LMFDB - Newform orbit 400.3.b.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,3,Mod(351,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	400.b (of order $2$ , degree $1$ , 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:	$10.8992105744$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 2x^{2} + x + 1$ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{7}\cdot 3$
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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.

$f(q)$	$=$	$q + \beta_1 q^{3} - \beta_{2} q^{7} + ( - \beta_{3} - 9) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{3} - 4) q^{13} - 18 q^{17} - 4 \beta_{2} q^{19} + (\beta_{3} - 6) q^{21} + (3 \beta_{2} + 2 \beta_1) q^{23}+ \cdots + (12 \beta_{2} + 42 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^3 - b2 * q^7 + (-b3 - 9) * q^9 - 2b1 q^11 + (-b3 - 4) * q^13 - 18 * q^17 - 4b2 q^19 + (b3 - 6) * q^21 + (3b2 + 2b1) * q^23 + (-6b2 - 12b1) * q^27 + (2b3 - 18) q^29 + (-2b2 + 8b1) * q^31 + (2b3 + 36) q^33 + (-b3 - 20) * q^37 + (-6b2 - 16b1) * q^39 + (-3b3 + 12) q^41 + (2b2 - 5b1) * q^43 + (-9b2 + 4b1) * q^47 + (3b3 + 7) q^49 - 18b1 q^51 + (3b3 - 12) q^53 + (4b3 - 24) q^57 - 8b1 q^59 + (-b3 - 64) * q^61 + (-3b2 + 6b1) * q^63 + (2b2 - 7b1) * q^67 + (-5b3 - 18) q^69 + (6b2 + 16b1) * q^71 + (-4b3 + 38) q^73 + (-2b3 + 12) q^77 + 20b1 q^79 + (9b3 + 99) q^81 + (6b2 + 15b1) * q^83 + (12b2 + 6b1) * q^87 + (-4b3 - 6) q^89 + (-8b2 + 6b1) * q^91 + (-6b3 - 156) q^93 + (-10b3 + 26) q^97 + (12b2 + 42b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 36 q^{9} - 16 q^{13} - 72 q^{17} - 24 q^{21} - 72 q^{29} + 144 q^{33} - 80 q^{37} + 48 q^{41} + 28 q^{49} - 48 q^{53} - 96 q^{57} - 256 q^{61} - 72 q^{69} + 152 q^{73} + 48 q^{77} + 396 q^{81} - 24 q^{89}+ \cdots + 104 q^{97}+O(q^{100})$ 4 * q - 36 * q^9 - 16 * q^13 - 72 * q^17 - 24 * q^21 - 72 * q^29 + 144 * q^33 - 80 * q^37 + 48 * q^41 + 28 * q^49 - 48 * q^53 - 96 * q^57 - 256 * q^61 - 72 * q^69 + 152 * q^73 + 48 * q^77 + 396 * q^81 - 24 * q^89 - 624 * q^93 + 104 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} + 2x^{2} + x + 1$ x^4 - x^3 + 2*x^2 + x + 1 :

$\beta_{1}$	$=$	$\nu^{3} + 4\nu + 1$ v^3 + 4*v + 1
$\beta_{2}$	$=$	$-5\nu^{3} + 8\nu^{2} - 12\nu - 1$ -5v^3 + 8v^2 - 12*v - 1
$\beta_{3}$	$=$	$-6\nu^{3} - 12$ -6*v^3 - 12

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

$\nu$	$=$	$( \beta_{3} + 6\beta _1 + 6 ) / 24$ (b3 + 6*b1 + 6) / 24
$\nu^{2}$	$=$	$( -\beta_{3} + 3\beta_{2} + 9\beta _1 - 18 ) / 24$ (-b3 + 3b2 + 9b1 - 18) / 24
$\nu^{3}$	$=$	$( -\beta_{3} - 12 ) / 6$ (-b3 - 12) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$1$	$-1$

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}

351.1

0.809017	−	1.40126i
−0.309017	−	0.535233i
−0.309017	+	0.535233i
0.809017	+	1.40126i

