Properties

Label 9800.2.a.da
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 18 x^{6} + 85 x^{4} - 38 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{9} + ( 1 + \beta_{3} + \beta_{6} ) q^{11} -\beta_{5} q^{13} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 3 \beta_{1} - \beta_{4} + \beta_{5} ) q^{27} + ( 2 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( \beta_{1} - 3 \beta_{4} + \beta_{7} ) q^{33} + ( \beta_{2} + \beta_{3} - \beta_{6} ) q^{37} + 2 q^{39} + ( -\beta_{1} - 2 \beta_{5} ) q^{41} + ( -2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{43} + ( 4 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{47} + ( 7 + \beta_{2} + \beta_{3} ) q^{51} + ( 4 + 2 \beta_{3} + \beta_{6} ) q^{53} + ( 1 + \beta_{2} - 5 \beta_{3} - \beta_{6} ) q^{57} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{61} + ( -5 + 2 \beta_{3} ) q^{67} + ( -4 \beta_{1} - 4 \beta_{4} - \beta_{5} + \beta_{7} ) q^{69} + ( 2 + \beta_{2} - \beta_{3} + \beta_{6} ) q^{71} + ( \beta_{1} + 5 \beta_{4} - \beta_{7} ) q^{73} + ( 2 - \beta_{2} - 3 \beta_{3} ) q^{79} + ( 7 + \beta_{3} ) q^{81} + ( -\beta_{1} - \beta_{7} ) q^{83} + ( 6 \beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{87} + ( -5 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{89} + ( 8 + 2 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 5 \beta_{4} - \beta_{5} + \beta_{7} ) q^{97} + ( 4 + \beta_{2} + 4 \beta_{3} - 2 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{9} + O(q^{10}) \) \( 8q + 12q^{9} + 4q^{11} + 4q^{23} + 8q^{29} + 16q^{39} - 16q^{43} + 52q^{51} + 28q^{53} + 8q^{57} - 40q^{67} + 8q^{71} + 20q^{79} + 56q^{81} + 56q^{93} + 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 18 x^{6} + 85 x^{4} - 38 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 9 \nu^{2} + 2 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 18 \nu^{5} + 83 \nu^{3} - 20 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 18 \nu^{5} + 85 \nu^{3} - 38 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\( \nu^{6} - 18 \nu^{4} + 83 \nu^{2} - 20 \)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} - 88 \nu^{5} + 397 \nu^{3} - 96 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 9 \beta_{2} + 43\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 9 \beta_{5} - 14 \beta_{4} + 79 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{6} + 18 \beta_{3} + 79 \beta_{2} + 379\)
\(\nu^{7}\)\(=\)\(18 \beta_{7} + 79 \beta_{5} - 167 \beta_{4} + 695 \beta_{1}\)

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
−3.04118
−2.87509
−0.568463
−0.402377
0.402377
0.568463
2.87509
3.04118
0 −3.04118 0 0 0 0 0 6.24878 0
1.2 0 −2.87509 0 0 0 0 0 5.26617 0
1.3 0 −0.568463 0 0 0 0 0 −2.67685 0
1.4 0 −0.402377 0 0 0 0 0 −2.83809 0
1.5 0 0.402377 0 0 0 0 0 −2.83809 0
1.6 0 0.568463 0 0 0 0 0 −2.67685 0
1.7 0 2.87509 0 0 0 0 0 5.26617 0
1.8 0 3.04118 0 0 0 0 0 6.24878 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.da yes 8
5.b even 2 1 9800.2.a.cz 8
7.b odd 2 1 inner 9800.2.a.da yes 8
35.c odd 2 1 9800.2.a.cz 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9800.2.a.cz 8 5.b even 2 1
9800.2.a.cz 8 35.c odd 2 1
9800.2.a.da yes 8 1.a even 1 1 trivial
9800.2.a.da yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{8} - 18 T_{3}^{6} + 85 T_{3}^{4} - 38 T_{3}^{2} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 38 T_{11}^{2} + 62 T_{11} + 257 \)
\( T_{13}^{8} - 38 T_{13}^{6} + 340 T_{13}^{4} - 288 T_{13}^{2} + 64 \)
\( T_{19}^{8} - 160 T_{19}^{6} + 7997 T_{19}^{4} - 119588 T_{19}^{2} + 84100 \)
\( T_{23}^{4} - 2 T_{23}^{3} - 47 T_{23}^{2} + 156 T_{23} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 - 38 T^{2} + 85 T^{4} - 18 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 257 + 62 T - 38 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 64 - 288 T^{2} + 340 T^{4} - 38 T^{6} + T^{8} \)
$17$ \( 40000 - 16600 T^{2} + 2089 T^{4} - 84 T^{6} + T^{8} \)
$19$ \( 84100 - 119588 T^{2} + 7997 T^{4} - 160 T^{6} + T^{8} \)
$23$ \( ( -8 + 156 T - 47 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$29$ \( ( -292 + 328 T - 85 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$31$ \( 4096 - 21248 T^{2} + 3812 T^{4} - 118 T^{6} + T^{8} \)
$37$ \( ( 488 + 92 T - 105 T^{2} + T^{4} )^{2} \)
$41$ \( 24964 - 20590 T^{2} + 4501 T^{4} - 138 T^{6} + T^{8} \)
$43$ \( ( -100 - 480 T - 81 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$47$ \( 66716224 - 3457024 T^{2} + 58260 T^{4} - 404 T^{6} + T^{8} \)
$53$ \( ( 512 + 320 T - 4 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$59$ \( 2704 - 14952 T^{2} + 19760 T^{4} - 282 T^{6} + T^{8} \)
$61$ \( 1000000 - 260000 T^{2} + 13300 T^{4} - 222 T^{6} + T^{8} \)
$67$ \( ( -1207 - 252 T + 78 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$71$ \( ( -340 + 444 T - 105 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$73$ \( 676 - 23134 T^{2} + 21717 T^{4} - 314 T^{6} + T^{8} \)
$79$ \( ( -7724 + 2552 T - 179 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$83$ \( 532900 - 204902 T^{2} + 10293 T^{4} - 178 T^{6} + T^{8} \)
$89$ \( 41860900 - 2808558 T^{2} + 59893 T^{4} - 442 T^{6} + T^{8} \)
$97$ \( 38416 - 238984 T^{2} + 27456 T^{4} - 386 T^{6} + T^{8} \)
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