LMFDB - Newform orbit 1584.2.cd.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,2,Mod(17,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 5, 9]))

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

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

chi := DirichletCharacter("1584.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1584.cd (of order $10$, degree $4$, 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:	$12.6483036802$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 15x^{12} - 62x^{10} - 371x^{8} - 558x^{6} + 1215x^{4} + 5832x^{2} + 6561 $ x^16 + 8x^14 + 15x^12 - 62x^10 - 371x^8 - 558x^6 + 1215x^4 + 5832*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{4}) q^{5}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots + 1) q^{7}+O(q^{10}) $ q + (-b15 + b13 + 2b5 - b4) q^5 + (-2b11 - b10 + b9 + b3 - b2 + 1) q^7	$ q + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{4}) q^{5}+ \cdots + (2 \beta_{14} + \beta_{12} - 9 \beta_{11} + \cdots + 7) q^{97}+O(q^{100}) $ q + (-b15 + b13 + 2b5 - b4) q^5 + (-2b11 - b10 + b9 + b3 - b2 + 1) q^7 + (-b15 - b8 - 3b6 - 2b4 + 2b1) q^11 + (2b12 + b11 + 2b10 + b9 - b3) * q^13 + (-b15 - b13 + b8 - 3b7 - 2b6 - b4) * q^17 + (-2b10 + b9 + b2) q^19 + (b15 + b13 - 2b6 - b5 - b4 + 2b1) * q^23 + (2b14 - 2b12 + 4b11 + b10 - 4b9 - 2b3 + 4b2) * q^25 + (-2b15 + b13 - 2b8 - b7 - b6 + 2b5 - 2b4) * q^29 + (2b14 - b12 + 2b11 + b10 - 2b9 - b3 + 2b2 - 2) * q^31 + (-3b15 - 6b13 - b8 - 2b7 - b6 - b5 - 3b1) * q^35 + (b14 + b12 + 3b9 - 2b3 + 3) * q^37 + (-b15 - b13 + b7 + 2b6 + b5) q^41 + (2b14 - 2b11 + 2b10 + 2b9 - 2b3 + 1) q^43 + (4b13 - b7 + 4b6 - b5 + b4 + b1) * q^47 + (2b14 - 2b12 - 2b11 + 2b3 + 2) * q^49 + (-b15 + 3b8 - 3b5) * q^53 + (3b14 - 2b12 + 3b11 + 2b10 - 4b9 - b3 + 3b2 + 3) * q^55 + (b13 + 2b8 - 2b7 + 2b6 + 3b5 - 2b4 + b1) q^59 + (b12 - 2b11 - 2b9 + 4b2 - 6) q^61 + (8b15 - b8 + 4b7 + 8b6 - 2b5 + 2b4 + b1) q^65 + (-b14 + 2b12 + 4b11 + b10 + b3 + 4b2 + 2) q^67 + (2b15 + b13 + 2b8 + 2b7 + 7b6 + 6b5 - b4 - 4b1) * q^71 + (3b14 - 3b12 + 2b11 - b10 + b3 + b2 - 2) q^73 + (-b15 - 2b8 - 5b7 - 3b6 - 2b5 - 2b4 + b1) q^77 + (2b12 - b11 + 2b10 - b9 - 3b3 + 2b2 + 2) * q^79 + (b15 + 3b13 + 3b8 + 3b7 + 5b6) * q^83 + (-3b11 - 4b10 + 2b9 - 4b2 + 6) * q^85 + (-6b15 - 6b13 - 6b7 - 6b6 - 3b5 - 3b4 + 6b1) q^89 + (3b14 + 8b11 + 3b10 + 4b9 - 6b3 - 4b2) * q^91 + (-16b15 - 5b13 - 5b8 - 8b7 - 10b6 + 3b5 - 5b4 + 2b1) * q^95 + (2b14 + b12 - 9b11 - b10 + 9b9 - b3 - 7b2 + 7) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 48 q^{25} - 8 q^{31} + 36 q^{37} + 24 q^{49} + 88 q^{55} - 80 q^{61} + 64 q^{67} - 20 q^{73} + 40 q^{79} + 60 q^{85} + 12 q^{97}+O(q^{100}) $ 16 * q + 48 * q^25 - 8 * q^31 + 36 * q^37 + 24 * q^49 + 88 * q^55 - 80 * q^61 + 64 * q^67 - 20 * q^73 + 40 * q^79 + 60 * q^85 + 12 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 8x^{14} + 15x^{12} - 62x^{10} - 371x^{8} - 558x^{6} + 1215x^{4} + 5832x^{2} + 6561 $ x^16 + 8*x^14 + 15*x^12 - 62*x^10 - 371*x^8 - 558*x^6 + 1215*x^4 + 5832*x^2 + 6561 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -26\nu^{14} - 121\nu^{12} - 126\nu^{10} + 2620\nu^{8} + 8464\nu^{6} + 2940\nu^{4} - 67122\nu^{2} - 130248 ) / 82863 $ (-26v^14 - 121v^12 - 126v^10 + 2620v^8 + 8464v^6 + 2940v^4 - 67122*v^2 - 130248) / 82863
