Properties

Label 338.2.a.b
Level $338$
Weight $2$
Character orbit 338.a
Self dual yes
Analytic conductor $2.699$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.69894358832\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + 3q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + 3q^{7} - q^{8} - 2q^{9} - 3q^{10} - q^{12} - 3q^{14} - 3q^{15} + q^{16} - 3q^{17} + 2q^{18} + 6q^{19} + 3q^{20} - 3q^{21} + 6q^{23} + q^{24} + 4q^{25} + 5q^{27} + 3q^{28} + 3q^{30} - q^{32} + 3q^{34} + 9q^{35} - 2q^{36} + 3q^{37} - 6q^{38} - 3q^{40} + 3q^{42} + q^{43} - 6q^{45} - 6q^{46} + 3q^{47} - q^{48} + 2q^{49} - 4q^{50} + 3q^{51} - 6q^{53} - 5q^{54} - 3q^{56} - 6q^{57} - 6q^{59} - 3q^{60} - 8q^{61} - 6q^{63} + q^{64} + 12q^{67} - 3q^{68} - 6q^{69} - 9q^{70} - 15q^{71} + 2q^{72} + 6q^{73} - 3q^{74} - 4q^{75} + 6q^{76} + 10q^{79} + 3q^{80} + q^{81} - 6q^{83} - 3q^{84} - 9q^{85} - q^{86} - 6q^{89} + 6q^{90} + 6q^{92} - 3q^{94} + 18q^{95} + q^{96} + 12q^{97} - 2q^{98} + O(q^{100}) \)

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.

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
−1.00000 −1.00000 1.00000 3.00000 1.00000 3.00000 −1.00000 −2.00000 −3.00000
\(n\): 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 338.2.a.b 1
3.b odd 2 1 3042.2.a.j 1
4.b odd 2 1 2704.2.a.k 1
5.b even 2 1 8450.2.a.u 1
13.b even 2 1 338.2.a.d 1
13.c even 3 2 338.2.c.f 2
13.d odd 4 2 26.2.b.a 2
13.e even 6 2 338.2.c.b 2
13.f odd 12 4 338.2.e.c 4
39.d odd 2 1 3042.2.a.g 1
39.f even 4 2 234.2.b.b 2
52.b odd 2 1 2704.2.a.j 1
52.f even 4 2 208.2.f.a 2
65.d even 2 1 8450.2.a.h 1
65.f even 4 2 650.2.c.d 2
65.g odd 4 2 650.2.d.b 2
65.k even 4 2 650.2.c.a 2
91.i even 4 2 1274.2.d.c 2
91.z odd 12 4 1274.2.n.d 4
91.bb even 12 4 1274.2.n.c 4
104.j odd 4 2 832.2.f.d 2
104.m even 4 2 832.2.f.b 2
156.l odd 4 2 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 13.d odd 4 2
208.2.f.a 2 52.f even 4 2
234.2.b.b 2 39.f even 4 2
338.2.a.b 1 1.a even 1 1 trivial
338.2.a.d 1 13.b even 2 1
338.2.c.b 2 13.e even 6 2
338.2.c.f 2 13.c even 3 2
338.2.e.c 4 13.f odd 12 4
650.2.c.a 2 65.k even 4 2
650.2.c.d 2 65.f even 4 2
650.2.d.b 2 65.g odd 4 2
832.2.f.b 2 104.m even 4 2
832.2.f.d 2 104.j odd 4 2
1274.2.d.c 2 91.i even 4 2
1274.2.n.c 4 91.bb even 12 4
1274.2.n.d 4 91.z odd 12 4
1872.2.c.f 2 156.l odd 4 2
2704.2.a.j 1 52.b odd 2 1
2704.2.a.k 1 4.b odd 2 1
3042.2.a.g 1 39.d odd 2 1
3042.2.a.j 1 3.b odd 2 1
8450.2.a.h 1 65.d even 2 1
8450.2.a.u 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} + 1 \)
\( T_{5} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( -3 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 3 + T \)
$19$ \( -6 + T \)
$23$ \( -6 + T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( -3 + T \)
$41$ \( T \)
$43$ \( -1 + T \)
$47$ \( -3 + T \)
$53$ \( 6 + T \)
$59$ \( 6 + T \)
$61$ \( 8 + T \)
$67$ \( -12 + T \)
$71$ \( 15 + T \)
$73$ \( -6 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( 6 + T \)
$97$ \( -12 + T \)
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