Properties

Label 9800.2.a.bw
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 5 q^{9} -4 q^{11} -\beta q^{13} -2 \beta q^{17} + \beta q^{19} + 2 \beta q^{27} + 2 q^{29} + 2 \beta q^{31} -4 \beta q^{33} -10 q^{37} -8 q^{39} -2 \beta q^{41} + 4 q^{43} + 2 \beta q^{47} -16 q^{51} -6 q^{53} + 8 q^{57} + \beta q^{59} -5 \beta q^{61} -12 q^{67} + 8 q^{79} + q^{81} -5 \beta q^{83} + 2 \beta q^{87} + 16 q^{93} + 2 \beta q^{97} -20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} - 8q^{11} + 4q^{29} - 20q^{37} - 16q^{39} + 8q^{43} - 32q^{51} - 12q^{53} + 16q^{57} - 24q^{67} + 16q^{79} + 2q^{81} + 32q^{93} - 40q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.bw 2
5.b even 2 1 392.2.a.h 2
7.b odd 2 1 inner 9800.2.a.bw 2
15.d odd 2 1 3528.2.a.bj 2
20.d odd 2 1 784.2.a.n 2
35.c odd 2 1 392.2.a.h 2
35.i odd 6 2 392.2.i.g 4
35.j even 6 2 392.2.i.g 4
40.e odd 2 1 3136.2.a.bq 2
40.f even 2 1 3136.2.a.bt 2
60.h even 2 1 7056.2.a.cj 2
105.g even 2 1 3528.2.a.bj 2
105.o odd 6 2 3528.2.s.be 4
105.p even 6 2 3528.2.s.be 4
140.c even 2 1 784.2.a.n 2
140.p odd 6 2 784.2.i.k 4
140.s even 6 2 784.2.i.k 4
280.c odd 2 1 3136.2.a.bt 2
280.n even 2 1 3136.2.a.bq 2
420.o odd 2 1 7056.2.a.cj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h 2 5.b even 2 1
392.2.a.h 2 35.c odd 2 1
392.2.i.g 4 35.i odd 6 2
392.2.i.g 4 35.j even 6 2
784.2.a.n 2 20.d odd 2 1
784.2.a.n 2 140.c even 2 1
784.2.i.k 4 140.p odd 6 2
784.2.i.k 4 140.s even 6 2
3136.2.a.bq 2 40.e odd 2 1
3136.2.a.bq 2 280.n even 2 1
3136.2.a.bt 2 40.f even 2 1
3136.2.a.bt 2 280.c odd 2 1
3528.2.a.bj 2 15.d odd 2 1
3528.2.a.bj 2 105.g even 2 1
3528.2.s.be 4 105.o odd 6 2
3528.2.s.be 4 105.p even 6 2
7056.2.a.cj 2 60.h even 2 1
7056.2.a.cj 2 420.o odd 2 1
9800.2.a.bw 2 1.a even 1 1 trivial
9800.2.a.bw 2 7.b odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( -32 + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( -32 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -8 + T^{2} \)
$61$ \( -200 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( -200 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( -32 + T^{2} \)
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