Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8} - q^{10} + ( - 2 i + 3) q^{13} - 4 q^{14} - q^{16} - 4 q^{17} - 4 i q^{19} + i q^{20} - 8 q^{23} - q^{25} + (3 i + 2) q^{26} + 4 i q^{28} + 8 q^{29} + 4 i q^{31} + 5 i q^{32} - 4 i q^{34} - 4 q^{35} + 4 i q^{37} + 4 q^{38} - 3 q^{40} - 2 i q^{41} + 8 q^{43} - 8 i q^{46} + 8 i q^{47} - 9 q^{49} - i q^{50} + ( - 2 i + 3) q^{52} + 12 q^{53} - 12 q^{56} + 8 i q^{58} - 2 q^{61} - 4 q^{62} - 7 q^{64} + (3 i + 2) q^{65} - 12 i q^{67} - 4 q^{68} - 4 i q^{70} - 4 i q^{71} + 4 i q^{73} - 4 q^{74} - 4 i q^{76} - 8 q^{79} - i q^{80} + 2 q^{82} - 12 i q^{83} - 4 i q^{85} + 8 i q^{86} - 18 i q^{89} + (12 i + 8) q^{91} - 8 q^{92} - 8 q^{94} + 4 q^{95} + 4 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{10} + 6 q^{13} - 8 q^{14} - 2 q^{16} - 8 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 16 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} - 18 q^{49} + 6 q^{52} + 24 q^{53} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{74} - 16 q^{79} + 4 q^{82} + 16 q^{91} - 16 q^{92} - 16 q^{94} + 8 q^{95}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
1.00000i 0 1.00000 1.00000i 0 4.00000i 3.00000i 0 −1.00000
181.2 1.00000i 0 1.00000 1.00000i 0 4.00000i 3.00000i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b 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.b.b 2
3.b odd 2 1 585.2.b.c yes 2
13.b even 2 1 inner 585.2.b.b 2
13.d odd 4 1 7605.2.a.e 1
13.d odd 4 1 7605.2.a.o 1
39.d odd 2 1 585.2.b.c yes 2
39.f even 4 1 7605.2.a.b 1
39.f even 4 1 7605.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.b.b 2 1.a even 1 1 trivial
585.2.b.b 2 13.b even 2 1 inner
585.2.b.c yes 2 3.b odd 2 1
585.2.b.c yes 2 39.d odd 2 1
7605.2.a.b 1 39.f even 4 1
7605.2.a.e 1 13.d odd 4 1
7605.2.a.o 1 13.d odd 4 1
7605.2.a.r 1 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 \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 16 \) Copy content Toggle raw display
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