Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.bn (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{2} -\zeta_{16}^{2} q^{4} + ( -\zeta_{16}^{2} + \zeta_{16}^{5} ) q^{5} -\zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} +O(q^{10})\) \( q + \zeta_{16}^{5} q^{2} -\zeta_{16}^{2} q^{4} + ( -\zeta_{16}^{2} + \zeta_{16}^{5} ) q^{5} -\zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} + ( -\zeta_{16}^{2} - \zeta_{16}^{7} ) q^{10} + ( -1 - \zeta_{16}^{4} ) q^{13} + \zeta_{16}^{4} q^{16} -\zeta_{16}^{5} q^{17} - q^{18} + ( \zeta_{16}^{4} - \zeta_{16}^{7} ) q^{20} + ( -\zeta_{16}^{2} + \zeta_{16}^{4} - \zeta_{16}^{7} ) q^{25} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{26} + ( -\zeta_{16} + \zeta_{16}^{6} ) q^{29} -\zeta_{16} q^{32} + \zeta_{16}^{2} q^{34} -\zeta_{16}^{5} q^{36} + ( 1 - \zeta_{16}^{3} ) q^{37} + ( -\zeta_{16} + \zeta_{16}^{4} ) q^{40} + ( -\zeta_{16}^{4} + \zeta_{16}^{5} ) q^{41} + ( -1 - \zeta_{16}^{5} ) q^{45} + ( -\zeta_{16} + \zeta_{16}^{4} - \zeta_{16}^{7} ) q^{50} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{52} + ( -\zeta_{16}^{4} - \zeta_{16}^{6} ) q^{53} + ( -\zeta_{16}^{3} - \zeta_{16}^{6} ) q^{58} + ( -\zeta_{16}^{4} - \zeta_{16}^{5} ) q^{61} -\zeta_{16}^{6} q^{64} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{65} + \zeta_{16}^{7} q^{68} + \zeta_{16}^{2} q^{72} + ( \zeta_{16} - \zeta_{16}^{6} ) q^{73} + ( 1 + \zeta_{16}^{5} ) q^{74} + ( -\zeta_{16} - \zeta_{16}^{6} ) q^{80} + \zeta_{16}^{6} q^{81} + ( \zeta_{16} - \zeta_{16}^{2} ) q^{82} + ( \zeta_{16}^{2} + \zeta_{16}^{7} ) q^{85} + 2 \zeta_{16}^{6} q^{89} + ( \zeta_{16}^{2} - \zeta_{16}^{5} ) q^{90} + ( -\zeta_{16}^{2} - \zeta_{16}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{13} - 8 q^{18} + 8 q^{37} - 8 q^{45} + 8 q^{74} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-\zeta_{16}^{3}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
979.1
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 0.382683i
−0.382683 0.923880i
0.382683 0.923880i
0.923880 + 0.382683i
0.382683 0.923880i 0 −0.707107 0.707107i −0.324423 1.63099i 0 0 −0.923880 + 0.382683i −0.382683 0.923880i −1.63099 0.324423i
1371.1 −0.923880 0.382683i 0 0.707107 + 0.707107i −0.216773 + 0.324423i 0 0 −0.382683 0.923880i 0.923880 0.382683i 0.324423 0.216773i
1567.1 0.923880 0.382683i 0 0.707107 0.707107i 1.63099 1.08979i 0 0 0.382683 0.923880i −0.923880 0.382683i 1.08979 1.63099i
1763.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i −0.324423 + 1.63099i 0 0 −0.923880 0.382683i −0.382683 + 0.923880i −1.63099 + 0.324423i
2351.1 −0.382683 0.923880i 0 −0.707107 + 0.707107i −1.08979 0.216773i 0 0 0.923880 + 0.382683i 0.382683 0.923880i 0.216773 + 1.08979i
2547.1 −0.923880 + 0.382683i 0 0.707107 0.707107i −0.216773 0.324423i 0 0 −0.382683 + 0.923880i 0.923880 + 0.382683i 0.324423 + 0.216773i
2743.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i 1.63099 + 1.08979i 0 0 0.382683 + 0.923880i −0.923880 + 0.382683i 1.08979 + 1.63099i
3135.1 −0.382683 + 0.923880i 0 −0.707107 0.707107i −1.08979 + 0.216773i 0 0 0.923880 0.382683i 0.382683 + 0.923880i 0.216773 1.08979i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3135.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
119.p even 16 1 inner
476.bf odd 16 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2 T_{5}^{4} + 16 T_{5}^{3} + 20 T_{5}^{2} + 8 T_{5} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 2 + 2 T + T^{2} )^{4} \)
$17$ \( 1 + T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( 2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( 2 - 8 T + 28 T^{2} - 56 T^{3} + 70 T^{4} - 56 T^{5} + 28 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( 2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 2 + 4 T + 2 T^{2} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( 2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 2 + 8 T + 12 T^{2} + 2 T^{4} - 8 T^{5} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 16 + T^{4} )^{2} \)
$97$ \( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} \)
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