Properties

Label 840.1.bp.d
Level $840$
Weight $1$
Character orbit 840.bp
Analytic conductor $0.419$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 840.bp (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.419214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.1778112000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{16}^{2} q^{2} + \zeta_{16}^{3} q^{3} + \zeta_{16}^{4} q^{4} + \zeta_{16} q^{5} -\zeta_{16}^{5} q^{6} + \zeta_{16}^{6} q^{7} -\zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} +O(q^{10})\) \( q -\zeta_{16}^{2} q^{2} + \zeta_{16}^{3} q^{3} + \zeta_{16}^{4} q^{4} + \zeta_{16} q^{5} -\zeta_{16}^{5} q^{6} + \zeta_{16}^{6} q^{7} -\zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} -\zeta_{16}^{3} q^{10} + \zeta_{16}^{7} q^{12} + ( \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} + q^{14} + \zeta_{16}^{4} q^{15} - q^{16} + q^{18} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{19} + \zeta_{16}^{5} q^{20} -\zeta_{16} q^{21} + ( -1 + \zeta_{16}^{4} ) q^{23} + \zeta_{16} q^{24} + \zeta_{16}^{2} q^{25} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{26} -\zeta_{16} q^{27} -\zeta_{16}^{2} q^{28} -\zeta_{16}^{6} q^{30} + \zeta_{16}^{2} q^{32} + \zeta_{16}^{7} q^{35} -\zeta_{16}^{2} q^{36} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{38} + ( -1 + \zeta_{16}^{2} ) q^{39} -\zeta_{16}^{7} q^{40} + \zeta_{16}^{3} q^{42} + \zeta_{16}^{7} q^{45} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{46} -\zeta_{16}^{3} q^{48} -\zeta_{16}^{4} q^{49} -\zeta_{16}^{4} q^{50} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{52} + \zeta_{16}^{3} q^{54} + \zeta_{16}^{4} q^{56} + ( 1 - \zeta_{16}^{6} ) q^{57} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{59} - q^{60} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{61} -\zeta_{16}^{4} q^{63} -\zeta_{16}^{4} q^{64} + ( 1 + \zeta_{16}^{6} ) q^{65} + ( -\zeta_{16}^{3} + \zeta_{16}^{7} ) q^{69} + \zeta_{16} q^{70} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{71} + \zeta_{16}^{4} q^{72} + \zeta_{16}^{5} q^{75} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{76} + ( \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{78} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{79} -\zeta_{16} q^{80} -\zeta_{16}^{4} q^{81} + ( -\zeta_{16}^{5} - \zeta_{16}^{7} ) q^{83} -\zeta_{16}^{5} q^{84} + \zeta_{16} q^{90} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{91} + ( -1 - \zeta_{16}^{4} ) q^{92} + ( -\zeta_{16}^{4} - \zeta_{16}^{6} ) q^{95} + \zeta_{16}^{5} q^{96} + \zeta_{16}^{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{14} - 8q^{16} + 8q^{18} - 8q^{23} - 8q^{39} + 8q^{57} - 8q^{60} + 8q^{65} - 8q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{16}^{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.923880 0.382683i
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
−0.707107 0.707107i −0.382683 0.923880i 1.00000i −0.923880 0.382683i −0.382683 + 0.923880i −0.707107 + 0.707107i 0.707107 0.707107i −0.707107 + 0.707107i 0.382683 + 0.923880i
293.2 −0.707107 0.707107i 0.382683 + 0.923880i 1.00000i 0.923880 + 0.382683i 0.382683 0.923880i −0.707107 + 0.707107i 0.707107 0.707107i −0.707107 + 0.707107i −0.382683 0.923880i
293.3 0.707107 + 0.707107i −0.923880 + 0.382683i 1.00000i 0.382683 0.923880i −0.923880 0.382683i 0.707107 0.707107i −0.707107 + 0.707107i 0.707107 0.707107i 0.923880 0.382683i
