Properties

Label 3330.2.a.o
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2q^{11} - 2q^{13} + q^{16} - 2q^{17} - 8q^{19} - q^{20} + 2q^{22} - 4q^{23} + q^{25} - 2q^{26} - 10q^{29} + 2q^{31} + q^{32} - 2q^{34} + q^{37} - 8q^{38} - q^{40} + 8q^{41} - 6q^{43} + 2q^{44} - 4q^{46} + 6q^{47} - 7q^{49} + q^{50} - 2q^{52} - 2q^{55} - 10q^{58} - 4q^{59} - 4q^{61} + 2q^{62} + q^{64} + 2q^{65} - 16q^{67} - 2q^{68} + 4q^{71} + 6q^{73} + q^{74} - 8q^{76} + 10q^{79} - q^{80} + 8q^{82} + 16q^{83} + 2q^{85} - 6q^{86} + 2q^{88} + 2q^{89} - 4q^{92} + 6q^{94} + 8q^{95} - 8q^{97} - 7q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(37\) \(-1\)

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 3330.2.a.o yes 1
3.b odd 2 1 3330.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.a.i 1 3.b odd 2 1
3330.2.a.o yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7} \)
\( T_{11} - 2 \)
\( T_{13} + 2 \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( 2 + T \)
$17$ \( 2 + T \)
$19$ \( 8 + T \)
$23$ \( 4 + T \)
$29$ \( 10 + T \)
$31$ \( -2 + T \)
$37$ \( -1 + T \)
$41$ \( -8 + T \)
$43$ \( 6 + T \)
$47$ \( -6 + T \)
$53$ \( T \)
$59$ \( 4 + T \)
$61$ \( 4 + T \)
$67$ \( 16 + T \)
$71$ \( -4 + T \)
$73$ \( -6 + T \)
$79$ \( -10 + T \)
$83$ \( -16 + T \)
$89$ \( -2 + T \)
$97$ \( 8 + T \)
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