Properties

Label 338.8.a.a
Level $338$
Weight $8$
Character orbit 338.a
Self dual yes
Analytic conductor $105.586$
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] = [338,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(105.586138614\)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} - 87 q^{3} + 64 q^{4} - 321 q^{5} + 696 q^{6} + 181 q^{7} - 512 q^{8} + 5382 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 87 q^{3} + 64 q^{4} - 321 q^{5} + 696 q^{6} + 181 q^{7} - 512 q^{8} + 5382 q^{9} + 2568 q^{10} - 7782 q^{11} - 5568 q^{12} - 1448 q^{14} + 27927 q^{15} + 4096 q^{16} + 9069 q^{17} - 43056 q^{18} + 37150 q^{19} - 20544 q^{20} - 15747 q^{21} + 62256 q^{22} + 19008 q^{23} + 44544 q^{24} + 24916 q^{25} - 277965 q^{27} + 11584 q^{28} + 174750 q^{29} - 223416 q^{30} - 29012 q^{31} - 32768 q^{32} + 677034 q^{33} - 72552 q^{34} - 58101 q^{35} + 344448 q^{36} - 323669 q^{37} - 297200 q^{38} + 164352 q^{40} - 795312 q^{41} + 125976 q^{42} - 314137 q^{43} - 498048 q^{44} - 1727622 q^{45} - 152064 q^{46} + 447441 q^{47} - 356352 q^{48} - 790782 q^{49} - 199328 q^{50} - 789003 q^{51} - 1469232 q^{53} + 2223720 q^{54} + 2498022 q^{55} - 92672 q^{56} - 3232050 q^{57} - 1398000 q^{58} - 1627770 q^{59} + 1787328 q^{60} - 2399608 q^{61} + 232096 q^{62} + 974142 q^{63} + 262144 q^{64} - 5416272 q^{66} + 64066 q^{67} + 580416 q^{68} - 1653696 q^{69} + 464808 q^{70} + 322383 q^{71} - 2755584 q^{72} + 4454782 q^{73} + 2589352 q^{74} - 2167692 q^{75} + 2377600 q^{76} - 1408542 q^{77} + 753560 q^{79} - 1314816 q^{80} + 12412521 q^{81} + 6362496 q^{82} + 1219092 q^{83} - 1007808 q^{84} - 2911149 q^{85} + 2513096 q^{86} - 15203250 q^{87} + 3984384 q^{88} - 3390330 q^{89} + 13820976 q^{90} + 1216512 q^{92} + 2524044 q^{93} - 3579528 q^{94} - 11925150 q^{95} + 2850816 q^{96} - 1628774 q^{97} + 6326256 q^{98} - 41882724 q^{99}+O(q^{100}) \) 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
−8.00000 −87.0000 64.0000 −321.000 696.000 181.000 −512.000 5382.00 2568.00
\(n\): 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 338.8.a.a 1
13.b even 2 1 26.8.a.b 1
13.d odd 4 2 338.8.b.a 2
39.d odd 2 1 234.8.a.a 1
52.b odd 2 1 208.8.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.b 1 13.b even 2 1
208.8.a.e 1 52.b odd 2 1
234.8.a.a 1 39.d odd 2 1
338.8.a.a 1 1.a even 1 1 trivial
338.8.b.a 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} + 87 \) Copy content Toggle raw display
\( T_{5} + 321 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 87 \) Copy content Toggle raw display
$5$ \( T + 321 \) Copy content Toggle raw display
$7$ \( T - 181 \) Copy content Toggle raw display
$11$ \( T + 7782 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 9069 \) Copy content Toggle raw display
$19$ \( T - 37150 \) Copy content Toggle raw display
$23$ \( T - 19008 \) Copy content Toggle raw display
$29$ \( T - 174750 \) Copy content Toggle raw display
$31$ \( T + 29012 \) Copy content Toggle raw display
$37$ \( T + 323669 \) Copy content Toggle raw display
$41$ \( T + 795312 \) Copy content Toggle raw display
$43$ \( T + 314137 \) Copy content Toggle raw display
$47$ \( T - 447441 \) Copy content Toggle raw display
$53$ \( T + 1469232 \) Copy content Toggle raw display
$59$ \( T + 1627770 \) Copy content Toggle raw display
$61$ \( T + 2399608 \) Copy content Toggle raw display
$67$ \( T - 64066 \) Copy content Toggle raw display
$71$ \( T - 322383 \) Copy content Toggle raw display
$73$ \( T - 4454782 \) Copy content Toggle raw display
$79$ \( T - 753560 \) Copy content Toggle raw display
$83$ \( T - 1219092 \) Copy content Toggle raw display
$89$ \( T + 3390330 \) Copy content Toggle raw display
$97$ \( T + 1628774 \) Copy content Toggle raw display
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