LMFDB - Newform orbit 775.1.w.a

Newspace parameters

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	775.w (of order $10$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.386775384791$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.375390625.1

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$-1$	$-\zeta_{10}$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

61.1

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

1.30902

−

0.951057i

0.500000

−

1.53884i

−0.809017

−

0.587785i

0.618034

−0.309017

−

0.951057i

−0.809017

−

0.587785i

−1.61803

216.1

1.30902

0.951057i

0.500000

1.53884i

−0.809017

0.587785i

0.618034

−0.309017

0.951057i

−0.809017

0.587785i

−1.61803

371.1

0.190983

−

0.587785i

0.500000

0.363271i

0.309017

0.951057i

−1.61803

0.809017

−

0.587785i

0.309017

0.951057i

0.618034

681.1

0.190983

0.587785i

0.500000

−

0.363271i

0.309017

−

0.951057i

−1.61803

0.809017

0.587785i

0.309017

−

0.951057i

0.618034

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	CM by $\Q(\sqrt{-31}) $
25.d	even	5	1	inner
775.w	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.1.w.a	✓	4
5.b	even	2	1		3875.1.w.a		4
5.c	odd	4	2		3875.1.z.c		8
25.d	even	5	1	inner	775.1.w.a	✓	4
25.e	even	10	1		3875.1.w.a		4
25.f	odd	20	2		3875.1.z.c		8
31.b	odd	2	1	CM	775.1.w.a	✓	4
155.c	odd	2	1		3875.1.w.a		4
155.f	even	4	2		3875.1.z.c		8
775.w	odd	10	1	inner	775.1.w.a	✓	4
775.z	odd	10	1		3875.1.w.a		4
775.ca	even	20	2		3875.1.z.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
775.1.w.a	✓	4	1.a	even	1	1	trivial
775.1.w.a	✓	4	25.d	even	5	1	inner
775.1.w.a	✓	4	31.b	odd	2	1	CM
775.1.w.a	✓	4	775.w	odd	10	1	inner
3875.1.w.a		4	5.b	even	2	1
3875.1.w.a		4	25.e	even	10	1
3875.1.w.a		4	155.c	odd	2	1
3875.1.w.a		4	775.z	odd	10	1
3875.1.z.c		8	5.c	odd	4	2
3875.1.z.c		8	25.f	odd	20	2
3875.1.z.c		8	155.f	even	4	2
3875.1.z.c		8	775.ca	even	20	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1 $ T2^4 - 3*T2^3 + 4*T2^2 - 2*T2 + 1 acting on $S_{1}^{\mathrm{new}}(775, [\chi])$.

Hecke characteristic polynomials

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	775.w (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9}+O(q^{10}) \) q + (-z + 1) * q^2 + (z^2 - z + 1) * q^4 + z^2 * q^5 + (z^4 - z) * q^7 + (-z^3 + z^2 - z + 1) * q^8 + z^2 * q^9	\( q + ( - \zeta_{10} + 1) q^{2} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}) q^{7} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{8} + \zeta_{10}^{2} q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{10} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{18} + ( - \zeta_{10}^{3} + 1) q^{19} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{20} + \zeta_{10}^{4} q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{28} + \zeta_{10}^{4} q^{31} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{32} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{35} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{36} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{40} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} + \zeta_{10}^{4} q^{45} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{49} + (\zeta_{10}^{4} + 1) q^{50} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{56} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{59} + (\zeta_{10}^{4} + 1) q^{62} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{63} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{64} + ( - \zeta_{10}^{3} + 1) q^{67} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{70} - \zeta_{10} q^{71} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{72} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{80} + \zeta_{10}^{4} q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{82} + (\zeta_{10}^{4} + 1) q^{90} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 1) q^{94} + (\zeta_{10}^{2} + 1) q^{95} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{97} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{98} +O(q^{100}) \) q + (-z + 1) * q^2 + (z^2 - z + 1) * q^4 + z^2 * q^5 + (z^4 - z) * q^7 + (-z^3 + z^2 - z + 1) * q^8 + z^2 * q^9 + (-z^3 + z^2) * q^10 + (z^4 + z^2 - z + 1) * q^14 + (z^4 - z^3 + z^2 - z + 1) * q^16 + (-z^3 + z^2) * q^18 + (-z^3 + 1) * q^19 + (z^4 - z^3 + z^2) * q^20 + z^4 * q^25 + (z^4 - z^3 + z^2 - z + 1) * q^28 + z^4 * q^31 + (z^4 - z^3 + z^2 - z + 2) * q^32 + (-z^3 - z) * q^35 + (z^4 - z^3 + z^2) * q^36 + (z^4 - z^3 - z + 1) * q^38 + (z^4 - z^3 + z^2 + 1) * q^40 + (-z^3 - z) * q^41 + z^4 * q^45 + (z^4 - z^3) * q^47 + (-z^3 + z^2 + 1) * q^49 + (z^4 + 1) * q^50 + (2z^4 - z^3 + z^2 - z + 1) q^56 + (-z^3 - z) * q^59 + (z^4 + 1) * q^62 + (-z^3 - z) * q^63 + (z^4 - z^3 + z^2 - 2z - 1) q^64 + (-z^3 + 1) * q^67 + (z^4 - z^3 + z^2 - z) * q^70 - z * q^71 + (z^4 - z^3 + z^2 + 1) * q^72 + (z^4 - z^3 + z^2 - z + 1) * q^76 + (z^4 - z^3 + z^2 - z + 1) * q^80 + z^4 * q^81 + (z^4 - z^3 + z^2 - z) * q^82 + (z^4 + 1) * q^90 + (2z^4 - z^3 + 1) q^94 + (z^2 + 1) * q^95 + (z^4 - z^3) * q^97 + (z^4 - 2z^3 + z^2 - z + 1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9}+O(q^{10}) \) 4 * q + 3 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + q^8 - q^9	\( 4 q + 3 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{9} - 2 q^{10} + q^{14} - 2 q^{18} + 3 q^{19} - 3 q^{20} - q^{25} - q^{28} - q^{31} + 4 q^{32} - 2 q^{35} - 3 q^{36} + q^{38} + q^{40} - 2 q^{41} - q^{45} - 2 q^{47} + 2 q^{49} + 3 q^{50} - 3 q^{56} - 2 q^{59} + 3 q^{62} - 2 q^{63} + 3 q^{64} + 3 q^{67} - 4 q^{70} - 2 q^{71} + q^{72} + 4 q^{76} - q^{81} - 4 q^{82} + 3 q^{90} + q^{94} + 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 + 2 * q^4 - q^5 - 2 * q^7 + q^8 - q^9 - 2 * q^10 + q^14 - 2 * q^18 + 3 * q^19 - 3 * q^20 - q^25 - q^28 - q^31 + 4 * q^32 - 2 * q^35 - 3 * q^36 + q^38 + q^40 - 2 * q^41 - q^45 - 2 * q^47 + 2 * q^49 + 3 * q^50 - 3 * q^56 - 2 * q^59 + 3 * q^62 - 2 * q^63 + 3 * q^64 + 3 * q^67 - 4 * q^70 - 2 * q^71 + q^72 + 4 * q^76 - q^81 - 4 * q^82 + 3 * q^90 + q^94 + 3 * q^95 - 2 * q^97 - q^98

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	775.1.w.a

Level	$775$
Weight	$1$
Character orbit	775.w
Analytic conductor	$0.387$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{5}$
CM discriminant	-31
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.386775384791\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.375390625.1

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

\(n\)	\(251\)	\(652\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{10}\)

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