LMFDB - Newform orbit 1368.2.s.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(505,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1368.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.s (of order $3$, degree $2$, 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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.2696112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 $ x^6 - x^5 + 5x^4 + 18x^2 - 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
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 basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7}+O(q^{10}) $ q + b1 * q^5 + (b3 - b2 - 1) * q^7	$ q + \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{2} - 1) q^{11} + (\beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{13}+ \cdots + (\beta_{4} + 2 \beta_1 - 1) q^{97}+O(q^{100}) $ q + b1 * q^5 + (b3 - b2 - 1) * q^7 + (b2 - 1) * q^11 + (b5 + b4 - b3 + 2b2 + 2b1) * q^13 + (-b5 - 3b4 + 2b1 + 3) * q^17 + (2b5 - b4 - b3 + b2 - b1 + 1) q^19 + (-b2 - b1) * q^23 + (b5 + 2b4 - b3) q^25 + (2b5 - 2b4 - 2b3 - 3b2 - 3b1) q^29 + (-3b3 - 3b2 - 3) * q^31 + (2b5 - 4b4 + 4) * q^35 + (-3b3 - 3b2 - 5) * q^37 + (-2b5 + 7b4 + 2b1 - 7) q^41 + (b5 + 3b4 + 4b1 - 3) * q^43 + (2b5 + 4b4 - 2b3 - 5b2 - 5b1) q^47 + (-2b2 + 3) q^49 + (-b5 - b4 + b3 + 2b2 + 2b1) * q^53 + (-b5 + 3b4 - b1 - 3) q^55 + (-b5 - 4b4 + b1 + 4) q^59 + (4b5 - 4b3 - 3b2 - 3b1) * q^61 + (-3b3 - 7) q^65 + (-b5 + 2b4 + b3 - 3b2 - 3b1) q^67 + (b5 + b4 - 1) * q^71 + (-3b4 + 2b1 + 3) * q^73 + (-3b3 + b2 - 3) q^77 + (-b5 - 7b4 - 2b1 + 7) * q^79 + (2b3 - 3b2 - 5) * q^83 + (b5 - 5b4 - b3 + 2b2 + 2b1) q^85 + (b5 + 3b4 - b3 + 6b2 + 6b1) q^89 + (2b5 - 14b4 - 2b3 + 2b2 + 2b1) q^91 + (-b5 + 5b4 - b3 + 3b2 + 2b1 - 4) q^95 + (b4 + 2b1 - 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} - 4 q^{7}+O(q^{10}) $ 6 * q + q^5 - 4 * q^7	$ 6 q + q^{5} - 4 q^{7} - 8 q^{11} + q^{13} + 11 q^{17} + q^{23} + 6 q^{25} - 3 q^{29} - 12 q^{31} + 12 q^{35} - 24 q^{37} - 19 q^{41} - 5 q^{43} + 17 q^{47} + 22 q^{49} - 5 q^{53} - 10 q^{55} + 13 q^{59} + 3 q^{61} - 42 q^{65} + 9 q^{67} - 3 q^{71} + 11 q^{73} - 20 q^{77} + 19 q^{79} - 24 q^{83} - 17 q^{85} + 3 q^{89} - 44 q^{91} - 13 q^{95} - q^{97}+O(q^{100}) $ 6 * q + q^5 - 4 * q^7 - 8 * q^11 + q^13 + 11 * q^17 + q^23 + 6 * q^25 - 3 * q^29 - 12 * q^31 + 12 * q^35 - 24 * q^37 - 19 * q^41 - 5 * q^43 + 17 * q^47 + 22 * q^49 - 5 * q^53 - 10 * q^55 + 13 * q^59 + 3 * q^61 - 42 * q^65 + 9 * q^67 - 3 * q^71 + 11 * q^73 - 20 * q^77 + 19 * q^79 - 24 * q^83 - 17 * q^85 + 3 * q^89 - 44 * q^91 - 13 * q^95 - q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 $ x^6 - x^5 + 5*x^4 + 18*x^2 - 8*x + 4 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 $ (v^5 - 5v^4 + 25v^3 - 18v^2 + 8v - 40) / 82
$\beta_{3}$	$=$	$ ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 $ (2v^5 - 10v^4 + 9v^3 - 36v^2 + 16*v - 121) / 41
$\beta_{4}$	$=$	$ ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 $ (-10v^5 + 9v^4 - 45v^3 - 25v^2 - 162*v + 72) / 82
$\beta_{5}$	$=$	$ ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 $ (-26v^5 + 7v^4 - 117v^3 - 65v^2 - 454*v - 26) / 82

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} - 3\beta_{4} - \beta_{3} $ b5 - 3*b4 - b3
$\nu^{3}$	$=$	$ -\beta_{3} + 4\beta_{2} - 1 $ -b3 + 4*b2 - 1
$\nu^{4}$	$=$	$ -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 $ -5b5 + 13b4 - 2*b1 - 13
$\nu^{5}$	$=$	$ -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 $ -7b5 + 11b4 + 7b3 - 18b2 - 18*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$-\beta_{4}$	$1$

