# Properties

 Label 66.2.b.a Level $66$ Weight $2$ Character orbit 66.b Analytic conductor $0.527$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [66,2,Mod(65,66)]

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([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("66.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 66.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.527012653340$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta - 1) q^{3} + q^{4} + \beta q^{5} + ( - \beta + 1) q^{6} + 3 \beta q^{7} - q^{8} + ( - 2 \beta - 1) q^{9}+O(q^{10})$$ q - q^2 + (b - 1) * q^3 + q^4 + b * q^5 + (-b + 1) * q^6 + 3*b * q^7 - q^8 + (-2*b - 1) * q^9 $$q - q^{2} + (\beta - 1) q^{3} + q^{4} + \beta q^{5} + ( - \beta + 1) q^{6} + 3 \beta q^{7} - q^{8} + ( - 2 \beta - 1) q^{9} - \beta q^{10} + ( - \beta + 3) q^{11} + (\beta - 1) q^{12} - 3 \beta q^{13} - 3 \beta q^{14} + ( - \beta - 2) q^{15} + q^{16} + (2 \beta + 1) q^{18} + \beta q^{20} + ( - 3 \beta - 6) q^{21} + (\beta - 3) q^{22} + \beta q^{23} + ( - \beta + 1) q^{24} + 3 q^{25} + 3 \beta q^{26} + (\beta + 5) q^{27} + 3 \beta q^{28} + 6 q^{29} + (\beta + 2) q^{30} - 4 q^{31} - q^{32} + (4 \beta - 1) q^{33} - 6 q^{35} + ( - 2 \beta - 1) q^{36} + 2 q^{37} + (3 \beta + 6) q^{39} - \beta q^{40} - 6 q^{41} + (3 \beta + 6) q^{42} - 6 \beta q^{43} + ( - \beta + 3) q^{44} + ( - \beta + 4) q^{45} - \beta q^{46} + 7 \beta q^{47} + (\beta - 1) q^{48} - 11 q^{49} - 3 q^{50} - 3 \beta q^{52} - 5 \beta q^{53} + ( - \beta - 5) q^{54} + (3 \beta + 2) q^{55} - 3 \beta q^{56} - 6 q^{58} - 8 \beta q^{59} + ( - \beta - 2) q^{60} + 3 \beta q^{61} + 4 q^{62} + ( - 3 \beta + 12) q^{63} + q^{64} + 6 q^{65} + ( - 4 \beta + 1) q^{66} - 4 q^{67} + ( - \beta - 2) q^{69} + 6 q^{70} - 5 \beta q^{71} + (2 \beta + 1) q^{72} - 2 q^{74} + (3 \beta - 3) q^{75} + (9 \beta + 6) q^{77} + ( - 3 \beta - 6) q^{78} + 3 \beta q^{79} + \beta q^{80} + (4 \beta - 7) q^{81} + 6 q^{82} - 12 q^{83} + ( - 3 \beta - 6) q^{84} + 6 \beta q^{86} + (6 \beta - 6) q^{87} + (\beta - 3) q^{88} + 4 \beta q^{89} + (\beta - 4) q^{90} + 18 q^{91} + \beta q^{92} + ( - 4 \beta + 4) q^{93} - 7 \beta q^{94} + ( - \beta + 1) q^{96} + 8 q^{97} + 11 q^{98} + ( - 5 \beta - 7) q^{99} +O(q^{100})$$ q - q^2 + (b - 1) * q^3 + q^4 + b * q^5 + (-b + 1) * q^6 + 3*b * q^7 - q^8 + (-2*b - 1) * q^9 - b * q^10 + (-b + 3) * q^11 + (b - 1) * q^12 - 3*b * q^13 - 3*b * q^14 + (-b - 2) * q^15 + q^16 + (2*b + 1) * q^18 + b * q^20 + (-3*b - 6) * q^21 + (b - 3) * q^22 + b * q^23 + (-b + 1) * q^24 + 3 * q^25 + 3*b * q^26 + (b + 5) * q^27 + 3*b * q^28 + 6 * q^29 + (b + 2) * q^30 - 4 * q^31 - q^32 + (4*b - 1) * q^33 - 6 * q^35 + (-2*b - 1) * q^36 + 2 * q^37 + (3*b + 6) * q^39 - b * q^40 - 6 * q^41 + (3*b + 6) * q^42 - 6*b * q^43 + (-b + 3) * q^44 + (-b + 4) * q^45 - b * q^46 + 7*b * q^47 + (b - 1) * q^48 - 11 * q^49 - 3 * q^50 - 3*b * q^52 - 5*b * q^53 + (-b - 5) * q^54 + (3*b + 2) * q^55 - 3*b * q^56 - 6 * q^58 - 8*b * q^59 + (-b - 2) * q^60 + 3*b * q^61 + 4 * q^62 + (-3*b + 12) * q^63 + q^64 + 6 * q^65 + (-4*b + 1) * q^66 - 4 * q^67 + (-b - 2) * q^69 + 6 * q^70 - 5*b * q^71 + (2*b + 1) * q^72 - 2 * q^74 + (3*b - 3) * q^75 + (9*b + 6) * q^77 + (-3*b - 6) * q^78 + 3*b * q^79 + b * q^80 + (4*b - 7) * q^81 + 6 * q^82 - 12 * q^83 + (-3*b - 6) * q^84 + 6*b * q^86 + (6*b - 6) * q^87 + (b - 3) * q^88 + 4*b * q^89 + (b - 4) * q^90 + 18 * q^91 + b * q^92 + (-4*b + 4) * q^93 - 7*b * q^94 + (-b + 1) * q^96 + 8 * q^97 + 11 * q^98 + (-5*b - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 2 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{15} + 2 q^{16} + 2 q^{18} - 12 q^{21} - 6 q^{22} + 2 q^{24} + 6 q^{25} + 10 q^{27} + 12 q^{29} + 4 q^{30} - 8 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} - 2 q^{36} + 4 q^{37} + 12 q^{39} - 12 q^{41} + 12 q^{42} + 6 q^{44} + 8 q^{45} - 2 q^{48} - 22 q^{49} - 6 q^{50} - 10 q^{54} + 4 q^{55} - 12 q^{58} - 4 q^{60} + 8 q^{62} + 24 q^{63} + 2 q^{64} + 12 q^{65} + 2 q^{66} - 8 q^{67} - 4 q^{69} + 12 q^{70} + 2 q^{72} - 4 q^{74} - 6 q^{75} + 12 q^{77} - 12 q^{78} - 14 q^{81} + 12 q^{82} - 24 q^{83} - 12 q^{84} - 12 q^{87} - 6 q^{88} - 8 q^{90} + 36 q^{91} + 8 q^{93} + 2 q^{96} + 16 q^{97} + 22 q^{98} - 14 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 2 * q^9 + 6 * q^11 - 2 * q^12 - 4 * q^15 + 2 * q^16 + 2 * q^18 - 12 * q^21 - 6 * q^22 + 2 * q^24 + 6 * q^25 + 10 * q^27 + 12 * q^29 + 4 * q^30 - 8 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^35 - 2 * q^36 + 4 * q^37 + 12 * q^39 - 12 * q^41 + 12 * q^42 + 6 * q^44 + 8 * q^45 - 2 * q^48 - 22 * q^49 - 6 * q^50 - 10 * q^54 + 4 * q^55 - 12 * q^58 - 4 * q^60 + 8 * q^62 + 24 * q^63 + 2 * q^64 + 12 * q^65 + 2 * q^66 - 8 * q^67 - 4 * q^69 + 12 * q^70 + 2 * q^72 - 4 * q^74 - 6 * q^75 + 12 * q^77 - 12 * q^78 - 14 * q^81 + 12 * q^82 - 24 * q^83 - 12 * q^84 - 12 * q^87 - 6 * q^88 - 8 * q^90 + 36 * q^91 + 8 * q^93 + 2 * q^96 + 16 * q^97 + 22 * q^98 - 14 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/66\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 1.41421i 1.41421i
−1.00000 −1.00000 1.41421i 1.00000 1.41421i 1.00000 + 1.41421i 4.24264i −1.00000 −1.00000 + 2.82843i 1.41421i
