# Properties

 Label 1110.2.i.i 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: $$\Q(\sqrt{-3}, \sqrt{145})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 37x^{2} + 36x + 1296$$ x^4 - x^3 + 37*x^2 + 36*x + 1296 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + q^{6} - \beta_{2} q^{7} + q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 - b2 * q^3 - b2 * q^4 + b2 * q^5 + q^6 - b2 * q^7 + q^8 + (b2 - 1) * q^9 $$q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{3} - \beta_{2} q^{4} + \beta_{2} q^{5} + q^{6} - \beta_{2} q^{7} + q^{8} + (\beta_{2} - 1) q^{9} - q^{10} - 2 q^{11} + (\beta_{2} - 1) q^{12} + \beta_{2} q^{13} + q^{14} + ( - \beta_{2} + 1) q^{15} + (\beta_{2} - 1) q^{16} + ( - 2 \beta_{2} + 2) q^{17} - \beta_{2} q^{18} + \beta_1 q^{19} + ( - \beta_{2} + 1) q^{20} + (\beta_{2} - 1) q^{21} + ( - 2 \beta_{2} + 2) q^{22} - \beta_{2} q^{24} + (\beta_{2} - 1) q^{25} - q^{26} + q^{27} + (\beta_{2} - 1) q^{28} + (\beta_{3} - 1) q^{29} + \beta_{2} q^{30} + ( - \beta_{3} + 4) q^{31} - \beta_{2} q^{32} + 2 \beta_{2} q^{33} + 2 \beta_{2} q^{34} + ( - \beta_{2} + 1) q^{35} + q^{36} + (\beta_1 - 1) q^{37} + (\beta_{3} - 1) q^{38} + ( - \beta_{2} + 1) q^{39} + \beta_{2} q^{40} - \beta_{2} q^{42} + ( - \beta_{3} - 4) q^{43} + 2 \beta_{2} q^{44} - q^{45} + ( - 2 \beta_{3} + 2) q^{47} + q^{48} + ( - 6 \beta_{2} + 6) q^{49} - \beta_{2} q^{50} - 2 q^{51} + ( - \beta_{2} + 1) q^{52} + (\beta_{2} - 1) q^{54} - 2 \beta_{2} q^{55} - \beta_{2} q^{56} + ( - \beta_{3} - \beta_1 + 1) q^{57} + ( - \beta_{3} - \beta_1 + 1) q^{58} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{59} - q^{60} - 10 \beta_{2} q^{61} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{62} + q^{63} + q^{64} + (\beta_{2} - 1) q^{65} - 2 q^{66} + (5 \beta_{2} + \beta_1) q^{67} - 2 q^{68} + \beta_{2} q^{70} + (6 \beta_{2} + \beta_1) q^{71} + (\beta_{2} - 1) q^{72} + ( - \beta_{3} + 4) q^{73} + (\beta_{3} - \beta_{2}) q^{74} + q^{75} + ( - \beta_{3} - \beta_1 + 1) q^{76} + 2 \beta_{2} q^{77} + \beta_{2} q^{78} + (3 \beta_{2} + \beta_1) q^{79} - q^{80} - \beta_{2} q^{81} + ( - \beta_{3} - 10 \beta_{2} + \cdots + 11) q^{83}+ \cdots + ( - 2 \beta_{2} + 2) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^2 - b2 * q^3 - b2 * q^4 + b2 * q^5 + q^6 - b2 * q^7 + q^8 + (b2 - 1) * q^9 - q^10 - 2 * q^11 + (b2 - 1) * q^12 + b2 * q^13 + q^14 + (-b2 + 1) * q^15 + (b2 - 1) * q^16 + (-2*b2 + 2) * q^17 - b2 * q^18 + b1 * q^19 + (-b2 + 1) * q^20 + (b2 - 1) * q^21 + (-2*b2 + 2) * q^22 - b2 * q^24 + (b2 - 1) * q^25 - q^26 + q^27 + (b2 - 1) * q^28 + (b3 - 1) * q^29 + b2 * q^30 + (-b3 + 4) * q^31 - b2 * q^32 + 2*b2 * q^33 + 2*b2 * q^34 + (-b2 + 1) * q^35 + q^36 + (b1 - 1) * q^37 + (b3 - 1) * q^38 + (-b2 + 1) * q^39 + b2 * q^40 - b2 * q^42 + (-b3 - 4) * q^43 + 2*b2 * q^44 - q^45 + (-2*b3 + 2) * q^47 + q^48 + (-6*b2 + 6) * q^49 - b2 * q^50 - 2 * q^51 + (-b2 + 1) * q^52 + (b2 - 1) * q^54 - 2*b2 * q^55 - b2 * q^56 + (-b3 - b1 + 1) * q^57 + (-b3 - b1 + 1) * q^58 + (2*b3 + 2*b2 + 2*b1 - 4) * q^59 - q^60 - 10*b2 * q^61 + (b3 + 3*b2 + b1 - 4) * q^62 + q^63 + q^64 + (b2 - 1) * q^65 - 2 * q^66 + (5*b2 + b1) * q^67 - 2 * q^68 + b2 * q^70 + (6*b2 + b1) * q^71 + (b2 - 1) * q^72 + (-b3 + 4) * q^73 + (b3 - b2) * q^74 + q^75 + (-b3 - b1 + 1) * q^76 + 2*b2 * q^77 + b2 * q^78 + (3*b2 + b1) * q^79 - q^80 - b2 * q^81 + (-b3 - 10*b2 - b1 + 11) * q^83 + q^84 + 2 * q^85 + (b3 - 5*b2 + b1 + 4) * q^86 + b1 * q^87 - 2 * q^88 + (2*b3 - 2*b2 + 2*b1) * q^89 + (-b2 + 1) * q^90 + (-b2 + 1) * q^91 + (-3*b2 - b1) * q^93 + (2*b3 + 2*b1 - 2) * q^94 + (b3 + b1 - 1) * q^95 + (b2 - 1) * q^96 - b3 * q^97 + 6*b2 * q^98 + (-2*b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} - 4 q^{10} - 8 q^{11} - 2 q^{12} + 2 q^{13} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 4 q^{17} - 2 q^{18} + q^{19} + 2 q^{20} - 2 q^{21} + 4 q^{22} - 2 q^{24} - 2 q^{25} - 4 q^{26} + 4 q^{27} - 2 q^{28} - 2 q^{29} + 2 q^{30} + 14 q^{31} - 2 q^{32} + 4 q^{33} + 4 q^{34} + 2 q^{35} + 4 q^{36} - 3 q^{37} - 2 q^{38} + 2 q^{39} + 2 q^{40} - 2 q^{42} - 18 q^{43} + 4 q^{44} - 4 q^{45} + 4 q^{47} + 4 q^{48} + 12 q^{49} - 2 q^{50} - 8 q^{51} + 2 q^{52} - 2 q^{54} - 4 q^{55} - 2 q^{56} + q^{57} + q^{58} - 6 q^{59} - 4 q^{60} - 20 q^{61} - 7 q^{62} + 4 q^{63} + 4 q^{64} - 2 q^{65} - 8 q^{66} + 11 q^{67} - 8 q^{68} + 2 q^{70} + 13 q^{71} - 2 q^{72} + 14 q^{73} + 4 q^{75} + q^{76} + 4 q^{77} + 2 q^{78} + 7 q^{79} - 4 q^{80} - 2 q^{81} + 21 q^{83} + 4 q^{84} + 8 q^{85} + 9 q^{86} + q^{87} - 8 q^{88} + 2 q^{89} + 2 q^{90} + 2 q^{91} - 7 q^{93} - 2 q^{94} - q^{95} - 2 q^{96} - 2 q^{97} + 12 q^{98} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 - 4 * q^10 - 8 * q^11 - 2 * q^12 + 2 * q^13 + 4 * q^14 + 2 * q^15 - 2 * q^16 + 4 * q^17 - 2 * q^18 + q^19 + 2 * q^20 - 2 * q^21 + 4 * q^22 - 2 * q^24 - 2 * q^25 - 4 * q^26 + 4 * q^27 - 2 * q^28 - 2 * q^29 + 2 * q^30 + 14 * q^31 - 2 * q^32 + 4 * q^33 + 4 * q^34 + 2 * q^35 + 4 * q^36 - 3 * q^37 - 2 * q^38 + 2 * q^39 + 2 * q^40 - 2 * q^42 - 18 * q^43 + 4 * q^44 - 4 * q^45 + 4 * q^47 + 4 * q^48 + 12 * q^49 - 2 * q^50 - 8 * q^51 + 2 * q^52 - 2 * q^54 - 4 * q^55 - 2 * q^56 + q^57 + q^58 - 6 * q^59 - 4 * q^60 - 20 * q^61 - 7 * q^62 + 4 * q^63 + 4 * q^64 - 2 * q^65 - 8 * q^66 + 11 * q^67 - 8 * q^68 + 2 * q^70 + 13 * q^71 - 2 * q^72 + 14 * q^73 + 4 * q^75 + q^76 + 4 * q^77 + 2 * q^78 + 7 * q^79 - 4 * q^80 - 2 * q^81 + 21 * q^83 + 4 * q^84 + 8 * q^85 + 9 * q^86 + q^87 - 8 * q^88 + 2 * q^89 + 2 * q^90 + 2 * q^91 - 7 * q^93 - 2 * q^94 - q^95 - 2 * q^96 - 2 * q^97 + 12 * q^98 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 37x^{2} + 36x + 1296$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 37\nu^{2} - 37\nu + 1296 ) / 1332$$ (-v^3 + 37*v^2 - 37*v + 1296) / 1332 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 73 ) / 37$$ (v^3 + 73) / 37
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 36\beta_{2} + \beta _1 - 37$$ b3 + 36*b2 + b1 - 37 $$\nu^{3}$$ $$=$$ $$37\beta_{3} - 73$$ 37*b3 - 73

