LMFDB - Newform orbit 1764.1.co.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,1,Mod(73,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 0, 37])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.73"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1764.co (of order $42$ , degree $12$ , minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.880350682285$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q - \zeta_{42}^{16} q^{7} + (\zeta_{42}^{10} - \zeta_{42}^{8}) q^{13} + ( - \zeta_{42}^{12} + \zeta_{42}^{2}) q^{19} - \zeta_{42}^{5} q^{25} + (\zeta_{42}^{4} + \zeta_{42}^{3}) q^{31} + ( - \zeta_{42}^{14} - \zeta_{42}^{6}) q^{37} + \cdots + (\zeta_{42}^{12} + \zeta_{42}^{9}) q^{97} +O(q^{100})$ q - z^16 * q^7 + (z^10 - z^8) * q^13 + (-z^12 + z^2) * q^19 - z^5 * q^25 + (z^4 + z^3) * q^31 + (-z^14 - z^6) * q^37 + (z^17 + z^13) * q^43 - z^11 * q^49 + (-z^13 + z^7) * q^61 + (z^6 - z) * q^67 + (-z^9 + z) * q^73 + (z^18 - z^17) * q^79 + (z^5 - z^3) * q^91 + (z^12 + z^9) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$12 q - q^{7} + 3 q^{19} + q^{25} + 3 q^{31} + 8 q^{37} - 2 q^{43} + q^{49} + 7 q^{61} - q^{67} - 3 q^{73} - q^{79} - 3 q^{91}+O(q^{100})$ 12 * q - q^7 + 3 * q^19 + q^25 + 3 * q^31 + 8 * q^37 - 2 * q^43 + q^49 + 7 * q^61 - q^67 - 3 * q^73 - q^79 - 3 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$ .

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$\zeta_{42}^{19}$

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}

73.1

0.365341	−	0.930874i
0.365341	+	0.930874i
0.826239	−	0.563320i
−0.988831	−	0.149042i
−0.733052	−	0.680173i
−0.733052	+	0.680173i
0.0747301	+	0.997204i
0.955573	+	0.294755i
0.826239	+	0.563320i
0.0747301	−	0.997204i
0.955573	−	0.294755i
−0.988831	+	0.149042i

−0.955573

0.294755i

145.1

−0.955573

−

0.294755i

397.1

0.988831

−

0.149042i

577.1

0.733052

−

0.680173i

649.1

−0.826239

0.563320i

829.1

−0.826239

−

0.563320i

1081.1

−0.365341

0.930874i

1153.1

−0.0747301

0.997204i

1333.1

0.988831

0.149042i

1405.1

−0.365341

−

0.930874i

1585.1

−0.0747301

−

0.997204i

1657.1

0.733052

0.680173i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
49.h	odd	42	1	inner
147.o	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.1.co.a	✓	12
3.b	odd	2	1	CM	1764.1.co.a	✓	12
49.h	odd	42	1	inner	1764.1.co.a	✓	12
147.o	even	42	1	inner	1764.1.co.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1764.1.co.a	✓	12	1.a	even	1	1	trivial
1764.1.co.a	✓	12	3.b	odd	2	1	CM
1764.1.co.a	✓	12	49.h	odd	42	1	inner
1764.1.co.a	✓	12	147.o	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1764, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12}$ T^12
$5$	$T^{12}$ T^12
$7$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$11$	$T^{12}$ T^12
$13$	$T^{12} - 3 T^{10} + \cdots + 1$ T^12 - 3T^10 - 7T^9 + 9T^8 + 7T^7 + 15T^6 - 21T^5 + 4T^4 + 14T^3 + 16T^2 + 7T + 1
$17$	$T^{12}$ T^12
$19$	$T^{12} - 3 T^{11} + \cdots + 1$ T^12 - 3T^11 - T^10 + 12T^9 + 2T^8 - 42T^7 + 36T^6 + 21T^5 - 24T^4 - 12T^3 + 10T^2 + 6T + 1
$23$	$T^{12}$ T^12
$29$	$T^{12}$ T^12
$31$	$T^{12} - 3 T^{11} + \cdots + 1$ T^12 - 3T^11 - T^10 + 12T^9 + 2T^8 - 42T^7 + 36T^6 + 21T^5 - 24T^4 - 12T^3 + 10T^2 + 6T + 1
$37$	$T^{12} - 8 T^{11} + \cdots + 1$ T^12 - 8T^11 + 35T^10 - 104T^9 + 230T^8 - 392T^7 + 519T^6 - 518T^5 + 349T^4 - 118T^3 + 6T + 1
$41$	$T^{12}$ T^12
$43$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 3T^10 + 4T^9 + 12T^8 + 6T^7 + 7T^6 + 6T^5 + 54T^4 - 94T^3 + 52T^2 - 5*T + 1
$47$	$T^{12}$ T^12
$53$	$T^{12}$ T^12
$59$	$T^{12}$ T^12
$61$	$T^{12} - 7 T^{11} + \cdots + 49$ T^12 - 7T^11 + 28T^10 - 77T^9 + 161T^8 - 266T^7 + 357T^6 - 392T^5 + 343T^4 - 245T^3 + 196T^2 - 147*T + 49
$67$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 + 7T^10 + 6T^9 + 34T^8 + 28T^7 + 78T^6 + 49T^5 + 118T^4 + 76T^3 + 56T^2 + 8T + 1
$71$	$T^{12}$ T^12
$73$	$T^{12} + 3 T^{11} + \cdots + 1$ T^12 + 3T^11 + 6T^10 + 9T^9 + 2T^8 + 7T^7 + 8T^6 + 25T^4 - 23T^3 + 3*T^2 + T + 1
$79$	$T^{12} + T^{11} + \cdots + 1$ T^12 + T^11 + 7T^10 + 6T^9 + 34T^8 + 28T^7 + 78T^6 + 49T^5 + 118T^4 + 76T^3 + 56T^2 + 8T + 1
$83$	$T^{12}$ T^12
$89$	$T^{12}$ T^12
$97$	$(T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2}$ (T^6 + 7T^4 + 14T^2 + 7)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 73.1
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more