LMFDB - Embedded newform 6038.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.49405 q^{3} +1.00000 q^{4} -0.329476 q^{5} +1.49405 q^{6} +1.19581 q^{7} -1.00000 q^{8} -0.767828 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.49405 q^{3} +1.00000 q^{4} -0.329476 q^{5} +1.49405 q^{6} +1.19581 q^{7} -1.00000 q^{8} -0.767828 q^{9} +0.329476 q^{10} -0.863718 q^{11} -1.49405 q^{12} -6.59984 q^{13} -1.19581 q^{14} +0.492252 q^{15} +1.00000 q^{16} -3.71248 q^{17} +0.767828 q^{18} -5.27917 q^{19} -0.329476 q^{20} -1.78659 q^{21} +0.863718 q^{22} -8.33345 q^{23} +1.49405 q^{24} -4.89145 q^{25} +6.59984 q^{26} +5.62931 q^{27} +1.19581 q^{28} -3.32682 q^{29} -0.492252 q^{30} -8.30093 q^{31} -1.00000 q^{32} +1.29043 q^{33} +3.71248 q^{34} -0.393989 q^{35} -0.767828 q^{36} -3.64260 q^{37} +5.27917 q^{38} +9.86046 q^{39} +0.329476 q^{40} +6.97238 q^{41} +1.78659 q^{42} +0.0669573 q^{43} -0.863718 q^{44} +0.252981 q^{45} +8.33345 q^{46} -10.3163 q^{47} -1.49405 q^{48} -5.57005 q^{49} +4.89145 q^{50} +5.54661 q^{51} -6.59984 q^{52} -5.13029 q^{53} -5.62931 q^{54} +0.284574 q^{55} -1.19581 q^{56} +7.88732 q^{57} +3.32682 q^{58} +0.992451 q^{59} +0.492252 q^{60} -4.25075 q^{61} +8.30093 q^{62} -0.918173 q^{63} +1.00000 q^{64} +2.17449 q^{65} -1.29043 q^{66} +12.2388 q^{67} -3.71248 q^{68} +12.4505 q^{69} +0.393989 q^{70} +12.7352 q^{71} +0.767828 q^{72} +3.14275 q^{73} +3.64260 q^{74} +7.30804 q^{75} -5.27917 q^{76} -1.03284 q^{77} -9.86046 q^{78} -3.12290 q^{79} -0.329476 q^{80} -6.10696 q^{81} -6.97238 q^{82} +4.56485 q^{83} -1.78659 q^{84} +1.22317 q^{85} -0.0669573 q^{86} +4.97042 q^{87} +0.863718 q^{88} -12.2224 q^{89} -0.252981 q^{90} -7.89213 q^{91} -8.33345 q^{92} +12.4020 q^{93} +10.3163 q^{94} +1.73936 q^{95} +1.49405 q^{96} +9.55303 q^{97} +5.57005 q^{98} +0.663186 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.49405	−0.862588	−0.431294	−	0.902212i	$-0.641943\pi$
$3$	−1.49405	−0.862588	−0.431294	+	0.902212i	$0.641943\pi$
$4$	1.00000	0.500000
$5$	−0.329476	−0.147346	−0.0736730	−	0.997282i	$-0.523472\pi$
$5$	−0.329476	−0.147346	−0.0736730	+	0.997282i	$0.523472\pi$
$6$	1.49405	0.609942
$7$	1.19581	0.451972	0.225986	−	0.974131i	$-0.427440\pi$
$7$	1.19581	0.451972	0.225986	+	0.974131i	$0.427440\pi$
$8$	−1.00000	−0.353553
$9$	−0.767828	−0.255943
$10$	0.329476	0.104189
$11$	−0.863718	−0.260421	−0.130210	−	0.991486i	$-0.541565\pi$
$11$	−0.863718	−0.260421	−0.130210	+	0.991486i	$0.541565\pi$
$12$	−1.49405	−0.431294
$13$	−6.59984	−1.83047	−0.915233	−	0.402924i	$-0.867994\pi$
$13$	−6.59984	−1.83047	−0.915233	+	0.402924i	$0.867994\pi$
$14$	−1.19581	−0.319593
$15$	0.492252	0.127099
$16$	1.00000	0.250000
$17$	−3.71248	−0.900408	−0.450204	−	0.892926i	$-0.648649\pi$
$17$	−3.71248	−0.900408	−0.450204	+	0.892926i	$0.648649\pi$
$18$	0.767828	0.180979
$19$	−5.27917	−1.21112	−0.605562	−	0.795798i	$-0.707052\pi$
$19$	−5.27917	−1.21112	−0.605562	+	0.795798i	$0.707052\pi$
$20$	−0.329476	−0.0736730
$21$	−1.78659	−0.389866
$22$	0.863718	0.184145
$23$	−8.33345	−1.73764	−0.868822	−	0.495125i	$-0.835122\pi$
$23$	−8.33345	−1.73764	−0.868822	+	0.495125i	$0.835122\pi$
$24$	1.49405	0.304971
$25$	−4.89145	−0.978289
$26$	6.59984	1.29434
$27$	5.62931	1.08336
$28$	1.19581	0.225986
$29$	−3.32682	−0.617775	−0.308888	−	0.951099i	$-0.599957\pi$
$29$	−3.32682	−0.617775	−0.308888	+	0.951099i	$0.599957\pi$
$30$	−0.492252	−0.0898725
$31$	−8.30093	−1.49089	−0.745446	−	0.666566i	$-0.767763\pi$
$31$	−8.30093	−1.49089	−0.745446	+	0.666566i	$0.767763\pi$
$32$	−1.00000	−0.176777
$33$	1.29043	0.224636
$34$	3.71248	0.636684
$35$	−0.393989	−0.0665964
$36$	−0.767828	−0.127971
$37$	−3.64260	−0.598839	−0.299420	−	0.954122i	$-0.596793\pi$
$37$	−3.64260	−0.598839	−0.299420	+	0.954122i	$0.596793\pi$
$38$	5.27917	0.856395
$39$	9.86046	1.57894
$40$	0.329476	0.0520947
$41$	6.97238	1.08890	0.544452	−	0.838792i	$-0.316738\pi$
$41$	6.97238	1.08890	0.544452	+	0.838792i	$0.316738\pi$
$42$	1.78659	0.275677
$43$	0.0669573	0.0102109	0.00510544	−	0.999987i	$-0.498375\pi$
$43$	0.0669573	0.0102109	0.00510544	+	0.999987i	$0.498375\pi$
$44$	−0.863718	−0.130210
$45$	0.252981	0.0377121
$46$	8.33345	1.22870
$47$	−10.3163	−1.50478	−0.752391	−	0.658716i	$-0.771100\pi$
$47$	−10.3163	−1.50478	−0.752391	+	0.658716i	$0.771100\pi$
$48$	−1.49405	−0.215647
$49$	−5.57005	−0.795721
$50$	4.89145	0.691755
$51$	5.54661	0.776681
$52$	−6.59984	−0.915233
$53$	−5.13029	−0.704699	−0.352350	−	0.935868i	$-0.614617\pi$
$53$	−5.13029	−0.704699	−0.352350	+	0.935868i	$0.614617\pi$
$54$	−5.62931	−0.766052
$55$	0.284574	0.0383720
$56$	−1.19581	−0.159796
$57$	7.88732	1.04470
$58$	3.32682	0.436833
$59$	0.992451	0.129206	0.0646030	−	0.997911i	$-0.479422\pi$
$59$	0.992451	0.129206	0.0646030	+	0.997911i	$0.479422\pi$
$60$	0.492252	0.0635494
$61$	−4.25075	−0.544252	−0.272126	−	0.962262i	$-0.587727\pi$
$61$	−4.25075	−0.544252	−0.272126	+	0.962262i	$0.587727\pi$
$62$	8.30093	1.05422
$63$	−0.918173	−0.115679
$64$	1.00000	0.125000
$65$	2.17449	0.269712
$66$	−1.29043	−0.158841
$67$	12.2388	1.49521	0.747604	−	0.664144i	$-0.231204\pi$
$67$	12.2388	1.49521	0.747604	+	0.664144i	$0.231204\pi$
$68$	−3.71248	−0.450204
$69$	12.4505	1.49887
$70$	0.393989	0.0470907
$71$	12.7352	1.51139	0.755694	−	0.654925i	$-0.227300\pi$
$71$	12.7352	1.51139	0.755694	+	0.654925i	$0.227300\pi$
$72$	0.767828	0.0904894
$73$	3.14275	0.367832	0.183916	−	0.982942i	$-0.441123\pi$
$73$	3.14275	0.367832	0.183916	+	0.982942i	$0.441123\pi$
$74$	3.64260	0.423443
$75$	7.30804	0.843860
$76$	−5.27917	−0.605562
$77$	−1.03284	−0.117703
