LMFDB - Newform orbit 2695.1.ca.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2695,1,Mod(509,2695)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2695, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 25, 24]))

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

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

chi := DirichletCharacter("2695.509");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2695 = 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2695.ca (of order $30$, degree $8$, not 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:	no
Analytic conductor:	$1.34498020905$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.17935225.1

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2695\mathbb{Z}\right)^\times$.

$n$	$981$	$1816$	$2157$
$\chi(n)$	$-\zeta_{30}^{9}$	$-\zeta_{30}^{10}$	$-1$

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} $

509.1

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

−0.413545

0.459289i

−0.978148

0.207912i

−0.913545

0.406737i

0.0646021

0.614648i

619.1

0.604528

0.128496i

0.669131

0.743145i

0.104528

−

0.994522i

−0.564602

−

0.251377i

999.1

1.47815

0.658114i

−0.104528

0.994522i

0.978148

0.207912i

1.08268

1.20243i

1109.1

−0.169131

−

1.60917i

0.913545

−

0.406737i

−0.669131

0.743145i

−1.58268

0.336408i

1489.1

0.604528

−

0.128496i

0.669131

−

0.743145i

0.104528

0.994522i

−0.564602

0.251377i

1599.1

−0.413545

−

0.459289i

−0.978148

−

0.207912i

−0.913545

−

0.406737i

0.0646021

−

0.614648i

2469.1

−0.169131

1.60917i

0.913545

0.406737i

−0.669131

−

0.743145i

−1.58268

−

0.336408i

2579.1

1.47815

−

0.658114i

−0.104528

−

0.994522i

0.978148

−

0.207912i

1.08268

−

1.20243i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.c	odd	2	1	CM by $\Q(\sqrt{-35}) $
7.c	even	3	1	inner
11.c	even	5	1	inner
35.i	odd	6	1	inner
77.m	even	15	1	inner
385.y	odd	10	1	inner
385.bn	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2695.1.ca.b		8
5.b	even	2	1		2695.1.ca.a		8
7.b	odd	2	1		2695.1.ca.a		8
7.c	even	3	1		385.1.y.a	✓	4
7.c	even	3	1	inner	2695.1.ca.b		8
7.d	odd	6	1		385.1.y.b	yes	4
7.d	odd	6	1		2695.1.ca.a		8
11.c	even	5	1	inner	2695.1.ca.b		8
21.g	even	6	1		3465.1.dt.b		4
21.h	odd	6	1		3465.1.dt.a		4
35.c	odd	2	1	CM	2695.1.ca.b		8
35.i	odd	6	1		385.1.y.a	✓	4
35.i	odd	6	1	inner	2695.1.ca.b		8
35.j	even	6	1		385.1.y.b	yes	4
35.j	even	6	1		2695.1.ca.a		8
35.k	even	12	2		1925.1.bn.c		8
35.l	odd	12	2		1925.1.bn.c		8
55.j	even	10	1		2695.1.ca.a		8
77.j	odd	10	1		2695.1.ca.a		8
77.m	even	15	1		385.1.y.a	✓	4
77.m	even	15	1	inner	2695.1.ca.b		8
77.p	odd	30	1		385.1.y.b	yes	4
77.p	odd	30	1		2695.1.ca.a		8
105.o	odd	6	1		3465.1.dt.b		4
105.p	even	6	1		3465.1.dt.a		4
231.z	odd	30	1		3465.1.dt.a		4
231.bc	even	30	1		3465.1.dt.b		4
385.y	odd	10	1	inner	2695.1.ca.b		8
385.bm	even	30	1		385.1.y.b	yes	4
385.bm	even	30	1		2695.1.ca.a		8
385.bn	odd	30	1		385.1.y.a	✓	4
385.bn	odd	30	1	inner	2695.1.ca.b		8
385.bt	odd	60	2		1925.1.bn.c		8
385.bu	even	60	2		1925.1.bn.c		8
1155.dd	even	30	1		3465.1.dt.a		4
1155.dj	odd	30	1		3465.1.dt.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
385.1.y.a	✓	4	7.c	even	3	1
385.1.y.a	✓	4	35.i	odd	6	1
385.1.y.a	✓	4	77.m	even	15	1
385.1.y.a	✓	4	385.bn	odd	30	1
385.1.y.b	yes	4	7.d	odd	6	1
385.1.y.b	yes	4	35.j	even	6	1
385.1.y.b	yes	4	77.p	odd	30	1
385.1.y.b	yes	4	385.bm	even	30	1
1925.1.bn.c		8	35.k	even	12	2
1925.1.bn.c		8	35.l	odd	12	2
1925.1.bn.c		8	385.bt	odd	60	2
1925.1.bn.c		8	385.bu	even	60	2
2695.1.ca.a		8	5.b	even	2	1
2695.1.ca.a		8	7.b	odd	2	1
2695.1.ca.a		8	7.d	odd	6	1
2695.1.ca.a		8	35.j	even	6	1
2695.1.ca.a		8	55.j	even	10	1
2695.1.ca.a		8	77.j	odd	10	1
2695.1.ca.a		8	77.p	odd	30	1
2695.1.ca.a		8	385.bm	even	30	1
2695.1.ca.b		8	1.a	even	1	1	trivial
2695.1.ca.b		8	7.c	even	3	1	inner
2695.1.ca.b		8	11.c	even	5	1	inner
2695.1.ca.b		8	35.c	odd	2	1	CM
2695.1.ca.b		8	35.i	odd	6	1	inner
2695.1.ca.b		8	77.m	even	15	1	inner
2695.1.ca.b		8	385.y	odd	10	1	inner
2695.1.ca.b		8	385.bn	odd	30	1	inner
3465.1.dt.a		4	21.h	odd	6	1
3465.1.dt.a		4	105.p	even	6	1
3465.1.dt.a		4	231.z	odd	30	1
3465.1.dt.a		4	1155.dd	even	30	1
3465.1.dt.b		4	21.g	even	6	1
