# Properties

 Label 48.3.e.b Level $48$ Weight $3$ Character orbit 48.e Analytic conductor $1.308$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,3,Mod(17,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (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: $$1.30790526893$$ 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: $$2$$ Twist minimal: no (minimal twist has level 24) 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 $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + 2 \beta q^{5} + 6 q^{7} + ( - 2 \beta - 7) q^{9}+O(q^{10})$$ q + (b - 1) * q^3 + 2*b * q^5 + 6 * q^7 + (-2*b - 7) * q^9 $$q + (\beta - 1) q^{3} + 2 \beta q^{5} + 6 q^{7} + ( - 2 \beta - 7) q^{9} - 2 \beta q^{11} + 10 q^{13} + ( - 2 \beta - 16) q^{15} - 8 \beta q^{17} - 2 q^{19} + (6 \beta - 6) q^{21} + 4 \beta q^{23} - 7 q^{25} + ( - 5 \beta + 23) q^{27} + 6 \beta q^{29} + 22 q^{31} + (2 \beta + 16) q^{33} + 12 \beta q^{35} - 6 q^{37} + (10 \beta - 10) q^{39} + 12 \beta q^{41} - 82 q^{43} + ( - 14 \beta + 32) q^{45} - 24 \beta q^{47} - 13 q^{49} + (8 \beta + 64) q^{51} - 22 \beta q^{53} + 32 q^{55} + ( - 2 \beta + 2) q^{57} - 26 \beta q^{59} - 86 q^{61} + ( - 12 \beta - 42) q^{63} + 20 \beta q^{65} - 2 q^{67} + ( - 4 \beta - 32) q^{69} + 44 \beta q^{71} + 82 q^{73} + ( - 7 \beta + 7) q^{75} - 12 \beta q^{77} - 10 q^{79} + (28 \beta + 17) q^{81} + 26 \beta q^{83} + 128 q^{85} + ( - 6 \beta - 48) q^{87} - 12 \beta q^{89} + 60 q^{91} + (22 \beta - 22) q^{93} - 4 \beta q^{95} - 94 q^{97} + (14 \beta - 32) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + 2*b * q^5 + 6 * q^7 + (-2*b - 7) * q^9 - 2*b * q^11 + 10 * q^13 + (-2*b - 16) * q^15 - 8*b * q^17 - 2 * q^19 + (6*b - 6) * q^21 + 4*b * q^23 - 7 * q^25 + (-5*b + 23) * q^27 + 6*b * q^29 + 22 * q^31 + (2*b + 16) * q^33 + 12*b * q^35 - 6 * q^37 + (10*b - 10) * q^39 + 12*b * q^41 - 82 * q^43 + (-14*b + 32) * q^45 - 24*b * q^47 - 13 * q^49 + (8*b + 64) * q^51 - 22*b * q^53 + 32 * q^55 + (-2*b + 2) * q^57 - 26*b * q^59 - 86 * q^61 + (-12*b - 42) * q^63 + 20*b * q^65 - 2 * q^67 + (-4*b - 32) * q^69 + 44*b * q^71 + 82 * q^73 + (-7*b + 7) * q^75 - 12*b * q^77 - 10 * q^79 + (28*b + 17) * q^81 + 26*b * q^83 + 128 * q^85 + (-6*b - 48) * q^87 - 12*b * q^89 + 60 * q^91 + (22*b - 22) * q^93 - 4*b * q^95 - 94 * q^97 + (14*b - 32) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 12 q^{7} - 14 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 12 * q^7 - 14 * q^9 $$2 q - 2 q^{3} + 12 q^{7} - 14 q^{9} + 20 q^{13} - 32 q^{15} - 4 q^{19} - 12 q^{21} - 14 q^{25} + 46 q^{27} + 44 q^{31} + 32 q^{33} - 12 q^{37} - 20 q^{39} - 164 q^{43} + 64 q^{45} - 26 q^{49} + 128 q^{51} + 64 q^{55} + 4 q^{57} - 172 q^{61} - 84 q^{63} - 4 q^{67} - 64 q^{69} + 164 q^{73} + 14 q^{75} - 20 q^{79} + 34 q^{81} + 