
$577$	45.1123	1.87805	0.939024	−	0.343851i	$-0.111732\pi$
$577$	45.1123	1.87805	0.939024	+	0.343851i	$0.111732\pi$
$578$	4.87799	0.202897
$579$	−10.4720	−0.435203
$580$	−34.0974	−1.41582
$581$	−40.4981	−1.68014
$582$	9.13510	0.378662
$583$	−35.8867	−1.48627
$584$	0.161516	0.00668359
$585$	−85.4332	−3.53223
$586$	3.61928	0.149511
$587$	−6.44525	−0.266024	−0.133012	−	0.991114i	$-0.542465\pi$
$587$	−6.44525	−0.266024	−0.133012	+	0.991114i	$0.542465\pi$
$588$	−7.65215	−0.315570
$589$	7.04304	0.290203
$590$	19.4538	0.800899
$591$	−33.1202	−1.36238
$592$	10.8529	0.446051
$593$	−39.1431	−1.60742	−0.803708	−	0.595024i	$-0.797142\pi$
$593$	−39.1431	−1.60742	−0.803708	+	0.595024i	$0.797142\pi$
$594$	−4.31723	−0.177138
$595$	−63.6276	−2.60848
$596$	4.61825	0.189171
$597$	22.3995	0.916749
$598$	−22.5341	−0.921489
$599$	35.0444	1.43188	0.715938	−	0.698163i	$-0.245999\pi$
$599$	35.0444	1.43188	0.715938	+	0.698163i	$0.245999\pi$
$600$	−33.8403	−1.38153
$601$	29.8349	1.21699	0.608496	−	0.793557i	$-0.291773\pi$
$601$	29.8349	1.21699	0.608496	+	0.793557i	$0.291773\pi$
$602$	2.63410	0.107358
$603$	33.3933	1.35988
$604$	15.1252	0.615435
$605$	−58.9739	−2.39763
$606$	0.0484890	0.00196973
$607$	3.21664	0.130559	0.0652796	−	0.997867i	$-0.479206\pi$
$607$	3.21664	0.130559	0.0652796	+	0.997867i	$0.479206\pi$
$608$	−0.937762	−0.0380313
$609$	−63.3764	−2.56814
$610$	34.9013	1.41311
$611$	−25.1660	−1.01811
$612$	15.6441	0.632375
$613$	12.8740	0.519978	0.259989	−	0.965612i	$-0.416281\pi$
$613$	12.8740	0.519978	0.259989	+	0.965612i	$0.416281\pi$
$614$	25.8314	1.04247
$615$	62.1049	2.50431
$616$	15.7573	0.634880
$617$	23.9885	0.965740	0.482870	−	0.875692i	$-0.339594\pi$
$617$	23.9885	0.965740	0.482870	+	0.875692i	$0.339594\pi$
$618$	3.05190	0.122765
$619$	−22.3854	−0.899745	−0.449873	−	0.893093i	$-0.648531\pi$
$619$	−22.3854	−0.899745	−0.449873	+	0.893093i	$0.648531\pi$
$620$	32.2468	1.29506
$621$	3.28796	0.131941
$622$	−6.46948	−0.259402
$623$	26.6223	1.06660
$624$	−14.9852	−0.599889
$625$	88.3203	3.53281
$626$	−24.4925	−0.978916
$627$	11.7477	0.469160
$628$	24.4915	0.977316
$629$	50.7632	2.02406
$630$	−45.4977	−1.81267
$631$	26.0681	1.03776	0.518878	−	0.854848i	$-0.326350\pi$
$631$	26.0681	1.03776	0.518878	+	0.854848i	$0.326350\pi$
$632$	12.8153	0.509765
$633$	−8.78000	−0.348973
$634$	−5.13868	−0.204083
$635$	−37.6382	−1.49363
$636$	18.1751	0.720689
$637$	18.0735	0.716096
$638$	39.4968	1.56369
$639$	8.93935	0.353635
$640$	−4.29358	−0.169719
$641$	−30.3688	−1.19949	−0.599747	−	0.800190i	$-0.704732\pi$
$641$	−30.3688	−1.19949	−0.599747	+	0.800190i	$0.704732\pi$
