LMFDB - Newform orbit 4.16.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4,16,Mod(1,4)] 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("4.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	4.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$5.70774020400$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - 276 q^{3} - 132210 q^{5} - 3585736 q^{7} - 14272731 q^{9} + 47801700 q^{11} + 247784966 q^{13} + 36489960 q^{15} - 2127682062 q^{17} - 1074862756 q^{19} + 989663136 q^{21} + 24982896168 q^{23} - 13038094025 q^{25}+ \cdots - 682260805442700 q^{99}+O(q^{100}) $

q - 276 * q^3 - 132210 * q^5 - 3585736 * q^7 - 14272731 * q^9 + 47801700 * q^11 + 247784966 * q^13 + 36489960 * q^15 - 2127682062 * q^17 - 1074862756 * q^19 + 989663136 * q^21 + 24982896168 * q^23 - 13038094025 * q^25 + 7899572088 * q^27 - 165099671946 * q^29 + 100736332256 * q^31 - 13193269200 * q^33 + 474070156560 * q^35 + 42490420334 * q^37 - 68388650616 * q^39 - 1388779245414 * q^41 - 1168783477180 * q^43 + 1886997765510 * q^45 - 1645655322672 * q^47 + 8109941151753 * q^49 + 587240249112 * q^51 - 4469627500578 * q^53 - 6319862757000 * q^55 + 296662120656 * q^57 - 28794808426572 * q^59 + 15719941145942 * q^61 + 51178245365016 * q^63 - 32759650354860 * q^65 + 61627103890604 * q^67 - 6895279342368 * q^69 - 66780412989192 * q^71 - 57749646345094 * q^73 + 3598513950900 * q^75 - 171404276551200 * q^77 + 198700138788272 * q^79 + 202617807858729 * q^81 - 113345193514212 * q^83 + 281300845417020 * q^85 + 45567509457096 * q^87 - 48230883277974 * q^89 - 888491472844976 * q^91 - 27803227702656 * q^93 + 142107604970760 * q^95 + 95121696327074 * q^97 - 682260805442700 * 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

−276.000

−132210.

−3.58574e6

−1.42727e7

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	4.16.a.a	✓	1
3.b	odd	2	1		36.16.a.b		1
4.b	odd	2	1		16.16.a.c		1
5.b	even	2	1		100.16.a.a		1
5.c	odd	4	2		100.16.c.a		2
8.b	even	2	1		64.16.a.g		1
8.d	odd	2	1		64.16.a.e		1
12.b	even	2	1		144.16.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.16.a.a	✓	1	1.a	even	1	1	trivial
16.16.a.c		1	4.b	odd	2	1
36.16.a.b		1	3.b	odd	2	1
64.16.a.e		1	8.d	odd	2	1
64.16.a.g		1	8.b	even	2	1
100.16.a.a		1	5.b	even	2	1
100.16.c.a		2	5.c	odd	4	2
144.16.a.k		1	12.b	even	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{16}^{\mathrm{new}}(\Gamma_0(4))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 276 $ T + 276
$5$	$ T + 132210 $ T + 132210
$7$	$ T + 3585736 $ T + 3585736
$11$	$ T - 47801700 $ T - 47801700
$13$	$ T - 247784966 $ T - 247784966
$17$	$ T + 2127682062 $ T + 2127682062
$19$	$ T + 1074862756 $ T + 1074862756
$23$	$ T - 24982896168 $ T - 24982896168
$29$	$ T + 165099671946 $ T + 165099671946
$31$	$ T - 100736332256 $ T - 100736332256
$37$	$ T - 42490420334 $ T - 42490420334
$41$	$ T + 1388779245414 $ T + 1388779245414
$43$	$ T + 1168783477180 $ T + 1168783477180
$47$	$ T + 1645655322672 $ T + 1645655322672
$53$	$ T + 4469627500578 $ T + 4469627500578
$59$	$ T + 28794808426572 $ T + 28794808426572
$61$	$ T - 15719941145942 $ T - 15719941145942
$67$	$ T - 61627103890604 $ T - 61627103890604
$71$	$ T + 66780412989192 $ T + 66780412989192
$73$	$ T + 57749646345094 $ T + 57749646345094
$79$	$ T - 198700138788272 $ T - 198700138788272
$83$	$ T + 113345193514212 $ T + 113345193514212
$89$	$ T + 48230883277974 $ T + 48230883277974
$97$	$ T - 95121696327074 $ T - 95121696327074
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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