LMFDB - Newform orbit 4000.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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, 0]))

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

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

chi := DirichletCharacter("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.26208800000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5$ x^8 - 11x^6 + 34x^4 - 30*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
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 a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{4} q^{3} - \beta_{6} q^{7} + ( - \beta_{5} + 1) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{11} + (\beta_{3} + 1) q^{13} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{17} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_1) q^{19}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100})$ q - b4 * q^3 - b6 * q^7 + (-b5 + 1) * q^9 + (-b7 - b6 - b4) * q^11 + (b3 + 1) * q^13 + (-b5 - b3 - b2 + 2) * q^17 + (b7 - b6 - b4 - b1) * q^19 + (b5 - b2 - 2) * q^21 + (-2b7 - b6 + 2b4 - b1) * q^23 + (b6 - b4) * q^27 + (b5 + b3 - 2) * q^29 + (-b7 - b6 - b4 + 2b1) q^31 + (-b5 - b3 + 3b2 + 6) q^33 + (-b5 - b2 + 3) * q^37 + (3b7 + b6 - 3b4 - b1) * q^39 + (-b5 + b3 + 2b2 + 2) q^41 + (b7 + b6 - 2b4 + b1) q^43 + (2b7 - b4 + 2b1) * q^47 + (b5 - b3 - 2b2 + 1) q^49 + (-2b7 - 4b4 + b1) * q^51 + (4b2 + 8) q^53 + (b5 + 2b3 - 7b2 - 1) * q^57 + (3b7 - b6 - 5b4 + 2b1) q^59 + (-2b5 - 2b3 + b2 - 3) * q^61 + (b7 + 2b6 + 4b4) * q^63 + (-5b7 + b6 + 3b4 - 2b1) q^67 + (b5 - b3 + 5b2 - 1) q^69 + (-b7 + b6 + b4 - 3b1) q^71 + (2b5 + b3 + 6b2 + 3) * q^73 + (3b5 - 5b2 + 3) * q^77 + (-2b7 - b1) q^79 + (b5 + b2 + 3) * q^81 + (4b7 - b6 - 2b4 + b1) * q^83 + (3b7 + 3b4 - b1) * q^87 + (-3b2 + 1) q^89 + (-4b7 + 2b6 + 6b4 - 3b1) * q^91 + (-b5 - 3b3 + 7b2 + 4) * q^93 + (2b5 - b3 - 2b2 + 1) * q^97 + (-3b7 + 3b6 - b4 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 8 q^{9} + 8 q^{13} + 20 q^{17} - 12 q^{21} - 16 q^{29} + 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} + 20 q^{57} - 28 q^{61} - 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93}+ \cdots + 16 q^{97}+O(q^{100})$ 8 * q + 8 * q^9 + 8 * q^13 + 20 * q^17 - 12 * q^21 - 16 * q^29 + 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 + 20 * q^57 - 28 * q^61 - 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 + 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5$ x^8 - 11*x^6 + 34*x^4 - 30*x^2 + 5 :

$\beta_{1}$	$=$	$2\nu$ 2*v
$\beta_{2}$	$=$	$( -\nu^{6} + 12\nu^{4} - 37\nu^{2} + 13 ) / 9$ (-v^6 + 12v^4 - 37v^2 + 13) / 9
$\beta_{3}$	$=$	$( -2\nu^{6} + 18\nu^{4} - 32\nu^{2} - 1 ) / 3$ (-2v^6 + 18v^4 - 32*v^2 - 1) / 3
$\beta_{4}$	$=$	$( 2\nu^{7} - 24\nu^{5} + 83\nu^{3} - 71\nu ) / 9$ (2v^7 - 24v^5 + 83v^3 - 71v) / 9
$\beta_{5}$	$=$	$( 7\nu^{6} - 66\nu^{4} + 151\nu^{2} - 64 ) / 9$ (7v^6 - 66v^4 + 151*v^2 - 64) / 9
$\beta_{6}$	$=$	$( -2\nu^{7} + 21\nu^{5} - 56\nu^{3} + 26\nu ) / 3$ (-2v^7 + 21v^5 - 56v^3 + 26v) / 3
$\beta_{7}$	$=$	$( 2\nu^{7} - 21\nu^{5} + 59\nu^{3} - 41\nu ) / 3$ (2v^7 - 21v^5 + 59v^3 - 41v) / 3

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

$\nu$	$=$	$( \beta_1 ) / 2$ (b1) / 2
$\nu^{2}$	$=$	$( \beta_{5} + \beta_{3} + \beta_{2} + 6 ) / 2$ (b5 + b3 + b2 + 6) / 2
$\nu^{3}$	$=$	$( 2\beta_{7} + 2\beta_{6} + 5\beta_1 ) / 2$ (2b7 + 2b6 + 5*b1) / 2
$\nu^{4}$	$=$	$( 7\beta_{5} + 6\beta_{3} + 13\beta_{2} + 33 ) / 2$ (7b5 + 6b3 + 13*b2 + 33) / 2
$\nu^{5}$	$=$	$9\beta_{7} + 8\beta_{6} - 3\beta_{4} + 15\beta_1$ 9b7 + 8b6 - 3b4 + 15b1
$\nu^{6}$	$=$	$( 47\beta_{5} + 35\beta_{3} + 101\beta_{2} + 200 ) / 2$ (47b5 + 35b3 + 101*b2 + 200) / 2
$\nu^{7}$	$=$	$( 133\beta_{7} + 109\beta_{6} - 63\beta_{4} + 188\beta_1 ) / 2$ (133b7 + 109b6 - 63b4 + 188b1) / 2