$642$	9.67668	0.381908
$643$	18.1799	0.716945	0.358473	−	0.933540i	$-0.383298\pi$
$643$	18.1799	0.716945	0.358473	+	0.933540i	$0.383298\pi$
$644$	−12.0006	−0.472890
$645$	8.99149	0.354040
$646$	−4.38628	−0.172576
$647$	32.5901	1.28125	0.640625	−	0.767854i	$-0.278675\pi$
$647$	32.5901	1.28125	0.640625	+	0.767854i	$0.278675\pi$
$648$	−7.84737	−0.308274
$649$	−22.5343	−0.884547
$650$	79.9268	3.13499
$651$	59.9367	2.34911
$652$	−19.8103	−0.775831
$653$	7.68066	0.300567	0.150284	−	0.988643i	$-0.451981\pi$
$653$	7.68066	0.300567	0.150284	+	0.988643i	$0.451981\pi$
$654$	15.0950	0.590263
$655$	−81.4491	−3.18248
$656$	5.74253	0.224208
$657$	0.540211	0.0210756
$658$	−13.4022	−0.522472
$659$	3.31820	0.129259	0.0646293	−	0.997909i	$-0.479413\pi$
$659$	3.31820	0.129259	0.0646293	+	0.997909i	$0.479413\pi$
$660$	53.7875	2.09368
$661$	−18.7137	−0.727880	−0.363940	−	0.931422i	$-0.618569\pi$
$661$	−18.7137	−0.727880	−0.363940	+	0.931422i	$0.618569\pi$
$662$	19.1733	0.745191
$663$	−70.0918	−2.72214
$664$	−12.7824	−0.496053
$665$	12.7566	0.494679
$666$	36.2988	1.40655
$667$	−30.0803	−1.16472
$668$	−12.8968	−0.498991
$669$	50.4909	1.95209
$670$	−42.8679	−1.65613
$671$	−40.4279	−1.56070
$672$	−7.98041	−0.307851
$673$	6.78008	0.261353	0.130676	−	0.991425i	$-0.458285\pi$
$673$	6.78008	0.261353	0.130676	+	0.991425i	$0.458285\pi$
$674$	−18.6397	−0.717974
$675$	−11.6621	−0.448875
$676$	22.3933	0.861280
$677$	20.2377	0.777797	0.388898	−	0.921281i	$-0.372856\pi$
$677$	20.2377	0.777797	0.388898	+	0.921281i	$0.372856\pi$
$678$	22.8463	0.877408
$679$	−11.4903	−0.440959
$680$	−20.0827	−0.770138
$681$	59.7381	2.28917
$682$	−37.3531	−1.43033
$683$	38.2215	1.46251	0.731253	−	0.682106i	$-0.238936\pi$
$683$	38.2215	1.46251	0.731253	+	0.682106i	$0.238936\pi$
$684$	−3.13646	−0.119925
$685$	46.9589	1.79421
$686$	−12.5529	−0.479270
$687$	−63.2359	−2.41260
$688$	0.831399	0.0316968
$689$	−42.9274	−1.63540
$690$	−40.9640	−1.55947
$691$	7.61556	0.289710	0.144855	−	0.989453i	$-0.453728\pi$
$691$	7.61556	0.289710	0.144855	+	0.989453i	$0.453728\pi$
$692$	2.09580	0.0796703
$693$	52.7022	2.00199
$694$	6.23534	0.236690
$695$	−13.3839	−0.507681
$696$	−20.0035	−0.758229
$697$	26.8601	1.01740
$698$	1.83984	0.0696389
$699$	−46.4452	−1.75672
$700$	42.5652	1.60881
$701$	0.313306	0.0118334	0.00591670	−	0.999982i	$-0.498117\pi$
$701$	0.313306	0.0118334	0.00591670	+	0.999982i	$0.498117\pi$
$702$	−5.16423	−0.194911
$703$	−10.1774	−0.383849
$704$	4.97347	0.187445
$705$	−45.7484	−1.72298
$706$	24.1797	0.910014
$707$	−0.0609906	−0.00229379
$708$	11.4127	0.428914
