Properties

Label 2496.1.cb.b
Level $2496$
Weight $1$
Character orbit 2496.cb
Analytic conductor $1.246$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.cb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.160398576.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} + ( 1 - \zeta_{6}^{2} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} + ( 1 - \zeta_{6}^{2} ) q^{7} -\zeta_{6} q^{9} -\zeta_{6} q^{13} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{21} - q^{25} - q^{27} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{31} - q^{39} + \zeta_{6} q^{43} + ( 1 - \zeta_{6} - \zeta_{6}^{2} ) q^{49} + \zeta_{6} q^{61} + ( -1 - \zeta_{6} ) q^{63} + ( 1 + \zeta_{6} ) q^{67} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{73} + \zeta_{6}^{2} q^{75} - q^{79} + \zeta_{6}^{2} q^{81} + ( -1 - \zeta_{6} ) q^{91} + ( 1 + \zeta_{6} ) q^{93} + ( -1 + \zeta_{6}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 3q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 3q^{7} - q^{9} - q^{13} - 2q^{25} - 2q^{27} - 2q^{39} + q^{43} + 2q^{49} + q^{61} - 3q^{63} + 3q^{67} - q^{75} - 2q^{79} - q^{81} - 3q^{91} + 3q^{93} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
641.1 0 0.500000 0.866025i 0 0 0 1.50000 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.cb.b 2
3.b odd 2 1 CM 2496.1.cb.b 2
4.b odd 2 1 2496.1.cb.a 2
8.b even 2 1 624.1.cb.a 2
8.d odd 2 1 156.1.s.a 2
12.b even 2 1 2496.1.cb.a 2
13.e even 6 1 inner 2496.1.cb.b 2
24.f even 2 1 156.1.s.a 2
24.h odd 2 1 624.1.cb.a 2
39.h odd 6 1 inner 2496.1.cb.b 2
40.e odd 2 1 3900.1.ca.b 2
40.k even 4 2 3900.1.br.b 4
52.i odd 6 1 2496.1.cb.a 2
104.h odd 2 1 2028.1.s.a 2
104.m even 4 2 2028.1.o.b 4
104.n odd 6 1 2028.1.g.a 2
104.n odd 6 1 2028.1.s.a 2
104.p odd 6 1 156.1.s.a 2
104.p odd 6 1 2028.1.g.a 2
104.s even 6 1 624.1.cb.a 2
104.u even 12 2 2028.1.d.c 2
104.u even 12 2 2028.1.o.b 4
120.m even 2 1 3900.1.ca.b 2
120.q odd 4 2 3900.1.br.b 4
156.r even 6 1 2496.1.cb.a 2
312.h even 2 1 2028.1.s.a 2
312.w odd 4 2 2028.1.o.b 4
312.ba even 6 1 156.1.s.a 2
312.ba even 6 1 2028.1.g.a 2
312.bg odd 6 1 624.1.cb.a 2
312.bn even 6 1 2028.1.g.a 2
312.bn even 6 1 2028.1.s.a 2
312.bq odd 12 2 2028.1.d.c 2
312.bq odd 12 2 2028.1.o.b 4
520.cd odd 6 1 3900.1.ca.b 2
520.cs even 12 2 3900.1.br.b 4
1560.dv even 6 1 3900.1.ca.b 2
1560.es odd 12 2 3900.1.br.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 8.d odd 2 1
156.1.s.a 2 24.f even 2 1
156.1.s.a 2 104.p odd 6 1
156.1.s.a 2 312.ba even 6 1
624.1.cb.a 2 8.b even 2 1
624.1.cb.a 2 24.h odd 2 1
624.1.cb.a 2 104.s even 6 1
624.1.cb.a 2 312.bg odd 6 1
2028.1.d.c 2 104.u even 12 2
2028.1.d.c 2 312.bq odd 12 2
2028.1.g.a 2 104.n odd 6 1
2028.1.g.a 2 104.p odd 6 1
2028.1.g.a 2 312.ba even 6 1
2028.1.g.a 2 312.bn even 6 1
2028.1.o.b 4 104.m even 4 2
2028.1.o.b 4 104.u even 12 2
2028.1.o.b 4 312.w odd 4 2
2028.1.o.b 4 312.bq odd 12 2
2028.1.s.a 2 104.h odd 2 1
2028.1.s.a 2 104.n odd 6 1
2028.1.s.a 2 312.h even 2 1
2028.1.s.a 2 312.bn even 6 1
2496.1.cb.a 2 4.b odd 2 1
2496.1.cb.a 2 12.b even 2 1
2496.1.cb.a 2 52.i odd 6 1
2496.1.cb.a 2 156.r even 6 1
2496.1.cb.b 2 1.a even 1 1 trivial
2496.1.cb.b 2 3.b odd 2 1 CM
2496.1.cb.b 2 13.e even 6 1 inner
2496.1.cb.b 2 39.h odd 6 1 inner
3900.1.br.b 4 40.k even 4 2
3900.1.br.b 4 120.q odd 4 2
3900.1.br.b 4 520.cs even 12 2
3900.1.br.b 4 1560.es odd 12 2
3900.1.ca.b 2 40.e odd 2 1
3900.1.ca.b 2 120.m even 2 1
3900.1.ca.b 2 520.cd odd 6 1
3900.1.ca.b 2 1560.dv even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 T_{7} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2496, [\chi])\).

Hecke characteristic polynomials

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