$78$	−9.86046	−1.11648
$79$	−3.12290	−0.351353	−0.175677	−	0.984448i	$-0.556211\pi$
$79$	−3.12290	−0.351353	−0.175677	+	0.984448i	$0.556211\pi$
$80$	−0.329476	−0.0368365
$81$	−6.10696	−0.678551
$82$	−6.97238	−0.769971
$83$	4.56485	0.501057	0.250528	−	0.968109i	$-0.419396\pi$
$83$	4.56485	0.501057	0.250528	+	0.968109i	$0.419396\pi$
$84$	−1.78659	−0.194933
$85$	1.22317	0.132672
$86$	−0.0669573	−0.00722019
$87$	4.97042	0.532885
$88$	0.863718	0.0920726
$89$	−12.2224	−1.29557	−0.647786	−	0.761823i	$-0.724305\pi$
$89$	−12.2224	−1.29557	−0.647786	+	0.761823i	$0.724305\pi$
$90$	−0.252981	−0.0266665
$91$	−7.89213	−0.827320
$92$	−8.33345	−0.868822
$93$	12.4020	1.28602
$94$	10.3163	1.06404
$95$	1.73936	0.178454
$96$	1.49405	0.152485
$97$	9.55303	0.969963	0.484982	−	0.874524i	$-0.338826\pi$
$97$	9.55303	0.969963	0.484982	+	0.874524i	$0.338826\pi$
$98$	5.57005	0.562660
$99$	0.663186	0.0666527
$100$	−4.89145	−0.489145
$101$	−9.61219	−0.956449	−0.478224	−	0.878238i	$-0.658719\pi$
$101$	−9.61219	−0.956449	−0.478224	+	0.878238i	$0.658719\pi$
$102$	−5.54661	−0.549196
$103$	17.0767	1.68262	0.841309	−	0.540554i	$-0.181785\pi$
$103$	17.0767	1.68262	0.841309	+	0.540554i	$0.181785\pi$
$104$	6.59984	0.647168
$105$	0.588638	0.0574452
$106$	5.13029	0.498298
$107$	11.8051	1.14124	0.570619	−	0.821215i	$-0.306703\pi$
$107$	11.8051	1.14124	0.570619	+	0.821215i	$0.306703\pi$
$108$	5.62931	0.541680
$109$	3.48248	0.333561	0.166780	−	0.985994i	$-0.446663\pi$
$109$	3.48248	0.333561	0.166780	+	0.985994i	$0.446663\pi$
$110$	−0.284574	−0.0271331
$111$	5.44220	0.516551
$112$	1.19581	0.112993
$113$	9.73871	0.916141	0.458070	−	0.888916i	$-0.348541\pi$
$113$	9.73871	0.916141	0.458070	+	0.888916i	$0.348541\pi$
$114$	−7.88732	−0.738715
$115$	2.74567	0.256035
$116$	−3.32682	−0.308888
$117$	5.06754	0.468494
$118$	−0.992451	−0.0913625
$119$	−4.43940	−0.406959
$120$	−0.492252	−0.0449362
$121$	−10.2540	−0.932181
$122$	4.25075	0.384844
$123$	−10.4171	−0.939275
$124$	−8.30093	−0.745446
$125$	3.25899	0.291493
$126$	0.918173	0.0817974
$127$	3.26455	0.289682	0.144841	−	0.989455i	$-0.453733\pi$
$127$	3.26455	0.289682	0.144841	+	0.989455i	$0.453733\pi$
$128$	−1.00000	−0.0883883
$129$	−0.100037	−0.00880779
$130$	−2.17449	−0.190715
$131$	−17.5842	−1.53634	−0.768169	−	0.640247i	$-0.778832\pi$
$131$	−17.5842	−1.53634	−0.768169	+	0.640247i	$0.778832\pi$
$132$	1.29043	0.112318
$133$	−6.31287	−0.547395
$134$	−12.2388	−1.05727
$135$	−1.85472	−0.159629
$136$	3.71248	0.318342
$137$	−8.43309	−0.720487	−0.360244	−	0.932858i	$-0.617306\pi$
$137$	−8.43309	−0.720487	−0.360244	+	0.932858i	$0.617306\pi$
$138$	−12.4505	−1.05986
$139$	−13.2016	−1.11975	−0.559873	−	0.828578i	$-0.689150\pi$
$139$	−13.2016	−1.11975	−0.559873	+	0.828578i	$0.689150\pi$
$140$	−0.393989	−0.0332982
$141$	15.4130	1.29801
$142$	−12.7352	−1.06871
$143$	5.70040	0.476691
$144$	−0.767828	−0.0639856
$145$	1.09611	0.0910268
$146$	−3.14275	−0.260096
$147$	8.32190	0.686379
$148$	−3.64260	−0.299420
$149$	−2.94643	−0.241381	−0.120690	−	0.992690i	$-0.538511\pi$
$149$	−2.94643	−0.241381	−0.120690	+	0.992690i	$0.538511\pi$
$150$	−7.30804	−0.596699
$151$	−13.8180	−1.12450	−0.562248	−	0.826969i	$-0.690063\pi$
$151$	−13.8180	−1.12450	−0.562248	+	0.826969i	$0.690063\pi$
$152$	5.27917	0.428197
$153$	2.85054	0.230453
$154$	1.03284	0.0832286
$155$	2.73496	0.219677
$156$	9.86046	0.789469
$157$	18.8674	1.50578	0.752890	−	0.658147i	$-0.228659\pi$
$157$	18.8674	1.50578	0.752890	+	0.658147i	$0.228659\pi$
$158$	3.12290	0.248444
$159$	7.66489	0.607865
$160$	0.329476	0.0260473
$161$	−9.96519	−0.785367
$162$	6.10696	0.479808
$163$	−15.8741	−1.24335	−0.621677	−	0.783274i	$-0.713548\pi$
$163$	−15.8741	−1.24335	−0.621677	+	0.783274i	$0.713548\pi$
$164$	6.97238	0.544452
$165$	−0.425167	−0.0330992
$166$	−4.56485	−0.354301
$167$	−11.6205	−0.899218	−0.449609	−	0.893226i	$-0.648437\pi$
$167$	−11.6205	−0.899218	−0.449609	+	0.893226i	$0.648437\pi$
$168$	1.78659	0.137838
$169$	30.5579	2.35061
$170$	−1.22317	−0.0938129
$171$	4.05349	0.309978
$172$	0.0669573	0.00510544
$173$	5.21605	0.396569	0.198284	−	0.980145i	$-0.436463\pi$
$173$	5.21605	0.396569	0.198284	+	0.980145i	$0.436463\pi$
$174$	−4.97042	−0.376807
$175$	−5.84922	−0.442160
$176$	−0.863718	−0.0651052
$177$	−1.48277	−0.111452
$178$	12.2224	0.916107
$179$	−19.1140	−1.42865	−0.714323	−	0.699816i	$-0.753265\pi$
$179$	−19.1140	−1.42865	−0.714323	+	0.699816i	$0.753265\pi$
$180$	0.252981	0.0188561
$181$	9.60158	0.713680	0.356840	−	0.934166i	$-0.383854\pi$
$181$	9.60158	0.713680	0.356840	+	0.934166i	$0.383854\pi$
$182$	7.89213	0.585004
$183$	6.35081	0.469465
$184$	8.33345	0.614350
$185$	1.20015	0.0882366
$186$	−12.4020	−0.909357
$187$	3.20653	0.234485
$188$	−10.3163	−0.752391
$189$	6.73156	0.489649
$190$	−1.73936	−0.126186
$191$	−12.2475	−0.886195	−0.443097	−	0.896474i	$-0.646120\pi$
$191$	−12.2475	−0.886195	−0.443097	+	0.896474i	$0.646120\pi$
$192$	−1.49405	−0.107823
$193$	−18.1203	−1.30433	−0.652163	−	0.758078i	$-0.726138\pi$
$193$	−18.1203	−1.30433	−0.652163	+	0.758078i	$0.726138\pi$
$194$	−9.55303	−0.685867
$195$	−3.24878	−0.232650
$196$	−5.57005	−0.397860
$197$	−10.3833	−0.739779	−0.369890	−	0.929076i	$-0.620604\pi$
$197$	−10.3833	−0.739779	−0.369890	+	0.929076i	$0.620604\pi$
$198$	−0.663186	−0.0471306
$199$	21.2539	1.50665	0.753323	−	0.657651i	$-0.228450\pi$
$199$	21.2539	1.50665	0.753323	+	0.657651i	$0.228450\pi$
$200$	4.89145	0.345877
$201$	−18.2853	−1.28975
$202$	9.61219	0.676311
$203$	−3.97824	−0.279217
$204$	5.54661	0.388340
$205$	−2.29723	−0.160446
