# 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 $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,4,Mod(1,98)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.78218718056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

comment: embeddings in the coefficient field

gp: mfembed(f)

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

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