Properties

Label 3528.1.cm.a
Level $3528$
Weight $1$
Character orbit 3528.cm
Analytic conductor $1.761$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
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.cm (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{24}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{48}^{12} q^{2} + \zeta_{48}^{5} q^{3} - q^{4} + \zeta_{48}^{17} q^{6} -\zeta_{48}^{12} q^{8} + \zeta_{48}^{10} q^{9} +O(q^{10})\) \( q + \zeta_{48}^{12} q^{2} + \zeta_{48}^{5} q^{3} - q^{4} + \zeta_{48}^{17} q^{6} -\zeta_{48}^{12} q^{8} + \zeta_{48}^{10} q^{9} + ( -\zeta_{48}^{18} - \zeta_{48}^{22} ) q^{11} -\zeta_{48}^{5} q^{12} + q^{16} + ( -\zeta_{48}^{9} - \zeta_{48}^{23} ) q^{17} + \zeta_{48}^{22} q^{18} + ( \zeta_{48}^{7} - \zeta_{48}^{9} ) q^{19} + ( \zeta_{48}^{6} + \zeta_{48}^{10} ) q^{22} -\zeta_{48}^{17} q^{24} -\zeta_{48}^{8} q^{25} + \zeta_{48}^{15} q^{27} + \zeta_{48}^{12} q^{32} + ( \zeta_{48}^{3} - \zeta_{48}^{23} ) q^{33} + ( \zeta_{48}^{11} - \zeta_{48}^{21} ) q^{34} -\zeta_{48}^{10} q^{36} + ( \zeta_{48}^{19} - \zeta_{48}^{21} ) q^{38} + ( \zeta_{48}^{3} + \zeta_{48}^{13} ) q^{41} + ( \zeta_{48}^{2} - \zeta_{48}^{6} ) q^{43} + ( \zeta_{48}^{18} + \zeta_{48}^{22} ) q^{44} + \zeta_{48}^{5} q^{48} -\zeta_{48}^{20} q^{50} + ( \zeta_{48}^{4} - \zeta_{48}^{14} ) q^{51} -\zeta_{48}^{3} q^{54} + ( \zeta_{48}^{12} - \zeta_{48}^{14} ) q^{57} + ( -\zeta_{48} + \zeta_{48}^{23} ) q^{59} - q^{64} + ( \zeta_{48}^{11} + \zeta_{48}^{15} ) q^{66} + ( \zeta_{48}^{4} - \zeta_{48}^{20} ) q^{67} + ( \zeta_{48}^{9} + \zeta_{48}^{23} ) q^{68} -\zeta_{48}^{22} q^{72} + ( -\zeta_{48}^{3} - \zeta_{48}^{5} ) q^{73} -\zeta_{48}^{13} q^{75} + ( -\zeta_{48}^{7} + \zeta_{48}^{9} ) q^{76} + \zeta_{48}^{20} q^{81} + ( -\zeta_{48} + \zeta_{48}^{15} ) q^{82} + ( -\zeta_{48} + \zeta_{48}^{7} ) q^{83} + ( \zeta_{48}^{14} - \zeta_{48}^{18} ) q^{86} + ( -\zeta_{48}^{6} - \zeta_{48}^{10} ) q^{88} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{89} + \zeta_{48}^{17} q^{96} + ( -\zeta_{48}^{9} + \zeta_{48}^{23} ) q^{97} + ( \zeta_{48}^{4} + \zeta_{48}^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} + O(q^{10}) \) \( 16q - 16q^{4} + 16q^{16} - 8q^{25} - 16q^{64} + 8q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(\zeta_{48}^{8}\) \(\zeta_{48}^{8}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
0.793353 0.608761i
0.608761 + 0.793353i
−0.608761 0.793353i
−0.793353 + 0.608761i
−0.991445 0.130526i
−0.130526 + 0.991445i
0.130526 0.991445i
0.991445 + 0.130526i
−0.991445 + 0.130526i
−0.130526 0.991445i
0.130526 + 0.991445i
0.991445 0.130526i
0.793353 + 0.608761i
0.608761 0.793353i
−0.608761 + 0.793353i
−0.793353 0.608761i
1.00000i −0.991445 + 0.130526i −1.00000 0 0.130526 + 0.991445i 0 1.00000i 0.965926 0.258819i 0
227.2 1.00000i −0.130526 0.991445i −1.00000 0 −0.991445 + 0.130526i 0 1.00000i −0.965926 + 0.258819i 0
227.3 1.00000i 0.130526 + 0.991445i −1.00000 0 0.991445 0.130526i 0 1.00000i −0.965926 + 0.258819i 0
227.4 1.00000i 0.991445 0.130526i −1.00000 0 −0.130526 0.991445i 0 1.00000i 0.965926 0.258819i 0
227.5 1.00000i −0.793353 0.608761i −1.00000 0 0.608761 0.793353i 0 1.00000i 0.258819 + 0.965926i 0
227.6 1.00000i −0.608761 + 0.793353i −1.00000 0 −0.793353 0.608761i 0 1.00000i −0.258819 0.965926i 0
227.7 1.00000i 0.608761 0.793353i −1.00000 0 0.793353 + 0.608761i 0 1.00000i −0.258819 0.965926i 0
227.8 1.00000i 0.793353 + 0.608761i −1.00000 0 −0.608761 + 0.793353i 0 1.00000i 0.258819 + 0.965926i 0