256 q^{85} - 96 q^{87} + 120 q^{91} - 44 q^{93} - 188 q^{97} - 64 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 12 * q^7 - 14 * q^9 + 20 * q^13 - 32 * q^15 - 4 * q^19 - 12 * q^21 - 14 * q^25 + 46 * q^27 + 44 * q^31 + 32 * q^33 - 12 * q^37 - 20 * q^39 - 164 * q^43 + 64 * q^45 - 26 * q^49 + 128 * q^51 + 64 * q^55 + 4 * q^57 - 172 * q^61 - 84 * q^63 - 4 * q^67 - 64 * q^69 + 164 * q^73 + 14 * q^75 - 20 * q^79 + 34 * q^81 + 256 * q^85 - 96 * q^87 + 120 * q^91 - 44 * q^93 - 188 * q^97 - 64 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 −1.00000 2.82843i 0 5.65685i 0 6.00000 0 −7.00000 + 5.65685i 0
17.2 0 −1.00000 + 2.82843i 0 5.65685i 0 6.00000 0 −7.00000 5.65685i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.e.b 2
3.b odd 2 1 inner 48.3.e.b 2
4.b odd 2 1 24.3.e.a 2
5.b even 2 1 1200.3.l.n 2
5.c odd 4 2 1200.3.c.i 4
8.b even 2 1 192.3.e.d 2
8.d odd 2 1 192.3.e.c 2
9.c even 3 2 1296.3.q.e 4
9.d odd 6 2 1296.3.q.e 4
12.b even 2 1 24.3.e.a 2
15.d odd 2 1 1200.3.l.n 2
15.e even 4 2 1200.3.c.i 4
16.e even 4 2 768.3.h.c 4
16.f odd 4 2 768.3.h.d 4
20.d odd 2 1 600.3.l.b 2
20.e even 4 2 600.3.c.a 4
24.f even 2 1 192.3.e.c 2
24.h odd 2 1 192.3.e.d 2
28.d even 2 1 1176.3.d.a 2
36.f odd 6 2 648.3.m.d 4
36.h even 6 2 648.3.m.d 4
48.i odd 4 2 768.3.h.c 4
48.k even 4 2 768.3.h.d 4
60.h even 2 1 600.3.l.b 2
60.l odd 4 2 600.3.c.a 4
84.h odd 2 1 1176.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 4.b odd 2 1
24.3.e.a 2 12.b even 2 1
48.3.e.b 2 1.a even 1 1 trivial
48.3.e.b 2 3.b odd 2 1 inner
192.3.e.c 2 8.d odd 2 1
192.3.e.c 2 24.f even 2 1
192.3.e.d 2 8.b even 2 1
192.3.e.d 2 24.h odd 2 1
600.3.c.a 4 20.e even 4 2
600.3.c.a 4 60.l odd 4 2
600.3.l.b 2 20.d odd 2 1
600.3.l.b 2 60.h even 2 1
648.3.m.d 4 36.f odd 6 2
648.3.m.d 4 36.h even 6 2
768.3.h.c 4 16.e even 4 2
768.3.h.c 4 48.i odd 4 2
768.3.h.d 4 16.f odd 4 2
768.3.h.d 4 48.k even 4 2
1176.3.d.a 2 28.d even 2 1
1176.3.d.a 2 84.h odd 2 1
1200.3.c.i 4 5.c odd 4 2
1200.3.c.i 4 15.e even 4 2
1200.3.l.n 2 5.b even 2 1
1200.3.l.n 2 15.d odd 2 1
1296.3.q.e 4 9.c even 3 2
1296.3.q.e 4 9.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 32$$ acting on $$S_{3}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 9$$
$5$ $$T^{2} + 32$$
$7$ $$(T - 6)^{2}$$
$11$ $$T^{2} + 32$$
$13$ $$(T - 10)^{2}$$
$17$ $$T^{2} + 512$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 128$$
$29$ $$T^{2} + 288$$
$31$ $$(T - 22)^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 1152$$
$43$ $$(T + 82)^{2}$$
$47$ $$T^{2} + 4608$$
$53$ $$T^{2} + 3872$$
$59$ $$T^{2} + 5408$$
$61$ $$(T + 86)^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$T^{2} + 15488$$
$73$ $$(T - 82)^{2}$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 5408$$
$89$ $$T^{2} + 1152$$
$97$ $$(T + 94)^{2}$$