LMFDB - Newform orbit 1700.1.cf.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1700,1,Mod(19,1700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 36, 35])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1700.19"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$1700 = 2^{2} \cdot 5^{2} \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1700.cf (of order $40$ , degree $16$ , minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$0.848410521476$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{12} + x^{8} - x^{4} + 1$ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

$q$ -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.

$f(q)$	$=$	$q + \zeta_{40}^{17} q^{2} - \zeta_{40}^{14} q^{4} - \zeta_{40}^{11} q^{5} + \zeta_{40}^{11} q^{8} + \zeta_{40}^{13} q^{9} + \zeta_{40}^{8} q^{10} + ( - \zeta_{40}^{9} - \zeta_{40}^{7}) q^{13} - \zeta_{40}^{8} q^{16} + \cdots - \zeta_{40}^{12} q^{98} +O(q^{100})$ q + z^17 * q^2 - z^14 * q^4 - z^11 * q^5 + z^11 * q^8 + z^13 * q^9 + z^8 * q^10 + (-z^9 - z^7) * q^13 - z^8 * q^16 - z^14 * q^17 - z^10 * q^18 - z^5 * q^20 - z^2 * q^25 + (z^6 + z^4) * q^26 + (z^12 + z) * q^29 + z^5 * q^32 + z^11 * q^34 + z^7 * q^36 + (-z^11 - z^10) * q^37 + z^2 * q^40 + (z^19 - z^12) * q^41 + z^4 * q^45 + z^15 * q^49 - z^19 * q^50 + (-z^3 - z) * q^52 + (-z^18 - 1) * q^53 + (z^18 - z^9) * q^58 + (-z^13 - z^6) * q^61 - z^2 * q^64 + (z^18 - 1) * q^65 - z^8 * q^68 - z^4 * q^72 + (z^16 - z^13) * q^73 + (z^8 + z^7) * q^74 + z^19 * q^80 - z^6 * q^81 + (-z^16 + z^9) * q^82 - z^5 * q^85 + (-z^15 + z^9) * q^89 - z * q^90 + (-z^7 + z^6) * q^97 - z^12 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 4 q^{10} + 4 q^{16} + 4 q^{26} + 4 q^{29} - 4 q^{41} + 4 q^{45} - 16 q^{53} - 16 q^{65} + 4 q^{68} - 4 q^{72} - 4 q^{73} - 4 q^{74} + 4 q^{82} - 4 q^{98}+O(q^{100})$ 16 * q - 4 * q^10 + 4 * q^16 + 4 * q^26 + 4 * q^29 - 4 * q^41 + 4 * q^45 - 16 * q^53 - 16 * q^65 + 4 * q^68 - 4 * q^72 - 4 * q^73 - 4 * q^74 + 4 * q^82 - 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$ .

$n$	$477$	$851$	$1601$
$\chi(n)$	$\zeta_{40}^{4}$	$-1$	$-\zeta_{40}^{5}$

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}

19.1

−0.987688	+	0.156434i
−0.891007	+	0.453990i
−0.987688	−	0.156434i
0.987688	−	0.156434i
−0.453990	−	0.891007i
−0.156434	−	0.987688i
0.453990	+	0.891007i
0.453990	−	0.891007i
0.891007	−	0.453990i
0.891007	+	0.453990i
0.987688	+	0.156434i
−0.891007	−	0.453990i
−0.156434	+	0.987688i
0.156434	+	0.987688i
−0.453990	+	0.891007i
0.156434	−	0.987688i

