LMFDB - Newform orbit 344.1.s.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [344,1,Mod(11,344)] 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("344.11"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(344, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 7, 10])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 344 = 2^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	344.s (of order $14$, degree $6$, 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:	$0.171678364346$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.3236537881088.1

$q$-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_{14}^{2} q^{2} + (\zeta_{14}^{2} - \zeta_{14}) q^{3} + \zeta_{14}^{4} q^{4} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{6} + \zeta_{14}^{6} q^{8} + (\zeta_{14}^{4} + \cdots + \zeta_{14}^{2}) q^{9}+ \cdots + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{99}+O(q^{100}) $ q + z^2 * q^2 + (z^2 - z) * q^3 + z^4 * q^4 + (z^4 - z^3) * q^6 + z^6 * q^8 + (z^4 - z^3 + z^2) * q^9 + (-z^5 - z) * q^11 + (z^6 - z^5) * q^12 - z * q^16 + (-z^5 + z^4) * q^17 + (z^6 - z^5 + z^4) * q^18 + (-z + 1) * q^19 + (-z^3 + 1) * q^22 + (-z + 1) * q^24 - z^5 * q^25 + (z^6 - z^5 + z^4 - z^3) * q^27 - z^3 * q^32 + (z^6 - z^3 + z^2 + 1) * q^33 + (z^6 + 1) * q^34 + (z^6 - z + 1) * q^36 + (-z^3 + z^2) * q^38 + (-z^3 - z) * q^41 + z^2 * q^43 + (-z^5 + z^2) * q^44 + (-z^3 + z^2) * q^48 + q^49 + q^50 + (2z^6 - z^5 + 1) q^51 + (z^6 - z^5 - z + 1) * q^54 + (-z^3 + 2z^2 - z) q^57 + (z^6 - z^3) * q^59 - z^5 * q^64 + (-z^5 + z^4 + z^2 - z) * q^66 + 2z^4 q^67 + (z^2 - z) * q^68 + (-z^3 + z^2 - z) * q^72 + 2z^6 q^73 + (z^6 + 1) * q^75 + (-z^5 + z^4) * q^76 + (z^6 - z^5 + z^4 - z + 1) * q^81 + (-z^5 - z^3) * q^82 + (-z^3 + 1) * q^83 + z^4 * q^86 + (z^4 + 1) * q^88 + (z^6 + z^4) * q^89 + (-z^5 + z^4) * q^96 + (-z^5 - z) * q^97 + z^2 * q^98 + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} + 5 q^{22} + 5 q^{24} - q^{25} - 4 q^{27} - q^{32} + 3 q^{33} + 5 q^{34} + 4 q^{36}+ \cdots + q^{99}+O(q^{100}) $ 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 + 5 * q^22 + 5 * q^24 - q^25 - 4 * q^27 - q^32 + 3 * q^33 + 5 * q^34 + 4 * q^36 - 2 * q^38 - 2 * q^41 - q^43 - 2 * q^44 - 2 * q^48 + 6 * q^49 + 6 * q^50 + 3 * q^51 + 3 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 + 5 * q^75 - 2 * q^76 + 2 * q^81 - 2 * q^82 + 5 * q^83 - q^86 + 5 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/344\mathbb{Z}\right)^\times$.

$n$	$87$	$89$	$173$
$\chi(n)$	$-1$	$-\zeta_{14}^{3}$	$-1$

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} $

11.1

0.900969	+	0.433884i
−0.623490	−	0.781831i
−0.623490	+	0.781831i
0.222521	−	0.974928i
0.900969	−	0.433884i
0.222521	+	0.974928i

