# Properties

 Label 1134.2.a.o Level $1134$ Weight $2$ Character orbit 1134.a Self dual yes Analytic conductor $9.055$ 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] = [1134,2,Mod(1,1134)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.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: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} + q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b * q^5 + q^7 + q^8 $$q + q^{2} + q^{4} + \beta q^{5} + q^{7} + q^{8} + \beta q^{10} + ( - \beta + 3) q^{11} - q^{13} + q^{14} + q^{16} + ( - 2 \beta + 3) q^{17} + (3 \beta - 1) q^{19} + \beta q^{20} + ( - \beta + 3) q^{22} + (\beta + 3) q^{23} - 2 q^{25} - q^{26} + q^{28} + ( - 2 \beta + 3) q^{29} + ( - 3 \beta - 1) q^{31} + q^{32} + ( - 2 \beta + 3) q^{34} + \beta q^{35} + (3 \beta + 2) q^{37} + (3 \beta - 1) q^{38} + \beta q^{40} + (2 \beta + 6) q^{41} + ( - 6 \beta + 2) q^{43} + ( - \beta + 3) q^{44} + (\beta + 3) q^{46} + ( - 3 \beta - 3) q^{47} + q^{49} - 2 q^{50} - q^{52} + ( - 2 \beta + 6) q^{53} + (3 \beta - 3) q^{55} + q^{56} + ( - 2 \beta + 3) q^{58} + (3 \beta - 3) q^{59} + ( - 6 \beta - 1) q^{61} + ( - 3 \beta - 1) q^{62} + q^{64} - \beta q^{65} + ( - 3 \beta - 1) q^{67} + ( - 2 \beta + 3) q^{68} + \beta q^{70} + (6 \beta + 6) q^{71} + (3 \beta - 4) q^{73} + (3 \beta + 2) q^{74} + (3 \beta - 1) q^{76} + ( - \beta + 3) q^{77} + (3 \beta - 1) q^{79} + \beta q^{80} + (2 \beta + 6) q^{82} + (\beta + 3) q^{83} + (3 \beta - 6) q^{85} + ( - 6 \beta + 2) q^{86} + ( - \beta + 3) q^{88} + (2 \beta - 9) q^{89} - q^{91} + (\beta + 3) q^{92} + ( - 3 \beta - 3) q^{94} + ( - \beta + 9) q^{95} - 16 q^{97} + q^{98} +O(q^{100})$$ q + q^2 + q^4 + b * q^5 + q^7 + q^8 + b * q^10 + (-b + 3) * q^11 - q^13 + q^14 + q^16 + (-2*b + 3) * q^17 + (3*b - 1) * q^19 + b * q^20 + (-b + 3) * q^22 + (b + 3) * q^23 - 2 * q^25 - q^26 + q^28 + (-2*b + 3) * q^29 + (-3*b - 1) * q^31 + q^32 + (-2*b + 3) * q^34 + b * q^35 + (3*b + 2) * q^37 + (3*b - 1) * q^38 + b * q^40 + (2*b + 6) * q^41 + (-6*b + 2) * q^43 + (-b + 3) * q^44 + (b + 3) * q^46 + (-3*b - 3) * q^47 + q^49 - 2 * q^50 - q^52 + (-2*b + 6) * q^53 + (3*b - 3) * q^55 + q^56 + (-2*b + 3) * q^58 + (3*b - 3) * q^59 + (-6*b - 1) * q^61 + (-3*b - 1) * q^62 + q^64 - b * q^65 + (-3*b - 1) * q^67 + (-2*b + 3) * q^68 + b * q^70 + (6*b + 6) * q^71 + (3*b - 4) * q^73 + (3*b + 2) * q^74 + (3*b - 1) * q^76 + (-b + 3) * q^77 + (3*b - 1) * q^79 + b * q^80 + (2*b + 6) * q^82 + (b + 3) * q^83 + (3*b - 6) * q^85 + (-6*b + 2) * q^86 + (-b + 3) * q^88 + (2*b - 9) * q^89 - q^91 + (b + 3) * q^92 + (-3*b - 3) * q^94 + (-b + 9) * q^95 - 16 * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 12 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} + 2 q^{49} - 4 q^{50} - 2 q^{52} + 12 q^{53} - 6 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{67} + 6 q^{68} + 12 q^{71} - 8 q^{73} + 4 q^{74} - 2 q^{76} + 6 q^{77} - 2 q^{79} + 12 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} - 6 q^{94} + 18 q^{95} - 32 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 + 2 * q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 12 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 4 * q^50 - 2 * q^52 + 12 * q^53 - 6 * q^55 + 2 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^67 + 6 * q^68 + 12 * q^71 - 8 * q^73 + 4 * q^74 - 2 * q^76 + 6 * q^77 - 2 * q^79 + 12 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 + 6 * q^88 - 18 * q^89 - 2 * q^91 + 6 * q^92 - 6 * q^94 + 18 * q^95 - 32 * q^97 + 2 * q^98

## 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
 −1.73205 1.73205
1.00000 0 1.00000 −1.73205 0 1.00000 1.00000 0 −1.73205
1.2 1.00000 0 1.00000 1.73205 0 1.00000 1.00000 0 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 1134.2.a.o yes 2
3.b odd 2 1 1134.2.a.j 2
4.b odd 2 1 9072.2.a.bf 2
7.b odd 2 1 7938.2.a.br 2
9.c even 3 2 1134.2.f.q 4
9.d odd 6 2 1134.2.f.t 4
12.b even 2 1 9072.2.a.bi 2
21.c even 2 1 7938.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.j 2 3.b odd 2 1
1134.2.a.o yes 2 1.a even 1 1 trivial
1134.2.f.q 4 9.c even 3 2
1134.2.f.t 4 9.d odd 6 2
7938.2.a.bi 2 21.c even 2 1
7938.2.a.br 2 7.b odd 2 1
9072.2.a.bf 2 4.b odd 2 1
9072.2.a.bi 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1134))$$:

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 6T + 6$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 6T - 3$$
$19$ $$T^{2} + 2T - 26$$
$23$ $$T^{2} - 6T + 6$$
$29$ $$T^{2} - 6T - 3$$
$31$ $$T^{2} + 2T - 26$$
$37$ $$T^{2} - 4T - 23$$
$41$ $$T^{2} - 12T + 24$$
$43$ $$T^{2} - 4T - 104$$
$47$ $$T^{2} + 6T - 18$$
$53$ $$T^{2} - 12T + 24$$
$59$ $$T^{2} + 6T - 18$$
$61$ $$T^{2} + 2T - 107$$
$67$ $$T^{2} + 2T - 26$$
$71$ $$T^{2} - 12T - 72$$
$73$ $$T^{2} + 8T - 11$$
$79$ $$T^{2} + 2T - 26$$
$83$ $$T^{2} - 6T + 6$$
$89$ $$T^{2} + 18T + 69$$
$97$ $$(T + 16)^{2}$$