293.4 0.707107 + 0.707107i 0.923880 0.382683i 1.00000i −0.382683 + 0.923880i 0.923880 + 0.382683i 0.707107 0.707107i −0.707107 + 0.707107i 0.707107 0.707107i −0.923880 + 0.382683i
797.1 −0.707107 + 0.707107i −0.382683 + 0.923880i 1.00000i −0.923880 + 0.382683i −0.382683 0.923880i −0.707107 0.707107i 0.707107 + 0.707107i −0.707107 0.707107i 0.382683 0.923880i
797.2 −0.707107 + 0.707107i 0.382683 0.923880i 1.00000i 0.923880 0.382683i 0.382683 + 0.923880i −0.707107 0.707107i 0.707107 + 0.707107i −0.707107 0.707107i −0.382683 + 0.923880i
797.3 0.707107 0.707107i −0.923880 0.382683i 1.00000i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.707107 + 0.707107i −0.707107 0.707107i 0.707107 + 0.707107i 0.923880 + 0.382683i
797.4 0.707107 0.707107i 0.923880 + 0.382683i 1.00000i −0.382683 0.923880i 0.923880 0.382683i 0.707107 + 0.707107i −0.707107 0.707107i 0.707107 + 0.707107i −0.923880 0.382683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 797.4
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
15.e even 4 1 inner
105.k odd 4 1 inner
120.w even 4 1 inner
840.bp odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.1.bp.d yes 8
3.b odd 2 1 840.1.bp.c 8
4.b odd 2 1 3360.1.cv.d 8
5.c odd 4 1 840.1.bp.c 8
7.b odd 2 1 inner 840.1.bp.d yes 8
8.b even 2 1 inner 840.1.bp.d yes 8
8.d odd 2 1 3360.1.cv.d 8
12.b even 2 1 3360.1.cv.c 8
15.e even 4 1 inner 840.1.bp.d yes 8
20.e even 4 1 3360.1.cv.c 8
21.c even 2 1 840.1.bp.c 8
24.f even 2 1 3360.1.cv.c 8
24.h odd 2 1 840.1.bp.c 8
28.d even 2 1 3360.1.cv.d 8
35.f even 4 1 840.1.bp.c 8
40.i odd 4 1 840.1.bp.c 8
40.k even 4 1 3360.1.cv.c 8
56.e even 2 1 3360.1.cv.d 8
56.h odd 2 1 CM 840.1.bp.d yes 8
60.l odd 4 1 3360.1.cv.d 8
84.h odd 2 1 3360.1.cv.c 8
105.k odd 4 1 inner 840.1.bp.d yes 8
120.q odd 4 1 3360.1.cv.d 8
120.w even 4 1 inner 840.1.bp.d yes 8
140.j odd 4 1 3360.1.cv.c 8
168.e odd 2 1 3360.1.cv.c 8
168.i even 2 1 840.1.bp.c 8
280.s even 4 1 840.1.bp.c 8
280.y odd 4 1 3360.1.cv.c 8
420.w even 4 1 3360.1.cv.d 8
840.bm even 4 1 3360.1.cv.d 8
840.bp odd 4 1 inner 840.1.bp.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.bp.c 8 3.b odd 2 1
840.1.bp.c 8 5.c odd 4 1
840.1.bp.c 8 21.c even 2 1
840.1.bp.c 8 24.h odd 2 1
840.1.bp.c 8 35.f even 4 1
840.1.bp.c 8 40.i odd 4 1
840.1.bp.c 8 168.i even 2 1
840.1.bp.c 8 280.s even 4 1
840.1.bp.d yes 8 1.a even 1 1 trivial
840.1.bp.d yes 8 7.b odd 2 1 inner
840.1.bp.d yes 8 8.b even 2 1 inner
840.1.bp.d yes 8 15.e even 4 1 inner
840.1.bp.d yes 8 56.h odd 2 1 CM
840.1.bp.d yes 8 105.k odd 4 1 inner
840.1.bp.d yes 8 120.w even 4 1 inner
840.1.bp.d yes 8 840.bp odd 4 1 inner
3360.1.cv.c 8 12.b even 2 1
3360.1.cv.c 8 20.e even 4 1
3360.1.cv.c 8 24.f even 2 1
3360.1.cv.c 8 40.k even 4 1
3360.1.cv.c 8 84.h odd 2 1
3360.1.cv.c 8 140.j odd 4 1
3360.1.cv.c 8 168.e odd 2 1
3360.1.cv.c 8 280.y odd 4 1
3360.1.cv.d 8 4.b odd 2 1
3360.1.cv.d 8 8.d odd 2 1
3360.1.cv.d 8 28.d even 2 1
3360.1.cv.d 8 56.e even 2 1
3360.1.cv.d 8 60.l odd 4 1
3360.1.cv.d 8 120.q odd 4 1
3360.1.cv.d 8 420.w even 4 1
3360.1.cv.d 8 840.bm even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(840, [\chi])\):

\( T_{11} \)
\( T_{23}^{2} + 2 T_{23} + 2 \)
\( T_{73} \)

Hecke characteristic polynomials

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