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Properties

Label 336.7.o.e
Level $336$
Weight $7$
Character orbit 336.o
Analytic conductor $77.298$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,7,Mod(335,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.335");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.o (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: \(77.2981720963\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 5000 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 5000 q^{9} - 10696 q^{21} + 173928 q^{25} - 165744 q^{37} - 276360 q^{49} - 98864 q^{57} - 5471816 q^{81} + 528960 q^{85} + 4184144 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
335.1 0 −20.0330 − 18.1019i 0 −138.405 0 272.716 + 208.027i 0 73.6404 + 725.271i 0
335.2 0 −20.0330 − 18.1019i 0 138.405 0 272.716 − 208.027i 0 73.6404 + 725.271i 0
335.3 0 −20.0330 + 18.1019i 0 −138.405 0 272.716 − 208.027i 0 73.6404 − 725.271i 0
335.4 0 −20.0330 + 18.1019i 0 138.405 0 272.716 + 208.027i 0 73.6404 − 725.271i 0
335.5 0 −18.7134 − 19.4630i 0 −214.780 0 −147.649 − 309.595i 0 −28.6171 + 728.438i 0
335.6 0 −18.7134 − 19.4630i 0 214.780 0 −147.649 + 309.595i 0 −28.6171 + 728.438i 0
335.7 0 −18.7134 + 19.4630i 0 −214.780 0 −147.649 + 309.595i 0 −28.6171 − 728.438i 0
335.8 0 −18.7134 + 19.4630i 0 214.780 0 −147.649 − 309.595i 0 −28.6171 − 728.438i 0
335.9 0 −5.43032 − 26.4483i 0 −57.7046 0 −251.052 + 233.713i 0 −670.023 + 287.245i 0
335.10 0 −5.43032 − 26.4483i 0 57.7046 0 −251.052 − 233.713i 0 −670.023 + 287.245i 0
335.11 0 −5.43032 + 26.4483i 0 −57.7046 0 −251.052 − 233.713i 0 −670.023 − 287.245i 0
335.12 0 −5.43032 + 26.4483i 0 57.7046 0 −251.052 + 233.713i 0 −670.023 − 287.245i 0
335.13 0 5.43032 − 26.4483i 0 −57.7046 0 251.052 + 233.713i 0 −670.023 − 287.245i 0
335.14 0 5.43032 − 26.4483i 0 57.7046 0 251.052 − 233.713i 0 −670.023 − 287.245i 0
335.15 0 5.43032 + 26.4483i 0 −57.7046 0 251.052 − 233.713i 0 −670.023 + 287.245i 0
335.16 0 5.43032 + 26.4483i 0 57.7046 0 251.052 + 233.713i 0 −670.023 + 287.245i 0
335.17 0 18.7134 − 19.4630i 0 −214.780 0 147.649 − 309.595i 0 −28.6171 − 728.438i 0
335.18 0 18.7134 − 19.4630i 0 214.780 0 147.649 + 309.595i 0 −28.6171 − 728.438i 0
335.19 0 18.7134 + 19.4630i 0 −214.780 0 147.649 + 309.595i 0 −28.6171 + 728.438i 0
335.20 0 18.7134 + 19.4630i 0 214.780 0 147.649 − 309.595i 0 −28.6171 + 728.438i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 335.24
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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.o.e ✓ 24
3.b odd 2 1 inner 336.7.o.e ✓ 24
4.b odd 2 1 inner 336.7.o.e ✓ 24
7.b odd 2 1 inner 336.7.o.e ✓ 24
12.b even 2 1 inner 336.7.o.e ✓ 24
21.c even 2 1 inner 336.7.o.e ✓ 24
28.d even 2 1 inner 336.7.o.e ✓ 24
84.h odd 2 1 inner 336.7.o.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.o.e ✓ 24 1.a even 1 1 trivial
336.7.o.e ✓ 24 3.b odd 2 1 inner
336.7.o.e ✓ 24 4.b odd 2 1 inner
336.7.o.e ✓ 24 7.b odd 2 1 inner
336.7.o.e ✓ 24 12.b even 2 1 inner
336.7.o.e ✓ 24 21.c even 2 1 inner
336.7.o.e ✓ 24 28.d even 2 1 inner
336.7.o.e ✓ 24 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{6} - 68616T_{5}^{4} + 1101059100T_{5}^{2} - 2942456220000 \) Copy content Toggle raw display
\( T_{19}^{6} - 95297796T_{19}^{4} + 2223702800731356T_{19}^{2} - 1827590823378704673520 \) Copy content Toggle raw display