# Properties

 Label 152.1.k.a Level $152$ Weight $1$ Character orbit 152.k Analytic conductor $0.076$ 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] = [152,1,Mod(11,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, 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("152.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$152 = 2^{3} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 152.k (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: $$0.0758578819202$$ 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 3.1.2888.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.184832.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{8} +O(q^{10})$$ q + z^2 * q^2 - z^2 * q^3 - z * q^4 + z * q^6 + q^8 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{8} - q^{11} - q^{12} + \zeta_{6}^{2} q^{16} + 2 \zeta_{6}^{2} q^{17} - \zeta_{6} q^{19} - \zeta_{6}^{2} q^{22} - \zeta_{6}^{2} q^{24} - \zeta_{6} q^{25} + q^{27} - \zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} - 2 \zeta_{6} q^{34} + q^{38} - \zeta_{6}^{2} q^{41} + 2 \zeta_{6}^{2} q^{43} + \zeta_{6} q^{44} + \zeta_{6} q^{48} + q^{49} + q^{50} + 2 \zeta_{6} q^{51} + \zeta_{6}^{2} q^{54} - q^{57} - \zeta_{6}^{2} q^{59} + q^{64} - \zeta_{6} q^{66} + \zeta_{6} q^{67} + 2 q^{68} - \zeta_{6}^{2} q^{73} - q^{75} + \zeta_{6}^{2} q^{76} - \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} - q^{83} - 2 \zeta_{6} q^{86} - q^{88} - 2 \zeta_{6} q^{89} - q^{96} - \zeta_{6}^{2} q^{97} + \zeta_{6}^{2} q^{98} +O(q^{100})$$ q + z^2 * q^2 - z^2 * q^3 - z * q^4 + z * q^6 + q^8 - q^11 - q^12 + z^2 * q^16 + 2*z^2 * q^17 - z * q^19 - z^2 * q^22 - z^2 * q^24 - z * q^25 + q^27 - z * q^32 + z^2 * q^33 - 2*z * q^34 + q^38 - z^2 * q^41 + 2*z^2 * q^43 + z * q^44 + z * q^48 + q^49 + q^50 + 2*z * q^51 + z^2 * q^54 - q^57 - z^2 * q^59 + q^64 - z * q^66 + z * q^67 + 2 * q^68 - z^2 * q^73 - q^75 + z^2 * q^76 - z^2 * q^81 + z * q^82 - q^83 - 2*z * q^86 - q^88 - 2*z * q^89 - q^96 - z^2 * q^97 + z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 + q^6 + 2 * q^8 $$2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{8} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - q^{19} + q^{22} + q^{24} - q^{25} + 2 q^{27} - q^{32} - q^{33} - 2 q^{34} + 2 q^{38} + q^{41} - 2 q^{43} + q^{44} + q^{48} + 2 q^{49} + 2 q^{50} + 2 q^{51} - q^{54} - 2 q^{57} + q^{59} + 2 q^{64} - q^{66} + q^{67} + 4 q^{68} + q^{73} - 2 q^{75} - q^{76} + q^{81} + q^{82} - 2 q^{83} - 2 q^{86} - 2 q^{88} - 2 q^{89} - 2 q^{96} + q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 + q^6 + 2 * q^8 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - q^19 + q^22 + q^24 - q^25 + 2 * q^27 - q^32 - q^33 - 2 * q^34 + 2 * q^38 + q^41 - 2 * q^43 + q^44 + q^48 + 2 * q^49 + 2 * q^50 + 2 * q^51 - q^54 - 2 * q^57 + q^59 + 2 * q^64 - q^66 + q^67 + 4 * q^68 + q^73 - 2 * q^75 - q^76 + q^81 + q^82 - 2 * q^83 - 2 * q^86 - 2 * q^88 - 2 * q^89 - 2 * q^96 + q^97 - q^98

## Character values

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

 $$n$$ $$39$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 0 1.00000 0 0
83.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 0 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
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 152.1.k.a 2
3.b odd 2 1 1368.1.bz.a 2
4.b odd 2 1 608.1.o.a 2
5.b even 2 1 3800.1.bd.c 2
5.c odd 4 2 3800.1.bn.b 4
8.b even 2 1 608.1.o.a 2
8.d odd 2 1 CM 152.1.k.a 2
19.b odd 2 1 2888.1.k.a 2
19.c even 3 1 inner 152.1.k.a 2
19.c even 3 1 2888.1.f.b 1
19.d odd 6 1 2888.1.f.a 1
19.d odd 6 1 2888.1.k.a 2
19.e even 9 6 2888.1.u.c 6
19.f odd 18 6 2888.1.u.d 6
24.f even 2 1 1368.1.bz.a 2
40.e odd 2 1 3800.1.bd.c 2
40.k even 4 2 3800.1.bn.b 4
57.h odd 6 1 1368.1.bz.a 2
76.g odd 6 1 608.1.o.a 2
95.i even 6 1 3800.1.bd.c 2
95.m odd 12 2 3800.1.bn.b 4
152.b even 2 1 2888.1.k.a 2
152.k odd 6 1 inner 152.1.k.a 2
152.k odd 6 1 2888.1.f.b 1
152.o even 6 1 2888.1.f.a 1
152.o even 6 1 2888.1.k.a 2
152.p even 6 1 608.1.o.a 2
152.u odd 18 6 2888.1.u.c 6
152.v even 18 6 2888.1.u.d 6
456.u even 6 1 1368.1.bz.a 2
760.bm odd 6 1 3800.1.bd.c 2
760.bw even 12 2 3800.1.bn.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 1.a even 1 1 trivial
152.1.k.a 2 8.d odd 2 1 CM
152.1.k.a 2 19.c even 3 1 inner
152.1.k.a 2 152.k odd 6 1 inner
608.1.o.a 2 4.b odd 2 1
608.1.o.a 2 8.b even 2 1
608.1.o.a 2 76.g odd 6 1
608.1.o.a 2 152.p even 6 1
1368.1.bz.a 2 3.b odd 2 1
1368.1.bz.a 2 24.f even 2 1
1368.1.bz.a 2 57.h odd 6 1
1368.1.bz.a 2 456.u even 6 1
2888.1.f.a 1 19.d odd 6 1
2888.1.f.a 1 152.o even 6 1
2888.1.f.b 1 19.c even 3 1
2888.1.f.b 1 152.k odd 6 1
2888.1.k.a 2 19.b odd 2 1
2888.1.k.a 2 19.d odd 6 1
2888.1.k.a 2 152.b even 2 1
2888.1.k.a 2 152.o even 6 1
2888.1.u.c 6 19.e even 9 6
2888.1.u.c 6 152.u odd 18 6
2888.1.u.d 6 19.f odd 18 6
2888.1.u.d 6 152.v even 18 6
3800.1.bd.c 2 5.b even 2 1
3800.1.bd.c 2 40.e odd 2 1
3800.1.bd.c 2 95.i even 6 1
3800.1.bd.c 2 760.bm odd 6 1
3800.1.bn.b 4 5.c odd 4 2
3800.1.bn.b 4 40.k even 4 2
3800.1.bn.b 4 95.m odd 12 2
3800.1.bn.b 4 760.bw even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(152, [\chi])$$.

## Hecke characteristic polynomials

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