LMFDB - Embedded newform 6045.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -3.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -3.56155 q^{7} +1.00000 q^{9} +2.00000 q^{10} -0.561553 q^{11} +2.00000 q^{12} +1.00000 q^{13} +7.12311 q^{14} -1.00000 q^{15} -4.00000 q^{16} +3.12311 q^{17} -2.00000 q^{18} -5.12311 q^{19} -2.00000 q^{20} -3.56155 q^{21} +1.12311 q^{22} -2.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -7.12311 q^{28} +4.68466 q^{29} +2.00000 q^{30} -1.00000 q^{31} +8.00000 q^{32} -0.561553 q^{33} -6.24621 q^{34} +3.56155 q^{35} +2.00000 q^{36} +8.56155 q^{37} +10.2462 q^{38} +1.00000 q^{39} +3.56155 q^{41} +7.12311 q^{42} -0.438447 q^{43} -1.12311 q^{44} -1.00000 q^{45} +4.00000 q^{46} -6.24621 q^{47} -4.00000 q^{48} +5.68466 q^{49} -2.00000 q^{50} +3.12311 q^{51} +2.00000 q^{52} -7.12311 q^{53} -2.00000 q^{54} +0.561553 q^{55} -5.12311 q^{57} -9.36932 q^{58} +5.80776 q^{59} -2.00000 q^{60} +6.00000 q^{61} +2.00000 q^{62} -3.56155 q^{63} -8.00000 q^{64} -1.00000 q^{65} +1.12311 q^{66} +0.684658 q^{67} +6.24621 q^{68} -2.00000 q^{69} -7.12311 q^{70} +2.24621 q^{71} +0.561553 q^{73} -17.1231 q^{74} +1.00000 q^{75} -10.2462 q^{76} +2.00000 q^{77} -2.00000 q^{78} +11.1231 q^{79} +4.00000 q^{80} +1.00000 q^{81} -7.12311 q^{82} -17.5616 q^{83} -7.12311 q^{84} -3.12311 q^{85} +0.876894 q^{86} +4.68466 q^{87} -12.5616 q^{89} +2.00000 q^{90} -3.56155 q^{91} -4.00000 q^{92} -1.00000 q^{93} +12.4924 q^{94} +5.12311 q^{95} +8.00000 q^{96} +15.5616 q^{97} -11.3693 q^{98} -0.561553 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 - 3 * q^7 + 2 * q^9	\( 2 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{9} + 4 q^{10} + 3 q^{11} + 4 q^{12} + 2 q^{13} + 6 q^{14} - 2 q^{15} - 8 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{19} - 4 q^{20} - 3 q^{21} - 6 q^{22} - 4 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 6 q^{28} - 3 q^{29} + 4 q^{30} - 2 q^{31} + 16 q^{32} + 3 q^{33} + 4 q^{34} + 3 q^{35} + 4 q^{36} + 13 q^{37} + 4 q^{38} + 2 q^{39} + 3 q^{41} + 6 q^{42} - 5 q^{43} + 6 q^{44} - 2 q^{45} + 8 q^{46} + 4 q^{47} - 8 q^{48} - q^{49} - 4 q^{50} - 2 q^{51} + 4 q^{52} - 6 q^{53} - 4 q^{54} - 3 q^{55} - 2 q^{57} + 6 q^{58} - 9 q^{59} - 4 q^{60} + 12 q^{61} + 4 q^{62} - 3 q^{63} - 16 q^{64} - 2 q^{65} - 6 q^{66} - 11 q^{67} - 4 q^{68} - 4 q^{69} - 6 q^{70} - 12 q^{71} - 3 q^{73} - 26 q^{74} + 2 q^{75} - 4 q^{76} + 4 q^{77} - 4 q^{78} + 14 q^{79} + 8 q^{80} + 2 q^{81} - 6 q^{82} - 31 q^{83} - 6 q^{84} + 2 q^{85} + 10 q^{86} - 3 q^{87} - 21 q^{89} + 4 q^{90} - 3 q^{91} - 8 q^{92} - 2 q^{93} - 8 q^{94} + 2 q^{95} + 16 q^{96} + 27 q^{97} + 2 q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 - 3 * q^7 + 2 * q^9 + 4 * q^10 + 3 * q^11 + 4 * q^12 + 2 * q^13 + 6 * q^14 - 2 * q^15 - 8 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^19 - 4 * q^20 - 3 * q^21 - 6 * q^22 - 4 * q^23 + 2 * q^25 - 4 * q^26 + 2 * q^27 - 6 * q^28 - 3 * q^29 + 4 * q^30 - 2 * q^31 + 16 * q^32 + 3 * q^33 + 4 * q^34 + 3 * q^35 + 4 * q^36 + 13 * q^37 + 4 * q^38 + 2 * q^39 + 3 * q^41 + 6 * q^42 - 5 * q^43 + 6 * q^44 - 2 * q^45 + 8 * q^46 + 4 * q^47 - 8 * q^48 - q^49 - 4 * q^50 - 2 * q^51 + 4 * q^52 - 6 * q^53 - 4 * q^54 - 3 * q^55 - 2 * q^57 + 6 * q^58 - 9 * q^59 - 4 * q^60 + 12 * q^61 + 4 * q^62 - 3 * q^63 - 16 * q^64 - 2 * q^65 - 6 * q^66 - 11 * q^67 - 4 * q^68 - 4 * q^69 - 6 * q^70 - 12 * q^71 - 3 * q^73 - 26 * q^74 + 2 * q^75 - 4 * q^76 + 4 * q^77 - 4 * q^78 + 14 * q^79 + 8 * q^80 + 2 * q^81 - 6 * q^82 - 31 * q^83 - 6 * q^84 + 2 * q^85 + 10 * q^86 - 3 * q^87 - 21 * q^89 + 4 * q^90 - 3 * q^91 - 8 * q^92 - 2 * q^93 - 8 * q^94 + 2 * q^95 + 16 * q^96 + 27 * q^97 + 2 * q^98 + 3 * 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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−3.56155	−1.34614	−0.673070	−	0.739579i	$-0.735025\pi$
$7$	−3.56155	−1.34614	−0.673070	+	0.739579i	$0.735025\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−0.561553	−0.169315	−0.0846573	−	0.996410i	$-0.526980\pi$
$11$	−0.561553	−0.169315	−0.0846573	+	0.996410i	$0.526980\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.12311	1.90373
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	−2.00000	−0.471405
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	−2.00000	−0.447214
$21$	−3.56155	−0.777195
$22$	1.12311	0.239447
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−7.12311	−1.34614
$29$	4.68466	0.869919	0.434960	−	0.900450i	$-0.356763\pi$
$29$	4.68466	0.869919	0.434960	+	0.900450i	$0.356763\pi$
$30$	2.00000	0.365148
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	8.00000	1.41421
$33$	−0.561553	−0.0977538
$34$	−6.24621	−1.07122
$35$	3.56155	0.602012
$36$	2.00000	0.333333
$37$	8.56155	1.40751	0.703755	−	0.710442i	$-0.251505\pi$
$37$	8.56155	1.40751	0.703755	+	0.710442i	$0.251505\pi$
$38$	10.2462	1.66215
$39$	1.00000	0.160128
$40$	0	0
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	7.12311	1.09912
$43$	−0.438447	−0.0668626	−0.0334313	−	0.999441i	$-0.510643\pi$
$43$	−0.438447	−0.0668626	−0.0334313	+	0.999441i	$0.510643\pi$
$44$	−1.12311	−0.169315
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	−4.00000	−0.577350
$49$	5.68466	0.812094
$50$	−2.00000	−0.282843
$51$	3.12311	0.437322
$52$	2.00000	0.277350
$53$	−7.12311	−0.978434	−0.489217	−	0.872162i	$-0.662717\pi$
$53$	−7.12311	−0.978434	−0.489217	+	0.872162i	$0.662717\pi$
$54$	−2.00000	−0.272166
$55$	0.561553	0.0757198
$56$	0	0
$57$	−5.12311	−0.678572
$58$	−9.36932	−1.23025
$59$	5.80776	0.756106	0.378053	−	0.925784i	$-0.376594\pi$
$59$	5.80776	0.756106	0.378053	+	0.925784i	$0.376594\pi$
