# Properties

 Label 1110.2.a.d 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^5 + q^6 + 3 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} - q^{10} - 5 q^{11} - q^{12} - 2 q^{13} - 3 q^{14} - q^{15} + q^{16} - 7 q^{17} - q^{18} - 2 q^{19} + q^{20} - 3 q^{21} + 5 q^{22} + 4 q^{23} + q^{24} + q^{25} + 2 q^{26} - q^{27} + 3 q^{28} - 5 q^{29} + q^{30} - 7 q^{31} - q^{32} + 5 q^{33} + 7 q^{34} + 3 q^{35} + q^{36} + q^{37} + 2 q^{38} + 2 q^{39} - q^{40} + q^{41} + 3 q^{42} + 9 q^{43} - 5 q^{44} + q^{45} - 4 q^{46} - q^{48} + 2 q^{49} - q^{50} + 7 q^{51} - 2 q^{52} + 3 q^{53} + q^{54} - 5 q^{55} - 3 q^{56} + 2 q^{57} + 5 q^{58} - 14 q^{59} - q^{60} - 7 q^{61} + 7 q^{62} + 3 q^{63} + q^{64} - 2 q^{65} - 5 q^{66} - 4 q^{67} - 7 q^{68} - 4 q^{69} - 3 q^{70} + 8 q^{71} - q^{72} - 12 q^{73} - q^{74} - q^{75} - 2 q^{76} - 15 q^{77} - 2 q^{78} + 4 q^{79} + q^{80} + q^{81} - q^{82} - 10 q^{83} - 3 q^{84} - 7 q^{85} - 9 q^{86} + 5 q^{87} + 5 q^{88} - 14 q^{89} - q^{90} - 6 q^{91} + 4 q^{92} + 7 q^{93} - 2 q^{95} + q^{96} + q^{97} - 2 q^{98} - 5 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^5 + q^6 + 3 * q^7 - q^8 + q^9 - q^10 - 5 * q^11 - q^12 - 2 * q^13 - 3 * q^14 - q^15 + q^16 - 7 * q^17 - q^18 - 2 * q^19 + q^20 - 3 * q^21 + 5 * q^22 + 4 * q^23 + q^24 + q^25 + 2 * q^26 - q^27 + 3 * q^28 - 5 * q^29 + q^30 - 7 * q^31 - q^32 + 5 * q^33 + 7 * q^34 + 3 * q^35 + q^36 + q^37 + 2 * q^38 + 2 * q^39 - q^40 + q^41 + 3 * q^42 + 9 * q^43 - 5 * q^44 + q^45 - 4 * q^46 - q^48 + 2 * q^49 - q^50 + 7 * q^51 - 2 * q^52 + 3 * q^53 + q^54 - 5 * q^55 - 3 * q^56 + 2 * q^57 + 5 * q^58 - 14 * q^59 - q^60 - 7 * q^61 + 7 * q^62 + 3 * q^63 + q^64 - 2 * q^65 - 5 * q^66 - 4 * q^67 - 7 * q^68 - 4 * q^69 - 3 * q^70 + 8 * q^71 - q^72 - 12 * q^73 - q^74 - q^75 - 2 * q^76 - 15 * q^77 - 2 * q^78 + 4 * q^79 + q^80 + q^81 - q^82 - 10 * q^83 - 3 * q^84 - 7 * q^85 - 9 * q^86 + 5 * q^87 + 5 * q^88 - 14 * q^89 - q^90 - 6 * q^91 + 4 * q^92 + 7 * q^93 - 2 * q^95 + q^96 + q^97 - 2 * 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 3.00000 −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.d 1
3.b odd 2 1 3330.2.a.q 1
4.b odd 2 1 8880.2.a.z 1
5.b even 2 1 5550.2.a.bf 1

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

 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 5$$ T11 + 5 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 3$$
$11$ $$T + 5$$
$13$ $$T + 2$$
$17$ $$T + 7$$
$19$ $$T + 2$$
$23$ $$T - 4$$
$29$ $$T + 5$$
$31$ $$T + 7$$
$37$ $$T - 1$$
$41$ $$T - 1$$
$43$ $$T - 9$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T + 14$$
$61$ $$T + 7$$
$67$ $$T + 4$$
$71$ $$T - 8$$
$73$ $$T + 12$$
$79$ $$T - 4$$
$83$ $$T + 10$$
$89$ $$T + 14$$
$97$ $$T - 1$$