# Properties

 Label 1134.2.e.h Level $1134$ Weight $2$ Character orbit 1134.e Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + ( - \zeta_{6} - 2) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (-4*z + 4) * q^5 + (-z - 2) * q^7 - q^8 $$q - q^{2} + q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + ( - \zeta_{6} - 2) q^{7} - q^{8} + (4 \zeta_{6} - 4) q^{10} + 2 \zeta_{6} q^{11} - 6 \zeta_{6} q^{13} + (\zeta_{6} + 2) q^{14} + q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + 4 \zeta_{6} q^{19} + ( - 4 \zeta_{6} + 4) q^{20} - 2 \zeta_{6} q^{22} + (\zeta_{6} - 1) q^{23} - 11 \zeta_{6} q^{25} + 6 \zeta_{6} q^{26} + ( - \zeta_{6} - 2) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} - 9 q^{31} - q^{32} + (2 \zeta_{6} - 2) q^{34} + (8 \zeta_{6} - 12) q^{35} - 8 \zeta_{6} q^{37} - 4 \zeta_{6} q^{38} + (4 \zeta_{6} - 4) q^{40} + 3 \zeta_{6} q^{41} + (2 \zeta_{6} - 2) q^{43} + 2 \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{46} + 9 q^{47} + (5 \zeta_{6} + 3) q^{49} + 11 \zeta_{6} q^{50} - 6 \zeta_{6} q^{52} + (12 \zeta_{6} - 12) q^{53} + 8 q^{55} + (\zeta_{6} + 2) q^{56} + (4 \zeta_{6} - 4) q^{58} - 4 q^{59} + 6 q^{61} + 9 q^{62} + q^{64} - 24 q^{65} - 14 q^{67} + ( - 2 \zeta_{6} + 2) q^{68} + ( - 8 \zeta_{6} + 12) q^{70} + q^{71} + (7 \zeta_{6} - 7) q^{73} + 8 \zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + ( - 6 \zeta_{6} + 2) q^{77} - 3 q^{79} + ( - 4 \zeta_{6} + 4) q^{80} - 3 \zeta_{6} q^{82} + ( - 14 \zeta_{6} + 14) q^{83} - 8 \zeta_{6} q^{85} + ( - 2 \zeta_{6} + 2) q^{86} - 2 \zeta_{6} q^{88} - 3 \zeta_{6} q^{89} + (18 \zeta_{6} - 6) q^{91} + (\zeta_{6} - 1) q^{92} - 9 q^{94} + 16 q^{95} + (10 \zeta_{6} - 10) q^{97} + ( - 5 \zeta_{6} - 3) q^{98} +O(q^{100})$$ q - q^2 + q^4 + (-4*z + 4) * q^5 + (-z - 2) * q^7 - q^8 + (4*z - 4) * q^10 + 2*z * q^11 - 6*z * q^13 + (z + 2) * q^14 + q^16 + (-2*z + 2) * q^17 + 4*z * q^19 + (-4*z + 4) * q^20 - 2*z * q^22 + (z - 1) * q^23 - 11*z * q^25 + 6*z * q^26 + (-z - 2) * q^28 + (-4*z + 4) * q^29 - 9 * q^31 - q^32 + (2*z - 2) * q^34 + (8*z - 12) * q^35 - 8*z * q^37 - 4*z * q^38 + (4*z - 4) * q^40 + 3*z * q^41 + (2*z - 2) * q^43 + 2*z * q^44 + (-z + 1) * q^46 + 9 * q^47 + (5*z + 3) * q^49 + 11*z * q^50 - 6*z * q^52 + (12*z - 12) * q^53 + 8 * q^55 + (z + 2) * q^56 + (4*z - 4) * q^58 - 4 * q^59 + 6 * q^61 + 9 * q^62 + q^64 - 24 * q^65 - 14 * q^67 + (-2*z + 2) * q^68 + (-8*z + 12) * q^70 + q^71 + (7*z - 7) * q^73 + 8*z * q^74 + 4*z * q^76 + (-6*z + 2) * q^77 - 3 * q^79 + (-4*z + 4) * q^80 - 3*z * q^82 + (-14*z + 14) * q^83 - 8*z * q^85 + (-2*z + 2) * q^86 - 2*z * q^88 - 3*z * q^89 + (18*z - 6) * q^91 + (z - 1) * q^92 - 9 * q^94 + 16 * q^95 + (10*z - 10) * q^97 + (-5*z - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 5 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 5 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 5 q^{7} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 6 q^{13} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} - 2 q^{22} - q^{23} - 11 q^{25} + 6 q^{26} - 5 q^{28} + 4 q^{29} - 18 q^{31} - 