$60$	−2.00000	−0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	2.00000	0.254000
$63$	−3.56155	−0.448713
$64$	−8.00000	−1.00000
$65$	−1.00000	−0.124035
$66$	1.12311	0.138245
$67$	0.684658	0.0836443	0.0418222	−	0.999125i	$-0.486684\pi$
$67$	0.684658	0.0836443	0.0418222	+	0.999125i	$0.486684\pi$
$68$	6.24621	0.757464
$69$	−2.00000	−0.240772
$70$	−7.12311	−0.851374
$71$	2.24621	0.266576	0.133288	−	0.991077i	$-0.457446\pi$
$71$	2.24621	0.266576	0.133288	+	0.991077i	$0.457446\pi$
$72$	0	0
$73$	0.561553	0.0657248	0.0328624	−	0.999460i	$-0.489538\pi$
$73$	0.561553	0.0657248	0.0328624	+	0.999460i	$0.489538\pi$
$74$	−17.1231	−1.99052
$75$	1.00000	0.115470
$76$	−10.2462	−1.17532
$77$	2.00000	0.227921
$78$	−2.00000	−0.226455
$79$	11.1231	1.25145	0.625724	−	0.780045i	$-0.284804\pi$
$79$	11.1231	1.25145	0.625724	+	0.780045i	$0.284804\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−7.12311	−0.786615
$83$	−17.5616	−1.92763	−0.963815	−	0.266570i	$-0.914109\pi$
$83$	−17.5616	−1.92763	−0.963815	+	0.266570i	$0.914109\pi$
$84$	−7.12311	−0.777195
$85$	−3.12311	−0.338748
$86$	0.876894	0.0945580
$87$	4.68466	0.502248
$88$	0	0
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	2.00000	0.210819
$91$	−3.56155	−0.373352
$92$	−4.00000	−0.417029
$93$	−1.00000	−0.103695
$94$	12.4924	1.28849
$95$	5.12311	0.525620
$96$	8.00000	0.816497
$97$	15.5616	1.58004	0.790018	−	0.613083i	$-0.210071\pi$
$97$	15.5616	1.58004	0.790018	+	0.613083i	$0.210071\pi$
$98$	−11.3693	−1.14847
$99$	−0.561553	−0.0564382
$100$	2.00000	0.200000
$101$	−8.56155	−0.851906	−0.425953	−	0.904745i	$-0.640061\pi$
$101$	−8.56155	−0.851906	−0.425953	+	0.904745i	$0.640061\pi$
$102$	−6.24621	−0.618467
$103$	10.8078	1.06492	0.532460	−	0.846455i	$-0.321268\pi$
$103$	10.8078	1.06492	0.532460	+	0.846455i	$0.321268\pi$
$104$	0	0
$105$	3.56155	0.347572
$106$	14.2462	1.38371
$107$	−10.3693	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$107$	−10.3693	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$108$	2.00000	0.192450
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.12311	−0.107084
$111$	8.56155	0.812627
$112$	14.2462	1.34614
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	10.2462	0.959646
$115$	2.00000	0.186501
$116$	9.36932	0.869919
$117$	1.00000	0.0924500
$118$	−11.6155	−1.06930
$119$	−11.1231	−1.01965
$120$	0	0
$121$	−10.6847	−0.971333
$122$	−12.0000	−1.08643
$123$	3.56155	0.321134
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	7.12311	0.634577
$127$	18.0540	1.60203	0.801016	−	0.598643i	$-0.204293\pi$
$127$	18.0540	1.60203	0.801016	+	0.598643i	$0.204293\pi$
$128$	0	0
$129$	−0.438447	−0.0386031
$130$	2.00000	0.175412
$131$	8.56155	0.748026	0.374013	−	0.927423i	$-0.377981\pi$
$131$	8.56155	0.748026	0.374013	+	0.927423i	$0.377981\pi$
$132$	−1.12311	−0.0977538
$133$	18.2462	1.58215
$134$	−1.36932	−0.118291
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	16.2462	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$138$	4.00000	0.340503
$139$	−2.24621	−0.190521	−0.0952606	−	0.995452i	$-0.530368\pi$
$139$	−2.24621	−0.190521	−0.0952606	+	0.995452i	$0.530368\pi$
$140$	7.12311	0.602012
$141$	−6.24621	−0.526026
$142$	−4.49242	−0.376996
$143$	−0.561553	−0.0469594
$144$	−4.00000	−0.333333
$145$	−4.68466	−0.389040
$146$	−1.12311	−0.0929489
$147$	5.68466	0.468863
$148$	17.1231	1.40751
$149$	−17.5616	−1.43870	−0.719349	−	0.694649i	$-0.755560\pi$
$149$	−17.5616	−1.43870	−0.719349	+	0.694649i	$0.755560\pi$
$150$	−2.00000	−0.163299
$151$	4.75379	0.386858	0.193429	−	0.981114i	$-0.438039\pi$
$151$	4.75379	0.386858	0.193429	+	0.981114i	$0.438039\pi$
$152$	0	0
$153$	3.12311	0.252488
$154$	−4.00000	−0.322329
$155$	1.00000	0.0803219
$156$	2.00000	0.160128
$157$	11.4384	0.912887	0.456444	−	0.889752i	$-0.349123\pi$
$157$	11.4384	0.912887	0.456444	+	0.889752i	$0.349123\pi$
$158$	−22.2462	−1.76981
$159$	−7.12311	−0.564899
$160$	−8.00000	−0.632456
$161$	7.12311	0.561379
$162$	−2.00000	−0.157135
$163$	−24.4924	−1.91839	−0.959197	−	0.282738i	$-0.908757\pi$
$163$	−24.4924	−1.91839	−0.959197	+	0.282738i	$0.908757\pi$
$164$	7.12311	0.556221
$165$	0.561553	0.0437168
$166$	35.1231	2.72608
$167$	−12.8078	−0.991095	−0.495547	−	0.868581i	$-0.665032\pi$
$167$	−12.8078	−0.991095	−0.495547	+	0.868581i	$0.665032\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.24621	0.479063
$171$	−5.12311	−0.391774
$172$	−0.876894	−0.0668626
$173$	25.7386	1.95687	0.978436	−	0.206550i	$-0.0662236\pi$
$173$	25.7386	1.95687	0.978436	+	0.206550i	$0.0662236\pi$
$174$	−9.36932	−0.710286
$175$	−3.56155	−0.269228
$176$	2.24621	0.169315
$177$	5.80776	0.436538
$178$	25.1231	1.88306
$179$	−15.8078	−1.18153	−0.590764	−	0.806844i	$-0.701174\pi$
$179$	−15.8078	−1.18153	−0.590764	+	0.806844i	$0.701174\pi$
$180$	−2.00000	−0.149071
$181$	−2.24621	−0.166960	−0.0834798	−	0.996509i	$-0.526603\pi$
$181$	−2.24621	−0.166960	−0.0834798	+	0.996509i	$0.526603\pi$
$182$	7.12311	0.528000
$183$	6.00000	0.443533
$184$	0	0
$185$	−8.56155	−0.629458
$186$	2.00000	0.146647
$187$	−1.75379	−0.128250
$188$	−12.4924	−0.911104
$189$	−3.56155	−0.259065
$190$	−10.2462	−0.743338
$191$	−2.31534	−0.167532	−0.0837661	−	0.996485i	$-0.526695\pi$
$191$	−2.31534	−0.167532	−0.0837661	+	0.996485i	$0.526695\pi$
$192$	−8.00000	−0.577350
$193$	−6.43845	−0.463450	−0.231725	−	0.972781i	$-0.574437\pi$
$193$	−6.43845	−0.463450	−0.231725	+	0.972781i	$0.574437\pi$
$194$	−31.1231	−2.23451
$195$	−1.00000	−0.0716115
$196$	11.3693	0.812094
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	1.12311	0.0798156
$199$	−15.3693	−1.08950	−0.544751	−	0.838598i	$-0.683376\pi$
$199$	−15.3693	−1.08950	−0.544751	+	0.838598i	$0.683376\pi$
$200$	0	0
$201$	0.684658	0.0482921
