LMFDB - Newform orbit 1775.1.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1775,1,Mod(851,1775)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1775.851");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1775 = 5^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1775.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$0.885840397424$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{28})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} + 14x^{2} - 7 $ x^6 - 7x^4 + 14x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 355)
Projective image:	$D_{14}$
Projective field:	Galois closure of 14.2.10007834681328125.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{6} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{4} + 1) q^{9}+O(q^{10}) $ q - b1 * q^2 + b3 * q^3 + (b2 + 1) * q^4 + (-b4 - 2b2 + 1) q^6 + (-b3 - b1) * q^8 + (b4 + 1) * q^9	\( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{6} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{4} + 1) q^{9} + (\beta_{5} + \beta_{3} + \beta_1) q^{12} + (\beta_{4} + 2 \beta_{2}) q^{16} + ( - \beta_{5} - \beta_1) q^{18} - \beta_{4} q^{19} + ( - 2 \beta_{4} - 2 \beta_{2} - 1) q^{24} + ( - \beta_{5} + \beta_{3}) q^{27} + (\beta_{4} + \beta_{2} - 1) q^{29} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{32} + (\beta_{4} + 2 \beta_{2}) q^{36} + \beta_1 q^{37} + \beta_{5} q^{38} + \beta_{5} q^{43} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{48} + q^{49} + (\beta_{4} - \beta_{2}) q^{54} + (\beta_{5} - \beta_{3}) q^{57} + ( - \beta_{5} - \beta_{3}) q^{58} + (2 \beta_{4} + 2 \beta_{2}) q^{64} - q^{71} + ( - 2 \beta_{3} - \beta_1) q^{72} - \beta_{5} q^{73} + ( - \beta_{2} - 2) q^{74} + ( - \beta_{4} - \beta_{2} + 1) q^{76} - \beta_{2} q^{79} + (\beta_{4} - \beta_{2} + 1) q^{81} + \beta_1 q^{83} + ( - 2 \beta_{4} - \beta_{2} + 1) q^{86} + \beta_1 q^{87} + ( - \beta_{4} - \beta_{2} + 1) q^{89} + ( - \beta_{4} - 3 \beta_{2} - 1) q^{96} - \beta_1 q^{98}+O(q^{100}) \) q - b1 * q^2 + b3 * q^3 + (b2 + 1) * q^4 + (-b4 - 2b2 + 1) q^6 + (-b3 - b1) * q^8 + (b4 + 1) * q^9 + (b5 + b3 + b1) * q^12 + (b4 + 2b2) q^16 + (-b5 - b1) * q^18 - b4 * q^19 + (-2b4 - 2b2 - 1) * q^24 + (-b5 + b3) * q^27 + (b4 + b2 - 1) * q^29 + (-b5 - b3 - b1) * q^32 + (b4 + 2b2) q^36 + b1 * q^37 + b5 * q^38 + b5 * q^43 + (b5 + b3 + 2b1) q^48 + q^49 + (b4 - b2) * q^54 + (b5 - b3) * q^57 + (-b5 - b3) * q^58 + (2b4 + 2b2) * q^64 - q^71 + (-2b3 - b1) q^72 - b5 * q^73 + (-b2 - 2) * q^74 + (-b4 - b2 + 1) * q^76 - b2 * q^79 + (b4 - b2 + 1) * q^81 + b1 * q^83 + (-2b4 - b2 + 1) q^86 + b1 * q^87 + (-b4 - b2 + 1) * q^89 + (-b4 - 3b2 - 1) q^96 - b1 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} + 8 q^{9}+O(q^{10}) $ 6 * q + 8 * q^4 + 8 * q^9	$ 6 q + 8 q^{4} + 8 q^{9} + 6 q^{16} - 2 q^{19} - 14 q^{24} - 2 q^{29} + 6 q^{36} + 6 q^{49} + 8 q^{64} - 6 q^{71} - 14 q^{74} + 2 q^{76} - 2 q^{79} + 6 q^{81} + 2 q^{89} - 14 q^{96}+O(q^{100}) $ 6 * q + 8 * q^4 + 8 * q^9 + 6 * q^16 - 2 * q^19 - 14 * q^24 - 2 * q^29 + 6 * q^36 + 6 * q^49 + 8 * q^64 - 6 * q^71 - 14 * q^74 + 2 * q^76 - 2 * q^79 + 6 * q^81 + 2 * q^89 - 14 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of $\nu = \zeta_{28} + \zeta_{28}^{-1}$:

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 2 $ v^2 - 2
$\beta_{3}$	$=$	$ \nu^{3} - 3\nu $ v^3 - 3*v
$\beta_{4}$	$=$	$ \nu^{4} - 5\nu^{2} + 5 $ v^4 - 5*v^2 + 5
$\beta_{5}$	$=$	$ \nu^{5} - 5\nu^{3} + 5\nu $ v^5 - 5v^3 + 5v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 2 $ b2 + 2
$\nu^{3}$	$=$	$ \beta_{3} + 3\beta_1 $ b3 + 3*b1
$\nu^{4}$	$=$	$ \beta_{4} + 5\beta_{2} + 5 $ b4 + 5*b2 + 5
$\nu^{5}$	$=$	$ \beta_{5} + 5\beta_{3} + 10\beta_1 $ b5 + 5b3 + 10b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1775\mathbb{Z}\right)^\times$.

