Properties

Label 9800.2.a.ce
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1944.1
Defining polynomial: \(x^{3} - 9 x - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} -2 q^{17} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{23} + ( 6 + 3 \beta_{1} ) q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} ) q^{31} + ( 6 + 4 \beta_{1} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} ) q^{37} + ( -6 - 2 \beta_{2} ) q^{39} + ( 3 + 2 \beta_{2} ) q^{41} + ( -4 + \beta_{1} ) q^{43} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{47} -2 \beta_{1} q^{51} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{57} + 8 q^{59} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( -2 - 3 \beta_{1} ) q^{67} + ( \beta_{1} - \beta_{2} ) q^{69} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 9 + 6 \beta_{1} ) q^{81} + ( 10 - \beta_{1} ) q^{83} + ( -6 + \beta_{1} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{89} + ( -12 + 2 \beta_{1} - 2 \beta_{2} ) q^{93} + 2 q^{97} + ( 21 + 7 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{9} + O(q^{10}) \) \( 3q + 9q^{9} + 3q^{11} - 3q^{13} - 6q^{17} + 3q^{19} - 3q^{23} + 18q^{27} + 12q^{29} + 12q^{31} + 18q^{33} - 9q^{37} - 18q^{39} + 9q^{41} - 12q^{43} + 15q^{47} - 9q^{53} - 18q^{57} + 24q^{59} - 6q^{61} - 6q^{67} - 18q^{79} + 27q^{81} + 30q^{83} - 18q^{87} - 36q^{93} + 6q^{97} + 63q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 9 x - 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 6\)

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
−2.58423
−0.705720
3.28995
0 −2.58423 0 0 0 0 0 3.67822 0
1.2 0 −0.705720 0 0 0 0 0 −2.50196 0
1.3 0 3.28995 0 0 0 0 0 7.82374 0
\(n\): e.g. 2-40 or 990-1000
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.ce 3
5.b even 2 1 1960.2.a.w 3
7.b odd 2 1 9800.2.a.cf 3
7.c even 3 2 1400.2.q.j 6
20.d odd 2 1 3920.2.a.cc 3
35.c odd 2 1 1960.2.a.v 3
35.i odd 6 2 1960.2.q.w 6
35.j even 6 2 280.2.q.e 6
35.l odd 12 4 1400.2.bh.i 12
105.o odd 6 2 2520.2.bi.q 6
140.c even 2 1 3920.2.a.cb 3
140.p odd 6 2 560.2.q.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 35.j even 6 2
560.2.q.l 6 140.p odd 6 2
1400.2.q.j 6 7.c even 3 2
1400.2.bh.i 12 35.l odd 12 4
1960.2.a.v 3 35.c odd 2 1
1960.2.a.w 3 5.b even 2 1
1960.2.q.w 6 35.i odd 6 2
2520.2.bi.q 6 105.o odd 6 2
3920.2.a.cb 3 140.c even 2 1
3920.2.a.cc 3 20.d odd 2 1
9800.2.a.ce 3 1.a even 1 1 trivial
9800.2.a.cf 3 7.b odd 2 1

Hecke kernels

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

\( T_{3}^{3} - 9 T_{3} - 6 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 24 T_{11} + 44 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 24 T_{13} - 68 \)
\( T_{19}^{3} - 3 T_{19}^{2} - 24 T_{19} - 16 \)
\( T_{23}^{3} + 3 T_{23}^{2} - 15 T_{23} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -6 - 9 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( T^{3} \)
$11$ \( 44 - 24 T - 3 T^{2} + T^{3} \)
$13$ \( -68 - 24 T + 3 T^{2} + T^{3} \)
$17$ \( ( 2 + T )^{3} \)
$19$ \( -16 - 24 T - 3 T^{2} + T^{3} \)
$23$ \( 7 - 15 T + 3 T^{2} + T^{3} \)
$29$ \( 26 + 21 T - 12 T^{2} + T^{3} \)
$31$ \( 128 + 12 T - 12 T^{2} + T^{3} \)
$37$ \( -96 + 9 T^{2} + T^{3} \)
$41$ \( 381 - 45 T - 9 T^{2} + T^{3} \)
$43$ \( 22 + 39 T + 12 T^{2} + T^{3} \)
$47$ \( 1588 - 96 T - 15 T^{2} + T^{3} \)
$53$ \( -624 - 72 T + 9 T^{2} + T^{3} \)
$59$ \( ( -8 + T )^{3} \)
$61$ \( -544 - 87 T + 6 T^{2} + T^{3} \)
$67$ \( 8 - 69 T + 6 T^{2} + T^{3} \)
$71$ \( T^{3} \)
$73$ \( -336 - 108 T + T^{3} \)
$79$ \( -768 + 18 T^{2} + T^{3} \)
$83$ \( -904 + 291 T - 30 T^{2} + T^{3} \)
$89$ \( 42 - 27 T + T^{3} \)
$97$ \( ( -2 + T )^{3} \)
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