LMFDB - Newform orbit 4000.2.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16$ x^12 + 22x^10 + 179x^8 + 646x^6 + 929x^4 + 252*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} - \beta_{11} q^{7} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{9} + (\beta_{4} + \beta_{3} + \beta_{2} - 2) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{13} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{17}+ \cdots + ( - 6 \beta_{6} - \beta_{4} - 4 \beta_{3} + \cdots + 4) q^{99}+O(q^{100})$ q + b1 * q^3 - b11 * q^7 + (b6 + b4 + b3) * q^9 + (b4 + b3 + b2 - 2) * q^11 + (-b11 + b10 + b9 + b7 - b1) * q^13 + (b11 + b10 + b7 - b5 + b1) * q^17 + (-b8 - 2b6 - b3 + b2 + 1) q^19 + (-b8 + b6 - b2 + 1) * q^21 + (-b10 - 2b9 - b1) q^23 + (b7 + 2b5) q^27 + (-b8 - b4 + b3 + b2) * q^29 + (-2b8 - 2b6 - b3 - b2 - 2) * q^31 + (-b11 - 3b9 + b7 - 4b1) * q^33 + (-b11 + 3b7 + 4b5) * q^37 + (-4b6 - 2b4 + b3 - b2 + 2) * q^39 + (-b4 + b3 + 2b2 - 1) q^41 + (2b10 + 2b9 - b7) * q^43 + (2b11 + b10 - b7 - 4b5) * q^47 + (b8 - 2b6 - 2b3 + b2 - 1) * q^49 + (2b8 - 4b6 + 3b3 - 6) q^51 + (-b11 - b10 + b9 + 3b7 - b5 - 3b1) * q^53 + (b10 - b9 - 5b5 + 3b1) * q^57 + (b8 - 2b6 + 2b4 + b3 + 2b2 + 3) q^59 + (3b8 - b6 - 2b3 - b2 - 3) * q^61 + (-b11 + 4b9 + b7 + 2b5 - b1) * q^63 + (2b11 + 2b9 - 3b7 + b1) q^67 + (-4b3 - 2b2 + 2) * q^69 + (-4b6 - 2b4 - 4b3 - b2 - 8) q^71 + (2b11 + b10 + b9 + 2b7 + 3b5 - b1) q^73 + (2b11 - 2b10 + 4b9 + 2b7 + 3b5) q^77 + (2b8 - 4b6 + 2b4 - 3b2) * q^79 + (b8 + 3b4 + b3 - b2 - 3) q^81 + (2b11 - 2b10 - 2b5 - 2b1) * q^83 + (-2b10 - 4b9 + 2b7 + 2b5 - b1) * q^87 + (-3b4 - b3 - 4b2) * q^89 + (b8 - 8b6 - b4 + 3b2 - 9) * q^91 + (3b11 + b10 + 6b9 - b7 - b5 - b1) * q^93 + (-2b11 - b10 - 5b9 + 2b7 - 5b5 - b1) * q^97 + (-6b6 - b4 - 4b3 - 2b2 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 8 q^{9} - 26 q^{11} + 18 q^{19} - 4 q^{29} - 24 q^{31} + 52 q^{39} - 10 q^{41} + 6 q^{49} - 36 q^{51} + 50 q^{59} - 12 q^{61} + 24 q^{69} - 68 q^{71} + 32 q^{79} - 36 q^{81} + 6 q^{89} - 52 q^{91}+ \cdots + 86 q^{99}+O(q^{100})$ 12 * q - 8 * q^9 - 26 * q^11 + 18 * q^19 - 4 * q^29 - 24 * q^31 + 52 * q^39 - 10 * q^41 + 6 * q^49 - 36 * q^51 + 50 * q^59 - 12 * q^61 + 24 * q^69 - 68 * q^71 + 32 * q^79 - 36 * q^81 + 6 * q^89 - 52 * q^91 + 86 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16$ x^12 + 22*x^10 + 179*x^8 + 646*x^6 + 929*x^4 + 252*x^2 + 16 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -2\nu^{10} - 33\nu^{8} - 165\nu^{6} - 189\nu^{4} + 274\nu^{2} + 36 ) / 46$ (-2v^10 - 33v^8 - 165v^6 - 189v^4 + 274*v^2 + 36) / 46
$\beta_{3}$	$=$	$( 4\nu^{10} + 66\nu^{8} + 353\nu^{6} + 631\nu^{4} + 119\nu^{2} + 20 ) / 46$ (4v^10 + 66v^8 + 353v^6 + 631v^4 + 119*v^2 + 20) / 46
$\beta_{4}$	$=$	$( -3\nu^{10} - 38\nu^{8} - 75\nu^{6} + 556\nu^{4} + 1883\nu^{2} + 560 ) / 92$ (-3v^10 - 38v^8 - 75v^6 + 556v^4 + 1883*v^2 + 560) / 92
$\beta_{5}$	$=$	$( 5\nu^{11} + 94\nu^{9} + 631\nu^{7} + 1818\nu^{5} + 2121\nu^{3} + 784\nu ) / 184$ (5v^11 + 94v^9 + 631v^7 + 1818v^5 + 2121v^3 + 784v) / 184
$\beta_{6}$	$=$	$( -5\nu^{10} - 94\nu^{8} - 631\nu^{6} - 1818\nu^{4} - 2029\nu^{2} - 324 ) / 92$ (-5v^10 - 94v^8 - 631v^6 - 1818v^4 - 2029*v^2 - 324) / 92
$\beta_{7}$	$=$	$( -5\nu^{11} - 94\nu^{9} - 631\nu^{7} - 1818\nu^{5} - 2029\nu^{3} - 232\nu ) / 92$ (-5v^11 - 94v^9 - 631v^7 - 1818v^5 - 2029v^3 - 232v) / 92
$\beta_{8}$	$=$	$( -3\nu^{10} - 84\nu^{8} - 811\nu^{6} - 3216\nu^{4} - 4511\nu^{2} - 544 ) / 92$ (-3v^10 - 84v^8 - 811v^6 - 3216v^4 - 4511*v^2 - 544) / 92
$\beta_{9}$	$=$	$( -7\nu^{11} - 150\nu^{9} - 1187\nu^{7} - 4146\nu^{5} - 5619\nu^{3} - 978\nu ) / 92$ (-7v^11 - 150v^9 - 1187v^7 - 4146v^5 - 5619v^3 - 978v) / 92
$\beta_{10}$	$=$	$( 13\nu^{11} + 318\nu^{9} + 2855\nu^{7} + 11222\nu^{5} + 17125\nu^{3} + 4044\nu ) / 184$ (13v^11 + 318v^9 + 2855v^7 + 11222v^5 + 17125v^3 + 4044v) / 184
$\beta_{11}$	$=$	$( 15\nu^{11} + 328\nu^{9} + 2629\nu^{7} + 9180\nu^{5} + 12159\nu^{3} + 1846\nu ) / 92$ (15v^11 + 328v^9 + 2629v^7 + 9180v^5 + 12159v^3 + 1846v) / 92

