Properties

Label 9800.2.a.bv
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: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{9} + 6 q^{11} -4 \beta q^{13} + \beta q^{17} + 3 \beta q^{19} -4 q^{23} -4 \beta q^{27} -6 q^{29} -2 \beta q^{31} + 6 \beta q^{33} -2 q^{37} -8 q^{39} + \beta q^{41} -10 q^{43} -2 \beta q^{47} + 2 q^{51} + 2 q^{53} + 6 q^{57} -\beta q^{59} + 6 \beta q^{61} -4 q^{67} -4 \beta q^{69} -12 q^{71} -7 \beta q^{73} -4 q^{79} -5 q^{81} + \beta q^{83} -6 \beta q^{87} + 3 \beta q^{89} -4 q^{93} + 9 \beta q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 12q^{11} - 8q^{23} - 12q^{29} - 4q^{37} - 16q^{39} - 20q^{43} + 4q^{51} + 4q^{53} + 12q^{57} - 8q^{67} - 24q^{71} - 8q^{79} - 10q^{81} - 8q^{93} - 12q^{99} + O(q^{100}) \)

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 −1.41421 0 0 0 0 0 −1.00000 0
1.2 0 1.41421 0 0 0 0 0 −1.00000 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.bv 2
5.b even 2 1 392.2.a.g 2
7.b odd 2 1 inner 9800.2.a.bv 2
15.d odd 2 1 3528.2.a.be 2
20.d odd 2 1 784.2.a.k 2
35.c odd 2 1 392.2.a.g 2
35.i odd 6 2 392.2.i.h 4
35.j even 6 2 392.2.i.h 4
40.e odd 2 1 3136.2.a.bp 2
40.f even 2 1 3136.2.a.bk 2
60.h even 2 1 7056.2.a.ct 2
105.g even 2 1 3528.2.a.be 2
105.o odd 6 2 3528.2.s.bj 4
105.p even 6 2 3528.2.s.bj 4
140.c even 2 1 784.2.a.k 2
140.p odd 6 2 784.2.i.n 4
140.s even 6 2 784.2.i.n 4
280.c odd 2 1 3136.2.a.bk 2
280.n even 2 1 3136.2.a.bp 2
420.o odd 2 1 7056.2.a.ct 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 5.b even 2 1
392.2.a.g 2 35.c odd 2 1
392.2.i.h 4 35.i odd 6 2
392.2.i.h 4 35.j even 6 2
784.2.a.k 2 20.d odd 2 1
784.2.a.k 2 140.c even 2 1
784.2.i.n 4 140.p odd 6 2
784.2.i.n 4 140.s even 6 2
3136.2.a.bk 2 40.f even 2 1
3136.2.a.bk 2 280.c odd 2 1
3136.2.a.bp 2 40.e odd 2 1
3136.2.a.bp 2 280.n even 2 1
3528.2.a.be 2 15.d odd 2 1
3528.2.a.be 2 105.g even 2 1
3528.2.s.bj 4 105.o odd 6 2
3528.2.s.bj 4 105.p even 6 2
7056.2.a.ct 2 60.h even 2 1
7056.2.a.ct 2 420.o odd 2 1
9800.2.a.bv 2 1.a even 1 1 trivial
9800.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{2} - 2 \)
\( T_{11} - 6 \)
\( T_{13}^{2} - 32 \)
\( T_{19}^{2} - 18 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -18 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -8 + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -2 + T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -2 + T^{2} \)
$61$ \( -72 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -98 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -2 + T^{2} \)
$89$ \( -18 + T^{2} \)
$97$ \( -162 + T^{2} \)
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