LMFDB - Newform orbit 693.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] 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("693.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-3,0,1,-12] 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:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 3 q^{2} + q^{4} - 12 q^{5} + 7 q^{7} + 21 q^{8} + 36 q^{10} - 11 q^{11} + 38 q^{13} - 21 q^{14} - 71 q^{16} + 48 q^{17} - 70 q^{19} - 12 q^{20} + 33 q^{22} - 12 q^{23} + 19 q^{25} - 114 q^{26} + 7 q^{28}+ \cdots - 147 q^{98}+O(q^{100}) $

q - 3 * q^2 + q^4 - 12 * q^5 + 7 * q^7 + 21 * q^8 + 36 * q^10 - 11 * q^11 + 38 * q^13 - 21 * q^14 - 71 * q^16 + 48 * q^17 - 70 * q^19 - 12 * q^20 + 33 * q^22 - 12 * q^23 + 19 * q^25 - 114 * q^26 + 7 * q^28 - 126 * q^29 - 70 * q^31 + 45 * q^32 - 144 * q^34 - 84 * q^35 - 358 * q^37 + 210 * q^38 - 252 * q^40 + 216 * q^41 + 344 * q^43 - 11 * q^44 + 36 * q^46 - 390 * q^47 + 49 * q^49 - 57 * q^50 + 38 * q^52 - 438 * q^53 + 132 * q^55 + 147 * q^56 + 378 * q^58 + 552 * q^59 + 830 * q^61 + 210 * q^62 + 433 * q^64 - 456 * q^65 - 196 * q^67 + 48 * q^68 + 252 * q^70 - 648 * q^71 - 16 * q^73 + 1074 * q^74 - 70 * q^76 - 77 * q^77 + 1352 * q^79 + 852 * q^80 - 648 * q^82 - 90 * q^83 - 576 * q^85 - 1032 * q^86 - 231 * q^88 - 1146 * q^89 + 266 * q^91 - 12 * q^92 + 1170 * q^94 + 840 * q^95 - 70 * q^97 - 147 * q^98

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} $

1.1

−3.00000

1.00000

−12.0000

7.00000

21.0000

36.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ -1 $
$11$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.4.a.b		1
3.b	odd	2	1		77.4.a.a	✓	1
12.b	even	2	1		1232.4.a.d		1
15.d	odd	2	1		1925.4.a.a		1
21.c	even	2	1		539.4.a.a		1
33.d	even	2	1		847.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.4.a.a	✓	1	3.b	odd	2	1
539.4.a.a		1	21.c	even	2	1
693.4.a.b		1	1.a	even	1	1	trivial
847.4.a.a		1	33.d	even	2	1
1232.4.a.d		1	12.b	even	2	1
1925.4.a.a		1	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(693))$:

$ T_{2} + 3 $

T2 + 3

$ T_{5} + 12 $

T5 + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 3 $ T + 3
$3$	$ T $ T
$5$	$ T + 12 $ T + 12
$7$	$ T - 7 $ T - 7
$11$	$ T + 11 $ T + 11
$13$	$ T - 38 $ T - 38
$17$	$ T - 48 $ T - 48
$19$	$ T + 70 $ T + 70
$23$	$ T + 12 $ T + 12
$29$	$ T + 126 $ T + 126
$31$	$ T + 70 $ T + 70
$37$	$ T + 358 $ T + 358
$41$	$ T - 216 $ T - 216
$43$	$ T - 344 $ T - 344
$47$	$ T + 390 $ T + 390
$53$	$ T + 438 $ T + 438
$59$	$ T - 552 $ T - 552
$61$	$ T - 830 $ T - 830
$67$	$ T + 196 $ T + 196
$71$	$ T + 648 $ T + 648
$73$	$ T + 16 $ T + 16
$79$	$ T - 1352 $ T - 1352
$83$	$ T + 90 $ T + 90
$89$	$ T + 1146 $ T + 1146
$97$	$ T + 70 $ T + 70
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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more