Properties

Label 585.2.a.m
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( 2 + 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} - q^{5} + ( 2 + 2 \beta ) q^{7} + ( 3 + \beta ) q^{8} + ( -1 - \beta ) q^{10} + ( -2 - \beta ) q^{11} - q^{13} + ( 6 + 4 \beta ) q^{14} + 3 q^{16} + ( 2 - 2 \beta ) q^{17} + ( 2 - \beta ) q^{19} + ( -1 - 2 \beta ) q^{20} + ( -4 - 3 \beta ) q^{22} -\beta q^{23} + q^{25} + ( -1 - \beta ) q^{26} + ( 10 + 6 \beta ) q^{28} + 4 \beta q^{29} + ( 6 - 3 \beta ) q^{31} + ( -3 + \beta ) q^{32} -2 q^{34} + ( -2 - 2 \beta ) q^{35} -6 \beta q^{37} + \beta q^{38} + ( -3 - \beta ) q^{40} + ( 6 - 2 \beta ) q^{41} + ( -4 - 5 \beta ) q^{43} + ( -6 - 5 \beta ) q^{44} + ( -2 - \beta ) q^{46} + ( 2 + 2 \beta ) q^{47} + ( 5 + 8 \beta ) q^{49} + ( 1 + \beta ) q^{50} + ( -1 - 2 \beta ) q^{52} + ( 6 - 6 \beta ) q^{53} + ( 2 + \beta ) q^{55} + ( 10 + 8 \beta ) q^{56} + ( 8 + 4 \beta ) q^{58} + ( -6 + 3 \beta ) q^{59} -8 q^{61} + 3 \beta q^{62} + ( -7 - 2 \beta ) q^{64} + q^{65} -2 q^{67} + ( -6 + 2 \beta ) q^{68} + ( -6 - 4 \beta ) q^{70} + ( -2 - 7 \beta ) q^{71} + 6 \beta q^{73} + ( -12 - 6 \beta ) q^{74} + ( -2 + 3 \beta ) q^{76} + ( -8 - 6 \beta ) q^{77} -6 \beta q^{79} -3 q^{80} + ( 2 + 4 \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( -2 + 2 \beta ) q^{85} + ( -14 - 9 \beta ) q^{86} + ( -8 - 5 \beta ) q^{88} -6 q^{89} + ( -2 - 2 \beta ) q^{91} + ( -4 - \beta ) q^{92} + ( 6 + 4 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( -2 - 4 \beta ) q^{97} + ( 21 + 13 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 0 −1.82843 −1.00000 0 −0.828427 1.58579 0 0.414214
1.2 2.41421 0 3.82843 −1.00000 0 4.82843 4.41421 0 −2.41421
\(n\): 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.m 2
3.b odd 2 1 65.2.a.b 2
4.b odd 2 1 9360.2.a.cd 2
5.b even 2 1 2925.2.a.u 2
5.c odd 4 2 2925.2.c.r 4
12.b even 2 1 1040.2.a.j 2
13.b even 2 1 7605.2.a.x 2
15.d odd 2 1 325.2.a.i 2
15.e even 4 2 325.2.b.f 4
21.c even 2 1 3185.2.a.j 2
24.f even 2 1 4160.2.a.z 2
24.h odd 2 1 4160.2.a.bf 2
33.d even 2 1 7865.2.a.j 2
39.d odd 2 1 845.2.a.g 2
39.f even 4 2 845.2.c.b 4
39.h odd 6 2 845.2.e.c 4
39.i odd 6 2 845.2.e.h 4
39.k even 12 4 845.2.m.f 8
60.h even 2 1 5200.2.a.bu 2
195.e odd 2 1 4225.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 3.b odd 2 1
325.2.a.i 2 15.d odd 2 1
325.2.b.f 4 15.e even 4 2
585.2.a.m 2 1.a even 1 1 trivial
845.2.a.g 2 39.d odd 2 1
845.2.c.b 4 39.f even 4 2
845.2.e.c 4 39.h odd 6 2
845.2.e.h 4 39.i odd 6 2
845.2.m.f 8 39.k even 12 4
1040.2.a.j 2 12.b even 2 1
2925.2.a.u 2 5.b even 2 1
2925.2.c.r 4 5.c odd 4 2
3185.2.a.j 2 21.c even 2 1
4160.2.a.z 2 24.f even 2 1
4160.2.a.bf 2 24.h odd 2 1
4225.2.a.r 2 195.e odd 2 1
5200.2.a.bu 2 60.h even 2 1
7605.2.a.x 2 13.b even 2 1
7865.2.a.j 2 33.d even 2 1
9360.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{7}^{2} - 4 T_{7} - 4 \)
\( T_{11}^{2} + 4 T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - 4 T + T^{2} \)
$11$ \( 2 + 4 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( 2 - 4 T + T^{2} \)
$23$ \( -2 + T^{2} \)
$29$ \( -32 + T^{2} \)
$31$ \( 18 - 12 T + T^{2} \)
$37$ \( -72 + T^{2} \)
$41$ \( 28 - 12 T + T^{2} \)
$43$ \( -34 + 8 T + T^{2} \)
$47$ \( -4 - 4 T + T^{2} \)
$53$ \( -36 - 12 T + T^{2} \)
$59$ \( 18 + 12 T + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( -94 + 4 T + T^{2} \)
$73$ \( -72 + T^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( 28 - 12 T + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
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