LMFDB - Embedded newform 6033.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6033,2,Mod(1,6033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6033, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6033 = 3 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6033.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:	$48.1737475394$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15116 q^{2} +1.00000 q^{3} +2.62747 q^{4} +2.30807 q^{5} -2.15116 q^{6} -1.03213 q^{7} -1.34980 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.15116 q^{2} +1.00000 q^{3} +2.62747 q^{4} +2.30807 q^{5} -2.15116 q^{6} -1.03213 q^{7} -1.34980 q^{8} +1.00000 q^{9} -4.96503 q^{10} +5.39561 q^{11} +2.62747 q^{12} -0.183225 q^{13} +2.22027 q^{14} +2.30807 q^{15} -2.35133 q^{16} -7.06883 q^{17} -2.15116 q^{18} +0.829029 q^{19} +6.06441 q^{20} -1.03213 q^{21} -11.6068 q^{22} -4.66371 q^{23} -1.34980 q^{24} +0.327209 q^{25} +0.394145 q^{26} +1.00000 q^{27} -2.71189 q^{28} -8.08230 q^{29} -4.96503 q^{30} +8.85645 q^{31} +7.75766 q^{32} +5.39561 q^{33} +15.2062 q^{34} -2.38223 q^{35} +2.62747 q^{36} +5.41916 q^{37} -1.78337 q^{38} -0.183225 q^{39} -3.11543 q^{40} -10.5159 q^{41} +2.22027 q^{42} -12.1844 q^{43} +14.1768 q^{44} +2.30807 q^{45} +10.0324 q^{46} +6.59423 q^{47} -2.35133 q^{48} -5.93471 q^{49} -0.703877 q^{50} -7.06883 q^{51} -0.481418 q^{52} +1.31319 q^{53} -2.15116 q^{54} +12.4535 q^{55} +1.39316 q^{56} +0.829029 q^{57} +17.3863 q^{58} -3.26395 q^{59} +6.06441 q^{60} -12.9389 q^{61} -19.0516 q^{62} -1.03213 q^{63} -11.9853 q^{64} -0.422896 q^{65} -11.6068 q^{66} -9.81729 q^{67} -18.5732 q^{68} -4.66371 q^{69} +5.12455 q^{70} -11.0561 q^{71} -1.34980 q^{72} -14.9940 q^{73} -11.6575 q^{74} +0.327209 q^{75} +2.17825 q^{76} -5.56896 q^{77} +0.394145 q^{78} +17.5203 q^{79} -5.42704 q^{80} +1.00000 q^{81} +22.6213 q^{82} +5.39144 q^{83} -2.71189 q^{84} -16.3154 q^{85} +26.2105 q^{86} -8.08230 q^{87} -7.28297 q^{88} -2.16814 q^{89} -4.96503 q^{90} +0.189111 q^{91} -12.2538 q^{92} +8.85645 q^{93} -14.1852 q^{94} +1.91346 q^{95} +7.75766 q^{96} -17.1524 q^{97} +12.7665 q^{98} +5.39561 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9}+O(q^{10}) $ 71 * q - 11 * q^2 + 71 * q^3 + 53 * q^4 - 8 * q^5 - 11 * q^6 - 46 * q^7 - 33 * q^8 + 71 * q^9	\( 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9} - 41 q^{10} - 18 q^{11} + 53 q^{12} - 67 q^{13} - 7 q^{14} - 8 q^{15} + 21 q^{16} - 25 q^{17} - 11 q^{18} - 43 q^{19} - 8 q^{20} - 46 q^{21} - 49 q^{22} - 75 q^{23} - 33 q^{24} + 19 q^{25} + 71 q^{27} - 89 q^{28} - 35 q^{29} - 41 q^{30} - 82 q^{31} - 62 q^{32} - 18 q^{33} - 28 q^{34} - 51 q^{35} + 53 q^{36} - 66 q^{37} - 29 q^{38} - 67 q^{39} - 102 q^{40} + q^{41} - 7 q^{42} - 112 q^{43} - 25 q^{44} - 8 q^{45} - 36 q^{46} - 67 q^{47} + 21 q^{48} + 7 q^{49} - 24 q^{50} - 25 q^{51} - 134 q^{52} - 40 q^{53} - 11 q^{54} - 112 q^{55} + 9 q^{56} - 43 q^{57} - 47 q^{58} - 18 q^{59} - 8 q^{60} - 144 q^{61} - 19 q^{62} - 46 q^{63} - 17 q^{64} - 31 q^{65} - 49 q^{66} - 85 q^{67} - 22 q^{68} - 75 q^{69} - 11 q^{70} - 44 q^{71} - 33 q^{72} - 98 q^{73} + 6 q^{74} + 19 q^{75} - 85 q^{76} - 39 q^{77} - 126 q^{79} + 21 q^{80} + 71 q^{81} - 69 q^{82} - 43 q^{83} - 89 q^{84} - 112 q^{85} + 32 q^{86} - 35 q^{87} - 85 q^{88} + 8 q^{89} - 41 q^{90} - 40 q^{91} - 96 q^{92} - 82 q^{93} - 99 q^{94} - 103 q^{95} - 62 q^{96} - 67 q^{97} - 11 q^{98} - 18 q^{99}+O(q^{100}) \) 71 * q - 11 * q^2 + 71 * q^3 + 53 * q^4 - 8 * q^5 - 11 * q^6 - 46 * q^7 - 33 * q^8 + 71 * q^9 - 41 * q^10 - 18 * q^11 + 53 * q^12 - 67 * q^13 - 7 * q^14 - 8 * q^15 + 21 * q^16 - 25 * q^17 - 11 * q^18 - 43 * q^19 - 8 * q^20 - 46 * q^21 - 49 * q^22 - 75 * q^23 - 33 * q^24 + 19 * q^25 + 71 * q^27 - 89 * q^28 - 35 * q^29 - 41 * q^30 - 82 * q^31 - 62 * q^32 - 18 * q^33 - 28 * q^34 - 51 * q^35 + 53 * q^36 - 66 * q^37 - 29 * q^38 - 67 * q^39 - 102 * q^40 + q^41 - 7 * q^42 - 112 * q^43 - 25 * q^44 - 8 * q^45 - 36 * q^46 - 67 * q^47 + 21 * q^48 + 7 * q^49 - 24 * q^50 - 25 * q^51 - 134 * q^52 - 40 * q^53 - 11 * q^54 - 112 * q^55 + 9 * q^56 - 43 * q^57 - 47 * q^58 - 18 * q^59 - 8 * q^60 - 144 * q^61 - 19 * q^62 - 46 * q^63 - 17 * q^64 - 31 * q^65 - 49 * q^66 - 85 * q^67 - 22 * q^68 - 75 * q^69 - 11 * q^70 - 44 * q^71 - 33 * q^72 - 98 * q^73 + 6 * q^74 + 19 * q^75 - 85 * q^76 - 39 * q^77 - 126 * q^79 + 21 * q^80 + 71 * q^81 - 69 * q^82 - 43 * q^83 - 89 * q^84 - 112 * q^85 + 32 * q^86 - 35 * q^87 - 85 * q^88 + 8 * q^89 - 41 * q^90 - 40 * q^91 - 96 * q^92 - 82 * q^93 - 99 * q^94 - 103 * q^95 - 62 * q^96 - 67 * q^97 - 11 * q^98 - 18 * 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 $

$2$	−2.15116	−1.52110	−0.760549	−	0.649281i	$-0.775070\pi$
$2$	−2.15116	−1.52110	−0.760549	+	0.649281i	$0.775070\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.62747	1.31374
$5$	2.30807	1.03220	0.516101	−	0.856528i	$-0.327383\pi$
$5$	2.30807	1.03220	0.516101	+	0.856528i	$0.327383\pi$
$6$	−2.15116	−0.878206
$7$	−1.03213	−0.390108	−0.195054	−	0.980793i	$-0.562488\pi$
$7$	−1.03213	−0.390108	−0.195054	+	0.980793i	$0.562488\pi$
$8$	−1.34980	−0.477225
$9$	1.00000	0.333333
$10$	−4.96503	−1.57008
$11$	5.39561	1.62684	0.813419	−	0.581679i	$-0.197604\pi$
$11$	5.39561	1.62684	0.813419	+	0.581679i	$0.197604\pi$
$12$	2.62747	0.758487
$13$	−0.183225	−0.0508173	−0.0254087	−	0.999677i	$-0.508089\pi$
$13$	−0.183225	−0.0508173	−0.0254087	+	0.999677i	$0.508089\pi$
$14$	2.22027	0.593392
$15$	2.30807	0.595942
$16$	−2.35133	−0.587832
$17$	−7.06883	−1.71444	−0.857222	−	0.514947i	$-0.827812\pi$
$17$	−7.06883	−1.71444	−0.857222	+	0.514947i	$0.827812\pi$
$18$	−2.15116	−0.507032
$19$	0.829029	0.190192	0.0950962	−	0.995468i	$-0.469684\pi$
$19$	0.829029	0.190192	0.0950962	+	0.995468i	$0.469684\pi$
$20$	6.06441	1.35604
$21$	−1.03213	−0.225229
$22$	−11.6068	−2.47458
$23$	−4.66371	−0.972451	−0.486225	−	0.873833i	$-0.661627\pi$
$23$	−4.66371	−0.972451	−0.486225	+	0.873833i	$0.661627\pi$
$24$	−1.34980	−0.275526
$25$	0.327209	0.0654417
$26$	0.394145	0.0772981
$27$	1.00000	0.192450
$28$	−2.71189	−0.512499
$29$	−8.08230	−1.50085	−0.750423	−	0.660958i	$-0.770150\pi$
$29$	−8.08230	−1.50085	−0.750423	+	0.660958i	$0.770150\pi$
