Properties

Label 8400.2.a.dl
Level $8400$
Weight $2$
Character orbit 8400.a
Self dual yes
Analytic conductor $67.074$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{7} + q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + ( 3 - \beta_{1} ) q^{13} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{19} + q^{21} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{23} + q^{27} -2 q^{29} + ( 5 + \beta_{1} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( 4 + 2 \beta_{2} ) q^{37} + ( 3 - \beta_{1} ) q^{39} + ( -3 + \beta_{1} ) q^{41} + ( -2 - 2 \beta_{1} ) q^{43} + q^{49} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{51} + ( 5 + \beta_{1} ) q^{53} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 + 4 \beta_{1} ) q^{61} + q^{63} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{69} + ( 7 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 7 - \beta_{1} - 2 \beta_{2} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} ) q^{79} + q^{81} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} -2 q^{87} + ( -3 - 3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 3 - \beta_{1} ) q^{91} + ( 5 + \beta_{1} ) q^{93} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{7} + 3q^{9} + 2q^{11} + 10q^{13} + 8q^{17} + 2q^{19} + 3q^{21} - 8q^{23} + 3q^{27} - 6q^{29} + 14q^{31} + 2q^{33} + 12q^{37} + 10q^{39} - 10q^{41} - 4q^{43} + 3q^{49} + 8q^{51} + 14q^{53} + 2q^{57} + 8q^{59} + 2q^{61} + 3q^{63} + 16q^{67} - 8q^{69} + 18q^{71} + 22q^{73} + 2q^{77} - 4q^{79} + 3q^{81} - 8q^{83} - 6q^{87} - 6q^{89} + 10q^{91} + 14q^{93} + 2q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
2.17009
0.311108
0 1.00000 0 0 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 0 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8400.2.a.dl 3
4.b odd 2 1 4200.2.a.bn 3
5.b even 2 1 8400.2.a.di 3
5.c odd 4 2 1680.2.t.j 6
15.e even 4 2 5040.2.t.z 6
20.d odd 2 1 4200.2.a.bp 3
20.e even 4 2 840.2.t.d 6
60.l odd 4 2 2520.2.t.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.d 6 20.e even 4 2
1680.2.t.j 6 5.c odd 4 2
2520.2.t.k 6 60.l odd 4 2
4200.2.a.bn 3 4.b odd 2 1
4200.2.a.bp 3 20.d odd 2 1
5040.2.t.z 6 15.e even 4 2
8400.2.a.di 3 5.b even 2 1
8400.2.a.dl 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11}^{3} - 2 T_{11}^{2} - 36 T_{11} + 104 \)
\( T_{13}^{3} - 10 T_{13}^{2} + 20 T_{13} + 8 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 32 T_{17} + 272 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200 \)
\( T_{23}^{3} + 8 T_{23}^{2} - 40 T_{23} - 304 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 104 - 36 T - 2 T^{2} + T^{3} \)
$13$ \( 8 + 20 T - 10 T^{2} + T^{3} \)
$17$ \( 272 - 32 T - 8 T^{2} + T^{3} \)
$19$ \( 200 - 60 T - 2 T^{2} + T^{3} \)
$23$ \( -304 - 40 T + 8 T^{2} + T^{3} \)
$29$ \( ( 2 + T )^{3} \)
$31$ \( -40 + 52 T - 14 T^{2} + T^{3} \)
$37$ \( 320 - 16 T - 12 T^{2} + T^{3} \)
$41$ \( -8 + 20 T + 10 T^{2} + T^{3} \)
$43$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$47$ \( T^{3} \)
$53$ \( -40 + 52 T - 14 T^{2} + T^{3} \)
$59$ \( 256 - 64 T - 8 T^{2} + T^{3} \)
$61$ \( 104 - 212 T - 2 T^{2} + T^{3} \)
$67$ \( 256 - 16 T^{2} + T^{3} \)
$71$ \( 1352 - 52 T - 18 T^{2} + T^{3} \)
$73$ \( -104 + 100 T - 22 T^{2} + T^{3} \)
$79$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$83$ \( -256 - 64 T + 8 T^{2} + T^{3} \)
$89$ \( 232 - 124 T + 6 T^{2} + T^{3} \)
$97$ \( 200 - 60 T - 2 T^{2} + T^{3} \)
show more
show less