LMFDB - Newform orbit 3400.1.cs.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3400,1,Mod(611,3400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3400, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 8, 5])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3400.611"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3400 = 2^{3} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3400.cs (of order $10$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,2,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.69682104295$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{10} + \cdots)$

$q$-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_{20}^{4} q^{2} + \zeta_{20}^{8} q^{4} + \zeta_{20}^{5} q^{5} + (\zeta_{20}^{9} - \zeta_{20}) q^{7} + \zeta_{20}^{2} q^{8} - \zeta_{20}^{6} q^{9} - \zeta_{20}^{9} q^{10} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{14} + \cdots + ( - \zeta_{20}^{6} + \cdots - \zeta_{20}^{2}) q^{98} +O(q^{100}) $ q - z^4 * q^2 + z^8 * q^4 + z^5 * q^5 + (z^9 - z) * q^7 + z^2 * q^8 - z^6 * q^9 - z^9 * q^10 + (z^5 + z^3) * q^14 - z^6 * q^16 + z^2 * q^17 - q^18 + (-z^4 - 1) * q^19 - z^3 * q^20 + (z^7 + z) * q^23 - q^25 + (-z^9 - z^7) * q^28 + (z^9 - z^5) * q^31 - q^32 - z^6 * q^34 + (-z^6 - z^4) * q^35 + z^4 * q^36 + (-z^7 - z^5) * q^37 + (z^8 + z^4) * q^38 + z^7 * q^40 + 2 * q^43 + z * q^45 + (-z^5 + z) * q^46 + (-z^8 + z^2 + 1) * q^49 + z^4 * q^50 + (-z^3 - z) * q^56 + (-z^2 + 1) * q^59 + (-z^7 - z) * q^61 + (z^9 + z^3) * q^62 + (z^7 + z^5) * q^63 + z^4 * q^64 + (-z^4 - 1) * q^67 - q^68 + (z^8 - 1) * q^70 + (-z^9 - z^7) * q^71 - z^8 * q^72 + (z^9 - z) * q^74 + (-z^8 + z^2) * q^76 + z * q^80 - z^2 * q^81 + (z^8 - z^6) * q^83 + z^7 * q^85 - 2z^4 q^86 + (-z^8 - 1) * q^89 - z^5 * q^90 + (z^9 - z^5) * q^92 + (-z^9 - z^5) * q^95 + (-z^6 - z^4 - z^2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{16} + 2 q^{17} - 8 q^{18} - 6 q^{19} - 8 q^{25} - 8 q^{32} - 2 q^{34} - 2 q^{36} - 4 q^{38} + 16 q^{43} + 12 q^{49} - 2 q^{50} + 6 q^{59} - 2 q^{64} - 6 q^{67}+ \cdots - 2 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^16 + 2 * q^17 - 8 * q^18 - 6 * q^19 - 8 * q^25 - 8 * q^32 - 2 * q^34 - 2 * q^36 - 4 * q^38 + 16 * q^43 + 12 * q^49 - 2 * q^50 + 6 * q^59 - 2 * q^64 - 6 * q^67 - 8 * q^68 - 10 * q^70 + 2 * q^72 + 4 * q^76 - 2 * q^81 - 4 * q^83 + 4 * q^86 - 6 * q^89 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times$.

$n$	$1601$	$1701$	$2177$	$2551$
$\chi(n)$	$-1$	$-1$	$\zeta_{20}^{8}$	$-1$

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} $

611.1

−0.587785	+	0.809017i
0.587785	−	0.809017i
0.587785	+	0.809017i
−0.587785	−	0.809017i
0.951057	−	0.309017i
−0.951057	+	0.309017i
−0.951057	−	0.309017i
0.951057	+	0.309017i

