LMFDB - Newform orbit 1488.1.dc.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1488,1,Mod(239,1488)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1488, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 0, 15, 17])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1488.239"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1488 = 2^{4} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1488.dc (of order $30$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$0.742608738798$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

$q$-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_{30}^{14} q^{3} + ( - \zeta_{30}^{11} - \zeta_{30}^{8}) q^{7} - \zeta_{30}^{13} q^{9} + (\zeta_{30}^{6} - \zeta_{30}^{2}) q^{13} + ( - \zeta_{30}^{4} - \zeta_{30}^{3}) q^{19} + (\zeta_{30}^{10} + \zeta_{30}^{7}) q^{21} + \cdots + ( - \zeta_{30}^{14} - \zeta_{30}^{10}) q^{97} +O(q^{100}) $ q + z^14 * q^3 + (-z^11 - z^8) * q^7 - z^13 * q^9 + (z^6 - z^2) * q^13 + (-z^4 - z^3) * q^19 + (z^10 + z^7) * q^21 - z^5 * q^25 + z^12 * q^27 + z^9 * q^31 + (-z^9 + z) * q^37 + (-z^5 + z) * q^39 + (-z^12 - z^10) * q^43 + (-z^7 - z^4 - z) * q^49 + (z^3 + z^2) * q^57 + (-z^12 - z^3) * q^61 + (-z^9 - z^6) * q^63 + (-z^13 + z^7) * q^67 + (z^8 - z^6) * q^73 + z^4 * q^75 + (z^11 + z^5) * q^79 - z^11 * q^81 + (-z^14 + z^13 + z^10 + z^2) * q^91 - z^8 * q^93 + (-z^14 - z^10) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} + q^{9} - 3 q^{13} - 3 q^{19} - 5 q^{21} - 4 q^{25} - 2 q^{27} + 2 q^{31} - 3 q^{37} - 5 q^{39} + 6 q^{43} + q^{49} + 3 q^{57} + 3 q^{73} + q^{75} + 3 q^{79} + q^{81} - 5 q^{91} - q^{93}+ \cdots + 3 q^{97}+O(q^{100}) $ 8 * q + q^3 + q^9 - 3 * q^13 - 3 * q^19 - 5 * q^21 - 4 * q^25 - 2 * q^27 + 2 * q^31 - 3 * q^37 - 5 * q^39 + 6 * q^43 + q^49 + 3 * q^57 + 3 * q^73 + q^75 + 3 * q^79 + q^81 - 5 * q^91 - q^93 + 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1488\mathbb{Z}\right)^\times$.

$n$	$373$	$497$	$559$	$1057$
$\chi(n)$	$1$	$-1$	$-1$	$\zeta_{30}^{13}$

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} $

239.1

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

−0.978148

−

0.207912i

0.773659

1.73767i

0.913545

0.406737i

383.1

0.669131

0.743145i

−1.89169

0.198825i

−0.104528

0.994522i

575.1

0.669131

−

0.743145i

−1.89169

−

0.198825i

−0.104528

−

0.994522i

623.1

−0.104528

0.994522i

0.244415

1.14988i

−0.978148

−

0.207912i

911.1

0.913545

0.406737i

0.873619

0.786610i

0.669131

0.743145i

1199.1

−0.104528

−

0.994522i

0.244415

−

1.14988i

−0.978148

0.207912i

1295.1

−0.978148

0.207912i

0.773659

−

1.73767i

0.913545

−

0.406737i

1439.1

0.913545

−

0.406737i

0.873619

−

0.786610i

0.669131

−

0.743145i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
124.p	even	30	1	inner
372.bc	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1488.1.dc.b	yes	8
3.b	odd	2	1	CM	1488.1.dc.b	yes	8
4.b	odd	2	1		1488.1.dc.a	✓	8
12.b	even	2	1		1488.1.dc.a	✓	8
31.h	odd	30	1		1488.1.dc.a	✓	8
93.p	even	30	1		1488.1.dc.a	✓	8
124.p	even	30	1	inner	1488.1.dc.b	yes	8
372.bc	odd	30	1	inner	1488.1.dc.b	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1488.1.dc.a	✓	8	4.b	odd	2	1
1488.1.dc.a	✓	8	12.b	even	2	1
1488.1.dc.a	✓	8	31.h	odd	30	1
1488.1.dc.a	✓	8	93.p	even	30	1
1488.1.dc.b	yes	8	1.a	even	1	1	trivial
1488.1.dc.b	yes	8	3.b	odd	2	1	CM
1488.1.dc.b	yes	8	124.p	even	30	1	inner
1488.1.dc.b	yes	8	372.bc	odd	30	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} + 10T_{7}^{5} - 5T_{7}^{4} + 25T_{7}^{2} - 25T_{7} + 25 $ T7^8 + 10*T7^5 - 5*T7^4 + 25*T7^2 - 25*T7 + 25 acting on $S_{1}^{\mathrm{new}}(1488, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 10 T^{5} + \cdots + 25 $ T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + 6T^6 + 14T^5 + 19T^4 + 12T^3 + 4T^2 + T + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + 6T^6 + 4T^5 + 4T^4 - 3T^3 + 4T^2 - 4*T + 1
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$37$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + T^6 - 6T^5 - T^4 + 12T^3 + 14T^2 + 6*T + 1
$41$	$ T^{8} $ T^8
$43$	$ T^{8} - 6 T^{7} + \cdots + 1 $ T^8 - 6T^7 + 20T^6 - 44T^5 + 69T^4 - 74T^3 + 45T^2 - 11*T + 1
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} + 5 T^{2} + 5)^{2} $ (T^4 + 5*T^2 + 5)^2
$67$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$71$	$ T^{8} $ T^8
$73$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 6T^6 - 4T^5 + 4T^4 + 3T^3 + 4T^2 + 4*T + 1
$79$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 8T^6 - 11T^5 + 15T^4 - 11T^3 + 18T^2 + 7*T + 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