$30$	−4.96503	−0.906486
$31$	8.85645	1.59066	0.795332	−	0.606174i	$-0.207296\pi$
$31$	8.85645	1.59066	0.795332	+	0.606174i	$0.207296\pi$
$32$	7.75766	1.37137
$33$	5.39561	0.939255
$34$	15.2062	2.60784
$35$	−2.38223	−0.402670
$36$	2.62747	0.437912
$37$	5.41916	0.890905	0.445452	−	0.895306i	$-0.353043\pi$
$37$	5.41916	0.890905	0.445452	+	0.895306i	$0.353043\pi$
$38$	−1.78337	−0.289301
$39$	−0.183225	−0.0293394
$40$	−3.11543	−0.492593
$41$	−10.5159	−1.64230	−0.821151	−	0.570712i	$-0.806667\pi$
$41$	−10.5159	−1.64230	−0.821151	+	0.570712i	$0.806667\pi$
$42$	2.22027	0.342595
$43$	−12.1844	−1.85810	−0.929048	−	0.369958i	$-0.879372\pi$
$43$	−12.1844	−1.85810	−0.929048	+	0.369958i	$0.879372\pi$
$44$	14.1768	2.13724
$45$	2.30807	0.344067
$46$	10.0324	1.47919
$47$	6.59423	0.961867	0.480933	−	0.876757i	$-0.340298\pi$
$47$	6.59423	0.961867	0.480933	+	0.876757i	$0.340298\pi$
$48$	−2.35133	−0.339385
$49$	−5.93471	−0.847816
$50$	−0.703877	−0.0995433
$51$	−7.06883	−0.989835
$52$	−0.481418	−0.0667606
$53$	1.31319	0.180381	0.0901906	−	0.995925i	$-0.471252\pi$
$53$	1.31319	0.180381	0.0901906	+	0.995925i	$0.471252\pi$
$54$	−2.15116	−0.292735
$55$	12.4535	1.67923
$56$	1.39316	0.186169
$57$	0.829029	0.109808
$58$	17.3863	2.28293
$59$	−3.26395	−0.424930	−0.212465	−	0.977169i	$-0.568149\pi$
$59$	−3.26395	−0.424930	−0.212465	+	0.977169i	$0.568149\pi$
$60$	6.06441	0.782912
$61$	−12.9389	−1.65666	−0.828329	−	0.560241i	$-0.810708\pi$
$61$	−12.9389	−1.65666	−0.828329	+	0.560241i	$0.810708\pi$
$62$	−19.0516	−2.41956
$63$	−1.03213	−0.130036
$64$	−11.9853	−1.49816
$65$	−0.422896	−0.0524538
$66$	−11.6068	−1.42870
$67$	−9.81729	−1.19937	−0.599686	−	0.800235i	$-0.704708\pi$
$67$	−9.81729	−1.19937	−0.599686	+	0.800235i	$0.704708\pi$
$68$	−18.5732	−2.25233
$69$	−4.66371	−0.561445
$70$	5.12455	0.612501
$71$	−11.0561	−1.31212	−0.656062	−	0.754707i	$-0.727779\pi$
$71$	−11.0561	−1.31212	−0.656062	+	0.754707i	$0.727779\pi$
$72$	−1.34980	−0.159075
$73$	−14.9940	−1.75491	−0.877456	−	0.479657i	$-0.840761\pi$
$73$	−14.9940	−1.75491	−0.877456	+	0.479657i	$0.840761\pi$
$74$	−11.6575	−1.35515
$75$	0.327209	0.0377828
$76$	2.17825	0.249863
$77$	−5.56896	−0.634642
$78$	0.394145	0.0446281
$79$	17.5203	1.97119	0.985593	−	0.169134i	$-0.0540969\pi$
$79$	17.5203	1.97119	0.985593	+	0.169134i	$0.0540969\pi$
$80$	−5.42704	−0.606761
$81$	1.00000	0.111111
$82$	22.6213	2.49810
$83$	5.39144	0.591788	0.295894	−	0.955221i	$-0.404383\pi$
$83$	5.39144	0.591788	0.295894	+	0.955221i	$0.404383\pi$
$84$	−2.71189	−0.295892
$85$	−16.3154	−1.76965
$86$	26.2105	2.82635
$87$	−8.08230	−0.866514
$88$	−7.28297	−0.776367
$89$	−2.16814	−0.229822	−0.114911	−	0.993376i	$-0.536658\pi$
$89$	−2.16814	−0.229822	−0.114911	+	0.993376i	$0.536658\pi$
$90$	−4.96503	−0.523360
$91$	0.189111	0.0198243
$92$	−12.2538	−1.27754
$93$	8.85645	0.918371
$94$	−14.1852	−1.46309
$95$	1.91346	0.196317
$96$	7.75766	0.791763
$97$	−17.1524	−1.74156	−0.870780	−	0.491672i	$-0.836386\pi$
$97$	−17.1524	−1.74156	−0.870780	+	0.491672i	$0.836386\pi$
$98$	12.7665	1.28961
$99$	5.39561	0.542279
$100$	0.859732	0.0859732
$101$	3.93991	0.392036	0.196018	−	0.980600i	$-0.437199\pi$
$101$	3.93991	0.392036	0.196018	+	0.980600i	$0.437199\pi$
$102$	15.2062	1.50564
$103$	6.89088	0.678978	0.339489	−	0.940610i	$-0.389746\pi$
$103$	6.89088	0.678978	0.339489	+	0.940610i	$0.389746\pi$
$104$	0.247316	0.0242513
$105$	−2.38223	−0.232482
$106$	−2.82489	−0.274377
$107$	10.9188	1.05556	0.527779	−	0.849382i	$-0.323025\pi$
$107$	10.9188	1.05556	0.527779	+	0.849382i	$0.323025\pi$
$108$	2.62747	0.252829
$109$	−9.51645	−0.911511	−0.455755	−	0.890105i	$-0.650631\pi$
$109$	−9.51645	−0.911511	−0.455755	+	0.890105i	$0.650631\pi$
$110$	−26.7894	−2.55427
$111$	5.41916	0.514364
$112$	2.42687	0.229318
$113$	−10.9698	−1.03196	−0.515978	−	0.856602i	$-0.672571\pi$
$113$	−10.9698	−1.03196	−0.515978	+	0.856602i	$0.672571\pi$
$114$	−1.78337	−0.167028
$115$	−10.7642	−1.00377
$116$	−21.2360	−1.97172
$117$	−0.183225	−0.0169391
$118$	7.02126	0.646360
$119$	7.29595	0.668818
$120$	−3.11543	−0.284399
$121$	18.1126	1.64660
$122$	27.8336	2.51994
$123$	−10.5159	−0.948183
$124$	23.2701	2.08972
$125$	−10.7852	−0.964653
$126$	2.22027	0.197797
$127$	−19.9910	−1.77392	−0.886958	−	0.461850i	$-0.847186\pi$
$127$	−19.9910	−1.77392	−0.886958	+	0.461850i	$0.847186\pi$
$128$	10.2669	0.907476
$129$	−12.1844	−1.07277
$130$	0.909715	0.0797873
$131$	4.94847	0.432350	0.216175	−	0.976355i	$-0.430642\pi$
$131$	4.94847	0.432350	0.216175	+	0.976355i	$0.430642\pi$
$132$	14.1768	1.23393
$133$	−0.855665	−0.0741956
$134$	21.1185	1.82436
$135$	2.30807	0.198647
$136$	9.54149	0.818176
$137$	3.03713	0.259480	0.129740	−	0.991548i	$-0.458586\pi$
$137$	3.03713	0.259480	0.129740	+	0.991548i	$0.458586\pi$
$138$	10.0324	0.854012
$139$	−7.33726	−0.622338	−0.311169	−	0.950355i	$-0.600721\pi$
$139$	−7.33726	−0.622338	−0.311169	+	0.950355i	$0.600721\pi$
$140$	−6.25925	−0.529003
$141$	6.59423	0.555334
$142$	23.7835	1.99587
$143$	−0.988608	−0.0826715
$144$	−2.35133	−0.195944
$145$	−18.6546	−1.54918
$146$	32.2544	2.66939
$147$	−5.93471	−0.489487
$148$	14.2387	1.17042
$149$	9.57155	0.784132	0.392066	−	0.919937i	$-0.371760\pi$
$149$	9.57155	0.784132	0.392066	+	0.919937i	$0.371760\pi$
$150$	−0.703877	−0.0574713
$151$	−4.10878	−0.334367	−0.167184	−	0.985926i	$-0.553467\pi$
$151$	−4.10878	−0.334367	−0.167184	+	0.985926i	$0.553467\pi$
$152$	−1.11902	−0.0907645
$153$	−7.06883	−0.571481
$154$	11.9797	0.965353
$155$	20.4413	1.64189
$156$	−0.481418	−0.0385443
$157$	22.1196	1.76534	0.882668	−	0.469998i	$-0.155745\pi$
$157$	22.1196	1.76534	0.882668	+	0.469998i	$0.155745\pi$
$158$	−37.6889	−2.99837
$159$	1.31319	0.104143
$160$	17.9053	1.41554
$161$	4.81355	0.379361
$162$	−2.15116	−0.169011
$163$	−2.28585	−0.179042	−0.0895209	−	0.995985i	$-0.528534\pi$
$163$	−2.28585	−0.179042	−0.0895209	+	0.995985i	$0.528534\pi$
$164$	−27.6301	−2.15755
$165$	12.4535	0.969501
$166$	−11.5978	−0.900167
$167$	−17.5811	−1.36046	−0.680231	−	0.732997i	$-0.738121\pi$
$167$	−17.5811	−1.36046	−0.680231	+	0.732997i	$0.738121\pi$
$168$	1.39316	0.107485
$169$	−12.9664	−0.997418
$170$	35.0970	2.69182
$171$	0.829029	0.0633974
$172$	−32.0141	−2.44105
