Properties

Label 3330.2.a.bg
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - \beta_1 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} - \beta_1 q^{7} - q^{8} - q^{10} + ( - \beta_{2} + \beta_1 - 4) q^{11} + 2 \beta_{2} q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{2} - \beta_1) q^{17} - \beta_{2} q^{19} + q^{20} + (\beta_{2} - \beta_1 + 4) q^{22} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} + q^{25} - 2 \beta_{2} q^{26} - \beta_1 q^{28} + ( - \beta_{2} - \beta_1 + 2) q^{29} + 3 \beta_1 q^{31} - q^{32} + ( - \beta_{2} + \beta_1) q^{34} - \beta_1 q^{35} - q^{37} + \beta_{2} q^{38} - q^{40} + ( - \beta_{2} + 3 \beta_1 + 2) q^{41} + (\beta_{2} + \beta_1 - 4) q^{43} + ( - \beta_{2} + \beta_1 - 4) q^{44} + (2 \beta_{2} - 2 \beta_1) q^{46} + (\beta_{2} - 2 \beta_1) q^{47} + (\beta_{2} - \beta_1 - 1) q^{49} - q^{50} + 2 \beta_{2} q^{52} + (\beta_{2} - 3 \beta_1 - 6) q^{53} + ( - \beta_{2} + \beta_1 - 4) q^{55} + \beta_1 q^{56} + (\beta_{2} + \beta_1 - 2) q^{58} + (\beta_{2} - 2 \beta_1 - 4) q^{59} + (\beta_{2} + \beta_1 + 2) q^{61} - 3 \beta_1 q^{62} + q^{64} + 2 \beta_{2} q^{65} + ( - \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - \beta_1) q^{68} + \beta_1 q^{70} + (2 \beta_{2} + 2 \beta_1 - 8) q^{71} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{73} + q^{74} - \beta_{2} q^{76} + (\beta_{2} + 5 \beta_1 - 4) q^{77} + ( - \beta_{2} - 2 \beta_1 - 8) q^{79} + q^{80} + (\beta_{2} - 3 \beta_1 - 2) q^{82} + ( - 3 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{2} - \beta_1) q^{85} + ( - \beta_{2} - \beta_1 + 4) q^{86} + (\beta_{2} - \beta_1 + 4) q^{88} - 6 q^{89} + ( - 4 \beta_{2} - 4) q^{91} + ( - 2 \beta_{2} + 2 \beta_1) q^{92} + ( - \beta_{2} + 2 \beta_1) q^{94} - \beta_{2} q^{95} + (\beta_{2} - 3 \beta_1 - 6) q^{97} + ( - \beta_{2} + \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34} - q^{35} - 3 q^{37} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} - 3 q^{50} - 21 q^{53} - 11 q^{55} + q^{56} - 5 q^{58} - 14 q^{59} + 7 q^{61} - 3 q^{62} + 3 q^{64} + 2 q^{67} - q^{68} + q^{70} - 22 q^{71} - 16 q^{73} + 3 q^{74} - 7 q^{77} - 26 q^{79} + 3 q^{80} - 9 q^{82} - 2 q^{83} - q^{85} + 11 q^{86} + 11 q^{88} - 18 q^{89} - 12 q^{91} + 2 q^{92} + 2 q^{94} - 21 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 8x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 6 \) Copy content Toggle raw display

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
2.59774
1.31955
−2.91729
−1.00000 0 1.00000 1.00000 0 −2.59774 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 −1.31955 −1.00000 0 −1.00000
1.3 −1.00000 0 1.00000 1.00000 0 2.91729 −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.bg 3
3.b odd 2 1 370.2.a.g 3
12.b even 2 1 2960.2.a.u 3
15.d odd 2 1 1850.2.a.z 3
15.e even 4 2 1850.2.b.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 3.b odd 2 1
1850.2.a.z 3 15.d odd 2 1
1850.2.b.o 6 15.e even 4 2
2960.2.a.u 3 12.b even 2 1
3330.2.a.bg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7}^{3} + T_{7}^{2} - 8T_{7} - 10 \) Copy content Toggle raw display
\( T_{11}^{3} + 11T_{11}^{2} + 28T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{3} - 40T_{13} - 32 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 12T_{17} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 8T - 10 \) Copy content Toggle raw display
$11$ \( T^{3} + 11 T^{2} + 28 T - 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 40T - 32 \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 12T + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 10T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} - 5 T^{2} - 16 T + 76 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 72 T + 270 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 9 T^{2} - 40 T + 364 \) Copy content Toggle raw display
$43$ \( T^{3} + 11 T^{2} + 16 T - 80 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 30 T - 56 \) Copy content Toggle raw display
$53$ \( T^{3} + 21 T^{2} + 80 T - 316 \) Copy content Toggle raw display
$59$ \( T^{3} + 14 T^{2} + 34 T - 80 \) Copy content Toggle raw display
$61$ \( T^{3} - 7 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 30 T + 56 \) Copy content Toggle raw display
$71$ \( T^{3} + 22 T^{2} + 64 T - 640 \) Copy content Toggle raw display
$73$ \( T^{3} + 16 T^{2} + 36 T - 208 \) Copy content Toggle raw display
$79$ \( T^{3} + 26 T^{2} + 170 T + 224 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} - 158 T + 664 \) Copy content Toggle raw display
$89$ \( (T + 6)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 21 T^{2} + 80 T - 316 \) Copy content Toggle raw display
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