LMFDB - Newform orbit 1296.3.o.bf

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(271,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

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

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

chi := DirichletCharacter("1296.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.o (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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{5} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{7}+O(q^{10}) $ q + (b3 - b2 - 6b1 + 6) q^5 + (4b3 - 2b2 - 2b1 + 4) q^7	$ q + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{5} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{7} + ( - 4 \beta_{3} + 2 \beta_{2} + \cdots - 12) q^{11}+ \cdots + (24 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100}) $ q + (b3 - b2 - 6b1 + 6) q^5 + (4b3 - 2b2 - 2b1 + 4) q^7 + (-4b3 + 2b2 + 6b1 - 12) q^11 + (-6b3 + 6b2 + 7b1 - 7) q^13 + (-5b3 - 12) q^17 + (10b3 - 20b2 - 4b1 + 2) q^19 + (-6b3 - 6b2 + 6b1 + 6) q^23 + (-12b2 - 14b1) * q^25 + (-b2 - 6b1) q^29 + (-4b3 - 4b2 - 12b1 - 12) q^31 + (14b3 - 28b2 - 36b1 + 18) q^35 + (-18b3 + 5) q^37 + (-24b3 + 24b2) * q^41 + (4b3 - 2b2 - 18b1 + 36) q^43 + (-16b3 + 8b2) * q^47 + (24b3 - 24b2 + b1 - 1) * q^49 + (-12b3 + 36) q^53 + (-18b3 + 36b2 + 84b1 - 42) q^55 + (-32b3 - 32b2) * q^59 + (30b2 - 17b1) * q^61 + (43b2 + 60b1) * q^65 + (6b3 + 6b2 - 34b1 - 34) q^67 + (14b3 - 28b2 + 108b1 - 54) q^71 + (48b3 + 49) q^73 + (-48b3 + 48b2 + 72b1 - 72) q^77 + (-20b3 + 10b2 - 46b1 + 92) q^79 + (-8b3 + 4b2 - 60b1 + 120) q^83 + (-42b3 + 42b2 + 87b1 - 87) q^85 + (-31b3 - 24) q^89 + (-26b3 + 52b2 + 100b1 - 50) q^91 + (-62b3 - 62b2 - 42b1 - 42) q^95 + (24b2 + 2b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} + 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^5 + 12 * q^7	$ 4 q + 12 q^{5} + 12 q^{7} - 36 q^{11} - 14 q^{13} - 48 q^{17} + 36 q^{23} - 28 q^{25} - 12 q^{29} - 72 q^{31} + 20 q^{37} + 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} - 204 q^{67} + 196 q^{73} - 144 q^{77} + 276 q^{79} + 360 q^{83} - 174 q^{85} - 96 q^{89} - 252 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 + 12 * q^7 - 36 * q^11 - 14 * q^13 - 48 * q^17 + 36 * q^23 - 28 * q^25 - 12 * q^29 - 72 * q^31 + 20 * q^37 + 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 - 204 * q^67 + 196 * q^73 - 144 * q^77 + 276 * q^79 + 360 * q^83 - 174 * q^85 - 96 * q^89 - 252 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{12}^{2} $ v^2
$\beta_{2}$	$=$	$ \zeta_{12}^{3} + \zeta_{12} $ v^3 + v
$\beta_{3}$	$=$	$ -\zeta_{12}^{3} + 2\zeta_{12} $ -v^3 + 2*v

Display $\zeta_{12}^j$ in terms of $\beta_i$

$\zeta_{12}$	$=$	$ ( \beta_{3} + \beta_{2} ) / 3 $ (b3 + b2) / 3
$\zeta_{12}^{2}$	$=$	$ \beta_1 $ b1
$\zeta_{12}^{3}$	$=$	$ ( -\beta_{3} + 2\beta_{2} ) / 3 $ (-b3 + 2*b2) / 3

Display $\beta_i$ in terms of $\zeta_{12}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$-1 + \beta_{1}$

