LMFDB - Newform orbit 1225.4.a.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1225,4,Mod(1,1225)] 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("1225.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [1,4,2,8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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 5)
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} + 8 q^{6} - 23 q^{9} + 32 q^{11} + 16 q^{12} - 38 q^{13} - 64 q^{16} + 26 q^{17} - 92 q^{18} - 100 q^{19} + 128 q^{22} + 78 q^{23} - 152 q^{26} - 100 q^{27} - 50 q^{29}+ \cdots - 736 q^{99}+O(q^{100}) $

q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 8 * q^6 - 23 * q^9 + 32 * q^11 + 16 * q^12 - 38 * q^13 - 64 * q^16 + 26 * q^17 - 92 * q^18 - 100 * q^19 + 128 * q^22 + 78 * q^23 - 152 * q^26 - 100 * q^27 - 50 * q^29 + 108 * q^31 - 256 * q^32 + 64 * q^33 + 104 * q^34 - 184 * q^36 - 266 * q^37 - 400 * q^38 - 76 * q^39 - 22 * q^41 - 442 * q^43 + 256 * q^44 + 312 * q^46 - 514 * q^47 - 128 * q^48 + 52 * q^51 - 304 * q^52 - 2 * q^53 - 400 * q^54 - 200 * q^57 - 200 * q^58 - 500 * q^59 + 518 * q^61 + 432 * q^62 - 512 * q^64 + 256 * q^66 - 126 * q^67 + 208 * q^68 + 156 * q^69 + 412 * q^71 - 878 * q^73 - 1064 * q^74 - 800 * q^76 - 304 * q^78 + 600 * q^79 + 421 * q^81 - 88 * q^82 + 282 * q^83 - 1768 * q^86 - 100 * q^87 + 150 * q^89 + 624 * q^92 + 216 * q^93 - 2056 * q^94 - 512 * q^96 + 386 * q^97 - 736 * q^99

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

−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.k		1
5.b	even	2	1		245.4.a.a		1
7.b	odd	2	1		25.4.a.c		1
15.d	odd	2	1		2205.4.a.q		1
21.c	even	2	1		225.4.a.b		1
28.d	even	2	1		400.4.a.m		1
35.c	odd	2	1		5.4.a.a	✓	1
35.f	even	4	2		25.4.b.a		2
35.i	odd	6	2		245.4.e.f		2
35.j	even	6	2		245.4.e.g		2
56.e	even	2	1		1600.4.a.s		1
56.h	odd	2	1		1600.4.a.bi		1
105.g	even	2	1		45.4.a.d		1
105.k	odd	4	2		225.4.b.c		2
140.c	even	2	1		80.4.a.d		1
140.j	odd	4	2		400.4.c.k		2
280.c	odd	2	1		320.4.a.g		1
280.n	even	2	1		320.4.a.h		1
315.z	even	6	2		405.4.e.c		2
315.bg	odd	6	2		405.4.e.l		2
385.h	even	2	1		605.4.a.d		1
420.o	odd	2	1		720.4.a.u		1
455.h	odd	2	1		845.4.a.b		1
560.be	even	4	2		1280.4.d.l		2
560.bf	odd	4	2		1280.4.d.e		2
595.b	odd	2	1		1445.4.a.a		1
665.g	even	2	1		1805.4.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	35.c	odd	2	1
25.4.a.c		1	7.b	odd	2	1
25.4.b.a		2	35.f	even	4	2
45.4.a.d		1	105.g	even	2	1
80.4.a.d		1	140.c	even	2	1
225.4.a.b		1	21.c	even	2	1
225.4.b.c		2	105.k	odd	4	2
245.4.a.a		1	5.b	even	2	1
245.4.e.f		2	35.i	odd	6	2
245.4.e.g		2	35.j	even	6	2
320.4.a.g		1	280.c	odd	2	1
320.4.a.h		1	280.n	even	2	1
400.4.a.m		1	28.d	even	2	1
400.4.c.k		2	140.j	odd	4	2
405.4.e.c		2	315.z	even	6	2
405.4.e.l		2	315.bg	odd	6	2
605.4.a.d		1	385.h	even	2	1
720.4.a.u		1	420.o	odd	2	1
845.4.a.b		1	455.h	odd	2	1
1225.4.a.k		1	1.a	even	1	1	trivial
1280.4.d.e		2	560.bf	odd	4	2
1280.4.d.l		2	560.be	even	4	2
1445.4.a.a		1	595.b	odd	2	1
1600.4.a.s		1	56.e	even	2	1
1600.4.a.bi		1	56.h	odd	2	1
1805.4.a.h		1	665.g	even	2	1
2205.4.a.q		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} - 4 $

T2 - 4

$ T_{3} - 2 $

T3 - 2

$ T_{19} + 100 $

T19 + 100

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 4 $ T - 4
$3$	$ T - 2 $ T - 2
$5$	$ T $ T
$7$	$ T $ T
$11$	$ 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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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:
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