Properties

Label 98.4.a.g
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - 2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} - 10 \beta q^{6} - 8 q^{8} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} - 10 \beta q^{6} - 8 q^{8} + 23 q^{9} - 28 \beta q^{10} - 14 q^{11} + 20 \beta q^{12} - 36 \beta q^{13} + 140 q^{15} + 16 q^{16} - \beta q^{17} - 46 q^{18} + \beta q^{19} + 56 \beta q^{20} + 28 q^{22} + 140 q^{23} - 40 \beta q^{24} + 267 q^{25} + 72 \beta q^{26} - 20 \beta q^{27} - 286 q^{29} - 280 q^{30} + 66 \beta q^{31} - 32 q^{32} - 70 \beta q^{33} + 2 \beta q^{34} + 92 q^{36} - 38 q^{37} - 2 \beta q^{38} - 360 q^{39} - 112 \beta q^{40} + 89 \beta q^{41} - 34 q^{43} - 56 q^{44} + 322 \beta q^{45} - 280 q^{46} - 370 \beta q^{47} + 80 \beta q^{48} - 534 q^{50} - 10 q^{51} - 144 \beta q^{52} - 74 q^{53} + 40 \beta q^{54} - 196 \beta q^{55} + 10 q^{57} + 572 q^{58} - 307 \beta q^{59} + 560 q^{60} - 10 \beta q^{61} - 132 \beta q^{62} + 64 q^{64} - 1008 q^{65} + 140 \beta q^{66} + 684 q^{67} - 4 \beta q^{68} + 700 \beta q^{69} + 588 q^{71} - 184 q^{72} + 191 \beta q^{73} + 76 q^{74} + 1335 \beta q^{75} + 4 \beta q^{76} + 720 q^{78} + 1220 q^{79} + 224 \beta q^{80} - 821 q^{81} - 178 \beta q^{82} - 299 \beta q^{83} - 28 q^{85} + 68 q^{86} - 1430 \beta q^{87} + 112 q^{88} - 437 \beta q^{89} - 644 \beta q^{90} + 560 q^{92} + 660 q^{93} + 740 \beta q^{94} + 28 q^{95} - 160 \beta q^{96} - 1049 \beta q^{97} - 322 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9} - 28 q^{11} + 280 q^{15} + 32 q^{16} - 92 q^{18} + 56 q^{22} + 280 q^{23} + 534 q^{25} - 572 q^{29} - 560 q^{30} - 64 q^{32} + 184 q^{36} - 76 q^{37} - 720 q^{39} - 68 q^{43} - 112 q^{44} - 560 q^{46} - 1068 q^{50} - 20 q^{51} - 148 q^{53} + 20 q^{57} + 1144 q^{58} + 1120 q^{60} + 128 q^{64} - 2016 q^{65} + 1368 q^{67} + 1176 q^{71} - 368 q^{72} + 152 q^{74} + 1440 q^{78} + 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 136 q^{86} + 224 q^{88} + 1120 q^{92} + 1320 q^{93} + 56 q^{95} - 644 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−2.00000 −7.07107 4.00000 −19.7990 14.1421 0 −8.00000 23.0000 39.5980
1.2 −2.00000 7.07107 4.00000 19.7990 −14.1421 0 −8.00000 23.0000 −39.5980
\(n\): 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.g 2
3.b odd 2 1 882.4.a.bg 2
4.b odd 2 1 784.4.a.y 2
5.b even 2 1 2450.4.a.bx 2
7.b odd 2 1 inner 98.4.a.g 2
7.c even 3 2 98.4.c.h 4
7.d odd 6 2 98.4.c.h 4
21.c even 2 1 882.4.a.bg 2
21.g even 6 2 882.4.g.ba 4
21.h odd 6 2 882.4.g.ba 4
28.d even 2 1 784.4.a.y 2
35.c odd 2 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 1.a even 1 1 trivial
98.4.a.g 2 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 2
98.4.c.h 4 7.d odd 6 2
784.4.a.y 2 4.b odd 2 1
784.4.a.y 2 28.d even 2 1
882.4.a.bg 2 3.b odd 2 1
882.4.a.bg 2 21.c even 2 1
882.4.g.ba 4 21.g even 6 2
882.4.g.ba 4 21.h odd 6 2
2450.4.a.bx 2 5.b even 2 1
2450.4.a.bx 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 50 \) 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
$3$ \( T^{2} - 50 \) Copy content Toggle raw display
$5$ \( T^{2} - 392 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2592 \) Copy content Toggle raw display
$17$ \( T^{2} - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 2 \) Copy content Toggle raw display
$23$ \( (T - 140)^{2} \) Copy content Toggle raw display
$29$ \( (T + 286)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8712 \) Copy content Toggle raw display
$37$ \( (T + 38)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 15842 \) Copy content Toggle raw display
$43$ \( (T + 34)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 273800 \) Copy content Toggle raw display
$53$ \( (T + 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 188498 \) Copy content Toggle raw display
$61$ \( T^{2} - 200 \) Copy content Toggle raw display
$67$ \( (T - 684)^{2} \) Copy content Toggle raw display
$71$ \( (T - 588)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 72962 \) Copy content Toggle raw display
$79$ \( (T - 1220)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 178802 \) Copy content Toggle raw display
$89$ \( T^{2} - 381938 \) Copy content Toggle raw display
$97$ \( T^{2} - 2200802 \) Copy content Toggle raw display
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