LMFDB - Newform orbit 16.22.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	16.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:	$44.7163750859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
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 - 71604 q^{3} - 28693770 q^{5} + 853202392 q^{7} - 5333220387 q^{9}+O(q^{10}) $

$ q - 71604 q^{3} - 28693770 q^{5} + 853202392 q^{7} - 5333220387 q^{9} - 86731179612 q^{11} - 895323442786 q^{13} + 2054588707080 q^{15} + 3257566804818 q^{17} - 23032467644420 q^{19} - 61092704076768 q^{21} - 146495714575224 q^{23} + 346495278609775 q^{25} + 11\!\cdots\!60 q^{27}+ \cdots + 46\!\cdots\!44 q^{99}+O(q^{100}) $

q - 71604 * q^3 - 28693770 * q^5 + 853202392 * q^7 - 5333220387 * q^9 - 86731179612 * q^11 - 895323442786 * q^13 + 2054588707080 * q^15 + 3257566804818 * q^17 - 23032467644420 * q^19 - 61092704076768 * q^21 - 146495714575224 * q^23 + 346495278609775 * q^25 + 1130883043338360 * q^27 - 734051633521170 * q^29 + 3146664162057568 * q^31 + 6210299384937648 * q^33 - 24481593199497840 * q^35 - 12963813600992362 * q^37 + 64108739797248744 * q^39 + 45714648841476042 * q^41 + 24073607797047556 * q^43 + 153030199143888990 * q^45 + 449991905173684752 * q^47 + 169408457631237657 * q^49 - 233254813492188072 * q^51 + 2064837217091540454 * q^53 + 2488644519615417240 * q^55 + 1649216813211049680 * q^57 + 3780497099978396340 * q^59 - 7619813346829729138 * q^61 - 4550316391251565704 * q^63 + 25690204942909643220 * q^65 + 18791158016925310732 * q^67 + 10489679146444339296 * q^69 + 4526486567453771928 * q^71 - 25571455286910443926 * q^73 - 24810447929574329100 * q^75 - 73999249905940031904 * q^77 - 99336442530925070480 * q^79 - 25188380477739579879 * q^81 - 2958180217887529284 * q^83 - 93471872657082583860 * q^85 + 52561033166649856680 * q^87 + 118802976167736540090 * q^89 - 763892102998690344112 * q^91 - 225313740659970099072 * q^93 + 660888329121429263400 * q^95 - 569053013925353654302 * q^97 + 462556495295277149844 * 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

−71604.0

−2.86938e7

8.53202e8

−5.33322e9

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	16.22.a.a		1
4.b	odd	2	1		2.22.a.a	✓	1
8.b	even	2	1		64.22.a.f		1
8.d	odd	2	1		64.22.a.b		1
12.b	even	2	1		18.22.a.e		1
20.d	odd	2	1		50.22.a.c		1
20.e	even	4	2		50.22.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.22.a.a	✓	1	4.b	odd	2	1
16.22.a.a		1	1.a	even	1	1	trivial
18.22.a.e		1	12.b	even	2	1
50.22.a.c		1	20.d	odd	2	1
50.22.b.a		2	20.e	even	4	2
64.22.a.b		1	8.d	odd	2	1
64.22.a.f		1	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 71604 $ T3 + 71604 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(16))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 71604 $ T + 71604
$5$	$ T + 28693770 $ T + 28693770
$7$	$ T - 853202392 $ T - 853202392
$11$	$ T + 86731179612 $ T + 86731179612
$13$	$ T + 895323442786 $ T + 895323442786
$17$	$ T - 3257566804818 $ T - 3257566804818
$19$	$ T + 23032467644420 $ T + 23032467644420
$23$	$ T + 146495714575224 $ T + 146495714575224
$29$	$ T + 734051633521170 $ T + 734051633521170
$31$	$ T - 3146664162057568 $ T - 3146664162057568
$37$	$ T + 12\!\cdots\!62 $ T + 12963813600992362
$41$	$ T - 45\!\cdots\!42 $ T - 45714648841476042
$43$	$ T - 24\!\cdots\!56 $ T - 24073607797047556
$47$	$ T - 44\!\cdots\!52 $ T - 449991905173684752
$53$	$ T - 20\!\cdots\!54 $ T - 2064837217091540454
$59$	$ T - 37\!\cdots\!40 $ T - 3780497099978396340
$61$	$ T + 76\!\cdots\!38 $ T + 7619813346829729138
$67$	$ T - 18\!\cdots\!32 $ T - 18791158016925310732
$71$	$ T - 45\!\cdots\!28 $ T - 4526486567453771928
$73$	$ T + 25\!\cdots\!26 $ T + 25571455286910443926
$79$	$ T + 99\!\cdots\!80 $ T + 99336442530925070480
$83$	$ T + 29\!\cdots\!84 $ T + 2958180217887529284
$89$	$ T - 11\!\cdots\!90 $ T - 118802976167736540090
$97$	$ T + 56\!\cdots\!02 $ T + 569053013925353654302
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

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