# Properties

 Label 3332.1.g.i Level $3332$ Weight $1$ Character orbit 3332.g Analytic conductor $1.663$ Analytic rank $0$ Dimension $8$ Projective image $D_{10}$ CM discriminant -119 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(883,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("3332.883");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.g (of order $$2$$, degree $$1$$, 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: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.0.205346735104.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{20}^{6} q^{2} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{3} - \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{6} + \zeta_{20}^{8} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} + 1) q^{9} +O(q^{10})$$ q - z^6 * q^2 + (-z^9 + z) * q^3 - z^2 * q^4 + (-z^7 - z^3) * q^5 + (-z^7 - z^5) * q^6 + z^8 * q^8 + (-z^8 + z^2 + 1) * q^9 $$q - \zeta_{20}^{6} q^{2} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{3} - \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{5}) q^{6} + \zeta_{20}^{8} q^{8} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} + 1) q^{9} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{10} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{12} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{15} + \zeta_{20}^{4} q^{16} + \zeta_{20}^{5} q^{17} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4}) q^{18} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{20} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{24} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{25} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} + \zeta_{20}^{3} + \zeta_{20}) q^{27} + (\zeta_{20}^{8} - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{30} + (\zeta_{20}^{7} - \zeta_{20}^{3}) q^{31} + q^{32} + \zeta_{20} q^{34} + ( - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{36} + (\zeta_{20}^{5} + \zeta_{20}) q^{40} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{41} + (\zeta_{20}^{6} + \zeta_{20}^{4}) q^{43} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{45} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{48} + (\zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{50} + (\zeta_{20}^{6} + \zeta_{20}^{4}) q^{51} + (\zeta_{20}^{8} - \zeta_{20}^{2}) q^{53} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{54} + (\zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{4} - 1) q^{60} + (\zeta_{20}^{9} + \zeta_{20}) q^{61} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{62} - \zeta_{20}^{6} q^{64} + (\zeta_{20}^{8} + \zeta_{20}^{2}) q^{67} - \zeta_{20}^{7} q^{68} + (\zeta_{20}^{8} + \zeta_{20}^{6} - 1) q^{72} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{73} + (\zeta_{20}^{9} + \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{75} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{80} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{81} + (\zeta_{20}^{7} - \zeta_{20}^{5}) q^{82} + ( - \zeta_{20}^{8} + \zeta_{20}^{2}) q^{85} + (\zeta_{20}^{2} + 1) q^{86} + (\zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - 2 \zeta_{20}) q^{90} + (\zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{93} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{96} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{97} +O(q^{100})$$ q - z^6 * q^2 + (-z^9 + z) * q^3 - z^2 * q^4 + (-z^7 - z^3) * q^5 + (-z^7 - z^5) * q^6 + z^8 * q^8 + (-z^8 + z^2 + 1) * q^9 + (z^9 - z^3) * q^10 + (-z^3 - z) * q^12 + (-z^8 - z^6 - z^4 - z^2) * q^15 + z^4 * q^16 + z^5 * q^17 + (-z^8 - z^6 - z^4) * q^18 + (z^9 + z^5) * q^20 + (z^9 + z^7) * q^24 + (z^6 - z^4 - 1) * q^25 + (-z^9 - z^7 + z^3 + z) * q^27 + (z^8 - z^4 - z^2 - 1) * q^30 + (z^7 - z^3) * q^31 + q^32 + z * q^34 + (-z^4 - z^2 - 1) * q^36 + (z^5 + z) * q^40 + (-z^9 - z) * q^41 + (z^6 + z^4) * q^43 + (-z^9 - z^7 - z^5 - z^3 - z) * q^45 + (z^5 + z^3) * q^48 + (z^6 + z^2 - 1) * q^50 + (z^6 + z^4) * q^51 + (z^8 - z^2) * q^53 + (-z^9 - z^7 - z^5 - z^3) * q^54 + (z^8 + z^6 + z^4 - 1) * q^60 + (z^9 + z) * q^61 + (z^9 + z^3) * q^62 - z^6 * q^64 + (z^8 + z^2) * q^67 - z^7 * q^68 + (z^8 + z^6 - 1) * q^72 + (-z^9 - z) * q^73 + (z^9 + z^7 + z^5 - z^3 - z) * q^75 + (-z^7 + z) * q^80 + (-z^8 - z^6 + z^4 + z^2 + 1) * q^81 + (z^7 - z^5) * q^82 + (-z^8 + z^2) * q^85 + (z^2 + 1) * q^86 + (z^9 + z^7 - z^5 - z^3 - 2*z) * q^90 + (z^8 + z^6 - z^4 - z^2) * q^93 + (-z^9 + z) * q^96 + (-z^7 - z^3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9 $$8 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 12 q^{9} - 2 q^{16} + 2 q^{18} - 4 q^{25} - 10 q^{30} + 8 q^{32} - 8 q^{36} - 4 q^{50} - 4 q^{53} - 10 q^{60} - 2 q^{64} - 8 q^{72} + 8 q^{81} + 4 q^{85} + 10 q^{86}+O(q^{100})$$ 8 * q - 2 * q^2 - 2 * q^4 - 2 * q^8 + 12 * q^9 - 2 * q^16 + 2 * q^18 - 4 * q^25 - 10 * q^30 + 8 * q^32 - 8 * q^36 - 4 * q^50 - 4 * q^53 - 10 * q^60 - 2 * q^64 - 8 * q^72 + 8 * q^81 + 4 * q^85 + 10 * q^86

