Properties

Label 3330.2.a.bi
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + ( -1 + 2 \beta_{1} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} + ( -1 + 2 \beta_{1} ) q^{7} + q^{8} - q^{10} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( -1 + 2 \beta_{1} ) q^{14} + q^{16} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 - \beta_{2} ) q^{19} - q^{20} + ( 2 - \beta_{1} + \beta_{2} ) q^{22} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( 1 - \beta_{2} ) q^{26} + ( -1 + 2 \beta_{1} ) q^{28} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( 3 + 3 \beta_{1} ) q^{31} + q^{32} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 1 - 2 \beta_{1} ) q^{35} - q^{37} + ( 1 - \beta_{2} ) q^{38} - q^{40} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} ) q^{44} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{46} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{47} + ( 6 - 4 \beta_{1} + 4 \beta_{2} ) q^{49} + q^{50} + ( 1 - \beta_{2} ) q^{52} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -2 + \beta_{1} - \beta_{2} ) q^{55} + ( -1 + 2 \beta_{1} ) q^{56} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{58} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( 3 + 3 \beta_{1} ) q^{62} + q^{64} + ( -1 + \beta_{2} ) q^{65} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 1 - 2 \beta_{1} ) q^{70} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 7 - 4 \beta_{1} + 7 \beta_{2} ) q^{73} - q^{74} + ( 1 - \beta_{2} ) q^{76} + ( -6 + 7 \beta_{1} - \beta_{2} ) q^{77} + ( 12 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} - q^{80} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -7 + 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{86} + ( 2 - \beta_{1} + \beta_{2} ) q^{88} + ( 3 - 3 \beta_{2} ) q^{89} + ( -3 - \beta_{2} ) q^{91} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{92} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{94} + ( -1 + \beta_{2} ) q^{95} + ( 5 - 3 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 6 - 4 \beta_{1} + 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} - q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} - q^{7} + 3q^{8} - 3q^{10} + 5q^{11} + 3q^{13} - q^{14} + 3q^{16} - 5q^{17} + 3q^{19} - 3q^{20} + 5q^{22} + q^{23} + 3q^{25} + 3q^{26} - q^{28} + 10q^{29} + 12q^{31} + 3q^{32} - 5q^{34} + q^{35} - 3q^{37} + 3q^{38} - 3q^{40} + 6q^{41} + 16q^{43} + 5q^{44} + q^{46} - 16q^{47} + 14q^{49} + 3q^{50} + 3q^{52} + 9q^{53} - 5q^{55} - q^{56} + 10q^{58} + 14q^{59} - 2q^{61} + 12q^{62} + 3q^{64} - 3q^{65} + 20q^{67} - 5q^{68} + q^{70} + 4q^{71} + 17q^{73} - 3q^{74} + 3q^{76} - 11q^{77} + 34q^{79} - 3q^{80} + 6q^{82} - 19q^{83} + 5q^{85} + 16q^{86} + 5q^{88} + 9q^{89} - 9q^{91} + q^{92} - 16q^{94} - 3q^{95} + 12q^{97} + 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
0.470683
2.34292
1.00000 0 1.00000 −1.00000 0 −4.62721 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −0.0586332 1.00000 0 −1.00000
1.3 1.00000 0 1.00000 −1.00000 0 3.68585 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.bi yes 3
3.b odd 2 1 3330.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.a.bh 3 3.b odd 2 1
3330.2.a.bi yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7}^{3} + T_{7}^{2} - 17 T_{7} - 1 \)
\( T_{11}^{3} - 5 T_{11}^{2} + T_{11} + 11 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 4 T_{13} + 4 \)
\( T_{17}^{3} + 5 T_{17}^{2} - 21 T_{17} - 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -1 - 17 T + T^{2} + T^{3} \)
$11$ \( 11 + T - 5 T^{2} + T^{3} \)
$13$ \( 4 - 4 T - 3 T^{2} + T^{3} \)
$17$ \( -17 - 21 T + 5 T^{2} + T^{3} \)
$19$ \( 4 - 4 T - 3 T^{2} + T^{3} \)
$23$ \( 184 - 60 T - T^{2} + T^{3} \)
$29$ \( -4 + 17 T - 10 T^{2} + T^{3} \)
$31$ \( 108 + 9 T - 12 T^{2} + T^{3} \)
$37$ \( ( 1 + T )^{3} \)
$41$ \( 506 - 79 T - 6 T^{2} + T^{3} \)
$43$ \( -68 + 61 T - 16 T^{2} + T^{3} \)
$47$ \( -736 - 12 T + 16 T^{2} + T^{3} \)
$53$ \( 271 - 31 T - 9 T^{2} + T^{3} \)
$59$ \( 8 + 4 T - 14 T^{2} + T^{3} \)
$61$ \( -62 - 23 T + 2 T^{2} + T^{3} \)
$67$ \( -64 + 72 T - 20 T^{2} + T^{3} \)
$71$ \( 256 - 56 T - 4 T^{2} + T^{3} \)
$73$ \( 3512 - 204 T - 17 T^{2} + T^{3} \)
$79$ \( -1096 + 356 T - 34 T^{2} + T^{3} \)
$83$ \( -724 + 16 T + 19 T^{2} + T^{3} \)
$89$ \( 108 - 36 T - 9 T^{2} + T^{3} \)
$97$ \( 632 - 43 T - 12 T^{2} + T^{3} \)
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