Properties

Label 3648.2.a.bx
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1824)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + \beta_{2} q^{17} - q^{19} + ( 2 - \beta_{2} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} ) q^{23} + ( 5 - \beta_{2} ) q^{25} - q^{27} + ( -2 + 2 \beta_{2} ) q^{29} + ( -3 - \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{33} + ( -4 + 2 \beta_{1} + 3 \beta_{2} ) q^{35} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} ) q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 - 2 \beta_{1} - 3 \beta_{2} ) q^{49} -\beta_{2} q^{51} + 2 q^{53} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{55} + q^{57} + ( -4 + 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( -2 + \beta_{2} ) q^{63} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 - 2 \beta_{1} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} ) q^{69} + ( 4 + 2 \beta_{2} ) q^{71} + ( 12 + \beta_{2} ) q^{73} + ( -5 + \beta_{2} ) q^{75} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{77} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 4 - 4 \beta_{2} ) q^{83} + ( -2 + 3 \beta_{2} ) q^{85} + ( 2 - 2 \beta_{2} ) q^{87} + 10 q^{89} + 8 q^{91} + ( 3 + \beta_{1} - \beta_{2} ) q^{93} + ( -1 + \beta_{1} ) q^{95} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{5} - 5q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{5} - 5q^{7} + 3q^{9} - 7q^{11} - 2q^{13} - 3q^{15} + q^{17} - 3q^{19} + 5q^{21} + 2q^{23} + 14q^{25} - 3q^{27} - 4q^{29} - 8q^{31} + 7q^{33} - 9q^{35} - 6q^{37} + 2q^{39} + 12q^{41} + q^{43} + 3q^{45} + 11q^{47} + 12q^{49} - q^{51} + 6q^{53} - 3q^{55} + 3q^{57} - 10q^{59} - 7q^{61} - 5q^{63} + 20q^{65} + 6q^{67} - 2q^{69} + 14q^{71} + 37q^{73} - 14q^{75} - 13q^{77} - 4q^{79} + 3q^{81} + 8q^{83} - 3q^{85} + 4q^{87} + 30q^{89} + 24q^{91} + 8q^{93} - 3q^{95} + 8q^{97} - 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 7\)\()/2\)

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
2.39138
−2.16425
0.772866
0 −1.00000 0 −3.11009 0 −1.67267 0 1.00000 0
1.2 0 −1.00000 0 2.48028 0 1.84822 0 1.00000 0
1.3 0 −1.00000 0 3.62981 0 −5.17554 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

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 3648.2.a.bx 3
4.b odd 2 1 3648.2.a.bz 3
8.b even 2 1 1824.2.a.u yes 3
8.d odd 2 1 1824.2.a.s 3
24.f even 2 1 5472.2.a.bp 3
24.h odd 2 1 5472.2.a.bo 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.s 3 8.d odd 2 1
1824.2.a.u yes 3 8.b even 2 1
3648.2.a.bx 3 1.a even 1 1 trivial
3648.2.a.bz 3 4.b odd 2 1
5472.2.a.bo 3 24.h odd 2 1
5472.2.a.bp 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5}^{3} - 3 T_{5}^{2} - 10 T_{5} + 28 \)
\( T_{7}^{3} + 5 T_{7}^{2} - 4 T_{7} - 16 \)
\( T_{11}^{3} + 7 T_{11}^{2} + 4 T_{11} - 16 \)
\( T_{23}^{3} - 2 T_{23}^{2} - 28 T_{23} - 32 \)
\( T_{31}^{3} + 8 T_{31}^{2} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 28 - 10 T - 3 T^{2} + T^{3} \)
$7$ \( -16 - 4 T + 5 T^{2} + T^{3} \)
$11$ \( -16 + 4 T + 7 T^{2} + T^{3} \)
$13$ \( -32 - 20 T + 2 T^{2} + T^{3} \)
$17$ \( 4 - 12 T - T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -32 - 28 T - 2 T^{2} + T^{3} \)
$29$ \( -64 - 44 T + 4 T^{2} + T^{3} \)
$31$ \( -56 + 8 T^{2} + T^{3} \)
$37$ \( -752 - 124 T + 6 T^{2} + T^{3} \)
$41$ \( 272 - 4 T - 12 T^{2} + T^{3} \)
$43$ \( 112 - 56 T - T^{2} + T^{3} \)
$47$ \( -4 - 14 T - 11 T^{2} + T^{3} \)
$53$ \( ( -2 + T )^{3} \)
$59$ \( -128 - 16 T + 10 T^{2} + T^{3} \)
$61$ \( -212 - 40 T + 7 T^{2} + T^{3} \)
$67$ \( 224 - 40 T - 6 T^{2} + T^{3} \)
$71$ \( 128 + 16 T - 14 T^{2} + T^{3} \)
$73$ \( -1724 + 444 T - 37 T^{2} + T^{3} \)
$79$ \( -56 - 136 T + 4 T^{2} + T^{3} \)
$83$ \( 512 - 176 T - 8 T^{2} + T^{3} \)
$89$ \( ( -10 + T )^{3} \)
$97$ \( 1792 - 196 T - 8 T^{2} + T^{3} \)
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