# Properties

 Label 1134.2.e.q Level $1134$ Weight $2$ Character orbit 1134.e Analytic conductor $9.055$ 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] = [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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_1 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (-b2 + b1 - 1) * q^5 - b1 * q^7 - q^8 $$q - q^{2} + q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_1 q^{7} - q^{8} + (\beta_{2} - \beta_1 + 1) q^{10} + (\beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{2} - \beta_1 + 1) q^{17} + 2 \beta_{2} q^{19} + ( - \beta_{2} + \beta_1 - 1) q^{20} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{22} + (4 \beta_{2} + 2 \beta_1 + 4) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{26} - \beta_1 q^{28} + (5 \beta_{2} + \beta_1 + 5) q^{29} + ( - \beta_{3} - 2) q^{31} - q^{32} + ( - \beta_{2} + \beta_1 - 1) q^{34} + (\beta_{3} - 7 \beta_{2} + \beta_1) q^{35} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{37} - 2 \beta_{2} q^{38} + (\beta_{2} - \beta_1 + 1) q^{40} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{41} + ( - 5 \beta_{2} - 5) q^{43} + (\beta_{3} - \beta_{2} + \beta_1) q^{44} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{46} + ( - 3 \beta_{3} + 3) q^{47} + 7 \beta_{2} q^{49} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{50} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{52} + (6 \beta_{2} + 6) q^{53} + ( - 2 \beta_{3} - 8) q^{55} + \beta_1 q^{56} + ( - 5 \beta_{2} - \beta_1 - 5) q^{58} + (\beta_{3} - 11) q^{59} + ( - \beta_{3} + 10) q^{61} + (\beta_{3} + 2) q^{62} + q^{64} + ( - 3 \beta_{3} - 9) q^{65} + ( - 2 \beta_{3} + 3) q^{67} + (\beta_{2} - \beta_1 + 1) q^{68} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{70} + ( - \beta_{3} - 13) q^{71} + 4 \beta_1 q^{73} + ( - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{74} + 2 \beta_{2} q^{76} + (\beta_{3} + 7) q^{77} + (5 \beta_{3} - 2) q^{79} + ( - \beta_{2} + \beta_1 - 1) q^{80} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{82} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{83} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{85} + (5 \beta_{2} + 5) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{88} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{89} + (2 \beta_{3} + 7) q^{91} + (4 \beta_{2} + 2 \beta_1 + 4) q^{92} + (3 \beta_{3} - 3) q^{94} + (2 \beta_{3} + 2) q^{95} + ( - 3 \beta_{2} + 4 \beta_1 - 3) q^{97} - 7 \beta_{2} q^{98}+O(q^{100})$$ q - q^2 + q^4 + (-b2 + b1 - 1) * q^5 - b1 * q^7 - q^8 + (b2 - b1 + 1) * q^10 + (b3 - b2 + b1) * q^11 + (b3 - 2*b2 + b1) * q^13 + b1 * q^14 + q^16 + (b2 - b1 + 1) * q^17 + 2*b2 * q^19 + (-b2 + b1 - 1) * q^20 + (-b3 + b2 - b1) * q^22 + (4*b2 + 2*b1 + 4) * q^23 + (-2*b3 + 3*b2 - 2*b1) * q^25 + (-b3 + 2*b2 - b1) * q^26 - b1 * q^28 + (5*b2 + b1 + 5) * q^29 + (-b3 - 2) * q^31 - q^32 + (-b2 + b1 - 1) * q^34 + (b3 - 7*b2 + b1) * q^35 + (3*b3 - 4*b2 + 3*b1) * q^37 - 2*b2 * q^38 + (b2 - b1 + 1) * q^40 + (3*b3 - 3*b2 + 3*b1) * q^41 + (-5*b2 - 5) * q^43 + (b3 - b2 + b1) * q^44 + (-4*b2 - 2*b1 - 4) * q^46 + (-3*b3 + 3) * q^47 + 7*b2 * q^49 + (2*b3 - 3*b2 + 2*b1) * q^50 + (b3 - 2*b2 + b1) * q^52 + (6*b2 + 6) * q^53 + (-2*b3 - 8) * q^55 + b1 * q^56 + (-5*b2 - b1 - 5) * q^58 + (b3 - 11) * q^59 + (-b3 + 10) * q^61 + (b3 + 2) * q^62 + q^64 + (-3*b3 - 9) * q^65 + (-2*b3 + 3) * q^67 + (b2 - b1 + 1) * q^68 + (-b3 + 7*b2 - b1) * q^70 + (-b3 - 13) * q^71 + 4*b1 * q^73 + (-3*b3 + 4*b2 - 3*b1) * q^74 + 2*b2 * q^76 + (b3 + 7) * q^77 + (5*b3 - 2) * q^79 + (-b2 + b1 - 1) * q^80 + (-3*b3 + 3*b2 - 3*b1) * q^82 + (-8*b2 + 2*b1 - 8) * q^83 + (2*b3 - 8*b2 + 2*b1) * q^85 + (5*b2 + 5) * q^86 + (-b3 + b2 - b1) * q^88 + (3*b3 + 3*b2 + 3*b1) * q^89 + (2*b3 + 7) * q^91 + (4*b2 + 2*b1 + 4) * q^92 + (3*b3 - 3) * q^94 + (2*b3 + 2) * q^95 + (-3*b2 + 4*b1 - 3) * q^97 - 7*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^8 $$4 q - 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} - 2 q^{22} + 8 q^{23} - 6 q^{25} - 4 q^{26} + 10 q^{29} - 8 q^{31} - 4 q^{32} - 2 q^{34} + 14 q^{35} + 8 q^{37} + 4 q^{38} + 2 q^{40} + 6 q^{41} - 10 q^{43} + 2 q^{44} - 8 q^{46} + 12 q^{47} - 14 q^{49} + 6 q^{50} + 4 q^{52} + 12 q^{53} - 32 q^{55} - 10 q^{58} - 44 q^{59} + 40 q^{61} + 8 q^{62} + 4 q^{64} - 36 q^{65} + 12 q^{67} + 2 q^{68} - 14 q^{70} - 52 q^{71} - 8 q^{74} - 4 q^{76} + 28 q^{77} - 8 q^{79} - 2 q^{80} - 6 q^{82} - 16 q^{83} + 16 q^{85} + 10 q^{86} - 2 q^{88} - 6 q^{89} + 28 q^{91} + 8 q^{92} - 12 q^{94} + 8 q^{95} - 6 q^{97} + 14 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^16 + 2 * q^17 - 4 * q^19 - 2 * q^20 - 2 * q^22 + 8 * q^23 - 6 * q^25 - 4 * q^26 + 10 * q^29 - 8 * q^31 - 4 * q^32 - 2 * q^34 + 14 * q^35 + 8 * q^37 + 4 * q^38 + 2 * q^40 + 6 * q^41 - 10 * q^43 + 2 * q^44 - 8 * q^46 + 12 * q^47 - 14 * q^49 + 6 * q^50 + 4 * q^52 + 12 * q^53 - 32 * q^55 - 10 * q^58 - 44 * q^59 + 40 * q^61 + 8 * q^62 + 4 * q^64 - 36 * q^65 + 12 * q^67 + 2 * q^68 - 14 * q^70 - 52 * q^71 - 8 * q^74 - 4 * q^76 + 28 * q^77 - 8 * q^79 - 2 * q^80 - 6 * q^82 - 16 * q^83 + 16 * q^85 + 10 * q^86 - 2 * q^88 - 6 * q^89 + 28 * q^91 + 8 * q^92 - 12 * q^94 + 8 * q^95 - 6 * q^97 + 14 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$\beta_{2}$$ $$\beta_{2}$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
