Properties

Label 3744.2.a.q
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{5} -\beta q^{7} +O(q^{10})\) \( q -3 q^{5} -\beta q^{7} -2 \beta q^{11} - q^{13} + 3 q^{17} + 2 \beta q^{19} -4 \beta q^{23} + 4 q^{25} -10 q^{29} + 3 \beta q^{35} + 3 q^{37} -3 \beta q^{43} + \beta q^{47} -2 q^{49} -4 q^{53} + 6 \beta q^{55} -2 \beta q^{59} + 3 q^{65} -6 \beta q^{67} + 3 \beta q^{71} + 14 q^{73} + 10 q^{77} + 4 \beta q^{79} + 8 \beta q^{83} -9 q^{85} + 10 q^{89} + \beta q^{91} -6 \beta q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + O(q^{10}) \) \( 2q - 6q^{5} - 2q^{13} + 6q^{17} + 8q^{25} - 20q^{29} + 6q^{37} - 4q^{49} - 8q^{53} + 6q^{65} + 28q^{73} + 20q^{77} - 18q^{85} + 20q^{89} - 4q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
0 0 0 −3.00000 0 −2.23607 0 0 0
1.2 0 0 0 −3.00000 0 2.23607 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.q 2
3.b odd 2 1 416.2.a.d 2
4.b odd 2 1 inner 3744.2.a.q 2
8.b even 2 1 7488.2.a.cw 2
8.d odd 2 1 7488.2.a.cw 2
12.b even 2 1 416.2.a.d 2
24.f even 2 1 832.2.a.m 2
24.h odd 2 1 832.2.a.m 2
39.d odd 2 1 5408.2.a.q 2
48.i odd 4 2 3328.2.b.x 4
48.k even 4 2 3328.2.b.x 4
156.h even 2 1 5408.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.d 2 3.b odd 2 1
416.2.a.d 2 12.b even 2 1
832.2.a.m 2 24.f even 2 1
832.2.a.m 2 24.h odd 2 1
3328.2.b.x 4 48.i odd 4 2
3328.2.b.x 4 48.k even 4 2
3744.2.a.q 2 1.a even 1 1 trivial
3744.2.a.q 2 4.b odd 2 1 inner
5408.2.a.q 2 39.d odd 2 1
5408.2.a.q 2 156.h even 2 1
7488.2.a.cw 2 8.b even 2 1
7488.2.a.cw 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5} + 3 \)
\( T_{7}^{2} - 5 \)
\( T_{11}^{2} - 20 \)
\( T_{29} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( -5 + T^{2} \)
$11$ \( -20 + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( -80 + T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -3 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( -45 + T^{2} \)
$47$ \( -5 + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( -20 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( -180 + T^{2} \)
$71$ \( -45 + T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -320 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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