$173$	10.9667	0.833786	0.416893	−	0.908956i	$-0.363119\pi$
$173$	10.9667	0.833786	0.416893	+	0.908956i	$0.363119\pi$
$174$	17.3863	1.31805
$175$	−0.337722	−0.0255293
$176$	−12.6868	−0.956306
$177$	−3.26395	−0.245333
$178$	4.66401	0.349582
$179$	20.2012	1.50991	0.754955	−	0.655777i	$-0.227659\pi$
$179$	20.2012	1.50991	0.754955	+	0.655777i	$0.227659\pi$
$180$	6.06441	0.452014
$181$	−12.0824	−0.898081	−0.449040	−	0.893511i	$-0.648234\pi$
$181$	−12.0824	−0.898081	−0.449040	+	0.893511i	$0.648234\pi$
$182$	−0.406808	−0.0301546
$183$	−12.9389	−0.956472
$184$	6.29506	0.464078
$185$	12.5078	0.919594
$186$	−19.0516	−1.39693
$187$	−38.1407	−2.78912
$188$	17.3262	1.26364
$189$	−1.03213	−0.0750763
$190$	−4.11615	−0.298617
$191$	−1.88127	−0.136124	−0.0680621	−	0.997681i	$-0.521682\pi$
$191$	−1.88127	−0.136124	−0.0680621	+	0.997681i	$0.521682\pi$
$192$	−11.9853	−0.864964
$193$	−4.06257	−0.292430	−0.146215	−	0.989253i	$-0.546709\pi$
$193$	−4.06257	−0.292430	−0.146215	+	0.989253i	$0.546709\pi$
$194$	36.8975	2.64908
$195$	−0.422896	−0.0302842
$196$	−15.5933	−1.11381
$197$	1.47589	0.105152	0.0525762	−	0.998617i	$-0.483257\pi$
$197$	1.47589	0.105152	0.0525762	+	0.998617i	$0.483257\pi$
$198$	−11.6068	−0.824859
$199$	16.1873	1.14749	0.573744	−	0.819035i	$-0.305491\pi$
$199$	16.1873	1.14749	0.573744	+	0.819035i	$0.305491\pi$
$200$	−0.441665	−0.0312304
$201$	−9.81729	−0.692458
$202$	−8.47536	−0.596324
$203$	8.34198	0.585492
$204$	−18.5732	−1.30038
$205$	−24.2714	−1.69519
$206$	−14.8234	−1.03279
$207$	−4.66371	−0.324150
$208$	0.430821	0.0298720
$209$	4.47312	0.309412
$210$	5.12455	0.353628
$211$	−24.4354	−1.68220	−0.841102	−	0.540877i	$-0.818093\pi$
$211$	−24.4354	−1.68220	−0.841102	+	0.540877i	$0.818093\pi$
$212$	3.45038	0.236973
$213$	−11.0561	−0.757555
$214$	−23.4880	−1.60561
$215$	−28.1224	−1.91793
$216$	−1.34980	−0.0918420
$217$	−9.14099	−0.620531
$218$	20.4714	1.38650
$219$	−14.9940	−1.01320
$220$	32.7212	2.20606
$221$	1.29518	0.0871235
$222$	−11.6575	−0.782398
$223$	5.54187	0.371111	0.185556	−	0.982634i	$-0.440591\pi$
$223$	5.54187	0.371111	0.185556	+	0.982634i	$0.440591\pi$
$224$	−8.00691	−0.534984
$225$	0.327209	0.0218139
$226$	23.5978	1.56970
$227$	−9.42560	−0.625599	−0.312799	−	0.949819i	$-0.601267\pi$
$227$	−9.42560	−0.625599	−0.312799	+	0.949819i	$0.601267\pi$
$228$	2.17825	0.144258
$229$	15.0923	0.997327	0.498663	−	0.866796i	$-0.333824\pi$
$229$	15.0923	0.997327	0.498663	+	0.866796i	$0.333824\pi$
$230$	23.1555	1.52683
$231$	−5.56896	−0.366411
$232$	10.9095	0.716241
$233$	−18.8135	−1.23251	−0.616256	−	0.787546i	$-0.711352\pi$
$233$	−18.8135	−1.23251	−0.616256	+	0.787546i	$0.711352\pi$
$234$	0.394145	0.0257660
$235$	15.2200	0.992841
$236$	−8.57594	−0.558246
$237$	17.5203	1.13806
$238$	−15.6947	−1.01734
$239$	23.4838	1.51904	0.759519	−	0.650485i	$-0.225434\pi$
$239$	23.4838	1.51904	0.759519	+	0.650485i	$0.225434\pi$
$240$	−5.42704	−0.350314
$241$	−4.15332	−0.267539	−0.133769	−	0.991012i	$-0.542708\pi$
$241$	−4.15332	−0.267539	−0.133769	+	0.991012i	$0.542708\pi$
$242$	−38.9630	−2.50464
$243$	1.00000	0.0641500
$244$	−33.9967	−2.17641
$245$	−13.6978	−0.875117
$246$	22.6213	1.44228
$247$	−0.151898	−0.00966507
$248$	−11.9544	−0.759105
$249$	5.39144	0.341669
$250$	23.2005	1.46733
$251$	18.1115	1.14319	0.571594	−	0.820536i	$-0.306325\pi$
$251$	18.1115	1.14319	0.571594	+	0.820536i	$0.306325\pi$
$252$	−2.71189	−0.170833
$253$	−25.1635	−1.58202
$254$	43.0038	2.69830
$255$	−16.3154	−1.02171
$256$	1.88484	0.117802
$257$	16.3637	1.02074	0.510369	−	0.859955i	$-0.329509\pi$
$257$	16.3637	1.02074	0.510369	+	0.859955i	$0.329509\pi$
$258$	26.2105	1.63179
$259$	−5.59327	−0.347549
$260$	−1.11115	−0.0689105
$261$	−8.08230	−0.500282
$262$	−10.6449	−0.657646
$263$	−4.50800	−0.277975	−0.138988	−	0.990294i	$-0.544385\pi$
$263$	−4.50800	−0.277975	−0.138988	+	0.990294i	$0.544385\pi$
$264$	−7.28297	−0.448236
$265$	3.03095	0.186190
$266$	1.84067	0.112859
$267$	−2.16814	−0.132688
$268$	−25.7947	−1.57566
$269$	7.28272	0.444035	0.222018	−	0.975043i	$-0.428736\pi$
$269$	7.28272	0.444035	0.222018	+	0.975043i	$0.428736\pi$
$270$	−4.96503	−0.302162
$271$	13.5945	0.825808	0.412904	−	0.910775i	$-0.364515\pi$
$271$	13.5945	0.825808	0.412904	+	0.910775i	$0.364515\pi$
$272$	16.6211	1.00780
$273$	0.189111	0.0114455
$274$	−6.53334	−0.394694
$275$	1.76549	0.106463
$276$	−12.2538	−0.737591
$277$	−3.34581	−0.201030	−0.100515	−	0.994936i	$-0.532049\pi$
$277$	−3.34581	−0.201030	−0.100515	+	0.994936i	$0.532049\pi$
$278$	15.7836	0.946637
$279$	8.85645	0.530222
$280$	3.21553	0.192164
$281$	29.2991	1.74784	0.873918	−	0.486073i	$-0.161571\pi$
$281$	29.2991	1.74784	0.873918	+	0.486073i	$0.161571\pi$
$282$	−14.1852	−0.844717
$283$	15.3959	0.915190	0.457595	−	0.889161i	$-0.348711\pi$
$283$	15.3959	0.915190	0.457595	+	0.889161i	$0.348711\pi$
$284$	−29.0498	−1.72379
$285$	1.91346	0.113344
$286$	2.12665	0.125751
$287$	10.8537	0.640675
$288$	7.75766	0.457125
$289$	32.9684	1.93932
$290$	40.1289	2.35645
$291$	−17.1524	−1.00549
$292$	−39.3963	−2.30549
$293$	9.55853	0.558415	0.279208	−	0.960231i	$-0.409928\pi$
$293$	9.55853	0.558415	0.279208	+	0.960231i	$0.409928\pi$
$294$	12.7665	0.744557
$295$	−7.53344	−0.438614
$296$	−7.31477	−0.425162
$297$	5.39561	0.313085
$298$	−20.5899	−1.19274
$299$	0.854506	0.0494174
$300$	0.859732	0.0496367
$301$	12.5758	0.724859
$302$	8.83862	0.508605
$303$	3.93991	0.226342
$304$	−1.94932	−0.111801
$305$	−29.8640	−1.71001
$306$	15.2062	0.869279
$307$	12.1170	0.691554	0.345777	−	0.938317i	$-0.387615\pi$
$307$	12.1170	0.691554	0.345777	+	0.938317i	$0.387615\pi$
$308$	−14.6323	−0.833753
$309$	6.89088	0.392008
$310$	−43.9725	−2.49747
$311$	3.43783	0.194941	0.0974706	−	0.995238i	$-0.468925\pi$
$311$	3.43783	0.194941	0.0974706	+	0.995238i	$0.468925\pi$
$312$	0.247316	0.0140015
$313$	−29.9439	−1.69253	−0.846264	−	0.532763i	$-0.821154\pi$
$313$	−29.9439	−1.69253	−0.846264	+	0.532763i	$0.821154\pi$
$314$	−47.5827	−2.68525
$315$	−2.38223	−0.134223
$316$	46.0341	2.58962
$317$	3.28916	0.184738	0.0923689	−	0.995725i	$-0.470556\pi$
$317$	3.28916	0.184738	0.0923689	+	0.995725i	$0.470556\pi$
$318$	−2.82489	−0.158412
$319$	−43.6089	−2.44163
