LMFDB - Newform orbit 245.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,4,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$245 = 5 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	245.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.4554679514$
Analytic rank:	$0$
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} + 5 q^{5} + 8 q^{6} - 23 q^{9} - 20 q^{10} + 32 q^{11} - 16 q^{12} + 38 q^{13} - 10 q^{15} - 64 q^{16} - 26 q^{17} + 92 q^{18} - 100 q^{19} + 40 q^{20} - 128 q^{22} - 78 q^{23}+ \cdots - 736 q^{99}+O(q^{100})

q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 - 23 * q^9 - 20 * q^10 + 32 * q^11 - 16 * q^12 + 38 * q^13 - 10 * q^15 - 64 * q^16 - 26 * q^17 + 92 * q^18 - 100 * q^19 + 40 * q^20 - 128 * q^22 - 78 * q^23 + 25 * q^25 - 152 * q^26 + 100 * q^27 - 50 * q^29 + 40 * q^30 + 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 - 115 * q^45 + 312 * q^46 + 514 * q^47 + 128 * q^48 - 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 - 512 * q^64 + 190 * q^65 + 256 * q^66 + 126 * q^67 - 208 * q^68 + 156 * q^69 + 412 * q^71 + 878 * q^73 - 1064 * q^74 - 50 * q^75 - 800 * q^76 + 304 * q^78 + 600 * q^79 - 320 * q^80 + 421 * q^81 + 88 * q^82 - 282 * q^83 - 130 * q^85 - 1768 * q^86 + 100 * q^87 + 150 * q^89 + 460 * q^90 - 624 * q^92 - 216 * q^93 - 2056 * q^94 - 500 * q^95 - 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

5.00000

8.00000

−23.0000

−20.0000

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	245.4.a.a		1
3.b	odd	2	1		2205.4.a.q		1
5.b	even	2	1		1225.4.a.k		1
7.b	odd	2	1		5.4.a.a	✓	1
7.c	even	3	2		245.4.e.g		2
7.d	odd	6	2		245.4.e.f		2
21.c	even	2	1		45.4.a.d		1
28.d	even	2	1		80.4.a.d		1
35.c	odd	2	1		25.4.a.c		1
35.f	even	4	2		25.4.b.a		2
56.e	even	2	1		320.4.a.h		1
56.h	odd	2	1		320.4.a.g		1
63.l	odd	6	2		405.4.e.l		2
63.o	even	6	2		405.4.e.c		2
77.b	even	2	1		605.4.a.d		1
84.h	odd	2	1		720.4.a.u		1
91.b	odd	2	1		845.4.a.b		1
105.g	even	2	1		225.4.a.b		1
105.k	odd	4	2		225.4.b.c		2
112.j	even	4	2		1280.4.d.l		2
112.l	odd	4	2		1280.4.d.e		2
119.d	odd	2	1		1445.4.a.a		1
133.c	even	2	1		1805.4.a.h		1
140.c	even	2	1		400.4.a.m		1
140.j	odd	4	2		400.4.c.k		2
280.c	odd	2	1		1600.4.a.bi		1
280.n	even	2	1		1600.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	7.b	odd	2	1
25.4.a.c		1	35.c	odd	2	1
25.4.b.a		2	35.f	even	4	2
45.4.a.d		1	21.c	even	2	1
80.4.a.d		1	28.d	even	2	1
225.4.a.b		1	105.g	even	2	1
225.4.b.c		2	105.k	odd	4	2
245.4.a.a		1	1.a	even	1	1	trivial
245.4.e.f		2	7.d	odd	6	2
245.4.e.g		2	7.c	even	3	2
320.4.a.g		1	56.h	odd	2	1
320.4.a.h		1	56.e	even	2	1
400.4.a.m		1	140.c	even	2	1
400.4.c.k		2	140.j	odd	4	2
405.4.e.c		2	63.o	even	6	2
405.4.e.l		2	63.l	odd	6	2
605.4.a.d		1	77.b	even	2	1
720.4.a.u		1	84.h	odd	2	1
845.4.a.b		1	91.b	odd	2	1
1225.4.a.k		1	5.b	even	2	1
1280.4.d.e		2	112.l	odd	4	2
1280.4.d.l		2	112.j	even	4	2
1445.4.a.a		1	119.d	odd	2	1
1600.4.a.s		1	280.n	even	2	1
1600.4.a.bi		1	280.c	odd	2	1
1805.4.a.h		1	133.c	even	2	1
2205.4.a.q		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(245))$ :

T_{2} + 4

T2 + 4

T_{3} + 2

T3 + 2

Hecke characteristic polynomials

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