# Properties

 Label 3648.1.cn.a Level $3648$ Weight $1$ Character orbit 3648.cn Analytic conductor $1.821$ Analytic rank $0$ Dimension $12$ Projective image $D_{18}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,1,Mod(161,3648)]

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, 9, 9, 8]))

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

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

chi := DirichletCharacter("3648.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3648.cn (of order $$18$$, degree $$6$$, 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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9} +O(q^{10})$$ q - z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9 $$q - \zeta_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9} + (\zeta_{36}^{12} + \zeta_{36}^{10}) q^{13} + \zeta_{36}^{15} q^{19} + (\zeta_{36}^{6} - \zeta_{36}^{2}) q^{21} + \zeta_{36}^{2} q^{25} + \zeta_{36}^{3} q^{27} + (\zeta_{36}^{11} + \zeta_{36}) q^{31} + (\zeta_{36}^{14} + \zeta_{36}^{4}) q^{37} + ( - \zeta_{36}^{17} + \zeta_{36}) q^{39} + ( - \zeta_{36}^{15} + \zeta_{36}^{5}) q^{43} + ( - \zeta_{36}^{16} + \zeta_{36}^{12} - \zeta_{36}^{8}) q^{49} + \zeta_{36}^{4} q^{57} + (\zeta_{36}^{16} - 1) q^{61} + ( - \zeta_{36}^{13} + \zeta_{36}^{9}) q^{63} + ( - \zeta_{36}^{7} - \zeta_{36}^{3}) q^{67} + (\zeta_{36}^{8} - \zeta_{36}^{6}) q^{73} - \zeta_{36}^{9} q^{75} + ( - \zeta_{36}^{9} + \zeta_{36}^{5}) q^{79} - \zeta_{36}^{10} q^{81} + ( - \zeta_{36}^{11} - \zeta_{36}^{9} + \zeta_{36}^{7} + \zeta_{36}^{5}) q^{91} + ( - \zeta_{36}^{8} + 1) q^{93} + \zeta_{36}^{4} q^{97} +O(q^{100})$$ q - z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9 + (z^12 + z^10) * q^13 + z^15 * q^19 + (z^6 - z^2) * q^21 + z^2 * q^25 + z^3 * q^27 + (z^11 + z) * q^31 + (z^14 + z^4) * q^37 + (-z^17 + z) * q^39 + (-z^15 + z^5) * q^43 + (-z^16 + z^12 - z^8) * q^49 + z^4 * q^57 + (z^16 - 1) * q^61 + (-z^13 + z^9) * q^63 + (-z^7 - z^3) * q^67 + (z^8 - z^6) * q^73 - z^9 * q^75 + (-z^9 + z^5) * q^79 - z^10 * q^81 + (-z^11 - z^9 + z^7 + z^5) * q^91 + (-z^8 + 1) * q^93 + z^4 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q - 6 q^{13} + 6 q^{21} - 6 q^{49} - 12 q^{61} - 6 q^{73} + 12 q^{93}+O(q^{100})$$ 12 * q - 6 * q^13 + 6 * q^21 - 6 * q^49 - 12 * q^61 - 6 * q^73 + 12 * q^93

## Character values

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

 $$n$$ $$1217$$ $$1921$$ $$2053$$ $$2623$$ $$\chi(n)$$ $$-1$$ $$\zeta_{36}^{8}$$ $$-1$$ $$1$$

## 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
 −0.342020 + 0.939693i 0.342020 − 0.939693i −0.342020 − 0.939693i 0.342020 + 0.939693i 0.642788 + 0.766044i −0.642788 − 0.766044i 0.984808 − 0.173648i −0.984808 + 0.173648i 0.984808 + 0.173648i −0.984808 − 0.173648i 0.642788 − 0.766044i −0.642788 + 0.766044i
0 −0.642788 0.766044i 0 0 0 −0.642788 + 1.11334i 0 −0.173648 + 0.984808i 0
161.2 0 0.642788 + 0.766044i 0 0 0 0.642788 1.11334i 0 −0.173648 + 0.984808i 0
929.1 0 −0.642788 + 0.766044i 0 0 0 −0.642788 1.11334i 0 −0.173648 0.984808i 0
929.2 0 0.642788 0.766044i 0 0 0 0.642788 + 1.11334i 0 −0.173648 0.984808i 0
1505.1 0 −0.984808 + 0.173648i 0 0 0 −0.984808 + 1.70574i 0 0.939693 0.342020i 0
1505.2 0 0.984808 0.173648i 0 0 0 0.984808 1.70574i 0 0.939693 0.342020i 0
1697.1 0 −0.342020 + 0.939693i 0 0 0 −0.342020 + 0.592396i 0 −0.766044 0.642788i 0
1697.2 0 0.342020 0.939693i 0 0 0 0.342020 0.592396i 0 −0.766044 0.642788i 0
2657.1 0 −0.342020 0.939693i 0 0 0 −0.342020 0.592396i 0 −0.766044 + 0.642788i 0
2657.2 0 0.342020 + 0.939693i 0 0 0 0.342020 + 0.592396i 0 −0.766044 + 0.642788i 0
3425.1 0 −0.984808 0.173648i 0 0 0 −0.984808 1.70574i 0 0.939693 + 0.342020i 0
3425.2 0 0.984808 + 0.173648i 0 0 0 0.984808 + 1.70574i 0 0.939693 + 0.342020i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.2 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 $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner
152.t even 18 1 inner
152.u odd 18 1 inner
456.bh odd 18 1 inner
456.bu even 18 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3648, [\chi])$$:

 $$T_{7}^{12} + 6T_{7}^{10} + 27T_{7}^{8} + 48T_{7}^{6} + 63T_{7}^{4} + 27T_{7}^{2} + 9$$ T7^12 + 6*T7^10 + 27*T7^8 + 48*T7^6 + 63*T7^4 + 27*T7^2 + 9 $$T_{13}^{6} + 3T_{13}^{5} + 6T_{13}^{4} + 6T_{13}^{3} + 3$$ T13^6 + 3*T13^5 + 6*T13^4 + 6*T13^3 + 3 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - T^{6} + 1$$
$5$ $$T^{12}$$
$7$ $$T^{12} + 6 T^{10} + 27 T^{8} + 48 T^{6} + \cdots + 9$$
$11$ $$T^{12}$$
$13$ $$(T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + 3)^{2}$$
$17$ $$T^{12}$$
$19$ $$(T^{4} - T^{2} + 1)^{3}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12} + 6 T^{10} + 27 T^{8} + 48 T^{6} + \cdots + 9$$
$37$ $$(T^{6} + 6 T^{4} + 9 T^{2} + 3)^{2}$$
$41$ $$T^{12}$$
$43$ $$T^{12} - 3 T^{10} + 12 T^{8} - 46 T^{6} + \cdots + 1$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$(T^{6} + 6 T^{5} + 15 T^{4} + 21 T^{3} + \cdots + 3)^{2}$$
$67$ $$T^{12} - 3 T^{10} + 12 T^{8} - 46 T^{6} + \cdots + 1$$
$71$ $$T^{12}$$
$73$ $$(T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + 12 T^{2} + \cdots + 1)^{2}$$
$79$ $$T^{12} + 6 T^{10} + 9 T^{8} + 3 T^{6} + \cdots + 9$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$(T^{6} + T^{3} + 1)^{2}$$