# Properties

 Label 66.4.a.a Level $66$ Weight $4$ Character orbit 66.a Self dual yes Analytic conductor $3.894$ 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] = [66,4,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, 4, names="a")

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

chi := DirichletCharacter("66.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$3.89412606038$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} + 14 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 + 14 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} + 14 q^{7} - 8 q^{8} + 9 q^{9} + 11 q^{11} + 12 q^{12} + 80 q^{13} - 28 q^{14} + 16 q^{16} + 30 q^{17} - 18 q^{18} + 56 q^{19} + 42 q^{21} - 22 q^{22} - 126 q^{23} - 24 q^{24} - 125 q^{25} - 160 q^{26} + 27 q^{27} + 56 q^{28} - 222 q^{29} - 16 q^{31} - 32 q^{32} + 33 q^{33} - 60 q^{34} + 36 q^{36} - 106 q^{37} - 112 q^{38} + 240 q^{39} + 114 q^{41} - 84 q^{42} - 52 q^{43} + 44 q^{44} + 252 q^{46} + 246 q^{47} + 48 q^{48} - 147 q^{49} + 250 q^{50} + 90 q^{51} + 320 q^{52} - 264 q^{53} - 54 q^{54} - 112 q^{56} + 168 q^{57} + 444 q^{58} + 264 q^{59} + 92 q^{61} + 32 q^{62} + 126 q^{63} + 64 q^{64} - 66 q^{66} - 796 q^{67} + 120 q^{68} - 378 q^{69} + 426 q^{71} - 72 q^{72} - 1174 q^{73} + 212 q^{74} - 375 q^{75} + 224 q^{76} + 154 q^{77} - 480 q^{78} + 842 q^{79} + 81 q^{81} - 228 q^{82} + 852 q^{83} + 168 q^{84} + 104 q^{86} - 666 q^{87} - 88 q^{88} - 1062 q^{89} + 1120 q^{91} - 504 q^{92} - 48 q^{93} - 492 q^{94} - 96 q^{96} - 1282 q^{97} + 294 q^{98} + 99 q^{99}+O(q^{100})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 + 14 * q^7 - 8 * q^8 + 9 * q^9 + 11 * q^11 + 12 * q^12 + 80 * q^13 - 28 * q^14 + 16 * q^16 + 30 * q^17 - 18 * q^18 + 56 * q^19 + 42 * q^21 - 22 * q^22 - 126 * q^23 - 24 * q^24 - 125 * q^25 - 160 * q^26 + 27 * q^27 + 56 * q^28 - 222 * q^29 - 16 * q^31 - 32 * q^32 + 33 * q^33 - 60 * q^34 + 36 * q^36 - 106 * q^37 - 112 * q^38 + 240 * q^39 + 114 * q^41 - 84 * q^42 - 52 * q^43 + 44 * q^44 + 252 * q^46 + 246 * q^47 + 48 * q^48 - 147 * q^49 + 250 * q^50 + 90 * q^51 + 320 * q^52 - 264 * q^53 - 54 * q^54 - 112 * q^56 + 168 * q^57 + 444 * q^58 + 264 * q^59 + 92 * q^61 + 32 * q^62 + 126 * q^63 + 64 * q^64 - 66 * q^66 - 796 * q^67 + 120 * q^68 - 378 * q^69 + 426 * q^71 - 72 * q^72 - 1174 * q^73 + 212 * q^74 - 375 * q^75 + 224 * q^76 + 154 * q^77 - 480 * q^78 + 842 * q^79 + 81 * q^81 - 228 * q^82 + 852 * q^83 + 168 * q^84 + 104 * q^86 - 666 * q^87 - 88 * q^88 - 1062 * q^89 + 1120 * q^91 - 504 * q^92 - 48 * q^93 - 492 * q^94 - 96 * q^96 - 1282 * q^97 + 294 * q^98 + 99 * 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
−2.00000 3.00000 4.00000 0 −6.00000 14.0000 −8.00000 9.00000 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 66.4.a.a 1
3.b odd 2 1 198.4.a.f 1
4.b odd 2 1 528.4.a.d 1
5.b even 2 1 1650.4.a.h 1
5.c odd 4 2 1650.4.c.g 2
8.b even 2 1 2112.4.a.g 1
8.d odd 2 1 2112.4.a.s 1
11.b odd 2 1 726.4.a.h 1
12.b even 2 1 1584.4.a.i 1
33.d even 2 1 2178.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.a 1 1.a even 1 1 trivial
198.4.a.f 1 3.b odd 2 1
528.4.a.d 1 4.b odd 2 1
726.4.a.h 1 11.b odd 2 1
1584.4.a.i 1 12.b even 2 1
1650.4.a.h 1 5.b even 2 1
1650.4.c.g 2 5.c odd 4 2
2112.4.a.g 1 8.b even 2 1
2112.4.a.s 1 8.d odd 2 1
2178.4.a.g 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(66))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 14$$
$11$ $$T - 11$$
$13$ $$T - 80$$
$17$ $$T - 30$$
$19$ $$T - 56$$
$23$ $$T + 126$$
$29$ $$T + 222$$
$31$ $$T + 16$$
$37$ $$T + 106$$
$41$ $$T - 114$$
$43$ $$T + 52$$
$47$ $$T - 246$$
$53$ $$T + 264$$
$59$ $$T - 264$$
$61$ $$T - 92$$
$67$ $$T + 796$$
$71$ $$T - 426$$
$73$ $$T + 1174$$
$79$ $$T - 842$$
$83$ $$T - 852$$
$89$ $$T + 1062$$
$97$ $$T + 1282$$