Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2070,2,Mod(1,2070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,3,3,0,3,-3,0,-3,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.43163\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.08719 q^{7} -1.00000 q^{8} -1.00000 q^{10} +6.46926 q^{11} +3.95044 q^{13} -3.08719 q^{14} +1.00000 q^{16} +3.43163 q^{17} +3.08719 q^{19} +1.00000 q^{20} -6.46926 q^{22} +1.00000 q^{23} +1.00000 q^{25} -3.95044 q^{26} +3.08719 q^{28} -0.863254 q^{29} -5.95044 q^{31} -1.00000 q^{32} -3.43163 q^{34} +3.08719 q^{35} -7.03763 q^{37} -3.08719 q^{38} -1.00000 q^{40} -5.60601 q^{41} +8.00000 q^{43} +6.46926 q^{44} -1.00000 q^{46} -3.90089 q^{47} +2.53074 q^{49} -1.00000 q^{50} +3.95044 q^{52} +6.00000 q^{53} +6.46926 q^{55} -3.08719 q^{56} +0.863254 q^{58} -6.86325 q^{59} -13.5069 q^{61} +5.95044 q^{62} +1.00000 q^{64} +3.95044 q^{65} -10.0753 q^{67} +3.43163 q^{68} -3.08719 q^{70} -2.56837 q^{71} +5.90089 q^{73} +7.03763 q^{74} +3.08719 q^{76} +19.9718 q^{77} +15.8018 q^{79} +1.00000 q^{80} +5.60601 q^{82} -9.03763 q^{83} +3.43163 q^{85} -8.00000 q^{86} -6.46926 q^{88} -16.7641 q^{89} +12.1958 q^{91} +1.00000 q^{92} +3.90089 q^{94} +3.08719 q^{95} -14.2949 q^{97} -2.53074 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.08719 1.16685 0.583424 − 0.812168i \(-0.301713\pi\)
0.583424 + 0.812168i \(0.301713\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 6.46926 1.95056 0.975278 − 0.220983i \(-0.0709265\pi\)
0.975278 + 0.220983i \(0.0709265\pi\)
\(12\) 0 0
\(13\) 3.95044 1.09566 0.547828 − 0.836591i \(-0.315455\pi\)
0.547828 + 0.836591i \(0.315455\pi\)
\(14\) −3.08719 −0.825086
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.43163 0.832292 0.416146 − 0.909298i \(-0.363381\pi\)
0.416146 + 0.909298i \(0.363381\pi\)
\(18\) 0 0
\(19\) 3.08719 0.708250 0.354125 − 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −6.46926 −1.37925
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.95044 −0.774746
\(27\) 0 0
\(28\) 3.08719 0.583424
\(29\) −0.863254 −0.160302 −0.0801511 − 0.996783i \(-0.525540\pi\)
−0.0801511 + 0.996783i \(0.525540\pi\)
\(30\) 0 0
\(31\) −5.95044 −1.06873 −0.534366 − 0.845253i \(-0.679449\pi\)
−0.534366 + 0.845253i \(0.679449\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.43163 −0.588519
\(35\) 3.08719 0.521830
\(36\) 0 0
\(37\) −7.03763 −1.15698 −0.578490 − 0.815690i \(-0.696358\pi\)
−0.578490 + 0.815690i \(0.696358\pi\)
\(38\) −3.08719 −0.500808
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −5.60601 −0.875511 −0.437756 − 0.899094i \(-0.644226\pi\)
−0.437756 + 0.899094i \(0.644226\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.46926 0.975278
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −3.90089 −0.569003 −0.284501 − 0.958676i \(-0.591828\pi\)
−0.284501 + 0.958676i \(0.591828\pi\)
\(48\) 0 0
\(49\) 2.53074 0.361534
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.95044 0.547828
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 6.46926 0.872315
\(56\) −3.08719 −0.412543
\(57\) 0 0
\(58\) 0.863254 0.113351
\(59\) −6.86325 −0.893520 −0.446760 − 0.894654i \(-0.647422\pi\)
−0.446760 + 0.894654i \(0.647422\pi\)
\(60\) 0 0
\(61\) −13.5069 −1.72938 −0.864690 − 0.502305i \(-0.832485\pi\)
−0.864690 + 0.502305i \(0.832485\pi\)
\(62\) 5.95044 0.755707
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.95044 0.489992
\(66\) 0 0
\(67\) −10.0753 −1.23089 −0.615445 − 0.788180i \(-0.711024\pi\)
−0.615445 + 0.788180i \(0.711024\pi\)
\(68\) 3.43163 0.416146
\(69\) 0 0
\(70\) −3.08719 −0.368990
\(71\) −2.56837 −0.304810 −0.152405 − 0.988318i \(-0.548702\pi\)
−0.152405 + 0.988318i \(0.548702\pi\)
\(72\) 0 0
\(73\) 5.90089 0.690647 0.345323 − 0.938484i \(-0.387769\pi\)
0.345323 + 0.938484i \(0.387769\pi\)
\(74\) 7.03763 0.818108
\(75\) 0 0
\(76\) 3.08719 0.354125
\(77\) 19.9718 2.27600
\(78\) 0 0
\(79\) 15.8018 1.77784 0.888919 − 0.458064i \(-0.151457\pi\)
0.888919 + 0.458064i \(0.151457\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 5.60601 0.619080
\(83\) −9.03763 −0.992009 −0.496005 − 0.868320i \(-0.665200\pi\)
−0.496005 + 0.868320i \(0.665200\pi\)
\(84\) 0 0
\(85\) 3.43163 0.372212
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −6.46926 −0.689625
\(89\) −16.7641 −1.77700 −0.888498 − 0.458881i \(-0.848250\pi\)
−0.888498 + 0.458881i \(0.848250\pi\)
\(90\) 0 0
\(91\) 12.1958 1.27846
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 3.90089 0.402346
\(95\) 3.08719 0.316739
\(96\) 0 0
\(97\) −14.2949 −1.45143 −0.725713 − 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(98\) −2.53074 −0.255643
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2070.2.a.z.1.2 3
3.2 odd 2 230.2.a.d.1.2 ✓ 3
12.11 even 2 1840.2.a.r.1.2 3
15.2 even 4 1150.2.b.j.599.5 6
15.8 even 4 1150.2.b.j.599.2 6
15.14 odd 2 1150.2.a.q.1.2 3
24.5 odd 2 7360.2.a.bz.1.2 3
24.11 even 2 7360.2.a.ce.1.2 3
60.59 even 2 9200.2.a.cf.1.2 3
69.68 even 2 5290.2.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 ✓ 3 3.2 odd 2
1150.2.a.q.1.2 3 15.14 odd 2
1150.2.b.j.599.2 6 15.8 even 4
1150.2.b.j.599.5 6 15.2 even 4
1840.2.a.r.1.2 3 12.11 even 2
2070.2.a.z.1.2 3 1.1 even 1 trivial
5290.2.a.r.1.2 3 69.68 even 2
7360.2.a.bz.1.2 3 24.5 odd 2
7360.2.a.ce.1.2 3 24.11 even 2
9200.2.a.cf.1.2 3 60.59 even 2