LMFDB - Newform orbit 1872.2.t.v

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.t (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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.27870912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 $ x^6 - x^5 + 7x^4 + 2x^3 + 38x^2 - 12x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 936)
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_{2} + 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) $ q + (b2 + 1) * q^5 + (-b2 - b1) * q^7	$ q + (\beta_{2} + 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{5} - \beta_{3}) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + \cdots - 2) q^{97}+O(q^{100}) $ q + (b2 + 1) * q^5 + (-b2 - b1) * q^7 + (b5 - b3) * q^11 + (-b5 - b4 - b2) * q^13 + (-b5 - b4 + b2 + b1 + 1) * q^17 - b5 * q^19 + (b5 + 2b4 - b3 + 2b1) * q^23 + (-b3 + b2) * q^25 + (-2b5 + b4 + 2b3 - b1) * q^29 + (-b3 + b2 + 2) * q^31 + (b5 + 4b4 - 4) q^35 + (b4 - 3b1) q^37 + (-b5 - b4 + b3 - 3b1) q^41 + (-2b5 - 2b4 + b2 + b1 + 2) * q^43 + (-b3 - 2b2 - 2) q^47 + (-b5 + 3b4 + b3 - b1) q^49 + (b2 + 5) * q^53 + (b5 + 2b4 - b3 - 2b1) * q^55 + (2b5 - 4b2 - 4b1) q^59 + (-b5 + 3b4 - 4b2 - 4b1 - 3) q^61 + (-b5 - 3b4 + b3 + 2b2 + 3b1 - 2) q^65 + (-2b5 + 6b4 + 2b3 - 5b1) * q^67 + (b5 - 6b4 + 2b2 + 2b1 + 6) q^71 + (-2b3 + 2b2 + 3) * q^73 + (-2b2 - 2) q^77 + (-b3 - 5b2 - 2) q^79 + (-b3 - 4b2 - 8) q^83 + (-2b5 - 7b4 + 3b2 + 3b1 + 7) * q^85 + 4b4 q^89 + (-b5 - 2b4 - 3b1 + 4) * q^91 + (-b5 - 2b4 + 2b2 + 2b1 + 2) q^95 + (-b5 + 2b4 - b2 - b1 - 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} + q^{7}+O(q^{10}) $ 6 * q + 4 * q^5 + q^7	$ 6 q + 4 q^{5} + q^{7} - q^{13} + 2 q^{17} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 10 q^{31} - 12 q^{35} - 6 q^{41} + 5 q^{43} - 8 q^{47} + 8 q^{49} + 28 q^{53} + 4 q^{55} + 4 q^{59} - 5 q^{61} - 22 q^{65} + 13 q^{67} + 16 q^{71} + 14 q^{73} - 8 q^{77} - 2 q^{79} - 40 q^{83} + 18 q^{85} + 12 q^{89} + 15 q^{91} + 4 q^{95} - 5 q^{97}+O(q^{100}) $ 6 * q + 4 * q^5 + q^7 - q^13 + 2 * q^17 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 10 * q^31 - 12 * q^35 - 6 * q^41 + 5 * q^43 - 8 * q^47 + 8 * q^49 + 28 * q^53 + 4 * q^55 + 4 * q^59 - 5 * q^61 - 22 * q^65 + 13 * q^67 + 16 * q^71 + 14 * q^73 - 8 * q^77 - 2 * q^79 - 40 * q^83 + 18 * q^85 + 12 * q^89 + 15 * q^91 + 4 * q^95 - 5 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 $ x^6 - x^5 + 7*x^4 + 2*x^3 + 38*x^2 - 12*x + 4 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 $ (v^5 - 7v^4 + 49v^3 - 38v^2 + 12v - 84) / 254
$\beta_{3}$	$=$	$ ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 $ (-7v^5 + 49v^4 - 89v^3 + 266v^2 - 84*v + 1096) / 254
$\beta_{4}$	$=$	$ ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 $ (21v^5 - 20v^4 + 140v^3 + 91v^2 + 760*v + 14) / 254
$\beta_{5}$	$=$	$ ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 $ (-85v^5 + 87v^4 - 609v^3 - 72v^2 - 3306*v + 1044) / 254

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 $ b5 + 4*b4 + b2 + b1 - 4
$\nu^{3}$	$=$	$ \beta_{3} + 7\beta_{2} - 2 $ b3 + 7*b2 - 2
$\nu^{4}$	$=$	$ -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 $ -7b5 - 26b4 + 7b3 - 11b1
$\nu^{5}$	$=$	$ -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 $ -11b5 - 30b4 - 51b2 - 51b1 + 30

