# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(889,1110)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1110.889");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.d (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - i q^{3} - q^{4} + ( - 2 i + 1) q^{5} + q^{6} + i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - i * q^3 - q^4 + (-2*i + 1) * q^5 + q^6 + i * q^7 - i * q^8 - q^9 $$q + i q^{2} - i q^{3} - q^{4} + ( - 2 i + 1) q^{5} + q^{6} + i q^{7} - i q^{8} - q^{9} + (i + 2) q^{10} + 5 q^{11} + i q^{12} - q^{14} + ( - i - 2) q^{15} + q^{16} + i q^{17} - i q^{18} + (2 i - 1) q^{20} + q^{21} + 5 i q^{22} - 4 i q^{23} - q^{24} + ( - 4 i - 3) q^{25} + i q^{27} - i q^{28} + 3 q^{29} + ( - 2 i + 1) q^{30} + q^{31} + i q^{32} - 5 i q^{33} - q^{34} + (i + 2) q^{35} + q^{36} - i q^{37} + ( - i - 2) q^{40} - q^{41} + i q^{42} - 7 i q^{43} - 5 q^{44} + (2 i - 1) q^{45} + 4 q^{46} - 4 i q^{47} - i q^{48} + 6 q^{49} + ( - 3 i + 4) q^{50} + q^{51} - 3 i q^{53} - q^{54} + ( - 10 i + 5) q^{55} + q^{56} + 3 i q^{58} + 8 q^{59} + (i + 2) q^{60} + 5 q^{61} + i q^{62} - i q^{63} - q^{64} + 5 q^{66} - 4 i q^{67} - i q^{68} - 4 q^{69} + (2 i - 1) q^{70} - 6 q^{71} + i q^{72} - 10 i q^{73} + q^{74} + (3 i - 4) q^{75} + 5 i q^{77} + ( - 2 i + 1) q^{80} + q^{81} - i q^{82} + 8 i q^{83} - q^{84} + (i + 2) q^{85} + 7 q^{86} - 3 i q^{87} - 5 i q^{88} - 8 q^{89} + ( - i - 2) q^{90} + 4 i q^{92} - i q^{93} + 4 q^{94} + q^{96} - i q^{97} + 6 i q^{98} - 5 q^{99} +O(q^{100})$$ q + i * q^2 - i * q^3 - q^4 + (-2*i + 1) * q^5 + q^6 + i * q^7 - i * q^8 - q^9 + (i + 2) * q^10 + 5 * q^11 + i * q^12 - q^14 + (-i - 2) * q^15 + q^16 + i * q^17 - i * q^18 + (2*i - 1) * q^20 + q^21 + 5*i * q^22 - 4*i * q^23 - q^24 + (-4*i - 3) * q^25 + i * q^27 - i * q^28 + 3 * q^29 + (-2*i + 1) * q^30 + q^31 + i * q^32 - 5*i * q^33 - q^34 + (i + 2) * q^35 + q^36 - i * q^37 + (-i - 2) * q^40 - q^41 + i * q^42 - 7*i * q^43 - 5 * q^44 + (2*i - 1) * q^45 + 4 * q^46 - 4*i * q^47 - i * q^48 + 6 * q^49 + (-3*i + 4) * q^50 + q^51 - 3*i * q^53 - q^54 + (-10*i + 5) * q^55 + q^56 + 3*i * q^58 + 8 * q^59 + (i + 2) * q^60 + 5 * q^61 + i * q^62 - i * q^63 - q^64 + 5 * q^66 - 4*i * q^67 - i * q^68 - 4 * q^69 + (2*i - 1) * q^70 - 6 * q^71 + i * q^72 - 10*i * q^73 + q^74 + (3*i - 4) * q^75 + 5*i * q^77 + (-2*i + 1) * q^80 + q^81 - i * q^82 + 8*i * q^83 - q^84 + (i + 2) * q^85 + 7 * q^86 - 3*i * q^87 - 5*i * q^88 - 8 * q^89 + (-i - 2) * q^90 + 4*i * q^92 - i * q^93 + 4 * q^94 + q^96 - i * q^97 + 6*i * q^98 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 10 q^{11} - 2 q^{14} - 4 q^{15} + 2 q^{16} - 2 q^{20} + 2 q^{21} - 2 q^{24} - 6 q^{25} + 6 q^{29} + 2 q^{30} + 2 q^{31} - 2 q^{34} + 4 q^{35} + 2 q^{36} - 4 q^{40} - 2 q^{41} - 10 q^{44} - 2 q^{45} + 8 q^{46} + 12 q^{49} + 8 q^{50} + 2 q^{51} - 2 q^{54} + 10 q^{55} + 2 q^{56} + 16 q^{59} + 4 q^{60} + 10 q^{61} - 2 q^{64} + 10 q^{66} - 8 q^{69} - 2 q^{70} - 12 q^{71} + 2 q^{74} - 8 q^{75} + 2 q^{80} + 2 q^{81} - 2 q^{84} + 4 q^{85} + 14 q^{86} - 16 q^{89} - 4 q^{90} + 8 q^{94} + 2 q^{96} - 10 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 + 10 * q^11 - 2 * q^14 - 4 * q^15 + 2 * q^16 - 2 * q^20 + 2 * q^21 - 2 * q^24 - 6 * q^25 + 6 * q^29 + 2 * q^30 + 2 * q^31 - 2 * q^34 + 4 * q^35 + 2 * q^36 - 4 * q^40 - 2 * q^41 - 10 * q^44 - 2 * q^45 + 8 * q^46 + 12 * q^49 + 8 * q^50 + 2 * q^51 - 2 * q^54 + 10 * q^55 + 2 * q^56 + 16 * q^59 + 4 * q^60 + 10 * q^61 - 2 * q^64 + 10 * q^66 - 8 * q^69 - 2 * q^70 - 12 * q^71 + 2 * q^74 - 8 * q^75 + 2 * q^80 + 2 * q^81 - 2 * q^84 + 4 * q^85 + 14 * q^86 - 16 * q^89 - 4 * q^90 + 8 * q^94 + 2 * q^96 - 10 * q^99

## Character values

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

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
889.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 1.00000 + 2.00000i 1.00000 1.00000i 1.00000i −1.00000 2.00000 1.00000i
889.2 1.00000i 1.00000i −1.00000 1.00000 2.00000i 1.00000 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.d.c 2
3.b odd 2 1 3330.2.d.d 2
5.b even 2 1 inner 1110.2.d.c 2
5.c odd 4 1 5550.2.a.d 1
5.c odd 4 1 5550.2.a.bo 1
15.d odd 2 1 3330.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.c 2 1.a even 1 1 trivial
1110.2.d.c 2 5.b even 2 1 inner
3330.2.d.d 2 3.b odd 2 1
3330.2.d.d 2 15.d odd 2 1
5550.2.a.d 1 5.c odd 4 1
5550.2.a.bo 1 5.c odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 1)^{2}$$
$43$ $$T^{2} + 49$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 9$$
$59$ $$(T - 8)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T + 8)^{2}$$
$97$ $$T^{2} + 1$$