Properties

Label 3744.2.a.bb
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.8959905168\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + q^{13} + ( -2 - 2 \beta_{3} ) q^{17} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( 1 - 2 \beta_{3} ) q^{25} -4 q^{29} -\beta_{2} q^{31} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -2 - 4 \beta_{3} ) q^{37} + ( -3 - \beta_{3} ) q^{41} + 2 \beta_{1} q^{43} + ( -3 \beta_{1} + \beta_{2} ) q^{47} + ( 3 - 2 \beta_{3} ) q^{49} + ( -4 + 4 \beta_{3} ) q^{53} + 2 \beta_{1} q^{55} + ( 3 \beta_{1} - \beta_{2} ) q^{59} + 2 \beta_{3} q^{61} + ( -1 + \beta_{3} ) q^{65} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{2} ) q^{71} + 6 q^{73} + ( -10 - 2 \beta_{3} ) q^{77} -4 \beta_{1} q^{79} + ( \beta_{1} - 3 \beta_{2} ) q^{83} -8 q^{85} + ( -9 - 3 \beta_{3} ) q^{89} -\beta_{2} q^{91} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{95} + ( 2 + 4 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{13} - 8q^{17} + 4q^{25} - 16q^{29} - 8q^{37} - 12q^{41} + 12q^{49} - 16q^{53} - 4q^{65} + 24q^{73} - 40q^{77} - 32q^{85} - 36q^{89} + 8q^{97} + O(q^{100}) \)

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
−1.17557
1.17557
1.90211
−1.90211
0 0 0 −3.23607 0 −3.80423 0 0 0
1.2 0 0 0 −3.23607 0 3.80423 0 0 0
1.3 0 0 0 1.23607 0 −2.35114 0 0 0
1.4 0 0 0 1.23607 0 2.35114 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.a.bb 4
3.b odd 2 1 3744.2.a.bf yes 4
4.b odd 2 1 inner 3744.2.a.bb 4
8.b even 2 1 7488.2.a.dd 4
8.d odd 2 1 7488.2.a.dd 4
12.b even 2 1 3744.2.a.bf yes 4
24.f even 2 1 7488.2.a.cz 4
24.h odd 2 1 7488.2.a.cz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.2.a.bb 4 1.a even 1 1 trivial
3744.2.a.bb 4 4.b odd 2 1 inner
3744.2.a.bf yes 4 3.b odd 2 1
3744.2.a.bf yes 4 12.b even 2 1
7488.2.a.cz 4 24.f even 2 1
7488.2.a.cz 4 24.h odd 2 1
7488.2.a.dd 4 8.b even 2 1
7488.2.a.dd 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{7}^{4} - 20 T_{7}^{2} + 80 \)
\( T_{11}^{4} - 40 T_{11}^{2} + 80 \)
\( T_{29} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -4 + 2 T + T^{2} )^{2} \)
$7$ \( 80 - 20 T^{2} + T^{4} \)
$11$ \( 80 - 40 T^{2} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( -16 + 4 T + T^{2} )^{2} \)
$19$ \( 2000 - 100 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( 80 - 20 T^{2} + T^{4} \)
$37$ \( ( -76 + 4 T + T^{2} )^{2} \)
$41$ \( ( 4 + 6 T + T^{2} )^{2} \)
$43$ \( 1280 - 80 T^{2} + T^{4} \)
$47$ \( 9680 - 200 T^{2} + T^{4} \)
$53$ \( ( -64 + 8 T + T^{2} )^{2} \)
$59$ \( 9680 - 200 T^{2} + T^{4} \)
$61$ \( ( -20 + T^{2} )^{2} \)
$67$ \( 9680 - 260 T^{2} + T^{4} \)
$71$ \( 80 - 40 T^{2} + T^{4} \)
$73$ \( ( -6 + T )^{4} \)
$79$ \( 20480 - 320 T^{2} + T^{4} \)
$83$ \( 2000 - 200 T^{2} + T^{4} \)
$89$ \( ( 36 + 18 T + T^{2} )^{2} \)
$97$ \( ( -76 - 4 T + T^{2} )^{2} \)
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