Properties

Label 468.1.br.a
Level $468$
Weight $1$
Character orbit 468.br
Analytic conductor $0.234$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 468.br (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.233562425912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 12.0.85282689024.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + q^{5} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + q^{5} - q^{8} + \zeta_{6} q^{10} -\zeta_{6} q^{13} -\zeta_{6} q^{16} + \zeta_{6}^{2} q^{17} + \zeta_{6}^{2} q^{20} -\zeta_{6}^{2} q^{26} -\zeta_{6} q^{29} -\zeta_{6}^{2} q^{32} - q^{34} + \zeta_{6} q^{37} - q^{40} -\zeta_{6} q^{41} -\zeta_{6} q^{49} + q^{52} + q^{53} -\zeta_{6}^{2} q^{58} -\zeta_{6}^{2} q^{61} + q^{64} -\zeta_{6} q^{65} -\zeta_{6} q^{68} - q^{73} + \zeta_{6}^{2} q^{74} -\zeta_{6} q^{80} -\zeta_{6}^{2} q^{82} + \zeta_{6}^{2} q^{85} + 2 \zeta_{6} q^{89} + 2 \zeta_{6}^{2} q^{97} -\zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{8} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{8} + q^{10} - q^{13} - q^{16} - q^{17} - q^{20} + q^{26} - q^{29} + q^{32} - 2 q^{34} + q^{37} - 2 q^{40} - q^{41} - q^{49} + 2 q^{52} + 2 q^{53} + q^{58} + q^{61} + 2 q^{64} - q^{65} - q^{68} - 2 q^{73} - q^{74} - q^{80} + q^{82} - q^{85} + 2 q^{89} - 2 q^{97} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 0 −1.00000 0 0.500000 + 0.866025i
451.1 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 0 −1.00000 0 0.500000 0.866025i
\(n\): 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 468.1.br.a 2
3.b odd 2 1 52.1.j.a 2
4.b odd 2 1 CM 468.1.br.a 2
12.b even 2 1 52.1.j.a 2
13.c even 3 1 inner 468.1.br.a 2
15.d odd 2 1 1300.1.bc.a 2
15.e even 4 2 1300.1.w.a 4
21.c even 2 1 2548.1.bn.a 2
21.g even 6 1 2548.1.q.a 2
21.g even 6 1 2548.1.bi.a 2
21.h odd 6 1 2548.1.q.b 2
21.h odd 6 1 2548.1.bi.b 2
24.f even 2 1 832.1.bb.a 2
24.h odd 2 1 832.1.bb.a 2
39.d odd 2 1 676.1.j.a 2
39.f even 4 2 676.1.i.a 4
39.h odd 6 1 676.1.c.a 1
39.h odd 6 1 676.1.j.a 2
39.i odd 6 1 52.1.j.a 2
39.i odd 6 1 676.1.c.b 1
39.k even 12 2 676.1.b.a 2
39.k even 12 2 676.1.i.a 4
48.i odd 4 2 3328.1.v.b 4
48.k even 4 2 3328.1.v.b 4
52.j odd 6 1 inner 468.1.br.a 2
60.h even 2 1 1300.1.bc.a 2
60.l odd 4 2 1300.1.w.a 4
84.h odd 2 1 2548.1.bn.a 2
84.j odd 6 1 2548.1.q.a 2
84.j odd 6 1 2548.1.bi.a 2
84.n even 6 1 2548.1.q.b 2
84.n even 6 1 2548.1.bi.b 2
156.h even 2 1 676.1.j.a 2
156.l odd 4 2 676.1.i.a 4
