Properties

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

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - 3q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} - 3q^{7} - q^{8} - q^{10} + q^{11} + q^{13} + 3q^{14} + q^{16} - q^{17} + q^{19} + q^{20} - q^{22} - 5q^{23} + q^{25} - q^{26} - 3q^{28} + 4q^{29} - 4q^{31} - q^{32} + q^{34} - 3q^{35} + q^{37} - q^{38} - q^{40} + 10q^{41} - 12q^{43} + q^{44} + 5q^{46} + 2q^{49} - q^{50} + q^{52} + 9q^{53} + q^{55} + 3q^{56} - 4q^{58} - 14q^{59} + 2q^{61} + 4q^{62} + q^{64} + q^{65} - 4q^{67} - q^{68} + 3q^{70} + 8q^{71} - 15q^{73} - q^{74} + q^{76} - 3q^{77} - 14q^{79} + q^{80} - 10q^{82} + 17q^{83} - q^{85} + 12q^{86} - q^{88} + 7q^{89} - 3q^{91} - 5q^{92} + q^{95} - 8q^{97} - 2q^{98} + O(q^{100}) \)

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 0 1.00000 1.00000 0 −3.00000 −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.g 1
3.b odd 2 1 3330.2.a.n yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.a.g 1 1.a even 1 1 trivial
3330.2.a.n yes 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7} + 3 \)
\( T_{11} - 1 \)
\( T_{13} - 1 \)
\( T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( 3 + T \)
$11$ \( -1 + T \)
$13$ \( -1 + T \)
$17$ \( 1 + T \)
$19$ \( -1 + T \)
$23$ \( 5 + T \)
$29$ \( -4 + T \)
$31$ \( 4 + T \)
$37$ \( -1 + T \)
$41$ \( -10 + T \)
$43$ \( 12 + T \)
$47$ \( T \)
$53$ \( -9 + T \)
$59$ \( 14 + T \)
$61$ \( -2 + T \)
$67$ \( 4 + T \)
$71$ \( -8 + T \)
$73$ \( 15 + T \)
$79$ \( 14 + T \)
$83$ \( -17 + T \)
$89$ \( -7 + T \)
$97$ \( 8 + T \)
show more
show less