# Properties

 Label 648.2.l.b Level $648$ Weight $2$ Character orbit 648.l Analytic conductor $5.174$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(107,648)]

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([3, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("648.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 648.l (of order $$6$$, 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: $$5.17430605098$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 2*b2 * q^4 + 2*b3 * q^8 $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} + 2 \beta_1 q^{11} + (4 \beta_{2} - 4) q^{16} + 4 \beta_{3} q^{17} + 2 q^{19} + 4 \beta_{2} q^{22} + ( - 5 \beta_{2} + 5) q^{25} + (4 \beta_{3} - 4 \beta_1) q^{32} + (8 \beta_{2} - 8) q^{34} + 2 \beta_1 q^{38} + (8 \beta_{3} - 8 \beta_1) q^{41} + ( - 10 \beta_{2} + 10) q^{43} + 4 \beta_{3} q^{44} - 7 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{50} + ( - 10 \beta_{3} + 10 \beta_1) q^{59} - 8 q^{64} - 14 \beta_{2} q^{67} + (8 \beta_{3} - 8 \beta_1) q^{68} + 2 q^{73} + 4 \beta_{2} q^{76} - 16 q^{82} + 2 \beta_1 q^{83} + ( - 10 \beta_{3} + 10 \beta_1) q^{86} + (8 \beta_{2} - 8) q^{88} + 4 \beta_{3} q^{89} + ( - 10 \beta_{2} + 10) q^{97} - 7 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 2*b2 * q^4 + 2*b3 * q^8 + 2*b1 * q^11 + (4*b2 - 4) * q^16 + 4*b3 * q^17 + 2 * q^19 + 4*b2 * q^22 + (-5*b2 + 5) * q^25 + (4*b3 - 4*b1) * q^32 + (8*b2 - 8) * q^34 + 2*b1 * q^38 + (8*b3 - 8*b1) * q^41 + (-10*b2 + 10) * q^43 + 4*b3 * q^44 - 7*b2 * q^49 + (-5*b3 + 5*b1) * q^50 + (-10*b3 + 10*b1) * q^59 - 8 * q^64 - 14*b2 * q^67 + (8*b3 - 8*b1) * q^68 + 2 * q^73 + 4*b2 * q^76 - 16 * q^82 + 2*b1 * q^83 + (-10*b3 + 10*b1) * q^86 + (8*b2 - 8) * q^88 + 4*b3 * q^89 + (-10*b2 + 10) * q^97 - 7*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} - 8 q^{16} + 8 q^{19} + 8 q^{22} + 10 q^{25} - 16 q^{34} + 20 q^{43} - 14 q^{49} - 32 q^{64} - 28 q^{67} + 8 q^{73} + 8 q^{76} - 64 q^{82} - 16 q^{88} + 20 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 - 8 * q^16 + 8 * q^19 + 8 * q^22 + 10 * q^25 - 16 * q^34 + 20 * q^43 - 14 * q^49 - 32 * q^64 - 28 * q^67 + 8 * q^73 + 8 * q^76 - 64 * q^82 - 16 * q^88 + 20 * q^97

