# Properties

 Label 648.3.e.b Level $648$ Weight $3$ Character orbit 648.e Analytic conductor $17.657$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,3,Mod(161,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 648.e (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: $$17.6567211305$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7}+O(q^{10})$$ q + (-b2 + b1) * q^5 + (b3 + 3) * q^7 $$q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7} + ( - 3 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} + 7) q^{13} + ( - 8 \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{3} + 2) q^{19} + ( - 5 \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} - 10) q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + (\beta_{3} - 37) q^{31} + 29 \beta_{2} q^{35} + (2 \beta_{3} - 30) q^{37} + 23 \beta_{2} q^{41} + (4 \beta_{3} - 5) q^{43} + (29 \beta_{2} + 3 \beta_1) q^{47} + (6 \beta_{3} + 56) q^{49} + ( - 32 \beta_{2} - 8 \beta_1) q^{53} + (\beta_{3} + 55) q^{55} + (3 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{3} + 31) q^{61} + ( - 39 \beta_{2} + 10 \beta_1) q^{65} + (10 \beta_{3} + 11) q^{67} + (24 \beta_{2} + 2 \beta_1) q^{71} + (6 \beta_{3} + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1) q^{77} + ( - 7 \beta_{3} + 43) q^{79} + ( - 11 \beta_{2} + 14 \beta_1) q^{83} + (10 \beta_{3} - 88) q^{85} + ( - 24 \beta_{2} + 6 \beta_1) q^{89} + (4 \beta_{3} - 75) q^{91} + (62 \beta_{2} - 4 \beta_1) q^{95} + (2 \beta_{3} - 121) q^{97}+O(q^{100})$$ q + (-b2 + b1) * q^5 + (b3 + 3) * q^7 + (-3*b2 - 2*b1) * q^11 + (-b3 + 7) * q^13 + (-8*b2 + 2*b1) * q^17 + (2*b3 + 2) * q^19 + (-5*b2 + b1) * q^23 + (2*b3 - 10) * q^25 + (b2 - 3*b1) * q^29 + (b3 - 37) * q^31 + 29*b2 * q^35 + (2*b3 - 30) * q^37 + 23*b2 * q^41 + (4*b3 - 5) * q^43 + (29*b2 + 3*b1) * q^47 + (6*b3 + 56) * q^49 + (-32*b2 - 8*b1) * q^53 + (b3 + 55) * q^55 + (3*b2 - 2*b1) * q^59 + (b3 + 31) * q^61 + (-39*b2 + 10*b1) * q^65 + (10*b3 + 11) * q^67 + (24*b2 + 2*b1) * q^71 + (6*b3 + 10) * q^73 + (-73*b2 - 15*b1) * q^77 + (-7*b3 + 43) * q^79 + (-11*b2 + 14*b1) * q^83 + (10*b3 - 88) * q^85 + (-24*b2 + 6*b1) * q^89 + (4*b3 - 75) * q^91 + (62*b2 - 4*b1) * q^95 + (2*b3 - 121) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7}+O(q^{10})$$ 4 * q + 12 * q^7 $$4 q + 12 q^{7} + 28 q^{13} + 8 q^{19} - 40 q^{25} - 148 q^{31} - 120 q^{37} - 20 q^{43} + 224 q^{49} + 220 q^{55} + 124 q^{61} + 44 q^{67} + 40 q^{73} + 172 q^{79} - 352 q^{85} - 300 q^{91} - 484 q^{97}+O(q^{100})$$ 4 * q + 12 * q^7 + 28 * q^13 + 8 * q^19 - 40 * q^25 - 148 * q^31 - 120 * q^37 - 20 * q^43 + 224 * q^49 + 220 * q^55 + 124 * q^61 + 44 * q^67 + 40 * q^73 + 172 * q^79 - 352 * q^85 - 300 * q^91 - 484 * q^97