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{6} + \beta_{4} + \beta_{3} - 3$ b6 + b4 + b3 - 3
$\nu^{3}$	$=$	$\beta_{7} + 2\beta_{5} - 6\beta_1$ b7 + 2b5 - 6b1
$\nu^{4}$	$=$	$\beta_{8} - 9\beta_{6} - 6\beta_{4} - 8\beta_{3} - \beta_{2} + 15$ b8 - 9b6 - 6b4 - 8*b3 - b2 + 15
$\nu^{5}$	$=$	$2\beta_{10} + 4\beta_{9} - 11\beta_{7} - 16\beta_{5} + 39\beta_1$ 2b10 + 4b9 - 11b7 - 16b5 + 39*b1
$\nu^{6}$	$=$	$-11\beta_{8} + 70\beta_{6} + 37\beta_{4} + 61\beta_{3} + 15\beta_{2} - 82$ -11b8 + 70b6 + 37b4 + 61b3 + 15*b2 - 82
$\nu^{7}$	$=$	$-4\beta_{11} - 24\beta_{10} - 58\beta_{9} + 96\beta_{7} + 116\beta_{5} - 261\beta_1$ -4b11 - 24b10 - 58b9 + 96b7 + 116b5 - 261b1
$\nu^{8}$	$=$	$92\beta_{8} - 521\beta_{6} - 237\beta_{4} - 459\beta_{3} - 158\beta_{2} + 475$ 92b8 - 521b6 - 237b4 - 459b3 - 158*b2 + 475
$\nu^{9}$	$=$	$66\beta_{11} + 222\beta_{10} + 604\beta_{9} - 771\beta_{7} - 824\beta_{5} + 1784\beta_1$ 66b11 + 222b10 + 604b9 - 771b7 - 824b5 + 1784b1
$\nu^{10}$	$=$	$-705\beta_{8} + 3809\beta_{6} + 1562\beta_{4} + 3434\beta_{3} + 1441\beta_{2} - 2883$ -705b8 + 3809b6 + 1562b4 + 3434b3 + 1441*b2 - 2883
$\nu^{11}$	$=$	$-736\beta_{11} - 1872\beta_{10} - 5490\beta_{9} + 5955\beta_{7} + 5858\beta_{5} - 12393\beta_1$ -736b11 - 1872b10 - 5490b9 + 5955b7 + 5858b5 - 12393b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$ .

