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Properties

Label 2370.2.d.f
Level $2370$
Weight $2$
Character orbit 2370.d
Analytic conductor $18.925$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2370,2,Mod(949,2370)] 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("2370.949"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2370 = 2 \cdot 3 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2370.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,-24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.9245452790\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} - 2 q^{10} + 16 q^{11} - 8 q^{14} - 2 q^{15} + 24 q^{16} - 2 q^{19} - 8 q^{21} + 24 q^{24} - 6 q^{25} + 18 q^{26} - 40 q^{29} - 6 q^{35} + 24 q^{36} + 18 q^{39} + 2 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1 − 1.00000i − 1.00000i −1.00000 −2.23590 + 0.0277372i −1.00000 3.93532i 1.00000i −1.00000 0.0277372 + 2.23590i
949.2 − 1.00000i − 1.00000i −1.00000 −2.05940 − 0.871133i −1.00000 − 3.42851i 1.00000i −1.00000 −0.871133 + 2.05940i
949.3 − 1.00000i − 1.00000i −1.00000 −1.83569 + 1.27681i −1.00000 2.44522i 1.00000i −1.00000 1.27681 + 1.83569i
949.4 − 1.00000i − 1.00000i −1.00000 −1.15652 + 1.91376i −1.00000 − 0.907474i 1.00000i −1.00000 1.91376 + 1.15652i
949.5 − 1.00000i − 1.00000i −1.00000 −0.507244 − 2.17777i −1.00000 0.701682i 1.00000i −1.00000 −2.17777 + 0.507244i
949.6 − 1.00000i − 1.00000i −1.00000 −0.0997844 − 2.23384i −1.00000 − 2.82824i 1.00000i −1.00000 −2.23384 + 0.0997844i
949.7 − 1.00000i − 1.00000i −1.00000 0.128302 + 2.23238i −1.00000 − 4.74001i 1.00000i −1.00000 2.23238 − 0.128302i
949.8 − 1.00000i − 1.00000i −1.00000 0.355189 − 2.20768i −1.00000 − 4.52679i 1.00000i −1.00000 −2.20768 − 0.355189i
949.9 − 1.00000i − 1.00000i −1.00000 1.48516 + 1.67162i −1.00000 0.463883i 1.00000i −1.00000 1.67162 − 1.48516i
949.10 − 1.00000i − 1.00000i −1.00000 1.58821 − 1.57404i −1.00000 4.60468i 1.00000i −1.00000 −1.57404 − 1.58821i
949.11 − 1.00000i − 1.00000i −1.00000 2.11101 + 0.737327i −1.00000 1.37453i 1.00000i −1.00000 0.737327 − 2.11101i
949.12 − 1.00000i − 1.00000i −1.00000 2.22667 + 0.204823i −1.00000 − 1.09429i 1.00000i −1.00000 0.204823 − 2.22667i
949.13 1.00000i 1.00000i −1.00000 −2.23590 − 0.0277372i −1.00000 − 3.93532i − 1.00000i −1.00000 0.0277372 − 2.23590i
949.14 1.00000i 1.00000i −1.00000 −2.05940 + 0.871133i −1.00000 3.42851i − 1.00000i −1.00000 −0.871133 − 2.05940i
949.15 1.00000i 1.00000i −1.00000 −1.83569 − 1.27681i −1.00000 − 2.44522i − 1.00000i −1.00000 1.27681 − 1.83569i
949.16 1.00000i 1.00000i −1.00000 −1.15652 − 1.91376i −1.00000 0.907474i − 1.00000i −1.00000 1.91376 − 1.15652i
949.17 1.00000i 1.00000i −1.00000 −0.507244 + 2.17777i −1.00000 − 0.701682i − 1.00000i −1.00000 −2.17777 − 0.507244i
949.18 1.00000i 1.00000i −1.00000 −0.0997844 + 2.23384i −1.00000 2.82824i − 1.00000i −1.00000 −2.23384 − 0.0997844i
949.19 1.00000i 1.00000i −1.00000 0.128302 − 2.23238i −1.00000 4.74001i − 1.00000i −1.00000 2.23238 + 0.128302i
949.20 1.00000i 1.00000i −1.00000 0.355189 + 2.20768i −1.00000 4.52679i − 1.00000i −1.00000 −2.20768 + 0.355189i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 949.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2370.2.d.f ✓ 24
5.b even 2 1 inner 2370.2.d.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2370.2.d.f ✓ 24 1.a even 1 1 trivial
2370.2.d.f ✓ 24 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2370, [\chi])\):

\( T_{7}^{24} + 110 T_{7}^{22} + 5121 T_{7}^{20} + 131494 T_{7}^{18} + 2036244 T_{7}^{16} + 19566638 T_{7}^{14} + \cdots + 16777216 \) Copy content Toggle raw display
\( T_{11}^{12} - 8 T_{11}^{11} - 52 T_{11}^{10} + 554 T_{11}^{9} + 291 T_{11}^{8} - 12370 T_{11}^{7} + \cdots + 61440 \) Copy content Toggle raw display