# Properties

 Label 98.4.a.f Level $98$ Weight $4$ Character orbit 98.a Self dual yes Analytic conductor $5.782$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 5 q^{3} + 4 q^{4} + 9 q^{5} + 10 q^{6} + 8 q^{8} - 2 q^{9}+O(q^{10})$$ q + 2 * q^2 + 5 * q^3 + 4 * q^4 + 9 * q^5 + 10 * q^6 + 8 * q^8 - 2 * q^9 $$q + 2 q^{2} + 5 q^{3} + 4 q^{4} + 9 q^{5} + 10 q^{6} + 8 q^{8} - 2 q^{9} + 18 q^{10} - 57 q^{11} + 20 q^{12} + 70 q^{13} + 45 q^{15} + 16 q^{16} - 51 q^{17} - 4 q^{18} - 5 q^{19} + 36 q^{20} - 114 q^{22} + 69 q^{23} + 40 q^{24} - 44 q^{25} + 140 q^{26} - 145 q^{27} + 114 q^{29} + 90 q^{30} - 23 q^{31} + 32 q^{32} - 285 q^{33} - 102 q^{34} - 8 q^{36} - 253 q^{37} - 10 q^{38} + 350 q^{39} + 72 q^{40} + 42 q^{41} - 124 q^{43} - 228 q^{44} - 18 q^{45} + 138 q^{46} - 201 q^{47} + 80 q^{48} - 88 q^{50} - 255 q^{51} + 280 q^{52} - 393 q^{53} - 290 q^{54} - 513 q^{55} - 25 q^{57} + 228 q^{58} - 219 q^{59} + 180 q^{60} + 709 q^{61} - 46 q^{62} + 64 q^{64} + 630 q^{65} - 570 q^{66} + 419 q^{67} - 204 q^{68} + 345 q^{69} - 96 q^{71} - 16 q^{72} + 313 q^{73} - 506 q^{74} - 220 q^{75} - 20 q^{76} + 700 q^{78} + 461 q^{79} + 144 q^{80} - 671 q^{81} + 84 q^{82} + 588 q^{83} - 459 q^{85} - 248 q^{86} + 570 q^{87} - 456 q^{88} + 1017 q^{89} - 36 q^{90} + 276 q^{92} - 115 q^{93} - 402 q^{94} - 45 q^{95} + 160 q^{96} + 1834 q^{97} + 114 q^{99}+O(q^{100})$$ q + 2 * q^2 + 5 * q^3 + 4 * q^4 + 9 * q^5 + 10 * q^6 + 8 * q^8 - 2 * q^9 + 18 * q^10 - 57 * q^11 + 20 * q^12 + 70 * q^13 + 45 * q^15 + 16 * q^16 - 51 * q^17 - 4 * q^18 - 5 * q^19 + 36 * q^20 - 114 * q^22 + 69 * q^23 + 40 * q^24 - 44 * q^25 + 140 * q^26 - 145 * q^27 + 114 * q^29 + 90 * q^30 - 23 * q^31 + 32 * q^32 - 285 * q^33 - 102 * q^34 - 8 * q^36 - 253 * q^37 - 10 * q^38 + 350 * q^39 + 72 * q^40 + 42 * q^41 - 124 * q^43 - 228 * q^44 - 18 * q^45 + 138 * q^46 - 201 * q^47 + 80 * q^48 - 88 * q^50 - 255 * q^51 + 280 * q^52 - 393 * q^53 - 290 * q^54 - 513 * q^55 - 25 * q^57 + 228 * q^58 - 219 * q^59 + 180 * q^60 + 709 * q^61 - 46 * q^62 + 64 * q^64 + 630 * q^65 - 570 * q^66 + 419 * q^67 - 204 * q^68 + 345 * q^69 - 96 * q^71 - 16 * q^72 + 313 * q^73 - 506 * q^74 - 220 * q^75 - 20 * q^76 + 700 * q^78 + 461 * q^79 + 144 * q^80 - 671 * q^81 + 84 * q^82 + 588 * q^83 - 459 * q^85 - 248 * q^86 + 570 * q^87 - 456 * q^88 + 1017 * q^89 - 36 * q^90 + 276 * q^92 - 115 * q^93 - 402 * q^94 - 45 * q^95 + 160 * q^96 + 1834 * q^97 + 114 * 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
 0
2.00000 5.00000 4.00000 9.00000 10.0000 0 8.00000 −2.00000 18.0000
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.a.f 1
3.b odd 2 1 882.4.a.c 1
4.b odd 2 1 784.4.a.c 1
5.b even 2 1 2450.4.a.d 1
7.b odd 2 1 98.4.a.d 1
7.c even 3 2 98.4.c.a 2
7.d odd 6 2 14.4.c.a 2
21.c even 2 1 882.4.a.f 1
21.g even 6 2 126.4.g.d 2
21.h odd 6 2 882.4.g.u 2
28.d even 2 1 784.4.a.p 1
28.f even 6 2 112.4.i.a 2
35.c odd 2 1 2450.4.a.q 1
35.i odd 6 2 350.4.e.e 2
35.k even 12 4 350.4.j.b 4
56.j odd 6 2 448.4.i.b 2
56.m even 6 2 448.4.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 7.d odd 6 2
98.4.a.d 1 7.b odd 2 1
98.4.a.f 1 1.a even 1 1 trivial
98.4.c.a 2 7.c even 3 2
112.4.i.a 2 28.f even 6 2
126.4.g.d 2 21.g even 6 2
350.4.e.e 2 35.i odd 6 2
350.4.j.b 4 35.k even 12 4
448.4.i.b 2 56.j odd 6 2
448.4.i.e 2 56.m even 6 2
784.4.a.c 1 4.b odd 2 1
784.4.a.p 1 28.d even 2 1
882.4.a.c 1 3.b odd 2 1
882.4.a.f 1 21.c even 2 1
882.4.g.u 2 21.h odd 6 2
2450.4.a.d 1 5.b even 2 1
2450.4.a.q 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 5$$
$5$ $$T - 9$$
$7$ $$T$$
$11$ $$T + 57$$
$13$ $$T - 70$$
$17$ $$T + 51$$
$19$ $$T + 5$$
$23$ $$T - 69$$
$29$ $$T - 114$$
$31$ $$T + 23$$
$37$ $$T + 253$$
$41$ $$T - 42$$
$43$ $$T + 124$$
$47$ $$T + 201$$
$53$ $$T + 393$$
$59$ $$T + 219$$
$61$ $$T - 709$$
$67$ $$T - 419$$
$71$ $$T + 96$$
$73$ $$T - 313$$
$79$ $$T - 461$$
$83$ $$T - 588$$
$89$ $$T - 1017$$
$97$ $$T - 1834$$