Properties

Label 98.4.a.h
Level $98$
Weight $4$
Character orbit 98.a
Self dual yes
Analytic conductor $5.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} -\beta q^{5} + 2 \beta q^{6} + 8 q^{8} + 61 q^{9} +O(q^{10})\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} -\beta q^{5} + 2 \beta q^{6} + 8 q^{8} + 61 q^{9} -2 \beta q^{10} + 20 q^{11} + 4 \beta q^{12} -7 \beta q^{13} -88 q^{15} + 16 q^{16} -6 \beta q^{17} + 122 q^{18} -\beta q^{19} -4 \beta q^{20} + 40 q^{22} + 48 q^{23} + 8 \beta q^{24} -37 q^{25} -14 \beta q^{26} + 34 \beta q^{27} -166 q^{29} -176 q^{30} + 22 \beta q^{31} + 32 q^{32} + 20 \beta q^{33} -12 \beta q^{34} + 244 q^{36} -78 q^{37} -2 \beta q^{38} -616 q^{39} -8 \beta q^{40} -42 \beta q^{41} + 436 q^{43} + 80 q^{44} -61 \beta q^{45} + 96 q^{46} -22 \beta q^{47} + 16 \beta q^{48} -74 q^{50} -528 q^{51} -28 \beta q^{52} + 62 q^{53} + 68 \beta q^{54} -20 \beta q^{55} -88 q^{57} -332 q^{58} + 71 \beta q^{59} -352 q^{60} -29 \beta q^{61} + 44 \beta q^{62} + 64 q^{64} + 616 q^{65} + 40 \beta q^{66} + 580 q^{67} -24 \beta q^{68} + 48 \beta q^{69} -544 q^{71} + 488 q^{72} + 64 \beta q^{73} -156 q^{74} -37 \beta q^{75} -4 \beta q^{76} -1232 q^{78} -680 q^{79} -16 \beta q^{80} + 1345 q^{81} -84 \beta q^{82} -21 \beta q^{83} + 528 q^{85} + 872 q^{86} -166 \beta q^{87} + 160 q^{88} + 160 \beta q^{89} -122 \beta q^{90} + 192 q^{92} + 1936 q^{93} -44 \beta q^{94} + 88 q^{95} + 32 \beta q^{96} + 70 \beta q^{97} + 1220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + 122q^{9} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 16q^{8} + 122q^{9} + 40q^{11} - 176q^{15} + 32q^{16} + 244q^{18} + 80q^{22} + 96q^{23} - 74q^{25} - 332q^{29} - 352q^{30} + 64q^{32} + 488q^{36} - 156q^{37} - 1232q^{39} + 872q^{43} + 160q^{44} + 192q^{46} - 148q^{50} - 1056q^{51} + 124q^{53} - 176q^{57} - 664q^{58} - 704q^{60} + 128q^{64} + 1232q^{65} + 1160q^{67} - 1088q^{71} + 976q^{72} - 312q^{74} - 2464q^{78} - 1360q^{79} + 2690q^{81} + 1056q^{85} + 1744q^{86} + 320q^{88} + 384q^{92} + 3872q^{93} + 176q^{95} + 2440q^{99} + O(q^{100}) \)

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
−4.69042
4.69042
2.00000 −9.38083 4.00000 9.38083 −18.7617 0 8.00000 61.0000 18.7617
1.2 2.00000 9.38083 4.00000 −9.38083 18.7617 0 8.00000 61.0000 −18.7617
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.a.h 2
3.b odd 2 1 882.4.a.w 2
4.b odd 2 1 784.4.a.z 2
5.b even 2 1 2450.4.a.bs 2
7.b odd 2 1 inner 98.4.a.h 2
7.c even 3 2 98.4.c.g 4
7.d odd 6 2 98.4.c.g 4
21.c even 2 1 882.4.a.w 2
21.g even 6 2 882.4.g.bi 4
21.h odd 6 2 882.4.g.bi 4
28.d even 2 1 784.4.a.z 2
35.c odd 2 1 2450.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 1.a even 1 1 trivial
98.4.a.h 2 7.b odd 2 1 inner
98.4.c.g 4 7.c even 3 2
98.4.c.g 4 7.d odd 6 2
784.4.a.z 2 4.b odd 2 1
784.4.a.z 2 28.d even 2 1
882.4.a.w 2 3.b odd 2 1
882.4.a.w 2 21.c even 2 1
882.4.g.bi 4 21.g even 6 2
882.4.g.bi 4 21.h odd 6 2
2450.4.a.bs 2 5.b even 2 1
2450.4.a.bs 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 88 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( -88 + T^{2} \)
$5$ \( -88 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -20 + T )^{2} \)
$13$ \( -4312 + T^{2} \)
$17$ \( -3168 + T^{2} \)
$19$ \( -88 + T^{2} \)
$23$ \( ( -48 + T )^{2} \)
$29$ \( ( 166 + T )^{2} \)
$31$ \( -42592 + T^{2} \)
$37$ \( ( 78 + T )^{2} \)
$41$ \( -155232 + T^{2} \)
$43$ \( ( -436 + T )^{2} \)
$47$ \( -42592 + T^{2} \)
$53$ \( ( -62 + T )^{2} \)
$59$ \( -443608 + T^{2} \)
$61$ \( -74008 + T^{2} \)
$67$ \( ( -580 + T )^{2} \)
$71$ \( ( 544 + T )^{2} \)
$73$ \( -360448 + T^{2} \)
$79$ \( ( 680 + T )^{2} \)
$83$ \( -38808 + T^{2} \)
$89$ \( -2252800 + T^{2} \)
$97$ \( -431200 + T^{2} \)
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