LMFDB - Newform orbit 1872.2.by.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(433,1872)]

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, 1]))

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

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

chi := DirichletCharacter("1872.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.by (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:	$14.9479952584$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
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 primitive root of unity $\zeta_{12}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{7} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{11} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13}+ \cdots - 6 \zeta_{12} q^{97} +O(q^{100}) $ q + (2z^3 + 2z^2 - 1) * q^5 + (-z^2 - z - 1) * q^7 + (-3z^3 + z^2 + 3z - 2) * q^11 + (-z^3 - 3z) q^13 + (-2z^3 - 4z^2 + z + 4) * q^17 + (-z^2 - 3z - 1) q^19 + (3z^3 - z^2 + 3z) * q^23 + (4z^3 - 8z - 2) * q^25 + (-2z^3 - z^2 - 2z) * q^29 + (-2z^3 + 4z^2 - 2) * q^31 + (-6z^3 - 5z^2 + 3z + 5) q^35 + (-7z^3 + 2z^2 + 7z - 4) q^37 + (-z^3 + 6z^2 + z - 12) q^41 + (-10z^3 - z^2 + 5z + 1) * q^43 + (-3z^3 + 6z^2 - 3) * q^47 + (2z^3 - 3z^2 + 2z) q^49 + (-2z^3 + 4z + 3) * q^53 + (z^3 + 3z^2 + z) q^55 - 8z q^59 + (-6z^3 - 4z^2 + 3z + 4) q^61 + (-7z^3 - 6z^2 + 5z + 8) q^65 + (-z^3 - 7z^2 + z + 14) q^67 + (z^2 + 3z + 1) q^71 + (-8z^3 + 2z^2 - 1) * q^73 + (2z^3 - 4z) * q^77 + (2z^3 - 4z + 6) * q^79 + (-5z^3 + 6z^2 - 3) * q^83 + (6z^2 + 11z + 6) * q^85 + (-6z^3 - 2z^2 + 6z + 4) q^89 + (5z^3 + 4z^2 + 2z - 1) q^91 + (-10z^3 - 9z^2 + 5z + 9) q^95 - 6z q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7} - 6 q^{11} + 8 q^{17} - 6 q^{19} - 2 q^{23} - 8 q^{25} - 2 q^{29} + 10 q^{35} - 12 q^{37} - 36 q^{41} + 2 q^{43} - 6 q^{49} + 12 q^{53} + 6 q^{55} + 8 q^{61} + 20 q^{65} + 42 q^{67} + 6 q^{71}+ \cdots + 18 q^{95}+O(q^{100}) $ 4 * q - 6 * q^7 - 6 * q^11 + 8 * q^17 - 6 * q^19 - 2 * q^23 - 8 * q^25 - 2 * q^29 + 10 * q^35 - 12 * q^37 - 36 * q^41 + 2 * q^43 - 6 * q^49 + 12 * q^53 + 6 * q^55 + 8 * q^61 + 20 * q^65 + 42 * q^67 + 6 * q^71 + 24 * q^79 + 36 * q^85 + 12 * q^89 + 4 * q^91 + 18 * q^95

Character values

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

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$1 - \zeta_{12}^{2}$	$1$	$1$	$1$

