# Properties

 Label 648.4.i.a Level $648$ Weight $4$ Character orbit 648.i Analytic conductor $38.233$ 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] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.i (of order $$3$$, degree $$2$$, 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: $$38.2332376837$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 16 \zeta_{6} q^{5} + ( - 12 \zeta_{6} + 12) q^{7} +O(q^{10})$$ q - 16*z * q^5 + (-12*z + 12) * q^7 $$q - 16 \zeta_{6} q^{5} + ( - 12 \zeta_{6} + 12) q^{7} + (64 \zeta_{6} - 64) q^{11} - 58 \zeta_{6} q^{13} + 32 q^{17} - 136 q^{19} + 128 \zeta_{6} q^{23} + (131 \zeta_{6} - 131) q^{25} + ( - 144 \zeta_{6} + 144) q^{29} - 20 \zeta_{6} q^{31} - 192 q^{35} - 18 q^{37} + 288 \zeta_{6} q^{41} + ( - 200 \zeta_{6} + 200) q^{43} + (384 \zeta_{6} - 384) q^{47} + 199 \zeta_{6} q^{49} + 496 q^{53} + 1024 q^{55} + 128 \zeta_{6} q^{59} + ( - 458 \zeta_{6} + 458) q^{61} + (928 \zeta_{6} - 928) q^{65} + 496 \zeta_{6} q^{67} + 512 q^{71} - 602 q^{73} + 768 \zeta_{6} q^{77} + (1108 \zeta_{6} - 1108) q^{79} + (704 \zeta_{6} - 704) q^{83} - 512 \zeta_{6} q^{85} - 960 q^{89} - 696 q^{91} + 2176 \zeta_{6} q^{95} + (206 \zeta_{6} - 206) q^{97} +O(q^{100})$$ q - 16*z * q^5 + (-12*z + 12) * q^7 + (64*z - 64) * q^11 - 58*z * q^13 + 32 * q^17 - 136 * q^19 + 128*z * q^23 + (131*z - 131) * q^25 + (-144*z + 144) * q^29 - 20*z * q^31 - 192 * q^35 - 18 * q^37 + 288*z * q^41 + (-200*z + 200) * q^43 + (384*z - 384) * q^47 + 199*z * q^49 + 496 * q^53 + 1024 * q^55 + 128*z * q^59 + (-458*z + 458) * q^61 + (928*z - 928) * q^65 + 496*z * q^67 + 512 * q^71 - 602 * q^73 + 768*z * q^77 + (1108*z - 1108) * q^79 + (704*z - 704) * q^83 - 512*z * q^85 - 960 * q^89 - 696 * q^91 + 2176*z * q^95 + (206*z - 206) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{5} + 12 q^{7}+O(q^{10})$$ 2 * q - 16 * q^5 + 12 * q^7 $$2 q - 16 q^{5} + 12 q^{7} - 64 q^{11} - 58 q^{13} + 64 q^{17} - 272 q^{19} + 128 q^{23} - 131 q^{25} + 144 q^{29} - 20 q^{31} - 384 q^{35} - 36 q^{37} + 288 q^{41} + 200 q^{43} - 384 q^{47} + 199 q^{49} + 992 q^{53} + 2048 q^{55} + 128 q^{59} + 458 q^{61} - 928 q^{65} + 496 q^{67} + 1024 q^{71} - 1204 q^{73} + 768 q^{77} - 1108 q^{79} - 704 q^{83} - 512 q^{85} - 1920 q^{89} - 1392 q^{91} + 2176 q^{95} - 206 q^{97}+O(q^{100})$$ 2 * q - 16 * q^5 + 12 * q^7 - 64 * q^11 - 58 * q^13 + 64 * q^17 - 272 * q^19 + 128 * q^23 - 131 * q^25 + 144 * q^29 - 20 * q^31 - 384 * q^35 - 36 * q^37 + 288 * q^41 + 200 * q^43 - 384 * q^47 + 199 * q^49 + 992 * q^53 + 2048 * q^55 + 128 * q^59 + 458 * q^61 - 928 * q^65 + 496 * q^67 + 1024 * q^71 - 1204 * q^73 + 768 * q^77 - 1108 * q^79 - 704 * q^83 - 512 * q^85 - 1920 * q^89 - 1392 * q^91 + 2176 * q^95 - 206 * q^97

## Character values

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

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −8.00000 13.8564i 0 6.00000 10.3923i 0 0 0
433.1 0 0 0 −8.00000 + 13.8564i 0 6.00000 + 10.3923i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.i.a 2
3.b odd 2 1 648.4.i.l 2
9.c even 3 1 72.4.a.d yes 1
9.c even 3 1 inner 648.4.i.a 2
9.d odd 6 1 72.4.a.a 1
9.d odd 6 1 648.4.i.l 2
36.f odd 6 1 144.4.a.f 1
36.h even 6 1 144.4.a.a 1
45.h odd 6 1 1800.4.a.z 1
45.j even 6 1 1800.4.a.ba 1
45.k odd 12 2 1800.4.f.x 2
45.l even 12 2 1800.4.f.b 2
72.j odd 6 1 576.4.a.w 1
72.l even 6 1 576.4.a.x 1
72.n even 6 1 576.4.a.c 1
72.p odd 6 1 576.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 9.d odd 6 1
72.4.a.d yes 1 9.c even 3 1
144.4.a.a 1 36.h even 6 1
144.4.a.f 1 36.f odd 6 1
576.4.a.c 1 72.n even 6 1
576.4.a.d 1 72.p odd 6 1
576.4.a.w 1 72.j odd 6 1
576.4.a.x 1 72.l even 6 1
648.4.i.a 2 1.a even 1 1 trivial
648.4.i.a 2 9.c even 3 1 inner
648.4.i.l 2 3.b odd 2 1
648.4.i.l 2 9.d odd 6 1
1800.4.a.z 1 45.h odd 6 1
1800.4.a.ba 1 45.j even 6 1
1800.4.f.b 2 45.l even 12 2
1800.4.f.x 2 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 16T_{5} + 256$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 16T + 256$$
$7$ $$T^{2} - 12T + 144$$
$11$ $$T^{2} + 64T + 4096$$
$13$ $$T^{2} + 58T + 3364$$
$17$ $$(T - 32)^{2}$$
$19$ $$(T + 136)^{2}$$
$23$ $$T^{2} - 128T + 16384$$
$29$ $$T^{2} - 144T + 20736$$
$31$ $$T^{2} + 20T + 400$$
$37$ $$(T + 18)^{2}$$
$41$ $$T^{2} - 288T + 82944$$
$43$ $$T^{2} - 200T + 40000$$
$47$ $$T^{2} + 384T + 147456$$
$53$ $$(T - 496)^{2}$$
$59$ $$T^{2} - 128T + 16384$$
$61$ $$T^{2} - 458T + 209764$$
$67$ $$T^{2} - 496T + 246016$$
$71$ $$(T - 512)^{2}$$
$73$ $$(T + 602)^{2}$$
$79$ $$T^{2} + 1108 T + 1227664$$
$83$ $$T^{2} + 704T + 495616$$
$89$ $$(T + 960)^{2}$$
$97$ $$T^{2} + 206T + 42436$$