# Properties

 Label 48.8.c.b Level $48$ Weight $8$ Character orbit 48.c Analytic conductor $14.994$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,8,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, 8, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$14.9944812232$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{30}, \sqrt{-123})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 3x^{2} - 2x + 3691$$ x^4 - 2*x^3 + 3*x^2 - 2*x + 3691 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - 3 \beta_{2} - \beta_1) q^{7} + (9 \beta_{3} - 27) q^{9}+O(q^{10})$$ q - b2 * q^3 + b3 * q^5 + (-3*b2 - b1) * q^7 + (9*b3 - 27) * q^9 $$q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - 3 \beta_{2} - \beta_1) q^{7} + (9 \beta_{3} - 27) q^{9} + (45 \beta_{2} - 15 \beta_1) q^{11} + 310 q^{13} + (3 \beta_{2} - 81 \beta_1) q^{15} - 100 \beta_{3} q^{17} + ( - 705 \beta_{2} - 235 \beta_1) q^{19} + (27 \beta_{3} - 6642) q^{21} + (1602 \beta_{2} - 534 \beta_1) q^{23} + 19085 q^{25} + (54 \beta_{2} - 729 \beta_1) q^{27} + 719 \beta_{3} q^{29} + ( - 2895 \beta_{2} - 965 \beta_1) q^{31} + ( - 405 \beta_{3} - 97200) q^{33} + (738 \beta_{2} - 246 \beta_1) q^{35} + 188030 q^{37} - 310 \beta_{2} q^{39} - 1758 \beta_{3} q^{41} + (8619 \beta_{2} + 2873 \beta_1) q^{43} + ( - 27 \beta_{3} - 531360) q^{45} + ( - 8172 \beta_{2} + 2724 \beta_1) q^{47} + 783691 q^{49} + ( - 300 \beta_{2} + 8100 \beta_1) q^{51} + 1245 \beta_{3} q^{53} + (10800 \beta_{2} + 3600 \beta_1) q^{55} + (6345 \beta_{3} - 1560870) q^{57} + ( - 33975 \beta_{2} + 11325 \beta_1) q^{59} + 2183462 q^{61} + (6723 \beta_{2} - 2187 \beta_1) q^{63} + 310 \beta_{3} q^{65} + (25983 \beta_{2} + 8661 \beta_1) q^{67} + ( - 14418 \beta_{3} - 3460320) q^{69} + (56790 \beta_{2} - 18930 \beta_1) q^{71} + 3790330 q^{73} - 19085 \beta_{2} q^{75} - 2430 \beta_{3} q^{77} + ( - 116535 \beta_{2} - 38845 \beta_1) q^{79} + ( - 486 \beta_{3} - 4781511) q^{81} + (13347 \beta_{2} - 4449 \beta_1) q^{83} + 5904000 q^{85} + (2157 \beta_{2} - 58239 \beta_1) q^{87} + 17542 \beta_{3} q^{89} + ( - 930 \beta_{2} - 310 \beta_1) q^{91} + (26055 \beta_{3} - 6409530) q^{93} + (173430 \beta_{2} - 57810 \beta_1) q^{95} + 4439890 q^{97} + (95985 \beta_{2} + 32805 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 + b3 * q^5 + (-3*b2 - b1) * q^7 + (9*b3 - 27) * q^9 + (45*b2 - 15*b1) * q^11 + 310 * q^13 + (3*b2 - 81*b1) * q^15 - 100*b3 * q^17 + (-705*b2 - 235*b1) * q^19 + (27*b3 - 6642) * q^21 + (1602*b2 - 534*b1) * q^23 + 19085 * q^25 + (54*b2 - 729*b1) * q^27 + 719*b3 * q^29 + (-2895*b2 - 965*b1) * q^31 + (-405*b3 - 97200) * q^33 + (738*b2 - 246*b1) * q^35 + 188030 * q^37 - 310*b2 * q^39 - 1758*b3 * q^41 + (8619*b2 + 2873*b1) * q^43 + (-27*b3 - 531360) * q^45 + (-8172*b2 + 2724*b1) * q^47 + 783691 * q^49 + (-300*b2 + 8100*b1) * q^51 + 1245*b3 * q^53 + (10800*b2 + 3600*b1) * q^55 + (6345*b3 - 1560870) * q^57 + (-33975*b2 + 11325*b1) * q^59 + 2183462 * q^61 + (6723*b2 - 2187*b1) * q^63 + 310*b3 * q^65 + (25983*b2 + 8661*b1) * q^67 + (-14418*b3 - 3460320) * q^69 + (56790*b2 - 18930*b1) * q^71 + 3790330 * q^73 - 19085*b2 * q^75 - 2430*b3 * q^77 + (-116535*b2 - 38845*b1) * q^79 + (-486*b3 - 4781511) * q^81 + (13347*b2 - 4449*b1) * q^83 + 5904000 * q^85 + (2157*b2 - 58239*b1) * q^87 + 17542*b3 * q^89 + (-930*b2 - 310*b1) * q^91 + (26055*b3 - 6409530) * q^93 + (173430*b2 - 57810*b1) * q^95 + 4439890 * q^97 + (95985*b2 + 32805*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 108 q^{9}+O(q^{10})$$ 4 * q - 108 * q^9 $$4 q - 108 q^{9} + 1240 q^{13} - 26568 q^{21} + 76340 q^{25} - 388800 q^{33} + 752120 q^{37} - 2125440 q^{45} + 3134764 q^{49} - 6243480 q^{57} + 8733848 q^{61} - 13841280 q^{69} + 15161320 q^{73} - 19126044 q^{81} + 23616000 q^{85} - 25638120 q^{93} + 17759560 q^{97}+O(q^{100})$$ 4 * q - 108 * q^9 + 1240 * q^13 - 26568 * q^21 + 76340 * q^25 - 388800 * q^33 + 752120 * q^37 - 2125440 * q^45 + 3134764 * q^49 - 6243480 * q^57 + 8733848 * q^61 - 13841280 * q^69 + 15161320 * q^73 - 19126044 * q^81 + 23616000 * q^85 - 25638120 * q^93 + 17759560 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 3x^{2} - 2x + 3691$$ :

