LMFDB - Newform orbit 2450.4.a.z

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)

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")

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);

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,-4,4,0,-8,0,8,-11,0,30] 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 350)
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} - 4 q^{3} + 4 q^{4} - 8 q^{6} + 8 q^{8} - 11 q^{9} + 30 q^{11} - 16 q^{12} + 4 q^{13} + 16 q^{16} - 9 q^{17} - 22 q^{18} - 88 q^{19} + 60 q^{22} - 33 q^{23} - 32 q^{24} + 8 q^{26} + 152 q^{27}+ \cdots - 330 q^{99}+O(q^{100}) $

q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 8 * q^6 + 8 * q^8 - 11 * q^9 + 30 * q^11 - 16 * q^12 + 4 * q^13 + 16 * q^16 - 9 * q^17 - 22 * q^18 - 88 * q^19 + 60 * q^22 - 33 * q^23 - 32 * q^24 + 8 * q^26 + 152 * q^27 + 126 * q^29 + 155 * q^31 + 32 * q^32 - 120 * q^33 - 18 * q^34 - 44 * q^36 - 116 * q^37 - 176 * q^38 - 16 * q^39 - 423 * q^41 + 340 * q^43 + 120 * q^44 - 66 * q^46 - 339 * q^47 - 64 * q^48 + 36 * q^51 + 16 * q^52 + 312 * q^53 + 304 * q^54 + 352 * q^57 + 252 * q^58 - 462 * q^59 + 326 * q^61 + 310 * q^62 + 64 * q^64 - 240 * q^66 - 704 * q^67 - 36 * q^68 + 132 * q^69 + 621 * q^71 - 88 * q^72 + 250 * q^73 - 232 * q^74 - 352 * q^76 - 32 * q^78 - 1105 * q^79 - 311 * q^81 - 846 * q^82 + 198 * q^83 + 680 * q^86 - 504 * q^87 + 240 * q^88 - 873 * q^89 - 132 * q^92 - 620 * q^93 - 678 * q^94 - 128 * q^96 - 905 * q^97 - 330 * 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

4.00000

−8.00000

8.00000

−11.0000

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.z		1
5.b	even	2	1		2450.4.a.p		1
7.b	odd	2	1		2450.4.a.bl		1
7.c	even	3	2		350.4.e.c	✓	2
35.c	odd	2	1		2450.4.a.f		1
35.j	even	6	2		350.4.e.f	yes	2
35.l	odd	12	4		350.4.j.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.4.e.c	✓	2	7.c	even	3	2
350.4.e.f	yes	2	35.j	even	6	2
350.4.j.c		4	35.l	odd	12	4
2450.4.a.f		1	35.c	odd	2	1
2450.4.a.p		1	5.b	even	2	1
2450.4.a.z		1	1.a	even	1	1	trivial
2450.4.a.bl		1	7.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 $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$:

$ T_{3} + 4 $

T3 + 4

$ T_{11} - 30 $

T11 - 30

$ T_{19} + 88 $

T19 + 88

$ T_{23} + 33 $

T23 + 33

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 4 $ T + 4
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 30 $ T - 30
$13$	$ T - 4 $ T - 4
$17$	$ T + 9 $ T + 9
$19$	$ T + 88 $ T + 88
$23$	$ T + 33 $ T + 33
$29$	$ T - 126 $ T - 126
$31$	$ T - 155 $ T - 155
$37$	$ T + 116 $ T + 116
$41$	$ T + 423 $ T + 423
$43$	$ T - 340 $ T - 340
$47$	$ T + 339 $ T + 339
$53$	$ T - 312 $ T - 312
$59$	$ T + 462 $ T + 462
$61$	$ T - 326 $ T - 326
$67$	$ T + 704 $ T + 704
$71$	$ T - 621 $ T - 621
$73$	$ T - 250 $ T - 250
$79$	$ T + 1105 $ T + 1105
$83$	$ T - 198 $ T - 198
$89$	$ T + 873 $ T + 873
$97$	$ T + 905 $ T + 905
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Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

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