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-1 + \beta_{4}$	$1$	$1$	$1$

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} $

289.1

1.42789	+	2.47317i
0.160819	+	0.278546i
−1.08870	−	1.88569i
1.42789	−	2.47317i
0.160819	−	0.278546i
−1.08870	+	1.88569i

−1.85577

1.42789

−

2.47317i

289.2

0.678363

0.160819

−

0.278546i

289.3

3.17741

−1.08870

1.88569i

1153.1

−1.85577

1.42789

2.47317i

1153.2

0.678363

0.160819

0.278546i

1153.3

3.17741

−1.08870

−

1.88569i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1872.2.t.v		6
3.b	odd	2	1		1872.2.t.t		6
4.b	odd	2	1		936.2.t.i	yes	6
12.b	even	2	1		936.2.t.g	✓	6
13.c	even	3	1	inner	1872.2.t.v		6
39.i	odd	6	1		1872.2.t.t		6
52.j	odd	6	1		936.2.t.i	yes	6
156.p	even	6	1		936.2.t.g	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
936.2.t.g	✓	6	12.b	even	2	1
936.2.t.g	✓	6	156.p	even	6	1
936.2.t.i	yes	6	4.b	odd	2	1
936.2.t.i	yes	6	52.j	odd	6	1
1872.2.t.t		6	3.b	odd	2	1
1872.2.t.t		6	39.i	odd	6	1
1872.2.t.v		6	1.a	even	1	1	trivial
1872.2.t.v		6	13.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1872, [\chi])$:

