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Properties

Label 126.10.d.a
Level $126$
Weight $10$
Character orbit 126.d
Analytic conductor $64.895$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,10,Mod(125,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.125");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.8945153566\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6144 q^{4} + 11352 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6144 q^{4} + 11352 q^{7} + 1572864 q^{16} - 2810880 q^{22} + 5727336 q^{25} - 2906112 q^{28} + 22697664 q^{37} - 37034832 q^{43} - 54125568 q^{46} - 7045176 q^{49} + 277714944 q^{58} - 402653184 q^{64} - 559614336 q^{67} - 234547200 q^{70} + 1451237088 q^{79} - 3592219680 q^{85} + 719585280 q^{88} - 3875613408 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 − 16.0000i 0 −256.000 −1538.46 0 −4139.71 − 4818.34i 4096.00i 0 24615.3i
125.2 16.0000i 0 −256.000 −1538.46 0 −4139.71 + 4818.34i − 4096.00i 0 − 24615.3i
125.3 − 16.0000i 0 −256.000 −1407.89 0 −2831.57 + 5686.46i 4096.00i 0 22526.2i
125.4 16.0000i 0 −256.000 −1407.89 0 −2831.57 − 5686.46i − 4096.00i 0 − 22526.2i
125.5 − 16.0000i 0 −256.000 579.753 0 2389.25 + 5886.01i 4096.00i 0 − 9276.05i
125.6 16.0000i 0 −256.000 579.753 0 2389.25 − 5886.01i − 4096.00i 0 9276.05i
125.7 − 16.0000i 0 −256.000 −282.526 0 −4386.62 − 4594.69i 4096.00i 0 4520.42i
125.8 16.0000i 0 −256.000 −282.526 0 −4386.62 + 4594.69i − 4096.00i 0 − 4520.42i
125.9 − 16.0000i 0 −256.000 −1298.97 0 6336.84 − 445.060i 4096.00i 0 20783.4i
125.10 16.0000i 0 −256.000 −1298.97 0 6336.84 + 445.060i − 4096.00i 0 − 20783.4i
125.11 − 16.0000i 0 −256.000 2588.12 0 5469.81 − 3230.30i 4096.00i 0 − 41409.8i
125.12 16.0000i 0 −256.000 2588.12 0 5469.81 + 3230.30i − 4096.00i 0 41409.8i
125.13 − 16.0000i 0 −256.000 −2588.12 0 5469.81 + 3230.30i 4096.00i 0 41409.8i
125.14 16.0000i 0 −256.000 −2588.12 0 5469.81 − 3230.30i − 4096.00i 0 − 41409.8i
125.15 − 16.0000i 0 −256.000 1298.97 0 6336.84 + 445.060i 4096.00i 0 − 20783.4i
125.16 16.0000i 0 −256.000 1298.97 0 6336.84 − 445.060i − 4096.00i 0 20783.4i
125.17 − 16.0000i 0 −256.000 282.526 0 −4386.62 + 4594.69i 4096.00i 0 − 4520.42i
125.18 16.0000i 0 −256.000 282.526 0 −4386.62 − 4594.69i − 4096.00i 0 4520.42i
125.19 − 16.0000i 0 −256.000 −579.753 0 2389.25 − 5886.01i 4096.00i 0 9276.05i
125.20 16.0000i 0 −256.000 −579.753 0 2389.25 + 5886.01i − 4096.00i 0 − 9276.05i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.10.d.a ✓ 24
3.b odd 2 1 inner 126.10.d.a ✓ 24
7.b odd 2 1 inner 126.10.d.a ✓ 24
21.c even 2 1 inner 126.10.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.10.d.a ✓ 24 1.a even 1 1 trivial
126.10.d.a ✓ 24 3.b odd 2 1 inner
126.10.d.a ✓ 24 7.b odd 2 1 inner
126.10.d.a ✓ 24 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(126, [\chi])\).