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Properties

Label 336.4.k.e
Level $336$
Weight $4$
Character orbit 336.k
Analytic conductor $19.825$
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] = [336,4,Mod(209,336)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.k (of order \(2\), degree \(1\), not 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} + 28 q^{9} + 44 q^{15} + 208 q^{21} + 816 q^{25} - 192 q^{37} - 388 q^{39} + 432 q^{43} - 240 q^{49} + 744 q^{51} + 524 q^{57} + 1096 q^{63} + 384 q^{67} + 1176 q^{79} + 768 q^{81} - 624 q^{85} + 1368 q^{91} - 1912 q^{93} + 896 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −5.18837 − 0.284208i 0 −11.2714 0 17.8595 − 4.90299i 0 26.8385 + 2.94915i 0
209.2 0 −5.18837 + 0.284208i 0 −11.2714 0 17.8595 + 4.90299i 0 26.8385 − 2.94915i 0
209.3 0 −5.02512 − 1.32218i 0 17.8550 0 1.81257 + 18.4313i 0 23.5037 + 13.2883i 0
209.4 0 −5.02512 + 1.32218i 0 17.8550 0 1.81257 − 18.4313i 0 23.5037 − 13.2883i 0
209.5 0 −4.24923 − 2.99066i 0 −0.465833 0 −18.4057 − 2.05669i 0 9.11192 + 25.4160i 0
209.6 0 −4.24923 + 2.99066i 0 −0.465833 0 −18.4057 + 2.05669i 0 9.11192 − 25.4160i 0
209.7 0 −2.45998 − 4.57695i 0 −21.1672 0 −12.8058 + 13.3795i 0 −14.8970 + 22.5185i 0
209.8 0 −2.45998 + 4.57695i 0 −21.1672 0 −12.8058 − 13.3795i 0 −14.8970 − 22.5185i 0
209.9 0 −2.04902 − 4.77509i 0 −1.64044 0 13.1134 + 13.0782i 0 −18.6030 + 19.5686i 0
209.10 0 −2.04902 + 4.77509i 0 −1.64044 0 13.1134 − 13.0782i 0 −18.6030 − 19.5686i 0
209.11 0 −2.00573 − 4.79344i 0 7.56258 0 1.42610 − 18.4653i 0 −18.9541 + 19.2287i 0
209.12 0 −2.00573 + 4.79344i 0 7.56258 0 1.42610 + 18.4653i 0 −18.9541 − 19.2287i 0
209.13 0 2.00573 − 4.79344i 0 −7.56258 0 1.42610 − 18.4653i 0 −18.9541 − 19.2287i 0
209.14 0 2.00573 + 4.79344i 0 −7.56258 0 1.42610 + 18.4653i 0 −18.9541 + 19.2287i 0
209.15 0 2.04902 − 4.77509i 0 1.64044 0 13.1134 + 13.0782i 0 −18.6030 − 19.5686i 0
209.16 0 2.04902 + 4.77509i 0 1.64044 0 13.1134 − 13.0782i 0 −18.6030 + 19.5686i 0
209.17 0 2.45998 − 4.57695i 0 21.1672 0 −12.8058 + 13.3795i 0 −14.8970 − 22.5185i 0
209.18 0 2.45998 + 4.57695i 0 21.1672 0 −12.8058 − 13.3795i 0 −14.8970 + 22.5185i 0
209.19 0 4.24923 − 2.99066i 0 0.465833 0 −18.4057 − 2.05669i 0 9.11192 − 25.4160i 0
209.20 0 4.24923 + 2.99066i 0 0.465833 0 −18.4057 + 2.05669i 0 9.11192 + 25.4160i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.24
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 336.4.k.e 24
3.b odd 2 1 inner 336.4.k.e 24
4.b odd 2 1 168.4.k.a ✓ 24
7.b odd 2 1 inner 336.4.k.e 24
12.b even 2 1 168.4.k.a ✓ 24
21.c even 2 1 inner 336.4.k.e 24
28.d even 2 1 168.4.k.a ✓ 24
84.h odd 2 1 168.4.k.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.k.a ✓ 24 4.b odd 2 1
168.4.k.a ✓ 24 12.b even 2 1
168.4.k.a ✓ 24 28.d even 2 1
168.4.k.a ✓ 24 84.h odd 2 1
336.4.k.e 24 1.a even 1 1 trivial
336.4.k.e 24 3.b odd 2 1 inner
336.4.k.e 24 7.b odd 2 1 inner
336.4.k.e 24 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 954T_{5}^{10} + 294156T_{5}^{8} - 32736472T_{5}^{6} + 1130786688T_{5}^{4} - 3036819840T_{5}^{2} + 606076928 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display