LMFDB - Newform orbit 1800.1.dg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(211,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 15, 10, 24]))

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

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

chi := DirichletCharacter("1800.211");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1800.dg (of order $30$ , degree $8$ , 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:	$0.898317022739$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.2025000000.9

$q$ -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.

$f(q)$	$=$	$q + \zeta_{60}^{29} q^{2} + \zeta_{60}^{18} q^{3} - \zeta_{60}^{28} q^{4} + \zeta_{60}^{13} q^{5} - \zeta_{60}^{17} q^{6} + \zeta_{60}^{5} q^{7} + \zeta_{60}^{27} q^{8} - \zeta_{60}^{6} q^{9} - \zeta_{60}^{12} q^{10} + \cdots + (\zeta_{60}^{26} + \zeta_{60}^{14}) q^{99} +O(q^{100})$ q + z^29 * q^2 + z^18 * q^3 - z^28 * q^4 + z^13 * q^5 - z^17 * q^6 + z^5 * q^7 + z^27 * q^8 - z^6 * q^9 - z^12 * q^10 + (-z^20 - z^8) * q^11 + z^16 * q^12 + (z^19 - z^13) * q^13 - z^4 * q^14 - z * q^15 - z^26 * q^16 + z^12 * q^17 + z^5 * q^18 + z^12 * q^19 + z^11 * q^20 + z^23 * q^21 + (z^19 + z^7) * q^22 + z^19 * q^23 - z^15 * q^24 + z^26 * q^25 + (-z^18 + z^12) * q^26 - z^24 * q^27 + z^3 * q^28 + (z^29 + z^17) * q^29 + q^30 + z^7 * q^31 + z^25 * q^32 + (-z^26 + z^8) * q^33 - z^11 * q^34 + z^18 * q^35 - z^4 * q^36 - z^11 * q^38 + (-z^7 + z) * q^39 - z^10 * q^40 + (z^28 + z^4) * q^41 - z^22 * q^42 + z^20 * q^43 + (-z^18 - z^6) * q^44 - z^19 * q^45 - z^18 * q^46 + (-z^29 - z^17) * q^47 + z^14 * q^48 - z^25 * q^50 - q^51 + (z^17 - z^11) * q^52 + (z^21 - z^15) * q^53 + z^23 * q^54 + (-z^21 + z^3) * q^55 - z^2 * q^56 - q^57 + (-z^28 - z^16) * q^58 + z^29 * q^60 + z^29 * q^61 - z^6 * q^62 - z^11 * q^63 - z^24 * q^64 + (-z^26 - z^2) * q^65 + (z^25 - z^7) * q^66 + (z^10 - z^4) * q^67 + z^10 * q^68 - z^7 * q^69 - z^17 * q^70 + z^3 * q^72 - z^14 * q^75 + z^10 * q^76 + (-z^25 - z^13) * q^77 + (z^6 - 1) * q^78 + (-z^29 - z^17) * q^79 + z^9 * q^80 + z^12 * q^81 + (-z^27 - z^3) * q^82 + (-z^26 + z^8) * q^83 + z^21 * q^84 + z^25 * q^85 - z^19 * q^86 + (-z^17 - z^5) * q^87 + (z^17 + z^5) * q^88 + z^24 * q^89 + z^18 * q^90 + (z^24 - z^18) * q^91 + z^17 * q^92 + z^25 * q^93 + (z^28 + z^16) * q^94 + z^25 * q^95 - z^13 * q^96 + (z^26 + z^14) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 4 q^{3} - 2 q^{4} - 4 q^{9} + 4 q^{10} + 6 q^{11} + 2 q^{12} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 4 q^{19} - 2 q^{25} - 8 q^{26} + 4 q^{27} + 16 q^{30} + 4 q^{33} + 4 q^{35} - 2 q^{36} - 8 q^{40} + 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100})$ 16 * q + 4 * q^3 - 2 * q^4 - 4 * q^9 + 4 * q^10 + 6 * q^11 + 2 * q^12 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 4 * q^19 - 2 * q^25 - 8 * q^26 + 4 * q^27 + 16 * q^30 + 4 * q^33 + 4 * q^35 - 2 * q^36 - 8 * q^40 + 4 * q^41 + 2 * q^42 - 8 * q^43 - 8 * q^44 - 4 * q^46 - 2 * q^48 - 16 * q^51 + 2 * q^56 - 16 * q^57 - 4 * q^58 - 4 * q^62 + 4 * q^64 + 4 * q^65 + 6 * q^67 + 8 * q^68 + 2 * q^75 + 8 * q^76 - 12 * q^78 - 4 * q^81 + 4 * q^83 - 4 * q^89 + 4 * q^90 - 8 * q^91 + 4 * q^94 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-\zeta_{60}^{18}$	$-1$	$-\zeta_{60}^{10}$	$-1$

