LMFDB - Newform orbit 416.2.k.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(31,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("416.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	416.k (of order $4$ , degree $2$ , 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16$ x^8 + 17x^6 + 84x^4 + 100*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_{4} q^{3} - \beta_{6} q^{5} + (\beta_{6} - \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 3) q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{11} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + (2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 9) q^{99}+O(q^{100})$ q - b4 * q^3 - b6 * q^5 + (b6 - b2) * q^7 + (-b5 + b3 - b2 - 3) * q^9 + (-b2 - b1 - 1) * q^11 + (-b6 - b4 - b3 - b1) * q^13 + (-b7 + b5 - b4 - 2b3 - 3b1 + 3) * q^15 + (-b4 - 2b1) q^17 + (-b5 + b4 + b3 + b1 - 1) * q^19 + (-b7 - b5 + b4 + 2b3 + b1 - 1) q^21 + (-b7 + b6 + 2b5 + 2) q^23 + (-b4 + b3 + b2 + 3b1) q^25 + (-2b7 - 2b6 + b4 + b3 + b2 + 2b1) q^27 + (-b7 + b6 + 2b5 + b3 - b2) q^29 + (b5 - b4 - b3 + 3b1 - 3) q^31 + (-2b7 - b5 + b4 - 2b1 + 2) * q^33 + (-b4 - 4b1) q^35 + (b7 - 2b5 + 2b4) * q^37 + (-b7 + 2b6 - b5 - b4 - b3 - b2 - b1 - 1) q^39 + (-2b6 - b5 - b4 - 3b1 - 3) * q^41 + (2b7 - 2b6 - b5 - b3 + b2 + 6) * q^43 + (2b6 - 3b5 - 3b4 + b1 + 1) q^45 + (-b6 - b2 - 2b1 - 2) q^47 + (2b7 + 2b6 + 3b4 - 2b3 - 2b2 + b1) q^49 + (-3b5 + b3 - b2 - 6) q^51 + (b7 - b6) * q^53 + (b7 + b6 - 2b4 + b3 + b2 + 4b1) * q^55 + (-2b6 + 2b5 + 2b4 + 2b2 + 4b1 + 4) q^57 + (2b6 + b5 + b4 + b2 + 3b1 + 3) * q^59 + (4b5 - 2b3 + 2b2 - 2) q^61 + (3b5 + 3b4 + b2 + 5b1 + 5) q^63 + (3b5 - 2b4 + 5b1 - 1) q^65 + (2b7 - 2b5 + 2b4 + 3b3 + b1 - 1) * q^67 + (b7 + b6 - 2b4 - 6b1) * q^69 + (3b7 - 2b5 + 2b4 - b3 - 2b1 + 2) * q^71 + (b5 - b4 - 2b3 - 3b1 + 3) * q^73 + (2b7 - 2b6 + 2b5 + b3 - b2 - 10) q^75 + (b7 + b6 + 2b4 - b3 - b2 + 4b1) * q^77 + (b7 + b6 + 2b4 + b3 + b2 - 4b1) * q^79 + (4b5 - 2b3 + 2b2 + 5) q^81 + (2b5 - 2b4 - b3 - b1 + 1) * q^83 + (b7 + b5 - b4 - 2b3 - 3b1 + 3) * q^85 + (-b7 - b6 - 10b1) q^87 + (2b7 - b5 + b4 - b1 + 1) q^89 + (-2b6 - 3b5 + b3 + 2b2 - 3b1 - 5) * q^91 + (2b6 + 2b5 + 2b4 - 2b2 - 4b1 - 4) q^93 + (-4b5 + b3 - b2 - 2) q^95 + (2b6 + 2b5 + 2b4 - b1 - 1) q^97 + (2b6 + b5 + b4 + 3b2 + 9b1 + 9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 2 q^{7} - 20 q^{9} - 6 q^{11} - 2 q^{13} + 20 q^{15} - 6 q^{19} - 4 q^{21} + 16 q^{23} + 4 q^{29} - 26 q^{31} + 16 q^{33} - 8 q^{39} - 24 q^{41} + 44 q^{43} + 8 q^{45} - 14 q^{47} - 44 q^{51} + 28 q^{57}+ \cdots + 66 q^{99}+O(q^{100})$ 8 * q + 2 * q^7 - 20 * q^9 - 6 * q^11 - 2 * q^13 + 20 * q^15 - 6 * q^19 - 4 * q^21 + 16 * q^23 + 4 * q^29 - 26 * q^31 + 16 * q^33 - 8 * q^39 - 24 * q^41 + 44 * q^43 + 8 * q^45 - 14 * q^47 - 44 * q^51 + 28 * q^57 + 22 * q^59 - 24 * q^61 + 38 * q^63 - 8 * q^65 - 2 * q^67 + 14 * q^71 + 20 * q^73 - 76 * q^75 + 32 * q^81 + 6 * q^83 + 20 * q^85 + 8 * q^89 - 42 * q^91 - 28 * q^93 - 12 * q^95 - 8 * q^97 + 66 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16$ x^8 + 17*x^6 + 84*x^4 + 100*x^2 + 16 :

