LMFDB - Newform orbit 18.22.a.d

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:	$1$
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} - 26444550 q^{5} + 166115864 q^{7} + 1073741824 q^{8}+O(q^{10}) $

$ q + 1024 q^{2} + 1048576 q^{4} - 26444550 q^{5} + 166115864 q^{7} + 1073741824 q^{8} - 27079219200 q^{10} + 104878761780 q^{11} + 335591325758 q^{13} + 170102644736 q^{14} + 1099511627776 q^{16} - 14596144763634 q^{17} + 3569529974996 q^{19} - 27729120460800 q^{20} + 107395852062720 q^{22} - 222369240588600 q^{23} + 222477066499375 q^{25} + 343645517576192 q^{26} + 174185108209664 q^{28} - 21\!\cdots\!54 q^{29}+ \cdots - 54\!\cdots\!64 q^{98}+O(q^{100}) $

q + 1024 * q^2 + 1048576 * q^4 - 26444550 * q^5 + 166115864 * q^7 + 1073741824 * q^8 - 27079219200 * q^10 + 104878761780 * q^11 + 335591325758 * q^13 + 170102644736 * q^14 + 1099511627776 * q^16 - 14596144763634 * q^17 + 3569529974996 * q^19 - 27729120460800 * q^20 + 107395852062720 * q^22 - 222369240588600 * q^23 + 222477066499375 * q^25 + 343645517576192 * q^26 + 174185108209664 * q^28 - 2194109701319454 * q^29 - 8723627187590032 * q^31 + 1125899906842624 * q^32 - 14946452237961216 * q^34 - 4392859271341200 * q^35 + 37470891663324758 * q^37 + 3655198694395904 * q^38 - 28394619351859200 * q^40 - 86616741616565034 * q^41 + 131416928813078444 * q^43 + 109973352512225280 * q^44 - 227706102362726400 * q^46 - 339041180377015440 * q^47 - 530951383810817511 * q^49 + 227816516095360000 * q^50 + 351893009998020608 * q^52 + 1571494796445297834 * q^53 - 2773471659829299000 * q^55 + 178365550806695936 * q^56 - 2246768334151120896 * q^58 - 5232984701774509020 * q^59 - 4788384962739867250 * q^61 - 8932994240092192768 * q^62 + 1152921504606846976 * q^64 - 8874561593573718900 * q^65 - 15480328743911983516 * q^67 - 15305167091672285184 * q^68 - 4498287893853388800 * q^70 + 12930906477499746840 * q^71 - 44257184658687636502 * q^73 + 38370193063244552192 * q^74 + 3742923463061405696 * q^76 + 17422026128334877920 * q^77 - 14888578935758942752 * q^79 - 29076090216303820800 * q^80 - 88695543415362594816 * q^82 - 37085068910999181588 * q^83 + 385988480009157494700 * q^85 + 134570935104592326656 * q^86 + 112612712972518686720 * q^88 + 105572017962561697542 * q^89 + 55747043029195624912 * q^91 - 233171048819431833600 * q^92 - 347178168706063810560 * q^94 - 94394613900280471800 * q^95 + 1381092294370554379298 * q^97 - 543694217022277131264 * 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

−2.64446e7

1.66116e8

1.07374e9

−2.70792e10

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.d		1
3.b	odd	2	1		6.22.a.a	✓	1
12.b	even	2	1		48.22.a.c		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 26444550 $ T5 + 26444550 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 + 26444550 $ T + 26444550
$7$	$ T - 166115864 $ T - 166115864
$11$	$ T - 104878761780 $ T - 104878761780
$13$	$ T - 335591325758 $ T - 335591325758
$17$	$ T + 14596144763634 $ T + 14596144763634
$19$	$ T - 3569529974996 $ T - 3569529974996
$23$	$ T + 222369240588600 $ T + 222369240588600
$29$	$ T + 2194109701319454 $ T + 2194109701319454
$31$	$ T + 8723627187590032 $ T + 8723627187590032
$37$	$ T - 37\!\cdots\!58 $ T - 37470891663324758
$41$	$ T + 86\!\cdots\!34 $ T + 86616741616565034
$43$	$ T - 13\!\cdots\!44 $ T - 131416928813078444
$47$	$ T + 33\!\cdots\!40 $ T + 339041180377015440
$53$	$ T - 15\!\cdots\!34 $ T - 1571494796445297834
$59$	$ T + 52\!\cdots\!20 $ T + 5232984701774509020
$61$	$ T + 47\!\cdots\!50 $ T + 4788384962739867250
$67$	$ T + 15\!\cdots\!16 $ T + 15480328743911983516
$71$	$ T - 12\!\cdots\!40 $ T - 12930906477499746840
$73$	$ T + 44\!\cdots\!02 $ T + 44257184658687636502
$79$	$ T + 14\!\cdots\!52 $ T + 14888578935758942752
$83$	$ T + 37\!\cdots\!88 $ T + 37085068910999181588
$89$	$ T - 10\!\cdots\!42 $ T - 105572017962561697542
$97$	$ T - 13\!\cdots\!98 $ T - 1381092294370554379298
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Classical	Maass
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Elliptic curves over $\Q$
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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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