# Properties

 Label 110.2.b.c 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 3 i q^{3} - q^{4} + ( - i + 2) q^{5} - 3 q^{6} - i q^{7} - i q^{8} - 6 q^{9} +O(q^{10})$$ q + i * q^2 + 3*i * q^3 - q^4 + (-i + 2) * q^5 - 3 * q^6 - i * q^7 - i * q^8 - 6 * q^9 $$q + i q^{2} + 3 i q^{3} - q^{4} + ( - i + 2) q^{5} - 3 q^{6} - i q^{7} - i q^{8} - 6 q^{9} + (2 i + 1) q^{10} - q^{11} - 3 i q^{12} + q^{14} + (6 i + 3) q^{15} + q^{16} - 5 i q^{17} - 6 i q^{18} + 7 q^{19} + (i - 2) q^{20} + 3 q^{21} - i q^{22} + 8 i q^{23} + 3 q^{24} + ( - 4 i + 3) q^{25} - 9 i q^{27} + i q^{28} - 3 q^{29} + (3 i - 6) q^{30} - 5 q^{31} + i q^{32} - 3 i q^{33} + 5 q^{34} + ( - 2 i - 1) q^{35} + 6 q^{36} - i q^{37} + 7 i q^{38} + ( - 2 i - 1) q^{40} - 8 q^{41} + 3 i q^{42} - 10 i q^{43} + q^{44} + (6 i - 12) q^{45} - 8 q^{46} + 3 i q^{48} + 6 q^{49} + (3 i + 4) q^{50} + 15 q^{51} + i q^{53} + 9 q^{54} + (i - 2) q^{55} - q^{56} + 21 i q^{57} - 3 i q^{58} - 12 q^{59} + ( - 6 i - 3) q^{60} + 5 q^{61} - 5 i q^{62} + 6 i q^{63} - q^{64} + 3 q^{66} - 4 i q^{67} + 5 i q^{68} - 24 q^{69} + ( - i + 2) q^{70} - 7 q^{71} + 6 i q^{72} - 2 i q^{73} + q^{74} + (9 i + 12) q^{75} - 7 q^{76} + i q^{77} + 4 q^{79} + ( - i + 2) q^{80} + 9 q^{81} - 8 i q^{82} - 3 q^{84} + ( - 10 i - 5) q^{85} + 10 q^{86} - 9 i q^{87} + i q^{88} + 7 q^{89} + ( - 12 i - 6) q^{90} - 8 i q^{92} - 15 i q^{93} + ( - 7 i + 14) q^{95} - 3 q^{96} + 8 i q^{97} + 6 i q^{98} + 6 q^{99} +O(q^{100})$$ q + i * q^2 + 3*i * q^3 - q^4 + (-i + 2) * q^5 - 3 * q^6 - i * q^7 - i * q^8 - 6 * q^9 + (2*i + 1) * q^10 - q^11 - 3*i * q^12 + q^14 + (6*i + 3) * q^15 + q^16 - 5*i * q^17 - 6*i * q^18 + 7 * q^19 + (i - 2) * q^20 + 3 * q^21 - i * q^22 + 8*i * q^23 + 3 * q^24 + (-4*i + 3) * q^25 - 9*i * q^27 + i * q^28 - 3 * q^29 + (3*i - 6) * q^30 - 5 * q^31 + i * q^32 - 3*i * q^33 + 5 * q^34 + (-2*i - 1) * q^35 + 6 * q^36 - i * q^37 + 7*i * q^38 + (-2*i - 1) * q^40 - 8 * q^41 + 3*i * q^42 - 10*i * q^43 + q^44 + (6*i - 12) * q^45 - 8 * q^46 + 3*i * q^48 + 6 * q^49 + (3*i + 4) * q^50 + 15 * q^51 + i * q^53 + 9 * q^54 + (i - 2) * q^55 - q^56 + 21*i * q^57 - 3*i * q^58 - 12 * q^59 + (-6*i - 3) * q^60 + 5 * q^61 - 5*i * q^62 + 6*i * q^63 - q^64 + 3 * q^66 - 4*i * q^67 + 5*i * q^68 - 24 * q^69 + (-i + 2) * q^70 - 7 * q^71 + 6*i * q^72 - 2*i * q^73 + q^74 + (9*i + 12) * q^75 - 7 * q^76 + i * q^77 + 4 * q^79 + (-i + 2) * q^80 + 9 * q^81 - 8*i * q^82 - 3 * q^84 + (-10*i - 5) * q^85 + 10 * q^86 - 9*i * q^87 + i * q^88 + 7 * q^89 + (-12*i - 6) * q^90 - 8*i * q^92 - 15*i * q^93 + (-7*i + 14) * q^95 - 3 * q^96 + 8*i * q^97 + 6*i * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{5} - 6 q^{6} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^5 - 6 * q^6 - 12 * q^9 $$2 q - 