$\beta_{3}$	$=$	$ ( 20\nu^{14} + 77\nu^{12} - 85\nu^{10} - 901\nu^{8} + 129\nu^{6} + 1525\nu^{4} + 13761\nu^{2} + 3483 ) / 27621 $ (20v^14 + 77v^12 - 85v^10 - 901v^8 + 129v^6 + 1525v^4 + 13761*v^2 + 3483) / 27621
$\beta_{4}$	$=$	$ ( 76\nu^{15} + 242\nu^{13} - 411\nu^{11} - 401\nu^{9} - 6233\nu^{7} + 13836\nu^{5} + 62964\nu^{3} + 31590\nu ) / 248589 $ (76v^15 + 242v^13 - 411v^11 - 401v^9 - 6233v^7 + 13836v^5 + 62964v^3 + 31590v) / 248589
$\beta_{5}$	$=$	$ ( - 26 \nu^{15} - 121 \nu^{13} - 126 \nu^{11} + 2620 \nu^{9} + 8464 \nu^{7} + 2940 \nu^{5} + \cdots - 130248 \nu ) / 82863 $ (-26v^15 - 121v^13 - 126v^11 + 2620v^9 + 8464v^7 + 2940v^5 - 67122v^3 - 130248v) / 82863
$\beta_{6}$	$=$	$ ( 86 \nu^{15} + 352 \nu^{13} - 129 \nu^{11} - 5323 \nu^{9} - 8077 \nu^{7} + 1635 \nu^{5} + \cdots + 223560 \nu ) / 248589 $ (86v^15 + 352v^13 - 129v^11 - 5323v^9 - 8077v^7 + 1635v^5 + 108405v^3 + 223560v) / 248589
$\beta_{7}$	$=$	$ ( 278 \nu^{15} + 2530 \nu^{13} + 5160 \nu^{11} - 20665 \nu^{9} - 103885 \nu^{7} - 75465 \nu^{5} + \cdots + 1388745 \nu ) / 745767 $ (278v^15 + 2530v^13 + 5160v^11 - 20665v^9 - 103885v^7 - 75465v^5 + 405405v^3 + 1388745v) / 745767
$\beta_{8}$	$=$	$ ( \nu^{15} + 8\nu^{13} + 15\nu^{11} - 62\nu^{9} - 371\nu^{7} - 558\nu^{5} + 1215\nu^{3} + 5832\nu ) / 2187 $ (v^15 + 8v^13 + 15v^11 - 62v^9 - 371v^7 - 558v^5 + 1215v^3 + 5832*v) / 2187
$\beta_{9}$	$=$	$ ( - 214 \nu^{14} - 836 \nu^{12} + 288 \nu^{10} + 10964 \nu^{8} + 31238 \nu^{6} - 9591 \nu^{4} + \cdots - 349920 ) / 82863 $ (-214v^14 - 836v^12 + 288v^10 + 10964v^8 + 31238v^6 - 9591v^4 - 222750*v^2 - 349920) / 82863
$\beta_{10}$	$=$	$ ( 100\nu^{14} + 396\nu^{12} - 139\nu^{10} - 5429\nu^{8} - 13105\nu^{6} + 3742\nu^{4} + 82764\nu^{2} + 135918 ) / 27621 $ (100v^14 + 396v^12 - 139v^10 - 5429v^8 - 13105v^6 + 3742v^4 + 82764*v^2 + 135918) / 27621
$\beta_{11}$	$=$	$ ( 980 \nu^{14} + 5500 \nu^{12} + 840 \nu^{10} - 66457 \nu^{8} - 209752 \nu^{6} - 50004 \nu^{4} + \cdots + 2700216 ) / 248589 $ (980v^14 + 5500v^12 + 840v^10 - 66457v^8 - 209752v^6 - 50004v^4 + 1343628*v^2 + 2700216) / 248589
$\beta_{12}$	$=$	$ ( - 1258 \nu^{14} - 8030 \nu^{12} - 6000 \nu^{10} + 87122 \nu^{8} + 313637 \nu^{6} + 125469 \nu^{4} + \cdots - 3840372 ) / 248589 $ (-1258v^14 - 8030v^12 - 6000v^10 + 87122v^8 + 313637v^6 + 125469v^4 - 1749033*v^2 - 3840372) / 248589
$\beta_{13}$	$=$	$ ( - 488 \nu^{15} - 1903 \nu^{13} + 831 \nu^{11} + 24631 \nu^{9} + 62089 \nu^{7} - 23757 \nu^{5} + \cdots - 627426 \nu ) / 248589 $ (-488v^15 - 1903v^13 + 831v^11 + 24631v^9 + 62089v^7 - 23757v^5 - 403920v^3 - 627426v) / 248589
$\beta_{14}$	$=$	$ ( 2183 \nu^{14} + 10417 \nu^{12} - 4677 \nu^{10} - 136804 \nu^{8} - 361315 \nu^{6} + 127863 \nu^{4} + \cdots + 4070736 ) / 248589 $ (2183v^14 + 10417v^12 - 4677v^10 - 136804v^8 - 361315v^6 + 127863v^4 + 2739825*v^2 + 4070736) / 248589
$\beta_{15}$	$=$	$ ( 3805 \nu^{15} + 18425 \nu^{13} - 4701 \nu^{11} - 236153 \nu^{9} - 664781 \nu^{7} + \cdots + 7323534 \nu ) / 745767 $ (3805v^15 + 18425v^13 - 4701v^11 - 236153v^9 - 664781v^7 + 106632v^5 + 4503114v^3 + 7323534v) / 745767

