Properties

Label 9800.2.a.cc
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: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
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.

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

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
−1.53209
−0.347296
1.87939
0 −2.53209 0 0 0 0 0 3.41147 0
1.2 0 −1.34730 0 0 0 0 0 −1.18479 0
1.3 0 0.879385 0 0 0 0 0 −2.22668 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.cc 3
5.b even 2 1 9800.2.a.ci 3
7.b odd 2 1 9800.2.a.ch 3
7.c even 3 2 1400.2.q.k yes 6
35.c odd 2 1 9800.2.a.cb 3
35.j even 6 2 1400.2.q.i 6
35.l odd 12 4 1400.2.bh.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.i 6 35.j even 6 2
1400.2.q.k yes 6 7.c even 3 2
1400.2.bh.h 12 35.l odd 12 4
9800.2.a.cb 3 35.c odd 2 1
9800.2.a.cc 3 1.a even 1 1 trivial
9800.2.a.ch 3 7.b odd 2 1
9800.2.a.ci 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{3} + 3 T_{3}^{2} - 3 \)
\( T_{11}^{3} + 6 T_{11}^{2} + 3 T_{11} - 19 \)
\( T_{13}^{3} - 39 T_{13} - 19 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 9 T_{19} + 17 \)
\( T_{23}^{3} + 3 T_{23}^{2} - 18 T_{23} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -3 + 3 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( T^{3} \)
$11$ \( -19 + 3 T + 6 T^{2} + T^{3} \)
$13$ \( -19 - 39 T + T^{3} \)
$17$ \( 19 - 15 T - 6 T^{2} + T^{3} \)
$19$ \( 17 - 9 T - 6 T^{2} + T^{3} \)
$23$ \( 17 - 18 T + 3 T^{2} + T^{3} \)
$29$ \( -19 + 39 T - 12 T^{2} + T^{3} \)
$31$ \( -19 - 36 T + 3 T^{2} + T^{3} \)
$37$ \( 51 - 27 T - 6 T^{2} + T^{3} \)
$41$ \( 57 - 18 T - 3 T^{2} + T^{3} \)
$43$ \( 179 - 12 T - 9 T^{2} + T^{3} \)
$47$ \( 107 + 87 T + 18 T^{2} + T^{3} \)
$53$ \( -1077 - 171 T + 6 T^{2} + T^{3} \)
$59$ \( -53 - 24 T + 3 T^{2} + T^{3} \)
$61$ \( -37 - 66 T - 9 T^{2} + T^{3} \)
$67$ \( -584 - 60 T + 12 T^{2} + T^{3} \)
$71$ \( 513 - 81 T - 9 T^{2} + T^{3} \)
$73$ \( 192 - 12 T^{2} + T^{3} \)
$79$ \( -3 + 3 T^{2} + T^{3} \)
$83$ \( 163 - 57 T + T^{3} \)
$89$ \( 699 - 117 T - 6 T^{2} + T^{3} \)
$97$ \( 467 + 66 T - 21 T^{2} + T^{3} \)
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