$320$	−27.6630	−1.54641
$321$	10.9188	0.609427
$322$	−10.3547	−0.577045
$323$	−5.86027	−0.326074
$324$	2.62747	0.145971
$325$	−0.0599527	−0.00332557
$326$	4.91723	0.272340
$327$	−9.51645	−0.526261
$328$	14.1943	0.783747
$329$	−6.80609	−0.375232
$330$	−26.7894	−1.47471
$331$	−7.43104	−0.408447	−0.204223	−	0.978924i	$-0.565467\pi$
$331$	−7.43104	−0.408447	−0.204223	+	0.978924i	$0.565467\pi$
$332$	14.1659	0.777454
$333$	5.41916	0.296968
$334$	37.8196	2.06940
$335$	−22.6590	−1.23800
$336$	2.42687	0.132397
$337$	12.0480	0.656295	0.328147	−	0.944627i	$-0.393576\pi$
$337$	12.0480	0.656295	0.328147	+	0.944627i	$0.393576\pi$
$338$	27.8928	1.51717
$339$	−10.9698	−0.595800
$340$	−42.8683	−2.32486
$341$	47.7859	2.58775
$342$	−1.78337	−0.0964337
$343$	13.3503	0.720848
$344$	16.4464	0.886730
$345$	−10.7642	−0.579524
$346$	−23.5912	−1.26827
$347$	9.85229	0.528898	0.264449	−	0.964400i	$-0.414810\pi$
$347$	9.85229	0.528898	0.264449	+	0.964400i	$0.414810\pi$
$348$	−21.2360	−1.13837
$349$	7.59997	0.406817	0.203409	−	0.979094i	$-0.434798\pi$
$349$	7.59997	0.406817	0.203409	+	0.979094i	$0.434798\pi$
$350$	0.726492	0.0388326
$351$	−0.183225	−0.00977980
$352$	41.8573	2.23100
$353$	3.04971	0.162320	0.0811599	−	0.996701i	$-0.474138\pi$
$353$	3.04971	0.162320	0.0811599	+	0.996701i	$0.474138\pi$
$354$	7.02126	0.373176
$355$	−25.5184	−1.35438
$356$	−5.69673	−0.301926
$357$	7.29595	0.386143
$358$	−43.4560	−2.29672
$359$	20.4409	1.07883	0.539415	−	0.842040i	$-0.318646\pi$
$359$	20.4409	1.07883	0.539415	+	0.842040i	$0.318646\pi$
$360$	−3.11543	−0.164198
$361$	−18.3127	−0.963827
$362$	25.9912	1.36607
$363$	18.1126	0.950664
$364$	0.496885	0.0260439
$365$	−34.6072	−1.81142
$366$	27.8336	1.45489
$367$	32.1661	1.67906	0.839529	−	0.543316i	$-0.182831\pi$
$367$	32.1661	1.67906	0.839529	+	0.543316i	$0.182831\pi$
$368$	10.9659	0.571637
$369$	−10.5159	−0.547434
$370$	−26.9063	−1.39879
$371$	−1.35539	−0.0703681
$372$	23.2701	1.20650
$373$	−26.9162	−1.39367	−0.696835	−	0.717232i	$-0.745409\pi$
$373$	−26.9162	−1.39367	−0.696835	+	0.717232i	$0.745409\pi$
$374$	82.0465	4.24253
$375$	−10.7852	−0.556943
$376$	−8.90086	−0.459027
$377$	1.48088	0.0762690
$378$	2.22027	0.114198
$379$	−7.11025	−0.365229	−0.182614	−	0.983185i	$-0.558456\pi$
$379$	−7.11025	−0.365229	−0.182614	+	0.983185i	$0.558456\pi$
$380$	5.02757	0.257909
$381$	−19.9910	−1.02417
$382$	4.04691	0.207058
$383$	−29.2384	−1.49401	−0.747006	−	0.664817i	$-0.768509\pi$
$383$	−29.2384	−1.49401	−0.747006	+	0.664817i	$0.768509\pi$
$384$	10.2669	0.523932
$385$	−12.8536	−0.655079
$386$	8.73922	0.444815
$387$	−12.1844	−0.619366
$388$	−45.0675	−2.28795
$389$	36.8865	1.87022	0.935109	−	0.354359i	$-0.115301\pi$
$389$	36.8865	1.87022	0.935109	+	0.354359i	$0.115301\pi$
$390$	0.909715	0.0460652
$391$	32.9670	1.66721
$392$	8.01065	0.404599
$393$	4.94847	0.249617
$394$	−3.17486	−0.159947
$395$	40.4381	2.03466
$396$	14.1768	0.712412
$397$	−23.2995	−1.16937	−0.584684	−	0.811261i	$-0.698781\pi$
$397$	−23.2995	−1.16937	−0.584684	+	0.811261i	$0.698781\pi$
$398$	−34.8214	−1.74544
$399$	−0.855665	−0.0428368
$400$	−0.769374	−0.0384687
$401$	21.7313	1.08521	0.542605	−	0.839988i	$-0.317438\pi$
$401$	21.7313	1.08521	0.542605	+	0.839988i	$0.317438\pi$
$402$	21.1185	1.05330
$403$	−1.62272	−0.0808333
$404$	10.3520	0.515032
$405$	2.30807	0.114689
$406$	−17.9449	−0.890590
$407$	29.2397	1.44936
$408$	9.54149	0.472374
$409$	5.75540	0.284586	0.142293	−	0.989825i	$-0.454552\pi$
$409$	5.75540	0.284586	0.142293	+	0.989825i	$0.454552\pi$
$410$	52.2115	2.57854
$411$	3.03713	0.149811
$412$	18.1056	0.891999
$413$	3.36882	0.165769
$414$	10.0324	0.493064
$415$	12.4439	0.610845
$416$	−1.42139	−0.0696896
$417$	−7.33726	−0.359307
$418$	−9.62238	−0.470646
$419$	−11.8960	−0.581160	−0.290580	−	0.956851i	$-0.593848\pi$
$419$	−11.8960	−0.581160	−0.290580	+	0.956851i	$0.593848\pi$
$420$	−6.25925	−0.305420
$421$	34.1703	1.66536	0.832680	−	0.553755i	$-0.186806\pi$
$421$	34.1703	1.66536	0.832680	+	0.553755i	$0.186806\pi$
$422$	52.5644	2.55880
$423$	6.59423	0.320622
$424$	−1.77254	−0.0860824
$425$	−2.31298	−0.112196
$426$	23.7835	1.15231
$427$	13.3546	0.646276
$428$	28.6888	1.38673
$429$	−0.988608	−0.0477304
$430$	60.4957	2.91736
$431$	−1.27549	−0.0614380	−0.0307190	−	0.999528i	$-0.509780\pi$
$431$	−1.27549	−0.0614380	−0.0307190	+	0.999528i	$0.509780\pi$
$432$	−2.35133	−0.113128
$433$	−10.2676	−0.493430	−0.246715	−	0.969088i	$-0.579351\pi$
$433$	−10.2676	−0.493430	−0.246715	+	0.969088i	$0.579351\pi$
$434$	19.6637	0.943888
$435$	−18.6546	−0.894418
$436$	−25.0042	−1.19749
$437$	−3.86635	−0.184953
$438$	32.2544	1.54117
$439$	−0.238622	−0.0113888	−0.00569440	−	0.999984i	$-0.501813\pi$
$439$	−0.238622	−0.0113888	−0.00569440	+	0.999984i	$0.501813\pi$
$440$	−16.8096	−0.801368
$441$	−5.93471	−0.282605
$442$	−2.78614	−0.132523
$443$	38.0586	1.80822	0.904110	−	0.427300i	$-0.140535\pi$
$443$	38.0586	1.80822	0.904110	+	0.427300i	$0.140535\pi$
$444$	14.2387	0.675739
$445$	−5.00423	−0.237223
$446$	−11.9214	−0.564496
$447$	9.57155	0.452719
$448$	12.3704	0.584445
$449$	10.8961	0.514220	0.257110	−	0.966382i	$-0.417230\pi$
$449$	10.8961	0.514220	0.257110	+	0.966382i	$0.417230\pi$
$450$	−0.703877	−0.0331811
$451$	−56.7394	−2.67176
$452$	−28.8230	−1.35572
$453$	−4.10878	−0.193047
$454$	20.2759	0.951597
$455$	0.436483	0.0204626
$456$	−1.11902	−0.0524029
$457$	−21.0299	−0.983737	−0.491868	−	0.870670i	$-0.663686\pi$
$457$	−21.0299	−0.983737	−0.491868	+	0.870670i	$0.663686\pi$
$458$	−32.4659	−1.51703
$459$	−7.06883	−0.329945
$460$	−28.2826	−1.31868
$461$	−13.2508	−0.617149	−0.308575	−	0.951200i	$-0.599852\pi$
$461$	−13.2508	−0.617149	−0.308575	+	0.951200i	$0.599852\pi$
$462$	11.9797	0.557347
$463$	−0.459254	−0.0213433	−0.0106717	−	0.999943i	$-0.503397\pi$
$463$	−0.459254	−0.0213433	−0.0106717	+	0.999943i	$0.503397\pi$
$464$	19.0041	0.882245
$465$	20.4413	0.947944
$466$	40.4708	1.87477
$467$	−26.2528	−1.21484	−0.607418	−	0.794383i	$-0.707795\pi$
$467$	−26.2528	−1.21484	−0.607418	+	0.794383i	$0.707795\pi$
$468$	−0.481418	−0.0222535
$469$	10.1327	0.467885
$470$	−32.7405	−1.51021
$471$	22.1196	1.01922
$472$	4.40567	0.202787
$473$	−65.7420	−3.02282
$474$	−37.6889	−1.73111
