Properties

Label 3528.1.cw.a
Level $3528$
Weight $1$
Character orbit 3528.cw
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -56
Inner twists $4$

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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: \(2\)
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 504)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.4536.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.697019904.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} - q^{5} -\zeta_{6} q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} - q^{5} -\zeta_{6} q^{6} + q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{10} + q^{12} + 2 \zeta_{6}^{2} q^{13} -\zeta_{6}^{2} q^{15} + \zeta_{6}^{2} q^{16} + q^{18} + \zeta_{6} q^{19} + \zeta_{6} q^{20} - q^{23} + \zeta_{6}^{2} q^{24} -2 \zeta_{6} q^{26} + q^{27} + \zeta_{6} q^{30} -\zeta_{6} q^{32} + \zeta_{6}^{2} q^{36} - q^{38} -2 \zeta_{6} q^{39} - q^{40} + \zeta_{6} q^{45} -\zeta_{6}^{2} q^{46} -\zeta_{6} q^{48} + 2 q^{52} + \zeta_{6}^{2} q^{54} - q^{57} -2 \zeta_{6} q^{59} - q^{60} -\zeta_{6}^{2} q^{61} + q^{64} -2 \zeta_{6}^{2} q^{65} -\zeta_{6}^{2} q^{69} - q^{71} -\zeta_{6} q^{72} -\zeta_{6}^{2} q^{76} + 2 q^{78} -\zeta_{6}^{2} q^{79} -\zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} -2 \zeta_{6} q^{83} - q^{90} + \zeta_{6} q^{92} -\zeta_{6} q^{95} + q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{8} - q^{9} + q^{10} + 2q^{12} - 2q^{13} + q^{15} - q^{16} + 2q^{18} + q^{19} + q^{20} - 2q^{23} - q^{24} - 2q^{26} + 2q^{27} + q^{30} - q^{32} - q^{36} - 2q^{38} - 2q^{39} - 2q^{40} + q^{45} + q^{46} - q^{48} + 4q^{52} - q^{54} - 2q^{57} - 2q^{59} - 2q^{60} + q^{61} + 2q^{64} + 2q^{65} + q^{69} - 2q^{71} - q^{72} + q^{76} + 4q^{78} + q^{79} + q^{80} - q^{81} - 2q^{83} - 2q^{90} + q^{92} - q^{95} + 2q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(\zeta_{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2077.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
2677.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 0.866025i 0 1.00000 −0.500000 0.866025i 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
63.g even 3 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.a 2
7.b odd 2 1 3528.1.cw.b 2
7.c even 3 1 504.1.bn.b yes 2
7.c even 3 1 3528.1.bp.a 2
7.d odd 6 1 504.1.bn.a 2
7.d odd 6 1 3528.1.bp.b 2
8.b even 2 1 3528.1.cw.b 2
9.c even 3 1 3528.1.bp.a 2
21.g even 6 1 1512.1.bn.b 2
21.h odd 6 1 1512.1.bn.a 2
28.f even 6 1 2016.1.bv.b 2
28.g odd 6 1 2016.1.bv.a 2
56.h odd 2 1 CM 3528.1.cw.a 2
56.j odd 6 1 504.1.bn.b yes 2
56.j odd 6 1 3528.1.bp.a 2
56.k odd 6 1 2016.1.bv.b 2
56.m even 6 1 2016.1.bv.a 2
56.p even 6 1 504.1.bn.a 2
56.p even 6 1 3528.1.bp.b 2
63.g even 3 1 inner 3528.1.cw.a 2
63.h even 3 1 504.1.bn.b yes 2
63.i even 6 1 1512.1.bn.b 2
63.j odd 6 1 1512.1.bn.a 2
63.k odd 6 1 3528.1.cw.b 2
63.l odd 6 1 3528.1.bp.b 2
63.t odd 6 1 504.1.bn.a 2
72.n even 6 1 3528.1.bp.b 2
168.s odd 6 1 1512.1.bn.b 2
168.ba even 6 1 1512.1.bn.a 2
252.u odd 6 1 2016.1.bv.a 2
252.bj even 6 1 2016.1.bv.b 2
504.w even 6 1 3528.1.cw.b 2
504.bf even 6 1 2016.1.bv.a 2
504.bi odd 6 1 1512.1.bn.b 2
504.bn odd 6 1 3528.1.bp.a 2
504.bp odd 6 1 504.1.bn.b yes 2
504.ca even 6 1 1512.1.bn.a 2
504.ce odd 6 1 2016.1.bv.b 2
504.cq even 6 1 504.1.bn.a 2
504.cw odd 6 1 inner 3528.1.cw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bn.a 2 7.d odd 6 1
504.1.bn.a 2 56.p even 6 1
504.1.bn.a 2 63.t odd 6 1
504.1.bn.a 2 504.cq even 6 1
504.1.bn.b yes 2 7.c even 3 1
504.1.bn.b yes 2 56.j odd 6 1
504.1.bn.b yes 2 63.h even 3 1
504.1.bn.b yes 2 504.bp odd 6 1
1512.1.bn.a 2 21.h odd 6 1
1512.1.bn.a 2 63.j odd 6 1
1512.1.bn.a 2 168.ba even 6 1
1512.1.bn.a 2 504.ca even 6 1
1512.1.bn.b 2 21.g even 6 1
1512.1.bn.b 2 63.i even 6 1
1512.1.bn.b 2 168.s odd 6 1
1512.1.bn.b 2 504.bi odd 6 1
2016.1.bv.a 2 28.g odd 6 1
2016.1.bv.a 2 56.m even 6 1
2016.1.bv.a 2 252.u odd 6 1
2016.1.bv.a 2 504.bf even 6 1
2016.1.bv.b 2 28.f even 6 1
2016.1.bv.b 2 56.k odd 6 1
2016.1.bv.b 2 252.bj even 6 1
2016.1.bv.b 2 504.ce odd 6 1
3528.1.bp.a 2 7.c even 3 1
3528.1.bp.a 2 9.c even 3 1
3528.1.bp.a 2 56.j odd 6 1
3528.1.bp.a 2 504.bn odd 6 1
3528.1.bp.b 2 7.d odd 6 1
3528.1.bp.b 2 56.p even 6 1
3528.1.bp.b 2 63.l odd 6 1
3528.1.bp.b 2 72.n even 6 1
3528.1.cw.a 2 1.a even 1 1 trivial
3528.1.cw.a 2 56.h odd 2 1 CM
3528.1.cw.a 2 63.g even 3 1 inner
3528.1.cw.a 2 504.cw odd 6 1 inner
3528.1.cw.b 2 7.b odd 2 1
3528.1.cw.b 2 8.b even 2 1
3528.1.cw.b 2 63.k odd 6 1
3528.1.cw.b 2 504.w even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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