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

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

−0.464606
1.81062
−1.05053
2.53025
−2.53025
1.05053
−1.81062
0.464606

−2.79703

2.30294

4.82335

1.2

−2.74204

1.42258

4.51880

1.3

−0.693685

−4.52199

−2.51880

1.4

−0.420293

−2.86781

−2.82335

1.5

0.420293

2.86781

−2.82335

1.6

0.693685

4.52199

−2.51880

1.7

2.74204

−1.42258

4.51880

1.8

2.79703

−2.30294

4.82335

Atkin-Lehner signs

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

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.2.a.p	yes	8
4.b	odd	2	1	inner	4000.2.a.p	yes	8
5.b	even	2	1		4000.2.a.o	✓	8
5.c	odd	4	2		4000.2.c.i		16
8.b	even	2	1		8000.2.a.by		8
8.d	odd	2	1		8000.2.a.by		8
20.d	odd	2	1		4000.2.a.o	✓	8
20.e	even	4	2		4000.2.c.i		16
40.e	odd	2	1		8000.2.a.bz		8
40.f	even	2	1		8000.2.a.bz		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.o	✓	8	5.b	even	2	1
4000.2.a.o	✓	8	20.d	odd	2	1
4000.2.a.p	yes	8	1.a	even	1	1	trivial
4000.2.a.p	yes	8	4.b	odd	2	1	inner
4000.2.c.i		16	5.c	odd	4	2
4000.2.c.i		16	20.e	even	4	2
8000.2.a.by		8	8.b	even	2	1
8000.2.a.by		8	8.d	odd	2	1
8000.2.a.bz		8	40.e	odd	2	1
8000.2.a.bz		8	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4000))$ :

T_{3}^{8} - 16T_{3}^{6} + 69T_{3}^{4} - 40T_{3}^{2} + 5

T_{7}^{8} - 36T_{7}^{6} + 389T_{7}^{4} - 1540T_{7}^{2} + 1805

T_{11}^{8} - 64T_{11}^{6} + 1344T_{11}^{4} - 9280T_{11}^{2} + 1280

T_{13}^{4} - 4T_{13}^{3} - 32T_{13}^{2} + 72T_{13} + 144

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} - 16 T^{6} + \cdots + 5$ T^8 - 16T^6 + 69T^4 - 40*T^2 + 5
$5$	$T^{8}$ T^8
$7$	$T^{8} - 36 T^{6} + \cdots + 1805$ T^8 - 36T^6 + 389T^4 - 1540*T^2 + 1805
$11$	$T^{8} - 64 T^{6} + \cdots + 1280$ T^8 - 64T^6 + 1344T^4 - 9280*T^2 + 1280
$13$	$(T^{4} - 4 T^{3} + \cdots + 144)^{2}$ (T^4 - 4T^3 - 32T^2 + 72*T + 144)^2
$17$	$(T^{4} - 10 T^{3} + \cdots - 304)^{2}$ (T^4 - 10T^3 - 8T^2 + 240*T - 304)^2
$19$	$T^{8} - 140 T^{6} + \cdots + 800000$ T^8 - 140T^6 + 6720T^4 - 128000*T^2 + 800000
$23$	$T^{8} - 116 T^{6} + \cdots + 5$ T^8 - 116T^6 + 3189T^4 - 260*T^2 + 5
$29$	$(T^{4} + 8 T^{3} - 19 T^{2} + \cdots + 25)^{2}$ (T^4 + 8T^3 - 19T^2 - 140*T + 25)^2
$31$	$T^{8} - 224 T^{6} + \cdots + 8398080$ T^8 - 224T^6 + 18304T^4 - 648000*T^2 + 8398080
$37$	$(T^{4} - 14 T^{3} + \cdots - 80)^{2}$ (T^4 - 14T^3 + 44T^2 + 40*T - 80)^2
$41$	$(T^{4} - 4 T^{3} + \cdots - 695)^{2}$ (T^4 - 4T^3 - 91T^2 + 560*T - 695)^2
$43$	$T^{8} - 104 T^{6} + \cdots + 10125$ T^8 - 104T^6 + 3109T^4 - 28800*T^2 + 10125
$47$	$T^{8} - 176 T^{6} + \cdots + 96605$ T^8 - 176T^6 + 7269T^4 - 86960*T^2 + 96605
$53$	$(T^{2} - 12 T + 16)^{4}$ (T^2 - 12*T + 16)^4
$59$	$T^{8} - 336 T^{6} + \cdots + 1280$ T^8 - 336T^6 + 33344T^4 - 964160*T^2 + 1280
$61$	$(T^{4} + 14 T^{3} + \cdots - 355)^{2}$ (T^4 + 14T^3 - 101T^2 - 1350*T - 355)^2
$67$	$T^{8} - 536 T^{6} + \cdots + 205825280$ T^8 - 536T^6 + 100384T^4 - 7704000*T^2 + 205825280
$71$	$T^{8} - 364 T^{6} + \cdots + 2151680$ T^8 - 364T^6 + 41664T^4 - 1454080*T^2 + 2151680
$73$	$(T^{4} - 192 T^{2} + \cdots - 2384)^{2}$ (T^4 - 192T^2 - 1320T - 2384)^2
$79$	$T^{8} - 116 T^{6} + \cdots + 1280$ T^8 - 116T^6 + 3664T^4 - 24000*T^2 + 1280
$83$	$T^{8} - 356 T^{6} + \cdots + 15753125$ T^8 - 356T^6 + 38829T^4 - 1500500*T^2 + 15753125
$89$	$(T^{2} - 5 T - 5)^{4}$ (T^2 - 5*T - 5)^4
$97$	$(T^{4} - 8 T^{3} + \cdots + 4016)^{2}$ (T^4 - 8T^3 - 176T^2 + 168*T + 4016)^2
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