# Properties

 Label 110.2.b.b Level $110$ Weight $2$ Character orbit 110.b Analytic conductor $0.878$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,2,Mod(89,110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("110.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$110 = 2 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 110.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: $$0.878354422234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 2 i q^{3} - q^{4} + ( - 2 i + 1) q^{5} + 2 q^{6} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - 2*i * q^3 - q^4 + (-2*i + 1) * q^5 + 2 * q^6 - i * q^8 - q^9 $$q + i q^{2} - 2 i q^{3} - q^{4} + ( - 2 i + 1) q^{5} + 2 q^{6} - i q^{8} - q^{9} + (i + 2) q^{10} + q^{11} + 2 i q^{12} + 2 i q^{13} + ( - 2 i - 4) q^{15} + q^{16} + 6 i q^{17} - i q^{18} - 4 q^{19} + (2 i - 1) q^{20} + i q^{22} + 2 i q^{23} - 2 q^{24} + ( - 4 i - 3) q^{25} - 2 q^{26} - 4 i q^{27} + 10 q^{29} + ( - 4 i + 2) q^{30} - 8 q^{31} + i q^{32} - 2 i q^{33} - 6 q^{34} + q^{36} + 8 i q^{37} - 4 i q^{38} + 4 q^{39} + ( - i - 2) q^{40} - 2 q^{41} - q^{44} + (2 i - 1) q^{45} - 2 q^{46} + 2 i q^{47} - 2 i q^{48} + 7 q^{49} + ( - 3 i + 4) q^{50} + 12 q^{51} - 2 i q^{52} + 4 q^{54} + ( - 2 i + 1) q^{55} + 8 i q^{57} + 10 i q^{58} + 12 q^{59} + (2 i + 4) q^{60} - 10 q^{61} - 8 i q^{62} - q^{64} + (2 i + 4) q^{65} + 2 q^{66} - 6 i q^{67} - 6 i q^{68} + 4 q^{69} + i q^{72} - 6 i q^{73} - 8 q^{74} + (6 i - 8) q^{75} + 4 q^{76} + 4 i q^{78} - 12 q^{79} + ( - 2 i + 1) q^{80} - 11 q^{81} - 2 i q^{82} - 16 i q^{83} + (6 i + 12) q^{85} - 20 i q^{87} - i q^{88} - 18 q^{89} + ( - i - 2) q^{90} - 2 i q^{92} + 16 i q^{93} - 2 q^{94} + (8 i - 4) q^{95} + 2 q^{96} - 12 i q^{97} + 7 i q^{98} - q^{99} +O(q^{100})$$ q + i * q^2 - 2*i * q^3 - q^4 + (-2*i + 1) * q^5 + 2 * q^6 - i * q^8 - q^9 + (i + 2) * q^10 + q^11 + 2*i * q^12 + 2*i * q^13 + (-2*i - 4) * q^15 + q^16 + 6*i * q^17 - i * q^18 - 4 * q^19 + (2*i - 1) * q^20 + i * q^22 + 2*i * q^23 - 2 * q^24 + (-4*i - 3) * q^25 - 2 * q^26 - 4*i * q^27 + 10 * q^29 + (-4*i + 2) * q^30 - 8 * q^31 + i * q^32 - 2*i * q^33 - 6 * q^34 + q^36 + 8*i * q^37 - 4*i * q^38 + 4 * q^39 + (-i - 2) * q^40 - 2 * q^41 - q^44 + (2*i - 1) * q^45 - 2 * q^46 + 2*i * q^47 - 2*i * q^48 + 7 * q^49 + (-3*i + 4) * q^50 + 12 * q^51 - 2*i * q^52 + 4 * q^54 + (-2*i + 1) * q^55 + 8*i * q^57 + 10*i * q^58 + 12 * q^59 + (2*i + 4) * q^60 - 10 * q^61 - 8*i * q^62 - q^64 + (2*i + 4) * q^65 + 2 * q^66 - 6*i * q^67 - 6*i * q^68 + 4 * q^69 + i * q^72 - 6*i * q^73 - 8 * q^74 + (6*i - 8) * q^75 + 4 * q^76 + 4*i * q^78 - 12 * q^79 + (-2*i + 1) * q^80 - 11 * q^81 - 2*i * q^82 - 16*i * q^83 + (6*i + 12) * q^85 - 20*i * q^87 - i * q^88 - 18 * q^89 + (-i - 2) * q^90 - 2*i * q^92 + 16*i * q^93 - 2 * q^94 + (8*i - 4) * q^95 + 2 * q^96 - 12*i * q^97 + 7*i * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 4 q^{10} + 2 q^{11} - 8 q^{15} + 2 q^{16} - 8 q^{19} - 2 q^{20} - 4 q^{24} - 6 q^{25} - 4 q^{26} + 20 q^{29} + 4 q^{30} - 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 2 q^{45} - 4 q^{46} + 14 q^{49} + 8 q^{50} + 24 q^{51} + 8 q^{54} + 2 q^{55} + 24 q^{59} + 8 q^{60} - 20 q^{61} - 2 q^{64} + 8 q^{65} + 4 q^{66} + 8 q^{69} - 16 q^{74} - 16 q^{75} + 8 q^{76} - 24 q^{79} + 2 q^{80} - 22 q^{81} + 24 q^{85} - 36 q^{89} - 4 q^{90} - 4 q^{94} - 8 q^{95} + 4 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 + 4 * q^10 + 2 * q^11 - 8 * q^15 + 2 * q^16 - 8 * q^19 - 2 * q^20 - 4 * q^24 - 6 * q^25 - 4 * q^26 + 20 * q^29 + 4 * q^30 - 16 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 2 * q^45 - 4 * q^46 + 14 * q^49 + 8 * q^50 + 24 * q^51 + 8 * q^54 + 2 * q^55 + 24 * q^59 + 8 * q^60 - 20 * q^61 - 2 * q^64 + 8 * q^65 + 4 * q^66 + 8 * q^69 - 16 * q^74 - 16 * q^75 + 8 * q^76 - 24 * q^79 + 2 * q^80 - 22 * q^81 + 24 * q^85 - 36 * q^89 - 4 * q^90 - 4 * q^94 - 8 * q^95 + 4 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/110\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 1.00000 + 2.00000i 2.00000 0 1.00000i −1.00000 2.00000 1.00000i
89.2 1.00000i 2.00000i −1.00000 1.00000 2.00000i 2.00000 0 1.00000i −1.00000 2.00000 + 1.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.2.b.b 2
3.b odd 2 1 990.2.c.c 2
4.b odd 2 1 880.2.b.e 2
5.b even 2 1 inner 110.2.b.b 2
5.c odd 4 1 550.2.a.c 1
5.c odd 4 1 550.2.a.k 1
11.b odd 2 1 1210.2.b.d 2
15.d odd 2 1 990.2.c.c 2
15.e even 4 1 4950.2.a.j 1
15.e even 4 1 4950.2.a.bj 1
20.d odd 2 1 880.2.b.e 2
20.e even 4 1 4400.2.a.f 1
20.e even 4 1 4400.2.a.ba 1
55.d odd 2 1 1210.2.b.d 2
55.e even 4 1 6050.2.a.q 1
55.e even 4 1 6050.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.b 2 1.a even 1 1 trivial
110.2.b.b 2 5.b even 2 1 inner
550.2.a.c 1 5.c odd 4 1
550.2.a.k 1 5.c odd 4 1
880.2.b.e 2 4.b odd 2 1
880.2.b.e 2 20.d odd 2 1
990.2.c.c 2 3.b odd 2 1
990.2.c.c 2 15.d odd 2 1
1210.2.b.d 2 11.b odd 2 1
1210.2.b.d 2 55.d odd 2 1
4400.2.a.f 1 20.e even 4 1
4400.2.a.ba 1 20.e even 4 1
4950.2.a.j 1 15.e even 4 1
4950.2.a.bj 1 15.e even 4 1
6050.2.a.q 1 55.e even 4 1
6050.2.a.x 1 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(110, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2}$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 36$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T + 18)^{2}$$
$97$ $$T^{2} + 144$$