$475$	0.271266	0.0124465
$476$	19.1699	0.878652
$477$	1.31319	0.0601270
$478$	−50.5172	−2.31060
$479$	3.44091	0.157219	0.0786096	−	0.996905i	$-0.474952\pi$
$479$	3.44091	0.157219	0.0786096	+	0.996905i	$0.474952\pi$
$480$	17.9053	0.817260
$481$	−0.992924	−0.0452734
$482$	8.93445	0.406953
$483$	4.81355	0.219024
$484$	47.5904	2.16320
$485$	−39.5890	−1.79764
$486$	−2.15116	−0.0975784
$487$	5.27321	0.238952	0.119476	−	0.992837i	$-0.461879\pi$
$487$	5.27321	0.238952	0.119476	+	0.992837i	$0.461879\pi$
$488$	17.4649	0.790599
$489$	−2.28585	−0.103370
$490$	29.4660	1.33114
$491$	−20.1376	−0.908797	−0.454398	−	0.890799i	$-0.650146\pi$
$491$	−20.1376	−0.908797	−0.454398	+	0.890799i	$0.650146\pi$
$492$	−27.6301	−1.24566
$493$	57.1325	2.57312
$494$	0.326757	0.0147015
$495$	12.4535	0.559742
$496$	−20.8244	−0.935043
$497$	11.4114	0.511870
$498$	−11.5978	−0.519712
$499$	11.1137	0.497517	0.248758	−	0.968566i	$-0.419977\pi$
$499$	11.1137	0.497517	0.248758	+	0.968566i	$0.419977\pi$
$500$	−28.3377	−1.26730
$501$	−17.5811	−0.785464
$502$	−38.9607	−1.73890
$503$	−32.9235	−1.46799	−0.733993	−	0.679157i	$-0.762346\pi$
$503$	−32.9235	−1.46799	−0.733993	+	0.679157i	$0.762346\pi$
$504$	1.39316	0.0620564
$505$	9.09360	0.404660
$506$	54.1307	2.40640
$507$	−12.9664	−0.575859
$508$	−52.5259	−2.33046
$509$	31.7009	1.40512	0.702559	−	0.711625i	$-0.252041\pi$
$509$	31.7009	1.40512	0.702559	+	0.711625i	$0.252041\pi$
$510$	35.0970	1.55412
$511$	15.4757	0.684605
$512$	−24.5884	−1.08666
$513$	0.829029	0.0366025
$514$	−35.2009	−1.55264
$515$	15.9047	0.700843
$516$	−32.0141	−1.40934
$517$	35.5799	1.56480
$518$	12.0320	0.528656
$519$	10.9667	0.481387
$520$	0.570823	0.0250323
$521$	−23.2367	−1.01802	−0.509009	−	0.860761i	$-0.669988\pi$
$521$	−23.2367	−1.01802	−0.509009	+	0.860761i	$0.669988\pi$
$522$	17.3863	0.760978
$523$	−2.91907	−0.127642	−0.0638209	−	0.997961i	$-0.520329\pi$
$523$	−2.91907	−0.127642	−0.0638209	+	0.997961i	$0.520329\pi$
$524$	13.0020	0.567994
$525$	−0.337722	−0.0147394
$526$	9.69742	0.422827
$527$	−62.6048	−2.72711
$528$	−12.6868	−0.552124
$529$	−1.24982	−0.0543399
$530$	−6.52005	−0.283213
$531$	−3.26395	−0.141643
$532$	−2.24824	−0.0974735
$533$	1.92676	0.0834574
$534$	4.66401	0.201831
$535$	25.2014	1.08955
$536$	13.2513	0.572371
$537$	20.2012	0.871747
$538$	−15.6663	−0.675420
$539$	−32.0214	−1.37926
$540$	6.06441	0.260971
$541$	1.00749	0.0433153	0.0216577	−	0.999765i	$-0.493106\pi$
$541$	1.00749	0.0433153	0.0216577	+	0.999765i	$0.493106\pi$
$542$	−29.2439	−1.25613
$543$	−12.0824	−0.518507
$544$	−54.8376	−2.35114
$545$	−21.9647	−0.940864
$546$	−0.406808	−0.0174098
$547$	19.0963	0.816501	0.408250	−	0.912870i	$-0.366139\pi$
$547$	19.0963	0.816501	0.408250	+	0.912870i	$0.366139\pi$
$548$	7.97998	0.340888
$549$	−12.9389	−0.552220
$550$	−3.79785	−0.161941
$551$	−6.70047	−0.285449
$552$	6.29506	0.267935
$553$	−18.0832	−0.768976
$554$	7.19736	0.305787
$555$	12.5078	0.530928
$556$	−19.2785	−0.817589
$557$	−20.8332	−0.882730	−0.441365	−	0.897328i	$-0.645506\pi$
$557$	−20.8332	−0.882730	−0.441365	+	0.897328i	$0.645506\pi$
$558$	−19.0516	−0.806519
$559$	2.23247	0.0944235
$560$	5.60140	0.236702
$561$	−38.1407	−1.61030
$562$	−63.0269	−2.65863
$563$	29.5536	1.24554	0.622769	−	0.782406i	$-0.286008\pi$
$563$	29.5536	1.24554	0.622769	+	0.782406i	$0.286008\pi$
$564$	17.3262	0.729563
$565$	−25.3192	−1.06519
$566$	−33.1189	−1.39209
$567$	−1.03213	−0.0433453
$568$	14.9235	0.626178
$569$	−17.8966	−0.750265	−0.375133	−	0.926971i	$-0.622403\pi$
$569$	−17.8966	−0.750265	−0.375133	+	0.926971i	$0.622403\pi$
$570$	−4.11615	−0.172407
$571$	−6.61621	−0.276880	−0.138440	−	0.990371i	$-0.544209\pi$
$571$	−6.61621	−0.276880	−0.138440	+	0.990371i	$0.544209\pi$
$572$	−2.59754	−0.108609
$573$	−1.88127	−0.0785913
$574$	−23.3481	−0.974529
$575$	−1.52601	−0.0636389
$576$	−11.9853	−0.499387
$577$	−40.9287	−1.70388	−0.851942	−	0.523637i	$-0.824575\pi$
$577$	−40.9287	−1.70388	−0.851942	+	0.523637i	$0.824575\pi$
$578$	−70.9202	−2.94989
$579$	−4.06257	−0.168835
$580$	−49.0144	−2.03521
$581$	−5.56466	−0.230861
$582$	36.8975	1.52945
$583$	7.08548	0.293451
$584$	20.2388	0.837488
$585$	−0.422896	−0.0174846
$586$	−20.5619	−0.849404
$587$	−30.4910	−1.25850	−0.629249	−	0.777204i	$-0.716637\pi$
$587$	−30.4910	−1.25850	−0.629249	+	0.777204i	$0.716637\pi$
$588$	−15.5933	−0.643057
$589$	7.34225	0.302532
$590$	16.2056	0.667174
$591$	1.47589	0.0607098
$592$	−12.7422	−0.523702
$593$	22.4668	0.922599	0.461299	−	0.887244i	$-0.347383\pi$
$593$	22.4668	0.922599	0.461299	+	0.887244i	$0.347383\pi$
$594$	−11.6068	−0.476233
$595$	16.8396	0.690356
$596$	25.1490	1.03014
$597$	16.1873	0.662502
$598$	−1.83818	−0.0751686
$599$	−30.0270	−1.22687	−0.613434	−	0.789746i	$-0.710212\pi$
$599$	−30.0270	−1.22687	−0.613434	+	0.789746i	$0.710212\pi$
$600$	−0.441665	−0.0180309
$601$	−3.52484	−0.143781	−0.0718906	−	0.997413i	$-0.522903\pi$
$601$	−3.52484	−0.143781	−0.0718906	+	0.997413i	$0.522903\pi$
$602$	−27.0526	−1.10258
$603$	−9.81729	−0.399791
$604$	−10.7957	−0.439271
$605$	41.8052	1.69962
$606$	−8.47536	−0.344288
$607$	−39.1441	−1.58881	−0.794404	−	0.607389i	$-0.792217\pi$
$607$	−39.1441	−1.58881	−0.794404	+	0.607389i	$0.792217\pi$
$608$	6.43133	0.260825
$609$	8.34198	0.338034
$610$	64.2421	2.60109
$611$	−1.20822	−0.0488795
$612$	−18.5732	−0.750776
$613$	−7.86063	−0.317488	−0.158744	−	0.987320i	$-0.550744\pi$
$613$	−7.86063	−0.317488	−0.158744	+	0.987320i	$0.550744\pi$
$614$	−26.0656	−1.05192
$615$	−24.2714	−0.978717
$616$	7.51697	0.302867
$617$	−20.4029	−0.821388	−0.410694	−	0.911773i	$-0.634714\pi$
$617$	−20.4029	−0.821388	−0.410694	+	0.911773i	$0.634714\pi$
$618$	−14.8234	−0.596283
$619$	24.4769	0.983809	0.491905	−	0.870649i	$-0.336301\pi$
$619$	24.4769	0.983809	0.491905	+	0.870649i	$0.336301\pi$
$620$	53.7091	2.15701
$621$	−4.66371	−0.187148
$622$	−7.39530	−0.296525
$623$	2.23780	0.0896555
$624$	0.430821	0.0172466
$625$	−26.5290	−1.06116
$626$	64.4140	2.57450
$627$	4.47312	0.178639
$628$	58.1186	2.31919
$629$	−38.3072	−1.52741
$630$	5.12455	0.204167
$631$	12.4474	0.495524	0.247762	−	0.968821i	$-0.420305\pi$
$631$	12.4474	0.495524	0.247762	+	0.968821i	$0.420305\pi$
$632$	−23.6488	−0.940699
