Properties

Label 840.2.bg.j
Level $840$
Weight $2$
Character orbit 840.bg
Analytic conductor $6.707$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.29428272.1
Defining polynomial: \( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{3} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{3} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{9} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{13} - q^{15} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{3} - 1) q^{19} + ( - \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{23} + \beta_{3} q^{25} + q^{27} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{29} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{33} + (\beta_{5} + 1) q^{35} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{37} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{39} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{41} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{43} - \beta_{3} q^{45} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{47} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{49} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{51} + ( - \beta_{5} - 5 \beta_{3} + \beta_{2} - 1) q^{53} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{55} + q^{57} + (\beta_{5} - \beta_{4} + 11 \beta_{3} + \beta_1) q^{59} + ( - \beta_{5} - 1) q^{63} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{65} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{67} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{69} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{71} + (\beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{73} + ( - \beta_{3} - 1) q^{75} + ( - 5 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} + 5) q^{77} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 1) q^{79} + \beta_{3} q^{81} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{83} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{87} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1 - 2) q^{89} + ( - \beta_{5} + 2 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 4) q^{91} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{93} - \beta_{3} q^{95} + 8 q^{97} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{5} - 3 q^{9} + 3 q^{11} - 6 q^{13} - 6 q^{15} - 6 q^{17} - 3 q^{19} + 3 q^{21} + 3 q^{23} - 3 q^{25} + 6 q^{27} + 12 q^{29} - 12 q^{31} + 3 q^{33} + 3 q^{35} - 3 q^{37} + 3 q^{39} - 18 q^{41} + 3 q^{45} - 3 q^{47} + 12 q^{49} - 6 q^{51} + 15 q^{53} + 6 q^{55} + 6 q^{57} - 30 q^{59} - 3 q^{63} - 3 q^{65} - 6 q^{69} - 24 q^{71} - 18 q^{73} - 3 q^{75} + 33 q^{77} - 6 q^{79} - 3 q^{81} + 12 q^{83} - 12 q^{85} - 6 q^{87} + 30 q^{91} - 12 q^{93} + 3 q^{95} + 48 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10 \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\) \(1\) \(1\)

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
−2.56022 + 0.667305i
−0.0741344 2.64471i
2.63435 + 0.245357i
−2.56022 0.667305i
−0.0741344 + 2.64471i
2.63435 0.245357i
0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −2.56022 0.667305i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.0741344 + 2.64471i 0 −0.500000 + 0.866025i 0
121.3 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 2.63435 0.245357i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −2.56022 + 0.667305i 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −0.0741344 2.64471i 0 −0.500000 0.866025i 0
361.3 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 2.63435 + 0.245357i 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
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.j 6
3.b odd 2 1 2520.2.bi.n 6
4.b odd 2 1 1680.2.bg.v 6
7.c even 3 1 inner 840.2.bg.j 6
7.c even 3 1 5880.2.a.bv 3
7.d odd 6 1 5880.2.a.bu 3
21.h odd 6 1 2520.2.bi.n 6
28.g odd 6 1 1680.2.bg.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.j 6 1.a even 1 1 trivial
840.2.bg.j 6 7.c even 3 1 inner
1680.2.bg.v 6 4.b odd 2 1
1680.2.bg.v 6 28.g odd 6 1
2520.2.bi.n 6 3.b odd 2 1
2520.2.bi.n 6 21.h odd 6 1
5880.2.a.bu 3 7.d odd 6 1
5880.2.a.bv 3 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{6} - 3T_{11}^{5} + 45T_{11}^{4} - 144T_{11}^{3} + 1674T_{11}^{2} - 4536T_{11} + 15876 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{4} - 4 T^{3} - 42 T^{2} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 15876 \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 15 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + 90 T^{4} + \cdots + 102400 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 33 T^{4} + 28 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} - 6 T + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 12 T^{5} + 171 T^{4} + \cdots + 202500 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{5} + 96 T^{4} + \cdots + 145161 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} - 58)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 39 T + 88)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + 81 T^{4} + \cdots + 19600 \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} + 165 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$59$ \( T^{6} + 30 T^{5} + 618 T^{4} + \cdots + 705600 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 39 T^{4} + 176 T^{3} + \cdots + 7744 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 186 T - 2100)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 18 T^{5} + 243 T^{4} + \cdots + 3364 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + 123 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( (T^{3} - 6 T^{2} - 6 T + 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 90 T^{4} + 584 T^{3} + \cdots + 85264 \) Copy content Toggle raw display
$97$ \( (T - 8)^{6} \) Copy content Toggle raw display
show more
show less