Properties

Label 3528.1.bi.a
Level $3528$
Weight $1$
Character orbit 3528.bi
Analytic conductor $1.761$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -56
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.bi (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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{6} q^{2} -\zeta_{24}^{7} q^{3} - q^{4} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} -\zeta_{24} q^{6} + \zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{6} q^{2} -\zeta_{24}^{7} q^{3} - q^{4} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} -\zeta_{24} q^{6} + \zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{9} + ( \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{10} + \zeta_{24}^{7} q^{12} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{13} + ( -1 + \zeta_{24}^{10} ) q^{15} + q^{16} + \zeta_{24}^{8} q^{18} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{19} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{20} + ( 1 - \zeta_{24}^{8} ) q^{23} + \zeta_{24} q^{24} + ( \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{25} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{26} + \zeta_{24}^{9} q^{27} + ( \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{30} -\zeta_{24}^{6} q^{32} + \zeta_{24}^{2} q^{36} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{38} + ( -1 + \zeta_{24}^{6} ) q^{39} + ( -\zeta_{24}^{9} - \zeta_{24}^{11} ) q^{40} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{45} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{46} -\zeta_{24}^{7} q^{48} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{50} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{52} + \zeta_{24}^{3} q^{54} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} ) q^{57} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{59} + ( 1 - \zeta_{24}^{10} ) q^{60} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{61} - q^{64} + ( \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{65} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{69} -\zeta_{24}^{6} q^{71} -\zeta_{24}^{8} q^{72} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{75} + ( \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{76} + ( 1 + \zeta_{24}^{6} ) q^{78} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{79} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{80} + \zeta_{24}^{4} q^{81} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{83} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{90} + ( -1 + \zeta_{24}^{8} ) q^{92} + ( \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{95} -\zeta_{24} q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} - 8q^{15} + 8q^{16} - 4q^{18} + 12q^{23} - 4q^{25} + 4q^{30} - 8q^{39} + 12q^{50} + 4q^{57} + 8q^{60} - 8q^{64} + 4q^{72} + 8q^{78} + 4q^{81} - 12q^{92} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1157.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
1.00000i −0.258819 + 0.965926i −1.00000 0.965926 + 1.67303i 0.965926 + 0.258819i 0 1.00000i −0.866025 0.500000i 1.67303 0.965926i
1157.2 1.00000i 0.258819 0.965926i −1.00000 −0.965926 1.67303i −0.965926 0.258819i 0 1.00000i −0.866025 0.500000i −1.67303 + 0.965926i
1157.3 1.00000i −0.965926 0.258819i −1.00000 0.258819 + 0.448288i 0.258819 0.965926i 0 1.00000i 0.866025 + 0.500000i −0.448288 + 0.258819i
1157.4 1.00000i 0.965926 + 0.258819i −1.00000 −0.258819 0.448288i −0.258819 + 0.965926i 0 1.00000i 0.866025 + 0.500000i 0.448288 0.258819i
2909.1 1.00000i −0.965926 + 0.258819i −1.00000 0.258819 0.448288i 0.258819 + 0.965926i 0 1.00000i 0.866025 0.500000i −0.448288 0.258819i
2909.2 1.00000i 0.965926 0.258819i −1.00000 −0.258819 + 0.448288i −0.258819 0.965926i 0 1.00000i 0.866025 0.500000i 0.448288 + 0.258819i
2909.3 1.00000i −0.258819 0.965926i −1.00000 0.965926 1.67303i 0.965926 0.258819i 0 1.00000i −0.866025 + 0.500000i 1.67303 + 0.965926i
2909.4 1.00000i 0.258819 + 0.965926i −1.00000 −0.965926 + 1.67303i −0.965926 + 0.258819i 0 1.00000i −0.866025 + 0.500000i −1.67303 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2909.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner
504.bi odd 6 1 inner
504.ca 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.bi.a 8
7.b odd 2 1 inner 3528.1.bi.a 8
7.c even 3 1 3528.1.bg.a 8
7.c even 3 1 3528.1.db.a 8
7.d odd 6 1 3528.1.bg.a 8
7.d odd 6 1 3528.1.db.a 8
8.b even 2 1 inner 3528.1.bi.a 8
9.d odd 6 1 3528.1.db.a 8
56.h odd 2 1 CM 3528.1.bi.a 8
56.j odd 6 1 3528.1.bg.a 8
56.j odd 6 1 3528.1.db.a 8
56.p even 6 1 3528.1.bg.a 8
56.p even 6 1 3528.1.db.a 8
63.i even 6 1 inner 3528.1.bi.a 8
63.j odd 6 1 inner 3528.1.bi.a 8
63.n odd 6 1 3528.1.bg.a 8
63.o even 6 1 3528.1.db.a 8
63.s even 6 1 3528.1.bg.a 8
72.j odd 6 1 3528.1.db.a 8
504.y even 6 1 3528.1.bg.a 8
504.bi odd 6 1 inner 3528.1.bi.a 8
504.ca even 6 1 inner 3528.1.bi.a 8
504.cc even 6 1 3528.1.db.a 8
504.db odd 6 1 3528.1.bg.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.bg.a 8 7.c even 3 1
3528.1.bg.a 8 7.d odd 6 1
3528.1.bg.a 8 56.j odd 6 1
3528.1.bg.a 8 56.p even 6 1
3528.1.bg.a 8 63.n odd 6 1
3528.1.bg.a 8 63.s even 6 1
3528.1.bg.a 8 504.y even 6 1
3528.1.bg.a 8 504.db odd 6 1
3528.1.bi.a 8 1.a even 1 1 trivial
3528.1.bi.a 8 7.b odd 2 1 inner
3528.1.bi.a 8 8.b even 2 1 inner
3528.1.bi.a 8 56.h odd 2 1 CM
3528.1.bi.a 8 63.i even 6 1 inner
3528.1.bi.a 8 63.j odd 6 1 inner
3528.1.bi.a 8 504.bi odd 6 1 inner
3528.1.bi.a 8 504.ca even 6 1 inner
3528.1.db.a 8 7.c even 3 1
3528.1.db.a 8 7.d odd 6 1
3528.1.db.a 8 9.d odd 6 1
3528.1.db.a 8 56.j odd 6 1
3528.1.db.a 8 56.p even 6 1
3528.1.db.a 8 63.o even 6 1
3528.1.db.a 8 72.j odd 6 1
3528.1.db.a 8 504.cc even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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