Properties

Label 585.2.j.a
Level $585$
Weight $2$
Character orbit 585.j
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{7} +O(q^{10})\) \( q + 2 \zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} + ( 4 - 3 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} + ( -6 + 6 \zeta_{6} ) q^{23} + q^{25} + ( -2 + 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + 5 q^{31} + \zeta_{6} q^{35} + ( -2 + 2 \zeta_{6} ) q^{37} -11 \zeta_{6} q^{43} + 12 q^{44} -6 q^{47} + ( 6 - 6 \zeta_{6} ) q^{49} + ( 6 + 2 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{55} + 6 \zeta_{6} q^{59} + \zeta_{6} q^{61} -8 q^{64} + ( 4 - 3 \zeta_{6} ) q^{65} + ( -11 + 11 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{71} + 5 q^{73} + ( -8 + 8 \zeta_{6} ) q^{76} + 6 q^{77} + 11 q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} -12 q^{83} + ( 12 - 12 \zeta_{6} ) q^{89} + ( 3 + \zeta_{6} ) q^{91} -12 q^{92} + 4 \zeta_{6} q^{95} -17 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + q^{7} + O(q^{10}) \) \( 2 q + 2 q^{4} + 2 q^{5} + q^{7} + 6 q^{11} + 5 q^{13} - 4 q^{16} + 4 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} - 2 q^{28} - 6 q^{29} + 10 q^{31} + q^{35} - 2 q^{37} - 11 q^{43} + 24 q^{44} - 12 q^{47} + 6 q^{49} + 14 q^{52} + 6 q^{55} + 6 q^{59} + q^{61} - 16 q^{64} + 5 q^{65} - 11 q^{67} - 6 q^{71} + 10 q^{73} - 8 q^{76} + 12 q^{77} + 22 q^{79} - 4 q^{80} - 24 q^{83} + 12 q^{89} + 7 q^{91} - 24 q^{92} + 4 q^{95} - 17 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 1.73205i 1.00000 0 0.500000 0.866025i 0 0 0
451.1 0 0 1.00000 + 1.73205i 1.00000 0 0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.a 2
3.b odd 2 1 195.2.i.b 2
13.c even 3 1 inner 585.2.j.a 2
13.c even 3 1 7605.2.a.l 1
13.e even 6 1 7605.2.a.k 1
15.d odd 2 1 975.2.i.d 2
15.e even 4 2 975.2.bb.b 4
39.h odd 6 1 2535.2.a.i 1
39.i odd 6 1 195.2.i.b 2
39.i odd 6 1 2535.2.a.h 1
195.x odd 6 1 975.2.i.d 2
195.bl even 12 2 975.2.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b 2 3.b odd 2 1
195.2.i.b 2 39.i odd 6 1
585.2.j.a 2 1.a even 1 1 trivial
585.2.j.a 2 13.c even 3 1 inner
975.2.i.d 2 15.d odd 2 1
975.2.i.d 2 195.x odd 6 1
975.2.bb.b 4 15.e even 4 2
975.2.bb.b 4 195.bl even 12 2
2535.2.a.h 1 39.i odd 6 1
2535.2.a.i 1 39.h odd 6 1
7605.2.a.k 1 13.e even 6 1
7605.2.a.l 1 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 36 - 6 T + T^{2} \)
$13$ \( 13 - 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 121 + 11 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( ( -5 + T )^{2} \)
$79$ \( ( -11 + T )^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 144 - 12 T + T^{2} \)
$97$ \( 289 + 17 T + T^{2} \)
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