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

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

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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