LMFDB - Newform orbit 1764.3.z.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (of order $6$, 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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5}+O(q^{10}) $ q + (b6 + b5 + 2b2) q^5	$ q + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{7} + 3 \beta_{6} - 20 \beta_{5} + \cdots + 28) q^{97}+O(q^{100}) $ q + (b6 + b5 + 2b2) q^5 + (2b7 - b5 - b2 + b1) q^11 + (2b7 - b6 - 8b5 - b3 - 4b2 + 2b1) * q^13 + (-3b5 - 4b4 + 3b3 + 3b2 + 4) * q^17 + (-b7 - 2b6 - 2b5 + 8b4 - 4b2 + 16) * q^19 + (b7 - 2b6 - 9b5 + 2b4 - 4b3 + 2b1) q^23 + (-4b7 - b5 - 9b4 - b2 - 2b1 - 9) q^25 + (-b7 - 6b6 + 6b3 - 5b2 + b1 - 10) q^29 + (14b5 + 4b4 + 4b3 - 14b2 + 3b1 - 4) q^31 + (-6b7 + 4b6 - 9b5 + 16b4 + 8b3 - 12b1) * q^37 + (5b7 - 7b6 + 26b5 - 28b4 - 7b3 + 13b2 + 5b1 - 14) q^41 + (2b7 + 8b6 - 8b3 - 8b2 - 2b1 - 14) q^43 + (7b7 - 8b6 + 2b5 - 22b4 + 4b2 - 44) q^47 + (12b7 - 8b6 + 26b5 - 18b4 - 4b3 + 26b2 + 6b1 - 18) q^53 + (b7 - 6b6 - 20b5 - 6b3 - 10b2 + b1) * q^55 + (-22b5 - 14b4 + 12b3 + 22b2 + 5b1 + 14) q^59 + (8b7 + 5b6 - 14b5 + 12b4 - 28b2 + 24) q^61 + (5b7 - 4b6 - 19b5 - 30b4 - 8b3 + 10b1) * q^65 + (-12b7 - 8b6 + 12b5 + 8b4 - 4b3 + 12b2 - 6b1 + 8) q^67 + (b7 - 8b6 + 8b3 + 43b2 - b1 - 28) q^71 + (6b5 + 28b4 + 15b3 - 6b2 - 8b1 - 28) q^73 + (4b7 - 28b5 - 54b4 + 8b1) * q^79 + (14b7 - 6b6 + 12b5 + 32b4 - 6b3 + 6b2 + 14b1 + 16) q^83 + (4b6 - 4b3 + 15b2 - 12) q^85 + (16b7 + 11b6 + 23b5 + 8b4 + 46b2 + 16) q^89 + (8b7 + 20b6 + 36b5 - 34b4 + 10b3 + 36b2 + 4b1 - 34) q^95 + (4b7 + 3b6 - 20b5 + 56b4 + 3b3 - 10b2 + 4b1 + 28) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) $ 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4*x^6 + 14*x^4 - 8*x^2 + 4 :

$\beta_{1}$	$=$	$ ( \nu^{7} + 76\nu ) / 14 $ (v^7 + 76*v) / 14
$\beta_{2}$	$=$	$ ( \nu^{6} + 20 ) / 14 $ (v^6 + 20) / 14
$\beta_{3}$	$=$	$ ( -\nu^{7} - 27\nu ) / 7 $ (-v^7 - 27*v) / 7
$\beta_{4}$	$=$	$ ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 $ (-2v^6 + 7v^4 - 28*v^2 + 2) / 14
$\beta_{5}$	$=$	$ ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 $ (-2v^6 + 7v^4 - 21*v^2 + 2) / 7
$\beta_{6}$	$=$	$ ( -8\nu^{7} + 35\nu^{5} - 112\nu^{3} + 64\nu ) / 14 $ (-8v^7 + 35v^5 - 112v^3 + 64v) / 14
$\beta_{7}$	$=$	$ ( 11\nu^{7} - 42\nu^{5} + 154\nu^{3} - 88\nu ) / 14 $ (11v^7 - 42v^5 + 154v^3 - 88v) / 14

