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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8} +O(q^{10})\) \( q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8} - q^{10} + ( 3 - 2 i ) q^{13} -4 q^{14} - q^{16} -4 q^{17} -4 i q^{19} + i q^{20} -8 q^{23} - q^{25} + ( 2 + 3 i ) q^{26} + 4 i q^{28} + 8 q^{29} + 4 i q^{31} + 5 i q^{32} -4 i q^{34} -4 q^{35} + 4 i q^{37} + 4 q^{38} -3 q^{40} -2 i q^{41} + 8 q^{43} -8 i q^{46} + 8 i q^{47} -9 q^{49} -i q^{50} + ( 3 - 2 i ) q^{52} + 12 q^{53} -12 q^{56} + 8 i q^{58} -2 q^{61} -4 q^{62} -7 q^{64} + ( 2 + 3 i ) q^{65} -12 i q^{67} -4 q^{68} -4 i q^{70} -4 i q^{71} + 4 i q^{73} -4 q^{74} -4 i q^{76} -8 q^{79} -i q^{80} + 2 q^{82} -12 i q^{83} -4 i q^{85} + 8 i q^{86} -18 i q^{89} + ( 8 + 12 i ) q^{91} -8 q^{92} -8 q^{94} + 4 q^{95} + 4 i q^{97} -9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + O(q^{10}) \) \( 2 q + 2 q^{4} - 2 q^{10} + 6 q^{13} - 8 q^{14} - 2 q^{16} - 8 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 16 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} - 18 q^{49} + 6 q^{52} + 24 q^{53} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{74} - 16 q^{79} + 4 q^{82} + 16 q^{91} - 16 q^{92} - 16 q^{94} + 8 q^{95} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 16 \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

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