Properties

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

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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: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{3} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{7} + q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{11} + (\beta_1 + 1) q^{13} + \beta_{2} q^{17} + ( - \beta_1 - 1) q^{19} + q^{21} + ( - \beta_1 - 3) q^{23} - q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_1 + 1) q^{31} + (\beta_{2} + \beta_1 + 1) q^{33} + ( - 2 \beta_1 - 2) q^{37} + ( - \beta_1 - 1) q^{39} + (\beta_1 + 5) q^{41} + ( - 2 \beta_{2} - 4) q^{43} + (2 \beta_{2} + 2 \beta_1 - 2) q^{47} + q^{49} - \beta_{2} q^{51} + ( - 2 \beta_{2} + \beta_1 + 5) q^{53} + (\beta_1 + 1) q^{57} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (2 \beta_{2} - 2 \beta_1) q^{61} - q^{63} + (\beta_1 + 3) q^{69} + (\beta_{2} + 3 \beta_1 + 3) q^{71} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{73} + (\beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_1 + 6) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + (2 \beta_{2} + 2 \beta_1) q^{87} + (2 \beta_{2} + \beta_1 + 1) q^{89} + ( - \beta_1 - 1) q^{91} + ( - \beta_1 - 1) q^{93} + ( - 2 \beta_{2} - \beta_1 - 5) q^{97} + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 4 q^{37} - 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} + 14 q^{53} + 2 q^{57} - 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} + 6 q^{71} + 6 q^{73} + 2 q^{77} + 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} - 2 q^{91} - 2 q^{93} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display

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
2.17009
−1.48119
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.dh 3
4.b odd 2 1 4200.2.a.bq 3
5.b even 2 1 8400.2.a.dk 3
5.c odd 4 2 1680.2.t.i 6
15.e even 4 2 5040.2.t.ba 6
20.d odd 2 1 4200.2.a.bo 3
20.e even 4 2 840.2.t.e 6
60.l odd 4 2 2520.2.t.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 20.e even 4 2
1680.2.t.i 6 5.c odd 4 2
2520.2.t.j 6 60.l odd 4 2
4200.2.a.bo 3 20.d odd 2 1
4200.2.a.bq 3 4.b odd 2 1
5040.2.t.ba 6 15.e even 4 2
8400.2.a.dh 3 1.a even 1 1 trivial
8400.2.a.dk 3 5.b even 2 1

Hecke kernels

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

\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 12T_{13} + 8 \) Copy content Toggle raw display
\( T_{17}^{3} - 16T_{17} + 16 \) Copy content Toggle raw display
\( T_{19}^{3} + 2T_{19}^{2} - 12T_{19} - 8 \) Copy content Toggle raw display
\( T_{23}^{3} + 8T_{23}^{2} + 8T_{23} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 12 T - 8 \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} + 8 T - 16 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 84 T + 104 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 52 T - 40 \) Copy content Toggle raw display
$43$ \( T^{3} + 12 T^{2} - 16 T - 320 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 64 T - 256 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} - 28 T + 472 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 64 T - 256 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 148 T - 536 \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 100 T - 200 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 124 T + 760 \) Copy content Toggle raw display
$79$ \( T^{3} - 20 T^{2} + 80 T + 64 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 64 T - 256 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 60 T + 200 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} + 4 T - 344 \) Copy content Toggle raw display
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