156.p even 6 1 52.1.j.a 2
156.p even 6 1 676.1.c.b 1
156.r even 6 1 676.1.c.a 1
156.r even 6 1 676.1.j.a 2
156.v odd 12 2 676.1.b.a 2
156.v odd 12 2 676.1.i.a 4
195.x odd 6 1 1300.1.bc.a 2
195.bl even 12 2 1300.1.w.a 4
273.r even 6 1 2548.1.q.a 2
273.s odd 6 1 2548.1.q.b 2
273.bf even 6 1 2548.1.bi.a 2
273.bm odd 6 1 2548.1.bi.b 2
273.bn even 6 1 2548.1.bn.a 2
312.bh odd 6 1 832.1.bb.a 2
312.bn even 6 1 832.1.bb.a 2
624.cl even 12 2 3328.1.v.b 4
624.cw odd 12 2 3328.1.v.b 4
780.br even 6 1 1300.1.bc.a 2
780.cj odd 12 2 1300.1.w.a 4
1092.bt odd 6 1 2548.1.q.a 2
1092.cc odd 6 1 2548.1.bi.a 2
1092.ck even 6 1 2548.1.q.b 2
1092.dc even 6 1 2548.1.bi.b 2
1092.dd odd 6 1 2548.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 3.b odd 2 1
52.1.j.a 2 12.b even 2 1
52.1.j.a 2 39.i odd 6 1
52.1.j.a 2 156.p even 6 1
468.1.br.a 2 1.a even 1 1 trivial
468.1.br.a 2 4.b odd 2 1 CM
468.1.br.a 2 13.c even 3 1 inner
468.1.br.a 2 52.j odd 6 1 inner
676.1.b.a 2 39.k even 12 2
676.1.b.a 2 156.v odd 12 2
676.1.c.a 1 39.h odd 6 1
676.1.c.a 1 156.r even 6 1
676.1.c.b 1 39.i odd 6 1
676.1.c.b 1 156.p even 6 1
676.1.i.a 4 39.f even 4 2
676.1.i.a 4 39.k even 12 2
676.1.i.a 4 156.l odd 4 2
676.1.i.a 4 156.v odd 12 2
676.1.j.a 2 39.d odd 2 1
676.1.j.a 2 39.h odd 6 1
676.1.j.a 2 156.h even 2 1
676.1.j.a 2 156.r even 6 1
832.1.bb.a 2 24.f even 2 1
832.1.bb.a 2 24.h odd 2 1
832.1.bb.a 2 312.bh odd 6 1
832.1.bb.a 2 312.bn even 6 1
1300.1.w.a 4 15.e even 4 2
1300.1.w.a 4 60.l odd 4 2
1300.1.w.a 4 195.bl even 12 2
1300.1.w.a 4 780.cj odd 12 2
1300.1.bc.a 2 15.d odd 2 1
1300.1.bc.a 2 60.h even 2 1
1300.1.bc.a 2 195.x odd 6 1
1300.1.bc.a 2 780.br even 6 1
2548.1.q.a 2 21.g even 6 1
2548.1.q.a 2 84.j odd 6 1
2548.1.q.a 2 273.r even 6 1
2548.1.q.a 2 1092.bt odd 6 1
2548.1.q.b 2 21.h odd 6 1
2548.1.q.b 2 84.n even 6 1
2548.1.q.b 2 273.s odd 6 1
2548.1.q.b 2 1092.ck even 6 1
2548.1.bi.a 2 21.g even 6 1
2548.1.bi.a 2 84.j odd 6 1
2548.1.bi.a 2 273.bf even 6 1
2548.1.bi.a 2 1092.cc odd 6 1
2548.1.bi.b 2 21.h odd 6 1
2548.1.bi.b 2 84.n even 6 1
2548.1.bi.b 2 273.bm odd 6 1
2548.1.bi.b 2 1092.dc even 6 1
2548.1.bn.a 2 21.c even 2 1
2548.1.bn.a 2 84.h odd 2 1
2548.1.bn.a 2 273.bn even 6 1
2548.1.bn.a 2 1092.dd odd 6 1
3328.1.v.b 4 48.i odd 4 2
3328.1.v.b 4 48.k even 4 2
3328.1.v.b 4 624.cl even 12 2
3328.1.v.b 4 624.cw odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

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