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

$\nu$	$=$	$ ( \beta_{3} + 2\beta_1 ) / 7 $ (b3 + 2*b1) / 7
$\nu^{2}$	$=$	$ \beta_{5} - 2\beta_{4} $ b5 - 2*b4
$\nu^{3}$	$=$	$ ( 5\beta_{7} + 6\beta_{6} + 6\beta_{3} + 5\beta_1 ) / 7 $ (5b7 + 6b6 + 6b3 + 5b1) / 7
$\nu^{4}$	$=$	$ 4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6 $ 4b5 - 6b4 + 4*b2 - 6
$\nu^{5}$	$=$	$ ( 16\beta_{7} + 22\beta_{6} ) / 7 $ (16b7 + 22b6) / 7
$\nu^{6}$	$=$	$ 14\beta_{2} - 20 $ 14*b2 - 20
$\nu^{7}$	$=$	$ ( -76\beta_{3} - 54\beta_1 ) / 7 $ (-76b3 - 54b1) / 7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$-\beta_{4}$

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} $

325.1

1.60021	+	0.923880i
−0.662827	−	0.382683i
−1.60021	−	0.923880i
0.662827	+	0.382683i
1.60021	−	0.923880i
−0.662827	+	0.382683i
−1.60021	+	0.923880i
0.662827	−	0.382683i

−5.04718

2.91399i

325.2

−0.416265

0.240331i

325.3

0.804540

−

0.464502i

325.4

4.65891

−

2.68982i

901.1

−5.04718

−

2.91399i

901.2

−0.416265

−

0.240331i

901.3

0.804540

0.464502i

901.4

4.65891

2.68982i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.3.z.m		8
3.b	odd	2	1		588.3.m.e		8
7.b	odd	2	1		1764.3.z.l		8
7.c	even	3	1		1764.3.d.h		8
7.c	even	3	1		1764.3.z.l		8
7.d	odd	6	1		1764.3.d.h		8
7.d	odd	6	1	inner	1764.3.z.m		8
21.c	even	2	1		588.3.m.f		8
21.g	even	6	1		588.3.d.c	✓	8
21.g	even	6	1		588.3.m.e		8
21.h	odd	6	1		588.3.d.c	✓	8
21.h	odd	6	1		588.3.m.f		8
84.j	odd	6	1		2352.3.f.j		8
84.n	even	6	1		2352.3.f.j		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.3.d.c	✓	8	21.g	even	6	1
588.3.d.c	✓	8	21.h	odd	6	1
588.3.m.e		8	3.b	odd	2	1
588.3.m.e		8	21.g	even	6	1
588.3.m.f		8	21.c	even	2	1
588.3.m.f		8	21.h	odd	6	1
1764.3.d.h		8	7.c	even	3	1
1764.3.d.h		8	7.d	odd	6	1
1764.3.z.l		8	7.b	odd	2	1
1764.3.z.l		8	7.c	even	3	1
1764.3.z.m		8	1.a	even	1	1	trivial
1764.3.z.m		8	7.d	odd	6	1	inner
2352.3.f.j		8	84.j	odd	6	1
2352.3.f.j		8	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1764, [\chi])$:

$ T_{5}^{8} - 32T_{5}^{6} + 1010T_{5}^{4} - 768T_{5}^{3} - 256T_{5}^{2} + 336T_{5} + 196 $

$ T_{11}^{8} + 124T_{11}^{6} + 96T_{11}^{5} + 12084T_{11}^{4} + 5952T_{11}^{3} + 410512T_{11}^{2} - 158016T_{11} + 10837264 $

