LMFDB - Newform orbit 372.1.be.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [372,1,Mod(41,372)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("372.41"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 15, 14])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 372 = 2^{2} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	372.be (of order $30$, degree $8$, 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:	$0.185652184699$
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:	$D_{15}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{15} - \cdots)$

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/372\mathbb{Z}\right)^\times$.

$n$	$125$	$187$	$313$
$\chi(n)$	$-1$	$1$	$\zeta_{30}^{14}$

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} $

41.1

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

−0.104528

−

0.994522i

1.58268

0.336408i

−0.978148

0.207912i

113.1

0.669131

0.743145i

−0.0646021

−

0.614648i

−0.104528

0.994522i

173.1

0.913545

−

0.406737i

−1.08268

−

1.20243i

0.669131

−

0.743145i

245.1

−0.104528

0.994522i

1.58268

−

0.336408i

−0.978148

−

0.207912i

257.1

−0.978148

−

0.207912i

0.564602

−

0.251377i

0.913545

0.406737i

293.1

0.669131

−

0.743145i

−0.0646021

0.614648i

−0.104528

−

0.994522i

317.1

−0.978148

0.207912i

0.564602

0.251377i

0.913545

−

0.406737i

329.1

0.913545

0.406737i

−1.08268

1.20243i

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}) $
31.g	even	15	1	inner
93.o	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	372.1.be.a	✓	8
3.b	odd	2	1	CM	372.1.be.a	✓	8
4.b	odd	2	1		1488.1.dh.a		8
12.b	even	2	1		1488.1.dh.a		8
31.g	even	15	1	inner	372.1.be.a	✓	8
93.o	odd	30	1	inner	372.1.be.a	✓	8
124.n	odd	30	1		1488.1.dh.a		8
372.bd	even	30	1		1488.1.dh.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
372.1.be.a	✓	8	1.a	even	1	1	trivial
372.1.be.a	✓	8	3.b	odd	2	1	CM
372.1.be.a	✓	8	31.g	even	15	1	inner
372.1.be.a	✓	8	93.o	odd	30	1	inner
1488.1.dh.a		8	4.b	odd	2	1
1488.1.dh.a		8	12.b	even	2	1
1488.1.dh.a		8	124.n	odd	30	1
1488.1.dh.a		8	372.bd	even	30	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(372, [\chi])$.

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} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + T^{7} + 4 T^{5} + \cdots + 1 $ T^8 + T^7 + 4T^5 + 9T^4 + 4T^3 + 20T^2 - 9*T + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} + T^{7} + 4 T^{5} + \cdots + 1 $ T^8 + T^7 + 4T^5 - 6T^4 - 11T^3 + 20T^2 + 6*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} + T^{7} + 5 T^{6} + \cdots + 1 $ T^8 + T^7 + 5T^6 + 4T^5 + 19T^4 + 14T^3 + 20T^2 - 4T + 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^{2} + T - 1)^{4} $ (T^2 + T - 1)^4
$67$	$ (T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + 2*T^2 + T + 1)^2
$71$	$ T^{8} $ T^8
$73$	$ T^{8} + T^{7} + 4 T^{5} + \cdots + 1 $ T^8 + T^7 + 4T^5 - 6T^4 - 11T^3 + 20T^2 + 6*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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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	372.be (of order \(30\), degree \(8\), minimal)

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

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	372.1.be.a

Level	$372$
Weight	$1$
Character orbit	372.be
Analytic conductor	$0.186$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{15}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.185652184699\)
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:	\(D_{15}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

\(n\)	\(125\)	\(187\)	\(313\)
\(\chi(n)\)	\(-1\)	\(1\)	\(\zeta_{30}^{14}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 41.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
31.g	even	15	1	inner
93.o	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
$7$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) T^8 + T^7 + 4T^5 + 9T^4 + 4T^3 + 20T^2 - 9*T + 1
$17$	\( T^{8} \) T^8
$19$	\( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) T^8 + T^7 + 4T^5 - 6T^4 - 11T^3 + 20T^2 + 6*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
$37$	\( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) T^8 + T^7 + 5T^6 + 4T^5 + 19T^4 + 14T^3 + 20T^2 - 4T + 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
$61$	\( (T^{2} + T - 1)^{4} \) (T^2 + T - 1)^4
$67$	\( (T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + 2*T^2 + T + 1)^2
$71$	\( T^{8} \) T^8
$73$	\( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) T^8 + T^7 + 4T^5 - 6T^4 - 11T^3 + 20T^2 + 6*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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