Properties

Label 3360.2.a.bb
Level $3360$
Weight $2$
Character orbit 3360.a
Self dual yes
Analytic conductor $26.830$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + q^{7} + q^{9} + ( -2 + \beta ) q^{11} -\beta q^{13} + q^{15} + 2 q^{17} + ( -2 + \beta ) q^{19} - q^{21} -2 \beta q^{23} + q^{25} - q^{27} -2 q^{29} + ( 2 - \beta ) q^{31} + ( 2 - \beta ) q^{33} - q^{35} + 2 q^{37} + \beta q^{39} + ( 2 + 2 \beta ) q^{41} -4 q^{43} - q^{45} + ( -4 + 2 \beta ) q^{47} + q^{49} -2 q^{51} + ( -4 - \beta ) q^{53} + ( 2 - \beta ) q^{55} + ( 2 - \beta ) q^{57} -4 \beta q^{59} + ( 2 + 2 \beta ) q^{61} + q^{63} + \beta q^{65} -2 \beta q^{67} + 2 \beta q^{69} + ( -6 + 3 \beta ) q^{71} + ( 4 + \beta ) q^{73} - q^{75} + ( -2 + \beta ) q^{77} -4 q^{79} + q^{81} + ( -8 + 2 \beta ) q^{83} -2 q^{85} + 2 q^{87} + ( -6 + 2 \beta ) q^{89} -\beta q^{91} + ( -2 + \beta ) q^{93} + ( 2 - \beta ) q^{95} + ( 16 - \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{15} + 4q^{17} - 4q^{19} - 2q^{21} + 2q^{25} - 2q^{27} - 4q^{29} + 4q^{31} + 4q^{33} - 2q^{35} + 4q^{37} + 4q^{41} - 8q^{43} - 2q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 8q^{53} + 4q^{55} + 4q^{57} + 4q^{61} + 2q^{63} - 12q^{71} + 8q^{73} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 16q^{83} - 4q^{85} + 4q^{87} - 12q^{89} - 4q^{93} + 4q^{95} + 32q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
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
\(n\): 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.bb 2
4.b odd 2 1 3360.2.a.bf yes 2
8.b even 2 1 6720.2.a.cz 2
8.d odd 2 1 6720.2.a.cq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bb 2 1.a even 1 1 trivial
3360.2.a.bf yes 2 4.b odd 2 1
6720.2.a.cq 2 8.d odd 2 1
6720.2.a.cz 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3360))\):

\( T_{11}^{2} + 4 T_{11} - 8 \)
\( T_{13}^{2} - 12 \)
\( T_{17} - 2 \)
\( T_{19}^{2} + 4 T_{19} - 8 \)
\( T_{23}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( -8 + 4 T + T^{2} \)
$23$ \( -48 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -44 - 4 T + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -32 + 8 T + T^{2} \)
$53$ \( 4 + 8 T + T^{2} \)
$59$ \( -192 + T^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( -48 + T^{2} \)
$71$ \( -72 + 12 T + T^{2} \)
$73$ \( 4 - 8 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 16 + 16 T + T^{2} \)
$89$ \( -12 + 12 T + T^{2} \)
$97$ \( 244 - 32 T + T^{2} \)
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