Properties

Label 5472.2.a.bc
Level $5472$
Weight $2$
Character orbit 5472.a
Self dual yes
Analytic conductor $43.694$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.6941399860\)
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 608)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} -3 q^{7} +O(q^{10})\) \( q + \beta q^{5} -3 q^{7} + ( -2 + \beta ) q^{11} + ( 1 - \beta ) q^{13} + ( 5 - 2 \beta ) q^{17} - q^{19} + ( 3 + \beta ) q^{23} + ( -1 + \beta ) q^{25} + ( 3 - 3 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} -3 \beta q^{35} + ( 2 + 2 \beta ) q^{37} -4 q^{41} + ( -8 + \beta ) q^{43} + ( 4 - 3 \beta ) q^{47} + 2 q^{49} + ( 7 - \beta ) q^{53} + ( 4 - \beta ) q^{55} + ( -7 + \beta ) q^{59} + ( 2 + 5 \beta ) q^{61} -4 q^{65} + ( 3 - \beta ) q^{67} + ( 6 - 4 \beta ) q^{71} + ( 1 - 4 \beta ) q^{73} + ( 6 - 3 \beta ) q^{77} -10 q^{79} + ( -2 - 6 \beta ) q^{83} + ( -8 + 3 \beta ) q^{85} + 6 \beta q^{89} + ( -3 + 3 \beta ) q^{91} -\beta q^{95} + ( 2 + 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 6q^{7} + O(q^{10}) \) \( 2q + q^{5} - 6q^{7} - 3q^{11} + q^{13} + 8q^{17} - 2q^{19} + 7q^{23} - q^{25} + 3q^{29} - 10q^{31} - 3q^{35} + 6q^{37} - 8q^{41} - 15q^{43} + 5q^{47} + 4q^{49} + 13q^{53} + 7q^{55} - 13q^{59} + 9q^{61} - 8q^{65} + 5q^{67} + 8q^{71} - 2q^{73} + 9q^{77} - 20q^{79} - 10q^{83} - 13q^{85} + 6q^{89} - 3q^{91} - q^{95} + 6q^{97} + O(q^{100}) \)

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 −1.56155 0 −3.00000 0 0 0
1.2 0 0 0 2.56155 0 −3.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

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 5472.2.a.bc 2
3.b odd 2 1 608.2.a.g 2
4.b odd 2 1 5472.2.a.bf 2
12.b even 2 1 608.2.a.h yes 2
24.f even 2 1 1216.2.a.s 2
24.h odd 2 1 1216.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 3.b odd 2 1
608.2.a.h yes 2 12.b even 2 1
1216.2.a.s 2 24.f even 2 1
1216.2.a.t 2 24.h odd 2 1
5472.2.a.bc 2 1.a even 1 1 trivial
5472.2.a.bf 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5472))\):

\( T_{5}^{2} - T_{5} - 4 \)
\( T_{7} + 3 \)
\( T_{11}^{2} + 3 T_{11} - 2 \)
\( T_{13}^{2} - T_{13} - 4 \)
\( T_{23}^{2} - 7 T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 - T + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( -2 + 3 T + T^{2} \)
$13$ \( -4 - T + T^{2} \)
$17$ \( -1 - 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 8 - 7 T + T^{2} \)
$29$ \( -36 - 3 T + T^{2} \)
$31$ \( 8 + 10 T + T^{2} \)
$37$ \( -8 - 6 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( 52 + 15 T + T^{2} \)
$47$ \( -32 - 5 T + T^{2} \)
$53$ \( 38 - 13 T + T^{2} \)
$59$ \( 38 + 13 T + T^{2} \)
$61$ \( -86 - 9 T + T^{2} \)
$67$ \( 2 - 5 T + T^{2} \)
$71$ \( -52 - 8 T + T^{2} \)
$73$ \( -67 + 2 T + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( -128 + 10 T + T^{2} \)
$89$ \( -144 - 6 T + T^{2} \)
$97$ \( -8 - 6 T + T^{2} \)
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