Properties

Label 9800.2.a.bx
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( 1 + \beta ) q^{9} + ( -3 + 2 \beta ) q^{11} + 2 q^{13} -\beta q^{17} + ( -2 + \beta ) q^{19} + ( -3 - \beta ) q^{23} + ( 4 - \beta ) q^{27} + ( 5 + \beta ) q^{29} + ( 6 - 2 \beta ) q^{31} + ( 8 - \beta ) q^{33} + ( -1 + 5 \beta ) q^{37} + 2 \beta q^{39} + ( 4 + \beta ) q^{41} + ( -5 + \beta ) q^{43} + ( -2 + 4 \beta ) q^{47} + ( -4 - \beta ) q^{51} + ( -2 - 2 \beta ) q^{53} + ( 4 - \beta ) q^{57} + ( -2 + 6 \beta ) q^{59} + ( -8 + 2 \beta ) q^{61} + ( -11 - 2 \beta ) q^{67} + ( -4 - 4 \beta ) q^{69} + ( -3 - 3 \beta ) q^{71} + ( 8 + \beta ) q^{73} + ( -1 + 3 \beta ) q^{79} -7 q^{81} + ( -6 - 3 \beta ) q^{83} + ( 4 + 6 \beta ) q^{87} + ( -2 + 5 \beta ) q^{89} + ( -8 + 4 \beta ) q^{93} + 6 q^{97} + ( 5 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + 3q^{9} - 4q^{11} + 4q^{13} - q^{17} - 3q^{19} - 7q^{23} + 7q^{27} + 11q^{29} + 10q^{31} + 15q^{33} + 3q^{37} + 2q^{39} + 9q^{41} - 9q^{43} - 9q^{51} - 6q^{53} + 7q^{57} + 2q^{59} - 14q^{61} - 24q^{67} - 12q^{69} - 9q^{71} + 17q^{73} + q^{79} - 14q^{81} - 15q^{83} + 14q^{87} + q^{89} - 12q^{93} + 12q^{97} + 11q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
0 −1.56155 0 0 0 0 0 −0.561553 0
1.2 0 2.56155 0 0 0 0 0 3.56155 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.bx 2
5.b even 2 1 9800.2.a.bt 2
7.b odd 2 1 1400.2.a.o 2
28.d even 2 1 2800.2.a.bo 2
35.c odd 2 1 1400.2.a.q yes 2
35.f even 4 2 1400.2.g.j 4
140.c even 2 1 2800.2.a.bj 2
140.j odd 4 2 2800.2.g.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.o 2 7.b odd 2 1
1400.2.a.q yes 2 35.c odd 2 1
1400.2.g.j 4 35.f even 4 2
2800.2.a.bj 2 140.c even 2 1
2800.2.a.bo 2 28.d even 2 1
2800.2.g.v 4 140.j odd 4 2
9800.2.a.bt 2 5.b even 2 1
9800.2.a.bx 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{11}^{2} + 4 T_{11} - 13 \)
\( T_{13} - 2 \)
\( T_{19}^{2} + 3 T_{19} - 2 \)
\( T_{23}^{2} + 7 T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -13 + 4 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -4 + T + T^{2} \)
$19$ \( -2 + 3 T + T^{2} \)
$23$ \( 8 + 7 T + T^{2} \)
$29$ \( 26 - 11 T + T^{2} \)
$31$ \( 8 - 10 T + T^{2} \)
$37$ \( -104 - 3 T + T^{2} \)
$41$ \( 16 - 9 T + T^{2} \)
$43$ \( 16 + 9 T + T^{2} \)
$47$ \( -68 + T^{2} \)
$53$ \( -8 + 6 T + T^{2} \)
$59$ \( -152 - 2 T + T^{2} \)
$61$ \( 32 + 14 T + T^{2} \)
$67$ \( 127 + 24 T + T^{2} \)
$71$ \( -18 + 9 T + T^{2} \)
$73$ \( 68 - 17 T + T^{2} \)
$79$ \( -38 - T + T^{2} \)
$83$ \( 18 + 15 T + T^{2} \)
$89$ \( -106 - T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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