# Properties

 Label 3332.1.bp.c Level $3332$ Weight $1$ Character orbit 3332.bp Analytic conductor $1.663$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(263,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(24))

chi = DirichletCharacter(H, H._module([12, 16, 15]))

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

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

chi := DirichletCharacter("3332.263");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.bp (of order $$24$$, degree $$8$$, 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: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.3089659810545728.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (\zeta_{24}^{10} + \zeta_{24}^{7}) q^{5} + \zeta_{24}^{9} q^{8} - \zeta_{24} q^{9} +O(q^{10})$$ q - z^7 * q^2 - z^2 * q^4 + (z^10 + z^7) * q^5 + z^9 * q^8 - z * q^9 $$q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (\zeta_{24}^{10} + \zeta_{24}^{7}) q^{5} + \zeta_{24}^{9} q^{8} - \zeta_{24} q^{9} + (\zeta_{24}^{5} + \zeta_{24}^{2}) q^{10} + \zeta_{24}^{4} q^{16} - \zeta_{24}^{11} q^{17} + \zeta_{24}^{8} q^{18} + ( - \zeta_{24}^{9} + 1) q^{20} + ( - \zeta_{24}^{8} - \zeta_{24}^{5} - \zeta_{24}^{2}) q^{25} + ( - \zeta_{24}^{6} - \zeta_{24}^{3}) q^{29} - \zeta_{24}^{11} q^{32} - \zeta_{24}^{6} q^{34} + \zeta_{24}^{3} q^{36} + ( - \zeta_{24}^{4} - \zeta_{24}) q^{37} + ( - \zeta_{24}^{7} - \zeta_{24}^{4}) q^{40} + ( - \zeta_{24}^{3} + 1) q^{41} + ( - \zeta_{24}^{11} - \zeta_{24}^{8}) q^{45} + (\zeta_{24}^{9} - \zeta_{24}^{3} - 1) q^{50} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{53} + (\zeta_{24}^{10} - \zeta_{24}) q^{58} + (\zeta_{24}^{7} - \zeta_{24}^{4}) q^{61} - \zeta_{24}^{6} q^{64} - \zeta_{24} q^{68} - \zeta_{24}^{10} q^{72} + (\zeta_{24}^{11} - \zeta_{24}^{2}) q^{73} + (\zeta_{24}^{11} + \zeta_{24}^{8}) q^{74} + (\zeta_{24}^{11} - \zeta_{24}^{2}) q^{80} + \zeta_{24}^{2} q^{81} + (\zeta_{24}^{10} - \zeta_{24}^{7}) q^{82} + (\zeta_{24}^{9} + \zeta_{24}^{6}) q^{85} - \zeta_{24}^{10} q^{89} + ( - \zeta_{24}^{6} - \zeta_{24}^{3}) q^{90} + ( - \zeta_{24}^{6} - \zeta_{24}^{3}) q^{97} +O(q^{100})$$ q - z^7 * q^2 - z^2 * q^4 + (z^10 + z^7) * q^5 + z^9 * q^8 - z * q^9 + (z^5 + z^2) * q^10 + z^4 * q^16 - z^11 * q^17 + z^8 * q^18 + (-z^9 + 1) * q^20 + (-z^8 - z^5 - z^2) * q^25 + (-z^6 - z^3) * q^29 - z^11 * q^32 - z^6 * q^34 + z^3 * q^36 + (-z^4 - z) * q^37 + (-z^7 - z^4) * q^40 + (-z^3 + 1) * q^41 + (-z^11 - z^8) * q^45 + (z^9 - z^3 - 1) * q^50 + (-z^8 + z^2) * q^53 + (z^10 - z) * q^58 + (z^7 - z^4) * q^61 - z^6 * q^64 - z * q^68 - z^10 * q^72 + (z^11 - z^2) * q^73 + (z^11 + z^8) * q^74 + (z^11 - z^2) * q^80 + z^2 * q^81 + (z^10 - z^7) * q^82 + (z^9 + z^6) * q^85 - z^10 * q^89 + (-z^6 - z^3) * q^90 + (-z^6 - z^3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 4 q^{16} - 4 q^{18} + 8 q^{20} + 4 q^{25} - 4 q^{37} - 4 q^{40} + 8 q^{41} + 4 q^{45} - 8 q^{50} + 4 q^{53} - 4 q^{61} - 4 q^{74}+O(q^{100})$$ 8 * q + 4 * q^16 - 4 * q^18 + 8 * q^20 + 4 * q^25 - 4 * q^37 - 4 * q^40 + 8 * q^41 + 4 * q^45 - 8 * q^50 + 4 * q^53 - 4 * q^61 - 4 * q^74