$ T_{5}^{3} - 2T_{5}^{2} - 5T_{5} + 4 $

$ T_{7}^{6} - T_{7}^{5} + 7T_{7}^{4} + 2T_{7}^{3} + 38T_{7}^{2} - 12T_{7} + 4 $

$ T_{11}^{6} + 14T_{11}^{4} - 32T_{11}^{3} + 196T_{11}^{2} - 224T_{11} + 256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ (T^{3} - 2 T^{2} - 5 T + 4)^{2} $ (T^3 - 2T^2 - 5T + 4)^2
$7$	$ T^{6} - T^{5} + 7 T^{4} + \cdots + 4 $ T^6 - T^5 + 7T^4 + 2T^3 + 38T^2 - 12T + 4
$11$	$ T^{6} + 14 T^{4} + \cdots + 256 $ T^6 + 14T^4 - 32T^3 + 196T^2 - 224T + 256
$13$	$ T^{6} + T^{5} + \cdots + 2197 $ T^6 + T^5 + 8T^4 - 41T^3 + 104T^2 + 169T + 2197
$17$	$ T^{6} - 2 T^{5} + \cdots + 144 $ T^6 - 2T^5 + 27T^4 + 70T^3 + 505T^2 + 276*T + 144
$19$	$ T^{6} + 14 T^{4} + \cdots + 256 $ T^6 + 14T^4 - 32T^3 + 196T^2 - 224T + 256
$23$	$ T^{6} - 8 T^{5} + \cdots + 64 $ T^6 - 8T^5 + 74T^4 + 64T^3 + 164T^2 - 80*T + 64
$29$	$ T^{6} - 2 T^{5} + \cdots + 11664 $ T^6 - 2T^5 + 57T^4 + 322T^3 + 2593T^2 + 5724*T + 11664
$31$	$ (T^{3} - 5 T^{2} - 16 T + 8)^{2} $ (T^3 - 5T^2 - 16T + 8)^2
$37$	$ T^{6} + 57 T^{4} + \cdots + 4 $ T^6 + 57T^4 - 4T^3 + 3249T^2 - 114T + 4
$41$	$ T^{6} + 6 T^{5} + \cdots + 4096 $ T^6 + 6T^5 + 83T^4 - 410T^3 + 1825T^2 - 3008*T + 4096
$43$	$ T^{6} - 5 T^{5} + \cdots + 13924 $ T^6 - 5T^5 + 87T^4 + 74T^3 + 4434T^2 - 7316*T + 13924
$47$	$ (T^{3} + 4 T^{2} - 26 T - 96)^{2} $ (T^3 + 4T^2 - 26T - 96)^2
$53$	$ (T^{3} - 14 T^{2} + \cdots - 72)^{2} $ (T^3 - 14T^2 + 59T - 72)^2
$59$	$ T^{6} - 4 T^{5} + \cdots + 1557504 $ T^6 - 4T^5 + 200T^4 - 1760T^3 + 38848T^2 - 229632*T + 1557504
$61$	$ T^{6} + 5 T^{5} + \cdots + 257049 $ T^6 + 5T^5 + 116T^4 + 559T^3 + 10816T^2 + 46137*T + 257049
$67$	$ T^{6} - 13 T^{5} + \cdots + 2377764 $ T^6 - 13T^5 + 287T^4 - 1550T^3 + 33970T^2 - 181956*T + 2377764
$71$	$ T^{6} - 16 T^{5} + \cdots + 5184 $ T^6 - 16T^5 + 202T^4 - 1008T^3 + 4068T^2 + 3888*T + 5184
$73$	$ (T^{3} - 7 T^{2} - 81 T - 9)^{2} $ (T^3 - 7T^2 - 81T - 9)^2
$79$	$ (T^{3} + T^{2} - 152 T - 624)^{2} $ (T^3 + T^2 - 152*T - 624)^2
$83$	$ (T^{3} + 20 T^{2} + \cdots - 712)^{2} $ (T^3 + 20T^2 + 34T - 712)^2
$89$	$ (T^{2} - 4 T + 16)^{3} $ (T^2 - 4*T + 16)^3
$97$	$ T^{6} + 5 T^{5} + \cdots + 256 $ T^6 + 5T^5 + 33T^4 - 8T^3 + 144T^2 + 128*T + 256
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Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{2} + 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) q + (b2 + 1) * q^5 + (-b2 - b1) * q^7	\( q + (\beta_{2} + 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{5} - \beta_{3}) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + \cdots - 2) q^{97}+O(q^{100}) \) q + (b2 + 1) * q^5 + (-b2 - b1) * q^7 + (b5 - b3) * q^11 + (-b5 - b4 - b2) * q^13 + (-b5 - b4 + b2 + b1 + 1) * q^17 - b5 * q^19 + (b5 + 2b4 - b3 + 2b1) * q^23 + (-b3 + b2) * q^25 + (-2b5 + b4 + 2b3 - b1) * q^29 + (-b3 + b2 + 2) * q^31 + (b5 + 4b4 - 4) q^35 + (b4 - 3b1) q^37 + (-b5 - b4 + b3 - 3b1) q^41 + (-2b5 - 2b4 + b2 + b1 + 2) * q^43 + (-b3 - 2b2 - 2) q^47 + (-b5 + 3b4 + b3 - b1) q^49 + (b2 + 5) * q^53 + (b5 + 2b4 - b3 - 2b1) * q^55 + (2b5 - 4b2 - 4b1) q^59 + (-b5 + 3b4 - 4b2 - 4b1 - 3) q^61 + (-b5 - 3b4 + b3 + 2b2 + 3b1 - 2) q^65 + (-2b5 + 6b4 + 2b3 - 5b1) * q^67 + (b5 - 6b4 + 2b2 + 2b1 + 6) q^71 + (-2b3 + 2b2 + 3) * q^73 + (-2b2 - 2) q^77 + (-b3 - 5b2 - 2) q^79 + (-b3 - 4b2 - 8) q^83 + (-2b5 - 7b4 + 3b2 + 3b1 + 7) * q^85 + 4b4 q^89 + (-b5 - 2b4 - 3b1 + 4) * q^91 + (-b5 - 2b4 + 2b2 + 2b1 + 2) q^95 + (-b5 + 2b4 - b2 - b1 - 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} + q^{7}+O(q^{10}) \) 6 * q + 4 * q^5 + q^7	\( 6 q + 4 q^{5} + q^{7} - q^{13} + 2 q^{17} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 10 q^{31} - 12 q^{35} - 6 q^{41} + 5 q^{43} - 8 q^{47} + 8 q^{49} + 28 q^{53} + 4 q^{55} + 4 q^{59} - 5 q^{61} - 22 q^{65} + 13 q^{67} + 16 q^{71} + 14 q^{73} - 8 q^{77} - 2 q^{79} - 40 q^{83} + 18 q^{85} + 12 q^{89} + 15 q^{91} + 4 q^{95} - 5 q^{97}+O(q^{100}) \) 6 * q + 4 * q^5 + q^7 - q^13 + 2 * q^17 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 10 * q^31 - 12 * q^35 - 6 * q^41 + 5 * q^43 - 8 * q^47 + 8 * q^49 + 28 * q^53 + 4 * q^55 + 4 * q^59 - 5 * q^61 - 22 * q^65 + 13 * q^67 + 16 * q^71 + 14 * q^73 - 8 * q^77 - 2 * q^79 - 40 * q^83 + 18 * q^85 + 12 * q^89 + 15 * q^91 + 4 * q^95 - 5 * q^97

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