LMFDB - Newform orbit 1225.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.a (trivial)

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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
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} - 2 q^{3} + q^{4} + 6 q^{6} + 21 q^{8} - 23 q^{9}+O(q^{10}) $

$ q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} + 21 q^{8} - 23 q^{9} - 45 q^{11} - 2 q^{12} + 59 q^{13} - 71 q^{16} - 54 q^{17} + 69 q^{18} + 121 q^{19} + 135 q^{22} - 69 q^{23} - 42 q^{24} - 177 q^{26} + 100 q^{27} - 162 q^{29} + 88 q^{31} + 45 q^{32} + 90 q^{33} + 162 q^{34} - 23 q^{36} + 259 q^{37} - 363 q^{38} - 118 q^{39} - 195 q^{41} + 286 q^{43} - 45 q^{44} + 207 q^{46} + 45 q^{47} + 142 q^{48} + 108 q^{51} + 59 q^{52} - 597 q^{53} - 300 q^{54} - 242 q^{57} + 486 q^{58} + 360 q^{59} - 392 q^{61} - 264 q^{62} + 433 q^{64} - 270 q^{66} + 280 q^{67} - 54 q^{68} + 138 q^{69} + 48 q^{71} - 483 q^{72} + 668 q^{73} - 777 q^{74} + 121 q^{76} + 354 q^{78} + 782 q^{79} + 421 q^{81} + 585 q^{82} + 768 q^{83} - 858 q^{86} + 324 q^{87} - 945 q^{88} + 1194 q^{89} - 69 q^{92} - 176 q^{93} - 135 q^{94} - 90 q^{96} + 902 q^{97} + 1035 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

−2.00000

1.00000

6.00000

21.0000

−23.0000

Atkin-Lehner signs

$ p $	Sign
$5$	$1$
$7$	$-1$

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	1225.4.a.a		1
5.b	even	2	1		245.4.a.f		1
7.b	odd	2	1		1225.4.a.b		1
7.d	odd	6	2		175.4.e.b		2
15.d	odd	2	1		2205.4.a.g		1
35.c	odd	2	1		245.4.a.e		1
35.i	odd	6	2		35.4.e.a	✓	2
35.j	even	6	2		245.4.e.a		2
35.k	even	12	4		175.4.k.b		4
105.g	even	2	1		2205.4.a.e		1
105.p	even	6	2		315.4.j.b		2
140.s	even	6	2		560.4.q.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.e.a	✓	2	35.i	odd	6	2
175.4.e.b		2	7.d	odd	6	2
175.4.k.b		4	35.k	even	12	4
245.4.a.e		1	35.c	odd	2	1
245.4.a.f		1	5.b	even	2	1
245.4.e.a		2	35.j	even	6	2
315.4.j.b		2	105.p	even	6	2
560.4.q.b		2	140.s	even	6	2
1225.4.a.a		1	1.a	even	1	1	trivial
1225.4.a.b		1	7.b	odd	2	1
2205.4.a.e		1	105.g	even	2	1
2205.4.a.g		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 $S_{4}^{\mathrm{new}}(\Gamma_0(1225))$:

$ T_{2} + 3 $

$ T_{3} + 2 $

$ T_{19} - 121 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 3 $ T + 3
$3$	$ T + 2 $ T + 2
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 45 $ T + 45
$13$	$ T - 59 $ T - 59
$17$	$ T + 54 $ T + 54
$19$	$ T - 121 $ T - 121
$23$	$ T + 69 $ T + 69
$29$	$ T + 162 $ T + 162
$31$	$ T - 88 $ T - 88
$37$	$ T - 259 $ T - 259
$41$	$ T + 195 $ T + 195
$43$	$ T - 286 $ T - 286
$47$	$ T - 45 $ T - 45
$53$	$ T + 597 $ T + 597
$59$	$ T - 360 $ T - 360
$61$	$ T + 392 $ T + 392
$67$	$ T - 280 $ T - 280
$71$	$ T - 48 $ T - 48
$73$	$ T - 668 $ T - 668
$79$	$ T - 782 $ T - 782
$83$	$ T - 768 $ T - 768
$89$	$ T - 1194 $ T - 1194
$97$	$ T - 902 $ T - 902
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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.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