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Properties

Label 1512.2.c.g
Level $1512$
Weight $2$
Character orbit 1512.c
Analytic conductor $12.073$
Analytic rank $0$
Dimension $24$
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] = [1512,2,Mod(757,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) 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("1512.757"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0733807856\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} + 24 q^{7} - 16 q^{10} + 2 q^{16} + 16 q^{22} - 24 q^{25} + 6 q^{28} + 8 q^{31} + 22 q^{34} + 26 q^{46} + 24 q^{49} - 6 q^{52} + 16 q^{55} - 58 q^{58} + 6 q^{64} - 16 q^{70} + 60 q^{76} + 8 q^{79}+ \cdots + 36 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1 −1.40920 − 0.118957i 0 1.97170 + 0.335269i − 3.46300i 0 1.00000 −2.73864 − 0.707009i 0 −0.411948 + 4.88007i
757.2 −1.40920 + 0.118957i 0 1.97170 − 0.335269i 3.46300i 0 1.00000 −2.73864 + 0.707009i 0 −0.411948 − 4.88007i
757.3 −1.35918 − 0.390688i 0 1.69473 + 1.06203i 1.82986i 0 1.00000 −1.88851 − 2.10559i 0 0.714906 − 2.48711i
757.4 −1.35918 + 0.390688i 0 1.69473 − 1.06203i − 1.82986i 0 1.00000 −1.88851 + 2.10559i 0 0.714906 + 2.48711i
757.5 −1.13123 − 0.848721i 0 0.559347 + 1.92019i 0.940450i 0 1.00000 0.996957 − 2.64690i 0 0.798179 − 1.06386i
757.6 −1.13123 + 0.848721i 0 0.559347 − 1.92019i − 0.940450i 0 1.00000 0.996957 + 2.64690i 0 0.798179 + 1.06386i
757.7 −1.10240 − 0.885841i 0 0.430570 + 1.95310i − 2.99001i 0 1.00000 1.25548 − 2.53452i 0 −2.64867 + 3.29619i
757.8 −1.10240 + 0.885841i 0 0.430570 − 1.95310i 2.99001i 0 1.00000 1.25548 + 2.53452i 0 −2.64867 − 3.29619i
757.9 −0.497658 − 1.32376i 0 −1.50467 + 1.31756i 1.25392i 0 1.00000 2.49294 + 1.33613i 0 1.65989 − 0.624024i
757.10 −0.497658 + 1.32376i 0 −1.50467 − 1.31756i − 1.25392i 0 1.00000 2.49294 − 1.33613i 0 1.65989 + 0.624024i
757.11 −0.417332 − 1.35123i 0 −1.65167 + 1.12783i − 3.04340i 0 1.00000 2.21325 + 1.76111i 0 −4.11235 + 1.27011i
757.12 −0.417332 + 1.35123i 0 −1.65167 − 1.12783i 3.04340i 0 1.00000 2.21325 − 1.76111i 0 −4.11235 − 1.27011i
757.13 0.417332 − 1.35123i 0 −1.65167 − 1.12783i − 3.04340i 0 1.00000 −2.21325 + 1.76111i 0 −4.11235 − 1.27011i
757.14 0.417332 + 1.35123i 0 −1.65167 + 1.12783i 3.04340i 0 1.00000 −2.21325 − 1.76111i 0 −4.11235 + 1.27011i
757.15 0.497658 − 1.32376i 0 −1.50467 − 1.31756i 1.25392i 0 1.00000 −2.49294 + 1.33613i 0 1.65989 + 0.624024i
757.16 0.497658 + 1.32376i 0 −1.50467 + 1.31756i − 1.25392i 0 1.00000 −2.49294 − 1.33613i 0 1.65989 − 0.624024i
757.17 1.10240 − 0.885841i 0 0.430570 − 1.95310i − 2.99001i 0 1.00000 −1.25548 − 2.53452i 0 −2.64867 − 3.29619i
757.18 1.10240 + 0.885841i 0 0.430570 + 1.95310i 2.99001i 0 1.00000 −1.25548 + 2.53452i 0 −2.64867 + 3.29619i
757.19 1.13123 − 0.848721i 0 0.559347 − 1.92019i 0.940450i 0 1.00000 −0.996957 − 2.64690i 0 0.798179 + 1.06386i
757.20 1.13123 + 0.848721i 0 0.559347 + 1.92019i − 0.940450i 0 1.00000 −0.996957 + 2.64690i 0 0.798179 − 1.06386i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.c.g ✓ 24
3.b odd 2 1 inner 1512.2.c.g ✓ 24
4.b odd 2 1 6048.2.c.f 24
8.b even 2 1 inner 1512.2.c.g ✓ 24
8.d odd 2 1 6048.2.c.f 24
12.b even 2 1 6048.2.c.f 24
24.f even 2 1 6048.2.c.f 24
24.h odd 2 1 inner 1512.2.c.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.g ✓ 24 1.a even 1 1 trivial
1512.2.c.g ✓ 24 3.b odd 2 1 inner
1512.2.c.g ✓ 24 8.b even 2 1 inner
1512.2.c.g ✓ 24 24.h odd 2 1 inner
6048.2.c.f 24 4.b odd 2 1
6048.2.c.f 24 8.d odd 2 1
6048.2.c.f 24 12.b even 2 1
6048.2.c.f 24 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 486T_{5}^{8} + 3036T_{5}^{6} + 8801T_{5}^{4} + 10952T_{5}^{2} + 4624 \) Copy content Toggle raw display
\( T_{17}^{12} - 100T_{17}^{10} + 3253T_{17}^{8} - 41296T_{17}^{6} + 197515T_{17}^{4} - 252108T_{17}^{2} + 63 \) Copy content Toggle raw display