# Properties

 Label 1110.2.a.p Level $1110$ Weight $2$ Character orbit 1110.a Self dual yes Analytic conductor $8.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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, 0]))

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

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

chi := DirichletCharacter("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: yes Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.37228 −2.37228
−1.00000 −1.00000 1.00000 1.00000 1.00000 −3.37228 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.37228 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$+1$$
$$5$$ $$-1$$
$$37$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.a.p 2
3.b odd 2 1 3330.2.a.be 2
4.b odd 2 1 8880.2.a.br 2
5.b even 2 1 5550.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.p 2 1.a even 1 1 trivial
3330.2.a.be 2 3.b odd 2 1
5550.2.a.ca 2 5.b even 2 1
8880.2.a.br 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1110))$$:

 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{11}^{2} - 3T_{11} - 6$$ T11^2 - 3*T11 - 6 $$T_{13}^{2} + 3T_{13} - 6$$ T13^2 + 3*T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + T - 8$$
$11$ $$T^{2} - 3T - 6$$
$13$ $$T^{2} + 3T - 6$$
$17$ $$T^{2} - 3T - 6$$
$19$ $$T^{2} - 3T - 6$$
$23$ $$T^{2} - T - 8$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 6T - 24$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 6T - 24$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 2T - 32$$
$53$ $$T^{2} - 5T - 2$$
$59$ $$T^{2} - 18T + 48$$
$61$ $$T^{2} - 6T - 24$$
$67$ $$T^{2} + 6T - 24$$
$71$ $$T^{2} - 14T + 16$$
$73$ $$T^{2} - 9T - 54$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} - 7T + 4$$
$89$ $$T^{2} - 21T + 102$$
$97$ $$T^{2} - 4T - 128$$