Properties

Label 585.2.h.e
Level $585$
Weight $2$
Character orbit 585.h
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM discriminant -195
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{13})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 69\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 q^{4} -\beta_{1} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q -2 q^{4} -\beta_{1} q^{5} + \beta_{3} q^{7} + \beta_{1} q^{11} -\beta_{3} q^{13} + 4 q^{16} -\beta_{2} q^{17} + 2 \beta_{1} q^{20} -\beta_{2} q^{23} -5 q^{25} -2 \beta_{3} q^{28} -\beta_{2} q^{35} + \beta_{3} q^{37} -5 \beta_{1} q^{41} -2 \beta_{1} q^{44} + 6 q^{49} + 2 \beta_{3} q^{52} -\beta_{2} q^{53} + 5 q^{55} + 4 \beta_{1} q^{59} -7 q^{61} -8 q^{64} + \beta_{2} q^{65} + 4 \beta_{3} q^{67} + 2 \beta_{2} q^{68} + 7 \beta_{1} q^{71} -2 \beta_{3} q^{73} + \beta_{2} q^{77} + 11 q^{79} -4 \beta_{1} q^{80} -5 \beta_{3} q^{85} + \beta_{1} q^{89} -13 q^{91} + 2 \beta_{2} q^{92} + \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 69\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} - 3 \nu^{2} + 25 \nu - 12 \)\()/33\)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 16 \nu - 9 \)\()/33\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{3} + 3 \beta_{2} + 11 \beta_{1} - 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−1.30278 + 2.23607i
2.30278 + 2.23607i
−1.30278 2.23607i
2.30278 2.23607i
0 0 −2.00000 2.23607i 0 −3.60555 0 0 0
64.2 0 0 −2.00000 2.23607i 0 3.60555 0 0 0
64.3 0 0 −2.00000 2.23607i 0 −3.60555 0 0 0
64.4 0 0 −2.00000 2.23607i 0 3.60555 0 0 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
195.e odd 2 1 CM by \(\Q(\sqrt{-195}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
13.b even 2 1 inner
15.d odd 2 1 inner
39.d odd 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.h.e 4
3.b odd 2 1 inner 585.2.h.e 4
5.b even 2 1 inner 585.2.h.e 4
13.b even 2 1 inner 585.2.h.e 4
15.d odd 2 1 inner 585.2.h.e 4
39.d odd 2 1 inner 585.2.h.e 4
65.d even 2 1 inner 585.2.h.e 4
195.e odd 2 1 CM 585.2.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.h.e 4 1.a even 1 1 trivial
585.2.h.e 4 3.b odd 2 1 inner
585.2.h.e 4 5.b even 2 1 inner
585.2.h.e 4 13.b even 2 1 inner
585.2.h.e 4 15.d odd 2 1 inner
585.2.h.e 4 39.d odd 2 1 inner
585.2.h.e 4 65.d even 2 1 inner
585.2.h.e 4 195.e odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( -13 + T^{2} )^{2} \)
$11$ \( ( 5 + T^{2} )^{2} \)
$13$ \( ( -13 + T^{2} )^{2} \)
$17$ \( ( 65 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 65 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -13 + T^{2} )^{2} \)
$41$ \( ( 125 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 65 + T^{2} )^{2} \)
$59$ \( ( 80 + T^{2} )^{2} \)
$61$ \( ( 7 + T )^{4} \)
$67$ \( ( -208 + T^{2} )^{2} \)
$71$ \( ( 245 + T^{2} )^{2} \)
$73$ \( ( -52 + T^{2} )^{2} \)
$79$ \( ( -11 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 5 + T^{2} )^{2} \)
$97$ \( ( -13 + T^{2} )^{2} \)
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