Properties

Label 7448.2.a.bc
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} + ( -1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} + ( -1 + 3 \beta ) q^{9} + ( -2 + 3 \beta ) q^{11} + ( 5 - 2 \beta ) q^{13} + ( 1 + 3 \beta ) q^{15} + ( 4 - 3 \beta ) q^{17} - q^{19} + ( 3 + 2 \beta ) q^{23} + ( -1 + 2 \beta ) q^{27} + ( -6 + \beta ) q^{29} + ( 4 - \beta ) q^{31} + ( 1 + 4 \beta ) q^{33} + ( -5 + 4 \beta ) q^{37} + ( 3 + \beta ) q^{39} + ( -4 + 9 \beta ) q^{41} + ( 6 - 8 \beta ) q^{43} + ( 7 + \beta ) q^{45} -7 q^{47} + ( 1 - 2 \beta ) q^{51} + ( 3 - 3 \beta ) q^{53} + ( 8 - \beta ) q^{55} + ( -1 - \beta ) q^{57} + ( 1 + 6 \beta ) q^{59} + ( 3 + 6 \beta ) q^{61} + ( -9 + 8 \beta ) q^{65} + ( -4 - \beta ) q^{67} + ( 5 + 7 \beta ) q^{69} + ( -9 + 8 \beta ) q^{71} + ( 5 - 3 \beta ) q^{73} + ( 2 - 4 \beta ) q^{79} + ( 4 - 6 \beta ) q^{81} + ( 13 - \beta ) q^{83} + ( -10 + 5 \beta ) q^{85} + ( -5 - 4 \beta ) q^{87} + ( -4 + 2 \beta ) q^{89} + ( 3 + 2 \beta ) q^{93} + ( 1 - 2 \beta ) q^{95} + ( 7 - 2 \beta ) q^{97} + 11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{9} + O(q^{10}) \) \( 2 q + 3 q^{3} + q^{9} - q^{11} + 8 q^{13} + 5 q^{15} + 5 q^{17} - 2 q^{19} + 8 q^{23} - 11 q^{29} + 7 q^{31} + 6 q^{33} - 6 q^{37} + 7 q^{39} + q^{41} + 4 q^{43} + 15 q^{45} - 14 q^{47} + 3 q^{53} + 15 q^{55} - 3 q^{57} + 8 q^{59} + 12 q^{61} - 10 q^{65} - 9 q^{67} + 17 q^{69} - 10 q^{71} + 7 q^{73} + 2 q^{81} + 25 q^{83} - 15 q^{85} - 14 q^{87} - 6 q^{89} + 8 q^{93} + 12 q^{97} + 22 q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 0.381966 0 −2.23607 0 0 0 −2.85410 0
1.2 0 2.61803 0 2.23607 0 0 0 3.85410 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.bc 2
7.b odd 2 1 1064.2.a.a 2
21.c even 2 1 9576.2.a.bm 2
28.d even 2 1 2128.2.a.n 2
56.e even 2 1 8512.2.a.i 2
56.h odd 2 1 8512.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.a 2 7.b odd 2 1
2128.2.a.n 2 28.d even 2 1
7448.2.a.bc 2 1.a even 1 1 trivial
8512.2.a.i 2 56.e even 2 1
8512.2.a.bf 2 56.h odd 2 1
9576.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{5}^{2} - 5 \)
\( T_{11}^{2} + T_{11} - 11 \)
\( T_{13}^{2} - 8 T_{13} + 11 \)
\( T_{17}^{2} - 5 T_{17} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - 3 T + T^{2} \)
$5$ \( -5 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -11 + T + T^{2} \)
$13$ \( 11 - 8 T + T^{2} \)
$17$ \( -5 - 5 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 11 - 8 T + T^{2} \)
$29$ \( 29 + 11 T + T^{2} \)
$31$ \( 11 - 7 T + T^{2} \)
$37$ \( -11 + 6 T + T^{2} \)
$41$ \( -101 - T + T^{2} \)
$43$ \( -76 - 4 T + T^{2} \)
$47$ \( ( 7 + T )^{2} \)
$53$ \( -9 - 3 T + T^{2} \)
$59$ \( -29 - 8 T + T^{2} \)
$61$ \( -9 - 12 T + T^{2} \)
$67$ \( 19 + 9 T + T^{2} \)
$71$ \( -55 + 10 T + T^{2} \)
$73$ \( 1 - 7 T + T^{2} \)
$79$ \( -20 + T^{2} \)
$83$ \( 155 - 25 T + T^{2} \)
$89$ \( 4 + 6 T + T^{2} \)
$97$ \( 31 - 12 T + T^{2} \)
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