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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}

1249.1

	−	2.71210i
	−	2.40554i
	−	2.29226i
	−	1.81030i
	−	0.481965i
	−	0.306558i
		0.306558i
		0.481965i
		1.81030i
		2.29226i
		2.40554i
		2.71210i

−

2.71210i

0.0156607i

−4.35547

1249.2

−

2.40554i

−

2.20893i

−2.78662

1249.3

−

2.29226i

−

0.938845i

−2.25447

1249.4

−

1.81030i

4.37760i

−0.277175

1249.5

−

0.481965i

−

2.69841i

2.76771

1249.6

−

0.306558i

2.60656i

2.90602

1249.7

0.306558i

−

2.60656i

2.90602

1249.8

0.481965i

2.69841i

2.76771

1249.9

1.81030i

−

4.37760i

−0.277175

1249.10

2.29226i

0.938845i

−2.25447

1249.11

2.40554i

2.20893i

−2.78662

1249.12

2.71210i

−

0.0156607i

−4.35547

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	4000.2.c.f		12
4.b	odd	2	1		4000.2.c.g		12
5.b	even	2	1	inner	4000.2.c.f		12
5.c	odd	4	1		4000.2.a.k	✓	6
5.c	odd	4	1		4000.2.a.m	yes	6
20.d	odd	2	1		4000.2.c.g		12
20.e	even	4	1		4000.2.a.l	yes	6
20.e	even	4	1		4000.2.a.n	yes	6
40.i	odd	4	1		8000.2.a.bv		6
40.i	odd	4	1		8000.2.a.bx		6
40.k	even	4	1		8000.2.a.bu		6
40.k	even	4	1		8000.2.a.bw		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.k	✓	6	5.c	odd	4	1
4000.2.a.l	yes	6	20.e	even	4	1
4000.2.a.m	yes	6	5.c	odd	4	1
4000.2.a.n	yes	6	20.e	even	4	1
4000.2.c.f		12	1.a	even	1	1	trivial
4000.2.c.f		12	5.b	even	2	1	inner
4000.2.c.g		12	4.b	odd	2	1
4000.2.c.g		12	20.d	odd	2	1
8000.2.a.bu		6	40.k	even	4	1
8000.2.a.bv		6	40.i	odd	4	1
8000.2.a.bw		6	40.k	even	4	1
8000.2.a.bx		6	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4000, [\chi])$ :