## Character values

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

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

## 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
 −2.76040 − 4.78115i 3.26040 + 5.64718i −2.76040 + 4.78115i 3.26040 − 5.64718i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 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 −0.500000 0.866025i 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 −0.500000 + 0.866025i 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 −0.500000 + 0.866025i 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.i 4
37.c even 3 1 inner 1110.2.i.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.i 4 1.a even 1 1 trivial
1110.2.i.i 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$(T + 2)^{4}$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} - T^{3} + \cdots + 1296$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + T - 36)^{2}$$
$31$ $$(T^{2} - 7 T - 24)^{2}$$
$37$ $$T^{4} + 3 T^{3} + \cdots + 1369$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 9 T - 16)^{2}$$
$47$ $$(T^{2} - 2 T - 144)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 6 T^{3} + \cdots + 18496$$
$61$ $$(T^{2} + 10 T + 100)^{2}$$
$67$ $$T^{4} - 11 T^{3} + \cdots + 36$$
$71$ $$T^{4} - 13 T^{3} + \cdots + 36$$
$73$ $$(T^{2} - 7 T - 24)^{2}$$
$79$ $$T^{4} - 7 T^{3} + \cdots + 576$$
$83$ $$T^{4} - 21 T^{3} + \cdots + 5476$$
$89$ $$T^{4} - 2 T^{3} + \cdots + 20736$$
$97$ $$(T^{2} + T - 36)^{2}$$