Properties

Label 7448.2.a.be
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} + q^{5} + q^{9} +O(q^{10})\) \( q + 2 q^{3} + q^{5} + q^{9} + ( -2 + \beta ) q^{11} -2 q^{13} + 2 q^{15} + ( -3 - \beta ) q^{17} + q^{19} + ( 1 - 2 \beta ) q^{23} -4 q^{25} -4 q^{27} -4 q^{29} + 4 q^{31} + ( -4 + 2 \beta ) q^{33} + 2 q^{37} -4 q^{39} + ( -4 + 2 \beta ) q^{41} - q^{43} + q^{45} + ( -2 + \beta ) q^{47} + ( -6 - 2 \beta ) q^{51} + ( 2 - 2 \beta ) q^{53} + ( -2 + \beta ) q^{55} + 2 q^{57} + 4 q^{59} + ( -8 - \beta ) q^{61} -2 q^{65} + ( 2 - 4 \beta ) q^{69} + ( 2 + 2 \beta ) q^{71} + ( -8 + \beta ) q^{73} -8 q^{75} + ( -8 + 2 \beta ) q^{79} -11 q^{81} + ( -3 - 2 \beta ) q^{83} + ( -3 - \beta ) q^{85} -8 q^{87} + ( -8 - 2 \beta ) q^{89} + 8 q^{93} + q^{95} -2 \beta q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{5} + 2q^{9} - 3q^{11} - 4q^{13} + 4q^{15} - 7q^{17} + 2q^{19} - 8q^{25} - 8q^{27} - 8q^{29} + 8q^{31} - 6q^{33} + 4q^{37} - 8q^{39} - 6q^{41} - 2q^{43} + 2q^{45} - 3q^{47} - 14q^{51} + 2q^{53} - 3q^{55} + 4q^{57} + 8q^{59} - 17q^{61} - 4q^{65} + 6q^{71} - 15q^{73} - 16q^{75} - 14q^{79} - 22q^{81} - 8q^{83} - 7q^{85} - 16q^{87} - 18q^{89} + 16q^{93} + 2q^{95} - 2q^{97} - 3q^{99} + O(q^{100}) \)

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
−3.27492
4.27492
0 2.00000 0 1.00000 0 0 0 1.00000 0
1.2 0 2.00000 0 1.00000 0 0 0 1.00000 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.be 2
7.b odd 2 1 7448.2.a.w 2
7.d odd 6 2 1064.2.q.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.l 4 7.d odd 6 2
7448.2.a.w 2 7.b odd 2 1
7448.2.a.be 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3} - 2 \)
\( T_{5} - 1 \)
\( T_{11}^{2} + 3 T_{11} - 12 \)
\( T_{13} + 2 \)
\( T_{17}^{2} + 7 T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -12 + 3 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -2 + 7 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -57 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -48 + 6 T + T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -12 + 3 T + T^{2} \)
$53$ \( -56 - 2 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 58 + 17 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( -48 - 6 T + T^{2} \)
$73$ \( 42 + 15 T + T^{2} \)
$79$ \( -8 + 14 T + T^{2} \)
$83$ \( -41 + 8 T + T^{2} \)
$89$ \( 24 + 18 T + T^{2} \)
$97$ \( -56 + 2 T + T^{2} \)
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