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{10} - \beta_{9} + \beta_{3} - \beta_{2} - 1 $ -b10 - b9 + b3 - b2 - 1
$\nu^{3}$	$=$	$ \beta_{13} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_1 $ b13 + 3b6 - b5 + 2b4 - 2*b1
$\nu^{4}$	$=$	$ 2\beta_{14} - 2\beta_{12} - 6\beta_{11} + \beta_{9} - 2\beta_{3} + 6 $ 2b14 - 2b12 - 6b11 + b9 - 2b3 + 6
$\nu^{5}$	$=$	$ -\beta_{13} - 6\beta_{8} + 6\beta_{7} - 6\beta_{6} - \beta_{5} + \beta_{4} + 6\beta_1 $ -b13 - 6b8 + 6b7 - 6b6 - b5 + b4 + 6b1
$\nu^{6}$	$=$	$ -\beta_{12} - 2\beta_{11} - \beta_{10} + 2\beta_{9} + 13\beta_{3} - 7\beta_{2} + 7 $ -b12 - 2b11 - b10 + 2b9 + 13b3 - 7b2 + 7
$\nu^{7}$	$=$	$ -\beta_{15} + 3\beta_{13} - \beta_{8} + 3\beta_{7} + 38\beta_{6} + 8\beta_{5} - \beta_{4} - 7\beta_1 $ -b15 + 3b13 - b8 + 3b7 + 38b6 + 8b5 - b4 - 7*b1
$\nu^{8}$	$=$	$ -4\beta_{14} + 7\beta_{11} - 28\beta_{10} - 48\beta_{9} + 48\beta_{2} $ -4b14 + 7b11 - 28b10 - 48b9 + 48*b2
$\nu^{9}$	$=$	$ -\beta_{15} + 11\beta_{13} + 11\beta_{8} - \beta_{6} + 72\beta_{4} $ -b15 + 11b13 + 11b8 - b6 + 72*b4
$\nu^{10}$	$=$	$ 12\beta_{14} - 73\beta_{12} - 155\beta_{11} + 12\beta_{10} + 61\beta_{3} - 155\beta_{2} + 49 $ 12b14 - 73b12 - 155b11 + 12b10 + 61b3 - 155b2 + 49
$\nu^{11}$	$=$	$ -58\beta_{15} - 94\beta_{13} - 94\beta_{8} + 219\beta_{7} + 125\beta_{6} - 94\beta_{5} - 85\beta_1 $ -58b15 - 94b13 - 94b8 + 219b7 + 125b6 - 94b5 - 85*b1
$\nu^{12}$	$=$	$ 58\beta_{14} - 58\beta_{12} - 134\beta_{10} + 90\beta_{9} + 134\beta_{3} + 581\beta_{2} + 90 $ 58b14 - 58b12 - 134b10 + 90b9 + 134b3 + 581b2 + 90
$\nu^{13}$	$=$	$ 174\beta_{15} + 474\beta_{13} + 174\beta_{7} + 576\beta_{6} + 805\beta_{5} + 102\beta_{4} - 102\beta_1 $ 174b15 + 474b13 + 174b7 + 576b6 + 805b5 + 102b4 - 102*b1
$\nu^{14}$	$=$	$ -505\beta_{14} + 72\beta_{12} + 127\beta_{11} - 1910\beta_{9} + 505\beta_{3} - 127 $ -505b14 + 72b12 + 127b11 - 1910b9 + 505*b3 - 127
$\nu^{15}$	$=$	$ - 955 \beta_{15} - 2360 \beta_{13} + 560 \beta_{8} - 216 \beta_{7} + 560 \beta_{6} - 1405 \beta_{5} + \cdots - 560 \beta_1 $ -955b15 - 2360b13 + 560b8 - 216b7 + 560b6 - 1405b5 + 1405b4 - 560b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$\beta_{2}$	$-1$	$1$	$1$

