# Properties

 Label 1134.2.a.k 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: $$\Q(\sqrt{33})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 1) q^{5} + q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 1) * q^5 + q^7 - q^8 $$q - q^{2} + q^{4} + (\beta + 1) q^{5} + q^{7} - q^{8} + ( - \beta - 1) q^{10} + ( - \beta + 2) q^{11} + 2 q^{13} - q^{14} + q^{16} + (\beta - 2) q^{17} + 5 q^{19} + (\beta + 1) q^{20} + (\beta - 2) q^{22} + (\beta - 5) q^{23} + (3 \beta + 4) q^{25} - 2 q^{26} + q^{28} + ( - 2 \beta - 2) q^{29} + 2 q^{31} - q^{32} + ( - \beta + 2) q^{34} + (\beta + 1) q^{35} + 2 q^{37} - 5 q^{38} + ( - \beta - 1) q^{40} + ( - \beta + 8) q^{41} + ( - 3 \beta + 2) q^{43} + ( - \beta + 2) q^{44} + ( - \beta + 5) q^{46} + q^{49} + ( - 3 \beta - 4) q^{50} + 2 q^{52} + ( - 2 \beta - 2) q^{53} - 6 q^{55} - q^{56} + (2 \beta + 2) q^{58} - 3 \beta q^{59} + (3 \beta - 7) q^{61} - 2 q^{62} + q^{64} + (2 \beta + 2) q^{65} + ( - 3 \beta + 8) q^{67} + (\beta - 2) q^{68} + ( - \beta - 1) q^{70} + ( - 3 \beta + 3) q^{71} + (3 \beta + 2) q^{73} - 2 q^{74} + 5 q^{76} + ( - \beta + 2) q^{77} + ( - 3 \beta + 5) q^{79} + (\beta + 1) q^{80} + (\beta - 8) q^{82} + (4 \beta + 4) q^{83} + 6 q^{85} + (3 \beta - 2) q^{86} + (\beta - 2) q^{88} + (2 \beta + 8) q^{89} + 2 q^{91} + (\beta - 5) q^{92} + (5 \beta + 5) q^{95} + ( - 3 \beta + 2) q^{97} - q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b + 1) * q^5 + q^7 - q^8 + (-b - 1) * q^10 + (-b + 2) * q^11 + 2 * q^13 - q^14 + q^16 + (b - 2) * q^17 + 5 * q^19 + (b + 1) * q^20 + (b - 2) * q^22 + (b - 5) * q^23 + (3*b + 4) * q^25 - 2 * q^26 + q^28 + (-2*b - 2) * q^29 + 2 * q^31 - q^32 + (-b + 2) * q^34 + (b + 1) * q^35 + 2 * q^37 - 5 * q^38 + (-b - 1) * q^40 + (-b + 8) * q^41 + (-3*b + 2) * q^43 + (-b + 2) * q^44 + (-b + 5) * q^46 + q^49 + (-3*b - 4) * q^50 + 2 * q^52 + (-2*b - 2) * q^53 - 6 * q^55 - q^56 + (2*b + 2) * q^58 - 3*b * q^59 + (3*b - 7) * q^61 - 2 * q^62 + q^64 + (2*b + 2) * q^65 + (-3*b + 8) * q^67 + (b - 2) * q^68 + (-b - 1) * q^70 + (-3*b + 3) * q^71 + (3*b + 2) * q^73 - 2 * q^74 + 5 * q^76 + (-b + 2) * q^77 + (-3*b + 5) * q^79 + (b + 1) * q^80 + (b - 8) * q^82 + (4*b + 4) * q^83 + 6 * q^85 + (3*b - 2) * q^86 + (b - 2) * q^88 + (2*b + 8) * q^89 + 2 * q^91 + (b - 5) * q^92 + (5*b + 5) * q^95 + (-3*b + 2) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{7} - 2 q^{8} - 3 q^{10} + 3 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 3 q^{17} + 10 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} + 11 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - 2 q^{32} + 3 q^{34} + 3 q^{35} + 4 q^{37} - 10 q^{38} - 3 q^{40} + 15 q^{41} + q^{43} + 3 q^{44} + 9 q^{46} + 2 q^{49} - 11 q^{50} + 4 q^{52} - 6 q^{53} - 12 q^{55} - 2 q^{56} + 6 q^{58} - 3 q^{59} - 11 q^{61} - 4 q^{62} + 2 q^{64} + 6 q^{65} + 13 q^{67} - 3 q^{68} - 3 q^{70} + 3 q^{71} + 7 q^{73} - 4 