# Properties

 Label 1110.2.i.j Level $1110$ Weight $2$ Character orbit 1110.i Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(121,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1110.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.i (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ x^4 - x^3 + 19*x^2 + 18*x + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b2 + 1) * q^3 + (b2 - 1) * q^4 + (b2 - 1) * q^5 - q^6 + q^8 - b2 * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} - q^{6} + q^{8} - \beta_{2} q^{9} + q^{10} + q^{11} + \beta_{2} q^{12} + (\beta_{2} - 1) q^{13} + \beta_{2} q^{15} - \beta_{2} q^{16} + (3 \beta_{2} + \beta_1) q^{17} + (\beta_{2} - 1) q^{18} + (\beta_{3} + \beta_1 - 1) q^{19} - \beta_{2} q^{20} - \beta_{2} q^{22} + (\beta_{3} + 1) q^{23} + ( - \beta_{2} + 1) q^{24} - \beta_{2} q^{25} + q^{26} - q^{27} + (\beta_{3} + 4) q^{29} + ( - \beta_{2} + 1) q^{30} + ( - 2 \beta_{3} + 2) q^{31} + (\beta_{2} - 1) q^{32} + ( - \beta_{2} + 1) q^{33} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{34} + q^{36} + (\beta_{3} - 5 \beta_{2} + 2) q^{37} + ( - \beta_{3} + 1) q^{38} + \beta_{2} q^{39} + (\beta_{2} - 1) q^{40} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{2} - 1) q^{44} + q^{45} + ( - 2 \beta_{2} + \beta_1) q^{46} + 3 q^{47} - q^{48} + 7 \beta_{2} q^{49} + (\beta_{2} - 1) q^{50} + ( - \beta_{3} + 4) q^{51} - \beta_{2} q^{52} + (4 \beta_{2} + 2 \beta_1) q^{53} + \beta_{2} q^{54} + (\beta_{2} - 1) q^{55} + \beta_1 q^{57} + ( - 5 \beta_{2} + \beta_1) q^{58} + (3 \beta_{2} + 2 \beta_1) q^{59} - q^{60} + ( - 10 \beta_{2} + 10) q^{61} - 2 \beta_1 q^{62} + q^{64} - \beta_{2} q^{65} - q^{66} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{67} + (\beta_{3} - 4) q^{68} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{69} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} + \beta_1 - 5) q^{74} - q^{75} - \beta_1 q^{76} + ( - \beta_{2} + 1) q^{78} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{79} + q^{80} + (\beta_{2} - 1) q^{81} + ( - 2 \beta_{3} - 2) q^{82} - 2 \beta_1 q^{83} + (\beta_{3} - 4) q^{85} + (\beta_{3} - 5 \beta_{2} + \beta_1 + 4) q^{87} + q^{88} + (12 \beta_{2} - \beta_1) q^{89} - \beta_{2} q^{90} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{92} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{93} - 3 \beta_{2} q^{94} - \beta_1 q^{95} + \beta_{2} q^{96} + ( - 2 \beta_{3} - 8) q^{97} + ( - 7 \beta_{2} + 7) q^{98} - \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b2 + 1) * q^3 + (b2 - 1) * q^4 + (b2 - 1) * q^5 - q^6 + q^8 - b2 * q^9 + q^10 + q^11 + b2 * q^12 + (b2 - 1) * q^13 + b2 * q^15 - b2 * q^16 + (3*b2 + b1) * q^17 + (b2 - 1) * q^18 + (b3 + b1 - 1) * q^19 - b2 * q^20 - b2 * q^22 + (b3 + 1) * q^23 + (-b2 + 1) * q^24 - b2 * q^25 + q^26 - q^27 + (b3 + 4) * q^29 + (-b2 + 1) * q^30 + (-2*b3 + 2) * q^31 + (b2 - 1) * q^32 + (-b2 + 1) * q^33 + (-b3 - 3*b2 - b1 + 4) * q^34 + q^36 + (b3 - 5*b2 + 2) * q^37 + (-b3 + 1) * q^38 + b2 * q^39 + (b2 - 1) * q^40 + (2*b3 - 4*b2 + 2*b1 + 2) * q^41 + (b2 - 1) * q^44 + q^45 + (-2*b2 + b1) * q^46 + 3 * q^47 - q^48 + 7*b2 * q^49 + (b2 - 1) * q^50 + (-b3 + 4) * q^51 - b2 * q^52 + (4*b2 + 2*b1) * q^53 + b2 * q^54 + (b2 - 1) * q^55 + b1 * q^57 + (-5*b2 + b1) * q^58 + (3*b2 + 2*b1) * q^59 - q^60 + (-10*b2 + 10) * q^61 - 2*b1 * q^62 + q^64 - b2 * q^65 - q^66 + (-3*b3 - b2 - 3*b1 + 4) * q^67 + (b3 - 4) * q^68 + (b3 - 2*b2 + b1 + 1) * q^69 + (2*b3 - 6*b2 + 2*b1 + 4) * q^71 - b2 * q^72 + (2*b2 + b1 - 5) * q^74 - q^75 - b1 * q^76 + (-b2 + 1) * q^78 + (2*b3 - 4*b2 + 2*b1 + 2) * q^79 + q^80 + (b2 - 1) * q^81 + (-2*b3 - 2) * q^82 - 2*b1 * q^83 + (b3 - 4) * q^85 + (b3 - 5*b2 + b1 + 4) * q^87 + q^88 + (12*b2 - b1) * q^89 - b2 * q^90 + (-b3 + 2*b2 - b1 - 1) * q^92 + (-2*b3 - 2*b1 + 2) * q^93 - 3*b2 * q^94 - b1 * q^95 + b2 * q^96 + (-2*b3 - 8) * q^97 + (-7*b2 + 7) * q^98 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{15} - 2 q^{16} + 7 q^{17} - 2 q^{18} - q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{24} - 2 q^{25} + 4 q^{26} - 4 q^{27} + 18 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 2 q^{33} + 7 q^{34} + 4 q^{36} + 2 q^{38} + 2 q^{39} - 2 q^{40} + 6 q^{41} - 2 q^{44} + 4 q^{45} - 3 q^{46} + 12 q^{47} - 4 q^{48} + 14 q^{49} - 2 q^{50} + 14 q^{51} - 2 q^{52} + 10 q^{53} + 2 q^{54} - 2 q^{55} + q^{57} - 9 q^{58} + 8 q^{59} - 4 q^{60} + 20 q^{61} - 2 q^{62} + 4 q^{64} - 2 q^{65} - 4 q^{66} + 5 q^{67} - 14 q^{68} + 3 q^{69} + 10 q^{71} - 2 q^{72} - 15 q^{74} - 4 q^{75} - q^{76} + 2 q^{78} + 6 q^{79} + 4 q^{80} - 2 q^{81} - 12 q^{82} - 2 q^{83} - 14 q^{85} + 9 q^{87} + 4 q^{88} + 23 q^{89} - 2 q^{90} - 3 q^{92} + 2 q^{93} - 6 q^{94} - q^{95} + 2 q^{96} - 36 q^{97} + 14 q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 + 4 * q^8 - 2 * q^9 + 4 * q^10 + 4 * q^11 + 2 * q^12 - 2 * q^13 + 2 * q^15 - 2 * q^16 + 7 * q^17 - 2 * q^18 - q^19 - 2 * q^20 - 2 * q^22 + 6 * q^23 + 2 * q^24 - 2 * q^25 + 4 * q^26 - 4 * q^27 + 18 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 + 2 * q^33 + 7 * q^34 + 4 * q^36 + 2 * q^38 + 2 * q^39 - 2 * q^40 + 6 * q^41 - 2 * q^44 + 4 * q^45 - 3 * q^46 + 12 * q^47 - 4 * q^48 + 14 * q^49 - 2 * q^50 + 14 * q^51 - 2 * q^52 + 10 * q^53 + 2 * q^54 - 2 * q^55 + q^57 - 9 * q^58 + 8 * q^59 - 4 * q^60 + 20 * q^61 - 2 * q^62 + 4 * q^64 - 2 * q^65 - 4 * q^66 + 5 * q^67 - 14 * q^68 + 3 * q^69 + 10 * q^71 - 2 * q^72 - 15 * q^74 - 4 * q^75 - q^76 + 2 * q^78 + 6 * q^79 + 4 * q^80 - 2 * q^81 - 12 * q^82 - 2 * q^83 - 14 * q^85 + 9 * q^87 + 4 * q^88 + 23 * q^89 - 2 * q^90 - 3 * q^92 + 2 * q^93 - 6 * q^94 - q^95 + 2 * q^96 - 36 * q^97 + 14 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342$$ (-v^3 + 19*v^2 - 19*v + 324) / 342 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 37 ) / 19$$ (v^3 + 37) / 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 18\beta_{2} + \beta _1 - 19$$ b3 + 18*b2 + b1 - 19 $$\nu^{3}$$ $$=$$ $$19\beta_{3} - 37$$ 19*b3 - 37

