Properties

Label 9200.2.a.cf
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: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} ) q^{13} + ( 2 + \beta_{1} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} ) q^{19} + ( -9 + 2 \beta_{1} - 3 \beta_{2} ) q^{21} - q^{23} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{27} + ( -2 + 2 \beta_{1} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} ) q^{31} + ( 3 - 3 \beta_{1} - 3 \beta_{2} ) q^{33} -2 \beta_{2} q^{37} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} ) q^{41} + 8 q^{43} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 11 - \beta_{1} + 2 \beta_{2} ) q^{49} + ( 7 + \beta_{1} + \beta_{2} ) q^{51} + 6 q^{53} + ( 9 - 2 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -4 - 2 \beta_{1} ) q^{59} + ( 2 - \beta_{1} + 4 \beta_{2} ) q^{61} + ( 5 - 8 \beta_{1} - \beta_{2} ) q^{63} + ( 4 + 4 \beta_{2} ) q^{67} -\beta_{1} q^{69} + ( -4 + \beta_{1} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 5 + 8 \beta_{1} - \beta_{2} ) q^{77} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{79} + ( 4 - 4 \beta_{1} + \beta_{2} ) q^{81} + ( 2 - 2 \beta_{2} ) q^{83} + ( 14 - 4 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 4 + 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 - 5 \beta_{1} + 2 \beta_{2} ) q^{91} + ( 5 - \beta_{2} ) q^{93} + ( 10 + 3 \beta_{1} ) q^{97} + ( -21 + 3 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 3q^{7} + 10q^{9} + O(q^{10}) \) \( 3q + q^{3} + 3q^{7} + 10q^{9} - 3q^{11} + q^{13} + 7q^{17} - 3q^{19} - 22q^{21} - 3q^{23} - 14q^{27} - 4q^{29} + 5q^{31} + 9q^{33} + 2q^{37} - 14q^{39} + q^{41} + 24q^{43} - 14q^{47} + 30q^{49} + 21q^{51} + 18q^{53} + 22q^{57} - 14q^{59} + q^{61} + 8q^{63} + 8q^{67} - q^{69} - 11q^{71} + 8q^{73} + 24q^{77} + 4q^{79} + 7q^{81} + 8q^{83} + 36q^{87} + 18q^{89} - q^{91} + 16q^{93} + 33q^{97} - 57q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 9 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 7\)

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
−3.11903
1.43163
2.68740
0 −3.11903 0 0 0 4.50973 0 6.72833 0
1.2 0 1.43163 0 0 0 3.08719 0 −0.950444 0
1.3 0 2.68740 0 0 0 −4.59692 0 4.22212 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.cf 3
4.b odd 2 1 1150.2.a.q 3
5.b even 2 1 1840.2.a.r 3
20.d odd 2 1 230.2.a.d 3
20.e even 4 2 1150.2.b.j 6
40.e odd 2 1 7360.2.a.bz 3
40.f even 2 1 7360.2.a.ce 3
60.h even 2 1 2070.2.a.z 3
460.g even 2 1 5290.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 20.d odd 2 1
1150.2.a.q 3 4.b odd 2 1
1150.2.b.j 6 20.e even 4 2
1840.2.a.r 3 5.b even 2 1
2070.2.a.z 3 60.h even 2 1
5290.2.a.r 3 460.g even 2 1
7360.2.a.bz 3 40.e odd 2 1
7360.2.a.ce 3 40.f even 2 1
9200.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{3} - T_{3}^{2} - 9 T_{3} + 12 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 21 T_{7} + 64 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 39 T_{11} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 12 - 9 T - T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 64 - 21 T - 3 T^{2} + T^{3} \)
$11$ \( -144 - 39 T + 3 T^{2} + T^{3} \)
$13$ \( 18 - 15 T - T^{2} + T^{3} \)
$17$ \( 18 + 7 T - 7 T^{2} + T^{3} \)
$19$ \( -64 - 21 T + 3 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 24 - 32 T + 4 T^{2} + T^{3} \)
$31$ \( 8 - 7 T - 5 T^{2} + T^{3} \)
$37$ \( 32 - 40 T - 2 T^{2} + T^{3} \)
$41$ \( 186 - 59 T - T^{2} + T^{3} \)
$43$ \( ( -8 + T )^{3} \)
$47$ \( -288 + 4 T + 14 T^{2} + T^{3} \)
$53$ \( ( -6 + T )^{3} \)
$59$ \( -144 + 28 T + 14 T^{2} + T^{3} \)
$61$ \( 526 - 157 T - T^{2} + T^{3} \)
$67$ \( 384 - 144 T - 8 T^{2} + T^{3} \)
$71$ \( 24 + 31 T + 11 T^{2} + T^{3} \)
$73$ \( 248 - 40 T - 8 T^{2} + T^{3} \)
$79$ \( 1152 - 240 T - 4 T^{2} + T^{3} \)
$83$ \( 96 - 20 T - 8 T^{2} + T^{3} \)
$89$ \( 1152 - 48 T - 18 T^{2} + T^{3} \)
$97$ \( -166 + 279 T - 33 T^{2} + T^{3} \)
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