Properties

Label 9800.2.a.cx
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.239575536.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 8 x^{3} + 35 x^{2} - 15 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{5} ) q^{11} + ( 1 + \beta_{2} ) q^{13} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{27} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{31} + ( 2 + 2 \beta_{2} - \beta_{4} ) q^{33} + ( -2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{39} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{41} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{43} + ( 5 - \beta_{2} + \beta_{4} ) q^{47} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{51} + ( 1 + \beta_{2} + \beta_{4} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{61} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{67} + ( 3 - 4 \beta_{1} - \beta_{3} - \beta_{5} ) q^{69} + ( 1 - 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{71} + ( 3 + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{73} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{79} + ( -2 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{81} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{83} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{87} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{89} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{97} + ( 2 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{3} + 7q^{9} + O(q^{10}) \) \( 6q + q^{3} + 7q^{9} + q^{11} + 7q^{13} + 4q^{17} - 5q^{19} - 6q^{23} + 7q^{27} - 3q^{29} - 2q^{31} + 16q^{33} - 9q^{37} + 10q^{39} + 6q^{41} + 3q^{43} + 27q^{47} + 5q^{53} + 26q^{57} - 24q^{59} + 9q^{61} - 17q^{67} + 15q^{69} + 4q^{71} + 18q^{73} - 22q^{79} - 6q^{81} + 9q^{83} + 39q^{87} + 15q^{89} - 10q^{93} + 12q^{97} + 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 12 x^{4} + 8 x^{3} + 35 x^{2} - 15 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 10 \nu^{3} - 4 \nu^{2} + 17 \nu + 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 14 \nu^{2} + 11 \nu - 16 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 14 \nu^{3} - 45 \nu + 4 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3} + \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 10 \beta_{2} + 4 \beta_{1} + 25\)
\(\nu^{5}\)\(=\)\(10 \beta_{5} + 14 \beta_{3} + 14 \beta_{2} + 53 \beta_{1} + 18\)

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
−2.54431
−2.03837
−0.319986
0.751428
2.10821
3.04302
0 −2.54431 0 0 0 0 0 3.47349 0
1.2 0 −2.03837 0 0 0 0 0 1.15495 0
1.3 0 −0.319986 0 0 0 0 0 −2.89761 0
1.4 0 0.751428 0 0 0 0 0 −2.43536 0
1.5 0 2.10821 0 0 0 0 0 1.44457 0
1.6 0 3.04302 0 0 0 0 0 6.25996 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
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 9800.2.a.cx 6
5.b even 2 1 9800.2.a.cv 6
5.c odd 4 2 1960.2.g.f 12
7.b odd 2 1 9800.2.a.cw 6
7.c even 3 2 1400.2.q.n 12
35.c odd 2 1 9800.2.a.cy 6
35.f even 4 2 1960.2.g.e 12
35.j even 6 2 1400.2.q.o 12
35.l odd 12 4 280.2.bg.a 24
140.w even 12 4 560.2.bw.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bg.a 24 35.l odd 12 4
560.2.bw.f 24 140.w even 12 4
1400.2.q.n 12 7.c even 3 2
1400.2.q.o 12 35.j even 6 2
1960.2.g.e 12 35.f even 4 2
1960.2.g.f 12 5.c odd 4 2
9800.2.a.cv 6 5.b even 2 1
9800.2.a.cw 6 7.b odd 2 1
9800.2.a.cx 6 1.a even 1 1 trivial
9800.2.a.cy 6 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{6} - T_{3}^{5} - 12 T_{3}^{4} + 8 T_{3}^{3} + 35 T_{3}^{2} - 15 T_{3} - 8 \)
\( T_{11}^{6} - T_{11}^{5} - 37 T_{11}^{4} + 69 T_{11}^{3} + 296 T_{11}^{2} - 832 T_{11} + 512 \)
\( T_{13}^{6} - 7 T_{13}^{5} - 10 T_{13}^{4} + 100 T_{13}^{3} - 8 T_{13}^{2} - 320 T_{13} + 256 \)
\( T_{19}^{6} + 5 T_{19}^{5} - 47 T_{19}^{4} - 301 T_{19}^{3} - 94 T_{19}^{2} + 1300 T_{19} + 568 \)
\( T_{23}^{6} + 6 T_{23}^{5} - 19 T_{23}^{4} - 108 T_{23}^{3} + 123 T_{23}^{2} + 478 T_{23} - 337 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -8 - 15 T + 35 T^{2} + 8 T^{3} - 12 T^{4} - T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( 512 - 832 T + 296 T^{2} + 69 T^{3} - 37 T^{4} - T^{5} + T^{6} \)
$13$ \( 256 - 320 T - 8 T^{2} + 100 T^{3} - 10 T^{4} - 7 T^{5} + T^{6} \)
$17$ \( -9552 - 3552 T + 1656 T^{2} + 244 T^{3} - 75 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( 568 + 1300 T - 94 T^{2} - 301 T^{3} - 47 T^{4} + 5 T^{5} + T^{6} \)
$23$ \( -337 + 478 T + 123 T^{2} - 108 T^{3} - 19 T^{4} + 6 T^{5} + T^{6} \)
$29$ \( -7344 - 2700 T + 2652 T^{2} - 35 T^{3} - 105 T^{4} + 3 T^{5} + T^{6} \)
$31$ \( -1968 - 1728 T + 2016 T^{2} - 124 T^{3} - 115 T^{4} + 2 T^{5} + T^{6} \)
$37$ \( -3456 + 8064 T + 1392 T^{2} - 547 T^{3} - 75 T^{4} + 9 T^{5} + T^{6} \)
$41$ \( 2764 + 5372 T + 2871 T^{2} + 266 T^{3} - 100 T^{4} - 6 T^{5} + T^{6} \)
$43$ \( -95736 - 2292 T + 6990 T^{2} + 203 T^{3} - 151 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( -1392 - 1320 T + 2604 T^{2} - 1271 T^{3} + 273 T^{4} - 27 T^{5} + T^{6} \)
$53$ \( 344 - 1268 T + 1030 T^{2} + 267 T^{3} - 77 T^{4} - 5 T^{5} + T^{6} \)
$59$ \( -124704 - 93384 T - 23820 T^{2} - 2042 T^{3} + 93 T^{4} + 24 T^{5} + T^{6} \)
$61$ \( -105806 - 36181 T + 5289 T^{2} + 1550 T^{3} - 172 T^{4} - 9 T^{5} + T^{6} \)
$67$ \( 3842 + 1883 T - 1201 T^{2} - 396 T^{3} + 38 T^{4} + 17 T^{5} + T^{6} \)
$71$ \( -183296 - 33536 T + 11984 T^{2} + 1280 T^{3} - 272 T^{4} - 4 T^{5} + T^{6} \)
$73$ \( -179072 + 130568 T - 35340 T^{2} + 3998 T^{3} - 79 T^{4} - 18 T^{5} + T^{6} \)
$79$ \( -272 + 1808 T - 1720 T^{2} - 872 T^{3} + 61 T^{4} + 22 T^{5} + T^{6} \)
$83$ \( -136 + 1556 T - 4230 T^{2} + 1389 T^{3} - 103 T^{4} - 9 T^{5} + T^{6} \)
$89$ \( 115554 - 42867 T - 11367 T^{2} + 4498 T^{3} - 268 T^{4} - 15 T^{5} + T^{6} \)
$97$ \( -8192 - 14336 T + 13824 T^{2} + 1600 T^{3} - 208 T^{4} - 12 T^{5} + T^{6} \)
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