Properties

Label 9800.2.a.bu
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 5 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 5 + \beta ) q^{9} + ( 4 - \beta ) q^{11} + ( 2 - \beta ) q^{13} + ( 2 + \beta ) q^{17} -2 \beta q^{19} -2 \beta q^{23} + ( -8 - 3 \beta ) q^{27} + ( -2 + \beta ) q^{29} + 8 q^{31} + ( 8 - 3 \beta ) q^{33} + 2 q^{37} + ( 8 - \beta ) q^{39} + ( -2 + 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + 3 \beta q^{47} + ( -8 - 3 \beta ) q^{51} + ( -6 + 2 \beta ) q^{53} + ( 16 + 2 \beta ) q^{57} -8 q^{59} + ( -2 - 2 \beta ) q^{61} + 4 q^{67} + ( 16 + 2 \beta ) q^{69} + 8 q^{71} -6 q^{73} + ( 8 - 3 \beta ) q^{79} + ( 9 + 8 \beta ) q^{81} + 4 \beta q^{83} + ( -8 + \beta ) q^{87} + ( -10 + 2 \beta ) q^{89} -8 \beta q^{93} + ( 2 + 5 \beta ) q^{97} + ( 12 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 11q^{9} + O(q^{10}) \) \( 2q - q^{3} + 11q^{9} + 7q^{11} + 3q^{13} + 5q^{17} - 2q^{19} - 2q^{23} - 19q^{27} - 3q^{29} + 16q^{31} + 13q^{33} + 4q^{37} + 15q^{39} - 2q^{41} + 6q^{43} + 3q^{47} - 19q^{51} - 10q^{53} + 34q^{57} - 16q^{59} - 6q^{61} + 8q^{67} + 34q^{69} + 16q^{71} - 12q^{73} + 13q^{79} + 26q^{81} + 4q^{83} - 15q^{87} - 18q^{89} - 8q^{93} + 9q^{97} + 22q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
0 −3.37228 0 0 0 0 0 8.37228 0
1.2 0 2.37228 0 0 0 0 0 2.62772 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.bu 2
5.b even 2 1 1960.2.a.s 2
7.b odd 2 1 1400.2.a.r 2
20.d odd 2 1 3920.2.a.bt 2
28.d even 2 1 2800.2.a.bk 2
35.c odd 2 1 280.2.a.c 2
35.f even 4 2 1400.2.g.i 4
35.i odd 6 2 1960.2.q.t 4
35.j even 6 2 1960.2.q.r 4
105.g even 2 1 2520.2.a.x 2
140.c even 2 1 560.2.a.h 2
140.j odd 4 2 2800.2.g.r 4
280.c odd 2 1 2240.2.a.bk 2
280.n even 2 1 2240.2.a.bg 2
420.o odd 2 1 5040.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 35.c odd 2 1
560.2.a.h 2 140.c even 2 1
1400.2.a.r 2 7.b odd 2 1
1400.2.g.i 4 35.f even 4 2
1960.2.a.s 2 5.b even 2 1
1960.2.q.r 4 35.j even 6 2
1960.2.q.t 4 35.i odd 6 2
2240.2.a.bg 2 280.n even 2 1
2240.2.a.bk 2 280.c odd 2 1
2520.2.a.x 2 105.g even 2 1
2800.2.a.bk 2 28.d even 2 1
2800.2.g.r 4 140.j odd 4 2
3920.2.a.bt 2 20.d odd 2 1
5040.2.a.by 2 420.o odd 2 1
9800.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{2} + T_{3} - 8 \)
\( T_{11}^{2} - 7 T_{11} + 4 \)
\( T_{13}^{2} - 3 T_{13} - 6 \)
\( T_{19}^{2} + 2 T_{19} - 32 \)
\( T_{23}^{2} + 2 T_{23} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 - 7 T + T^{2} \)
$13$ \( -6 - 3 T + T^{2} \)
$17$ \( -2 - 5 T + T^{2} \)
$19$ \( -32 + 2 T + T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -6 + 3 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -32 + 2 T + T^{2} \)
$43$ \( -24 - 6 T + T^{2} \)
$47$ \( -72 - 3 T + T^{2} \)
$53$ \( -8 + 10 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( -24 + 6 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( -32 - 13 T + T^{2} \)
$83$ \( -128 - 4 T + T^{2} \)
$89$ \( 48 + 18 T + T^{2} \)
$97$ \( -186 - 9 T + T^{2} \)
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