# Properties

 Label 1110.2.e.b Level $1110$ Weight $2$ Character orbit 1110.e 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(739,1110)]

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, 1]))

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

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

chi := DirichletCharacter("1110.739");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.e (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} - i q^{3} + q^{4} + (i - 2) q^{5} - i q^{6} + 3 i q^{7} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 - i * q^3 + q^4 + (i - 2) * q^5 - i * q^6 + 3*i * q^7 + q^8 - q^9 $$q + q^{2} - i q^{3} + q^{4} + (i - 2) q^{5} - i q^{6} + 3 i q^{7} + q^{8} - q^{9} + (i - 2) q^{10} - 3 q^{11} - i q^{12} + 4 q^{13} + 3 i q^{14} + (2 i + 1) q^{15} + q^{16} - 7 q^{17} - q^{18} + 4 i q^{19} + (i - 2) q^{20} + 3 q^{21} - 3 q^{22} - 6 q^{23} - i q^{24} + ( - 4 i + 3) q^{25} + 4 q^{26} + i q^{27} + 3 i q^{28} + 9 i q^{29} + (2 i + 1) q^{30} + 5 i q^{31} + q^{32} + 3 i q^{33} - 7 q^{34} + ( - 6 i - 3) q^{35} - q^{36} + ( - i - 6) q^{37} + 4 i q^{38} - 4 i q^{39} + (i - 2) q^{40} + 7 q^{41} + 3 q^{42} - q^{43} - 3 q^{44} + ( - i + 2) q^{45} - 6 q^{46} + 8 i q^{47} - i q^{48} - 2 q^{49} + ( - 4 i + 3) q^{50} + 7 i q^{51} + 4 q^{52} + i q^{53} + i q^{54} + ( - 3 i + 6) q^{55} + 3 i q^{56} + 4 q^{57} + 9 i q^{58} - 6 i q^{59} + (2 i + 1) q^{60} + 5 i q^{61} + 5 i q^{62} - 3 i q^{63} + q^{64} + (4 i - 8) q^{65} + 3 i q^{66} - 2 i q^{67} - 7 q^{68} + 6 i q^{69} + ( - 6 i - 3) q^{70} + 12 q^{71} - q^{72} - 4 i q^{73} + ( - i - 6) q^{74} + ( - 3 i - 4) q^{75} + 4 i q^{76} - 9 i q^{77} - 4 i q^{78} + 4 i q^{79} + (i - 2) q^{80} + q^{81} + 7 q^{82} - 14 i q^{83} + 3 q^{84} + ( - 7 i + 14) q^{85} - q^{86} + 9 q^{87} - 3 q^{88} - 6 i q^{89} + ( - i + 2) q^{90} + 12 i q^{91} - 6 q^{92} + 5 q^{93} + 8 i q^{94} + ( - 8 i - 4) q^{95} - i q^{96} - 7 q^{97} - 2 q^{98} + 3 q^{99} +O(q^{100})$$ q + q^2 - i * q^3 + q^4 + (i - 2) * q^5 - i * q^6 + 3*i * q^7 + q^8 - q^9 + (i - 2) * q^10 - 3 * q^11 - i * q^12 + 4 * q^13 + 3*i * q^14 + (2*i + 1) * q^15 + q^16 - 7 * q^17 - q^18 + 4*i * q^19 + (i - 2) * q^20 + 3 * q^21 - 3 * q^22 - 6 * q^23 - i * q^24 + (-4*i + 3) * q^25 + 4 * q^26 + i * q^27 + 3*i * q^28 + 9*i * q^29 + (2*i + 1) * q^30 + 5*i * q^31 + q^32 + 3*i * q^33 - 7 * q^34 + (-6*i - 3) * q^35 - q^36 + (-i - 6) * q^37 + 4*i * q^38 - 4*i * q^39 + (i - 2) * q^40 + 7 * q^41 + 3 * q^42 - q^43 - 3 * q^44 + (-i + 2) * q^45 - 6 * q^46 + 8*i * q^47 - i * q^48 - 2 * q^49 + (-4*i + 3) * q^50 + 7*i * q^51 + 4 * q^52 + i * q^53 + i * q^54 + (-3*i + 6) * q^55 + 3*i * q^56 + 4 * q^57 + 9*i * q^58 - 6*i * q^59 + (2*i + 1) * q^60 + 5*i * q^61 + 5*i * q^62 - 3*i * q^63 + q^64 + (4*i - 8) * q^65 + 3*i * q^66 - 2*i * q^67 - 7 * q^68 + 6*i * q^69 + (-6*i - 3) * q^70 + 12 * q^71 - q^72 - 4*i * q^73 + (-i - 6) * q^74 + (-3*i - 4) * q^75 + 4*i * q^76 - 9*i * q^77 - 4*i * q^78 + 4*i * q^79 + (i - 2) * q^80 + q^81 + 7 * q^82 - 14*i * q^83 + 3 * q^84 + (-7*i + 14) * q^85 - q^86 + 9 * q^87 - 3 * q^88 - 6*i * q^89 + (-i + 2) * q^90 + 12*i * q^91 - 6 * q^92 + 5 * q^93 + 8*i * q^94 + (-8*i - 4) * q^95 - i * q^96 - 7 * q^97 - 2 * q^98 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{8} - 2 q^{9} - 4 q^{10} - 6 q^{11} + 8 q^{13} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 2 q^{18} - 4 q^{20} + 6 q^{21} - 6 q^{22} - 12 q^{23} + 6 q^{25} + 8 q^{26} + 2 q^{30} + 2 q^{32} - 14 q^{34} - 6 q^{35} - 2 q^{36} - 12 q^{37} - 4 q^{40} + 14 q^{41} + 6 q^{42} - 2 q^{43} - 6 q^{44} + 4 q^{45} - 12 q^{46} - 4 q^{49} + 6 q^{50} + 8 q^{52} + 12 q^{55} + 8 q^{57} + 2 q^{60} + 2 q^{64} - 16 q^{65} - 14 q^{68} - 6 q^{70} + 24 q^{71} - 2 q^{72} - 12 q^{74} - 8 q^{75} - 4 q^{80} + 2 q^{81} + 14 q^{82} + 6 q^{84} + 28 q^{85} - 2 q^{86} + 18 q^{87} - 6 q^{88} + 4 q^{90} - 12 q^{92} + 10 q^{93} - 8 q^{95} - 14 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^8 - 2 * q^9 - 4 * q^10 - 6 * q^11 + 8 * q^13 + 2 * q^15 + 2 * q^16 - 14 * q^17 - 2 * q^18 - 4 * q^20 + 6 * q^21 - 6 * q^22 - 12 * q^23 + 6 * q^25 + 8 * q^26 + 2 * q^30 + 2 * q^32 - 14 * q^34 - 6 * q^35 - 2 * q^36 - 12 * q^37 - 4 * q^40 + 14 * q^41 + 6 * q^42 - 2 * q^43 - 6 * q^44 + 4 * q^45 - 12 * q^46 - 4 * q^49 + 6 * q^50 + 8 * q^52 + 12 * q^55 + 8 * q^57 + 2 * q^60 + 2 * q^64 - 16 * q^65 - 14 * q^68 - 6 * q^70 + 24 * q^71 - 2 * q^72 - 12 * q^74 - 8 * q^75 - 4 * q^80 + 2 * q^81 + 14 * q^82 + 6 * q^84 + 28 * q^85 - 2 * q^86 + 18 * q^87 - 6 * q^88 + 4 * q^90 - 12 * q^92 + 10 * q^93 - 8 * q^95 - 14 * q^97 - 4 * q^98 + 6 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 1.00000i − 1.00000i
1.00000 1.00000i 1.00000 −2.00000 + 1.00000i 1.00000i 3.00000i 1.00000 −1.00000 −2.00000 + 1.00000i
739.2 1.00000 1.00000i 1.00000 −2.00000 1.00000i 1.00000i 3.00000i 1.00000 −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
185.d 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.e.b yes 2
3.b odd 2 1 3330.2.e.a 2
5.b even 2 1 1110.2.e.a 2
15.d odd 2 1 3330.2.e.b 2
37.b even 2 1 1110.2.e.a 2
111.d odd 2 1 3330.2.e.b 2
185.d even 2 1 inner 1110.2.e.b yes 2
555.b odd 2 1 3330.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.a 2 5.b even 2 1
1110.2.e.a 2 37.b even 2 1
1110.2.e.b yes 2 1.a even 1 1 trivial
1110.2.e.b yes 2 185.d even 2 1 inner
3330.2.e.a 2 3.b odd 2 1
3330.2.e.a 2 555.b odd 2 1
3330.2.e.b 2 15.d odd 2 1
3330.2.e.b 2 111.d odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 3)^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T + 7)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 81$$
$31$ $$T^{2} + 25$$
$37$ $$T^{2} + 12T + 37$$
$41$ $$(T - 7)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 1$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2} + 25$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2} + 16$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2} + 36$$
$97$ $$(T + 7)^{2}$$