LMFDB - Newform orbit 48.22.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

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:	yes
Analytic conductor:	$134.149125258$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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 - 59049 q^{3} - 23245050 q^{5} + 1322977768 q^{7} + 3486784401 q^{9}+O(q^{10}) $

$ q - 59049 q^{3} - 23245050 q^{5} + 1322977768 q^{7} + 3486784401 q^{9} + 109174443828 q^{11} + 468325115966 q^{13} + 1372596957450 q^{15} + 2654798072562 q^{17} + 43712786306860 q^{19} - 78120514222632 q^{21} + 216861233964744 q^{23} + 63495191299375 q^{25} - 205891132094649 q^{27} + 25\!\cdots\!10 q^{29}+ \cdots + 38\!\cdots\!28 q^{99}+O(q^{100}) $

q - 59049 * q^3 - 23245050 * q^5 + 1322977768 * q^7 + 3486784401 * q^9 + 109174443828 * q^11 + 468325115966 * q^13 + 1372596957450 * q^15 + 2654798072562 * q^17 + 43712786306860 * q^19 - 78120514222632 * q^21 + 216861233964744 * q^23 + 63495191299375 * q^25 - 205891132094649 * q^27 + 2535247265345310 * q^29 - 5132915444930672 * q^31 - 6446641733599572 * q^33 - 30752684366048400 * q^35 - 8126962096433578 * q^37 - 27654129772676334 * q^39 - 28546174551317718 * q^41 - 60426656396902316 * q^43 - 81050477740465050 * q^45 + 316578527337771888 * q^47 + 1191724310538977817 * q^49 - 156763171386713538 * q^51 + 237962956198086486 * q^53 - 2537765405504051400 * q^55 - 2581196318633776140 * q^57 + 1932323017293179940 * q^59 - 5540333075504971378 * q^61 + 4612938244332196968 * q^63 - 10886240736885468300 * q^65 - 19977690098489923172 * q^67 - 12805439004384168456 * q^69 + 43257338948428460568 * q^71 + 1683404149334166506 * q^73 - 3749327551036794375 * q^75 + 144435362018208815904 * q^77 - 159081535530354452960 * q^79 + 12157665459056928801 * q^81 + 265853420659068579564 * q^83 - 61710913936607318100 * q^85 - 149703815771375210190 * q^87 - 17496064778396968710 * q^89 + 619583716619039843888 * q^91 + 303093524107711250928 * q^93 - 1016105903342276043000 * q^95 - 361656712539028124638 * q^97 + 380667747727321127028 * q^99

Display 100 coefficients 10 coefficients

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

−59049.0

−2.32450e7

1.32298e9

3.48678e9

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$

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	48.22.a.a		1
4.b	odd	2	1		6.22.a.c	✓	1
12.b	even	2	1		18.22.a.c		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 23245050 $ T5 + 23245050 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(48))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 59049 $ T + 59049
$5$	$ T + 23245050 $ T + 23245050
$7$	$ T - 1322977768 $ T - 1322977768
$11$	$ T - 109174443828 $ T - 109174443828
$13$	$ T - 468325115966 $ T - 468325115966
$17$	$ T - 2654798072562 $ T - 2654798072562
$19$	$ T - 43712786306860 $ T - 43712786306860
$23$	$ T - 216861233964744 $ T - 216861233964744
$29$	$ T - 2535247265345310 $ T - 2535247265345310
$31$	$ T + 5132915444930672 $ T + 5132915444930672
$37$	$ T + 8126962096433578 $ T + 8126962096433578
$41$	$ T + 28\!\cdots\!18 $ T + 28546174551317718
$43$	$ T + 60\!\cdots\!16 $ T + 60426656396902316
$47$	$ T - 31\!\cdots\!88 $ T - 316578527337771888
$53$	$ T - 23\!\cdots\!86 $ T - 237962956198086486
$59$	$ T - 19\!\cdots\!40 $ T - 1932323017293179940
$61$	$ T + 55\!\cdots\!78 $ T + 5540333075504971378
$67$	$ T + 19\!\cdots\!72 $ T + 19977690098489923172
$71$	$ T - 43\!\cdots\!68 $ T - 43257338948428460568
$73$	$ T - 16\!\cdots\!06 $ T - 1683404149334166506
$79$	$ T + 15\!\cdots\!60 $ T + 159081535530354452960
$83$	$ T - 26\!\cdots\!64 $ T - 265853420659068579564
$89$	$ T + 17\!\cdots\!10 $ T + 17496064778396968710
$97$	$ T + 36\!\cdots\!38 $ T + 361656712539028124638
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

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