# Properties

 Label 608.1.bg.a Level $608$ Weight $1$ Character orbit 608.bg Analytic conductor $0.303$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ 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(47,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 8]))

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

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

chi := DirichletCharacter("608.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 608.bg (of order $$18$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.69564674215936.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{18}^{6} + \zeta_{18}) q^{3} + ( - \zeta_{18}^{7} + \cdots + \zeta_{18}^{2}) q^{9}+O(q^{10})$$ q + (-z^6 + z) * q^3 + (-z^7 - z^3 + z^2) * q^9 $$q + ( - \zeta_{18}^{6} + \zeta_{18}) q^{3} + ( - \zeta_{18}^{7} + \cdots + \zeta_{18}^{2}) q^{9}+ \cdots + ( - \zeta_{18}^{8} + \zeta_{18}^{7} + \cdots - 1) q^{99}+O(q^{100})$$ q + (-z^6 + z) * q^3 + (-z^7 - z^3 + z^2) * q^9 + (z^7 + z^5) * q^11 - z^2 * q^17 - z^4 * q^19 - z * q^25 + (-z^8 + z^4 + z^3 - 1) * q^27 + (z^8 + z^6 + z^4 + z^2) * q^33 + (z^4 - z^3) * q^41 - z^5 * q^43 + z^6 * q^49 + (z^8 - z^3) * q^51 + (-z^5 - z) * q^57 + (z^3 + z) * q^59 + (z^3 - z^2) * q^67 + (-z^7 + 1) * q^73 + (z^7 - z^2) * q^75 + (z^6 + z^5 + z^4 + z - 1) * q^81 + (-z^8 + z^7) * q^83 + z * q^89 + (-z^7 + z^6) * q^97 + (-z^8 + z^7 + z^5 + z^3 + z - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 3 * q^9 $$6 q + 3 q^{3} - 3 q^{9} - 3 q^{27} - 3 q^{33} - 3 q^{41} - 3 q^{49} - 3 q^{51} + 3 q^{59} + 3 q^{67} + 6 q^{73} + 3 q^{81} - 3 q^{97} - 3 q^{99}+O(q^{100})$$ 6 * q + 3 * q^3 - 3 * q^9 - 3 * q^27 - 3 * q^33 - 3 * q^41 - 3 * q^49 - 3 * q^51 + 3 * q^59 + 3 * q^67 + 6 * q^73 + 3 * q^81 - 3 * q^97 - 3 * q^99