$\beta_{1}$	$=$	$( \nu^{7} + 15\nu^{5} + 62\nu^{3} + 48\nu ) / 16$ (v^7 + 15v^5 + 62v^3 + 48*v) / 16
$\beta_{2}$	$=$	$( \nu^{6} + 11\nu^{4} + 26\nu^{2} + 8\nu + 8 ) / 8$ (v^6 + 11v^4 + 26v^2 + 8*v + 8) / 8
$\beta_{3}$	$=$	$( -\nu^{6} - 11\nu^{4} - 26\nu^{2} + 8\nu - 8 ) / 8$ (-v^6 - 11v^4 - 26v^2 + 8*v - 8) / 8
$\beta_{4}$	$=$	$( \nu^{7} + 19\nu^{5} + 106\nu^{3} + 136\nu ) / 16$ (v^7 + 19v^5 + 106v^3 + 136*v) / 16
$\beta_{5}$	$=$	$( \nu^{4} + 9\nu^{2} + 8 ) / 2$ (v^4 + 9*v^2 + 8) / 2
$\beta_{6}$	$=$	$( -\nu^{7} + \nu^{6} - 19\nu^{5} + 11\nu^{4} - 98\nu^{3} + 18\nu^{2} - 80\nu - 24 ) / 16$ (-v^7 + v^6 - 19v^5 + 11v^4 - 98v^3 + 18v^2 - 80*v - 24) / 16
$\beta_{7}$	$=$	$( -\nu^{7} - \nu^{6} - 19\nu^{5} - 11\nu^{4} - 98\nu^{3} - 18\nu^{2} - 80\nu + 24 ) / 16$ (-v^7 - v^6 - 19v^5 - 11v^4 - 98v^3 - 18v^2 - 80*v + 24) / 16

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

$\nu$	$=$	$( \beta_{3} + \beta_{2} ) / 2$ (b3 + b2) / 2
$\nu^{2}$	$=$	$( 2\beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} - 8 ) / 2$ (2b7 - 2b6 - b3 + b2 - 8) / 2
$\nu^{3}$	$=$	$( 2\beta_{7} + 2\beta_{6} + 4\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2$ (2b7 + 2b6 + 4b4 - 7b3 - 7*b2) / 2
$\nu^{4}$	$=$	$( -18\beta_{7} + 18\beta_{6} + 4\beta_{5} + 9\beta_{3} - 9\beta_{2} + 56 ) / 2$ (-18b7 + 18b6 + 4b5 + 9b3 - 9*b2 + 56) / 2
$\nu^{5}$	$=$	$( -22\beta_{7} - 22\beta_{6} - 36\beta_{4} + 55\beta_{3} + 55\beta_{2} - 8\beta_1 ) / 2$ (-22b7 - 22b6 - 36b4 + 55b3 + 55b2 - 8b1) / 2
$\nu^{6}$	$=$	$( 146\beta_{7} - 146\beta_{6} - 44\beta_{5} - 81\beta_{3} + 81\beta_{2} - 424 ) / 2$ (146b7 - 146b6 - 44b5 - 81b3 + 81*b2 - 424) / 2
$\nu^{7}$	$=$	$( 206\beta_{7} + 206\beta_{6} + 292\beta_{4} - 439\beta_{3} - 439\beta_{2} + 152\beta_1 ) / 2$ (206b7 + 206b6 + 292b4 - 439b3 - 439b2 + 152b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$ .

