LMFDB - Embedded newform 6015.2.a.g.1.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.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:	$48.0300168158$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60730 q^{2} -1.00000 q^{3} +0.583405 q^{4} +1.00000 q^{5} -1.60730 q^{6} +0.620888 q^{7} -2.27689 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.60730 q^{2} -1.00000 q^{3} +0.583405 q^{4} +1.00000 q^{5} -1.60730 q^{6} +0.620888 q^{7} -2.27689 q^{8} +1.00000 q^{9} +1.60730 q^{10} +4.20498 q^{11} -0.583405 q^{12} +1.55179 q^{13} +0.997951 q^{14} -1.00000 q^{15} -4.82645 q^{16} -1.20810 q^{17} +1.60730 q^{18} +8.14460 q^{19} +0.583405 q^{20} -0.620888 q^{21} +6.75866 q^{22} -2.59878 q^{23} +2.27689 q^{24} +1.00000 q^{25} +2.49418 q^{26} -1.00000 q^{27} +0.362229 q^{28} -2.80203 q^{29} -1.60730 q^{30} +1.41765 q^{31} -3.20376 q^{32} -4.20498 q^{33} -1.94177 q^{34} +0.620888 q^{35} +0.583405 q^{36} -2.48108 q^{37} +13.0908 q^{38} -1.55179 q^{39} -2.27689 q^{40} +4.20947 q^{41} -0.997951 q^{42} +3.10947 q^{43} +2.45321 q^{44} +1.00000 q^{45} -4.17701 q^{46} +1.78404 q^{47} +4.82645 q^{48} -6.61450 q^{49} +1.60730 q^{50} +1.20810 q^{51} +0.905319 q^{52} +7.70325 q^{53} -1.60730 q^{54} +4.20498 q^{55} -1.41369 q^{56} -8.14460 q^{57} -4.50369 q^{58} -1.69621 q^{59} -0.583405 q^{60} -11.8730 q^{61} +2.27858 q^{62} +0.620888 q^{63} +4.50350 q^{64} +1.55179 q^{65} -6.75866 q^{66} +12.7938 q^{67} -0.704809 q^{68} +2.59878 q^{69} +0.997951 q^{70} -4.30104 q^{71} -2.27689 q^{72} -14.5395 q^{73} -3.98783 q^{74} -1.00000 q^{75} +4.75160 q^{76} +2.61082 q^{77} -2.49418 q^{78} +8.16958 q^{79} -4.82645 q^{80} +1.00000 q^{81} +6.76587 q^{82} +11.2674 q^{83} -0.362229 q^{84} -1.20810 q^{85} +4.99785 q^{86} +2.80203 q^{87} -9.57428 q^{88} +15.9520 q^{89} +1.60730 q^{90} +0.963485 q^{91} -1.51614 q^{92} -1.41765 q^{93} +2.86748 q^{94} +8.14460 q^{95} +3.20376 q^{96} +0.268348 q^{97} -10.6315 q^{98} +4.20498 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9}+O(q^{10}) $ 36 * q - 36 * q^3 + 36 * q^4 + 36 * q^5 - 4 * q^7 - 3 * q^8 + 36 * q^9	\( 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9} + 15 q^{11} - 36 q^{12} + 8 q^{13} + 10 q^{14} - 36 q^{15} + 36 q^{16} + 32 q^{17} + 5 q^{19} + 36 q^{20} + 4 q^{21} + q^{22} - 10 q^{23} + 3 q^{24} + 36 q^{25} + 22 q^{26} - 36 q^{27} + 61 q^{29} + 15 q^{31} - q^{32} - 15 q^{33} + 26 q^{34} - 4 q^{35} + 36 q^{36} + 16 q^{37} + 20 q^{38} - 8 q^{39} - 3 q^{40} + 61 q^{41} - 10 q^{42} - 35 q^{43} + 39 q^{44} + 36 q^{45} + 11 q^{46} - 28 q^{47} - 36 q^{48} + 68 q^{49} - 32 q^{51} + 4 q^{52} + 33 q^{53} + 15 q^{55} + 23 q^{56} - 5 q^{57} + 6 q^{58} + 35 q^{59} - 36 q^{60} + 55 q^{61} + q^{62} - 4 q^{63} + 15 q^{64} + 8 q^{65} - q^{66} - 34 q^{67} + 60 q^{68} + 10 q^{69} + 10 q^{70} + 42 q^{71} - 3 q^{72} + 53 q^{73} + 54 q^{74} - 36 q^{75} + 20 q^{76} + 20 q^{77} - 22 q^{78} + 25 q^{79} + 36 q^{80} + 36 q^{81} - 15 q^{82} - 11 q^{83} + 32 q^{85} + 53 q^{86} - 61 q^{87} + 27 q^{88} + 81 q^{89} + 15 q^{91} - 7 q^{92} - 15 q^{93} + 56 q^{94} + 5 q^{95} + q^{96} + 47 q^{97} - 3 q^{98} + 15 q^{99}+O(q^{100}) \) 36 * q - 36 * q^3 + 36 * q^4 + 36 * q^5 - 4 * q^7 - 3 * q^8 + 36 * q^9 + 15 * q^11 - 36 * q^12 + 8 * q^13 + 10 * q^14 - 36 * q^15 + 36 * q^16 + 32 * q^17 + 5 * q^19 + 36 * q^20 + 4 * q^21 + q^22 - 10 * q^23 + 3 * q^24 + 36 * q^25 + 22 * q^26 - 36 * q^27 + 61 * q^29 + 15 * q^31 - q^32 - 15 * q^33 + 26 * q^34 - 4 * q^35 + 36 * q^36 + 16 * q^37 + 20 * q^38 - 8 * q^39 - 3 * q^40 + 61 * q^41 - 10 * q^42 - 35 * q^43 + 39 * q^44 + 36 * q^45 + 11 * q^46 - 28 * q^47 - 36 * q^48 + 68 * q^49 - 32 * q^51 + 4 * q^52 + 33 * q^53 + 15 * q^55 + 23 * q^56 - 5 * q^57 + 6 * q^58 + 35 * q^59 - 36 * q^60 + 55 * q^61 + q^62 - 4 * q^63 + 15 * q^64 + 8 * q^65 - q^66 - 34 * q^67 + 60 * q^68 + 10 * q^69 + 10 * q^70 + 42 * q^71 - 3 * q^72 + 53 * q^73 + 54 * q^74 - 36 * q^75 + 20 * q^76 + 20 * q^77 - 22 * q^78 + 25 * q^79 + 36 * q^80 + 36 * q^81 - 15 * q^82 - 11 * q^83 + 32 * q^85 + 53 * q^86 - 61 * q^87 + 27 * q^88 + 81 * q^89 + 15 * q^91 - 7 * q^92 - 15 * q^93 + 56 * q^94 + 5 * q^95 + q^96 + 47 * q^97 - 3 * q^98 + 15 * 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 $

$2$	1.60730	1.13653	0.568265	−	0.822845i	$-0.307615\pi$
$2$	1.60730	1.13653	0.568265	+	0.822845i	$0.307615\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.583405	0.291702
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.60730	−0.656176
$7$	0.620888	0.234673	0.117337	−	0.993092i	$-0.462564\pi$
$7$	0.620888	0.234673	0.117337	+	0.993092i	$0.462564\pi$
$8$	−2.27689	−0.805002
$9$	1.00000	0.333333
$10$	1.60730	0.508272
$11$	4.20498	1.26785	0.633925	−	0.773394i	$-0.281443\pi$
$11$	4.20498	1.26785	0.633925	+	0.773394i	$0.281443\pi$
$12$	−0.583405	−0.168414
$13$	1.55179	0.430388	0.215194	−	0.976571i	$-0.430962\pi$
$13$	1.55179	0.430388	0.215194	+	0.976571i	$0.430962\pi$
$14$	0.997951	0.266714
$15$	−1.00000	−0.258199
$16$	−4.82645	−1.20661
$17$	−1.20810	−0.293006	−0.146503	−	0.989210i	$-0.546802\pi$
$17$	−1.20810	−0.293006	−0.146503	+	0.989210i	$0.546802\pi$
$18$	1.60730	0.378844
$19$	8.14460	1.86850	0.934249	−	0.356620i	$-0.116071\pi$
$19$	8.14460	1.86850	0.934249	+	0.356620i	$0.116071\pi$
$20$	0.583405	0.130453
$21$	−0.620888	−0.135489
$22$	6.75866	1.44095
$23$	−2.59878	−0.541883	−0.270941	−	0.962596i	$-0.587335\pi$
$23$	−2.59878	−0.541883	−0.270941	+	0.962596i	$0.587335\pi$
$24$	2.27689	0.464768
$25$	1.00000	0.200000
$26$	2.49418	0.489149
$27$	−1.00000	−0.192450
$28$	0.362229	0.0684548
$29$	−2.80203	−0.520324	−0.260162	−	0.965565i	$-0.583776\pi$
$29$	−2.80203	−0.520324	−0.260162	+	0.965565i	$0.583776\pi$
$30$	−1.60730	−0.293451
$31$	1.41765	0.254617	0.127308	−	0.991863i	$-0.459366\pi$
$31$	1.41765	0.254617	0.127308	+	0.991863i	$0.459366\pi$
$32$	−3.20376	−0.566350
$33$	−4.20498	−0.731994
$34$	−1.94177	−0.333011
$35$	0.620888	0.104949
$36$	0.583405	0.0972341
$37$	−2.48108	−0.407886	−0.203943	−	0.978983i	$-0.565376\pi$
$37$	−2.48108	−0.407886	−0.203943	+	0.978983i	$0.565376\pi$
$38$	13.0908	2.12361
$39$	−1.55179	−0.248485