$633$	−24.4354	−0.971221
$634$	−7.07550	−0.281004
$635$	−46.1408	−1.83104
$636$	3.45038	0.136817
$637$	1.08738	0.0430837
$638$	93.8097	3.71396
$639$	−11.0561	−0.437375
$640$	23.6968	0.936699
$641$	2.41817	0.0955119	0.0477559	−	0.998859i	$-0.484793\pi$
$641$	2.41817	0.0955119	0.0477559	+	0.998859i	$0.484793\pi$
$642$	−23.4880	−0.926998
$643$	−2.29334	−0.0904407	−0.0452203	−	0.998977i	$-0.514399\pi$
$643$	−2.29334	−0.0904407	−0.0452203	+	0.998977i	$0.514399\pi$
$644$	12.6475	0.498380
$645$	−28.1224	−1.10732
$646$	12.6064	0.495991
$647$	5.16251	0.202959	0.101480	−	0.994838i	$-0.467642\pi$
$647$	5.16251	0.202959	0.101480	+	0.994838i	$0.467642\pi$
$648$	−1.34980	−0.0530250
$649$	−17.6110	−0.691292
$650$	0.128968	0.00505852
$651$	−9.14099	−0.358264
$652$	−6.00602	−0.235214
$653$	−22.9085	−0.896481	−0.448240	−	0.893913i	$-0.647949\pi$
$653$	−22.9085	−0.896481	−0.448240	+	0.893913i	$0.647949\pi$
$654$	20.4714	0.800494
$655$	11.4214	0.446273
$656$	24.7262	0.965396
$657$	−14.9940	−0.584971
$658$	14.6410	0.570764
$659$	30.6490	1.19392	0.596958	−	0.802273i	$-0.296376\pi$
$659$	30.6490	1.19392	0.596958	+	0.802273i	$0.296376\pi$
$660$	32.7212	1.27367
$661$	−4.20065	−0.163386	−0.0816932	−	0.996658i	$-0.526033\pi$
$661$	−4.20065	−0.163386	−0.0816932	+	0.996658i	$0.526033\pi$
$662$	15.9853	0.621287
$663$	1.29518	0.0503008
$664$	−7.27735	−0.282416
$665$	−1.97494	−0.0765848
$666$	−11.6575	−0.451718
$667$	37.6935	1.45950
$668$	−46.1938	−1.78729
$669$	5.54187	0.214261
$670$	48.7431	1.88311
$671$	−69.8133	−2.69511
$672$	−8.00691	−0.308873
$673$	44.1935	1.70353	0.851767	−	0.523922i	$-0.175531\pi$
$673$	44.1935	1.70353	0.851767	+	0.523922i	$0.175531\pi$
$674$	−25.9171	−0.998288
$675$	0.327209	0.0125943
$676$	−34.0690	−1.31034
$677$	−0.101232	−0.00389066	−0.00194533	−	0.999998i	$-0.500619\pi$
$677$	−0.101232	−0.00389066	−0.00194533	+	0.999998i	$0.500619\pi$
$678$	23.5978	0.906269
$679$	17.7035	0.679397
$680$	22.0225	0.844523
$681$	−9.42560	−0.361190
$682$	−102.795	−3.93622
$683$	15.7582	0.602970	0.301485	−	0.953471i	$-0.402518\pi$
$683$	15.7582	0.602970	0.301485	+	0.953471i	$0.402518\pi$
$684$	2.17825	0.0832876
$685$	7.00993	0.267835
$686$	−28.7186	−1.09648
$687$	15.0923	0.575807
$688$	28.6494	1.09225
$689$	−0.240609	−0.00916649
$690$	23.1555	0.881513
$691$	27.4987	1.04610	0.523050	−	0.852302i	$-0.324794\pi$
$691$	27.4987	1.04610	0.523050	+	0.852302i	$0.324794\pi$
$692$	28.8148	1.09538
$693$	−5.56896	−0.211547
$694$	−21.1938	−0.804506
$695$	−16.9349	−0.642379
$696$	10.9095	0.413522
$697$	74.3349	2.81563
$698$	−16.3487	−0.618809
$699$	−18.8135	−0.711592
$700$	−0.887355	−0.0335389
$701$	15.8667	0.599278	0.299639	−	0.954053i	$-0.403134\pi$
$701$	15.8667	0.599278	0.299639	+	0.954053i	$0.403134\pi$
$702$	0.394145	0.0148760
$703$	4.49264	0.169443
$704$	−64.6680	−2.43727
$705$	15.2200	0.573217
$706$	−6.56041	−0.246904
$707$	−4.06649	−0.152936
$708$	−8.57594	−0.322304
$709$	−17.1610	−0.644496	−0.322248	−	0.946655i	$-0.604438\pi$
$709$	−17.1610	−0.644496	−0.322248	+	0.946655i	$0.604438\pi$
$710$	54.8941	2.06014
$711$	17.5203	0.657062
$712$	2.92655	0.109677
$713$	−41.3039	−1.54684
$714$	−15.6947	−0.587360
$715$	−2.28178	−0.0853338
$716$	53.0782	1.98363
$717$	23.4838	0.877017
$718$	−43.9716	−1.64100
$719$	−20.1840	−0.752737	−0.376369	−	0.926470i	$-0.622827\pi$
$719$	−20.1840	−0.752737	−0.376369	+	0.926470i	$0.622827\pi$
$720$	−5.42704	−0.202254
$721$	−7.11227	−0.264875
$722$	39.3935	1.46607
$723$	−4.15332	−0.154464
$724$	−31.7463	−1.17984
$725$	−2.64460	−0.0982180
$726$	−38.9630	−1.44605
$727$	4.22039	0.156525	0.0782627	−	0.996933i	$-0.475063\pi$
$727$	4.22039	0.156525	0.0782627	+	0.996933i	$0.475063\pi$
$728$	−0.255262	−0.00946063
$729$	1.00000	0.0370370
$730$	74.4456	2.75535
$731$	86.1292	3.18560
$732$	−33.9967	−1.25655
$733$	−35.6481	−1.31669	−0.658346	−	0.752716i	$-0.728744\pi$
$733$	−35.6481	−1.31669	−0.658346	+	0.752716i	$0.728744\pi$
$734$	−69.1943	−2.55401
$735$	−13.6978	−0.505249
$736$	−36.1795	−1.33359
$737$	−52.9702	−1.95118
$738$	22.6213	0.832700
$739$	−6.52703	−0.240101	−0.120050	−	0.992768i	$-0.538306\pi$
$739$	−6.52703	−0.240101	−0.120050	+	0.992768i	$0.538306\pi$
$740$	32.8640	1.20811
$741$	−0.151898	−0.00558013
$742$	2.91565	0.107037
$743$	−32.7062	−1.19987	−0.599937	−	0.800048i	$-0.704808\pi$
$743$	−32.7062	−1.19987	−0.599937	+	0.800048i	$0.704808\pi$
$744$	−11.9544	−0.438269
$745$	22.0919	0.809383
$746$	57.9010	2.11991
$747$	5.39144	0.197263
$748$	−100.214	−3.66417
$749$	−11.2696	−0.411782
$750$	23.2005	0.847164
$751$	−49.0081	−1.78833	−0.894166	−	0.447736i	$-0.852230\pi$
$751$	−49.0081	−1.78833	−0.894166	+	0.447736i	$0.852230\pi$
$752$	−15.5052	−0.565416
$753$	18.1115	0.660020
$754$	−3.18560	−0.116013
$755$	−9.48336	−0.345135
$756$	−2.71189	−0.0986306
$757$	−37.3390	−1.35711	−0.678554	−	0.734550i	$-0.737393\pi$
$757$	−37.3390	−1.35711	−0.678554	+	0.734550i	$0.737393\pi$
$758$	15.2953	0.555549
$759$	−25.1635	−0.913379
$760$	−2.58278	−0.0936874
$761$	−14.6549	−0.531239	−0.265620	−	0.964078i	$-0.585576\pi$
$761$	−14.6549	−0.531239	−0.265620	+	0.964078i	$0.585576\pi$
$762$	43.0038	1.55786
$763$	9.82221	0.355588
$764$	−4.94300	−0.178831
$765$	−16.3154	−0.589884
$766$	62.8964	2.27254
$767$	0.598035	0.0215938
$768$	1.88484	0.0680132
$769$	−8.12267	−0.292911	−0.146455	−	0.989217i	$-0.546787\pi$
$769$	−8.12267	−0.292911	−0.146455	+	0.989217i	$0.546787\pi$
$770$	27.6501	0.996439
$771$	16.3637	0.589324
$772$	−10.6743	−0.384176
$773$	−27.6327	−0.993878	−0.496939	−	0.867785i	$-0.665543\pi$
$773$	−27.6327	−0.993878	−0.496939	+	0.867785i	$0.665543\pi$
$774$	26.2105	0.942115
$775$	2.89791	0.104096
$776$	23.1522	0.831116
$777$	−5.59327	−0.200658
$778$	−79.3486	−2.84479
$779$	−8.71795	−0.312353
$780$	−1.11115	−0.0397855
$781$	−59.6547	−2.13461
$782$	−70.9172	−2.53599
$783$	−8.08230	−0.288838
$784$	13.9544	0.498373
$785$	51.0537	1.82218
$786$	−10.6449	−0.379692
$787$	2.86693	0.102195	0.0510975	−	0.998694i	$-0.483728\pi$
$787$	2.86693	0.102195	0.0510975	+	0.998694i	$0.483728\pi$
$788$	3.87785	0.138143
$789$	−4.50800	−0.160489
$790$	−86.9888	−3.09492
$791$	11.3223	0.402574
$792$	−7.28297	−0.258789
$793$	2.37073	0.0841870
$794$	50.1208	1.77872
