Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3042,2,Mod(1,3042)] 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("3042.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,2,0,-2,1,0,2,0,0,0,-2,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3042.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{28} +10.0000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +8.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{53} -2.00000 q^{56} +10.0000 q^{58} -4.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +2.00000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -4.00000 q^{73} +8.00000 q^{74} +6.00000 q^{76} +2.00000 q^{80} +10.0000 q^{82} -4.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} +12.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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 3042.2.a.n.1.1 1
3.2 odd 2 1014.2.a.b.1.1 1
12.11 even 2 8112.2.a.g.1.1 1
13.5 odd 4 234.2.b.a.181.1 2
13.8 odd 4 234.2.b.a.181.2 2
13.12 even 2 3042.2.a.c.1.1 1
39.2 even 12 1014.2.i.c.823.1 4
39.5 even 4 78.2.b.a.25.2 yes 2
39.8 even 4 78.2.b.a.25.1 ✓ 2
39.11 even 12 1014.2.i.c.823.2 4
39.17 odd 6 1014.2.e.b.991.1 2
39.20 even 12 1014.2.i.c.361.1 4
39.23 odd 6 1014.2.e.b.529.1 2
39.29 odd 6 1014.2.e.e.529.1 2
39.32 even 12 1014.2.i.c.361.2 4
39.35 odd 6 1014.2.e.e.991.1 2
39.38 odd 2 1014.2.a.g.1.1 1
52.31 even 4 1872.2.c.b.1585.1 2
52.47 even 4 1872.2.c.b.1585.2 2
156.47 odd 4 624.2.c.a.337.1 2
156.83 odd 4 624.2.c.a.337.2 2
156.155 even 2 8112.2.a.j.1.1 1
195.8 odd 4 1950.2.f.d.649.2 2
195.44 even 4 1950.2.b.c.1351.1 2
195.47 odd 4 1950.2.f.g.649.1 2
195.83 odd 4 1950.2.f.g.649.2 2
195.122 odd 4 1950.2.f.d.649.1 2
195.164 even 4 1950.2.b.c.1351.2 2
273.83 odd 4 3822.2.c.d.883.2 2
273.125 odd 4 3822.2.c.d.883.1 2
312.5 even 4 2496.2.c.f.961.1 2
312.83 odd 4 2496.2.c.m.961.1 2
312.125 even 4 2496.2.c.f.961.2 2
312.203 odd 4 2496.2.c.m.961.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 39.8 even 4
78.2.b.a.25.2 yes 2 39.5 even 4
234.2.b.a.181.1 2 13.5 odd 4
234.2.b.a.181.2 2 13.8 odd 4
624.2.c.a.337.1 2 156.47 odd 4
624.2.c.a.337.2 2 156.83 odd 4
1014.2.a.b.1.1 1 3.2 odd 2
1014.2.a.g.1.1 1 39.38 odd 2
1014.2.e.b.529.1 2 39.23 odd 6
1014.2.e.b.991.1 2 39.17 odd 6
1014.2.e.e.529.1 2 39.29 odd 6
1014.2.e.e.991.1 2 39.35 odd 6
1014.2.i.c.361.1 4 39.20 even 12
1014.2.i.c.361.2 4 39.32 even 12
1014.2.i.c.823.1 4 39.2 even 12
1014.2.i.c.823.2 4 39.11 even 12
1872.2.c.b.1585.1 2 52.31 even 4
1872.2.c.b.1585.2 2 52.47 even 4
1950.2.b.c.1351.1 2 195.44 even 4
1950.2.b.c.1351.2 2 195.164 even 4
1950.2.f.d.649.1 2 195.122 odd 4
1950.2.f.d.649.2 2 195.8 odd 4
1950.2.f.g.649.1 2 195.47 odd 4
1950.2.f.g.649.2 2 195.83 odd 4
2496.2.c.f.961.1 2 312.5 even 4
2496.2.c.f.961.2 2 312.125 even 4
2496.2.c.m.961.1 2 312.83 odd 4
2496.2.c.m.961.2 2 312.203 odd 4
3042.2.a.c.1.1 1 13.12 even 2
3042.2.a.n.1.1 1 1.1 even 1 trivial
3822.2.c.d.883.1 2 273.125 odd 4
3822.2.c.d.883.2 2 273.83 odd 4
8112.2.a.g.1.1 1 12.11 even 2
8112.2.a.j.1.1 1 156.155 even 2