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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + (\beta + 1) q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + (\beta + 1) q^{5} - 3 q^{8} + (\beta + 1) q^{10} + \beta q^{11} + (\beta + 3) q^{13} - q^{16} + 3 \beta q^{19} + ( - \beta - 1) q^{20} + \beta q^{22} + 3 \beta q^{23} + (2 \beta - 3) q^{25} + (\beta + 3) q^{26} - 6 q^{29} - 3 \beta q^{31} + 5 q^{32} + 6 q^{37} + 3 \beta q^{38} + ( - 3 \beta - 3) q^{40} - 4 \beta q^{41} + 3 \beta q^{43} - \beta q^{44} + 3 \beta q^{46} + 8 q^{47} - 7 q^{49} + (2 \beta - 3) q^{50} + ( - \beta - 3) q^{52} - 6 \beta q^{53} + (\beta - 4) q^{55} - 6 q^{58} + \beta q^{59} + 6 q^{61} - 3 \beta q^{62} + 7 q^{64} + (4 \beta - 1) q^{65} + 12 q^{67} - \beta q^{71} - 6 q^{73} + 6 q^{74} - 3 \beta q^{76} + ( - \beta - 1) q^{80} - 4 \beta q^{82} + 4 q^{83} + 3 \beta q^{86} - 3 \beta q^{88} - 4 \beta q^{89} - 3 \beta q^{92} + 8 q^{94} + (3 \beta - 12) q^{95} - 6 q^{97} - 7 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} + 6 q^{13} - 2 q^{16} - 2 q^{20} - 6 q^{25} + 6 q^{26} - 12 q^{29} + 10 q^{32} + 12 q^{37} - 6 q^{40} + 16 q^{47} - 14 q^{49} - 6 q^{50} - 6 q^{52} - 8 q^{55} - 12 q^{58} + 12 q^{61} + 14 q^{64} - 2 q^{65} + 24 q^{67} - 12 q^{73} + 12 q^{74} - 2 q^{80} + 8 q^{83} + 16 q^{94} - 24 q^{95} - 12 q^{97} - 14 q^{98}+O(q^{100}) \) 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
1.00000i
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 \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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