# Properties

 Label 96.3.e.b Level $96$ Weight $3$ Character orbit 96.e Analytic conductor $2.616$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,3,Mod(65,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("96.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 96.e (of order $$2$$, degree $$1$$, 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: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 8x^{2} + 9$$ x^4 + 8*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{3} - \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_1) q^{7} + ( - \beta_{2} + 5) q^{9}+O(q^{10})$$ q + (b3 + b1) * q^3 - b2 * q^5 + (-2*b3 - b1) * q^7 + (-b2 + 5) * q^9 $$q + (\beta_{3} + \beta_1) q^{3} - \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_1) q^{7} + ( - \beta_{2} + 5) q^{9} - 5 \beta_1 q^{11} + 10 q^{13} + ( - 4 \beta_{3} + 5 \beta_1) q^{15} + (10 \beta_{3} + 5 \beta_1) q^{19} + (\beta_{2} - 14) q^{21} - 6 \beta_1 q^{23} - 31 q^{25} + (\beta_{3} + 10 \beta_1) q^{27} + 5 \beta_{2} q^{29} + ( - 10 \beta_{3} - 5 \beta_1) q^{31} + (5 \beta_{2} + 20) q^{33} - 14 \beta_1 q^{35} + 10 q^{37} + (10 \beta_{3} + 10 \beta_1) q^{39} + 2 \beta_{2} q^{41} + ( - 22 \beta_{3} - 11 \beta_1) q^{43} + ( - 5 \beta_{2} - 56) q^{45} - 4 \beta_1 q^{47} - 21 q^{49} - 5 \beta_{2} q^{53} + (40 \beta_{3} + 20 \beta_1) q^{55} + ( - 5 \beta_{2} + 70) q^{57} + 35 \beta_1 q^{59} + 90 q^{61} + ( - 10 \beta_{3} - 19 \beta_1) q^{63} - 10 \beta_{2} q^{65} + (2 \beta_{3} + \beta_1) q^{67} + (6 \beta_{2} + 24) q^{69} - 10 \beta_1 q^{71} - 30 q^{73} + ( - 31 \beta_{3} - 31 \beta_1) q^{75} - 10 \beta_{2} q^{77} + ( - 10 \beta_{3} - 5 \beta_1) q^{79} + ( - 10 \beta_{2} - 31) q^{81} + 9 \beta_1 q^{83} + (20 \beta_{3} - 25 \beta_1) q^{87} + 10 \beta_{2} q^{89} + ( - 20 \beta_{3} - 10 \beta_1) q^{91} + (5 \beta_{2} - 70) q^{93} + 70 \beta_1 q^{95} + 10 q^{97} + (40 \beta_{3} - 5 \beta_1) q^{99}+O(q^{100})$$ q + (b3 + b1) * q^3 - b2 * q^5 + (-2*b3 - b1) * q^7 + (-b2 + 5) * q^9 - 5*b1 * q^11 + 10 * q^13 + (-4*b3 + 5*b1) * q^15 + (10*b3 + 5*b1) * q^19 + (b2 - 14) * q^21 - 6*b1 * q^23 - 31 * q^25 + (b3 + 10*b1) * q^27 + 5*b2 * q^29 + (-10*b3 - 5*b1) * q^31 + (5*b2 + 20) * q^33 - 14*b1 * q^35 + 10 * q^37 + (10*b3 + 10*b1) * q^39 + 2*b2 * q^41 + (-22*b3 - 11*b1) * q^43 + (-5*b2 - 56) * q^45 - 4*b1 * q^47 - 21 * q^49 - 5*b2 * q^53 + (40*b3 + 20*b1) * q^55 + (-5*b2 + 70) * q^57 + 35*b1 * q^59 + 90 * q^61 + (-10*b3 - 19*b1) * q^63 - 10*b2 * q^65 + (2*b3 + b1) * q^67 + (6*b2 + 24) * q^69 - 10*b1 * q^71 - 30 * q^73 + (-31*b3 - 31*b1) * q^75 - 10*b2 * q^77 + (-10*b3 - 5*b1) * q^79 + (-10*b2 - 31) * q^81 + 9*b1 * q^83 + (20*b3 - 25*b1) * q^87 + 10*b2 * q^89 + (-20*b3 - 10*b1) * q^91 + (5*b2 - 70) * q^93 + 70*b1 * q^95 + 10 * q^97 + (40*b3 - 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 