$202$	17.1231	1.20478
$203$	−16.6847	−1.17103
$204$	6.24621	0.437322
$205$	−3.56155	−0.248750
$206$	−21.6155	−1.50603
$207$	−2.00000	−0.139010
$208$	−4.00000	−0.277350
$209$	2.87689	0.198999
$210$	−7.12311	−0.491541
$211$	−20.6155	−1.41923	−0.709616	−	0.704589i	$-0.751131\pi$
$211$	−20.6155	−1.41923	−0.709616	+	0.704589i	$0.751131\pi$
$212$	−14.2462	−0.978434
$213$	2.24621	0.153908
$214$	20.7386	1.41766
$215$	0.438447	0.0299018
$216$	0	0
$217$	3.56155	0.241774
$218$	20.0000	1.35457
$219$	0.561553	0.0379462
$220$	1.12311	0.0757198
$221$	3.12311	0.210083
$222$	−17.1231	−1.14923
$223$	−17.0540	−1.14202	−0.571009	−	0.820944i	$-0.693448\pi$
$223$	−17.0540	−1.14202	−0.571009	+	0.820944i	$0.693448\pi$
$224$	−28.4924	−1.90373
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	4.87689	0.323691	0.161845	−	0.986816i	$-0.448255\pi$
$227$	4.87689	0.323691	0.161845	+	0.986816i	$0.448255\pi$
$228$	−10.2462	−0.678572
$229$	11.9309	0.788414	0.394207	−	0.919022i	$-0.371019\pi$
$229$	11.9309	0.788414	0.394207	+	0.919022i	$0.371019\pi$
$230$	−4.00000	−0.263752
$231$	2.00000	0.131590
$232$	0	0
$233$	16.6155	1.08852	0.544260	−	0.838917i	$-0.316811\pi$
$233$	16.6155	1.08852	0.544260	+	0.838917i	$0.316811\pi$
$234$	−2.00000	−0.130744
$235$	6.24621	0.407458
$236$	11.6155	0.756106
$237$	11.1231	0.722523
$238$	22.2462	1.44201
$239$	−25.3693	−1.64100	−0.820502	−	0.571643i	$-0.806306\pi$
$239$	−25.3693	−1.64100	−0.820502	+	0.571643i	$0.806306\pi$
$240$	4.00000	0.258199
$241$	6.68466	0.430597	0.215298	−	0.976548i	$-0.430928\pi$
$241$	6.68466	0.430597	0.215298	+	0.976548i	$0.430928\pi$
$242$	21.3693	1.37367
$243$	1.00000	0.0641500
$244$	12.0000	0.768221
$245$	−5.68466	−0.363180
$246$	−7.12311	−0.454153
$247$	−5.12311	−0.325975
$248$	0	0
$249$	−17.5616	−1.11292
$250$	2.00000	0.126491
$251$	−17.8078	−1.12402	−0.562008	−	0.827132i	$-0.689971\pi$
$251$	−17.8078	−1.12402	−0.562008	+	0.827132i	$0.689971\pi$
$252$	−7.12311	−0.448713
$253$	1.12311	0.0706090
$254$	−36.1080	−2.26561
$255$	−3.12311	−0.195576
$256$	16.0000	1.00000
$257$	−12.5616	−0.783568	−0.391784	−	0.920057i	$-0.628142\pi$
$257$	−12.5616	−0.783568	−0.391784	+	0.920057i	$0.628142\pi$
$258$	0.876894	0.0545931
$259$	−30.4924	−1.89471
$260$	−2.00000	−0.124035
$261$	4.68466	0.289973
$262$	−17.1231	−1.05787
$263$	−11.3693	−0.701062	−0.350531	−	0.936551i	$-0.613999\pi$
$263$	−11.3693	−0.701062	−0.350531	+	0.936551i	$0.613999\pi$
$264$	0	0
$265$	7.12311	0.437569
$266$	−36.4924	−2.23749
$267$	−12.5616	−0.768755
$268$	1.36932	0.0836443
$269$	−20.8769	−1.27289	−0.636443	−	0.771323i	$-0.719595\pi$
$269$	−20.8769	−1.27289	−0.636443	+	0.771323i	$0.719595\pi$
$270$	2.00000	0.121716
$271$	−17.8769	−1.08594	−0.542972	−	0.839751i	$-0.682701\pi$
$271$	−17.8769	−1.08594	−0.542972	+	0.839751i	$0.682701\pi$
$272$	−12.4924	−0.757464
$273$	−3.56155	−0.215555
$274$	−32.4924	−1.96294
$275$	−0.561553	−0.0338629
$276$	−4.00000	−0.240772
$277$	8.43845	0.507017	0.253509	−	0.967333i	$-0.418415\pi$
$277$	8.43845	0.507017	0.253509	+	0.967333i	$0.418415\pi$
$278$	4.49242	0.269438
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−0.192236	−0.0114678	−0.00573392	−	0.999984i	$-0.501825\pi$
$281$	−0.192236	−0.0114678	−0.00573392	+	0.999984i	$0.501825\pi$
$282$	12.4924	0.743913
$283$	12.5616	0.746707	0.373353	−	0.927689i	$-0.378208\pi$
$283$	12.5616	0.746707	0.373353	+	0.927689i	$0.378208\pi$
$284$	4.49242	0.266576
$285$	5.12311	0.303467
$286$	1.12311	0.0664106
$287$	−12.6847	−0.748752
$288$	8.00000	0.471405
$289$	−7.24621	−0.426248
$290$	9.36932	0.550185
$291$	15.5616	0.912234
$292$	1.12311	0.0657248
$293$	−1.12311	−0.0656125	−0.0328063	−	0.999462i	$-0.510444\pi$
$293$	−1.12311	−0.0656125	−0.0328063	+	0.999462i	$0.510444\pi$
$294$	−11.3693	−0.663072
$295$	−5.80776	−0.338141
$296$	0	0
$297$	−0.561553	−0.0325846
$298$	35.1231	2.03463
$299$	−2.00000	−0.115663
$300$	2.00000	0.115470
$301$	1.56155	0.0900064
$302$	−9.50758	−0.547100
$303$	−8.56155	−0.491848
$304$	20.4924	1.17532
$305$	−6.00000	−0.343559
$306$	−6.24621	−0.357072
$307$	14.2462	0.813074	0.406537	−	0.913634i	$-0.366736\pi$
$307$	14.2462	0.813074	0.406537	+	0.913634i	$0.366736\pi$
$308$	4.00000	0.227921
$309$	10.8078	0.614832
$310$	−2.00000	−0.113592
$311$	13.9309	0.789947	0.394974	−	0.918692i	$-0.370754\pi$
$311$	13.9309	0.789947	0.394974	+	0.918692i	$0.370754\pi$
$312$	0	0
$313$	9.80776	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$313$	9.80776	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$314$	−22.8769	−1.29102
$315$	3.56155	0.200671
$316$	22.2462	1.25145
$317$	−27.3693	−1.53721	−0.768607	−	0.639721i	$-0.779050\pi$
$317$	−27.3693	−1.53721	−0.768607	+	0.639721i	$0.779050\pi$
$318$	14.2462	0.798888
$319$	−2.63068	−0.147290
$320$	8.00000	0.447214
$321$	−10.3693	−0.578759
$322$	−14.2462	−0.793910
$323$	−16.0000	−0.890264
$324$	2.00000	0.111111
$325$	1.00000	0.0554700
$326$	48.9848	2.71302
$327$	−10.0000	−0.553001
$328$	0	0
$329$	22.2462	1.22647
$330$	−1.12311	−0.0618249
$331$	−14.9309	−0.820675	−0.410337	−	0.911934i	$-0.634589\pi$
$331$	−14.9309	−0.820675	−0.410337	+	0.911934i	$0.634589\pi$
$332$	−35.1231	−1.92763
$333$	8.56155	0.469170
$334$	25.6155	1.40162
$335$	−0.684658	−0.0374069
$336$	14.2462	0.777195
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−2.00000	−0.108786
$339$	9.00000	0.488813
$340$	−6.24621	−0.338748
$341$	0.561553	0.0304098
$342$	10.2462	0.554052
$343$	4.68466	0.252948
$344$	0	0
$345$	2.00000	0.107676
$346$	−51.4773	−2.76744
$347$	31.3693	1.68399	0.841997	−	0.539483i	$-0.181380\pi$
$347$	31.3693	1.68399	0.841997	+	0.539483i	$0.181380\pi$
$348$	9.36932	0.502248