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} $

271.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

2.13397

−

3.69615i

−2.19615

1.26795i

271.2

3.86603

−

6.69615i

8.19615

−

4.73205i

703.1

2.13397

3.69615i

−2.19615

−

1.26795i

703.2

3.86603

6.69615i

8.19615

4.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1296.3.o.bf		4
3.b	odd	2	1		1296.3.o.r		4
4.b	odd	2	1		1296.3.o.be		4
9.c	even	3	1		1296.3.g.c	✓	4
9.c	even	3	1		1296.3.o.be		4
9.d	odd	6	1		1296.3.g.i	yes	4
9.d	odd	6	1		1296.3.o.q		4
12.b	even	2	1		1296.3.o.q		4
36.f	odd	6	1		1296.3.g.c	✓	4
36.f	odd	6	1	inner	1296.3.o.bf		4
36.h	even	6	1		1296.3.g.i	yes	4
36.h	even	6	1		1296.3.o.r		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1296.3.g.c	✓	4	9.c	even	3	1
1296.3.g.c	✓	4	36.f	odd	6	1
1296.3.g.i	yes	4	9.d	odd	6	1
1296.3.g.i	yes	4	36.h	even	6	1
1296.3.o.q		4	9.d	odd	6	1
1296.3.o.q		4	12.b	even	2	1
1296.3.o.r		4	3.b	odd	2	1
1296.3.o.r		4	36.h	even	6	1
1296.3.o.be		4	4.b	odd	2	1
1296.3.o.be		4	9.c	even	3	1
1296.3.o.bf		4	1.a	even	1	1	trivial
1296.3.o.bf		4	36.f	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1296, [\chi])$:

