Properties

Label 98.10.a.a
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 16q^{2} + 6q^{3} + 256q^{4} - 560q^{5} - 96q^{6} - 4096q^{8} - 19647q^{9} + O(q^{10}) \) \( q - 16q^{2} + 6q^{3} + 256q^{4} - 560q^{5} - 96q^{6} - 4096q^{8} - 19647q^{9} + 8960q^{10} - 54152q^{11} + 1536q^{12} + 113172q^{13} - 3360q^{15} + 65536q^{16} - 6262q^{17} + 314352q^{18} - 257078q^{19} - 143360q^{20} + 866432q^{22} - 266000q^{23} - 24576q^{24} - 1639525q^{25} - 1810752q^{26} - 235980q^{27} + 1574714q^{29} + 53760q^{30} + 4637484q^{31} - 1048576q^{32} - 324912q^{33} + 100192q^{34} - 5029632q^{36} - 11946238q^{37} + 4113248q^{38} + 679032q^{39} + 2293760q^{40} - 21909126q^{41} + 27520592q^{43} - 13862912q^{44} + 11002320q^{45} + 4256000q^{46} - 52927836q^{47} + 393216q^{48} + 26232400q^{50} - 37572q^{51} + 28972032q^{52} + 16221222q^{53} + 3775680q^{54} + 30325120q^{55} - 1542468q^{57} - 25195424q^{58} + 140509618q^{59} - 860160q^{60} + 202963560q^{61} - 74199744q^{62} + 16777216q^{64} - 63376320q^{65} + 5198592q^{66} + 153734572q^{67} - 1603072q^{68} - 1596000q^{69} + 279655936q^{71} + 80474112q^{72} + 404022830q^{73} + 191139808q^{74} - 9837150q^{75} - 65811968q^{76} - 10864512q^{78} - 130689816q^{79} - 36700160q^{80} + 385296021q^{81} + 350546016q^{82} - 420134014q^{83} + 3506720q^{85} - 440329472q^{86} + 9448284q^{87} + 221806592q^{88} + 469542390q^{89} - 176037120q^{90} - 68096000q^{92} + 27824904q^{93} + 846845376q^{94} + 143963680q^{95} - 6291456q^{96} + 872501690q^{97} + 1063924344q^{99} + O(q^{100}) \)

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
0
−16.0000 6.00000 256.000 −560.000 −96.0000 0 −4096.00 −19647.0 8960.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

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 98.10.a.a 1
7.b odd 2 1 14.10.a.a 1
7.c even 3 2 98.10.c.e 2
7.d odd 6 2 98.10.c.f 2
21.c even 2 1 126.10.a.e 1
28.d even 2 1 112.10.a.b 1
35.c odd 2 1 350.10.a.c 1
35.f even 4 2 350.10.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 7.b odd 2 1
98.10.a.a 1 1.a even 1 1 trivial
98.10.c.e 2 7.c even 3 2
98.10.c.f 2 7.d odd 6 2
112.10.a.b 1 28.d even 2 1
126.10.a.e 1 21.c even 2 1
350.10.a.c 1 35.c odd 2 1
350.10.c.b 2 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T \)
$3$ \( -6 + T \)
$5$ \( 560 + T \)
$7$ \( T \)
$11$ \( 54152 + T \)
$13$ \( -113172 + T \)
$17$ \( 6262 + T \)
$19$ \( 257078 + T \)
$23$ \( 266000 + T \)
$29$ \( -1574714 + T \)
$31$ \( -4637484 + T \)
$37$ \( 11946238 + T \)
$41$ \( 21909126 + T \)
$43$ \( -27520592 + T \)
$47$ \( 52927836 + T \)
$53$ \( -16221222 + T \)
$59$ \( -140509618 + T \)
$61$ \( -202963560 + T \)
$67$ \( -153734572 + T \)
$71$ \( -279655936 + T \)
$73$ \( -404022830 + T \)
$79$ \( 130689816 + T \)
$83$ \( 420134014 + T \)
$89$ \( -469542390 + T \)
$97$ \( -872501690 + T \)
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