−1.00000 0 1.00000 −1.82288 + 3.15731i 0 1.32288 2.29129i −1.00000 0 1.82288 3.15731i
865.2 −1.00000 0 1.00000 0.822876 1.42526i 0 −1.32288 + 2.29129i −1.00000 0 −0.822876 + 1.42526i
919.1 −1.00000 0 1.00000 −1.82288 3.15731i 0 1.32288 + 2.29129i −1.00000 0 1.82288 + 3.15731i
919.2 −1.00000 0 1.00000 0.822876 + 1.42526i 0 −1.32288 2.29129i −1.00000 0 −0.822876 1.42526i
 $$n$$: 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.q 4
3.b odd 2 1 1134.2.e.t 4
7.c even 3 1 1134.2.h.t 4
9.c even 3 1 378.2.g.h yes 4
9.c even 3 1 1134.2.h.t 4
9.d odd 6 1 378.2.g.g 4
9.d odd 6 1 1134.2.h.q 4
21.h odd 6 1 1134.2.h.q 4
63.g even 3 1 378.2.g.h yes 4
63.h even 3 1 inner 1134.2.e.q 4
63.h even 3 1 2646.2.a.bi 2
63.i even 6 1 2646.2.a.bo 2
63.j odd 6 1 1134.2.e.t 4
63.j odd 6 1 2646.2.a.bl 2
63.n odd 6 1 378.2.g.g 4
63.t odd 6 1 2646.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 9.d odd 6 1
378.2.g.g 4 63.n odd 6 1
378.2.g.h yes 4 9.c even 3 1
378.2.g.h yes 4 63.g even 3 1
1134.2.e.q 4 1.a even 1 1 trivial
1134.2.e.q 4 63.h even 3 1 inner
1134.2.e.t 4 3.b odd 2 1
1134.2.e.t 4 63.j odd 6 1
1134.2.h.q 4 9.d odd 6 1
1134.2.h.q 4 21.h odd 6 1
1134.2.h.t 4 7.c even 3 1
1134.2.h.t 4 9.c even 3 1
2646.2.a.bf 2 63.t odd 6 1
2646.2.a.bi 2 63.h even 3 1
2646.2.a.bl 2 63.j odd 6 1
2646.2.a.bo 2 63.i even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 2T_{5}^{3} + 10T_{5}^{2} - 12T_{5} + 36$$ T5^4 + 2*T5^3 + 10*T5^2 - 12*T5 + 36 $$T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36$$ T11^4 - 2*T11^3 + 10*T11^2 + 12*T11 + 36 $$T_{17}^{4} - 2T_{17}^{3} + 10T_{17}^{2} + 12T_{17} + 36$$ T17^4 - 2*T17^3 + 10*T17^2 + 12*T17 + 36 $$T_{23}^{4} - 8T_{23}^{3} + 76T_{23}^{2} + 96T_{23} + 144$$ T23^4 - 8*T23^3 + 76*T23^2 + 96*T23 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$13$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$17$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} - 8 T^{3} + 76 T^{2} + 96 T + 144$$
$29$ $$T^{4} - 10 T^{3} + 82 T^{2} + \cdots + 324$$
$31$ $$(T^{2} + 4 T - 3)^{2}$$
$37$ $$T^{4} - 8 T^{3} + 111 T^{2} + \cdots + 2209$$
$41$ $$T^{4} - 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$43$ $$(T^{2} + 5 T + 25)^{2}$$
$47$ $$(T^{2} - 6 T - 54)^{2}$$
$53$ $$(T^{2} - 6 T + 36)^{2}$$
$59$ $$(T^{2} + 22 T + 114)^{2}$$
$61$ $$(T^{2} - 20 T + 93)^{2}$$
$67$ $$(T^{2} - 6 T - 19)^{2}$$
$71$ $$(T^{2} + 26 T + 162)^{2}$$
$73$ $$T^{4} + 112 T^{2} + 12544$$
$79$ $$(T^{2} + 4 T - 171)^{2}$$
$83$ $$T^{4} + 16 T^{3} + 220 T^{2} + \cdots + 1296$$
$89$ $$T^{4} + 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$97$ $$T^{4} + 6 T^{3} + 139 T^{2} + \cdots + 10609$$