# Properties

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

# Learn more

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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^3 - z * q^4 + z * q^5 - q^6 + z * q^7 - q^8 + (z - 1) * q^9 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} + q^{10} + 6 q^{11} + (\zeta_{6} - 1) q^{12} + \zeta_{6} q^{13} + q^{14} + ( - \zeta_{6} + 1) q^{15} + (\zeta_{6} - 1) q^{16} + (6 \zeta_{6} - 6) q^{17} + \zeta_{6} q^{18} + 7 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{20} + ( - \zeta_{6} + 1) q^{21} + ( - 6 \zeta_{6} + 6) q^{22} - 6 q^{23} + \zeta_{6} q^{24} + (\zeta_{6} - 1) q^{25} + q^{26} + q^{27} + ( - \zeta_{6} + 1) q^{28} + 9 q^{29} - \zeta_{6} q^{30} - 4 q^{31} + \zeta_{6} q^{32} - 6 \zeta_{6} q^{33} + 6 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{35} + q^{36} + ( - 3 \zeta_{6} + 7) q^{37} + 7 q^{38} + ( - \zeta_{6} + 1) q^{39} - \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} - \zeta_{6} q^{42} + 2 q^{43} - 6 \zeta_{6} q^{44} - q^{45} + (6 \zeta_{6} - 6) q^{46} + 12 q^{47} + q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + \zeta_{6} q^{50} + 6 q^{51} + ( - \zeta_{6} + 1) q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - \zeta_{6} + 1) q^{54} + 6 \zeta_{6} q^{55} - \zeta_{6} q^{56} + ( - 7 \zeta_{6} + 7) q^{57} + ( - 9 \zeta_{6} + 9) q^{58} + ( - 6 \zeta_{6} + 6) q^{59} - q^{60} - 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} - q^{63} + q^{64} + (\zeta_{6} - 1) q^{65} - 6 q^{66} + 10 \zeta_{6} q^{67} + 6 q^{68} + 6 \zeta_{6} q^{69} + \zeta_{6} q^{70} + 3 \zeta_{6} q^{71} + ( - \zeta_{6} + 1) q^{72} - 10 q^{73} + ( - 7 \zeta_{6} + 4) q^{74} + q^{75} + ( - 7 \zeta_{6} + 7) q^{76} + 6 \zeta_{6} q^{77} - \zeta_{6} q^{78} - 14 \zeta_{6} q^{79} - q^{80} - \zeta_{6} q^{81} - 6 q^{82} + ( - 3 \zeta_{6} + 3) q^{83} - q^{84} - 6 q^{85} + ( - 2 \zeta_{6} + 2) q^{86} - 9 \zeta_{6} q^{87} - 6 q^{88} + (\zeta_{6} - 1) q^{90} + (\zeta_{6} - 1) q^{91} + 6 \zeta_{6} q^{92} + 4 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + (7 \zeta_{6} - 7) q^{95} + ( - \zeta_{6} + 1) q^{96} - 10 q^{97} - 6 \zeta_{6} q^{98} + (6 \zeta_{6} - 6) q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^3 - z * q^4 + z * q^5 - q^6 + z * q^7 - q^8 + (z - 1) * q^9 + q^10 + 6 * q^11 + (z - 1) * q^12 + z * q^13 + q^14 + (-z + 1) * q^15 + (z - 1) * q^16 + (6*z - 6) * q^17 + z * q^18 + 7*z * q^19 + (-z + 1) * q^20 + (-z + 1) * q^21 + (-6*z + 6) * q^22 - 6 * q^23 + z * q^24 + (z - 1) * q^25 + q^26 + q^27 + (-z + 1) * q^28 + 9 * q^29 - z * q^30 - 4 * q^31 + z * q^32 - 6*z * q^33 + 6*z * q^34 + (z - 1) * q^35 + q^36 + (-3*z + 7) * q^37 + 7 * q^38 + (-z + 1) * q^39 - z * q^40 - 6*z * q^41 - z * q^42 + 2 * q^43 - 6*z * q^44 - q^45 + (6*z - 6) * q^46 + 12 * q^47 + q^48 + (-6*z + 6) * q^49 + z * q^50 + 6 * q^51 + (-z + 1) * q^52 + (6*z - 6) * q^53 + (-z + 1) * q^54 + 6*z * q^55 - z * q^56 + (-7*z + 7) * q^57 + (-9*z + 9) * q^58 + (-6*z + 6) * q^59 - q^60 - 8*z * q^61 + (4*z - 4) * q^62 - q^63 + q^64 + (z - 1) * q^65 - 6 * q^66 + 10*z * q^67 + 6 * q^68 + 6*z * q^69 + z * q^70 + 3*z * q^71 + (-z + 1) * q^72 - 10 * q^73 + (-7*z + 4) * q^74 + q^75 + (-7*z + 7) * q^76 + 6*z * q^77 - z * q^78 - 14*z * q^79 - q^80 - z * q^81 - 6 * q^82 + (-3*z + 3) * q^83 - q^84 - 6 * q^85 + (-2*z + 2) * q^86 - 9*z * q^87 - 6 * q^88 + (z - 1) * q^90 + (z - 1) * q^91 + 6*z * q^92 + 4*z * q^93 + (-12*z + 12) * q^94 + (7*z - 7) * q^95 + (-z + 1) * q^96 - 10 * q^97 - 6*z * q^98 + (6*z - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 - q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} - q^{9} + 2 q^{10} + 12 q^{11} - q^{12} + q^{13} + 2 q^{14} + q^{15} - q^{16} - 6 q^{17} + q^{18} + 7 q^{19} + q^{20} + q^{21} + 6 q^{22} - 12 q^{23} + q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} + 18 q^{29} - q^{30} - 8 q^{31} + q^{32} - 6 q^{33} + 6 q^{34} - q^{35} + 2 q^{36} + 11 q^{37} + 14 q^{38} + q^{39} - q^{40} - 6 q^{41} - q^{42} + 4 q^{43} - 6 q^{44} - 2 q^{45} - 6 q^{46} + 24 q^{47} + 2 q^{48} + 6 q^{49} + q^{50} + 12 q^{51} + q^{52} - 6 q^{53} + q^{54} + 6 q^{55} - q^{56} + 7 q^{57} + 9 q^{58} + 6 q^{59} - 2 q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} - q^{65} - 12 q^{66} + 10 q^{67} + 12 q^{68} + 6 q^{69} + q^{70} + 3 q^{71} + q^{72} - 20 q^{73} + q^{74} + 2 q^{75} + 7 q^{76} + 6 q^{77} - q^{78} - 14 q^{79} - 2 q^{80} - q^{81} - 12 q^{82} + 3 q^{83} - 2 q^{84} - 12 q^{85} + 2 q^{86} - 9 q^{87} - 12 q^{88} - q^{90} - q^{91} + 6 q^{92} + 4 q^{93} + 12 q^{94} - 7 q^{95} + q^{96} - 20 q^{97} - 6 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 - q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 - q^9 + 2 * q^10 + 12 * q^11 - q^12 + q^13 + 2 * q^14 + q^15 - q^16 - 6 * q^17 + q^18 + 7 * q^19 + q^20 + q^21 + 6 * q^22 - 12 * q^23 + q^24 - q^25 + 2 * q^26 + 2 * q^27 + q^28 + 18 * q^29 - q^30 - 8 * q^31 + q^32 - 6 * q^33 + 6 * q^34 - q^35 + 2 * q^36 + 11 * q^37 + 14 * q^38 + q^39 - q^40 - 6 * q^41 - q^42 + 4 * q^43 - 6 * q^44 - 2 * q^45 - 6 * q^46 + 24 * q^47 + 2 * q^48 + 6 * q^49 + q^50 + 12 * q^51 + q^52 - 6 * q^53 + q^54 + 6 * q^55 - q^56 + 7 * q^57 + 9 * q^58 + 6 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 - q^65 - 12 * q^66 + 10 * q^67 + 12 * q^68 + 6 * q^69 + q^70 + 3 * q^71 + q^72 - 20 * q^73 + q^74 + 2 * q^75 + 7 * q^76 + 6 * q^77 - q^78 - 14 * q^79 - 2 * q^80 - q^81 - 12 * q^82 + 3 * q^83 - 2 * q^84 - 12 * q^85 + 2 * q^86 - 9 * q^87 - 12 * q^88 - q^90 - q^91 + 6 * q^92 + 4 * q^93 + 12 * q^94 - 7 * q^95 + q^96 - 20 * q^97 - 6 * q^98 - 6 * q^99

## Character values

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

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
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
 $$n$$: 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.d 2
37.c even 3 1 inner 1110.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.d 2 1.a even 1 1 trivial
1110.2.i.d 2 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 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 11T + 37$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$(T - 2)^{2}$$
$47$ $$(T - 12)^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$T^{2} - 3T + 9$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} - 3T + 9$$
$89$ $$T^{2}$$
$97$ $$(T + 10)^{2}$$
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