LMFDB - Newform orbit 5.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5,4,Mod(1,5)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	5.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.295009550029$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 - 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} - 8 q^{6} + 6 q^{7} - 23 q^{9} + 20 q^{10} + 32 q^{11} + 16 q^{12} - 38 q^{13} - 24 q^{14} - 10 q^{15} - 64 q^{16} + 26 q^{17} + 92 q^{18} + 100 q^{19} - 40 q^{20}+ \cdots - 736 q^{99}+O(q^{100})

q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 5 * q^5 - 8 * q^6 + 6 * q^7 - 23 * q^9 + 20 * q^10 + 32 * q^11 + 16 * q^12 - 38 * q^13 - 24 * q^14 - 10 * q^15 - 64 * q^16 + 26 * q^17 + 92 * q^18 + 100 * q^19 - 40 * q^20 + 12 * q^21 - 128 * q^22 - 78 * q^23 + 25 * q^25 + 152 * q^26 - 100 * q^27 + 48 * q^28 - 50 * q^29 + 40 * q^30 - 108 * q^31 + 256 * q^32 + 64 * q^33 - 104 * q^34 - 30 * q^35 - 184 * q^36 + 266 * q^37 - 400 * q^38 - 76 * q^39 + 22 * q^41 - 48 * q^42 + 442 * q^43 + 256 * q^44 + 115 * q^45 + 312 * q^46 - 514 * q^47 - 128 * q^48 - 307 * q^49 - 100 * q^50 + 52 * q^51 - 304 * q^52 + 2 * q^53 + 400 * q^54 - 160 * q^55 + 200 * q^57 + 200 * q^58 + 500 * q^59 - 80 * q^60 - 518 * q^61 + 432 * q^62 - 138 * q^63 - 512 * q^64 + 190 * q^65 - 256 * q^66 + 126 * q^67 + 208 * q^68 - 156 * q^69 + 120 * q^70 + 412 * q^71 - 878 * q^73 - 1064 * q^74 + 50 * q^75 + 800 * q^76 + 192 * q^77 + 304 * q^78 + 600 * q^79 + 320 * q^80 + 421 * q^81 - 88 * q^82 + 282 * q^83 + 96 * q^84 - 130 * q^85 - 1768 * q^86 - 100 * q^87 - 150 * q^89 - 460 * q^90 - 228 * q^91 - 624 * q^92 - 216 * q^93 + 2056 * q^94 - 500 * q^95 + 512 * q^96 + 386 * q^97 + 1228 * q^98 - 736 * q^99

Expression as an eta quotient

$f(z) = \eta(z)^{4}\eta(5z)^{4}=q\prod_{n=1}^\infty(1 - q^{n})^{4}(1 - q^{5n})^{4}$

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

−4.00000

2.00000

8.00000

−5.00000

−8.00000

6.00000

−23.0000

20.0000

Atkin-Lehner signs

$p$	Sign
$5$	$+1$

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	5.4.a.a	✓	1
3.b	odd	2	1		45.4.a.d		1
4.b	odd	2	1		80.4.a.d		1
5.b	even	2	1		25.4.a.c		1
5.c	odd	4	2		25.4.b.a		2
7.b	odd	2	1		245.4.a.a		1
7.c	even	3	2		245.4.e.f		2
7.d	odd	6	2		245.4.e.g		2
8.b	even	2	1		320.4.a.g		1
8.d	odd	2	1		320.4.a.h		1
9.c	even	3	2		405.4.e.l		2
9.d	odd	6	2		405.4.e.c		2
11.b	odd	2	1		605.4.a.d		1
12.b	even	2	1		720.4.a.u		1
13.b	even	2	1		845.4.a.b		1
15.d	odd	2	1		225.4.a.b		1
15.e	even	4	2		225.4.b.c		2
16.e	even	4	2		1280.4.d.e		2
16.f	odd	4	2		1280.4.d.l		2
17.b	even	2	1		1445.4.a.a		1
19.b	odd	2	1		1805.4.a.h		1
20.d	odd	2	1		400.4.a.m		1
20.e	even	4	2		400.4.c.k		2
21.c	even	2	1		2205.4.a.q		1
35.c	odd	2	1		1225.4.a.k		1
40.e	odd	2	1		1600.4.a.s		1
40.f	even	2	1		1600.4.a.bi		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	1.a	even	1	1	trivial
25.4.a.c		1	5.b	even	2	1
25.4.b.a		2	5.c	odd	4	2
45.4.a.d		1	3.b	odd	2	1
80.4.a.d		1	4.b	odd	2	1
225.4.a.b		1	15.d	odd	2	1
225.4.b.c		2	15.e	even	4	2
245.4.a.a		1	7.b	odd	2	1
245.4.e.f		2	7.c	even	3	2
245.4.e.g		2	7.d	odd	6	2
320.4.a.g		1	8.b	even	2	1
320.4.a.h		1	8.d	odd	2	1
400.4.a.m		1	20.d	odd	2	1
400.4.c.k		2	20.e	even	4	2
405.4.e.c		2	9.d	odd	6	2
405.4.e.l		2	9.c	even	3	2
605.4.a.d		1	11.b	odd	2	1
720.4.a.u		1	12.b	even	2	1
845.4.a.b		1	13.b	even	2	1
1225.4.a.k		1	35.c	odd	2	1
1280.4.d.e		2	16.e	even	4	2
1280.4.d.l		2	16.f	odd	4	2
1445.4.a.a		1	17.b	even	2	1
1600.4.a.s		1	40.e	odd	2	1
1600.4.a.bi		1	40.f	even	2	1
1805.4.a.h		1	19.b	odd	2	1
2205.4.a.q		1	21.c	even	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(\Gamma_0(5))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 4$ T + 4
$3$	$T - 2$ T - 2
$5$	$T + 5$ T + 5
$7$	$T - 6$ T - 6
$11$	$T - 32$ T - 32
$13$	$T + 38$ T + 38
$17$	$T - 26$ T - 26
$19$	$T - 100$ T - 100
$23$	$T + 78$ T + 78
$29$	$T + 50$ T + 50
$31$	$T + 108$ T + 108
$37$	$T - 266$ T - 266
$41$	$T - 22$ T - 22
$43$	$T - 442$ T - 442
$47$	$T + 514$ T + 514
$53$	$T - 2$ T - 2
$59$	$T - 500$ T - 500
$61$	$T + 518$ T + 518
$67$	$T - 126$ T - 126
$71$	$T - 412$ T - 412
$73$	$T + 878$ T + 878
$79$	$T - 600$ T - 600
$83$	$T - 282$ T - 282
$89$	$T + 150$ T + 150
$97$	$T - 386$ T - 386
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