# Properties

 Label 1764.1.q.b Level $1764$ Weight $1$ Character orbit 1764.q Analytic conductor $0.880$ Analytic rank $0$ Dimension $16$ Projective image $D_{8}$ CM discriminant -4 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,1,Mod(215,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("1764.215");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.q (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: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{48})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.38423222208.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{48}^{20} q^{2} - \zeta_{48}^{16} q^{4} + ( - \zeta_{48}^{11} - \zeta_{48}^{5}) q^{5} - \zeta_{48}^{12} q^{8} +O(q^{10})$$ q - z^20 * q^2 - z^16 * q^4 + (-z^11 - z^5) * q^5 - z^12 * q^8 $$q - \zeta_{48}^{20} q^{2} - \zeta_{48}^{16} q^{4} + ( - \zeta_{48}^{11} - \zeta_{48}^{5}) q^{5} - \zeta_{48}^{12} q^{8} + ( - \zeta_{48}^{7} - \zeta_{48}) q^{10} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{13} - \zeta_{48}^{8} q^{16} + ( - \zeta_{48}^{7} + \zeta_{48}) q^{17} + (\zeta_{48}^{21} - \zeta_{48}^{3}) q^{20} + (\zeta_{48}^{22} + \zeta_{48}^{16} + \zeta_{48}^{10}) q^{25} + ( - \zeta_{48}^{23} + \zeta_{48}^{17}) q^{26} - \zeta_{48}^{4} q^{32} + ( - \zeta_{48}^{21} - \zeta_{48}^{3}) q^{34} + ( - \zeta_{48}^{14} - \zeta_{48}^{2}) q^{37} + (\zeta_{48}^{23} + \zeta_{48}^{17}) q^{40} + (\zeta_{48}^{15} - \zeta_{48}^{9}) q^{41} + (\zeta_{48}^{18} + \zeta_{48}^{12} + \zeta_{48}^{6}) q^{50} + ( - \zeta_{48}^{19} + \zeta_{48}^{13}) q^{52} + ( - \zeta_{48}^{22} + \zeta_{48}^{10}) q^{53} + ( - \zeta_{48}^{23} - \zeta_{48}^{17}) q^{61} - q^{64} + ( - \zeta_{48}^{14} + \zeta_{48}^{8} + \zeta_{48}^{2}) q^{65} + (\zeta_{48}^{23} - \zeta_{48}^{17}) q^{68} + (\zeta_{48}^{7} + \zeta_{48}) q^{73} + (\zeta_{48}^{22} - \zeta_{48}^{10}) q^{74} + (\zeta_{48}^{19} + \zeta_{48}^{13}) q^{80} + (\zeta_{48}^{11} - \zeta_{48}^{5}) q^{82} + (\zeta_{48}^{18} - \zeta_{48}^{6}) q^{85} + (\zeta_{48}^{11} + \zeta_{48}^{5}) q^{89} + (\zeta_{48}^{15} + \zeta_{48}^{9}) q^{97} +O(q^{100})$$ q - z^20 * q^2 - z^16 * q^4 + (-z^11 - z^5) * q^5 - z^12 * q^8 + (-z^7 - z) * q^10 + (z^21 + z^3) * q^13 - z^8 * q^16 + (-z^7 + z) * q^17 + (z^21 - z^3) * q^20 + (z^22 + z^16 + z^10) * q^25 + (-z^23 + z^17) * q^26 - z^4 * q^32 + (-z^21 - z^3) * q^34 + (-z^14 - z^2) * q^37 + (z^23 + z^17) * q^40 + (z^15 - z^9) * q^41 + (z^18 + z^12 + z^6) * q^50 + (-z^19 + z^13) * q^52 + (-z^22 + z^10) * q^53 + (-z^23 - z^17) * q^61 - q^64 + (-z^14 + z^8 + z^2) * q^65 + (z^23 - z^17) * q^68 + (z^7 + z) * q^73 + (z^22 - z^10) * q^74 + (z^19 + z^13) * q^80 + (z^11 - z^5) * q^82 + (z^18 - z^6) * q^85 + (z^11 + z^5) * q^89 + (z^15 + z^9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4}+O(q^{10})$$ 16 * q + 8 * q^4 $$16 q + 8 q^{4} - 8 q^{16} - 8 q^{25} - 16 q^{64}+O(q^{100})$$ 16 * q + 8 * q^4 - 8 * q^16 - 8 * q^25 - 16 * q^64