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}

211.1

0.406737	+	0.913545i
−0.406737	−	0.913545i
0.207912	+	0.978148i
−0.207912	−	0.978148i
0.207912	−	0.978148i
−0.207912	+	0.978148i
0.406737	−	0.913545i
−0.406737	+	0.913545i
0.743145	−	0.669131i
−0.743145	+	0.669131i
0.994522	−	0.104528i
−0.994522	+	0.104528i
0.994522	+	0.104528i
−0.994522	−	0.104528i
0.743145	+	0.669131i
−0.743145	−	0.669131i

−0.406737

0.913545i

−0.309017

0.951057i

−0.669131

−

0.743145i

−0.743145

0.669131i

−0.743145

−

0.669131i

0.866025

−

0.500000i

0.951057

−

0.309017i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

211.2

0.406737

−

0.913545i

−0.309017

0.951057i

−0.669131

−

0.743145i

0.743145

−

0.669131i

0.743145

0.669131i

−0.866025

0.500000i

−0.951057

0.309017i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

331.1

−0.207912

0.978148i

0.809017

−

0.587785i

−0.913545

−

0.406737i

0.406737

−

0.913545i

0.406737

0.913545i

0.866025

0.500000i

0.587785

−

0.809017i

0.309017

−

0.951057i

0.809017

0.587785i

331.2

0.207912

−

0.978148i

0.809017

−

0.587785i

−0.913545

−

0.406737i

−0.406737

0.913545i

−0.406737

−

0.913545i

−0.866025

−

0.500000i

−0.587785

0.809017i

0.309017

−

0.951057i

0.809017

0.587785i

571.1

−0.207912

−

0.978148i

0.809017

0.587785i

−0.913545

0.406737i

0.406737

0.913545i

0.406737

−

0.913545i

0.866025

−

0.500000i

0.587785

0.809017i

0.309017

0.951057i

0.809017

−

0.587785i

571.2

0.207912

0.978148i

0.809017

0.587785i

−0.913545

0.406737i

−0.406737

−

0.913545i

−0.406737

0.913545i

−0.866025

0.500000i

−0.587785

−

0.809017i

0.309017

0.951057i

0.809017

−

0.587785i

691.1

−0.406737

−

0.913545i

−0.309017

−

0.951057i

−0.669131

0.743145i

−0.743145

−

0.669131i

−0.743145

0.669131i

0.866025

0.500000i

0.951057

0.309017i

−0.809017

0.587785i

−0.309017

0.951057i

691.2

0.406737

0.913545i

−0.309017

−

0.951057i

−0.669131

0.743145i

0.743145

0.669131i

0.743145

−

0.669131i

−0.866025

−

0.500000i

−0.951057

−

0.309017i

−0.809017

0.587785i

−0.309017

0.951057i

931.1

−0.743145

−

0.669131i

0.809017

−

0.587785i

0.104528

0.994522i

−0.994522

0.104528i

−0.994522

−

0.104528i

−0.866025

0.500000i

0.587785

−

0.809017i

0.309017

−

0.951057i

0.809017

0.587785i

931.2

0.743145

0.669131i

0.809017

−

0.587785i

0.104528

0.994522i

0.994522

−

0.104528i

0.994522

0.104528i

0.866025

−

0.500000i

−0.587785

0.809017i

0.309017

−

0.951057i

0.809017

0.587785i

1291.1

−0.994522

−

0.104528i

−0.309017

−

0.951057i

0.978148

0.207912i

0.207912

−

0.978148i

0.207912

0.978148i

0.866025

−

0.500000i

−0.951057

−

0.309017i

−0.809017

0.587785i

−0.309017

0.951057i

1291.2

0.994522

0.104528i

−0.309017

−

0.951057i

0.978148

0.207912i

−0.207912

0.978148i

−0.207912

−

0.978148i

−0.866025

0.500000i

0.951057

0.309017i

−0.809017

0.587785i

−0.309017

0.951057i

1411.1

−0.994522

0.104528i

−0.309017

0.951057i

0.978148

−

0.207912i

0.207912

0.978148i

0.207912

−

0.978148i

0.866025

0.500000i

−0.951057

0.309017i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

1411.2

0.994522

−

0.104528i

−0.309017

0.951057i

0.978148

−

0.207912i

−0.207912

−

0.978148i

−0.207912

0.978148i

−0.866025

