Properties

Label 9800.2.a.db
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 22 x^{8} - 16 x^{7} + 146 x^{6} + 200 x^{5} - 206 x^{4} - 440 x^{3} - 124 x^{2} + 72 x + 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( 1 - \beta_{4} - \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} + ( 1 - \beta_{4} - \beta_{8} ) q^{9} + ( -1 + \beta_{1} ) q^{11} + ( -\beta_{3} - \beta_{5} + \beta_{6} ) q^{13} + ( \beta_{3} - \beta_{7} + \beta_{9} ) q^{17} + ( \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{23} + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{27} + ( 1 - \beta_{1} + 2 \beta_{8} ) q^{29} + ( \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{31} + ( 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{33} + ( -4 - \beta_{2} + \beta_{8} ) q^{37} + ( -2 - \beta_{2} + \beta_{4} - \beta_{8} ) q^{39} + ( \beta_{3} - 2 \beta_{9} ) q^{41} + ( -2 - \beta_{2} + 2 \beta_{4} + 2 \beta_{8} ) q^{43} + ( \beta_{5} - \beta_{6} - \beta_{9} ) q^{47} + ( -3 - \beta_{1} + 2 \beta_{4} + 2 \beta_{8} ) q^{51} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} ) q^{57} + ( -\beta_{3} + \beta_{5} - 4 \beta_{7} ) q^{59} + ( 2 \beta_{3} + \beta_{5} - 2 \beta_{9} ) q^{61} + ( -4 - 2 \beta_{1} + \beta_{2} + \beta_{8} ) q^{67} + ( -3 \beta_{3} - \beta_{6} - 5 \beta_{7} + 3 \beta_{9} ) q^{69} + ( -1 + \beta_{1} - \beta_{4} + \beta_{8} ) q^{71} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{8} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{8} ) q^{81} + ( -2 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{83} + ( 8 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{87} + ( -2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{89} + ( -4 + 2 \beta_{1} + \beta_{2} + 3 \beta_{8} ) q^{93} + ( -3 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{97} + ( -6 + \beta_{2} + 2 \beta_{4} + 6 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 14q^{9} + O(q^{10}) \) \( 10q + 14q^{9} - 12q^{11} - 16q^{23} + 12q^{29} - 36q^{37} - 20q^{39} - 24q^{43} - 36q^{51} - 8q^{53} - 16q^{57} - 40q^{67} - 8q^{71} - 4q^{79} + 50q^{81} - 48q^{93} - 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 22 x^{8} - 16 x^{7} + 146 x^{6} + 200 x^{5} - 206 x^{4} - 440 x^{3} - 124 x^{2} + 72 x + 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 134 \nu^{9} - 1139 \nu^{8} - 724 \nu^{7} + 18925 \nu^{6} - 5572 \nu^{5} - 98852 \nu^{4} + 19160 \nu^{3} + 154038 \nu^{2} + 1496 \nu - 15000 \)\()/5966\)
\(\beta_{2}\)\(=\)\((\)\( -96 \nu^{9} - 2167 \nu^{8} + 7108 \nu^{7} + 39557 \nu^{6} - 71340 \nu^{5} - 236118 \nu^{4} + 138540 \nu^{3} + 403878 \nu^{2} + 6408 \nu - 73312 \)\()/2983\)
\(\beta_{3}\)\(=\)\((\)\( -424 \nu^{9} + 621 \nu^{8} + 8524 \nu^{7} - 6010 \nu^{6} - 55564 \nu^{5} - 296 \nu^{4} + 107758 \nu^{3} + 31282 \nu^{2} - 37324 \nu - 5608 \)\()/5966\)