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	7.12311	0.380746
$351$	1.00000	0.0533761
$352$	−4.49242	−0.239447
$353$	−5.63068	−0.299691	−0.149846	−	0.988709i	$-0.547878\pi$
$353$	−5.63068	−0.299691	−0.149846	+	0.988709i	$0.547878\pi$
$354$	−11.6155	−0.617358
$355$	−2.24621	−0.119217
$356$	−25.1231	−1.33152
$357$	−11.1231	−0.588697
$358$	31.6155	1.67093
$359$	2.68466	0.141691	0.0708454	−	0.997487i	$-0.477430\pi$
$359$	2.68466	0.141691	0.0708454	+	0.997487i	$0.477430\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	4.49242	0.236116
$363$	−10.6847	−0.560799
$364$	−7.12311	−0.373352
$365$	−0.561553	−0.0293930
$366$	−12.0000	−0.627250
$367$	5.75379	0.300345	0.150173	−	0.988660i	$-0.452017\pi$
$367$	5.75379	0.300345	0.150173	+	0.988660i	$0.452017\pi$
$368$	8.00000	0.417029
$369$	3.56155	0.185407
$370$	17.1231	0.890188
$371$	25.3693	1.31711
$372$	−2.00000	−0.103695
$373$	−2.31534	−0.119884	−0.0599419	−	0.998202i	$-0.519092\pi$
$373$	−2.31534	−0.119884	−0.0599419	+	0.998202i	$0.519092\pi$
$374$	3.50758	0.181373
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.68466	0.241272
$378$	7.12311	0.366373
$379$	−12.2462	−0.629046	−0.314523	−	0.949250i	$-0.601844\pi$
$379$	−12.2462	−0.629046	−0.314523	+	0.949250i	$0.601844\pi$
$380$	10.2462	0.525620
$381$	18.0540	0.924933
$382$	4.63068	0.236926
$383$	22.6155	1.15560	0.577800	−	0.816179i	$-0.303911\pi$
$383$	22.6155	1.15560	0.577800	+	0.816179i	$0.303911\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	12.8769	0.655417
$387$	−0.438447	−0.0222875
$388$	31.1231	1.58004
$389$	−25.1771	−1.27653	−0.638264	−	0.769817i	$-0.720347\pi$
$389$	−25.1771	−1.27653	−0.638264	+	0.769817i	$0.720347\pi$
$390$	2.00000	0.101274
$391$	−6.24621	−0.315884
$392$	0	0
$393$	8.56155	0.431873
$394$	2.00000	0.100759
$395$	−11.1231	−0.559664
$396$	−1.12311	−0.0564382
$397$	−6.49242	−0.325845	−0.162923	−	0.986639i	$-0.552092\pi$
$397$	−6.49242	−0.325845	−0.162923	+	0.986639i	$0.552092\pi$
$398$	30.7386	1.54079
$399$	18.2462	0.913453
$400$	−4.00000	−0.200000
$401$	−11.6847	−0.583504	−0.291752	−	0.956494i	$-0.594238\pi$
$401$	−11.6847	−0.583504	−0.291752	+	0.956494i	$0.594238\pi$
$402$	−1.36932	−0.0682953
$403$	−1.00000	−0.0498135
$404$	−17.1231	−0.851906
$405$	−1.00000	−0.0496904
$406$	33.3693	1.65609
$407$	−4.80776	−0.238312
$408$	0	0
$409$	12.1231	0.599449	0.299724	−	0.954026i	$-0.403105\pi$
$409$	12.1231	0.599449	0.299724	+	0.954026i	$0.403105\pi$
$410$	7.12311	0.351785
$411$	16.2462	0.801367
$412$	21.6155	1.06492
$413$	−20.6847	−1.01783
$414$	4.00000	0.196589
$415$	17.5616	0.862063
$416$	8.00000	0.392232
$417$	−2.24621	−0.109997
$418$	−5.75379	−0.281427
$419$	36.1080	1.76399	0.881994	−	0.471260i	$-0.156201\pi$
$419$	36.1080	1.76399	0.881994	+	0.471260i	$0.156201\pi$
$420$	7.12311	0.347572
$421$	38.1080	1.85727	0.928634	−	0.370997i	$-0.120984\pi$
$421$	38.1080	1.85727	0.928634	+	0.370997i	$0.120984\pi$
$422$	41.2311	2.00710
$423$	−6.24621	−0.303701
$424$	0	0
$425$	3.12311	0.151493
$426$	−4.49242	−0.217659
$427$	−21.3693	−1.03413
$428$	−20.7386	−1.00244
$429$	−0.561553	−0.0271120
$430$	−0.876894	−0.0422876
$431$	−30.3002	−1.45951	−0.729754	−	0.683709i	$-0.760366\pi$
$431$	−30.3002	−1.45951	−0.729754	+	0.683709i	$0.760366\pi$
$432$	−4.00000	−0.192450
$433$	−34.0540	−1.63653	−0.818265	−	0.574841i	$-0.805064\pi$
$433$	−34.0540	−1.63653	−0.818265	+	0.574841i	$0.805064\pi$
$434$	−7.12311	−0.341920
$435$	−4.68466	−0.224612
$436$	−20.0000	−0.957826
$437$	10.2462	0.490143
$438$	−1.12311	−0.0536641
$439$	−26.0540	−1.24349	−0.621744	−	0.783220i	$-0.713576\pi$
$439$	−26.0540	−1.24349	−0.621744	+	0.783220i	$0.713576\pi$
$440$	0	0
$441$	5.68466	0.270698
$442$	−6.24621	−0.297102
$443$	−12.3693	−0.587684	−0.293842	−	0.955854i	$-0.594934\pi$
$443$	−12.3693	−0.587684	−0.293842	+	0.955854i	$0.594934\pi$
$444$	17.1231	0.812627
$445$	12.5616	0.595475
$446$	34.1080	1.61506
$447$	−17.5616	−0.830633
$448$	28.4924	1.34614
$449$	−12.2462	−0.577934	−0.288967	−	0.957339i	$-0.593312\pi$
$449$	−12.2462	−0.577934	−0.288967	+	0.957339i	$0.593312\pi$
$450$	−2.00000	−0.0942809
$451$	−2.00000	−0.0941763
$452$	18.0000	0.846649
$453$	4.75379	0.223352
$454$	−9.75379	−0.457768
$455$	3.56155	0.166968
$456$	0	0
$457$	−28.1771	−1.31807	−0.659034	−	0.752113i	$-0.729035\pi$
$457$	−28.1771	−1.31807	−0.659034	+	0.752113i	$0.729035\pi$
$458$	−23.8617	−1.11499
$459$	3.12311	0.145774
$460$	4.00000	0.186501
$461$	−20.3153	−0.946180	−0.473090	−	0.881014i	$-0.656861\pi$
$461$	−20.3153	−0.946180	−0.473090	+	0.881014i	$0.656861\pi$
$462$	−4.00000	−0.186097
$463$	−0.807764	−0.0375400	−0.0187700	−	0.999824i	$-0.505975\pi$
$463$	−0.807764	−0.0375400	−0.0187700	+	0.999824i	$0.505975\pi$
$464$	−18.7386	−0.869919
$465$	1.00000	0.0463739
$466$	−33.2311	−1.53940
$467$	−13.5616	−0.627554	−0.313777	−	0.949497i	$-0.601594\pi$
$467$	−13.5616	−0.627554	−0.313777	+	0.949497i	$0.601594\pi$
$468$	2.00000	0.0924500
$469$	−2.43845	−0.112597
$470$	−12.4924	−0.576232
$471$	11.4384	0.527056
$472$	0	0
$473$	0.246211	0.0113208
$474$	−22.2462	−1.02180
$475$	−5.12311	−0.235064
$476$	−22.2462	−1.01965
$477$	−7.12311	−0.326145
$478$	50.7386	2.32073
$479$	37.1771	1.69866	0.849332	−	0.527859i	$-0.177005\pi$
$479$	37.1771	1.69866	0.849332	+	0.527859i	$0.177005\pi$
$480$	−8.00000	−0.365148
$481$	8.56155	0.390373
$482$	−13.3693	−0.608956
$483$	7.12311	0.324113
$484$	−21.3693	−0.971333
$485$	−15.5616	−0.706614
$486$	−2.00000	−0.0907218
$487$	32.9848	1.49469	0.747343	−	0.664438i	$-0.231329\pi$
$487$	32.9848	1.49469	0.747343	+	0.664438i	$0.231329\pi$
$488$	0	0
$489$	−24.4924	−1.10759
$490$	11.3693	0.513613
$491$	−6.68466	−0.301674	−0.150837	−	0.988559i	$-0.548197\pi$