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} $

433.1

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

−

3.73205i

−2.36603

1.36603i

433.2

0.267949i

−0.633975

0.366025i

1297.1

−

0.267949i

−0.633975

−

0.366025i

1297.2

3.73205i

−2.36603

−

1.36603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1872.2.by.h		4
3.b	odd	2	1		624.2.bv.e		4
4.b	odd	2	1		234.2.l.c		4
12.b	even	2	1		78.2.i.a	✓	4
13.e	even	6	1	inner	1872.2.by.h		4
39.h	odd	6	1		624.2.bv.e		4
39.k	even	12	1		8112.2.a.bj		2
39.k	even	12	1		8112.2.a.bp		2
52.i	odd	6	1		234.2.l.c		4
52.i	odd	6	1		3042.2.b.i		4
52.j	odd	6	1		3042.2.b.i		4
52.l	even	12	1		3042.2.a.p		2
52.l	even	12	1		3042.2.a.y		2
60.h	even	2	1		1950.2.bc.d		4
60.l	odd	4	1		1950.2.y.b		4
60.l	odd	4	1		1950.2.y.g		4
156.h	even	2	1		1014.2.i.a		4
156.l	odd	4	1		1014.2.e.g		4
156.l	odd	4	1		1014.2.e.i		4
156.p	even	6	1		1014.2.b.e		4
156.p	even	6	1		1014.2.i.a		4
156.r	even	6	1		78.2.i.a	✓	4
156.r	even	6	1		1014.2.b.e		4
156.v	odd	12	1		1014.2.a.i		2
156.v	odd	12	1		1014.2.a.k		2
156.v	odd	12	1		1014.2.e.g		4
156.v	odd	12	1		1014.2.e.i		4
780.cb	even	6	1		1950.2.bc.d		4
780.cw	odd	12	1		1950.2.y.b		4
780.cw	odd	12	1		1950.2.y.g		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.i.a	✓	4	12.b	even	2	1
78.2.i.a	✓	4	156.r	even	6	1
234.2.l.c		4	4.b	odd	2	1
234.2.l.c		4	52.i	odd	6	1
624.2.bv.e		4	3.b	odd	2	1
624.2.bv.e		4	39.h	odd	6	1
1014.2.a.i		2	156.v	odd	12	1
1014.2.a.k		2	156.v	odd	12	1
1014.2.b.e		4	156.p	even	6	1
1014.2.b.e		4	156.r	even	6	1
1014.2.e.g		4	156.l	odd	4	1
1014.2.e.g		4	156.v	odd	12	1
1014.2.e.i		4	156.l	odd	4	1
1014.2.e.i		4	156.v	odd	12	1
1014.2.i.a		4	156.h	even	2	1
1014.2.i.a		4	156.p	even	6	1
1872.2.by.h		4	1.a	even	1	1	trivial
1872.2.by.h		4	13.e	even	6	1	inner
1950.2.y.b		4	60.l	odd	4	1
1950.2.y.b		4	780.cw	odd	12	1
1950.2.y.g		4	60.l	odd	4	1
1950.2.y.g		4	780.cw	odd	12	1
1950.2.bc.d		4	60.h	even	2	1
1950.2.bc.d		4	780.cb	even	6	1
3042.2.a.p		2	52.l	even	12	1
3042.2.a.y		2	52.l	even	12	1
3042.2.b.i		4	52.i	odd	6	1
3042.2.b.i		4	52.j	odd	6	1
8112.2.a.bj		2	39.k	even	12	1
8112.2.a.bp		2	39.k	even	12	1

