Properties

Label 3528.1.ba.e
Level $3528$
Weight $1$
Character orbit 3528.ba
Analytic conductor $1.761$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{12}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{8} q^{2} + \zeta_{24}^{9} q^{3} -\zeta_{24}^{4} q^{4} + \zeta_{24}^{5} q^{6} - q^{8} -\zeta_{24}^{6} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{8} q^{2} + \zeta_{24}^{9} q^{3} -\zeta_{24}^{4} q^{4} + \zeta_{24}^{5} q^{6} - q^{8} -\zeta_{24}^{6} q^{9} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{11} + \zeta_{24} q^{12} + \zeta_{24}^{8} q^{16} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{17} -\zeta_{24}^{2} q^{18} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{19} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{22} -\zeta_{24}^{9} q^{24} + q^{25} + \zeta_{24}^{3} q^{27} + \zeta_{24}^{4} q^{32} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{33} + ( \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{34} + \zeta_{24}^{10} q^{36} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{38} + ( \zeta_{24} - \zeta_{24}^{3} ) q^{41} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{43} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{44} -\zeta_{24}^{5} q^{48} -\zeta_{24}^{8} q^{50} + ( -1 - \zeta_{24}^{10} ) q^{51} -\zeta_{24}^{11} q^{54} + ( -1 - \zeta_{24}^{2} ) q^{57} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{59} + q^{64} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{66} -\zeta_{24}^{4} q^{67} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{68} + \zeta_{24}^{6} q^{72} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{73} + \zeta_{24}^{9} q^{75} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{76} - q^{81} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{82} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{83} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{86} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{88} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{89} -\zeta_{24} q^{96} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{97} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 4q^{4} - 8q^{8} + O(q^{10}) \) \( 8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{16} + 8q^{25} + 4q^{32} + 4q^{50} - 8q^{51} - 8q^{57} + 8q^{64} - 4q^{67} - 8q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(-\zeta_{24}^{4}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.500000 0.866025i −0.707107 0.707107i −0.500000 0.866025i 0 −0.965926 + 0.258819i 0 −1.00000 1.00000i 0
67.2 0.500000 0.866025i −0.707107 + 0.707107i −0.500000 0.866025i 0 0.258819 + 0.965926i 0 −1.00000 1.00000i 0
67.3 0.500000 0.866025i 0.707107 0.707107i −0.500000 0.866025i 0 −0.258819 0.965926i 0 −1.00000 1.00000i 0
67.4 0.500000 0.866025i 0.707107 + 0.707107i −0.500000 0.866025i 0 0.965926 0.258819i 0 −1.00000 1.00000i 0
1843.1 0.500000 + 0.866025i −0.707107 0.707107i −0.500000 + 0.866025i 0 0.258819 0.965926i 0 −1.00000 1.00000i 0
1843.2 0.500000 + 0.866025i −0.707107 + 0.707107i −0.500000 + 0.866025i 0 −0.965926 0.258819i 0 −1.00000 1.00000i 0
1843.3 0.500000 + 0.866025i 0.707107 0.707107i −0.500000 + 0.866025i 0 0.965926 + 0.258819i 0 −1.00000 1.00000i 0
1843.4 0.500000 + 0.866025i 0.707107 + 0.707107i −0.500000 + 0.866025i 0 −0.258819 + 0.965926i 0 −1.00000 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1843.4
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.e 8
7.b odd 2 1 inner 3528.1.ba.e 8
7.c even 3 1 3528.1.ce.e 8
7.c even 3 1 3528.1.cg.e 8
7.d odd 6 1 3528.1.ce.e 8
7.d odd 6 1 3528.1.cg.e 8
8.d odd 2 1 CM 3528.1.ba.e 8
9.c even 3 1 3528.1.ce.e 8
56.e even 2 1 inner 3528.1.ba.e 8
56.k odd 6 1 3528.1.ce.e 8
56.k odd 6 1 3528.1.cg.e 8
56.m even 6 1 3528.1.ce.e 8
56.m even 6 1 3528.1.cg.e 8
63.g even 3 1 inner 3528.1.ba.e 8
63.h even 3 1 3528.1.cg.e 8
63.k odd 6 1 inner 3528.1.ba.e 8
63.l odd 6 1 3528.1.ce.e 8
63.t odd 6 1 3528.1.cg.e 8
72.p odd 6 1 3528.1.ce.e 8
504.ba odd 6 1 inner 3528.1.ba.e 8
504.be even 6 1 3528.1.ce.e 8
504.bf even 6 1 3528.1.cg.e 8
504.ce odd 6 1 3528.1.cg.e 8
504.cz even 6 1 inner 3528.1.ba.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.ba.e 8 1.a even 1 1 trivial
3528.1.ba.e 8 7.b odd 2 1 inner
3528.1.ba.e 8 8.d odd 2 1 CM
3528.1.ba.e 8 56.e even 2 1 inner
3528.1.ba.e 8 63.g even 3 1 inner
3528.1.ba.e 8 63.k odd 6 1 inner
3528.1.ba.e 8 504.ba odd 6 1 inner
3528.1.ba.e 8 504.cz even 6 1 inner
3528.1.ce.e 8 7.c even 3 1
3528.1.ce.e 8 7.d odd 6 1
3528.1.ce.e 8 9.c even 3 1
3528.1.ce.e 8 56.k odd 6 1
3528.1.ce.e 8 56.m even 6 1
3528.1.ce.e 8 63.l odd 6 1
3528.1.ce.e 8 72.p odd 6 1
3528.1.ce.e 8 504.be even 6 1
3528.1.cg.e 8 7.c even 3 1
3528.1.cg.e 8 7.d odd 6 1
3528.1.cg.e 8 56.k odd 6 1
3528.1.cg.e 8 56.m even 6 1
3528.1.cg.e 8 63.h even 3 1
3528.1.cg.e 8 63.t odd 6 1
3528.1.cg.e 8 504.bf even 6 1
3528.1.cg.e 8 504.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 \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - 3 \)
\( T_{17}^{8} + 4 T_{17}^{6} + 15 T_{17}^{4} + 4 T_{17}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( -3 + T^{2} )^{4} \)
$13$ \( T^{8} \)
$17$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$19$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$43$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 1 + T + T^{2} )^{4} \)
$71$ \( T^{8} \)
$73$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$89$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$97$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
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