LMFDB - Newform orbit 648.4.i.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 2 \zeta_{6} q^{5} + (24 \zeta_{6} - 24) q^{7} + (44 \zeta_{6} - 44) q^{11} - 22 \zeta_{6} q^{13} - 50 q^{17} + 44 q^{19} - 56 \zeta_{6} q^{23} + ( - 121 \zeta_{6} + 121) q^{25} + ( - 198 \zeta_{6} + 198) q^{29} + \cdots + ( - 478 \zeta_{6} + 478) q^{97} +O(q^{100}) $ q - 2z q^5 + (24z - 24) q^7 + (44z - 44) q^11 - 22z q^13 - 50 * q^17 + 44 * q^19 - 56z q^23 + (-121z + 121) q^25 + (-198z + 198) q^29 + 160z q^31 + 48 * q^35 - 162 * q^37 - 198z q^41 + (52z - 52) q^43 + (-528z + 528) q^47 - 233z q^49 + 242 * q^53 + 88 * q^55 - 668z q^59 + (550z - 550) q^61 + (44z - 44) q^65 - 188z q^67 - 728 * q^71 + 154 * q^73 - 1056z q^77 + (-656z + 656) q^79 + (-236z + 236) q^83 + 100z q^85 - 714 * q^89 + 528 * q^91 - 88z q^95 + (-478z + 478) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 24 q^{7} - 44 q^{11} - 22 q^{13} - 100 q^{17} + 88 q^{19} - 56 q^{23} + 121 q^{25} + 198 q^{29} + 160 q^{31} + 96 q^{35} - 324 q^{37} - 198 q^{41} - 52 q^{43} + 528 q^{47} - 233 q^{49} + 484 q^{53}+ \cdots + 478 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 24 * q^7 - 44 * q^11 - 22 * q^13 - 100 * q^17 + 88 * q^19 - 56 * q^23 + 121 * q^25 + 198 * q^29 + 160 * q^31 + 96 * q^35 - 324 * q^37 - 198 * q^41 - 52 * q^43 + 528 * q^47 - 233 * q^49 + 484 * q^53 + 176 * q^55 - 668 * q^59 - 550 * q^61 - 44 * q^65 - 188 * q^67 - 1456 * q^71 + 308 * q^73 - 1056 * q^77 + 656 * q^79 + 236 * q^83 + 100 * q^85 - 1428 * q^89 + 1056 * q^91 - 88 * q^95 + 478 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$-\zeta_{6}$