$n$	$427$	$1001$
$\chi(n)$	$1$	$-1$

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} $

851.1

1.94986
1.56366
0.867767
−0.867767
−1.56366
−1.94986

−1.94986

1.56366

2.80194

−3.04892

−3.51352

1.44504

851.2

−1.56366

−0.867767

1.44504

1.35690

−0.695895

−0.246980

851.3

−0.867767

−1.94986

−0.246980

1.69202

1.08209

2.80194

851.4

0.867767

1.94986

−0.246980

1.69202

−1.08209

2.80194

851.5

1.56366

0.867767

1.44504

1.35690

0.695895

−0.246980

851.6

1.94986

−1.56366

2.80194

−3.04892

3.51352

1.44504

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.b	odd	2	1	CM by $\Q(\sqrt{-71}) $
5.b	even	2	1	inner
355.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1775.1.d.c		6
5.b	even	2	1	inner	1775.1.d.c		6
5.c	odd	4	2		355.1.c.b	✓	6
15.e	even	4	2		3195.1.e.d		6
71.b	odd	2	1	CM	1775.1.d.c		6
355.c	odd	2	1	inner	1775.1.d.c		6
355.e	even	4	2		355.1.c.b	✓	6
1065.l	odd	4	2		3195.1.e.d		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
355.1.c.b	✓	6	5.c	odd	4	2
355.1.c.b	✓	6	355.e	even	4	2
1775.1.d.c		6	1.a	even	1	1	trivial
1775.1.d.c		6	5.b	even	2	1	inner
1775.1.d.c		6	71.b	odd	2	1	CM
1775.1.d.c		6	355.c	odd	2	1	inner
3195.1.e.d		6	15.e	even	4	2
3195.1.e.d		6	1065.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 7T_{2}^{4} + 14T_{2}^{2} - 7 $ T2^6 - 7*T2^4 + 14*T2^2 - 7 acting on $S_{1}^{\mathrm{new}}(1775, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$3$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} $ T^6
$17$	$ T^{6} $ T^6
$19$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$23$	$ T^{6} $ T^6
$29$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$31$	$ T^{6} $ T^6
$37$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$41$	$ T^{6} $ T^6
$43$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$47$	$ T^{6} $ T^6
$53$	$ T^{6} $ T^6
$59$	$ T^{6} $ T^6
$61$	$ T^{6} $ T^6
$67$	$ T^{6} $ T^6
$71$	$ (T + 1)^{6} $ (T + 1)^6
$73$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$79$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$83$	$ T^{6} - 7 T^{4} + \cdots - 7 $ T^6 - 7T^4 + 14T^2 - 7
$89$	$ (T^{3} - T^{2} - 2 T + 1)^{2} $ (T^3 - T^2 - 2*T + 1)^2
$97$	$ T^{6} $ T^6
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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:	\( N \)	\(=\)	\( 1775 = 5^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1775.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{6} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) q - b1 * q^2 + b3 * q^3 + (b2 + 1) * q^4 + (-b4 - 2b2 + 1) q^6 + (-b3 - b1) * q^8 + (b4 + 1) * q^9	\( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{6} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{4} + 1) q^{9} + (\beta_{5} + \beta_{3} + \beta_1) q^{12} + (\beta_{4} + 2 \beta_{2}) q^{16} + ( - \beta_{5} - \beta_1) q^{18} - \beta_{4} q^{19} + ( - 2 \beta_{4} - 2 \beta_{2} - 1) q^{24} + ( - \beta_{5} + \beta_{3}) q^{27} + (\beta_{4} + \beta_{2} - 1) q^{29} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{32} + (\beta_{4} + 2 \beta_{2}) q^{36} + \beta_1 q^{37} + \beta_{5} q^{38} + \beta_{5} q^{43} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{48} + q^{49} + (\beta_{4} - \beta_{2}) q^{54} + (\beta_{5} - \beta_{3}) q^{57} + ( - \beta_{5} - \beta_{3}) q^{58} + (2 \beta_{4} + 2 \beta_{2}) q^{64} - q^{71} + ( - 2 \beta_{3} - \beta_1) q^{72} - \beta_{5} q^{73} + ( - \beta_{2} - 2) q^{74} + ( - \beta_{4} - \beta_{2} + 1) q^{76} - \beta_{2} q^{79} + (\beta_{4} - \beta_{2} + 1) q^{81} + \beta_1 q^{83} + ( - 2 \beta_{4} - \beta_{2} + 1) q^{86} + \beta_1 q^{87} + ( - \beta_{4} - \beta_{2} + 1) q^{89} + ( - \beta_{4} - 3 \beta_{2} - 1) q^{96} - \beta_1 q^{98}+O(q^{100}) \) q - b1 * q^2 + b3 * q^3 + (b2 + 1) * q^4 + (-b4 - 2b2 + 1) q^6 + (-b3 - b1) * q^8 + (b4 + 1) * q^9 + (b5 + b3 + b1) * q^12 + (b4 + 2b2) q^16 + (-b5 - b1) * q^18 - b4 * q^19 + (-2b4 - 2b2 - 1) * q^24 + (-b5 + b3) * q^27 + (b4 + b2 - 1) * q^29 + (-b5 - b3 - b1) * q^32 + (b4 + 2b2) q^36 + b1 * q^37 + b5 * q^38 + b5 * q^43 + (b5 + b3 + 2b1) q^48 + q^49 + (b4 - b2) * q^54 + (b5 - b3) * q^57 + (-b5 - b3) * q^58 + (2b4 + 2b2) * q^64 - q^71 + (-2b3 - b1) q^72 - b5 * q^73 + (-b2 - 2) * q^74 + (-b4 - b2 + 1) * q^76 - b2 * q^79 + (b4 - b2 + 1) * q^81 + b1 * q^83 + (-2b4 - b2 + 1) q^86 + b1 * q^87 + (-b4 - b2 + 1) * q^89 + (-b4 - 3b2 - 1) q^96 - b1 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} + 8 q^{9}+O(q^{10}) \) 6 * q + 8 * q^4 + 8 * q^9	\( 6 q + 8 q^{4} + 8 q^{9} + 6 q^{16} - 2 q^{19} - 14 q^{24} - 2 q^{29} + 6 q^{36} + 6 q^{49} + 8 q^{64} - 6 q^{71} - 14 q^{74} + 2 q^{76} - 2 q^{79} + 6 q^{81} + 2 q^{89} - 14 q^{96}+O(q^{100}) \) 6 * q + 8 * q^4 + 8 * q^9 + 6 * q^16 - 2 * q^19 - 14 * q^24 - 2 * q^29 + 6 * q^36 + 6 * q^49 + 8 * q^64 - 6 * q^71 - 14 * q^74 + 2 * q^76 - 2 * q^79 + 6 * q^81 + 2 * q^89 - 14 * q^96