0.623490

0.781831i

−0.277479

0.347948i

−0.222521

0.974928i

−0.445042

−0.900969

0.433884i

0.178448

0.781831i

35.1

−0.222521

0.974928i

0.400969

1.75676i

−0.900969

−

0.433884i

−1.80194

0.623490

−

0.781831i

−2.02446

0.974928i

59.1

−0.222521

−

0.974928i

0.400969

−

1.75676i

−0.900969

0.433884i

−1.80194

0.623490

0.781831i

−2.02446

−

0.974928i

107.1

−0.900969

−

0.433884i

−1.12349

0.541044i

0.623490

0.781831i

1.24698

−0.222521

−

0.974928i

0.346011

−

0.433884i

219.1

0.623490

−

0.781831i

−0.277479

−

0.347948i

−0.222521

−

0.974928i

−0.445042

−0.900969

−

0.433884i

0.178448

−

0.781831i

299.1

−0.900969

0.433884i

−1.12349

−

0.541044i

0.623490

−

0.781831i

1.24698

−0.222521

0.974928i

0.346011

0.433884i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
43.e	even	7	1	inner
344.s	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	344.1.s.a	✓	6
3.b	odd	2	1		3096.1.dm.a		6
4.b	odd	2	1		1376.1.be.a		6
8.b	even	2	1		1376.1.be.a		6
8.d	odd	2	1	CM	344.1.s.a	✓	6
24.f	even	2	1		3096.1.dm.a		6
43.e	even	7	1	inner	344.1.s.a	✓	6
129.l	odd	14	1		3096.1.dm.a		6
172.k	odd	14	1		1376.1.be.a		6
344.s	odd	14	1	inner	344.1.s.a	✓	6
344.x	even	14	1		1376.1.be.a		6
1032.bk	even	14	1		3096.1.dm.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
344.1.s.a	✓	6	1.a	even	1	1	trivial
344.1.s.a	✓	6	8.d	odd	2	1	CM
344.1.s.a	✓	6	43.e	even	7	1	inner
344.1.s.a	✓	6	344.s	odd	14	1	inner
1376.1.be.a		6	4.b	odd	2	1
1376.1.be.a		6	8.b	even	2	1
1376.1.be.a		6	172.k	odd	14	1
1376.1.be.a		6	344.x	even	14	1
3096.1.dm.a		6	3.b	odd	2	1
3096.1.dm.a		6	24.f	even	2	1
3096.1.dm.a		6	129.l	odd	14	1
3096.1.dm.a		6	1032.bk	even	14	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(344, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{5} + T^{4} + \cdots + 1 $ T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$13$	$ T^{6} $ T^6
$17$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$19$	$ T^{6} - 5 T^{5} + \cdots + 1 $ T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ T^{6} $ T^6
$37$	$ T^{6} $ T^6
$41$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$43$	$ T^{6} + T^{5} + T^{4} + \cdots + 1 $ T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$47$	$ T^{6} $ T^6
$53$	$ T^{6} $ T^6
$59$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1
$61$	$ T^{6} $ T^6
$67$	$ T^{6} + 2 T^{5} + \cdots + 64 $ T^6 + 2T^5 + 4T^4 + 8T^3 + 16T^2 + 32*T + 64
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + 2 T^{5} + \cdots + 64 $ T^6 + 2T^5 + 4T^4 + 8T^3 + 16T^2 + 32*T + 64
$79$	$ T^{6} $ T^6
$83$	$ T^{6} - 5 T^{5} + \cdots + 1 $ T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$89$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$97$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	344.s (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{14}^{2} q^{2} + (\zeta_{14}^{2} - \zeta_{14}) q^{3} + \zeta_{14}^{4} q^{4} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{6} + \zeta_{14}^{6} q^{8} + (\zeta_{14}^{4} + \cdots + \zeta_{14}^{2}) q^{9}+ \cdots + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{99}+O(q^{100}) \) q + z^2 * q^2 + (z^2 - z) * q^3 + z^4 * q^4 + (z^4 - z^3) * q^6 + z^6 * q^8 + (z^4 - z^3 + z^2) * q^9 + (-z^5 - z) * q^11 + (z^6 - z^5) * q^12 - z * q^16 + (-z^5 + z^4) * q^17 + (z^6 - z^5 + z^4) * q^18 + (-z + 1) * q^19 + (-z^3 + 1) * q^22 + (-z + 1) * q^24 - z^5 * q^25 + (z^6 - z^5 + z^4 - z^3) * q^27 - z^3 * q^32 + (z^6 - z^3 + z^2 + 1) * q^33 + (z^6 + 1) * q^34 + (z^6 - z + 1) * q^36 + (-z^3 + z^2) * q^38 + (-z^3 - z) * q^41 + z^2 * q^43 + (-z^5 + z^2) * q^44 + (-z^3 + z^2) * q^48 + q^49 + q^50 + (2z^6 - z^5 + 1) q^51 + (z^6 - z^5 - z + 1) * q^54 + (-z^3 + 2z^2 - z) q^57 + (z^6 - z^3) * q^59 - z^5 * q^64 + (-z^5 + z^4 + z^2 - z) * q^66 + 2z^4 q^67 + (z^2 - z) * q^68 + (-z^3 + z^2 - z) * q^72 + 2z^6 q^73 + (z^6 + 1) * q^75 + (-z^5 + z^4) * q^76 + (z^6 - z^5 + z^4 - z + 1) * q^81 + (-z^5 - z^3) * q^82 + (-z^3 + 1) * q^83 + z^4 * q^86 + (z^4 + 1) * q^88 + (z^6 + z^4) * q^89 + (-z^5 + z^4) * q^96 + (-z^5 - z) * q^97 + z^2 * q^98 + (-z^5 + z^4 - z^3 + z^2 - z + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} + 5 q^{22} + 5 q^{24} - q^{25} - 4 q^{27} - q^{32} + 3 q^{33} + 5 q^{34} + 4 q^{36}+ \cdots + q^{99}+O(q^{100}) \) 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 + 5 * q^22 + 5 * q^24 - q^25 - 4 * q^27 - q^32 + 3 * q^33 + 5 * q^34 + 4 * q^36 - 2 * q^38 - 2 * q^41 - q^43 - 2 * q^44 - 2 * q^48 + 6 * q^49 + 6 * q^50 + 3 * q^51 + 3 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 + 5 * q^75 - 2 * q^76 + 2 * q^81 - 2 * q^82 + 5 * q^83 - q^86 + 5 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 + q^99

