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: \( x^{4} - 2x^{3} + 5x^{2} - 4x + 69 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - 2 q^{4} - \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - \beta_1 q^{5} + \beta_{3} q^{7} + \beta_1 q^{11} - \beta_{3} q^{13} + 4 q^{16} - \beta_{2} q^{17} + 2 \beta_1 q^{20} - \beta_{2} q^{23} - 5 q^{25} - 2 \beta_{3} q^{28} - \beta_{2} q^{35} + \beta_{3} q^{37} - 5 \beta_1 q^{41} - 2 \beta_1 q^{44} + 6 q^{49} + 2 \beta_{3} q^{52} - \beta_{2} q^{53} + 5 q^{55} + 4 \beta_1 q^{59} - 7 q^{61} - 8 q^{64} + \beta_{2} q^{65} + 4 \beta_{3} q^{67} + 2 \beta_{2} q^{68} + 7 \beta_1 q^{71} - 2 \beta_{3} q^{73} + \beta_{2} q^{77} + 11 q^{79} - 4 \beta_1 q^{80} - 5 \beta_{3} q^{85} + \beta_1 q^{89} - 13 q^{91} + 2 \beta_{2} q^{92} + \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 25\nu - 12 ) / 33 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 16\nu - 9 ) / 33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{3} + 3\beta_{2} + 11\beta _1 - 5 ) / 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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