2 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} - 4 q^{38} - 4 q^{40} + 3 q^{41} - 2 q^{43} + 2 q^{44} + q^{46} + 18 q^{47} + 11 q^{49} + 11 q^{50} - 6 q^{52} - 12 q^{53} + 16 q^{55} + 5 q^{56} - 4 q^{58} - 8 q^{59} + 12 q^{61} + 18 q^{62} + 2 q^{64} - 48 q^{65} - 28 q^{67} + 2 q^{68} + 16 q^{70} + 2 q^{71} - 7 q^{73} + 8 q^{74} + 4 q^{76} - 2 q^{77} - 6 q^{79} + 4 q^{80} - 3 q^{82} + 14 q^{83} - 8 q^{85} + 2 q^{86} - 2 q^{88} - 3 q^{89} + 6 q^{91} - q^{92} - 18 q^{94} + 32 q^{95} - 10 q^{97} - 11 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 5 * q^7 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 6 * q^13 + 5 * q^14 + 2 * q^16 + 2 * q^17 + 4 * q^19 + 4 * q^20 - 2 * q^22 - q^23 - 11 * q^25 + 6 * q^26 - 5 * q^28 + 4 * q^29 - 18 * q^31 - 2 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 - 4 * q^38 - 4 * q^40 + 3 * q^41 - 2 * q^43 + 2 * q^44 + q^46 + 18 * q^47 + 11 * q^49 + 11 * q^50 - 6 * q^52 - 12 * q^53 + 16 * q^55 + 5 * q^56 - 4 * q^58 - 8 * q^59 + 12 * q^61 + 18 * q^62 + 2 * q^64 - 48 * q^65 - 28 * q^67 + 2 * q^68 + 16 * q^70 + 2 * q^71 - 7 * q^73 + 8 * q^74 + 4 * q^76 - 2 * q^77 - 6 * q^79 + 4 * q^80 - 3 * q^82 + 14 * q^83 - 8 * q^85 + 2 * q^86 - 2 * q^88 - 3 * q^89 + 6 * q^91 - q^92 - 18 * q^94 + 32 * q^95 - 10 * q^97 - 11 * 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 2.00000 3.46410i 0 −2.50000 0.866025i −1.00000 0 −2.00000 + 3.46410i
919.1 −1.00000 0 1.00000 2.00000 + 3.46410i 0 −2.50000 + 0.866025i −1.00000 0 −2.00000 3.46410i
 $$n$$: 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.h 2
3.b odd 2 1 1134.2.e.i 2
7.c even 3 1 1134.2.h.i 2
9.c even 3 1 1134.2.g.h yes 2
9.c even 3 1 1134.2.h.i 2
9.d odd 6 1 1134.2.g.a 2
9.d odd 6 1 1134.2.h.h 2
21.h odd 6 1 1134.2.h.h 2
63.g even 3 1 1134.2.g.h yes 2
63.h even 3 1 inner 1134.2.e.h 2
63.h even 3 1 7938.2.a.a 1
63.i even 6 1 7938.2.a.q 1
63.j odd 6 1 1134.2.e.i 2
63.j odd 6 1 7938.2.a.bf 1
63.n odd 6 1 1134.2.g.a 2
63.t odd 6 1 7938.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.h 2 1.a even 1 1 trivial
1134.2.e.h 2 63.h even 3 1 inner
1134.2.e.i 2 3.b odd 2 1
1134.2.e.i 2 63.j odd 6 1
1134.2.g.a 2 9.d odd 6 1
1134.2.g.a 2 63.n odd 6 1
1134.2.g.h yes 2 9.c even 3 1
1134.2.g.h yes 2 63.g even 3 1
1134.2.h.h 2 9.d odd 6 1
1134.2.h.h 2 21.h odd 6 1
1134.2.h.i 2 7.c even 3 1
1134.2.h.i 2 9.c even 3 1
7938.2.a.a 1 63.h even 3 1
7938.2.a.p 1 63.t odd 6 1
7938.2.a.q 1 63.i even 6 1
7938.2.a.bf 1 63.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 4T_{5} + 16$$ T5^2 - 4*T5 + 16 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4 $$T_{23}^{2} + T_{23} + 1$$ T23^2 + T23 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} - 4T + 16$$
$31$ $$(T + 9)^{2}$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$(T - 9)^{2}$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$(T + 14)^{2}$$
$71$ $$(T - 1)^{2}$$
$73$ $$T^{2} + 7T + 49$$
$79$ $$(T + 3)^{2}$$
$83$ $$T^{2} - 14T + 196$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} + 10T + 100$$
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