# Properties

 Label 152.1.g.a Level $152$ Weight $1$ Character orbit 152.g Self dual yes Analytic conductor $0.076$ 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] = [152,1,Mod(37,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, 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("152.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$152 = 2^{3} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 152.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: yes Analytic conductor: $$0.0758578819202$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.184832.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - q^7 - q^8 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{12} + q^{13} + q^{14} + q^{16} - q^{17} - q^{19} - q^{21} - q^{23} - q^{24} + q^{25} - q^{26} - q^{27} - q^{28} + q^{29} - q^{32} + q^{34} - 2 q^{37} + q^{38} + q^{39} + q^{42} + q^{46} + 2 q^{47} + q^{48} - q^{50} - q^{51} + q^{52} + q^{53} + q^{54} + q^{56} - q^{57} - q^{58} + q^{59} + q^{64} + q^{67} - q^{68} - q^{69} - q^{73} + 2 q^{74} + q^{75} - q^{76} - q^{78} - q^{81} - q^{84} + q^{87} - q^{91} - q^{92} - 2 q^{94} - q^{96}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^7 - q^8 + q^12 + q^13 + q^14 + q^16 - q^17 - q^19 - q^21 - q^23 - q^24 + q^25 - q^26 - q^27 - q^28 + q^29 - q^32 + q^34 - 2 * q^37 + q^38 + q^39 + q^42 + q^46 + 2 * q^47 + q^48 - q^50 - q^51 + q^52 + q^53 + q^54 + q^56 - q^57 - q^58 + q^59 + q^64 + q^67 - q^68 - q^69 - q^73 + 2 * q^74 + q^75 - q^76 - q^78 - q^81 - q^84 + q^87 - q^91 - q^92 - 2 * q^94 - q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/152\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0
−1.00000 1.00000 1.00000 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
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 152.1.g.a 1
3.b odd 2 1 1368.1.i.b 1
4.b odd 2 1 608.1.g.a 1
5.b even 2 1 3800.1.o.b 1
5.c odd 4 2 3800.1.b.a 2
8.b even 2 1 152.1.g.b yes 1
8.d odd 2 1 608.1.g.b 1
19.b odd 2 1 152.1.g.b yes 1
19.c even 3 2 2888.1.l.b 2
19.d odd 6 2 2888.1.l.a 2
19.e even 9 6 2888.1.s.b 6
19.f odd 18 6 2888.1.s.a 6
24.h odd 2 1 1368.1.i.a 1
40.f even 2 1 3800.1.o.a 1
40.i odd 4 2 3800.1.b.b 2
57.d even 2 1 1368.1.i.a 1
76.d even 2 1 608.1.g.b 1
95.d odd 2 1 3800.1.o.a 1
95.g even 4 2 3800.1.b.b 2
152.b even 2 1 608.1.g.a 1
152.g odd 2 1 CM 152.1.g.a 1
152.l odd 6 2 2888.1.l.b 2
152.p even 6 2 2888.1.l.a 2
152.s odd 18 6 2888.1.s.b 6
152.t even 18 6 2888.1.s.a 6
456.p even 2 1 1368.1.i.b 1
760.b odd 2 1 3800.1.o.b 1
760.t even 4 2 3800.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 1.a even 1 1 trivial
152.1.g.a 1 152.g odd 2 1 CM
152.1.g.b yes 1 8.b even 2 1
152.1.g.b yes 1 19.b odd 2 1
608.1.g.a 1 4.b odd 2 1
608.1.g.a 1 152.b even 2 1
608.1.g.b 1 8.d odd 2 1
608.1.g.b 1 76.d even 2 1
1368.1.i.a 1 24.h odd 2 1
1368.1.i.a 1 57.d even 2 1
1368.1.i.b 1 3.b odd 2 1
1368.1.i.b 1 456.p even 2 1
2888.1.l.a 2 19.d odd 6 2
2888.1.l.a 2 152.p even 6 2
2888.1.l.b 2 19.c even 3 2
2888.1.l.b 2 152.l odd 6 2
2888.1.s.a 6 19.f odd 18 6
2888.1.s.a 6 152.t even 18 6
2888.1.s.b 6 19.e even 9 6
2888.1.s.b 6 152.s odd 18 6
3800.1.b.a 2 5.c odd 4 2
3800.1.b.a 2 760.t even 4 2
3800.1.b.b 2 40.i odd 4 2
3800.1.b.b 2 95.g even 4 2
3800.1.o.a 1 40.f even 2 1
3800.1.o.a 1 95.d odd 2 1
3800.1.o.b 1 5.b even 2 1
3800.1.o.b 1 760.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{1}^{\mathrm{new}}(152, [\chi])$$.

## Hecke characteristic polynomials

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