$ T_{5}^{4} - 12T_{5}^{3} + 111T_{5}^{2} - 396T_{5} + 1089 $

$ T_{7}^{4} - 12T_{7}^{3} + 24T_{7}^{2} + 288T_{7} + 576 $

$ T_{11}^{4} + 36T_{11}^{3} + 504T_{11}^{2} + 2592T_{11} + 5184 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} - 12 T^{3} + \cdots + 1089 $ T^4 - 12T^3 + 111T^2 - 396*T + 1089
$7$	$ T^{4} - 12 T^{3} + \cdots + 576 $ T^4 - 12T^3 + 24T^2 + 288*T + 576
$11$	$ T^{4} + 36 T^{3} + \cdots + 5184 $ T^4 + 36T^3 + 504T^2 + 2592*T + 5184
$13$	$ T^{4} + 14 T^{3} + \cdots + 3481 $ T^4 + 14T^3 + 255T^2 - 826*T + 3481
$17$	$ (T^{2} + 24 T + 69)^{2} $ (T^2 + 24*T + 69)^2
$19$	$ T^{4} + 1824 T^{2} + 788544 $ T^4 + 1824*T^2 + 788544
$23$	$ T^{4} - 36 T^{3} + \cdots + 46656 $ T^4 - 36T^3 + 216T^2 + 7776*T + 46656
$29$	$ T^{4} + 12 T^{3} + \cdots + 1089 $ T^4 + 12T^3 + 111T^2 + 396*T + 1089
$31$	$ T^{4} + 72 T^{3} + \cdots + 82944 $ T^4 + 72T^3 + 2016T^2 + 20736*T + 82944
$37$	$ (T^{2} - 10 T - 947)^{2} $ (T^2 - 10*T - 947)^2
$41$	$ T^{4} + 1728 T^{2} + 2985984 $ T^4 + 1728*T^2 + 2985984
$43$	$ T^{4} - 108 T^{3} + \cdots + 876096 $ T^4 - 108T^3 + 4824T^2 - 101088*T + 876096
$47$	$ T^{4} - 576 T^{2} + 331776 $ T^4 - 576*T^2 + 331776
$53$	$ (T^{2} - 72 T + 864)^{2} $ (T^2 - 72*T + 864)^2
$59$	$ T^{4} - 9216 T^{2} + 84934656 $ T^4 - 9216*T^2 + 84934656
$61$	$ T^{4} + 34 T^{3} + \cdots + 5812921 $ T^4 + 34T^3 + 3567T^2 - 81974*T + 5812921
$67$	$ T^{4} + 204 T^{3} + \cdots + 9884736 $ T^4 + 204T^3 + 17016T^2 + 641376*T + 9884736
$71$	$ T^{4} + 21024 T^{2} + 48776256 $ T^4 + 21024*T^2 + 48776256
$73$	$ (T^{2} - 98 T - 4511)^{2} $ (T^2 - 98*T - 4511)^2
$79$	$ T^{4} - 276 T^{3} + \cdots + 29680704 $ T^4 - 276T^3 + 30840T^2 - 1503648*T + 29680704
$83$	$ T^{4} - 360 T^{3} + \cdots + 113550336 $ T^4 - 360T^3 + 53856T^2 - 3836160*T + 113550336
$89$	$ (T^{2} + 48 T - 2307)^{2} $ (T^2 + 48*T - 2307)^2
$97$	$ T^{4} - 4 T^{3} + \cdots + 2972176 $ T^4 - 4T^3 + 1740T^2 + 6896*T + 2972176
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Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{5} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{7}+O(q^{10}) \) q + (b3 - b2 - 6b1 + 6) q^5 + (4b3 - 2b2 - 2b1 + 4) q^7	\( q + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{5} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{7} + ( - 4 \beta_{3} + 2 \beta_{2} + \cdots - 12) q^{11}+ \cdots + (24 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100}) \) q + (b3 - b2 - 6b1 + 6) q^5 + (4b3 - 2b2 - 2b1 + 4) q^7 + (-4b3 + 2b2 + 6b1 - 12) q^11 + (-6b3 + 6b2 + 7b1 - 7) q^13 + (-5b3 - 12) q^17 + (10b3 - 20b2 - 4b1 + 2) q^19 + (-6b3 - 6b2 + 6b1 + 6) q^23 + (-12b2 - 14b1) * q^25 + (-b2 - 6b1) q^29 + (-4b3 - 4b2 - 12b1 - 12) q^31 + (14b3 - 28b2 - 36b1 + 18) q^35 + (-18b3 + 5) q^37 + (-24b3 + 24b2) * q^41 + (4b3 - 2b2 - 18b1 + 36) q^43 + (-16b3 + 8b2) * q^47 + (24b3 - 24b2 + b1 - 1) * q^49 + (-12b3 + 36) q^53 + (-18b3 + 36b2 + 84b1 - 42) q^55 + (-32b3 - 32b2) * q^59 + (30b2 - 17b1) * q^61 + (43b2 + 60b1) * q^65 + (6b3 + 6b2 - 34b1 - 34) q^67 + (14b3 - 28b2 + 108b1 - 54) q^71 + (48b3 + 49) q^73 + (-48b3 + 48b2 + 72b1 - 72) q^77 + (-20b3 + 10b2 - 46b1 + 92) q^79 + (-8b3 + 4b2 - 60b1 + 120) q^83 + (-42b3 + 42b2 + 87b1 - 87) q^85 + (-31b3 - 24) q^89 + (-26b3 + 52b2 + 100b1 - 50) q^91 + (-62b3 - 62b2 - 42b1 - 42) q^95 + (24b2 + 2b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} + 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^5 + 12 * q^7	\( 4 q + 12 q^{5} + 12 q^{7} - 36 q^{11} - 14 q^{13} - 48 q^{17} + 36 q^{23} - 28 q^{25} - 12 q^{29} - 72 q^{31} + 20 q^{37} + 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} - 204 q^{67} + 196 q^{73} - 144 q^{77} + 276 q^{79} + 360 q^{83} - 174 q^{85} - 96 q^{89} - 252 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 + 12 * q^7 - 36 * q^11 - 14 * q^13 - 48 * q^17 + 36 * q^23 - 28 * q^25 - 12 * q^29 - 72 * q^31 + 20 * q^37 + 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 - 204 * q^67 + 196 * q^73 - 144 * q^77 + 276 * q^79 + 360 * q^83 - 174 * q^85 - 96 * q^89 - 252 * q^95 + 4 * q^97

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