# Properties

 Label 98.4.a.e 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

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 2q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 8q^{8} - 23q^{9} + O(q^{10})$$ $$q + 2q^{2} + 2q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 8q^{8} - 23q^{9} + 24q^{10} + 48q^{11} + 8q^{12} - 56q^{13} + 24q^{15} + 16q^{16} + 114q^{17} - 46q^{18} - 2q^{19} + 48q^{20} + 96q^{22} - 120q^{23} + 16q^{24} + 19q^{25} - 112q^{26} - 100q^{27} - 54q^{29} + 48q^{30} - 236q^{31} + 32q^{32} + 96q^{33} + 228q^{34} - 92q^{36} + 146q^{37} - 4q^{38} - 112q^{39} + 96q^{40} - 126q^{41} - 376q^{43} + 192q^{44} - 276q^{45} - 240q^{46} + 12q^{47} + 32q^{48} + 38q^{50} + 228q^{51} - 224q^{52} + 174q^{53} - 200q^{54} + 576q^{55} - 4q^{57} - 108q^{58} - 138q^{59} + 96q^{60} - 380q^{61} - 472q^{62} + 64q^{64} - 672q^{65} + 192q^{66} - 484q^{67} + 456q^{68} - 240q^{69} + 576q^{71} - 184q^{72} + 1150q^{73} + 292q^{74} + 38q^{75} - 8q^{76} - 224q^{78} + 776q^{79} + 192q^{80} + 421q^{81} - 252q^{82} - 378q^{83} + 1368q^{85} - 752q^{86} - 108q^{87} + 384q^{88} + 390q^{89} - 552q^{90} - 480q^{92} - 472q^{93} + 24q^{94} - 24q^{95} + 64q^{96} + 1330q^{97} - 1104q^{99} + O(q^{100})$$

## 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 2.00000 4.00000 12.0000 4.00000 0 8.00000 −23.0000 24.0000
 $$n$$: 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.e 1
3.b odd 2 1 882.4.a.b 1
4.b odd 2 1 784.4.a.h 1
5.b even 2 1 2450.4.a.i 1
7.b odd 2 1 14.4.a.b 1
7.c even 3 2 98.4.c.b 2
7.d odd 6 2 98.4.c.c 2
21.c even 2 1 126.4.a.d 1
21.g even 6 2 882.4.g.p 2
21.h odd 6 2 882.4.g.v 2
28.d even 2 1 112.4.a.e 1
35.c odd 2 1 350.4.a.f 1
35.f even 4 2 350.4.c.g 2
56.e even 2 1 448.4.a.g 1
56.h odd 2 1 448.4.a.k 1
77.b even 2 1 1694.4.a.b 1
84.h odd 2 1 1008.4.a.r 1
91.b odd 2 1 2366.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 7.b odd 2 1
98.4.a.e 1 1.a even 1 1 trivial
98.4.c.b 2 7.c even 3 2
98.4.c.c 2 7.d odd 6 2
112.4.a.e 1 28.d even 2 1
126.4.a.d 1 21.c even 2 1
350.4.a.f 1 35.c odd 2 1
350.4.c.g 2 35.f even 4 2
448.4.a.g 1 56.e even 2 1
448.4.a.k 1 56.h odd 2 1
784.4.a.h 1 4.b odd 2 1
882.4.a.b 1 3.b odd 2 1
882.4.g.p 2 21.g even 6 2
882.4.g.v 2 21.h odd 6 2
1008.4.a.r 1 84.h odd 2 1
1694.4.a.b 1 77.b even 2 1
2366.4.a.c 1 91.b odd 2 1
2450.4.a.i 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$-2 + T$$
$5$ $$-12 + T$$
$7$ $$T$$
$11$ $$-48 + T$$
$13$ $$56 + T$$
$17$ $$-114 + T$$
$19$ $$2 + T$$
$23$ $$120 + T$$
$29$ $$54 + T$$
$31$ $$236 + T$$
$37$ $$-146 + T$$
$41$ $$126 + T$$
$43$ $$376 + T$$
$47$ $$-12 + T$$
$53$ $$-174 + T$$
$59$ $$138 + T$$
$61$ $$380 + T$$
$67$ $$484 + T$$
$71$ $$-576 + T$$
$73$ $$-1150 + T$$
$79$ $$-776 + T$$
$83$ $$378 + T$$
$89$ $$-390 + T$$
$97$ $$-1330 + T$$