Properties

Label 338.6.a.c
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 13 q^{3} + 16 q^{4} - 51 q^{5} - 52 q^{6} + 105 q^{7} + 64 q^{8} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 13 q^{3} + 16 q^{4} - 51 q^{5} - 52 q^{6} + 105 q^{7} + 64 q^{8} - 74 q^{9} - 204 q^{10} + 120 q^{11} - 208 q^{12} + 420 q^{14} + 663 q^{15} + 256 q^{16} + 1101 q^{17} - 296 q^{18} + 1170 q^{19} - 816 q^{20} - 1365 q^{21} + 480 q^{22} - 1050 q^{23} - 832 q^{24} - 524 q^{25} + 4121 q^{27} + 1680 q^{28} - 4104 q^{29} + 2652 q^{30} - 9624 q^{31} + 1024 q^{32} - 1560 q^{33} + 4404 q^{34} - 5355 q^{35} - 1184 q^{36} + 8709 q^{37} + 4680 q^{38} - 3264 q^{40} + 9480 q^{41} - 5460 q^{42} - 9995 q^{43} + 1920 q^{44} + 3774 q^{45} - 4200 q^{46} - 2943 q^{47} - 3328 q^{48} - 5782 q^{49} - 2096 q^{50} - 14313 q^{51} - 750 q^{53} + 16484 q^{54} - 6120 q^{55} + 6720 q^{56} - 15210 q^{57} - 16416 q^{58} - 40938 q^{59} + 10608 q^{60} - 57920 q^{61} - 38496 q^{62} - 7770 q^{63} + 4096 q^{64} - 6240 q^{66} - 22812 q^{67} + 17616 q^{68} + 13650 q^{69} - 21420 q^{70} - 63741 q^{71} - 4736 q^{72} + 58866 q^{73} + 34836 q^{74} + 6812 q^{75} + 18720 q^{76} + 12600 q^{77} + 63202 q^{79} - 13056 q^{80} - 35591 q^{81} + 37920 q^{82} - 55458 q^{83} - 21840 q^{84} - 56151 q^{85} - 39980 q^{86} + 53352 q^{87} + 7680 q^{88} - 104778 q^{89} + 15096 q^{90} - 16800 q^{92} + 125112 q^{93} - 11772 q^{94} - 59670 q^{95} - 13312 q^{96} - 160452 q^{97} - 23128 q^{98} - 8880 q^{99}+O(q^{100}) \) 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 −13.0000 16.0000 −51.0000 −52.0000 105.000 64.0000 −74.0000 −204.000
\(n\): 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.c 1
13.b even 2 1 338.6.a.a 1
13.d odd 4 2 26.6.b.a 2
39.f even 4 2 234.6.b.b 2
52.f even 4 2 208.6.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 13.d odd 4 2
208.6.f.b 2 52.f even 4 2
234.6.b.b 2 39.f even 4 2
338.6.a.a 1 13.b even 2 1
338.6.a.c 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} + 13 \) Copy content Toggle raw display
\( T_{5} + 51 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 13 \) Copy content Toggle raw display
$5$ \( T + 51 \) Copy content Toggle raw display
$7$ \( T - 105 \) Copy content Toggle raw display
$11$ \( T - 120 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 1101 \) Copy content Toggle raw display
$19$ \( T - 1170 \) Copy content Toggle raw display
$23$ \( T + 1050 \) Copy content Toggle raw display
$29$ \( T + 4104 \) Copy content Toggle raw display
$31$ \( T + 9624 \) Copy content Toggle raw display
$37$ \( T - 8709 \) Copy content Toggle raw display
$41$ \( T - 9480 \) Copy content Toggle raw display
$43$ \( T + 9995 \) Copy content Toggle raw display
$47$ \( T + 2943 \) Copy content Toggle raw display
$53$ \( T + 750 \) Copy content Toggle raw display
$59$ \( T + 40938 \) Copy content Toggle raw display
$61$ \( T + 57920 \) Copy content Toggle raw display
$67$ \( T + 22812 \) Copy content Toggle raw display
$71$ \( T + 63741 \) Copy content Toggle raw display
$73$ \( T - 58866 \) Copy content Toggle raw display
$79$ \( T - 63202 \) Copy content Toggle raw display
$83$ \( T + 55458 \) Copy content Toggle raw display
$89$ \( T + 104778 \) Copy content Toggle raw display
$97$ \( T + 160452 \) Copy content Toggle raw display
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