$206$	−17.0767	−1.18979
$207$	6.39865	0.444737
$208$	−6.59984	−0.457617
$209$	4.55971	0.315402
$210$	−0.588638	−0.0406199
$211$	−14.4558	−0.995180	−0.497590	−	0.867412i	$-0.665782\pi$
$211$	−14.4558	−0.995180	−0.497590	+	0.867412i	$0.665782\pi$
$212$	−5.13029	−0.352350
$213$	−19.0269	−1.30371
$214$	−11.8051	−0.806977
$215$	−0.0220608	−0.00150453
$216$	−5.62931	−0.383026
$217$	−9.92631	−0.673842
$218$	−3.48248	−0.235863
$219$	−4.69542	−0.317287
$220$	0.284574	0.0191860
$221$	24.5018	1.64817
$222$	−5.44220	−0.365257
$223$	22.0418	1.47603	0.738015	−	0.674784i	$-0.235763\pi$
$223$	22.0418	1.47603	0.738015	+	0.674784i	$0.235763\pi$
$224$	−1.19581	−0.0798982
$225$	3.75579	0.250386
$226$	−9.73871	−0.647809
$227$	12.0595	0.800417	0.400209	−	0.916424i	$-0.368938\pi$
$227$	12.0595	0.800417	0.400209	+	0.916424i	$0.368938\pi$
$228$	7.88732	0.522351
$229$	−6.43756	−0.425406	−0.212703	−	0.977117i	$-0.568227\pi$
$229$	−6.43756	−0.425406	−0.212703	+	0.977117i	$0.568227\pi$
$230$	−2.74567	−0.181044
$231$	1.54311	0.101529
$232$	3.32682	0.218417
$233$	−16.9583	−1.11098	−0.555488	−	0.831525i	$-0.687468\pi$
$233$	−16.9583	−1.11098	−0.555488	+	0.831525i	$0.687468\pi$
$234$	−5.06754	−0.331275
$235$	3.39896	0.221724
$236$	0.992451	0.0646030
$237$	4.66575	0.303073
$238$	4.43940	0.287764
$239$	−16.6047	−1.07407	−0.537033	−	0.843561i	$-0.680455\pi$
$239$	−16.6047	−1.07407	−0.537033	+	0.843561i	$0.680455\pi$
$240$	0.492252	0.0317747
$241$	−0.977158	−0.0629443	−0.0314721	−	0.999505i	$-0.510020\pi$
$241$	−0.977158	−0.0629443	−0.0314721	+	0.999505i	$0.510020\pi$
$242$	10.2540	0.659152
$243$	−7.76385	−0.498051
$244$	−4.25075	−0.272126
$245$	1.83520	0.117246
$246$	10.4171	0.664168
$247$	34.8417	2.21692
$248$	8.30093	0.527110
$249$	−6.82009	−0.432205
$250$	−3.25899	−0.206117
$251$	−15.0206	−0.948093	−0.474047	−	0.880500i	$-0.657207\pi$
$251$	−15.0206	−0.948093	−0.474047	+	0.880500i	$0.657207\pi$
$252$	−0.918173	−0.0578395
$253$	7.19775	0.452518
$254$	−3.26455	−0.204836
$255$	−1.82747	−0.114441
$256$	1.00000	0.0625000
$257$	−8.15310	−0.508576	−0.254288	−	0.967128i	$-0.581841\pi$
$257$	−8.15310	−0.508576	−0.254288	+	0.967128i	$0.581841\pi$
$258$	0.100037	0.00622805
$259$	−4.35584	−0.270659
$260$	2.17449	0.134856
$261$	2.55443	0.158115
$262$	17.5842	1.08635
$263$	8.61772	0.531391	0.265696	−	0.964057i	$-0.414398\pi$
$263$	8.61772	0.531391	0.265696	+	0.964057i	$0.414398\pi$
$264$	−1.29043	−0.0794207
$265$	1.69031	0.103835
$266$	6.31287	0.387067
$267$	18.2608	1.11754
$268$	12.2388	0.747604
$269$	2.90393	0.177056	0.0885278	−	0.996074i	$-0.471784\pi$
$269$	2.90393	0.177056	0.0885278	+	0.996074i	$0.471784\pi$
$270$	1.85472	0.112875
$271$	26.3532	1.60085	0.800423	−	0.599436i	$-0.204608\pi$
$271$	26.3532	1.60085	0.800423	+	0.599436i	$0.204608\pi$
$272$	−3.71248	−0.225102
$273$	11.7912	0.713636
$274$	8.43309	0.509461
$275$	4.22483	0.254767
$276$	12.4505	0.749435
$277$	4.21349	0.253164	0.126582	−	0.991956i	$-0.459599\pi$
$277$	4.21349	0.253164	0.126582	+	0.991956i	$0.459599\pi$
$278$	13.2016	0.791780
$279$	6.37369	0.381583
$280$	0.393989	0.0235454
$281$	18.3075	1.09213	0.546066	−	0.837742i	$-0.316125\pi$
$281$	18.3075	1.09213	0.546066	+	0.837742i	$0.316125\pi$
$282$	−15.4130	−0.917830
$283$	−12.4874	−0.742299	−0.371149	−	0.928573i	$-0.621036\pi$
$283$	−12.4874	−0.742299	−0.371149	+	0.928573i	$0.621036\pi$
$284$	12.7352	0.755694
$285$	−2.59868	−0.153933
$286$	−5.70040	−0.337072
$287$	8.33762	0.492154
$288$	0.767828	0.0452447
$289$	−3.21752	−0.189266
$290$	−1.09611	−0.0643657
$291$	−14.2727	−0.836678
$292$	3.14275	0.183916
$293$	−7.40868	−0.432820	−0.216410	−	0.976303i	$-0.569435\pi$
$293$	−7.40868	−0.432820	−0.216410	+	0.976303i	$0.569435\pi$
$294$	−8.32190	−0.485343
$295$	−0.326989	−0.0190380
$296$	3.64260	0.211722
$297$	−4.86213	−0.282130
$298$	2.94643	0.170682
$299$	54.9994	3.18070
$300$	7.30804	0.421930
$301$	0.0800680	0.00461504
$302$	13.8180	0.795139
$303$	14.3611	0.825021
$304$	−5.27917	−0.302781
$305$	1.40052	0.0801934
$306$	−2.85054	−0.162955
$307$	−25.3648	−1.44765	−0.723824	−	0.689985i	$-0.757617\pi$
$307$	−25.3648	−1.44765	−0.723824	+	0.689985i	$0.757617\pi$
$308$	−1.03284	−0.0588515
$309$	−25.5134	−1.45141
$310$	−2.73496	−0.155335
$311$	−13.7711	−0.780885	−0.390442	−	0.920627i	$-0.627678\pi$
$311$	−13.7711	−0.780885	−0.390442	+	0.920627i	$0.627678\pi$
$312$	−9.86046	−0.558239
$313$	−8.95514	−0.506175	−0.253087	−	0.967443i	$-0.581446\pi$
$313$	−8.95514	−0.506175	−0.253087	+	0.967443i	$0.581446\pi$
$314$	−18.8674	−1.06475
$315$	0.302516	0.0170448
$316$	−3.12290	−0.175677
$317$	−15.1628	−0.851627	−0.425813	−	0.904811i	$-0.640012\pi$
$317$	−15.1628	−0.851627	−0.425813	+	0.904811i	$0.640012\pi$
$318$	−7.66489	−0.429825
$319$	2.87344	0.160882
$320$	−0.329476	−0.0184183
$321$	−17.6373	−0.984418
$322$	9.96519	0.555338
$323$	19.5988	1.09051
$324$	−6.10696	−0.339275
$325$	32.2828	1.79073
$326$	15.8741	0.879184
$327$	−5.20298	−0.287725
$328$	−6.97238	−0.384986
$329$	−12.3363	−0.680120
$330$	0.425167	0.0234047
$331$	−22.2340	−1.22209	−0.611046	−	0.791595i	$-0.709251\pi$
$331$	−22.2340	−1.22209	−0.611046	+	0.791595i	$0.709251\pi$
$332$	4.56485	0.250528
$333$	2.79689	0.153268
$334$	11.6205	0.635843
$335$	−4.03239	−0.220313
$336$	−1.78659	−0.0974665
$337$	13.0577	0.711296	0.355648	−	0.934620i	$-0.384260\pi$
$337$	13.0577	0.711296	0.355648	+	0.934620i	$0.384260\pi$
$338$	−30.5579	−1.66213
$339$	−14.5501	−0.790252
$340$	1.22317	0.0663358
$341$	7.16966	0.388259
$342$	−4.05349	−0.219188
$343$	−15.0313	−0.811616
$344$	−0.0669573	−0.00361009
