# Properties

 Label 1134.2.e.o Level $1134$ Weight $2$ Character orbit 1134.e Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(865,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 4]))

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

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

chi := DirichletCharacter("1134.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.e (of order $$3$$, degree $$2$$, not 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: $$9.05503558921$$ 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + (2 \zeta_{6} + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (-2*z + 2) * q^5 + (2*z + 1) * q^7 + q^8 $$q + q^{2} + q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + (2 \zeta_{6} + 1) q^{7} + q^{8} + ( - 2 \zeta_{6} + 2) q^{10} - 5 \zeta_{6} q^{11} - 6 \zeta_{6} q^{13} + (2 \zeta_{6} + 1) q^{14} + q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + 4 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 2) q^{20} - 5 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} + \zeta_{6} q^{25} - 6 \zeta_{6} q^{26} + (2 \zeta_{6} + 1) q^{28} + (7 \zeta_{6} - 7) q^{29} + 3 q^{31} + q^{32} + ( - 4 \zeta_{6} + 4) q^{34} + ( - 2 \zeta_{6} + 6) q^{35} - 8 \zeta_{6} q^{37} + 4 \zeta_{6} q^{38} + ( - 2 \zeta_{6} + 2) q^{40} + 6 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} - 5 \zeta_{6} q^{44} + ( - 4 \zeta_{6} + 4) q^{46} + 6 q^{47} + (8 \zeta_{6} - 3) q^{49} + \zeta_{6} q^{50} - 6 \zeta_{6} q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 10 q^{55} + (2 \zeta_{6} + 1) q^{56} + (7 \zeta_{6} - 7) q^{58} + 7 q^{59} + 3 q^{62} + q^{64} - 12 q^{65} + 10 q^{67} + ( - 4 \zeta_{6} + 4) q^{68} + ( - 2 \zeta_{6} + 6) q^{70} - 4 q^{71} + (13 \zeta_{6} - 13) q^{73} - 8 \zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + ( - 15 \zeta_{6} + 10) q^{77} - 3 q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + 6 \zeta_{6} q^{82} + ( - 7 \zeta_{6} + 7) q^{83} - 8 \zeta_{6} q^{85} + (8 \zeta_{6} - 8) q^{86} - 5 \zeta_{6} q^{88} - 6 \zeta_{6} q^{89} + ( - 18 \zeta_{6} + 12) q^{91} + ( - 4 \zeta_{6} + 4) q^{92} + 6 q^{94} + 8 q^{95} + ( - 5 \zeta_{6} + 5) q^{97} + (8 \zeta_{6} - 3) q^{98} +O(q^{100})$$ q + q^2 + q^4 + (-2*z + 2) * q^5 + (2*z + 1) * q^7 + q^8 + (-2*z + 2) * q^10 - 5*z * q^11 - 6*z * q^13 + (2*z + 1) * q^14 + q^16 + (-4*z + 4) * q^17 + 4*z * q^19 + (-2*z + 2) * q^20 - 5*z * q^22 + (-4*z + 4) * q^23 + z * q^25 - 6*z * q^26 + (2*z + 1) * q^28 + (7*z - 7) * q^29 + 3 * q^31 + q^32 + (-4*z + 4) * q^34 + (-2*z + 6) * q^35 - 8*z * q^37 + 4*z * q^38 + (-2*z + 2) * q^40 + 6*z * q^41 + (8*z - 8) * q^43 - 5*z * q^44 + (-4*z + 4) * q^46 + 6 * q^47 + (8*z - 3) * q^49 + z * q^50 - 6*z * q^52 + (-6*z + 6) * q^53 - 10 * q^55 + (2*z + 1) * q^56 + (7*z - 7) * q^58 + 7 * q^59 + 3 * q^62 + q^64 - 12 * q^65 + 10 * q^67 + (-4*z + 4) * q^68 + (-2*z + 6) * q^70 - 4 * q^71 + (13*z - 13) * q^73 - 8*z * q^74 + 4*z * q^76 + (-15*z + 10) * q^77 - 3 * q^79 + (-2*z + 2) * q^80 + 6*z * q^82 + (-7*z + 7) * q^83 - 8*z * q^85 + (8*z - 8) * q^86 - 5*z * q^88 - 6*z * q^89 + (-18*z + 12) * q^91 + (-4*z + 4) * q^92 + 6 * q^94 + 8 * q^95 + (-5*z + 5) * q^97 + (8*z - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 5 q^{11} - 6 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 2 q^{20} - 5 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} + 4 q^{28} - 7 q^{29} + 6 q^{31} + 2 q^{32} + 4 q^{34} + 10 q^{35} - 8 q^{37} + 4 q^{38} + 2 q^{40} + 6 q^{41} - 8 q^{43} - 5 q^{44} + 4 q^{46} + 12 q^{47} + 2 q^{49} + q^{50} - 6 q^{52} + 6 q^{53} - 20 q^{55} + 4 q^{56} - 7 q^{58} + 14 q^{59} + 6 q^{62} + 2 q^{64} - 24 q^{65} + 20 q^{67} + 4 q^{68} + 10 q^{70} - 8 q^{71} - 13 q^{73} - 8 q^{74} + 4 q^{76} + 5 q^{77} - 6 q^{79} + 2 q^{80} + 6 q^{82} + 7 q^{83} - 8 q^{85} - 8 q^{86} - 5 q^{88} - 6 q^{89} + 6 q^{91} + 4 q^{92} + 12 q^{94} + 16 q^{95} + 5 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 - 5 * q^11 - 6 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 2 * q^20 - 5 * q^22 + 4 * q^23 + q^25 - 6 * q^26 + 4 * q^28 - 7 * q^29 + 6 * q^31 + 2 * q^32 + 4 * q^34 + 10 * q^35 - 8 * q^37 + 4 * q^38 + 2 * q^40 + 6 * q^41 - 8 * q^43 - 5 * q^44 + 4 * q^46 + 12 * q^47 + 2 * q^49 + q^50 - 6 * q^52 + 6 * q^53 - 20 * q^55 + 4 * q^56 - 7 * q^58 + 14 * q^59 + 6 * q^62 + 2 * q^64 - 24 * q^65 + 20 * q^67 + 4 * q^68 + 10 * q^70 - 8 * q^71 - 13 * q^73 - 8 * q^74 + 4 * q^76 + 5 * q^77 - 6 * q^79 + 2 * q^80 + 6 * q^82 + 7 * q^83 - 8 * q^85 - 8 * q^86 - 5 * q^88 - 6 * q^89 + 6 * q^91 + 4 * q^92 + 12 * q^94 + 16 * q^95 + 5 * q^97 + 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 0 1.00000 1.00000 1.73205i 0 2.00000 + 1.73205i 1.00000 0 1.00000 1.73205i
919.1 1.00000 0 1.00000 1.00000 + 1.73205i 0 2.00000 1.73205i 1.00000 0 1.00000 + 1.73205i
 $$n$$: 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
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.e.o 2
3.b odd 2 1 1134.2.e.b 2
7.c even 3 1 1134.2.h.b 2
9.c even 3 1 378.2.g.c 2
9.c even 3 1 1134.2.h.b 2
9.d odd 6 1 378.2.g.d yes 2
9.d odd 6 1 1134.2.h.o 2
21.h odd 6 1 1134.2.h.o 2
63.g even 3 1 378.2.g.c 2
63.h even 3 1 inner 1134.2.e.o 2
63.h even 3 1 2646.2.a.t 1
63.i even 6 1 2646.2.a.c 1
63.j odd 6 1 1134.2.e.b 2
63.j odd 6 1 2646.2.a.k 1
63.n odd 6 1 378.2.g.d yes 2
63.t odd 6 1 2646.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.c 2 9.c even 3 1
378.2.g.c 2 63.g even 3 1
378.2.g.d yes 2 9.d odd 6 1
378.2.g.d yes 2 63.n odd 6 1
1134.2.e.b 2 3.b odd 2 1
1134.2.e.b 2 63.j odd 6 1
1134.2.e.o 2 1.a even 1 1 trivial
1134.2.e.o 2 63.h even 3 1 inner
1134.2.h.b 2 7.c even 3 1
1134.2.h.b 2 9.c even 3 1
1134.2.h.o 2 9.d odd 6 1
1134.2.h.o 2 21.h odd 6 1
2646.2.a.c 1 63.i even 6 1
2646.2.a.k 1 63.j odd 6 1
2646.2.a.t 1 63.h even 3 1
2646.2.a.bb 1 63.t odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25 $$T_{17}^{2} - 4T_{17} + 16$$ T17^2 - 4*T17 + 16 $$T_{23}^{2} - 4T_{23} + 16$$ T23^2 - 4*T23 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} + 7T + 49$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$(T - 7)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T - 10)^{2}$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 13T + 169$$
$79$ $$(T + 3)^{2}$$
$83$ $$T^{2} - 7T + 49$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} - 5T + 25$$