# Properties

 Label 98.2.a.b Level $98$ Weight $2$ Character orbit 98.a Self dual yes Analytic conductor $0.783$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,2,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, 2, names="a")

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.782533939809$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} - 2 \beta q^{5} + \beta q^{6} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 + b * q^3 + q^4 - 2*b * q^5 + b * q^6 + q^8 - q^9 $$q + q^{2} + \beta q^{3} + q^{4} - 2 \beta q^{5} + \beta q^{6} + q^{8} - q^{9} - 2 \beta q^{10} - 2 q^{11} + \beta q^{12} - 4 q^{15} + q^{16} + \beta q^{17} - q^{18} + 5 \beta q^{19} - 2 \beta q^{20} - 2 q^{22} - 4 q^{23} + \beta q^{24} + 3 q^{25} - 4 \beta q^{27} + 2 q^{29} - 4 q^{30} - 6 \beta q^{31} + q^{32} - 2 \beta q^{33} + \beta q^{34} - q^{36} + 10 q^{37} + 5 \beta q^{38} - 2 \beta q^{40} + 7 \beta q^{41} + 2 q^{43} - 2 q^{44} + 2 \beta q^{45} - 4 q^{46} - 2 \beta q^{47} + \beta q^{48} + 3 q^{50} + 2 q^{51} - 2 q^{53} - 4 \beta q^{54} + 4 \beta q^{55} + 10 q^{57} + 2 q^{58} + \beta q^{59} - 4 q^{60} - 2 \beta q^{61} - 6 \beta q^{62} + q^{64} - 2 \beta q^{66} + 12 q^{67} + \beta q^{68} - 4 \beta q^{69} - 12 q^{71} - q^{72} + \beta q^{73} + 10 q^{74} + 3 \beta q^{75} + 5 \beta q^{76} - 4 q^{79} - 2 \beta q^{80} - 5 q^{81} + 7 \beta q^{82} - 7 \beta q^{83} - 4 q^{85} + 2 q^{86} + 2 \beta q^{87} - 2 q^{88} + 5 \beta q^{89} + 2 \beta q^{90} - 4 q^{92} - 12 q^{93} - 2 \beta q^{94} - 20 q^{95} + \beta q^{96} - 7 \beta q^{97} + 2 q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 - 2*b * q^5 + b * q^6 + q^8 - q^9 - 2*b * q^10 - 2 * q^11 + b * q^12 - 4 * q^15 + q^16 + b * q^17 - q^18 + 5*b * q^19 - 2*b * q^20 - 2 * q^22 - 4 * q^23 + b * q^24 + 3 * q^25 - 4*b * q^27 + 2 * q^29 - 4 * q^30 - 6*b * q^31 + q^32 - 2*b * q^33 + b * q^34 - q^36 + 10 * q^37 + 5*b * q^38 - 2*b * q^40 + 7*b * q^41 + 2 * q^43 - 2 * q^44 + 2*b * q^45 - 4 * q^46 - 2*b * q^47 + b * q^48 + 3 * q^50 + 2 * q^51 - 2 * q^53 - 4*b * q^54 + 4*b * q^55 + 10 * q^57 + 2 * q^58 + b * q^59 - 4 * q^60 - 2*b * q^61 - 6*b * q^62 + q^64 - 2*b * q^66 + 12 * q^67 + b * q^68 - 4*b * q^69 - 12 * q^71 - q^72 + b * q^73 + 10 * q^74 + 3*b * q^75 + 5*b * q^76 - 4 * q^79 - 2*b * q^80 - 5 * q^81 + 7*b * q^82 - 7*b * q^83 - 4 * q^85 + 2 * q^86 + 2*b * q^87 - 2 * q^88 + 5*b * q^89 + 2*b * q^90 - 4 * q^92 - 12 * q^93 - 2*b * q^94 - 20 * q^95 + b * q^96 - 7*b * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 4 q^{11} - 8 q^{15} + 2 q^{16} - 2 q^{18} - 4 q^{22} - 8 q^{23} + 6 q^{25} + 4 q^{29} - 