Properties

Label 3744.2.a.u
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta q^{7} +O(q^{10})\) \( q + \beta q^{7} + \beta q^{11} - q^{13} + \beta q^{19} + 2 \beta q^{23} -5 q^{25} + 4 q^{29} -3 \beta q^{31} + 6 q^{37} + 12 q^{41} + \beta q^{47} + q^{49} + 4 q^{53} -5 \beta q^{59} + 6 q^{61} + 3 \beta q^{67} -3 \beta q^{71} -10 q^{73} + 8 q^{77} -4 \beta q^{79} + 5 \beta q^{83} -4 q^{89} -\beta q^{91} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{13} - 10q^{25} + 8q^{29} + 12q^{37} + 24q^{41} + 2q^{49} + 8q^{53} + 12q^{61} - 20q^{73} + 16q^{77} - 8q^{89} - 4q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 0 0 −2.82843 0 0 0
1.2 0 0 0 0 0 2.82843 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.u yes 2
3.b odd 2 1 3744.2.a.t 2
4.b odd 2 1 inner 3744.2.a.u yes 2
8.b even 2 1 7488.2.a.cm 2
8.d odd 2 1 7488.2.a.cm 2
12.b even 2 1 3744.2.a.t 2
24.f even 2 1 7488.2.a.cn 2
24.h odd 2 1 7488.2.a.cn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.2.a.t 2 3.b odd 2 1
3744.2.a.t 2 12.b even 2 1
3744.2.a.u yes 2 1.a even 1 1 trivial
3744.2.a.u yes 2 4.b odd 2 1 inner
7488.2.a.cm 2 8.b even 2 1
7488.2.a.cm 2 8.d odd 2 1
7488.2.a.cn 2 24.f even 2 1
7488.2.a.cn 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5} \)
\( T_{7}^{2} - 8 \)
\( T_{11}^{2} - 8 \)
\( T_{29} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( -32 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( ( -12 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -4 + T )^{2} \)
$59$ \( -200 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -72 + T^{2} \)
$71$ \( -72 + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -200 + T^{2} \)
$89$ \( ( 4 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
show more
show less