$40$	−2.27689	−0.360008
$41$	4.20947	0.657408	0.328704	−	0.944433i	$-0.393388\pi$
$41$	4.20947	0.657408	0.328704	+	0.944433i	$0.393388\pi$
$42$	−0.997951	−0.153987
$43$	3.10947	0.474190	0.237095	−	0.971486i	$-0.423805\pi$
$43$	3.10947	0.474190	0.237095	+	0.971486i	$0.423805\pi$
$44$	2.45321	0.369835
$45$	1.00000	0.149071
$46$	−4.17701	−0.615867
$47$	1.78404	0.260229	0.130115	−	0.991499i	$-0.458465\pi$
$47$	1.78404	0.260229	0.130115	+	0.991499i	$0.458465\pi$
$48$	4.82645	0.696638
$49$	−6.61450	−0.944928
$50$	1.60730	0.227306
$51$	1.20810	0.169167
$52$	0.905319	0.125545
$53$	7.70325	1.05812	0.529062	−	0.848583i	$-0.322544\pi$
$53$	7.70325	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$54$	−1.60730	−0.218725
$55$	4.20498	0.567000
$56$	−1.41369	−0.188913
$57$	−8.14460	−1.07878
$58$	−4.50369	−0.591364
$59$	−1.69621	−0.220827	−0.110414	−	0.993886i	$-0.535218\pi$
$59$	−1.69621	−0.220827	−0.110414	+	0.993886i	$0.535218\pi$
$60$	−0.583405	−0.0753172
$61$	−11.8730	−1.52018	−0.760092	−	0.649816i	$-0.774846\pi$
$61$	−11.8730	−1.52018	−0.760092	+	0.649816i	$0.774846\pi$
$62$	2.27858	0.289380
$63$	0.620888	0.0782245
$64$	4.50350	0.562938
$65$	1.55179	0.192475
$66$	−6.75866	−0.831933
$67$	12.7938	1.56301	0.781505	−	0.623898i	$-0.214452\pi$
$67$	12.7938	1.56301	0.781505	+	0.623898i	$0.214452\pi$
$68$	−0.704809	−0.0854706
$69$	2.59878	0.312856
$70$	0.997951	0.119278
$71$	−4.30104	−0.510440	−0.255220	−	0.966883i	$-0.582148\pi$
$71$	−4.30104	−0.510440	−0.255220	+	0.966883i	$0.582148\pi$
$72$	−2.27689	−0.268334
$73$	−14.5395	−1.70172	−0.850859	−	0.525394i	$-0.823918\pi$
$73$	−14.5395	−1.70172	−0.850859	+	0.525394i	$0.823918\pi$
$74$	−3.98783	−0.463575
$75$	−1.00000	−0.115470
$76$	4.75160	0.545046
$77$	2.61082	0.297531
$78$	−2.49418	−0.282410
$79$	8.16958	0.919150	0.459575	−	0.888139i	$-0.348002\pi$
$79$	8.16958	0.919150	0.459575	+	0.888139i	$0.348002\pi$
$80$	−4.82645	−0.539613
$81$	1.00000	0.111111
$82$	6.76587	0.747165
$83$	11.2674	1.23676	0.618379	−	0.785880i	$-0.287790\pi$
$83$	11.2674	1.23676	0.618379	+	0.785880i	$0.287790\pi$
$84$	−0.362229	−0.0395224
$85$	−1.20810	−0.131036
$86$	4.99785	0.538932
$87$	2.80203	0.300409
$88$	−9.57428	−1.02062
$89$	15.9520	1.69091	0.845453	−	0.534049i	$-0.179330\pi$
$89$	15.9520	1.69091	0.845453	+	0.534049i	$0.179330\pi$
$90$	1.60730	0.169424
$91$	0.963485	0.101001
$92$	−1.51614	−0.158069
$93$	−1.41765	−0.147003
$94$	2.86748	0.295758
$95$	8.14460	0.835618
$96$	3.20376	0.326982
$97$	0.268348	0.0272466	0.0136233	−	0.999907i	$-0.495663\pi$
$97$	0.268348	0.0272466	0.0136233	+	0.999907i	$0.495663\pi$
$98$	−10.6315	−1.07394
$99$	4.20498	0.422617
$100$	0.583405	0.0583405
$101$	4.29721	0.427588	0.213794	−	0.976879i	$-0.431418\pi$
$101$	4.29721	0.427588	0.213794	+	0.976879i	$0.431418\pi$
$102$	1.94177	0.192264
$103$	−3.48395	−0.343284	−0.171642	−	0.985159i	$-0.554907\pi$
$103$	−3.48395	−0.343284	−0.171642	+	0.985159i	$0.554907\pi$
$104$	−3.53325	−0.346463
$105$	−0.620888	−0.0605924
$106$	12.3814	1.20259
$107$	0.192122	0.0185731	0.00928657	−	0.999957i	$-0.497044\pi$
$107$	0.192122	0.0185731	0.00928657	+	0.999957i	$0.497044\pi$
$108$	−0.583405	−0.0561381
$109$	11.1346	1.06650	0.533252	−	0.845956i	$-0.320970\pi$
$109$	11.1346	1.06650	0.533252	+	0.845956i	$0.320970\pi$
$110$	6.75866	0.644413
$111$	2.48108	0.235493
$112$	−2.99668	−0.283160
$113$	−1.71176	−0.161029	−0.0805143	−	0.996753i	$-0.525656\pi$
$113$	−1.71176	−0.161029	−0.0805143	+	0.996753i	$0.525656\pi$
$114$	−13.0908	−1.22606
$115$	−2.59878	−0.242337
$116$	−1.63472	−0.151780
$117$	1.55179	0.143463
$118$	−2.72631	−0.250977
$119$	−0.750092	−0.0687608
$120$	2.27689	0.207851
$121$	6.68188	0.607444
$122$	−19.0835	−1.72774
$123$	−4.20947	−0.379555
$124$	0.827061	0.0742723
$125$	1.00000	0.0894427
$126$	0.997951	0.0889046
$127$	−17.1849	−1.52491	−0.762455	−	0.647041i	$-0.776006\pi$
$127$	−17.1849	−1.52491	−0.762455	+	0.647041i	$0.776006\pi$
$128$	13.6460	1.20615
$129$	−3.10947	−0.273774
$130$	2.49418	0.218754
$131$	2.59744	0.226940	0.113470	−	0.993541i	$-0.463803\pi$
$131$	2.59744	0.226940	0.113470	+	0.993541i	$0.463803\pi$
$132$	−2.45321	−0.213524
$133$	5.05688	0.438487
$134$	20.5634	1.77641
$135$	−1.00000	−0.0860663
$136$	2.75070	0.235871
$137$	−3.18666	−0.272254	−0.136127	−	0.990691i	$-0.543466\pi$
$137$	−3.18666	−0.272254	−0.136127	+	0.990691i	$0.543466\pi$
$138$	4.17701	0.355571
$139$	16.3508	1.38685	0.693426	−	0.720528i	$-0.256100\pi$
$139$	16.3508	1.38685	0.693426	+	0.720528i	$0.256100\pi$
$140$	0.362229	0.0306139
$141$	−1.78404	−0.150243
$142$	−6.91305	−0.580131
$143$	6.52523	0.545668
$144$	−4.82645	−0.402204
$145$	−2.80203	−0.232696
$146$	−23.3693	−1.93405
$147$	6.61450	0.545555
$148$	−1.44747	−0.118981
$149$	19.0866	1.56364	0.781820	−	0.623505i	$-0.214292\pi$
$149$	19.0866	1.56364	0.781820	+	0.623505i	$0.214292\pi$
$150$	−1.60730	−0.131235
$151$	18.7002	1.52180	0.760902	−	0.648867i	$-0.224757\pi$
$151$	18.7002	1.52180	0.760902	+	0.648867i	$0.224757\pi$
$152$	−18.5444	−1.50415
$153$	−1.20810	−0.0976688
$154$	4.19637	0.338153
$155$	1.41765	0.113868
$156$	−0.905319	−0.0724836
$157$	0.650762	0.0519365	0.0259682	−	0.999663i	$-0.491733\pi$
$157$	0.650762	0.0519365	0.0259682	+	0.999663i	$0.491733\pi$
$158$	13.1309	1.04464
$159$	−7.70325	−0.610908
$160$	−3.20376	−0.253279
$161$	−1.61355	−0.127166
$162$	1.60730	0.126281
$163$	5.67991	0.444885	0.222442	−	0.974946i	$-0.428597\pi$
$163$	5.67991	0.444885	0.222442	+	0.974946i	$0.428597\pi$
$164$	2.45582	0.191768
$165$	−4.20498	−0.327357
$166$	18.1101	1.40561
$167$	−12.9603	−1.00290	−0.501451	−	0.865186i	$-0.667200\pi$
$167$	−12.9603	−1.00290	−0.501451	+	0.865186i	$0.667200\pi$
$168$	1.41369	0.109069
$169$	−10.5920	−0.814766
$170$	−1.94177	−0.148927
$171$	8.14460	0.622833
$172$	1.81408	0.138322
$173$	10.4278	0.792814	0.396407	−	0.918075i	$-0.370257\pi$
$173$	10.4278	0.792814	0.396407	+	0.918075i	$0.370257\pi$
$174$	4.50369	0.341424
$175$	0.620888	0.0469347
$176$	−20.2951	−1.52980
$177$	1.69621	0.127495
$178$	25.6396	1.92177
$179$	8.27835	0.618753	0.309376	−	0.950940i	$-0.399880\pi$
