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Properties

Label 371.2.g.e
Level $371$
Weight $2$
Character orbit 371.g
Analytic conductor $2.962$
Analytic rank $0$
Dimension $52$
Inner twists $4$

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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] = [371,2,Mod(76,371)] 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("371.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [52,-8,0,0,0] 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: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{2} + 20 q^{8} - 50 q^{14} - 24 q^{15} - 64 q^{16} - 8 q^{18} - 30 q^{21} + 40 q^{22} - 20 q^{23} + 80 q^{30} + 12 q^{32} - 24 q^{35} - 16 q^{36} + 4 q^{39} - 80 q^{42} + 64 q^{44} + 88 q^{46}+ \cdots + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −1.95826 + 1.95826i −0.496243 − 0.496243i − 5.66960i 1.61321 − 1.61321i 1.94355 −2.64570 + 0.0165586i 7.18605 + 7.18605i − 2.50749i 6.31817i
76.2 −1.95826 + 1.95826i 0.496243 + 0.496243i − 5.66960i −1.61321 + 1.61321i −1.94355 2.64570 + 0.0165586i 7.18605 + 7.18605i − 2.50749i − 6.31817i
76.3 −1.64562 + 1.64562i −2.31665 − 2.31665i − 3.41613i 0.901960 − 0.901960i 7.62466 1.63697 + 2.07854i 2.33042 + 2.33042i 7.73375i 2.96857i
76.4 −1.64562 + 1.64562i 2.31665 + 2.31665i − 3.41613i −0.901960 + 0.901960i −7.62466 −1.63697 + 2.07854i 2.33042 + 2.33042i 7.73375i − 2.96857i
76.5 −1.43389 + 1.43389i −0.867405 − 0.867405i − 2.11208i 0.743483 − 0.743483i 2.48753 −2.00974 + 1.72074i 0.160714 + 0.160714i − 1.49522i 2.13215i
76.6 −1.43389 + 1.43389i 0.867405 + 0.867405i − 2.11208i −0.743483 + 0.743483i −2.48753 2.00974 + 1.72074i 0.160714 + 0.160714i − 1.49522i − 2.13215i
76.7 −1.11032 + 1.11032i −0.129800 − 0.129800i − 0.465639i −2.62208 + 2.62208i 0.288240 −1.65924 + 2.06080i −1.70364 − 1.70364i − 2.96630i − 5.82272i
76.8 −1.11032 + 1.11032i 0.129800 + 0.129800i − 0.465639i 2.62208 − 2.62208i −0.288240 1.65924 + 2.06080i −1.70364 − 1.70364i − 2.96630i 5.82272i
76.9 −0.786981 + 0.786981i −1.83124 − 1.83124i 0.761322i 2.54246 − 2.54246i 2.88231 −0.989787 − 2.45363i −2.17311 − 2.17311i 3.70691i 4.00174i
76.10 −0.786981 + 0.786981i 1.83124 + 1.83124i 0.761322i −2.54246 + 2.54246i −2.88231 0.989787 − 2.45363i −2.17311 − 2.17311i 3.70691i − 4.00174i
76.11 −0.574858 + 0.574858i −1.67128 − 1.67128i 1.33908i −1.93555 + 1.93555i 1.92150 2.28962 + 1.32577i −1.91949 − 1.91949i 2.58636i − 2.22533i
76.12 −0.574858 + 0.574858i 1.67128 + 1.67128i 1.33908i 1.93555 − 1.93555i −1.92150 −2.28962 + 1.32577i −1.91949 − 1.91949i 2.58636i 2.22533i
76.13 −0.216432 + 0.216432i −1.60791 − 1.60791i 1.90631i 0.976675 − 0.976675i 0.696006 −0.983051 + 2.45634i −0.845450 − 0.845450i 2.17077i 0.422766i
76.14 −0.216432 + 0.216432i 1.60791 + 1.60791i 1.90631i −0.976675 + 0.976675i −0.696006 0.983051 + 2.45634i −0.845450 − 0.845450i 2.17077i − 0.422766i
76.15 0.286724 − 0.286724i −1.19905 − 1.19905i 1.83558i −0.104585 + 0.104585i −0.687596 −2.64286 + 0.123581i 1.09975 + 1.09975i − 0.124543i 0.0599743i
76.16 0.286724 − 0.286724i 1.19905 + 1.19905i 1.83558i 0.104585 − 0.104585i 0.687596 2.64286 + 0.123581i 1.09975 + 1.09975i − 0.124543i − 0.0599743i
76.17 0.419884 − 0.419884i −2.26807 − 2.26807i 1.64740i −1.60799 + 1.60799i −1.90465 0.287913 − 2.63004i 1.53148 + 1.53148i 7.28828i 1.35034i
76.18 0.419884 − 0.419884i 2.26807 + 2.26807i 1.64740i 1.60799 − 1.60799i 1.90465 −0.287913 − 2.63004i 1.53148 + 1.53148i 7.28828i − 1.35034i
76.19 0.890212 − 0.890212i −1.59119 − 1.59119i 0.415046i 2.14326 − 2.14326i −2.83300 2.63997 − 0.174863i 2.14990 + 2.14990i 2.06379i − 3.81592i
76.20 0.890212 − 0.890212i 1.59119 + 1.59119i 0.415046i −2.14326 + 2.14326i 2.83300 −2.63997 − 0.174863i 2.14990 + 2.14990i 2.06379i 3.81592i
See all 52 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 76.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
53.c odd 4 1 inner
371.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.2.g.e ✓ 52
7.b odd 2 1 inner 371.2.g.e ✓ 52
53.c odd 4 1 inner 371.2.g.e ✓ 52
371.g even 4 1 inner 371.2.g.e ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.2.g.e ✓ 52 1.a even 1 1 trivial
371.2.g.e ✓ 52 7.b odd 2 1 inner
371.2.g.e ✓ 52 53.c odd 4 1 inner
371.2.g.e ✓ 52 371.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(371, [\chi])\):

\( T_{2}^{26} + 4 T_{2}^{25} + 8 T_{2}^{24} + 2 T_{2}^{23} + 66 T_{2}^{22} + 266 T_{2}^{21} + 538 T_{2}^{20} + \cdots + 128 \) Copy content Toggle raw display
\( T_{5}^{52} + 605 T_{5}^{48} + 144280 T_{5}^{44} + 17586938 T_{5}^{40} + 1200843950 T_{5}^{36} + \cdots + 20694548736 \) Copy content Toggle raw display