Properties

Label 840.2.bg.d
Level $840$
Weight $2$
Character orbit 840.bg
Analytic conductor $6.707$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + q^{13} + q^{15} + 7 \zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} + 5 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} - q^{27} + ( -6 + 6 \zeta_{6} ) q^{31} + 3 \zeta_{6} q^{33} + ( -2 - \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} -3 q^{41} + 8 q^{43} + ( 1 - \zeta_{6} ) q^{45} + \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -5 + 5 \zeta_{6} ) q^{53} -3 q^{55} + 7 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + ( 2 + \zeta_{6} ) q^{63} + \zeta_{6} q^{65} + 5 q^{69} -6 q^{71} + ( 14 - 14 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 3 - 9 \zeta_{6} ) q^{77} + 16 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -16 q^{83} -6 \zeta_{6} q^{89} + ( -3 + 2 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{93} + ( -7 + 7 \zeta_{6} ) q^{95} + 16 q^{97} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} - 4q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} - 4q^{7} - q^{9} - 3q^{11} + 2q^{13} + 2q^{15} + 7q^{19} + q^{21} + 5q^{23} - q^{25} - 2q^{27} - 6q^{31} + 3q^{33} - 5q^{35} - 3q^{37} + q^{39} - 6q^{41} + 16q^{43} + q^{45} + q^{47} + 2q^{49} - 5q^{53} - 6q^{55} + 14q^{57} + 4q^{59} + 8q^{61} + 5q^{63} + q^{65} + 10q^{69} - 12q^{71} + 14q^{73} + q^{75} - 3q^{77} + 16q^{79} - q^{81} - 32q^{83} - 6q^{89} - 4q^{91} + 6q^{93} - 7q^{95} + 32q^{97} + 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −2.00000 1.73205i 0 −0.500000 + 0.866025i 0
361.1 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −2.00000 + 1.73205i 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.bg.d 2
3.b odd 2 1 2520.2.bi.b 2
4.b odd 2 1 1680.2.bg.i 2
7.c even 3 1 inner 840.2.bg.d 2
7.c even 3 1 5880.2.a.f 1
7.d odd 6 1 5880.2.a.bh 1
21.h odd 6 1 2520.2.bi.b 2
28.g odd 6 1 1680.2.bg.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.d 2 1.a even 1 1 trivial
840.2.bg.d 2 7.c even 3 1 inner
1680.2.bg.i 2 4.b odd 2 1
1680.2.bg.i 2 28.g odd 6 1
2520.2.bi.b 2 3.b odd 2 1
2520.2.bi.b 2 21.h odd 6 1
5880.2.a.f 1 7.c even 3 1
5880.2.a.bh 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 3 T_{11} + 9 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( 25 - 5 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 36 + 6 T + T^{2} \)
$37$ \( 9 + 3 T + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( 25 + 5 T + T^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 196 - 14 T + T^{2} \)
$79$ \( 256 - 16 T + T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( -16 + T )^{2} \)
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