$345$	−4.10215	−0.220853
$346$	−5.21605	−0.280416
$347$	4.41711	0.237123	0.118562	−	0.992947i	$-0.462172\pi$
$347$	4.41711	0.237123	0.118562	+	0.992947i	$0.462172\pi$
$348$	4.97042	0.266443
$349$	13.1222	0.702416	0.351208	−	0.936297i	$-0.385771\pi$
$349$	13.1222	0.702416	0.351208	+	0.936297i	$0.385771\pi$
$350$	5.84922	0.312654
$351$	−37.1525	−1.98306
$352$	0.863718	0.0460363
$353$	−26.3013	−1.39988	−0.699939	−	0.714203i	$-0.746789\pi$
$353$	−26.3013	−1.39988	−0.699939	+	0.714203i	$0.746789\pi$
$354$	1.48277	0.0788082
$355$	−4.19594	−0.222697
$356$	−12.2224	−0.647786
$357$	6.63267	0.351038
$358$	19.1140	1.01021
$359$	−19.8674	−1.04856	−0.524281	−	0.851545i	$-0.675666\pi$
$359$	−19.8674	−1.04856	−0.524281	+	0.851545i	$0.675666\pi$
$360$	−0.252981	−0.0133332
$361$	8.86965	0.466824
$362$	−9.60158	−0.504648
$363$	15.3199	0.804088
$364$	−7.89213	−0.413660
$365$	−1.03546	−0.0541985
$366$	−6.35081	−0.331962
$367$	24.7863	1.29383	0.646917	−	0.762560i	$-0.276058\pi$
$367$	24.7863	1.29383	0.646917	+	0.762560i	$0.276058\pi$
$368$	−8.33345	−0.434411
$369$	−5.35359	−0.278697
$370$	−1.20015	−0.0623927
$371$	−6.13483	−0.318505
$372$	12.4020	0.643012
$373$	3.15488	0.163354	0.0816768	−	0.996659i	$-0.473972\pi$
$373$	3.15488	0.163354	0.0816768	+	0.996659i	$0.473972\pi$
$374$	−3.20653	−0.165806
$375$	−4.86908	−0.251438
$376$	10.3163	0.532021
$377$	21.9565	1.13082
$378$	−6.73156	−0.346234
$379$	4.22656	0.217104	0.108552	−	0.994091i	$-0.465379\pi$
$379$	4.22656	0.217104	0.108552	+	0.994091i	$0.465379\pi$
$380$	1.73936	0.0892272
$381$	−4.87738	−0.249876
$382$	12.2475	0.626634
$383$	0.207644	0.0106101	0.00530507	−	0.999986i	$-0.498311\pi$
$383$	0.207644	0.0106101	0.00530507	+	0.999986i	$0.498311\pi$
$384$	1.49405	0.0762427
$385$	0.340296	0.0173431
$386$	18.1203	0.922298
$387$	−0.0514116	−0.00261340
$388$	9.55303	0.484982
$389$	−13.2797	−0.673310	−0.336655	−	0.941628i	$-0.609296\pi$
$389$	−13.2797	−0.673310	−0.336655	+	0.941628i	$0.609296\pi$
$390$	3.24878	0.164509
$391$	30.9377	1.56459
$392$	5.57005	0.281330
$393$	26.2716	1.32523
$394$	10.3833	0.523103
$395$	1.02892	0.0517705
$396$	0.663186	0.0333264
$397$	23.1132	1.16002	0.580008	−	0.814611i	$-0.303049\pi$
$397$	23.1132	1.16002	0.580008	+	0.814611i	$0.303049\pi$
$398$	−21.2539	−1.06536
$399$	9.43171	0.472176
$400$	−4.89145	−0.244572
$401$	−29.0011	−1.44825	−0.724123	−	0.689671i	$-0.757755\pi$
$401$	−29.0011	−1.44825	−0.724123	+	0.689671i	$0.757755\pi$
$402$	18.2853	0.911990
$403$	54.7848	2.72903
$404$	−9.61219	−0.478224
$405$	2.01209	0.0999818
$406$	3.97824	0.197437
$407$	3.14618	0.155950
$408$	−5.54661	−0.274598
$409$	−15.2858	−0.755834	−0.377917	−	0.925840i	$-0.623359\pi$
$409$	−15.2858	−0.755834	−0.377917	+	0.925840i	$0.623359\pi$
$410$	2.29723	0.113452
$411$	12.5994	0.621483
$412$	17.0767	0.841309
$413$	1.18678	0.0583976
$414$	−6.39865	−0.314476
$415$	−1.50401	−0.0738288
$416$	6.59984	0.323584
$417$	19.7238	0.965879
$418$	−4.55971	−0.223023
$419$	−7.41968	−0.362475	−0.181238	−	0.983439i	$-0.558010\pi$
$419$	−7.41968	−0.362475	−0.181238	+	0.983439i	$0.558010\pi$
$420$	0.588638	0.0287226
$421$	8.72342	0.425154	0.212577	−	0.977144i	$-0.431814\pi$
$421$	8.72342	0.425154	0.212577	+	0.977144i	$0.431814\pi$
$422$	14.4558	0.703699
$423$	7.92112	0.385138
$424$	5.13029	0.249149
$425$	18.1594	0.880859
$426$	19.0269	0.921859
$427$	−5.08307	−0.245987
$428$	11.8051	0.570619
$429$	−8.51666	−0.411188
$430$	0.0220608	0.00106387
$431$	−21.0242	−1.01270	−0.506350	−	0.862328i	$-0.669006\pi$
$431$	−21.0242	−1.01270	−0.506350	+	0.862328i	$0.669006\pi$
$432$	5.62931	0.270840
$433$	−24.8435	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$433$	−24.8435	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$434$	9.92631	0.476478
$435$	−1.63763	−0.0785186
$436$	3.48248	0.166780
$437$	43.9937	2.10450
$438$	4.69542	0.224356
$439$	37.0519	1.76839	0.884195	−	0.467118i	$-0.154708\pi$
$439$	37.0519	1.76839	0.884195	+	0.467118i	$0.154708\pi$
$440$	−0.284574	−0.0135665
$441$	4.27684	0.203659
$442$	−24.5018	−1.16543
$443$	13.8323	0.657194	0.328597	−	0.944470i	$-0.393424\pi$
$443$	13.8323	0.657194	0.328597	+	0.944470i	$0.393424\pi$
$444$	5.44220	0.258276
$445$	4.02698	0.190897
$446$	−22.0418	−1.04371
$447$	4.40210	0.208212
$448$	1.19581	0.0564966
$449$	33.2481	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$449$	33.2481	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$450$	−3.75579	−0.177050
$451$	−6.02217	−0.283573
$452$	9.73871	0.458070
$453$	20.6448	0.969977
$454$	−12.0595	−0.565981
$455$	2.60027	0.121902
$456$	−7.88732	−0.369358
$457$	31.1641	1.45780	0.728898	−	0.684623i	$-0.240033\pi$
$457$	31.1641	1.45780	0.728898	+	0.684623i	$0.240033\pi$
$458$	6.43756	0.300807
$459$	−20.8987	−0.975466
$460$	2.74567	0.128017
$461$	24.3523	1.13420	0.567101	−	0.823648i	$-0.308065\pi$
$461$	24.3523	1.13420	0.567101	+	0.823648i	$0.308065\pi$
$462$	−1.54311	−0.0717919
$463$	−12.8425	−0.596843	−0.298421	−	0.954434i	$-0.596460\pi$
$463$	−12.8425	−0.596843	−0.298421	+	0.954434i	$0.596460\pi$
$464$	−3.32682	−0.154444
$465$	−4.08615	−0.189491
$466$	16.9583	0.785578
$467$	10.2403	0.473866	0.236933	−	0.971526i	$-0.423858\pi$
$467$	10.2403	0.473866	0.236933	+	0.971526i	$0.423858\pi$
$468$	5.06754	0.234247
$469$	14.6352	0.675793
$470$	−3.39896	−0.156782
$471$	−28.1887	−1.29887
$472$	−0.992451	−0.0456813
$473$	−0.0578322	−0.00265913
$474$	−4.66575	−0.214305
$475$	25.8228	1.18483
$476$	−4.43940	−0.203480
$477$	3.93918	0.180363
$478$	16.6047	0.759480
$479$	−42.4426	−1.93925	−0.969627	−	0.244588i	$-0.921347\pi$
