Properties

Label 9800.2.a.cr
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\sqrt{7}, \sqrt{15})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{9} + q^{11} + ( \beta_{1} + \beta_{2} ) q^{13} + \beta_{1} q^{17} + ( -\beta_{1} + 2 \beta_{2} ) q^{19} + ( 2 - \beta_{3} ) q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + \beta_{3} q^{29} + ( -\beta_{1} + \beta_{2} ) q^{31} + \beta_{2} q^{33} + ( -2 + \beta_{3} ) q^{37} + ( 6 - 2 \beta_{3} ) q^{39} -3 \beta_{2} q^{41} + ( -6 + \beta_{3} ) q^{43} + 2 \beta_{2} q^{47} + ( 1 - \beta_{3} ) q^{51} -2 \beta_{3} q^{53} + ( 9 - \beta_{3} ) q^{57} + ( \beta_{1} + \beta_{2} ) q^{59} + ( \beta_{1} - 3 \beta_{2} ) q^{61} + ( 3 + 2 \beta_{3} ) q^{67} + ( \beta_{1} + 7 \beta_{2} ) q^{69} + ( 8 + \beta_{3} ) q^{71} -5 \beta_{2} q^{73} -\beta_{3} q^{79} + ( 15 - 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} + \beta_{2} ) q^{83} + ( -\beta_{1} - 5 \beta_{2} ) q^{87} + ( 2 \beta_{1} - \beta_{2} ) q^{89} + 4 q^{93} + ( -\beta_{1} + 5 \beta_{2} ) q^{97} + ( 2 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{9} + O(q^{10}) \) \( 4q + 10q^{9} + 4q^{11} + 10q^{23} - 2q^{29} - 10q^{37} + 28q^{39} - 26q^{43} + 6q^{51} + 4q^{53} + 38q^{57} + 8q^{67} + 30q^{71} + 2q^{79} + 64q^{81} + 16q^{93} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 11 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 7 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 11 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 6\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{2} + 11 \beta_{1}\)\()/2\)

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
0.613616
3.25937
−3.25937
−0.613616
0 −3.25937 0 0 0 0 0 7.62348 0
1.2 0 −0.613616 0 0 0 0 0 −2.62348 0
1.3 0 0.613616 0 0 0 0 0 −2.62348 0
1.4 0 3.25937 0 0 0 0 0 7.62348 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.cr yes 4
5.b even 2 1 9800.2.a.cq 4
7.b odd 2 1 inner 9800.2.a.cr yes 4
35.c odd 2 1 9800.2.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9800.2.a.cq 4 5.b even 2 1
9800.2.a.cq 4 35.c odd 2 1
9800.2.a.cr yes 4 1.a even 1 1 trivial
9800.2.a.cr yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{4} - 11 T_{3}^{2} + 4 \)
\( T_{11} - 1 \)
\( T_{13}^{2} - 28 \)
\( T_{19}^{4} - 71 T_{19}^{2} + 1024 \)
\( T_{23}^{2} - 5 T_{23} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 - 11 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( -28 + T^{2} )^{2} \)
$17$ \( 144 - 39 T^{2} + T^{4} \)
$19$ \( 1024 - 71 T^{2} + T^{4} \)
$23$ \( ( -20 - 5 T + T^{2} )^{2} \)
$29$ \( ( -26 + T + T^{2} )^{2} \)
$31$ \( 64 - 44 T^{2} + T^{4} \)
$37$ \( ( -20 + 5 T + T^{2} )^{2} \)
$41$ \( 324 - 99 T^{2} + T^{4} \)
$43$ \( ( 16 + 13 T + T^{2} )^{2} \)
$47$ \( 64 - 44 T^{2} + T^{4} \)
$53$ \( ( -104 - 2 T + T^{2} )^{2} \)
$59$ \( ( -28 + T^{2} )^{2} \)
$61$ \( ( -60 + T^{2} )^{2} \)
$67$ \( ( -101 - 4 T + T^{2} )^{2} \)
$71$ \( ( 30 - 15 T + T^{2} )^{2} \)
$73$ \( 2500 - 275 T^{2} + T^{4} \)
$79$ \( ( -26 - T + T^{2} )^{2} \)
$83$ \( 100 - 155 T^{2} + T^{4} \)
$89$ \( 100 - 155 T^{2} + T^{4} \)
$97$ \( 16384 - 284 T^{2} + T^{4} \)
show more
show less