Properties

Label 585.2.a.n
Level $585$
Weight $2$
Character orbit 585.a
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + q^{5} - \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + q^{5} - \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8} + \beta_1 q^{10} + \beta_{2} q^{11} + q^{13} + ( - 2 \beta_1 + 2) q^{14} + (\beta_{2} - 2 \beta_1 + 9) q^{16} + ( - \beta_{2} - 2 \beta_1) q^{17} + (2 \beta_1 + 2) q^{19} + (\beta_{2} + 3) q^{20} + (2 \beta_1 - 2) q^{22} + ( - \beta_{2} - 2 \beta_1 + 2) q^{23} + q^{25} + \beta_1 q^{26} + (2 \beta_1 - 10) q^{28} - 6 q^{29} + ( - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{32} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{34} - \beta_{2} q^{35} + ( - \beta_{2} + 2 \beta_1 + 4) q^{37} + (2 \beta_{2} + 2 \beta_1 + 10) q^{38} + (3 \beta_1 - 2) q^{40} + (\beta_{2} + 2 \beta_1) q^{41} - 4 \beta_1 q^{43} + ( - 2 \beta_1 + 10) q^{44} + ( - 2 \beta_{2} - 8) q^{46} + ( - 2 \beta_1 + 6) q^{47} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{49} + \beta_1 q^{50} + (\beta_{2} + 3) q^{52} + ( - \beta_{2} - 2 \beta_1 - 4) q^{53} + \beta_{2} q^{55} + (2 \beta_{2} - 6 \beta_1 + 6) q^{56} - 6 \beta_1 q^{58} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + (3 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{62} + (3 \beta_{2} - 8 \beta_1 + 11) q^{64} + q^{65} + (2 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 8 \beta_1 - 6) q^{68} + ( - 2 \beta_1 + 2) q^{70} + (\beta_{2} + 4) q^{71} + ( - 4 \beta_1 - 2) q^{73} + (2 \beta_{2} + 2 \beta_1 + 12) q^{74} + (2 \beta_{2} + 10 \beta_1 + 2) q^{76} + (3 \beta_{2} + 2 \beta_1 - 10) q^{77} + (\beta_{2} - 2 \beta_1 + 2) q^{79} + (\beta_{2} - 2 \beta_1 + 9) q^{80} + (2 \beta_{2} + 2 \beta_1 + 8) q^{82} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + ( - \beta_{2} - 2 \beta_1) q^{85} + ( - 4 \beta_{2} - 20) q^{86} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{88} + ( - \beta_{2} - 2 \beta_1 - 4) q^{89} - \beta_{2} q^{91} + (2 \beta_{2} - 8 \beta_1) q^{92} + ( - 2 \beta_{2} + 6 \beta_1 - 10) q^{94} + (2 \beta_1 + 2) q^{95} + (\beta_{2} + 2 \beta_1 - 8) q^{97} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8} - q^{11} + 3 q^{13} + 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} + 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} - 30 q^{28} - 18 q^{29} + 6 q^{31} - 22 q^{32} - 22 q^{34} + q^{35} + 13 q^{37} + 28 q^{38} - 6 q^{40} - q^{41} + 30 q^{44} - 22 q^{46} + 18 q^{47} + 12 q^{49} + 8 q^{52} - 11 q^{53} - q^{55} + 16 q^{56} + 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} + 3 q^{65} + 4 q^{67} - 18 q^{68} + 6 q^{70} + 11 q^{71} - 6 q^{73} + 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 26 q^{80} + 22 q^{82} - 8 q^{83} + q^{85} - 56 q^{86} - 16 q^{88} - 11 q^{89} + q^{91} - 2 q^{92} - 28 q^{94} + 6 q^{95} - 25 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) 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
0.470683
−1.81361
2.34292
−2.77846 0 5.71982 1.00000 0 −2.71982 −10.3354 0 −2.77846
1.2 0.289169 0 −1.91638 1.00000 0 4.91638 −1.13249 0 0.289169
1.3 2.48929 0 4.19656 1.00000 0 −1.19656 5.46787 0 2.48929
\(n\): 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.n 3
3.b odd 2 1 195.2.a.e 3
4.b odd 2 1 9360.2.a.dd 3
5.b even 2 1 2925.2.a.bh 3
5.c odd 4 2 2925.2.c.w 6
12.b even 2 1 3120.2.a.bj 3
13.b even 2 1 7605.2.a.bx 3
15.d odd 2 1 975.2.a.o 3
15.e even 4 2 975.2.c.i 6
21.c even 2 1 9555.2.a.bq 3
39.d odd 2 1 2535.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 3.b odd 2 1
585.2.a.n 3 1.a even 1 1 trivial
975.2.a.o 3 15.d odd 2 1
975.2.c.i 6 15.e even 4 2
2535.2.a.bc 3 39.d odd 2 1
2925.2.a.bh 3 5.b even 2 1
2925.2.c.w 6 5.c odd 4 2
3120.2.a.bj 3 12.b even 2 1
7605.2.a.bx 3 13.b even 2 1
9360.2.a.dd 3 4.b odd 2 1
9555.2.a.bq 3 21.c even 2 1

Hecke kernels

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

\( T_{2}^{3} - 7T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 16T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 7T + 2 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 16 T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 16 T + 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 32 T + 76 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 16 T + 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 7 T^{2} - 16 T + 128 \) Copy content Toggle raw display
$29$ \( (T + 6)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 13T^{2} + 316 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 32 T - 76 \) Copy content Toggle raw display
$43$ \( T^{3} - 112T - 128 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + 80 T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 11 T^{2} + 8 T - 4 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 48 T + 128 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 112 T + 844 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 64 T + 128 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} + 24 T + 32 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 100 T - 344 \) Copy content Toggle raw display
$79$ \( T^{3} - 5 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 48 T - 128 \) Copy content Toggle raw display
$89$ \( T^{3} + 11 T^{2} + 8 T - 4 \) Copy content Toggle raw display
$97$ \( T^{3} + 25 T^{2} + 176 T + 244 \) Copy content Toggle raw display
show more
show less