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: \( x^{3} - 9x - 6 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1 + 3) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} - \beta_1 - 1) q^{13} - 2 q^{17} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} - 1) q^{23} + (3 \beta_1 + 6) q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + ( - 2 \beta_1 + 4) q^{31} + (4 \beta_1 + 6) q^{33} + (\beta_{2} - \beta_1 - 3) q^{37} + ( - 2 \beta_{2} - 6) q^{39} + (2 \beta_{2} + 3) q^{41} + (\beta_1 - 4) q^{43} + (3 \beta_{2} + \beta_1 + 5) q^{47} - 2 \beta_1 q^{51} + (\beta_{2} + 3 \beta_1 - 3) q^{53} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{57} + 8 q^{59} + (\beta_{2} + 3 \beta_1 - 2) q^{61} + ( - 3 \beta_1 - 2) q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (2 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{2} - 2 \beta_1 - 6) q^{79} + (6 \beta_1 + 9) q^{81} + ( - \beta_1 + 10) q^{83} + (\beta_1 - 6) q^{87} + ( - \beta_{2} + \beta_1) q^{89} + ( - 2 \beta_{2} + 2 \beta_1 - 12) q^{93} + 2 q^{97} + (\beta_{2} + 7 \beta_1 + 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 9 q^{9} + 3 q^{11} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{23} + 18 q^{27} + 12 q^{29} + 12 q^{31} + 18 q^{33} - 9 q^{37} - 18 q^{39} + 9 q^{41} - 12 q^{43} + 15 q^{47} - 9 q^{53} - 18 q^{57} + 24 q^{59} - 6 q^{61} - 6 q^{67} - 18 q^{79} + 27 q^{81} + 30 q^{83} - 18 q^{87} - 36 q^{93} + 6 q^{97} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

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

Hecke characteristic polynomials

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