Properties

Label 9800.2.a.by
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 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( 1 + \beta ) q^{9} -\beta q^{11} + ( 2 - 3 \beta ) q^{13} + ( 6 - \beta ) q^{17} + ( 4 - 2 \beta ) q^{19} + 2 \beta q^{23} + ( 4 - \beta ) q^{27} + ( 2 + \beta ) q^{29} + 4 \beta q^{31} + ( -4 - \beta ) q^{33} + ( 2 - 4 \beta ) q^{37} + ( -12 - \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( 4 + \beta ) q^{47} + ( -4 + 5 \beta ) q^{51} + ( 10 - 2 \beta ) q^{53} + ( -8 + 2 \beta ) q^{57} + 4 q^{59} + ( 10 + 2 \beta ) q^{61} + ( 4 + 4 \beta ) q^{67} + ( 8 + 2 \beta ) q^{69} + ( 2 + 4 \beta ) q^{73} + ( -4 - 3 \beta ) q^{79} -7 q^{81} + 12 q^{83} + ( 4 + 3 \beta ) q^{87} + ( -2 + 2 \beta ) q^{89} + ( 16 + 4 \beta ) q^{93} + ( 6 + 3 \beta ) q^{97} + ( -4 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + 3q^{9} - q^{11} + q^{13} + 11q^{17} + 6q^{19} + 2q^{23} + 7q^{27} + 5q^{29} + 4q^{31} - 9q^{33} - 25q^{39} - 6q^{41} + 6q^{43} + 9q^{47} - 3q^{51} + 18q^{53} - 14q^{57} + 8q^{59} + 22q^{61} + 12q^{67} + 18q^{69} + 8q^{73} - 11q^{79} - 14q^{81} + 24q^{83} + 11q^{87} - 2q^{89} + 36q^{93} + 15q^{97} - 10q^{99} + O(q^{100}) \)

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.by 2
5.b even 2 1 1960.2.a.r 2
7.b odd 2 1 1400.2.a.p 2
20.d odd 2 1 3920.2.a.bu 2
28.d even 2 1 2800.2.a.bn 2
35.c odd 2 1 280.2.a.d 2
35.f even 4 2 1400.2.g.k 4
35.i odd 6 2 1960.2.q.s 4
35.j even 6 2 1960.2.q.u 4
105.g even 2 1 2520.2.a.w 2
140.c even 2 1 560.2.a.g 2
140.j odd 4 2 2800.2.g.u 4
280.c odd 2 1 2240.2.a.be 2
280.n even 2 1 2240.2.a.bi 2
420.o odd 2 1 5040.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 35.c odd 2 1
560.2.a.g 2 140.c even 2 1
1400.2.a.p 2 7.b odd 2 1
1400.2.g.k 4 35.f even 4 2
1960.2.a.r 2 5.b even 2 1
1960.2.q.s 4 35.i odd 6 2
1960.2.q.u 4 35.j even 6 2
2240.2.a.be 2 280.c odd 2 1
2240.2.a.bi 2 280.n even 2 1
2520.2.a.w 2 105.g even 2 1
2800.2.a.bn 2 28.d even 2 1
2800.2.g.u 4 140.j odd 4 2
3920.2.a.bu 2 20.d odd 2 1
5040.2.a.bq 2 420.o odd 2 1
9800.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{11}^{2} + T_{11} - 4 \)
\( T_{13}^{2} - T_{13} - 38 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)
\( T_{23}^{2} - 2 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( -38 - T + T^{2} \)
$17$ \( 26 - 11 T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( -16 - 2 T + T^{2} \)
$29$ \( 2 - 5 T + T^{2} \)
$31$ \( -64 - 4 T + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( -8 + 6 T + T^{2} \)
$43$ \( -8 - 6 T + T^{2} \)
$47$ \( 16 - 9 T + T^{2} \)
$53$ \( 64 - 18 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 104 - 22 T + T^{2} \)
$67$ \( -32 - 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( -8 + 11 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( -16 + 2 T + T^{2} \)
$97$ \( 18 - 15 T + T^{2} \)
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