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} $

217.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

−

1.73205i

−12.0000

20.7846i

433.1

−1.00000

1.73205i

−12.0000

−

20.7846i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.4.i.e		2
3.b	odd	2	1		648.4.i.h		2
9.c	even	3	1		72.4.a.c		1
9.c	even	3	1	inner	648.4.i.e		2
9.d	odd	6	1		8.4.a.a	✓	1
9.d	odd	6	1		648.4.i.h		2
36.f	odd	6	1		144.4.a.e		1
36.h	even	6	1		16.4.a.a		1
45.h	odd	6	1		200.4.a.g		1
45.j	even	6	1		1800.4.a.d		1
45.k	odd	12	2		1800.4.f.u		2
45.l	even	12	2		200.4.c.e		2
63.i	even	6	1		392.4.i.b		2
63.j	odd	6	1		392.4.i.g		2
63.n	odd	6	1		392.4.i.g		2
63.o	even	6	1		392.4.a.e		1
63.s	even	6	1		392.4.i.b		2
72.j	odd	6	1		64.4.a.d		1
72.l	even	6	1		64.4.a.b		1
72.n	even	6	1		576.4.a.k		1
72.p	odd	6	1		576.4.a.j		1
99.g	even	6	1		968.4.a.a		1
117.n	odd	6	1		1352.4.a.a		1
144.u	even	12	2		256.4.b.g		2
144.w	odd	12	2		256.4.b.a		2
153.i	odd	6	1		2312.4.a.a		1
180.n	even	6	1		400.4.a.g		1
180.v	odd	12	2		400.4.c.i		2
252.s	odd	6	1		784.4.a.e		1
360.bd	even	6	1		1600.4.a.bm		1
360.bh	odd	6	1		1600.4.a.o		1
396.o	odd	6	1		1936.4.a.l		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.4.a.a	✓	1	9.d	odd	6	1
16.4.a.a		1	36.h	even	6	1
64.4.a.b		1	72.l	even	6	1
64.4.a.d		1	72.j	odd	6	1
72.4.a.c		1	9.c	even	3	1
144.4.a.e		1	36.f	odd	6	1
200.4.a.g		1	45.h	odd	6	1
200.4.c.e		2	45.l	even	12	2
256.4.b.a		2	144.w	odd	12	2
256.4.b.g		2	144.u	even	12	2
392.4.a.e		1	63.o	even	6	1
392.4.i.b		2	63.i	even	6	1
392.4.i.b		2	63.s	even	6	1
392.4.i.g		2	63.j	odd	6	1
392.4.i.g		2	63.n	odd	6	1
400.4.a.g		1	180.n	even	6	1
400.4.c.i		2	180.v	odd	12	2
576.4.a.j		1	72.p	odd	6	1
576.4.a.k		1	72.n	even	6	1
648.4.i.e		2	1.a	even	1	1	trivial
648.4.i.e		2	9.c	even	3	1	inner
648.4.i.h		2	3.b	odd	2	1
648.4.i.h		2	9.d	odd	6	1
784.4.a.e		1	252.s	odd	6	1
968.4.a.a		1	99.g	even	6	1
1352.4.a.a		1	117.n	odd	6	1
1600.4.a.o		1	360.bh	odd	6	1
1600.4.a.bm		1	360.bd	even	6	1
1800.4.a.d		1	45.j	even	6	1
1800.4.f.u		2	45.k	odd	12	2
1936.4.a.l		1	396.o	odd	6	1
2312.4.a.a		1	153.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 2T_{5} + 4 $ T5^2 + 2*T5 + 4 acting on $S_{4}^{\mathrm{new}}(648, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$7$	$ T^{2} + 24T + 576 $ T^2 + 24*T + 576
$11$	$ T^{2} + 44T + 1936 $ T^2 + 44*T + 1936
$13$	$ T^{2} + 22T + 484 $ T^2 + 22*T + 484
$17$	$ (T + 50)^{2} $ (T + 50)^2
$19$	$ (T - 44)^{2} $ (T - 44)^2
$23$	$ T^{2} + 56T + 3136 $ T^2 + 56*T + 3136
$29$	$ T^{2} - 198T + 39204 $ T^2 - 198*T + 39204
$31$	$ T^{2} - 160T + 25600 $ T^2 - 160*T + 25600
$37$	$ (T + 162)^{2} $ (T + 162)^2
$41$	$ T^{2} + 198T + 39204 $ T^2 + 198*T + 39204
$43$	$ T^{2} + 52T + 2704 $ T^2 + 52*T + 2704
$47$	$ T^{2} - 528T + 278784 $ T^2 - 528*T + 278784
$53$	$ (T - 242)^{2} $ (T - 242)^2
$59$	$ T^{2} + 668T + 446224 $ T^2 + 668*T + 446224
$61$	$ T^{2} + 550T + 302500 $ T^2 + 550*T + 302500
$67$	$ T^{2} + 188T + 35344 $ T^2 + 188*T + 35344
$71$	$ (T + 728)^{2} $ (T + 728)^2
$73$	$ (T - 154)^{2} $ (T - 154)^2
$79$	$ T^{2} - 656T + 430336 $ T^2 - 656*T + 430336
$83$	$ T^{2} - 236T + 55696 $ T^2 - 236*T + 55696
$89$	$ (T + 714)^{2} $ (T + 714)^2
$97$	$ T^{2} - 478T + 228484 $ T^2 - 478*T + 228484
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Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 2 \zeta_{6} q^{5} + (24 \zeta_{6} - 24) q^{7} + (44 \zeta_{6} - 44) q^{11} - 22 \zeta_{6} q^{13} - 50 q^{17} + 44 q^{19} - 56 \zeta_{6} q^{23} + ( - 121 \zeta_{6} + 121) q^{25} + ( - 198 \zeta_{6} + 198) q^{29} + \cdots + ( - 478 \zeta_{6} + 478) q^{97} +O(q^{100}) \) q - 2z q^5 + (24z - 24) q^7 + (44z - 44) q^11 - 22z q^13 - 50 * q^17 + 44 * q^19 - 56z q^23 + (-121z + 121) q^25 + (-198z + 198) q^29 + 160z q^31 + 48 * q^35 - 162 * q^37 - 198z q^41 + (52z - 52) q^43 + (-528z + 528) q^47 - 233z q^49 + 242 * q^53 + 88 * q^55 - 668z q^59 + (550z - 550) q^61 + (44z - 44) q^65 - 188z q^67 - 728 * q^71 + 154 * q^73 - 1056z q^77 + (-656z + 656) q^79 + (-236z + 236) q^83 + 100z q^85 - 714 * q^89 + 528 * q^91 - 88z q^95 + (-478z + 478) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 24 q^{7} - 44 q^{11} - 22 q^{13} - 100 q^{17} + 88 q^{19} - 56 q^{23} + 121 q^{25} + 198 q^{29} + 160 q^{31} + 96 q^{35} - 324 q^{37} - 198 q^{41} - 52 q^{43} + 528 q^{47} - 233 q^{49} + 484 q^{53}+ \cdots + 478 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 24 * q^7 - 44 * q^11 - 22 * q^13 - 100 * q^17 + 88 * q^19 - 56 * q^23 + 121 * q^25 + 198 * q^29 + 160 * q^31 + 96 * q^35 - 324 * q^37 - 198 * q^41 - 52 * q^43 + 528 * q^47 - 233 * q^49 + 484 * q^53 + 176 * q^55 - 668 * q^59 - 550 * q^61 - 44 * q^65 - 188 * q^67 - 1456 * q^71 + 308 * q^73 - 1056 * q^77 + 656 * q^79 + 236 * q^83 + 100 * q^85 - 1428 * q^89 + 1056 * q^91 - 88 * q^95 + 478 * q^97

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