Properties

Label 585.2.b.f
Level $585$
Weight $2$
Character orbit 585.b
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 10 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 5 \beta_{1}\)

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.30278i
2.30278i
2.30278i
1.30278i
2.00000i 0 −2.00000 1.00000i 0 3.60555i 0 0 2.00000
181.2 2.00000i 0 −2.00000 1.00000i 0 3.60555i 0 0 2.00000
181.3 2.00000i 0 −2.00000 1.00000i 0 3.60555i 0 0 2.00000
181.4 2.00000i 0 −2.00000 1.00000i 0 3.60555i 0 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.b.f 4
3.b odd 2 1 inner 585.2.b.f 4
13.b even 2 1 inner 585.2.b.f 4
13.d odd 4 1 7605.2.a.w 2
13.d odd 4 1 7605.2.a.bl 2
39.d odd 2 1 inner 585.2.b.f 4
39.f even 4 1 7605.2.a.w 2
39.f even 4 1 7605.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.b.f 4 1.a even 1 1 trivial
585.2.b.f 4 3.b odd 2 1 inner
585.2.b.f 4 13.b even 2 1 inner
585.2.b.f 4 39.d odd 2 1 inner
7605.2.a.w 2 13.d odd 4 1
7605.2.a.w 2 39.f even 4 1
7605.2.a.bl 2 13.d odd 4 1
7605.2.a.bl 2 39.f even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \)
\( T_{7}^{2} + 13 \)
\( T_{17}^{2} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 13 + T^{2} )^{2} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( ( 13 + T^{2} )^{2} \)
$17$ \( ( -13 + T^{2} )^{2} \)
$19$ \( ( 52 + T^{2} )^{2} \)
$23$ \( ( -13 + T^{2} )^{2} \)
$29$ \( ( -52 + T^{2} )^{2} \)
$31$ \( ( 52 + T^{2} )^{2} \)
$37$ \( ( 13 + T^{2} )^{2} \)
$41$ \( ( 121 + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( ( 16 + T^{2} )^{2} \)
$53$ \( ( -117 + T^{2} )^{2} \)
$59$ \( ( 144 + T^{2} )^{2} \)
$61$ \( ( -13 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 25 + T^{2} )^{2} \)
$73$ \( ( 52 + T^{2} )^{2} \)
$79$ \( ( -13 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 9 + T^{2} )^{2} \)
$97$ \( ( 13 + T^{2} )^{2} \)
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