Properties

Label 3360.2.a.bj
Level $3360$
Weight $2$
Character orbit 3360.a
Self dual yes
Analytic conductor $26.830$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -2 \nu^{2} + 4 \nu + 3 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 5\)\()/2\)

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
2.17009
−1.48119
0.311108
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.bj yes 3
4.b odd 2 1 3360.2.a.bi 3
8.b even 2 1 6720.2.a.da 3
8.d odd 2 1 6720.2.a.db 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bi 3 4.b odd 2 1
3360.2.a.bj yes 3 1.a even 1 1 trivial
6720.2.a.da 3 8.b even 2 1
6720.2.a.db 3 8.d odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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