# Properties

 Label 66.2.a.b Level $66$ Weight $2$ Character orbit 66.a Self dual yes Analytic conductor $0.527$ 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] = [66,2,Mod(1,66)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(66, 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("66.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 66.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: $$0.527012653340$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{10} - q^{11} - q^{12} - 6 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + 2 q^{20} + 4 q^{21} - q^{22} + 4 q^{23} - q^{24} - q^{25} - 6 q^{26} - q^{27} - 4 q^{28} + 6 q^{29} - 2 q^{30} + q^{32} + q^{33} + 2 q^{34} - 8 q^{35} + q^{36} + 6 q^{37} + 4 q^{38} + 6 q^{39} + 2 q^{40} - 6 q^{41} + 4 q^{42} + 4 q^{43} - q^{44} + 2 q^{45} + 4 q^{46} - 12 q^{47} - q^{48} + 9 q^{49} - q^{50} - 2 q^{51} - 6 q^{52} + 2 q^{53} - q^{54} - 2 q^{55} - 4 q^{56} - 4 q^{57} + 6 q^{58} + 12 q^{59} - 2 q^{60} - 14 q^{61} - 4 q^{63} + q^{64} - 12 q^{65} + q^{66} + 4 q^{67} + 2 q^{68} - 4 q^{69} - 8 q^{70} - 12 q^{71} + q^{72} - 6 q^{73} + 6 q^{74} + q^{75} + 4 q^{76} + 4 q^{77} + 6 q^{78} - 4 q^{79} + 2 q^{80} + q^{81} - 6 q^{82} + 4 q^{83} + 4 q^{84} + 4 q^{85} + 4 q^{86} - 6 q^{87} - q^{88} + 10 q^{89} + 2 q^{90} + 24 q^{91} + 4 q^{92} - 12 q^{94} + 8 q^{95} - q^{96} - 14 q^{97} + 9 q^{98} - q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 - 4 * q^7 + q^8 + q^9 + 2 * q^10 - q^11 - q^12 - 6 * q^13 - 4 * q^14 - 2 * q^15 + q^16 + 2 * q^17 + q^18 + 4 * q^19 + 2 * q^20 + 4 * q^21 - q^22 + 4 * q^23 - q^24 - q^25 - 6 * q^26 - q^27 - 4 * q^28 + 6 * q^29 - 2 * q^30 + q^32 + q^33 + 2 * q^34 - 8 * q^35 + q^36 + 6 * q^37 + 4 * q^38 + 6 * q^39 + 2 * q^40 - 6 * q^41 + 4 * q^42 + 4 * q^43 - q^44 + 2 * q^45 + 4 * q^46 - 12 * q^47 - q^48 + 9 * q^49 - q^50 - 2 * q^51 - 6 * q^52 + 2 * q^53 - q^54 - 2 * q^55 - 4 * q^56 - 4 * q^57 + 6 * q^58 + 12 * q^59 - 2 * q^60 - 14 * q^61 - 4 * q^63 + q^64 - 12 * q^65 + q^66 + 4 * q^67 + 2 * q^68 - 4 * q^69 - 8 * q^70 - 12 * q^71 + q^72 - 6 * q^73 + 6 * q^74 + q^75 + 4 * q^76 + 4 * q^77 + 6 * q^78 - 4 * q^79 + 2 * q^80 + q^81 - 6 * q^82 + 4 * q^83 + 4 * q^84 + 4 * q^85 + 4 * q^86 - 6 * q^87 - q^88 + 10 * q^89 + 2 * q^90 + 24 * q^91 + 4 * q^92 - 12 * q^94 + 8 * q^95 - q^96 - 14 * q^97 + 9 * q^98 - 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
 0
1.00000 −1.00000 1.00000 2.00000 −1.00000 −4.00000 1.00000 1.00000 2.00000
 $$n$$: 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 66.2.a.b 1
3.b odd 2 1 198.2.a.a 1
4.b odd 2 1 528.2.a.j 1
5.b even 2 1 1650.2.a.k 1
5.c odd 4 2 1650.2.c.e 2
7.b odd 2 1 3234.2.a.t 1
8.b even 2 1 2112.2.a.r 1
8.d odd 2 1 2112.2.a.e 1
9.c even 3 2 1782.2.e.e 2
9.d odd 6 2 1782.2.e.v 2
11.b odd 2 1 726.2.a.c 1
11.c even 5 4 726.2.e.g 4
11.d odd 10 4 726.2.e.o 4
12.b even 2 1 1584.2.a.f 1
15.d odd 2 1 4950.2.a.bu 1
15.e even 4 2 4950.2.c.p 2
21.c even 2 1 9702.2.a.x 1
24.f even 2 1 6336.2.a.cj 1
24.h odd 2 1 6336.2.a.bw 1
33.d even 2 1 2178.2.a.g 1
44.c even 2 1 5808.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 1.a even 1 1 trivial
198.2.a.a 1 3.b odd 2 1
528.2.a.j 1 4.b odd 2 1
726.2.a.c 1 11.b odd 2 1
726.2.e.g 4 11.c even 5 4
726.2.e.o 4 11.d odd 10 4
1584.2.a.f 1 12.b even 2 1
1650.2.a.k 1 5.b even 2 1
1650.2.c.e 2 5.c odd 4 2
1782.2.e.e 2 9.c even 3 2
1782.2.e.v 2 9.d odd 6 2
2112.2.a.e 1 8.d odd 2 1
2112.2.a.r 1 8.b even 2 1
2178.2.a.g 1 33.d even 2 1
3234.2.a.t 1 7.b odd 2 1
4950.2.a.bu 1 15.d odd 2 1
4950.2.c.p 2 15.e even 4 2
5808.2.a.bc 1 44.c even 2 1
6336.2.a.bw 1 24.h odd 2 1
6336.2.a.cj 1 24.f even 2 1
9702.2.a.x 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(66))$$.

## Hecke characteristic polynomials

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