Properties

Label 585.2.a.k
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + q^{5} + 2 q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{4} + q^{5} + 2 q^{7} - \beta q^{8} + \beta q^{10} + (\beta + 3) q^{11} + q^{13} + 2 \beta q^{14} - 5 q^{16} + 2 \beta q^{17} + ( - 3 \beta - 1) q^{19} + q^{20} + (3 \beta + 3) q^{22} + (\beta - 3) q^{23} + q^{25} + \beta q^{26} + 2 q^{28} + ( - 2 \beta + 6) q^{29} + (3 \beta + 5) q^{31} - 3 \beta q^{32} + 6 q^{34} + 2 q^{35} - 4 q^{37} + ( - \beta - 9) q^{38} - \beta q^{40} - 2 \beta q^{41} + ( - 3 \beta + 5) q^{43} + (\beta + 3) q^{44} + ( - 3 \beta + 3) q^{46} - 6 q^{47} - 3 q^{49} + \beta q^{50} + q^{52} - 6 \beta q^{53} + (\beta + 3) q^{55} - 2 \beta q^{56} + (6 \beta - 6) q^{58} + ( - 7 \beta + 3) q^{59} + ( - 6 \beta + 2) q^{61} + (5 \beta + 9) q^{62} + q^{64} + q^{65} + (6 \beta - 4) q^{67} + 2 \beta q^{68} + 2 \beta q^{70} + ( - \beta - 3) q^{71} - 4 q^{73} - 4 \beta q^{74} + ( - 3 \beta - 1) q^{76} + (2 \beta + 6) q^{77} + ( - 6 \beta + 2) q^{79} - 5 q^{80} - 6 q^{82} + 6 q^{83} + 2 \beta q^{85} + (5 \beta - 9) q^{86} + ( - 3 \beta - 3) q^{88} + (4 \beta + 6) q^{89} + 2 q^{91} + (\beta - 3) q^{92} - 6 \beta q^{94} + ( - 3 \beta - 1) q^{95} + 2 q^{97} - 3 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 0 1.00000 1.00000 0 2.00000 1.73205 0 −1.73205
1.2 1.73205 0 1.00000 1.00000 0 2.00000 −1.73205 0 1.73205
\(n\): 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.k 2
3.b odd 2 1 65.2.a.c 2
4.b odd 2 1 9360.2.a.cm 2
5.b even 2 1 2925.2.a.z 2
5.c odd 4 2 2925.2.c.v 4
12.b even 2 1 1040.2.a.h 2
13.b even 2 1 7605.2.a.be 2
15.d odd 2 1 325.2.a.g 2
15.e even 4 2 325.2.b.e 4
21.c even 2 1 3185.2.a.k 2
24.f even 2 1 4160.2.a.bj 2
24.h odd 2 1 4160.2.a.y 2
33.d even 2 1 7865.2.a.h 2
39.d odd 2 1 845.2.a.d 2
39.f even 4 2 845.2.c.e 4
39.h odd 6 2 845.2.e.f 4
39.i odd 6 2 845.2.e.e 4
39.k even 12 2 845.2.m.a 4
39.k even 12 2 845.2.m.c 4
60.h even 2 1 5200.2.a.ca 2
195.e odd 2 1 4225.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 3.b odd 2 1
325.2.a.g 2 15.d odd 2 1
325.2.b.e 4 15.e even 4 2
585.2.a.k 2 1.a even 1 1 trivial
845.2.a.d 2 39.d odd 2 1
845.2.c.e 4 39.f even 4 2
845.2.e.e 4 39.i odd 6 2
845.2.e.f 4 39.h odd 6 2
845.2.m.a 4 39.k even 12 2
845.2.m.c 4 39.k even 12 2
1040.2.a.h 2 12.b even 2 1
2925.2.a.z 2 5.b even 2 1
2925.2.c.v 4 5.c odd 4 2
3185.2.a.k 2 21.c even 2 1
4160.2.a.y 2 24.h odd 2 1
4160.2.a.bj 2 24.f even 2 1
4225.2.a.w 2 195.e odd 2 1
5200.2.a.ca 2 60.h even 2 1
7605.2.a.be 2 13.b even 2 1
7865.2.a.h 2 33.d even 2 1
9360.2.a.cm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 108 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 138 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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