Properties

 Label 1152.2.k.e Level $1152$ Weight $2$ Character orbit 1152.k Analytic conductor $9.199$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("1152.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.k (of order $$4$$, 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: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16$$ x^8 - 2*x^6 + 2*x^4 - 8*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + (\beta_{7} - \beta_{3}) q^{7}+O(q^{10})$$ q + b2 * q^5 + (b7 - b3) * q^7 $$q + \beta_{2} q^{5} + (\beta_{7} - \beta_{3}) q^{7} + (\beta_{2} + \beta_1) q^{11} + (\beta_{7} - \beta_{6}) q^{13} + ( - \beta_{5} - \beta_1) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{3} + 1) q^{19} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{23}+ \cdots - 4 \beta_{6} q^{97}+O(q^{100})$$ q + b2 * q^5 + (b7 - b3) * q^7 + (b2 + b1) * q^11 + (b7 - b6) * q^13 + (-b5 - b1) * q^17 + (-b7 + b6 - b3 + 1) * q^19 + (-b5 + b4 - b2 + b1) * q^23 + (-2*b7 + b3) * q^25 + (-2*b5 - b4) * q^29 + (-b6 - 3) * q^31 + (-b5 + 3*b4) * q^35 + (b7 + b6 - 2*b3 - 2) * q^37 + (-b5 - 2*b4 + 2*b2 + b1) * q^41 + (-b7 - b6 + 3*b3 + 3) * q^43 + (b5 + 3*b4 + 3*b2 + b1) * q^47 + (2*b6 - 1) * q^49 + (b2 - 2*b1) * q^53 + 4*b3 * q^55 + (-2*b2 + 2*b1) * q^59 + (-b7 + b6 + 2*b3 - 2) * q^61 + (-b5 + 2*b4 + 2*b2 - b1) * q^65 + (4*b3 - 4) * q^67 + (-2*b5 - 2*b4 + 2*b2 + 2*b1) * q^71 + (2*b7 - 2*b3) * q^73 - 2*b5 * q^77 + (b6 + 7) * q^79 + (-b5 - b4) * q^83 + (-2*b7 - 2*b6 + 2*b3 + 2) * q^85 + (-2*b4 + 2*b2) * q^89 + (b7 + b6 - 7*b3 - 7) * q^91 + (b5 - b4 - b2 + b1) * q^95 - 4*b6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{19} - 24 q^{31} - 16 q^{37} + 24 q^{43} - 8 q^{49} - 16 q^{61} - 32 q^{67} + 56 q^{79} + 16 q^{85} - 56 q^{91}+O(q^{100})$$ 8 * q + 8 * q^19 - 24 * q^31 - 16 * q^37 + 24 * q^43 - 8 * q^49 - 16 * q^61 - 32 * q^67 + 56 * q^79 + 16 * q^85 - 56 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 2\nu ) / 2$$ (v^5 + 2*v) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 4\nu^{5} + 10\nu^{3} - 4\nu ) / 12$$ (v^7 - 4*v^5 + 10*v^3 - 4*v) / 12 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 12$$ (v^6 + 2*v^4 - 2*v^2 - 4) / 12 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + \nu^{5} + 2\nu^{3} - 2\nu ) / 6$$ (-v^7 + v^5 + 2*v^3 - 2*v) / 6 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} - 2\nu^{3} + 12\nu ) / 4$$ (-v^7 - 2*v^3 + 12*v) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 2\nu^{4} + 2\nu^{2} + 4 ) / 4$$ (-v^6 + 2*v^4 + 2*v^2 + 4) / 4 $$\beta_{7}$$ $$=$$ $$( \nu^{6} - \nu^{4} + 4\nu^{2} - 7 ) / 3$$ (v^6 - v^4 + 4*v^2 - 7) / 3
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 ) / 4$$ (b5 - b4 + b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{3} + 1 ) / 2$$ (b7 + b6 - b3 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{4} + 2\beta_{2} + \beta_1 ) / 2$$ (b4 + 2*b2 + b1) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{6} + 3\beta_{3}$$ b6 + 3*b3 $$\nu^{5}$$ $$=$$ $$( -\beta_{5} + \beta_{4} - \beta_{2} + 3\beta_1 ) / 2$$ (-b5 + b4 - b2 + 3*b1) / 2 $$\nu^{6}$$ $$=$$ $$\beta_{7} - \beta_{6} + 5\beta_{3} + 5$$ b7 - b6 + 5*b3 + 5 $$\nu^{7}$$ $$=$$ $$-\beta_{5} - 4\beta_{4} + \beta_{2} + 2\beta_1$$ -b5 - 4*b4 + b2 + 2*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.767178 + 1.18804i −1.38255 + 0.297594i 1.38255 − 0.297594i −0.767178 − 1.18804i 0.767178 − 1.18804i −1.38255 − 0.297594i 1.38255 + 0.297594i −0.767178 + 1.18804i
