# Properties

 Label 198.6.a.i 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + 51 q^{5} - 166 q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + 51 * q^5 - 166 * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} + 51 q^{5} - 166 q^{7} + 64 q^{8} + 204 q^{10} + 121 q^{11} + 692 q^{13} - 664 q^{14} + 256 q^{16} + 738 q^{17} + 1424 q^{19} + 816 q^{20} + 484 q^{22} + 1779 q^{23} - 524 q^{25} + 2768 q^{26} - 2656 q^{28} + 2064 q^{29} + 6245 q^{31} + 1024 q^{32} + 2952 q^{34} - 8466 q^{35} - 14785 q^{37} + 5696 q^{38} + 3264 q^{40} - 5304 q^{41} + 17798 q^{43} + 1936 q^{44} + 7116 q^{46} + 17184 q^{47} + 10749 q^{49} - 2096 q^{50} + 11072 q^{52} + 30726 q^{53} + 6171 q^{55} - 10624 q^{56} + 8256 q^{58} + 34989 q^{59} - 45940 q^{61} + 24980 q^{62} + 4096 q^{64} + 35292 q^{65} + 25343 q^{67} + 11808 q^{68} - 33864 q^{70} - 13311 q^{71} - 53260 q^{73} - 59140 q^{74} + 22784 q^{76} - 20086 q^{77} + 77234 q^{79} + 13056 q^{80} - 21216 q^{82} - 55014 q^{83} + 37638 q^{85} + 71192 q^{86} + 7744 q^{88} - 125415 q^{89} - 114872 q^{91} + 28464 q^{92} + 68736 q^{94} + 72624 q^{95} - 88807 q^{97} + 42996 q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + 51 * q^5 - 166 * q^7 + 64 * q^8 + 204 * q^10 + 121 * q^11 + 692 * q^13 - 664 * q^14 + 256 * q^16 + 738 * q^17 + 1424 * q^19 + 816 * q^20 + 484 * q^22 + 1779 * q^23 - 524 * q^25 + 2768 * q^26 - 2656 * q^28 + 2064 * q^29 + 6245 * q^31 + 1024 * q^32 + 2952 * q^34 - 8466 * q^35 - 14785 * q^37 + 5696 * q^38 + 3264 * q^40 - 5304 * q^41 + 17798 * q^43 + 1936 * q^44 + 7116 * q^46 + 17184 * q^47 + 10749 * q^49 - 2096 * q^50 + 11072 * q^52 + 30726 * q^53 + 6171 * q^55 - 10624 * q^56 + 8256 * q^58 + 34989 * q^59 - 45940 * q^61 + 24980 * q^62 + 4096 * q^64 + 35292 * q^65 + 25343 * q^67 + 11808 * q^68 - 33864 * q^70 - 13311 * q^71 - 53260 * q^73 - 59140 * q^74 + 22784 * q^76 - 20086 * q^77 + 77234 * q^79 + 13056 * q^80 - 21216 * q^82 - 55014 * q^83 + 37638 * q^85 + 71192 * q^86 + 7744 * q^88 - 125415 * q^89 - 114872 * q^91 + 28464 * q^92 + 68736 * q^94 + 72624 * q^95 - 88807 * q^97 + 42996 * 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 51.0000 0 −166.000 64.0000 0 204.000
 $$n$$: 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.i 1
3.b odd 2 1 22.6.a.b 1
12.b even 2 1 176.6.a.b 1
15.d odd 2 1 550.6.a.f 1
15.e even 4 2 550.6.b.f 2
21.c even 2 1 1078.6.a.a 1
24.f even 2 1 704.6.a.f 1
24.h odd 2 1 704.6.a.e 1
33.d even 2 1 242.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 3.b odd 2 1
176.6.a.b 1 12.b even 2 1
198.6.a.i 1 1.a even 1 1 trivial
242.6.a.d 1 33.d even 2 1
550.6.a.f 1 15.d odd 2 1
550.6.b.f 2 15.e even 4 2
704.6.a.e 1 24.h odd 2 1
704.6.a.f 1 24.f even 2 1
1078.6.a.a 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(198))$$:

 $$T_{5} - 51$$ T5 - 51 $$T_{7} + 166$$ T7 + 166

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T - 51$$
$7$ $$T + 166$$
$11$ $$T - 121$$
$13$ $$T - 692$$
$17$ $$T - 738$$
$19$ $$T - 1424$$
$23$ $$T - 1779$$
$29$ $$T - 2064$$
$31$ $$T - 6245$$
$37$ $$T + 14785$$
$41$ $$T + 5304$$
$43$ $$T - 17798$$
$47$ $$T - 17184$$
$53$ $$T - 30726$$
$59$ $$T - 34989$$
$61$ $$T + 45940$$
$67$ $$T - 25343$$
$71$ $$T + 13311$$
$73$ $$T + 53260$$
$79$ $$T - 77234$$
$83$ $$T + 55014$$
$89$ $$T + 125415$$
$97$ $$T + 88807$$