\(\beta_{4}\)\(=\)\((\)\( 1151 \nu^{9} - 2326 \nu^{8} - 20466 \nu^{7} + 24293 \nu^{6} + 115180 \nu^{5} - 26944 \nu^{4} - 163198 \nu^{3} - 52936 \nu^{2} - 38796 \nu - 4982 \)\()/5966\)
\(\beta_{5}\)\(=\)\((\)\( -1431 \nu^{9} - 1260 \nu^{8} + 36226 \nu^{7} + 40122 \nu^{6} - 272544 \nu^{5} - 355976 \nu^{4} + 522528 \nu^{3} + 722312 \nu^{2} - 55868 \nu - 135264 \)\()/5966\)
\(\beta_{6}\)\(=\)\((\)\( -1450 \nu^{9} + 393 \nu^{8} + 33034 \nu^{7} + 10881 \nu^{6} - 234088 \nu^{5} - 185508 \nu^{4} + 443678 \nu^{3} + 407558 \nu^{2} - 71752 \nu - 49346 \)\()/5966\)
\(\beta_{7}\)\(=\)\((\)\( 2516 \nu^{9} - 3488 \nu^{8} - 49568 \nu^{7} + 26095 \nu^{6} + 314068 \nu^{5} + 94736 \nu^{4} - 547972 \nu^{3} - 392804 \nu^{2} + 61748 \nu + 61982 \)\()/5966\)
\(\beta_{8}\)\(=\)\((\)\( 2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} - 603030 \nu^{3} - 398834 \nu^{2} + 113948 \nu + 84236 \)\()/5966\)
\(\beta_{9}\)\(=\)\((\)\( 2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} - 603030 \nu^{3} - 398834 \nu^{2} + 102016 \nu + 84236 \)\()/5966\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + \beta_{8}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{4} + \beta_{2} - 3 \beta_{1} + 17\)\()/4\)
\(\nu^{3}\)\(=\)\(-5 \beta_{9} + 4 \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\((\)\(-28 \beta_{9} + 10 \beta_{8} + 24 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + 7 \beta_{2} - 16 \beta_{1} + 74\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-111 \beta_{9} + 81 \beta_{8} + 32 \beta_{7} + 2 \beta_{6} - 27 \beta_{5} - 3 \beta_{4} + 45 \beta_{3} + 29 \beta_{2} - 37 \beta_{1} + 155\)\()/2\)
\(\nu^{6}\)\(=\)\(-176 \beta_{9} + 75 \beta_{8} + 130 \beta_{7} + 28 \beta_{6} - 34 \beta_{5} - 29 \beta_{4} + 2 \beta_{3} + 45 \beta_{2} - 91 \beta_{1} + 391\)
\(\nu^{7}\)\(=\)\(-648 \beta_{9} + 444 \beta_{8} + 223 \beta_{7} + 31 \beta_{6} - 164 \beta_{5} - 22 \beta_{4} + 213 \beta_{3} + 179 \beta_{2} - 262 \beta_{1} + 1024\)
\(\nu^{8}\)\(=\)\((\)\(-4304 \beta_{9} + 2055 \beta_{8} + 2784 \beta_{7} + 672 \beta_{6} - 944 \beta_{5} - 523 \beta_{4} + 80 \beta_{3} + 1127 \beta_{2} - 2181 \beta_{1} + 8839\)\()/2\)
\(\nu^{9}\)\(=\)\(-7702 \beta_{9} + 5048 \beta_{8} + 2902 \beta_{7} + 586 \beta_{6} - 1990 \beta_{5} - 234 \beta_{4} + 1858 \beta_{3} + 2156 \beta_{2} - 3454 \beta_{1} + 12906\)

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
−1.84294
0.422497
−0.609577
3.50994
1.61569
−0.432569
−1.92302
−1.30040
3.39389
−2.83351
0 −3.30674 0 0 0 0 0 7.93455 0
1.2 0 −2.69859 0 0 0 0 0 4.28239 0
1.3 0 −1.35056 0 0 0 0 0 −1.17599 0
1.4 0 −1.12212 0 0 0 0 0 −1.74084 0
1.5 0 −0.836591 0 0 0 0 0 −2.30012 0
1.6 0 0.836591 0 0 0 0 0 −2.30012 0
1.7 0 1.12212 0 0 0 0 0 −1.74084 0
1.8 0 1.35056 0 0 0 0 0 −1.17599 0
1.9 0 2.69859 0 0 0 0 0 4.28239 0
1.10 0 3.30674 0 0 0 0 0 7.93455 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.db 10
