# Properties

 Label 96.3.e.a 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - 3 \beta_{2} - 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + b2 * q^5 + (-b3 + 2*b1) * q^7 + (-3*b2 - 3) * q^9 $$q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - 3 \beta_{2} - 3) q^{9} + (\beta_{3} + 4 \beta_1) q^{11} - 6 q^{13} - \beta_{3} q^{15} + 8 \beta_{2} q^{17} + (\beta_{3} - 2 \beta_1) q^{19} + ( - 9 \beta_{2} + 18) q^{21} + ( - 2 \beta_{3} - 8 \beta_1) q^{23} + 17 q^{25} + (3 \beta_{3} - 3 \beta_1) q^{27} + 11 \beta_{2} q^{29} + (3 \beta_{3} - 6 \beta_1) q^{31} + ( - 9 \beta_{2} - 36) q^{33} + ( - 2 \beta_{3} - 8 \beta_1) q^{35} - 38 q^{37} - 6 \beta_1 q^{39} - 2 \beta_{2} q^{41} + (\beta_{3} - 2 \beta_1) q^{43} + ( - 3 \beta_{2} + 24) q^{45} + (4 \beta_{3} + 16 \beta_1) q^{47} + 59 q^{49} - 8 \beta_{3} q^{51} + 5 \beta_{2} q^{53} + ( - 4 \beta_{3} + 8 \beta_1) q^{55} + (9 \beta_{2} - 18) q^{57} + (\beta_{3} + 4 \beta_1) q^{59} - 22 q^{61} + (9 \beta_{3} + 18 \beta_1) q^{63} - 6 \beta_{2} q^{65} + ( - 11 \beta_{3} + 22 \beta_1) q^{67} + (18 \beta_{2} + 72) q^{69} + (2 \beta_{3} + 8 \beta_1) q^{71} - 30 q^{73} + 17 \beta_1 q^{75} - 54 \beta_{2} q^{77} + (3 \beta_{3} - 6 \beta_1) q^{79} + (18 \beta_{2} - 63) q^{81} + ( - 5 \beta_{3} - 20 \beta_1) q^{83} - 64 q^{85} - 11 \beta_{3} q^{87} - 2 \beta_{2} q^{89} + (6 \beta_{3} - 12 \beta_1) q^{91} + (27 \beta_{2} - 54) q^{93} + (2 \beta_{3} + 8 \beta_1) q^{95} + 90 q^{97} + (9 \beta_{3} - 36 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + b2 * q^5 + (-b3 + 2*b1) * q^7 + (-3*b2 - 3) * q^9 + (b3 + 4*b1) * q^11 - 6 * q^13 - b3 * q^15 + 8*b2 * q^17 + (b3 - 2*b1) * q^19 + (-9*b2 + 18) * q^21 + (-2*b3 - 8*b1) * q^23 + 17 * q^25 + (3*b3 - 3*b1) * q^27 + 11*b2 * q^29 + (3*b3 - 6*b1) * q^31 + (-9*b2 - 36) * q^33 + (-2*b3 - 8*b1) * q^35 - 38 * q^37 - 6*b1 * q^39 - 2*b2 * q^41 + (b3 - 2*b1) * q^43 + (-3*b2 + 24) * q^45 + (4*b3 + 16*b1) * q^47 + 59 * q^49 - 8*b3 * q^51 + 5*b2 * q^53 + (-4*b3 + 8*b1) * q^55 + (9*b2 - 18) * q^57 + (b3 + 4*b1) * q^59 - 22 * q^61 + (9*b3 + 18*b1) * q^63 - 6*b2 * q^65 + (-11*b3 + 22*b1) * q^67 + (18*b2 + 72) * q^69 + (2*b3 + 8*b1) * q^71 - 30 * q^73 + 17*b1 * q^75 - 54*b2 * q^77 + (3*b3 - 6*b1) * q^79 + (18*b2 - 63) * q^81 + (-5*b3 - 20*b1) * q^83 - 64 * q^85 - 11*b3 * q^87 - 2*b2 * q^89 + (6*b3 - 12*b1) * q^91 + (27*b2 - 54) * q^93 + (2*b3 + 8*b1) * q^95 + 90 * q^97 + (9*b3 - 36*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} - 