# Properties

 Label 48.9.g.b Level $48$ Weight $9$ Character orbit 48.g Analytic conductor $19.554$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,9,Mod(31,48)]

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, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (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: $$19.5541732829$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ 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 $$\beta = 9\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \beta q^{3} + 726 q^{5} - 196 \beta q^{7} - 2187 q^{9} +O(q^{10})$$ q - 3*b * q^3 + 726 * q^5 - 196*b * q^7 - 2187 * q^9 $$q - 3 \beta q^{3} + 726 q^{5} - 196 \beta q^{7} - 2187 q^{9} + 852 \beta q^{11} + 39034 q^{13} - 2178 \beta q^{15} - 65814 q^{17} - 8356 \beta q^{19} - 142884 q^{21} - 32208 \beta q^{23} + 136451 q^{25} + 6561 \beta q^{27} + 202062 q^{29} - 76700 \beta q^{31} + 621108 q^{33} - 142296 \beta q^{35} - 1876030 q^{37} - 117102 \beta q^{39} + 3091050 q^{41} - 145228 \beta q^{43} - 1587762 q^{45} + 407784 \beta q^{47} - 3570287 q^{49} + 197442 \beta q^{51} - 1066482 q^{53} + 618552 \beta q^{55} - 6091524 q^{57} - 369732 \beta q^{59} + 17154194 q^{61} + 428652 \beta q^{63} + 28338684 q^{65} + 1759476 \beta q^{67} - 23479632 q^{69} + 2555328 \beta q^{71} - 53286014 q^{73} - 409353 \beta q^{75} + 40579056 q^{77} - 1171996 \beta q^{79} + 4782969 q^{81} - 499668 \beta q^{83} - 47780964 q^{85} - 606186 \beta q^{87} + 86667234 q^{89} - 7650664 \beta q^{91} - 55914300 q^{93} - 6066456 \beta q^{95} - 73901822 q^{97} - 1863324 \beta q^{99} +O(q^{100})$$ q - 3*b * q^3 + 726 * q^5 - 196*b * q^7 - 2187 * q^9 + 852*b * q^11 + 39034 * q^13 - 2178*b * q^15 - 65814 * q^17 - 8356*b * q^19 - 142884 * q^21 - 32208*b * q^23 + 136451 * q^25 + 6561*b * q^27 + 202062 * q^29 - 76700*b * q^31 + 621108 * q^33 - 142296*b * q^35 - 1876030 * q^37 - 117102*b * q^39 + 3091050 * q^41 - 145228*b * q^43 - 1587762 * q^45 + 407784*b * q^47 - 3570287 * q^49 + 197442*b * q^51 - 1066482 * q^53 + 618552*b * q^55 - 6091524 * q^57 - 369732*b * q^59 + 17154194 * q^61 + 428652*b * q^63 + 28338684 * q^65 + 1759476*b * q^67 - 23479632 * q^69 + 2555328*b * q^71 - 53286014 * q^73 - 409353*b * q^75 + 40579056 * q^77 - 1171996*b * q^79 + 4782969 * q^81 - 499668*b * q^83 - 47780964 * q^85 - 606186*b * q^87 + 86667234 * q^89 - 7650664*b * q^91 - 55914300 * q^93 - 6066456*b * q^95 - 73901822 * q^97 - 1863324*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 1452 q^{5} - 4374 q^{9}+O(q^{10})$$ 2 * q + 1452 * q^5 - 4374 * q^9 $$2 q + 1452 q^{5} - 4374 q^{9} + 78068 q^{13} - 131628 q^{17} - 285768 q^{21} + 272902 q^{25} + 404124 q^{29} + 1242216 q^{33} - 3752060 q^{37} + 6182100 q^{41} - 3175524 q^{45} - 7140574 q^{49} - 2132964 q^{53} - 12183048 q^{57} + 34308388 q^{61} + 56677368 q^{65} - 46959264 q^{69} - 106572028 q^{73} + 81158112 q^{77} + 9565938 q^{81} - 95561928 q^{85} + 173334468 q^{89} - 111828600 q^{93} - 147803644 q^{97}+O(q^{100})$$ 2 * q + 1452 * q^5 - 4374 * q^9 + 78068 * q^13 - 131628 * q^17 - 285768 * q^21 + 272902 * q^25 + 404124 * q^29 + 1242216 * q^33 - 3752060 * q^37 + 6182100 * q^41 - 3175524 * q^45 - 7140574 * q^49 - 2132964 * q^53 - 12183048 * q^57 + 34308388 * q^61 + 56677368 * q^65 - 46959264 * q^69 - 106572028 * q^73 + 81158112 * q^77 + 9565938 * q^81 - 95561928 * q^85 + 173334468 * q^89 - 111828600 * q^93 - 147803644 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 46.7654i 0 726.000 0 3055.34i 0 −2187.00 0
31.2 0 46.7654i 0 726.000 0 3055.34i 0 −2187.00 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.9.g.b 2
3.b odd 2 1 144.9.g.c 2
4.b odd 2 1 inner 48.9.g.b 2
8.b even 2 1 192.9.g.a 2
8.d odd 2 1 192.9.g.a 2
12.b even 2 1 144.9.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.g.b 2 1.a even 1 1 trivial
48.9.g.b 2 4.b odd 2 1 inner
144.9.g.c 2 3.b odd 2 1
144.9.g.c 2 12.b even 2 1
192.9.g.a 2 8.b even 2 1
192.9.g.a 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 726$$ acting on $$S_{9}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2187$$
$5$ $$(T - 726)^{2}$$
$7$ $$T^{2} + 9335088$$
$11$ $$T^{2} + 176394672$$
$13$ $$(T - 39034)^{2}$$
$17$ $$(T + 65814)^{2}$$
$19$ $$T^{2} + 16966924848$$
$23$ $$T^{2} + 252077329152$$
$29$ $$(T - 202062)^{2}$$
$31$ $$T^{2} + 1429542270000$$
$37$ $$(T + 1876030)^{2}$$
$41$ $$(T - 3091050)^{2}$$
$43$ $$T^{2} + 5125154792112$$
$47$ $$T^{2} + 40407933129408$$
$53$ $$(T + 1066482)^{2}$$
$59$ $$T^{2} + 33218525693232$$
$61$ $$(T - 17154194)^{2}$$
$67$ $$T^{2} + 752268658081968$$
$71$ $$T^{2} + 15\!\cdots\!12$$
$73$ $$(T + 53286014)^{2}$$
$79$ $$T^{2} + 333778633635888$$
$83$ $$T^{2} + 60669350784432$$
$89$ $$(T - 86667234)^{2}$$
$97$ $$(T + 73901822)^{2}$$