0 0 0 −2.37608 + 2.37608i 0 3.64575i 0 0 0
289.2 0 0 0 −0.595188 + 0.595188i 0 1.64575i 0 0 0
289.3 0 0 0 0.595188 0.595188i 0 1.64575i 0 0 0
289.4 0 0 0 2.37608 2.37608i 0 3.64575i 0 0 0
865.1 0 0 0 −2.37608 2.37608i 0 3.64575i 0 0 0
865.2 0 0 0 −0.595188 0.595188i 0 1.64575i 0 0 0
865.3 0 0 0 0.595188 + 0.595188i 0 1.64575i 0 0 0
865.4 0 0 0 2.37608 + 2.37608i 0 3.64575i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.k.e 8
3.b odd 2 1 inner 1152.2.k.e 8
4.b odd 2 1 1152.2.k.d 8
8.b even 2 1 144.2.k.c 8
8.d odd 2 1 576.2.k.c 8
12.b even 2 1 1152.2.k.d 8
16.e even 4 1 144.2.k.c 8
16.e even 4 1 inner 1152.2.k.e 8
16.f odd 4 1 576.2.k.c 8
16.f odd 4 1 1152.2.k.d 8
24.f even 2 1 576.2.k.c 8
24.h odd 2 1 144.2.k.c 8
32.g even 8 2 9216.2.a.bq 8
32.h odd 8 2 9216.2.a.bt 8
48.i odd 4 1 144.2.k.c 8
48.i odd 4 1 inner 1152.2.k.e 8
48.k even 4 1 576.2.k.c 8
48.k even 4 1 1152.2.k.d 8
96.o even 8 2 9216.2.a.bt 8
96.p odd 8 2 9216.2.a.bq 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 8.b even 2 1
144.2.k.c 8 16.e even 4 1
144.2.k.c 8 24.h odd 2 1
144.2.k.c 8 48.i odd 4 1
576.2.k.c 8 8.d odd 2 1
576.2.k.c 8 16.f odd 4 1
576.2.k.c 8 24.f even 2 1
576.2.k.c 8 48.k even 4 1
1152.2.k.d 8 4.b odd 2 1
1152.2.k.d 8 12.b even 2 1
1152.2.k.d 8 16.f odd 4 1
1152.2.k.d 8 48.k even 4 1
1152.2.k.e 8 1.a even 1 1 trivial
1152.2.k.e 8 3.b odd 2 1 inner
1152.2.k.e 8 16.e even 4 1 inner
1152.2.k.e 8 48.i odd 4 1 inner
9216.2.a.bq 8 32.g even 8 2
9216.2.a.bq 8 96.p odd 8 2
9216.2.a.bt 8 32.h odd 8 2
9216.2.a.bt 8 96.o even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{8} + 128T_{5}^{4} + 64$$ T5^8 + 128*T5^4 + 64 $$T_{11}^{8} + 512T_{11}^{4} + 1024$$ T11^8 + 512*T11^4 + 1024 $$T_{19}^{4} - 4T_{19}^{3} + 8T_{19}^{2} + 48T_{19} + 144$$ T19^4 - 4*T19^3 + 8*T19^2 + 48*T19 + 144

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 128T^{4} + 64$$
$7$ $$(T^{4} + 16 T^{2} + 36)^{2}$$
$11$ $$T^{8} + 512T^{4} + 1024$$
$13$ $$(T^{4} + 196)^{2}$$
$17$ $$(T^{4} - 40 T^{2} + 288)^{2}$$
$19$ $$(T^{4} - 4 T^{3} + \cdots + 144)^{2}$$
$23$ $$(T^{4} + 80 T^{2} + 1152)^{2}$$
$29$ $$T^{8} + 5632 T^{4} + 5184$$
$31$ $$(T^{2} + 6 T + 2)^{4}$$
$37$ $$(T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 36)^{2}$$
$41$ $$(T^{4} + 104 T^{2} + 2592)^{2}$$
$43$ $$(T^{4} - 12 T^{3} + \cdots + 16)^{2}$$
$47$ $$(T^{4} - 208 T^{2} + 10368)^{2}$$
$53$ $$T^{8} + 5888 T^{4} + 8340544$$
$59$ $$T^{8} + 16384 T^{4} + 21233664$$
$61$ $$(T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 36)^{2}$$
$67$ $$(T^{2} + 8 T + 32)^{4}$$
$71$ $$(T^{4} + 192 T^{2} + 2048)^{2}$$
$73$ $$(T^{4} + 64 T^{2} + 576)^{2}$$
$79$ $$(T^{2} - 14 T + 42)^{4}$$
$83$ $$T^{8} + 512T^{4} + 1024$$
$89$ $$(T^{4} + 96 T^{2} + 512)^{2}$$
$97$ $$(T^{2} - 112)^{4}$$
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