LMFDB - Newform orbit 450.4.c.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	450.c (of order $2$ , degree $1$ , 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:	$26.5508595026$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 50)
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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 i q^{2} - 4 q^{4} + 34 i q^{7} - 8 i q^{8} - 27 q^{11} - 28 i q^{13} - 68 q^{14} + 16 q^{16} + 21 i q^{17} - 35 q^{19} - 54 i q^{22} + 78 i q^{23} + 56 q^{26} - 136 i q^{28} - 120 q^{29} + 182 q^{31} + \cdots - 1626 i q^{98} +O(q^{100})$ q + 2i q^2 - 4 * q^4 + 34i q^7 - 8i q^8 - 27 * q^11 - 28i q^13 - 68 * q^14 + 16 * q^16 + 21i q^17 - 35 * q^19 - 54i q^22 + 78i q^23 + 56 * q^26 - 136i q^28 - 120 * q^29 + 182 * q^31 + 32i q^32 - 42 * q^34 - 146i q^37 - 70i q^38 - 357 * q^41 - 148i q^43 + 108 * q^44 - 156 * q^46 - 84i q^47 - 813 * q^49 + 112i q^52 - 702i q^53 + 272 * q^56 - 240i q^58 - 840 * q^59 - 238 * q^61 + 364i q^62 - 64 * q^64 - 461i q^67 - 84i q^68 + 708 * q^71 - 133i q^73 + 292 * q^74 + 140 * q^76 - 918i q^77 - 650 * q^79 - 714i q^82 + 903i q^83 + 296 * q^86 + 216i q^88 + 735 * q^89 + 952 * q^91 - 312i q^92 + 168 * q^94 - 1106i q^97 - 1626i q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} - 54 q^{11} - 136 q^{14} + 32 q^{16} - 70 q^{19} + 112 q^{26} - 240 q^{29} + 364 q^{31} - 84 q^{34} - 714 q^{41} + 216 q^{44} - 312 q^{46} - 1626 q^{49} + 544 q^{56} - 1680 q^{59} - 476 q^{61}+ \cdots + 336 q^{94}+O(q^{100})$ 2 * q - 8 * q^4 - 54 * q^11 - 136 * q^14 + 32 * q^16 - 70 * q^19 + 112 * q^26 - 240 * q^29 + 364 * q^31 - 84 * q^34 - 714 * q^41 + 216 * q^44 - 312 * q^46 - 1626 * q^49 + 544 * q^56 - 1680 * q^59 - 476 * q^61 - 128 * q^64 + 1416 * q^71 + 584 * q^74 + 280 * q^76 - 1300 * q^79 + 592 * q^86 + 1470 * q^89 + 1904 * q^91 + 336 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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}

199.1

	−	1.00000i
		1.00000i

−

2.00000i

−4.00000

−

34.0000i

8.00000i

199.2

2.00000i

−4.00000

34.0000i

−

8.00000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.4.c.c		2
3.b	odd	2	1		50.4.b.b		2
5.b	even	2	1	inner	450.4.c.c		2
5.c	odd	4	1		450.4.a.a		1
5.c	odd	4	1		450.4.a.t		1
12.b	even	2	1		400.4.c.d		2
15.d	odd	2	1		50.4.b.b		2
15.e	even	4	1		50.4.a.a	✓	1
15.e	even	4	1		50.4.a.e	yes	1
60.h	even	2	1		400.4.c.d		2
60.l	odd	4	1		400.4.a.d		1
60.l	odd	4	1		400.4.a.r		1
105.k	odd	4	1		2450.4.a.t		1
105.k	odd	4	1		2450.4.a.y		1
120.q	odd	4	1		1600.4.a.g		1
120.q	odd	4	1		1600.4.a.bv		1
120.w	even	4	1		1600.4.a.f		1
120.w	even	4	1		1600.4.a.bu		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.4.a.a	✓	1	15.e	even	4	1
50.4.a.e	yes	1	15.e	even	4	1
50.4.b.b		2	3.b	odd	2	1
50.4.b.b		2	15.d	odd	2	1
400.4.a.d		1	60.l	odd	4	1
400.4.a.r		1	60.l	odd	4	1
400.4.c.d		2	12.b	even	2	1
400.4.c.d		2	60.h	even	2	1
450.4.a.a		1	5.c	odd	4	1
450.4.a.t		1	5.c	odd	4	1
450.4.c.c		2	1.a	even	1	1	trivial
450.4.c.c		2	5.b	even	2	1	inner
1600.4.a.f		1	120.w	even	4	1
1600.4.a.g		1	120.q	odd	4	1
1600.4.a.bu		1	120.w	even	4	1
1600.4.a.bv		1	120.q	odd	4	1
2450.4.a.t		1	105.k	odd	4	1
2450.4.a.y		1	105.k	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(450, [\chi])$ :

T_{7}^{2} + 1156

T_{11} + 27

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$ T^2 + 4
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1156$ T^2 + 1156
$11$	$(T + 27)^{2}$ (T + 27)^2
$13$	$T^{2} + 784$ T^2 + 784
$17$	$T^{2} + 441$ T^2 + 441
$19$	$(T + 35)^{2}$ (T + 35)^2
$23$	$T^{2} + 6084$ T^2 + 6084
$29$	$(T + 120)^{2}$ (T + 120)^2
$31$	$(T - 182)^{2}$ (T - 182)^2
$37$	$T^{2} + 21316$ T^2 + 21316
$41$	$(T + 357)^{2}$ (T + 357)^2
$43$	$T^{2} + 21904$ T^2 + 21904
$47$	$T^{2} + 7056$ T^2 + 7056
$53$	$T^{2} + 492804$ T^2 + 492804
$59$	$(T + 840)^{2}$ (T + 840)^2
$61$	$(T + 238)^{2}$ (T + 238)^2
$67$	$T^{2} + 212521$ T^2 + 212521
$71$	$(T - 708)^{2}$ (T - 708)^2
$73$	$T^{2} + 17689$ T^2 + 17689
$79$	$(T + 650)^{2}$ (T + 650)^2
$83$	$T^{2} + 815409$ T^2 + 815409
$89$	$(T - 735)^{2}$ (T - 735)^2
$97$	$T^{2} + 1223236$ T^2 + 1223236
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