$709$	21.3186	0.800635	0.400318	−	0.916376i	$-0.368900\pi$
$709$	21.3186	0.800635	0.400318	+	0.916376i	$0.368900\pi$
$710$	−11.4757	−0.430675
$711$	42.8623	1.60746
$712$	8.40279	0.314908
$713$	28.4478	1.06538
$714$	−37.3275	−1.39695
$715$	−127.040	−4.75101
$716$	−6.94468	−0.259535
$717$	55.2318	2.06267
$718$	−15.9147	−0.593933
$719$	−36.6910	−1.36834	−0.684172	−	0.729321i	$-0.739836\pi$
$719$	−36.6910	−1.36834	−0.684172	+	0.729321i	$0.739836\pi$
$720$	−14.3604	−0.535180
$721$	−3.83875	−0.142962
$722$	−18.1206	−0.674379
$723$	−38.0835	−1.41634
$724$	−12.1786	−0.452615
$725$	106.693	3.96246
$726$	−34.5974	−1.28403
$727$	−12.3174	−0.456827	−0.228414	−	0.973564i	$-0.573354\pi$
$727$	−12.3174	−0.456827	−0.228414	+	0.973564i	$0.573354\pi$
$728$	18.8488	0.698582
$729$	−32.8060	−1.21504
$730$	−0.693483	−0.0256669
$731$	3.88878	0.143832
$732$	20.4750	0.756779
$733$	0.428019	0.0158092	0.00790461	−	0.999969i	$-0.497484\pi$
$733$	0.428019	0.0158092	0.00790461	+	0.999969i	$0.497484\pi$
$734$	−3.07107	−0.113355
$735$	32.8551	1.21188
$736$	−3.78774	−0.139618
$737$	49.6560	1.82910
$738$	19.2066	0.707005
$739$	11.8226	0.434902	0.217451	−	0.976071i	$-0.430226\pi$
$739$	11.8226	0.434902	0.217451	+	0.976071i	$0.430226\pi$
$740$	−46.5977	−1.71297
$741$	14.0526	0.516234
$742$	−22.8611	−0.839256
$743$	−23.3268	−0.855776	−0.427888	−	0.903832i	$-0.640742\pi$
$743$	−23.3268	−0.855776	−0.427888	+	0.903832i	$0.640742\pi$
$744$	18.9178	0.693560
$745$	−19.8288	−0.726472
$746$	−31.2257	−1.14325
$747$	−42.7522	−1.56422
$748$	23.2628	0.850574
$749$	−12.1716	−0.444739
$750$	91.2217	3.33094
$751$	14.6661	0.535172	0.267586	−	0.963534i	$-0.413774\pi$
$751$	14.6661	0.535172	0.267586	+	0.963534i	$0.413774\pi$
$752$	−4.23012	−0.154257
$753$	−74.5257	−2.71587
$754$	47.2457	1.72059
$755$	−64.9412	−2.36345
$756$	−2.75022	−0.100025
$757$	49.5804	1.80203	0.901015	−	0.433787i	$-0.142823\pi$
$757$	49.5804	1.80203	0.901015	+	0.433787i	$0.142823\pi$
$758$	−27.3045	−0.991745
$759$	47.4507	1.72235
$760$	4.02635	0.146051
$761$	5.46901	0.198252	0.0991258	−	0.995075i	$-0.468395\pi$
$761$	5.46901	0.198252	0.0991258	+	0.995075i	$0.468395\pi$
$762$	−22.0807	−0.799898
$763$	−18.9869	−0.687371
$764$	−21.8928	−0.792053
$765$	−67.1692	−2.42851
$766$	−20.4956	−0.740536
$767$	−26.9553	−0.973300
$768$	−2.51885	−0.0908913
$769$	5.42054	0.195470	0.0977349	−	0.995212i	$-0.468840\pi$
$769$	5.42054	0.195470	0.0977349	+	0.995212i	$0.468840\pi$
$770$	−67.6552	−2.43812
$771$	−0.834790	−0.0300642
$772$	4.15746	0.149630
$773$	7.23177	0.260109	0.130054	−	0.991507i	$-0.458485\pi$
$773$	7.23177	0.260109	0.130054	+	0.991507i	$0.458485\pi$