20 q^{9}+O(q^{10})$$ 4 * q + 20 * q^9 $$4 q + 20 q^{9} + 40 q^{13} - 56 q^{21} - 124 q^{25} + 80 q^{33} + 40 q^{37} - 224 q^{45} - 84 q^{49} + 280 q^{57} + 360 q^{61} + 96 q^{69} - 120 q^{73} - 124 q^{81} - 280 q^{93} + 40 q^{97}+O(q^{100})$$ 4 * q + 20 * q^9 + 40 * q^13 - 56 * q^21 - 124 * q^25 + 80 * q^33 + 40 * q^37 - 224 * q^45 - 84 * q^49 + 280 * q^57 + 360 * q^61 + 96 * q^69 - 120 * q^73 - 124 * q^81 - 280 * q^93 + 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} - 10\nu ) / 3$$ (-2*v^3 - 10*v) / 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 22\nu ) / 3$$ (2*v^3 + 22*v) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 3\nu^{2} + 5\nu + 12 ) / 3$$ (v^3 + 3*v^2 + 5*v + 12) / 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 4$$ (b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta _1 - 8 ) / 2$$ (2*b3 + b1 - 8) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{2} - 11\beta_1 ) / 4$$ (-5*b2 - 11*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 2.57794i 2.57794i 1.16372i − 1.16372i
0 −2.64575 1.41421i 0 7.48331i 0 5.29150 0 5.00000 + 7.48331i 0
65.2 0 −2.64575 + 1.41421i 0 7.48331i 0 5.29150 0 5.00000 7.48331i 0
65.3 0 2.64575 1.41421i 0 7.48331i 0 −5.29150 0 5.00000 7.48331i 0
65.4 0 2.64575 + 1.41421i 0 7.48331i 0 −5.29150 0 5.00000 + 7.48331i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.e.b 4
3.b odd 2 1 inner 96.3.e.b 4
4.b odd 2 1 inner 96.3.e.b 4
8.b even 2 1 192.3.e.f 4
8.d odd 2 1 192.3.e.f 4
12.b even 2 1 inner 96.3.e.b 4
16.e even 4 2 768.3.h.e 8
16.f odd 4 2 768.3.h.e 8
24.f even 2 1 192.3.e.f 4
24.h odd 2 1 192.3.e.f 4
48.i odd 4 2 768.3.h.e 8
48.k even 4 2 768.3.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.b 4 1.a even 1 1 trivial
96.3.e.b 4 3.b odd 2 1 inner
96.3.e.b 4 4.b odd 2 1 inner
96.3.e.b 4 12.b even 2 1 inner
192.3.e.f 4 8.b even 2 1
192.3.e.f 4 8.d odd 2 1
192.3.e.f 4 24.f even 2 1
192.3.e.f 4 24.h odd 2 1
768.3.h.e 8 16.e even 4 2
768.3.h.e 8 16.f odd 4 2
768.3.h.e 8 48.i odd 4 2
768.3.h.e 8 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 56$$ acting on $$S_{3}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 10T^{2} + 81$$
$5$ $$(T^{2} + 56)^{2}$$
$7$ $$(T^{2} - 28)^{2}$$
$11$ $$(T^{2} + 200)^{2}$$
$13$ $$(T - 10)^{4}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 700)^{2}$$
$23$ $$(T^{2} + 288)^{2}$$
$29$ $$(T^{2} + 1400)^{2}$$
$31$ $$(T^{2} - 700)^{2}$$
$37$ $$(T - 10)^{4}$$
$41$ $$(T^{2} + 224)^{2}$$
$43$ $$(T^{2} - 3388)^{2}$$
$47$ $$(T^{2} + 128)^{2}$$
$53$ $$(T^{2} + 1400)^{2}$$
$59$ $$(T^{2} + 9800)^{2}$$
$61$ $$(T - 90)^{4}$$
$67$ $$(T^{2} - 28)^{2}$$
$71$ $$(T^{2} + 800)^{2}$$
$73$ $$(T + 30)^{4}$$
$79$ $$(T^{2} - 700)^{2}$$
$83$ $$(T^{2} + 648)^{2}$$
$89$ $$(T^{2} + 5600)^{2}$$
$97$ $$(T - 10)^{4}$$