# Properties

 Label 1110.2.a.a Level $1110$ Weight $2$ Character orbit 1110.a Self dual yes Analytic conductor $8.863$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - 4 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + 4 q^{14} + q^{15} + q^{16} - 2 q^{17} - q^{18} + 8 q^{19} - q^{20} + 4 q^{21} - 4 q^{22} + q^{24} + q^{25} - 2 q^{26} - q^{27} - 4 q^{28} - 6 q^{29} - q^{30} - 8 q^{31} - q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{35} + q^{36} - q^{37} - 8 q^{38} - 2 q^{39} + q^{40} - 6 q^{41} - 4 q^{42} + 4 q^{43} + 4 q^{44} - q^{45} - 12 q^{47} - q^{48} + 9 q^{49} - q^{50} + 2 q^{51} + 2 q^{52} - 6 q^{53} + q^{54} - 4 q^{55} + 4 q^{56} - 8 q^{57} + 6 q^{58} + q^{60} - 6 q^{61} + 8 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} + 4 q^{66} + 12 q^{67} - 2 q^{68} - 4 q^{70} - q^{72} - 6 q^{73} + q^{74} - q^{75} + 8 q^{76} - 16 q^{77} + 2 q^{78} - 8 q^{79} - q^{80} + q^{81} + 6 q^{82} + 4 q^{83} + 4 q^{84} + 2 q^{85} - 4 q^{86} + 6 q^{87} - 4 q^{88} - 14 q^{89} + q^{90} - 8 q^{91} + 8 q^{93} + 12 q^{94} - 8 q^{95} + q^{96} + 6 q^{97} - 9 q^{98} + 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - 4 * q^7 - q^8 + q^9 + q^10 + 4 * q^11 - q^12 + 2 * q^13 + 4 * q^14 + q^15 + q^16 - 2 * q^17 - q^18 + 8 * q^19 - q^20 + 4 * q^21 - 4 * q^22 + q^24 + q^25 - 2 * q^26 - q^27 - 4 * q^28 - 6 * q^29 - q^30 - 8 * q^31 - q^32 - 4 * q^33 + 2 * q^34 + 4 * q^35 + q^36 - q^37 - 8 * q^38 - 2 * q^39 + q^40 - 6 * q^41 - 4 * q^42 + 4 * q^43 + 4 * q^44 - q^45 - 12 * q^47 - q^48 + 9 * q^49 - q^50 + 2 * q^51 + 2 * q^52 - 6 * q^53 + q^54 - 4 * q^55 + 4 * q^56 - 8 * q^57 + 6 * q^58 + q^60 - 6 * q^61 + 8 * q^62 - 4 * q^63 + q^64 - 2 * q^65 + 4 * q^66 + 12 * q^67 - 2 * q^68 - 4 * q^70 - q^72 - 6 * q^73 + q^74 - q^75 + 8 * q^76 - 16 * q^77 + 2 * q^78 - 8 * q^79 - q^80 + q^81 + 6 * q^82 + 4 * q^83 + 4 * q^84 + 2 * q^85 - 4 * q^86 + 6 * q^87 - 4 * q^88 - 14 * q^89 + q^90 - 8 * q^91 + 8 * q^93 + 12 * q^94 - 8 * q^95 + q^96 + 6 * q^97 - 9 * q^98 + 4 * 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 −4.00000 −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.a 1
3.b odd 2 1 3330.2.a.s 1
4.b odd 2 1 8880.2.a.w 1
5.b even 2 1 5550.2.a.br 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.a 1 1.a even 1 1 trivial
3330.2.a.s 1 3.b odd 2 1
5550.2.a.br 1 5.b even 2 1
8880.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1110))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 4$$ T11 - 4 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T - 4$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T + 1$$
$41$ $$T + 6$$
$43$ $$T - 4$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T + 6$$
$67$ $$T - 12$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T + 14$$
$97$ $$T - 6$$