$479$	−42.4426	−1.93925	−0.969627	+	0.244588i	$0.921347\pi$
$480$	−0.492252	−0.0224681
$481$	24.0406	1.09615
$482$	0.977158	0.0445083
$483$	14.8884	0.677448
$484$	−10.2540	−0.466091
$485$	−3.14749	−0.142920
$486$	7.76385	0.352175
$487$	−41.2457	−1.86902	−0.934511	−	0.355933i	$-0.884163\pi$
$487$	−41.2457	−1.86902	−0.934511	+	0.355933i	$0.884163\pi$
$488$	4.25075	0.192422
$489$	23.7166	1.07250
$490$	−1.83520	−0.0829057
$491$	20.8294	0.940018	0.470009	−	0.882662i	$-0.344251\pi$
$491$	20.8294	0.940018	0.470009	+	0.882662i	$0.344251\pi$
$492$	−10.4171	−0.469637
$493$	12.3508	0.556250
$494$	−34.8417	−1.56760
$495$	−0.218504	−0.00982102
$496$	−8.30093	−0.372723
$497$	15.2288	0.683106
$498$	6.82009	0.305615
$499$	20.9348	0.937171	0.468585	−	0.883418i	$-0.344764\pi$
$499$	20.9348	0.937171	0.468585	+	0.883418i	$0.344764\pi$
$500$	3.25899	0.145747
$501$	17.3615	0.775654
$502$	15.0206	0.670403
$503$	−4.91836	−0.219299	−0.109649	−	0.993970i	$-0.534973\pi$
$503$	−4.91836	−0.219299	−0.109649	+	0.993970i	$0.534973\pi$
$504$	0.918173	0.0408987
$505$	3.16698	0.140929
$506$	−7.19775	−0.319979
$507$	−45.6549	−2.02761
$508$	3.26455	0.144841
$509$	33.0720	1.46589	0.732945	−	0.680288i	$-0.238145\pi$
$509$	33.0720	1.46589	0.732945	+	0.680288i	$0.238145\pi$
$510$	1.82747	0.0809219
$511$	3.75813	0.166250
$512$	−1.00000	−0.0441942
$513$	−29.7181	−1.31208
$514$	8.15310	0.359618
$515$	−5.62636	−0.247927
$516$	−0.100037	−0.00440389
$517$	8.91035	0.391877
$518$	4.35584	0.191385
$519$	−7.79301	−0.342075
$520$	−2.17449	−0.0953576
$521$	7.66491	0.335806	0.167903	−	0.985804i	$-0.446300\pi$
$521$	7.66491	0.335806	0.167903	+	0.985804i	$0.446300\pi$
$522$	−2.55443	−0.111804
$523$	−15.6861	−0.685904	−0.342952	−	0.939353i	$-0.611427\pi$
$523$	−15.6861	−0.685904	−0.342952	+	0.939353i	$0.611427\pi$
$524$	−17.5842	−0.768169
$525$	8.73901	0.381401
$526$	−8.61772	−0.375750
$527$	30.8170	1.34241
$528$	1.29043	0.0561589
$529$	46.4463	2.01941
$530$	−1.69031	−0.0734222
$531$	−0.762031	−0.0330693
$532$	−6.31287	−0.273698
$533$	−46.0166	−1.99320
$534$	−18.2608	−0.790223
$535$	−3.88948	−0.168157
$536$	−12.2388	−0.528636
$537$	28.5572	1.23233
$538$	−2.90393	−0.125197
$539$	4.81095	0.207222
$540$	−1.85472	−0.0798145
$541$	−23.0927	−0.992832	−0.496416	−	0.868085i	$-0.665351\pi$
$541$	−23.0927	−0.992832	−0.496416	+	0.868085i	$0.665351\pi$
$542$	−26.3532	−1.13197
$543$	−14.3452	−0.615612
$544$	3.71248	0.159171
$545$	−1.14739	−0.0491489
$546$	−11.7912	−0.504617
$547$	−17.7089	−0.757176	−0.378588	−	0.925565i	$-0.623590\pi$
$547$	−17.7089	−0.757176	−0.378588	+	0.925565i	$0.623590\pi$
$548$	−8.43309	−0.360244
$549$	3.26384	0.139297
$550$	−4.22483	−0.180147
$551$	17.5629	0.748203
$552$	−12.4505	−0.529931
$553$	−3.73438	−0.158802
$554$	−4.21349	−0.179014
$555$	−1.79307	−0.0761118
$556$	−13.2016	−0.559873
$557$	−3.21982	−0.136428	−0.0682141	−	0.997671i	$-0.521730\pi$
$557$	−3.21982	−0.136428	−0.0682141	+	0.997671i	$0.521730\pi$
$558$	−6.37369	−0.269820
$559$	−0.441907	−0.0186907
$560$	−0.393989	−0.0166491
$561$	−4.79071	−0.202264
$562$	−18.3075	−0.772254
$563$	35.4477	1.49394	0.746971	−	0.664857i	$-0.231508\pi$
$563$	35.4477	1.49394	0.746971	+	0.664857i	$0.231508\pi$
$564$	15.4130	0.649004
$565$	−3.20867	−0.134990
$566$	12.4874	0.524884
$567$	−7.30274	−0.306686
$568$	−12.7352	−0.534357
$569$	−26.5999	−1.11513	−0.557564	−	0.830134i	$-0.688264\pi$
$569$	−26.5999	−1.11513	−0.557564	+	0.830134i	$0.688264\pi$
$570$	2.59868	0.108847
$571$	27.9177	1.16832	0.584159	−	0.811639i	$-0.301425\pi$
$571$	27.9177	1.16832	0.584159	+	0.811639i	$0.301425\pi$
$572$	5.70040	0.238346
$573$	18.2983	0.764421
$574$	−8.33762	−0.348006
$575$	40.7626	1.69992
$576$	−0.767828	−0.0319928
$577$	−12.9505	−0.539137	−0.269569	−	0.962981i	$-0.586881\pi$
$577$	−12.9505	−0.539137	−0.269569	+	0.962981i	$0.586881\pi$
$578$	3.21752	0.133831
$579$	27.0725	1.12510
$580$	1.09611	0.0455134
$581$	5.45867	0.226464
$582$	14.2727	0.591621
$583$	4.43112	0.183518
$584$	−3.14275	−0.130048
$585$	−1.66963	−0.0690308
$586$	7.40868	0.306050
$587$	−13.5668	−0.559964	−0.279982	−	0.960005i	$-0.590328\pi$
$587$	−13.5668	−0.559964	−0.279982	+	0.960005i	$0.590328\pi$
$588$	8.32190	0.343190
$589$	43.8220	1.80566
$590$	0.326989	0.0134619
$591$	15.5131	0.638124
$592$	−3.64260	−0.149710
$593$	−28.0711	−1.15274	−0.576372	−	0.817188i	$-0.695532\pi$
$593$	−28.0711	−1.15274	−0.576372	+	0.817188i	$0.695532\pi$
$594$	4.86213	0.199496
$595$	1.46268	0.0599639
$596$	−2.94643	−0.120690
$597$	−31.7542	−1.29961
$598$	−54.9994	−2.24909
$599$	−25.9963	−1.06218	−0.531090	−	0.847315i	$-0.678217\pi$
$599$	−25.9963	−1.06218	−0.531090	+	0.847315i	$0.678217\pi$
$600$	−7.30804	−0.298350
$601$	24.2996	0.991203	0.495601	−	0.868550i	$-0.334948\pi$
$601$	24.2996	0.991203	0.495601	+	0.868550i	$0.334948\pi$
$602$	−0.0800680	−0.00326333
$603$	−9.39730	−0.382687
$604$	−13.8180	−0.562248
$605$	3.37844	0.137353
$606$	−14.3611	−0.583378
$607$	−1.03903	−0.0421730	−0.0210865	−	0.999778i	$-0.506713\pi$
$607$	−1.03903	−0.0421730	−0.0210865	+	0.999778i	$0.506713\pi$
$608$	5.27917	0.214099
$609$	5.94367	0.240850
$610$	−1.40052	−0.0567053
$611$	68.0858	2.75445
$612$	2.85054	0.115226
$613$	2.08051	0.0840310	0.0420155	−	0.999117i	$-0.486622\pi$
$613$	2.08051	0.0840310	0.0420155	+	0.999117i	$0.486622\pi$
$614$	25.3648	1.02364
$615$	3.43217	0.138398
$616$	1.03284	0.0416143
$617$	2.97988	0.119965	0.0599827	−	0.998199i	$-0.480895\pi$
$617$	2.97988	0.119965	0.0599827	+	0.998199i	$0.480895\pi$
$618$	25.5134	1.02630