65.2 −1.00000 −1.00000 + 1.41421i 1.00000 1.41421i 1.00000 1.41421i 4.24264i −1.00000 −1.00000 2.82843i 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.2.b.a 2
3.b odd 2 1 66.2.b.b yes 2
4.b odd 2 1 528.2.b.b 2
5.b even 2 1 1650.2.d.b 2
5.c odd 4 2 1650.2.f.b 4
8.b even 2 1 2112.2.b.g 2
8.d odd 2 1 2112.2.b.d 2
9.c even 3 2 1782.2.i.f 4
9.d odd 6 2 1782.2.i.c 4
11.b odd 2 1 66.2.b.b yes 2
11.c even 5 4 726.2.h.i 8
11.d odd 10 4 726.2.h.e 8
12.b even 2 1 528.2.b.c 2
15.d odd 2 1 1650.2.d.a 2
15.e even 4 2 1650.2.f.a 4
24.f even 2 1 2112.2.b.b 2
24.h odd 2 1 2112.2.b.i 2
33.d even 2 1 inner 66.2.b.a 2
33.f even 10 4 726.2.h.i 8
33.h odd 10 4 726.2.h.e 8
44.c even 2 1 528.2.b.c 2
55.d odd 2 1 1650.2.d.a 2
55.e even 4 2 1650.2.f.a 4
88.b odd 2 1 2112.2.b.i 2
88.g even 2 1 2112.2.b.b 2
99.g even 6 2 1782.2.i.f 4
99.h odd 6 2 1782.2.i.c 4
132.d odd 2 1 528.2.b.b 2
165.d even 2 1 1650.2.d.b 2
165.l odd 4 2 1650.2.f.b 4
264.m even 2 1 2112.2.b.g 2
264.p odd 2 1 2112.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.b.a 2 1.a even 1 1 trivial
66.2.b.a 2 33.d even 2 1 inner
66.2.b.b yes 2 3.b odd 2 1
66.2.b.b yes 2 11.b odd 2 1
528.2.b.b 2 4.b odd 2 1
528.2.b.b 2 132.d odd 2 1
528.2.b.c 2 12.b even 2 1
528.2.b.c 2 44.c even 2 1
726.2.h.e 8 11.d odd 10 4
726.2.h.e 8 33.h odd 10 4
726.2.h.i 8 11.c even 5 4
726.2.h.i 8 33.f even 10 4
1650.2.d.a 2 15.d odd 2 1
1650.2.d.a 2 55.d odd 2 1
1650.2.d.b 2 5.b even 2 1
1650.2.d.b 2 165.d even 2 1
1650.2.f.a 4 15.e even 4 2
1650.2.f.a 4 55.e even 4 2
1650.2.f.b 4 5.c odd 4 2
1650.2.f.b 4 165.l odd 4 2
1782.2.i.c 4 9.d odd 6 2
1782.2.i.c 4 99.h odd 6 2
1782.2.i.f 4 9.c even 3 2
1782.2.i.f 4 99.g even 6 2
2112.2.b.b 2 24.f even 2 1
2112.2.b.b 2 88.g even 2 1
2112.2.b.d 2 8.d odd 2 1
2112.2.b.d 2 264.p odd 2 1
2112.2.b.g 2 8.b even 2 1
2112.2.b.g 2 264.m even 2 1
2112.2.b.i 2 24.h odd 2 1
2112.2.b.i 2 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{29} - 6$$ acting on $$S_{2}^{\mathrm{new}}(66, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 2T + 3$$
$5$ $$T^{2} + 2$$
$7$ $$T^{2} + 18$$
$11$ $$T^{2} - 6T + 11$$
$13$ $$T^{2} + 18$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 2$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 72$$
$47$ $$T^{2} + 98$$
$53$ $$T^{2} + 50$$
$59$ $$T^{2} + 128$$
$61$ $$T^{2} + 18$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 50$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 18$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 32$$
$97$ $$(T - 8)^{2}$$