3465.1.dt.b		4	105.o	odd	6	1
3465.1.dt.b		4	231.bc	even	30	1
3465.1.dt.b		4	1155.dj	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 3T_{3}^{7} + 5T_{3}^{6} - 8T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} - 2T_{3} + 1 $ T3^8 - 3*T3^7 + 5*T3^6 - 8*T3^5 + 9*T3^4 - 2*T3^3 - 2*T3 + 1 acting on $S_{1}^{\mathrm{new}}(2695, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$5$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$7$	$ T^{8} $ T^8
$11$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$13$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$17$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 2T^5 + 9T^4 + 8T^3 + 5T^2 + 3T + 1
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$73$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 2T^5 + 9T^4 + 8T^3 + 5T^2 + 3T + 1
$79$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1
$83$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$89$	$ T^{8} $ T^8
$97$	$ (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
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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:	\( N \)	\(=\)	\( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2695.ca (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{30}^{5} - \zeta_{30}^{2}) q^{3} + \zeta_{30}^{14} q^{4} + \zeta_{30}^{13} q^{5} + (\zeta_{30}^{10} + \cdots + \zeta_{30}^{4}) q^{9}+O(q^{10}) \) q + (z^5 - z^2) * q^3 + z^14 * q^4 + z^13 * q^5 + (z^10 - z^7 + z^4) * q^9	\( q + (\zeta_{30}^{5} - \zeta_{30}^{2}) q^{3} + \zeta_{30}^{14} q^{4} + \zeta_{30}^{13} q^{5} + (\zeta_{30}^{10} + \cdots + \zeta_{30}^{4}) q^{9}+ \cdots + (\zeta_{30}^{12} + \cdots + \zeta_{30}^{6}) q^{99}+O(q^{100}) \) q + (z^5 - z^2) * q^3 + z^14 * q^4 + z^13 * q^5 + (z^10 - z^7 + z^4) * q^9 + z^2 * q^11 + (-z^4 + z) * q^12 + (-z^6 + z^3) * q^13 + (-z^3 + 1) * q^15 - z^13 * q^16 + (z^11 + z^5) * q^17 - z^12 * q^20 - z^11 * q^25 + (-z^12 + z^9 - z^6 - 1) * q^27 + (z^12 + z^6) * q^29 + (z^7 - z^4) * q^33 + (-z^9 + z^6 - z^3) * q^36 + (-z^11 + 2z^8 - z^5) q^39 - z * q^44 + (-z^8 + z^5 - z^2) * q^45 + (z^13 - z^4) * q^47 + (z^3 - 1) * q^48 + (-z^13 + z^10 - z^7 - z) * q^51 + (z^5 - z^2) * q^52 - q^55 + (z^14 + z^2) * q^60 + z^12 * q^64 + (z^4 - z) * q^65 + (-z^10 - z^4) * q^68 + (z^12 - z^9) * q^71 + (z^11 - z^2) * q^73 + (z^13 + z) * q^75 + (-z^13 - z) * q^79 + z^11 * q^80 + (z^14 - z^11 + z^8 - z^5 - z^2) * q^81 + (-z^12 + z^9) * q^83 + (-z^9 - z^3) * q^85 + (-z^14 + z^11 - z^8 - z^2) * q^87 + (z^9 - 1) * q^97 + (z^12 - z^9 + z^6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + q^{4} - q^{5} - 2 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + q^4 - q^5 - 2 * q^9	\( 8 q + 3 q^{3} + q^{4} - q^{5} - 2 q^{9} + q^{11} - 2 q^{12} + 4 q^{13} + 6 q^{15} + q^{16} + 3 q^{17} + 2 q^{20} + q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{33} - 6 q^{36} - q^{39} + q^{44} + 2 q^{45} - 2 q^{47} - 6 q^{48} - q^{51} + 3 q^{52} - 8 q^{55} + 2 q^{60} - 2 q^{64} + 2 q^{65} + 3 q^{68} - 4 q^{71} - 2 q^{73} - 2 q^{75} + 2 q^{79} - q^{80} + 4 q^{83} - 4 q^{85} - 4 q^{87} - 6 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + q^4 - q^5 - 2 * q^9 + q^11 - 2 * q^12 + 4 * q^13 + 6 * q^15 + q^16 + 3 * q^17 + 2 * q^20 + q^25 - 2 * q^27 - 4 * q^29 - 2 * q^33 - 6 * q^36 - q^39 + q^44 + 2 * q^45 - 2 * q^47 - 6 * q^48 - q^51 + 3 * q^52 - 8 * q^55 + 2 * q^60 - 2 * q^64 + 2 * q^65 + 3 * q^68 - 4 * q^71 - 2 * q^73 - 2 * q^75 + 2 * q^79 - q^80 + 4 * q^83 - 4 * q^85 - 4 * q^87 - 6 * q^97 - 6 * q^99

Introduction

L-functions

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	2695.1.ca.b

Level	$2695$
Weight	$1$
Character orbit	2695.ca
Analytic conductor	$1.345$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{5}$
CM discriminant	-35
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.34498020905\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 385)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.17935225.1

\(n\)	\(981\)	\(1816\)	\(2157\)
\(\chi(n)\)	\(-\zeta_{30}^{9}\)	\(-\zeta_{30}^{10}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 509.1
Significant digits:
Format:

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