0.891007

0.453990i

0.587785

0.809017i

−0.156434

−

0.987688i

0.156434

0.987688i

0.453990

0.891007i

0.309017

−

0.951057i

59.1

0.156434

0.987688i

−0.951057

0.309017i

0.453990

0.891007i

−0.453990

−

0.891007i

−0.987688

−

0.156434i

−0.809017

0.587785i

179.1

0.891007

−

0.453990i

0.587785

−

0.809017i

−0.156434

0.987688i

0.156434

−

0.987688i

0.453990

−

0.891007i

0.309017

0.951057i

219.1

−0.891007

−

0.453990i

0.587785

0.809017i

0.156434

0.987688i

−0.156434

−

0.987688i

−0.453990

−

0.891007i

0.309017

−

0.951057i

359.1

−0.987688

0.156434i

0.951057

−

0.309017i

0.891007

−

0.453990i

−0.891007

0.453990i

0.156434

−

0.987688i

−0.809017

0.587785i

519.1

−0.453990

0.891007i

−0.587785

−

0.809017i

−0.987688

0.156434i

0.987688

−

0.156434i

−0.891007

0.453990i

0.309017

−

0.951057i

559.1

0.987688

−

0.156434i

0.951057

−

0.309017i

−0.891007

0.453990i

0.891007

−

0.453990i

−0.156434

0.987688i

−0.809017

0.587785i

739.1

0.987688

0.156434i

0.951057

0.309017i

−0.891007

−

0.453990i

0.891007

0.453990i

−0.156434

−

0.987688i

−0.809017

−

0.587785i

859.1

−0.156434

−

0.987688i

−0.951057

0.309017i

−0.453990

−

0.891007i

0.453990

0.891007i

0.987688

0.156434i

−0.809017

0.587785i

1039.1

−0.156434

0.987688i

−0.951057

−

0.309017i

−0.453990

0.891007i

0.453990

−

0.891007i

0.987688

−

0.156434i

−0.809017

−

0.587785i

1079.1

−0.891007

0.453990i

0.587785

−

0.809017i

0.156434

−

0.987688i

−0.156434

0.987688i

−0.453990

0.891007i

0.309017

0.951057i

1239.1

0.156434

−

0.987688i

−0.951057

−

0.309017i

0.453990

−

0.891007i

−0.453990

0.891007i

−0.987688

0.156434i

−0.809017

−

0.587785i

1379.1

−0.453990

−

0.891007i

−0.587785

0.809017i

−0.987688

−

0.156434i

0.987688

0.156434i

−0.891007

−

0.453990i

0.309017

0.951057i

1419.1

0.453990

−

0.891007i

−0.587785

−

0.809017i

0.987688

−

0.156434i

−0.987688

0.156434i

0.891007

−

0.453990i

0.309017

−

0.951057i

1539.1

−0.987688

−

0.156434i

0.951057

0.309017i

0.891007

0.453990i

−0.891007

−

0.453990i

0.156434

0.987688i

−0.809017

−

0.587785i

1579.1

0.453990

0.891007i

−0.587785

0.809017i

0.987688

0.156434i

−0.987688

−

0.156434i

0.891007

0.453990i

0.309017

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
425.bd	even	40	1	inner
1700.cf	odd	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1700.1.cf.a	✓	16
4.b	odd	2	1	CM	1700.1.cf.a	✓	16
17.d	even	8	1		1700.1.cf.b	yes	16
25.e	even	10	1		1700.1.cf.b	yes	16
68.g	odd	8	1		1700.1.cf.b	yes	16
100.h	odd	10	1		1700.1.cf.b	yes	16
425.bd	even	40	1	inner	1700.1.cf.a	✓	16
1700.cf	odd	40	1	inner	1700.1.cf.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1700.1.cf.a	✓	16	1.a	even	1	1	trivial
1700.1.cf.a	✓	16	4.b	odd	2	1	CM