$619$	−38.9186	−1.56427	−0.782135	−	0.623109i	$-0.785869\pi$
$619$	−38.9186	−1.56427	−0.782135	+	0.623109i	$0.785869\pi$
$620$	2.73496	0.109839
$621$	−46.9115	−1.88249
$622$	13.7711	0.552169
$623$	−14.6156	−0.585562
$624$	9.86046	0.394734
$625$	23.3835	0.935339
$626$	8.95514	0.357919
$627$	−6.81242	−0.272062
$628$	18.8674	0.752890
$629$	13.5231	0.539199
$630$	−0.302516	−0.0120525
$631$	23.3320	0.928834	0.464417	−	0.885617i	$-0.346264\pi$
$631$	23.3320	0.928834	0.464417	+	0.885617i	$0.346264\pi$
$632$	3.12290	0.124222
$633$	21.5977	0.858430
$634$	15.1628	0.602191
$635$	−1.07559	−0.0426835
$636$	7.66489	0.303933
$637$	36.7614	1.45654
$638$	−2.87344	−0.113760
$639$	−9.77843	−0.386829
$640$	0.329476	0.0130237
$641$	−39.8363	−1.57344	−0.786719	−	0.617311i	$-0.788222\pi$
$641$	−39.8363	−1.57344	−0.786719	+	0.617311i	$0.788222\pi$
$642$	17.6373	0.696088
$643$	−21.3507	−0.841988	−0.420994	−	0.907063i	$-0.638319\pi$
$643$	−21.3507	−0.841988	−0.420994	+	0.907063i	$0.638319\pi$
$644$	−9.96519	−0.392683
$645$	0.0329598	0.00129779
$646$	−19.5988	−0.771104
$647$	15.3928	0.605154	0.302577	−	0.953125i	$-0.402153\pi$
$647$	15.3928	0.605154	0.302577	+	0.953125i	$0.402153\pi$
$648$	6.10696	0.239904
$649$	−0.857198	−0.0336479
$650$	−32.2828	−1.26623
$651$	14.8304	0.581248
$652$	−15.8741	−0.621677
$653$	−2.65100	−0.103742	−0.0518708	−	0.998654i	$-0.516518\pi$
$653$	−2.65100	−0.103742	−0.0518708	+	0.998654i	$0.516518\pi$
$654$	5.20298	0.203453
$655$	5.79357	0.226373
$656$	6.97238	0.272226
$657$	−2.41309	−0.0941438
$658$	12.3363	0.480918
$659$	−13.4683	−0.524650	−0.262325	−	0.964980i	$-0.584489\pi$
$659$	−13.4683	−0.524650	−0.262325	+	0.964980i	$0.584489\pi$
$660$	−0.425167	−0.0165496
$661$	8.97456	0.349070	0.174535	−	0.984651i	$-0.444158\pi$
$661$	8.97456	0.349070	0.174535	+	0.984651i	$0.444158\pi$
$662$	22.2340	0.864149
$663$	−36.6067	−1.42169
$664$	−4.56485	−0.177150
$665$	2.07994	0.0806565
$666$	−2.79689	−0.108377
$667$	27.7239	1.07347
$668$	−11.6205	−0.449609
$669$	−32.9315	−1.27321
$670$	4.03239	0.155785
$671$	3.67144	0.141735
$672$	1.78659	0.0689192
$673$	−31.7459	−1.22371	−0.611857	−	0.790968i	$-0.709577\pi$
$673$	−31.7459	−1.22371	−0.611857	+	0.790968i	$0.709577\pi$
$674$	−13.0577	−0.502962
$675$	−27.5354	−1.05984
$676$	30.5579	1.17530
$677$	25.7592	0.990006	0.495003	−	0.868891i	$-0.335167\pi$
$677$	25.7592	0.990006	0.495003	+	0.868891i	$0.335167\pi$
$678$	14.5501	0.558792
$679$	11.4236	0.438397
$680$	−1.22317	−0.0469065
$681$	−18.0174	−0.690430
$682$	−7.16966	−0.274541
$683$	−35.1546	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$683$	−35.1546	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$684$	4.05349	0.154989
$685$	2.77850	0.106161
$686$	15.0313	0.573899
$687$	9.61801	0.366950
$688$	0.0669573	0.00255272
$689$	33.8591	1.28993
$690$	4.10215	0.156166
$691$	−13.4674	−0.512323	−0.256162	−	0.966634i	$-0.582458\pi$
$691$	−13.4674	−0.512323	−0.256162	+	0.966634i	$0.582458\pi$
$692$	5.21605	0.198284
$693$	0.793043	0.0301252
$694$	−4.41711	−0.167671
$695$	4.34961	0.164990
$696$	−4.97042	−0.188403
$697$	−25.8848	−0.980457
$698$	−13.1222	−0.496683
$699$	25.3365	0.958314
$700$	−5.84922	−0.221080
$701$	−4.97849	−0.188035	−0.0940175	−	0.995571i	$-0.529971\pi$
$701$	−4.97849	−0.188035	−0.0940175	+	0.995571i	$0.529971\pi$
$702$	37.1525	1.40223
$703$	19.2299	0.725269
$704$	−0.863718	−0.0325526
$705$	−5.07820	−0.191256
$706$	26.3013	0.989863
$707$	−11.4943	−0.432288
$708$	−1.48277	−0.0557258
$709$	43.3475	1.62795	0.813974	−	0.580901i	$-0.197300\pi$
$709$	43.3475	1.62795	0.813974	+	0.580901i	$0.197300\pi$
$710$	4.19594	0.157471
$711$	2.39785	0.0899263
$712$	12.2224	0.458054
$713$	69.1754	2.59064
$714$	−6.63267	−0.248222
$715$	−1.87814	−0.0702386
$716$	−19.1140	−0.714323
$717$	24.8081	0.926477
$718$	19.8674	0.741445
$719$	−38.5391	−1.43726	−0.718632	−	0.695390i	$-0.755231\pi$
$719$	−38.5391	−1.43726	−0.718632	+	0.695390i	$0.755231\pi$
$720$	0.252981	0.00942803
$721$	20.4204	0.760497
$722$	−8.86965	−0.330094
$723$	1.45992	0.0542950
$724$	9.60158	0.356840
$725$	16.2730	0.604363
$726$	−15.3199	−0.568576
$727$	−30.4122	−1.12793	−0.563964	−	0.825800i	$-0.690724\pi$
$727$	−30.4122	−1.12793	−0.563964	+	0.825800i	$0.690724\pi$
$728$	7.89213	0.292502
$729$	29.9204	1.10816
$730$	1.03546	0.0383242
$731$	−0.248577	−0.00919396
$732$	6.35081	0.234733
$733$	16.1509	0.596547	0.298274	−	0.954480i	$-0.403589\pi$
$733$	16.1509	0.596547	0.298274	+	0.954480i	$0.403589\pi$
$734$	−24.7863	−0.914879
$735$	−2.74187	−0.101135
$736$	8.33345	0.307175
$737$	−10.5709	−0.389383
$738$	5.35359	0.197068
$739$	24.1972	0.890109	0.445055	−	0.895503i	$-0.353184\pi$
$739$	24.1972	0.890109	0.445055	+	0.895503i	$0.353184\pi$
$740$	1.20015	0.0441183
$741$	−52.0551	−1.91229
$742$	6.13483	0.225217
$743$	−36.7329	−1.34760	−0.673800	−	0.738914i	$-0.735339\pi$
$743$	−36.7329	−1.34760	−0.673800	+	0.738914i	$0.735339\pi$
$744$	−12.4020	−0.454678
$745$	0.970777	0.0355665
$746$	−3.15488	−0.115508
$747$	−3.50501	−0.128242
$748$	3.20653	0.117242
$749$	14.1166	0.515808
$750$	4.86908	0.177794
$751$	−45.1131	−1.64620	−0.823100	−	0.567896i	$-0.807758\pi$
$751$	−45.1131	−1.64620	−0.823100	+	0.567896i	$0.807758\pi$
$752$	−10.3163	−0.376196
$753$	22.4415	0.817813
$754$	−21.9565	−0.799609
$755$	4.55271	0.165690
$756$	6.73156	0.244825
$757$	11.1396	0.404877	0.202439	−	0.979295i	$-0.435113\pi$
$757$	11.1396	0.404877	0.202439	+	0.979295i	$0.435113\pi$
$758$	−4.22656	−0.153515
$759$	−10.7538	−0.390337
$760$	−1.73936	−0.0630932
$761$	−52.0802	−1.88791	−0.943953	−	0.330080i	$-0.892924\pi$
