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Properties

Label 98.4.a.f
Level 98
Weight 4
Character orbit 98.a
Self dual yes
Analytic conductor 5.782
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.78218718056\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 5q^{3} + 4q^{4} + 9q^{5} + 10q^{6} + 8q^{8} - 2q^{9} + O(q^{10}) \) \( q + 2q^{2} + 5q^{3} + 4q^{4} + 9q^{5} + 10q^{6} + 8q^{8} - 2q^{9} + 18q^{10} - 57q^{11} + 20q^{12} + 70q^{13} + 45q^{15} + 16q^{16} - 51q^{17} - 4q^{18} - 5q^{19} + 36q^{20} - 114q^{22} + 69q^{23} + 40q^{24} - 44q^{25} + 140q^{26} - 145q^{27} + 114q^{29} + 90q^{30} - 23q^{31} + 32q^{32} - 285q^{33} - 102q^{34} - 8q^{36} - 253q^{37} - 10q^{38} + 350q^{39} + 72q^{40} + 42q^{41} - 124q^{43} - 228q^{44} - 18q^{45} + 138q^{46} - 201q^{47} + 80q^{48} - 88q^{50} - 255q^{51} + 280q^{52} - 393q^{53} - 290q^{54} - 513q^{55} - 25q^{57} + 228q^{58} - 219q^{59} + 180q^{60} + 709q^{61} - 46q^{62} + 64q^{64} + 630q^{65} - 570q^{66} + 419q^{67} - 204q^{68} + 345q^{69} - 96q^{71} - 16q^{72} + 313q^{73} - 506q^{74} - 220q^{75} - 20q^{76} + 700q^{78} + 461q^{79} + 144q^{80} - 671q^{81} + 84q^{82} + 588q^{83} - 459q^{85} - 248q^{86} + 570q^{87} - 456q^{88} + 1017q^{89} - 36q^{90} + 276q^{92} - 115q^{93} - 402q^{94} - 45q^{95} + 160q^{96} + 1834q^{97} + 114q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
2.00000 5.00000 4.00000 9.00000 10.0000 0 8.00000 −2.00000 18.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 98.4.a.f 1
3.b odd 2 1 882.4.a.c 1
4.b odd 2 1 784.4.a.c 1
5.b even 2 1 2450.4.a.d 1
7.b odd 2 1 98.4.a.d 1
7.c even 3 2 98.4.c.a 2
7.d odd 6 2 14.4.c.a ✓ 2
21.c even 2 1 882.4.a.f 1
21.g even 6 2 126.4.g.d 2
21.h odd 6 2 882.4.g.u 2
28.d even 2 1 784.4.a.p 1
28.f even 6 2 112.4.i.a 2
35.c odd 2 1 2450.4.a.q 1
35.i odd 6 2 350.4.e.e 2
35.k even 12 4 350.4.j.b 4
56.j odd 6 2 448.4.i.b 2
56.m even 6 2 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a ✓ 2 7.d odd 6 2
98.4.a.d 1 7.b odd 2 1
98.4.a.f 1 1.a even 1 1 trivial
98.4.c.a 2 7.c even 3 2
112.4.i.a 2 28.f even 6 2
126.4.g.d 2 21.g even 6 2
350.4.e.e 2 35.i odd 6 2
350.4.j.b 4 35.k even 12 4
448.4.i.b 2 56.j odd 6 2
448.4.i.e 2 56.m even 6 2
784.4.a.c 1 4.b odd 2 1
784.4.a.p 1 28.d even 2 1
882.4.a.c 1 3.b odd 2 1
882.4.a.f 1 21.c even 2 1
882.4.g.u 2 21.h odd 6 2
2450.4.a.d 1 5.b even 2 1
2450.4.a.q 1 35.c odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 - 5 T + 27 T^{2} \)
$5$ \( 1 - 9 T + 125 T^{2} \)
$7$ \( \)
$11$ \( 1 + 57 T + 1331 T^{2} \)
$13$ \( 1 - 70 T + 2197 T^{2} \)
$17$ \( 1 + 51 T + 4913 T^{2} \)
$19$ \( 1 + 5 T + 6859 T^{2} \)
$23$ \( 1 - 69 T + 12167 T^{2} \)
$29$ \( 1 - 114 T + 24389 T^{2} \)
$31$ \( 1 + 23 T + 29791 T^{2} \)
$37$ \( 1 + 253 T + 50653 T^{2} \)
$41$ \( 1 - 42 T + 68921 T^{2} \)
$43$ \( 1 + 124 T + 79507 T^{2} \)
$47$ \( 1 + 201 T + 103823 T^{2} \)
$53$ \( 1 + 393 T + 148877 T^{2} \)
$59$ \( 1 + 219 T + 205379 T^{2} \)
$61$ \( 1 - 709 T + 226981 T^{2} \)
$67$ \( 1 - 419 T + 300763 T^{2} \)
$71$ \( 1 + 96 T + 357911 T^{2} \)
$73$ \( 1 - 313 T + 389017 T^{2} \)
$79$ \( 1 - 461 T + 493039 T^{2} \)
$83$ \( 1 - 588 T + 571787 T^{2} \)
$89$ \( 1 - 1017 T + 704969 T^{2} \)
$97$ \( 1 - 1834 T + 912673 T^{2} \)
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