Properties

Label 3528.1.bw.a
Level $3528$
Weight $1$
Character orbit 3528.bw
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -7, -56, 8
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: \(2\)
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 56)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image $C_3\times D_4$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{8} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{8} + \zeta_{6}^{2} q^{16} -2 \zeta_{6}^{2} q^{23} + \zeta_{6} q^{25} -\zeta_{6} q^{32} + 2 \zeta_{6} q^{46} - q^{50} + q^{64} + 2 q^{71} -2 \zeta_{6}^{2} q^{79} -2 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{8} - q^{16} + 2q^{23} + q^{25} - q^{32} + 2q^{46} - 2q^{50} + 2q^{64} + 4q^{71} + 2q^{79} - 4q^{92} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 0 0
901.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
56.j odd 6 1 inner
56.p 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.a 2
3.b odd 2 1 392.1.j.a 2
7.b odd 2 1 CM 3528.1.bw.a 2
7.c even 3 1 504.1.l.a 1
7.c even 3 1 inner 3528.1.bw.a 2
7.d odd 6 1 504.1.l.a 1
7.d odd 6 1 inner 3528.1.bw.a 2
8.b even 2 1 RM 3528.1.bw.a 2
12.b even 2 1 1568.1.n.a 2
21.c even 2 1 392.1.j.a 2
21.g even 6 1 56.1.h.a 1
21.g even 6 1 392.1.j.a 2
21.h odd 6 1 56.1.h.a 1
21.h odd 6 1 392.1.j.a 2
24.f even 2 1 1568.1.n.a 2
24.h odd 2 1 392.1.j.a 2
28.f even 6 1 2016.1.l.a 1
28.g odd 6 1 2016.1.l.a 1
56.h odd 2 1 CM 3528.1.bw.a 2
56.j odd 6 1 504.1.l.a 1
56.j odd 6 1 inner 3528.1.bw.a 2
56.k odd 6 1 2016.1.l.a 1
56.m even 6 1 2016.1.l.a 1
56.p even 6 1 504.1.l.a 1
56.p even 6 1 inner 3528.1.bw.a 2
84.h odd 2 1 1568.1.n.a 2
84.j odd 6 1 224.1.h.a 1
84.j odd 6 1 1568.1.n.a 2
84.n even 6 1 224.1.h.a 1
84.n even 6 1 1568.1.n.a 2
105.o odd 6 1 1400.1.m.a 1
105.p even 6 1 1400.1.m.a 1
105.w odd 12 2 1400.1.c.a 2
105.x even 12 2 1400.1.c.a 2
168.e odd 2 1 1568.1.n.a 2
168.i even 2 1 392.1.j.a 2
168.s odd 6 1 56.1.h.a 1
168.s odd 6 1 392.1.j.a 2
168.v even 6 1 224.1.h.a 1
168.v even 6 1 1568.1.n.a 2
168.ba even 6 1 56.1.h.a 1
168.ba even 6 1 392.1.j.a 2
168.be odd 6 1 224.1.h.a 1
168.be odd 6 1 1568.1.n.a 2
336.bo even 12 2 1792.1.c.b 1
336.br odd 12 2 1792.1.c.a 1
336.bt odd 12 2 1792.1.c.b 1
336.bu even 12 2 1792.1.c.a 1
840.cb even 6 1 1400.1.m.a 1
840.cg odd 6 1 1400.1.m.a 1
840.dc even 12 2 1400.1.c.a 2
840.dh odd 12 2 1400.1.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 21.g even 6 1
56.1.h.a 1 21.h odd 6 1
56.1.h.a 1 168.s odd 6 1
56.1.h.a 1 168.ba even 6 1
224.1.h.a 1 84.j odd 6 1
224.1.h.a 1 84.n even 6 1
224.1.h.a 1 168.v even 6 1
224.1.h.a 1 168.be odd 6 1
392.1.j.a 2 3.b odd 2 1
392.1.j.a 2 21.c even 2 1
392.1.j.a 2 21.g even 6 1
392.1.j.a 2 21.h odd 6 1
392.1.j.a 2 24.h odd 2 1
392.1.j.a 2 168.i even 2 1
392.1.j.a 2 168.s odd 6 1
392.1.j.a 2 168.ba even 6 1
504.1.l.a 1 7.c even 3 1
504.1.l.a 1 7.d odd 6 1
504.1.l.a 1 56.j odd 6 1
504.1.l.a 1 56.p even 6 1
1400.1.c.a 2 105.w odd 12 2
1400.1.c.a 2 105.x even 12 2
1400.1.c.a 2 840.dc even 12 2
1400.1.c.a 2 840.dh odd 12 2
1400.1.m.a 1 105.o odd 6 1
1400.1.m.a 1 105.p even 6 1
1400.1.m.a 1 840.cb even 6 1
1400.1.m.a 1 840.cg odd 6 1
1568.1.n.a 2 12.b even 2 1
1568.1.n.a 2 24.f even 2 1
1568.1.n.a 2 84.h odd 2 1
1568.1.n.a 2 84.j odd 6 1
1568.1.n.a 2 84.n even 6 1
1568.1.n.a 2 168.e odd 2 1
1568.1.n.a 2 168.v even 6 1
1568.1.n.a 2 168.be odd 6 1
1792.1.c.a 1 336.br odd 12 2
1792.1.c.a 1 336.bu even 12 2
1792.1.c.b 1 336.bo even 12 2
1792.1.c.b 1 336.bt odd 12 2
2016.1.l.a 1 28.f even 6 1
2016.1.l.a 1 28.g odd 6 1
2016.1.l.a 1 56.k odd 6 1
2016.1.l.a 1 56.m even 6 1
3528.1.bw.a 2 1.a even 1 1 trivial
3528.1.bw.a 2 7.b odd 2 1 CM
3528.1.bw.a 2 7.c even 3 1 inner
3528.1.bw.a 2 7.d odd 6 1 inner
3528.1.bw.a 2 8.b even 2 1 RM
3528.1.bw.a 2 56.h odd 2 1 CM
3528.1.bw.a 2 56.j odd 6 1 inner
3528.1.bw.a 2 56.p 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 \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5} \)
\( T_{11} \)

Hecke characteristic polynomials

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