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

## 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

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