Properties

Label 338.2.a.f
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} + q^{8} - 2 q^{9} + 3 q^{10} - 6 q^{11} + q^{12} + q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 2 q^{19} + 3 q^{20} + q^{21} - 6 q^{22} + q^{24} + 4 q^{25} - 5 q^{27} + q^{28} + 6 q^{29} + 3 q^{30} + 4 q^{31} + q^{32} - 6 q^{33} - 3 q^{34} + 3 q^{35} - 2 q^{36} + 7 q^{37} - 2 q^{38} + 3 q^{40} + q^{42} - q^{43} - 6 q^{44} - 6 q^{45} - 3 q^{47} + q^{48} - 6 q^{49} + 4 q^{50} - 3 q^{51} - 5 q^{54} - 18 q^{55} + q^{56} - 2 q^{57} + 6 q^{58} + 6 q^{59} + 3 q^{60} + 8 q^{61} + 4 q^{62} - 2 q^{63} + q^{64} - 6 q^{66} - 14 q^{67} - 3 q^{68} + 3 q^{70} + 3 q^{71} - 2 q^{72} - 2 q^{73} + 7 q^{74} + 4 q^{75} - 2 q^{76} - 6 q^{77} + 8 q^{79} + 3 q^{80} + q^{81} - 12 q^{83} + q^{84} - 9 q^{85} - q^{86} + 6 q^{87} - 6 q^{88} + 6 q^{89} - 6 q^{90} + 4 q^{93} - 3 q^{94} - 6 q^{95} + q^{96} + 10 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100}) \) 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.

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 1.00000 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.f 1
3.b odd 2 1 3042.2.a.a 1
4.b odd 2 1 2704.2.a.f 1
5.b even 2 1 8450.2.a.c 1
13.b even 2 1 26.2.a.a 1
13.c even 3 2 338.2.c.a 2
13.d odd 4 2 338.2.b.c 2
13.e even 6 2 338.2.c.d 2
13.f odd 12 4 338.2.e.a 4
39.d odd 2 1 234.2.a.e 1
39.f even 4 2 3042.2.b.a 2
52.b odd 2 1 208.2.a.a 1
52.f even 4 2 2704.2.f.d 2
65.d even 2 1 650.2.a.j 1
65.h odd 4 2 650.2.b.d 2
91.b odd 2 1 1274.2.a.d 1
91.r even 6 2 1274.2.f.p 2
91.s odd 6 2 1274.2.f.r 2
104.e even 2 1 832.2.a.d 1
104.h odd 2 1 832.2.a.i 1
117.n odd 6 2 2106.2.e.b 2
117.t even 6 2 2106.2.e.ba 2
143.d odd 2 1 3146.2.a.n 1
156.h even 2 1 1872.2.a.q 1
195.e odd 2 1 5850.2.a.p 1
195.s even 4 2 5850.2.e.a 2
208.o odd 4 2 3328.2.b.j 2
208.p even 4 2 3328.2.b.m 2
221.b even 2 1 7514.2.a.c 1
247.d odd 2 1 9386.2.a.j 1
260.g odd 2 1 5200.2.a.x 1
312.b odd 2 1 7488.2.a.g 1
312.h even 2 1 7488.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 13.b even 2 1
208.2.a.a 1 52.b odd 2 1
234.2.a.e 1 39.d odd 2 1
338.2.a.f 1 1.a even 1 1 trivial
338.2.b.c 2 13.d odd 4 2
338.2.c.a 2 13.c even 3 2
338.2.c.d 2 13.e even 6 2
338.2.e.a 4 13.f odd 12 4
650.2.a.j 1 65.d even 2 1
650.2.b.d 2 65.h odd 4 2
832.2.a.d 1 104.e even 2 1
832.2.a.i 1 104.h odd 2 1
1274.2.a.d 1 91.b odd 2 1
1274.2.f.p 2 91.r even 6 2
1274.2.f.r 2 91.s odd 6 2
1872.2.a.q 1 156.h even 2 1
2106.2.e.b 2 117.n odd 6 2
2106.2.e.ba 2 117.t even 6 2
2704.2.a.f 1 4.b odd 2 1
2704.2.f.d 2 52.f even 4 2
3042.2.a.a 1 3.b odd 2 1
3042.2.b.a 2 39.f even 4 2
3146.2.a.n 1 143.d odd 2 1
3328.2.b.j 2 208.o odd 4 2
3328.2.b.m 2 208.p even 4 2
5200.2.a.x 1 260.g odd 2 1
5850.2.a.p 1 195.e odd 2 1
5850.2.e.a 2 195.s even 4 2
7488.2.a.g 1 312.b odd 2 1
7488.2.a.h 1 312.h even 2 1
7514.2.a.c 1 221.b even 2 1
8450.2.a.c 1 5.b even 2 1
9386.2.a.j 1 247.d odd 2 1

Hecke kernels

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

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 14 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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