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i
−0.809017 0.587785i −1.17557 0.309017 + 0.951057i 0.618034i 0.951057 + 0.690983i 0 0.309017 0.951057i 0.381966 −0.363271 + 0.500000i
883.2 −0.809017 0.587785i 1.17557 0.309017 + 0.951057i 0.618034i −0.951057 0.690983i 0 0.309017 0.951057i 0.381966 0.363271 0.500000i
883.3 −0.809017 + 0.587785i −1.17557 0.309017 0.951057i 0.618034i 0.951057 0.690983i 0 0.309017 + 0.951057i 0.381966 −0.363271 0.500000i
883.4 −0.809017 + 0.587785i 1.17557 0.309017 0.951057i 0.618034i −0.951057 + 0.690983i 0 0.309017 + 0.951057i 0.381966 0.363271 + 0.500000i
883.5 0.309017 0.951057i −1.90211 −0.809017 0.587785i 1.61803i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i 2.61803 1.53884 + 0.500000i
883.6 0.309017 0.951057i 1.90211 −0.809017 0.587785i 1.61803i 0.587785 1.80902i 0 −0.809017 + 0.587785i 2.61803 −1.53884 0.500000i
883.7 0.309017 + 0.951057i −1.90211 −0.809017 + 0.587785i 1.61803i −0.587785 1.80902i 0 −0.809017 0.587785i 2.61803 1.53884 0.500000i
883.8 0.309017 + 0.951057i 1.90211 −0.809017 + 0.587785i 1.61803i 0.587785 + 1.80902i 0 −0.809017 0.587785i 2.61803 −1.53884 + 0.500000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by $$\Q(\sqrt{-119})$$
4.b odd 2 1 inner
7.b odd 2 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
68.d odd 2 1 inner
476.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.i 8
4.b odd 2 1 inner 3332.1.g.i 8
7.b odd 2 1 inner 3332.1.g.i 8
7.c even 3 2 3332.1.o.g 16
7.d odd 6 2 3332.1.o.g 16
17.b even 2 1 inner 3332.1.g.i 8
28.d even 2 1 inner 3332.1.g.i 8
28.f even 6 2 3332.1.o.g 16
28.g odd 6 2 3332.1.o.g 16
68.d odd 2 1 inner 3332.1.g.i 8
119.d odd 2 1 CM 3332.1.g.i 8
119.h odd 6 2 3332.1.o.g 16
119.j even 6 2 3332.1.o.g 16
476.e even 2 1 inner 3332.1.g.i 8
476.o odd 6 2 3332.1.o.g 16
476.q even 6 2 3332.1.o.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.i 8 1.a even 1 1 trivial
3332.1.g.i 8 4.b odd 2 1 inner
3332.1.g.i 8 7.b odd 2 1 inner
3332.1.g.i 8 17.b even 2 1 inner
3332.1.g.i 8 28.d even 2 1 inner
3332.1.g.i 8 68.d odd 2 1 inner
3332.1.g.i 8 119.d odd 2 1 CM
3332.1.g.i 8 476.e even 2 1 inner
3332.1.o.g 16 7.c even 3 2
3332.1.o.g 16 7.d odd 6 2
3332.1.o.g 16 28.f even 6 2
3332.1.o.g 16 28.g odd 6 2
3332.1.o.g 16 119.h odd 6 2
3332.1.o.g 16 119.j even 6 2
3332.1.o.g 16 476.o odd 6 2
3332.1.o.g 16 476.q even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}^{4} - 5T_{3}^{2} + 5$$ T3^4 - 5*T3^2 + 5 $$T_{5}^{4} + 3T_{5}^{2} + 1$$ T5^4 + 3*T5^2 + 1 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

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