$ T_{13}^{8} + 712T_{13}^{6} + 123284T_{13}^{4} + 1723280T_{13}^{2} + 2979076 $

$ T_{19}^{8} - 96 T_{19}^{7} + 4040 T_{19}^{6} - 92928 T_{19}^{5} + 1297976 T_{19}^{4} - 11430144 T_{19}^{3} + \cdots + 285745216 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 32 T^{6} + \cdots + 196 $ T^8 - 32T^6 + 1010T^4 - 768T^3 - 256T^2 + 336*T + 196
$7$	$ T^{8} $ T^8
$11$	$ T^{8} + 124 T^{6} + \cdots + 10837264 $ T^8 + 124T^6 + 96T^5 + 12084T^4 + 5952T^3 + 410512T^2 - 158016T + 10837264
$13$	$ T^{8} + 712 T^{6} + \cdots + 2979076 $ T^8 + 712T^6 + 123284T^4 + 1723280*T^2 + 2979076
$17$	$ T^{8} - 48 T^{7} + \cdots + 168428484 $ T^8 - 48T^7 + 768T^6 - 71118T^4 + 9208512T^2 - 68212368*T + 168428484
$19$	$ T^{8} - 96 T^{7} + \cdots + 285745216 $ T^8 - 96T^7 + 4040T^6 - 92928T^5 + 1297976T^4 - 11430144T^3 + 62839360T^2 - 199602432*T + 285745216
$23$	$ T^{8} + \cdots + 1443088144 $ T^8 + 8T^7 + 676T^6 + 14816T^5 + 491380T^4 + 6639680T^3 + 73892080T^2 + 374409728*T + 1443088144
$29$	$ (T^{4} + 40 T^{3} + \cdots + 468892)^{2} $ (T^4 + 40T^3 - 1924T^2 - 65440*T + 468892)^2
$31$	$ T^{8} + \cdots + 2052452416 $ T^8 + 48T^7 - 2072T^6 - 136320T^5 + 5473592T^4 + 468122880T^3 + 9185192768T^2 + 7467548928*T + 2052452416
$37$	$ T^{8} + \cdots + 298373767696 $ T^8 + 64T^7 + 7588T^6 + 234880T^5 + 27408076T^4 + 870228736T^3 + 50617849744T^2 + 125188551424*T + 298373767696
$41$	$ T^{8} + \cdots + 26697826996036 $ T^8 + 9808T^6 + 34716572T^4 + 51769047200*T^2 + 26697826996036
$43$	$ (T^{4} + 56 T^{3} + \cdots + 192784)^{2} $ (T^4 + 56T^3 - 3784T^2 - 203168*T + 192784)^2
$47$	$ T^{8} + \cdots + 8881401308224 $ T^8 + 264T^7 + 29104T^6 + 1550208T^5 + 34157384T^4 - 220411392T^3 - 17029896064T^2 + 111863586048*T + 8881401308224
$53$	$ T^{8} + \cdots + 71641191948544 $ T^8 + 72T^7 + 10648T^6 + 582528T^5 + 73453104T^4 + 3885089280T^3 + 191864861056T^2 + 4130215804416*T + 71641191948544
$59$	$ T^{8} + \cdots + 372481662446656 $ T^8 - 168T^7 + 2768T^6 + 1115520T^5 - 20126776T^4 - 9902630400T^3 + 869535448960T^2 - 28782925866240*T + 372481662446656
$61$	$ T^{8} - 144 T^{7} + \cdots + 454276 $ T^8 - 144T^7 + 3772T^6 + 452160T^5 + 8405102T^4 - 95104320T^3 + 303671288T^2 + 20414112*T + 454276
$67$	$ T^{8} + \cdots + 306756468736 $ T^8 - 32T^7 + 7072T^6 - 335360T^5 + 45594496T^4 - 1634828288T^3 + 66583023616T^2 - 146466111488*T + 306756468736