$761$	−52.0802	−1.88791	−0.943953	+	0.330080i	$0.892924\pi$
$762$	4.87738	0.176689
$763$	4.16437	0.150760
$764$	−12.2475	−0.443097
$765$	−0.939185	−0.0339563
$766$	−0.207644	−0.00750250
$767$	−6.55002	−0.236507
$768$	−1.49405	−0.0539117
$769$	1.26475	0.0456079	0.0228039	−	0.999740i	$-0.492741\pi$
$769$	1.26475	0.0456079	0.0228039	+	0.999740i	$0.492741\pi$
$770$	−0.340296	−0.0122634
$771$	12.1811	0.438692
$772$	−18.1203	−0.652163
$773$	6.22905	0.224043	0.112022	−	0.993706i	$-0.464267\pi$
$773$	6.22905	0.224043	0.112022	+	0.993706i	$0.464267\pi$
$774$	0.0514116	0.00184795
$775$	40.6036	1.45852
$776$	−9.55303	−0.342934
$777$	6.50782	0.233467
$778$	13.2797	0.476102
$779$	−36.8084	−1.31880
$780$	−3.24878	−0.116325
$781$	−10.9996	−0.393597
$782$	−30.9377	−1.10633
$783$	−18.7277	−0.669274
$784$	−5.57005	−0.198930
$785$	−6.21634	−0.221871
$786$	−26.2716	−0.937076
$787$	12.3561	0.440447	0.220224	−	0.975449i	$-0.429321\pi$
$787$	12.3561	0.440447	0.220224	+	0.975449i	$0.429321\pi$
$788$	−10.3833	−0.369890
$789$	−12.8753	−0.458372
$790$	−1.02892	−0.0366073
$791$	11.6456	0.414070
$792$	−0.663186	−0.0235653
$793$	28.0542	0.996235
$794$	−23.1132	−0.820256
$795$	−2.52539	−0.0895665
$796$	21.2539	0.753323
$797$	−2.19838	−0.0778707	−0.0389354	−	0.999242i	$-0.512397\pi$
$797$	−2.19838	−0.0778707	−0.0389354	+	0.999242i	$0.512397\pi$
$798$	−9.43171	−0.333879
$799$	38.2989	1.35492
$800$	4.89145	0.172939
$801$	9.38469	0.331592
$802$	29.0011	1.02406
$803$	−2.71445	−0.0957910
$804$	−18.2853	−0.644874
$805$	3.28329	0.115721
$806$	−54.7848	−1.92971
$807$	−4.33860	−0.152726
$808$	9.61219	0.338156
$809$	52.1558	1.83370	0.916851	−	0.399230i	$-0.130722\pi$
$809$	52.1558	1.83370	0.916851	+	0.399230i	$0.130722\pi$
$810$	−2.01209	−0.0706978
$811$	−5.36835	−0.188508	−0.0942542	−	0.995548i	$-0.530047\pi$
$811$	−5.36835	−0.188508	−0.0942542	+	0.995548i	$0.530047\pi$
$812$	−3.97824	−0.139609
$813$	−39.3729	−1.38087
$814$	−3.14618	−0.110273
$815$	5.23012	0.183203
$816$	5.54661	0.194170
$817$	−0.353479	−0.0123667
$818$	15.2858	0.534455
$819$	6.05980	0.211746
$820$	−2.29723	−0.0802228
$821$	3.21177	0.112091	0.0560457	−	0.998428i	$-0.482151\pi$
$821$	3.21177	0.112091	0.0560457	+	0.998428i	$0.482151\pi$
$822$	−12.5994	−0.439455
$823$	−44.8900	−1.56477	−0.782384	−	0.622796i	$-0.785997\pi$
$823$	−44.8900	−1.56477	−0.782384	+	0.622796i	$0.785997\pi$
$824$	−17.0767	−0.594896
$825$	−6.31209	−0.219759
$826$	−1.18678	−0.0412933
$827$	18.1869	0.632420	0.316210	−	0.948689i	$-0.397590\pi$
$827$	18.1869	0.632420	0.316210	+	0.948689i	$0.397590\pi$
$828$	6.39865	0.222368
$829$	49.9427	1.73458	0.867291	−	0.497802i	$-0.165859\pi$
$829$	49.9427	1.73458	0.867291	+	0.497802i	$0.165859\pi$
$830$	1.50401	0.0522048
$831$	−6.29515	−0.218376
$832$	−6.59984	−0.228808
$833$	20.6787	0.716473
$834$	−19.7238	−0.682980
$835$	3.82866	0.132496
$836$	4.55971	0.157701
$837$	−46.7285	−1.61517
$838$	7.41968	0.256309
$839$	−4.02006	−0.138788	−0.0693940	−	0.997589i	$-0.522107\pi$
$839$	−4.02006	−0.138788	−0.0693940	+	0.997589i	$0.522107\pi$
$840$	−0.588638	−0.0203099
$841$	−17.9323	−0.618353
$842$	−8.72342	−0.300629
$843$	−27.3522	−0.942060
$844$	−14.4558	−0.497590
$845$	−10.0681	−0.346353
$846$	−7.92112	−0.272334
$847$	−12.2618	−0.421320
$848$	−5.13029	−0.176175
$849$	18.6567	0.640298
$850$	−18.1594	−0.622862
$851$	30.3554	1.04057
$852$	−19.0269	−0.651853
$853$	−20.0009	−0.684817	−0.342409	−	0.939551i	$-0.611243\pi$
$853$	−20.0009	−0.684817	−0.342409	+	0.939551i	$0.611243\pi$
$854$	5.08307	0.173939
$855$	−1.33553	−0.0456741
$856$	−11.8051	−0.403489
$857$	−13.8809	−0.474162	−0.237081	−	0.971490i	$-0.576191\pi$
$857$	−13.8809	−0.474162	−0.237081	+	0.971490i	$0.576191\pi$
$858$	8.51666	0.290754
$859$	−27.3001	−0.931468	−0.465734	−	0.884925i	$-0.654210\pi$
$859$	−27.3001	−0.931468	−0.465734	+	0.884925i	$0.654210\pi$
$860$	−0.0220608	−0.000752267 0
$861$	−12.4568	−0.424526
$862$	21.0242	0.716088
$863$	45.6270	1.55316	0.776581	−	0.630017i	$-0.216952\pi$
$863$	45.6270	1.55316	0.776581	+	0.630017i	$0.216952\pi$
$864$	−5.62931	−0.191513
$865$	−1.71856	−0.0584328
$866$	24.8435	0.844216
$867$	4.80712	0.163258
$868$	−9.92631	−0.336921
$869$	2.69730	0.0914997
$870$	1.63763	0.0555210
$871$	−80.7742	−2.73693
$872$	−3.48248	−0.117932
$873$	−7.33508	−0.248255
$874$	−43.9937	−1.48811
$875$	3.89712	0.131747
$876$	−4.69542	−0.158644
$877$	30.4654	1.02874	0.514372	−	0.857567i	$-0.328025\pi$
$877$	30.4654	1.02874	0.514372	+	0.857567i	$0.328025\pi$
$878$	−37.0519	−1.25044
$879$	11.0689	0.373345
$880$	0.284574	0.00959299
$881$	−14.0239	−0.472478	−0.236239	−	0.971695i	$-0.575915\pi$
$881$	−14.0239	−0.472478	−0.236239	+	0.971695i	$0.575915\pi$
$882$	−4.27684	−0.144009
$883$	33.5728	1.12982	0.564908	−	0.825154i	$-0.308912\pi$
$883$	33.5728	1.12982	0.564908	+	0.825154i	$0.308912\pi$
$884$	24.5018	0.824083
$885$	0.488536	0.0164220
$886$	−13.8323	−0.464706
$887$	35.6610	1.19738	0.598689	−	0.800982i	$-0.295689\pi$
$887$	35.6610	1.19738	0.598689	+	0.800982i	$0.295689\pi$
$888$	−5.44220	−0.182628
$889$	3.90377	0.130928
$890$	−4.02698	−0.134985
$891$	5.27469	0.176709
$892$	22.0418	0.738015
$893$	54.4614	1.82248
$894$	−4.40210	−0.147228
$895$	6.29759	0.210505
$896$	−1.19581	−0.0399491
$897$	−82.1716	−2.74363
$898$	−33.2481	−1.10950
$899$	27.6157	0.921036
$900$	3.75579	0.125193
$901$	19.0461	0.634517
$902$	6.02217	0.200516
$903$	−0.119625	−0.00398088
$904$	−9.73871	−0.323905
$905$	−3.16349	−0.105158
$906$	−20.6448	−0.685877
$907$	28.9761	0.962136	0.481068	−	0.876683i	$-0.340249\pi$
