Properties

Label 4560.2.a.bf
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} + ( -2 + 3 \beta ) q^{11} + ( -2 - \beta ) q^{13} + q^{15} + ( 4 - 2 \beta ) q^{17} + q^{19} -\beta q^{21} + ( -2 - 4 \beta ) q^{23} + q^{25} - q^{27} + \beta q^{29} + ( 6 - 2 \beta ) q^{31} + ( 2 - 3 \beta ) q^{33} -\beta q^{35} + ( -2 - \beta ) q^{37} + ( 2 + \beta ) q^{39} + ( 4 - 3 \beta ) q^{41} + ( -8 - 3 \beta ) q^{43} - q^{45} + ( 2 + 4 \beta ) q^{47} -5 q^{49} + ( -4 + 2 \beta ) q^{51} + 8 q^{53} + ( 2 - 3 \beta ) q^{55} - q^{57} + ( -4 - 6 \beta ) q^{59} + ( -4 + 8 \beta ) q^{61} + \beta q^{63} + ( 2 + \beta ) q^{65} + ( 4 + 4 \beta ) q^{67} + ( 2 + 4 \beta ) q^{69} + ( 8 + 2 \beta ) q^{71} + ( -2 - 4 \beta ) q^{73} - q^{75} + ( 6 - 2 \beta ) q^{77} + q^{81} + ( -10 - 2 \beta ) q^{83} + ( -4 + 2 \beta ) q^{85} -\beta q^{87} + ( -4 + 7 \beta ) q^{89} + ( -2 - 2 \beta ) q^{91} + ( -6 + 2 \beta ) q^{93} - q^{95} + ( -14 + 3 \beta ) q^{97} + ( -2 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} - 4q^{11} - 4q^{13} + 2q^{15} + 8q^{17} + 2q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 12q^{31} + 4q^{33} - 4q^{37} + 4q^{39} + 8q^{41} - 16q^{43} - 2q^{45} + 4q^{47} - 10q^{49} - 8q^{51} + 16q^{53} + 4q^{55} - 2q^{57} - 8q^{59} - 8q^{61} + 4q^{65} + 8q^{67} + 4q^{69} + 16q^{71} - 4q^{73} - 2q^{75} + 12q^{77} + 2q^{81} - 20q^{83} - 8q^{85} - 8q^{89} - 4q^{91} - 12q^{93} - 2q^{95} - 28q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.00000 0 −1.00000 0 −1.41421 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.41421 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.bf 2
4.b odd 2 1 285.2.a.g 2
12.b even 2 1 855.2.a.d 2
20.d odd 2 1 1425.2.a.k 2
20.e even 4 2 1425.2.c.l 4
60.h even 2 1 4275.2.a.y 2
76.d even 2 1 5415.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 4.b odd 2 1
855.2.a.d 2 12.b even 2 1
1425.2.a.k 2 20.d odd 2 1
1425.2.c.l 4 20.e even 4 2
4275.2.a.y 2 60.h even 2 1
4560.2.a.bf 2 1.a even 1 1 trivial
5415.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{2} - 2 \)
\( T_{11}^{2} + 4 T_{11} - 14 \)
\( T_{13}^{2} + 4 T_{13} + 2 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -14 + 4 T + T^{2} \)
$13$ \( 2 + 4 T + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -28 + 4 T + T^{2} \)
$29$ \( -2 + T^{2} \)
$31$ \( 28 - 12 T + T^{2} \)
$37$ \( 2 + 4 T + T^{2} \)
$41$ \( -2 - 8 T + T^{2} \)
$43$ \( 46 + 16 T + T^{2} \)
$47$ \( -28 - 4 T + T^{2} \)
$53$ \( ( -8 + T )^{2} \)
$59$ \( -56 + 8 T + T^{2} \)
$61$ \( -112 + 8 T + T^{2} \)
$67$ \( -16 - 8 T + T^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 92 + 20 T + T^{2} \)
$89$ \( -82 + 8 T + T^{2} \)
$97$ \( 178 + 28 T + T^{2} \)
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