# Properties

 Label 1110.2.d.g Level $1110$ Weight $2$ Character orbit 1110.d Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} - \zeta_{8}^{2} q^{3} - q^{4} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + q^{6} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{8} - q^{9} +O(q^{10})$$ q + z^2 * q^2 - z^2 * q^3 - q^4 + (-2*z^3 - z) * q^5 + q^6 + (z^3 + z^2 + z) * q^7 - z^2 * q^8 - q^9 $$q + \zeta_{8}^{2} q^{2} - \zeta_{8}^{2} q^{3} - q^{4} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + q^{6} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{8} - q^{9} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{10} + (\zeta_{8}^{3} - \zeta_{8} - 5) q^{11} + \zeta_{8}^{2} q^{12} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + 3 \zeta_{8}) q^{13} + (\zeta_{8}^{3} - \zeta_{8} - 1) q^{14} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{15} + q^{16} + (\zeta_{8}^{3} - 5 \zeta_{8}^{2} + \zeta_{8}) q^{17} - \zeta_{8}^{2} q^{18} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 5) q^{19} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{20} + ( - \zeta_{8}^{3} + \zeta_{8} + 1) q^{21} + ( - \zeta_{8}^{3} - 5 \zeta_{8}^{2} - \zeta_{8}) q^{22} + 3 \zeta_{8}^{2} q^{23} - q^{24} + ( - 3 \zeta_{8}^{2} - 4) q^{25} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 1) q^{26} + \zeta_{8}^{2} q^{27} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{28} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{29} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{30} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 2) q^{31} + \zeta_{8}^{2} q^{32} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} + \zeta_{8}) q^{33} + (\zeta_{8}^{3} - \zeta_{8} + 5) q^{34} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8} + 3) q^{35} + q^{36} + \zeta_{8}^{2} q^{37} + ( - 2 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 2 \zeta_{8}) q^{38} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 1) q^{39} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{40} + ( - \zeta_{8}^{3} + \zeta_{8} - 8) q^{41} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{42} + (6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 6 \zeta_{8}) q^{43} + ( - \zeta_{8}^{3} + \zeta_{8} + 5) q^{44} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{45} - 3 q^{46} + (7 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 7 \zeta_{8}) q^{47} - \zeta_{8}^{2} q^{48} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 4) q^{49} + ( - 4 \zeta_{8}^{2} + 3) q^{50} + ( - \zeta_{8}^{3} + \zeta_{8} - 5) q^{51} + ( - 3 \zeta_{8}^{3} - \zeta_{8}^{2} - 3 \zeta_{8}) q^{52} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{53} - q^{54} + (10 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 5 \zeta_{8} - 1) q^{55} + ( - \zeta_{8}^{3} + \zeta_{8} + 1) q^{56} + (2 \zeta_{8}^{3} - 5 \zeta_{8}^{2} + 2 \zeta_{8}) q^{57} + ( - 5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{58} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8} + 6) q^{59} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{60} + (6 \zeta_{8}^{3} - 6 \zeta_{8} + 2) q^{61} + ( - 4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 4 \zeta_{8}) q^{62} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{63} - q^{64} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8} + 9) q^{65} + (\zeta_{8}^{3} - \zeta_{8} - 5) q^{66} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{67} + ( - \zeta_{8}^{3} + 