Properties

Label 9200.2.a.cd
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{19} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{21} + q^{23} + 3 q^{27} + ( 1 + \beta_{2} ) q^{29} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{31} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{33} -4 q^{37} + ( -5 - 2 \beta_{2} ) q^{39} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 6 + 2 \beta_{2} ) q^{43} + ( 5 + \beta_{2} ) q^{47} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{49} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -2 - 4 \beta_{2} ) q^{53} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 9 - \beta_{1} + \beta_{2} ) q^{61} + ( 4 + 4 \beta_{1} + \beta_{2} ) q^{63} + ( 4 + 4 \beta_{1} - 4 \beta_{2} ) q^{67} + \beta_{1} q^{69} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 5 + 4 \beta_{1} - \beta_{2} ) q^{73} + ( 4 + \beta_{2} ) q^{77} + ( -3 - 3 \beta_{2} ) q^{81} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{87} + ( -10 - 2 \beta_{2} ) q^{89} + ( -5 - \beta_{1} - \beta_{2} ) q^{91} + ( 11 + 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{97} + ( -3 + 3 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{17} - 3 q^{19} + 12 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 6 q^{31} + 15 q^{33} - 12 q^{37} - 15 q^{39} - 6 q^{41} + 18 q^{43} + 15 q^{47} - 6 q^{49} - 15 q^{51} - 6 q^{53} + 6 q^{57} - 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} - 6 q^{71} + 15 q^{73} + 12 q^{77} - 9 q^{81} - 12 q^{83} - 3 q^{87} - 30 q^{89} - 15 q^{91} + 33 q^{93} - 9 q^{97} - 9 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 3\):

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

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
−2.14510
−0.523976
2.66908
0 −2.14510 0 0 0 −1.14510 0 1.60147 0
1.2 0 −0.523976 0 0 0 0.476024 0 −2.72545 0
1.3 0 2.66908 0 0 0 3.66908 0 4.12398 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 9200.2.a.cd 3
4.b odd 2 1 4600.2.a.y 3
5.b even 2 1 1840.2.a.t 3
20.d odd 2 1 920.2.a.g 3
20.e even 4 2 4600.2.e.r 6
40.e odd 2 1 7360.2.a.cb 3
40.f even 2 1 7360.2.a.ca 3
60.h even 2 1 8280.2.a.bo 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 20.d odd 2 1
1840.2.a.t 3 5.b even 2 1
4600.2.a.y 3 4.b odd 2 1
4600.2.e.r 6 20.e even 4 2
7360.2.a.ca 3 40.f even 2 1
7360.2.a.cb 3 40.e odd 2 1
8280.2.a.bo 3 60.h even 2 1
9200.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{3} - 6 T_{3} - 3 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 3 T_{7} + 2 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 15 T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -3 - 6 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 2 - 3 T - 3 T^{2} + T^{3} \)
$11$ \( 12 - 15 T + 3 T^{2} + T^{3} \)
$13$ \( -29 - 18 T + T^{3} \)
$17$ \( -12 - 15 T - 3 T^{2} + T^{3} \)
$19$ \( 48 - 33 T + 3 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 12 - 6 T - 3 T^{2} + T^{3} \)
$31$ \( -249 - 42 T + 6 T^{2} + T^{3} \)
$37$ \( ( 4 + T )^{3} \)
$41$ \( -69 - 60 T + 6 T^{2} + T^{3} \)
$43$ \( 32 + 72 T - 18 T^{2} + T^{3} \)
$47$ \( -76 + 66 T - 15 T^{2} + T^{3} \)
$53$ \( -536 - 132 T + 6 T^{2} + T^{3} \)
$59$ \( -32 - 36 T + 6 T^{2} + T^{3} \)
$61$ \( -596 + 225 T - 27 T^{2} + T^{3} \)
$67$ \( 2944 - 240 T - 12 T^{2} + T^{3} \)
$71$ \( -203 - 60 T + 6 T^{2} + T^{3} \)
$73$ \( 588 - 42 T - 15 T^{2} + T^{3} \)
$79$ \( T^{3} \)
$83$ \( -64 - 60 T + 12 T^{2} + T^{3} \)
$89$ \( 608 + 264 T + 30 T^{2} + T^{3} \)
$97$ \( -138 - 69 T + 9 T^{2} + T^{3} \)
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