$491$	−6.68466	−0.301674	−0.150837	+	0.988559i	$0.548197\pi$
$492$	7.12311	0.321134
$493$	14.6307	0.658933
$494$	10.2462	0.460999
$495$	0.561553	0.0252399
$496$	4.00000	0.179605
$497$	−8.00000	−0.358849
$498$	35.1231	1.57390
$499$	14.0540	0.629142	0.314571	−	0.949234i	$-0.398139\pi$
$499$	14.0540	0.629142	0.314571	+	0.949234i	$0.398139\pi$
$500$	−2.00000	−0.0894427
$501$	−12.8078	−0.572209
$502$	35.6155	1.58960
$503$	−35.4233	−1.57945	−0.789723	−	0.613463i	$-0.789776\pi$
$503$	−35.4233	−1.57945	−0.789723	+	0.613463i	$0.789776\pi$
$504$	0	0
$505$	8.56155	0.380984
$506$	−2.24621	−0.0998563
$507$	1.00000	0.0444116
$508$	36.1080	1.60203
$509$	−36.2462	−1.60659	−0.803293	−	0.595585i	$-0.796920\pi$
$509$	−36.2462	−1.60659	−0.803293	+	0.595585i	$0.796920\pi$
$510$	6.24621	0.276587
$511$	−2.00000	−0.0884748
$512$	−32.0000	−1.41421
$513$	−5.12311	−0.226191
$514$	25.1231	1.10813
$515$	−10.8078	−0.476247
$516$	−0.876894	−0.0386031
$517$	3.50758	0.154263
$518$	60.9848	2.67952
$519$	25.7386	1.12980
$520$	0	0
$521$	−18.1771	−0.796352	−0.398176	−	0.917309i	$-0.630357\pi$
$521$	−18.1771	−0.796352	−0.398176	+	0.917309i	$0.630357\pi$
$522$	−9.36932	−0.410084
$523$	−41.8078	−1.82813	−0.914063	−	0.405572i	$-0.867072\pi$
$523$	−41.8078	−1.82813	−0.914063	+	0.405572i	$0.867072\pi$
$524$	17.1231	0.748026
$525$	−3.56155	−0.155439
$526$	22.7386	0.991452
$527$	−3.12311	−0.136045
$528$	2.24621	0.0977538
$529$	−19.0000	−0.826087
$530$	−14.2462	−0.618816
$531$	5.80776	0.252035
$532$	36.4924	1.58215
$533$	3.56155	0.154268
$534$	25.1231	1.08718
$535$	10.3693	0.448305
$536$	0	0
$537$	−15.8078	−0.682155
$538$	41.7538	1.80013
$539$	−3.19224	−0.137499
$540$	−2.00000	−0.0860663
$541$	34.4924	1.48295	0.741473	−	0.670983i	$-0.234128\pi$
$541$	34.4924	1.48295	0.741473	+	0.670983i	$0.234128\pi$
$542$	35.7538	1.53576
$543$	−2.24621	−0.0963942
$544$	24.9848	1.07122
$545$	10.0000	0.428353
$546$	7.12311	0.304841
$547$	−45.3693	−1.93985	−0.969926	−	0.243400i	$-0.921737\pi$
$547$	−45.3693	−1.93985	−0.969926	+	0.243400i	$0.921737\pi$
$548$	32.4924	1.38801
$549$	6.00000	0.256074
$550$	1.12311	0.0478894
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−39.6155	−1.68462
$554$	−16.8769	−0.717031
$555$	−8.56155	−0.363418
$556$	−4.49242	−0.190521
$557$	18.8078	0.796911	0.398455	−	0.917188i	$-0.369546\pi$
$557$	18.8078	0.796911	0.398455	+	0.917188i	$0.369546\pi$
$558$	2.00000	0.0846668
$559$	−0.438447	−0.0185443
$560$	−14.2462	−0.602012
$561$	−1.75379	−0.0740450
$562$	0.384472	0.0162180
$563$	−21.4384	−0.903523	−0.451761	−	0.892139i	$-0.649204\pi$
$563$	−21.4384	−0.903523	−0.451761	+	0.892139i	$0.649204\pi$
$564$	−12.4924	−0.526026
$565$	−9.00000	−0.378633
$566$	−25.1231	−1.05600
$567$	−3.56155	−0.149571
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−10.2462	−0.429167
$571$	−2.87689	−0.120394	−0.0601971	−	0.998187i	$-0.519173\pi$
$571$	−2.87689	−0.120394	−0.0601971	+	0.998187i	$0.519173\pi$
$572$	−1.12311	−0.0469594
$573$	−2.31534	−0.0967248
$574$	25.3693	1.05889
$575$	−2.00000	−0.0834058
$576$	−8.00000	−0.333333
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	14.4924	0.602805
$579$	−6.43845	−0.267573
$580$	−9.36932	−0.389040
$581$	62.5464	2.59486
$582$	−31.1231	−1.29009
$583$	4.00000	0.165663
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	2.24621	0.0927901
$587$	−11.7386	−0.484505	−0.242253	−	0.970213i	$-0.577886\pi$
$587$	−11.7386	−0.484505	−0.242253	+	0.970213i	$0.577886\pi$
$588$	11.3693	0.468863
$589$	5.12311	0.211094
$590$	11.6155	0.478204
$591$	−1.00000	−0.0411345
$592$	−34.2462	−1.40751
$593$	−38.1080	−1.56491	−0.782453	−	0.622710i	$-0.786032\pi$
$593$	−38.1080	−1.56491	−0.782453	+	0.622710i	$0.786032\pi$
$594$	1.12311	0.0460816
$595$	11.1231	0.456003
$596$	−35.1231	−1.43870
$597$	−15.3693	−0.629024
$598$	4.00000	0.163572
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	14.7386	0.601201	0.300601	−	0.953750i	$-0.402813\pi$
$601$	14.7386	0.601201	0.300601	+	0.953750i	$0.402813\pi$
$602$	−3.12311	−0.127288
$603$	0.684658	0.0278814
$604$	9.50758	0.386858
$605$	10.6847	0.434393
$606$	17.1231	0.695579
$607$	5.68466	0.230733	0.115367	−	0.993323i	$-0.463196\pi$
$607$	5.68466	0.230733	0.115367	+	0.993323i	$0.463196\pi$
$608$	−40.9848	−1.66215
$609$	−16.6847	−0.676096
$610$	12.0000	0.485866
$611$	−6.24621	−0.252695
$612$	6.24621	0.252488
$613$	20.7386	0.837626	0.418813	−	0.908073i	$-0.362446\pi$
$613$	20.7386	0.837626	0.418813	+	0.908073i	$0.362446\pi$
$614$	−28.4924	−1.14986
$615$	−3.56155	−0.143616
$616$	0	0
$617$	−41.1231	−1.65555	−0.827777	−	0.561057i	$-0.810395\pi$
$617$	−41.1231	−1.65555	−0.827777	+	0.561057i	$0.810395\pi$
$618$	−21.6155	−0.869504
$619$	−6.50758	−0.261562	−0.130781	−	0.991411i	$-0.541748\pi$
$619$	−6.50758	−0.261562	−0.130781	+	0.991411i	$0.541748\pi$
$620$	2.00000	0.0803219
$621$	−2.00000	−0.0802572
$622$	−27.8617	−1.11715
$623$	44.7386	1.79242
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−19.6155	−0.783994
$627$	2.87689	0.114892
$628$	22.8769	0.912887
$629$	26.7386	1.06614
$630$	−7.12311	−0.283791
$631$	−2.06913	−0.0823708	−0.0411854	−	0.999152i	$-0.513113\pi$
$631$	−2.06913	−0.0823708	−0.0411854	+	0.999152i	$0.513113\pi$
$632$	0	0
$633$	−20.6155	−0.819394
$634$	54.7386	2.17395
$635$	−18.0540	−0.716450
$636$	−14.2462	−0.564899
$637$	5.68466	0.225234
$638$	5.26137	0.208299
$639$	2.24621	0.0888587
$640$	0	0
$641$	−29.1771	−1.15243	−0.576213	−	0.817300i	$-0.695470\pi$
$641$	−29.1771	−1.15243	−0.576213	+	0.817300i	$0.695470\pi$
$642$	20.7386	0.818489
$643$	−13.0540	−0.514799	−0.257399	−	0.966305i	$-0.582866\pi$
$643$	−13.0540	−0.514799	−0.257399	+	0.966305i	$0.582866\pi$
$644$	14.2462	0.561379
