Properties

Label 9200.2.a.cx
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.143376304.1
Defining polynomial: \(x^{6} - 12 x^{4} + 22 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 460)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -2 + \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -2 + \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{13} + ( 1 - \beta_{2} - \beta_{5} ) q^{17} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{19} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} - q^{23} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{27} + ( 2 + 2 \beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{33} + ( 2 - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 1 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{43} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{49} + ( 2 - \beta_{3} - \beta_{4} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{53} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{57} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{59} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{61} + ( -3 + \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{63} + ( -4 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{67} + ( 1 + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{71} + ( -3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{73} + ( 4 - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( 1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{81} + ( -3 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{83} + ( -11 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{87} + ( -5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( 1 - 2 \beta_{1} + 7 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{91} + ( -1 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{93} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{97} + ( 4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{3} - 9q^{7} + 10q^{9} + O(q^{10}) \) \( 6q - 4q^{3} - 9q^{7} + 10q^{9} - 2q^{11} + 8q^{13} + 5q^{17} - 4q^{19} - 6q^{23} - 22q^{27} + 5q^{29} - 9q^{31} - 10q^{33} + 21q^{37} + 8q^{39} - q^{41} - 16q^{43} - 16q^{47} + 19q^{49} + 12q^{51} - q^{53} + 12q^{57} + 11q^{59} - 4q^{61} - 19q^{63} - 25q^{67} + 4q^{69} + 17q^{71} - 14q^{73} + 20q^{77} - 10q^{79} + 14q^{81} - 21q^{83} - 64q^{87} - 24q^{89} + 4q^{91} + 4q^{97} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 12 x^{4} + 22 x^{2} - 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{4} + 11 \nu^{3} + 33 \nu^{2} - 3 \nu - 27 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{5} + \nu^{4} - 33 \nu^{3} - 11 \nu^{2} + 41 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{5} + 3 \nu^{4} + 85 \nu^{3} - 25 \nu^{2} - 157 \nu + 43 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( 9 \nu^{5} + 3 \nu^{4} - 107 \nu^{3} - 41 \nu^{2} + 195 \nu + 11 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{5} - 3 \nu^{4} + 107 \nu^{3} + 25 \nu^{2} - 179 \nu + 53 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{1} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-22 \beta_{5} - 7 \beta_{4} + 15 \beta_{3} - 2 \beta_{2} + 9 \beta_{1} + 66\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(22 \beta_{5} + 51 \beta_{4} + 73 \beta_{3} + 72 \beta_{2} + 75 \beta_{1} + 12\)\()/2\)

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
3.16223
−1.65047
0.420790
−0.116918
−3.08006
1.26443
0 −3.21923 0 0 0 2.43185 0 7.36343 0
1.2 0 −2.80150 0 0 0 −4.50896 0 4.84843 0
1.3 0 −1.73961 0 0 0 −3.32224 0 0.0262434 0
1.4 0 0.486391 0 0 0 1.80495 0 −2.76342 0
1.5 0 0.873449 0 0 0 −0.992530 0 −2.23709 0
1.6 0 2.40050 0 0 0 −4.41307 0 2.76241 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cx 6
4.b odd 2 1 2300.2.a.o 6
5.b even 2 1 9200.2.a.cy 6
5.c odd 4 2 1840.2.e.f 12
20.d odd 2 1 2300.2.a.n 6
20.e even 4 2 460.2.c.a 12
60.l odd 4 2 4140.2.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 20.e even 4 2
1840.2.e.f 12 5.c odd 4 2
2300.2.a.n 6 20.d odd 2 1
2300.2.a.o 6 4.b odd 2 1
4140.2.f.b 12 60.l odd 4 2
9200.2.a.cx 6 1.a even 1 1 trivial
9200.2.a.cy 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{6} + 4 T_{3}^{5} - 6 T_{3}^{4} - 30 T_{3}^{3} + 5 T_{3}^{2} + 38 T_{3} - 16 \)
\( T_{7}^{6} + 9 T_{7}^{5} + 10 T_{7}^{4} - 88 T_{7}^{3} - 152 T_{7}^{2} + 228 T_{7} + 288 \)
\( T_{11}^{6} + 2 T_{11}^{5} - 32 T_{11}^{4} - 28 T_{11}^{3} + 212 T_{11}^{2} + 144 T_{11} - 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -16 + 38 T + 5 T^{2} - 30 T^{3} - 6 T^{4} + 4 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 288 + 228 T - 152 T^{2} - 88 T^{3} + 10 T^{4} + 9 T^{5} + T^{6} \)
$11$ \( -256 + 144 T + 212 T^{2} - 28 T^{3} - 32 T^{4} + 2 T^{5} + T^{6} \)
$13$ \( 1184 - 1354 T + 45 T^{2} + 222 T^{3} - 22 T^{4} - 8 T^{5} + T^{6} \)
$17$ \( 16 - 60 T + 36 T^{2} + 30 T^{3} - 16 T^{4} - 5 T^{5} + T^{6} \)
$19$ \( 256 + 848 T + 236 T^{2} - 140 T^{3} - 42 T^{4} + 4 T^{5} + T^{6} \)
$23$ \( ( 1 + T )^{6} \)
$29$ \( 11862 - 9453 T + 1229 T^{2} + 450 T^{3} - 84 T^{4} - 5 T^{5} + T^{6} \)
$31$ \( 916 - 1987 T - 2651 T^{2} - 838 T^{3} - 58 T^{4} + 9 T^{5} + T^{6} \)
$37$ \( 63216 - 12012 T - 7096 T^{2} + 1444 T^{3} + 42 T^{4} - 21 T^{5} + T^{6} \)
$41$ \( -2 + 477 T + 257 T^{2} - 158 T^{3} - 64 T^{4} + T^{5} + T^{6} \)
$43$ \( 91648 + 17072 T - 6612 T^{2} - 1764 T^{3} - 42 T^{4} + 16 T^{5} + T^{6} \)
$47$ \( -464 + 2178 T - 1959 T^{2} - 678 T^{3} + 14 T^{4} + 16 T^{5} + T^{6} \)
$53$ \( 12224 - 13552 T + 3648 T^{2} + 280 T^{3} - 136 T^{4} + T^{5} + T^{6} \)
$59$ \( -360576 - 63024 T + 12832 T^{2} + 1692 T^{3} - 188 T^{4} - 11 T^{5} + T^{6} \)
$61$ \( -171088 + 10976 T + 10996 T^{2} - 532 T^{3} - 198 T^{4} + 4 T^{5} + T^{6} \)
$67$ \( 7424 - 2772 T - 6472 T^{2} - 692 T^{3} + 142 T^{4} + 25 T^{5} + T^{6} \)
$71$ \( -16108 - 20769 T + 569 T^{2} + 1570 T^{3} - 62 T^{4} - 17 T^{5} + T^{6} \)
$73$ \( 5504 + 13624 T - 4847 T^{2} - 1914 T^{3} - 90 T^{4} + 14 T^{5} + T^{6} \)
$79$ \( -128 + 584 T + 84 T^{2} - 208 T^{3} - 10 T^{4} + 10 T^{5} + T^{6} \)
$83$ \( -633344 + 145852 T + 15572 T^{2} - 3726 T^{3} - 168 T^{4} + 21 T^{5} + T^{6} \)
$89$ \( 180432 + 167568 T - 3212 T^{2} - 5172 T^{3} - 142 T^{4} + 24 T^{5} + T^{6} \)
$97$ \( 45728 - 43992 T + 10148 T^{2} + 932 T^{3} - 344 T^{4} - 4 T^{5} + T^{6} \)
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