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: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 2) q^{7} + (\beta + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 2) q^{7} + (\beta + 3) q^{8} + ( - \beta - 1) q^{10} + ( - \beta - 2) q^{11} - q^{13} + (4 \beta + 6) q^{14} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + ( - \beta + 2) q^{19} + ( - 2 \beta - 1) q^{20} + ( - 3 \beta - 4) q^{22} - \beta q^{23} + q^{25} + ( - \beta - 1) q^{26} + (6 \beta + 10) q^{28} + 4 \beta q^{29} + ( - 3 \beta + 6) q^{31} + (\beta - 3) q^{32} - 2 q^{34} + ( - 2 \beta - 2) q^{35} - 6 \beta q^{37} + \beta q^{38} + ( - \beta - 3) q^{40} + ( - 2 \beta + 6) q^{41} + ( - 5 \beta - 4) q^{43} + ( - 5 \beta - 6) q^{44} + ( - \beta - 2) q^{46} + (2 \beta + 2) q^{47} + (8 \beta + 5) q^{49} + (\beta + 1) q^{50} + ( - 2 \beta - 1) q^{52} + ( - 6 \beta + 6) q^{53} + (\beta + 2) q^{55} + (8 \beta + 10) q^{56} + (4 \beta + 8) q^{58} + (3 \beta - 6) q^{59} - 8 q^{61} + 3 \beta q^{62} + ( - 2 \beta - 7) q^{64} + q^{65} - 2 q^{67} + (2 \beta - 6) q^{68} + ( - 4 \beta - 6) q^{70} + ( - 7 \beta - 2) q^{71} + 6 \beta q^{73} + ( - 6 \beta - 12) q^{74} + (3 \beta - 2) q^{76} + ( - 6 \beta - 8) q^{77} - 6 \beta q^{79} - 3 q^{80} + (4 \beta + 2) q^{82} + ( - 2 \beta + 6) q^{83} + (2 \beta - 2) q^{85} + ( - 9 \beta - 14) q^{86} + ( - 5 \beta - 8) q^{88} - 6 q^{89} + ( - 2 \beta - 2) q^{91} + ( - \beta - 4) q^{92} + (4 \beta + 6) q^{94} + (\beta - 2) q^{95} + ( - 4 \beta - 2) q^{97} + (13 \beta + 21) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100}) \) 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
−1.41421
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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