Properties

Label 4560.2.a.bi
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + ( 3 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + ( 3 + \beta ) q^{7} + q^{9} + ( 1 - 3 \beta ) q^{11} + ( -3 + \beta ) q^{13} - q^{15} -4 \beta q^{17} - q^{19} + ( -3 - \beta ) q^{21} + 6 q^{23} + q^{25} - q^{27} + ( -5 - 3 \beta ) q^{29} + 2 q^{31} + ( -1 + 3 \beta ) q^{33} + ( 3 + \beta ) q^{35} + ( 1 + \beta ) q^{37} + ( 3 - \beta ) q^{39} + ( 3 + 5 \beta ) q^{41} + ( 3 + \beta ) q^{43} + q^{45} + 10 q^{47} + ( 5 + 6 \beta ) q^{49} + 4 \beta q^{51} + ( 1 - 3 \beta ) q^{55} + q^{57} + ( 6 + 2 \beta ) q^{59} + ( 2 + 2 \beta ) q^{61} + ( 3 + \beta ) q^{63} + ( -3 + \beta ) q^{65} + ( 8 + 4 \beta ) q^{67} -6 q^{69} + ( -2 - 6 \beta ) q^{71} + ( -6 + 4 \beta ) q^{73} - q^{75} + ( -6 - 8 \beta ) q^{77} + 8 q^{79} + q^{81} + ( -12 + 2 \beta ) q^{83} -4 \beta q^{85} + ( 5 + 3 \beta ) q^{87} + ( 1 - 5 \beta ) q^{89} -6 q^{91} -2 q^{93} - q^{95} + ( -7 + 5 \beta ) q^{97} + ( 1 - 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + 6q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 6q^{7} + 2q^{9} + 2q^{11} - 6q^{13} - 2q^{15} - 2q^{19} - 6q^{21} + 12q^{23} + 2q^{25} - 2q^{27} - 10q^{29} + 4q^{31} - 2q^{33} + 6q^{35} + 2q^{37} + 6q^{39} + 6q^{41} + 6q^{43} + 2q^{45} + 20q^{47} + 10q^{49} + 2q^{55} + 2q^{57} + 12q^{59} + 4q^{61} + 6q^{63} - 6q^{65} + 16q^{67} - 12q^{69} - 4q^{71} - 12q^{73} - 2q^{75} - 12q^{77} + 16q^{79} + 2q^{81} - 24q^{83} + 10q^{87} + 2q^{89} - 12q^{91} - 4q^{93} - 2q^{95} - 14q^{97} + 2q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −1.00000 0 1.00000 0 1.26795 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 4.73205 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.bi 2
4.b odd 2 1 2280.2.a.q 2
12.b even 2 1 6840.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.q 2 4.b odd 2 1
4560.2.a.bi 2 1.a even 1 1 trivial
6840.2.a.v 2 12.b even 2 1

Hecke kernels

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

\( T_{7}^{2} - 6 T_{7} + 6 \)
\( T_{11}^{2} - 2 T_{11} - 26 \)
\( T_{13}^{2} + 6 T_{13} + 6 \)
\( T_{17}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 6 - 6 T + T^{2} \)
$11$ \( -26 - 2 T + T^{2} \)
$13$ \( 6 + 6 T + T^{2} \)
$17$ \( -48 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( -2 + 10 T + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( -2 - 2 T + T^{2} \)
$41$ \( -66 - 6 T + T^{2} \)
$43$ \( 6 - 6 T + T^{2} \)
$47$ \( ( -10 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( 24 - 12 T + T^{2} \)
$61$ \( -8 - 4 T + T^{2} \)
$67$ \( 16 - 16 T + T^{2} \)
$71$ \( -104 + 4 T + T^{2} \)
$73$ \( -12 + 12 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 132 + 24 T + T^{2} \)
$89$ \( -74 - 2 T + T^{2} \)
$97$ \( -26 + 14 T + T^{2} \)
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