 $$\beta_{1}$$ $$=$$ $$( 8\nu^{3} - 12\nu^{2} + 18\nu - 7 ) / 27$$ (8*v^3 - 12*v^2 + 18*v - 7) / 27 $$\beta_{2}$$ $$=$$ $$6\nu - 3$$ 6*v - 3 $$\beta_{3}$$ $$=$$ $$4\nu^{2} - 4\nu + 4$$ 4*v^2 - 4*v + 4
 $$\nu$$ $$=$$ $$( \beta_{2} + 3 ) / 6$$ (b2 + 3) / 6 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} + 2\beta_{2} - 6 ) / 12$$ (3*b3 + 2*b2 - 6) / 12 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} - \beta_{2} + 27\beta _1 - 8 ) / 8$$ (3*b3 - b2 + 27*b1 - 8) / 8

## 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
 5.97723 + 5.54527i 5.97723 − 5.54527i −4.97723 + 5.54527i −4.97723 − 5.54527i
0 −32.8634 33.2716i 0 242.981i 0 199.630i 0 −27.0000 + 2186.83i 0
47.2 0 −32.8634 + 33.2716i 0 242.981i 0 199.630i 0 −27.0000 2186.83i 0
47.3 0 32.8634 33.2716i 0 242.981i 0 199.630i 0 −27.0000 2186.83i 0
47.4 0 32.8634 + 33.2716i 0 242.981i 0 199.630i 0 −27.0000 + 2186.83i 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 48.8.c.b 4
3.b odd 2 1 inner 48.8.c.b 4
4.b odd 2 1 inner 48.8.c.b 4
8.b even 2 1 192.8.c.c 4
8.d odd 2 1 192.8.c.c 4
12.b even 2 1 inner 48.8.c.b 4
24.f even 2 1 192.8.c.c 4
24.h odd 2 1 192.8.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.8.c.b 4 1.a even 1 1 trivial
48.8.c.b 4 3.b odd 2 1 inner
48.8.c.b 4 4.b odd 2 1 inner
48.8.c.b 4 12.b even 2 1 inner
192.8.c.c 4 8.b even 2 1
192.8.c.c 4 8.d odd 2 1
192.8.c.c 4 24.f even 2 1
192.8.c.c 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 59040$$ acting on $$S_{8}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 54 T^{2} + 4782969$$
$5$ $$(T^{2} + 59040)^{2}$$
$7$ $$(T^{2} + 39852)^{2}$$
$11$ $$(T^{2} - 8748000)^{2}$$
$13$ $$(T - 310)^{4}$$
$17$ $$(T^{2} + 590400000)^{2}$$
$19$ $$(T^{2} + 2200826700)^{2}$$
$23$ $$(T^{2} - 11086865280)^{2}$$
$29$ $$(T^{2} + 30521377440)^{2}$$
$31$ $$(T^{2} + 37111178700)^{2}$$
$37$ $$(T - 188030)^{4}$$
$41$ $$(T^{2} + 182466898560)^{2}$$
$43$ $$(T^{2} + 328943548908)^{2}$$
$47$ $$(T^{2} - 288496442880)^{2}$$
$53$ $$(T^{2} + 91513476000)^{2}$$
$59$ $$(T^{2} - 4986578700000)^{2}$$
$61$ $$(T - 2183462)^{4}$$
$67$ $$(T^{2} + 2989414927692)^{2}$$
$71$ $$(T^{2} - 13932449712000)^{2}$$
$73$ $$(T - 3790330)^{4}$$
$79$ $$(T^{2} + 60134038764300)^{2}$$
$83$ $$(T^{2} - 769575206880)^{2}$$
$89$ $$(T^{2} + 18167892946560)^{2}$$
$97$ $$(T - 4439890)^{4}$$