Properties

Label 840.2.t.d
Level $840$
Weight $2$
Character orbit 840.t
Analytic conductor $6.707$
Analytic rank $0$
Dimension $6$
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.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} + ( 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{13} + \beta_{4} q^{15} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + q^{21} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{25} + \beta_{1} q^{27} + 2 q^{29} + ( -4 - \beta_{2} + \beta_{5} ) q^{31} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{33} -\beta_{4} q^{35} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{37} + ( 4 - \beta_{2} + \beta_{5} ) q^{39} + ( -4 + \beta_{2} - \beta_{5} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{43} + \beta_{2} q^{45} - q^{49} + ( -4 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{51} + ( 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{53} + ( 6 - 8 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{57} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( -2 + 4 \beta_{2} - 4 \beta_{5} ) q^{61} -\beta_{1} q^{63} + ( 2 + 4 \beta_{1} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{69} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{71} + ( 8 \beta_{1} - 2 \beta_{2} + \beta_{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} + ( -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} + ( 2 - 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{85} -2 \beta_{1} q^{87} + ( 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{89} + ( -4 + \beta_{2} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{93} + ( -10 \beta_{1} - \beta_{2} + 2 \beta_{4} + 5 \beta_{5} ) q^{95} + ( -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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{5} - 6q^{9} + O(q^{10}) \) \( 6q - 2q^{5} - 6q^{9} - 4q^{11} + 4q^{19} + 6q^{21} - 2q^{25} + 12q^{29} - 28q^{31} + 20q^{39} - 20q^{41} + 2q^{45} - 6q^{49} - 16q^{51} + 36q^{55} + 16q^{59} + 4q^{61} + 8q^{65} + 16q^{69} - 36q^{71} - 8q^{75} - 8q^{79} + 6q^{81} + 16q^{85} + 12q^{89} - 20q^{91} - 12q^{95} + 4q^{99} + O(q^{100}) \)

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.d 6
3.b odd 2 1 2520.2.t.k 6
4.b odd 2 1 1680.2.t.j 6
5.b even 2 1 inner 840.2.t.d 6
5.c odd 4 1 4200.2.a.bn 3
5.c odd 4 1 4200.2.a.bp 3
12.b even 2 1 5040.2.t.z 6
15.d odd 2 1 2520.2.t.k 6
20.d odd 2 1 1680.2.t.j 6
20.e even 4 1 8400.2.a.di 3
20.e even 4 1 8400.2.a.dl 3
60.h even 2 1 5040.2.t.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.d 6 1.a even 1 1 trivial
840.2.t.d 6 5.b even 2 1 inner
1680.2.t.j 6 4.b odd 2 1
1680.2.t.j 6 20.d odd 2 1
2520.2.t.k 6 3.b odd 2 1
2520.2.t.k 6 15.d odd 2 1
4200.2.a.bn 3 5.c odd 4 1
4200.2.a.bp 3 5.c odd 4 1
5040.2.t.z 6 12.b even 2 1
5040.2.t.z 6 60.h even 2 1
8400.2.a.di 3 20.e even 4 1
8400.2.a.dl 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 \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{3} + 2 T_{11}^{2} - 36 T_{11} - 104 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 50 T + 15 T^{2} + 12 T^{3} + 3 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( -104 - 36 T + 2 T^{2} + T^{3} )^{2} \)
$13$ \( 64 + 560 T^{2} + 60 T^{4} + T^{6} \)
$17$ \( 73984 + 5376 T^{2} + 128 T^{4} + T^{6} \)
$19$ \( ( 200 - 60 T - 2 T^{2} + T^{3} )^{2} \)
$23$ \( 92416 + 6464 T^{2} + 144 T^{4} + T^{6} \)
$29$ \( ( -2 + T )^{6} \)
$31$ \( ( 40 + 52 T + 14 T^{2} + T^{3} )^{2} \)
$37$ \( 102400 + 7936 T^{2} + 176 T^{4} + T^{6} \)
$41$ \( ( -8 + 20 T + 10 T^{2} + T^{3} )^{2} \)
$43$ \( 4096 + 2816 T^{2} + 112 T^{4} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( 1600 + 1584 T^{2} + 92 T^{4} + T^{6} \)
$59$ \( ( 256 - 64 T - 8 T^{2} + T^{3} )^{2} \)
$61$ \( ( 104 - 212 T - 2 T^{2} + T^{3} )^{2} \)
$67$ \( 65536 + 8192 T^{2} + 256 T^{4} + T^{6} \)
$71$ \( ( -1352 - 52 T + 18 T^{2} + T^{3} )^{2} \)
$73$ \( 10816 + 5424 T^{2} + 284 T^{4} + T^{6} \)
$79$ \( ( -64 - 48 T + 4 T^{2} + T^{3} )^{2} \)
$83$ \( 65536 + 8192 T^{2} + 192 T^{4} + T^{6} \)
$89$ \( ( -232 - 124 T - 6 T^{2} + T^{3} )^{2} \)
$97$ \( 40000 + 4400 T^{2} + 124 T^{4} + T^{6} \)
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