q^{74} + 10 q^{76} + 3 q^{77} + 7 q^{79} + 3 q^{80} - 15 q^{82} + 12 q^{83} + 12 q^{85} - q^{86} - 3 q^{88} + 18 q^{89} + 4 q^{91} - 9 q^{92} + 15 q^{95} + q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^7 - 2 * q^8 - 3 * q^10 + 3 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 3 * q^17 + 10 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 + 11 * q^25 - 4 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 3 * q^34 + 3 * q^35 + 4 * q^37 - 10 * q^38 - 3 * q^40 + 15 * q^41 + q^43 + 3 * q^44 + 9 * q^46 + 2 * q^49 - 11 * q^50 + 4 * q^52 - 6 * q^53 - 12 * q^55 - 2 * q^56 + 6 * q^58 - 3 * q^59 - 11 * q^61 - 4 * q^62 + 2 * q^64 + 6 * q^65 + 13 * q^67 - 3 * q^68 - 3 * q^70 + 3 * q^71 + 7 * q^73 - 4 * q^74 + 10 * q^76 + 3 * q^77 + 7 * q^79 + 3 * q^80 - 15 * q^82 + 12 * q^83 + 12 * q^85 - q^86 - 3 * q^88 + 18 * q^89 + 4 * q^91 - 9 * q^92 + 15 * q^95 + 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
 −2.37228 3.37228
−1.00000 0 1.00000 −1.37228 0 1.00000 −1.00000 0 1.37228
1.2 −1.00000 0 1.00000 4.37228 0 1.00000 −1.00000 0 −4.37228
 $$n$$: 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.k 2
3.b odd 2 1 1134.2.a.n 2
4.b odd 2 1 9072.2.a.bm 2
7.b odd 2 1 7938.2.a.bh 2
9.c even 3 2 126.2.f.d 4
9.d odd 6 2 378.2.f.c 4
12.b even 2 1 9072.2.a.bb 2
21.c even 2 1 7938.2.a.bs 2
36.f odd 6 2 1008.2.r.f 4
36.h even 6 2 3024.2.r.f 4
63.g even 3 2 882.2.h.m 4
63.h even 3 2 882.2.e.l 4
63.i even 6 2 2646.2.e.m 4
63.j odd 6 2 2646.2.e.n 4
63.k odd 6 2 882.2.h.n 4
63.l odd 6 2 882.2.f.k 4
63.n odd 6 2 2646.2.h.k 4
63.o even 6 2 2646.2.f.j 4
63.s even 6 2 2646.2.h.l 4
63.t odd 6 2 882.2.e.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 9.c even 3 2
378.2.f.c 4 9.d odd 6 2
882.2.e.k 4 63.t odd 6 2
882.2.e.l 4 63.h even 3 2
882.2.f.k 4 63.l odd 6 2
882.2.h.m 4 63.g even 3 2
882.2.h.n 4 63.k odd 6 2
1008.2.r.f 4 36.f odd 6 2
1134.2.a.k 2 1.a even 1 1 trivial
1134.2.a.n 2 3.b odd 2 1
2646.2.e.m 4 63.i even 6 2
2646.2.e.n 4 63.j odd 6 2
2646.2.f.j 4 63.o even 6 2
2646.2.h.k 4 63.n odd 6 2
2646.2.h.l 4 63.s even 6 2
3024.2.r.f 4 36.h even 6 2
7938.2.a.bh 2 7.b odd 2 1
7938.2.a.bs 2 21.c even 2 1
9072.2.a.bb 2 12.b even 2 1
9072.2.a.bm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1134))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T - 6$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 3T - 6$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 3T - 6$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 9T + 12$$
$29$ $$T^{2} + 6T - 24$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 15T + 48$$
$43$ $$T^{2} - T - 74$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T - 24$$
$59$ $$T^{2} + 3T - 72$$
$61$ $$T^{2} + 11T - 44$$
$67$ $$T^{2} - 13T - 32$$
$71$ $$T^{2} - 3T - 72$$
$73$ $$T^{2} - 7T - 62$$
$79$ $$T^{2} - 7T - 62$$
$83$ $$T^{2} - 12T - 96$$
$89$ $$T^{2} - 18T + 48$$
$97$ $$T^{2} - T - 74$$