# Properties

 Label 93.1.l.a Level $93$ Weight $1$ Character orbit 93.l Analytic conductor $0.046$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [93,1,Mod(2,93)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(93, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 8]))

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

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

chi := DirichletCharacter("93.2");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$93 = 3 \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 93.l (of order $$10$$, degree $$4$$, 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: $$0.0464130461749$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.8311689.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{10} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7} + \zeta_{10}^{2} q^{9} +O(q^{10})$$ q - z * q^3 - z^3 * q^4 + (z^4 + z^2) * q^7 + z^2 * q^9 $$q - \zeta_{10} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7} + \zeta_{10}^{2} q^{9} + \zeta_{10}^{4} q^{12} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{13} - \zeta_{10} q^{16} + (\zeta_{10}^{2} - \zeta_{10}) q^{19} + ( - \zeta_{10}^{3} + 1) q^{21} + q^{25} - \zeta_{10}^{3} q^{27} + (\zeta_{10}^{2} + 1) q^{28} - \zeta_{10} q^{31} + q^{36} + (\zeta_{10}^{4} - \zeta_{10}) q^{37} + (\zeta_{10}^{4} + 1) q^{39} + ( - \zeta_{10}^{3} + 1) q^{43} + \zeta_{10}^{2} q^{48} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{49} + (\zeta_{10}^{2} - \zeta_{10}) q^{52} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{57} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{61} + (\zeta_{10}^{4} - \zeta_{10}) q^{63} + \zeta_{10}^{4} q^{64} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{67} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{73} - \zeta_{10} q^{75} + (\zeta_{10}^{4} + 1) q^{76} + (\zeta_{10}^{4} + 1) q^{79} + \zeta_{10}^{4} q^{81} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{84} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{91} + \zeta_{10}^{2} q^{93} + ( - \zeta_{10} + 1) q^{97} +O(q^{100})$$ q - z * q^3 - z^3 * q^4 + (z^4 + z^2) * q^7 + z^2 * q^9 + z^4 * q^12 + (z^4 - z^3) * q^13 - z * q^16 + (z^2 - z) * q^19 + (-z^3 + 1) * q^21 + q^25 - z^3 * q^27 + (z^2 + 1) * q^28 - z * q^31 + q^36 + (z^4 - z) * q^37 + (z^4 + 1) * q^39 + (-z^3 + 1) * q^43 + z^2 * q^48 + (z^4 - z^3 - z) * q^49 + (z^2 - z) * q^52 + (-z^3 + z^2) * q^57 + (-z^3 + z^2) * q^61 + (z^4 - z) * q^63 + z^4 * q^64 + (-z^3 + z^2) * q^67 + (z^4 + z^2) * q^73 - z * q^75 + (z^4 + 1) * q^76 + (z^4 + 1) * q^79 + z^4 * q^81 + (-z^3 - z) * q^84 + (-z^3 + z^2 - z + 1) * q^91 + z^2 * q^93 + (-z + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10})$$ 4 * q - q^3 - q^4 - 2 * q^7 - q^9 $$4 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} - 2 q^{13} - q^{16} - 2 q^{19} + 3 q^{21} + 4 q^{25} - q^{27} + 3 q^{28} - q^{31} + 4 q^{36} - 2 q^{37} + 3 q^{39} + 3 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} + 3 q^{76} + 3 q^{79} - q^{81} - 2 q^{84} + q^{91} - q^{93} + 3 q^{97}+O(q^{100})$$ 4 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 - 2 * q^13 - q^16 - 2 * q^19 + 3 * q^21 + 4 * q^25 - q^27 + 3 * q^28 - q^31 + 4 * q^36 - 2 * q^37 + 3 * q^39 + 3 * q^43 - q^48 - 3 * q^49 - 2 * q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 + 3 * q^76 + 3 * q^79 - q^81 - 2 * q^84 + q^91 - q^93 + 3 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/93\mathbb{Z}\right)^\times$$.

