# Properties

 Label 608.1.o.a Level $608$ Weight $1$ Character orbit 608.o Analytic conductor $0.303$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,1,Mod(239,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("608.239");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 608.o (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: $$0.303431527681$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.2888.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.2186423566336.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{3}+O(q^{10})$$ q + z^2 * q^3 $$q + \zeta_{6}^{2} q^{3} + q^{11} + \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} - \zeta_{6} q^{25} - q^{27} + \zeta_{6}^{2} q^{33} - \zeta_{6}^{2} q^{41} - \zeta_{6}^{2} q^{43} + q^{49} - 2 \zeta_{6} q^{51} - q^{57} + \zeta_{6}^{2} q^{59} - \zeta_{6} q^{67} - \zeta_{6}^{2} q^{73} + q^{75} - \zeta_{6}^{2} q^{81} + q^{83} - \zeta_{6} q^{89} - \zeta_{6}^{2} q^{97} +O(q^{100})$$ q + z^2 * q^3 + q^11 + z^2 * q^17 + z * q^19 - z * q^25 - q^27 + z^2 * q^33 - z^2 * q^41 - z^2 * q^43 + q^49 - 2*z * q^51 - q^57 + z^2 * q^59 - z * q^67 - z^2 * q^73 + q^75 - z^2 * q^81 + q^83 - z * q^89 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3}+O(q^{10})$$ 2 * q - q^3 $$2 q - q^{3} + 2 q^{11} - 2 q^{17} + q^{19} - q^{25} - 2 q^{27} - q^{33} + q^{41} + 2 q^{43} + 2 q^{49} - 2 q^{51} - 2 q^{57} - q^{59} - q^{67} + q^{73} + 2 q^{75} + q^{81} + 2 q^{83} - 2 q^{89} + q^{97}+O(q^{100})$$ 2 * q - q^3 + 2 * q^11 - 2 * q^17 + q^19 - q^25 - 2 * q^27 - q^33 + q^41 + 2 * q^43 + 2 * q^49 - 2 * q^51 - 2 * q^57 - q^59 - q^67 + q^73 + 2 * q^75 + q^81 + 2 * q^83 - 2 * q^89 + q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 0 0
463.1 0 −0.500000 0.866025i 0 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.c even 3 1 inner
152.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.o.a 2
4.b odd 2 1 152.1.k.a 2
8.b even 2 1 152.1.k.a 2
8.d odd 2 1 CM 608.1.o.a 2
12.b even 2 1 1368.1.bz.a 2
19.c even 3 1 inner 608.1.o.a 2
20.d odd 2 1 3800.1.bd.c 2
20.e even 4 2 3800.1.bn.b 4
24.h odd 2 1 1368.1.bz.a 2
40.f even 2 1 3800.1.bd.c 2
40.i odd 4 2 3800.1.bn.b 4
76.d even 2 1 2888.1.k.a 2
76.f even 6 1 2888.1.f.a 1
76.f even 6 1 2888.1.k.a 2
76.g odd 6 1 152.1.k.a 2
76.g odd 6 1 2888.1.f.b 1
76.k even 18 6 2888.1.u.d 6
76.l odd 18 6 2888.1.u.c 6
152.g odd 2 1 2888.1.k.a 2
152.k odd 6 1 inner 608.1.o.a 2
152.l odd 6 1 2888.1.f.a 1
152.l odd 6 1 2888.1.k.a 2
152.p even 6 1 152.1.k.a 2
152.p even 6 1 2888.1.f.b 1
152.s odd 18 6 2888.1.u.d 6
152.t even 18 6 2888.1.u.c 6
228.m even 6 1 1368.1.bz.a 2
380.p odd 6 1 3800.1.bd.c 2
380.v even 12 2 3800.1.bn.b 4
456.x odd 6 1 1368.1.bz.a 2
760.z even 6 1 3800.1.bd.c 2
760.br odd 12 2 3800.1.bn.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 4.b odd 2 1
152.1.k.a 2 8.b even 2 1
152.1.k.a 2 76.g odd 6 1
152.1.k.a 2 152.p even 6 1
608.1.o.a 2 1.a even 1 1 trivial
608.1.o.a 2 8.d odd 2 1 CM
608.1.o.a 2 19.c even 3 1 inner
608.1.o.a 2 152.k odd 6 1 inner
1368.1.bz.a 2 12.b even 2 1
1368.1.bz.a 2 24.h odd 2 1
1368.1.bz.a 2 228.m even 6 1
1368.1.bz.a 2 456.x odd 6 1
2888.1.f.a 1 76.f even 6 1
2888.1.f.a 1 152.l odd 6 1
2888.1.f.b 1 76.g odd 6 1
2888.1.f.b 1 152.p even 6 1
2888.1.k.a 2 76.d even 2 1
2888.1.k.a 2 76.f even 6 1
2888.1.k.a 2 152.g odd 2 1
2888.1.k.a 2 152.l odd 6 1
2888.1.u.c 6 76.l odd 18 6
2888.1.u.c 6 152.t even 18 6
2888.1.u.d 6 76.k even 18 6
2888.1.u.d 6 152.s odd 18 6
3800.1.bd.c 2 20.d odd 2 1
3800.1.bd.c 2 40.f even 2 1
3800.1.bd.c 2 380.p odd 6 1
3800.1.bd.c 2 760.z even 6 1
3800.1.bn.b 4 20.e even 4 2
3800.1.bn.b 4 40.i odd 4 2
3800.1.bn.b 4 380.v even 12 2
3800.1.bn.b 4 760.br odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(608, [\chi])$$.

## Hecke characteristic polynomials

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