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} $

17.1

−0.102289	−	1.72903i
−0.933543	−	1.45894i
0.933543	+	1.45894i
0.102289	+	1.72903i
1.72453	−	0.161272i
0.379529	+	1.68996i
−0.379529	−	1.68996i
−1.72453	+	0.161272i
1.72453	+	0.161272i
0.379529	−	1.68996i
−0.379529	+	1.68996i
−1.72453	−	0.161272i
−0.102289	+	1.72903i
−0.933543	+	1.45894i
0.933543	−	1.45894i
0.102289	−	1.72903i

−4.12006

−

1.33869i

−1.97907

−

2.72396i

17.2

−1.94381

−

0.631582i

−0.256994

−

0.353722i

17.3

1.94381

0.631582i

−0.256994

−

0.353722i

17.4

4.12006

1.33869i

−1.97907

−

2.72396i

161.1

−1.15541

−

1.59029i

3.94798

−

1.28278i

161.2

−0.641668

−

0.883180i

−1.71191

0.556235i

161.3

0.641668

0.883180i

−1.71191

0.556235i

161.4

1.15541

1.59029i

3.94798

−

1.28278i

305.1

−1.15541

1.59029i

3.94798

1.28278i

305.2

−0.641668

0.883180i

−1.71191

−

0.556235i

305.3

0.641668

−

0.883180i

−1.71191

−

0.556235i

305.4

1.15541

−

1.59029i

3.94798

1.28278i

1025.1

−4.12006

1.33869i

−1.97907

2.72396i

1025.2

−1.94381

0.631582i

−0.256994

0.353722i

1025.3

1.94381

−

0.631582i

−0.256994

0.353722i

1025.4

4.12006

−

1.33869i

−1.97907

2.72396i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.d	odd	10	1	inner
33.f	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1584.2.cd.d		16
3.b	odd	2	1	inner	1584.2.cd.d		16
4.b	odd	2	1		396.2.x.a	✓	16
11.d	odd	10	1	inner	1584.2.cd.d		16
12.b	even	2	1		396.2.x.a	✓	16
33.f	even	10	1	inner	1584.2.cd.d		16
44.g	even	10	1		396.2.x.a	✓	16
44.g	even	10	1		4356.2.b.e		16
44.h	odd	10	1		4356.2.b.e		16
132.n	odd	10	1		396.2.x.a	✓	16
132.n	odd	10	1		4356.2.b.e		16
132.o	even	10	1		4356.2.b.e		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
396.2.x.a	✓	16	4.b	odd	2	1
396.2.x.a	✓	16	12.b	even	2	1
396.2.x.a	✓	16	44.g	even	10	1
396.2.x.a	✓	16	132.n	odd	10	1
1584.2.cd.d		16	1.a	even	1	1	trivial
1584.2.cd.d		16	3.b	odd	2	1	inner
1584.2.cd.d		16	11.d	odd	10	1	inner
1584.2.cd.d		16	33.f	even	10	1	inner
4356.2.b.e		16	44.g	even	10	1
4356.2.b.e		16	44.h	odd	10	1
4356.2.b.e		16	132.n	odd	10	1
4356.2.b.e		16	132.o	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{16} - 34 T_{5}^{14} + 477 T_{5}^{12} - 1772 T_{5}^{10} + 6950 T_{5}^{8} - 26018 T_{5}^{6} + \cdots + 130321 $ T5^16 - 34*T5^14 + 477*T5^12 - 1772*T5^10 + 6950*T5^8 - 26018*T5^6 + 81612*T5^4 + 26714*T5^2 + 130321 acting on $S_{2}^{\mathrm{new}}(1584, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - 34 T^{14} + \cdots + 130321 $ T^16 - 34T^14 + 477T^12 - 1772T^10 + 6950T^8 - 26018T^6 + 81612T^4 + 26714*T^2 + 130321
$7$	$ (T^{8} - 13 T^{6} + \cdots + 121)^{2} $ (T^8 - 13T^6 - 50T^5 + 89T^4 + 650T^3 + 963T^2 + 440T + 121)^2
$11$	$ T^{16} + \cdots + 214358881 $ T^16 + 44T^14 + 995T^12 + 15646T^10 + 192109T^8 + 1893166T^6 + 14567795T^4 + 77948684*T^2 + 214358881
$13$	$ (T^{8} - 35 T^{6} + \cdots + 126025)^{2} $ (T^8 - 35T^6 - 210T^5 + 835T^4 + 7350T^3 + 2425T^2 - 21300T + 126025)^2
$17$	$ T^{16} + 2 T^{14} + \cdots + 923521 $ T^16 + 2T^14 + 2148T^12 + 81454T^10 + 1310750T^8 + 8775124T^6 + 24959373T^4 - 2708098*T^2 + 923521
$19$	$ (T^{8} - 8 T^{6} + \cdots + 52441)^{2} $ (T^8 - 8T^6 + 210T^5 + 754T^4 - 1680T^3 - 5747T^2 - 6870T + 52441)^2
$23$	$ (T^{8} + 58 T^{6} + \cdots + 5041)^{2} $ (T^8 + 58T^6 + 894T^4 + 3962*T^2 + 5041)^2
$29$	$ T^{16} + \cdots + 236421376 $ T^16 - 8T^14 + 2128T^12 + 51664T^10 + 774240T^8 + 6798464T^6 + 183344448T^4 + 333843712*T^2 + 236421376
$31$	$ (T^{8} + 4 T^{7} + \cdots + 121)^{2} $ (T^8 + 4T^7 + 42T^6 + 272T^5 + 1200T^4 + 2168T^3 + 3997T^2 - 1144*T + 121)^2
$37$	$ (T^{8} - 18 T^{7} + \cdots + 346921)^{2} $ (T^8 - 18T^7 + 203T^6 - 1446T^5 + 7375T^4 - 26226T^3 + 91743T^2 - 236778*T + 346921)^2
$41$	$ T^{16} + 82 T^{14} + \cdots + 25411681 $ T^16 + 82T^14 + 2963T^12 + 45534T^10 + 777065T^8 + 11445414T^6 + 104686283T^4 - 29630998*T^2 + 25411681
