Properties

Label 840.2.t.c
Level $840$
Weight $2$
Character orbit 840.t
Analytic conductor $6.707$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - q^{9} - 2 q^{11} + 2 \beta_{2} q^{13} + \beta_{2} q^{15} + (2 \beta_{2} - 2 \beta_1) q^{17} - 2 q^{19} - q^{21} + 4 \beta_1 q^{23} + 5 q^{25} + \beta_1 q^{27} + ( - 2 \beta_{3} + 4) q^{29} + (2 \beta_{3} + 4) q^{31} + 2 \beta_1 q^{33} + \beta_{2} q^{35} + (2 \beta_{2} + 2 \beta_1) q^{37} + 2 \beta_{3} q^{39} + ( - 2 \beta_{3} - 8) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + \beta_{3} q^{45} + (2 \beta_{2} - 2 \beta_1) q^{47} - q^{49} + (2 \beta_{3} - 2) q^{51} + 2 \beta_1 q^{53} + 2 \beta_{3} q^{55} + 2 \beta_1 q^{57} + ( - 2 \beta_{3} - 8) q^{61} + \beta_1 q^{63} - 10 \beta_1 q^{65} + ( - 2 \beta_{2} - 6 \beta_1) q^{67} + 4 q^{69} + ( - 2 \beta_{3} + 8) q^{71} + (2 \beta_{2} + 12 \beta_1) q^{73} - 5 \beta_1 q^{75} + 2 \beta_1 q^{77} + 4 \beta_{3} q^{79} + q^{81} + (4 \beta_{2} + 4 \beta_1) q^{83} + (2 \beta_{2} - 10 \beta_1) q^{85} + (2 \beta_{2} - 4 \beta_1) q^{87} + ( - 6 \beta_{3} + 4) q^{89} + 2 \beta_{3} q^{91} + ( - 2 \beta_{2} - 4 \beta_1) q^{93} + 2 \beta_{3} q^{95} + ( - 2 \beta_{2} - 8 \beta_1) q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 16 q^{31} - 32 q^{41} - 4 q^{49} - 8 q^{51} - 32 q^{61} + 16 q^{69} + 32 q^{71} + 4 q^{81} + 16 q^{89} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

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

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

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
0.618034i
1.61803i
0.618034i
1.61803i
0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 0
169.3 0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
169.4 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 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
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.c 4
3.b odd 2 1 2520.2.t.f 4
4.b odd 2 1 1680.2.t.h 4
5.b even 2 1 inner 840.2.t.c 4
5.c odd 4 1 4200.2.a.bj 2
5.c odd 4 1 4200.2.a.bk 2
12.b even 2 1 5040.2.t.u 4
15.d odd 2 1 2520.2.t.f 4
20.d odd 2 1 1680.2.t.h 4
20.e even 4 1 8400.2.a.cz 2
20.e even 4 1 8400.2.a.db 2
60.h even 2 1 5040.2.t.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 1.a even 1 1 trivial
840.2.t.c 4 5.b even 2 1 inner
1680.2.t.h 4 4.b odd 2 1
1680.2.t.h 4 20.d odd 2 1
2520.2.t.f 4 3.b odd 2 1
2520.2.t.f 4 15.d odd 2 1
4200.2.a.bj 2 5.c odd 4 1
4200.2.a.bk 2 5.c odd 4 1
5040.2.t.u 4 12.b even 2 1
5040.2.t.u 4 60.h even 2 1
8400.2.a.cz 2 20.e even 4 1
8400.2.a.db 2 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 \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16 T + 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 16 T + 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 328 T^{2} + 15376 \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 164)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
show more
show less