# Properties

 Label 198.6.a.b Level $198$ Weight $6$ Character orbit 198.a Self dual yes Analytic conductor $31.756$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,6,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("198.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$198 = 2 \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 198.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: $$31.7559963230$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 16 q^{4} + 31 q^{5} - 230 q^{7} - 64 q^{8}+O(q^{10})$$ q - 4 * q^2 + 16 * q^4 + 31 * q^5 - 230 * q^7 - 64 * q^8 $$q - 4 q^{2} + 16 q^{4} + 31 q^{5} - 230 q^{7} - 64 q^{8} - 124 q^{10} - 121 q^{11} + 112 q^{13} + 920 q^{14} + 256 q^{16} + 1142 q^{17} - 612 q^{19} + 496 q^{20} + 484 q^{22} + 1941 q^{23} - 2164 q^{25} - 448 q^{26} - 3680 q^{28} - 1192 q^{29} - 1037 q^{31} - 1024 q^{32} - 4568 q^{34} - 7130 q^{35} + 8083 q^{37} + 2448 q^{38} - 1984 q^{40} + 10444 q^{41} + 58 q^{43} - 1936 q^{44} - 7764 q^{46} - 8656 q^{47} + 36093 q^{49} + 8656 q^{50} + 1792 q^{52} + 20318 q^{53} - 3751 q^{55} + 14720 q^{56} + 4768 q^{58} + 21351 q^{59} + 47044 q^{61} + 4148 q^{62} + 4096 q^{64} + 3472 q^{65} + 48093 q^{67} + 18272 q^{68} + 28520 q^{70} + 24967 q^{71} - 42288 q^{73} - 32332 q^{74} - 9792 q^{76} + 27830 q^{77} - 72410 q^{79} + 7936 q^{80} - 41776 q^{82} + 15806 q^{83} + 35402 q^{85} - 232 q^{86} + 7744 q^{88} + 114761 q^{89} - 25760 q^{91} + 31056 q^{92} + 34624 q^{94} - 18972 q^{95} - 5159 q^{97} - 144372 q^{98}+O(q^{100})$$ q - 4 * q^2 + 16 * q^4 + 31 * q^5 - 230 * q^7 - 64 * q^8 - 124 * q^10 - 121 * q^11 + 112 * q^13 + 920 * q^14 + 256 * q^16 + 1142 * q^17 - 612 * q^19 + 496 * q^20 + 484 * q^22 + 1941 * q^23 - 2164 * q^25 - 448 * q^26 - 3680 * q^28 - 1192 * q^29 - 1037 * q^31 - 1024 * q^32 - 4568 * q^34 - 7130 * q^35 + 8083 * q^37 + 2448 * q^38 - 1984 * q^40 + 10444 * q^41 + 58 * q^43 - 1936 * q^44 - 7764 * q^46 - 8656 * q^47 + 36093 * q^49 + 8656 * q^50 + 1792 * q^52 + 20318 * q^53 - 3751 * q^55 + 14720 * q^56 + 4768 * q^58 + 21351 * q^59 + 47044 * q^61 + 4148 * q^62 + 4096 * q^64 + 3472 * q^65 + 48093 * q^67 + 18272 * q^68 + 28520 * q^70 + 24967 * q^71 - 42288 * q^73 - 32332 * q^74 - 9792 * q^76 + 27830 * q^77 - 72410 * q^79 + 7936 * q^80 - 41776 * q^82 + 15806 * q^83 + 35402 * q^85 - 232 * q^86 + 7744 * q^88 + 114761 * q^89 - 25760 * q^91 + 31056 * q^92 + 34624 * q^94 - 18972 * q^95 - 5159 * q^97 - 144372 * q^98

## 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
 0
−4.00000 0 16.0000 31.0000 0 −230.000 −64.0000 0 −124.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 198.6.a.b 1
3.b odd 2 1 22.6.a.c 1
12.b even 2 1 176.6.a.e 1
15.d odd 2 1 550.6.a.c 1
15.e even 4 2 550.6.b.a 2
21.c even 2 1 1078.6.a.f 1
24.f even 2 1 704.6.a.a 1
24.h odd 2 1 704.6.a.j 1
33.d even 2 1 242.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.c 1 3.b odd 2 1
176.6.a.e 1 12.b even 2 1
198.6.a.b 1 1.a even 1 1 trivial
242.6.a.a 1 33.d even 2 1
550.6.a.c 1 15.d odd 2 1
550.6.b.a 2 15.e even 4 2
704.6.a.a 1 24.f even 2 1
704.6.a.j 1 24.h odd 2 1
1078.6.a.f 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(198))$$:

 $$T_{5} - 31$$ T5 - 31 $$T_{7} + 230$$ T7 + 230

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T - 31$$
$7$ $$T + 230$$
$11$ $$T + 121$$
$13$ $$T - 112$$
$17$ $$T - 1142$$
$19$ $$T + 612$$
$23$ $$T - 1941$$
$29$ $$T + 1192$$
$31$ $$T + 1037$$
$37$ $$T - 8083$$
$41$ $$T - 10444$$
$43$ $$T - 58$$
$47$ $$T + 8656$$
$53$ $$T - 20318$$
$59$ $$T - 21351$$
$61$ $$T - 47044$$
$67$ $$T - 48093$$
$71$ $$T - 24967$$
$73$ $$T + 42288$$
$79$ $$T + 72410$$
$83$ $$T - 15806$$
$89$ $$T - 114761$$
$97$ $$T + 5159$$