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: \( x^{2} - 7 \) 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 \(\beta = \sqrt{7}\). 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 \( q + q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9} + ( - \beta - 3) q^{11} + ( - \beta - 3) q^{13} + q^{15} - 4 q^{17} + q^{19} + (\beta + 1) q^{21} + (2 \beta - 4) q^{23} + q^{25} + q^{27} + ( - 3 \beta + 1) q^{29} - 6 q^{31} + ( - \beta - 3) q^{33} + (\beta + 1) q^{35} + ( - \beta + 1) q^{37} + ( - \beta - 3) q^{39} + (\beta - 7) q^{41} + (\beta - 3) q^{43} + q^{45} + ( - 2 \beta - 4) q^{47} + (2 \beta + 1) q^{49} - 4 q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta - 3) q^{55} + q^{57} + (2 \beta - 6) q^{59} + (2 \beta - 6) q^{61} + (\beta + 1) q^{63} + ( - \beta - 3) q^{65} + ( - 4 \beta - 4) q^{67} + (2 \beta - 4) q^{69} + (2 \beta - 2) q^{71} + 10 q^{73} + q^{75} + ( - 4 \beta - 10) q^{77} + (4 \beta + 4) q^{79} + q^{81} - 6 q^{83} - 4 q^{85} + ( - 3 \beta + 1) q^{87} + (3 \beta - 9) q^{89} + ( - 4 \beta - 10) q^{91} - 6 q^{93} + q^{95} + (3 \beta + 5) q^{97} + ( - \beta - 3) q^{99}+O(q^{100}) \) 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 \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

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