Properties

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

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta - 2) q^{7} + q^{9} - \beta q^{11} + (\beta + 4) q^{13} - q^{15} + (2 \beta - 4) q^{17} - q^{19} + (\beta - 2) q^{21} + ( - 4 \beta - 2) q^{23} + q^{25} + q^{27} + ( - 5 \beta - 2) q^{29} + ( - 6 \beta + 2) q^{31} - \beta q^{33} + ( - \beta + 2) q^{35} + (5 \beta + 4) q^{37} + (\beta + 4) q^{39} + ( - \beta - 6) q^{41} + (\beta - 2) q^{43} - q^{45} + (4 \beta - 6) q^{47} + ( - 4 \beta - 1) q^{49} + (2 \beta - 4) q^{51} + 4 q^{53} + \beta q^{55} - q^{57} - 6 \beta q^{59} + 4 \beta q^{61} + (\beta - 2) q^{63} + ( - \beta - 4) q^{65} - 12 q^{67} + ( - 4 \beta - 2) q^{69} + (6 \beta - 4) q^{71} - 2 q^{73} + q^{75} + (2 \beta - 2) q^{77} + 8 \beta q^{79} + q^{81} + (6 \beta + 2) q^{83} + ( - 2 \beta + 4) q^{85} + ( - 5 \beta - 2) q^{87} + (9 \beta - 2) q^{89} + (2 \beta - 6) q^{91} + ( - 6 \beta + 2) q^{93} + q^{95} - 3 \beta q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 8 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 8 q^{39} - 12 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 2 q^{49} - 8 q^{51} + 8 q^{53} - 2 q^{57} - 4 q^{63} - 8 q^{65} - 24 q^{67} - 4 q^{69} - 8 q^{71} - 4 q^{73} + 2 q^{75} - 4 q^{77} + 2 q^{81} + 4 q^{83} + 8 q^{85} - 4 q^{87} - 4 q^{89} - 12 q^{91} + 4 q^{93} + 2 q^{95}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 −1.00000 0 −3.41421 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.585786 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.bj 2
4.b odd 2 1 285.2.a.f 2
12.b even 2 1 855.2.a.e 2
20.d odd 2 1 1425.2.a.l 2
20.e even 4 2 1425.2.c.j 4
60.h even 2 1 4275.2.a.x 2
76.d even 2 1 5415.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.f 2 4.b odd 2 1
855.2.a.e 2 12.b even 2 1
1425.2.a.l 2 20.d odd 2 1
1425.2.c.j 4 20.e even 4 2
4275.2.a.x 2 60.h even 2 1
4560.2.a.bj 2 1.a even 1 1 trivial
5415.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} + 14 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 8 \) 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
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( T^{2} - 32 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 158 \) Copy content Toggle raw display
$97$ \( T^{2} - 18 \) Copy content Toggle raw display
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