Properties

Label 832.1.bb.a
Level $832$
Weight $1$
Character orbit 832.bb
Analytic conductor $0.415$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 832.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.415222090511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.479174066176.4

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + q^{5} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{13} + \zeta_{6} q^{17} + \zeta_{6}^{2} q^{29} + \zeta_{6}^{2} q^{37} -\zeta_{6}^{2} q^{41} -\zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} + q^{53} -\zeta_{6} q^{61} -\zeta_{6}^{2} q^{65} - q^{73} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{85} + 2 \zeta_{6}^{2} q^{89} -2 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - q^{9} + q^{13} + q^{17} - q^{29} - q^{37} + q^{41} - q^{45} - q^{49} + 2q^{53} - q^{61} + q^{65} - 2q^{73} - q^{81} + q^{85} - 2q^{89} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.00000 0 0 0 −0.500000 0.866025i 0
575.1 0 0 0 1.00000 0 0 0 −0.500000 + 0.866025i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.1.bb.a 2
4.b odd 2 1 CM 832.1.bb.a 2
8.b even 2 1 52.1.j.a 2
8.d odd 2 1 52.1.j.a 2
13.c even 3 1 inner 832.1.bb.a 2
16.e even 4 2 3328.1.v.b 4
16.f odd 4 2 3328.1.v.b 4
24.f even 2 1 468.1.br.a 2
24.h odd 2 1 468.1.br.a 2
40.e odd 2 1 1300.1.bc.a 2
40.f even 2 1 1300.1.bc.a 2
40.i odd 4 2 1300.1.w.a 4
40.k even 4 2 1300.1.w.a 4
52.j odd 6 1 inner 832.1.bb.a 2
56.e even 2 1 2548.1.bn.a 2
56.h odd 2 1 2548.1.bn.a 2
56.j odd 6 1 2548.1.q.a 2
56.j odd 6 1 2548.1.bi.a 2
56.k odd 6 1 2548.1.q.b 2
56.k odd 6 1 2548.1.bi.b 2
56.m even 6 1 2548.1.q.a 2
56.m even 6 1 2548.1.bi.a 2
56.p even 6 1 2548.1.q.b 2
56.p even 6 1 2548.1.bi.b 2
104.e even 2 1 676.1.j.a 2
104.h odd 2 1 676.1.j.a 2
104.j odd 4 2 676.1.i.a 4
104.m even 4 2 676.1.i.a 4
104.n odd 6 1 52.1.j.a 2
104.n odd 6 1 676.1.c.b 1
104.p odd 6 1 676.1.c.a 1
104.p odd 6 1 676.1.j.a 2
104.r even 6 1 52.1.j.a 2
104.r even 6 1 676.1.c.b 1
104.s even 6 1 676.1.c.a 1
104.s even 6 1 676.1.j.a 2
104.u even 12 2 676.1.b.a 2
104.u even 12 2 676.1.i.a 4
104.x odd 12 2 676.1.b.a 2
104.x odd 12 2 676.1.i.a 4
208.bg odd 12 2 3328.1.v.b 4
208.bj even 12 2 3328.1.v.b 4
312.bh odd 6 1 468.1.br.a 2
312.bn even 6 1 468.1.br.a 2
520.bv even 6 1 1300.1.bc.a 2
520.bx odd 6 1 1300.1.bc.a 2
520.cm even 12 2 1300.1.w.a 4
520.cq odd 12 2 1300.1.w.a 4
728.bf even 6 1 2548.1.bi.a 2
728.bg even 6 1 2548.1.bi.b 2
728.bq odd 6 1 2548.1.q.a 2
728.bx odd 6 1 2548.1.q.b 2
728.ce odd 6 1 2548.1.bn.a 2
728.cl even 6 1 2548.1.q.a 2
728.cw even 6 1 2548.1.q.b 2
728.da even 6 1 2548.1.bn.a 2
728.de odd 6 1 2548.1.bi.a 2
728.di odd 6 1 2548.1.bi.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 8.b even 2 1
52.1.j.a 2 8.d odd 2 1
52.1.j.a 2 104.n odd 6 1
52.1.j.a 2 104.r even 6 1
468.1.br.a 2 24.f even 2 1
468.1.br.a 2 24.h odd 2 1
468.1.br.a 2 312.bh odd 6 1
468.1.br.a 2 312.bn even 6 1
676.1.b.a 2 104.u even 12 2
676.1.b.a 2 104.x odd 12 2
676.1.c.a 1 104.p odd 6 1
676.1.c.a 1 104.s even 6 1
676.1.c.b 1 104.n odd 6 1
676.1.c.b 1 104.r even 6 1
676.1.i.a 4 104.j odd 4 2
676.1.i.a 4 104.m even 4 2
676.1.i.a 4 104.u even 12 2
676.1.i.a 4 104.x odd 12 2
676.1.j.a 2 104.e even 2 1
676.1.j.a 2 104.h odd 2 1
676.1.j.a 2 104.p odd 6 1
676.1.j.a 2 104.s even 6 1
832.1.bb.a 2 1.a even 1 1 trivial
832.1.bb.a 2 4.b odd 2 1 CM
832.1.bb.a 2 13.c even 3 1 inner
832.1.bb.a 2 52.j odd 6 1 inner
1300.1.w.a 4 40.i odd 4 2
1300.1.w.a 4 40.k even 4 2
1300.1.w.a 4 520.cm even 12 2
1300.1.w.a 4 520.cq odd 12 2
1300.1.bc.a 2 40.e odd 2 1
1300.1.bc.a 2 40.f even 2 1
1300.1.bc.a 2 520.bv even 6 1
1300.1.bc.a 2 520.bx odd 6 1
2548.1.q.a 2 56.j odd 6 1
2548.1.q.a 2 56.m even 6 1
2548.1.q.a 2 728.bq odd 6 1
2548.1.q.a 2 728.cl even 6 1
2548.1.q.b 2 56.k odd 6 1
2548.1.q.b 2 56.p even 6 1
2548.1.q.b 2 728.bx odd 6 1
2548.1.q.b 2 728.cw even 6 1
2548.1.bi.a 2 56.j odd 6 1
2548.1.bi.a 2 56.m even 6 1
2548.1.bi.a 2 728.bf even 6 1
2548.1.bi.a 2 728.de odd 6 1
2548.1.bi.b 2 56.k odd 6 1
2548.1.bi.b 2 56.p even 6 1
2548.1.bi.b 2 728.bg even 6 1
2548.1.bi.b 2 728.di odd 6 1
2548.1.bn.a 2 56.e even 2 1
2548.1.bn.a 2 56.h odd 2 1
2548.1.bn.a 2 728.ce odd 6 1
2548.1.bn.a 2 728.da even 6 1
3328.1.v.b 4 16.e even 4 2
3328.1.v.b 4 16.f odd 4 2
3328.1.v.b 4 208.bg odd 12 2
3328.1.v.b 4 208.bj even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(832, [\chi])\).

Hecke characteristic polynomials

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