Properties

Label 4200.2.a.bq
Level $4200$
Weight $2$
Character orbit 4200.a
Self dual yes
Analytic conductor $33.537$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.5371688489\)
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.

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

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
0.311108
−1.48119
2.17009
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 4200.2.a.bq 3
4.b odd 2 1 8400.2.a.dh 3
5.b even 2 1 4200.2.a.bo 3
5.c odd 4 2 840.2.t.e 6
15.e even 4 2 2520.2.t.j 6
20.d odd 2 1 8400.2.a.dk 3
20.e even 4 2 1680.2.t.i 6
60.l odd 4 2 5040.2.t.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 5.c odd 4 2
1680.2.t.i 6 20.e even 4 2
2520.2.t.j 6 15.e even 4 2
4200.2.a.bo 3 5.b even 2 1
4200.2.a.bq 3 1.a even 1 1 trivial
5040.2.t.ba 6 60.l odd 4 2
8400.2.a.dh 3 4.b odd 2 1
8400.2.a.dk 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\):

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

Hecke characteristic polynomials

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