Properties

 Label 2268.2.i.b Level $2268$ Weight $2$ Character orbit 2268.i Analytic conductor $18.110$ 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] = [2268,2,Mod(865,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("2268.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.i (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: $$18.1100711784$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{5} + (2 \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + (2*z - 2) * q^5 + (2*z + 1) * q^7 $$q + (2 \zeta_{6} - 2) q^{5} + (2 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{11} + 3 \zeta_{6} q^{13} + ( - 8 \zeta_{6} + 8) q^{17} + \zeta_{6} q^{19} + ( - 8 \zeta_{6} + 8) q^{23} + \zeta_{6} q^{25} + ( - 4 \zeta_{6} + 4) q^{29} + 3 q^{31} + (2 \zeta_{6} - 6) q^{35} + \zeta_{6} q^{37} + 6 \zeta_{6} q^{41} + (11 \zeta_{6} - 11) q^{43} - 6 q^{47} + (8 \zeta_{6} - 3) q^{49} + (12 \zeta_{6} - 12) q^{53} - 4 q^{55} - 4 q^{59} - 6 q^{61} - 6 q^{65} + 13 q^{67} + 10 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + (6 \zeta_{6} - 4) q^{77} - 3 q^{79} + ( - 2 \zeta_{6} + 2) q^{83} + 16 \zeta_{6} q^{85} + (9 \zeta_{6} - 6) q^{91} - 2 q^{95} + (10 \zeta_{6} - 10) q^{97}+O(q^{100})$$ q + (2*z - 2) * q^5 + (2*z + 1) * q^7 + 2*z * q^11 + 3*z * q^13 + (-8*z + 8) * q^17 + z * q^19 + (-8*z + 8) * q^23 + z * q^25 + (-4*z + 4) * q^29 + 3 * q^31 + (2*z - 6) * q^35 + z * q^37 + 6*z * q^41 + (11*z - 11) * q^43 - 6 * q^47 + (8*z - 3) * q^49 + (12*z - 12) * q^53 - 4 * q^55 - 4 * q^59 - 6 * q^61 - 6 * q^65 + 13 * q^67 + 10 * q^71 + (-11*z + 11) * q^73 + (6*z - 4) * q^77 - 3 * q^79 + (-2*z + 2) * q^83 + 16*z * q^85 + (9*z - 6) * q^91 - 2 * q^95 + (10*z - 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 4 * q^7 $$2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} + 3 q^{13} + 8 q^{17} + q^{19} + 8 q^{23} + q^{25} + 4 q^{29} + 6 q^{31} - 10 q^{35} + q^{37} + 6 q^{41} - 11 q^{43} - 12 q^{47} + 2 q^{49} - 12 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{61} - 12 q^{65} + 26 q^{67} + 20 q^{71} + 11 q^{73} - 2 q^{77} - 6 q^{79} + 2 q^{83} + 16 q^{85} - 3 q^{91} - 4 q^{95} - 10 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 4 * q^7 + 2 * q^11 + 3 * q^13 + 8 * q^17 + q^19 + 8 * q^23 + q^25 + 4 * q^29 + 6 * q^31 - 10 * q^35 + q^37 + 6 * q^41 - 11 * q^43 - 12 * q^47 + 2 * q^49 - 12 * q^53 - 8 * q^55 - 8 * q^59 - 12 * q^61 - 12 * q^65 + 26 * q^67 + 20 * q^71 + 11 * q^73 - 2 * q^77 - 6 * q^79 + 2 * q^83 + 16 * q^85 - 3 * q^91 - 4 * q^95 - 10 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$-\zeta_{6}$$

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
0 0 0 −1.00000 + 1.73205i 0 2.00000 + 1.73205i 0 0 0
2053.1 0 0 0 −1.00000 1.73205i 0 2.00000 1.73205i 0 0 0
 $$n$$: 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 2268.2.i.b 2
3.b odd 2 1 2268.2.i.g 2
