Properties

 Label 96.2.f.a Level $96$ Weight $2$ Character orbit 96.f Analytic conductor $0.767$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,2,Mod(47,96)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("96.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 96.f (of order $$2$$, degree $$1$$, 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: $$0.766563859404$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (2 \beta - 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (2*b - 1) * q^9 $$q + (\beta + 1) q^{3} + (2 \beta - 1) q^{9} - 2 \beta q^{11} - 4 \beta q^{17} - 2 q^{19} - 5 q^{25} + (\beta - 5) q^{27} + ( - 2 \beta + 4) q^{33} + 8 \beta q^{41} + 10 q^{43} + 7 q^{49} + ( - 4 \beta + 8) q^{51} + ( - 2 \beta - 2) q^{57} + 10 \beta q^{59} - 14 q^{67} + 2 q^{73} + ( - 5 \beta - 5) q^{75} + ( - 4 \beta - 7) q^{81} - 2 \beta q^{83} - 4 \beta q^{89} - 10 q^{97} + (2 \beta + 8) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (2*b - 1) * q^9 - 2*b * q^11 - 4*b * q^17 - 2 * q^19 - 5 * q^25 + (b - 5) * q^27 + (-2*b + 4) * q^33 + 8*b * q^41 + 10 * q^43 + 7 * q^49 + (-4*b + 8) * q^51 + (-2*b - 2) * q^57 + 10*b * q^59 - 14 * q^67 + 2 * q^73 + (-5*b - 5) * q^75 + (-4*b - 7) * q^81 - 2*b * q^83 - 4*b * q^89 - 10 * q^97 + (2*b + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^9 $$2 q + 2 q^{3} - 2 q^{9} - 4 q^{19} - 10 q^{25} - 10 q^{27} + 8 q^{33} + 20 q^{43} + 14 q^{49} + 16 q^{51} - 4 q^{57} - 28 q^{67} + 4 q^{73} - 10 q^{75} - 14 q^{81} - 20 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^9 - 4 * q^19 - 10 * q^25 - 10 * q^27 + 8 * q^33 + 20 * q^43 + 14 * q^49 + 16 * q^51 - 4 * q^57 - 28 * q^67 + 4 * q^73 - 10 * q^75 - 14 * q^81 - 20 * q^97 + 16 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 − 1.41421i 1.41421i
0 1.00000 1.41421i 0 0 0 0 0 −1.00000 2.82843i 0
47.2 0 1.00000 + 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
24.f even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.2.f.a 2
3.b odd 2 1 inner 96.2.f.a 2
4.b odd 2 1 24.2.f.a 2
5.b even 2 1 2400.2.b.a 2
5.c odd 4 2 2400.2.m.a 4
8.b even 2 1 24.2.f.a 2
8.d odd 2 1 CM 96.2.f.a 2
9.c even 3 2 2592.2.p.b 4
9.d odd 6 2 2592.2.p.b 4
12.b even 2 1 24.2.f.a 2
15.d odd 2 1 2400.2.b.a 2
15.e even 4 2 2400.2.m.a 4
16.e even 4 2 768.2.c.h 4
16.f odd 4 2 768.2.c.h 4
20.d odd 2 1 600.2.b.a 2
20.e even 4 2 600.2.m.a 4
24.f even 2 1 inner 96.2.f.a 2
24.h odd 2 1 24.2.f.a 2
36.f odd 6 2 648.2.l.b 4
36.h even 6 2 648.2.l.b 4
40.e odd 2 1 2400.2.b.a 2
40.f even 2 1 600.2.b.a 2
40.i odd 4 2 600.2.m.a 4
40.k even 4 2 2400.2.m.a 4
48.i odd 4 2 768.2.c.h 4
48.k even 4 2 768.2.c.h 4
60.h even 2 1 600.2.b.a 2
60.l odd 4 2 600.2.m.a 4
72.j odd 6 2 648.2.l.b 4
72.l even 6 2 2592.2.p.b 4
72.n even 6 2 648.2.l.b 4
72.p odd 6 2 2592.2.p.b 4
120.i odd 2 1 600.2.b.a 2
120.m even 2 1 2400.2.b.a 2
120.q odd 4 2 2400.2.m.a 4
120.w even 4 2 600.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 4.b odd 2 1
24.2.f.a 2 8.b even 2 1
24.2.f.a 2 12.b even 2 1
24.2.f.a 2 24.h odd 2 1
96.2.f.a 2 1.a even 1 1 trivial
96.2.f.a 2 3.b odd 2 1 inner
96.2.f.a 2 8.d odd 2 1 CM
96.2.f.a 2 24.f even 2 1 inner
600.2.b.a 2 20.d odd 2 1
600.2.b.a 2 40.f even 2 1
600.2.b.a 2 60.h even 2 1
600.2.b.a 2 120.i odd 2 1
600.2.m.a 4 20.e even 4 2
600.2.m.a 4 40.i odd 4 2
600.2.m.a 4 60.l odd 4 2
600.2.m.a 4 120.w even 4 2
648.2.l.b 4 36.f odd 6 2
648.2.l.b 4 36.h even 6 2
648.2.l.b 4 72.j odd 6 2
648.2.l.b 4 72.n even 6 2
768.2.c.h 4 16.e even 4 2
768.2.c.h 4 16.f odd 4 2
768.2.c.h 4 48.i odd 4 2
768.2.c.h 4 48.k even 4 2
2400.2.b.a 2 5.b even 2 1
2400.2.b.a 2 15.d odd 2 1
2400.2.b.a 2 40.e odd 2 1
2400.2.b.a 2 120.m even 2 1
2400.2.m.a 4 5.c odd 4 2
2400.2.m.a 4 15.e even 4 2
2400.2.m.a 4 40.k even 4 2
2400.2.m.a 4 120.q odd 4 2
2592.2.p.b 4 9.c even 3 2
2592.2.p.b 4 9.d odd 6 2
2592.2.p.b 4 72.l even 6 2
2592.2.p.b 4 72.p odd 6 2

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(96, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 32$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 128$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 200$$
$61$ $$T^{2}$$
$67$ $$(T + 14)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 8$$
$89$ $$T^{2} + 32$$
$97$ $$(T + 10)^{2}$$