Properties

Label 3528.1.bw.c
Level $3528$
Weight $1$
Character orbit 3528.bw
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.bw (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.9680832.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{5} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{5} + \zeta_{12}^{3} q^{8} + ( 1 + \zeta_{12}^{2} ) q^{10} -\zeta_{12} q^{11} -\zeta_{12}^{2} q^{16} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{20} + q^{22} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{25} -\zeta_{12}^{3} q^{29} + ( -1 - \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} + ( 1 - \zeta_{12}^{4} ) q^{40} + \zeta_{12}^{5} q^{44} + ( -\zeta_{12} - \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{50} + \zeta_{12} q^{53} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{55} + \zeta_{12}^{2} q^{58} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{59} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{62} - q^{64} -\zeta_{12}^{2} q^{79} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{80} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{83} -\zeta_{12}^{4} q^{88} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 6q^{10} - 2q^{16} + 4q^{22} - 4q^{25} - 6q^{31} + 6q^{40} + 2q^{58} - 4q^{64} - 2q^{79} + 2q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.866025 1.50000i 0 0 1.00000i 0 1.50000 + 0.866025i
325.2 0.866025 0.500000i 0 0.500000 0.866025i 0.866025 + 1.50000i 0 0 1.00000i 0 1.50000 + 0.866025i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.866025 + 1.50000i 0 0 1.00000i 0 1.50000 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 1.50000i 0 0 1.00000i 0 1.50000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.bw.c 4
3.b odd 2 1 inner 3528.1.bw.c 4
7.b odd 2 1 504.1.bw.a 4
7.c even 3 1 504.1.bw.a 4
7.c even 3 1 3528.1.l.a 4
7.d odd 6 1 3528.1.l.a 4
7.d odd 6 1 inner 3528.1.bw.c 4
8.b even 2 1 inner 3528.1.bw.c 4
21.c even 2 1 504.1.bw.a 4
21.g even 6 1 3528.1.l.a 4
21.g even 6 1 inner 3528.1.bw.c 4
21.h odd 6 1 504.1.bw.a 4
21.h odd 6 1 3528.1.l.a 4
24.h odd 2 1 CM 3528.1.bw.c 4
28.d even 2 1 2016.1.ce.a 4
28.g odd 6 1 2016.1.ce.a 4
56.e even 2 1 2016.1.ce.a 4
56.h odd 2 1 504.1.bw.a 4
56.j odd 6 1 3528.1.l.a 4
56.j odd 6 1 inner 3528.1.bw.c 4
56.k odd 6 1 2016.1.ce.a 4
56.p even 6 1 504.1.bw.a 4
56.p even 6 1 3528.1.l.a 4
84.h odd 2 1 2016.1.ce.a 4
84.n even 6 1 2016.1.ce.a 4
168.e odd 2 1 2016.1.ce.a 4
168.i even 2 1 504.1.bw.a 4
168.s odd 6 1 504.1.bw.a 4
168.s odd 6 1 3528.1.l.a 4
168.v even 6 1 2016.1.ce.a 4
168.ba even 6 1 3528.1.l.a 4
168.ba even 6 1 inner 3528.1.bw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bw.a 4 7.b odd 2 1
504.1.bw.a 4 7.c even 3 1
504.1.bw.a 4 21.c even 2 1
504.1.bw.a 4 21.h odd 6 1
504.1.bw.a 4 56.h odd 2 1
504.1.bw.a 4 56.p even 6 1
504.1.bw.a 4 168.i even 2 1
504.1.bw.a 4 168.s odd 6 1
2016.1.ce.a 4 28.d even 2 1
2016.1.ce.a 4 28.g odd 6 1
2016.1.ce.a 4 56.e even 2 1
2016.1.ce.a 4 56.k odd 6 1
2016.1.ce.a 4 84.h odd 2 1
2016.1.ce.a 4 84.n even 6 1
2016.1.ce.a 4 168.e odd 2 1
2016.1.ce.a 4 168.v even 6 1
3528.1.l.a 4 7.c even 3 1
3528.1.l.a 4 7.d odd 6 1
3528.1.l.a 4 21.g even 6 1
3528.1.l.a 4 21.h odd 6 1
3528.1.l.a 4 56.j odd 6 1
3528.1.l.a 4 56.p even 6 1
3528.1.l.a 4 168.s odd 6 1
3528.1.l.a 4 168.ba even 6 1
3528.1.bw.c 4 1.a even 1 1 trivial
3528.1.bw.c 4 3.b odd 2 1 inner
3528.1.bw.c 4 7.d odd 6 1 inner
3528.1.bw.c 4 8.b even 2 1 inner
3528.1.bw.c 4 21.g even 6 1 inner
3528.1.bw.c 4 24.h odd 2 1 CM
3528.1.bw.c 4 56.j odd 6 1 inner
3528.1.bw.c 4 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{4} + 3 T_{5}^{2} + 9 \)
\( T_{11}^{4} - T_{11}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 9 + 3 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( ( 3 + 3 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1 - T^{2} + T^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 1 + T + T^{2} )^{2} \)
$83$ \( ( -3 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 3 + T^{2} )^{2} \)
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