Introduction

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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	1775.1.d.c

Level	$1775$
Weight	$1$
Character orbit	1775.d
Self dual	yes
Analytic conductor	$0.886$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{14}$
CM discriminant	-71
Inner twists	$4$

Self dual:	yes
Analytic conductor:	\(0.885840397424\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{28})^+\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - 7x^{4} + 14x^{2} - 7 \) x^6 - 7x^4 + 14x^2 - 7
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 355)
Projective image:	\(D_{14}\)
Projective field:	Galois closure of 14.2.10007834681328125.1

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \) v^2 - 2
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 3\nu \) v^3 - 3*v
\(\beta_{4}\)	\(=\)	\( \nu^{4} - 5\nu^{2} + 5 \) v^4 - 5*v^2 + 5
\(\beta_{5}\)	\(=\)	\( \nu^{5} - 5\nu^{3} + 5\nu \) v^5 - 5v^3 + 5v

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 2 \) b2 + 2
\(\nu^{3}\)	\(=\)	\( \beta_{3} + 3\beta_1 \) b3 + 3*b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} + 5\beta_{2} + 5 \) b4 + 5*b2 + 5
\(\nu^{5}\)	\(=\)	\( \beta_{5} + 5\beta_{3} + 10\beta_1 \) b5 + 5b3 + 10b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 851.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$3$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$5$	\( T^{6} \) T^6
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( T^{6} \) T^6
$17$	\( T^{6} \) T^6
$19$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$23$	\( T^{6} \) T^6
$29$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$31$	\( T^{6} \) T^6
$37$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$41$	\( T^{6} \) T^6
$43$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$47$	\( T^{6} \) T^6
$53$	\( T^{6} \) T^6
$59$	\( T^{6} \) T^6
$61$	\( T^{6} \) T^6
$67$	\( T^{6} \) T^6
$71$	\( (T + 1)^{6} \) (T + 1)^6
$73$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$79$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$83$	\( T^{6} - 7 T^{4} + \cdots - 7 \) T^6 - 7T^4 + 14T^2 - 7
$89$	\( (T^{3} - T^{2} - 2 T + 1)^{2} \) (T^3 - T^2 - 2*T + 1)^2
$97$	\( T^{6} \) T^6
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\(n\)	\(427\)	\(1001\)
\(\chi(n)\)	\(1\)	\(-1\)