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Properties

Label 1332.2.g.d
Level $1332$
Weight $2$
Character orbit 1332.g
Analytic conductor $10.636$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1332,2,Mod(1331,1332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.1331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.g (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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{4} + 8 q^{10} - 56 q^{16} + 128 q^{25} + 8 q^{28} - 56 q^{34} + 16 q^{37} + 40 q^{40} - 48 q^{46} - 112 q^{49} - 32 q^{58} + 128 q^{64} + 72 q^{70} + 112 q^{73}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1331.1 −1.40286 − 0.178813i 0 1.93605 + 0.501699i −2.77245 0 − 3.68039i −2.62631 − 1.05001i 0 3.88936 + 0.495748i
1331.2 −1.40286 − 0.178813i 0 1.93605 + 0.501699i −2.77245 0 − 3.68039i −2.62631 − 1.05001i 0 3.88936 + 0.495748i
1331.3 −1.40286 + 0.178813i 0 1.93605 − 0.501699i −2.77245 0 3.68039i −2.62631 + 1.05001i 0 3.88936 − 0.495748i
1331.4 −1.40286 + 0.178813i 0 1.93605 − 0.501699i −2.77245 0 3.68039i −2.62631 + 1.05001i 0 3.88936 − 0.495748i
1331.5 −1.34556 − 0.435269i 0 1.62108 + 1.17136i 3.09629 0 − 0.608394i −1.67141 − 2.28175i 0 −4.16626 − 1.34772i
1331.6 −1.34556 − 0.435269i 0 1.62108 + 1.17136i 3.09629 0 − 0.608394i −1.67141 − 2.28175i 0 −4.16626 − 1.34772i
1331.7 −1.34556 + 0.435269i 0 1.62108 − 1.17136i 3.09629 0 0.608394i −1.67141 + 2.28175i 0 −4.16626 + 1.34772i
1331.8 −1.34556 + 0.435269i 0 1.62108 − 1.17136i 3.09629 0 0.608394i −1.67141 + 2.28175i 0 −4.16626 + 1.34772i
1331.9 −1.25153 − 0.658529i 0 1.13268 + 1.64834i −0.275745 0 2.85311i −0.332104 − 2.80886i 0 0.345105 + 0.181586i
1331.10 −1.25153 − 0.658529i 0 1.13268 + 1.64834i −0.275745 0 2.85311i −0.332104 − 2.80886i 0 0.345105 + 0.181586i
1331.11 −1.25153 + 0.658529i 0 1.13268 − 1.64834i −0.275745 0 − 2.85311i −0.332104 + 2.80886i 0 0.345105 − 0.181586i
1331.12 −1.25153 + 0.658529i 0 1.13268 − 1.64834i −0.275745 0 − 2.85311i −0.332104 + 2.80886i 0 0.345105 − 0.181586i
1331.13 −1.07305 − 0.921177i 0 0.302866 + 1.97693i −2.89717 0 1.41615i 1.49612 − 2.40034i 0 3.10880 + 2.66880i
1331.14 −1.07305 − 0.921177i 0 0.302866 + 1.97693i −2.89717 0 1.41615i 1.49612 − 2.40034i 0 3.10880 + 2.66880i
1331.15 −1.07305 + 0.921177i 0 0.302866 − 1.97693i −2.89717 0 − 1.41615i 1.49612 + 2.40034i 0 3.10880 − 2.66880i
1331.16 −1.07305 + 0.921177i 0 0.302866 − 1.97693i −2.89717 0 − 1.41615i 1.49612 + 2.40034i 0 3.10880 − 2.66880i
1331.17 −0.925993 − 1.06890i 0 −0.285075 + 1.97958i 0.884046 0 − 2.97420i 2.37994 − 1.52836i 0 −0.818620 − 0.944952i
1331.18 −0.925993 − 1.06890i 0 −0.285075 + 1.97958i 0.884046 0 − 2.97420i 2.37994 − 1.52836i 0 −0.818620 − 0.944952i
1331.19 −0.925993 + 1.06890i 0 −0.285075 − 1.97958i 0.884046 0 2.97420i 2.37994 + 1.52836i 0 −0.818620 + 0.944952i
1331.20 −0.925993 + 1.06890i 0 −0.285075 − 1.97958i 0.884046 0 2.97420i 2.37994 + 1.52836i 0 −0.818620 + 0.944952i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1331.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
37.b even 2 1 inner
111.d odd 2 1 inner
148.b odd 2 1 inner
444.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.g.d ✓ 64
3.b odd 2 1 inner 1332.2.g.d ✓ 64
4.b odd 2 1 inner 1332.2.g.d ✓ 64
12.b even 2 1 inner 1332.2.g.d ✓ 64
37.b even 2 1 inner 1332.2.g.d ✓ 64
111.d odd 2 1 inner 1332.2.g.d ✓ 64
148.b odd 2 1 inner 1332.2.g.d ✓ 64
444.g even 2 1 inner 1332.2.g.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.g.d ✓ 64 1.a even 1 1 trivial
1332.2.g.d ✓ 64 3.b odd 2 1 inner
1332.2.g.d ✓ 64 4.b odd 2 1 inner
1332.2.g.d ✓ 64 12.b even 2 1 inner
1332.2.g.d ✓ 64 37.b even 2 1 inner
1332.2.g.d ✓ 64 111.d odd 2 1 inner
1332.2.g.d ✓ 64 148.b odd 2 1 inner
1332.2.g.d ✓ 64 444.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 \(S_{2}^{\mathrm{new}}(1332, [\chi])\):

\( T_{5}^{16} - 56 T_{5}^{14} + 1276 T_{5}^{12} - 15136 T_{5}^{10} + 98892 T_{5}^{8} - 343544 T_{5}^{6} + \cdots + 18752 \) Copy content Toggle raw display
\( T_{19}^{16} - 154 T_{19}^{14} + 9465 T_{19}^{12} - 297540 T_{19}^{10} + 5121552 T_{19}^{8} + \cdots + 358896384 \) Copy content Toggle raw display