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Properties

Label 1170.2.q.c
Level $1170$
Weight $2$
Character orbit 1170.q
Analytic conductor $9.342$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1170,2,Mod(359,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.359"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,-20] 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: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{13} - 24 q^{16} - 48 q^{19} + 32 q^{25} - 8 q^{31} + 16 q^{34} - 32 q^{37} + 8 q^{40} + 80 q^{43} + 8 q^{46} - 12 q^{52} + 16 q^{55} - 24 q^{58} - 16 q^{61} - 8 q^{67} - 24 q^{70} + 48 q^{73}+ \cdots + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
359.1 −0.707107 − 0.707107i 0 1.00000i 1.15322 + 1.91575i 0 1.29303 + 1.29303i 0.707107 − 0.707107i 0 0.539189 − 2.17009i
359.2 −0.707107 − 0.707107i 0 1.00000i −2.23186 + 0.137134i 0 −3.64075 − 3.64075i 0.707107 − 0.707107i 0 1.67513 + 1.48119i
359.3 −0.707107 − 0.707107i 0 1.00000i 1.78575 − 1.34577i 0 2.75044 + 2.75044i 0.707107 − 0.707107i 0 −2.21432 − 0.311108i
359.4 −0.707107 − 0.707107i 0 1.00000i −2.23186 + 0.137134i 0 1.96562 + 1.96562i 0.707107 − 0.707107i 0 1.67513 + 1.48119i
359.5 −0.707107 − 0.707107i 0 1.00000i 1.78575 − 1.34577i 0 −0.536122 − 0.536122i 0.707107 − 0.707107i 0 −2.21432 − 0.311108i
359.6 −0.707107 − 0.707107i 0 1.00000i 1.15322 + 1.91575i 0 −1.83222 − 1.83222i 0.707107 − 0.707107i 0 0.539189 − 2.17009i
359.7 0.707107 + 0.707107i 0 1.00000i −1.78575 + 1.34577i 0 2.75044 + 2.75044i −0.707107 + 0.707107i 0 −2.21432 − 0.311108i
359.8 0.707107 + 0.707107i 0 1.00000i 2.23186 − 0.137134i 0 1.96562 + 1.96562i −0.707107 + 0.707107i 0 1.67513 + 1.48119i
359.9 0.707107 + 0.707107i 0 1.00000i −1.15322 − 1.91575i 0 1.29303 + 1.29303i −0.707107 + 0.707107i 0 0.539189 − 2.17009i
359.10 0.707107 + 0.707107i 0 1.00000i 2.23186 − 0.137134i 0 −3.64075 − 3.64075i −0.707107 + 0.707107i 0 1.67513 + 1.48119i
359.11 0.707107 + 0.707107i 0 1.00000i −1.15322 − 1.91575i 0 −1.83222 − 1.83222i −0.707107 + 0.707107i 0 0.539189 − 2.17009i
359.12 0.707107 + 0.707107i 0 1.00000i −1.78575 + 1.34577i 0 −0.536122 − 0.536122i −0.707107 + 0.707107i 0 −2.21432 − 0.311108i
629.1 −0.707107 + 0.707107i 0 − 1.00000i 1.15322 − 1.91575i 0 1.29303 − 1.29303i 0.707107 + 0.707107i 0 0.539189 + 2.17009i
629.2 −0.707107 + 0.707107i 0 − 1.00000i −2.23186 − 0.137134i 0 −3.64075 + 3.64075i 0.707107 + 0.707107i 0 1.67513 − 1.48119i
629.3 −0.707107 + 0.707107i 0 − 1.00000i 1.78575 + 1.34577i 0 2.75044 − 2.75044i 0.707107 + 0.707107i 0 −2.21432 + 0.311108i
629.4 −0.707107 + 0.707107i 0 − 1.00000i −2.23186 − 0.137134i 0 1.96562 − 1.96562i 0.707107 + 0.707107i 0 1.67513 − 1.48119i
629.5 −0.707107 + 0.707107i 0 − 1.00000i 1.78575 + 1.34577i 0 −0.536122 + 0.536122i 0.707107 + 0.707107i 0 −2.21432 + 0.311108i
629.6 −0.707107 + 0.707107i 0 − 1.00000i 1.15322 − 1.91575i 0 −1.83222 + 1.83222i 0.707107 + 0.707107i 0 0.539189 + 2.17009i
629.7 0.707107 − 0.707107i 0 − 1.00000i −1.78575 − 1.34577i 0 2.75044 − 2.75044i −0.707107 − 0.707107i 0 −2.21432 + 0.311108i
629.8 0.707107 − 0.707107i 0 − 1.00000i 2.23186 + 0.137134i 0 1.96562 − 1.96562i −0.707107 − 0.707107i 0 1.67513 − 1.48119i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 359.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
65.g odd 4 1 inner
195.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.q.c ✓ 24
3.b odd 2 1 inner 1170.2.q.c ✓ 24
5.b even 2 1 1170.2.q.d yes 24
13.d odd 4 1 1170.2.q.d yes 24
15.d odd 2 1 1170.2.q.d yes 24
39.f even 4 1 1170.2.q.d yes 24
65.g odd 4 1 inner 1170.2.q.c ✓ 24
195.n even 4 1 inner 1170.2.q.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.q.c ✓ 24 1.a even 1 1 trivial
1170.2.q.c ✓ 24 3.b odd 2 1 inner
1170.2.q.c ✓ 24 65.g odd 4 1 inner
1170.2.q.c ✓ 24 195.n even 4 1 inner
1170.2.q.d yes 24 5.b even 2 1
1170.2.q.d yes 24 13.d odd 4 1
1170.2.q.d yes 24 15.d odd 2 1
1170.2.q.d yes 24 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{12} - 32 T_{7}^{9} + 524 T_{7}^{8} - 752 T_{7}^{7} + 512 T_{7}^{6} - 224 T_{7}^{5} + 23344 T_{7}^{4} + \cdots + 40000 \) Copy content Toggle raw display
\( T_{37}^{12} + 16 T_{37}^{11} + 128 T_{37}^{10} + 176 T_{37}^{9} + 1548 T_{37}^{8} + 29088 T_{37}^{7} + \cdots + 7884864 \) Copy content Toggle raw display