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$-1$	$\beta_{1}$

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}

31.1

	−	2.65328i
		1.20241i
		2.88540i
	−	0.434529i
		0.434529i
	−	2.88540i
	−	1.20241i
		2.65328i

−

2.89949i

−2.84658

2.84658i

0.193303

−

0.193303i

−5.40706

31.2

−

1.46091i

1.87831

−

1.87831i

−0.675897

0.675897i

0.865736

31.3

1.19225i

−0.720056

0.720056i

3.60545

−

3.60545i

1.57854

31.4

3.16816i

1.68833

−

1.68833i

−2.12286

2.12286i

−7.03721

255.1

−

3.16816i

1.68833

1.68833i

−2.12286

−

2.12286i

−7.03721

255.2

−

1.19225i

−0.720056

−

0.720056i

3.60545

3.60545i

1.57854

255.3

1.46091i

1.87831

1.87831i

−0.675897

−

0.675897i

0.865736

255.4

2.89949i

−2.84658

−

2.84658i

0.193303

0.193303i

−5.40706

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
52.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.k.f	yes	8
4.b	odd	2	1		416.2.k.e	✓	8
8.b	even	2	1		832.2.k.i		8
8.d	odd	2	1		832.2.k.g		8
13.d	odd	4	1		416.2.k.e	✓	8
52.f	even	4	1	inner	416.2.k.f	yes	8
104.j	odd	4	1		832.2.k.g		8
104.m	even	4	1		832.2.k.i		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.k.e	✓	8	4.b	odd	2	1
416.2.k.e	✓	8	13.d	odd	4	1
416.2.k.f	yes	8	1.a	even	1	1	trivial
416.2.k.f	yes	8	52.f	even	4	1	inner
832.2.k.g		8	8.d	odd	2	1
832.2.k.g		8	104.j	odd	4	1
832.2.k.i		8	8.b	even	2	1
832.2.k.i		8	104.m	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(416, [\chi])$ :

