Properties

Label 585.2.i.a
Level $585$
Weight $2$
Character orbit 585.i
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.i (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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 2) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 2) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + (2 \zeta_{6} + 2) q^{12} - \zeta_{6} q^{13} + ( - \zeta_{6} - 1) q^{15} + (4 \zeta_{6} - 4) q^{16} - 3 q^{17} + 2 q^{19} + ( - 2 \zeta_{6} + 2) q^{20} + ( - 8 \zeta_{6} + 4) q^{21} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + 8 q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} - 4 q^{35} + 6 q^{36} + 8 q^{37} + ( - \zeta_{6} - 1) q^{39} + 12 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 q^{45} + (8 \zeta_{6} - 4) q^{48} - 9 \zeta_{6} q^{49} + (3 \zeta_{6} - 6) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} - 3 q^{53} + ( - 2 \zeta_{6} + 4) q^{57} - 6 \zeta_{6} q^{59} + ( - 4 \zeta_{6} + 2) q^{60} + (11 \zeta_{6} - 11) q^{61} - 12 \zeta_{6} q^{63} - 8 q^{64} + (\zeta_{6} - 1) q^{65} + 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} + ( - 3 \zeta_{6} - 3) q^{69} + 6 q^{71} + 8 q^{73} + (2 \zeta_{6} - 1) q^{75} + 4 \zeta_{6} q^{76} + ( - 7 \zeta_{6} + 7) q^{79} + 4 q^{80} - 9 \zeta_{6} q^{81} + (6 \zeta_{6} - 6) q^{83} + ( - 8 \zeta_{6} + 16) q^{84} + 3 \zeta_{6} q^{85} + (12 \zeta_{6} - 6) q^{87} - 18 q^{89} - 4 q^{91} + ( - 6 \zeta_{6} + 6) q^{92} + (4 \zeta_{6} + 4) q^{93} - 2 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 2 q^{4} - q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 2 q^{4} - q^{5} + 4 q^{7} + 3 q^{9} + 6 q^{12} - q^{13} - 3 q^{15} - 4 q^{16} - 6 q^{17} + 4 q^{19} + 2 q^{20} - 3 q^{23} - q^{25} + 16 q^{28} - 6 q^{29} + 4 q^{31} - 8 q^{35} + 12 q^{36} + 16 q^{37} - 3 q^{39} + 12 q^{41} + q^{43} - 6 q^{45} - 9 q^{49} - 9 q^{51} + 2 q^{52} - 6 q^{53} + 6 q^{57} - 6 q^{59} - 11 q^{61} - 12 q^{63} - 16 q^{64} - q^{65} + 4 q^{67} - 6 q^{68} - 9 q^{69} + 12 q^{71} + 16 q^{73} + 4 q^{76} + 7 q^{79} + 8 q^{80} - 9 q^{81} - 6 q^{83} + 24 q^{84} + 3 q^{85} - 36 q^{89} - 8 q^{91} + 6 q^{92} + 12 q^{93} - 2 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i 0 2.00000 3.46410i 0 1.50000 2.59808i 0
391.1 0 1.50000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0 2.00000 + 3.46410i 0 1.50000 + 2.59808i 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
9.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.i.a 2
3.b odd 2 1 1755.2.i.d 2
9.c even 3 1 inner 585.2.i.a 2
9.c even 3 1 5265.2.a.i 1
9.d odd 6 1 1755.2.i.d 2
9.d odd 6 1 5265.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.a 2 1.a even 1 1 trivial
585.2.i.a 2 9.c even 3 1 inner
1755.2.i.d 2 3.b odd 2 1
1755.2.i.d 2 9.d odd 6 1
5265.2.a.g 1 9.d odd 6 1
5265.2.a.i 1 9.c even 3 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( (T + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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