Properties

Label 3380.2.a.a
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $2$
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: \(2\)
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

\(f(q)\) \(=\) \( q - 3q^{3} - q^{5} - 3q^{7} + 6q^{9} + O(q^{10}) \) \( q - 3q^{3} - q^{5} - 3q^{7} + 6q^{9} - 3q^{11} + 3q^{15} - 7q^{17} - q^{19} + 9q^{21} - 7q^{23} + q^{25} - 9q^{27} - 5q^{29} + 4q^{31} + 9q^{33} + 3q^{35} + 3q^{37} - 7q^{41} - 9q^{43} - 6q^{45} - 8q^{47} + 2q^{49} + 21q^{51} - 6q^{53} + 3q^{55} + 3q^{57} - 5q^{59} - 5q^{61} - 18q^{63} - 13q^{67} + 21q^{69} + 3q^{71} + 14q^{73} - 3q^{75} + 9q^{77} - 8q^{79} + 9q^{81} - 12q^{83} + 7q^{85} + 15q^{87} - 7q^{89} - 12q^{93} + q^{95} + 11q^{97} - 18q^{99} + O(q^{100}) \)

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 −3.00000 0 −1.00000 0 −3.00000 0 6.00000 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.a 1
13.b even 2 1 3380.2.a.b 1
13.d odd 4 2 3380.2.f.a 2
13.e even 6 2 260.2.i.d 2
39.h odd 6 2 2340.2.q.a 2
52.i odd 6 2 1040.2.q.b 2
65.l even 6 2 1300.2.i.a 2
65.r odd 12 4 1300.2.bb.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 13.e even 6 2
1040.2.q.b 2 52.i odd 6 2
1300.2.i.a 2 65.l even 6 2
1300.2.bb.e 4 65.r odd 12 4
2340.2.q.a 2 39.h odd 6 2
3380.2.a.a 1 1.a even 1 1 trivial
3380.2.a.b 1 13.b even 2 1
3380.2.f.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 1 + T \)
$7$ \( 3 + T \)
$11$ \( 3 + T \)
$13$ \( T \)
$17$ \( 7 + T \)
$19$ \( 1 + T \)
$23$ \( 7 + T \)
$29$ \( 5 + T \)
$31$ \( -4 + T \)
$37$ \( -3 + T \)
$41$ \( 7 + T \)
$43$ \( 9 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( 5 + T \)
$61$ \( 5 + T \)
$67$ \( 13 + T \)
$71$ \( -3 + T \)
$73$ \( -14 + T \)
$79$ \( 8 + T \)
$83$ \( 12 + T \)
$89$ \( 7 + T \)
$97$ \( -11 + T \)
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