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: \(x^{2} - 2\)
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.

\(f(q)\) \(=\) \( q -2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} -10 \beta q^{6} -8 q^{8} + 23 q^{9} +O(q^{10})\) \( q -2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} -10 \beta q^{6} -8 q^{8} + 23 q^{9} -28 \beta q^{10} -14 q^{11} + 20 \beta q^{12} -36 \beta q^{13} + 140 q^{15} + 16 q^{16} -\beta q^{17} -46 q^{18} + \beta q^{19} + 56 \beta q^{20} + 28 q^{22} + 140 q^{23} -40 \beta q^{24} + 267 q^{25} + 72 \beta q^{26} -20 \beta q^{27} -286 q^{29} -280 q^{30} + 66 \beta q^{31} -32 q^{32} -70 \beta q^{33} + 2 \beta q^{34} + 92 q^{36} -38 q^{37} -2 \beta q^{38} -360 q^{39} -112 \beta q^{40} + 89 \beta q^{41} -34 q^{43} -56 q^{44} + 322 \beta q^{45} -280 q^{46} -370 \beta q^{47} + 80 \beta q^{48} -534 q^{50} -10 q^{51} -144 \beta q^{52} -74 q^{53} + 40 \beta q^{54} -196 \beta q^{55} + 10 q^{57} + 572 q^{58} -307 \beta q^{59} + 560 q^{60} -10 \beta q^{61} -132 \beta q^{62} + 64 q^{64} -1008 q^{65} + 140 \beta q^{66} + 684 q^{67} -4 \beta q^{68} + 700 \beta q^{69} + 588 q^{71} -184 q^{72} + 191 \beta q^{73} + 76 q^{74} + 1335 \beta q^{75} + 4 \beta q^{76} + 720 q^{78} + 1220 q^{79} + 224 \beta q^{80} -821 q^{81} -178 \beta q^{82} -299 \beta q^{83} -28 q^{85} + 68 q^{86} -1430 \beta q^{87} + 112 q^{88} -437 \beta q^{89} -644 \beta q^{90} + 560 q^{92} + 660 q^{93} + 740 \beta q^{94} + 28 q^{95} -160 \beta q^{96} -1049 \beta q^{97} -322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} - 28q^{11} + 280q^{15} + 32q^{16} - 92q^{18} + 56q^{22} + 280q^{23} + 534q^{25} - 572q^{29} - 560q^{30} - 64q^{32} + 184q^{36} - 76q^{37} - 720q^{39} - 68q^{43} - 112q^{44} - 560q^{46} - 1068q^{50} - 20q^{51} - 148q^{53} + 20q^{57} + 1144q^{58} + 1120q^{60} + 128q^{64} - 2016q^{65} + 1368q^{67} + 1176q^{71} - 368q^{72} + 152q^{74} + 1440q^{78} + 2440q^{79} - 1642q^{81} - 56q^{85} + 136q^{86} + 224q^{88} + 1120q^{92} + 1320q^{93} + 56q^{95} - 644q^{99} + O(q^{100}) \)

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
−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 \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( -50 + T^{2} \)
$5$ \( -392 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 14 + T )^{2} \)
$13$ \( -2592 + T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( ( -140 + T )^{2} \)
$29$ \( ( 286 + T )^{2} \)
$31$ \( -8712 + T^{2} \)
$37$ \( ( 38 + T )^{2} \)
$41$ \( -15842 + T^{2} \)
$43$ \( ( 34 + T )^{2} \)
$47$ \( -273800 + T^{2} \)
$53$ \( ( 74 + T )^{2} \)
$59$ \( -188498 + T^{2} \)
$61$ \( -200 + T^{2} \)
$67$ \( ( -684 + T )^{2} \)
$71$ \( ( -588 + T )^{2} \)
$73$ \( -72962 + T^{2} \)
$79$ \( ( -1220 + T )^{2} \)
$83$ \( -178802 + T^{2} \)
$89$ \( -381938 + T^{2} \)
$97$ \( -2200802 + T^{2} \)
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