−

5.60503i

1.32317i

−22.4164

351.2

−

2.14093i

−

9.06914i

4.41641

351.3

2.14093i

9.06914i

4.41641

351.4

5.60503i

−

1.32317i

−22.4164

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.3.b.g		4
3.b	odd	2	1		3600.3.e.bb		4
4.b	odd	2	1	inner	400.3.b.g		4
5.b	even	2	1		80.3.b.a	✓	4
5.c	odd	4	2		400.3.h.d		8
8.b	even	2	1		1600.3.b.k		4
8.d	odd	2	1		1600.3.b.k		4
12.b	even	2	1		3600.3.e.bb		4
15.d	odd	2	1		720.3.e.c		4
15.e	even	4	2		3600.3.j.k		8
20.d	odd	2	1		80.3.b.a	✓	4
20.e	even	4	2		400.3.h.d		8
40.e	odd	2	1		320.3.b.a		4
40.f	even	2	1		320.3.b.a		4
40.i	odd	4	2		1600.3.h.p		8
40.k	even	4	2		1600.3.h.p		8
60.h	even	2	1		720.3.e.c		4
60.l	odd	4	2		3600.3.j.k		8
80.k	odd	4	2		1280.3.g.f		8
80.q	even	4	2		1280.3.g.f		8
120.i	odd	2	1		2880.3.e.b		4
120.m	even	2	1		2880.3.e.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.3.b.a	✓	4	5.b	even	2	1
80.3.b.a	✓	4	20.d	odd	2	1
320.3.b.a		4	40.e	odd	2	1
320.3.b.a		4	40.f	even	2	1
400.3.b.g		4	1.a	even	1	1	trivial
400.3.b.g		4	4.b	odd	2	1	inner
400.3.h.d		8	5.c	odd	4	2
400.3.h.d		8	20.e	even	4	2
720.3.e.c		4	15.d	odd	2	1
720.3.e.c		4	60.h	even	2	1
1280.3.g.f		8	80.k	odd	4	2
1280.3.g.f		8	80.q	even	4	2
1600.3.b.k		4	8.b	even	2	1
1600.3.b.k		4	8.d	odd	2	1
1600.3.h.p		8	40.i	odd	4	2
1600.3.h.p		8	40.k	even	4	2
2880.3.e.b		4	120.i	odd	2	1
2880.3.e.b		4	120.m	even	2	1
3600.3.e.bb		4	3.b	odd	2	1
3600.3.e.bb		4	12.b	even	2	1
3600.3.j.k		8	15.e	even	4	2
3600.3.j.k		8	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(400, [\chi])$ :

T_{3}^{4} + 36T_{3}^{2} + 144

T_{13}^{2} + 8T_{13} - 164

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 36T^{2} + 144$ T^4 + 36*T^2 + 144
$5$	$T^{4}$ T^4
$7$	$T^{4} + 84T^{2} + 144$ T^4 + 84*T^2 + 144
$11$	$T^{4} + 144T^{2} + 2304$ T^4 + 144*T^2 + 2304
$13$	$(T^{2} + 8 T - 164)^{2}$ (T^2 + 8*T - 164)^2
$17$	$(T + 18)^{4}$ (T + 18)^4
$19$	$T^{4} + 1344 T^{2} + 36864$ T^4 + 1344*T^2 + 36864
$23$	$T^{4} + 756 T^{2} + 121104$ T^4 + 756*T^2 + 121104
$29$	$(T^{2} + 36 T - 396)^{2}$ (T^2 + 36*T - 396)^2
$31$	$T^{4} + 3024 T^{2} + 2214144$ T^4 + 3024*T^2 + 2214144
$37$	$(T^{2} + 40 T + 220)^{2}$ (T^2 + 40*T + 220)^2
$41$	$(T^{2} - 24 T - 1476)^{2}$ (T^2 - 24*T - 1476)^2
$43$	$T^{4} + 1476 T^{2} + 535824$ T^4 + 1476*T^2 + 535824
$47$	$T^{4} + 8244 T^{2} + 898704$ T^4 + 8244*T^2 + 898704
$53$	$(T^{2} + 24 T - 1476)^{2}$ (T^2 + 24*T - 1476)^2
$59$	$T^{4} + 2304 T^{2} + 589824$ T^4 + 2304*T^2 + 589824
$61$	$(T^{2} + 128 T + 3916)^{2}$ (T^2 + 128*T + 3916)^2
$67$	$T^{4} + 2436 T^{2} + 1468944$ T^4 + 2436*T^2 + 1468944
$71$	$T^{4} + 9936 T^{2} + 3873024$ T^4 + 9936*T^2 + 3873024
$73$	$(T^{2} - 76 T - 1436)^{2}$ (T^2 - 76*T - 1436)^2
$79$	$T^{4} + 14400 T^{2} + 23040000$ T^4 + 14400*T^2 + 23040000
$83$	$T^{4} + 8964 T^{2} + 4210704$ T^4 + 8964*T^2 + 4210704
$89$	$(T^{2} + 12 T - 2844)^{2}$ (T^2 + 12*T - 2844)^2
$97$	$(T^{2} - 52 T - 17324)^{2}$ (T^2 - 52*T - 17324)^2
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