$43$	$ (T^{8} + 92 T^{6} + \cdots + 28561)^{2} $ (T^8 + 92T^6 + 2134T^4 + 15548*T^2 + 28561)^2
$47$	$ T^{16} + \cdots + 32573512997041 $ T^16 - 356T^14 + 54597T^12 - 3304768T^10 + 145395890T^8 - 6552831652T^6 + 321765095352T^4 + 1416248876866*T^2 + 32573512997041
$53$	$ T^{16} + \cdots + 831789600625 $ T^16 - 70T^14 + 11820T^12 - 1501250T^10 + 116594150T^8 - 258363500T^6 + 22753696125T^4 - 215283501250*T^2 + 831789600625
$59$	$ T^{16} + \cdots + 10355301121 $ T^16 + 54T^14 + 18032T^12 - 668688T^10 + 10686730T^8 - 83770872T^6 + 386008217T^4 - 1602125184*T^2 + 10355301121
$61$	$ (T^{8} + 40 T^{7} + \cdots + 2307361)^{2} $ (T^8 + 40T^7 + 793T^6 + 9680T^5 + 77724T^4 + 415240T^3 + 1422862T^2 + 2764580*T + 2307361)^2
$67$	$ (T^{4} - 16 T^{3} + \cdots - 29)^{4} $ (T^4 - 16T^3 + T^2 + 44T - 29)^4
$71$	$ T^{16} + \cdots + 25401852000625 $ T^16 - 170T^14 + 25845T^12 - 3683900T^10 + 727194150T^8 + 8069173750T^6 + 1358132775500T^4 - 9535475298750*T^2 + 25401852000625
$73$	$ (T^{8} + 10 T^{7} + \cdots + 3857296)^{2} $ (T^8 + 10T^7 - 62T^6 - 320T^5 + 5424T^4 - 14360T^3 - 17328T^2 - 510640*T + 3857296)^2
$79$	$ (T^{8} - 20 T^{7} + \cdots + 78961)^{2} $ (T^8 - 20T^7 + 237T^6 - 1010T^5 + 2039T^4 - 8770T^3 + 45713T^2 - 101160*T + 78961)^2
$83$	$ T^{16} + \cdots + 1804403844961 $ T^16 - 12T^14 + 15383T^12 + 1246906T^10 + 44564205T^8 + 769189546T^6 + 11505429543T^4 + 166550724628*T^2 + 1804403844961
$89$	$ (T^{8} + 522 T^{6} + \cdots + 33074001)^{2} $ (T^8 + 522T^6 + 72414T^4 + 2888298*T^2 + 33074001)^2
$97$	$ (T^{8} - 6 T^{7} + \cdots + 27552001)^{2} $ (T^8 - 6T^7 + 117T^6 - 1148T^5 + 26340T^4 + 202798T^3 + 2974902T^2 + 13510926*T + 27552001)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.cd (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{4}) q^{5}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots + 1) q^{7}+O(q^{10}) \) q + (-b15 + b13 + 2b5 - b4) q^5 + (-2b11 - b10 + b9 + b3 - b2 + 1) q^7	\( q + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{4}) q^{5}+ \cdots + (2 \beta_{14} + \beta_{12} - 9 \beta_{11} + \cdots + 7) q^{97}+O(q^{100}) \) q + (-b15 + b13 + 2b5 - b4) q^5 + (-2b11 - b10 + b9 + b3 - b2 + 1) q^7 + (-b15 - b8 - 3b6 - 2b4 + 2b1) q^11 + (2b12 + b11 + 2b10 + b9 - b3) * q^13 + (-b15 - b13 + b8 - 3b7 - 2b6 - b4) * q^17 + (-2b10 + b9 + b2) q^19 + (b15 + b13 - 2b6 - b5 - b4 + 2b1) * q^23 + (2b14 - 2b12 + 4b11 + b10 - 4b9 - 2b3 + 4b2) * q^25 + (-2b15 + b13 - 2b8 - b7 - b6 + 2b5 - 2b4) * q^29 + (2b14 - b12 + 2b11 + b10 - 2b9 - b3 + 2b2 - 2) * q^31 + (-3b15 - 6b13 - b8 - 2b7 - b6 - b5 - 3b1) * q^35 + (b14 + b12 + 3b9 - 2b3 + 3) * q^37 + (-b15 - b13 + b7 + 2b6 + b5) q^41 + (2b14 - 2b11 + 2b10 + 2b9 - 2b3 + 1) q^43 + (4b13 - b7 + 4b6 - b5 + b4 + b1) * q^47 + (2b14 - 2b12 - 2b11 + 2b3 + 2) * q^49 + (-b15 + 3b8 - 3b5) * q^53 + (3b14 - 2b12 + 3b11 + 2b10 - 4b9 - b3 + 3b2 + 3) * q^55 + (b13 + 2b8 - 2b7 + 2b6 + 3b5 - 2b4 + b1) q^59 + (b12 - 2b11 - 2b9 + 4b2 - 6) q^61 + (8b15 - b8 + 4b7 + 8b6 - 2b5 + 2b4 + b1) q^65 + (-b14 + 2b12 + 4b11 + b10 + b3 + 4b2 + 2) q^67 + (2b15 + b13 + 2b8 + 2b7 + 7b6 + 6b5 - b4 - 4b1) * q^71 + (3b14 - 3b12 + 2b11 - b10 + b3 + b2 - 2) q^73 + (-b15 - 2b8 - 5b7 - 3b6 - 2b5 - 2b4 + b1) q^77 + (2b12 - b11 + 2b10 - b9 - 3b3 + 2b2 + 2) * q^79 + (b15 + 3b13 + 3b8 + 3b7 + 5b6) * q^83 + (-3b11 - 4b10 + 2b9 - 4b2 + 6) * q^85 + (-6b15 - 6b13 - 6b7 - 6b6 - 3b5 - 3b4 + 6b1) q^89 + (3b14 + 8b11 + 3b10 + 4b9 - 6b3 - 4b2) * q^91 + (-16b15 - 5b13 - 5b8 - 8b7 - 10b6 + 3b5 - 5b4 + 2b1) * q^95 + (2b14 + b12 - 9b11 - b10 + 9b9 - b3 - 7b2 + 7) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 48 q^{25} - 8 q^{31} + 36 q^{37} + 24 q^{49} + 88 q^{55} - 80 q^{61} + 64 q^{67} - 20 q^{73} + 40 q^{79} + 60 q^{85} + 12 q^{97}+O(q^{100}) \) 16 * q + 48 * q^25 - 8 * q^31 + 36 * q^37 + 24 * q^49 + 88 * q^55 - 80 * q^61 + 64 * q^67 - 20 * q^73 + 40 * q^79 + 60 * q^85 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1584.2.cd.d