$179$	8.27835	0.618753	0.309376	+	0.950940i	$0.399880\pi$
$180$	0.583405	0.0434844
$181$	3.56722	0.265149	0.132575	−	0.991173i	$-0.457676\pi$
$181$	3.56722	0.265149	0.132575	+	0.991173i	$0.457676\pi$
$182$	1.54861	0.114790
$183$	11.8730	0.877679
$184$	5.91713	0.436217
$185$	−2.48108	−0.182412
$186$	−2.27858	−0.167073
$187$	−5.08002	−0.371488
$188$	1.04082	0.0759094
$189$	−0.620888	−0.0451629
$190$	13.0908	0.949706
$191$	1.75717	0.127144	0.0635722	−	0.997977i	$-0.479751\pi$
$191$	1.75717	0.127144	0.0635722	+	0.997977i	$0.479751\pi$
$192$	−4.50350	−0.325012
$193$	−4.03386	−0.290364	−0.145182	−	0.989405i	$-0.546377\pi$
$193$	−4.03386	−0.290364	−0.145182	+	0.989405i	$0.546377\pi$
$194$	0.431315	0.0309666
$195$	−1.55179	−0.111126
$196$	−3.85893	−0.275638
$197$	8.69528	0.619513	0.309757	−	0.950816i	$-0.399752\pi$
$197$	8.69528	0.619513	0.309757	+	0.950816i	$0.399752\pi$
$198$	6.75866	0.480317
$199$	26.4154	1.87254	0.936270	−	0.351280i	$-0.114253\pi$
$199$	26.4154	1.87254	0.936270	+	0.351280i	$0.114253\pi$
$200$	−2.27689	−0.161000
$201$	−12.7938	−0.902405
$202$	6.90689	0.485967
$203$	−1.73975	−0.122106
$204$	0.704809	0.0493465
$205$	4.20947	0.294002
$206$	−5.59975	−0.390153
$207$	−2.59878	−0.180628
$208$	−7.48962	−0.519311
$209$	34.2479	2.36898
$210$	−0.997951	−0.0688652
$211$	2.41258	0.166089	0.0830446	−	0.996546i	$-0.473536\pi$
$211$	2.41258	0.166089	0.0830446	+	0.996546i	$0.473536\pi$
$212$	4.49412	0.308657
$213$	4.30104	0.294703
$214$	0.308797	0.0211089
$215$	3.10947	0.212064
$216$	2.27689	0.154923
$217$	0.880199	0.0597518
$218$	17.8967	1.21211
$219$	14.5395	0.982487
$220$	2.45321	0.165395
$221$	−1.87471	−0.126106
$222$	3.98783	0.267645
$223$	−0.674487	−0.0451670	−0.0225835	−	0.999745i	$-0.507189\pi$
$223$	−0.674487	−0.0451670	−0.0225835	+	0.999745i	$0.507189\pi$
$224$	−1.98917	−0.132907
$225$	1.00000	0.0666667
$226$	−2.75130	−0.183014
$227$	−6.05817	−0.402095	−0.201047	−	0.979582i	$-0.564435\pi$
$227$	−6.05817	−0.402095	−0.201047	+	0.979582i	$0.564435\pi$
$228$	−4.75160	−0.314682
$229$	20.8923	1.38061	0.690303	−	0.723521i	$-0.257477\pi$
$229$	20.8923	1.38061	0.690303	+	0.723521i	$0.257477\pi$
$230$	−4.17701	−0.275424
$231$	−2.61082	−0.171779
$232$	6.37991	0.418862
$233$	17.0036	1.11394	0.556970	−	0.830533i	$-0.311964\pi$
$233$	17.0036	1.11394	0.556970	+	0.830533i	$0.311964\pi$
$234$	2.49418	0.163050
$235$	1.78404	0.116378
$236$	−0.989574	−0.0644158
$237$	−8.16958	−0.530671
$238$	−1.20562	−0.0781488
$239$	−24.6067	−1.59167	−0.795837	−	0.605511i	$-0.792969\pi$
$239$	−24.6067	−1.59167	−0.795837	+	0.605511i	$0.792969\pi$
$240$	4.82645	0.311546
$241$	−8.91923	−0.574538	−0.287269	−	0.957850i	$-0.592747\pi$
$241$	−8.91923	−0.574538	−0.287269	+	0.957850i	$0.592747\pi$
$242$	10.7398	0.690379
$243$	−1.00000	−0.0641500
$244$	−6.92677	−0.443441
$245$	−6.61450	−0.422585
$246$	−6.76587	−0.431376
$247$	12.6387	0.804180
$248$	−3.22782	−0.204967
$249$	−11.2674	−0.714042
$250$	1.60730	0.101654
$251$	−14.6410	−0.924134	−0.462067	−	0.886845i	$-0.652892\pi$
$251$	−14.6410	−0.924134	−0.462067	+	0.886845i	$0.652892\pi$
$252$	0.362229	0.0228183
$253$	−10.9278	−0.687026
$254$	−27.6212	−1.73311
$255$	1.20810	0.0756539
$256$	12.9261	0.807884
$257$	−19.4622	−1.21402	−0.607010	−	0.794694i	$-0.707631\pi$
$257$	−19.4622	−1.21402	−0.607010	+	0.794694i	$0.707631\pi$
$258$	−4.99785	−0.311152
$259$	−1.54047	−0.0957201
$260$	0.905319	0.0561455
$261$	−2.80203	−0.173441
$262$	4.17486	0.257924
$263$	−12.7580	−0.786693	−0.393347	−	0.919390i	$-0.628683\pi$
$263$	−12.7580	−0.786693	−0.393347	+	0.919390i	$0.628683\pi$
$264$	9.57428	0.589256
$265$	7.70325	0.473207
$266$	8.12791	0.498354
$267$	−15.9520	−0.976245
$268$	7.46396	0.455934
$269$	−0.266211	−0.0162312	−0.00811558	−	0.999967i	$-0.502583\pi$
$269$	−0.266211	−0.0162312	−0.00811558	+	0.999967i	$0.502583\pi$
$270$	−1.60730	−0.0978170
$271$	−12.6939	−0.771100	−0.385550	−	0.922687i	$-0.625988\pi$
$271$	−12.6939	−0.771100	−0.385550	+	0.922687i	$0.625988\pi$
$272$	5.83081	0.353545
$273$	−0.963485	−0.0583128
$274$	−5.12190	−0.309425
$275$	4.20498	0.253570
$276$	1.51614	0.0912609
$277$	6.73629	0.404744	0.202372	−	0.979309i	$-0.435135\pi$
$277$	6.73629	0.404744	0.202372	+	0.979309i	$0.435135\pi$
$278$	26.2805	1.57620
$279$	1.41765	0.0848722
$280$	−1.41369	−0.0844843
$281$	1.73384	0.103432	0.0517160	−	0.998662i	$-0.483531\pi$
$281$	1.73384	0.103432	0.0517160	+	0.998662i	$0.483531\pi$
$282$	−2.86748	−0.170756
$283$	21.6116	1.28468	0.642338	−	0.766421i	$-0.277964\pi$
$283$	21.6116	1.28468	0.642338	+	0.766421i	$0.277964\pi$
$284$	−2.50925	−0.148897
$285$	−8.14460	−0.482444
$286$	10.4880	0.620168
$287$	2.61361	0.154276
$288$	−3.20376	−0.188783
$289$	−15.5405	−0.914147
$290$	−4.50369	−0.264466
$291$	−0.268348	−0.0157308
$292$	−8.48240	−0.496395
$293$	−3.02233	−0.176566	−0.0882831	−	0.996095i	$-0.528138\pi$
$293$	−3.02233	−0.176566	−0.0882831	+	0.996095i	$0.528138\pi$
$294$	10.6315	0.620040
$295$	−1.69621	−0.0987569
$296$	5.64914	0.328349
$297$	−4.20498	−0.243998
$298$	30.6779	1.77712
$299$	−4.03275	−0.233220
$300$	−0.583405	−0.0336829
$301$	1.93063	0.111280
$302$	30.0568	1.72958
$303$	−4.29721	−0.246868
$304$	−39.3095	−2.25455
$305$	−11.8730	−0.679847
$306$	−1.94177	−0.111004
$307$	−6.78296	−0.387124	−0.193562	−	0.981088i	$-0.562004\pi$
$307$	−6.78296	−0.387124	−0.193562	+	0.981088i	$0.562004\pi$
$308$	1.52317	0.0867904
$309$	3.48395	0.198195
$310$	2.27858	0.129415
$311$	18.0514	1.02360	0.511799	−	0.859105i	$-0.328979\pi$
$311$	18.0514	1.02360	0.511799	+	0.859105i	$0.328979\pi$
$312$	3.53325	0.200031
$313$	14.8049	0.836820	0.418410	−	0.908258i	$-0.362588\pi$
$313$	14.8049	0.836820	0.418410	+	0.908258i	$0.362588\pi$
$314$	1.04597	0.0590274
$315$	0.620888	0.0349831
$316$	4.76617	0.268118
$317$	25.4608	1.43002	0.715009	−	0.699115i	$-0.246423\pi$
$317$	25.4608	1.43002	0.715009	+	0.699115i	$0.246423\pi$
$318$	−12.3814	−0.694316
$319$	−11.7825	−0.659692
$320$	4.50350	0.251754
$321$	−0.192122	−0.0107232
$322$	−2.59345	−0.144528
$323$	−9.83945	−0.547482
$324$	0.583405	0.0324114
$325$	1.55179	0.0860776
$326$	9.12930	0.505625