$645$	0.438447	0.0172638
$646$	32.0000	1.25902
$647$	21.6155	0.849794	0.424897	−	0.905242i	$-0.360310\pi$
$647$	21.6155	0.849794	0.424897	+	0.905242i	$0.360310\pi$
$648$	0	0
$649$	−3.26137	−0.128020
$650$	−2.00000	−0.0784465
$651$	3.56155	0.139588
$652$	−48.9848	−1.91839
$653$	−36.3693	−1.42324	−0.711621	−	0.702564i	$-0.752039\pi$
$653$	−36.3693	−1.42324	−0.711621	+	0.702564i	$0.752039\pi$
$654$	20.0000	0.782062
$655$	−8.56155	−0.334528
$656$	−14.2462	−0.556221
$657$	0.561553	0.0219083
$658$	−44.4924	−1.73450
$659$	−24.4924	−0.954089	−0.477045	−	0.878879i	$-0.658292\pi$
$659$	−24.4924	−0.954089	−0.477045	+	0.878879i	$0.658292\pi$
$660$	1.12311	0.0437168
$661$	11.7538	0.457169	0.228585	−	0.973524i	$-0.426590\pi$
$661$	11.7538	0.457169	0.228585	+	0.973524i	$0.426590\pi$
$662$	29.8617	1.16061
$663$	3.12311	0.121291
$664$	0	0
$665$	−18.2462	−0.707558
$666$	−17.1231	−0.663507
$667$	−9.36932	−0.362781
$668$	−25.6155	−0.991095
$669$	−17.0540	−0.659345
$670$	1.36932	0.0529013
$671$	−3.36932	−0.130071
$672$	−28.4924	−1.09912
$673$	33.1771	1.27888	0.639441	−	0.768840i	$-0.279166\pi$
$673$	33.1771	1.27888	0.639441	+	0.768840i	$0.279166\pi$
$674$	−52.0000	−2.00297
$675$	1.00000	0.0384900
$676$	2.00000	0.0769231
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	−18.0000	−0.691286
$679$	−55.4233	−2.12695
$680$	0	0
$681$	4.87689	0.186883
$682$	−1.12311	−0.0430059
$683$	22.0000	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$683$	22.0000	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$684$	−10.2462	−0.391774
$685$	−16.2462	−0.620736
$686$	−9.36932	−0.357722
$687$	11.9309	0.455191
$688$	1.75379	0.0668626
$689$	−7.12311	−0.271369
$690$	−4.00000	−0.152277
$691$	−4.73863	−0.180266	−0.0901331	−	0.995930i	$-0.528729\pi$
$691$	−4.73863	−0.180266	−0.0901331	+	0.995930i	$0.528729\pi$
$692$	51.4773	1.95687
$693$	2.00000	0.0759737
$694$	−62.7386	−2.38153
$695$	2.24621	0.0852036
$696$	0	0
$697$	11.1231	0.421318
$698$	60.0000	2.27103
$699$	16.6155	0.628457
$700$	−7.12311	−0.269228
$701$	−0.561553	−0.0212096	−0.0106048	−	0.999944i	$-0.503376\pi$
$701$	−0.561553	−0.0212096	−0.0106048	+	0.999944i	$0.503376\pi$
$702$	−2.00000	−0.0754851
$703$	−43.8617	−1.65428
$704$	4.49242	0.169315
$705$	6.24621	0.235246
$706$	11.2614	0.423827
$707$	30.4924	1.14679
$708$	11.6155	0.436538
$709$	21.0540	0.790699	0.395349	−	0.918531i	$-0.370624\pi$
$709$	21.0540	0.790699	0.395349	+	0.918531i	$0.370624\pi$
$710$	4.49242	0.168598
$711$	11.1231	0.417149
$712$	0	0
$713$	2.00000	0.0749006
$714$	22.2462	0.832544
$715$	0.561553	0.0210009
$716$	−31.6155	−1.18153
$717$	−25.3693	−0.947435
$718$	−5.36932	−0.200381
$719$	−35.4233	−1.32107	−0.660533	−	0.750797i	$-0.729670\pi$
$719$	−35.4233	−1.32107	−0.660533	+	0.750797i	$0.729670\pi$
$720$	4.00000	0.149071
$721$	−38.4924	−1.43353
$722$	−14.4924	−0.539352
$723$	6.68466	0.248605
$724$	−4.49242	−0.166960
$725$	4.68466	0.173984
$726$	21.3693	0.793090
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.12311	0.0415680
$731$	−1.36932	−0.0506460
$732$	12.0000	0.443533
$733$	−46.7926	−1.72832	−0.864162	−	0.503213i	$-0.832151\pi$
$733$	−46.7926	−1.72832	−0.864162	+	0.503213i	$0.832151\pi$
$734$	−11.5076	−0.424752
$735$	−5.68466	−0.209682
$736$	−16.0000	−0.589768
$737$	−0.384472	−0.0141622
$738$	−7.12311	−0.262205
$739$	−24.3153	−0.894454	−0.447227	−	0.894420i	$-0.647588\pi$
$739$	−24.3153	−0.894454	−0.447227	+	0.894420i	$0.647588\pi$
$740$	−17.1231	−0.629458
$741$	−5.12311	−0.188202
$742$	−50.7386	−1.86267
$743$	4.12311	0.151262	0.0756310	−	0.997136i	$-0.475903\pi$
$743$	4.12311	0.151262	0.0756310	+	0.997136i	$0.475903\pi$
$744$	0	0
$745$	17.5616	0.643406
$746$	4.63068	0.169541
$747$	−17.5616	−0.642544
$748$	−3.50758	−0.128250
$749$	36.9309	1.34942
$750$	2.00000	0.0730297
$751$	10.8617	0.396350	0.198175	−	0.980167i	$-0.436498\pi$
$751$	10.8617	0.396350	0.198175	+	0.980167i	$0.436498\pi$
$752$	24.9848	0.911104
$753$	−17.8078	−0.648951
$754$	−9.36932	−0.341210
$755$	−4.75379	−0.173008
$756$	−7.12311	−0.259065
$757$	31.6695	1.15105	0.575524	−	0.817785i	$-0.304798\pi$
$757$	31.6695	1.15105	0.575524	+	0.817785i	$0.304798\pi$
$758$	24.4924	0.889605
$759$	1.12311	0.0407662
$760$	0	0
$761$	−28.2462	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$761$	−28.2462	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$762$	−36.1080	−1.30805
$763$	35.6155	1.28937
$764$	−4.63068	−0.167532
$765$	−3.12311	−0.112916
$766$	−45.2311	−1.63426
$767$	5.80776	0.209706
$768$	16.0000	0.577350
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	4.00000	0.144150
$771$	−12.5616	−0.452393
$772$	−12.8769	−0.463450
$773$	43.1080	1.55049	0.775243	−	0.631664i	$-0.217628\pi$
$773$	43.1080	1.55049	0.775243	+	0.631664i	$0.217628\pi$
$774$	0.876894	0.0315193
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−30.4924	−1.09391
$778$	50.3542	1.80528
$779$	−18.2462	−0.653738
$780$	−2.00000	−0.0716115
$781$	−1.26137	−0.0451352
$782$	12.4924	0.446728
$783$	4.68466	0.167416
$784$	−22.7386	−0.812094
$785$	−11.4384	−0.408256
$786$	−17.1231	−0.610761
$787$	−0.0691303	−0.00246423	−0.00123211	−	0.999999i	$-0.500392\pi$
$787$	−0.0691303	−0.00246423	−0.00123211	+	0.999999i	$0.500392\pi$
$788$	−2.00000	−0.0712470
$789$	−11.3693	−0.404758
$790$	22.2462	0.791485
$791$	−32.0540	−1.13971
$792$	0	0
$793$	6.00000	0.213066
$794$	12.9848	0.460815
$795$	7.12311	0.252631
$796$	−30.7386	−1.08950
$797$	−31.7538	−1.12478	−0.562388	−	0.826873i	$-0.690117\pi$
$797$	−31.7538	−1.12478	−0.562388	+	0.826873i	$0.690117\pi$
$798$	−36.4924	−1.29182
$799$	−19.5076	−0.690128
$800$	8.00000	0.282843
$801$	−12.5616	−0.443841
$802$	23.3693	0.825199
$803$	−0.315342	−0.0111282