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} $

505.1

−0.906803	+	1.57063i
0.235342	−	0.407624i
1.17146	−	2.02903i
−0.906803	−	1.57063i
0.235342	+	0.407624i
1.17146	+	2.02903i

−0.906803

1.57063i

−2.52444

505.2

0.235342

−

0.407624i

−3.30777

505.3

1.17146

−

2.02903i

3.83221

577.1

−0.906803

−

1.57063i

−2.52444

577.2

0.235342

0.407624i

−3.30777

577.3

1.17146

2.02903i

3.83221

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1368.2.s.k		6
3.b	odd	2	1		152.2.i.c	✓	6
4.b	odd	2	1		2736.2.s.y		6
12.b	even	2	1		304.2.i.f		6
19.c	even	3	1	inner	1368.2.s.k		6
24.f	even	2	1		1216.2.i.m		6
24.h	odd	2	1		1216.2.i.n		6
57.f	even	6	1		2888.2.a.n		3
57.h	odd	6	1		152.2.i.c	✓	6
57.h	odd	6	1		2888.2.a.r		3
76.g	odd	6	1		2736.2.s.y		6
228.m	even	6	1		304.2.i.f		6
228.m	even	6	1		5776.2.a.bk		3
228.n	odd	6	1		5776.2.a.bq		3
456.u	even	6	1		1216.2.i.m		6
456.x	odd	6	1		1216.2.i.n		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.i.c	✓	6	3.b	odd	2	1
152.2.i.c	✓	6	57.h	odd	6	1
304.2.i.f		6	12.b	even	2	1
304.2.i.f		6	228.m	even	6	1
1216.2.i.m		6	24.f	even	2	1
1216.2.i.m		6	456.u	even	6	1
1216.2.i.n		6	24.h	odd	2	1
1216.2.i.n		6	456.x	odd	6	1
1368.2.s.k		6	1.a	even	1	1	trivial
1368.2.s.k		6	19.c	even	3	1	inner
2736.2.s.y		6	4.b	odd	2	1
2736.2.s.y		6	76.g	odd	6	1
2888.2.a.n		3	57.f	even	6	1
2888.2.a.r		3	57.h	odd	6	1
5776.2.a.bk		3	228.m	even	6	1
5776.2.a.bq		3	228.n	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1368, [\chi])$:

$ T_{5}^{6} - T_{5}^{5} + 5T_{5}^{4} + 18T_{5}^{2} - 8T_{5} + 4 $

$ T_{7}^{3} + 2T_{7}^{2} - 14T_{7} - 32 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - T^{5} + 5 T^{4} + \cdots + 4 $ T^6 - T^5 + 5T^4 + 18T^2 - 8*T + 4
$7$	$ (T^{3} + 2 T^{2} - 14 T - 32)^{2} $ (T^3 + 2T^2 - 14T - 32)^2
$11$	$ (T^{3} + 4 T^{2} + T - 4)^{2} $ (T^3 + 4*T^2 + T - 4)^2
$13$	$ T^{6} - T^{5} + \cdots + 5776 $ T^6 - T^5 + 33T^4 - 120T^3 + 1100T^2 - 2432T + 5776
$17$	$ T^{6} - 11 T^{5} + \cdots + 1024 $ T^6 - 11T^5 + 97T^4 - 328T^3 + 928T^2 + 768*T + 1024
$19$	$ T^{6} + 18 T^{4} + \cdots + 6859 $ T^6 + 18T^4 - 16T^3 + 342*T^2 + 6859
$23$	$ T^{6} - T^{5} + 5 T^{4} + \cdots + 4 $ T^6 - T^5 + 5T^4 + 18T^2 - 8*T + 4
$29$	$ T^{6} + 3 T^{5} + \cdots + 4 $ T^6 + 3T^5 + 49T^4 - 124T^3 + 1594T^2 - 80*T + 4
$31$	$ (T^{3} + 6 T^{2} + \cdots - 216)^{2} $ (T^3 + 6T^2 - 54T - 216)^2
$37$	$ (T^{3} + 12 T^{2} + \cdots - 292)^{2} $ (T^3 + 12T^2 - 18T - 292)^2
$41$	$ T^{6} + 19 T^{5} + \cdots + 1681 $ T^6 + 19T^5 + 270T^4 + 1647T^3 + 7502T^2 + 3731*T + 1681
$43$	$ T^{6} + 5 T^{5} + \cdots + 118336 $ T^6 + 5T^5 + 109T^4 + 268T^3 + 8776T^2 + 28896*T + 118336
$47$	$ T^{6} - 17 T^{5} + \cdots + 521284 $ T^6 - 17T^5 + 289T^4 - 1444T^3 + 12274T^2 + 521284
$53$	$ T^{6} + 5 T^{5} + \cdots + 1936 $ T^6 + 5T^5 + 33T^4 + 48T^3 + 284T^2 + 352*T + 1936
$59$	$ T^{6} - 13 T^{5} + \cdots + 2809 $ T^6 - 13T^5 + 120T^4 - 531T^3 + 1712T^2 - 2597*T + 2809
$61$	$ T^{6} - 3 T^{5} + \cdots + 58564 $ T^6 - 3T^5 + 109T^4 + 784T^3 + 9274T^2 + 24200*T + 58564
$67$	$ T^{6} - 9 T^{5} + \cdots + 529 $ T^6 - 9T^5 + 112T^4 + 233T^3 + 1168T^2 - 713*T + 529
$71$	$ T^{6} + 3 T^{5} + \cdots + 16 $ T^6 + 3T^5 + 13T^4 - 4T^3 + 28T^2 + 16*T + 16
$73$	$ T^{6} - 11 T^{5} + \cdots + 361 $ T^6 - 11T^5 + 98T^4 - 291T^3 + 738T^2 + 437*T + 361
$79$	$ T^{6} - 19 T^{5} + \cdots + 256 $ T^6 - 19T^5 + 273T^4 - 1704T^3 + 8048T^2 + 1408*T + 256
$83$	$ (T^{3} + 12 T^{2} + \cdots - 632)^{2} $ (T^3 + 12T^2 - 43T - 632)^2
$89$	$ T^{6} - 3 T^{5} + \cdots + 295936 $ T^6 - 3T^5 + 193T^4 - 536T^3 + 35488T^2 - 100096*T + 295936
$97$	$ T^{6} + T^{5} + 18 T^{4} + \cdots + 1 $ T^6 + T^5 + 18T^4 - 15T^3 + 290T^2 + 17T + 1
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Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7}+O(q^{10}) \) q + b1 * q^5 + (b3 - b2 - 1) * q^7	\( q + \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{2} - 1) q^{11} + (\beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{13}+ \cdots + (\beta_{4} + 2 \beta_1 - 1) q^{97}+O(q^{100}) \) q + b1 * q^5 + (b3 - b2 - 1) * q^7 + (b2 - 1) * q^11 + (b5 + b4 - b3 + 2b2 + 2b1) * q^13 + (-b5 - 3b4 + 2b1 + 3) * q^17 + (2b5 - b4 - b3 + b2 - b1 + 1) q^19 + (-b2 - b1) * q^23 + (b5 + 2b4 - b3) q^25 + (2b5 - 2b4 - 2b3 - 3b2 - 3b1) q^29 + (-3b3 - 3b2 - 3) * q^31 + (2b5 - 4b4 + 4) * q^35 + (-3b3 - 3b2 - 5) * q^37 + (-2b5 + 7b4 + 2b1 - 7) q^41 + (b5 + 3b4 + 4b1 - 3) * q^43 + (2b5 + 4b4 - 2b3 - 5b2 - 5b1) q^47 + (-2b2 + 3) q^49 + (-b5 - b4 + b3 + 2b2 + 2b1) * q^53 + (-b5 + 3b4 - b1 - 3) q^55 + (-b5 - 4b4 + b1 + 4) q^59 + (4b5 - 4b3 - 3b2 - 3b1) * q^61 + (-3b3 - 7) q^65 + (-b5 + 2b4 + b3 - 3b2 - 3b1) q^67 + (b5 + b4 - 1) * q^71 + (-3b4 + 2b1 + 3) * q^73 + (-3b3 + b2 - 3) q^77 + (-b5 - 7b4 - 2b1 + 7) * q^79 + (2b3 - 3b2 - 5) * q^83 + (b5 - 5b4 - b3 + 2b2 + 2b1) q^85 + (b5 + 3b4 - b3 + 6b2 + 6b1) q^89 + (2b5 - 14b4 - 2b3 + 2b2 + 2b1) q^91 + (-b5 + 5b4 - b3 + 3b2 + 2b1 - 4) q^95 + (b4 + 2b1 - 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} - 4 q^{7}+O(q^{10}) \) 6 * q + q^5 - 4 * q^7	\( 6 q + q^{5} - 4 q^{7} - 8 q^{11} + q^{13} + 11 q^{17} + q^{23} + 6 q^{25} - 3 q^{29} - 12 q^{31} + 12 q^{35} - 24 q^{37} - 19 q^{41} - 5 q^{43} + 17 q^{47} + 22 q^{49} - 5 q^{53} - 10 q^{55} + 13 q^{59} + 3 q^{61} - 42 q^{65} + 9 q^{67} - 3 q^{71} + 11 q^{73} - 20 q^{77} + 19 q^{79} - 24 q^{83} - 17 q^{85} + 3 q^{89} - 44 q^{91} - 13 q^{95} - q^{97}+O(q^{100}) \) 6 * q + q^5 - 4 * q^7 - 8 * q^11 + q^13 + 11 * q^17 + q^23 + 6 * q^25 - 3 * q^29 - 12 * q^31 + 12 * q^35 - 24 * q^37 - 19 * q^41 - 5 * q^43 + 17 * q^47 + 22 * q^49 - 5 * q^53 - 10 * q^55 + 13 * q^59 + 3 * q^61 - 42 * q^65 + 9 * q^67 - 3 * q^71 + 11 * q^73 - 20 * q^77 + 19 * q^79 - 24 * q^83 - 17 * q^85 + 3 * q^89 - 44 * q^91 - 13 * q^95 - q^97

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