LMFDB - Newform orbit 550.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [550,4,Mod(1,550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, 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("550.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	550.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,9,4,0,-18,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.4510505032$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
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} + 9 q^{3} + 4 q^{4} - 18 q^{6} + 5 q^{7} - 8 q^{8} + 54 q^{9} + 11 q^{11} + 36 q^{12} + 36 q^{13} - 10 q^{14} + 16 q^{16} - 17 q^{17} - 108 q^{18} + 41 q^{19} + 45 q^{21} - 22 q^{22} - 44 q^{23}+ \cdots + 594 q^{99}+O(q^{100}) $

q - 2 * q^2 + 9 * q^3 + 4 * q^4 - 18 * q^6 + 5 * q^7 - 8 * q^8 + 54 * q^9 + 11 * q^11 + 36 * q^12 + 36 * q^13 - 10 * q^14 + 16 * q^16 - 17 * q^17 - 108 * q^18 + 41 * q^19 + 45 * q^21 - 22 * q^22 - 44 * q^23 - 72 * q^24 - 72 * q^26 + 243 * q^27 + 20 * q^28 + 285 * q^29 - 323 * q^31 - 32 * q^32 + 99 * q^33 + 34 * q^34 + 216 * q^36 + 29 * q^37 - 82 * q^38 + 324 * q^39 + 208 * q^41 - 90 * q^42 + 430 * q^43 + 44 * q^44 + 88 * q^46 + 336 * q^47 + 144 * q^48 - 318 * q^49 - 153 * q^51 + 144 * q^52 + 725 * q^53 - 486 * q^54 - 40 * q^56 + 369 * q^57 - 570 * q^58 - 648 * q^59 - 565 * q^61 + 646 * q^62 + 270 * q^63 + 64 * q^64 - 198 * q^66 - 748 * q^67 - 68 * q^68 - 396 * q^69 - 265 * q^71 - 432 * q^72 + 602 * q^73 - 58 * q^74 + 164 * q^76 + 55 * q^77 - 648 * q^78 + 8 * q^79 + 729 * q^81 - 416 * q^82 - 708 * q^83 + 180 * q^84 - 860 * q^86 + 2565 * q^87 - 88 * q^88 + 137 * q^89 + 180 * q^91 - 176 * q^92 - 2907 * q^93 - 672 * q^94 - 288 * q^96 + 44 * q^97 + 636 * q^98 + 594 * 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

9.00000

4.00000

−18.0000

5.00000

−8.00000

54.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$5$	$ -1 $
$11$	$ -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	550.4.a.h		1
5.b	even	2	1		550.4.a.i		1
5.c	odd	4	2		110.4.b.a	✓	2
15.e	even	4	2		990.4.c.b		2
20.e	even	4	2		880.4.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.4.b.a	✓	2	5.c	odd	4	2
550.4.a.h		1	1.a	even	1	1	trivial
550.4.a.i		1	5.b	even	2	1
880.4.b.a		2	20.e	even	4	2
990.4.c.b		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(550))$:

$ T_{3} - 9 $

T3 - 9

$ T_{7} - 5 $

T7 - 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T - 9 $ T - 9
$5$	$ T $ T
$7$	$ T - 5 $ T - 5
$11$	$ T - 11 $ T - 11
$13$	$ T - 36 $ T - 36
$17$	$ T + 17 $ T + 17
$19$	$ T - 41 $ T - 41
$23$	$ T + 44 $ T + 44
$29$	$ T - 285 $ T - 285
$31$	$ T + 323 $ T + 323
$37$	$ T - 29 $ T - 29
$41$	$ T - 208 $ T - 208
$43$	$ T - 430 $ T - 430
$47$	$ T - 336 $ T - 336
$53$	$ T - 725 $ T - 725
$59$	$ T + 648 $ T + 648
$61$	$ T + 565 $ T + 565
$67$	$ T + 748 $ T + 748
$71$	$ T + 265 $ T + 265
$73$	$ T - 602 $ T - 602
$79$	$ T - 8 $ T - 8
$83$	$ T + 708 $ T + 708
$89$	$ T - 137 $ T - 137
$97$	$ T - 44 $ T - 44
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Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	550.a (trivial)

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