Properties

Label 3528.1.cw.c
Level $3528$
Weight $1$
Character orbit 3528.cw
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -56
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.cw (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.144027072.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{4} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{5} -\zeta_{12} q^{6} - q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{4} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{5} -\zeta_{12} q^{6} - q^{8} -\zeta_{12}^{4} q^{9} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{10} -\zeta_{12}^{3} q^{12} + ( 1 - \zeta_{12}^{4} ) q^{15} -\zeta_{12}^{2} q^{16} + q^{18} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{19} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{20} - q^{23} -\zeta_{12}^{5} q^{24} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{25} + \zeta_{12}^{3} q^{27} + ( 1 + \zeta_{12}^{2} ) q^{30} -\zeta_{12}^{4} q^{32} + \zeta_{12}^{2} q^{36} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{38} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{40} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{45} -\zeta_{12}^{2} q^{46} + \zeta_{12} q^{48} + ( 1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{50} + \zeta_{12}^{5} q^{54} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{57} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{60} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{61} + q^{64} -\zeta_{12}^{5} q^{69} + q^{71} + \zeta_{12}^{4} q^{72} + ( -\zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{75} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{76} -\zeta_{12}^{2} q^{79} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{90} -\zeta_{12}^{4} q^{92} + ( -1 + \zeta_{12}^{2} + 2 \zeta_{12}^{4} ) q^{95} + \zeta_{12}^{3} q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} + 2q^{9} + 6q^{15} - 2q^{16} + 4q^{18} - 4q^{23} + 8q^{25} + 6q^{30} + 2q^{32} + 2q^{36} - 2q^{46} + 4q^{50} + 4q^{64} + 4q^{71} - 2q^{72} - 2q^{79} - 2q^{81} + 2q^{92} - 6q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2077.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −1.73205 −0.866025 0.500000i 0 −1.00000 0.500000 0.866025i −0.866025 1.50000i
2077.2 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 1.73205 0.866025 + 0.500000i 0 −1.00000 0.500000 0.866025i 0.866025 + 1.50000i
2677.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i −1.73205 −0.866025 + 0.500000i 0 −1.00000 0.500000 + 0.866025i −0.866025 + 1.50000i
2677.2 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 1.73205 0.866025 0.500000i 0 −1.00000 0.500000 + 0.866025i 0.866025 1.50000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner
504.w even 6 1 inner
504.cw odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.cw.c 4
7.b odd 2 1 inner 3528.1.cw.c 4
7.c even 3 1 504.1.bn.c 4
7.c even 3 1 3528.1.bp.c 4
7.d odd 6 1 504.1.bn.c 4
7.d odd 6 1 3528.1.bp.c 4
8.b even 2 1 inner 3528.1.cw.c 4
9.c even 3 1 3528.1.bp.c 4
21.g even 6 1 1512.1.bn.c 4
21.h odd 6 1 1512.1.bn.c 4
28.f even 6 1 2016.1.bv.c 4
28.g odd 6 1 2016.1.bv.c 4
56.h odd 2 1 CM 3528.1.cw.c 4
56.j odd 6 1 504.1.bn.c 4
56.j odd 6 1 3528.1.bp.c 4
56.k odd 6 1 2016.1.bv.c 4
56.m even 6 1 2016.1.bv.c 4
56.p even 6 1 504.1.bn.c 4
56.p even 6 1 3528.1.bp.c 4
63.g even 3 1 inner 3528.1.cw.c 4
63.h even 3 1 504.1.bn.c 4
63.i even 6 1 1512.1.bn.c 4
63.j odd 6 1 1512.1.bn.c 4
63.k odd 6 1 inner 3528.1.cw.c 4
63.l odd 6 1 3528.1.bp.c 4
63.t odd 6 1 504.1.bn.c 4
72.n even 6 1 3528.1.bp.c 4
168.s odd 6 1 1512.1.bn.c 4
168.ba even 6 1 1512.1.bn.c 4
252.u odd 6 1 2016.1.bv.c 4
252.bj even 6 1 2016.1.bv.c 4
504.w even 6 1 inner 3528.1.cw.c 4
504.bf even 6 1 2016.1.bv.c 4
504.bi odd 6 1 1512.1.bn.c 4
504.bn odd 6 1 3528.1.bp.c 4
504.bp odd 6 1 504.1.bn.c 4
504.ca even 6 1 1512.1.bn.c 4
504.ce odd 6 1 2016.1.bv.c 4
504.cq even 6 1 504.1.bn.c 4
504.cw odd 6 1 inner 3528.1.cw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bn.c 4 7.c even 3 1
504.1.bn.c 4 7.d odd 6 1
504.1.bn.c 4 56.j odd 6 1
504.1.bn.c 4 56.p even 6 1
504.1.bn.c 4 63.h even 3 1
504.1.bn.c 4 63.t odd 6 1
504.1.bn.c 4 504.bp odd 6 1
504.1.bn.c 4 504.cq even 6 1
1512.1.bn.c 4 21.g even 6 1
1512.1.bn.c 4 21.h odd 6 1
1512.1.bn.c 4 63.i even 6 1
1512.1.bn.c 4 63.j odd 6 1
1512.1.bn.c 4 168.s odd 6 1
1512.1.bn.c 4 168.ba even 6 1
1512.1.bn.c 4 504.bi odd 6 1
1512.1.bn.c 4 504.ca even 6 1
2016.1.bv.c 4 28.f even 6 1
2016.1.bv.c 4 28.g odd 6 1
2016.1.bv.c 4 56.k odd 6 1
2016.1.bv.c 4 56.m even 6 1
2016.1.bv.c 4 252.u odd 6 1
2016.1.bv.c 4 252.bj even 6 1
2016.1.bv.c 4 504.bf even 6 1
2016.1.bv.c 4 504.ce odd 6 1
3528.1.bp.c 4 7.c even 3 1
3528.1.bp.c 4 7.d odd 6 1
3528.1.bp.c 4 9.c even 3 1
3528.1.bp.c 4 56.j odd 6 1
3528.1.bp.c 4 56.p even 6 1
3528.1.bp.c 4 63.l odd 6 1
3528.1.bp.c 4 72.n even 6 1
3528.1.bp.c 4 504.bn odd 6 1
3528.1.cw.c 4 1.a even 1 1 trivial
3528.1.cw.c 4 7.b odd 2 1 inner
3528.1.cw.c 4 8.b even 2 1 inner
3528.1.cw.c 4 56.h odd 2 1 CM
3528.1.cw.c 4 63.g even 3 1 inner
3528.1.cw.c 4 63.k odd 6 1 inner
3528.1.cw.c 4 504.w even 6 1 inner
3528.1.cw.c 4 504.cw odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

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