Properties

Label 3332.1.ce.a
Level $3332$
Weight $1$
Character orbit 3332.ce
Analytic conductor $1.663$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.ce (of order \(48\), degree \(16\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + ( \zeta_{48}^{13} - \zeta_{48}^{22} ) q^{5} + \zeta_{48}^{3} q^{8} -\zeta_{48}^{7} q^{9} +O(q^{10})\) \( q + \zeta_{48} q^{2} + \zeta_{48}^{2} q^{4} + ( \zeta_{48}^{13} - \zeta_{48}^{22} ) q^{5} + \zeta_{48}^{3} q^{8} -\zeta_{48}^{7} q^{9} + ( \zeta_{48}^{14} - \zeta_{48}^{23} ) q^{10} + ( -1 + \zeta_{48}^{12} ) q^{13} + \zeta_{48}^{4} q^{16} + \zeta_{48}^{5} q^{17} -\zeta_{48}^{8} q^{18} + ( 1 + \zeta_{48}^{15} ) q^{20} + ( -\zeta_{48}^{2} + \zeta_{48}^{11} - \zeta_{48}^{20} ) q^{25} + ( -\zeta_{48} + \zeta_{48}^{13} ) q^{26} + ( \zeta_{48}^{6} + \zeta_{48}^{21} ) q^{29} + \zeta_{48}^{5} q^{32} + \zeta_{48}^{6} q^{34} -\zeta_{48}^{9} q^{36} + ( -\zeta_{48}^{7} - \zeta_{48}^{16} ) q^{37} + ( \zeta_{48} + \zeta_{48}^{16} ) q^{40} + ( 1 + \zeta_{48}^{21} ) q^{41} + ( -\zeta_{48}^{5} - \zeta_{48}^{20} ) q^{45} + ( -\zeta_{48}^{3} + \zeta_{48}^{12} - \zeta_{48}^{21} ) q^{50} + ( -\zeta_{48}^{2} + \zeta_{48}^{14} ) q^{52} + ( -\zeta_{48}^{14} - \zeta_{48}^{20} ) q^{53} + ( \zeta_{48}^{7} + \zeta_{48}^{22} ) q^{58} + ( -\zeta_{48}^{13} - \zeta_{48}^{16} ) q^{61} + \zeta_{48}^{6} q^{64} + ( -\zeta_{48} + \zeta_{48}^{10} - \zeta_{48}^{13} + \zeta_{48}^{22} ) q^{65} + \zeta_{48}^{7} q^{68} -\zeta_{48}^{10} q^{72} + ( -\zeta_{48}^{2} + \zeta_{48}^{17} ) q^{73} + ( -\zeta_{48}^{8} - \zeta_{48}^{17} ) q^{74} + ( \zeta_{48}^{2} + \zeta_{48}^{17} ) q^{80} + \zeta_{48}^{14} q^{81} + ( \zeta_{48} + \zeta_{48}^{22} ) q^{82} + ( \zeta_{48}^{3} + \zeta_{48}^{18} ) q^{85} + ( -\zeta_{48}^{6} - \zeta_{48}^{21} ) q^{90} + ( -\zeta_{48}^{6} + \zeta_{48}^{21} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 16 q^{13} - 8 q^{18} + 16 q^{20} + 8 q^{37} - 8 q^{40} + 16 q^{41} + 8 q^{61} - 8 q^{74} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{48}^{15}\) \(-\zeta_{48}^{16}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.793353 0.608761i
−0.793353 + 0.608761i
0.130526 + 0.991445i
0.130526 0.991445i
0.991445 + 0.130526i
−0.991445 + 0.130526i
−0.991445 0.130526i
−0.130526 + 0.991445i
0.793353 0.608761i
0.991445 0.130526i
0.608761 + 0.793353i
−0.130526 0.991445i
0.793353 + 0.608761i
−0.608761 + 0.793353i
−0.608761 0.793353i
0.608761 0.793353i
−0.793353 0.608761i 0 0.258819 + 0.965926i 0.867580 1.75928i 0 0 0.382683 0.923880i −0.130526 0.991445i −1.75928 + 0.867580i
215.1 −0.793353 + 0.608761i 0 0.258819 0.965926i 0.867580 + 1.75928i 0 0 0.382683 + 0.923880i −0.130526 + 0.991445i −1.75928 0.867580i
227.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i 0.0255190 0.389345i 0 0 −0.382683 0.923880i 0.793353 + 0.608761i 0.389345 0.0255190i
411.1 0.130526 0.991445i 0 −0.965926 0.258819i 0.0255190 + 0.389345i 0 0 −0.382683 + 0.923880i 0.793353 0.608761i 0.389345 + 0.0255190i
607.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i 0.835400 + 0.732626i 0 0 0.923880 + 0.382683i −0.608761 0.793353i 0.732626 + 0.835400i
619.1 −0.991445 + 0.130526i 0 0.965926 0.258819i 1.09645 + 1.25026i 0 0 −0.923880 + 0.382683i 0.608761 0.793353i −1.25026 1.09645i
1195.1 −0.991445 0.130526i 0 0.965926 + 0.258819i 1.09645 1.25026i 0 0 −0.923880 0.382683i 0.608761 + 0.793353i −1.25026 + 1.09645i
1391.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i −1.95737 + 0.128293i 0 0 0.382683 0.923880i −0.793353 + 0.608761i 0.128293 1.95737i
1587.1 0.793353 0.608761i 0 0.258819 0.965926i −0.349942 + 0.172572i 0 0 −0.382683 0.923880i 0.130526 0.991445i −0.172572 + 0.349942i
1795.1 0.991445 0.130526i 0 0.965926 0.258819i 0.835400 0.732626i 0 0 0.923880 0.382683i −0.608761 + 0.793353i 0.732626 0.835400i
1979.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i 0.534534 1.57469i 0 0 −0.923880 + 0.382683i −0.991445 0.130526i 1.57469 0.534534i
