Properties

Label 3744.2.a.r
Level $3744$
Weight $2$
Character orbit 3744.a
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{5} + ( -1 - \beta ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta ) q^{5} + ( -1 - \beta ) q^{7} + 2 q^{11} - q^{13} + 2 \beta q^{17} + ( -3 + \beta ) q^{19} + ( 2 + 2 \beta ) q^{23} + ( 1 + 2 \beta ) q^{25} + 2 \beta q^{29} + ( -1 + 3 \beta ) q^{31} + ( 6 + 2 \beta ) q^{35} + ( -4 - 2 \beta ) q^{37} + ( 1 + \beta ) q^{41} + ( -2 + 2 \beta ) q^{43} + ( -2 - 4 \beta ) q^{47} + ( -1 + 2 \beta ) q^{49} + ( -4 + 2 \beta ) q^{53} + ( -2 - 2 \beta ) q^{55} + ( 4 - 2 \beta ) q^{59} + ( -8 + 2 \beta ) q^{61} + ( 1 + \beta ) q^{65} + ( -9 - \beta ) q^{67} -2 \beta q^{71} + ( -4 - 2 \beta ) q^{73} + ( -2 - 2 \beta ) q^{77} -4 \beta q^{79} + ( 12 + 2 \beta ) q^{83} + ( -10 - 2 \beta ) q^{85} + ( -1 - 5 \beta ) q^{89} + ( 1 + \beta ) q^{91} + ( -2 + 2 \beta ) q^{95} + 2 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{7} + 4q^{11} - 2q^{13} - 6q^{19} + 4q^{23} + 2q^{25} - 2q^{31} + 12q^{35} - 8q^{37} + 2q^{41} - 4q^{43} - 4q^{47} - 2q^{49} - 8q^{53} - 4q^{55} + 8q^{59} - 16q^{61} + 2q^{65} - 18q^{67} - 8q^{73} - 4q^{77} + 24q^{83} - 20q^{85} - 2q^{89} + 2q^{91} - 4q^{95} + O(q^{100}) \)

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 −3.23607 0 −3.23607 0 0 0
1.2 0 0 0 1.23607 0 1.23607 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

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.r 2
3.b odd 2 1 1248.2.a.l 2
4.b odd 2 1 3744.2.a.s 2
8.b even 2 1 7488.2.a.cs 2
8.d odd 2 1 7488.2.a.ct 2
12.b even 2 1 1248.2.a.n yes 2
24.f even 2 1 2496.2.a.be 2
24.h odd 2 1 2496.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.l 2 3.b odd 2 1
1248.2.a.n yes 2 12.b even 2 1
2496.2.a.be 2 24.f even 2 1
2496.2.a.bh 2 24.h odd 2 1
3744.2.a.r 2 1.a even 1 1 trivial
3744.2.a.s 2 4.b odd 2 1
7488.2.a.cs 2 8.b even 2 1
7488.2.a.ct 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( 4 + 6 T + T^{2} \)
$23$ \( -16 - 4 T + T^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -44 + 2 T + T^{2} \)
$37$ \( -4 + 8 T + T^{2} \)
$41$ \( -4 - 2 T + T^{2} \)
$43$ \( -16 + 4 T + T^{2} \)
$47$ \( -76 + 4 T + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -4 - 8 T + T^{2} \)
$61$ \( 44 + 16 T + T^{2} \)
$67$ \( 76 + 18 T + T^{2} \)
$71$ \( -20 + T^{2} \)
$73$ \( -4 + 8 T + T^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( 124 - 24 T + T^{2} \)
$89$ \( -124 + 2 T + T^{2} \)
$97$ \( -20 + T^{2} \)
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