$774$	2.78072	0.0999507
$775$	−100.902	−3.62451
$776$	−3.62669	−0.130191
$777$	−86.6105	−3.10714
$778$	8.61229	0.308766
$779$	−5.38513	−0.192942
$780$	64.3402	2.30375
$781$	13.2929	0.475656
$782$	−17.7168	−0.633550
$783$	−6.89363	−0.246358
$784$	3.03795	0.108498
$785$	−105.156	−3.75318
$786$	−47.7826	−1.70435
$787$	13.1036	0.467093	0.233546	−	0.972346i	$-0.424967\pi$
$787$	13.1036	0.467093	0.233546	+	0.972346i	$0.424967\pi$
$788$	13.1489	0.468411
$789$	−31.0357	−1.10490
$790$	−55.0235	−1.95765
$791$	−28.7367	−1.02176
$792$	16.6344	0.591076
$793$	−48.3595	−1.71730
$794$	−12.5662	−0.445957
$795$	−78.0362	−2.76766
$796$	−8.89273	−0.315194
$797$	4.94125	0.175028	0.0875140	−	0.996163i	$-0.472108\pi$
$797$	4.94125	0.175028	0.0875140	+	0.996163i	$0.472108\pi$
$798$	7.48373	0.264921
$799$	−19.7859	−0.699977
$800$	13.4348	0.474993
$801$	28.1042	0.993012
$802$	−16.0809	−0.567836
$803$	0.803296	0.0283477
$804$	−25.1487	−0.886925
$805$	51.5255	1.81604
$806$	−44.6815	−1.57384
$807$	−44.0425	−1.55037
$808$	−0.0192504	−0.000677227 0
$809$	29.1766	1.02579	0.512897	−	0.858450i	$-0.328572\pi$
$809$	29.1766	1.02579	0.512897	+	0.858450i	$0.328572\pi$
$810$	33.6933	1.18386
$811$	4.53856	0.159371	0.0796853	−	0.996820i	$-0.474608\pi$
$811$	4.53856	0.159371	0.0796853	+	0.996820i	$0.474608\pi$
$812$	25.1608	0.882972
$813$	14.1841	0.497458
$814$	53.9765	1.89187
$815$	85.0570	2.97942
$816$	−11.7817	−0.412441
$817$	−0.779654	−0.0272766
$818$	11.5810	0.404920
$819$	63.0420	2.20287
$820$	−24.6560	−0.861026
$821$	−30.2019	−1.05406	−0.527028	−	0.849848i	$-0.676694\pi$
$821$	−30.2019	−1.05406	−0.527028	+	0.849848i	$0.676694\pi$
$822$	27.5487	0.960872
$823$	24.9500	0.869701	0.434851	−	0.900503i	$-0.356801\pi$
$823$	24.9500	0.869701	0.434851	+	0.900503i	$0.356801\pi$
$824$	−1.21162	−0.0422088
$825$	−168.304	−5.85959
$826$	−14.3551	−0.499478
$827$	35.0695	1.21949	0.609744	−	0.792599i	$-0.291272\pi$
$827$	35.0695	1.21949	0.609744	+	0.792599i	$0.291272\pi$
$828$	−12.6686	−0.440263
$829$	32.3010	1.12186	0.560930	−	0.827864i	$-0.310444\pi$
$829$	32.3010	1.12186	0.560930	+	0.827864i	$0.310444\pi$
$830$	54.8822	1.90499
$831$	−27.4507	−0.952256
$832$	5.94922	0.206252
$833$	14.2097	0.492336
$834$	−7.85176	−0.271884
$835$	55.3733	1.91627
$836$	−4.66393	−0.161305
$837$	6.51948	0.225346
$838$	19.5900	0.676725
$839$	−50.6991	−1.75033	−0.875163	−	0.483828i	$-0.839246\pi$
$839$	−50.6991	−1.75033	−0.875163	+	0.483828i	$0.839246\pi$
$840$	34.2645	1.18224
$841$	34.0674	1.17474
$842$	20.6745	0.712491
$843$	21.9570	0.756240
$844$	3.48571	0.119983
