Properties

Label 4200.2.a.bk
Level $4200$
Weight $2$
Character orbit 4200.a
Self dual yes
Analytic conductor $33.537$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{7} + q^{9} -2 q^{11} + \beta q^{13} + ( -2 - \beta ) q^{17} + 2 q^{19} - q^{21} -4 q^{23} + q^{27} + ( -4 - \beta ) q^{29} + ( 4 - \beta ) q^{31} -2 q^{33} + ( 2 - \beta ) q^{37} + \beta q^{39} + ( -8 + \beta ) q^{41} + ( -2 + \beta ) q^{43} + ( -2 - \beta ) q^{47} + q^{49} + ( -2 - \beta ) q^{51} -2 q^{53} + 2 q^{57} + ( -8 + \beta ) q^{61} - q^{63} + ( -6 + \beta ) q^{67} -4 q^{69} + ( 8 + \beta ) q^{71} + ( -12 + \beta ) q^{73} + 2 q^{77} + 2 \beta q^{79} + q^{81} + ( -4 + 2 \beta ) q^{83} + ( -4 - \beta ) q^{87} + ( -4 - 3 \beta ) q^{89} -\beta q^{91} + ( 4 - \beta ) q^{93} + ( -8 + \beta ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{33} + 4 q^{37} - 16 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{57} - 16 q^{61} - 2 q^{63} - 12 q^{67} - 8 q^{69} + 16 q^{71} - 24 q^{73} + 4 q^{77} + 2 q^{81} - 8 q^{83} - 8 q^{87} - 8 q^{89} + 8 q^{93} - 16 q^{97} - 4 q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
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
\(n\): 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.bk 2
4.b odd 2 1 8400.2.a.cz 2
5.b even 2 1 4200.2.a.bj 2
5.c odd 4 2 840.2.t.c 4
15.e even 4 2 2520.2.t.f 4
20.d odd 2 1 8400.2.a.db 2
20.e even 4 2 1680.2.t.h 4
60.l odd 4 2 5040.2.t.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 5.c odd 4 2
1680.2.t.h 4 20.e even 4 2
2520.2.t.f 4 15.e even 4 2
4200.2.a.bj 2 5.b even 2 1
4200.2.a.bk 2 1.a even 1 1 trivial
5040.2.t.u 4 60.l odd 4 2
8400.2.a.cz 2 4.b odd 2 1
8400.2.a.db 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\):

\( T_{11} + 2 \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} + 4 T_{17} - 16 \)
\( T_{19} - 2 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -4 + 8 T + T^{2} \)
$31$ \( -4 - 8 T + T^{2} \)
$37$ \( -16 - 4 T + T^{2} \)
$41$ \( 44 + 16 T + T^{2} \)
$43$ \( -16 + 4 T + T^{2} \)
$47$ \( -16 + 4 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 44 + 16 T + T^{2} \)
$67$ \( 16 + 12 T + T^{2} \)
$71$ \( 44 - 16 T + T^{2} \)
$73$ \( 124 + 24 T + T^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -64 + 8 T + T^{2} \)
$89$ \( -164 + 8 T + T^{2} \)
$97$ \( 44 + 16 T + T^{2} \)
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