$804$	1.36932	0.0482921
$805$	−7.12311	−0.251056
$806$	2.00000	0.0704470
$807$	−20.8769	−0.734901
$808$	0	0
$809$	−38.6847	−1.36008	−0.680040	−	0.733175i	$-0.738038\pi$
$809$	−38.6847	−1.36008	−0.680040	+	0.733175i	$0.738038\pi$
$810$	2.00000	0.0702728
$811$	−54.2462	−1.90484	−0.952421	−	0.304785i	$-0.901415\pi$
$811$	−54.2462	−1.90484	−0.952421	+	0.304785i	$0.901415\pi$
$812$	−33.3693	−1.17103
$813$	−17.8769	−0.626970
$814$	9.61553	0.337024
$815$	24.4924	0.857932
$816$	−12.4924	−0.437322
$817$	2.24621	0.0785850
$818$	−24.2462	−0.847749
$819$	−3.56155	−0.124451
$820$	−7.12311	−0.248750
$821$	44.7386	1.56139	0.780695	−	0.624913i	$-0.214865\pi$
$821$	44.7386	1.56139	0.780695	+	0.624913i	$0.214865\pi$
$822$	−32.4924	−1.13330
$823$	−8.05398	−0.280744	−0.140372	−	0.990099i	$-0.544830\pi$
$823$	−8.05398	−0.280744	−0.140372	+	0.990099i	$0.544830\pi$
$824$	0	0
$825$	−0.561553	−0.0195508
$826$	41.3693	1.43942
$827$	8.68466	0.301995	0.150998	−	0.988534i	$-0.451751\pi$
$827$	8.68466	0.301995	0.150998	+	0.988534i	$0.451751\pi$
$828$	−4.00000	−0.139010
$829$	−6.73863	−0.234042	−0.117021	−	0.993129i	$-0.537335\pi$
$829$	−6.73863	−0.234042	−0.117021	+	0.993129i	$0.537335\pi$
$830$	−35.1231	−1.21914
$831$	8.43845	0.292726
$832$	−8.00000	−0.277350
$833$	17.7538	0.615132
$834$	4.49242	0.155560
$835$	12.8078	0.443231
$836$	5.75379	0.198999
$837$	−1.00000	−0.0345651
$838$	−72.2159	−2.49466
$839$	2.05398	0.0709111	0.0354556	−	0.999371i	$-0.488712\pi$
$839$	2.05398	0.0709111	0.0354556	+	0.999371i	$0.488712\pi$
$840$	0	0
$841$	−7.05398	−0.243241
$842$	−76.2159	−2.62657
$843$	−0.192236	−0.00662096
$844$	−41.2311	−1.41923
$845$	−1.00000	−0.0344010
$846$	12.4924	0.429498
$847$	38.0540	1.30755
$848$	28.4924	0.978434
$849$	12.5616	0.431111
$850$	−6.24621	−0.214243
$851$	−17.1231	−0.586973
$852$	4.49242	0.153908
$853$	45.8078	1.56843	0.784214	−	0.620490i	$-0.213066\pi$
$853$	45.8078	1.56843	0.784214	+	0.620490i	$0.213066\pi$
$854$	42.7386	1.46249
$855$	5.12311	0.175207
$856$	0	0
$857$	35.9309	1.22738	0.613688	−	0.789549i	$-0.289685\pi$
$857$	35.9309	1.22738	0.613688	+	0.789549i	$0.289685\pi$
$858$	1.12311	0.0383422
$859$	31.8617	1.08711	0.543554	−	0.839374i	$-0.317078\pi$
$859$	31.8617	1.08711	0.543554	+	0.839374i	$0.317078\pi$
$860$	0.876894	0.0299018
$861$	−12.6847	−0.432292
$862$	60.6004	2.06406
$863$	−30.5464	−1.03981	−0.519906	−	0.854224i	$-0.674033\pi$
$863$	−30.5464	−1.03981	−0.519906	+	0.854224i	$0.674033\pi$
$864$	8.00000	0.272166
$865$	−25.7386	−0.875140
$866$	68.1080	2.31440
$867$	−7.24621	−0.246094
$868$	7.12311	0.241774
$869$	−6.24621	−0.211888
$870$	9.36932	0.317650
$871$	0.684658	0.0231988
$872$	0	0
$873$	15.5616	0.526679
$874$	−20.4924	−0.693167
$875$	3.56155	0.120402
$876$	1.12311	0.0379462
$877$	−14.9848	−0.506002	−0.253001	−	0.967466i	$-0.581418\pi$
$877$	−14.9848	−0.506002	−0.253001	+	0.967466i	$0.581418\pi$
$878$	52.1080	1.75856
$879$	−1.12311	−0.0378814
$880$	−2.24621	−0.0757198
$881$	7.80776	0.263050	0.131525	−	0.991313i	$-0.458013\pi$
$881$	7.80776	0.263050	0.131525	+	0.991313i	$0.458013\pi$
$882$	−11.3693	−0.382825
$883$	0.684658	0.0230406	0.0115203	−	0.999934i	$-0.496333\pi$
$883$	0.684658	0.0230406	0.0115203	+	0.999934i	$0.496333\pi$
$884$	6.24621	0.210083
$885$	−5.80776	−0.195226
$886$	24.7386	0.831111
$887$	−11.8078	−0.396466	−0.198233	−	0.980155i	$-0.563520\pi$
$887$	−11.8078	−0.396466	−0.198233	+	0.980155i	$0.563520\pi$
$888$	0	0
$889$	−64.3002	−2.15656
$890$	−25.1231	−0.842128
$891$	−0.561553	−0.0188127
$892$	−34.1080	−1.14202
$893$	32.0000	1.07084
$894$	35.1231	1.17469
$895$	15.8078	0.528395
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	24.4924	0.817323
$899$	−4.68466	−0.156242
$900$	2.00000	0.0666667
$901$	−22.2462	−0.741129
$902$	4.00000	0.133185
$903$	1.56155	0.0519652
$904$	0	0
$905$	2.24621	0.0746666
$906$	−9.50758	−0.315868
$907$	−8.31534	−0.276106	−0.138053	−	0.990425i	$-0.544084\pi$
$907$	−8.31534	−0.276106	−0.138053	+	0.990425i	$0.544084\pi$
$908$	9.75379	0.323691
$909$	−8.56155	−0.283969
$910$	−7.12311	−0.236129
$911$	−30.9309	−1.02479	−0.512393	−	0.858751i	$-0.671241\pi$
$911$	−30.9309	−1.02479	−0.512393	+	0.858751i	$0.671241\pi$
$912$	20.4924	0.678572
$913$	9.86174	0.326376
$914$	56.3542	1.86403
$915$	−6.00000	−0.198354
$916$	23.8617	0.788414
$917$	−30.4924	−1.00695
$918$	−6.24621	−0.206156
$919$	−41.0000	−1.35247	−0.676233	−	0.736688i	$-0.736389\pi$
$919$	−41.0000	−1.35247	−0.676233	+	0.736688i	$0.736389\pi$
$920$	0	0
$921$	14.2462	0.469429
$922$	40.6307	1.33810
$923$	2.24621	0.0739349
$924$	4.00000	0.131590
$925$	8.56155	0.281502
$926$	1.61553	0.0530895
$927$	10.8078	0.354974
$928$	37.4773	1.23025
$929$	−7.19224	−0.235970	−0.117985	−	0.993015i	$-0.537643\pi$
$929$	−7.19224	−0.235970	−0.117985	+	0.993015i	$0.537643\pi$
$930$	−2.00000	−0.0655826
$931$	−29.1231	−0.954471
$932$	33.2311	1.08852
$933$	13.9309	0.456076
$934$	27.1231	0.887495
$935$	1.75379	0.0573550
$936$	0	0
$937$	−1.82292	−0.0595522	−0.0297761	−	0.999557i	$-0.509479\pi$
$937$	−1.82292	−0.0595522	−0.0297761	+	0.999557i	$0.509479\pi$
$938$	4.87689	0.159236
$939$	9.80776	0.320064
$940$	12.4924	0.407458
$941$	−46.6695	−1.52138	−0.760691	−	0.649114i	$-0.775140\pi$
$941$	−46.6695	−1.52138	−0.760691	+	0.649114i	$0.775140\pi$
$942$	−22.8769	−0.745369
$943$	−7.12311	−0.231960
$944$	−23.2311	−0.756106
$945$	3.56155	0.115857
$946$	−0.492423	−0.0160100
$947$	37.9309	1.23259	0.616294	−	0.787516i	$-0.288633\pi$
$947$	37.9309	1.23259	0.616294	+	0.787516i	$0.288633\pi$
$948$	22.2462	0.722523
$949$	0.561553	0.0182288
$950$	10.2462	0.332431
$951$	−27.3693	−0.887511
$952$	0	0
$953$	−18.7386	−0.607004	−0.303502	−	0.952831i	$-0.598156\pi$
