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Properties

Label 315.5.e.g
Level $315$
Weight $5$
Character orbit 315.e
Analytic conductor $32.562$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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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] = [315,5,Mod(244,315)] 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("315.244"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,-256,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 256 q^{4} + 3152 q^{16} - 40 q^{25} + 3072 q^{46} + 4448 q^{49} + 32 q^{64} + 13560 q^{70} - 28256 q^{79} + 41560 q^{85} - 59600 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1 − 0.475469i 0 15.7739 −11.5269 + 22.1840i 0 30.4287 + 38.4070i − 15.1075i 0 10.5478 + 5.48070i
244.2 0.475469i 0 15.7739 −11.5269 − 22.1840i 0 30.4287 − 38.4070i 15.1075i 0 10.5478 − 5.48070i
244.3 − 7.33947i 0 −37.8679 22.3714 + 11.1589i 0 44.2569 − 21.0315i 160.499i 0 81.9001 − 164.194i
244.4 7.33947i 0 −37.8679 22.3714 − 11.1589i 0 44.2569 + 21.0315i − 160.499i 0 81.9001 + 164.194i
244.5 − 3.31656i 0 5.00044 24.0676 − 6.76400i 0 −40.5894 − 27.4499i − 69.6492i 0 −22.4332 − 79.8215i
244.6 3.31656i 0 5.00044 24.0676 + 6.76400i 0 −40.5894 + 27.4499i 69.6492i 0 −22.4332 + 79.8215i
244.7 − 3.31656i 0 5.00044 −24.0676 − 6.76400i 0 −40.5894 + 27.4499i − 69.6492i 0 −22.4332 + 79.8215i
244.8 3.31656i 0 5.00044 −24.0676 + 6.76400i 0 −40.5894 − 27.4499i 69.6492i 0 −22.4332 − 79.8215i
244.9 − 5.55936i 0 −14.9065 −5.90780 + 24.2919i 0 −23.4077 − 43.0474i − 6.07916i 0 135.048 + 32.8436i
244.10 5.55936i 0 −14.9065 −5.90780 − 24.2919i 0 −23.4077 + 43.0474i 6.07916i 0 135.048 − 32.8436i
244.11 − 7.33947i 0 −37.8679 −22.3714 − 11.1589i 0 −44.2569 − 21.0315i 160.499i 0 −81.9001 + 164.194i
244.12 7.33947i 0 −37.8679 −22.3714 + 11.1589i 0 −44.2569 + 21.0315i − 160.499i 0 −81.9001 − 164.194i
244.13 − 5.55936i 0 −14.9065 5.90780 − 24.2919i 0 23.4077 − 43.0474i − 6.07916i 0 −135.048 − 32.8436i
244.14 5.55936i 0 −14.9065 5.90780 + 24.2919i 0 23.4077 + 43.0474i 6.07916i 0 −135.048 + 32.8436i
244.15 − 0.475469i 0 15.7739 11.5269 + 22.1840i 0 30.4287 − 38.4070i − 15.1075i 0 10.5478 − 5.48070i
244.16 0.475469i 0 15.7739 11.5269 − 22.1840i 0 30.4287 + 38.4070i 15.1075i 0 10.5478 + 5.48070i
244.17 − 0.475469i 0 15.7739 11.5269 − 22.1840i 0 −30.4287 + 38.4070i − 15.1075i 0 −10.5478 − 5.48070i
244.18 0.475469i 0 15.7739 11.5269 + 22.1840i 0 −30.4287 − 38.4070i 15.1075i 0 −10.5478 + 5.48070i
244.19 − 5.55936i 0 −14.9065 5.90780 + 24.2919i 0 −23.4077 + 43.0474i − 6.07916i 0 135.048 − 32.8436i
244.20 5.55936i 0 −14.9065 5.90780 − 24.2919i 0 −23.4077 − 43.0474i 6.07916i 0 135.048 + 32.8436i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.g ✓ 32
3.b odd 2 1 inner 315.5.e.g ✓ 32
5.b even 2 1 inner 315.5.e.g ✓ 32
7.b odd 2 1 inner 315.5.e.g ✓ 32
15.d odd 2 1 inner 315.5.e.g ✓ 32
21.c even 2 1 inner 315.5.e.g ✓ 32
35.c odd 2 1 inner 315.5.e.g ✓ 32
105.g even 2 1 inner 315.5.e.g ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.5.e.g ✓ 32 1.a even 1 1 trivial
315.5.e.g ✓ 32 3.b odd 2 1 inner
315.5.e.g ✓ 32 5.b even 2 1 inner
315.5.e.g ✓ 32 7.b odd 2 1 inner
315.5.e.g ✓ 32 15.d odd 2 1 inner
315.5.e.g ✓ 32 21.c even 2 1 inner
315.5.e.g ✓ 32 35.c odd 2 1 inner
315.5.e.g ✓ 32 105.g 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 \(S_{5}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{8} + 96T_{2}^{6} + 2619T_{2}^{4} + 18900T_{2}^{2} + 4140 \) Copy content Toggle raw display
\( T_{13}^{8} - 106930T_{13}^{6} + 3637379760T_{13}^{4} - 39983264658400T_{13}^{2} + 644668884160000 \) Copy content Toggle raw display