$845$	−96.1473	−3.30757
$846$	−14.1482	−0.486424
$847$	43.5174	1.49528
$848$	−7.21562	−0.247786
$849$	−76.4568	−2.62399
$850$	62.8399	2.15539
$851$	−41.1079	−1.40916
$852$	−6.73228	−0.230644
$853$	51.7257	1.77105	0.885526	−	0.464589i	$-0.153798\pi$
$853$	51.7257	1.77105	0.885526	+	0.464589i	$0.153798\pi$
$854$	−25.7540	−0.881283
$855$	13.4666	0.460549
$856$	−3.84170	−0.131307
$857$	23.0561	0.787580	0.393790	−	0.919200i	$-0.371164\pi$
$857$	23.0561	0.787580	0.393790	+	0.919200i	$0.371164\pi$
$858$	−74.5285	−2.54436
$859$	3.23577	0.110403	0.0552015	−	0.998475i	$-0.482420\pi$
$859$	3.23577	0.110403	0.0552015	+	0.998475i	$0.482420\pi$
$860$	−3.56968	−0.121725
$861$	−45.8278	−1.56181
$862$	27.3937	0.933033
$863$	−50.6340	−1.72360	−0.861801	−	0.507246i	$-0.830664\pi$
$863$	−50.6340	−1.72360	−0.861801	+	0.507246i	$0.830664\pi$
$864$	−0.868051	−0.0295317
$865$	−8.99847	−0.305957
$866$	−29.3484	−0.997299
$867$	−12.2869	−0.417286
$868$	−23.7953	−0.807664
$869$	63.7365	2.16211
$870$	85.8864	2.91182
$871$	59.3981	2.01263
$872$	−5.99282	−0.202942
$873$	−12.1299	−0.410535
$874$	3.55200	0.120148
$875$	−114.741	−3.87895
$876$	−0.406836	−0.0137457
$877$	−48.3963	−1.63423	−0.817113	−	0.576477i	$-0.804427\pi$
$877$	−48.3963	−1.63423	−0.817113	+	0.576477i	$0.804427\pi$
$878$	−20.2339	−0.682861
$879$	−9.11643	−0.307490
$880$	−21.3540	−0.719842
$881$	25.5503	0.860811	0.430405	−	0.902636i	$-0.358371\pi$
$881$	25.5503	0.860811	0.430405	+	0.902636i	$0.358371\pi$
$882$	10.1608	0.342132
$883$	33.2751	1.11980	0.559898	−	0.828561i	$-0.310840\pi$
$883$	33.2751	1.11980	0.559898	+	0.828561i	$0.310840\pi$
$884$	27.8269	0.935918
$885$	−49.0012	−1.64716
$886$	5.56166	0.186847
$887$	−38.3759	−1.28854	−0.644269	−	0.764799i	$-0.722838\pi$
$887$	−38.3759	−1.28854	−0.644269	+	0.764799i	$0.722838\pi$
$888$	−27.3368	−0.917364
$889$	27.7736	0.931496
$890$	−36.0781	−1.20934
$891$	−39.0286	−1.30751
$892$	−20.0452	−0.671163
$893$	3.96685	0.132746
$894$	−11.6327	−0.389055
$895$	29.8175	0.996690
$896$	3.16827	0.105845
$897$	56.7602	1.89517
$898$	−3.46701	−0.115696
$899$	−59.6445	−1.98925
$900$	44.9344	1.49781
$901$	−33.7503	−1.12439
$902$	28.5603	0.950954
$903$	−6.63491	−0.220796
$904$	−9.07014	−0.301668
$905$	52.2899	1.73817
$906$	−38.0981	−1.26573
$907$	37.6881	1.25141	0.625706	−	0.780059i	$-0.284811\pi$
$907$	37.6881	1.25141	0.625706	+	0.780059i	$0.284811\pi$
$908$	−23.7164	−0.787056
$909$	−0.0643854	−0.00213553
$910$	−80.9287	−2.68276
$911$	−12.4730	−0.413250	−0.206625	−	0.978420i	$-0.566248\pi$
$911$	−12.4730	−0.413250	−0.206625	+	0.978420i	$0.566248\pi$
$912$	2.36208	0.0782164
