Properties

Label 4560.2.a.bh
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9} + ( - \beta - 3) q^{11} + (\beta - 1) q^{13} - q^{15} - q^{19} + ( - \beta - 1) q^{21} - 2 \beta q^{23} + q^{25} - q^{27} + (3 \beta + 3) q^{29} + ( - 4 \beta - 2) q^{31} + (\beta + 3) q^{33} + (\beta + 1) q^{35} + ( - 3 \beta - 1) q^{37} + ( - \beta + 1) q^{39} + ( - \beta + 3) q^{41} + ( - 3 \beta + 1) q^{43} + q^{45} + 2 \beta q^{47} + (2 \beta - 3) q^{49} + ( - 2 \beta - 6) q^{53} + ( - \beta - 3) q^{55} + q^{57} + (2 \beta - 6) q^{59} + (2 \beta - 10) q^{61} + (\beta + 1) q^{63} + (\beta - 1) q^{65} - 8 q^{67} + 2 \beta q^{69} + (6 \beta - 6) q^{71} + ( - 4 \beta - 10) q^{73} - q^{75} + ( - 4 \beta - 6) q^{77} + (4 \beta + 4) q^{79} + q^{81} + (4 \beta + 6) q^{83} + ( - 3 \beta - 3) q^{87} + (\beta + 9) q^{89} + 2 q^{91} + (4 \beta + 2) q^{93} - q^{95} + ( - 3 \beta - 1) q^{97} + ( - \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −0.732051 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 2.73205 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.bh 2
4.b odd 2 1 285.2.a.e 2
12.b even 2 1 855.2.a.f 2
20.d odd 2 1 1425.2.a.o 2
20.e even 4 2 1425.2.c.k 4
60.h even 2 1 4275.2.a.t 2
76.d even 2 1 5415.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.e 2 4.b odd 2 1
855.2.a.f 2 12.b even 2 1
1425.2.a.o 2 20.d odd 2 1
1425.2.c.k 4 20.e even 4 2
4275.2.a.t 2 60.h even 2 1
4560.2.a.bh 2 1.a even 1 1 trivial
5415.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{2} - 2T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$47$ \( T^{2} - 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$61$ \( T^{2} + 20T + 88 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 72 \) Copy content Toggle raw display
$73$ \( T^{2} + 20T + 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 78 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
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