2 q^{4} + 4 q^{5} - 6 q^{6} - 12 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{14} + 6 q^{15} + 2 q^{16} + 14 q^{19} - 4 q^{20} + 6 q^{21} + 6 q^{24} + 6 q^{25} - 6 q^{29} - 12 q^{30} - 10 q^{31} + 10 q^{34} - 2 q^{35} + 12 q^{36} - 2 q^{40} - 16 q^{41} + 2 q^{44} - 24 q^{45} - 16 q^{46} + 12 q^{49} + 8 q^{50} + 30 q^{51} + 18 q^{54} - 4 q^{55} - 2 q^{56} - 24 q^{59} - 6 q^{60} + 10 q^{61} - 2 q^{64} + 6 q^{66} - 48 q^{69} + 4 q^{70} - 14 q^{71} + 2 q^{74} + 24 q^{75} - 14 q^{76} + 8 q^{79} + 4 q^{80} + 18 q^{81} - 6 q^{84} - 10 q^{85} + 20 q^{86} + 14 q^{89} - 12 q^{90} + 28 q^{95} - 6 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^5 - 6 * q^6 - 12 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^14 + 6 * q^15 + 2 * q^16 + 14 * q^19 - 4 * q^20 + 6 * q^21 + 6 * q^24 + 6 * q^25 - 6 * q^29 - 12 * q^30 - 10 * q^31 + 10 * q^34 - 2 * q^35 + 12 * q^36 - 2 * q^40 - 16 * q^41 + 2 * q^44 - 24 * q^45 - 16 * q^46 + 12 * q^49 + 8 * q^50 + 30 * q^51 + 18 * q^54 - 4 * q^55 - 2 * q^56 - 24 * q^59 - 6 * q^60 + 10 * q^61 - 2 * q^64 + 6 * q^66 - 48 * q^69 + 4 * q^70 - 14 * q^71 + 2 * q^74 + 24 * q^75 - 14 * q^76 + 8 * q^79 + 4 * q^80 + 18 * q^81 - 6 * q^84 - 10 * q^85 + 20 * q^86 + 14 * q^89 - 12 * q^90 + 28 * q^95 - 6 * q^96 + 12 * 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 3.00000i −1.00000 2.00000 + 1.00000i −3.00000 1.00000i 1.00000i −6.00000 1.00000 2.00000i
89.2 1.00000i 3.00000i −1.00000 2.00000 1.00000i −3.00000 1.00000i 1.00000i −6.00000 1.00000 + 2.00000i
 $$n$$: 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.c 2
3.b odd 2 1 990.2.c.b 2
4.b odd 2 1 880.2.b.f 2
5.b even 2 1 inner 110.2.b.c 2
5.c odd 4 1 550.2.a.g 1
5.c odd 4 1 550.2.a.h 1
11.b odd 2 1 1210.2.b.e 2
15.d odd 2 1 990.2.c.b 2
15.e even 4 1 4950.2.a.h 1
15.e even 4 1 4950.2.a.bn 1
20.d odd 2 1 880.2.b.f 2
20.e even 4 1 4400.2.a.b 1
20.e even 4 1 4400.2.a.bd 1
55.d odd 2 1 1210.2.b.e 2
55.e even 4 1 6050.2.a.b 1
55.e even 4 1 6050.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.c 2 1.a even 1 1 trivial
110.2.b.c 2 5.b even 2 1 inner
550.2.a.g 1 5.c odd 4 1
550.2.a.h 1 5.c odd 4 1
880.2.b.f 2 4.b odd 2 1
880.2.b.f 2 20.d odd 2 1
990.2.c.b 2 3.b odd 2 1
990.2.c.b 2 15.d odd 2 1
1210.2.b.e 2 11.b odd 2 1
1210.2.b.e 2 55.d odd 2 1
4400.2.a.b 1 20.e even 4 1
4400.2.a.bd 1 20.e even 4 1
4950.2.a.h 1 15.e even 4 1
4950.2.a.bn 1 15.e even 4 1
6050.2.a.b 1 55.e even 4 1
6050.2.a.bo 1 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(110, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 25$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 1$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 7)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 7)^{2}$$
$97$ $$T^{2} + 64$$