5 \zeta_{8}^{2} - \zeta_{8}) q^{68} + 3 q^{69} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{70} + (\zeta_{8}^{3} - \zeta_{8} - 8) q^{71} + \zeta_{8}^{2} q^{72} + ( - 2 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 2 \zeta_{8}) q^{73} - q^{74} + (4 \zeta_{8}^{2} - 3) q^{75} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 5) q^{76} + ( - 6 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 6 \zeta_{8}) q^{77} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + 3 \zeta_{8}) q^{78} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 8) q^{79} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{80} + q^{81} + (\zeta_{8}^{3} - 8 \zeta_{8}^{2} + \zeta_{8}) q^{82} + (7 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 7 \zeta_{8}) q^{83} + (\zeta_{8}^{3} - \zeta_{8} - 1) q^{84} + (5 \zeta_{8}^{3} + \zeta_{8}^{2} - 10 \zeta_{8} + 3) q^{85} + (6 \zeta_{8}^{3} - 6 \zeta_{8} - 2) q^{86} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{87} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} + \zeta_{8}) q^{88} + (7 \zeta_{8}^{3} - 7 \zeta_{8} + 7) q^{89} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{90} + (4 \zeta_{8}^{3} - 4 \zeta_{8} - 7) q^{91} - 3 \zeta_{8}^{2} q^{92} + (4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{93} + (7 \zeta_{8}^{3} - 7 \zeta_{8} - 2) q^{94} + ( - 10 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 5 \zeta_{8} - 2) q^{95} + q^{96} + (9 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 9 \zeta_{8}) q^{97} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{98} + ( - \zeta_{8}^{3} + \zeta_{8} + 5) q^{99} +O(q^{100})$$ q + z^2 * q^2 - z^2 * q^3 - q^4 + (-2*z^3 - z) * q^5 + q^6 + (z^3 + z^2 + z) * q^7 - z^2 * q^8 - q^9 + (-z^3 + 2*z) * q^10 + (z^3 - z - 5) * q^11 + z^2 * q^12 + (3*z^3 + z^2 + 3*z) * q^13 + (z^3 - z - 1) * q^14 + (z^3 - 2*z) * q^15 + q^16 + (z^3 - 5*z^2 + z) * q^17 - z^2 * q^18 + (2*z^3 - 2*z + 5) * q^19 + (2*z^3 + z) * q^20 + (-z^3 + z + 1) * q^21 + (-z^3 - 5*z^2 - z) * q^22 + 3*z^2 * q^23 - q^24 + (-3*z^2 - 4) * q^25 + (3*z^3 - 3*z - 1) * q^26 + z^2 * q^27 + (-z^3 - z^2 - z) * q^28 + (5*z^3 - 5*z) * q^29 + (-2*z^3 - z) * q^30 + (4*z^3 - 4*z + 2) * q^31 + z^2 * q^32 + (z^3 + 5*z^2 + z) * q^33 + (z^3 - z + 5) * q^34 + (-z^3 + z^2 + 2*z + 3) * q^35 + q^36 + z^2 * q^37 + (-2*z^3 + 5*z^2 - 2*z) * q^38 + (-3*z^3 + 3*z + 1) * q^39 + (z^3 - 2*z) * q^40 + (-z^3 + z - 8) * q^41 + (z^3 + z^2 + z) * q^42 + (6*z^3 + 2*z^2 + 6*z) * q^43 + (-z^3 + z + 5) * q^44 + (2*z^3 + z) * q^45 - 3 * q^46 + (7*z^3 + 2*z^2 + 7*z) * q^47 - z^2 * q^48 + (2*z^3 - 2*z + 4) * q^49 + (-4*z^2 + 3) * q^50 + (-z^3 + z - 5) * q^51 + (-3*z^3 - z^2 - 3*z) * q^52 + (-2*z^3 - z^2 - 2*z) * q^53 - q^54 + (10*z^3 + 3*z^2 + 5*z - 1) * q^55 + (-z^3 + z + 1) * q^56 + (2*z^3 - 5*z^2 + 2*z) * q^57 + (-5*z^3 - 5*z) * q^58 + (-5*z^3 + 5*z + 6) * q^59 + (-z^3 + 2*z) * q^60 + (6*z^3 - 6*z + 2) * q^61 + (-4*z^3 + 2*z^2 - 4*z) * q^62 + (-z^3 - z^2 - z) * q^63 - q^64 + (-z^3 + 3*z^2 + 2*z + 9) * q^65 + (z^3 - z - 5) * q^66 + (2*z^3 - 2*z^2 + 2*z) * q^67 + (-z^3 + 5*z^2 - z) * q^68 + 3 * q^69 + (2*z^3 + 3*z^2 + z - 1) * q^70 + (z^3 - z - 8) * q^71 + z^2 * q^72 + (-2*z^3 + 5*z^2 - 2*z) * q^73 - q^74 + (4*z^2 - 3) * q^75 + (-2*z^3 + 2*z - 5) * q^76 + (-6*z^3 - 7*z^2 - 6*z) * q^77 + (3*z^3 + z^2 + 3*z) * q^78 + (3*z^3 - 3*z - 8) * q^79 + (-2*z^3 - z) * q^80 + q^81 + (z^3 - 8*z^2 + z) * q^82 + (7*z^3 - 3*z^2 + 7*z) * q^83 + (z^3 - z - 1) * q^84 + (5*z^3 + z^2 - 10*z + 3) * q^85 + (6*z^3 - 6*z - 2) * q^86 + (5*z^3 + 5*z) * q^87 + (z^3 + 5*z^2 + z) * q^88 + (7*z^3 - 7*z + 7) * q^89 + (z^3 - 2*z) * q^90 + (4*z^3 - 4*z - 7) * q^91 - 3*z^2 * q^92 + (4*z^3 - 2*z^2 + 4*z) * q^93 + (7*z^3 - 7*z - 2) * q^94 + (-10*z^3 + 6*z^2 - 5*z - 2) * q^95 + q^96 + (9*z^3 - 6*z^2 + 9*z) * q^97 + (-2*z^3 + 4*z^2 - 2*z) * q^98 + (-z^3 + z + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 20 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} + 4 q^{21} - 4 q^{24} - 16 q^{25} - 4 q^{26} + 8 q^{31} + 20 q^{34} + 12 q^{35} + 4 q^{36} + 4 q^{39} - 32 q^{41} + 20 q^{44} - 12 q^{46} + 16 q^{49} + 12 q^{50} - 20 q^{51} - 4 q^{54} - 4 q^{55} + 4 q^{56} + 24 q^{59} + 8 q^{61} - 4 q^{64} + 36 q^{65} - 20 q^{66} + 12 q^{69} - 4 q^{70} - 32 q^{71} - 4 q^{74} - 12 q^{75} - 20 q^{76} - 32 q^{79} + 4 q^{81} - 4 q^{84} + 12 q^{85} - 8 q^{86} + 28 q^{89} - 28 q^{91} - 8 q^{94} - 8 q^{95} + 4 q^{96} + 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 20 * q^11 - 4 * q^14 + 4 * q^16 + 20 * q^19 + 4 * q^21 - 4 * q^24 - 16 * q^25 - 4 * q^26 + 8 * q^31 + 20 * q^34 + 12 * q^35 + 4 * q^36 + 4 * q^39 - 32 * q^41 + 20 * q^44 - 12 * q^46 + 16 * q^49 + 12 * q^50 - 20 * q^51 - 4 * q^54 - 4 * q^55 + 4 * q^56 + 24 * q^59 + 8 * q^61 - 4 * q^64 + 36 * q^65 - 20 * q^66 + 12 * q^69 - 4 * q^70 - 32 * q^71 - 4 * q^74 - 12 * q^75 - 20 * q^76 - 32 * q^79 + 4 * q^81 - 4 * q^84 + 12 * q^85 - 8 * q^86 + 28 * q^89 - 28 * q^91 - 8 * q^94 - 8 * q^95 + 4 * q^96 + 20 * 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
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 −0.707107 2.12132i 1.00000 0.414214i 1.00000i −1.00000 −2.12132 + 0.707107i
889.2 1.00000i 1.00000i −1.00000 0.707107 + 2.12132i 1.00000 2.41421i 1.00000i −1.00000 2.12132 0.707107i
889.3 1.00000i 1.00000i −1.00000 −0.707107 + 2.12132i 1.00000 0.414214i 1.00000i −1.00000 −2.12132 0.707107i
889.4 1.00000i 1.00000i −1.00000 0.707107 2.12132i 1.00000 2.41421i 1.00000i −1.00000 2.12132 + 0.707107i
 $$n$$: 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.g 4
3.b odd 2 1 3330.2.d.k 4
5.b even 2 1 inner 1110.2.d.g 4
5.c odd 4 1 5550.2.a.bs 2
5.c odd 4 1 5550.2.a.cb 2
15.d odd 2 1 3330.2.d.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.g 4 1.a even 1 1 trivial
1110.2.d.g 4 5.b even 2 1 inner
3330.2.d.k 4 3.b odd 2 1
3330.2.d.k 4 15.d odd 2 1
5550.2.a.bs 2 5.c odd 4 1
5550.2.a.cb 2 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{4} + 6T_{7}^{2} + 1$$ T7^4 + 6*T7^2 + 1 $$T_{11}^{2} + 10T_{11} + 23$$ T11^2 + 10*T11 + 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4} + 8T^{2} + 25$$
$7$ $$T^{4} + 6T^{2} + 1$$
$11$ $$(T^{2} + 10 T + 23)^{2}$$
$13$ $$T^{4} + 38T^{2} + 289$$
$17$ $$T^{4} + 54T^{2} + 529$$
$19$ $$(T^{2} - 10 T + 17)^{2}$$
$23$ $$(T^{2} + 9)^{2}$$
$29$ $$(T^{2} - 50)^{2}$$
$31$ $$(T^{2} - 4 T - 28)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} + 16 T + 62)^{2}$$
$43$ $$T^{4} + 152T^{2} + 4624$$
$47$ $$T^{4} + 204T^{2} + 8836$$
$53$ $$T^{4} + 18T^{2} + 49$$
$59$ $$(T^{2} - 12 T - 14)^{2}$$
$61$ $$(T^{2} - 4 T - 68)^{2}$$
$67$ $$T^{4} + 24T^{2} + 16$$
$71$ $$(T^{2} + 16 T + 62)^{2}$$
$73$ $$T^{4} + 66T^{2} + 289$$
$79$ $$(T^{2} + 16 T + 46)^{2}$$
$83$ $$T^{4} + 214T^{2} + 7921$$
$89$ $$(T^{2} - 14 T - 49)^{2}$$
$97$ $$T^{4} + 396 T^{2} + 15876$$