$327$	−11.1346	−0.615746
$328$	−9.58449	−0.529215
$329$	1.10769	0.0610689
$330$	−6.75866	−0.372052
$331$	−24.9656	−1.37223	−0.686117	−	0.727491i	$-0.740686\pi$
$331$	−24.9656	−1.37223	−0.686117	+	0.727491i	$0.740686\pi$
$332$	6.57345	0.360765
$333$	−2.48108	−0.135962
$334$	−20.8311	−1.13983
$335$	12.7938	0.699000
$336$	2.99668	0.163482
$337$	21.8011	1.18758	0.593791	−	0.804620i	$-0.297631\pi$
$337$	21.8011	1.18758	0.593791	+	0.804620i	$0.297631\pi$
$338$	−17.0244	−0.926007
$339$	1.71176	0.0929700
$340$	−0.704809	−0.0382236
$341$	5.96118	0.322816
$342$	13.0908	0.707869
$343$	−8.45307	−0.456423
$344$	−7.07993	−0.381724
$345$	2.59878	0.139914
$346$	16.7606	0.901057
$347$	29.4275	1.57975	0.789877	−	0.613266i	$-0.210144\pi$
$347$	29.4275	1.57975	0.789877	+	0.613266i	$0.210144\pi$
$348$	1.63472	0.0876300
$349$	−11.7793	−0.630531	−0.315266	−	0.949004i	$-0.602094\pi$
$349$	−11.7793	−0.630531	−0.315266	+	0.949004i	$0.602094\pi$
$350$	0.997951	0.0533427
$351$	−1.55179	−0.0828282
$352$	−13.4717	−0.718047
$353$	6.59892	0.351225	0.175613	−	0.984459i	$-0.443809\pi$
$353$	6.59892	0.351225	0.175613	+	0.984459i	$0.443809\pi$
$354$	2.72631	0.144902
$355$	−4.30104	−0.228276
$356$	9.30646	0.493241
$357$	0.750092	0.0396991
$358$	13.3058	0.703231
$359$	9.77614	0.515965	0.257982	−	0.966150i	$-0.416942\pi$
$359$	9.77614	0.515965	0.257982	+	0.966150i	$0.416942\pi$
$360$	−2.27689	−0.120003
$361$	47.3345	2.49129
$362$	5.73358	0.301350
$363$	−6.68188	−0.350708
$364$	0.562102	0.0294621
$365$	−14.5395	−0.761031
$366$	19.0835	0.997509
$367$	−17.6852	−0.923160	−0.461580	−	0.887098i	$-0.652717\pi$
$367$	−17.6852	−0.923160	−0.461580	+	0.887098i	$0.652717\pi$
$368$	12.5429	0.653842
$369$	4.20947	0.219136
$370$	−3.98783	−0.207317
$371$	4.78286	0.248314
$372$	−0.827061	−0.0428811
$373$	−18.5031	−0.958055	−0.479027	−	0.877800i	$-0.659011\pi$
$373$	−18.5031	−0.958055	−0.479027	+	0.877800i	$0.659011\pi$
$374$	−8.16511	−0.422208
$375$	−1.00000	−0.0516398
$376$	−4.06206	−0.209485
$377$	−4.34815	−0.223941
$378$	−0.997951	−0.0513291
$379$	11.3176	0.581346	0.290673	−	0.956822i	$-0.406121\pi$
$379$	11.3176	0.581346	0.290673	+	0.956822i	$0.406121\pi$
$380$	4.75160	0.243752
$381$	17.1849	0.880407
$382$	2.82429	0.144503
$383$	−19.2706	−0.984682	−0.492341	−	0.870402i	$-0.663859\pi$
$383$	−19.2706	−0.984682	−0.492341	+	0.870402i	$0.663859\pi$
$384$	−13.6460	−0.696369
$385$	2.61082	0.133060
$386$	−6.48361	−0.330007
$387$	3.10947	0.158063
$388$	0.156556	0.00794790
$389$	32.5877	1.65226	0.826131	−	0.563478i	$-0.190537\pi$
$389$	32.5877	1.65226	0.826131	+	0.563478i	$0.190537\pi$
$390$	−2.49418	−0.126298
$391$	3.13957	0.158775
$392$	15.0605	0.760669
$393$	−2.59744	−0.131024
$394$	13.9759	0.704096
$395$	8.16958	0.411056
$396$	2.45321	0.123278
$397$	−36.2757	−1.82063	−0.910313	−	0.413921i	$-0.864159\pi$
$397$	−36.2757	−1.82063	−0.910313	+	0.413921i	$0.864159\pi$
$398$	42.4575	2.12820
$399$	−5.05688	−0.253161
$400$	−4.82645	−0.241322
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−20.5634	−1.02561
$403$	2.19988	0.109584
$404$	2.50701	0.124728
$405$	1.00000	0.0496904
$406$	−2.79629	−0.138777
$407$	−10.4329	−0.517139
$408$	−2.75070	−0.136180
$409$	0.170165	0.00841413	0.00420706	−	0.999991i	$-0.498661\pi$
$409$	0.170165	0.00841413	0.00420706	+	0.999991i	$0.498661\pi$
$410$	6.76587	0.334142
$411$	3.18666	0.157186
$412$	−2.03256	−0.100137
$413$	−1.05315	−0.0518223
$414$	−4.17701	−0.205289
$415$	11.2674	0.553095
$416$	−4.97155	−0.243750
$417$	−16.3508	−0.800700
$418$	55.0466	2.69241
$419$	9.95733	0.486447	0.243224	−	0.969970i	$-0.421795\pi$
$419$	9.95733	0.486447	0.243224	+	0.969970i	$0.421795\pi$
$420$	−0.362229	−0.0176750
$421$	−11.1731	−0.544542	−0.272271	−	0.962221i	$-0.587775\pi$
$421$	−11.1731	−0.544542	−0.272271	+	0.962221i	$0.587775\pi$
$422$	3.87774	0.188765
$423$	1.78404	0.0867430
$424$	−17.5395	−0.851792
$425$	−1.20810	−0.0586013
$426$	6.91305	0.334939
$427$	−7.37181	−0.356747
$428$	0.112085	0.00541783
$429$	−6.52523	−0.315041
$430$	4.99785	0.241018
$431$	−30.9733	−1.49193	−0.745966	−	0.665984i	$-0.768012\pi$
$431$	−30.9733	−1.49193	−0.745966	+	0.665984i	$0.768012\pi$
$432$	4.82645	0.232213
$433$	1.88573	0.0906225	0.0453112	−	0.998973i	$-0.485572\pi$
$433$	1.88573	0.0906225	0.0453112	+	0.998973i	$0.485572\pi$
$434$	1.41474	0.0679098
$435$	2.80203	0.134347
$436$	6.49599	0.311102
$437$	−21.1660	−1.01251
$438$	23.3693	1.11663
$439$	−30.8234	−1.47112	−0.735560	−	0.677459i	$-0.763081\pi$
$439$	−30.8234	−1.47112	−0.735560	+	0.677459i	$0.763081\pi$
$440$	−9.57428	−0.456436
$441$	−6.61450	−0.314976
$442$	−3.01321	−0.143324
$443$	2.53615	0.120496	0.0602481	−	0.998183i	$-0.480811\pi$
$443$	2.53615	0.120496	0.0602481	+	0.998183i	$0.480811\pi$
$444$	1.44747	0.0686939
$445$	15.9520	0.756196
$446$	−1.08410	−0.0513337
$447$	−19.0866	−0.902768
$448$	2.79617	0.132107
$449$	13.1374	0.619993	0.309997	−	0.950738i	$-0.399672\pi$
$449$	13.1374	0.619993	0.309997	+	0.950738i	$0.399672\pi$
$450$	1.60730	0.0757687
$451$	17.7007	0.833495
$452$	−0.998648	−0.0469725
$453$	−18.7002	−0.878614
$454$	−9.73728	−0.456993
$455$	0.963485	0.0451689
$456$	18.5444	0.868419
$457$	14.6520	0.685390	0.342695	−	0.939447i	$-0.388660\pi$
$457$	14.6520	0.685390	0.342695	+	0.939447i	$0.388660\pi$
$458$	33.5802	1.56910
$459$	1.20810	0.0563891
$460$	−1.51614	−0.0706904
$461$	16.1003	0.749867	0.374934	−	0.927052i	$-0.377665\pi$
$461$	16.1003	0.749867	0.374934	+	0.927052i	$0.377665\pi$
$462$	−4.19637	−0.195233
$463$	−22.7370	−1.05668	−0.528340	−	0.849033i	$-0.677185\pi$
$463$	−22.7370	−1.05668	−0.528340	+	0.849033i	$0.677185\pi$
$464$	13.5238	0.627829
$465$	−1.41765	−0.0657418
$466$	27.3298	1.26603
$467$	−22.5452	−1.04327	−0.521633	−	0.853170i	$-0.674677\pi$
$467$	−22.5452	−1.04327	−0.521633	+	0.853170i	$0.674677\pi$
$468$	0.905319	0.0418484
$469$	7.94351	0.366797
$470$	2.86748	0.132267
$471$	−0.650762	−0.0299855
$472$	3.86207	0.177766
$473$	13.0753	0.601202
$474$	−13.1309	−0.603124
$475$	8.14460	0.373700
$476$	−0.437607	−0.0200577
$477$	7.70325	0.352708
$478$	−39.5502	−1.80899
$479$	8.80375	0.402254	0.201127	−	0.979565i	$-0.435540\pi$