$913$	−63.5728	−2.10395
$914$	−1.45040	−0.0479751
$915$	−87.9112	−2.90625
$916$	25.1050	0.829493
$917$	60.1021	1.98475
$918$	−4.06022	−0.134007
$919$	11.8148	0.389735	0.194867	−	0.980830i	$-0.437572\pi$
$919$	11.8148	0.389735	0.194867	+	0.980830i	$0.437572\pi$
$920$	16.2630	0.536174
$921$	−65.0656	−2.14398
$922$	−21.1407	−0.696230
$923$	15.9008	0.523382
$924$	−39.6903	−1.30572
$925$	145.807	4.79409
$926$	22.1072	0.726488
$927$	−4.05241	−0.133099
$928$	7.94150	0.260692
$929$	27.2396	0.893703	0.446851	−	0.894608i	$-0.352545\pi$
$929$	27.2396	0.893703	0.446851	+	0.894608i	$0.352545\pi$
$930$	−81.2251	−2.66347
$931$	−2.84887	−0.0933681
$932$	18.4390	0.603990
$933$	16.2957	0.533496
$934$	6.15180	0.201293
$935$	−99.8809	−3.26645
$936$	19.8979	0.650383
$937$	10.8258	0.353664	0.176832	−	0.984241i	$-0.443415\pi$
$937$	10.8258	0.353664	0.176832	+	0.984241i	$0.443415\pi$
$938$	31.6326	1.03284
$939$	61.6929	2.01327
$940$	18.1624	0.592391
$941$	−28.4885	−0.928698	−0.464349	−	0.885652i	$-0.653712\pi$
$941$	−28.4885	−0.928698	−0.464349	+	0.885652i	$0.653712\pi$
$942$	−61.6904	−2.00998
$943$	−21.7512	−0.708318
$944$	−4.53090	−0.147468
$945$	11.8083	0.384124
$946$	4.13494	0.134438
$947$	−59.0341	−1.91835	−0.959175	−	0.282815i	$-0.908732\pi$
$947$	−59.0341	−1.91835	−0.959175	+	0.282815i	$0.908732\pi$
$948$	−32.2799	−1.04840
$949$	0.960896	0.0311920
$950$	−12.5987	−0.408754
$951$	12.9436	0.419725
$952$	14.8192	0.480295
$953$	1.14079	0.0369539	0.0184770	−	0.999829i	$-0.494118\pi$
$953$	1.14079	0.0369539	0.0184770	+	0.999829i	$0.494118\pi$
$954$	−24.1335	−0.781352
$955$	93.9984	3.04172
$956$	−21.9273	−0.709181
$957$	−99.4866	−3.21594
$958$	−34.9734	−1.12994
$959$	−34.6514	−1.11895
$960$	10.8149	0.349049
$961$	25.4073	0.819591
$962$	64.5663	2.08170
$963$	−12.8490	−0.414054
$964$	15.1194	0.486963
$965$	−17.8504	−0.574624
$966$	30.2278	0.972562
$967$	3.89362	0.125210	0.0626052	−	0.998038i	$-0.480059\pi$
$967$	3.89362	0.125210	0.0626052	+	0.998038i	$0.480059\pi$
$968$	13.7354	0.441472
$969$	11.0484	0.354925
$970$	15.5715	0.499970
$971$	1.74001	0.0558397	0.0279199	−	0.999610i	$-0.491112\pi$
$971$	1.74001	0.0558397	0.0279199	+	0.999610i	$0.491112\pi$
$972$	22.3705	0.717535
$973$	9.87613	0.316614
$974$	−11.0631	−0.354484
$975$	−201.324	−6.44752
$976$	−8.12871	−0.260194
$977$	3.18728	0.101970	0.0509851	−	0.998699i	$-0.483764\pi$
$977$	3.18728	0.101970	0.0509851	+	0.998699i	$0.483764\pi$
$978$	49.8992	1.59560
$979$	41.7910	1.33565
$980$	−13.0437	−0.416665
$981$	−20.0437	−0.639947
$982$	−22.2560	−0.710218
$983$	−4.53222	−0.144555	−0.0722777	−	0.997385i	$-0.523027\pi$
