Properties

Label 3528.1.bd.a
Level $3528$
Weight $1$
Character orbit 3528.bd
Analytic conductor $1.761$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -7
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.bd (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.84672.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{9} q^{8} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{11} + \zeta_{24}^{4} q^{16} + ( 1 + \zeta_{24}^{6} ) q^{22} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{8} q^{25} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{29} -\zeta_{24}^{11} q^{32} -2 \zeta_{24}^{10} q^{37} -2 \zeta_{24}^{6} q^{43} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{44} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{46} -\zeta_{24}^{3} q^{50} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{53} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{58} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{71} -2 \zeta_{24}^{5} q^{74} + 2 \zeta_{24}^{4} q^{79} -2 \zeta_{24} q^{86} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{88} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{16} + 8q^{22} + 4q^{25} + 4q^{46} - 4q^{58} + 8q^{79} + 4q^{88} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
557.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
557.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
557.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
3509.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0 0
3509.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0 0
3509.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0 0
3509.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0 0.707107 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3509.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
168.i even 2 1 inner
168.s odd 6 1 inner
168.ba 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.bd.a 8
3.b odd 2 1 inner 3528.1.bd.a 8
7.b odd 2 1 CM 3528.1.bd.a 8
7.c even 3 1 3528.1.n.a 4
7.c even 3 1 inner 3528.1.bd.a 8
7.d odd 6 1 3528.1.n.a 4
7.d odd 6 1 inner 3528.1.bd.a 8
8.b even 2 1 inner 3528.1.bd.a 8
21.c even 2 1 inner 3528.1.bd.a 8
21.g even 6 1 3528.1.n.a 4
21.g even 6 1 inner 3528.1.bd.a 8
21.h odd 6 1 3528.1.n.a 4
21.h odd 6 1 inner 3528.1.bd.a 8
24.h odd 2 1 inner 3528.1.bd.a 8
56.h odd 2 1 inner 3528.1.bd.a 8
56.j odd 6 1 3528.1.n.a 4
56.j odd 6 1 inner 3528.1.bd.a 8
56.p even 6 1 3528.1.n.a 4
56.p even 6 1 inner 3528.1.bd.a 8
168.i even 2 1 inner 3528.1.bd.a 8
168.s odd 6 1 3528.1.n.a 4
168.s odd 6 1 inner 3528.1.bd.a 8
168.ba even 6 1 3528.1.n.a 4
168.ba even 6 1 inner 3528.1.bd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.n.a 4 7.c even 3 1
3528.1.n.a 4 7.d odd 6 1
3528.1.n.a 4 21.g even 6 1
3528.1.n.a 4 21.h odd 6 1
3528.1.n.a 4 56.j odd 6 1
3528.1.n.a 4 56.p even 6 1
3528.1.n.a 4 168.s odd 6 1
3528.1.n.a 4 168.ba even 6 1
3528.1.bd.a 8 1.a even 1 1 trivial
3528.1.bd.a 8 3.b odd 2 1 inner
3528.1.bd.a 8 7.b odd 2 1 CM
3528.1.bd.a 8 7.c even 3 1 inner
3528.1.bd.a 8 7.d odd 6 1 inner
3528.1.bd.a 8 8.b even 2 1 inner
3528.1.bd.a 8 21.c even 2 1 inner
3528.1.bd.a 8 21.g even 6 1 inner
3528.1.bd.a 8 21.h odd 6 1 inner
3528.1.bd.a 8 24.h odd 2 1 inner
3528.1.bd.a 8 56.h odd 2 1 inner
3528.1.bd.a 8 56.j odd 6 1 inner
3528.1.bd.a 8 56.p even 6 1 inner
3528.1.bd.a 8 168.i even 2 1 inner
3528.1.bd.a 8 168.s odd 6 1 inner
3528.1.bd.a 8 168.ba even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$29$ \( ( -2 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( 4 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 2 + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( 4 - 2 T + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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