$71$	$ (T^{4} + 112 T^{3} + \cdots - 7722596)^{2} $ (T^4 + 112T^3 - 6460T^2 - 634384*T - 7722596)^2
$73$	$ T^{8} + \cdots + 3712242251524 $ T^8 + 336T^7 + 45212T^6 + 2546880T^5 + 58001486T^4 - 93506880T^3 - 14553796808T^2 + 23767993248*T + 3712242251524
$79$	$ T^{8} + \cdots + 49705658450176 $ T^8 - 216T^7 + 34216T^6 - 2562432T^5 + 148346160T^4 - 3820758528T^3 + 91586574976T^2 + 439257156096*T + 49705658450176
$83$	$ T^{8} + \cdots + 61585579131904 $ T^8 + 19712T^6 + 89759168T^4 + 132200402944*T^2 + 61585579131904
$89$	$ T^{8} + \cdots + 17\!\cdots\!76 $ T^8 - 96T^7 - 16192T^6 + 1849344T^5 + 308791826T^4 - 62684901888T^3 + 4335067503424T^2 - 136075027453008*T + 1748734585879876
$97$	$ T^{8} + \cdots + 5315948141956 $ T^8 + 13640T^6 + 36017108T^4 + 28759544464*T^2 + 5315948141956
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5}+O(q^{10}) \) q + (b6 + b5 + 2b2) q^5	\( q + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{7} + 3 \beta_{6} - 20 \beta_{5} + \cdots + 28) q^{97}+O(q^{100}) \) q + (b6 + b5 + 2b2) q^5 + (2b7 - b5 - b2 + b1) q^11 + (2b7 - b6 - 8b5 - b3 - 4b2 + 2b1) * q^13 + (-3b5 - 4b4 + 3b3 + 3b2 + 4) * q^17 + (-b7 - 2b6 - 2b5 + 8b4 - 4b2 + 16) * q^19 + (b7 - 2b6 - 9b5 + 2b4 - 4b3 + 2b1) q^23 + (-4b7 - b5 - 9b4 - b2 - 2b1 - 9) q^25 + (-b7 - 6b6 + 6b3 - 5b2 + b1 - 10) q^29 + (14b5 + 4b4 + 4b3 - 14b2 + 3b1 - 4) q^31 + (-6b7 + 4b6 - 9b5 + 16b4 + 8b3 - 12b1) * q^37 + (5b7 - 7b6 + 26b5 - 28b4 - 7b3 + 13b2 + 5b1 - 14) q^41 + (2b7 + 8b6 - 8b3 - 8b2 - 2b1 - 14) q^43 + (7b7 - 8b6 + 2b5 - 22b4 + 4b2 - 44) q^47 + (12b7 - 8b6 + 26b5 - 18b4 - 4b3 + 26b2 + 6b1 - 18) q^53 + (b7 - 6b6 - 20b5 - 6b3 - 10b2 + b1) * q^55 + (-22b5 - 14b4 + 12b3 + 22b2 + 5b1 + 14) q^59 + (8b7 + 5b6 - 14b5 + 12b4 - 28b2 + 24) q^61 + (5b7 - 4b6 - 19b5 - 30b4 - 8b3 + 10b1) * q^65 + (-12b7 - 8b6 + 12b5 + 8b4 - 4b3 + 12b2 - 6b1 + 8) q^67 + (b7 - 8b6 + 8b3 + 43b2 - b1 - 28) q^71 + (6b5 + 28b4 + 15b3 - 6b2 - 8b1 - 28) q^73 + (4b7 - 28b5 - 54b4 + 8b1) * q^79 + (14b7 - 6b6 + 12b5 + 32b4 - 6b3 + 6b2 + 14b1 + 16) q^83 + (4b6 - 4b3 + 15b2 - 12) q^85 + (16b7 + 11b6 + 23b5 + 8b4 + 46b2 + 16) q^89 + (8b7 + 20b6 + 36b5 - 34b4 + 10b3 + 36b2 + 4b1 - 34) q^95 + (4b7 + 3b6 - 20b5 + 56b4 + 3b3 - 10b2 + 4b1 + 28) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) \) 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

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