Introduction

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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	344.1.s.a

Level	$344$
Weight	$1$
Character orbit	344.s
Analytic conductor	$0.172$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{7}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.171678364346\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.3236537881088.1

\(n\)	\(87\)	\(89\)	\(173\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{14}^{3}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
43.e	even	7	1	inner
344.s	odd	14	1	inner

$p$	$F_p(T)$
$2$	\( T^{6} + T^{5} + T^{4} + \cdots + 1 \) T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1
$5$	\( T^{6} \) T^6
$7$	\( T^{6} \) T^6
$11$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$13$	\( T^{6} \) T^6
$17$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$19$	\( T^{6} - 5 T^{5} + \cdots + 1 \) T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$23$	\( T^{6} \) T^6
$29$	\( T^{6} \) T^6
$31$	\( T^{6} \) T^6
$37$	\( T^{6} \) T^6
$41$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$43$	\( T^{6} + T^{5} + T^{4} + \cdots + 1 \) T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$47$	\( T^{6} \) T^6
$53$	\( T^{6} \) T^6
$59$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1
$61$	\( T^{6} \) T^6
$67$	\( T^{6} + 2 T^{5} + \cdots + 64 \) T^6 + 2T^5 + 4T^4 + 8T^3 + 16T^2 + 32*T + 64
$71$	\( T^{6} \) T^6
$73$	\( T^{6} + 2 T^{5} + \cdots + 64 \) T^6 + 2T^5 + 4T^4 + 8T^3 + 16T^2 + 32*T + 64
$79$	\( T^{6} \) T^6
$83$	\( T^{6} - 5 T^{5} + \cdots + 1 \) T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$89$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$97$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
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