# Properties

 Label 48.4.c.b Level $48$ Weight $4$ Character orbit 48.c Analytic conductor $2.832$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(47,48)]

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([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (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: $$2.83209168028$$ 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: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - \beta_{3} q^{5} - 5 \beta_1 q^{7} + ( - \beta_{3} + 21) q^{9}+O(q^{10})$$ q - b2 * q^3 - b3 * q^5 - 5*b1 * q^7 + (-b3 + 21) * q^9 $$q - \beta_{2} q^{3} - \beta_{3} q^{5} - 5 \beta_1 q^{7} + ( - \beta_{3} + 21) q^{9} + ( - 6 \beta_{2} - 3 \beta_1) q^{11} - 26 q^{13} + (6 \beta_{2} + 27 \beta_1) q^{15} + 4 \beta_{3} q^{17} - 31 \beta_1 q^{19} + (5 \beta_{3} + 30) q^{21} + (36 \beta_{2} + 18 \beta_1) q^{23} - 163 q^{25} + ( - 15 \beta_{2} + 27 \beta_1) q^{27} + \beta_{3} q^{29} - 9 \beta_1 q^{31} + ( - 3 \beta_{3} + 144) q^{33} + ( - 60 \beta_{2} - 30 \beta_1) q^{35} + 206 q^{37} + 26 \beta_{2} q^{39} - 18 \beta_{3} q^{41} - 27 \beta_1 q^{43} + ( - 21 \beta_{3} - 288) q^{45} + ( - 24 \beta_{2} - 12 \beta_1) q^{47} + 43 q^{49} + ( - 24 \beta_{2} - 108 \beta_1) q^{51} + 3 \beta_{3} q^{53} + 144 \beta_1 q^{55} + (31 \beta_{3} + 186) q^{57} + (114 \beta_{2} + 57 \beta_1) q^{59} + 278 q^{61} + ( - 60 \beta_{2} - 135 \beta_1) q^{63} + 26 \beta_{3} q^{65} + 257 \beta_1 q^{67} + (18 \beta_{3} - 864) q^{69} + (12 \beta_{2} + 6 \beta_1) q^{71} - 422 q^{73} + 163 \beta_{2} q^{75} + 30 \beta_{3} q^{77} - 193 \beta_1 q^{79} + ( - 42 \beta_{3} + 153) q^{81} + (6 \beta_{2} + 3 \beta_1) q^{83} + 1152 q^{85} + ( - 6 \beta_{2} - 27 \beta_1) q^{87} - 22 \beta_{3} q^{89} + 130 \beta_1 q^{91} + (9 \beta_{3} + 54) q^{93} + ( - 372 \beta_{2} - 186 \beta_1) q^{95} - 1070 q^{97} + ( - 126 \beta_{2} + 81 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 - b3 * q^5 - 5*b1 * q^7 + (-b3 + 21) * q^9 + (-6*b2 - 3*b1) * q^11 - 26 * q^13 + (6*b2 + 27*b1) * q^15 + 4*b3 * q^17 - 31*b1 * q^19 + (5*b3 + 30) * q^21 + (36*b2 + 18*b1) * q^23 - 163 * q^25 + (-15*b2 + 27*b1) * q^27 + b3 * q^29 - 9*b1 * q^31 + (-3*b3 + 144) * q^33 + (-60*b2 - 30*b1) * q^35 + 206 * q^37 + 26*b2 * q^39 - 18*b3 * q^41 - 27*b1 * q^43 + (-21*b3 - 288) * q^45 + (-24*b2 - 12*b1) * q^47 + 43 * q^49 + (-24*b2 - 108*b1) * q^51 + 3*b3 * q^53 + 144*b1 * q^55 + (31*b3 + 186) * q^57 + (114*b2 + 57*b1) * q^59 + 278 * q^61 + (-60*b2 - 135*b1) * q^63 + 26*b3 * q^65 + 257*b1 * q^67 + (18*b3 - 864) * q^69 + (12*b2 + 6*b1) * q^71 - 422 * q^73 + 163*b2 * q^75 + 30*b3 * q^77 - 193*b1 * q^79 + (-42*b3 + 153) * q^81 + (6*b2 + 3*b1) * q^83 + 1152 * q^85 + (-6*b2 - 27*b1) * q^87 - 22*b3 * q^89 + 130*b1 * q^91 + (9*b3 + 54) * q^93 + (-372*b2 - 186*b1) * q^95 - 1070 * q^97 + (-126*b2 + 81*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 84 q^{9}+O(q^{10})$$ 4 * q + 84 * q^9 $$4 q + 84 q^{9} - 104 q^{13} + 120 q^{21} - 652 q^{25} + 576 q^{33} + 824 q^{37} - 1152 q^{45} + 172 q^{49} + 744 q^{57} + 1112 q^{61} - 3456 q^{69} - 1688 q^{73} + 612 q^{81} + 4608 q^{85} + 216 q^{93} - 4280 q^{97}+O(q^{100})$$ 4 * q + 84 * q^9 - 104 * q^13 + 120 * q^21 - 652 * q^25 + 576 * q^33 + 824 * q^37 - 1152 * q^45 + 172 * q^49 + 744 * q^57 + 1112 * q^61 - 3456 * q^69 - 1688 * q^73 + 612 * q^81 + 4608 * q^85 + 216 * q^93 - 4280 * q^97