Level:	\( N \)	\(=\)	\( 1488 = 2^{4} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1488.dc (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{30}^{14} q^{3} + ( - \zeta_{30}^{11} - \zeta_{30}^{8}) q^{7} - \zeta_{30}^{13} q^{9} + (\zeta_{30}^{6} - \zeta_{30}^{2}) q^{13} + ( - \zeta_{30}^{4} - \zeta_{30}^{3}) q^{19} + (\zeta_{30}^{10} + \zeta_{30}^{7}) q^{21} + \cdots + ( - \zeta_{30}^{14} - \zeta_{30}^{10}) q^{97} +O(q^{100}) \) q + z^14 * q^3 + (-z^11 - z^8) * q^7 - z^13 * q^9 + (z^6 - z^2) * q^13 + (-z^4 - z^3) * q^19 + (z^10 + z^7) * q^21 - z^5 * q^25 + z^12 * q^27 + z^9 * q^31 + (-z^9 + z) * q^37 + (-z^5 + z) * q^39 + (-z^12 - z^10) * q^43 + (-z^7 - z^4 - z) * q^49 + (z^3 + z^2) * q^57 + (-z^12 - z^3) * q^61 + (-z^9 - z^6) * q^63 + (-z^13 + z^7) * q^67 + (z^8 - z^6) * q^73 + z^4 * q^75 + (z^11 + z^5) * q^79 - z^11 * q^81 + (-z^14 + z^13 + z^10 + z^2) * q^91 - z^8 * q^93 + (-z^14 - z^10) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} + q^{9} - 3 q^{13} - 3 q^{19} - 5 q^{21} - 4 q^{25} - 2 q^{27} + 2 q^{31} - 3 q^{37} - 5 q^{39} + 6 q^{43} + q^{49} + 3 q^{57} + 3 q^{73} + q^{75} + 3 q^{79} + q^{81} - 5 q^{91} - q^{93}+ \cdots + 3 q^{97}+O(q^{100}) \) 8 * q + q^3 + q^9 - 3 * q^13 - 3 * q^19 - 5 * q^21 - 4 * q^25 - 2 * q^27 + 2 * q^31 - 3 * q^37 - 5 * q^39 + 6 * q^43 + q^49 + 3 * q^57 + 3 * q^73 + q^75 + 3 * q^79 + q^81 - 5 * q^91 - q^93 + 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	1488.1.dc.b

Level	$1488$
Weight	$1$
Character orbit	1488.dc
Analytic conductor	$0.743$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{30}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{30}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

\(n\)	\(373\)	\(497\)	\(559\)	\(1057\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(\zeta_{30}^{13}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 239.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
124.p	even	30	1	inner
372.bc	odd	30	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} - T^{7} + T^{5} + \cdots + 1 \) T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 10 T^{5} + \cdots + 25 \) T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + 6T^6 + 14T^5 + 19T^4 + 12T^3 + 4T^2 + T + 1
$17$	\( T^{8} \) T^8
$19$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + 6T^6 + 4T^5 + 4T^4 - 3T^3 + 4T^2 - 4*T + 1
$23$	\( T^{8} \) T^8
$29$	\( T^{8} \) T^8
$31$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$37$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + T^6 - 6T^5 - T^4 + 12T^3 + 14T^2 + 6*T + 1
$41$	\( T^{8} \) T^8
$43$	\( T^{8} - 6 T^{7} + \cdots + 1 \) T^8 - 6T^7 + 20T^6 - 44T^5 + 69T^4 - 74T^3 + 45T^2 - 11*T + 1
$47$	\( T^{8} \) T^8
$53$	\( T^{8} \) T^8
$59$	\( T^{8} \) T^8
$61$	\( (T^{4} + 5 T^{2} + 5)^{2} \) (T^4 + 5*T^2 + 5)^2
$67$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$71$	\( T^{8} \) T^8
$73$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 6T^6 - 4T^5 + 4T^4 + 3T^3 + 4T^2 + 4*T + 1
$79$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$83$	\( T^{8} \) T^8
$89$	\( T^{8} \) T^8
$97$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 8T^6 - 11T^5 + 15T^4 - 11T^3 + 18T^2 + 7*T + 1
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