LMFDB - Newform orbit 48.22.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(47,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$134.149125258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 9\sqrt{-3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} +O(q^{10}) $ q - 6561b q^3 + 63207986b q^7 - 10460353203 * q^9	$ q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} + 370076825230 q^{13} - 2560945382530 \beta q^{19} + 100773945863478 q^{21} + 476837158203125 q^{25} + 68630377364883 \beta q^{27} - 80612910215030 \beta q^{31} - 57\!\cdots\!90 q^{37} + \cdots + 11\!\cdots\!30 q^{97} +O(q^{100}) $ q - 6561b q^3 + 63207986b q^7 - 10460353203 * q^9 + 370076825230 * q^13 - 2560945382530b q^19 + 100773945863478 * q^21 + 476837158203125 * q^25 + 68630377364883b q^27 - 80612910215030b q^31 - 57776323439003290 * q^37 - 2428074050334030b q^39 + 6405893160250598b q^43 - 412299763001531621 * q^49 - 4082974125111377190 * q^57 - 10860691764464843938 * q^61 - 661177858810279158b q^63 + 1861754937027440574b q^67 + 39098623343480501290 * q^73 - 3128528594970703125b q^75 + 580541259680729690b q^79 + 109418989131512359209 * q^81 + 23391810788062286780b q^91 - 128523016852757274690 * q^93 + 1132068302544444351730 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20920706406 q^{9}+O(q^{10}) $ 2 * q - 20920706406 * q^9	$ 2 q - 20920706406 q^{9} + 740153650460 q^{13} + 201547891726956 q^{21} + 953674316406250 q^{25} - 11\!\cdots\!80 q^{37}+ \cdots + 22\!\cdots\!60 q^{97}+O(q^{100}) $ 2 * q - 20920706406 * q^9 + 740153650460 * q^13 + 201547891726956 * q^21 + 953674316406250 * q^25 - 115552646878006580 * q^37 - 824599526003063242 * q^49 - 8165948250222754380 * q^57 - 21721383528929687876 * q^61 + 78197246686961002580 * q^73 + 218837978263024718418 * q^81 - 257046033705514549380 * q^93 + 2264136605088888703460 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

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} $

47.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

102276.i

9.85315e8i

−1.04604e10

47.2

102276.i

−

9.85315e8i

−1.04604e10

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	48.22.c.a	✓	2
3.b	odd	2	1	CM	48.22.c.a	✓	2
4.b	odd	2	1	inner	48.22.c.a	✓	2
12.b	even	2	1	inner	48.22.c.a	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
48.22.c.a	✓	2	1.a	even	1	1	trivial
48.22.c.a	✓	2	3.b	odd	2	1	CM
48.22.c.a	✓	2	4.b	odd	2	1	inner
48.22.c.a	✓	2	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{22}^{\mathrm{new}}(48, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 10460353203 $ T^2 + 10460353203
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 97\!\cdots\!28 $ T^2 + 970845627084815628
$11$	$ T^{2} $ T^2
$13$	$ (T - 370076825230)^{2} $ (T - 370076825230)^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} + 15\!\cdots\!00 $ T^2 + 1593701224309319911095818700
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ T^{2} + 15\!\cdots\!00 $ T^2 + 1579121234280766613363609918700
$37$	$ (T + 57\!\cdots\!90)^{2} $ (T + 57776323439003290)^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} + 99\!\cdots\!72 $ T^2 + 9971618524872530651672564723897772
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ (T + 10\!\cdots\!38)^{2} $ (T + 10860691764464843938)^2
$67$	$ T^{2} + 84\!\cdots\!68 $ T^2 + 842269941267689959788829300647652222668
$71$	$ T^{2} $ T^2
$73$	$ (T - 39\!\cdots\!90)^{2} $ (T - 39098623343480501290)^2
$79$	$ T^{2} + 81\!\cdots\!00 $ T^2 + 81897841468580287033446416395941552300
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ (T - 11\!\cdots\!30)^{2} $ (T - 1132068302544444351730)^2
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Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} +O(q^{10}) \) q - 6561b q^3 + 63207986b q^7 - 10460353203 * q^9	\( q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} + 370076825230 q^{13} - 2560945382530 \beta q^{19} + 100773945863478 q^{21} + 476837158203125 q^{25} + 68630377364883 \beta q^{27} - 80612910215030 \beta q^{31} - 57\!\cdots\!90 q^{37} + \cdots + 11\!\cdots\!30 q^{97} +O(q^{100}) \) q - 6561b q^3 + 63207986b q^7 - 10460353203 * q^9 + 370076825230 * q^13 - 2560945382530b q^19 + 100773945863478 * q^21 + 476837158203125 * q^25 + 68630377364883b q^27 - 80612910215030b q^31 - 57776323439003290 * q^37 - 2428074050334030b q^39 + 6405893160250598b q^43 - 412299763001531621 * q^49 - 4082974125111377190 * q^57 - 10860691764464843938 * q^61 - 661177858810279158b q^63 + 1861754937027440574b q^67 + 39098623343480501290 * q^73 - 3128528594970703125b q^75 + 580541259680729690b q^79 + 109418989131512359209 * q^81 + 23391810788062286780b q^91 - 128523016852757274690 * q^93 + 1132068302544444351730 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20920706406 q^{9}+O(q^{10}) \) 2 * q - 20920706406 * q^9	\( 2 q - 20920706406 q^{9} + 740153650460 q^{13} + 201547891726956 q^{21} + 953674316406250 q^{25} - 11\!\cdots\!80 q^{37}+ \cdots + 22\!\cdots\!60 q^{97}+O(q^{100}) \) 2 * q - 20920706406 * q^9 + 740153650460 * q^13 + 201547891726956 * q^21 + 953674316406250 * q^25 - 115552646878006580 * q^37 - 824599526003063242 * q^49 - 8165948250222754380 * q^57 - 21721383528929687876 * q^61 + 78197246686961002580 * q^73 + 218837978263024718418 * q^81 - 257046033705514549380 * q^93 + 2264136605088888703460 * q^97

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