Properties

Label 3528.1.ba.c
Level $3528$
Weight $1$
Character orbit 3528.ba
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -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.ba (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: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.1152216576.12

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{2} -\zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12} q^{6} + q^{8} - q^{9} +O(q^{10})\) \( q + \zeta_{12}^{4} q^{2} -\zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12} q^{6} + q^{8} - q^{9} + q^{11} + \zeta_{12}^{5} q^{12} + \zeta_{12}^{4} q^{16} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{17} -\zeta_{12}^{4} q^{18} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{19} + \zeta_{12}^{4} q^{22} -\zeta_{12}^{3} q^{24} + q^{25} + \zeta_{12}^{3} q^{27} -\zeta_{12}^{2} q^{32} -\zeta_{12}^{3} q^{33} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{34} + \zeta_{12}^{2} q^{36} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{38} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{41} -\zeta_{12}^{2} q^{43} -\zeta_{12}^{2} q^{44} + \zeta_{12} q^{48} + \zeta_{12}^{4} q^{50} + ( 1 + \zeta_{12}^{2} ) q^{51} -\zeta_{12} q^{54} + ( 1 - \zeta_{12}^{4} ) q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + q^{64} + \zeta_{12} q^{66} + \zeta_{12}^{2} q^{67} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{68} - q^{72} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{73} -\zeta_{12}^{3} q^{75} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{76} + q^{81} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{82} + q^{86} + q^{88} + \zeta_{12}^{5} q^{96} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 4q^{8} - 4q^{9} + 4q^{11} - 2q^{16} + 2q^{18} - 2q^{22} + 4q^{25} - 2q^{32} + 2q^{36} - 2q^{43} - 2q^{44} - 2q^{50} + 6q^{51} + 6q^{57} + 4q^{64} + 2q^{67} - 4q^{72} + 4q^{81} + 4q^{86} + 4q^{88} - 4q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i 1.00000i −0.500000 0.866025i 0 0.866025 + 0.500000i 0 1.00000 −1.00000 0
67.2 −0.500000 + 0.866025i 1.00000i −0.500000 0.866025i 0 −0.866025 0.500000i 0 1.00000 −1.00000 0
1843.1 −0.500000 0.866025i 1.00000i −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 1.00000 −1.00000 0
1843.2 −0.500000 0.866025i 1.00000i −0.500000 + 0.866025i 0 0.866025 0.500000i 0 1.00000 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
56.e even 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner
504.ba odd 6 1 inner
504.cz 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.ba.c 4
7.b odd 2 1 inner 3528.1.ba.c 4
7.c even 3 1 3528.1.ce.d 4
7.c even 3 1 3528.1.cg.b 4
7.d odd 6 1 3528.1.ce.d 4
7.d odd 6 1 3528.1.cg.b 4
8.d odd 2 1 CM 3528.1.ba.c 4
9.c even 3 1 3528.1.ce.d 4
56.e even 2 1 inner 3528.1.ba.c 4
56.k odd 6 1 3528.1.ce.d 4
56.k odd 6 1 3528.1.cg.b 4
56.m even 6 1 3528.1.ce.d 4
56.m even 6 1 3528.1.cg.b 4
63.g even 3 1 inner 3528.1.ba.c 4
63.h even 3 1 3528.1.cg.b 4
63.k odd 6 1 inner 3528.1.ba.c 4
63.l odd 6 1 3528.1.ce.d 4
63.t odd 6 1 3528.1.cg.b 4
72.p odd 6 1 3528.1.ce.d 4
504.ba odd 6 1 inner 3528.1.ba.c 4
504.be even 6 1 3528.1.ce.d 4
504.bf even 6 1 3528.1.cg.b 4
504.ce odd 6 1 3528.1.cg.b 4
504.cz even 6 1 inner 3528.1.ba.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.ba.c 4 1.a even 1 1 trivial
3528.1.ba.c 4 7.b odd 2 1 inner
3528.1.ba.c 4 8.d odd 2 1 CM
3528.1.ba.c 4 56.e even 2 1 inner
3528.1.ba.c 4 63.g even 3 1 inner
3528.1.ba.c 4 63.k odd 6 1 inner
3528.1.ba.c 4 504.ba odd 6 1 inner
3528.1.ba.c 4 504.cz even 6 1 inner
3528.1.ce.d 4 7.c even 3 1
3528.1.ce.d 4 7.d odd 6 1
3528.1.ce.d 4 9.c even 3 1
3528.1.ce.d 4 56.k odd 6 1
3528.1.ce.d 4 56.m even 6 1
3528.1.ce.d 4 63.l odd 6 1
3528.1.ce.d 4 72.p odd 6 1
3528.1.ce.d 4 504.be even 6 1
3528.1.cg.b 4 7.c even 3 1
3528.1.cg.b 4 7.d odd 6 1
3528.1.cg.b 4 56.k odd 6 1
3528.1.cg.b 4 56.m even 6 1
3528.1.cg.b 4 63.h even 3 1
3528.1.cg.b 4 63.t odd 6 1
3528.1.cg.b 4 504.bf even 6 1
3528.1.cg.b 4 504.ce odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11} - 1 \)
\( T_{17}^{4} + 3 T_{17}^{2} + 9 \)

Hecke characteristic polynomials

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