# Properties

 Label 110.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 5 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + 5 * q^7 - q^8 - 2 * q^9 $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 5 q^{7} - q^{8} - 2 q^{9} + q^{10} + q^{11} + q^{12} + 2 q^{13} - 5 q^{14} - q^{15} + q^{16} + 3 q^{17} + 2 q^{18} - 7 q^{19} - q^{20} + 5 q^{21} - q^{22} - 6 q^{23} - q^{24} + q^{25} - 2 q^{26} - 5 q^{27} + 5 q^{28} - 3 q^{29} + q^{30} - 7 q^{31} - q^{32} + q^{33} - 3 q^{34} - 5 q^{35} - 2 q^{36} - 7 q^{37} + 7 q^{38} + 2 q^{39} + q^{40} + 6 q^{41} - 5 q^{42} + 8 q^{43} + q^{44} + 2 q^{45} + 6 q^{46} + 6 q^{47} + q^{48} + 18 q^{49} - q^{50} + 3 q^{51} + 2 q^{52} - 3 q^{53} + 5 q^{54} - q^{55} - 5 q^{56} - 7 q^{57} + 3 q^{58} - 6 q^{59} - q^{60} - q^{61} + 7 q^{62} - 10 q^{63} + q^{64} - 2 q^{65} - q^{66} + 8 q^{67} + 3 q^{68} - 6 q^{69} + 5 q^{70} + 3 q^{71} + 2 q^{72} + 2 q^{73} + 7 q^{74} + q^{75} - 7 q^{76} + 5 q^{77} - 2 q^{78} - 10 q^{79} - q^{80} + q^{81} - 6 q^{82} - 6 q^{83} + 5 q^{84} - 3 q^{85} - 8 q^{86} - 3 q^{87} - q^{88} + 9 q^{89} - 2 q^{90} + 10 q^{91} - 6 q^{92} - 7 q^{93} - 6 q^{94} + 7 q^{95} - q^{96} - 4 q^{97} - 18 q^{98} - 2 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + 5 * q^7 - q^8 - 2 * q^9 + q^10 + q^11 + q^12 + 2 * q^13 - 5 * q^14 - q^15 + q^16 + 3 * q^17 + 2 * q^18 - 7 * q^19 - q^20 + 5 * q^21 - q^22 - 6 * q^23 - q^24 + q^25 - 2 * q^26 - 5 * q^27 + 5 * q^28 - 3 * q^29 + q^30 - 7 * q^31 - q^32 + q^33 - 3 * q^34 - 5 * q^35 - 2 * q^36 - 7 * q^37 + 7 * q^38 + 2 * q^39 + q^40 + 6 * q^41 - 5 * q^42 + 8 * q^43 + q^44 + 2 * q^45 + 6 * q^46 + 6 * q^47 + q^48 + 18 * q^49 - q^50 + 3 * q^51 + 2 * q^52 - 3 * q^53 + 5 * q^54 - q^55 - 5 * q^56 - 7 * q^57 + 3 * q^58 - 6 * q^59 - q^60 - q^61 + 7 * q^62 - 10 * q^63 + q^64 - 2 * q^65 - q^66 + 8 * q^67 + 3 * q^68 - 6 * q^69 + 5 * q^70 + 3 * q^71 + 2 * q^72 + 2 * q^73 + 7 * q^74 + q^75 - 7 * q^76 + 5 * q^77 - 2 * q^78 - 10 * q^79 - q^80 + q^81 - 6 * q^82 - 6 * q^83 + 5 * q^84 - 3 * q^85 - 8 * q^86 - 3 * q^87 - q^88 + 9 * q^89 - 2 * q^90 + 10 * q^91 - 6 * q^92 - 7 * q^93 - 6 * q^94 + 7 * q^95 - q^96 - 4 * q^97 - 18 * 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 5.00000 −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.a 1
3.b odd 2 1 990.2.a.l 1
4.b odd 2 1 880.2.a.c 1
5.b even 2 1 550.2.a.i 1
5.c odd 4 2 550.2.b.b 2
7.b odd 2 1 5390.2.a.h 1
8.b even 2 1 3520.2.a.l 1
8.d odd 2 1 3520.2.a.z 1
11.b odd 2 1 1210.2.a.k 1
12.b even 2 1 7920.2.a.s 1
15.d odd 2 1 4950.2.a.a 1
15.e even 4 2 4950.2.c.a 2
20.d odd 2 1 4400.2.a.w 1
20.e even 4 2 4400.2.b.g 2
44.c even 2 1 9680.2.a.j 1
55.d odd 2 1 6050.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 1.a even 1 1 trivial
550.2.a.i 1 5.b even 2 1
550.2.b.b 2 5.c odd 4 2
880.2.a.c 1 4.b odd 2 1
990.2.a.l 1 3.b odd 2 1
1210.2.a.k 1 11.b odd 2 1
3520.2.a.l 1 8.b even 2 1
3520.2.a.z 1 8.d odd 2 1
4400.2.a.w 1 20.d odd 2 1
4400.2.b.g 2 20.e even 4 2
4950.2.a.a 1 15.d odd 2 1
4950.2.c.a 2 15.e even 4 2
5390.2.a.h 1 7.b odd 2 1
6050.2.a.i 1 55.d odd 2 1
7920.2.a.s 1 12.b even 2 1
9680.2.a.j 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 $$T_{7} - 5$$ T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 5$$
$11$ $$T - 1$$
$13$ $$T - 2$$
$17$ $$T - 3$$
$19$ $$T + 7$$
$23$ $$T + 6$$
$29$ $$T + 3$$
$31$ $$T + 7$$
$37$ $$T + 7$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T - 6$$
$53$ $$T + 3$$
$59$ $$T + 6$$
$61$ $$T + 1$$
$67$ $$T - 8$$
$71$ $$T - 3$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T + 6$$
$89$ $$T - 9$$
$97$ $$T + 4$$