Properties

Label 3744.2.a.z
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1248)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{5} -\beta_{2} q^{7} + ( -1 + \beta_{1} + \beta_{2} ) q^{11} + q^{13} -2 q^{17} + \beta_{2} q^{19} -2 \beta_{2} q^{23} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{25} -2 q^{29} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{31} + ( -2 + 2 \beta_{1} ) q^{35} -2 \beta_{1} q^{37} + ( -3 + \beta_{1} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 3 + \beta_{1} + \beta_{2} ) q^{47} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -4 + 2 \beta_{1} ) q^{53} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -7 - \beta_{1} + \beta_{2} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -1 - \beta_{1} ) q^{65} + ( -6 - 2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 3 + \beta_{1} - \beta_{2} ) q^{71} + ( 8 + 2 \beta_{1} ) q^{73} + ( -8 + 4 \beta_{2} ) q^{77} -12 q^{79} + ( -1 + \beta_{1} - \beta_{2} ) q^{83} + ( 2 + 2 \beta_{1} ) q^{85} + ( 3 + 3 \beta_{1} ) q^{89} -\beta_{2} q^{91} + ( 2 - 2 \beta_{1} ) q^{95} + 2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{5} + O(q^{10}) \) \( 3q - 2q^{5} - 4q^{11} + 3q^{13} - 6q^{17} + 13q^{25} - 6q^{29} - 8q^{31} - 8q^{35} + 2q^{37} - 10q^{41} + 4q^{43} + 8q^{47} + 11q^{49} - 14q^{53} - 16q^{55} - 20q^{59} + 2q^{61} - 2q^{65} - 16q^{67} + 8q^{71} + 22q^{73} - 24q^{77} - 36q^{79} - 4q^{83} + 4q^{85} + 6q^{89} + 8q^{95} - 2q^{97} + O(q^{100}) \)

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

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

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
0.311108
−1.48119
0 0 0 −4.34017 0 −1.07838 0 0 0
1.2 0 0 0 −0.622216 0 4.42864 0 0 0
1.3 0 0 0 2.96239 0 −3.35026 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

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 3744.2.a.z 3
3.b odd 2 1 1248.2.a.p yes 3
4.b odd 2 1 3744.2.a.ba 3
8.b even 2 1 7488.2.a.cy 3
8.d odd 2 1 7488.2.a.cx 3
12.b even 2 1 1248.2.a.o 3
24.f even 2 1 2496.2.a.bl 3
24.h odd 2 1 2496.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.o 3 12.b even 2 1
1248.2.a.p yes 3 3.b odd 2 1
2496.2.a.bk 3 24.h odd 2 1
2496.2.a.bl 3 24.f even 2 1
3744.2.a.z 3 1.a even 1 1 trivial
3744.2.a.ba 3 4.b odd 2 1
7488.2.a.cx 3 8.d odd 2 1
7488.2.a.cy 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5}^{3} + 2 T_{5}^{2} - 12 T_{5} - 8 \)
\( T_{7}^{3} - 16 T_{7} - 16 \)
\( T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32 \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$7$ \( -16 - 16 T + T^{3} \)
$11$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( ( 2 + T )^{3} \)
$19$ \( 16 - 16 T + T^{3} \)
$23$ \( -128 - 64 T + T^{3} \)
$29$ \( ( 2 + T )^{3} \)
$31$ \( -272 - 32 T + 8 T^{2} + T^{3} \)
$37$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$41$ \( -8 + 20 T + 10 T^{2} + T^{3} \)
$43$ \( 64 - 80 T - 4 T^{2} + T^{3} \)
$47$ \( 32 - 8 T^{2} + T^{3} \)
$53$ \( -152 + 12 T + 14 T^{2} + T^{3} \)
$59$ \( -32 + 96 T + 20 T^{2} + T^{3} \)
$61$ \( -536 - 148 T - 2 T^{2} + T^{3} \)
$67$ \( -1040 - 64 T + 16 T^{2} + T^{3} \)
$71$ \( 160 - 16 T - 8 T^{2} + T^{3} \)
$73$ \( -8 + 108 T - 22 T^{2} + T^{3} \)
$79$ \( ( 12 + T )^{3} \)
$83$ \( 32 - 32 T + 4 T^{2} + T^{3} \)
$89$ \( 216 - 108 T - 6 T^{2} + T^{3} \)
$97$ \( -40 - 52 T + 2 T^{2} + T^{3} \)
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