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −1.88600 + 3.26665i 2.38600 − 4.13267i −1.88600 − 3.26665i 2.38600 + 4.13267i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i 1.00000
121.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i 1.00000
211.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i 1.00000
211.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.j 4
37.c even 3 1 inner 1110.2.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.j 4 1.a even 1 1 trivial
1110.2.i.j 4 37.c even 3 1 inner

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + T + 1)^{2}$$
$17$ $$T^{4} - 7 T^{3} + \cdots + 36$$
$19$ $$T^{4} + T^{3} + \cdots + 324$$
$23$ $$(T^{2} - 3 T - 16)^{2}$$
$29$ $$(T^{2} - 9 T + 2)^{2}$$
$31$ $$(T^{2} - 2 T - 72)^{2}$$
$37$ $$T^{4} + T^{2} + 1369$$
$41$ $$T^{4} - 6 T^{3} + \cdots + 4096$$
$43$ $$T^{4}$$
$47$ $$(T - 3)^{4}$$
$53$ $$T^{4} - 10 T^{3} + \cdots + 2304$$
$59$ $$T^{4} - 8 T^{3} + \cdots + 3249$$
$61$ $$(T^{2} - 10 T + 100)^{2}$$
$67$ $$T^{4} - 5 T^{3} + \cdots + 24964$$
$71$ $$T^{4} - 10 T^{3} + \cdots + 2304$$
$73$ $$T^{4}$$
$79$ $$T^{4} - 6 T^{3} + \cdots + 4096$$
$83$ $$T^{4} + 2 T^{3} + \cdots + 5184$$
$89$ $$T^{4} - 23 T^{3} + \cdots + 12996$$
$97$ $$(T^{2} + 18 T + 8)^{2}$$