Properties

Label 840.1.bp.b
Level $840$
Weight $1$
Character orbit 840.bp
Analytic conductor $0.419$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -24
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.441000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{3} q^{3} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} - \zeta_{8}^{2} q^{6} + \zeta_{8}^{2} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{3} q^{3} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} - \zeta_{8}^{2} q^{6} + \zeta_{8}^{2} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{2} q^{9} + q^{10} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} - \zeta_{8} q^{12} + \zeta_{8} q^{14} + q^{15} - q^{16} - \zeta_{8} q^{18} - \zeta_{8}^{3} q^{20} + \zeta_{8} q^{21} + (\zeta_{8}^{2} - 1) q^{22} - q^{24} + \zeta_{8}^{2} q^{25} - \zeta_{8} q^{27} + q^{28} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{29} - \zeta_{8}^{3} q^{30} + \zeta_{8}^{2} q^{31} + \zeta_{8}^{3} q^{32} + (\zeta_{8}^{2} - 1) q^{33} + \zeta_{8}^{3} q^{35} - q^{36} - \zeta_{8}^{2} q^{40} + q^{42} + (\zeta_{8}^{3} + \zeta_{8}) q^{44} - \zeta_{8}^{3} q^{45} + \zeta_{8}^{3} q^{48} - q^{49} + \zeta_{8} q^{50} - q^{54} + ( - \zeta_{8}^{2} - 1) q^{55} - \zeta_{8}^{3} q^{56} + ( - \zeta_{8}^{2} - 1) q^{58} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} - \zeta_{8}^{2} q^{60} + 2 \zeta_{8} q^{62} + q^{63} + \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} + \zeta_{8}) q^{66} + \zeta_{8}^{2} q^{70} + \zeta_{8}^{3} q^{72} + ( - \zeta_{8}^{2} + 1) q^{73} + \zeta_{8} q^{75} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{77} - \zeta_{8} q^{80} - q^{81} - \zeta_{8}^{3} q^{84} + ( - \zeta_{8}^{2} - 1) q^{87} + (\zeta_{8}^{2} + 1) q^{88} - \zeta_{8}^{2} q^{90} + 2 \zeta_{8} q^{93} + \zeta_{8}^{2} q^{96} + (\zeta_{8}^{2} + 1) q^{97} + \zeta_{8}^{3} q^{98} + (\zeta_{8}^{3} + \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{10} + 4 q^{15} - 4 q^{16} - 4 q^{22} - 4 q^{24} + 4 q^{28} - 4 q^{33} - 4 q^{36} + 4 q^{42} - 4 q^{49} - 4 q^{54} - 4 q^{55} - 4 q^{58} + 4 q^{63} + 4 q^{73} - 4 q^{81} - 4 q^{87} + 4 q^{88} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{8}^{2}\) \(-1\) \(1\)

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
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −0.707107 0.707107i 1.00000i 1.00000i 0.707107 + 0.707107i 1.00000i 1.00000
293.2 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.707107 + 0.707107i 1.00000i 1.00000i −0.707107 0.707107i 1.00000i 1.00000
797.1 −0.707107 0.707107i −0.707107 0.707107i 1.00000i −0.707107 + 0.707107i 1.00000i 1.00000i 0.707107 0.707107i 1.00000i 1.00000
797.2 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.707107 0.707107i 1.00000i 1.00000i −0.707107 + 0.707107i 1.00000i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner
280.s 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.b yes 4
3.b odd 2 1 inner 840.1.bp.b yes 4
4.b odd 2 1 3360.1.cv.a 4
5.c odd 4 1 840.1.bp.a 4
7.b odd 2 1 840.1.bp.a 4
8.b even 2 1 inner 840.1.bp.b yes 4
8.d odd 2 1 3360.1.cv.a 4
12.b even 2 1 3360.1.cv.a 4
15.e even 4 1 840.1.bp.a 4
20.e even 4 1 3360.1.cv.b 4
21.c even 2 1 840.1.bp.a 4
24.f even 2 1 3360.1.cv.a 4
24.h odd 2 1 CM 840.1.bp.b yes 4
28.d even 2 1 3360.1.cv.b 4
35.f even 4 1 inner 840.1.bp.b yes 4
40.i odd 4 1 840.1.bp.a 4
40.k even 4 1 3360.1.cv.b 4
56.e even 2 1 3360.1.cv.b 4
56.h odd 2 1 840.1.bp.a 4
60.l odd 4 1 3360.1.cv.b 4
84.h odd 2 1 3360.1.cv.b 4
105.k odd 4 1 inner 840.1.bp.b yes 4
120.q odd 4 1 3360.1.cv.b 4
120.w even 4 1 840.1.bp.a 4
140.j odd 4 1 3360.1.cv.a 4
168.e odd 2 1 3360.1.cv.b 4
168.i even 2 1 840.1.bp.a 4
280.s even 4 1 inner 840.1.bp.b yes 4
280.y odd 4 1 3360.1.cv.a 4
420.w even 4 1 3360.1.cv.a 4
840.bm even 4 1 3360.1.cv.a 4
840.bp odd 4 1 inner 840.1.bp.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.bp.a 4 5.c odd 4 1
840.1.bp.a 4 7.b odd 2 1
840.1.bp.a 4 15.e even 4 1
840.1.bp.a 4 21.c even 2 1
840.1.bp.a 4 40.i odd 4 1
840.1.bp.a 4 56.h odd 2 1
840.1.bp.a 4 120.w even 4 1
840.1.bp.a 4 168.i even 2 1
840.1.bp.b yes 4 1.a even 1 1 trivial
840.1.bp.b yes 4 3.b odd 2 1 inner
840.1.bp.b yes 4 8.b even 2 1 inner
840.1.bp.b yes 4 24.h odd 2 1 CM
840.1.bp.b yes 4 35.f even 4 1 inner
840.1.bp.b yes 4 105.k odd 4 1 inner
840.1.bp.b yes 4 280.s even 4 1 inner
840.1.bp.b yes 4 840.bp odd 4 1 inner
3360.1.cv.a 4 4.b odd 2 1
3360.1.cv.a 4 8.d odd 2 1
3360.1.cv.a 4 12.b even 2 1
3360.1.cv.a 4 24.f even 2 1
3360.1.cv.a 4 140.j odd 4 1
3360.1.cv.a 4 280.y odd 4 1
3360.1.cv.a 4 420.w even 4 1
3360.1.cv.a 4 840.bm even 4 1
3360.1.cv.b 4 20.e even 4 1
3360.1.cv.b 4 28.d even 2 1
3360.1.cv.b 4 40.k even 4 1
3360.1.cv.b 4 56.e even 2 1
3360.1.cv.b 4 60.l odd 4 1
3360.1.cv.b 4 84.h odd 2 1
3360.1.cv.b 4 120.q odd 4 1
3360.1.cv.b 4 168.e odd 2 1

Hecke kernels

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

\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{73}^{2} - 2T_{73} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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