Properties

Label 585.2.a.l
Level $585$
Weight $2$
Character orbit 585.a
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta - 3) q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta - 3) q^{7} + (\beta + 4) q^{8} + \beta q^{10} + ( - \beta + 1) q^{11} - q^{13} + ( - 2 \beta + 4) q^{14} + 3 \beta q^{16} + (\beta - 1) q^{17} - 2 \beta q^{19} + (\beta + 2) q^{20} - 4 q^{22} + ( - \beta + 5) q^{23} + q^{25} - \beta q^{26} - 2 q^{28} + ( - 2 \beta - 2) q^{29} + 6 q^{31} + (\beta + 4) q^{32} + 4 q^{34} + (\beta - 3) q^{35} + ( - 3 \beta - 3) q^{37} + ( - 2 \beta - 8) q^{38} + (\beta + 4) q^{40} + (\beta + 1) q^{41} + (2 \beta - 2) q^{43} + ( - 2 \beta - 2) q^{44} + (4 \beta - 4) q^{46} + (2 \beta + 6) q^{47} + ( - 5 \beta + 6) q^{49} + \beta q^{50} + ( - \beta - 2) q^{52} + ( - 3 \beta + 3) q^{53} + ( - \beta + 1) q^{55} + (2 \beta - 8) q^{56} + ( - 4 \beta - 8) q^{58} + 12 q^{59} + ( - 3 \beta + 1) q^{61} + 6 \beta q^{62} + ( - \beta + 4) q^{64} - q^{65} + ( - 6 \beta + 4) q^{67} + (2 \beta + 2) q^{68} + ( - 2 \beta + 4) q^{70} + ( - \beta + 13) q^{71} - 6 q^{73} + ( - 6 \beta - 12) q^{74} + ( - 6 \beta - 8) q^{76} + (3 \beta - 7) q^{77} + (3 \beta - 3) q^{79} + 3 \beta q^{80} + (2 \beta + 4) q^{82} + ( - 8 \beta + 4) q^{83} + (\beta - 1) q^{85} + 8 q^{86} - 4 \beta q^{88} + (3 \beta + 3) q^{89} + ( - \beta + 3) q^{91} + (2 \beta + 6) q^{92} + (8 \beta + 8) q^{94} - 2 \beta q^{95} + (7 \beta - 1) q^{97} + (\beta - 20) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 1.00000 0 −4.56155 2.43845 0 −1.56155
1.2 2.56155 0 4.56155 1.00000 0 −0.438447 6.56155 0 2.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 585.2.a.l yes 2
3.b odd 2 1 585.2.a.j 2
4.b odd 2 1 9360.2.a.cw 2
5.b even 2 1 2925.2.a.x 2
5.c odd 4 2 2925.2.c.p 4
12.b even 2 1 9360.2.a.cl 2
13.b even 2 1 7605.2.a.bd 2
15.d odd 2 1 2925.2.a.bc 2
15.e even 4 2 2925.2.c.o 4
39.d odd 2 1 7605.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.j 2 3.b odd 2 1
585.2.a.l yes 2 1.a even 1 1 trivial
2925.2.a.x 2 5.b even 2 1
2925.2.a.bc 2 15.d odd 2 1
2925.2.c.o 4 15.e even 4 2
2925.2.c.p 4 5.c odd 4 2
7605.2.a.bd 2 13.b even 2 1
7605.2.a.bi 2 39.d odd 2 1
9360.2.a.cl 2 12.b even 2 1
9360.2.a.cw 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 5T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$71$ \( T^{2} - 25T + 152 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 272 \) Copy content Toggle raw display
$89$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T - 202 \) Copy content Toggle raw display
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