Properties

Label 9200.2.a.ct
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.521397.1
Defining polynomial: \(x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{4} q^{11} + ( 1 + \beta_{2} - \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{17} + \beta_{4} q^{19} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + q^{23} + ( -2 + \beta_{1} - \beta_{3} ) q^{27} + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{29} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{31} + ( 1 - \beta_{3} + \beta_{4} ) q^{33} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{37} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{39} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{43} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{47} + ( 3 + 2 \beta_{1} - \beta_{4} ) q^{49} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{53} + ( 1 - \beta_{3} + \beta_{4} ) q^{57} + ( -\beta_{3} - 2 \beta_{4} ) q^{59} + ( 1 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{61} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{63} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{67} -\beta_{1} q^{69} + ( -4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( -1 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{73} + ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{77} + ( -1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{79} + ( -6 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} ) q^{83} + ( -9 - \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{87} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 8 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{91} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{93} + ( 1 - \beta_{3} + 3 \beta_{4} ) q^{97} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 5q - 4q^{7} + 3q^{9} + 4q^{13} + 6q^{17} + 5q^{23} - 9q^{27} + 12q^{29} + 18q^{31} + 6q^{33} + 10q^{37} - 9q^{39} - 6q^{41} - 10q^{43} - 22q^{47} + 15q^{49} + 6q^{51} + 10q^{53} + 6q^{57} + q^{59} + 10q^{61} - 8q^{67} - 8q^{71} + 6q^{73} - 27q^{81} + 2q^{83} - 39q^{87} + 14q^{89} + 46q^{91} - 3q^{93} + 6q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 7 \nu^{2} + 4 \nu + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 21\)

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
2.61696
1.83957
−0.794805
−1.36629
−2.29544
0 −2.61696 0 0 0 −3.83744 0 3.84849 0
1.2 0 −1.83957 0 0 0 3.97272 0 0.384010 0
1.3 0 0.794805 0 0 0 −2.47193 0 −2.36829 0
1.4 0 1.36629 0 0 0 −3.28093 0 −1.13327 0
1.5 0 2.29544 0 0 0 1.61758 0 2.26905 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.ct 5
4.b odd 2 1 4600.2.a.bf yes 5
5.b even 2 1 9200.2.a.cv 5
20.d odd 2 1 4600.2.a.bd 5
20.e even 4 2 4600.2.e.w 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bd 5 20.d odd 2 1
4600.2.a.bf yes 5 4.b odd 2 1
4600.2.e.w 10 20.e even 4 2
9200.2.a.ct 5 1.a even 1 1 trivial
9200.2.a.cv 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{5} - 9 T_{3}^{3} + 3 T_{3}^{2} + 18 T_{3} - 12 \)
\( T_{7}^{5} + 4 T_{7}^{4} - 17 T_{7}^{3} - 76 T_{7}^{2} + 20 T_{7} + 200 \)
\( T_{11}^{5} - 15 T_{11}^{3} + 6 T_{11}^{2} + 48 T_{11} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -12 + 18 T + 3 T^{2} - 9 T^{3} + T^{5} \)
$5$ \( T^{5} \)
$7$ \( 200 + 20 T - 76 T^{2} - 17 T^{3} + 4 T^{4} + T^{5} \)
$11$ \( -24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5} \)
$13$ \( -347 + 287 T + 91 T^{2} - 32 T^{3} - 4 T^{4} + T^{5} \)
$17$ \( 384 - 672 T + 360 T^{2} - 48 T^{3} - 6 T^{4} + T^{5} \)
$19$ \( -24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( -2787 - 273 T + 339 T^{2} - 12 T^{4} + T^{5} \)
$31$ \( 228 - 516 T - 45 T^{2} + 93 T^{3} - 18 T^{4} + T^{5} \)
$37$ \( -608 - 928 T + 496 T^{2} - 32 T^{3} - 10 T^{4} + T^{5} \)
$41$ \( 453 + 1311 T - 237 T^{2} - 66 T^{3} + 6 T^{4} + T^{5} \)
$43$ \( 1472 - 112 T - 352 T^{2} - 23 T^{3} + 10 T^{4} + T^{5} \)
$47$ \( -4 - 334 T - 301 T^{2} + 97 T^{3} + 22 T^{4} + T^{5} \)
$53$ \( 256 + 704 T + 400 T^{2} - 92 T^{3} - 10 T^{4} + T^{5} \)
$59$ \( 3004 + 1322 T - 107 T^{2} - 83 T^{3} - T^{4} + T^{5} \)
$61$ \( 7648 + 12800 T + 1072 T^{2} - 200 T^{3} - 10 T^{4} + T^{5} \)
$67$ \( -9728 - 8704 T - 2408 T^{2} - 188 T^{3} + 8 T^{4} + T^{5} \)
$71$ \( -8084 + 5960 T - 689 T^{2} - 179 T^{3} + 8 T^{4} + T^{5} \)
$73$ \( 28569 + 10407 T + 39 T^{2} - 222 T^{3} - 6 T^{4} + T^{5} \)
$79$ \( 1728 + 5184 T + 36 T^{2} - 153 T^{3} + T^{5} \)
$83$ \( -2392 + 524 T + 1112 T^{2} - 191 T^{3} - 2 T^{4} + T^{5} \)
$89$ \( 3104 - 4864 T + 1280 T^{2} - 44 T^{3} - 14 T^{4} + T^{5} \)
$97$ \( 96 + 576 T + 816 T^{2} - 144 T^{3} - 6 T^{4} + T^{5} \)
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