7.c even 3 1 2268.2.l.g 2
9.c even 3 1 252.2.k.a 2
9.c even 3 1 2268.2.l.g 2
9.d odd 6 1 84.2.i.a 2
9.d odd 6 1 2268.2.l.b 2
21.h odd 6 1 2268.2.l.b 2
36.f odd 6 1 1008.2.s.c 2
36.h even 6 1 336.2.q.c 2
45.h odd 6 1 2100.2.q.b 2
45.l even 12 2 2100.2.bc.a 4
63.g even 3 1 252.2.k.a 2
63.h even 3 1 1764.2.a.h 1
63.h even 3 1 inner 2268.2.i.b 2
63.i even 6 1 588.2.a.f 1
63.j odd 6 1 588.2.a.a 1
63.j odd 6 1 2268.2.i.g 2
63.k odd 6 1 1764.2.k.j 2
63.l odd 6 1 1764.2.k.j 2
63.n odd 6 1 84.2.i.a 2
63.o even 6 1 588.2.i.b 2
63.s even 6 1 588.2.i.b 2
63.t odd 6 1 1764.2.a.c 1
72.j odd 6 1 1344.2.q.b 2
72.l even 6 1 1344.2.q.n 2
252.o even 6 1 336.2.q.c 2
252.r odd 6 1 2352.2.a.k 1
252.s odd 6 1 2352.2.q.q 2
252.u odd 6 1 7056.2.a.bs 1
252.bb even 6 1 2352.2.a.o 1
252.bj even 6 1 7056.2.a.o 1
252.bl odd 6 1 1008.2.s.c 2
252.bn odd 6 1 2352.2.q.q 2
315.v odd 6 1 2100.2.q.b 2
315.bx even 12 2 2100.2.bc.a 4
504.bi odd 6 1 9408.2.a.cx 1
504.bt even 6 1 9408.2.a.bi 1
504.ca even 6 1 9408.2.a.i 1
504.cm odd 6 1 9408.2.a.bx 1
504.cy even 6 1 1344.2.q.n 2
504.db odd 6 1 1344.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 9.d odd 6 1
84.2.i.a 2 63.n odd 6 1
252.2.k.a 2 9.c even 3 1
252.2.k.a 2 63.g even 3 1
336.2.q.c 2 36.h even 6 1
336.2.q.c 2 252.o even 6 1
588.2.a.a 1 63.j odd 6 1
588.2.a.f 1 63.i even 6 1
588.2.i.b 2 63.o even 6 1
588.2.i.b 2 63.s even 6 1
1008.2.s.c 2 36.f odd 6 1
1008.2.s.c 2 252.bl odd 6 1
1344.2.q.b 2 72.j odd 6 1
1344.2.q.b 2 504.db odd 6 1
1344.2.q.n 2 72.l even 6 1
1344.2.q.n 2 504.cy even 6 1
1764.2.a.c 1 63.t odd 6 1
1764.2.a.h 1 63.h even 3 1
1764.2.k.j 2 63.k odd 6 1
1764.2.k.j 2 63.l odd 6 1
2100.2.q.b 2 45.h odd 6 1
2100.2.q.b 2 315.v odd 6 1
2100.2.bc.a 4 45.l even 12 2
2100.2.bc.a 4 315.bx even 12 2
2268.2.i.b 2 1.a even 1 1 trivial
2268.2.i.b 2 63.h even 3 1 inner
2268.2.i.g 2 3.b odd 2 1
2268.2.i.g 2 63.j odd 6 1
2268.2.l.b 2 9.d odd 6 1
2268.2.l.b 2 21.h odd 6 1
2268.2.l.g 2 7.c even 3 1
2268.2.l.g 2 9.c even 3 1
2352.2.a.k 1 252.r odd 6 1
2352.2.a.o 1 252.bb even 6 1
2352.2.q.q 2 252.s odd 6 1
2352.2.q.q 2 252.bn odd 6 1
7056.2.a.o 1 252.bj even 6 1
7056.2.a.bs 1 252.u odd 6 1
9408.2.a.i 1 504.ca even 6 1
9408.2.a.bi 1 504.bt even 6 1
9408.2.a.bx 1 504.cm odd 6 1
9408.2.a.cx 1 504.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{13}^{2} - 3T_{13} + 9$$ T13^2 - 3*T13 + 9 $$T_{19}^{2} - T_{19} + 1$$ T19^2 - T19 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} - 3T + 9$$
$17$ $$T^{2} - 8T + 64$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$T^{2} - 4T + 16$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$(T - 13)^{2}$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$(T + 3)^{2}$$
$83$ $$T^{2} - 2T + 4$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 10T + 100$$
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