# Properties

 Label 110.2.a.c Level $110$ Weight $2$ Character orbit 110.a Self dual yes Analytic conductor $0.878$ 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] = [110,2,Mod(1,110)]

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([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("110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$110 = 2 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 110.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: $$0.878354422234$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 - q^7 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} - 2 q^{9} - q^{10} - q^{11} + q^{12} + 2 q^{13} - q^{14} - q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - q^{19} - q^{20} - q^{21} - q^{22} + 6 q^{23} + q^{24} + q^{25} + 2 q^{26} - 5 q^{27} - q^{28} - 9 q^{29} - q^{30} + 5 q^{31} + q^{32} - q^{33} - 3 q^{34} + q^{35} - 2 q^{36} + 5 q^{37} - q^{38} + 2 q^{39} - q^{40} - 6 q^{41} - q^{42} + 8 q^{43} - q^{44} + 2 q^{45} + 6 q^{46} + 6 q^{47} + q^{48} - 6 q^{49} + q^{50} - 3 q^{51} + 2 q^{52} + 9 q^{53} - 5 q^{54} + q^{55} - q^{56} - q^{57} - 9 q^{58} + 6 q^{59} - q^{60} + 5 q^{61} + 5 q^{62} + 2 q^{63} + q^{64} - 2 q^{65} - q^{66} + 8 q^{67} - 3 q^{68} + 6 q^{69} + q^{70} - 9 q^{71} - 2 q^{72} - 10 q^{73} + 5 q^{74} + q^{75} - q^{76} + q^{77} + 2 q^{78} + 14 q^{79} - q^{80} + q^{81} - 6 q^{82} - 6 q^{83} - q^{84} + 3 q^{85} + 8 q^{86} - 9 q^{87} - q^{88} - 15 q^{89} + 2 q^{90} - 2 q^{91} + 6 q^{92} + 5 q^{93} + 6 q^{94} + q^{95} + q^{96} + 8 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 - q^7 + q^8 - 2 * q^9 - q^10 - q^11 + q^12 + 2 * q^13 - q^14 - q^15 + q^16 - 3 * q^17 - 2 * q^18 - q^19 - q^20 - q^21 - q^22 + 6 * q^23 + q^24 + q^25 + 2 * q^26 - 5 * q^27 - q^28 - 9 * q^29 - q^30 + 5 * q^31 + q^32 - q^33 - 3 * q^34 + q^35 - 2 * q^36 + 5 * q^37 - q^38 + 2 * q^39 - q^40 - 6 * q^41 - q^42 + 8 * q^43 - q^44 + 2 * q^45 + 6 * q^46 + 6 * q^47 + q^48 - 6 * q^49 + q^50 - 3 * q^51 + 2 * q^52 + 9 * q^53 - 5 * q^54 + q^55 - q^56 - q^57 - 9 * q^58 + 6 * q^59 - q^60 + 5 * q^61 + 5 * q^62 + 2 * q^63 + q^64 - 2 * q^65 - q^66 + 8 * q^67 - 3 * q^68 + 6 * q^69 + q^70 - 9 * q^71 - 2 * q^72 - 10 * q^73 + 5 * q^74 + q^75 - q^76 + q^77 + 2 * q^78 + 14 * q^79 - q^80 + q^81 - 6 * q^82 - 6 * q^83 - q^84 + 3 * q^85 + 8 * q^86 - 9 * q^87 - q^88 - 15 * q^89 + 2 * q^90 - 2 * q^91 + 6 * q^92 + 5 * q^93 + 6 * q^94 + q^95 + q^96 + 8 * q^97 - 6 * q^98 + 2 * 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 −1.00000 1.00000 −1.00000 1.00000 −2.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$+1$$

## 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 110.2.a.c 1
3.b odd 2 1 990.2.a.f 1
4.b odd 2 1 880.2.a.d 1
5.b even 2 1 550.2.a.d 1
5.c odd 4 2 550.2.b.c 2
7.b odd 2 1 5390.2.a.x 1
8.b even 2 1 3520.2.a.k 1
8.d odd 2 1 3520.2.a.ba 1
11.b odd 2 1 1210.2.a.e 1
12.b even 2 1 7920.2.a.bc 1
15.d odd 2 1 4950.2.a.bm 1
15.e even 4 2 4950.2.c.s 2
20.d odd 2 1 4400.2.a.t 1
20.e even 4 2 4400.2.b.j 2
44.c even 2 1 9680.2.a.g 1
55.d odd 2 1 6050.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.c 1 1.a even 1 1 trivial
550.2.a.d 1 5.b even 2 1
550.2.b.c 2 5.c odd 4 2
880.2.a.d 1 4.b odd 2 1
990.2.a.f 1 3.b odd 2 1
1210.2.a.e 1 11.b odd 2 1
3520.2.a.k 1 8.b even 2 1
3520.2.a.ba 1 8.d odd 2 1
4400.2.a.t 1 20.d odd 2 1
4400.2.b.j 2 20.e even 4 2
4950.2.a.bm 1 15.d odd 2 1
4950.2.c.s 2 15.e even 4 2
5390.2.a.x 1 7.b odd 2 1
6050.2.a.bc 1 55.d odd 2 1
7920.2.a.bc 1 12.b even 2 1
9680.2.a.g 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(110))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T - 5$$
$37$ $$T - 5$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T - 6$$
$53$ $$T - 9$$
$59$ $$T - 6$$
$61$ $$T - 5$$
$67$ $$T - 8$$
$71$ $$T + 9$$
$73$ $$T + 10$$
$79$ $$T - 14$$
$83$ $$T + 6$$
$89$ $$T + 15$$
$97$ $$T - 8$$