LMFDB - Newform orbit 400.6.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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([0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 4)
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 $\beta = 2i$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 6 \beta q^{3} - 44 \beta q^{7} + 99 q^{9} - 540 q^{11} - 209 \beta q^{13} - 297 \beta q^{17} + 836 q^{19} + 1056 q^{21} + 2052 \beta q^{23} + 2052 \beta q^{27} + 594 q^{29} - 4256 q^{31} - 3240 \beta q^{33} + \cdots - 53460 q^{99} +O(q^{100})$ q + 6b q^3 - 44b q^7 + 99 * q^9 - 540 * q^11 - 209b q^13 - 297b q^17 + 836 * q^19 + 1056 * q^21 + 2052b q^23 + 2052b q^27 + 594 * q^29 - 4256 * q^31 - 3240b q^33 + 149b q^37 + 5016 * q^39 + 17226 * q^41 + 6050b q^43 - 648b q^47 + 9063 * q^49 + 7128 * q^51 + 9747b q^53 + 5016b q^57 - 7668 * q^59 - 34738 * q^61 - 4356b q^63 + 10906b q^67 - 49248 * q^69 + 46872 * q^71 + 33781b q^73 + 23760b q^77 - 76912 * q^79 - 25191 * q^81 - 33858b q^83 + 3564b q^87 - 29754 * q^89 - 36784 * q^91 - 25536b q^93 + 61199b q^97 - 53460 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 198 q^{9} - 1080 q^{11} + 1672 q^{19} + 2112 q^{21} + 1188 q^{29} - 8512 q^{31} + 10032 q^{39} + 34452 q^{41} + 18126 q^{49} + 14256 q^{51} - 15336 q^{59} - 69476 q^{61} - 98496 q^{69} + 93744 q^{71}+ \cdots - 106920 q^{99}+O(q^{100})$ 2 * q + 198 * q^9 - 1080 * q^11 + 1672 * q^19 + 2112 * q^21 + 1188 * q^29 - 8512 * q^31 + 10032 * q^39 + 34452 * q^41 + 18126 * q^49 + 14256 * q^51 - 15336 * q^59 - 69476 * q^61 - 98496 * q^69 + 93744 * q^71 - 153824 * q^79 - 50382 * q^81 - 59508 * q^89 - 73568 * q^91 - 106920 * q^99

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}

49.1

	−	1.00000i
		1.00000i

−

12.0000i

88.0000i

99.0000

49.2

12.0000i

−

88.0000i

99.0000

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	400.6.c.f		2
4.b	odd	2	1		100.6.c.b		2
5.b	even	2	1	inner	400.6.c.f		2
5.c	odd	4	1		16.6.a.b		1
5.c	odd	4	1		400.6.a.d		1
12.b	even	2	1		900.6.d.a		2
15.e	even	4	1		144.6.a.c		1
20.d	odd	2	1		100.6.c.b		2
20.e	even	4	1		4.6.a.a	✓	1
20.e	even	4	1		100.6.a.b		1
35.f	even	4	1		784.6.a.d		1
40.i	odd	4	1		64.6.a.b		1
40.k	even	4	1		64.6.a.f		1
60.h	even	2	1		900.6.d.a		2
60.l	odd	4	1		36.6.a.a		1
60.l	odd	4	1		900.6.a.h		1
80.i	odd	4	1		256.6.b.c		2
80.j	even	4	1		256.6.b.g		2
80.s	even	4	1		256.6.b.g		2
80.t	odd	4	1		256.6.b.c		2
120.q	odd	4	1		576.6.a.bc		1
120.w	even	4	1		576.6.a.bd		1
140.j	odd	4	1		196.6.a.e		1
140.w	even	12	2		196.6.e.g		2
140.x	odd	12	2		196.6.e.d		2
180.v	odd	12	2		324.6.e.d		2
180.x	even	12	2		324.6.e.a		2
220.i	odd	4	1		484.6.a.a		1
260.l	odd	4	1		676.6.d.a		2
260.p	even	4	1		676.6.a.a		1
260.s	odd	4	1		676.6.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.6.a.a	✓	1	20.e	even	4	1
16.6.a.b		1	5.c	odd	4	1
36.6.a.a		1	60.l	odd	4	1
64.6.a.b		1	40.i	odd	4	1
64.6.a.f		1	40.k	even	4	1
100.6.a.b		1	20.e	even	4	1
100.6.c.b		2	4.b	odd	2	1
100.6.c.b		2	20.d	odd	2	1
144.6.a.c		1	15.e	even	4	1
196.6.a.e		1	140.j	odd	4	1
196.6.e.d		2	140.x	odd	12	2
196.6.e.g		2	140.w	even	12	2
256.6.b.c		2	80.i	odd	4	1
256.6.b.c		2	80.t	odd	4	1
256.6.b.g		2	80.j	even	4	1
256.6.b.g		2	80.s	even	4	1
324.6.e.a		2	180.x	even	12	2
324.6.e.d		2	180.v	odd	12	2
400.6.a.d		1	5.c	odd	4	1
400.6.c.f		2	1.a	even	1	1	trivial
400.6.c.f		2	5.b	even	2	1	inner
484.6.a.a		1	220.i	odd	4	1
576.6.a.bc		1	120.q	odd	4	1
576.6.a.bd		1	120.w	even	4	1
676.6.a.a		1	260.p	even	4	1
676.6.d.a		2	260.l	odd	4	1
676.6.d.a		2	260.s	odd	4	1
784.6.a.d		1	35.f	even	4	1
900.6.a.h		1	60.l	odd	4	1
900.6.d.a		2	12.b	even	2	1
900.6.d.a		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 144$ T3^2 + 144 acting on $S_{6}^{\mathrm{new}}(400, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 144$ T^2 + 144
$5$	$T^{2}$ T^2
$7$	$T^{2} + 7744$ T^2 + 7744
$11$	$(T + 540)^{2}$ (T + 540)^2
$13$	$T^{2} + 174724$ T^2 + 174724
$17$	$T^{2} + 352836$ T^2 + 352836
$19$	$(T - 836)^{2}$ (T - 836)^2
$23$	$T^{2} + 16842816$ T^2 + 16842816
$29$	$(T - 594)^{2}$ (T - 594)^2
$31$	$(T + 4256)^{2}$ (T + 4256)^2
$37$	$T^{2} + 88804$ T^2 + 88804
$41$	$(T - 17226)^{2}$ (T - 17226)^2
$43$	$T^{2} + 146410000$ T^2 + 146410000
$47$	$T^{2} + 1679616$ T^2 + 1679616
$53$	$T^{2} + 380016036$ T^2 + 380016036
$59$	$(T + 7668)^{2}$ (T + 7668)^2
$61$	$(T + 34738)^{2}$ (T + 34738)^2
$67$	$T^{2} + 475763344$ T^2 + 475763344
$71$	$(T - 46872)^{2}$ (T - 46872)^2
$73$	$T^{2} + 4564623844$ T^2 + 4564623844
$79$	$(T + 76912)^{2}$ (T + 76912)^2
$83$	$T^{2} + 4585456656$ T^2 + 4585456656
$89$	$(T + 29754)^{2}$ (T + 29754)^2
$97$	$T^{2} + 14981270404$ T^2 + 14981270404
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