# Properties

 Label 1110.2.h.a Level $1110$ Weight $2$ Character orbit 1110.h 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(961,1110)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1110.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.h (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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} - q^{4} + i q^{5} - i q^{6} - q^{7} - i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 - q^3 - q^4 + i * q^5 - i * q^6 - q^7 - i * q^8 + q^9 $$q + i q^{2} - q^{3} - q^{4} + i q^{5} - i q^{6} - q^{7} - i q^{8} + q^{9} - q^{10} - q^{11} + q^{12} - i q^{13} - i q^{14} - i q^{15} + q^{16} - 5 i q^{17} + i q^{18} - 3 i q^{19} - i q^{20} + q^{21} - i q^{22} - 3 i q^{23} + i q^{24} - q^{25} + q^{26} - q^{27} + q^{28} + 10 i q^{29} + q^{30} - 2 i q^{31} + i q^{32} + q^{33} + 5 q^{34} - i q^{35} - q^{36} + (i + 6) q^{37} + 3 q^{38} + i q^{39} + q^{40} - 2 q^{41} + i q^{42} - 12 i q^{43} + q^{44} + i q^{45} + 3 q^{46} + 12 q^{47} - q^{48} - 6 q^{49} - i q^{50} + 5 i q^{51} + i q^{52} + 11 q^{53} - i q^{54} - i q^{55} + i q^{56} + 3 i q^{57} - 10 q^{58} - 10 i q^{59} + i q^{60} - 14 i q^{61} + 2 q^{62} - q^{63} - q^{64} + q^{65} + i q^{66} + 12 q^{67} + 5 i q^{68} + 3 i q^{69} + q^{70} + 6 q^{71} - i q^{72} - q^{73} + (6 i - 1) q^{74} + q^{75} + 3 i q^{76} + q^{77} - q^{78} + 10 i q^{79} + i q^{80} + q^{81} - 2 i q^{82} - 3 q^{83} - q^{84} + 5 q^{85} + 12 q^{86} - 10 i q^{87} + i q^{88} + 9 i q^{89} - q^{90} + i q^{91} + 3 i q^{92} + 2 i q^{93} + 12 i q^{94} + 3 q^{95} - i q^{96} + 12 i q^{97} - 6 i q^{98} - q^{99} +O(q^{100})$$ q + i * q^2 - q^3 - q^4 + i * q^5 - i * q^6 - q^7 - i * q^8 + q^9 - q^10 - q^11 + q^12 - i * q^13 - i * q^14 - i * q^15 + q^16 - 5*i * q^17 + i * q^18 - 3*i * q^19 - i * q^20 + q^21 - i * q^22 - 3*i * q^23 + i * q^24 - q^25 + q^26 - q^27 + q^28 + 10*i * q^29 + q^30 - 2*i * q^31 + i * q^32 + q^33 + 5 * q^34 - i * q^35 - q^36 + (i + 6) * q^37 + 3 * q^38 + i * q^39 + q^40 - 2 * q^41 + i * q^42 - 12*i * q^43 + q^44 + i * q^45 + 3 * q^46 + 12 * q^47 - q^48 - 6 * q^49 - i * q^50 + 5*i * q^51 + i * q^52 + 11 * q^53 - i * q^54 - i * q^55 + i * q^56 + 3*i * q^57 - 10 * q^58 - 10*i * q^59 + i * q^60 - 14*i * q^61 + 2 * q^62 - q^63 - q^64 + q^65 + i * q^66 + 12 * q^67 + 5*i * q^68 + 3*i * q^69 + q^70 + 6 * q^71 - i * q^72 - q^73 + (6*i - 1) * q^74 + q^75 + 3*i * q^76 + q^77 - q^78 + 10*i * q^79 + i * q^80 + q^81 - 2*i * q^82 - 3 * q^83 - q^84 + 5 * q^85 + 12 * q^86 - 10*i * q^87 + i * q^88 + 9*i * q^89 - q^90 + i * q^91 + 3*i * q^92 + 2*i * q^93 + 12*i * q^94 + 3 * q^95 - i * q^96 + 12*i * q^97 - 6*i * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{21} - 2 q^{25} + 2 q^{26} - 2 q^{27} + 2 q^{28} + 2 q^{30} + 2 q^{33} + 10 q^{34} - 2 q^{36} + 12 q^{37} + 6 q^{38} + 2 q^{40} - 4 q^{41} + 2 q^{44} + 6 q^{46} + 24 q^{47} - 2 q^{48} - 12 q^{49} + 22 q^{53} - 20 q^{58} + 4 q^{62} - 2 q^{63} - 2 q^{64} + 2 q^{65} + 24 q^{67} + 2 q^{70} + 12 q^{71} - 2 q^{73} - 2 q^{74} + 2 q^{75} + 2 q^{77} - 2 q^{78} + 2 q^{81} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 24 q^{86} - 2 q^{90} + 6 q^{95} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^16 + 2 * q^21 - 2 * q^25 + 2 * q^26 - 2 * q^27 + 2 * q^28 + 2 * q^30 + 2 * q^33 + 10 * q^34 - 2 * q^36 + 12 * q^37 + 6 * q^38 + 2 * q^40 - 4 * q^41 + 2 * q^44 + 6 * q^46 + 24 * q^47 - 2 * q^48 - 12 * q^49 + 22 * q^53 - 20 * q^58 + 4 * q^62 - 2 * q^63 - 2 * q^64 + 2 * q^65 + 24 * q^67 + 2 * q^70 + 12 * q^71 - 2 * q^73 - 2 * q^74 + 2 * q^75 + 2 * q^77 - 2 * q^78 + 2 * q^81 - 6 * q^83 - 2 * q^84 + 10 * q^85 + 24 * q^86 - 2 * q^90 + 6 * q^95 - 2 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 −1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 −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.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.h.a 2
3.b odd 2 1 3330.2.h.d 2
37.b even 2 1 inner 1110.2.h.a 2
111.d odd 2 1 3330.2.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.a 2 1.a even 1 1 trivial
1110.2.h.a 2 37.b even 2 1 inner
3330.2.h.d 2 3.b odd 2 1
3330.2.h.d 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{13}^{2} + 1$$ T13^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 25$$
$19$ $$T^{2} + 9$$
$23$ $$T^{2} + 9$$
$29$ $$T^{2} + 100$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} - 12T + 37$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$(T - 12)^{2}$$
$53$ $$(T - 11)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$T^{2} + 196$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T + 1)^{2}$$
$79$ $$T^{2} + 100$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} + 81$$
$97$ $$T^{2} + 144$$