T_{3}^{12} + 22T_{3}^{10} + 179T_{3}^{8} + 646T_{3}^{6} + 929T_{3}^{4} + 252T_{3}^{2} + 16

T_{7}^{12} + 39T_{7}^{10} + 515T_{7}^{8} + 2930T_{7}^{6} + 6835T_{7}^{4} + 4079T_{7}^{2} + 1

T_{11}^{6} + 13T_{11}^{5} + 41T_{11}^{4} - 50T_{11}^{3} - 325T_{11}^{2} - 375T_{11} - 125

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12} + 22 T^{10} + \cdots + 16$ T^12 + 22T^10 + 179T^8 + 646T^6 + 929T^4 + 252*T^2 + 16
$5$	$T^{12}$ T^12
$7$	$T^{12} + 39 T^{10} + \cdots + 1$ T^12 + 39T^10 + 515T^8 + 2930T^6 + 6835T^4 + 4079*T^2 + 1
$11$	$(T^{6} + 13 T^{5} + \cdots - 125)^{2}$ (T^6 + 13T^5 + 41T^4 - 50T^3 - 325T^2 - 375*T - 125)^2
$13$	$T^{12} + 117 T^{10} + \cdots + 390625$ T^12 + 117T^10 + 4886T^8 + 88225T^6 + 653750T^4 + 1703125*T^2 + 390625
$17$	$T^{12} + 148 T^{10} + \cdots + 4000000$ T^12 + 148T^10 + 8326T^8 + 216900T^6 + 2466625T^4 + 8150000*T^2 + 4000000
$19$	$(T^{6} - 9 T^{5} + \cdots + 125)^{2}$ (T^6 - 9T^5 - 16T^4 + 285T^3 - 300T^2 - 1125*T + 125)^2
$23$	$T^{12} + 71 T^{10} + \cdots + 4096$ T^12 + 71T^10 + 1625T^8 + 14160T^6 + 41920T^4 + 39936*T^2 + 4096
$29$	$(T^{6} + 2 T^{5} + \cdots - 2636)^{2}$ (T^6 + 2T^5 - 54T^4 - 64T^3 + 841T^2 + 310*T - 2636)^2
$31$	$(T^{6} + 12 T^{5} + \cdots - 29500)^{2}$ (T^6 + 12T^5 - 59T^4 - 700T^3 + 2125T^2 + 9750*T - 29500)^2
$37$	$T^{12} + 303 T^{10} + \cdots + 64000000$ T^12 + 303T^10 + 34601T^8 + 1785200T^6 + 37016000T^4 + 118400000*T^2 + 64000000
$41$	$(T^{6} + 5 T^{5} + \cdots + 10475)^{2}$ (T^6 + 5T^5 - 75T^4 - 450T^3 + 695T^2 + 7525*T + 10475)^2
$43$	$T^{12} + \cdots + 588353536$ T^12 + 212T^10 + 17134T^8 + 679716T^6 + 14074009T^4 + 145243072*T^2 + 588353536
$47$	$T^{12} + 249 T^{10} + \cdots + 6497401$ T^12 + 249T^10 + 21250T^8 + 742065T^6 + 9569570T^4 + 15368089*T^2 + 6497401
$53$	$T^{12} + 333 T^{10} + \cdots + 97515625$ T^12 + 333T^10 + 39086T^8 + 1876825T^6 + 31334750T^4 + 105428125*T^2 + 97515625
$59$	$(T^{6} - 25 T^{5} + \cdots - 59375)^{2}$ (T^6 - 25T^5 + 80T^4 + 2425T^3 - 24500T^2 + 76875*T - 59375)^2
$61$	$(T^{6} + 6 T^{5} + \cdots - 32220)^{2}$ (T^6 + 6T^5 - 181T^4 - 390T^3 + 7205T^2 + 5220*T - 32220)^2
$67$	$T^{12} + 318 T^{10} + \cdots + 28772496$ T^12 + 318T^10 + 32299T^8 + 1131214T^6 + 13464769T^4 + 54242028*T^2 + 28772496
$71$	$(T^{6} + 34 T^{5} + \cdots - 372500)^{2}$ (T^6 + 34T^5 + 274T^4 - 2280T^3 - 45275T^2 - 228250*T - 372500)^2
$73$	$T^{12} + 490 T^{10} + \cdots + 6250000$ T^12 + 490T^10 + 85275T^8 + 6191250T^6 + 157415625T^4 + 148437500*T^2 + 6250000
$79$	$(T^{6} - 16 T^{5} + \cdots + 174500)^{2}$ (T^6 - 16T^5 - 206T^4 + 4100T^3 - 2275T^2 - 139500*T + 174500)^2
$83$	$T^{12} + \cdots + 2123366400$ T^12 + 304T^10 + 32896T^8 + 1644544T^6 + 40800256T^4 + 480706560*T^2 + 2123366400
$89$	$(T^{6} - 3 T^{5} + \cdots - 142144)^{2}$ (T^6 - 3T^5 - 361T^4 + 136T^3 + 27344T^2 - 768*T - 142144)^2
$97$	$T^{12} + \cdots + 31506250000$ T^12 + 490T^10 + 87275T^8 + 7271250T^6 + 294915625T^4 + 5359687500*T^2 + 31506250000
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1249.12
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