$479$	8.80375	0.402254	0.201127	+	0.979565i	$0.435540\pi$
$480$	3.20376	0.146231
$481$	−3.85010	−0.175549
$482$	−14.3359	−0.652981
$483$	1.61355	0.0734191
$484$	3.89824	0.177193
$485$	0.268348	0.0121851
$486$	−1.60730	−0.0729085
$487$	−23.0511	−1.04455	−0.522273	−	0.852778i	$-0.674916\pi$
$487$	−23.0511	−1.04455	−0.522273	+	0.852778i	$0.674916\pi$
$488$	27.0335	1.22375
$489$	−5.67991	−0.256854
$490$	−10.6315	−0.480281
$491$	−5.22175	−0.235654	−0.117827	−	0.993034i	$-0.537593\pi$
$491$	−5.22175	−0.235654	−0.117827	+	0.993034i	$0.537593\pi$
$492$	−2.45582	−0.110717
$493$	3.38512	0.152458
$494$	20.3141	0.913975
$495$	4.20498	0.189000
$496$	−6.84219	−0.307224
$497$	−2.67046	−0.119787
$498$	−18.1101	−0.811531
$499$	22.2117	0.994331	0.497166	−	0.867656i	$-0.334374\pi$
$499$	22.2117	0.994331	0.497166	+	0.867656i	$0.334374\pi$
$500$	0.583405	0.0260907
$501$	12.9603	0.579025
$502$	−23.5325	−1.05031
$503$	13.3072	0.593340	0.296670	−	0.954980i	$-0.404124\pi$
$503$	13.3072	0.593340	0.296670	+	0.954980i	$0.404124\pi$
$504$	−1.41369	−0.0629709
$505$	4.29721	0.191223
$506$	−17.5643	−0.780826
$507$	10.5920	0.470405
$508$	−10.0257	−0.444820
$509$	−20.6577	−0.915635	−0.457818	−	0.889046i	$-0.651369\pi$
$509$	−20.6577	−0.915635	−0.457818	+	0.889046i	$0.651369\pi$
$510$	1.94177	0.0859830
$511$	−9.02739	−0.399348
$512$	−6.51581	−0.287961
$513$	−8.14460	−0.359593
$514$	−31.2816	−1.37977
$515$	−3.48395	−0.153521
$516$	−1.81408	−0.0798605
$517$	7.50186	0.329931
$518$	−2.47599	−0.108789
$519$	−10.4278	−0.457731
$520$	−3.53325	−0.154943
$521$	−12.4660	−0.546144	−0.273072	−	0.961994i	$-0.588040\pi$
$521$	−12.4660	−0.546144	−0.273072	+	0.961994i	$0.588040\pi$
$522$	−4.50369	−0.197121
$523$	−10.2114	−0.446514	−0.223257	−	0.974760i	$-0.571669\pi$
$523$	−10.2114	−0.446514	−0.223257	+	0.974760i	$0.571669\pi$
$524$	1.51536	0.0661988
$525$	−0.620888	−0.0270978
$526$	−20.5059	−0.894101
$527$	−1.71265	−0.0746043
$528$	20.2951	0.883232
$529$	−16.2463	−0.706363
$530$	12.3814	0.537815
$531$	−1.69621	−0.0736090
$532$	2.95021	0.127908
$533$	6.53219	0.282941
$534$	−25.6396	−1.10953
$535$	0.192122	0.00830616
$536$	−29.1301	−1.25823
$537$	−8.27835	−0.357237
$538$	−0.427880	−0.0184472
$539$	−27.8139	−1.19803
$540$	−0.583405	−0.0251057
$541$	12.6218	0.542653	0.271327	−	0.962487i	$-0.412538\pi$
$541$	12.6218	0.542653	0.271327	+	0.962487i	$0.412538\pi$
$542$	−20.4029	−0.876379
$543$	−3.56722	−0.153084
$544$	3.87045	0.165944
$545$	11.1346	0.476955
$546$	−1.54861	−0.0662742
$547$	−15.6796	−0.670409	−0.335205	−	0.942145i	$-0.608805\pi$
$547$	−15.6796	−0.670409	−0.335205	+	0.942145i	$0.608805\pi$
$548$	−1.85911	−0.0794172
$549$	−11.8730	−0.506728
$550$	6.75866	0.288190
$551$	−22.8214	−0.972224
$552$	−5.91713	−0.251850
$553$	5.07239	0.215700
$554$	10.8272	0.460005
$555$	2.48108	0.105316
$556$	9.53911	0.404548
$557$	−21.4088	−0.907119	−0.453560	−	0.891226i	$-0.649846\pi$
$557$	−21.4088	−0.907119	−0.453560	+	0.891226i	$0.649846\pi$
$558$	2.27858	0.0964599
$559$	4.82524	0.204086
$560$	−2.99668	−0.126633
$561$	5.08002	0.214479
$562$	2.78679	0.117554
$563$	−38.4659	−1.62115	−0.810573	−	0.585638i	$-0.800844\pi$
$563$	−38.4659	−1.62115	−0.810573	+	0.585638i	$0.800844\pi$
$564$	−1.04082	−0.0438263
$565$	−1.71176	−0.0720142
$566$	34.7363	1.46007
$567$	0.620888	0.0260748
$568$	9.79300	0.410905
$569$	−19.1936	−0.804636	−0.402318	−	0.915500i	$-0.631795\pi$
$569$	−19.1936	−0.804636	−0.402318	+	0.915500i	$0.631795\pi$
$570$	−13.0908	−0.548313
$571$	−39.0156	−1.63275	−0.816376	−	0.577521i	$-0.804020\pi$
$571$	−39.0156	−1.63275	−0.816376	+	0.577521i	$0.804020\pi$
$572$	3.80685	0.159173
$573$	−1.75717	−0.0734068
$574$	4.20084	0.175340
$575$	−2.59878	−0.108377
$576$	4.50350	0.187646
$577$	10.9485	0.455793	0.227896	−	0.973685i	$-0.426815\pi$
$577$	10.9485	0.455793	0.227896	+	0.973685i	$0.426815\pi$
$578$	−24.9782	−1.03896
$579$	4.03386	0.167642
$580$	−1.63472	−0.0678779
$581$	6.99579	0.290234
$582$	−0.431315	−0.0178786
$583$	32.3921	1.34154
$584$	33.1048	1.36989
$585$	1.55179	0.0641585
$586$	−4.85778	−0.200673
$587$	−29.7606	−1.22835	−0.614176	−	0.789169i	$-0.710512\pi$
$587$	−29.7606	−1.22835	−0.614176	+	0.789169i	$0.710512\pi$
$588$	3.85893	0.159140
$589$	11.5462	0.475751
$590$	−2.72631	−0.112240
$591$	−8.69528	−0.357676
$592$	11.9748	0.492161
$593$	16.4783	0.676681	0.338341	−	0.941024i	$-0.390134\pi$
$593$	16.4783	0.676681	0.338341	+	0.941024i	$0.390134\pi$
$594$	−6.75866	−0.277311
$595$	−0.750092	−0.0307508
$596$	11.1352	0.456117
$597$	−26.4154	−1.08111
$598$	−6.48183	−0.265062
$599$	−18.3094	−0.748101	−0.374051	−	0.927408i	$-0.622031\pi$
$599$	−18.3094	−0.748101	−0.374051	+	0.927408i	$0.622031\pi$
$600$	2.27689	0.0929536
$601$	−42.5703	−1.73648	−0.868240	−	0.496145i	$-0.834748\pi$
$601$	−42.5703	−1.73648	−0.868240	+	0.496145i	$0.834748\pi$
$602$	3.10310	0.126473
$603$	12.7938	0.521004
$604$	10.9098	0.443914
$605$	6.68188	0.271657
$606$	−6.90689	−0.280573
$607$	−0.218876	−0.00888390	−0.00444195	−	0.999990i	$-0.501414\pi$
$607$	−0.218876	−0.00888390	−0.00444195	+	0.999990i	$0.501414\pi$
$608$	−26.0933	−1.05822
$609$	1.73975	0.0704980
$610$	−19.0835	−0.772667
$611$	2.76845	0.111999
$612$	−0.704809	−0.0284902
$613$	13.4580	0.543564	0.271782	−	0.962359i	$-0.412387\pi$
$613$	13.4580	0.543564	0.271782	+	0.962359i	$0.412387\pi$
$614$	−10.9022	−0.439978
$615$	−4.20947	−0.169742
$616$	−5.94455	−0.239513
$617$	−27.4315	−1.10435	−0.552175	−	0.833728i	$-0.686202\pi$
$617$	−27.4315	−1.10435	−0.552175	+	0.833728i	$0.686202\pi$
$618$	5.59975	0.225255
$619$	39.8623	1.60220	0.801101	−	0.598529i	$-0.204248\pi$
$619$	39.8623	1.60220	0.801101	+	0.598529i	$0.204248\pi$
$620$	0.827061	0.0332156
$621$	2.59878	0.104285
$622$	29.0139	1.16335
$623$	9.90439	0.396811
$624$	7.48962	0.299825
$625$	1.00000	0.0400000
$626$	23.7958	0.951071
$627$	−34.2479	−1.36773
$628$	0.379658	0.0151500
$629$	2.99738	0.119513
$630$	0.997951	0.0397593
$631$	−24.5389	−0.976878	−0.488439	−	0.872598i	$-0.662434\pi$
$631$	−24.5389	−0.976878	−0.488439	+	0.872598i	$0.662434\pi$
$632$	−18.6012	−0.739917