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

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

## Character values

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

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
539.1 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 0 0
539.2 1.22474 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.f even 2 1 inner
72.l even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.l.b 4
3.b odd 2 1 inner 648.2.l.b 4
4.b odd 2 1 2592.2.p.b 4
8.b even 2 1 2592.2.p.b 4
8.d odd 2 1 CM 648.2.l.b 4
9.c even 3 1 24.2.f.a 2
9.c even 3 1 inner 648.2.l.b 4
9.d odd 6 1 24.2.f.a 2
9.d odd 6 1 inner 648.2.l.b 4
12.b even 2 1 2592.2.p.b 4
24.f even 2 1 inner 648.2.l.b 4
24.h odd 2 1 2592.2.p.b 4
36.f odd 6 1 96.2.f.a 2
36.f odd 6 1 2592.2.p.b 4
36.h even 6 1 96.2.f.a 2
36.h even 6 1 2592.2.p.b 4
45.h odd 6 1 600.2.b.a 2
45.j even 6 1 600.2.b.a 2
45.k odd 12 2 600.2.m.a 4
45.l even 12 2 600.2.m.a 4
72.j odd 6 1 96.2.f.a 2
72.j odd 6 1 2592.2.p.b 4
72.l even 6 1 24.2.f.a 2
72.l even 6 1 inner 648.2.l.b 4
72.n even 6 1 96.2.f.a 2
72.n even 6 1 2592.2.p.b 4
72.p odd 6 1 24.2.f.a 2
72.p odd 6 1 inner 648.2.l.b 4
144.u even 12 2 768.2.c.h 4
144.v odd 12 2 768.2.c.h 4
144.w odd 12 2 768.2.c.h 4
144.x even 12 2 768.2.c.h 4
180.n even 6 1 2400.2.b.a 2
180.p odd 6 1 2400.2.b.a 2
180.v odd 12 2 2400.2.m.a 4
180.x even 12 2 2400.2.m.a 4
360.z odd 6 1 600.2.b.a 2
360.bd even 6 1 600.2.b.a 2
360.bh odd 6 1 2400.2.b.a 2
360.bk even 6 1 2400.2.b.a 2
360.bo even 12 2 600.2.m.a 4
360.br even 12 2 2400.2.m.a 4
360.bt odd 12 2 600.2.m.a 4
360.bu odd 12 2 2400.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 9.c even 3 1
24.2.f.a 2 9.d odd 6 1
24.2.f.a 2 72.l even 6 1
24.2.f.a 2 72.p odd 6 1
96.2.f.a 2 36.f odd 6 1
96.2.f.a 2 36.h even 6 1
96.2.f.a 2 72.j odd 6 1
96.2.f.a 2 72.n even 6 1
600.2.b.a 2 45.h odd 6 1
600.2.b.a 2 45.j even 6 1
600.2.b.a 2 360.z odd 6 1
600.2.b.a 2 360.bd even 6 1
600.2.m.a 4 45.k odd 12 2
600.2.m.a 4 45.l even 12 2
600.2.m.a 4 360.bo even 12 2
600.2.m.a 4 360.bt odd 12 2
648.2.l.b 4 1.a even 1 1 trivial
648.2.l.b 4 3.b odd 2 1 inner
648.2.l.b 4 8.d odd 2 1 CM
648.2.l.b 4 9.c even 3 1 inner
648.2.l.b 4 9.d odd 6 1 inner
648.2.l.b 4 24.f even 2 1 inner
648.2.l.b 4 72.l even 6 1 inner
648.2.l.b 4 72.p odd 6 1 inner
768.2.c.h 4 144.u even 12 2
768.2.c.h 4 144.v odd 12 2
768.2.c.h 4 144.w odd 12 2
768.2.c.h 4 144.x even 12 2
2400.2.b.a 2 180.n even 6 1
2400.2.b.a 2 180.p odd 6 1
2400.2.b.a 2 360.bh odd 6 1
2400.2.b.a 2 360.bk even 6 1
2400.2.m.a 4 180.v odd 12 2
2400.2.m.a 4 180.x even 12 2
2400.2.m.a 4 360.br even 12 2
2400.2.m.a 4 360.bu odd 12 2
2592.2.p.b 4 4.b odd 2 1
2592.2.p.b 4 8.b even 2 1
2592.2.p.b 4 12.b even 2 1
2592.2.p.b 4 24.h odd 2 1
2592.2.p.b 4 36.f odd 6 1
2592.2.p.b 4 36.h even 6 1
2592.2.p.b 4 72.j odd 6 1
2592.2.p.b 4 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(648, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}$$ T7 $$T_{41}^{4} - 128T_{41}^{2} + 16384$$ T41^4 - 128*T41^2 + 16384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 8T^{2} + 64$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 32)^{2}$$
$19$ $$(T - 2)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4} - 128 T^{2} + 16384$$
$43$ $$(T^{2} - 10 T + 100)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} - 200 T^{2} + 40000$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 14 T + 196)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T - 2)^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} - 8T^{2} + 64$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T^{2} - 10 T + 100)^{2}$$