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{48}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.608761 + 0.793353i −0.793353 − 0.608761i 0.793353 + 0.608761i 0.608761 − 0.793353i 0.991445 − 0.130526i −0.130526 − 0.991445i 0.130526 + 0.991445i −0.991445 + 0.130526i −0.608761 − 0.793353i −0.793353 + 0.608761i 0.793353 − 0.608761i 0.608761 + 0.793353i 0.991445 + 0.130526i −0.130526 + 0.991445i 0.130526 − 0.991445i −0.991445 − 0.130526i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
215.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
1403.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
1403.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.5 0.866025 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.6 0.866025 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.7 0.866025 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.8 0.866025 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.h odd 2 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.q.b 16
3.b odd 2 1 inner 1764.1.q.b 16
4.b odd 2 1 CM 1764.1.q.b 16
7.b odd 2 1 inner 1764.1.q.b 16
7.c even 3 1 1764.1.h.a 8
7.c even 3 1 inner 1764.1.q.b 16
7.d odd 6 1 1764.1.h.a 8
7.d odd 6 1 inner 1764.1.q.b 16
12.b even 2 1 inner 1764.1.q.b 16
21.c even 2 1 inner 1764.1.q.b 16
21.g even 6 1 1764.1.h.a 8
21.g even 6 1 inner 1764.1.q.b 16
21.h odd 6 1 1764.1.h.a 8
21.h odd 6 1 inner 1764.1.q.b 16
28.d even 2 1 inner 1764.1.q.b 16
28.f even 6 1 1764.1.h.a 8
28.f even 6 1 inner 1764.1.q.b 16
28.g odd 6 1 1764.1.h.a 8
28.g odd 6 1 inner 1764.1.q.b 16
84.h odd 2 1 inner 1764.1.q.b 16
84.j odd 6 1 1764.1.h.a 8
84.j odd 6 1 inner 1764.1.q.b 16
84.n even 6 1 1764.1.h.a 8
84.n even 6 1 inner 1764.1.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.h.a 8 7.c even 3 1
1764.1.h.a 8 7.d odd 6 1
1764.1.h.a 8 21.g even 6 1
1764.1.h.a 8 21.h odd 6 1
1764.1.h.a 8 28.f even 6 1
1764.1.h.a 8 28.g odd 6 1
1764.1.h.a 8 84.j odd 6 1
1764.1.h.a 8 84.n even 6 1
1764.1.q.b 16 1.a even 1 1 trivial
1764.1.q.b 16 3.b odd 2 1 inner
1764.1.q.b 16 4.b odd 2 1 CM
1764.1.q.b 16 7.b odd 2 1 inner
1764.1.q.b 16 7.c even 3 1 inner
1764.1.q.b 16 7.d odd 6 1 inner
1764.1.q.b 16 12.b even 2 1 inner
1764.1.q.b 16 21.c even 2 1 inner
1764.1.q.b 16 21.g even 6 1 inner
1764.1.q.b 16 21.h odd 6 1 inner
1764.1.q.b 16 28.d even 2 1 inner
1764.1.q.b 16 28.f even 6 1 inner
1764.1.q.b 16 28.g odd 6 1 inner
1764.1.q.b 16 84.h odd 2 1 inner
1764.1.q.b 16 84.j odd 6 1 inner
1764.1.q.b 16 84.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 4T_{5}^{6} + 14T_{5}^{4} + 8T_{5}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$(T^{4} + 4 T^{2} + 2)^{4}$$
$17$ $$(T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$(T^{4} + 2 T^{2} + 4)^{4}$$
$41$ $$(T^{4} - 4 T^{2} + 2)^{4}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$(T^{4} - 2 T^{2} + 4)^{4}$$
$59$ $$T^{16}$$
$61$ $$(T^{8} - 4 T^{6} + 14 T^{4} - 8 T^{2} + 4)^{2}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$(T^{8} - 4 T^{6} + 14 T^{4} - 8 T^{2} + 4)^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$(T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2}$$
$97$ $$(T^{4} + 4 T^{2} + 2)^{4}$$