LMFDB - Newform orbit 400.6.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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, 0]))

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.a (trivial)

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:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 11$ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $\beta = 2\sqrt{11}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 \beta q^{3} - 9 \beta q^{7} + 153 q^{9} - 252 q^{11} - 18 \beta q^{13} + 104 \beta q^{17} - 220 q^{19} - 1188 q^{21} - 367 \beta q^{23} - 270 \beta q^{27} + 6930 q^{29} - 6752 q^{31} - 756 \beta q^{33} + \cdots - 38556 q^{99} +O(q^{100})$ q + 3b q^3 - 9b q^7 + 153 * q^9 - 252 * q^11 - 18b q^13 + 104b q^17 - 220 * q^19 - 1188 * q^21 - 367b q^23 - 270b q^27 + 6930 * q^29 - 6752 * q^31 - 756b q^33 - 2106b q^37 - 2376 * q^39 - 198 * q^41 + 63b q^43 - 1589b q^47 - 13243 * q^49 + 13728 * q^51 - 878b q^53 - 660b q^57 - 24660 * q^59 - 5698 * q^61 - 1377b q^63 - 6579b q^67 - 48444 * q^69 - 53352 * q^71 + 10692b q^73 + 2268b q^77 + 51920 * q^79 - 72819 * q^81 + 9323b q^83 + 20790b q^87 + 9990 * q^89 + 7128 * q^91 - 20256b q^93 + 15264b q^97 - 38556 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 306 q^{9} - 504 q^{11} - 440 q^{19} - 2376 q^{21} + 13860 q^{29} - 13504 q^{31} - 4752 q^{39} - 396 q^{41} - 26486 q^{49} + 27456 q^{51} - 49320 q^{59} - 11396 q^{61} - 96888 q^{69} - 106704 q^{71}+ \cdots - 77112 q^{99}+O(q^{100})$ 2 * q + 306 * q^9 - 504 * q^11 - 440 * q^19 - 2376 * q^21 + 13860 * q^29 - 13504 * q^31 - 4752 * q^39 - 396 * q^41 - 26486 * q^49 + 27456 * q^51 - 49320 * q^59 - 11396 * q^61 - 96888 * q^69 - 106704 * q^71 + 103840 * q^79 - 145638 * q^81 + 19980 * q^89 + 14256 * q^91 - 77112 * q^99

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}

1.1

−3.31662
3.31662

−19.8997

59.6992

153.000

1.2

19.8997

−59.6992

153.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$

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.a.t		2
4.b	odd	2	1		25.6.a.c		2
5.b	even	2	1	inner	400.6.a.t		2
5.c	odd	4	2		80.6.c.a		2
12.b	even	2	1		225.6.a.n		2
15.e	even	4	2		720.6.f.f		2
20.d	odd	2	1		25.6.a.c		2
20.e	even	4	2		5.6.b.a	✓	2
40.i	odd	4	2		320.6.c.g		2
40.k	even	4	2		320.6.c.f		2
60.h	even	2	1		225.6.a.n		2
60.l	odd	4	2		45.6.b.b		2
140.j	odd	4	2		245.6.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.6.b.a	✓	2	20.e	even	4	2
25.6.a.c		2	4.b	odd	2	1
25.6.a.c		2	20.d	odd	2	1
45.6.b.b		2	60.l	odd	4	2
80.6.c.a		2	5.c	odd	4	2
225.6.a.n		2	12.b	even	2	1
225.6.a.n		2	60.h	even	2	1
245.6.b.a		2	140.j	odd	4	2
320.6.c.f		2	40.k	even	4	2
320.6.c.g		2	40.i	odd	4	2
400.6.a.t		2	1.a	even	1	1	trivial
400.6.a.t		2	5.b	even	2	1	inner
720.6.f.f		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} - 396$ T3^2 - 396 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 396$ T^2 - 396
$5$	$T^{2}$ T^2
$7$	$T^{2} - 3564$ T^2 - 3564
$11$	$(T + 252)^{2}$ (T + 252)^2
$13$	$T^{2} - 14256$ T^2 - 14256
$17$	$T^{2} - 475904$ T^2 - 475904
$19$	$(T + 220)^{2}$ (T + 220)^2
$23$	$T^{2} - 5926316$ T^2 - 5926316
$29$	$(T - 6930)^{2}$ (T - 6930)^2
$31$	$(T + 6752)^{2}$ (T + 6752)^2
$37$	$T^{2} - 195150384$ T^2 - 195150384
$41$	$(T + 198)^{2}$ (T + 198)^2
$43$	$T^{2} - 174636$ T^2 - 174636
$47$	$T^{2} - 111096524$ T^2 - 111096524
$53$	$T^{2} - 33918896$ T^2 - 33918896
$59$	$(T + 24660)^{2}$ (T + 24660)^2
$61$	$(T + 5698)^{2}$ (T + 5698)^2
$67$	$T^{2} - 1904462604$ T^2 - 1904462604
$71$	$(T + 53352)^{2}$ (T + 53352)^2
$73$	$T^{2} - 5030030016$ T^2 - 5030030016
$79$	$(T - 51920)^{2}$ (T - 51920)^2
$83$	$T^{2} - 3824406476$ T^2 - 3824406476
$89$	$(T - 9990)^{2}$ (T - 9990)^2
$97$	$T^{2} - 10251546624$ T^2 - 10251546624
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