T_{3}^{8} + 22T_{3}^{6} + 153T_{3}^{4} + 356T_{3}^{2} + 256

T_{7}^{8} - 2T_{7}^{7} + 2T_{7}^{6} + 48T_{7}^{5} + 281T_{7}^{4} + 246T_{7}^{3} + 98T_{7}^{2} - 56T_{7} + 16

T_{11}^{8} + 6T_{11}^{7} + 18T_{11}^{6} - 24T_{11}^{5} + 84T_{11}^{4} + 312T_{11}^{3} + 648T_{11}^{2} - 288T_{11} + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} + 22 T^{6} + \cdots + 256$ T^8 + 22T^6 + 153T^4 + 356*T^2 + 256
$5$	$T^{8} - 16 T^{5} + \cdots + 676$ T^8 - 16T^5 + 173T^4 - 240T^3 + 128T^2 + 416*T + 676
$7$	$T^{8} - 2 T^{7} + \cdots + 16$ T^8 - 2T^7 + 2T^6 + 48T^5 + 281T^4 + 246T^3 + 98T^2 - 56*T + 16
$11$	$T^{8} + 6 T^{7} + \cdots + 64$ T^8 + 6T^7 + 18T^6 - 24T^5 + 84T^4 + 312T^3 + 648T^2 - 288*T + 64
$13$	$T^{8} + 2 T^{7} + \cdots + 28561$ T^8 + 2T^7 + 18T^6 + 106T^5 + 146T^4 + 1378T^3 + 3042T^2 + 4394*T + 28561
$17$	$T^{8} + 38 T^{6} + \cdots + 64$ T^8 + 38T^6 + 313T^4 + 308*T^2 + 64
$19$	$T^{8} + 6 T^{7} + \cdots + 256$ T^8 + 6T^7 + 18T^6 - 88T^5 + 224T^4 + 32T^3 + 32T^2 - 128*T + 256
$23$	$(T^{4} - 8 T^{3} + \cdots - 256)^{2}$ (T^4 - 8T^3 - 10T^2 + 176*T - 256)^2
$29$	$(T^{4} - 2 T^{3} + \cdots + 464)^{2}$ (T^4 - 2T^3 - 70T^2 - 76*T + 464)^2
$31$	$T^{8} + 26 T^{7} + \cdots + 43264$ T^8 + 26T^7 + 338T^6 + 2504T^5 + 11232T^4 + 26208T^3 + 20000T^2 - 41600*T + 43264
$37$	$T^{8} + 8 T^{5} + \cdots + 669124$ T^8 + 8T^5 + 2333T^4 + 504T^3 + 32T^2 - 6544*T + 669124
$41$	$T^{8} + 24 T^{7} + \cdots + 652864$ T^8 + 24T^7 + 288T^6 + 1792T^5 + 5972T^4 + 5664T^3 + 21632T^2 + 168064*T + 652864
$43$	$(T^{4} - 22 T^{3} + \cdots - 1424)^{2}$ (T^4 - 22T^3 + 97T^2 + 384*T - 1424)^2
$47$	$T^{8} + 14 T^{7} + \cdots + 16$ T^8 + 14T^7 + 98T^6 - 104T^5 + 73T^4 + 142T^3 + 242T^2 - 88*T + 16
$53$	$(T^{4} - 30 T^{2} + \cdots + 104)^{2}$ (T^4 - 30T^2 + 32T + 104)^2
$59$	$T^{8} - 22 T^{7} + \cdots + 1048576$ T^8 - 22T^7 + 242T^6 - 952T^5 + 2448T^4 - 12288T^3 + 131072T^2 - 524288*T + 1048576
$61$	$(T^{4} + 12 T^{3} + \cdots - 4112)^{2}$ (T^4 + 12T^3 - 128T^2 - 1840*T - 4112)^2
$67$	$T^{8} + 2 T^{7} + \cdots + 43983424$ T^8 + 2T^7 + 2T^6 + 24T^5 + 29172T^4 + 76552T^3 + 95048T^2 - 2891552*T + 43983424
$71$	$T^{8} - 14 T^{7} + \cdots + 204304$ T^8 - 14T^7 + 98T^6 + 432T^5 + 6233T^4 - 49398T^3 + 174050T^2 - 266680*T + 204304
$73$	$T^{8} - 20 T^{7} + \cdots + 43264$ T^8 - 20T^7 + 200T^6 - 664T^5 - 92T^4 + 9632T^3 + 46208T^2 + 63232*T + 43264
$79$	$T^{8} + 208 T^{6} + \cdots + 256$ T^8 + 208T^6 + 6356T^4 + 8912*T^2 + 256
$83$	$T^{8} - 6 T^{7} + \cdots + 10816$ T^8 - 6T^7 + 18T^6 + 536T^5 + 4148T^4 + 9864T^3 + 9800T^2 - 14560*T + 10816
$89$	$T^{8} - 8 T^{7} + \cdots + 7744$ T^8 - 8T^7 + 32T^6 + 224T^5 + 724T^4 + 224T^3 + 128T^2 + 1408*T + 7744
$97$	$T^{8} + 8 T^{7} + \cdots + 35344$ T^8 + 8T^7 + 32T^6 - 480T^5 + 2760T^4 - 3296T^3 + 512T^2 + 6016*T + 35344
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Elliptic curves over $\Q$
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