Properties

Label 3528.1.bp.b
Level $3528$
Weight $1$
Character orbit 3528.bp
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.bp (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: \(\Z[a_1, a_2, a_3]\)
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

$q$-expansion

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1501.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
3253.1 1.00000 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 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.h even 3 1 inner
504.bp 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.bp.b 2
7.b odd 2 1 3528.1.bp.a 2
7.c even 3 1 504.1.bn.a 2
7.c even 3 1 3528.1.cw.b 2
7.d odd 6 1 504.1.bn.b yes 2
7.d odd 6 1 3528.1.cw.a 2
8.b even 2 1 3528.1.bp.a 2
9.c even 3 1 3528.1.cw.b 2
21.g even 6 1 1512.1.bn.a 2
21.h odd 6 1 1512.1.bn.b 2
28.f even 6 1 2016.1.bv.a 2
28.g odd 6 1 2016.1.bv.b 2
56.h odd 2 1 CM 3528.1.bp.b 2
56.j odd 6 1 504.1.bn.a 2
56.j odd 6 1 3528.1.cw.b 2
56.k odd 6 1 2016.1.bv.a 2
56.m even 6 1 2016.1.bv.b 2
56.p even 6 1 504.1.bn.b yes 2
56.p even 6 1 3528.1.cw.a 2
63.g even 3 1 504.1.bn.a 2
63.h even 3 1 inner 3528.1.bp.b 2
63.k odd 6 1 504.1.bn.b yes 2
63.l odd 6 1 3528.1.cw.a 2
63.n odd 6 1 1512.1.bn.b 2
63.s even 6 1 1512.1.bn.a 2
63.t odd 6 1 3528.1.bp.a 2
72.n even 6 1 3528.1.cw.a 2
168.s odd 6 1 1512.1.bn.a 2
168.ba even 6 1 1512.1.bn.b 2
252.n even 6 1 2016.1.bv.a 2
252.bl odd 6 1 2016.1.bv.b 2
504.w even 6 1 504.1.bn.b yes 2
504.y even 6 1 1512.1.bn.b 2
504.ba odd 6 1 2016.1.bv.a 2
504.bn odd 6 1 3528.1.cw.b 2
504.bp odd 6 1 inner 3528.1.bp.b 2
504.cq even 6 1 3528.1.bp.a 2
504.cw odd 6 1 504.1.bn.a 2
504.cz even 6 1 2016.1.bv.b 2
504.db odd 6 1 1512.1.bn.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bn.a 2 7.c even 3 1
504.1.bn.a 2 56.j odd 6 1
504.1.bn.a 2 63.g even 3 1
504.1.bn.a 2 504.cw odd 6 1
504.1.bn.b yes 2 7.d odd 6 1
504.1.bn.b yes 2 56.p even 6 1
504.1.bn.b yes 2 63.k odd 6 1
504.1.bn.b yes 2 504.w even 6 1
1512.1.bn.a 2 21.g even 6 1
1512.1.bn.a 2 63.s even 6 1
1512.1.bn.a 2 168.s odd 6 1
1512.1.bn.a 2 504.db odd 6 1
1512.1.bn.b 2 21.h odd 6 1
1512.1.bn.b 2 63.n odd 6 1
1512.1.bn.b 2 168.ba even 6 1
1512.1.bn.b 2 504.y even 6 1
2016.1.bv.a 2 28.f even 6 1
2016.1.bv.a 2 56.k odd 6 1
2016.1.bv.a 2 252.n even 6 1
2016.1.bv.a 2 504.ba odd 6 1
2016.1.bv.b 2 28.g odd 6 1
2016.1.bv.b 2 56.m even 6 1
2016.1.bv.b 2 252.bl odd 6 1
2016.1.bv.b 2 504.cz even 6 1
3528.1.bp.a 2 7.b odd 2 1
3528.1.bp.a 2 8.b even 2 1
3528.1.bp.a 2 63.t odd 6 1
3528.1.bp.a 2 504.cq even 6 1
3528.1.bp.b 2 1.a even 1 1 trivial
3528.1.bp.b 2 56.h odd 2 1 CM
3528.1.bp.b 2 63.h even 3 1 inner
3528.1.bp.b 2 504.bp odd 6 1 inner
3528.1.cw.a 2 7.d odd 6 1
3528.1.cw.a 2 56.p even 6 1
3528.1.cw.a 2 63.l odd 6 1
3528.1.cw.a 2 72.n even 6 1
3528.1.cw.b 2 7.c even 3 1
3528.1.cw.b 2 9.c even 3 1
3528.1.cw.b 2 56.j odd 6 1
3528.1.cw.b 2 504.bn odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

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