24 q^{13} + 72 q^{21} + 68 q^{25} - 144 q^{33} - 152 q^{37} + 96 q^{45} + 236 q^{49} - 72 q^{57} - 88 q^{61} + 288 q^{69} - 120 q^{73} - 252 q^{81} - 256 q^{85} - 216 q^{93} + 360 q^{97}+O(q^{100})$$ 4 * q - 12 * q^9 - 24 * q^13 + 72 * q^21 + 68 * q^25 - 144 * q^33 - 152 * q^37 + 96 * q^45 + 236 * q^49 - 72 * q^57 - 88 * q^61 + 288 * q^69 - 120 * q^73 - 252 * q^81 - 256 * q^85 - 216 * q^93 + 360 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + \nu^{2} + 5\nu + 2$$ v^3 + v^2 + 5*v + 2 $$\beta_{2}$$ $$=$$ $$-2\nu^{3} - 6\nu$$ -2*v^3 - 6*v $$\beta_{3}$$ $$=$$ $$2\nu^{3} - 4\nu^{2} + 10\nu - 8$$ 2*v^3 - 4*v^2 + 10*v - 8
 $$\nu$$ $$=$$ $$( \beta_{3} + 3\beta_{2} + 4\beta_1 ) / 12$$ (b3 + 3*b2 + 4*b1) / 12 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta _1 - 12 ) / 6$$ (-b3 + 2*b1 - 12) / 6 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} - 5\beta_{2} - 4\beta_1 ) / 4$$ (-b3 - 5*b2 - 4*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
 − 1.93185i 1.93185i − 0.517638i 0.517638i
0 −1.73205 2.44949i 0 2.82843i 0 −10.3923 0 −3.00000 + 8.48528i 0
65.2 0 −1.73205 + 2.44949i 0 2.82843i 0 −10.3923 0 −3.00000 8.48528i 0
65.3 0 1.73205 2.44949i 0 2.82843i 0 10.3923 0 −3.00000 8.48528i 0
65.4 0 1.73205 + 2.44949i 0 2.82843i 0 10.3923 0 −3.00000 + 8.48528i 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.a 4
3.b odd 2 1 inner 96.3.e.a 4
4.b odd 2 1 inner 96.3.e.a 4
8.b even 2 1 192.3.e.e 4
8.d odd 2 1 192.3.e.e 4
12.b even 2 1 inner 96.3.e.a 4
16.e even 4 2 768.3.h.f 8
16.f odd 4 2 768.3.h.f 8
24.f even 2 1 192.3.e.e 4
24.h odd 2 1 192.3.e.e 4
48.i odd 4 2 768.3.h.f 8
48.k even 4 2 768.3.h.f 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8$$ acting on $$S_{3}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 81$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} - 108)^{2}$$
$11$ $$(T^{2} + 216)^{2}$$
$13$ $$(T + 6)^{4}$$
$17$ $$(T^{2} + 512)^{2}$$
$19$ $$(T^{2} - 108)^{2}$$
$23$ $$(T^{2} + 864)^{2}$$
$29$ $$(T^{2} + 968)^{2}$$
$31$ $$(T^{2} - 972)^{2}$$
$37$ $$(T + 38)^{4}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} - 108)^{2}$$
$47$ $$(T^{2} + 3456)^{2}$$
$53$ $$(T^{2} + 200)^{2}$$
$59$ $$(T^{2} + 216)^{2}$$
$61$ $$(T + 22)^{4}$$
$67$ $$(T^{2} - 13068)^{2}$$
$71$ $$(T^{2} + 864)^{2}$$
$73$ $$(T + 30)^{4}$$
$79$ $$(T^{2} - 972)^{2}$$
$83$ $$(T^{2} + 5400)^{2}$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T - 90)^{4}$$