3155.1 1.00000i −0.793353 + 0.608761i −1.00000 0 0.608761 + 0.793353i 0 1.00000i 0.258819 0.965926i 0
3155.2 1.00000i −0.608761 0.793353i −1.00000 0 −0.793353 + 0.608761i 0 1.00000i −0.258819 + 0.965926i 0
3155.3 1.00000i 0.608761 + 0.793353i −1.00000 0 0.793353 0.608761i 0 1.00000i −0.258819 + 0.965926i 0
3155.4 1.00000i 0.793353 0.608761i −1.00000 0 −0.608761 0.793353i 0 1.00000i 0.258819 0.965926i 0
3155.5 1.00000i −0.991445 0.130526i −1.00000 0 0.130526 0.991445i 0 1.00000i 0.965926 + 0.258819i 0
3155.6 1.00000i −0.130526 + 0.991445i −1.00000 0 −0.991445 0.130526i 0 1.00000i −0.965926 0.258819i 0
3155.7 1.00000i 0.130526 0.991445i −1.00000 0 0.991445 + 0.130526i 0 1.00000i −0.965926 0.258819i 0
3155.8 1.00000i 0.991445 + 0.130526i −1.00000 0 −0.130526 + 0.991445i 0 1.00000i 0.965926 + 0.258819i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3155.8
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.i even 6 1 inner
63.j odd 6 1 inner
504.bt even 6 1 inner
504.cm 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.cm.a 16
7.b odd 2 1 inner 3528.1.cm.a 16
7.c even 3 1 3528.1.u.a 16
7.c even 3 1 3528.1.co.a 16
7.d odd 6 1 3528.1.u.a 16
7.d odd 6 1 3528.1.co.a 16
8.d odd 2 1 CM 3528.1.cm.a 16
9.d odd 6 1 3528.1.u.a 16
56.e even 2 1 inner 3528.1.cm.a 16
56.k odd 6 1 3528.1.u.a 16
56.k odd 6 1 3528.1.co.a 16
56.m even 6 1 3528.1.u.a 16
56.m even 6 1 3528.1.co.a 16
63.i even 6 1 inner 3528.1.cm.a 16
63.j odd 6 1 inner 3528.1.cm.a 16
63.n odd 6 1 3528.1.co.a 16
63.o even 6 1 3528.1.u.a 16
63.s even 6 1 3528.1.co.a 16
72.l even 6 1 3528.1.u.a 16
504.u odd 6 1 3528.1.co.a 16
504.bt even 6 1 inner 3528.1.cm.a 16
504.cm odd 6 1 inner 3528.1.cm.a 16
504.co odd 6 1 3528.1.u.a 16
504.cy even 6 1 3528.1.co.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.u.a 16 7.c even 3 1
3528.1.u.a 16 7.d odd 6 1
3528.1.u.a 16 9.d odd 6 1
3528.1.u.a 16 56.k odd 6 1
3528.1.u.a 16 56.m even 6 1
3528.1.u.a 16 63.o even 6 1
3528.1.u.a 16 72.l even 6 1
3528.1.u.a 16 504.co odd 6 1
3528.1.cm.a 16 1.a even 1 1 trivial
3528.1.cm.a 16 7.b odd 2 1 inner
3528.1.cm.a 16 8.d odd 2 1 CM
3528.1.cm.a 16 56.e even 2 1 inner
3528.1.cm.a 16 63.i even 6 1 inner
3528.1.cm.a 16 63.j odd 6 1 inner
3528.1.cm.a 16 504.bt even 6 1 inner
3528.1.cm.a 16 504.cm odd 6 1 inner
3528.1.co.a 16 7.c even 3 1
3528.1.co.a 16 7.d odd 6 1
3528.1.co.a 16 56.k odd 6 1
3528.1.co.a 16 56.m even 6 1
3528.1.co.a 16 63.n odd 6 1
3528.1.co.a 16 63.s even 6 1
3528.1.co.a 16 504.u odd 6 1
3528.1.co.a 16 504.cy even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( 1 - T^{8} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( T^{16} \)
$11$ \( ( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} )^{2} \)
$13$ \( T^{16} \)
$17$ \( 1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16} \)
$19$ \( 1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( 1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16} \)
$43$ \( ( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( ( 1 - 16 T^{2} + 20 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$61$ \( T^{16} \)
$67$ \( ( -3 + T^{2} )^{8} \)
$71$ \( T^{16} \)
$73$ \( 1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( ( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$89$ \( ( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$97$ \( 1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16} \)
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