# Properties

 Label 198.2.a.e Level $198$ Weight $2$ Character orbit 198.a Self dual yes Analytic conductor $1.581$ 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] = [198,2,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, 2, names="a")

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

chi := DirichletCharacter("198.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$198 = 2 \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$1.58103796002$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 2 * q^7 + q^8 $$q + q^{2} + q^{4} + 2 q^{7} + q^{8} + q^{11} - 4 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + q^{22} - 6 q^{23} - 5 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29} + 8 q^{31} + q^{32} + 6 q^{34} - 10 q^{37} - 4 q^{38} - 6 q^{41} + 8 q^{43} + q^{44} - 6 q^{46} + 6 q^{47} - 3 q^{49} - 5 q^{50} - 4 q^{52} + 2 q^{56} - 6 q^{58} + 8 q^{61} + 8 q^{62} + q^{64} - 4 q^{67} + 6 q^{68} - 6 q^{71} + 2 q^{73} - 10 q^{74} - 4 q^{76} + 2 q^{77} + 14 q^{79} - 6 q^{82} + 12 q^{83} + 8 q^{86} + q^{88} + 6 q^{89} - 8 q^{91} - 6 q^{92} + 6 q^{94} + 14 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 2 * q^7 + q^8 + q^11 - 4 * q^13 + 2 * q^14 + q^16 + 6 * q^17 - 4 * q^19 + q^22 - 6 * q^23 - 5 * q^25 - 4 * q^26 + 2 * q^28 - 6 * q^29 + 8 * q^31 + q^32 + 6 * q^34 - 10 * q^37 - 4 * q^38 - 6 * q^41 + 8 * q^43 + q^44 - 6 * q^46 + 6 * q^47 - 3 * q^49 - 5 * q^50 - 4 * q^52 + 2 * q^56 - 6 * q^58 + 8 * q^61 + 8 * q^62 + q^64 - 4 * q^67 + 6 * q^68 - 6 * q^71 + 2 * q^73 - 10 * q^74 - 4 * q^76 + 2 * q^77 + 14 * q^79 - 6 * q^82 + 12 * q^83 + 8 * q^86 + q^88 + 6 * q^89 - 8 * q^91 - 6 * q^92 + 6 * q^94 + 14 * q^97 - 3 * 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
1.00000 0 1.00000 0 0 2.00000 1.00000 0 0
 $$n$$: 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 198.2.a.e 1
3.b odd 2 1 66.2.a.a 1
4.b odd 2 1 1584.2.a.h 1
5.b even 2 1 4950.2.a.g 1
5.c odd 4 2 4950.2.c.r 2
7.b odd 2 1 9702.2.a.bu 1
8.b even 2 1 6336.2.a.bj 1
8.d odd 2 1 6336.2.a.bf 1
9.c even 3 2 1782.2.e.f 2
9.d odd 6 2 1782.2.e.s 2
11.b odd 2 1 2178.2.a.b 1
12.b even 2 1 528.2.a.d 1
15.d odd 2 1 1650.2.a.m 1
15.e even 4 2 1650.2.c.d 2
21.c even 2 1 3234.2.a.d 1
24.f even 2 1 2112.2.a.v 1
24.h odd 2 1 2112.2.a.i 1
33.d even 2 1 726.2.a.i 1
33.f even 10 4 726.2.e.b 4
33.h odd 10 4 726.2.e.k 4
132.d odd 2 1 5808.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 3.b odd 2 1
198.2.a.e 1 1.a even 1 1 trivial
528.2.a.d 1 12.b even 2 1
726.2.a.i 1 33.d even 2 1
726.2.e.b 4 33.f even 10 4
726.2.e.k 4 33.h odd 10 4
1584.2.a.h 1 4.b odd 2 1
1650.2.a.m 1 15.d odd 2 1
1650.2.c.d 2 15.e even 4 2
1782.2.e.f 2 9.c even 3 2
1782.2.e.s 2 9.d odd 6 2
2112.2.a.i 1 24.h odd 2 1
2112.2.a.v 1 24.f even 2 1
2178.2.a.b 1 11.b odd 2 1
3234.2.a.d 1 21.c even 2 1
4950.2.a.g 1 5.b even 2 1
4950.2.c.r 2 5.c odd 4 2
5808.2.a.l 1 132.d odd 2 1
6336.2.a.bf 1 8.d odd 2 1
6336.2.a.bj 1 8.b even 2 1
9702.2.a.bu 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(198))$$:

 $$T_{5}$$ T5 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T - 6$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T - 2$$
$79$ $$T - 14$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T - 14$$