## Character values

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

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$\zeta_{18}^{4}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 + 0.984808i −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 − 0.342020i
0 −0.266044 + 0.223238i 0 0 0 0 0 −0.152704 + 0.866025i 0
111.1 0 1.43969 0.524005i 0 0 0 0 0 1.03209 0.866025i 0
175.1 0 0.326352 + 1.85083i 0 0 0 0 0 −2.37939 + 0.866025i 0
207.1 0 −0.266044 0.223238i 0 0 0 0 0 −0.152704 0.866025i 0
271.1 0 0.326352 1.85083i 0 0 0 0 0 −2.37939 0.866025i 0
367.1 0 1.43969 + 0.524005i 0 0 0 0 0 1.03209 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.e even 9 1 inner
152.u odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.bg.a 6
4.b odd 2 1 152.1.u.a 6
8.b even 2 1 152.1.u.a 6
8.d odd 2 1 CM 608.1.bg.a 6
12.b even 2 1 1368.1.eh.a 6
19.e even 9 1 inner 608.1.bg.a 6
20.d odd 2 1 3800.1.cv.c 6
20.e even 4 2 3800.1.cq.b 12
24.h odd 2 1 1368.1.eh.a 6
40.f even 2 1 3800.1.cv.c 6
40.i odd 4 2 3800.1.cq.b 12
76.d even 2 1 2888.1.u.e 6
76.f even 6 1 2888.1.u.a 6
76.f even 6 1 2888.1.u.f 6
76.g odd 6 1 2888.1.u.b 6
76.g odd 6 1 2888.1.u.g 6
76.k even 18 1 2888.1.f.c 3
76.k even 18 2 2888.1.k.c 6
76.k even 18 1 2888.1.u.a 6
76.k even 18 1 2888.1.u.e 6
76.k even 18 1 2888.1.u.f 6
76.l odd 18 1 152.1.u.a 6
76.l odd 18 1 2888.1.f.d 3
76.l odd 18 2 2888.1.k.b 6
76.l odd 18 1 2888.1.u.b 6
76.l odd 18 1 2888.1.u.g 6
152.g odd 2 1 2888.1.u.e 6
152.l odd 6 1 2888.1.u.a 6
152.l odd 6 1 2888.1.u.f 6
152.p even 6 1 2888.1.u.b 6
152.p even 6 1 2888.1.u.g 6
152.s odd 18 1 2888.1.f.c 3
152.s odd 18 2 2888.1.k.c 6
152.s odd 18 1 2888.1.u.a 6
152.s odd 18 1 2888.1.u.e 6
152.s odd 18 1 2888.1.u.f 6
152.t even 18 1 152.1.u.a 6
152.t even 18 1 2888.1.f.d 3
152.t even 18 2 2888.1.k.b 6
152.t even 18 1 2888.1.u.b 6
152.t even 18 1 2888.1.u.g 6
152.u odd 18 1 inner 608.1.bg.a 6
228.v even 18 1 1368.1.eh.a 6
380.ba odd 18 1 3800.1.cv.c 6
380.bj even 36 2 3800.1.cq.b 12
456.bh odd 18 1 1368.1.eh.a 6
760.cj even 18 1 3800.1.cv.c 6
760.cq odd 36 2 3800.1.cq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 4.b odd 2 1
152.1.u.a 6 8.b even 2 1
152.1.u.a 6 76.l odd 18 1
152.1.u.a 6 152.t even 18 1
608.1.bg.a 6 1.a even 1 1 trivial
608.1.bg.a 6 8.d odd 2 1 CM
608.1.bg.a 6 19.e even 9 1 inner
608.1.bg.a 6 152.u odd 18 1 inner
1368.1.eh.a 6 12.b even 2 1
1368.1.eh.a 6 24.h odd 2 1
1368.1.eh.a 6 228.v even 18 1
1368.1.eh.a 6 456.bh odd 18 1
2888.1.f.c 3 76.k even 18 1
2888.1.f.c 3 152.s odd 18 1
2888.1.f.d 3 76.l odd 18 1
2888.1.f.d 3 152.t even 18 1
2888.1.k.b 6 76.l odd 18 2
2888.1.k.b 6 152.t even 18 2
2888.1.k.c 6 76.k even 18 2
2888.1.k.c 6 152.s odd 18 2
2888.1.u.a 6 76.f even 6 1
2888.1.u.a 6 76.k even 18 1
2888.1.u.a 6 152.l odd 6 1
2888.1.u.a 6 152.s odd 18 1
2888.1.u.b 6 76.g odd 6 1
2888.1.u.b 6 76.l odd 18 1
2888.1.u.b 6 152.p even 6 1
2888.1.u.b 6 152.t even 18 1
2888.1.u.e 6 76.d even 2 1
2888.1.u.e 6 76.k even 18 1
2888.1.u.e 6 152.g odd 2 1
2888.1.u.e 6 152.s odd 18 1
2888.1.u.f 6 76.f even 6 1
2888.1.u.f 6 76.k even 18 1
2888.1.u.f 6 152.l odd 6 1
2888.1.u.f 6 152.s odd 18 1
2888.1.u.g 6 76.g odd 6 1
2888.1.u.g 6 76.l odd 18 1
2888.1.u.g 6 152.p even 6 1
2888.1.u.g 6 152.t even 18 1
3800.1.cq.b 12 20.e even 4 2
3800.1.cq.b 12 40.i odd 4 2
3800.1.cq.b 12 380.bj even 36 2
3800.1.cq.b 12 760.cq odd 36 2
3800.1.cv.c 6 20.d odd 2 1
3800.1.cv.c 6 40.f even 2 1
3800.1.cv.c 6 380.ba odd 18 1
3800.1.cv.c 6 760.cj even 18 1

## Hecke kernels

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

## Hecke characteristic polynomials

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