Properties

Label 3744.2.a.x
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
0 0 0 −0.561553 0 0.561553 0 0 0
1.2 0 0 0 3.56155 0 −3.56155 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.x 2
3.b odd 2 1 416.2.a.c 2
4.b odd 2 1 3744.2.a.y 2
8.b even 2 1 7488.2.a.ce 2
8.d odd 2 1 7488.2.a.cf 2
12.b even 2 1 416.2.a.e yes 2
24.f even 2 1 832.2.a.l 2
24.h odd 2 1 832.2.a.o 2
39.d odd 2 1 5408.2.a.p 2
48.i odd 4 2 3328.2.b.ba 4
48.k even 4 2 3328.2.b.u 4
156.h even 2 1 5408.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.c 2 3.b odd 2 1
416.2.a.e yes 2 12.b even 2 1
832.2.a.l 2 24.f even 2 1
832.2.a.o 2 24.h odd 2 1
3328.2.b.u 4 48.k even 4 2
3328.2.b.ba 4 48.i odd 4 2
3744.2.a.x 2 1.a even 1 1 trivial
3744.2.a.y 2 4.b odd 2 1
5408.2.a.p 2 39.d odd 2 1
5408.2.a.bd 2 156.h even 2 1
7488.2.a.ce 2 8.b even 2 1
7488.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3744))\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{7}^{2} + 3 T_{7} - 2 \)
\( T_{11} - 2 \)
\( T_{29}^{2} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( -2 + 3 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -2 + 3 T + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( -68 + T^{2} \)
$31$ \( -8 + 6 T + T^{2} \)
$37$ \( -26 + 7 T + T^{2} \)
$41$ \( -8 + 6 T + T^{2} \)
$43$ \( -104 + 3 T + T^{2} \)
$47$ \( -18 + 9 T + T^{2} \)
$53$ \( 64 - 18 T + T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( -152 - 2 T + T^{2} \)
$67$ \( ( 6 + T )^{2} \)
$71$ \( -18 - 9 T + T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( ( 12 + T )^{2} \)
$83$ \( -128 + 10 T + T^{2} \)
$89$ \( -68 + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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