Hecke kernels

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

$ T_{5}^{4} + 14T_{5}^{2} + 1 $

$ T_{7}^{4} + 6T_{7}^{3} + 14T_{7}^{2} + 12T_{7} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 14T^{2} + 1 $ T^4 + 14*T^2 + 1
$7$	$ T^{4} + 6 T^{3} + \cdots + 4 $ T^4 + 6T^3 + 14T^2 + 12*T + 4
$11$	$ T^{4} + 6 T^{3} + \cdots + 36 $ T^4 + 6T^3 + 6T^2 - 36*T + 36
$13$	$ T^{4} - T^{2} + 169 $ T^4 - T^2 + 169
$17$	$ T^{4} - 8 T^{3} + \cdots + 169 $ T^4 - 8T^3 + 51T^2 - 104*T + 169
$19$	$ T^{4} + 6 T^{3} + \cdots + 36 $ T^4 + 6T^3 + 6T^2 - 36*T + 36
$23$	$ T^{4} + 2 T^{3} + \cdots + 676 $ T^4 + 2T^3 + 30T^2 - 52*T + 676
$29$	$ T^{4} + 2 T^{3} + \cdots + 121 $ T^4 + 2T^3 + 15T^2 - 22*T + 121
$31$	$ T^{4} + 32T^{2} + 64 $ T^4 + 32*T^2 + 64
$37$	$ T^{4} + 12 T^{3} + \cdots + 1369 $ T^4 + 12T^3 + 11T^2 - 444*T + 1369
$41$	$ T^{4} + 36 T^{3} + \cdots + 11449 $ T^4 + 36T^3 + 539T^2 + 3852*T + 11449
$43$	$ T^{4} - 2 T^{3} + \cdots + 5476 $ T^4 - 2T^3 + 78T^2 + 148*T + 5476
$47$	$ T^{4} + 72T^{2} + 324 $ T^4 + 72*T^2 + 324
$53$	$ (T^{2} - 6 T - 3)^{2} $ (T^2 - 6*T - 3)^2
$59$	$ T^{4} - 64T^{2} + 4096 $ T^4 - 64*T^2 + 4096
$61$	$ T^{4} - 8 T^{3} + \cdots + 121 $ T^4 - 8T^3 + 75T^2 + 88*T + 121
$67$	$ T^{4} - 42 T^{3} + \cdots + 21316 $ T^4 - 42T^3 + 734T^2 - 6132*T + 21316
$71$	$ T^{4} - 6 T^{3} + \cdots + 36 $ T^4 - 6T^3 + 6T^2 + 36*T + 36
$73$	$ T^{4} + 134T^{2} + 3721 $ T^4 + 134*T^2 + 3721
$79$	$ (T^{2} - 12 T + 24)^{2} $ (T^2 - 12*T + 24)^2
$83$	$ T^{4} + 104T^{2} + 4 $ T^4 + 104*T^2 + 4
$89$	$ T^{4} - 12 T^{3} + \cdots + 576 $ T^4 - 12T^3 + 24T^2 + 288*T + 576
$97$	$ T^{4} - 36T^{2} + 1296 $ T^4 - 36*T^2 + 1296
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Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{5} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{7} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{11} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13}+ \cdots - 6 \zeta_{12} q^{97} +O(q^{100}) \) q + (2z^3 + 2z^2 - 1) * q^5 + (-z^2 - z - 1) * q^7 + (-3z^3 + z^2 + 3z - 2) * q^11 + (-z^3 - 3z) q^13 + (-2z^3 - 4z^2 + z + 4) * q^17 + (-z^2 - 3z - 1) q^19 + (3z^3 - z^2 + 3z) * q^23 + (4z^3 - 8z - 2) * q^25 + (-2z^3 - z^2 - 2z) * q^29 + (-2z^3 + 4z^2 - 2) * q^31 + (-6z^3 - 5z^2 + 3z + 5) q^35 + (-7z^3 + 2z^2 + 7z - 4) q^37 + (-z^3 + 6z^2 + z - 12) q^41 + (-10z^3 - z^2 + 5z + 1) * q^43 + (-3z^3 + 6z^2 - 3) * q^47 + (2z^3 - 3z^2 + 2z) q^49 + (-2z^3 + 4z + 3) * q^53 + (z^3 + 3z^2 + z) q^55 - 8z q^59 + (-6z^3 - 4z^2 + 3z + 4) q^61 + (-7z^3 - 6z^2 + 5z + 8) q^65 + (-z^3 - 7z^2 + z + 14) q^67 + (z^2 + 3z + 1) q^71 + (-8z^3 + 2z^2 - 1) * q^73 + (2z^3 - 4z) * q^77 + (2z^3 - 4z + 6) * q^79 + (-5z^3 + 6z^2 - 3) * q^83 + (6z^2 + 11z + 6) * q^85 + (-6z^3 - 2z^2 + 6z + 4) q^89 + (5z^3 + 4z^2 + 2z - 1) q^91 + (-10z^3 - 9z^2 + 5z + 9) q^95 - 6z q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7} - 6 q^{11} + 8 q^{17} - 6 q^{19} - 2 q^{23} - 8 q^{25} - 2 q^{29} + 10 q^{35} - 12 q^{37} - 36 q^{41} + 2 q^{43} - 6 q^{49} + 12 q^{53} + 6 q^{55} + 8 q^{61} + 20 q^{65} + 42 q^{67} + 6 q^{71}+ \cdots + 18 q^{95}+O(q^{100}) \) 4 * q - 6 * q^7 - 6 * q^11 + 8 * q^17 - 6 * q^19 - 2 * q^23 - 8 * q^25 - 2 * q^29 + 10 * q^35 - 12 * q^37 - 36 * q^41 + 2 * q^43 - 6 * q^49 + 12 * q^53 + 6 * q^55 + 8 * q^61 + 20 * q^65 + 42 * q^67 + 6 * q^71 + 24 * q^79 + 36 * q^85 + 12 * q^89 + 4 * q^91 + 18 * q^95

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