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-\zeta_{24}^{9}$$ $$-\zeta_{24}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i
0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.0999004 0.758819i 0 0 0.707107 + 0.707107i −0.258819 + 0.965926i 0.0999004 0.758819i
655.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 1.83195 0.241181i 0 0 −0.707107 0.707107i 0.258819 0.965926i −1.83195 0.241181i
1243.1 0.258819 0.965926i 0 −0.866025 0.500000i −1.12484 + 1.46593i 0 0 −0.707107 + 0.707107i −0.965926 0.258819i 1.12484 + 1.46593i
1647.1 0.965926 0.258819i 0 0.866025 0.500000i −0.0999004 + 0.758819i 0 0 0.707107 0.707107i −0.258819 0.965926i 0.0999004 + 0.758819i
2235.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −0.607206 + 0.465926i 0 0 0.707107 + 0.707107i 0.965926 0.258819i 0.607206 + 0.465926i
2627.1 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −1.12484 1.46593i 0 0 −0.707107 0.707107i −0.965926 + 0.258819i 1.12484 1.46593i
3007.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −0.607206 0.465926i 0 0 0.707107 0.707107i 0.965926 + 0.258819i 0.607206 0.465926i
3215.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 1.83195 + 0.241181i 0 0 −0.707107 + 0.707107i 0.258819 + 0.965926i −1.83195 + 0.241181i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3215.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.c even 3 1 inner
17.d even 8 1 inner
28.g odd 6 1 inner
68.g odd 8 1 inner
119.q even 24 1 inner
476.bg odd 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bp.c 8
4.b odd 2 1 CM 3332.1.bp.c 8
7.b odd 2 1 3332.1.bp.b 8
7.c even 3 1 3332.1.w.c yes 4
7.c even 3 1 inner 3332.1.bp.c 8
7.d odd 6 1 3332.1.w.b 4
7.d odd 6 1 3332.1.bp.b 8
17.d even 8 1 inner 3332.1.bp.c 8
28.d even 2 1 3332.1.bp.b 8
28.f even 6 1 3332.1.w.b 4
28.f even 6 1 3332.1.bp.b 8
28.g odd 6 1 3332.1.w.c yes 4
28.g odd 6 1 inner 3332.1.bp.c 8
68.g odd 8 1 inner 3332.1.bp.c 8
119.l odd 8 1 3332.1.bp.b 8
119.q even 24 1 3332.1.w.c yes 4
119.q even 24 1 inner 3332.1.bp.c 8
119.r odd 24 1 3332.1.w.b 4
119.r odd 24 1 3332.1.bp.b 8
476.w even 8 1 3332.1.bp.b 8
476.bg odd 24 1 3332.1.w.c yes 4
476.bg odd 24 1 inner 3332.1.bp.c 8
476.bj even 24 1 3332.1.w.b 4
476.bj even 24 1 3332.1.bp.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.w.b 4 7.d odd 6 1
3332.1.w.b 4 28.f even 6 1
3332.1.w.b 4 119.r odd 24 1
3332.1.w.b 4 476.bj even 24 1
3332.1.w.c yes 4 7.c even 3 1
3332.1.w.c yes 4 28.g odd 6 1
3332.1.w.c yes 4 119.q even 24 1
3332.1.w.c yes 4 476.bg odd 24 1
3332.1.bp.b 8 7.b odd 2 1
3332.1.bp.b 8 7.d odd 6 1
3332.1.bp.b 8 28.d even 2 1
3332.1.bp.b 8 28.f even 6 1
3332.1.bp.b 8 119.l odd 8 1
3332.1.bp.b 8 119.r odd 24 1
3332.1.bp.b 8 476.w even 8 1
3332.1.bp.b 8 476.bj even 24 1
3332.1.bp.c 8 1.a even 1 1 trivial
3332.1.bp.c 8 4.b odd 2 1 CM
3332.1.bp.c 8 7.c even 3 1 inner
3332.1.bp.c 8 17.d even 8 1 inner
3332.1.bp.c 8 28.g odd 6 1 inner
3332.1.bp.c 8 68.g odd 8 1 inner
3332.1.bp.c 8 119.q even 24 1 inner
3332.1.bp.c 8 476.bg odd 24 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 2T_{5}^{6} - 8T_{5}^{5} + 2T_{5}^{4} + 8T_{5}^{3} + 12T_{5}^{2} + 8T_{5} + 4$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{4} + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{6} - 8 T^{5} + 2 T^{4} + \cdots + 4$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8} - T^{4} + 1$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$(T^{4} + 2 T^{2} - 4 T + 2)^{2}$$
$31$ $$T^{8}$$
$37$ $$T^{8} + 4 T^{7} + 10 T^{6} + 16 T^{5} + \cdots + 4$$
$41$ $$(T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2)^{2}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8} + 4 T^{7} + 10 T^{6} + 16 T^{5} + \cdots + 4$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8} - 2 T^{6} + 8 T^{5} + 2 T^{4} + \cdots + 4$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$97$ $$(T^{4} + 2 T^{2} - 4 T + 2)^{2}$$