LMFDB - Newform orbit 64.12.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,12,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	64.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:	$49.1739635558$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
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 + 252 q^{3} - 4830 q^{5} + 16744 q^{7} - 113643 q^{9}+O(q^{10}) $

$ q + 252 q^{3} - 4830 q^{5} + 16744 q^{7} - 113643 q^{9} + 534612 q^{11} + 577738 q^{13} - 1217160 q^{15} - 6905934 q^{17} + 10661420 q^{19} + 4219488 q^{21} - 18643272 q^{23} - 25499225 q^{25} - 73279080 q^{27} - 128406630 q^{29} + 52843168 q^{31} + 134722224 q^{33} - 80873520 q^{35} + 182213314 q^{37} + 145589976 q^{39} + 308120442 q^{41} - 17125708 q^{43} + 548895690 q^{45} - 2687348496 q^{47} - 1696965207 q^{49} - 1740295368 q^{51} + 1596055698 q^{53} - 2582175960 q^{55} + 2686677840 q^{57} - 5189203740 q^{59} - 6956478662 q^{61} - 1902838392 q^{63} - 2790474540 q^{65} - 15481826884 q^{67} - 4698104544 q^{69} - 9791485272 q^{71} + 1463791322 q^{73} - 6425804700 q^{75} + 8951543328 q^{77} - 38116845680 q^{79} + 1665188361 q^{81} - 29335099668 q^{83} + 33355661220 q^{85} - 32358470760 q^{87} - 24992917110 q^{89} + 9673645072 q^{91} + 13316478336 q^{93} - 51494658600 q^{95} + 75013568546 q^{97} - 60754911516 q^{99}+O(q^{100}) $

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

252.000

−4830.00

16744.0

−113643.

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	64.12.a.f		1
4.b	odd	2	1		64.12.a.b		1
8.b	even	2	1		16.12.a.a		1
8.d	odd	2	1		1.12.a.a	✓	1
16.e	even	4	2		256.12.b.c		2
16.f	odd	4	2		256.12.b.e		2
24.f	even	2	1		9.12.a.b		1
24.h	odd	2	1		144.12.a.d		1
40.e	odd	2	1		25.12.a.b		1
40.k	even	4	2		25.12.b.b		2
56.e	even	2	1		49.12.a.a		1
56.k	odd	6	2		49.12.c.b		2
56.m	even	6	2		49.12.c.c		2
72.l	even	6	2		81.12.c.b		2
72.p	odd	6	2		81.12.c.d		2
88.g	even	2	1		121.12.a.b		1
104.h	odd	2	1		169.12.a.a		1
120.m	even	2	1		225.12.a.b		1
120.q	odd	4	2		225.12.b.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.12.a.a	✓	1	8.d	odd	2	1
9.12.a.b		1	24.f	even	2	1
16.12.a.a		1	8.b	even	2	1
25.12.a.b		1	40.e	odd	2	1
25.12.b.b		2	40.k	even	4	2
49.12.a.a		1	56.e	even	2	1
49.12.c.b		2	56.k	odd	6	2
49.12.c.c		2	56.m	even	6	2
64.12.a.b		1	4.b	odd	2	1
64.12.a.f		1	1.a	even	1	1	trivial
81.12.c.b		2	72.l	even	6	2
81.12.c.d		2	72.p	odd	6	2
121.12.a.b		1	88.g	even	2	1
144.12.a.d		1	24.h	odd	2	1
169.12.a.a		1	104.h	odd	2	1
225.12.a.b		1	120.m	even	2	1
225.12.b.d		2	120.q	odd	4	2
256.12.b.c		2	16.e	even	4	2
256.12.b.e		2	16.f	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 252 $ T3 - 252 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(64))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 252 $ T - 252
$5$	$ T + 4830 $ T + 4830
$7$	$ T - 16744 $ T - 16744
$11$	$ T - 534612 $ T - 534612
$13$	$ T - 577738 $ T - 577738
$17$	$ T + 6905934 $ T + 6905934
$19$	$ T - 10661420 $ T - 10661420
$23$	$ T + 18643272 $ T + 18643272
$29$	$ T + 128406630 $ T + 128406630
$31$	$ T - 52843168 $ T - 52843168
$37$	$ T - 182213314 $ T - 182213314
$41$	$ T - 308120442 $ T - 308120442
$43$	$ T + 17125708 $ T + 17125708
$47$	$ T + 2687348496 $ T + 2687348496
$53$	$ T - 1596055698 $ T - 1596055698
$59$	$ T + 5189203740 $ T + 5189203740
$61$	$ T + 6956478662 $ T + 6956478662
$67$	$ T + 15481826884 $ T + 15481826884
$71$	$ T + 9791485272 $ T + 9791485272
$73$	$ T - 1463791322 $ T - 1463791322
$79$	$ T + 38116845680 $ T + 38116845680
$83$	$ T + 29335099668 $ T + 29335099668
$89$	$ T + 24992917110 $ T + 24992917110
$97$	$ T - 75013568546 $ T - 75013568546
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Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

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