Properties

 Label 3648.1.cr.a Level $3648$ Weight $1$ Character orbit 3648.cr Analytic conductor $1.821$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -3 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,1,Mod(833,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 9, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3648.833");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3648.cr (of order $$18$$, degree $$6$$, 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: $$1.82058916609$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.88042790804544.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{4} q^{3} + ( - \zeta_{18}^{2} + \zeta_{18}) q^{7} + \zeta_{18}^{8} q^{9}+O(q^{10})$$ q + z^4 * q^3 + (-z^2 + z) * q^7 + z^8 * q^9 $$q + \zeta_{18}^{4} q^{3} + ( - \zeta_{18}^{2} + \zeta_{18}) q^{7} + \zeta_{18}^{8} q^{9} + (\zeta_{18}^{7} + \zeta_{18}^{3}) q^{13} + \zeta_{18}^{6} q^{19} + ( - \zeta_{18}^{6} + \zeta_{18}^{5}) q^{21} - \zeta_{18}^{5} q^{25} - \zeta_{18}^{3} q^{27} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{31} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{37} + (\zeta_{18}^{7} - \zeta_{18}^{2}) q^{39} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{43} + (\zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{49} - \zeta_{18} q^{57} + ( - \zeta_{18}^{4} - 1) q^{61} + (\zeta_{18} - 1) q^{63} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{67} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{73} + q^{75} + ( - \zeta_{18}^{8} - 1) q^{79} - \zeta_{18}^{7} q^{81} + (\zeta_{18}^{8} - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{91} + ( - \zeta_{18}^{2} - 1) q^{93} + \zeta_{18} q^{97} +O(q^{100})$$ q + z^4 * q^3 + (-z^2 + z) * q^7 + z^8 * q^9 + (z^7 + z^3) * q^13 + z^6 * q^19 + (-z^6 + z^5) * q^21 - z^5 * q^25 - z^3 * q^27 + (z^7 + z^5) * q^31 + (-z^8 + z) * q^37 + (z^7 - z^2) * q^39 + (z^8 + z^6) * q^43 + (z^4 - z^3 + z^2) * q^49 - z * q^57 + (-z^4 - 1) * q^61 + (z - 1) * q^63 + (z^4 - z^3) * q^67 + (z^6 + z^2) * q^73 + q^75 + (-z^8 - 1) * q^79 - z^7 * q^81 + (z^8 - z^5 + z^4 + 1) * q^91 + (-z^2 - 1) * q^93 + z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 3 q^{13} - 3 q^{19} + 3 q^{21} - 3 q^{27} - 3 q^{43} - 3 q^{49} - 6 q^{61} - 6 q^{63} - 3 q^{67} - 3 q^{73} + 6 q^{75} - 6 q^{79} + 6 q^{91} - 6 q^{93}+O(q^{100})$$ 6 * q + 3 * q^13 - 3 * q^19 + 3 * q^21 - 3 * q^27 - 3 * q^43 - 3 * q^49 - 6 * q^61 - 6 * q^63 - 3 * q^67 - 3 * q^73 + 6 * q^75 - 6 * q^79 + 6 * q^91 - 6 * q^93

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$$.

 $$n$$ $$1217$$ $$1921$$ $$2053$$ $$2623$$ $$\chi(n)$$ $$-1$$ $$\zeta_{18}^{2}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
833.1
 −0.766044 − 0.642788i 0.939693 − 0.342020i −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i
0 −0.939693 + 0.342020i 0 0 0 −0.939693 1.62760i 0 0.766044 0.642788i 0
1601.1 0 0.173648 0.984808i 0 0 0 0.173648 + 0.300767i 0 −0.939693 0.342020i 0
1985.1 0 0.766044 0.642788i 0 0 0 0.766044 1.32683i 0 0.173648 0.984808i 0
2753.1 0 0.766044 + 0.642788i 0 0 0 0.766044 + 1.32683i 0 0.173648 + 0.984808i 0
3329.1 0 0.173648 + 0.984808i 0 0 0 0.173648 0.300767i 0 −0.939693 + 0.342020i 0
3521.1 0 −0.939693 0.342020i 0 0 0 −0.939693 + 1.62760i 0 0.766044 + 0.642788i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 833.1 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 CM by $$\Q(\sqrt{-3})$$
19.e even 9 1 inner
57.l odd 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.1.cr.a 6
3.b odd 2 1 CM 3648.1.cr.a 6
4.b odd 2 1 3648.1.cr.b 6
8.b even 2 1 912.1.cb.a 6
8.d odd 2 1 228.1.s.a 6
12.b even 2 1 3648.1.cr.b 6
19.e even 9 1 inner 3648.1.cr.a 6
24.f even 2 1 228.1.s.a 6
24.h odd 2 1 912.1.cb.a 6
57.l odd 18 1 inner 3648.1.cr.a 6
76.l odd 18 1 3648.1.cr.b 6
152.t even 18 1 912.1.cb.a 6
152.u odd 18 1 228.1.s.a 6
228.v even 18 1 3648.1.cr.b 6
456.bh odd 18 1 912.1.cb.a 6
456.bu even 18 1 228.1.s.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.1.s.a 6 8.d odd 2 1
228.1.s.a 6 24.f even 2 1
228.1.s.a 6 152.u odd 18 1
228.1.s.a 6 456.bu even 18 1
912.1.cb.a 6 8.b even 2 1
912.1.cb.a 6 24.h odd 2 1
912.1.cb.a 6 152.t even 18 1
912.1.cb.a 6 456.bh odd 18 1
3648.1.cr.a 6 1.a even 1 1 trivial
3648.1.cr.a 6 3.b odd 2 1 CM
3648.1.cr.a 6 19.e even 9 1 inner
3648.1.cr.a 6 57.l odd 18 1 inner
3648.1.cr.b 6 4.b odd 2 1
3648.1.cr.b 6 12.b even 2 1
3648.1.cr.b 6 76.l odd 18 1
3648.1.cr.b 6 228.v even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 3T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} - 3T_{7} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3648, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$17$ $$T^{6}$$
$19$ $$(T^{2} + T + 1)^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$37$ $$(T^{3} - 3 T - 1)^{2}$$
$41$ $$T^{6}$$
$43$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$67$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$79$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6} - T^{3} + 1$$