Level	$1584$
Weight	$2$
Character orbit	1584.cd
Analytic conductor	$12.648$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(12.6483036802\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 8x^{14} + 15x^{12} - 62x^{10} - 371x^{8} - 558x^{6} + 1215x^{4} + 5832x^{2} + 6561 \) x^16 + 8x^14 + 15x^12 - 62x^10 - 371x^8 - 558x^6 + 1215x^4 + 5832*x^2 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -26\nu^{14} - 121\nu^{12} - 126\nu^{10} + 2620\nu^{8} + 8464\nu^{6} + 2940\nu^{4} - 67122\nu^{2} - 130248 ) / 82863 \) (-26v^14 - 121v^12 - 126v^10 + 2620v^8 + 8464v^6 + 2940v^4 - 67122*v^2 - 130248) / 82863
\(\beta_{3}\)	\(=\)	\( ( 20\nu^{14} + 77\nu^{12} - 85\nu^{10} - 901\nu^{8} + 129\nu^{6} + 1525\nu^{4} + 13761\nu^{2} + 3483 ) / 27621 \) (20v^14 + 77v^12 - 85v^10 - 901v^8 + 129v^6 + 1525v^4 + 13761*v^2 + 3483) / 27621
\(\beta_{4}\)	\(=\)	\( ( 76\nu^{15} + 242\nu^{13} - 411\nu^{11} - 401\nu^{9} - 6233\nu^{7} + 13836\nu^{5} + 62964\nu^{3} + 31590\nu ) / 248589 \) (76v^15 + 242v^13 - 411v^11 - 401v^9 - 6233v^7 + 13836v^5 + 62964v^3 + 31590v) / 248589
\(\beta_{5}\)	\(=\)	\( ( - 26 \nu^{15} - 121 \nu^{13} - 126 \nu^{11} + 2620 \nu^{9} + 8464 \nu^{7} + 2940 \nu^{5} + \cdots - 130248 \nu ) / 82863 \) (-26v^15 - 121v^13 - 126v^11 + 2620v^9 + 8464v^7 + 2940v^5 - 67122v^3 - 130248v) / 82863
\(\beta_{6}\)	\(=\)	\( ( 86 \nu^{15} + 352 \nu^{13} - 129 \nu^{11} - 5323 \nu^{9} - 8077 \nu^{7} + 1635 \nu^{5} + \cdots + 223560 \nu ) / 248589 \) (86v^15 + 352v^13 - 129v^11 - 5323v^9 - 8077v^7 + 1635v^5 + 108405v^3 + 223560v) / 248589
\(\beta_{7}\)	\(=\)	\( ( 278 \nu^{15} + 2530 \nu^{13} + 5160 \nu^{11} - 20665 \nu^{9} - 103885 \nu^{7} - 75465 \nu^{5} + \cdots + 1388745 \nu ) / 745767 \) (278v^15 + 2530v^13 + 5160v^11 - 20665v^9 - 103885v^7 - 75465v^5 + 405405v^3 + 1388745v) / 745767
\(\beta_{8}\)	\(=\)	\( ( \nu^{15} + 8\nu^{13} + 15\nu^{11} - 62\nu^{9} - 371\nu^{7} - 558\nu^{5} + 1215\nu^{3} + 5832\nu ) / 2187 \) (v^15 + 8v^13 + 15v^11 - 62v^9 - 371v^7 - 558v^5 + 1215v^3 + 5832*v) / 2187
\(\beta_{9}\)	\(=\)	\( ( - 214 \nu^{14} - 836 \nu^{12} + 288 \nu^{10} + 10964 \nu^{8} + 31238 \nu^{6} - 9591 \nu^{4} + \cdots - 349920 ) / 82863 \) (-214v^14 - 836v^12 + 288v^10 + 10964v^8 + 31238v^6 - 9591v^4 - 222750*v^2 - 349920) / 82863
\(\beta_{10}\)	\(=\)	\( ( 100\nu^{14} + 396\nu^{12} - 139\nu^{10} - 5429\nu^{8} - 13105\nu^{6} + 3742\nu^{4} + 82764\nu^{2} + 135918 ) / 27621 \) (100v^14 + 396v^12 - 139v^10 - 5429v^8 - 13105v^6 + 3742v^4 + 82764*v^2 + 135918) / 27621
\(\beta_{11}\)	\(=\)	\( ( 980 \nu^{14} + 5500 \nu^{12} + 840 \nu^{10} - 66457 \nu^{8} - 209752 \nu^{6} - 50004 \nu^{4} + \cdots + 2700216 ) / 248589 \) (980v^14 + 5500v^12 + 840v^10 - 66457v^8 - 209752v^6 - 50004v^4 + 1343628*v^2 + 2700216) / 248589
\(\beta_{12}\)	\(=\)	\( ( - 1258 \nu^{14} - 8030 \nu^{12} - 6000 \nu^{10} + 87122 \nu^{8} + 313637 \nu^{6} + 125469 \nu^{4} + \cdots - 3840372 ) / 248589 \) (-1258v^14 - 8030v^12 - 6000v^10 + 87122v^8 + 313637v^6 + 125469v^4 - 1749033*v^2 - 3840372) / 248589
\(\beta_{13}\)	\(=\)	\( ( - 488 \nu^{15} - 1903 \nu^{13} + 831 \nu^{11} + 24631 \nu^{9} + 62089 \nu^{7} - 23757 \nu^{5} + \cdots - 627426 \nu ) / 248589 \) (-488v^15 - 1903v^13 + 831v^11 + 24631v^9 + 62089v^7 - 23757v^5 - 403920v^3 - 627426v) / 248589
\(\beta_{14}\)	\(=\)	\( ( 2183 \nu^{14} + 10417 \nu^{12} - 4677 \nu^{10} - 136804 \nu^{8} - 361315 \nu^{6} + 127863 \nu^{4} + \cdots + 4070736 ) / 248589 \) (2183v^14 + 10417v^12 - 4677v^10 - 136804v^8 - 361315v^6 + 127863v^4 + 2739825*v^2 + 4070736) / 248589
\(\beta_{15}\)	\(=\)	\( ( 3805 \nu^{15} + 18425 \nu^{13} - 4701 \nu^{11} - 236153 \nu^{9} - 664781 \nu^{7} + \cdots + 7323534 \nu ) / 745767 \) (3805v^15 + 18425v^13 - 4701v^11 - 236153v^9 - 664781v^7 + 106632v^5 + 4503114v^3 + 7323534v) / 745767