0.809017

−

0.587785i

0.309017

−

0.951057i

−

1.00000i

1.17557

−0.309017

−

0.951057i

−0.809017

−

0.587785i

−0.587785

−

0.809017i

611.2

0.809017

−

0.587785i

0.309017

−

0.951057i

1.00000i

−1.17557

−0.309017

−

0.951057i

−0.809017

−

0.587785i

0.587785

0.809017i

1291.1

0.809017

0.587785i

0.309017

0.951057i

−

1.00000i

−1.17557

−0.309017

0.951057i

−0.809017

0.587785i

0.587785

−

0.809017i

1291.2

0.809017

0.587785i

0.309017

0.951057i

1.00000i

1.17557

−0.309017

0.951057i

−0.809017

0.587785i

−0.587785

0.809017i

1971.1

−0.309017

0.951057i

−0.809017

−

0.587785i

−

1.00000i

−1.90211

0.809017

−

0.587785i

0.309017

0.951057i

0.951057

0.309017i

1971.2

−0.309017

0.951057i

−0.809017

−

0.587785i

1.00000i

1.90211

0.809017

−

0.587785i

0.309017

0.951057i

−0.951057

−

0.309017i

3331.1

−0.309017

−

0.951057i

−0.809017

0.587785i

−

1.00000i

1.90211

0.809017

0.587785i

0.309017

−

0.951057i

−0.951057

0.309017i

3331.2

−0.309017

−

0.951057i

−0.809017

0.587785i

1.00000i

−1.90211

0.809017

0.587785i

0.309017

−

0.951057i

0.951057

−

0.309017i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
136.e	odd	2	1	CM by $\Q(\sqrt{-34}) $
8.d	odd	2	1	inner
17.b	even	2	1	inner
25.d	even	5	1	inner
200.n	odd	10	1	inner
425.p	even	10	1	inner
3400.cs	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3400.1.cs.c	✓	8
8.d	odd	2	1	inner	3400.1.cs.c	✓	8
17.b	even	2	1	inner	3400.1.cs.c	✓	8
25.d	even	5	1	inner	3400.1.cs.c	✓	8
136.e	odd	2	1	CM	3400.1.cs.c	✓	8
200.n	odd	10	1	inner	3400.1.cs.c	✓	8
425.p	even	10	1	inner	3400.1.cs.c	✓	8
3400.cs	odd	10	1	inner	3400.1.cs.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3400.1.cs.c	✓	8	1.a	even	1	1	trivial
3400.1.cs.c	✓	8	8.d	odd	2	1	inner
3400.1.cs.c	✓	8	17.b	even	2	1	inner
3400.1.cs.c	✓	8	25.d	even	5	1	inner
3400.1.cs.c	✓	8	136.e	odd	2	1	CM
3400.1.cs.c	✓	8	200.n	odd	10	1	inner
3400.1.cs.c	✓	8	425.p	even	10	1	inner
3400.1.cs.c	✓	8	3400.cs	odd	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} - 5T_{7}^{2} + 5 $ T7^4 - 5*T7^2 + 5 acting on $S_{1}^{\mathrm{new}}(3400, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{8} $ T^8
$5$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$7$	$ (T^{4} - 5 T^{2} + 5)^{2} $ (T^4 - 5*T^2 + 5)^2
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$19$	$ (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$23$	$ T^{8} + 10 T^{4} + \cdots + 25 $ T^8 + 10T^4 + 25T^2 + 25
$29$	$ T^{8} $ T^8
$31$	$ T^{8} + 5 T^{6} + \cdots + 25 $ T^8 + 5T^6 + 10T^4 + 25
$37$	$ T^{8} + 5 T^{6} + \cdots + 25 $ T^8 + 5T^6 + 10T^4 + 25
$41$	$ T^{8} $ T^8
$43$	$ (T - 2)^{8} $ (T - 2)^8
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$61$	$ T^{8} + 10 T^{4} + \cdots + 25 $ T^8 + 10T^4 + 25T^2 + 25
$67$	$ (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$71$	$ T^{8} + 10 T^{4} + \cdots + 25 $ T^8 + 10T^4 + 25T^2 + 25
$73$	$ T^{8} $ T^8
$79$	$ T^{8} $ T^8
$83$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$89$	$ (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$97$	$ T^{8} $ T^8
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Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3400.cs (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{20}^{4} q^{2} + \zeta_{20}^{8} q^{4} + \zeta_{20}^{5} q^{5} + (\zeta_{20}^{9} - \zeta_{20}) q^{7} + \zeta_{20}^{2} q^{8} - \zeta_{20}^{6} q^{9} - \zeta_{20}^{9} q^{10} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{14} + \cdots + ( - \zeta_{20}^{6} + \cdots - \zeta_{20}^{2}) q^{98} +O(q^{100}) \) q - z^4 * q^2 + z^8 * q^4 + z^5 * q^5 + (z^9 - z) * q^7 + z^2 * q^8 - z^6 * q^9 - z^9 * q^10 + (z^5 + z^3) * q^14 - z^6 * q^16 + z^2 * q^17 - q^18 + (-z^4 - 1) * q^19 - z^3 * q^20 + (z^7 + z) * q^23 - q^25 + (-z^9 - z^7) * q^28 + (z^9 - z^5) * q^31 - q^32 - z^6 * q^34 + (-z^6 - z^4) * q^35 + z^4 * q^36 + (-z^7 - z^5) * q^37 + (z^8 + z^4) * q^38 + z^7 * q^40 + 2 * q^43 + z * q^45 + (-z^5 + z) * q^46 + (-z^8 + z^2 + 1) * q^49 + z^4 * q^50 + (-z^3 - z) * q^56 + (-z^2 + 1) * q^59 + (-z^7 - z) * q^61 + (z^9 + z^3) * q^62 + (z^7 + z^5) * q^63 + z^4 * q^64 + (-z^4 - 1) * q^67 - q^68 + (z^8 - 1) * q^70 + (-z^9 - z^7) * q^71 - z^8 * q^72 + (z^9 - z) * q^74 + (-z^8 + z^2) * q^76 + z * q^80 - z^2 * q^81 + (z^8 - z^6) * q^83 + z^7 * q^85 - 2z^4 q^86 + (-z^8 - 1) * q^89 - z^5 * q^90 + (z^9 - z^5) * q^92 + (-z^9 - z^5) * q^95 + (-z^6 - z^4 - z^2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{16} + 2 q^{17} - 8 q^{18} - 6 q^{19} - 8 q^{25} - 8 q^{32} - 2 q^{34} - 2 q^{36} - 4 q^{38} + 16 q^{43} + 12 q^{49} - 2 q^{50} + 6 q^{59} - 2 q^{64} - 6 q^{67}+ \cdots - 2 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^16 + 2 * q^17 - 8 * q^18 - 6 * q^19 - 8 * q^25 - 8 * q^32 - 2 * q^34 - 2 * q^36 - 4 * q^38 + 16 * q^43 + 12 * q^49 - 2 * q^50 + 6 * q^59 - 2 * q^64 - 6 * q^67 - 8 * q^68 - 10 * q^70 + 2 * q^72 + 4 * q^76 - 2 * q^81 - 4 * q^83 + 4 * q^86 - 6 * q^89 - 2 * q^98

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3400.1.cs.c

Level	$3400$
Weight	$1$
Character orbit	3400.cs
Analytic conductor	$1.697$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{10}$
CM discriminant	-136
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.69682104295\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{10}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{10} + \cdots)\)

\(n\)	\(1601\)	\(1701\)	\(2177\)	\(2551\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{20}^{8}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 611.2
Significant digits:
Format:

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