$633$	−2.41258	−0.0958916
$634$	40.9230	1.62526
$635$	−17.1849	−0.681960
$636$	−4.49412	−0.178203
$637$	−10.2643	−0.406686
$638$	−18.9380	−0.749761
$639$	−4.30104	−0.170147
$640$	13.6460	0.539405
$641$	−11.9738	−0.472938	−0.236469	−	0.971639i	$-0.575990\pi$
$641$	−11.9738	−0.472938	−0.236469	+	0.971639i	$0.575990\pi$
$642$	−0.308797	−0.0121873
$643$	33.4931	1.32084	0.660419	−	0.750897i	$-0.270379\pi$
$643$	33.4931	1.32084	0.660419	+	0.750897i	$0.270379\pi$
$644$	−0.941353	−0.0370945
$645$	−3.10947	−0.122435
$646$	−15.8149	−0.622230
$647$	−2.28029	−0.0896473	−0.0448237	−	0.998995i	$-0.514273\pi$
$647$	−2.28029	−0.0896473	−0.0448237	+	0.998995i	$0.514273\pi$
$648$	−2.27689	−0.0894447
$649$	−7.13251	−0.279976
$650$	2.49418	0.0978299
$651$	−0.880199	−0.0344977
$652$	3.31368	0.129774
$653$	−37.5730	−1.47034	−0.735172	−	0.677880i	$-0.762899\pi$
$653$	−37.5730	−1.47034	−0.735172	+	0.677880i	$0.762899\pi$
$654$	−17.8967	−0.699815
$655$	2.59744	0.101491
$656$	−20.3168	−0.793237
$657$	−14.5395	−0.567239
$658$	1.78039	0.0694066
$659$	30.0821	1.17183	0.585917	−	0.810371i	$-0.300735\pi$
$659$	30.0821	1.17183	0.585917	+	0.810371i	$0.300735\pi$
$660$	−2.45321	−0.0954910
$661$	30.7201	1.19487	0.597437	−	0.801916i	$-0.296186\pi$
$661$	30.7201	1.19487	0.597437	+	0.801916i	$0.296186\pi$
$662$	−40.1272	−1.55959
$663$	1.87471	0.0728076
$664$	−25.6546	−0.995593
$665$	5.05688	0.196097
$666$	−3.98783	−0.154525
$667$	7.28185	0.281954
$668$	−7.56113	−0.292549
$669$	0.674487	0.0260772
$670$	20.5634	0.794435
$671$	−49.9258	−1.92737
$672$	1.98917	0.0767340
$673$	40.1545	1.54784	0.773921	−	0.633282i	$-0.218293\pi$
$673$	40.1545	1.54784	0.773921	+	0.633282i	$0.218293\pi$
$674$	35.0408	1.34972
$675$	−1.00000	−0.0384900
$676$	−6.17940	−0.237669
$677$	−13.3253	−0.512131	−0.256066	−	0.966659i	$-0.582426\pi$
$677$	−13.3253	−0.512131	−0.256066	+	0.966659i	$0.582426\pi$
$678$	2.75130	0.105663
$679$	0.166614	0.00639406
$680$	2.75070	0.105485
$681$	6.05817	0.232149
$682$	9.58138	0.366890
$683$	24.1688	0.924793	0.462396	−	0.886673i	$-0.346990\pi$
$683$	24.1688	0.924793	0.462396	+	0.886673i	$0.346990\pi$
$684$	4.75160	0.181682
$685$	−3.18666	−0.121756
$686$	−13.5866	−0.518739
$687$	−20.8923	−0.797093
$688$	−15.0077	−0.572164
$689$	11.9538	0.455404
$690$	4.17701	0.159016
$691$	7.46567	0.284008	0.142004	−	0.989866i	$-0.454645\pi$
$691$	7.46567	0.284008	0.142004	+	0.989866i	$0.454645\pi$
$692$	6.08365	0.231266
$693$	2.61082	0.0991769
$694$	47.2988	1.79544
$695$	16.3508	0.620219
$696$	−6.37991	−0.241830
$697$	−5.08544	−0.192625
$698$	−18.9328	−0.716618
$699$	−17.0036	−0.643133
$700$	0.362229	0.0136910
$701$	−35.0520	−1.32389	−0.661947	−	0.749551i	$-0.730270\pi$
$701$	−35.0520	−1.32389	−0.661947	+	0.749551i	$0.730270\pi$
$702$	−2.49418	−0.0941368
$703$	−20.2074	−0.762135
$704$	18.9372	0.713721
$705$	−1.78404	−0.0671909
$706$	10.6064	0.399178
$707$	2.66808	0.100344
$708$	0.989574	0.0371905
$709$	−43.2819	−1.62548	−0.812742	−	0.582624i	$-0.802026\pi$
$709$	−43.2819	−1.62548	−0.812742	+	0.582624i	$0.802026\pi$
$710$	−6.91305	−0.259442
$711$	8.16958	0.306383
$712$	−36.3209	−1.36118
$713$	−3.68415	−0.137972
$714$	1.20562	0.0451192
$715$	6.52523	0.244030
$716$	4.82963	0.180492
$717$	24.6067	0.918953
$718$	15.7132	0.586410
$719$	16.3811	0.610910	0.305455	−	0.952206i	$-0.401191\pi$
$719$	16.3811	0.610910	0.305455	+	0.952206i	$0.401191\pi$
$720$	−4.82645	−0.179871
$721$	−2.16314	−0.0805597
$722$	76.0806	2.83143
$723$	8.91923	0.331710
$724$	2.08113	0.0773447
$725$	−2.80203	−0.104065
$726$	−10.7398	−0.398590
$727$	−18.6615	−0.692118	−0.346059	−	0.938213i	$-0.612480\pi$
$727$	−18.6615	−0.692118	−0.346059	+	0.938213i	$0.612480\pi$
$728$	−2.19375	−0.0813057
$729$	1.00000	0.0370370
$730$	−23.3693	−0.864936
$731$	−3.75654	−0.138941
$732$	6.92677	0.256021
$733$	−28.3250	−1.04621	−0.523103	−	0.852269i	$-0.675226\pi$
$733$	−28.3250	−1.04621	−0.523103	+	0.852269i	$0.675226\pi$
$734$	−28.4254	−1.04920
$735$	6.61450	0.243979
$736$	8.32586	0.306895
$737$	53.7977	1.98166
$738$	6.76587	0.249055
$739$	−13.1309	−0.483028	−0.241514	−	0.970397i	$-0.577644\pi$
$739$	−13.1309	−0.483028	−0.241514	+	0.970397i	$0.577644\pi$
$740$	−1.44747	−0.0532101
$741$	−12.6387	−0.464293
$742$	7.68747	0.282216
$743$	51.1119	1.87512	0.937558	−	0.347830i	$-0.113081\pi$
$743$	51.1119	1.87512	0.937558	+	0.347830i	$0.113081\pi$
$744$	3.22782	0.118338
$745$	19.0866	0.699281
$746$	−29.7400	−1.08886
$747$	11.2674	0.412253
$748$	−2.96371	−0.108364
$749$	0.119286	0.00435862
$750$	−1.60730	−0.0586902
$751$	36.9167	1.34711	0.673554	−	0.739138i	$-0.264767\pi$
$751$	36.9167	1.34711	0.673554	+	0.739138i	$0.264767\pi$
$752$	−8.61058	−0.313996
$753$	14.6410	0.533549
$754$	−6.98877	−0.254516
$755$	18.7002	0.680572
$756$	−0.362229	−0.0131741
$757$	−0.598956	−0.0217694	−0.0108847	−	0.999941i	$-0.503465\pi$
$757$	−0.598956	−0.0217694	−0.0108847	+	0.999941i	$0.503465\pi$
$758$	18.1907	0.660718
$759$	10.9278	0.396655
$760$	−18.5444	−0.672674
$761$	34.6134	1.25474	0.627368	−	0.778723i	$-0.284132\pi$
$761$	34.6134	1.25474	0.627368	+	0.778723i	$0.284132\pi$
$762$	27.6212	1.00061
$763$	6.91335	0.250280
$764$	1.02514	0.0370883
$765$	−1.20810	−0.0436788
$766$	−30.9736	−1.11912
$767$	−2.63215	−0.0950413
$768$	−12.9261	−0.466432
$769$	−1.66458	−0.0600265	−0.0300132	−	0.999550i	$-0.509555\pi$
$769$	−1.66458	−0.0600265	−0.0300132	+	0.999550i	$0.509555\pi$
$770$	4.19637	0.151227
$771$	19.4622	0.700915
$772$	−2.35337	−0.0846998
$773$	−8.78378	−0.315931	−0.157965	−	0.987445i	$-0.550493\pi$
$773$	−8.78378	−0.315931	−0.157965	+	0.987445i	$0.550493\pi$
$774$	4.99785	0.179644
$775$	1.41765	0.0509233
$776$	−0.610999	−0.0219336
$777$	1.54047	0.0552640
$778$	52.3781	1.87785
$779$	34.2844	1.22837
$780$	−0.905319	−0.0324156
$781$	−18.0858	−0.647161
$782$	5.04623	0.180453
$783$	2.80203	0.100136
$784$	31.9245	1.14016
$785$	0.650762	0.0232267
$786$	−4.17486	−0.148912
$787$	−17.5150	−0.624342	−0.312171	−	0.950026i	$-0.601056\pi$
$787$	−17.5150	−0.624342	−0.312171	+	0.950026i	$0.601056\pi$
$788$	5.07287	0.180713
$789$	12.7580	0.454198
$790$	13.1309	0.467178