$907$	28.9761	0.962136	0.481068	+	0.876683i	$0.340249\pi$
$908$	12.0595	0.400209
$909$	7.38051	0.244796
$910$	−2.60027	−0.0861980
$911$	−24.0375	−0.796397	−0.398199	−	0.917299i	$-0.630364\pi$
$911$	−24.0375	−0.796397	−0.398199	+	0.917299i	$0.630364\pi$
$912$	7.88732	0.261175
$913$	−3.94274	−0.130486
$914$	−31.1641	−1.03082
$915$	−2.09244	−0.0691739
$916$	−6.43756	−0.212703
$917$	−21.0273	−0.694382
$918$	20.8987	0.689759
$919$	−48.8612	−1.61178	−0.805892	−	0.592063i	$-0.798314\pi$
$919$	−48.8612	−1.61178	−0.805892	+	0.592063i	$0.798314\pi$
$920$	−2.74567	−0.0905220
$921$	37.8962	1.24872
$922$	−24.3523	−0.802002
$923$	−84.0502	−2.76655
$924$	1.54311	0.0507646
$925$	17.8176	0.585838
$926$	12.8425	0.422031
$927$	−13.1120	−0.430654
$928$	3.32682	0.109208
$929$	−6.72550	−0.220656	−0.110328	−	0.993895i	$-0.535190\pi$
$929$	−6.72550	−0.220656	−0.110328	+	0.993895i	$0.535190\pi$
$930$	4.08615	0.133990
$931$	29.4052	0.963717
$932$	−16.9583	−0.555488
$933$	20.5746	0.673582
$934$	−10.2403	−0.335074
$935$	−1.05647	−0.0345504
$936$	−5.06754	−0.165638
$937$	49.4700	1.61611	0.808057	−	0.589105i	$-0.200519\pi$
$937$	49.4700	1.61611	0.808057	+	0.589105i	$0.200519\pi$
$938$	−14.6352	−0.477858
$939$	13.3794	0.436620
$940$	3.39896	0.110862
$941$	−17.5942	−0.573556	−0.286778	−	0.957997i	$-0.592584\pi$
$941$	−17.5942	−0.573556	−0.286778	+	0.957997i	$0.592584\pi$
$942$	28.1887	0.918437
$943$	−58.1040	−1.89213
$944$	0.992451	0.0323015
$945$	−2.21789	−0.0721479
$946$	0.0578322	0.00188029
$947$	−55.2273	−1.79465	−0.897323	−	0.441375i	$-0.854491\pi$
$947$	−55.2273	−1.79465	−0.897323	+	0.441375i	$0.854491\pi$
$948$	4.66575	0.151537
$949$	−20.7417	−0.673303
$950$	−25.8228	−0.837802
$951$	22.6539	0.734603
$952$	4.43940	0.143882
$953$	−10.9139	−0.353536	−0.176768	−	0.984253i	$-0.556564\pi$
$953$	−10.9139	−0.353536	−0.176768	+	0.984253i	$0.556564\pi$
$954$	−3.93918	−0.127536
$955$	4.03524	0.130577
$956$	−16.6047	−0.537033
$957$	−4.29304	−0.138774
$958$	42.4426	1.37126
$959$	−10.0843	−0.325640
$960$	0.492252	0.0158874
$961$	37.9055	1.22276
$962$	−24.0406	−0.775099
$963$	−9.06425	−0.292091
$964$	−0.977158	−0.0314721
$965$	5.97020	0.192187
$966$	−14.8884	−0.479028
$967$	−51.2088	−1.64676	−0.823382	−	0.567487i	$-0.807916\pi$
$967$	−51.2088	−1.64676	−0.823382	+	0.567487i	$0.807916\pi$
$968$	10.2540	0.329576
$969$	−29.2815	−0.940657
$970$	3.14749	0.101060
$971$	34.1571	1.09615	0.548077	−	0.836428i	$-0.315360\pi$
$971$	34.1571	1.09615	0.548077	+	0.836428i	$0.315360\pi$
$972$	−7.76385	−0.249025
$973$	−15.7866	−0.506094
$974$	41.2457	1.32160
$975$	−48.2319	−1.54466
$976$	−4.25075	−0.136063
$977$	2.58159	0.0825924	0.0412962	−	0.999147i	$-0.486851\pi$
$977$	2.58159	0.0825924	0.0412962	+	0.999147i	$0.486851\pi$
$978$	−23.7166	−0.758373
$979$	10.5567	0.337394
$980$	1.83520	0.0586232
$981$	−2.67394	−0.0853724
$982$	−20.8294	−0.664693
$983$	5.82306	0.185727	0.0928634	−	0.995679i	$-0.470398\pi$
$983$	5.82306	0.185727	0.0928634	+	0.995679i	$0.470398\pi$
$984$	10.4171	0.332084
$985$	3.42104	0.109004
$986$	−12.3508	−0.393328
$987$	18.4309	0.586663
$988$	34.8417	1.10846
$989$	−0.557985	−0.0177429
$990$	0.218504	0.00694451
$991$	−1.08552	−0.0344828	−0.0172414	−	0.999851i	$-0.505488\pi$
$991$	−1.08552	−0.0344828	−0.0172414	+	0.999851i	$0.505488\pi$
$992$	8.30093	0.263555
$993$	33.2186	1.05416
$994$	−15.2288	−0.483029
$995$	−7.00263	−0.221998
$996$	−6.82009	−0.216103
$997$	18.3073	0.579800	0.289900	−	0.957057i	$-0.406378\pi$
$997$	18.3073	0.579800	0.289900	+	0.957057i	$0.406378\pi$
$998$	−20.9348	−0.662680
$999$	−20.5053	−0.648759

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.20	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.20	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.49405 q^{3} +1.00000 q^{4} -0.329476 q^{5} +1.49405 q^{6} +1.19581 q^{7} -1.00000 q^{8} -0.767828 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.49405 q^{3} +1.00000 q^{4} -0.329476 q^{5} +1.49405 q^{6} +1.19581 q^{7} -1.00000 q^{8} -0.767828 q^{9} +0.329476 q^{10} -0.863718 q^{11} -1.49405 q^{12} -6.59984 q^{13} -1.19581 q^{14} +0.492252 q^{15} +1.00000 q^{16} -3.71248 q^{17} +0.767828 q^{18} -5.27917 q^{19} -0.329476 q^{20} -1.78659 q^{21} +0.863718 q^{22} -8.33345 q^{23} +1.49405 q^{24} -4.89145 q^{25} +6.59984 q^{26} +5.62931 q^{27} +1.19581 q^{28} -3.32682 q^{29} -0.492252 q^{30} -8.30093 q^{31} -1.00000 q^{32} +1.29043 q^{33} +3.71248 q^{34} -0.393989 q^{35} -0.767828 q^{36} -3.64260 q^{37} +5.27917 q^{38} +9.86046 q^{39} +0.329476 q^{40} +6.97238 q^{41} +1.78659 q^{42} +0.0669573 q^{43} -0.863718 q^{44} +0.252981 q^{45} +8.33345 q^{46} -10.3163 q^{47} -1.49405 q^{48} -5.57005 q^{49} +4.89145 q^{50} +5.54661 q^{51} -6.59984 q^{52} -5.13029 q^{53} -5.62931 q^{54} +0.284574 q^{55} -1.19581 q^{56} +7.88732 q^{57} +3.32682 q^{58} +0.992451 q^{59} +0.492252 q^{60} -4.25075 q^{61} +8.30093 q^{62} -0.918173 q^{63} +1.00000 q^{64} +2.17449 q^{65} -1.29043 q^{66} +12.2388 q^{67} -3.71248 q^{68} +12.4505 q^{69} +0.393989 q^{70} +12.7352 q^{71} +0.767828 q^{72} +3.14275 q^{73} +3.64260 q^{74} +7.30804 q^{75} -5.27917 q^{76} -1.03284 q^{77} -9.86046 q^{78} -3.12290 q^{79} -0.329476 q^{80} -6.10696 q^{81} -6.97238 q^{82} +4.56485 q^{83} -1.78659 q^{84} +1.22317 q^{85} -0.0669573 q^{86} +4.97042 q^{87} +0.863718 q^{88} -12.2224 q^{89} -0.252981 q^{90} -7.89213 q^{91} -8.33345 q^{92} +12.4020 q^{93} +10.3163 q^{94} +1.73936 q^{95} +1.49405 q^{96} +9.55303 q^{97} +5.57005 q^{98} +0.663186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more