# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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