Properties

Label 338.6.a.e
Level $338$
Weight $6$
Character orbit 338.a
Self dual yes
Analytic conductor $54.210$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 4 q^{3} + 16 q^{4} + 68 q^{5} + 16 q^{6} - 82 q^{7} + 64 q^{8} - 227 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 4 q^{3} + 16 q^{4} + 68 q^{5} + 16 q^{6} - 82 q^{7} + 64 q^{8} - 227 q^{9} + 272 q^{10} - 390 q^{11} + 64 q^{12} - 328 q^{14} + 272 q^{15} + 256 q^{16} - 1738 q^{17} - 908 q^{18} - 1074 q^{19} + 1088 q^{20} - 328 q^{21} - 1560 q^{22} - 2104 q^{23} + 256 q^{24} + 1499 q^{25} - 1880 q^{27} - 1312 q^{28} - 1690 q^{29} + 1088 q^{30} - 1430 q^{31} + 1024 q^{32} - 1560 q^{33} - 6952 q^{34} - 5576 q^{35} - 3632 q^{36} - 8852 q^{37} - 4296 q^{38} + 4352 q^{40} + 6760 q^{41} - 1312 q^{42} + 16916 q^{43} - 6240 q^{44} - 15436 q^{45} - 8416 q^{46} + 25158 q^{47} + 1024 q^{48} - 10083 q^{49} + 5996 q^{50} - 6952 q^{51} + 38214 q^{53} - 7520 q^{54} - 26520 q^{55} - 5248 q^{56} - 4296 q^{57} - 6760 q^{58} - 21286 q^{59} + 4352 q^{60} - 5458 q^{61} - 5720 q^{62} + 18614 q^{63} + 4096 q^{64} - 6240 q^{66} + 44542 q^{67} - 27808 q^{68} - 8416 q^{69} - 22304 q^{70} - 17790 q^{71} - 14528 q^{72} - 31064 q^{73} - 35408 q^{74} + 5996 q^{75} - 17184 q^{76} + 31980 q^{77} - 45360 q^{79} + 17408 q^{80} + 47641 q^{81} + 27040 q^{82} - 124546 q^{83} - 5248 q^{84} - 118184 q^{85} + 67664 q^{86} - 6760 q^{87} - 24960 q^{88} + 18744 q^{89} - 61744 q^{90} - 33664 q^{92} - 5720 q^{93} + 100632 q^{94} - 73032 q^{95} + 4096 q^{96} - 121488 q^{97} - 40332 q^{98} + 88530 q^{99}+O(q^{100}) \) 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 4.00000 16.0000 68.0000 16.0000 −82.0000 64.0000 −227.000 272.000
\(n\): 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.e 1
13.b even 2 1 338.6.a.b 1
13.d odd 4 2 26.6.b.b 2
39.f even 4 2 234.6.b.a 2
52.f even 4 2 208.6.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.b 2 13.d odd 4 2
208.6.f.a 2 52.f even 4 2
234.6.b.a 2 39.f even 4 2
338.6.a.b 1 13.b even 2 1
338.6.a.e 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3} - 4 \) Copy content Toggle raw display
\( T_{5} - 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T - 68 \) Copy content Toggle raw display
$7$ \( T + 82 \) Copy content Toggle raw display
$11$ \( T + 390 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 1738 \) Copy content Toggle raw display
$19$ \( T + 1074 \) Copy content Toggle raw display
$23$ \( T + 2104 \) Copy content Toggle raw display
$29$ \( T + 1690 \) Copy content Toggle raw display
$31$ \( T + 1430 \) Copy content Toggle raw display
$37$ \( T + 8852 \) Copy content Toggle raw display
$41$ \( T - 6760 \) Copy content Toggle raw display
$43$ \( T - 16916 \) Copy content Toggle raw display
$47$ \( T - 25158 \) Copy content Toggle raw display
$53$ \( T - 38214 \) Copy content Toggle raw display
$59$ \( T + 21286 \) Copy content Toggle raw display
$61$ \( T + 5458 \) Copy content Toggle raw display
$67$ \( T - 44542 \) Copy content Toggle raw display
$71$ \( T + 17790 \) Copy content Toggle raw display
$73$ \( T + 31064 \) Copy content Toggle raw display
$79$ \( T + 45360 \) Copy content Toggle raw display
$83$ \( T + 124546 \) Copy content Toggle raw display
$89$ \( T - 18744 \) Copy content Toggle raw display
$97$ \( T + 121488 \) Copy content Toggle raw display
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