5.b even 2 1 9800.2.a.dc 10
5.c odd 4 2 1960.2.g.g 20
7.b odd 2 1 inner 9800.2.a.db 10
35.c odd 2 1 9800.2.a.dc 10
35.f even 4 2 1960.2.g.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.g.g 20 5.c odd 4 2
1960.2.g.g 20 35.f even 4 2
9800.2.a.db 10 1.a even 1 1 trivial
9800.2.a.db 10 7.b odd 2 1 inner
9800.2.a.dc 10 5.b even 2 1
9800.2.a.dc 10 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{10} - 22 T_{3}^{8} + 153 T_{3}^{6} - 384 T_{3}^{4} + 384 T_{3}^{2} - 128 \)
\( T_{11}^{5} + 6 T_{11}^{4} - 15 T_{11}^{3} - 76 T_{11}^{2} + 112 T_{11} - 32 \)
\( T_{13}^{10} - 92 T_{13}^{8} + 2873 T_{13}^{6} - 34134 T_{13}^{4} + 110844 T_{13}^{2} - 95048 \)
\( T_{19}^{10} - 112 T_{19}^{8} + 3776 T_{19}^{6} - 39040 T_{19}^{4} + 153600 T_{19}^{2} - 204800 \)
\( T_{23}^{5} + 8 T_{23}^{4} - 70 T_{23}^{3} - 528 T_{23}^{2} + 1056 T_{23} + 7232 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( -128 + 384 T^{2} - 384 T^{4} + 153 T^{6} - 22 T^{8} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( T^{10} \)
$11$ \( ( -32 + 112 T - 76 T^{2} - 15 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$13$ \( -95048 + 110844 T^{2} - 34134 T^{4} + 2873 T^{6} - 92 T^{8} + T^{10} \)
$17$ \( -245000 + 131100 T^{2} - 24758 T^{4} + 2057 T^{6} - 76 T^{8} + T^{10} \)
$19$ \( -204800 + 153600 T^{2} - 39040 T^{4} + 3776 T^{6} - 112 T^{8} + T^{10} \)
$23$ \( ( 7232 + 1056 T - 528 T^{2} - 70 T^{3} + 8 T^{4} + T^{5} )^{2} \)
$29$ \( ( -2848 + 1072 T + 348 T^{2} - 71 T^{3} - 6 T^{4} + T^{5} )^{2} \)
$31$ \( -6422528 + 1765376 T^{2} - 178560 T^{4} + 8196 T^{6} - 164 T^{8} + T^{10} \)
$37$ \( ( -976 - 968 T - 132 T^{2} + 70 T^{3} + 18 T^{4} + T^{5} )^{2} \)
$41$ \( -3135008 + 1201744 T^{2} - 161872 T^{4} + 8872 T^{6} - 170 T^{8} + T^{10} \)
$43$ \( ( 2560 - 2560 T - 1440 T^{2} - 96 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$47$ \( -131072 + 143360 T^{2} - 48640 T^{4} + 5785 T^{6} - 150 T^{8} + T^{10} \)
$53$ \( ( -3904 + 3264 T - 88 T^{2} - 118 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$59$ \( -685388288 + 79078912 T^{2} - 2903552 T^{4} + 47364 T^{6} - 356 T^{8} + T^{10} \)
$61$ \( -8000000 + 8080000 T^{2} - 640200 T^{4} + 18700 T^{6} - 230 T^{8} + T^{10} \)
$67$ \( ( -11648 - 7456 T - 1192 T^{2} + 42 T^{3} + 20 T^{4} + T^{5} )^{2} \)
$71$ \( ( -640 - 1600 T - 800 T^{2} - 108 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$73$ \( -8388608 + 2146304 T^{2} - 194632 T^{4} + 8012 T^{6} - 150 T^{8} + T^{10} \)
$79$ \( ( -2144 + 3792 T - 1000 T^{2} - 223 T^{3} + 2 T^{4} + T^{5} )^{2} \)
$83$ \( -52428800 + 12288000 T^{2} - 970880 T^{4} + 32256 T^{6} - 416 T^{8} + T^{10} \)
$89$ \( -51200 + 108800 T^{2} - 68680 T^{4} + 11916 T^{6} - 214 T^{8} + T^{10} \)
$97$ \( -41332232 + 19890844 T^{2} - 2251766 T^{4} + 56553 T^{6} - 428 T^{8} + T^{10} \)
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