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{10} - \beta_{9} + \beta_{3} - \beta_{2} - 1 \) -b10 - b9 + b3 - b2 - 1
\(\nu^{3}\)	\(=\)	\( \beta_{13} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_1 \) b13 + 3b6 - b5 + 2b4 - 2*b1
\(\nu^{4}\)	\(=\)	\( 2\beta_{14} - 2\beta_{12} - 6\beta_{11} + \beta_{9} - 2\beta_{3} + 6 \) 2b14 - 2b12 - 6b11 + b9 - 2b3 + 6
\(\nu^{5}\)	\(=\)	\( -\beta_{13} - 6\beta_{8} + 6\beta_{7} - 6\beta_{6} - \beta_{5} + \beta_{4} + 6\beta_1 \) -b13 - 6b8 + 6b7 - 6b6 - b5 + b4 + 6b1
\(\nu^{6}\)	\(=\)	\( -\beta_{12} - 2\beta_{11} - \beta_{10} + 2\beta_{9} + 13\beta_{3} - 7\beta_{2} + 7 \) -b12 - 2b11 - b10 + 2b9 + 13b3 - 7b2 + 7
\(\nu^{7}\)	\(=\)	\( -\beta_{15} + 3\beta_{13} - \beta_{8} + 3\beta_{7} + 38\beta_{6} + 8\beta_{5} - \beta_{4} - 7\beta_1 \) -b15 + 3b13 - b8 + 3b7 + 38b6 + 8b5 - b4 - 7*b1
\(\nu^{8}\)	\(=\)	\( -4\beta_{14} + 7\beta_{11} - 28\beta_{10} - 48\beta_{9} + 48\beta_{2} \) -4b14 + 7b11 - 28b10 - 48b9 + 48*b2
\(\nu^{9}\)	\(=\)	\( -\beta_{15} + 11\beta_{13} + 11\beta_{8} - \beta_{6} + 72\beta_{4} \) -b15 + 11b13 + 11b8 - b6 + 72*b4
\(\nu^{10}\)	\(=\)	\( 12\beta_{14} - 73\beta_{12} - 155\beta_{11} + 12\beta_{10} + 61\beta_{3} - 155\beta_{2} + 49 \) 12b14 - 73b12 - 155b11 + 12b10 + 61b3 - 155b2 + 49
\(\nu^{11}\)	\(=\)	\( -58\beta_{15} - 94\beta_{13} - 94\beta_{8} + 219\beta_{7} + 125\beta_{6} - 94\beta_{5} - 85\beta_1 \) -58b15 - 94b13 - 94b8 + 219b7 + 125b6 - 94b5 - 85*b1
\(\nu^{12}\)	\(=\)	\( 58\beta_{14} - 58\beta_{12} - 134\beta_{10} + 90\beta_{9} + 134\beta_{3} + 581\beta_{2} + 90 \) 58b14 - 58b12 - 134b10 + 90b9 + 134b3 + 581b2 + 90
\(\nu^{13}\)	\(=\)	\( 174\beta_{15} + 474\beta_{13} + 174\beta_{7} + 576\beta_{6} + 805\beta_{5} + 102\beta_{4} - 102\beta_1 \) 174b15 + 474b13 + 174b7 + 576b6 + 805b5 + 102b4 - 102*b1
\(\nu^{14}\)	\(=\)	\( -505\beta_{14} + 72\beta_{12} + 127\beta_{11} - 1910\beta_{9} + 505\beta_{3} - 127 \) -505b14 + 72b12 + 127b11 - 1910b9 + 505*b3 - 127
\(\nu^{15}\)	\(=\)	\( - 955 \beta_{15} - 2360 \beta_{13} + 560 \beta_{8} - 216 \beta_{7} + 560 \beta_{6} - 1405 \beta_{5} + \cdots - 560 \beta_1 \) -955b15 - 2360b13 + 560b8 - 216b7 + 560b6 - 1405b5 + 1405b4 - 560b1