$791$	−1.06281	−0.0377892
$792$	−9.57428	−0.340207
$793$	−18.4244	−0.654269
$794$	−58.3058	−2.06920
$795$	−7.70325	−0.273206
$796$	15.4109	0.546225
$797$	18.0365	0.638884	0.319442	−	0.947606i	$-0.396505\pi$
$797$	18.0365	0.638884	0.319442	+	0.947606i	$0.396505\pi$
$798$	−8.12791	−0.287725
$799$	−2.15529	−0.0762487
$800$	−3.20376	−0.113270
$801$	15.9520	0.563636
$802$	−1.60730	−0.0567556
$803$	−61.1383	−2.15752
$804$	−7.46396	−0.263234
$805$	−1.61355	−0.0568702
$806$	3.53587	0.124546
$807$	0.266211	0.00937106
$808$	−9.78427	−0.344209
$809$	−8.96292	−0.315119	−0.157560	−	0.987509i	$-0.550363\pi$
$809$	−8.96292	−0.315119	−0.157560	+	0.987509i	$0.550363\pi$
$810$	1.60730	0.0564747
$811$	2.14276	0.0752424	0.0376212	−	0.999292i	$-0.488022\pi$
$811$	2.14276	0.0752424	0.0376212	+	0.999292i	$0.488022\pi$
$812$	−1.01498	−0.0356187
$813$	12.6939	0.445195
$814$	−16.7687	−0.587744
$815$	5.67991	0.198958
$816$	−5.83081	−0.204119
$817$	25.3254	0.886024
$818$	0.273506	0.00956291
$819$	0.963485	0.0336669
$820$	2.45582	0.0857611
$821$	15.6258	0.545344	0.272672	−	0.962107i	$-0.412093\pi$
$821$	15.6258	0.545344	0.272672	+	0.962107i	$0.412093\pi$
$822$	5.12190	0.178647
$823$	−4.70841	−0.164125	−0.0820624	−	0.996627i	$-0.526151\pi$
$823$	−4.70841	−0.164125	−0.0820624	+	0.996627i	$0.526151\pi$
$824$	7.93258	0.276345
$825$	−4.20498	−0.146399
$826$	−1.69273	−0.0588976
$827$	2.02730	0.0704961	0.0352480	−	0.999379i	$-0.488778\pi$
$827$	2.02730	0.0704961	0.0352480	+	0.999379i	$0.488778\pi$
$828$	−1.51614	−0.0526895
$829$	43.2184	1.50104	0.750518	−	0.660850i	$-0.229804\pi$
$829$	43.2184	1.50104	0.750518	+	0.660850i	$0.229804\pi$
$830$	18.1101	0.628609
$831$	−6.73629	−0.233679
$832$	6.98848	0.242282
$833$	7.99095	0.276870
$834$	−26.2805	−0.910020
$835$	−12.9603	−0.448511
$836$	19.9804	0.691036
$837$	−1.41765	−0.0490010
$838$	16.0044	0.552862
$839$	23.7912	0.821365	0.410682	−	0.911778i	$-0.365291\pi$
$839$	23.7912	0.821365	0.410682	+	0.911778i	$0.365291\pi$
$840$	1.41369	0.0487770
$841$	−21.1486	−0.729263
$842$	−17.9584	−0.618888
$843$	−1.73384	−0.0597165
$844$	1.40751	0.0484486
$845$	−10.5920	−0.364374
$846$	2.86748	0.0985861
$847$	4.14870	0.142551
$848$	−37.1794	−1.27674
$849$	−21.6116	−0.741709
$850$	−1.94177	−0.0666021
$851$	6.44777	0.221027
$852$	2.50925	0.0859654
$853$	35.3510	1.21040	0.605198	−	0.796075i	$-0.293094\pi$
$853$	35.3510	1.21040	0.605198	+	0.796075i	$0.293094\pi$
$854$	−11.8487	−0.405454
$855$	8.14460	0.278539
$856$	−0.437441	−0.0149514
$857$	4.97594	0.169975	0.0849874	−	0.996382i	$-0.472915\pi$
$857$	4.97594	0.169975	0.0849874	+	0.996382i	$0.472915\pi$
$858$	−10.4880	−0.358054
$859$	−37.1411	−1.26724	−0.633619	−	0.773645i	$-0.718431\pi$
$859$	−37.1411	−1.26724	−0.633619	+	0.773645i	$0.718431\pi$
$860$	1.81408	0.0618597
$861$	−2.61361	−0.0890715
$862$	−49.7833	−1.69563
$863$	35.7743	1.21777	0.608886	−	0.793258i	$-0.291617\pi$
$863$	35.7743	1.21777	0.608886	+	0.793258i	$0.291617\pi$
$864$	3.20376	0.108994
$865$	10.4278	0.354557
$866$	3.03093	0.102995
$867$	15.5405	0.527783
$868$	0.513512	0.0174297
$869$	34.3529	1.16534
$870$	4.50369	0.152690
$871$	19.8532	0.672701
$872$	−25.3523	−0.858538
$873$	0.268348	0.00908221
$874$	−34.0201	−1.15075
$875$	0.620888	0.0209898
$876$	8.48240	0.286594
$877$	−32.3902	−1.09374	−0.546869	−	0.837218i	$-0.684181\pi$
$877$	−32.3902	−1.09374	−0.546869	+	0.837218i	$0.684181\pi$
$878$	−49.5424	−1.67197
$879$	3.02233	0.101940
$880$	−20.2951	−0.684149
$881$	35.3864	1.19220	0.596099	−	0.802911i	$-0.296716\pi$
$881$	35.3864	1.19220	0.596099	+	0.802911i	$0.296716\pi$
$882$	−10.6315	−0.357980
$883$	29.2103	0.983006	0.491503	−	0.870876i	$-0.336448\pi$
$883$	29.2103	0.983006	0.491503	+	0.870876i	$0.336448\pi$
$884$	−1.09371	−0.0367855
$885$	1.69621	0.0570173
$886$	4.07635	0.136948
$887$	−31.3987	−1.05427	−0.527133	−	0.849783i	$-0.676733\pi$
$887$	−31.3987	−1.05427	−0.527133	+	0.849783i	$0.676733\pi$
$888$	−5.64914	−0.189573
$889$	−10.6699	−0.357856
$890$	25.6396	0.859441
$891$	4.20498	0.140872
$892$	−0.393499	−0.0131753
$893$	14.5303	0.486238
$894$	−30.6779	−1.02602
$895$	8.27835	0.276715
$896$	8.47263	0.283051
$897$	4.03275	0.134650
$898$	21.1157	0.704642
$899$	−3.97228	−0.132483
$900$	0.583405	0.0194468
$901$	−9.30627	−0.310037
$902$	28.4503	0.947293
$903$	−1.93063	−0.0642474
$904$	3.89749	0.129628
$905$	3.56722	0.118578
$906$	−30.0568	−0.998572
$907$	−43.3467	−1.43930	−0.719652	−	0.694335i	$-0.755699\pi$
$907$	−43.3467	−1.43930	−0.719652	+	0.694335i	$0.755699\pi$
$908$	−3.53436	−0.117292
$909$	4.29721	0.142529
$910$	1.54861	0.0513358
$911$	11.2389	0.372362	0.186181	−	0.982515i	$-0.440389\pi$
$911$	11.2389	0.372362	0.186181	+	0.982515i	$0.440389\pi$
$912$	39.3095	1.30167
$913$	47.3792	1.56802
$914$	23.5501	0.778967
$915$	11.8730	0.392510
$916$	12.1887	0.402726
$917$	1.61272	0.0532567
$918$	1.94177	0.0640879
$919$	−11.8059	−0.389440	−0.194720	−	0.980859i	$-0.562380\pi$
$919$	−11.8059	−0.389440	−0.194720	+	0.980859i	$0.562380\pi$
$920$	5.91713	0.195082
$921$	6.78296	0.223506
$922$	25.8780	0.852247
$923$	−6.67430	−0.219687
$924$	−1.52317	−0.0501085
$925$	−2.48108	−0.0815773
$926$	−36.5452	−1.20095
$927$	−3.48395	−0.114428
$928$	8.97702	0.294685
$929$	−40.4910	−1.32847	−0.664234	−	0.747525i	$-0.731242\pi$
$929$	−40.4910	−1.32847	−0.664234	+	0.747525i	$0.731242\pi$
$930$	−2.27858	−0.0747175
$931$	−53.8724	−1.76560
$932$	9.91995	0.324939
$933$	−18.0514	−0.590975
$934$	−36.2368	−1.18570
$935$	−5.08002	−0.166134
$936$	−3.53325	−0.115488
$937$	−51.8892	−1.69515	−0.847574	−	0.530678i	$-0.821937\pi$
$937$	−51.8892	−1.69515	−0.847574	+	0.530678i	$0.821937\pi$
$938$	12.7676	0.416876
$939$	−14.8049	−0.483138
$940$	1.04082	0.0339477
$941$	30.1276	0.982131	0.491065	−	0.871123i	$-0.336608\pi$
$941$	30.1276	0.982131	0.491065	+	0.871123i	$0.336608\pi$
$942$	−1.04597	−0.0340795
$943$	−10.9395	−0.356238
$944$	8.18665	0.266453
$945$	−0.620888	−0.0201975
$946$	21.0159	0.683285
$947$	2.26740	0.0736806	0.0368403	−	0.999321i	$-0.488271\pi$
$947$	2.26740	0.0736806	0.0368403	+	0.999321i	$0.488271\pi$
$948$	−4.76617	−0.154798
