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: \( x^{2} - 22 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} - \beta q^{5} + 2 \beta q^{6} + 8 q^{8} + 61 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + \beta q^{3} + 4 q^{4} - \beta q^{5} + 2 \beta q^{6} + 8 q^{8} + 61 q^{9} - 2 \beta q^{10} + 20 q^{11} + 4 \beta q^{12} - 7 \beta q^{13} - 88 q^{15} + 16 q^{16} - 6 \beta q^{17} + 122 q^{18} - \beta q^{19} - 4 \beta q^{20} + 40 q^{22} + 48 q^{23} + 8 \beta q^{24} - 37 q^{25} - 14 \beta q^{26} + 34 \beta q^{27} - 166 q^{29} - 176 q^{30} + 22 \beta q^{31} + 32 q^{32} + 20 \beta q^{33} - 12 \beta q^{34} + 244 q^{36} - 78 q^{37} - 2 \beta q^{38} - 616 q^{39} - 8 \beta q^{40} - 42 \beta q^{41} + 436 q^{43} + 80 q^{44} - 61 \beta q^{45} + 96 q^{46} - 22 \beta q^{47} + 16 \beta q^{48} - 74 q^{50} - 528 q^{51} - 28 \beta q^{52} + 62 q^{53} + 68 \beta q^{54} - 20 \beta q^{55} - 88 q^{57} - 332 q^{58} + 71 \beta q^{59} - 352 q^{60} - 29 \beta q^{61} + 44 \beta q^{62} + 64 q^{64} + 616 q^{65} + 40 \beta q^{66} + 580 q^{67} - 24 \beta q^{68} + 48 \beta q^{69} - 544 q^{71} + 488 q^{72} + 64 \beta q^{73} - 156 q^{74} - 37 \beta q^{75} - 4 \beta q^{76} - 1232 q^{78} - 680 q^{79} - 16 \beta q^{80} + 1345 q^{81} - 84 \beta q^{82} - 21 \beta q^{83} + 528 q^{85} + 872 q^{86} - 166 \beta q^{87} + 160 q^{88} + 160 \beta q^{89} - 122 \beta q^{90} + 192 q^{92} + 1936 q^{93} - 44 \beta q^{94} + 88 q^{95} + 32 \beta q^{96} + 70 \beta q^{97} + 1220 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9} + 40 q^{11} - 176 q^{15} + 32 q^{16} + 244 q^{18} + 80 q^{22} + 96 q^{23} - 74 q^{25} - 332 q^{29} - 352 q^{30} + 64 q^{32} + 488 q^{36} - 156 q^{37} - 1232 q^{39} + 872 q^{43} + 160 q^{44} + 192 q^{46} - 148 q^{50} - 1056 q^{51} + 124 q^{53} - 176 q^{57} - 664 q^{58} - 704 q^{60} + 128 q^{64} + 1232 q^{65} + 1160 q^{67} - 1088 q^{71} + 976 q^{72} - 312 q^{74} - 2464 q^{78} - 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 1744 q^{86} + 320 q^{88} + 384 q^{92} + 3872 q^{93} + 176 q^{95} + 2440 q^{99}+O(q^{100}) \) 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
−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 \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

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