LMFDB - Newform orbit 400.6.c.o

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:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 121x^{2} + 3600$ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 200)
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.

$f(q)$	$=$	$q + ( - \beta_{2} - 4 \beta_1) q^{3} + ( - 2 \beta_{2} - 4 \beta_1) q^{7} + ( - 8 \beta_{3} - 14) q^{9} + (21 \beta_{3} + 100) q^{11} + ( - 52 \beta_{2} - 296 \beta_1) q^{13} + (16 \beta_{2} - 139 \beta_1) q^{17}+ \cdots + ( - 1094 \beta_{3} - 41888) q^{99}+O(q^{100})$ q + (-b2 - 4b1) q^3 + (-2b2 - 4b1) * q^7 + (-8b3 - 14) q^9 + (21b3 + 100) q^11 + (-52b2 - 296b1) * q^13 + (16b2 - 139b1) * q^17 + (59b3 - 420) q^19 + (-12b3 - 498) q^21 + (98b2 + 976b1) * q^23 + (-197b2 + 1012b1) * q^27 + (-148b3 + 2340) q^29 + (354b3 + 2504) q^31 + (-184b2 - 5461b1) * q^33 + (-484b2 - 6250b1) * q^37 + (-504b3 - 13716) q^39 + (936b3 - 2667) q^41 + (-1076b2 - 112b1) * q^43 + 13036b1 q^47 + (-16b3 + 15827) q^49 + (-75b3 + 3300) q^51 + (360b2 - 23406b1) * q^53 + (184b2 - 12539b1) * q^57 + (-416b3 - 40888) q^59 + (44b3 - 23466) q^61 + (60b2 + 3912b1) * q^63 + (323b2 + 34404b1) * q^67 + (1368b3 + 27522) q^69 + (344b3 - 3724) q^71 + (72b2 - 54411b1) * q^73 + (-284b2 - 10522b1) * q^77 + (3062b3 - 54052) q^79 + (-1720b3 - 46831) q^81 + (3841b2 - 13612b1) * q^83 + (-1748b2 + 26308b1) * q^87 + (-5024b3 - 35495) q^89 + (-800b3 - 26248) q^91 + (-3920b2 - 95330b1) * q^93 + (3800b2 + 48426b1) * q^97 + (-1094b3 - 41888) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 56 q^{9} + 400 q^{11} - 1680 q^{19} - 1992 q^{21} + 9360 q^{29} + 10016 q^{31} - 54864 q^{39} - 10668 q^{41} + 63308 q^{49} + 13200 q^{51} - 163552 q^{59} - 93864 q^{61} + 110088 q^{69} - 14896 q^{71}+ \cdots - 167552 q^{99}+O(q^{100})$ 4 * q - 56 * q^9 + 400 * q^11 - 1680 * q^19 - 1992 * q^21 + 9360 * q^29 + 10016 * q^31 - 54864 * q^39 - 10668 * q^41 + 63308 * q^49 + 13200 * q^51 - 163552 * q^59 - 93864 * q^61 + 110088 * q^69 - 14896 * q^71 - 216208 * q^79 - 187324 * q^81 - 141980 * q^89 - 104992 * q^91 - 167552 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 121x^{2} + 3600$ x^4 + 121*x^2 + 3600 :

$\beta_{1}$	$=$	$( \nu^{3} + 61\nu ) / 60$ (v^3 + 61*v) / 60
$\beta_{2}$	$=$	$( \nu^{3} + 181\nu ) / 60$ (v^3 + 181*v) / 60
$\beta_{3}$	$=$	$2\nu^{2} + 121$ 2*v^2 + 121

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

$\nu$	$=$	$( \beta_{2} - \beta_1 ) / 2$ (b2 - b1) / 2
$\nu^{2}$	$=$	$( \beta_{3} - 121 ) / 2$ (b3 - 121) / 2
$\nu^{3}$	$=$	$( -61\beta_{2} + 181\beta_1 ) / 2$ (-61b2 + 181b1) / 2

