# Properties

 Label 98.4.a.a Level $98$ Weight $4$ Character orbit 98.a Self dual yes Analytic conductor $5.782$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 14 q^{5} + 16 q^{6} - 8 q^{8} + 37 q^{9}+O(q^{10})$$ q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 14 * q^5 + 16 * q^6 - 8 * q^8 + 37 * q^9 $$q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 14 q^{5} + 16 q^{6} - 8 q^{8} + 37 q^{9} - 28 q^{10} - 28 q^{11} - 32 q^{12} - 18 q^{13} - 112 q^{15} + 16 q^{16} - 74 q^{17} - 74 q^{18} - 80 q^{19} + 56 q^{20} + 56 q^{22} - 112 q^{23} + 64 q^{24} + 71 q^{25} + 36 q^{26} - 80 q^{27} + 190 q^{29} + 224 q^{30} - 72 q^{31} - 32 q^{32} + 224 q^{33} + 148 q^{34} + 148 q^{36} - 346 q^{37} + 160 q^{38} + 144 q^{39} - 112 q^{40} - 162 q^{41} - 412 q^{43} - 112 q^{44} + 518 q^{45} + 224 q^{46} - 24 q^{47} - 128 q^{48} - 142 q^{50} + 592 q^{51} - 72 q^{52} + 318 q^{53} + 160 q^{54} - 392 q^{55} + 640 q^{57} - 380 q^{58} + 200 q^{59} - 448 q^{60} + 198 q^{61} + 144 q^{62} + 64 q^{64} - 252 q^{65} - 448 q^{66} - 716 q^{67} - 296 q^{68} + 896 q^{69} + 392 q^{71} - 296 q^{72} - 538 q^{73} + 692 q^{74} - 568 q^{75} - 320 q^{76} - 288 q^{78} + 240 q^{79} + 224 q^{80} - 359 q^{81} + 324 q^{82} + 1072 q^{83} - 1036 q^{85} + 824 q^{86} - 1520 q^{87} + 224 q^{88} - 810 q^{89} - 1036 q^{90} - 448 q^{92} + 576 q^{93} + 48 q^{94} - 1120 q^{95} + 256 q^{96} - 1354 q^{97} - 1036 q^{99}+O(q^{100})$$ q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 14 * q^5 + 16 * q^6 - 8 * q^8 + 37 * q^9 - 28 * q^10 - 28 * q^11 - 32 * q^12 - 18 * q^13 - 112 * q^15 + 16 * q^16 - 74 * q^17 - 74 * q^18 - 80 * q^19 + 56 * q^20 + 56 * q^22 - 112 * q^23 + 64 * q^24 + 71 * q^25 + 36 * q^26 - 80 * q^27 + 190 * q^29 + 224 * q^30 - 72 * q^31 - 32 * q^32 + 224 * q^33 + 148 * q^34 + 148 * q^36 - 346 * q^37 + 160 * q^38 + 144 * q^39 - 112 * q^40 - 162 * q^41 - 412 * q^43 - 112 * q^44 + 518 * q^45 + 224 * q^46 - 24 * q^47 - 128 * q^48 - 142 * q^50 + 592 * q^51 - 72 * q^52 + 318 * q^53 + 160 * q^54 - 392 * q^55 + 640 * q^57 - 380 * q^58 + 200 * q^59 - 448 * q^60 + 198 * q^61 + 144 * q^62 + 64 * q^64 - 252 * q^65 - 448 * q^66 - 716 * q^67 - 296 * q^68 + 896 * q^69 + 392 * q^71 - 296 * q^72 - 538 * q^73 + 692 * q^74 - 568 * q^75 - 320 * q^76 - 288 * q^78 + 240 * q^79 + 224 * q^80 - 359 * q^81 + 324 * q^82 + 1072 * q^83 - 1036 * q^85 + 824 * q^86 - 1520 * q^87 + 224 * q^88 - 810 * q^89 - 1036 * q^90 - 448 * q^92 + 576 * q^93 + 48 * q^94 - 1120 * q^95 + 256 * q^96 - 1354 * q^97 - 1036 * 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 −8.00000 4.00000 14.0000 16.0000 0 −8.00000 37.0000 −28.0000
 $$n$$: 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.a 1
3.b odd 2 1 882.4.a.i 1
4.b odd 2 1 784.4.a.s 1
5.b even 2 1 2450.4.a.bo 1
7.b odd 2 1 14.4.a.a 1
7.c even 3 2 98.4.c.f 2
7.d odd 6 2 98.4.c.d 2
21.c even 2 1 126.4.a.h 1
21.g even 6 2 882.4.g.b 2
21.h odd 6 2 882.4.g.k 2
28.d even 2 1 112.4.a.a 1
35.c odd 2 1 350.4.a.l 1
35.f even 4 2 350.4.c.b 2
56.e even 2 1 448.4.a.o 1
56.h odd 2 1 448.4.a.b 1
77.b even 2 1 1694.4.a.g 1
84.h odd 2 1 1008.4.a.s 1
91.b odd 2 1 2366.4.a.h 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 8$$
$5$ $$T - 14$$
$7$ $$T$$
$11$ $$T + 28$$
$13$ $$T + 18$$
$17$ $$T + 74$$
$19$ $$T + 80$$
$23$ $$T + 112$$
$29$ $$T - 190$$
$31$ $$T + 72$$
$37$ $$T + 346$$
$41$ $$T + 162$$
$43$ $$T + 412$$
$47$ $$T + 24$$
$53$ $$T - 318$$
$59$ $$T - 200$$
$61$ $$T - 198$$
$67$ $$T + 716$$
$71$ $$T - 392$$
$73$ $$T + 538$$
$79$ $$T - 240$$
$83$ $$T - 1072$$
$89$ $$T + 810$$
$97$ $$T + 1354$$