# Properties

 Label 476.1.o.a Level $476$ Weight $1$ Character orbit 476.o Analytic conductor $0.238$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -68 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,1,Mod(67,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("476.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$476 = 2^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 476.o (of order $$6$$, degree $$2$$, 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: $$0.237554946013$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.3332.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} - q^{7} + q^{8} +O(q^{10})$$ q + z^2 * q^2 - z * q^3 - z * q^4 + q^6 - q^7 + q^8 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} - q^{7} + q^{8} - \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} - q^{13} - \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} - \zeta_{6} q^{17} + \zeta_{6} q^{21} + q^{22} - \zeta_{6}^{2} q^{23} - \zeta_{6} q^{24} - \zeta_{6} q^{25} - \zeta_{6}^{2} q^{26} - q^{27} + \zeta_{6} q^{28} + \zeta_{6} q^{31} - \zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} + q^{34} + \zeta_{6} q^{39} - q^{42} + \zeta_{6}^{2} q^{44} + 2 \zeta_{6} q^{46} + q^{48} + q^{49} + q^{50} + \zeta_{6}^{2} q^{51} + \zeta_{6} q^{52} + \zeta_{6} q^{53} - \zeta_{6}^{2} q^{54} - q^{56} - 2 q^{62} + q^{64} - \zeta_{6} q^{66} + \zeta_{6}^{2} q^{68} - 2 q^{69} + q^{71} + \zeta_{6}^{2} q^{75} + \zeta_{6} q^{77} - q^{78} + \zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} - \zeta_{6}^{2} q^{84} - \zeta_{6} q^{88} - \zeta_{6}^{2} q^{89} + q^{91} - 2 q^{92} - 2 \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} + \zeta_{6}^{2} q^{98} +O(q^{100})$$ q + z^2 * q^2 - z * q^3 - z * q^4 + q^6 - q^7 + q^8 - z * q^11 + z^2 * q^12 - q^13 - z^2 * q^14 + z^2 * q^16 - z * q^17 + z * q^21 + q^22 - z^2 * q^23 - z * q^24 - z * q^25 - z^2 * q^26 - q^27 + z * q^28 + z * q^31 - z * q^32 + z^2 * q^33 + q^34 + z * q^39 - q^42 + z^2 * q^44 + 2*z * q^46 + q^48 + q^49 + q^50 + z^2 * q^51 + z * q^52 + z * q^53 - z^2 * q^54 - q^56 - 2 * q^62 + q^64 - z * q^66 + z^2 * q^68 - 2 * q^69 + q^71 + z^2 * q^75 + z * q^77 - q^78 + z^2 * q^79 + z * q^81 - z^2 * q^84 - z * q^88 - z^2 * q^89 + q^91 - 2 * q^92 - 2*z^2 * q^93 + z^2 * q^96 + z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} - q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{16} - q^{17} + q^{21} + 2 q^{22} + 2 q^{23} - q^{24} - q^{25} + q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - q^{32} - q^{33} + 2 q^{34} + q^{39} - 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - q^{51} + q^{52} + q^{53} + q^{54} - 2 q^{56} - 4 q^{62} + 2 q^{64} - q^{66} - q^{68} - 4 q^{69} + 2 q^{71} - q^{75} + q^{77} - 2 q^{78} - q^{79} + q^{81} + q^{84} - q^{88} + q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{93} - q^{96} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 - q^11 - q^12 - 2 * q^13 + q^14 - q^16 - q^17 + q^21 + 2 * q^22 + 2 * q^23 - q^24 - q^25 + q^26 - 2 * q^27 + q^28 + 2 * q^31 - q^32 - q^33 + 2 * q^34 + q^39 - 2 * q^42 - q^44 + 2 * q^46 + 2 * q^48 + 2 * q^49 + 2 * q^50 - q^51 + q^52 + q^53 + q^54 - 2 * q^56 - 4 * q^62 + 2 * q^64 - q^66 - q^68 - 4 * q^69 + 2 * q^71 - q^75 + q^77 - 2 * q^78 - q^79 + q^81 + q^84 - q^88 + q^89 + 2 * q^91 - 4 * q^92 + 2 * q^93 - q^96 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$$.

 $$n$$ $$239$$ $$309$$ $$409$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 −1.00000 1.00000 0 0
135.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 −1.00000 1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
7.c even 3 1 inner
476.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.1.o.a 2
4.b odd 2 1 476.1.o.b yes 2
7.b odd 2 1 3332.1.o.b 2
7.c even 3 1 inner 476.1.o.a 2
7.c even 3 1 3332.1.g.e 1
7.d odd 6 1 3332.1.g.c 1
7.d odd 6 1 3332.1.o.b 2
17.b even 2 1 476.1.o.b yes 2
28.d even 2 1 3332.1.o.a 2
28.f even 6 1 3332.1.g.d 1
28.f even 6 1 3332.1.o.a 2
28.g odd 6 1 476.1.o.b yes 2
28.g odd 6 1 3332.1.g.b 1
68.d odd 2 1 CM 476.1.o.a 2
119.d odd 2 1 3332.1.o.a 2
119.h odd 6 1 3332.1.g.d 1
119.h odd 6 1 3332.1.o.a 2
119.j even 6 1 476.1.o.b yes 2
119.j even 6 1 3332.1.g.b 1
476.e even 2 1 3332.1.o.b 2
476.o odd 6 1 inner 476.1.o.a 2
476.o odd 6 1 3332.1.g.e 1
476.q even 6 1 3332.1.g.c 1
476.q even 6 1 3332.1.o.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(476, [\chi])$$.

## Hecke characteristic polynomials

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