# Properties

 Label 98.4.a.g 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} -10 \beta q^{6} -8 q^{8} + 23 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} -10 \beta q^{6} -8 q^{8} + 23 q^{9} -28 \beta q^{10} -14 q^{11} + 20 \beta q^{12} -36 \beta q^{13} + 140 q^{15} + 16 q^{16} -\beta q^{17} -46 q^{18} + \beta q^{19} + 56 \beta q^{20} + 28 q^{22} + 140 q^{23} -40 \beta q^{24} + 267 q^{25} + 72 \beta q^{26} -20 \beta q^{27} -286 q^{29} -280 q^{30} + 66 \beta q^{31} -32 q^{32} -70 \beta q^{33} + 2 \beta q^{34} + 92 q^{36} -38 q^{37} -2 \beta q^{38} -360 q^{39} -112 \beta q^{40} + 89 \beta q^{41} -34 q^{43} -56 q^{44} + 322 \beta q^{45} -280 q^{46} -370 \beta q^{47} + 80 \beta q^{48} -534 q^{50} -10 q^{51} -144 \beta q^{52} -74 q^{53} + 40 \beta q^{54} -196 \beta q^{55} + 10 q^{57} + 572 q^{58} -307 \beta q^{59} + 560 q^{60} -10 \beta q^{61} -132 \beta q^{62} + 64 q^{64} -1008 q^{65} + 140 \beta q^{66} + 684 q^{67} -4 \beta q^{68} + 700 \beta q^{69} + 588 q^{71} -184 q^{72} + 191 \beta q^{73} + 76 q^{74} + 1335 \beta q^{75} + 4 \beta q^{76} + 720 q^{78} + 1220 q^{79} + 224 \beta q^{80} -821 q^{81} -178 \beta q^{82} -299 \beta q^{83} -28 q^{85} + 68 q^{86} -1430 \beta q^{87} + 112 q^{88} -437 \beta q^{89} -644 \beta q^{90} + 560 q^{92} + 660 q^{93} + 740 \beta q^{94} + 28 q^{95} -160 \beta q^{96} -1049 \beta q^{97} -322 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} - 28q^{11} + 280q^{15} + 32q^{16} - 92q^{18} + 56q^{22} + 280q^{23} + 534q^{25} - 572q^{29} - 560q^{30} - 64q^{32} + 184q^{36} - 76q^{37} - 720q^{39} - 68q^{43} - 112q^{44} - 560q^{46} - 1068q^{50} - 20q^{51} - 148q^{53} + 20q^{57} + 1144q^{58} + 1120q^{60} + 128q^{64} - 2016q^{65} + 1368q^{67} + 1176q^{71} - 368q^{72} + 152q^{74} + 1440q^{78} + 2440q^{79} - 1642q^{81} - 56q^{85} + 136q^{86} + 224q^{88} + 1120q^{92} + 1320q^{93} + 56q^{95} - 644q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.00000 −7.07107 4.00000 −19.7990 14.1421 0 −8.00000 23.0000 39.5980
1.2 −2.00000 7.07107 4.00000 19.7990 −14.1421 0 −8.00000 23.0000 −39.5980
 $$n$$: 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.g 2
3.b odd 2 1 882.4.a.bg 2
4.b odd 2 1 784.4.a.y 2
5.b even 2 1 2450.4.a.bx 2
7.b odd 2 1 inner 98.4.a.g 2
7.c even 3 2 98.4.c.h 4
7.d odd 6 2 98.4.c.h 4
21.c even 2 1 882.4.a.bg 2
21.g even 6 2 882.4.g.ba 4
21.h odd 6 2 882.4.g.ba 4
28.d even 2 1 784.4.a.y 2
35.c odd 2 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 1.a even 1 1 trivial
98.4.a.g 2 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 2
98.4.c.h 4 7.d odd 6 2
784.4.a.y 2 4.b odd 2 1
784.4.a.y 2 28.d even 2 1
882.4.a.bg 2 3.b odd 2 1
882.4.a.bg 2 21.c even 2 1
882.4.g.ba 4 21.g even 6 2
882.4.g.ba 4 21.h odd 6 2
2450.4.a.bx 2 5.b even 2 1
2450.4.a.bx 2 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 50$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$-50 + T^{2}$$
$5$ $$-392 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$-2592 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-2 + T^{2}$$
$23$ $$( -140 + T )^{2}$$
$29$ $$( 286 + T )^{2}$$
$31$ $$-8712 + T^{2}$$
$37$ $$( 38 + T )^{2}$$
$41$ $$-15842 + T^{2}$$
$43$ $$( 34 + T )^{2}$$
$47$ $$-273800 + T^{2}$$
$53$ $$( 74 + T )^{2}$$
$59$ $$-188498 + T^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( -684 + T )^{2}$$
$71$ $$( -588 + T )^{2}$$
$73$ $$-72962 + T^{2}$$
$79$ $$( -1220 + T )^{2}$$
$83$ $$-178802 + T^{2}$$
$89$ $$-381938 + T^{2}$$
$97$ $$-2200802 + T^{2}$$