$795$	3.03095	0.107497
$796$	42.5317	1.50750
$797$	31.4152	1.11278	0.556391	−	0.830921i	$-0.312186\pi$
$797$	31.4152	1.11278	0.556391	+	0.830921i	$0.312186\pi$
$798$	1.84067	0.0651590
$799$	−46.6135	−1.64907
$800$	2.53837	0.0897451
$801$	−2.16814	−0.0766074
$802$	−46.7475	−1.65071
$803$	−80.9016	−2.85496
$804$	−25.7947	−0.909708
$805$	11.1100	0.391577
$806$	3.49072	0.122955
$807$	7.28272	0.256364
$808$	−5.31807	−0.187089
$809$	41.7346	1.46731	0.733655	−	0.679522i	$-0.237813\pi$
$809$	41.7346	1.46731	0.733655	+	0.679522i	$0.237813\pi$
$810$	−4.96503	−0.174453
$811$	−24.1190	−0.846931	−0.423466	−	0.905912i	$-0.639187\pi$
$811$	−24.1190	−0.846931	−0.423466	+	0.905912i	$0.639187\pi$
$812$	21.9183	0.769183
$813$	13.5945	0.476780
$814$	−62.8991	−2.20461
$815$	−5.27592	−0.184807
$816$	16.6211	0.581856
$817$	−10.1012	−0.353396
$818$	−12.3808	−0.432883
$819$	0.189111	0.00660808
$820$	−63.7724	−2.22703
$821$	7.27207	0.253797	0.126899	−	0.991916i	$-0.459498\pi$
$821$	7.27207	0.253797	0.126899	+	0.991916i	$0.459498\pi$
$822$	−6.53334	−0.227877
$823$	31.9131	1.11242	0.556210	−	0.831042i	$-0.312255\pi$
$823$	31.9131	1.11242	0.556210	+	0.831042i	$0.312255\pi$
$824$	−9.30128	−0.324025
$825$	1.76549	0.0614665
$826$	−7.24685	−0.252150
$827$	−12.7042	−0.441769	−0.220884	−	0.975300i	$-0.570894\pi$
$827$	−12.7042	−0.441769	−0.220884	+	0.975300i	$0.570894\pi$
$828$	−12.2538	−0.425848
$829$	−19.3704	−0.672762	−0.336381	−	0.941726i	$-0.609203\pi$
$829$	−19.3704	−0.672762	−0.336381	+	0.941726i	$0.609203\pi$
$830$	−26.7687	−0.929154
$831$	−3.34581	−0.116065
$832$	2.19600	0.0761326
$833$	41.9515	1.45353
$834$	15.7836	0.546541
$835$	−40.5784	−1.40427
$836$	11.7530	0.406486
$837$	8.85645	0.306124
$838$	25.5902	0.884000
$839$	25.9919	0.897339	0.448670	−	0.893698i	$-0.351898\pi$
$839$	25.9919	0.897339	0.448670	+	0.893698i	$0.351898\pi$
$840$	3.21553	0.110946
$841$	36.3236	1.25254
$842$	−73.5057	−2.53317
$843$	29.2991	1.00911
$844$	−64.2035	−2.20997
$845$	−29.9275	−1.02954
$846$	−14.1852	−0.487698
$847$	−18.6945	−0.642352
$848$	−3.08775	−0.106034
$849$	15.3959	0.528385
$850$	4.97559	0.170661
$851$	−25.2734	−0.866361
$852$	−29.0498	−0.995228
$853$	−28.6882	−0.982265	−0.491132	−	0.871085i	$-0.663417\pi$
$853$	−28.6882	−0.982265	−0.491132	+	0.871085i	$0.663417\pi$
$854$	−28.7279	−0.983049
$855$	1.91346	0.0654390
$856$	−14.7381	−0.503739
$857$	18.3984	0.628479	0.314239	−	0.949344i	$-0.398251\pi$
$857$	18.3984	0.628479	0.314239	+	0.949344i	$0.398251\pi$
$858$	2.12665	0.0726026
$859$	−44.5289	−1.51931	−0.759653	−	0.650329i	$-0.774631\pi$
$859$	−44.5289	−1.51931	−0.759653	+	0.650329i	$0.774631\pi$
$860$	−73.8909	−2.51966
$861$	10.8537	0.369894
$862$	2.74377	0.0934532
$863$	−39.5281	−1.34555	−0.672775	−	0.739847i	$-0.734898\pi$
$863$	−39.5281	−1.34555	−0.672775	+	0.739847i	$0.734898\pi$
$864$	7.75766	0.263921
$865$	25.3121	0.860636
$866$	22.0873	0.750556
$867$	32.9684	1.11967
$868$	−24.0177	−0.815215
$869$	94.5326	3.20680
$870$	40.1289	1.36050
$871$	1.79877	0.0609489
$872$	12.8453	0.434996
$873$	−17.1524	−0.580520
$874$	8.31713	0.281331
$875$	11.1317	0.376319
$876$	−39.3963	−1.33108
$877$	33.1048	1.11787	0.558935	−	0.829211i	$-0.311210\pi$
$877$	33.1048	1.11787	0.558935	+	0.829211i	$0.311210\pi$
$878$	0.513313	0.0173235
$879$	9.55853	0.322401
$880$	−29.2822	−0.987102
$881$	42.3655	1.42733	0.713665	−	0.700487i	$-0.247034\pi$
$881$	42.3655	1.42733	0.713665	+	0.700487i	$0.247034\pi$
$882$	12.7665	0.429870
$883$	−32.4775	−1.09295	−0.546477	−	0.837474i	$-0.684032\pi$
$883$	−32.4775	−1.09295	−0.546477	+	0.837474i	$0.684032\pi$
$884$	3.40306	0.114457
$885$	−7.53344	−0.253234
$886$	−81.8701	−2.75048
$887$	2.43679	0.0818193	0.0409097	−	0.999163i	$-0.486974\pi$
$887$	2.43679	0.0818193	0.0409097	+	0.999163i	$0.486974\pi$
$888$	−7.31477	−0.245467
$889$	20.6333	0.692019
$890$	10.7649	0.360839
$891$	5.39561	0.180760
$892$	14.5611	0.487543
$893$	5.46681	0.182940
$894$	−20.5899	−0.688629
$895$	46.6259	1.55853
$896$	−10.5968	−0.354014
$897$	0.854506	0.0285311
$898$	−23.4393	−0.782179
$899$	−71.5805	−2.38734
$900$	0.859732	0.0286577
$901$	−9.28275	−0.309253
$902$	122.055	4.06400
$903$	12.5758	0.418497
$904$	14.8070	0.492475
$905$	−27.8872	−0.927001
$906$	8.83862	0.293643
$907$	39.4743	1.31072	0.655361	−	0.755316i	$-0.272516\pi$
$907$	39.4743	1.31072	0.655361	+	0.755316i	$0.272516\pi$
$908$	−24.7655	−0.821873
$909$	3.93991	0.130679
$910$	−0.938943	−0.0311257
$911$	−47.3060	−1.56732	−0.783660	−	0.621190i	$-0.786649\pi$
$911$	−47.3060	−1.56732	−0.783660	+	0.621190i	$0.786649\pi$
$912$	−1.94932	−0.0645484
$913$	29.0901	0.962742
$914$	45.2386	1.49636
$915$	−29.8640	−0.987273
$916$	39.6546	1.31023
$917$	−5.10746	−0.168663
$918$	15.2062	0.501878
$919$	31.4621	1.03784	0.518919	−	0.854823i	$-0.326335\pi$
$919$	31.4621	1.03784	0.518919	+	0.854823i	$0.326335\pi$
$920$	14.5295	0.479022
$921$	12.1170	0.399269
$922$	28.5045	0.938744
$923$	2.02576	0.0666786
$924$	−14.6323	−0.481368
$925$	1.77320	0.0583024
$926$	0.987927	0.0324653
$927$	6.89088	0.226326
$928$	−62.6998	−2.05822
$929$	49.7786	1.63318	0.816592	−	0.577215i	$-0.195861\pi$
$929$	49.7786	1.63318	0.816592	+	0.577215i	$0.195861\pi$
$930$	−43.9725	−1.44192
$931$	−4.92005	−0.161248
$932$	−49.4320	−1.61920
$933$	3.43783	0.112549
$934$	56.4739	1.84788
$935$	−88.0315	−2.87894
$936$	0.247316	0.00808377
$937$	55.1293	1.80100	0.900498	−	0.434861i	$-0.143202\pi$
$937$	55.1293	1.80100	0.900498	+	0.434861i	$0.143202\pi$
$938$	−21.7970	−0.711698
$939$	−29.9439	−0.977182
$940$	39.9901	1.30433
$941$	−43.4557	−1.41661	−0.708307	−	0.705904i	$-0.750541\pi$
$941$	−43.4557	−1.41661	−0.708307	+	0.705904i	$0.750541\pi$
$942$	−47.5827	−1.55033
$943$	49.0429	1.59706
$944$	7.67461	0.249787
$945$	−2.38223	−0.0774940
$946$	141.421	4.59801
$947$	13.0404	0.423757	0.211879	−	0.977296i	$-0.432042\pi$
$947$	13.0404	0.423757	0.211879	+	0.977296i	$0.432042\pi$
$948$	46.0341	1.49512
$949$	2.74726	0.0891800
$950$	−0.583535	−0.0189324
$951$	3.28916	0.106658
$952$	−9.84804	−0.319177
$953$	6.53247	0.211607	0.105804	−	0.994387i	$-0.466258\pi$
$953$	6.53247	0.211607	0.105804	+	0.994387i	$0.466258\pi$
$954$	−2.82489	−0.0914591
