Properties

Label 3380.2.a.d
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + 5q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} + 5q^{7} - 2q^{9} - 5q^{11} + q^{15} - q^{17} - 3q^{19} - 5q^{21} + 3q^{23} + q^{25} + 5q^{27} - q^{29} + 5q^{33} - 5q^{35} + 7q^{37} - 5q^{41} + 5q^{43} + 2q^{45} + 12q^{47} + 18q^{49} + q^{51} + 2q^{53} + 5q^{55} + 3q^{57} - 11q^{59} - 13q^{61} - 10q^{63} + 3q^{67} - 3q^{69} + 13q^{71} - 2q^{73} - q^{75} - 25q^{77} - 4q^{79} + q^{81} + 12q^{83} + q^{85} + q^{87} + 7q^{89} + 3q^{95} + 11q^{97} + 10q^{99} + O(q^{100}) \)

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
0 −1.00000 0 −1.00000 0 5.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 3380.2.a.d 1
13.b even 2 1 3380.2.a.e 1
13.c even 3 2 260.2.i.c 2
13.d odd 4 2 3380.2.f.c 2
39.i odd 6 2 2340.2.q.c 2
52.j odd 6 2 1040.2.q.f 2
65.n even 6 2 1300.2.i.c 2
65.q odd 12 4 1300.2.bb.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.c 2 13.c even 3 2
1040.2.q.f 2 52.j odd 6 2
1300.2.i.c 2 65.n even 6 2
1300.2.bb.c 4 65.q odd 12 4
2340.2.q.c 2 39.i odd 6 2
3380.2.a.d 1 1.a even 1 1 trivial
3380.2.a.e 1 13.b even 2 1
3380.2.f.c 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\( T_{3} + 1 \)
\( T_{7} - 5 \)
\( T_{19} + 3 \)

Hecke characteristic polynomials

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