Properties

Label 98.4.a.b
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

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 4q^{4} + 7q^{5} + 2q^{6} - 8q^{8} - 26q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 4q^{4} + 7q^{5} + 2q^{6} - 8q^{8} - 26q^{9} - 14q^{10} + 35q^{11} - 4q^{12} + 66q^{13} - 7q^{15} + 16q^{16} + 59q^{17} + 52q^{18} + 137q^{19} + 28q^{20} - 70q^{22} - 7q^{23} + 8q^{24} - 76q^{25} - 132q^{26} + 53q^{27} + 106q^{29} + 14q^{30} + 75q^{31} - 32q^{32} - 35q^{33} - 118q^{34} - 104q^{36} + 11q^{37} - 274q^{38} - 66q^{39} - 56q^{40} - 498q^{41} + 260q^{43} + 140q^{44} - 182q^{45} + 14q^{46} - 171q^{47} - 16q^{48} + 152q^{50} - 59q^{51} + 264q^{52} - 417q^{53} - 106q^{54} + 245q^{55} - 137q^{57} - 212q^{58} - 17q^{59} - 28q^{60} + 51q^{61} - 150q^{62} + 64q^{64} + 462q^{65} + 70q^{66} + 439q^{67} + 236q^{68} + 7q^{69} - 784q^{71} + 208q^{72} + 295q^{73} - 22q^{74} + 76q^{75} + 548q^{76} + 132q^{78} - 495q^{79} + 112q^{80} + 649q^{81} + 996q^{82} + 932q^{83} + 413q^{85} - 520q^{86} - 106q^{87} - 280q^{88} - 873q^{89} + 364q^{90} - 28q^{92} - 75q^{93} + 342q^{94} + 959q^{95} + 32q^{96} - 290q^{97} - 910q^{99} + O(q^{100}) \)

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
−2.00000 −1.00000 4.00000 7.00000 2.00000 0 −8.00000 −26.0000 −14.0000
\(n\): 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.b 1
3.b odd 2 1 882.4.a.k 1
4.b odd 2 1 784.4.a.l 1
5.b even 2 1 2450.4.a.bh 1
7.b odd 2 1 98.4.a.c 1
7.c even 3 2 14.4.c.b 2
7.d odd 6 2 98.4.c.e 2
21.c even 2 1 882.4.a.p 1
21.g even 6 2 882.4.g.d 2
21.h odd 6 2 126.4.g.c 2
28.d even 2 1 784.4.a.j 1
28.g odd 6 2 112.4.i.b 2
35.c odd 2 1 2450.4.a.bf 1
35.j even 6 2 350.4.e.b 2
35.l odd 12 4 350.4.j.d 4
56.k odd 6 2 448.4.i.d 2
56.p even 6 2 448.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 7.c even 3 2
98.4.a.b 1 1.a even 1 1 trivial
98.4.a.c 1 7.b odd 2 1
98.4.c.e 2 7.d odd 6 2
112.4.i.b 2 28.g odd 6 2
126.4.g.c 2 21.h odd 6 2
350.4.e.b 2 35.j even 6 2
350.4.j.d 4 35.l odd 12 4
448.4.i.c 2 56.p even 6 2
448.4.i.d 2 56.k odd 6 2
784.4.a.j 1 28.d even 2 1
784.4.a.l 1 4.b odd 2 1
882.4.a.k 1 3.b odd 2 1
882.4.a.p 1 21.c even 2 1
882.4.g.d 2 21.g even 6 2
2450.4.a.bf 1 35.c odd 2 1
2450.4.a.bh 1 5.b even 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 \( T_{3} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 + T + 27 T^{2} \)
$5$ \( 1 - 7 T + 125 T^{2} \)
$7$ 1
$11$ \( 1 - 35 T + 1331 T^{2} \)
$13$ \( 1 - 66 T + 2197 T^{2} \)
$17$ \( 1 - 59 T + 4913 T^{2} \)
$19$ \( 1 - 137 T + 6859 T^{2} \)
$23$ \( 1 + 7 T + 12167 T^{2} \)
$29$ \( 1 - 106 T + 24389 T^{2} \)
$31$ \( 1 - 75 T + 29791 T^{2} \)
$37$ \( 1 - 11 T + 50653 T^{2} \)
$41$ \( 1 + 498 T + 68921 T^{2} \)
$43$ \( 1 - 260 T + 79507 T^{2} \)
$47$ \( 1 + 171 T + 103823 T^{2} \)
$53$ \( 1 + 417 T + 148877 T^{2} \)
$59$ \( 1 + 17 T + 205379 T^{2} \)
$61$ \( 1 - 51 T + 226981 T^{2} \)
$67$ \( 1 - 439 T + 300763 T^{2} \)
$71$ \( 1 + 784 T + 357911 T^{2} \)
$73$ \( 1 - 295 T + 389017 T^{2} \)
$79$ \( 1 + 495 T + 493039 T^{2} \)
$83$ \( 1 - 932 T + 571787 T^{2} \)
$89$ \( 1 + 873 T + 704969 T^{2} \)
$97$ \( 1 + 290 T + 912673 T^{2} \)
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