2187.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i −1.95737 0.128293i 0 0 0.382683 + 0.923880i −0.793353 0.608761i 0.128293 + 1.95737i
2383.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i −0.349942 0.172572i 0 0 −0.382683 + 0.923880i 0.130526 + 0.991445i −0.172572 0.349942i
2579.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i −1.05217 + 0.357164i 0 0 0.923880 + 0.382683i 0.991445 0.130526i 0.357164 1.05217i
3155.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i −1.05217 0.357164i 0 0 0.923880 0.382683i 0.991445 + 0.130526i 0.357164 + 1.05217i
3167.1 0.608761 0.793353i 0 −0.258819 0.965926i 0.534534 + 1.57469i 0 0 −0.923880 0.382683i −0.991445 + 0.130526i 1.57469 + 0.534534i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3167.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.c even 3 1 inner
28.g odd 6 1 inner
119.p even 16 1 inner
119.s even 48 1 inner
476.bf odd 16 1 inner
476.bk odd 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.ce.a 16
4.b odd 2 1 CM 3332.1.ce.a 16
7.b odd 2 1 3332.1.ce.c 16
7.c even 3 1 3332.1.bn.b 8
7.c even 3 1 inner 3332.1.ce.a 16
7.d odd 6 1 3332.1.bn.d yes 8
7.d odd 6 1 3332.1.ce.c 16
17.e odd 16 1 3332.1.ce.c 16
28.d even 2 1 3332.1.ce.c 16
28.f even 6 1 3332.1.bn.d yes 8
28.f even 6 1 3332.1.ce.c 16
28.g odd 6 1 3332.1.bn.b 8
28.g odd 6 1 inner 3332.1.ce.a 16
68.i even 16 1 3332.1.ce.c 16
119.p even 16 1 inner 3332.1.ce.a 16
119.s even 48 1 3332.1.bn.b 8
119.s even 48 1 inner 3332.1.ce.a 16
119.t odd 48 1 3332.1.bn.d yes 8
119.t odd 48 1 3332.1.ce.c 16
476.bf odd 16 1 inner 3332.1.ce.a 16
476.bk odd 48 1 3332.1.bn.b 8
476.bk odd 48 1 inner 3332.1.ce.a 16
476.bm even 48 1 3332.1.bn.d yes 8
476.bm even 48 1 3332.1.ce.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.bn.b 8 7.c even 3 1
3332.1.bn.b 8 28.g odd 6 1
3332.1.bn.b 8 119.s even 48 1
3332.1.bn.b 8 476.bk odd 48 1
3332.1.bn.d yes 8 7.d odd 6 1
3332.1.bn.d yes 8 28.f even 6 1
3332.1.bn.d yes 8 119.t odd 48 1
3332.1.bn.d yes 8 476.bm even 48 1
3332.1.ce.a 16 1.a even 1 1 trivial
3332.1.ce.a 16 4.b odd 2 1 CM
3332.1.ce.a 16 7.c even 3 1 inner
3332.1.ce.a 16 28.g odd 6 1 inner
3332.1.ce.a 16 119.p even 16 1 inner
3332.1.ce.a 16 119.s even 48 1 inner
3332.1.ce.a 16 476.bf odd 16 1 inner
3332.1.ce.a 16 476.bk odd 48 1 inner
3332.1.ce.c 16 7.b odd 2 1
3332.1.ce.c 16 7.d odd 6 1
3332.1.ce.c 16 17.e odd 16 1
3332.1.ce.c 16 28.d even 2 1
3332.1.ce.c 16 28.f even 6 1
3332.1.ce.c 16 68.i even 16 1
3332.1.ce.c 16 119.t odd 48 1
3332.1.ce.c 16 476.bm even 48 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{8} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 4 + 16 T + 40 T^{2} + 96 T^{3} + 140 T^{4} - 48 T^{5} + 40 T^{6} + 192 T^{7} + 2 T^{8} - 8 T^{9} + 88 T^{10} - 2 T^{12} + 16 T^{13} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 2 + 2 T + T^{2} )^{8} \)
$17$ \( 1 - T^{8} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( ( 2 + 8 T + 12 T^{2} + 2 T^{4} - 8 T^{5} + T^{8} )^{2} \)
$31$ \( T^{16} \)
$37$ \( 4 - 16 T + 8 T^{2} + 196 T^{4} - 560 T^{5} + 840 T^{6} - 1024 T^{7} + 1106 T^{8} - 1016 T^{9} + 784 T^{10} - 504 T^{11} + 266 T^{12} - 112 T^{13} + 36 T^{14} - 8 T^{15} + T^{16} \)
$41$ \( ( 2 - 8 T + 28 T^{2} - 56 T^{3} + 70 T^{4} - 56 T^{5} + 28 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( ( 4 - 8 T + 12 T^{2} - 8 T^{3} + 2 T^{4} + 8 T^{5} - 2 T^{6} + T^{8} )^{2} \)
$59$ \( T^{16} \)
$61$ \( 4 - 16 T + 8 T^{2} + 196 T^{4} - 560 T^{5} + 840 T^{6} - 1024 T^{7} + 1106 T^{8} - 1016 T^{9} + 784 T^{10} - 504 T^{11} + 266 T^{12} - 112 T^{13} + 36 T^{14} - 8 T^{15} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( 4 + 16 T + 24 T^{2} + 96 T^{3} + 268 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 2 T^{8} + 8 T^{9} + 40 T^{10} + 16 T^{11} - 2 T^{12} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( ( 2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8} )^{2} \)
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