LMFDB - Newform orbit 1470.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,4,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1470.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1470.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:	$86.7328077084$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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 - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} - 8 q^{8} + 9 q^{9}+O(q^{10}) $

$ q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} - 8 q^{8} + 9 q^{9} + 10 q^{10} - 60 q^{11} - 12 q^{12} + 34 q^{13} + 15 q^{15} + 16 q^{16} - 42 q^{17} - 18 q^{18} + 76 q^{19} - 20 q^{20} + 120 q^{22} + 24 q^{24} + 25 q^{25} - 68 q^{26} - 27 q^{27} + 6 q^{29} - 30 q^{30} + 232 q^{31} - 32 q^{32} + 180 q^{33} + 84 q^{34} + 36 q^{36} + 134 q^{37} - 152 q^{38} - 102 q^{39} + 40 q^{40} - 234 q^{41} - 412 q^{43} - 240 q^{44} - 45 q^{45} + 360 q^{47} - 48 q^{48} - 50 q^{50} + 126 q^{51} + 136 q^{52} + 222 q^{53} + 54 q^{54} + 300 q^{55} - 228 q^{57} - 12 q^{58} - 660 q^{59} + 60 q^{60} + 490 q^{61} - 464 q^{62} + 64 q^{64} - 170 q^{65} - 360 q^{66} + 812 q^{67} - 168 q^{68} + 120 q^{71} - 72 q^{72} - 746 q^{73} - 268 q^{74} - 75 q^{75} + 304 q^{76} + 204 q^{78} + 152 q^{79} - 80 q^{80} + 81 q^{81} + 468 q^{82} + 804 q^{83} + 210 q^{85} + 824 q^{86} - 18 q^{87} + 480 q^{88} + 678 q^{89} + 90 q^{90} - 696 q^{93} - 720 q^{94} - 380 q^{95} + 96 q^{96} - 194 q^{97} - 540 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

−2.00000

−3.00000

4.00000

−5.00000

6.00000

−8.00000

9.00000

10.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$5$	$ +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	1470.4.a.a		1
7.b	odd	2	1		30.4.a.a	✓	1
21.c	even	2	1		90.4.a.d		1
28.d	even	2	1		240.4.a.c		1
35.c	odd	2	1		150.4.a.e		1
35.f	even	4	2		150.4.c.a		2
56.e	even	2	1		960.4.a.s		1
56.h	odd	2	1		960.4.a.j		1
63.l	odd	6	2		810.4.e.m		2
63.o	even	6	2		810.4.e.e		2
84.h	odd	2	1		720.4.a.b		1
105.g	even	2	1		450.4.a.b		1
105.k	odd	4	2		450.4.c.k		2
140.c	even	2	1		1200.4.a.bk		1
140.j	odd	4	2		1200.4.f.u		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.4.a.a	✓	1	7.b	odd	2	1
90.4.a.d		1	21.c	even	2	1
150.4.a.e		1	35.c	odd	2	1
150.4.c.a		2	35.f	even	4	2
240.4.a.c		1	28.d	even	2	1
450.4.a.b		1	105.g	even	2	1
450.4.c.k		2	105.k	odd	4	2
720.4.a.b		1	84.h	odd	2	1
810.4.e.e		2	63.o	even	6	2
810.4.e.m		2	63.l	odd	6	2
960.4.a.j		1	56.h	odd	2	1
960.4.a.s		1	56.e	even	2	1
1200.4.a.bk		1	140.c	even	2	1
1200.4.f.u		2	140.j	odd	4	2
1470.4.a.a		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1470))$:

$ T_{11} + 60 $

$ T_{13} - 34 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T + 3 $ T + 3
$5$	$ T + 5 $ T + 5
$7$	$ T $ T
$11$	$ T + 60 $ T + 60
$13$	$ T - 34 $ T - 34
$17$	$ T + 42 $ T + 42
$19$	$ T - 76 $ T - 76
$23$	$ T $ T
$29$	$ T - 6 $ T - 6
$31$	$ T - 232 $ T - 232
$37$	$ T - 134 $ T - 134
$41$	$ T + 234 $ T + 234
$43$	$ T + 412 $ T + 412
$47$	$ T - 360 $ T - 360
$53$	$ T - 222 $ T - 222
$59$	$ T + 660 $ T + 660
$61$	$ T - 490 $ T - 490
$67$	$ T - 812 $ T - 812
$71$	$ T - 120 $ T - 120
$73$	$ T + 746 $ T + 746
$79$	$ T - 152 $ T - 152
$83$	$ T - 804 $ T - 804
$89$	$ T - 678 $ T - 678
$97$	$ T + 194 $ T + 194
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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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

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