$983$	−4.53222	−0.144555	−0.0722777	+	0.997385i	$0.523027\pi$
$984$	−14.4646	−0.461115
$985$	−56.4560	−1.79884
$986$	37.1455	1.18295
$987$	33.7581	1.07453
$988$	−5.57895	−0.177490
$989$	−3.14913	−0.100136
$990$	−71.4210	−2.26991
$991$	26.1663	0.831200	0.415600	−	0.909548i	$-0.363572\pi$
$991$	26.1663	0.831200	0.415600	+	0.909548i	$0.363572\pi$
$992$	−7.51048	−0.238458
$993$	−48.2947	−1.53259
$994$	8.46802	0.268589
$995$	38.1816	1.21044
$996$	32.1970	1.02020
$997$	−18.3907	−0.582438	−0.291219	−	0.956656i	$-0.594061\pi$
$997$	−18.3907	−0.582438	−0.291219	+	0.956656i	$0.594061\pi$
$998$	6.68878	0.211730
$999$	−9.42086	−0.298063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6002.2.a.d.1.11	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6002.2.a.d.1.11	✓	79	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:	\( N \)	\(=\)	\( 6002 = 2 \cdot 3001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.51885 q^{3} +1.00000 q^{4} -4.29358 q^{5} -2.51885 q^{6} +3.16827 q^{7} +1.00000 q^{8} +3.34462 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.51885 q^{3} +1.00000 q^{4} -4.29358 q^{5} -2.51885 q^{6} +3.16827 q^{7} +1.00000 q^{8} +3.34462 q^{9} -4.29358 q^{10} +4.97347 q^{11} -2.51885 q^{12} +5.94922 q^{13} +3.16827 q^{14} +10.8149 q^{15} +1.00000 q^{16} +4.67739 q^{17} +3.34462 q^{18} -0.937762 q^{19} -4.29358 q^{20} -7.98041 q^{21} +4.97347 q^{22} -3.78774 q^{23} -2.51885 q^{24} +13.4348 q^{25} +5.94922 q^{26} -0.868051 q^{27} +3.16827 q^{28} +7.94150 q^{29} +10.8149 q^{30} -7.51048 q^{31} +1.00000 q^{32} -12.5274 q^{33} +4.67739 q^{34} -13.6032 q^{35} +3.34462 q^{36} +10.8529 q^{37} -0.937762 q^{38} -14.9852 q^{39} -4.29358 q^{40} +5.74253 q^{41} -7.98041 q^{42} +0.831399 q^{43} +4.97347 q^{44} -14.3604 q^{45} -3.78774 q^{46} -4.23012 q^{47} -2.51885 q^{48} +3.03795 q^{49} +13.4348 q^{50} -11.7817 q^{51} +5.94922 q^{52} -7.21562 q^{53} -0.868051 q^{54} -21.3540 q^{55} +3.16827 q^{56} +2.36208 q^{57} +7.94150 q^{58} -4.53090 q^{59} +10.8149 q^{60} -8.12871 q^{61} -7.51048 q^{62} +10.5967 q^{63} +1.00000 q^{64} -25.5435 q^{65} -12.5274 q^{66} +9.98418 q^{67} +4.67739 q^{68} +9.54077 q^{69} -13.6032 q^{70} +2.67275 q^{71} +3.34462 q^{72} +0.161516 q^{73} +10.8529 q^{74} -33.8403 q^{75} -0.937762 q^{76} +15.7573 q^{77} -14.9852 q^{78} +12.8153 q^{79} -4.29358 q^{80} -7.84737 q^{81} +5.74253 q^{82} -12.7824 q^{83} -7.98041 q^{84} -20.0827 q^{85} +0.831399 q^{86} -20.0035 q^{87} +4.97347 q^{88} +8.40279 q^{89} -14.3604 q^{90} +18.8488 q^{91} -3.78774 q^{92} +18.9178 q^{93} -4.23012 q^{94} +4.02635 