−

0.500000i

0.951057

−

0.309017i

−0.809017

−

0.587785i

−0.309017

−

0.951057i

1771.1

−0.743145

0.669131i

0.809017

0.587785i

0.104528

−

0.994522i

−0.994522

−

0.104528i

−0.994522

0.104528i

−0.866025

−

0.500000i

0.587785

0.809017i

0.309017

0.951057i

0.809017

−

0.587785i

1771.2

0.743145

−

0.669131i

0.809017

0.587785i

0.104528

−

0.994522i

0.994522

0.104528i

0.994522

−

0.104528i

0.866025

0.500000i

−0.587785

−

0.809017i

0.309017

0.951057i

0.809017

−

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
9.c	even	3	1	inner
25.d	even	5	1	inner
72.p	odd	6	1	inner
200.n	odd	10	1	inner
225.q	even	15	1	inner
1800.dg	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1800.1.dg.a	✓	16
8.d	odd	2	1	inner	1800.1.dg.a	✓	16
9.c	even	3	1	inner	1800.1.dg.a	✓	16
25.d	even	5	1	inner	1800.1.dg.a	✓	16
72.p	odd	6	1	inner	1800.1.dg.a	✓	16
200.n	odd	10	1	inner	1800.1.dg.a	✓	16
225.q	even	15	1	inner	1800.1.dg.a	✓	16
1800.dg	odd	30	1	inner	1800.1.dg.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1800.1.dg.a	✓	16	1.a	even	1	1	trivial
1800.1.dg.a	✓	16	8.d	odd	2	1	inner
1800.1.dg.a	✓	16	9.c	even	3	1	inner
1800.1.dg.a	✓	16	25.d	even	5	1	inner
1800.1.dg.a	✓	16	72.p	odd	6	1	inner
1800.1.dg.a	✓	16	200.n	odd	10	1	inner
1800.1.dg.a	✓	16	225.q	even	15	1	inner
1800.1.dg.a	✓	16	1800.dg	odd	30	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1800, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$3$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 - T^3 + T^2 - T + 1)^4
$5$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$7$	$(T^{4} - T^{2} + 1)^{4}$ (T^4 - T^2 + 1)^4
$11$	$(T^{8} - 3 T^{7} + 5 T^{6} + \cdots + 1)^{2}$ (T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1)^2
$13$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$17$	$(T^{4} + T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 + T^3 + T^2 + T + 1)^4
$19$	$(T^{4} + T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 + T^3 + T^2 + T + 1)^4
$23$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$29$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$31$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$37$	$T^{16}$ T^16
$41$	$(T^{8} - 2 T^{7} - 2 T^{5} + \cdots + 1)^{2}$ (T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1)^2
$43$	$(T^{2} + T + 1)^{8}$ (T^2 + T + 1)^8
$47$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$53$	$(T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2}$ (T^8 + T^6 + 6T^4 - 4T^2 + 1)^2
$59$	$T^{16}$ T^16
$61$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$67$	$(T^{8} - 3 T^{7} + 5 T^{6} + \cdots + 1)^{2}$ (T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1)^2
$71$	$T^{16}$ T^16
$73$	$T^{16}$ T^16
$79$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$83$	$(T^{8} - 2 T^{7} - 2 T^{5} + \cdots + 1)^{2}$ (T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1)^2
$89$	$(T^{4} + T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 + T^3 + T^2 + T + 1)^4
$97$	$T^{16}$ T^16
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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 211.2
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1800.1.dg.a

Level	$1800$
Weight	$1$
Character orbit	1800.dg
Analytic conductor	$0.898$
Analytic rank	$0$
Dimension	$16$
Projective image	$A_{5}$
CM/RM	no
Inner twists	$8$