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

 $$\beta_{1}$$ $$=$$ $$2\nu^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\nu^{3} + 8\nu$$ -2*v^3 + 8*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 8$$ (b3 + b1) / 8 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ b2 + 1 $$\nu^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −1.22474 − 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i
0 0 0 7.38891i 0 −6.79796 0 0 0
161.2 0 0 0 3.92480i 0 12.7980 0 0 0
161.3 0 0 0 3.92480i 0 12.7980 0 0 0
161.4 0 0 0 7.38891i 0 −6.79796 0 0 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 648.3.e.b 4
3.b odd 2 1 inner 648.3.e.b 4
4.b odd 2 1 1296.3.e.c 4
9.c even 3 1 72.3.m.a 4
9.c even 3 1 216.3.m.a 4
9.d odd 6 1 72.3.m.a 4
9.d odd 6 1 216.3.m.a 4
12.b even 2 1 1296.3.e.c 4
36.f odd 6 1 144.3.q.d 4
36.f odd 6 1 432.3.q.c 4
36.h even 6 1 144.3.q.d 4
36.h even 6 1 432.3.q.c 4
72.j odd 6 1 576.3.q.h 4
72.j odd 6 1 1728.3.q.e 4
72.l even 6 1 576.3.q.c 4
72.l even 6 1 1728.3.q.f 4
72.n even 6 1 576.3.q.h 4
72.n even 6 1 1728.3.q.e 4
72.p odd 6 1 576.3.q.c 4
72.p odd 6 1 1728.3.q.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a 4 9.c even 3 1
72.3.m.a 4 9.d odd 6 1
144.3.q.d 4 36.f odd 6 1
144.3.q.d 4 36.h even 6 1
216.3.m.a 4 9.c even 3 1
216.3.m.a 4 9.d odd 6 1
432.3.q.c 4 36.f odd 6 1
432.3.q.c 4 36.h even 6 1
576.3.q.c 4 72.l even 6 1
576.3.q.c 4 72.p odd 6 1
576.3.q.h 4 72.j odd 6 1
576.3.q.h 4 72.n even 6 1
648.3.e.b 4 1.a even 1 1 trivial
648.3.e.b 4 3.b odd 2 1 inner
1296.3.e.c 4 4.b odd 2 1
1296.3.e.c 4 12.b even 2 1
1728.3.q.e 4 72.j odd 6 1
1728.3.q.e 4 72.n even 6 1
1728.3.q.f 4 72.l even 6 1
1728.3.q.f 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 70T_{5}^{2} + 841$$ acting on $$S_{3}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 70T^{2} + 841$$
$7$ $$(T^{2} - 6 T - 87)^{2}$$
$11$ $$T^{4} + 310 T^{2} + 10201$$
$13$ $$(T^{2} - 14 T - 47)^{2}$$
$17$ $$T^{4} + 640T^{2} + 4096$$
$19$ $$(T^{2} - 4 T - 380)^{2}$$
$23$ $$T^{4} + 214T^{2} + 1849$$
$29$ $$T^{4} + 582 T^{2} + 81225$$
$31$ $$(T^{2} + 74 T + 1273)^{2}$$
$37$ $$(T^{2} + 60 T + 516)^{2}$$
$41$ $$(T^{2} + 1587)^{2}$$
$43$ $$(T^{2} + 10 T - 1511)^{2}$$
$47$ $$T^{4} + 5622 T^{2} + \cdots + 4995225$$
$53$ $$T^{4} + 10240 T^{2} + \cdots + 1048576$$
$59$ $$T^{4} + 310 T^{2} + 10201$$
$61$ $$(T^{2} - 62 T + 865)^{2}$$
$67$ $$(T^{2} - 22 T - 9479)^{2}$$
$71$ $$T^{4} + 3712 T^{2} + \cdots + 2560000$$
$73$ $$(T^{2} - 20 T - 3356)^{2}$$
$79$ $$(T^{2} - 86 T - 2855)^{2}$$
$83$ $$T^{4} + 13270 T^{2} + \cdots + 34916281$$
$89$ $$T^{4} + 5760 T^{2} + 331776$$
$97$ $$(T^{2} + 242 T + 14257)^{2}$$