Properties

Label 3360.2.a.bc
Level $3360$
Weight $2$
Character orbit 3360.a
Self dual yes
Analytic conductor $26.830$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + q^{7} + q^{9} + \beta q^{11} + ( -2 + \beta ) q^{13} + q^{15} + ( 2 + 2 \beta ) q^{17} + \beta q^{19} - q^{21} + 4 q^{23} + q^{25} - q^{27} + ( -2 - 2 \beta ) q^{29} + \beta q^{31} -\beta q^{33} - q^{35} + ( -6 - 2 \beta ) q^{37} + ( 2 - \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} - q^{45} -4 \beta q^{47} + q^{49} + ( -2 - 2 \beta ) q^{51} + ( 2 + 3 \beta ) q^{53} -\beta q^{55} -\beta q^{57} + ( -2 + 4 \beta ) q^{61} + q^{63} + ( 2 - \beta ) q^{65} + 4 q^{67} -4 q^{69} + ( 8 + \beta ) q^{71} + ( -2 - \beta ) q^{73} - q^{75} + \beta q^{77} + ( 4 + 4 \beta ) q^{79} + q^{81} + ( 12 + 2 \beta ) q^{83} + ( -2 - 2 \beta ) q^{85} + ( 2 + 2 \beta ) q^{87} + ( 6 + 2 \beta ) q^{89} + ( -2 + \beta ) q^{91} -\beta q^{93} -\beta q^{95} + ( -10 + \beta ) q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{13} + 2q^{15} + 4q^{17} - 2q^{21} + 8q^{23} + 2q^{25} - 2q^{27} - 4q^{29} - 2q^{35} - 12q^{37} + 4q^{39} - 4q^{41} + 8q^{43} - 2q^{45} + 2q^{49} - 4q^{51} + 4q^{53} - 4q^{61} + 2q^{63} + 4q^{65} + 8q^{67} - 8q^{69} + 16q^{71} - 4q^{73} - 2q^{75} + 8q^{79} + 2q^{81} + 24q^{83} - 4q^{85} + 4q^{87} + 12q^{89} - 4q^{91} - 20q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
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.bc 2
4.b odd 2 1 3360.2.a.be yes 2
8.b even 2 1 6720.2.a.cy 2
8.d odd 2 1 6720.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bc 2 1.a even 1 1 trivial
3360.2.a.be yes 2 4.b odd 2 1
6720.2.a.cr 2 8.d odd 2 1
6720.2.a.cy 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3360))\):

\( T_{11}^{2} - 8 \)
\( T_{13}^{2} + 4 T_{13} - 4 \)
\( T_{17}^{2} - 4 T_{17} - 28 \)
\( T_{19}^{2} - 8 \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( -4 + 4 T + T^{2} \)
$17$ \( -28 - 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( -28 + 4 T + T^{2} \)
$31$ \( -8 + T^{2} \)
$37$ \( 4 + 12 T + T^{2} \)
$41$ \( -28 + 4 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( -128 + T^{2} \)
$53$ \( -68 - 4 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -4 + 4 T + T^{2} \)
$79$ \( -112 - 8 T + T^{2} \)
$83$ \( 112 - 24 T + T^{2} \)
$89$ \( 4 - 12 T + T^{2} \)
$97$ \( 92 + 20 T + T^{2} \)
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