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}

49.1

		7.26209i
		8.26209i
	−	8.26209i
	−	7.26209i

−

19.5242i

−

35.0483i

−138.193

49.2

−

11.5242i

−

27.0483i

110.193

49.3

11.5242i

27.0483i

110.193

49.4

19.5242i

35.0483i

−138.193

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.o		4
4.b	odd	2	1		200.6.c.f		4
5.b	even	2	1	inner	400.6.c.o		4
5.c	odd	4	1		400.6.a.r		2
5.c	odd	4	1		400.6.a.u		2
20.d	odd	2	1		200.6.c.f		4
20.e	even	4	1		200.6.a.e	✓	2
20.e	even	4	1		200.6.a.f	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.6.a.e	✓	2	20.e	even	4	1
200.6.a.f	yes	2	20.e	even	4	1
200.6.c.f		4	4.b	odd	2	1
200.6.c.f		4	20.d	odd	2	1
400.6.a.r		2	5.c	odd	4	1
400.6.a.u		2	5.c	odd	4	1
400.6.c.o		4	1.a	even	1	1	trivial
400.6.c.o		4	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} + 514T_{3}^{2} + 50625$ T3^4 + 514*T3^2 + 50625 acting on $S_{6}^{\mathrm{new}}(400, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 514 T^{2} + 50625$ T^4 + 514*T^2 + 50625
$5$	$T^{4}$ T^4
$7$	$T^{4} + 1960 T^{2} + 898704$ T^4 + 1960*T^2 + 898704
$11$	$(T^{2} - 200 T - 96281)^{2}$ (T^2 - 200*T - 96281)^2
$13$	$T^{4} + \cdots + 318150146304$ T^4 + 1478560*T^2 + 318150146304
$17$	$T^{4} + \cdots + 1795640625$ T^4 + 162034*T^2 + 1795640625
$19$	$(T^{2} + 840 T - 662521)^{2}$ (T^2 + 840*T - 662521)^2
$23$	$T^{4} + \cdots + 1855011312144$ T^4 + 6534280*T^2 + 1855011312144
$29$	$(T^{2} - 4680 T + 196736)^{2}$ (T^2 - 4680*T + 196736)^2
$31$	$(T^{2} - 5008 T - 23931140)^{2}$ (T^2 - 5008*T - 23931140)^2
$37$	$T^{4} + \cdots + 302523267094416$ T^4 + 191036392*T^2 + 302523267094416
$41$	$(T^{2} + 5334 T - 204026247)^{2}$ (T^2 + 5334*T - 204026247)^2
$43$	$T^{4} + \cdots + 77\!\cdots\!84$ T^4 + 558073120*T^2 + 77847401507606784
$47$	$(T^{2} + 169937296)^{2}$ (T^2 + 169937296)^2
$53$	$T^{4} + \cdots + 26\!\cdots\!96$ T^4 + 1158148872*T^2 + 266883036287559696
$59$	$(T^{2} + 81776 T + 1630122048)^{2}$ (T^2 + 81776*T + 1630122048)^2
$61$	$(T^{2} + 46932 T + 550186580)^{2}$ (T^2 + 46932*T + 550186580)^2
$67$	$T^{4} + \cdots + 13\!\cdots\!29$ T^4 + 2417557010*T^2 + 1342103544924173329
$71$	$(T^{2} + 7448 T - 14650800)^{2}$ (T^2 + 7448*T - 14650800)^2
$73$	$T^{4} + \cdots + 87\!\cdots\!29$ T^4 + 5923612530*T^2 + 8757501335289610929
$79$	$(T^{2} + 108104 T + 662040300)^{2}$ (T^2 + 108104*T + 662040300)^2
$83$	$T^{4} + \cdots + 11\!\cdots\!29$ T^4 + 7481654530*T^2 + 11358613217585947329
$89$	$(T^{2} + 70990 T - 4823083791)^{2}$ (T^2 + 70990*T - 4823083791)^2
$97$	$T^{4} + \cdots + 12\!\cdots\!76$ T^4 + 11650234952*T^2 + 1288139930884450576
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