# Properties

 Label 1110.2.a.f Level $1110$ Weight $2$ Character orbit 1110.a Self dual yes Analytic conductor $8.863$ 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] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.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: $$8.86339462436$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 5 q^{11} + q^{12} + 2 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + 6 q^{19} - q^{20} + q^{21} + 5 q^{22} - q^{24} + q^{25} - 2 q^{26} + q^{27} + q^{28} + 9 q^{29} + q^{30} + 3 q^{31} - q^{32} - 5 q^{33} - q^{34} - q^{35} + q^{36} - q^{37} - 6 q^{38} + 2 q^{39} + q^{40} + 9 q^{41} - q^{42} + q^{43} - 5 q^{44} - q^{45} + q^{48} - 6 q^{49} - q^{50} + q^{51} + 2 q^{52} + 9 q^{53} - q^{54} + 5 q^{55} - q^{56} + 6 q^{57} - 9 q^{58} - 10 q^{59} - q^{60} - 5 q^{61} - 3 q^{62} + q^{63} + q^{64} - 2 q^{65} + 5 q^{66} + 16 q^{67} + q^{68} + q^{70} - q^{72} + 12 q^{73} + q^{74} + q^{75} + 6 q^{76} - 5 q^{77} - 2 q^{78} - 12 q^{79} - q^{80} + q^{81} - 9 q^{82} + 2 q^{83} + q^{84} - q^{85} - q^{86} + 9 q^{87} + 5 q^{88} - 2 q^{89} + q^{90} + 2 q^{91} + 3 q^{93} - 6 q^{95} - q^{96} + 17 q^{97} + 6 q^{98} - 5 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + q^7 - q^8 + q^9 + q^10 - 5 * q^11 + q^12 + 2 * q^13 - q^14 - q^15 + q^16 + q^17 - q^18 + 6 * q^19 - q^20 + q^21 + 5 * q^22 - q^24 + q^25 - 2 * q^26 + q^27 + q^28 + 9 * q^29 + q^30 + 3 * q^31 - q^32 - 5 * q^33 - q^34 - q^35 + q^36 - q^37 - 6 * q^38 + 2 * q^39 + q^40 + 9 * q^41 - q^42 + q^43 - 5 * q^44 - q^45 + q^48 - 6 * q^49 - q^50 + q^51 + 2 * q^52 + 9 * q^53 - q^54 + 5 * q^55 - q^56 + 6 * q^57 - 9 * q^58 - 10 * q^59 - q^60 - 5 * q^61 - 3 * q^62 + q^63 + q^64 - 2 * q^65 + 5 * q^66 + 16 * q^67 + q^68 + q^70 - q^72 + 12 * q^73 + q^74 + q^75 + 6 * q^76 - 5 * q^77 - 2 * q^78 - 12 * q^79 - q^80 + q^81 - 9 * q^82 + 2 * q^83 + q^84 - q^85 - q^86 + 9 * q^87 + 5 * q^88 - 2 * q^89 + q^90 + 2 * q^91 + 3 * q^93 - 6 * q^95 - q^96 + 17 * q^97 + 6 * q^98 - 5 * 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 1.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$37$$ $$+1$$

## 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 1110.2.a.f 1
3.b odd 2 1 3330.2.a.y 1
4.b odd 2 1 8880.2.a.d 1
5.b even 2 1 5550.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.f 1 1.a even 1 1 trivial
3330.2.a.y 1 3.b odd 2 1
5550.2.a.y 1 5.b even 2 1
8880.2.a.d 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1110))$$:

 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 5$$ T11 + 5 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T - 2$$
$17$ $$T - 1$$
$19$ $$T - 6$$
$23$ $$T$$
$29$ $$T - 9$$
$31$ $$T - 3$$
$37$ $$T + 1$$
$41$ $$T - 9$$
$43$ $$T - 1$$
$47$ $$T$$
$53$ $$T - 9$$
$59$ $$T + 10$$
$61$ $$T + 5$$
$67$ $$T - 16$$
$71$ $$T$$
$73$ $$T - 12$$
$79$ $$T + 12$$
$83$ $$T - 2$$
$89$ $$T + 2$$
$97$ $$T - 17$$