Properties

Label 585.2.h.c
Level $585$
Weight $2$
Character orbit 585.h
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.h (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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + ( 1 + 2 i ) q^{5} -3 q^{8} +O(q^{10})\) \( q + q^{2} - q^{4} + ( 1 + 2 i ) q^{5} -3 q^{8} + ( 1 + 2 i ) q^{10} + 2 i q^{11} + ( 3 + 2 i ) q^{13} - q^{16} + 6 i q^{19} + ( -1 - 2 i ) q^{20} + 2 i q^{22} + 6 i q^{23} + ( -3 + 4 i ) q^{25} + ( 3 + 2 i ) q^{26} -6 q^{29} -6 i q^{31} + 5 q^{32} + 6 q^{37} + 6 i q^{38} + ( -3 - 6 i ) q^{40} -8 i q^{41} + 6 i q^{43} -2 i q^{44} + 6 i q^{46} + 8 q^{47} -7 q^{49} + ( -3 + 4 i ) q^{50} + ( -3 - 2 i ) q^{52} -12 i q^{53} + ( -4 + 2 i ) q^{55} -6 q^{58} + 2 i q^{59} + 6 q^{61} -6 i q^{62} + 7 q^{64} + ( -1 + 8 i ) q^{65} + 12 q^{67} -2 i q^{71} -6 q^{73} + 6 q^{74} -6 i q^{76} + ( -1 - 2 i ) q^{80} -8 i q^{82} + 4 q^{83} + 6 i q^{86} -6 i q^{88} -8 i q^{89} -6 i q^{92} + 8 q^{94} + ( -12 + 6 i ) q^{95} -6 q^{97} -7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} + 6 q^{13} - 2 q^{16} - 2 q^{20} - 6 q^{25} + 6 q^{26} - 12 q^{29} + 10 q^{32} + 12 q^{37} - 6 q^{40} + 16 q^{47} - 14 q^{49} - 6 q^{50} - 6 q^{52} - 8 q^{55} - 12 q^{58} + 12 q^{61} + 14 q^{64} - 2 q^{65} + 24 q^{67} - 12 q^{73} + 12 q^{74} - 2 q^{80} + 8 q^{83} + 16 q^{94} - 24 q^{95} - 12 q^{97} - 14 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
1.00000 0 −1.00000 1.00000 2.00000i 0 0 −3.00000 0 1.00000 2.00000i
64.2 1.00000 0 −1.00000 1.00000 + 2.00000i 0 0 −3.00000 0 1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 2
3.b odd 2 1 65.2.d.a 2
5.b even 2 1 585.2.h.b 2
12.b even 2 1 1040.2.f.a 2
13.b even 2 1 585.2.h.b 2
15.d odd 2 1 65.2.d.b yes 2
15.e even 4 1 325.2.c.b 2
15.e even 4 1 325.2.c.e 2
39.d odd 2 1 65.2.d.b yes 2
39.f even 4 1 845.2.b.a 2
39.f even 4 1 845.2.b.b 2
39.h odd 6 2 845.2.l.a 4
39.i odd 6 2 845.2.l.b 4
39.k even 12 2 845.2.n.a 4
39.k even 12 2 845.2.n.b 4
60.h even 2 1 1040.2.f.b 2
65.d even 2 1 inner 585.2.h.c 2
156.h even 2 1 1040.2.f.b 2
195.e odd 2 1 65.2.d.a 2
195.j odd 4 1 4225.2.a.e 1
195.j odd 4 1 4225.2.a.h 1
195.n even 4 1 845.2.b.a 2
195.n even 4 1 845.2.b.b 2
195.s even 4 1 325.2.c.b 2
195.s even 4 1 325.2.c.e 2
195.u odd 4 1 4225.2.a.k 1
195.u odd 4 1 4225.2.a.m 1
195.x odd 6 2 845.2.l.a 4
195.y odd 6 2 845.2.l.b 4
195.bh even 12 2 845.2.n.a 4
195.bh even 12 2 845.2.n.b 4
780.d even 2 1 1040.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 3.b odd 2 1
65.2.d.a 2 195.e odd 2 1
65.2.d.b yes 2 15.d odd 2 1
65.2.d.b yes 2 39.d odd 2 1
325.2.c.b 2 15.e even 4 1
325.2.c.b 2 195.s even 4 1
325.2.c.e 2 15.e even 4 1
325.2.c.e 2 195.s even 4 1
585.2.h.b 2 5.b even 2 1
585.2.h.b 2 13.b even 2 1
585.2.h.c 2 1.a even 1 1 trivial
585.2.h.c 2 65.d even 2 1 inner
845.2.b.a 2 39.f even 4 1
845.2.b.a 2 195.n even 4 1
845.2.b.b 2 39.f even 4 1
845.2.b.b 2 195.n even 4 1
845.2.l.a 4 39.h odd 6 2
845.2.l.a 4 195.x odd 6 2
845.2.l.b 4 39.i odd 6 2
845.2.l.b 4 195.y odd 6 2
845.2.n.a 4 39.k even 12 2
845.2.n.a 4 195.bh even 12 2
845.2.n.b 4 39.k even 12 2
845.2.n.b 4 195.bh even 12 2
1040.2.f.a 2 12.b even 2 1
1040.2.f.a 2 780.d even 2 1
1040.2.f.b 2 60.h even 2 1
1040.2.f.b 2 156.h even 2 1
4225.2.a.e 1 195.j odd 4 1
4225.2.a.h 1 195.j odd 4 1
4225.2.a.k 1 195.u odd 4 1
4225.2.a.m 1 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 13 - 6 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( 4 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( ( -12 + T )^{2} \)
$71$ \( 4 + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( 64 + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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