1700.1.cf.a	✓	16	425.bd	even	40	1	inner
1700.1.cf.a	✓	16	1700.cf	odd	40	1	inner
1700.1.cf.b	yes	16	17.d	even	8	1
1700.1.cf.b	yes	16	25.e	even	10	1
1700.1.cf.b	yes	16	68.g	odd	8	1
1700.1.cf.b	yes	16	100.h	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $T_{29}^{16} - 4 T_{29}^{15} + 10 T_{29}^{14} - 20 T_{29}^{13} + 34 T_{29}^{12} - 44 T_{29}^{11} + \cdots + 1$ T29^16 - 4*T29^15 + 10*T29^14 - 20*T29^13 + 34*T29^12 - 44*T29^11 + 32*T29^10 - 60*T29^9 + 156*T29^8 - 284*T29^7 + 382*T29^6 - 328*T29^5 + 269*T29^4 - 68*T29^3 - 4*T29^2 + 8*T29 + 1 acting on $S_{1}^{\mathrm{new}}(1700, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} - T^{12} + \cdots + 1$ T^16 - T^12 + T^8 - T^4 + 1
$3$	$T^{16}$ T^16
$5$	$T^{16} - T^{12} + \cdots + 1$ T^16 - T^12 + T^8 - T^4 + 1
$7$	$T^{16}$ T^16
$11$	$T^{16}$ T^16
$13$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 12T^12 + 32T^10 + 150T^8 - 32T^6 + 97T^4 + 16*T^2 + 1
$17$	$(T^{8} - T^{6} + T^{4} + \cdots + 1)^{2}$ (T^8 - T^6 + T^4 - T^2 + 1)^2
$19$	$T^{16}$ T^16
$23$	$T^{16}$ T^16
$29$	$T^{16} - 4 T^{15} + \cdots + 1$ T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 32T^10 - 60T^9 + 156T^8 - 284T^7 + 382T^6 - 328T^5 + 269T^4 - 68T^3 - 4T^2 + 8*T + 1
$31$	$T^{16}$ T^16
$37$	$T^{16} + 8 T^{14} + \cdots + 1$ T^16 + 8T^14 - 4T^13 + 27T^12 - 24T^11 + 66T^10 - 52T^9 + 120T^8 - 112T^7 + 88T^6 - 132T^5 + 242T^4 - 104T^3 - 12T^2 + 8T + 1
$41$	$T^{16} + 4 T^{15} + \cdots + 1$ T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 82T^10 + 160T^9 + 256T^8 + 304T^7 + 162T^6 - 72T^5 - 131T^4 - 32T^3 + 46T^2 - 8*T + 1
$43$	$T^{16}$ T^16
$47$	$T^{16}$ T^16
$53$	$(T^{8} + 8 T^{7} + 27 T^{6} + \cdots + 1)^{2}$ (T^8 + 8T^7 + 27T^6 + 50T^5 + 56T^4 + 40T^3 + 18T^2 + 4*T + 1)^2
$59$	$T^{16}$ T^16
$61$	$T^{16} - 2 T^{14} + \cdots + 1$ T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 32T^9 - 100T^8 - 32T^7 + 298T^6 + 628T^5 + 557T^4 + 256T^3 + 68T^2 + 8T + 1
$67$	$T^{16}$ T^16
$71$	$T^{16}$ T^16
$73$	$T^{16} + 4 T^{15} + \cdots + 1$ T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 32T^10 + 60T^9 + 156T^8 + 284T^7 + 382T^6 + 328T^5 + 269T^4 + 68T^3 - 4T^2 - 8*T + 1
$79$	$T^{16}$ T^16
$83$	$T^{16}$ T^16
$89$	$T^{16} - 4 T^{14} + \cdots + 1$ T^16 - 4T^14 + 17T^12 - 72T^10 + 230T^8 - 228T^6 + 92T^4 + 4*T^2 + 1
$97$	$T^{16} - 2 T^{14} + \cdots + 1$ T^16 - 2T^14 + 16T^13 + 2T^12 - 24T^11 + 86T^10 + 8T^9 - 80T^8 + 208T^7 - 42T^6 - 112T^5 + 237T^4 - 104T^3 + 58T^2 - 12T + 1
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Classical	Maass
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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 19.1
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	1700.1.cf.a

Level	$1700$
Weight	$1$
Character orbit	1700.cf
Analytic conductor	$0.848$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{40}$
CM discriminant	-4
Inner twists	$4$