Properties

Label 338.6.a.d
Level $338$
Weight $6$
Character orbit 338.a
Self dual yes
Analytic conductor $54.210$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,6,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

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: yes
Analytic conductor: \(54.2097310968\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 14 q^{5} + 170 q^{7} + 64 q^{8} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 14 q^{5} + 170 q^{7} + 64 q^{8} - 243 q^{9} + 56 q^{10} + 250 q^{11} + 680 q^{14} + 256 q^{16} + 1062 q^{17} - 972 q^{18} + 78 q^{19} + 224 q^{20} + 1000 q^{22} + 1576 q^{23} - 2929 q^{25} + 2720 q^{28} + 2578 q^{29} + 8654 q^{31} + 1024 q^{32} + 4248 q^{34} + 2380 q^{35} - 3888 q^{36} - 10986 q^{37} + 312 q^{38} + 896 q^{40} - 1050 q^{41} - 5900 q^{43} + 4000 q^{44} - 3402 q^{45} + 6304 q^{46} + 5962 q^{47} + 12093 q^{49} - 11716 q^{50} + 29046 q^{53} + 3500 q^{55} + 10880 q^{56} + 10312 q^{58} + 13922 q^{59} - 32882 q^{61} + 34616 q^{62} - 41310 q^{63} + 4096 q^{64} + 69566 q^{67} + 16992 q^{68} + 9520 q^{70} + 50542 q^{71} - 15552 q^{72} + 46750 q^{73} - 43944 q^{74} + 1248 q^{76} + 42500 q^{77} - 19348 q^{79} + 3584 q^{80} + 59049 q^{81} - 4200 q^{82} + 87438 q^{83} + 14868 q^{85} - 23600 q^{86} + 16000 q^{88} - 94170 q^{89} - 13608 q^{90} + 25216 q^{92} + 23848 q^{94} + 1092 q^{95} - 182786 q^{97} + 48372 q^{98} - 60750 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 0 16.0000 14.0000 0 170.000 64.0000 −243.000 56.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.6.a.d 1
13.b even 2 1 26.6.a.a 1
13.d odd 4 2 338.6.b.a 2
39.d odd 2 1 234.6.a.g 1
52.b odd 2 1 208.6.a.b 1
65.d even 2 1 650.6.a.a 1
65.h odd 4 2 650.6.b.a 2
104.e even 2 1 832.6.a.d 1
104.h odd 2 1 832.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.a 1 13.b even 2 1
208.6.a.b 1 52.b odd 2 1
234.6.a.g 1 39.d odd 2 1
338.6.a.d 1 1.a even 1 1 trivial
338.6.b.a 2 13.d odd 4 2
650.6.a.a 1 65.d even 2 1
650.6.b.a 2 65.h odd 4 2
832.6.a.d 1 104.e even 2 1
832.6.a.e 1 104.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 14 \) Copy content Toggle raw display
$7$ \( T - 170 \) Copy content Toggle raw display
$11$ \( T - 250 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 1062 \) Copy content Toggle raw display
$19$ \( T - 78 \) Copy content Toggle raw display
$23$ \( T - 1576 \) Copy content Toggle raw display
$29$ \( T - 2578 \) Copy content Toggle raw display
$31$ \( T - 8654 \) Copy content Toggle raw display
$37$ \( T + 10986 \) Copy content Toggle raw display
$41$ \( T + 1050 \) Copy content Toggle raw display
$43$ \( T + 5900 \) Copy content Toggle raw display
$47$ \( T - 5962 \) Copy content Toggle raw display
$53$ \( T - 29046 \) Copy content Toggle raw display
$59$ \( T - 13922 \) Copy content Toggle raw display
$61$ \( T + 32882 \) Copy content Toggle raw display
$67$ \( T - 69566 \) Copy content Toggle raw display
$71$ \( T - 50542 \) Copy content Toggle raw display
$73$ \( T - 46750 \) Copy content Toggle raw display
$79$ \( T + 19348 \) Copy content Toggle raw display
$83$ \( T - 87438 \) Copy content Toggle raw display
$89$ \( T + 94170 \) Copy content Toggle raw display
$97$ \( T + 182786 \) Copy content Toggle raw display
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