LMFDB - Newform orbit 2450.4.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2450,4,Mod(1,2450)] 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("2450.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$2450 = 2 \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,-2,4,0,4,0,-8,-23,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$144.554679514$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
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} - 2 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{8} - 23 q^{9} + 48 q^{11} - 8 q^{12} + 56 q^{13} + 16 q^{16} - 114 q^{17} + 46 q^{18} - 2 q^{19} - 96 q^{22} + 120 q^{23} + 16 q^{24} - 112 q^{26} + 100 q^{27}+ \cdots - 1104 q^{99}+O(q^{100})

q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^6 - 8 * q^8 - 23 * q^9 + 48 * q^11 - 8 * q^12 + 56 * q^13 + 16 * q^16 - 114 * q^17 + 46 * q^18 - 2 * q^19 - 96 * q^22 + 120 * q^23 + 16 * q^24 - 112 * q^26 + 100 * q^27 - 54 * q^29 - 236 * q^31 - 32 * q^32 - 96 * q^33 + 228 * q^34 - 92 * q^36 - 146 * q^37 + 4 * q^38 - 112 * q^39 - 126 * q^41 + 376 * q^43 + 192 * q^44 - 240 * q^46 - 12 * q^47 - 32 * q^48 + 228 * q^51 + 224 * q^52 - 174 * q^53 - 200 * q^54 + 4 * q^57 + 108 * q^58 - 138 * q^59 - 380 * q^61 + 472 * q^62 + 64 * q^64 + 192 * q^66 + 484 * q^67 - 456 * q^68 - 240 * q^69 + 576 * q^71 + 184 * q^72 - 1150 * q^73 + 292 * q^74 - 8 * q^76 + 224 * q^78 + 776 * q^79 + 421 * q^81 + 252 * q^82 + 378 * q^83 - 752 * q^86 + 108 * q^87 - 384 * q^88 + 390 * q^89 + 480 * q^92 + 472 * q^93 + 24 * q^94 + 64 * q^96 - 1330 * q^97 - 1104 * 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

−2.00000

4.00000

−8.00000

−23.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$7$	$-1$

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	2450.4.a.i		1
5.b	even	2	1		98.4.a.e		1
7.b	odd	2	1		350.4.a.f		1
15.d	odd	2	1		882.4.a.b		1
20.d	odd	2	1		784.4.a.h		1
35.c	odd	2	1		14.4.a.b	✓	1
35.f	even	4	2		350.4.c.g		2
35.i	odd	6	2		98.4.c.c		2
35.j	even	6	2		98.4.c.b		2
105.g	even	2	1		126.4.a.d		1
105.o	odd	6	2		882.4.g.v		2
105.p	even	6	2		882.4.g.p		2
140.c	even	2	1		112.4.a.e		1
280.c	odd	2	1		448.4.a.k		1
280.n	even	2	1		448.4.a.g		1
385.h	even	2	1		1694.4.a.b		1
420.o	odd	2	1		1008.4.a.r		1
455.h	odd	2	1		2366.4.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.a.b	✓	1	35.c	odd	2	1
98.4.a.e		1	5.b	even	2	1
98.4.c.b		2	35.j	even	6	2
98.4.c.c		2	35.i	odd	6	2
112.4.a.e		1	140.c	even	2	1
126.4.a.d		1	105.g	even	2	1
350.4.a.f		1	7.b	odd	2	1
350.4.c.g		2	35.f	even	4	2
448.4.a.g		1	280.n	even	2	1
448.4.a.k		1	280.c	odd	2	1
784.4.a.h		1	20.d	odd	2	1
882.4.a.b		1	15.d	odd	2	1
882.4.g.p		2	105.p	even	6	2
882.4.g.v		2	105.o	odd	6	2
1008.4.a.r		1	420.o	odd	2	1
1694.4.a.b		1	385.h	even	2	1
2366.4.a.c		1	455.h	odd	2	1
2450.4.a.i		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 $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$ :

T_{3} + 2

T3 + 2

T_{11} - 48

T11 - 48

T_{19} + 2

T19 + 2

T_{23} - 120

T23 - 120

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T + 2$ T + 2
$5$	$T$ T
$7$	$T$ T
$11$	$T - 48$ T - 48
$13$	$T - 56$ T - 56
$17$	$T + 114$ T + 114
$19$	$T + 2$ T + 2
$23$	$T - 120$ T - 120
$29$	$T + 54$ T + 54
$31$	$T + 236$ T + 236
$37$	$T + 146$ T + 146
$41$	$T + 126$ T + 126
$43$	$T - 376$ T - 376
$47$	$T + 12$ T + 12
$53$	$T + 174$ T + 174
$59$	$T + 138$ T + 138
$61$	$T + 380$ T + 380
$67$	$T - 484$ T - 484
$71$	$T - 576$ T - 576
$73$	$T + 1150$ T + 1150
$79$	$T - 776$ T - 776
$83$	$T - 378$ T - 378
$89$	$T - 390$ T - 390
$97$	$T + 1330$ T + 1330
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