q^{95} -2.51885 q^{96} -3.62669 q^{97} +3.03795 q^{98} +16.6344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9}+O(q^{10}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9	\( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9} + 18 q^{10} + 28 q^{11} + 17 q^{12} + 47 q^{13} + 19 q^{14} + 14 q^{15} + 79 q^{16} + 36 q^{17} + 118 q^{18} + 29 q^{19} + 18 q^{20} + 45 q^{21} + 28 q^{22} + 23 q^{23} + 17 q^{24} + 161 q^{25} + 47 q^{26} + 50 q^{27} + 19 q^{28} + 53 q^{29} + 14 q^{30} + 29 q^{31} + 79 q^{32} + 34 q^{33} + 36 q^{34} + 33 q^{35} + 118 q^{36} + 89 q^{37} + 29 q^{38} - 7 q^{39} + 18 q^{40} + 58 q^{41} + 45 q^{42} + 88 q^{43} + 28 q^{44} + 45 q^{45} + 23 q^{46} + 3 q^{47} + 17 q^{48} + 162 q^{49} + 161 q^{50} + 29 q^{51} + 47 q^{52} + 88 q^{53} + 50 q^{54} + 37 q^{55} + 19 q^{56} + 54 q^{57} + 53 q^{58} + 37 q^{59} + 14 q^{60} + 55 q^{61} + 29 q^{62} + 21 q^{63} + 79 q^{64} + 55 q^{65} + 34 q^{66} + 107 q^{67} + 36 q^{68} + 39 q^{69} + 33 q^{70} - 5 q^{71} + 118 q^{72} + 71 q^{73} + 89 q^{74} + 37 q^{75} + 29 q^{76} + 61 q^{77} - 7 q^{78} + 29 q^{79} + 18 q^{80} + 215 q^{81} + 58 q^{82} + 42 q^{83} + 45 q^{84} + 84 q^{85} + 88 q^{86} + 15 q^{87} + 28 q^{88} + 72 q^{89} + 45 q^{90} + 70 q^{91} + 23 q^{92} + 97 q^{93} + 3 q^{94} - 18 q^{95} + 17 q^{96} + 93 q^{97} + 162 q^{98} + 49 q^{99}+O(q^{100}) \) 79 * q + 79 * q^2 + 17 * q^3 + 79 * q^4 + 18 * q^5 + 17 * q^6 + 19 * q^7 + 79 * q^8 + 118 * q^9 + 18 * q^10 + 28 * q^11 + 17 * q^12 + 47 * q^13 + 19 * q^14 + 14 * q^15 + 79 * q^16 + 36 * q^17 + 118 * q^18 + 29 * q^19 + 18 * q^20 + 45 * q^21 + 28 * q^22 + 23 * q^23 + 17 * q^24 + 161 * q^25 + 47 * q^26 + 50 * q^27 + 19 * q^28 + 53 * q^29 + 14 * q^30 + 29 * q^31 + 79 * q^32 + 34 * q^33 + 36 * q^34 + 33 * q^35 + 118 * q^36 + 89 * q^37 + 29 * q^38 - 7 * q^39 + 18 * q^40 + 58 * q^41 + 45 * q^42 + 88 * q^43 + 28 * q^44 + 45 * q^45 + 23 * q^46 + 3 * q^47 + 17 * q^48 + 162 * q^49 + 161 * q^50 + 29 * q^51 + 47 * q^52 + 88 * q^53 + 50 * q^54 + 37 * q^55 + 19 * q^56 + 54 * q^57 + 53 * q^58 + 37 * q^59 + 14 * q^60 + 55 * q^61 + 29 * q^62 + 21 * q^63 + 79 * q^64 + 55 * q^65 + 34 * q^66 + 107 * q^67 + 36 * q^68 + 39 * q^69 + 33 * q^70 - 5 * q^71 + 118 * q^72 + 71 * q^73 + 89 * q^74 + 37 * q^75 + 29 * q^76 + 61 * q^77 - 7 * q^78 + 29 * q^79 + 18 * q^80 + 215 * q^81 + 58 * q^82 + 42 * q^83 + 45 * q^84 + 84 * q^85 + 88 * q^86 + 15 * q^87 + 28 * q^88 + 72 * q^89 + 45 * q^90 + 70 * q^91 + 23 * q^92 + 97 * q^93 + 3 * q^94 - 18 * q^95 + 17 * q^96 + 93 * q^97 + 162 * q^98 + 49 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more