LMFDB - Newform orbit 18.22.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,22,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	18.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:	$50.3059219717$
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 - 1024 q^{2} + 1048576 q^{4} - 12954174 q^{5} - 479513104 q^{7} - 1073741824 q^{8}+O(q^{10}) $

$ q - 1024 q^{2} + 1048576 q^{4} - 12954174 q^{5} - 479513104 q^{7} - 1073741824 q^{8} + 13265074176 q^{10} - 115657781700 q^{11} + 295658246702 q^{13} + 491021418496 q^{14} + 1099511627776 q^{16} - 6626983431906 q^{17} + 28576184164796 q^{19} - 13583435956224 q^{20} + 118433568460800 q^{22} - 335385196791000 q^{23} - 309026534180849 q^{25} - 302754044622848 q^{26} - 502805932539904 q^{28} + 699224214482106 q^{29} - 34\!\cdots\!92 q^{31}+ \cdots + 33\!\cdots\!84 q^{98}+O(q^{100}) $

q - 1024 * q^2 + 1048576 * q^4 - 12954174 * q^5 - 479513104 * q^7 - 1073741824 * q^8 + 13265074176 * q^10 - 115657781700 * q^11 + 295658246702 * q^13 + 491021418496 * q^14 + 1099511627776 * q^16 - 6626983431906 * q^17 + 28576184164796 * q^19 - 13583435956224 * q^20 + 118433568460800 * q^22 - 335385196791000 * q^23 - 309026534180849 * q^25 - 302754044622848 * q^26 - 502805932539904 * q^28 + 699224214482106 * q^29 - 3484957262657992 * q^31 - 1125899906842624 * q^32 + 6786031034271744 * q^34 + 6211696184496096 * q^35 - 35181531093012298 * q^37 - 29262012584751104 * q^38 + 13909438419173376 * q^40 - 6132056240639994 * q^41 + 233260850096910596 * q^43 - 121275974103859200 * q^44 + 343434441513984000 * q^46 + 580205712121346400 * q^47 - 328613047175569191 * q^49 + 316443171001189376 * q^50 + 310020141693796352 * q^52 + 1394471665941750306 * q^53 + 1498251028595815800 * q^55 + 514873274920861696 * q^56 - 716005595629676544 * q^58 - 2352476807159705700 * q^59 + 9920628300330384590 * q^61 + 3568596236961783808 * q^62 + 1152921504606846976 * q^64 - 3830008372312634148 * q^65 + 26068981808996843996 * q^67 - 6948895779094265856 * q^68 - 6360776892924002304 * q^70 + 13336955952504341400 * q^71 + 9037529597968684202 * q^73 + 36025887839244593152 * q^74 + 29964300886785130496 * q^76 + 55459421904721396800 * q^77 - 77283864571811027992 * q^79 - 14243264941233537024 * q^80 + 6279225590415353856 * q^82 + 155698418876868248388 * q^83 + 85847096472027475644 * q^85 - 238859110499236450304 * q^86 + 124186597482351820800 * q^88 - 253837312813381912218 * q^89 - 141772003599273783008 * q^91 - 351676868110319616000 * q^92 - 594130649212258713600 * q^94 - 370180861926812058504 * q^95 - 1030722092535365121598 * q^97 + 336499760307782851584 * q^98

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

−1024.00

1.04858e6

−1.29542e7

−4.79513e8

−1.07374e9

1.32651e10

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	18.22.a.a		1
3.b	odd	2	1		6.22.a.b	✓	1
12.b	even	2	1		48.22.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.22.a.b	✓	1	3.b	odd	2	1
18.22.a.a		1	1.a	even	1	1	trivial
48.22.a.f		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 12954174 $ T5 + 12954174 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(18))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1024 $ T + 1024
$3$	$ T $ T
$5$	$ T + 12954174 $ T + 12954174
$7$	$ T + 479513104 $ T + 479513104
$11$	$ T + 115657781700 $ T + 115657781700
$13$	$ T - 295658246702 $ T - 295658246702
$17$	$ T + 6626983431906 $ T + 6626983431906
$19$	$ T - 28576184164796 $ T - 28576184164796
$23$	$ T + 335385196791000 $ T + 335385196791000
$29$	$ T - 699224214482106 $ T - 699224214482106
$31$	$ T + 3484957262657992 $ T + 3484957262657992
$37$	$ T + 35\!\cdots\!98 $ T + 35181531093012298
$41$	$ T + 6132056240639994 $ T + 6132056240639994
$43$	$ T - 23\!\cdots\!96 $ T - 233260850096910596
$47$	$ T - 58\!\cdots\!00 $ T - 580205712121346400
$53$	$ T - 13\!\cdots\!06 $ T - 1394471665941750306
$59$	$ T + 23\!\cdots\!00 $ T + 2352476807159705700
$61$	$ T - 99\!\cdots\!90 $ T - 9920628300330384590
$67$	$ T - 26\!\cdots\!96 $ T - 26068981808996843996
$71$	$ T - 13\!\cdots\!00 $ T - 13336955952504341400
$73$	$ T - 90\!\cdots\!02 $ T - 9037529597968684202
$79$	$ T + 77\!\cdots\!92 $ T + 77283864571811027992
$83$	$ T - 15\!\cdots\!88 $ T - 155698418876868248388
$89$	$ T + 25\!\cdots\!18 $ T + 253837312813381912218
$97$	$ T + 10\!\cdots\!98 $ T + 1030722092535365121598
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

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