$955$	−4.34212	−0.140508
$956$	61.7030	1.99562
$957$	−43.6089	−1.40968
$958$	−7.40194	−0.239146
$959$	−3.13471	−0.101225
$960$	−27.6630	−0.892818
$961$	47.4366	1.53021
$962$	2.13593	0.0688653
$963$	10.9188	0.351853
$964$	−10.9127	−0.351476
$965$	−9.37671	−0.301847
$966$	−10.3547	−0.333157
$967$	−29.1108	−0.936139	−0.468070	−	0.883692i	$-0.655050\pi$
$967$	−29.1108	−0.936139	−0.468070	+	0.883692i	$0.655050\pi$
$968$	−24.4483	−0.785798
$969$	−5.86027	−0.188259
$970$	85.1621	2.73439
$971$	−26.2176	−0.841363	−0.420682	−	0.907208i	$-0.638209\pi$
$971$	−26.2176	−0.841363	−0.420682	+	0.907208i	$0.638209\pi$
$972$	2.62747	0.0842763
$973$	7.57299	0.242779
$974$	−11.3435	−0.363469
$975$	−0.0599527	−0.00192002
$976$	30.4236	0.973836
$977$	18.2094	0.582570	0.291285	−	0.956636i	$-0.405917\pi$
$977$	18.2094	0.582570	0.291285	+	0.956636i	$0.405917\pi$
$978$	4.91723	0.157236
$979$	−11.6984	−0.373883
$980$	−35.9905	−1.14967
$981$	−9.51645	−0.303837
$982$	43.3191	1.38237
$983$	40.4201	1.28920	0.644601	−	0.764519i	$-0.277024\pi$
$983$	40.4201	1.28920	0.644601	+	0.764519i	$0.277024\pi$
$984$	14.1943	0.452497
$985$	3.40645	0.108539
$986$	−122.901	−3.91396
$987$	−6.80609	−0.216640
$988$	−0.399109	−0.0126974
$989$	56.8243	1.80691
$990$	−26.7894	−0.851422
$991$	16.8763	0.536093	0.268047	−	0.963406i	$-0.413622\pi$
$991$	16.8763	0.536093	0.268047	+	0.963406i	$0.413622\pi$
$992$	68.7053	2.18140
$993$	−7.43104	−0.235817
$994$	−24.5476	−0.778604
$995$	37.3615	1.18444
$996$	14.1659	0.448863
$997$	−11.3551	−0.359619	−0.179809	−	0.983701i	$-0.557548\pi$
$997$	−11.3551	−0.359619	−0.179809	+	0.983701i	$0.557548\pi$
$998$	−23.9073	−0.756772
$999$	5.41916	0.171455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6033.2.a.b.1.11	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6033.2.a.b.1.11	✓	71	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:	\( N \)	\(=\)	\( 6033 = 3 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15116 q^{2} +1.00000 q^{3} +2.62747 q^{4} +2.30807 q^{5} -2.15116 q^{6} -1.03213 q^{7} -1.34980 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.15116 q^{2} +1.00000 q^{3} +2.62747 q^{4} +2.30807 q^{5} -2.15116 q^{6} -1.03213 q^{7} -1.34980 q^{8} +1.00000 q^{9} -4.96503 q^{10} +5.39561 q^{11} +2.62747 q^{12} -0.183225 q^{13} +2.22027 q^{14} +2.30807 q^{15} -2.35133 q^{16} -7.06883 q^{17} -2.15116 q^{18} +0.829029 q^{19} +6.06441 q^{20} -1.03213 q^{21} -11.6068 q^{22} -4.66371 q^{23} -1.34980 q^{24} +0.327209 q^{25} +0.394145 q^{26} +1.00000 q^{27} -2.71189 q^{28} -8.08230 q^{29} -4.96503 q^{30} +8.85645 q^{31} +7.75766 q^{32} +5.39561 q^{33} +15.2062 q^{34} -2.38223 q^{35} +2.62747 q^{36} +5.41916 q^{37} -1.78337 q^{38} -0.183225 q^{39} -3.11543 q^{40} -10.5159 q^{41} +2.22027 q^{42} -12.1844 q^{43} +14.1768 q^{44} +2.30807 q^{45} +10.0324 q^{46} +6.59423 q^{47} -2.35133 q^{48} -5.93471 q^{49} -0.703877 q^{50} -7.06883 q^{51} -0.481418 q^{52} +1.31319 q^{53} -2.15116 q^{54} +12.4535 q^{55} +1.39316 q^{56} +0.829029 q^{57} +17.3863 q^{58} -3.26395 q^{59} +6.06441 q^{60} -12.9389 q^{61} -19.0516 q^{62} -1.03213 q^{63} -11.9853 q^{64} -0.422896 q^{65} -11.6068 q^{66} -9.81729 q^{67} -18.5732 q^{68} -4.66371 q^{69} +5.12455 q^{70} -11.0561 q^{71} -1.34980 q^{72} -14.9940 q^{73} -11.6575 q^{74} +0.327209 q^{75} +2.17825 q^{76} -5.56896 q^{77} +0.394145 q^{78} +17.5203 q^{79} -5.42704 q^{80} +1.00000 q^{81} +22.6213 q^{82} +5.39144 q^{83} -2.71189 q^{84} -16.3154 q^{85} +26.2105 q^{86} -8.08230 q^{87} -7.28297 q^{88} -2.16814 q^{89} -4.96503 q^{90} +0.189111 q^{91} -12.2538 q^{92} +8.85645 q^{93} -14.1852 q^{94} +1.91346 q^{95} +7.75766 q^{96} -17.1524 q^{97} +12.7665 q^{98} +5.39561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9}+O(q^{10}) \) 71 * q - 11 * q^2 + 71 * q^3 + 53 * q^4 - 8 * q^5 - 11 * q^6 - 46 * q^7 - 33 * q^8 + 71 * q^9	\( 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9} - 41 q^{10} - 18 q^{11} + 53 q^{12} - 67 q^{13} - 7 q^{14} - 8 q^{15} + 21 q^{16} - 25 q^{17} - 11 q^{18} - 43 q^{19} - 8 q^{20} - 46 q^{21} - 49 q^{22} - 75 q^{23} - 33 q^{24} + 19 q^{25} + 71 q^{27} - 89 q^{28} - 35 q^{29} - 41 q^{30} - 82 q^{31} - 62 q^{32} - 18 q^{33} - 28 q^{34} - 51 q^{35} + 53 q^{36} - 66 q^{37} - 29 q^{38} - 67 q^{39} - 102 q^{40} + q^{41} - 7 q^{42} - 112 q^{43} - 25 q^{44} - 8 q^{45} - 36 q^{46} - 67 q^{47} + 21 q^{48} + 7 q^{49} - 24 q^{50} - 25 q^{51} - 134 q^{52} - 40 q^{53} - 11 q^{54} - 112 q^{55} + 9 q^{56} - 43 q^{57} - 47 q^{58} - 18 q^{59} - 8 q^{60} - 144 q^{61} - 19 q^{62} - 46 q^{63} - 17 q^{64} - 31 q^{65} - 49 q^{66} - 85 q^{67} - 22 q^{68} - 75 q^{69} - 11 q^{70} - 44 q^{71} - 33 q^{72} - 98 q^{73} + 6 q^{74} + 19 q^{75} - 85 q^{76} - 39 q^{77} - 126 q^{79} + 21 q^{80} + 71 q^{81} - 69 q^{82} - 43 q^{83} - 89 q^{84} - 112 q^{85} + 32 q^{86} - 35 q^{87} - 85 q^{88} + 8 q^{89} - 41 q^{90} - 40 q^{91} - 96 q^{92} - 82 q^{93} - 99 q^{94} - 103 q^{95} - 62 q^{96} - 67 q^{97} - 11 q^{98} - 18 q^{99}+O(q^{100}) \) 71 * q - 11 * q^2 + 71 * q^3 + 53 * q^4 - 8 * q^5 - 11 * q^6 - 46 * q^7 - 33 * q^8 + 71 * q^9 - 41 * q^10 - 18 * q^11 + 53 * q^12 - 67 * q^13 - 7 * q^14 - 8 * q^15 + 21 * q^16 - 25 * q^17 - 11 * q^18 - 43 * q^19 - 8 * q^20 - 46 * q^21 - 49 * q^22 - 75 * q^23 - 33 * q^24 + 19 * q^25 + 71 * q^27 - 89 * q^28 - 35 * q^29 - 41 * q^30 - 82 * q^31 - 62 * q^32 - 18 * q^33 - 28 * q^34 - 51 * q^35 + 53 * q^36 - 66 * q^37 - 29 * q^38 - 67 * q^39 - 102 * q^40 + q^41 - 7 * q^42 - 112 * q^43 - 25 * q^44 - 8 * q^45 - 36 * q^46 - 67 * q^47 + 21 * q^48 + 7 * q^49 - 24 * q^50 - 25 * q^51 - 134 * q^52 - 40 * q^53 - 11 * q^54 - 112 * q^55 + 9 * q^56 - 43 * q^57 - 47 * q^58 - 18 * q^59 - 8 * q^60 - 144 * q^61 - 19 * q^62 - 46 * q^63 - 17 * q^64 - 31 * q^65 - 49 * q^66 - 85 * q^67 - 22 * q^68 - 75 * q^69 - 11 * q^70 - 44 * q^71 - 33 * q^72 - 98 * q^73 + 6 * q^74 + 19 * q^75 - 85 * q^76 - 39 * q^77 - 126 * q^79 + 21 * q^80 + 71 * q^81 - 69 * q^82 - 43 * q^83 - 89 * q^84 - 112 * q^85 + 32 * q^86 - 35 * q^87 - 85 * q^88 + 8 * q^89 - 41 * q^90 - 40 * q^91 - 96 * q^92 - 82 * q^93 - 99 * q^94 - 103 * q^95 - 62 * q^96 - 67 * q^97 - 11 * q^98 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more