Properties

Label 98.4.a.a
Level $98$
Weight $4$
Character orbit 98.a
Self dual yes
Analytic conductor $5.782$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 14 q^{5} + 16 q^{6} - 8 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 14 q^{5} + 16 q^{6} - 8 q^{8} + 37 q^{9} - 28 q^{10} - 28 q^{11} - 32 q^{12} - 18 q^{13} - 112 q^{15} + 16 q^{16} - 74 q^{17} - 74 q^{18} - 80 q^{19} + 56 q^{20} + 56 q^{22} - 112 q^{23} + 64 q^{24} + 71 q^{25} + 36 q^{26} - 80 q^{27} + 190 q^{29} + 224 q^{30} - 72 q^{31} - 32 q^{32} + 224 q^{33} + 148 q^{34} + 148 q^{36} - 346 q^{37} + 160 q^{38} + 144 q^{39} - 112 q^{40} - 162 q^{41} - 412 q^{43} - 112 q^{44} + 518 q^{45} + 224 q^{46} - 24 q^{47} - 128 q^{48} - 142 q^{50} + 592 q^{51} - 72 q^{52} + 318 q^{53} + 160 q^{54} - 392 q^{55} + 640 q^{57} - 380 q^{58} + 200 q^{59} - 448 q^{60} + 198 q^{61} + 144 q^{62} + 64 q^{64} - 252 q^{65} - 448 q^{66} - 716 q^{67} - 296 q^{68} + 896 q^{69} + 392 q^{71} - 296 q^{72} - 538 q^{73} + 692 q^{74} - 568 q^{75} - 320 q^{76} - 288 q^{78} + 240 q^{79} + 224 q^{80} - 359 q^{81} + 324 q^{82} + 1072 q^{83} - 1036 q^{85} + 824 q^{86} - 1520 q^{87} + 224 q^{88} - 810 q^{89} - 1036 q^{90} - 448 q^{92} + 576 q^{93} + 48 q^{94} - 1120 q^{95} + 256 q^{96} - 1354 q^{97} - 1036 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −8.00000 4.00000 14.0000 16.0000 0 −8.00000 37.0000 −28.0000
\(n\): 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 98.4.a.a 1
3.b odd 2 1 882.4.a.i 1
4.b odd 2 1 784.4.a.s 1
5.b even 2 1 2450.4.a.bo 1
7.b odd 2 1 14.4.a.a 1
7.c even 3 2 98.4.c.f 2
7.d odd 6 2 98.4.c.d 2
21.c even 2 1 126.4.a.h 1
21.g even 6 2 882.4.g.b 2
21.h odd 6 2 882.4.g.k 2
28.d even 2 1 112.4.a.a 1
35.c odd 2 1 350.4.a.l 1
35.f even 4 2 350.4.c.b 2
56.e even 2 1 448.4.a.o 1
56.h odd 2 1 448.4.a.b 1
77.b even 2 1 1694.4.a.g 1
84.h odd 2 1 1008.4.a.s 1
91.b odd 2 1 2366.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 7.b odd 2 1
98.4.a.a 1 1.a even 1 1 trivial
98.4.c.d 2 7.d odd 6 2
98.4.c.f 2 7.c even 3 2
112.4.a.a 1 28.d even 2 1
126.4.a.h 1 21.c even 2 1
350.4.a.l 1 35.c odd 2 1
350.4.c.b 2 35.f even 4 2
448.4.a.b 1 56.h odd 2 1
448.4.a.o 1 56.e even 2 1
784.4.a.s 1 4.b odd 2 1
882.4.a.i 1 3.b odd 2 1
882.4.g.b 2 21.g even 6 2
882.4.g.k 2 21.h odd 6 2
1008.4.a.s 1 84.h odd 2 1
1694.4.a.g 1 77.b even 2 1
2366.4.a.h 1 91.b odd 2 1
2450.4.a.bo 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T - 14 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 28 \) Copy content Toggle raw display
$13$ \( T + 18 \) Copy content Toggle raw display
$17$ \( T + 74 \) Copy content Toggle raw display
$19$ \( T + 80 \) Copy content Toggle raw display
$23$ \( T + 112 \) Copy content Toggle raw display
$29$ \( T - 190 \) Copy content Toggle raw display
$31$ \( T + 72 \) Copy content Toggle raw display
$37$ \( T + 346 \) Copy content Toggle raw display
$41$ \( T + 162 \) Copy content Toggle raw display
$43$ \( T + 412 \) Copy content Toggle raw display
$47$ \( T + 24 \) Copy content Toggle raw display
$53$ \( T - 318 \) Copy content Toggle raw display
$59$ \( T - 200 \) Copy content Toggle raw display
$61$ \( T - 198 \) Copy content Toggle raw display
$67$ \( T + 716 \) Copy content Toggle raw display
$71$ \( T - 392 \) Copy content Toggle raw display
$73$ \( T + 538 \) Copy content Toggle raw display
$79$ \( T - 240 \) Copy content Toggle raw display
$83$ \( T - 1072 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T + 1354 \) Copy content Toggle raw display
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