Properties

Label 3360.2.a.bi
Level $3360$
Weight $2$
Character orbit 3360.a
Self dual yes
Analytic conductor $26.830$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.8297350792\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - q^{7} + q^{9} + (\beta_{2} - 1) q^{11} + (\beta_1 + 1) q^{13} - q^{15} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 3) q^{19} + q^{21} + (\beta_{2} - \beta_1) q^{23} + q^{25} - q^{27} + ( - \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{2} + \beta_1 - 3) q^{31} + ( - \beta_{2} + 1) q^{33} - q^{35} + (\beta_{2} + \beta_1) q^{37} + ( - \beta_1 - 1) q^{39} + 2 \beta_{2} q^{41} + ( - \beta_{2} - \beta_1 - 2) q^{43} + q^{45} + ( - \beta_{2} + \beta_1) q^{47} + q^{49} + (\beta_{2} + \beta_1) q^{51} + (\beta_{2} - 3) q^{53} + (\beta_{2} - 1) q^{55} + (\beta_{2} + 3) q^{57} - 8 q^{59} + (\beta_{2} - \beta_1 - 2) q^{61} - q^{63} + (\beta_1 + 1) q^{65} + (3 \beta_{2} + \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (\beta_1 - 1) q^{71} + ( - \beta_1 + 3) q^{73} - q^{75} + ( - \beta_{2} + 1) q^{77} - 4 q^{79} + q^{81} + ( - 2 \beta_{2} - 2) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + (\beta_{2} + \beta_1) q^{87} + 2 \beta_1 q^{89} + ( - \beta_1 - 1) q^{91} + (2 \beta_{2} - \beta_1 + 3) q^{93} + ( - \beta_{2} - 3) q^{95} + (\beta_1 + 1) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - 3 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 3 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{61} - 3 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{71} + 10 q^{73} - 3 q^{75} + 4 q^{77} - 12 q^{79} + 3 q^{81} - 4 q^{83} + 2 q^{85} - 2 q^{87} - 2 q^{89} - 2 q^{91} + 8 q^{93} - 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 5 ) / 2 \) 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
0.311108
−1.48119
2.17009
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 3360.2.a.bi 3
4.b odd 2 1 3360.2.a.bj yes 3
8.b even 2 1 6720.2.a.db 3
8.d odd 2 1 6720.2.a.da 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bi 3 1.a even 1 1 trivial
3360.2.a.bj yes 3 4.b odd 2 1
6720.2.a.da 3 8.d odd 2 1
6720.2.a.db 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3360))\):

\( T_{11}^{3} + 4T_{11}^{2} - 16T_{11} - 32 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 36T_{13} + 104 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 52T_{17} + 40 \) Copy content Toggle raw display
\( T_{19}^{3} + 8T_{19}^{2} - 32 \) Copy content Toggle raw display
\( T_{23}^{3} - 64T_{23} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 16 T - 32 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 52 T + 40 \) Copy content Toggle raw display
$19$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 52 T + 40 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 112 T - 928 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 52 T - 40 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 84 T - 104 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$47$ \( T^{3} - 64T - 128 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$59$ \( (T + 8)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 52 T + 8 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 208 T - 1472 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$79$ \( (T + 4)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 148 T + 536 \) Copy content Toggle raw display
$97$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
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