$949$	−22.5622	−0.732399
$950$	13.0908	0.424721
$951$	−25.4608	−0.825621
$952$	1.70788	0.0553526
$953$	−36.7215	−1.18953	−0.594763	−	0.803901i	$-0.702754\pi$
$953$	−36.7215	−1.18953	−0.594763	+	0.803901i	$0.702754\pi$
$954$	12.3814	0.400863
$955$	1.75717	0.0568607
$956$	−14.3556	−0.464295
$957$	11.7825	0.380874
$958$	14.1503	0.457174
$959$	−1.97856	−0.0638909
$960$	−4.50350	−0.145350
$961$	−28.9903	−0.935170
$962$	−6.18825	−0.199517
$963$	0.192122	0.00619105
$964$	−5.20352	−0.167594
$965$	−4.03386	−0.129855
$966$	2.59345	0.0834430
$967$	−19.2573	−0.619272	−0.309636	−	0.950855i	$-0.600207\pi$
$967$	−19.2573	−0.619272	−0.309636	+	0.950855i	$0.600207\pi$
$968$	−15.2139	−0.488994
$969$	9.83945	0.316089
$970$	0.431315	0.0138487
$971$	38.0950	1.22253	0.611264	−	0.791427i	$-0.290661\pi$
$971$	38.0950	1.22253	0.611264	+	0.791427i	$0.290661\pi$
$972$	−0.583405	−0.0187127
$973$	10.1520	0.325458
$974$	−37.0500	−1.18716
$975$	−1.55179	−0.0496969
$976$	57.3045	1.83427
$977$	−20.1867	−0.645831	−0.322915	−	0.946428i	$-0.604663\pi$
$977$	−20.1867	−0.645831	−0.322915	+	0.946428i	$0.604663\pi$
$978$	−9.12930	−0.291923
$979$	67.0778	2.14382
$980$	−3.85893	−0.123269
$981$	11.1346	0.355501
$982$	−8.39290	−0.267828
$983$	−45.3311	−1.44584	−0.722918	−	0.690934i	$-0.757200\pi$
$983$	−45.3311	−1.44584	−0.722918	+	0.690934i	$0.757200\pi$
$984$	9.58449	0.305542
$985$	8.69528	0.277055
$986$	5.44089	0.173273
$987$	−1.10769	−0.0352581
$988$	7.37346	0.234581
$989$	−8.08083	−0.256955
$990$	6.75866	0.214804
$991$	−33.2959	−1.05768	−0.528839	−	0.848722i	$-0.677373\pi$
$991$	−33.2959	−1.05768	−0.528839	+	0.848722i	$0.677373\pi$
$992$	−4.54179	−0.144202
$993$	24.9656	0.792260
$994$	−4.29223	−0.136141
$995$	26.4154	0.837426
$996$	−6.57345	−0.208288
$997$	36.8047	1.16562	0.582808	−	0.812610i	$-0.301954\pi$
$997$	36.8047	1.16562	0.582808	+	0.812610i	$0.301954\pi$
$998$	35.7008	1.13009
$999$	2.48108	0.0784978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.g.1.27	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.g.1.27	✓	36	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q+1.60730 q^{2} -1.00000 q^{3} +0.583405 q^{4} +1.00000 q^{5} -1.60730 q^{6} +0.620888 q^{7} -2.27689 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.60730 q^{2} -1.00000 q^{3} +0.583405 q^{4} +1.00000 q^{5} -1.60730 q^{6} +0.620888 q^{7} -2.27689 q^{8} +1.00000 q^{9} +1.60730 q^{10} +4.20498 q^{11} -0.583405 q^{12} +1.55179 q^{13} +0.997951 q^{14} -1.00000 q^{15} -4.82645 q^{16} -1.20810 q^{17} +1.60730 q^{18} +8.14460 q^{19} +0.583405 q^{20} -0.620888 q^{21} +6.75866 q^{22} -2.59878 q^{23} +2.27689 q^{24} +1.00000 q^{25} +2.49418 q^{26} -1.00000 q^{27} +0.362229 q^{28} -2.80203 q^{29} -1.60730 q^{30} +1.41765 q^{31} -3.20376 q^{32} -4.20498 q^{33} -1.94177 q^{34} +0.620888 q^{35} +0.583405 q^{36} -2.48108 q^{37} +13.0908 q^{38} -1.55179 q^{39} -2.27689 q^{40} +4.20947 q^{41} -0.997951 q^{42} +3.10947 q^{43} +2.45321 q^{44} +1.00000 q^{45} -4.17701 q^{46} +1.78404 q^{47} +4.82645 q^{48} -6.61450 q^{49} +1.60730 q^{50} +1.20810 q^{51} +0.905319 q^{52} +7.70325 q^{53} -1.60730 q^{54} +4.20498 q^{55} -1.41369 q^{56} -8.14460 q^{57} -4.50369 q^{58} -1.69621 q^{59} -0.583405 q^{60} -11.8730 q^{61} +2.27858 q^{62} +0.620888 q^{63} +4.50350 q^{64} +1.55179 q^{65} -6.75866 q^{66} +12.7938 q^{67} -0.704809 q^{68} +2.59878 q^{69} +0.997951 q^{70} -4.30104 q^{71} -2.27689 q^{72} -14.5395 q^{73} -3.98783 q^{74} -1.00000 q^{75} +4.75160 q^{76} +2.61082 q^{77} -2.49418 q^{78} +8.16958 q^{79} -4.82645 q^{80} +1.00000 q^{81} +6.76587 q^{82} +11.2674 q^{83} -0.362229 q^{84} -1.20810 q^{85} +4.99785 q^{86} +2.80203 q^{87} -9.57428 q^{88} +15.9520 q^{89} +1.60730 q^{90} +0.963485 q^{91} -1.51614 q^{92} -1.41765 q^{93} +2.86748 q^{94} +8.14460 q^{95} +3.20376 q^{96} +0.268348 q^{97} -10.6315 q^{98} +4.20498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9}+O(q^{10}) \) 36 * q - 36 * q^3 + 36 * q^4 + 36 * q^5 - 4 * q^7 - 3 * q^8 + 36 * q^9	\( 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9} + 15 q^{11} - 36 q^{12} + 8 q^{13} + 10 q^{14} - 36 q^{15} + 36 q^{16} + 32 q^{17} + 5 q^{19} + 36 q^{20} + 4 q^{21} + q^{22} - 10 q^{23} + 3 q^{24} + 36 q^{25} + 22 q^{26} - 36 q^{27} + 61 q^{29} + 15 q^{31} - q^{32} - 15 q^{33} + 26 q^{34} - 4 q^{35} + 36 q^{36} + 16 q^{37} + 20 q^{38} - 8 q^{39} - 3 q^{40} + 61 q^{41} - 10 q^{42} - 35 q^{43} + 39 q^{44} + 36 q^{45} + 11 q^{46} - 28 q^{47} - 36 q^{48} + 68 q^{49} - 32 q^{51} + 4 q^{52} + 33 q^{53} + 15 q^{55} + 23 q^{56} - 5 q^{57} + 6 q^{58} + 35 q^{59} - 36 q^{60} + 55 q^{61} + q^{62} - 4 q^{63} + 15 q^{64} + 8 q^{65} - q^{66} - 34 q^{67} + 60 q^{68} + 10 q^{69} + 10 q^{70} + 42 q^{71} - 3 q^{72} + 53 q^{73} + 54 q^{74} - 36 q^{75} + 20 q^{76} + 20 q^{77} - 22 q^{78} + 25 q^{79} + 36 q^{80} + 36 q^{81} - 15 q^{82} - 11 q^{83} + 32 q^{85} + 53 q^{86} - 61 q^{87} + 27 q^{88} + 81 q^{89} + 15 q^{91} - 7 q^{92} - 15 q^{93} + 56 q^{94} + 5 q^{95} + q^{96} + 47 q^{97} - 3 q^{98} + 15 q^{99}+O(q^{100}) \) 36 * q - 36 * q^3 + 36 * q^4 + 36 * q^5 - 4 * q^7 - 3 * q^8 + 36 * q^9 + 15 * q^11 - 36 * q^12 + 8 * q^13 + 10 * q^14 - 36 * q^15 + 36 * q^16 + 32 * q^17 + 5 * q^19 + 36 * q^20 + 4 * q^21 + q^22 - 10 * q^23 + 3 * q^24 + 36 * q^25 + 22 * q^26 - 36 * q^27 + 61 * q^29 + 15 * q^31 - q^32 - 15 * q^33 + 26 * q^34 - 4 * q^35 + 36 * q^36 + 16 * q^37 + 20 * q^38 - 8 * q^39 - 3 * q^40 + 61 * q^41 - 10 * q^42 - 35 * q^43 + 39 * q^44 + 36 * q^45 + 11 * q^46 - 28 * q^47 - 36 * q^48 + 68 * q^49 - 32 * q^51 + 4 * q^52 + 33 * q^53 + 15 * q^55 + 23 * q^56 - 5 * q^57 + 6 * q^58 + 35 * q^59 - 36 * q^60 + 55 * q^61 + q^62 - 4 * q^63 + 15 * q^64 + 8 * q^65 - q^66 - 34 * q^67 + 60 * q^68 + 10 * q^69 + 10 * q^70 + 42 * q^71 - 3 * q^72 + 53 * q^73 + 54 * q^74 - 36 * q^75 + 20 * q^76 + 20 * q^77 - 22 * q^78 + 25 * q^79 + 36 * q^80 + 36 * q^81 - 15 * q^82 - 11 * q^83 + 32 * q^85 + 53 * q^86 - 61 * q^87 + 27 * q^88 + 81 * q^89 + 15 * q^91 - 7 * q^92 - 15 * q^93 + 56 * q^94 + 5 * q^95 + q^96 + 47 * q^97 - 3 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more