$953$	−18.7386	−0.607004	−0.303502	+	0.952831i	$0.598156\pi$
$954$	14.2462	0.461238
$955$	2.31534	0.0749227
$956$	−50.7386	−1.64100
$957$	−2.63068	−0.0850379
$958$	−74.3542	−2.40227
$959$	−57.8617	−1.86845
$960$	8.00000	0.258199
$961$	1.00000	0.0322581
$962$	−17.1231	−0.552071
$963$	−10.3693	−0.334147
$964$	13.3693	0.430597
$965$	6.43845	0.207261
$966$	−14.2462	−0.458364
$967$	33.3002	1.07086	0.535431	−	0.844579i	$-0.320149\pi$
$967$	33.3002	1.07086	0.535431	+	0.844579i	$0.320149\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	31.1231	0.999303
$971$	−22.8078	−0.731936	−0.365968	−	0.930627i	$-0.619262\pi$
$971$	−22.8078	−0.731936	−0.365968	+	0.930627i	$0.619262\pi$
$972$	2.00000	0.0641500
$973$	8.00000	0.256468
$974$	−65.9697	−2.11381
$975$	1.00000	0.0320256
$976$	−24.0000	−0.768221
$977$	8.63068	0.276120	0.138060	−	0.990424i	$-0.455913\pi$
$977$	8.63068	0.276120	0.138060	+	0.990424i	$0.455913\pi$
$978$	48.9848	1.56636
$979$	7.05398	0.225446
$980$	−11.3693	−0.363180
$981$	−10.0000	−0.319275
$982$	13.3693	0.426632
$983$	0.192236	0.00613137	0.00306569	−	0.999995i	$-0.499024\pi$
$983$	0.192236	0.00613137	0.00306569	+	0.999995i	$0.499024\pi$
$984$	0	0
$985$	1.00000	0.0318626
$986$	−29.2614	−0.931872
$987$	22.2462	0.708105
$988$	−10.2462	−0.325975
$989$	0.876894	0.0278836
$990$	−1.12311	−0.0356946
$991$	−2.63068	−0.0835664	−0.0417832	−	0.999127i	$-0.513304\pi$
$991$	−2.63068	−0.0835664	−0.0417832	+	0.999127i	$0.513304\pi$
$992$	−8.00000	−0.254000
$993$	−14.9309	−0.473817
$994$	16.0000	0.507489
$995$	15.3693	0.487240
$996$	−35.1231	−1.11292
$997$	−49.7926	−1.57695	−0.788474	−	0.615068i	$-0.789128\pi$
$997$	−49.7926	−1.57695	−0.788474	+	0.615068i	$0.789128\pi$
$998$	−28.1080	−0.889742
$999$	8.56155	0.270876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.m.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.m.1.1	✓	2	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -3.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -3.56155 q^{7} +1.00000 q^{9} +2.00000 q^{10} -0.561553 q^{11} +2.00000 q^{12} +1.00000 q^{13} +7.12311 q^{14} -1.00000 q^{15} -4.00000 q^{16} +3.12311 q^{17} -2.00000 q^{18} -5.12311 q^{19} -2.00000 q^{20} -3.56155 q^{21} +1.12311 q^{22} -2.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -7.12311 q^{28} +4.68466 q^{29} +2.00000 q^{30} -1.00000 q^{31} +8.00000 q^{32} -0.561553 q^{33} -6.24621 q^{34} +3.56155 q^{35} +2.00000 q^{36} +8.56155 q^{37} +10.2462 q^{38} +1.00000 q^{39} +3.56155 q^{41} +7.12311 q^{42} -0.438447 q^{43} -1.12311 q^{44} -1.00000 q^{45} +4.00000 q^{46} -6.24621 q^{47} -4.00000 q^{48} +5.68466 q^{49} -2.00000 q^{50} +3.12311 q^{51} +2.00000 q^{52} -7.12311 q^{53} -2.00000 q^{54} +0.561553 q^{55} -5.12311 q^{57} -9.36932 q^{58} +5.80776 q^{59} -2.00000 q^{60} +6.00000 q^{61} +2.00000 q^{62} -3.56155 q^{63} -8.00000 q^{64} -1.00000 q^{65} +1.12311 q^{66} +0.684658 q^{67} +6.24621 q^{68} -2.00000 q^{69} -7.12311 q^{70} +2.24621 q^{71} +0.561553 q^{73} -17.1231 q^{74} +1.00000 q^{75} -10.2462 q^{76} +2.00000 q^{77} -2.00000 q^{78} +11.1231 q^{79} +4.00000 q^{80} +1.00000 q^{81} -7.12311 q^{82} -17.5616 q^{83} -7.12311 q^{84} -3.12311 q^{85} +0.876894 q^{86} +4.68466 q^{87} -12.5616 q^{89} +2.00000 q^{90} -3.56155 q^{91} -4.00000 q^{92} -1.00000 q^{93} +12.4924 q^{94} +5.12311 q^{95} +8.00000 q^{96} +15.5616 q^{97} -11.3693 q^{98} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 - 3 * q^7 + 2 * q^9	\( 2 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{9} + 4 q^{10} + 3 q^{11} + 4 q^{12} + 2 q^{13} + 6 q^{14} - 2 q^{15} - 8 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{19} - 4 q^{20} - 3 q^{21} - 6 q^{22} - 4 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 6 q^{28} - 3 q^{29} + 4 q^{30} - 2 q^{31} + 16 q^{32} + 3 q^{33} + 4 q^{34} + 3 q^{35} + 4 q^{36} + 13 q^{37} + 4 q^{38} + 2 q^{39} + 3 q^{41} + 6 q^{42} - 5 q^{43} + 6 q^{44} - 2 q^{45} + 8 q^{46} + 4 q^{47} - 8 q^{48} - q^{49} - 4 q^{50} - 2 q^{51} + 4 q^{52} - 6 q^{53} - 4 q^{54} - 3 q^{55} - 2 q^{57} + 6 q^{58} - 9 q^{59} - 4 q^{60} + 12 q^{61} + 4 q^{62} - 3 q^{63} - 16 q^{64} - 2 q^{65} - 6 q^{66} - 11 q^{67} - 4 q^{68} - 4 q^{69} - 6 q^{70} - 12 q^{71} - 3 q^{73} - 26 q^{74} + 2 q^{75} - 4 q^{76} + 4 q^{77} - 4 q^{78} + 14 q^{79} + 8 q^{80} + 2 q^{81} - 6 q^{82} - 31 q^{83} - 6 q^{84} + 2 q^{85} + 10 q^{86} - 3 q^{87} - 21 q^{89} + 4 q^{90} - 3 q^{91} - 8 q^{92} - 2 q^{93} - 8 q^{94} + 2 q^{95} + 16 q^{96} + 27 q^{97} + 2 q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 4 * q^6 - 3 * q^7 + 2 * q^9 + 4 * q^10 + 3 * q^11 + 4 * q^12 + 2 * q^13 + 6 * q^14 - 2 * q^15 - 8 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^19 - 4 * q^20 - 3 * q^21 - 6 * q^22 - 4 * q^23 + 2 * q^25 - 4 * q^26 + 2 * q^27 - 6 * q^28 - 3 * q^29 + 4 * q^30 - 2 * q^31 + 16 * q^32 + 3 * q^33 + 4 * q^34 + 3 * q^35 + 4 * q^36 + 13 * q^37 + 4 * q^38 + 2 * q^39 + 3 * q^41 + 6 * q^42 - 5 * q^43 + 6 * q^44 - 2 * q^45 + 8 * q^46 + 4 * q^47 - 8 * q^48 - q^49 - 4 * q^50 - 2 * q^51 + 4 * q^52 - 6 * q^53 - 4 * q^54 - 3 * q^55 - 2 * q^57 + 6 * q^58 - 9 * q^59 - 4 * q^60 + 12 * q^61 + 4 * q^62 - 3 * q^63 - 16 * q^64 - 2 * q^65 - 6 * q^66 - 11 * q^67 - 4 * q^68 - 4 * q^69 - 6 * q^70 - 12 * q^71 - 3 * q^73 - 26 * q^74 + 2 * q^75 - 4 * q^76 + 4 * q^77 - 4 * q^78 + 14 * q^79 + 8 * q^80 + 2 * q^81 - 6 * q^82 - 31 * q^83 - 6 * q^84 + 2 * q^85 + 10 * q^86 - 3 * q^87 - 21 * q^89 + 4 * q^90 - 3 * q^91 - 8 * q^92 - 2 * q^93 - 8 * q^94 + 2 * q^95 + 16 * q^96 + 27 * q^97 + 2 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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