Properties

Label 4560.2.a.bl
Level $4560$
Weight $2$
Character orbit 4560.a
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( -1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( -1 - \beta ) q^{7} + q^{9} + ( 1 - \beta ) q^{11} + ( 1 - \beta ) q^{13} - q^{15} + 2 q^{17} + q^{19} + ( -1 - \beta ) q^{21} + 2 q^{23} + q^{25} + q^{27} + ( -1 - \beta ) q^{29} -4 q^{31} + ( 1 - \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( 7 + \beta ) q^{37} + ( 1 - \beta ) q^{39} + ( 3 - \beta ) q^{41} + ( -7 + \beta ) q^{43} - q^{45} -6 q^{47} + ( 7 + 2 \beta ) q^{49} + 2 q^{51} + ( -1 + \beta ) q^{55} + q^{57} + ( 2 + 2 \beta ) q^{59} + 2 \beta q^{61} + ( -1 - \beta ) q^{63} + ( -1 + \beta ) q^{65} -4 q^{67} + 2 q^{69} + ( 2 - 2 \beta ) q^{71} + 6 q^{73} + q^{75} + 12 q^{77} -8 q^{79} + q^{81} + ( 4 - 2 \beta ) q^{83} -2 q^{85} + ( -1 - \beta ) q^{87} + ( 3 - \beta ) q^{89} + 12 q^{91} -4 q^{93} - q^{95} + ( 13 - \beta ) q^{97} + ( 1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + 2q^{11} + 2q^{13} - 2q^{15} + 4q^{17} + 2q^{19} - 2q^{21} + 4q^{23} + 2q^{25} + 2q^{27} - 2q^{29} - 8q^{31} + 2q^{33} + 2q^{35} + 14q^{37} + 2q^{39} + 6q^{41} - 14q^{43} - 2q^{45} - 12q^{47} + 14q^{49} + 4q^{51} - 2q^{55} + 2q^{57} + 4q^{59} - 2q^{63} - 2q^{65} - 8q^{67} + 4q^{69} + 4q^{71} + 12q^{73} + 2q^{75} + 24q^{77} - 16q^{79} + 2q^{81} + 8q^{83} - 4q^{85} - 2q^{87} + 6q^{89} + 24q^{91} - 8q^{93} - 2q^{95} + 26q^{97} + 2q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 1.00000 0 −1.00000 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.60555 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 4560.2.a.bl 2
4.b odd 2 1 1140.2.a.e 2
12.b even 2 1 3420.2.a.i 2
20.d odd 2 1 5700.2.a.u 2
20.e even 4 2 5700.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 4.b odd 2 1
3420.2.a.i 2 12.b even 2 1
4560.2.a.bl 2 1.a even 1 1 trivial
5700.2.a.u 2 20.d odd 2 1
5700.2.f.n 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -12 + 2 T + T^{2} \)
$11$ \( -12 - 2 T + T^{2} \)
$13$ \( -12 - 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( ( -2 + T )^{2} \)
$29$ \( -12 + 2 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 36 - 14 T + T^{2} \)
$41$ \( -4 - 6 T + T^{2} \)
$43$ \( 36 + 14 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( -48 - 4 T + T^{2} \)
$61$ \( -52 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -48 - 4 T + T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -36 - 8 T + T^{2} \)
$89$ \( -4 - 6 T + T^{2} \)
$97$ \( 156 - 26 T + T^{2} \)
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