# 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

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

## 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
 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

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