Properties

Label 4560.2.a.bo
Level $4560$
Weight $2$
Character orbit 4560.a
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( 1 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( 1 + \beta ) q^{7} + q^{9} + ( -3 - \beta ) q^{11} + ( -3 - \beta ) q^{13} + q^{15} -4 q^{17} + q^{19} + ( 1 + \beta ) q^{21} + ( -4 + 2 \beta ) q^{23} + q^{25} + q^{27} + ( 1 - 3 \beta ) q^{29} -6 q^{31} + ( -3 - \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( 1 - \beta ) q^{37} + ( -3 - \beta ) q^{39} + ( -7 + \beta ) q^{41} + ( -3 + \beta ) q^{43} + q^{45} + ( -4 - 2 \beta ) q^{47} + ( 1 + 2 \beta ) q^{49} -4 q^{51} + ( 6 - 2 \beta ) q^{53} + ( -3 - \beta ) q^{55} + q^{57} + ( -6 + 2 \beta ) q^{59} + ( -6 + 2 \beta ) q^{61} + ( 1 + \beta ) q^{63} + ( -3 - \beta ) q^{65} + ( -4 - 4 \beta ) q^{67} + ( -4 + 2 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + 10 q^{73} + q^{75} + ( -10 - 4 \beta ) q^{77} + ( 4 + 4 \beta ) q^{79} + q^{81} -6 q^{83} -4 q^{85} + ( 1 - 3 \beta ) q^{87} + ( -9 + 3 \beta ) q^{89} + ( -10 - 4 \beta ) q^{91} -6 q^{93} + q^{95} + ( 5 + 3 \beta ) q^{97} + ( -3 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} - 6q^{11} - 6q^{13} + 2q^{15} - 8q^{17} + 2q^{19} + 2q^{21} - 8q^{23} + 2q^{25} + 2q^{27} + 2q^{29} - 12q^{31} - 6q^{33} + 2q^{35} + 2q^{37} - 6q^{39} - 14q^{41} - 6q^{43} + 2q^{45} - 8q^{47} + 2q^{49} - 8q^{51} + 12q^{53} - 6q^{55} + 2q^{57} - 12q^{59} - 12q^{61} + 2q^{63} - 6q^{65} - 8q^{67} - 8q^{69} - 4q^{71} + 20q^{73} + 2q^{75} - 20q^{77} + 8q^{79} + 2q^{81} - 12q^{83} - 8q^{85} + 2q^{87} - 18q^{89} - 20q^{91} - 12q^{93} + 2q^{95} + 10q^{97} - 6q^{99} + O(q^{100}) \)

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
−2.64575
2.64575
0 1.00000 0 1.00000 0 −1.64575 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.64575 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.bo 2
4.b odd 2 1 285.2.a.d 2
12.b even 2 1 855.2.a.g 2
20.d odd 2 1 1425.2.a.p 2
20.e even 4 2 1425.2.c.i 4
60.h even 2 1 4275.2.a.u 2
76.d even 2 1 5415.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 4.b odd 2 1
855.2.a.g 2 12.b even 2 1
1425.2.a.p 2 20.d odd 2 1
1425.2.c.i 4 20.e even 4 2
4275.2.a.u 2 60.h even 2 1
4560.2.a.bo 2 1.a even 1 1 trivial
5415.2.a.s 2 76.d even 2 1

Hecke kernels

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

\( T_{7}^{2} - 2 T_{7} - 6 \)
\( T_{11}^{2} + 6 T_{11} + 2 \)
\( T_{13}^{2} + 6 T_{13} + 2 \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -6 - 2 T + T^{2} \)
$11$ \( 2 + 6 T + T^{2} \)
$13$ \( 2 + 6 T + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -12 + 8 T + T^{2} \)
$29$ \( -62 - 2 T + T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( -6 - 2 T + T^{2} \)
$41$ \( 42 + 14 T + T^{2} \)
$43$ \( 2 + 6 T + T^{2} \)
$47$ \( -12 + 8 T + T^{2} \)
$53$ \( 8 - 12 T + T^{2} \)
$59$ \( 8 + 12 T + T^{2} \)
$61$ \( 8 + 12 T + T^{2} \)
$67$ \( -96 + 8 T + T^{2} \)
$71$ \( -24 + 4 T + T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( -96 - 8 T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 18 + 18 T + T^{2} \)
$97$ \( -38 - 10 T + T^{2} \)
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