Properties

Label 840.2.t.e
Level $840$
Weight $2$
Character orbit 840.t
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.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{1} q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{1} q^{7} - q^{9} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + ( -\beta_{3} + \beta_{4} ) q^{13} -\beta_{4} q^{15} + ( -\beta_{2} - \beta_{5} ) q^{17} + ( -\beta_{2} + \beta_{5} ) q^{19} + q^{21} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{25} -\beta_{1} q^{27} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -\beta_{2} + \beta_{5} ) q^{31} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{33} + \beta_{4} q^{35} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( -\beta_{2} + \beta_{5} ) q^{39} + ( 4 + \beta_{2} - \beta_{5} ) q^{41} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{43} + \beta_{2} q^{45} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} - q^{49} + ( -\beta_{3} - \beta_{4} ) q^{51} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{53} + ( -2 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{55} + ( \beta_{3} - \beta_{4} ) q^{57} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{61} + \beta_{1} q^{63} + ( -2 - 4 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{65} + ( -2 - \beta_{2} + \beta_{5} ) q^{69} + ( -3 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{71} + ( 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{75} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{77} + ( 8 - 2 \beta_{2} + 2 \beta_{5} ) q^{79} + q^{81} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -6 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( \beta_{2} - \beta_{5} ) q^{91} + ( \beta_{3} - \beta_{4} ) q^{93} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{95} + ( 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{97} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{5} - 6q^{9} + O(q^{10}) \) \( 6q - 2q^{5} - 6q^{9} + 4q^{11} - 4q^{19} + 6q^{21} - 2q^{25} - 4q^{29} - 4q^{31} - 4q^{39} + 28q^{41} + 2q^{45} - 6q^{49} - 20q^{55} - 16q^{59} + 4q^{61} - 8q^{65} - 16q^{69} - 12q^{71} + 8q^{75} + 40q^{79} + 6q^{81} - 32q^{85} - 4q^{89} + 4q^{91} + 28q^{95} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -7 \nu^{5} + 10 \nu^{4} - 5 \nu^{3} - 30 \nu^{2} - 32 \nu + 13 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{5} + 3 \nu^{4} + 10 \nu^{3} - 32 \nu^{2} - 74 \nu - 3 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( -10 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} - 10 \nu^{2} - 72 \nu - 11 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( 12 \nu^{5} - 27 \nu^{4} + 25 \nu^{3} + 12 \nu^{2} + 68 \nu - 65 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -19 \nu^{5} + 37 \nu^{4} - 30 \nu^{3} - 42 \nu^{2} - 54 \nu + 55 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{2} - 4 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} - 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} - 5 \beta_{4} - 5 \beta_{3} + \beta_{2} - 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 18 \beta_{1} - 18\)\()/2\)

Character values

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

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 0.403032i
0 1.00000i 0 −2.17009 + 0.539189i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 −0.311108 2.21432i 0 1.00000i 0 −1.00000 0
169.3 0 1.00000i 0 1.48119 + 1.67513i 0 1.00000i 0 −1.00000 0
169.4 0 1.00000i 0 −2.17009 0.539189i 0 1.00000i 0 −1.00000 0
169.5 0 1.00000i 0 −0.311108 + 2.21432i 0 1.00000i 0 −1.00000 0
169.6 0 1.00000i 0 1.48119 1.67513i 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.t.e 6
3.b odd 2 1 2520.2.t.j 6
4.b odd 2 1 1680.2.t.i 6
5.b even 2 1 inner 840.2.t.e 6
5.c odd 4 1 4200.2.a.bo 3
5.c odd 4 1 4200.2.a.bq 3
12.b even 2 1 5040.2.t.ba 6
15.d odd 2 1 2520.2.t.j 6
20.d odd 2 1 1680.2.t.i 6
20.e even 4 1 8400.2.a.dh 3
20.e even 4 1 8400.2.a.dk 3
60.h even 2 1 5040.2.t.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 1.a even 1 1 trivial
840.2.t.e 6 5.b even 2 1 inner
1680.2.t.i 6 4.b odd 2 1
1680.2.t.i 6 20.d odd 2 1
2520.2.t.j 6 3.b odd 2 1
2520.2.t.j 6 15.d odd 2 1
4200.2.a.bo 3 5.c odd 4 1
4200.2.a.bq 3 5.c odd 4 1
5040.2.t.ba 6 12.b even 2 1
5040.2.t.ba 6 60.h even 2 1
8400.2.a.dh 3 20.e even 4 1
8400.2.a.dk 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{3} - 2 T_{11}^{2} - 20 T_{11} + 8 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 12 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 50 T + 15 T^{2} + 12 T^{3} + 3 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 8 - 20 T - 2 T^{2} + T^{3} )^{2} \)
$13$ \( 64 + 176 T^{2} + 28 T^{4} + T^{6} \)
$17$ \( 256 + 256 T^{2} + 32 T^{4} + T^{6} \)
$19$ \( ( -8 - 12 T + 2 T^{2} + T^{3} )^{2} \)
$23$ \( 256 + 320 T^{2} + 48 T^{4} + T^{6} \)
$29$ \( ( -104 - 84 T + 2 T^{2} + T^{3} )^{2} \)
$31$ \( ( -8 - 12 T + 2 T^{2} + T^{3} )^{2} \)
$37$ \( 4096 + 2816 T^{2} + 112 T^{4} + T^{6} \)
$41$ \( ( -40 + 52 T - 14 T^{2} + T^{3} )^{2} \)
$43$ \( 102400 + 7936 T^{2} + 176 T^{4} + T^{6} \)
$47$ \( 65536 + 8192 T^{2} + 192 T^{4} + T^{6} \)
$53$ \( 222784 + 14000 T^{2} + 252 T^{4} + T^{6} \)
$59$ \( ( -256 - 64 T + 8 T^{2} + T^{3} )^{2} \)
$61$ \( ( -536 - 148 T - 2 T^{2} + T^{3} )^{2} \)
$67$ \( T^{6} \)
$71$ \( ( 200 - 100 T + 6 T^{2} + T^{3} )^{2} \)
$73$ \( 577600 + 24496 T^{2} + 284 T^{4} + T^{6} \)
$79$ \( ( 64 + 80 T - 20 T^{2} + T^{3} )^{2} \)
$83$ \( 65536 + 8192 T^{2} + 192 T^{4} + T^{6} \)
$89$ \( ( -200 - 60 T + 2 T^{2} + T^{3} )^{2} \)
$97$ \( 118336 + 9648 T^{2} + 188 T^{4} + T^{6} \)
show more
show less