# Properties

 Label 608.1.g.a Level $608$ Weight $1$ Character orbit 608.g Self dual yes Analytic conductor $0.303$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -152 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,1,Mod(113,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("608.113");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 608.g (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.303431527681$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.739328.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7}+O(q^{10})$$ q - q^3 + q^7 $$q - q^{3} + q^{7} + q^{13} - q^{17} + q^{19} - q^{21} + q^{23} + q^{25} + q^{27} + q^{29} - 2 q^{37} - q^{39} - 2 q^{47} + q^{51} + q^{53} - q^{57} - q^{59} - q^{67} - q^{69} - q^{73} - q^{75} - q^{81} - q^{87} + q^{91}+O(q^{100})$$ q - q^3 + q^7 + q^13 - q^17 + q^19 - q^21 + q^23 + q^25 + q^27 + q^29 - 2 * q^37 - q^39 - 2 * q^47 + q^51 + q^53 - q^57 - q^59 - q^67 - q^69 - q^73 - q^75 - q^81 - q^87 + q^91

## Character values

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

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 0
0 −1.00000 0 0 0 1.00000 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
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.g.a 1
4.b odd 2 1 152.1.g.a 1
8.b even 2 1 608.1.g.b 1
8.d odd 2 1 152.1.g.b yes 1
12.b even 2 1 1368.1.i.b 1
19.b odd 2 1 608.1.g.b 1
20.d odd 2 1 3800.1.o.b 1
20.e even 4 2 3800.1.b.a 2
24.f even 2 1 1368.1.i.a 1
40.e odd 2 1 3800.1.o.a 1
40.k even 4 2 3800.1.b.b 2
76.d even 2 1 152.1.g.b yes 1
76.f even 6 2 2888.1.l.a 2
76.g odd 6 2 2888.1.l.b 2
76.k even 18 6 2888.1.s.a 6
76.l odd 18 6 2888.1.s.b 6
152.b even 2 1 152.1.g.a 1
152.g odd 2 1 CM 608.1.g.a 1
152.k odd 6 2 2888.1.l.a 2
152.o even 6 2 2888.1.l.b 2
152.u odd 18 6 2888.1.s.a 6
152.v even 18 6 2888.1.s.b 6
228.b odd 2 1 1368.1.i.a 1
380.d even 2 1 3800.1.o.a 1
380.j odd 4 2 3800.1.b.b 2
456.l odd 2 1 1368.1.i.b 1
760.p even 2 1 3800.1.o.b 1
760.y odd 4 2 3800.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 4.b odd 2 1
152.1.g.a 1 152.b even 2 1
152.1.g.b yes 1 8.d odd 2 1
152.1.g.b yes 1 76.d even 2 1
608.1.g.a 1 1.a even 1 1 trivial
608.1.g.a 1 152.g odd 2 1 CM
608.1.g.b 1 8.b even 2 1
608.1.g.b 1 19.b odd 2 1
1368.1.i.a 1 24.f even 2 1
1368.1.i.a 1 228.b odd 2 1
1368.1.i.b 1 12.b even 2 1
1368.1.i.b 1 456.l odd 2 1
2888.1.l.a 2 76.f even 6 2
2888.1.l.a 2 152.k odd 6 2
2888.1.l.b 2 76.g odd 6 2
2888.1.l.b 2 152.o even 6 2
2888.1.s.a 6 76.k even 18 6
2888.1.s.a 6 152.u odd 18 6
2888.1.s.b 6 76.l odd 18 6
2888.1.s.b 6 152.v even 18 6
3800.1.b.a 2 20.e even 4 2
3800.1.b.a 2 760.y odd 4 2
3800.1.b.b 2 40.k even 4 2
3800.1.b.b 2 380.j odd 4 2
3800.1.o.a 1 40.e odd 2 1
3800.1.o.a 1 380.d even 2 1
3800.1.o.b 1 20.d odd 2 1
3800.1.o.b 1 760.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(608, [\chi])$$.

## Hecke characteristic polynomials

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