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

 $$\beta_{1}$$ $$=$$ $$2\nu^{2} - 2$$ 2*v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{3} - \nu^{2} + 4\nu + 1$$ -v^3 - v^2 + 4*v + 1 $$\beta_{3}$$ $$=$$ $$6\nu^{3}$$ 6*v^3
 $$\nu$$ $$=$$ $$( \beta_{3} + 6\beta_{2} + 3\beta_1 ) / 24$$ (b3 + 6*b2 + 3*b1) / 24 $$\nu^{2}$$ $$=$$ $$( \beta _1 + 2 ) / 2$$ (b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} ) / 6$$ (b3) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i
0 −4.89898 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 + 16.9706i 0
47.2 0 −4.89898 + 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 16.9706i 0
47.3 0 4.89898 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 16.9706i 0
47.4 0 4.89898 + 1.73205i 0 16.9706i 0 17.3205i 0 21.0000 + 16.9706i 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 48.4.c.b 4
3.b odd 2 1 inner 48.4.c.b 4
4.b odd 2 1 inner 48.4.c.b 4
8.b even 2 1 192.4.c.c 4
8.d odd 2 1 192.4.c.c 4
12.b even 2 1 inner 48.4.c.b 4
16.e even 4 2 768.4.f.b 8
16.f odd 4 2 768.4.f.b 8
24.f even 2 1 192.4.c.c 4
24.h odd 2 1 192.4.c.c 4
48.i odd 4 2 768.4.f.b 8
48.k even 4 2 768.4.f.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 288$$ acting on $$S_{4}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 42T^{2} + 729$$
$5$ $$(T^{2} + 288)^{2}$$
$7$ $$(T^{2} + 300)^{2}$$
$11$ $$(T^{2} - 864)^{2}$$
$13$ $$(T + 26)^{4}$$
$17$ $$(T^{2} + 4608)^{2}$$
$19$ $$(T^{2} + 11532)^{2}$$
$23$ $$(T^{2} - 31104)^{2}$$
$29$ $$(T^{2} + 288)^{2}$$
$31$ $$(T^{2} + 972)^{2}$$
$37$ $$(T - 206)^{4}$$
$41$ $$(T^{2} + 93312)^{2}$$
$43$ $$(T^{2} + 8748)^{2}$$
$47$ $$(T^{2} - 13824)^{2}$$
$53$ $$(T^{2} + 2592)^{2}$$
$59$ $$(T^{2} - 311904)^{2}$$
$61$ $$(T - 278)^{4}$$
$67$ $$(T^{2} + 792588)^{2}$$
$71$ $$(T^{2} - 3456)^{2}$$
$73$ $$(T + 422)^{4}$$
$79$ $$(T^{2} + 446988)^{2}$$
$83$ $$(T^{2} - 864)^{2}$$
$89$ $$(T^{2} + 139392)^{2}$$
$97$ $$(T + 1070)^{4}$$
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