\(n\)	\(145\)	\(353\)	\(991\)	\(1189\)
\(\chi(n)\)	\(\beta_{2}\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 17.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 34 T^{14} + \cdots + 130321 \) T^16 - 34T^14 + 477T^12 - 1772T^10 + 6950T^8 - 26018T^6 + 81612T^4 + 26714*T^2 + 130321
$7$	\( (T^{8} - 13 T^{6} + \cdots + 121)^{2} \) (T^8 - 13T^6 - 50T^5 + 89T^4 + 650T^3 + 963T^2 + 440T + 121)^2
$11$	\( T^{16} + \cdots + 214358881 \) T^16 + 44T^14 + 995T^12 + 15646T^10 + 192109T^8 + 1893166T^6 + 14567795T^4 + 77948684*T^2 + 214358881
$13$	\( (T^{8} - 35 T^{6} + \cdots + 126025)^{2} \) (T^8 - 35T^6 - 210T^5 + 835T^4 + 7350T^3 + 2425T^2 - 21300T + 126025)^2
$17$	\( T^{16} + 2 T^{14} + \cdots + 923521 \) T^16 + 2T^14 + 2148T^12 + 81454T^10 + 1310750T^8 + 8775124T^6 + 24959373T^4 - 2708098*T^2 + 923521
$19$	\( (T^{8} - 8 T^{6} + \cdots + 52441)^{2} \) (T^8 - 8T^6 + 210T^5 + 754T^4 - 1680T^3 - 5747T^2 - 6870T + 52441)^2
$23$	\( (T^{8} + 58 T^{6} + \cdots + 5041)^{2} \) (T^8 + 58T^6 + 894T^4 + 3962*T^2 + 5041)^2
$29$	\( T^{16} + \cdots + 236421376 \) T^16 - 8T^14 + 2128T^12 + 51664T^10 + 774240T^8 + 6798464T^6 + 183344448T^4 + 333843712*T^2 + 236421376
$31$	\( (T^{8} + 4 T^{7} + \cdots + 121)^{2} \) (T^8 + 4T^7 + 42T^6 + 272T^5 + 1200T^4 + 2168T^3 + 3997T^2 - 1144*T + 121)^2
$37$	\( (T^{8} - 18 T^{7} + \cdots + 346921)^{2} \) (T^8 - 18T^7 + 203T^6 - 1446T^5 + 7375T^4 - 26226T^3 + 91743T^2 - 236778*T + 346921)^2
$41$	\( T^{16} + 82 T^{14} + \cdots + 25411681 \) T^16 + 82T^14 + 2963T^12 + 45534T^10 + 777065T^8 + 11445414T^6 + 104686283T^4 - 29630998*T^2 + 25411681
$43$	\( (T^{8} + 92 T^{6} + \cdots + 28561)^{2} \) (T^8 + 92T^6 + 2134T^4 + 15548*T^2 + 28561)^2
$47$	\( T^{16} + \cdots + 32573512997041 \) T^16 - 356T^14 + 54597T^12 - 3304768T^10 + 145395890T^8 - 6552831652T^6 + 321765095352T^4 + 1416248876866*T^2 + 32573512997041
$53$	\( T^{16} + \cdots + 831789600625 \) T^16 - 70T^14 + 11820T^12 - 1501250T^10 + 116594150T^8 - 258363500T^6 + 22753696125T^4 - 215283501250*T^2 + 831789600625
$59$	\( T^{16} + \cdots + 10355301121 \) T^16 + 54T^14 + 18032T^12 - 668688T^10 + 10686730T^8 - 83770872T^6 + 386008217T^4 - 1602125184*T^2 + 10355301121
$61$	\( (T^{8} + 40 T^{7} + \cdots + 2307361)^{2} \) (T^8 + 40T^7 + 793T^6 + 9680T^5 + 77724T^4 + 415240T^3 + 1422862T^2 + 2764580*T + 2307361)^2
$67$	\( (T^{4} - 16 T^{3} + \cdots - 29)^{4} \) (T^4 - 16T^3 + T^2 + 44T - 29)^4
$71$	\( T^{16} + \cdots + 25401852000625 \) T^16 - 170T^14 + 25845T^12 - 3683900T^10 + 727194150T^8 + 8069173750T^6 + 1358132775500T^4 - 9535475298750*T^2 + 25401852000625
$73$	\( (T^{8} + 10 T^{7} + \cdots + 3857296)^{2} \) (T^8 + 10T^7 - 62T^6 - 320T^5 + 5424T^4 - 14360T^3 - 17328T^2 - 510640*T + 3857296)^2
$79$	\( (T^{8} - 20 T^{7} + \cdots + 78961)^{2} \) (T^8 - 20T^7 + 237T^6 - 1010T^5 + 2039T^4 - 8770T^3 + 45713T^2 - 101160*T + 78961)^2
$83$	\( T^{16} + \cdots + 1804403844961 \) T^16 - 12T^14 + 15383T^12 + 1246906T^10 + 44564205T^8 + 769189546T^6 + 11505429543T^4 + 166550724628*T^2 + 1804403844961
$89$	\( (T^{8} + 522 T^{6} + \cdots + 33074001)^{2} \) (T^8 + 522T^6 + 72414T^4 + 2888298*T^2 + 33074001)^2
$97$	\( (T^{8} - 6 T^{7} + \cdots + 27552001)^{2} \) (T^8 - 6T^7 + 117T^6 - 1148T^5 + 26340T^4 + 202798T^3 + 2974902T^2 + 13510926*T + 27552001)^2
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