8 q^{30} + 2 q^{32} - 2 q^{36} + 20 q^{37} + 4 q^{43} - 4 q^{44} - 8 q^{46} + 6 q^{50} + 4 q^{51} - 4 q^{53} + 20 q^{57} + 4 q^{58} - 8 q^{60} + 2 q^{64} + 24 q^{67} - 24 q^{71} - 2 q^{72} + 20 q^{74} - 8 q^{79} - 10 q^{81} - 8 q^{85} + 4 q^{86} - 4 q^{88} - 8 q^{92} - 24 q^{93} - 40 q^{95} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 4 * q^11 - 8 * q^15 + 2 * q^16 - 2 * q^18 - 4 * q^22 - 8 * q^23 + 6 * q^25 + 4 * q^29 - 8 * q^30 + 2 * q^32 - 2 * q^36 + 20 * q^37 + 4 * q^43 - 4 * q^44 - 8 * q^46 + 6 * q^50 + 4 * q^51 - 4 * q^53 + 20 * q^57 + 4 * q^58 - 8 * q^60 + 2 * q^64 + 24 * q^67 - 24 * q^71 - 2 * q^72 + 20 * q^74 - 8 * q^79 - 10 * q^81 - 8 * q^85 + 4 * q^86 - 4 * q^88 - 8 * q^92 - 24 * q^93 - 40 * q^95 + 4 * 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
1.00000 −1.41421 1.00000 2.82843 −1.41421 0 1.00000 −1.00000 2.82843
1.2 1.00000 1.41421 1.00000 −2.82843 1.41421 0 1.00000 −1.00000 −2.82843
 $$n$$: 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 98.2.a.b 2
3.b odd 2 1 882.2.a.n 2
4.b odd 2 1 784.2.a.l 2
5.b even 2 1 2450.2.a.bj 2
5.c odd 4 2 2450.2.c.v 4
7.b odd 2 1 inner 98.2.a.b 2
7.c even 3 2 98.2.c.c 4
7.d odd 6 2 98.2.c.c 4
8.b even 2 1 3136.2.a.bn 2
8.d odd 2 1 3136.2.a.bm 2
12.b even 2 1 7056.2.a.cl 2
21.c even 2 1 882.2.a.n 2
21.g even 6 2 882.2.g.l 4
21.h odd 6 2 882.2.g.l 4
28.d even 2 1 784.2.a.l 2
28.f even 6 2 784.2.i.m 4
28.g odd 6 2 784.2.i.m 4
35.c odd 2 1 2450.2.a.bj 2
35.f even 4 2 2450.2.c.v 4
56.e even 2 1 3136.2.a.bm 2
56.h odd 2 1 3136.2.a.bn 2
84.h odd 2 1 7056.2.a.cl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 1.a even 1 1 trivial
98.2.a.b 2 7.b odd 2 1 inner
98.2.c.c 4 7.c even 3 2
98.2.c.c 4 7.d odd 6 2
784.2.a.l 2 4.b odd 2 1
784.2.a.l 2 28.d even 2 1
784.2.i.m 4 28.f even 6 2
784.2.i.m 4 28.g odd 6 2
882.2.a.n 2 3.b odd 2 1
882.2.a.n 2 21.c even 2 1
882.2.g.l 4 21.g even 6 2
882.2.g.l 4 21.h odd 6 2
2450.2.a.bj 2 5.b even 2 1
2450.2.a.bj 2 35.c odd 2 1
2450.2.c.v 4 5.c odd 4 2
2450.2.c.v 4 35.f even 4 2
3136.2.a.bm 2 8.d odd 2 1
3136.2.a.bm 2 56.e even 2 1
3136.2.a.bn 2 8.b even 2 1
3136.2.a.bn 2 56.h odd 2 1
7056.2.a.cl 2 12.b even 2 1
7056.2.a.cl 2 84.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 50$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 72$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 98$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 2$$
$61$ $$T^{2} - 8$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 2$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 98$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 98$$