 $$n$$ $$32$$ $$34$$ $$\chi(n)$$ $$-1$$ $$\zeta_{10}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
0 −0.809017 + 0.587785i 0.309017 + 0.951057i 0 0 −0.500000 1.53884i 0 0.309017 0.951057i 0
8.1 0 0.309017 + 0.951057i −0.809017 0.587785i 0 0 −0.500000 0.363271i 0 −0.809017 + 0.587785i 0
35.1 0 0.309017 0.951057i −0.809017 + 0.587785i 0 0 −0.500000 + 0.363271i 0 −0.809017 0.587785i 0
47.1 0 −0.809017 0.587785i 0.309017 0.951057i 0 0 −0.500000 + 1.53884i 0 0.309017 + 0.951057i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
31.d even 5 1 inner
93.l odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.1.l.a 4
3.b odd 2 1 CM 93.1.l.a 4
4.b odd 2 1 1488.1.br.a 4
5.b even 2 1 2325.1.ca.a 4
5.c odd 4 2 2325.1.bq.a 8
9.c even 3 2 2511.1.bu.a 8
9.d odd 6 2 2511.1.bu.a 8
12.b even 2 1 1488.1.br.a 4
15.d odd 2 1 2325.1.ca.a 4
15.e even 4 2 2325.1.bq.a 8
31.b odd 2 1 2883.1.l.b 4
31.c even 3 2 2883.1.o.d 8
31.d even 5 1 inner 93.1.l.a 4
31.d even 5 1 2883.1.b.b 2
31.d even 5 2 2883.1.l.a 4
31.e odd 6 2 2883.1.o.b 8
31.f odd 10 1 2883.1.b.a 2
31.f odd 10 1 2883.1.l.b 4
31.f odd 10 2 2883.1.l.c 4
31.g even 15 2 2883.1.h.a 4
31.g even 15 4 2883.1.o.c 8
31.g even 15 2 2883.1.o.d 8
31.h odd 30 2 2883.1.h.b 4
31.h odd 30 4 2883.1.o.a 8
31.h odd 30 2 2883.1.o.b 8
93.c even 2 1 2883.1.l.b 4
93.g even 6 2 2883.1.o.b 8
93.h odd 6 2 2883.1.o.d 8
93.k even 10 1 2883.1.b.a 2
93.k even 10 1 2883.1.l.b 4
93.k even 10 2 2883.1.l.c 4
93.l odd 10 1 inner 93.1.l.a 4
93.l odd 10 1 2883.1.b.b 2
93.l odd 10 2 2883.1.l.a 4
93.o odd 30 2 2883.1.h.a 4
93.o odd 30 4 2883.1.o.c 8
93.o odd 30 2 2883.1.o.d 8
93.p even 30 2 2883.1.h.b 4
93.p even 30 4 2883.1.o.a 8
93.p even 30 2 2883.1.o.b 8
124.l odd 10 1 1488.1.br.a 4
155.n even 10 1 2325.1.ca.a 4
155.s odd 20 2 2325.1.bq.a 8
279.z even 15 2 2511.1.bu.a 8
279.bf odd 30 2 2511.1.bu.a 8
372.t even 10 1 1488.1.br.a 4
465.x odd 10 1 2325.1.ca.a 4
465.bj even 20 2 2325.1.bq.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.1.l.a 4 1.a even 1 1 trivial
93.1.l.a 4 3.b odd 2 1 CM
93.1.l.a 4 31.d even 5 1 inner
93.1.l.a 4 93.l odd 10 1 inner
1488.1.br.a 4 4.b odd 2 1
1488.1.br.a 4 12.b even 2 1
1488.1.br.a 4 124.l odd 10 1
1488.1.br.a 4 372.t even 10 1
2325.1.bq.a 8 5.c odd 4 2
2325.1.bq.a 8 15.e even 4 2
2325.1.bq.a 8 155.s odd 20 2
2325.1.bq.a 8 465.bj even 20 2
2325.1.ca.a 4 5.b even 2 1
2325.1.ca.a 4 15.d odd 2 1
2325.1.ca.a 4 155.n even 10 1
2325.1.ca.a 4 465.x odd 10 1
2511.1.bu.a 8 9.c even 3 2
2511.1.bu.a 8 9.d odd 6 2
2511.1.bu.a 8 279.z even 15 2
2511.1.bu.a 8 279.bf odd 30 2
2883.1.b.a 2 31.f odd 10 1
2883.1.b.a 2 93.k even 10 1
2883.1.b.b 2 31.d even 5 1
2883.1.b.b 2 93.l odd 10 1
2883.1.h.a 4 31.g even 15 2
2883.1.h.a 4 93.o odd 30 2
2883.1.h.b 4 31.h odd 30 2
2883.1.h.b 4 93.p even 30 2
2883.1.l.a 4 31.d even 5 2
2883.1.l.a 4 93.l odd 10 2
2883.1.l.b 4 31.b odd 2 1
2883.1.l.b 4 31.f odd 10 1
2883.1.l.b 4 93.c even 2 1
2883.1.l.b 4 93.k even 10 1
2883.1.l.c 4 31.f odd 10 2
2883.1.l.c 4 93.k even 10 2
2883.1.o.a 8 31.h odd 30 4
2883.1.o.a 8 93.p even 30 4
2883.1.o.b 8 31.e odd 6 2
2883.1.o.b 8 31.h odd 30 2
2883.1.o.b 8 93.g even 6 2
2883.1.o.b 8 93.p even 30 2
2883.1.o.c 8 31.g even 15 4
2883.1.o.c 8 93.o odd 30 4
2883.1.o.d 8 31.c even 3 2
2883.1.o.d 8 31.g even 15 2
2883.1.o.d 8 93.h odd 6 2
2883.1.o.d 8 93.o odd 30 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(93, [\chi])$$.

## Hecke characteristic polynomials

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