Properties

Label 7448.2.a.bb
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + \beta q^{9} +O(q^{10})\) \( q + \beta q^{3} + q^{5} + \beta q^{9} + ( -1 + \beta ) q^{11} + ( 1 - 2 \beta ) q^{13} + \beta q^{15} + ( -3 - \beta ) q^{17} - q^{19} + ( -3 - 2 \beta ) q^{23} -4 q^{25} + ( 3 - 2 \beta ) q^{27} + ( -3 + \beta ) q^{29} + ( 7 - \beta ) q^{31} + 3 q^{33} + ( 1 - 2 \beta ) q^{37} + ( -6 - \beta ) q^{39} + ( 1 + \beta ) q^{41} + ( 2 - 4 \beta ) q^{43} + \beta q^{45} + ( 5 - 2 \beta ) q^{47} + ( -3 - 4 \beta ) q^{51} + 3 \beta q^{53} + ( -1 + \beta ) q^{55} -\beta q^{57} + ( -3 + 2 \beta ) q^{59} + ( -3 + 4 \beta ) q^{61} + ( 1 - 2 \beta ) q^{65} + ( -7 + 3 \beta ) q^{67} + ( -6 - 5 \beta ) q^{69} + ( 1 - 6 \beta ) q^{71} + ( 2 - 3 \beta ) q^{73} -4 \beta q^{75} -14 q^{79} + ( -6 - 2 \beta ) q^{81} + ( 4 - 5 \beta ) q^{83} + ( -3 - \beta ) q^{85} + ( 3 - 2 \beta ) q^{87} + ( -6 + 6 \beta ) q^{89} + ( -3 + 6 \beta ) q^{93} - q^{95} + ( -9 + 6 \beta ) q^{97} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} + q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + q^{9} - q^{11} + q^{15} - 7q^{17} - 2q^{19} - 8q^{23} - 8q^{25} + 4q^{27} - 5q^{29} + 13q^{31} + 6q^{33} - 13q^{39} + 3q^{41} + q^{45} + 8q^{47} - 10q^{51} + 3q^{53} - q^{55} - q^{57} - 4q^{59} - 2q^{61} - 11q^{67} - 17q^{69} - 4q^{71} + q^{73} - 4q^{75} - 28q^{79} - 14q^{81} + 3q^{83} - 7q^{85} + 4q^{87} - 6q^{89} - 2q^{95} - 12q^{97} + 6q^{99} + O(q^{100}) \)

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
−1.30278
2.30278
0 −1.30278 0 1.00000 0 0 0 −1.30278 0
1.2 0 2.30278 0 1.00000 0 0 0 2.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(19\) \(1\)

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 7448.2.a.bb 2
7.b odd 2 1 1064.2.a.b 2
21.c even 2 1 9576.2.a.bs 2
28.d even 2 1 2128.2.a.i 2
56.e even 2 1 8512.2.a.r 2
56.h odd 2 1 8512.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.b 2 7.b odd 2 1
2128.2.a.i 2 28.d even 2 1
7448.2.a.bb 2 1.a even 1 1 trivial
8512.2.a.r 2 56.e even 2 1
8512.2.a.x 2 56.h odd 2 1
9576.2.a.bs 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{2} - T_{3} - 3 \)
\( T_{5} - 1 \)
\( T_{11}^{2} + T_{11} - 3 \)
\( T_{13}^{2} - 13 \)
\( T_{17}^{2} + 7 T_{17} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -3 - T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -3 + T + T^{2} \)
$13$ \( -13 + T^{2} \)
$17$ \( 9 + 7 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 3 + 8 T + T^{2} \)
$29$ \( 3 + 5 T + T^{2} \)
$31$ \( 39 - 13 T + T^{2} \)
$37$ \( -13 + T^{2} \)
$41$ \( -1 - 3 T + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( 3 - 8 T + T^{2} \)
$53$ \( -27 - 3 T + T^{2} \)
$59$ \( -9 + 4 T + T^{2} \)
$61$ \( -51 + 2 T + T^{2} \)
$67$ \( 1 + 11 T + T^{2} \)
$71$ \( -113 + 4 T + T^{2} \)
$73$ \( -29 - T + T^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( -79 - 3 T + T^{2} \)
$89$ \( -108 + 6 T + T^{2} \)
$97$ \( -81 + 12 T + T^{2} \)
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