LMFDB - Newform orbit 2016.2.bs.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(271,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))

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

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

chi := DirichletCharacter("2016.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.bs (of order $6$, degree $2$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.144054149089536.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 $ x^12 - 3x^11 + x^9 + 48x^8 - 189x^7 + 431x^6 - 654x^5 + 624x^4 - 340x^3 + 96x^2 - 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{7} q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{7}+O(q^{10}) $ q - b7 * q^5 + (-b11 + b10 + b5) * q^7	$ q - \beta_{7} q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{11} - \beta_{7} + 2 \beta_{5}) q^{13} + (\beta_{9} + \beta_{3} - \beta_1 + 1) q^{17} + ( - 2 \beta_{8} + \beta_{6} + \beta_{3} + \cdots + 1) q^{19}+ \cdots + (7 \beta_{9} - 7 \beta_{8} + 2 \beta_{6} + \cdots + 6) q^{97}+O(q^{100}) $ q - b7 * q^5 + (-b11 + b10 + b5) * q^7 + (-b6 + b3 + b1) * q^11 + (-b11 - b7 + 2b5) q^13 + (b9 + b3 - b1 + 1) * q^17 + (-2b8 + b6 + b3 + 2b1 + 1) * q^19 + (-b10 + b7 + 2b4 + b2) q^23 + (-2b3 - 2b1) * q^25 + (-b11 + 2b10 + b7) q^29 + (-b11 - 2b10 + b7 - b5 + b2) q^31 + (b9 - 3b8 + b6 + b3 + 3b1 + 3) * q^35 + (2b10 + 2b7 - 3b5 + b4 - 2b2) * q^37 + (3b9 - 3b8 + 2b6 - 2b3 - b1 + 2) * q^41 + (-2b11 + b10 + 2b7 + b5 - b4 + b2) * q^47 + (b9 - 3b8 + 2b3 - b1 + 1) * q^49 + (-b11 - b7 + b5 - 4b2) q^53 + (b10 + b4 - 2b2) q^55 + (b9 - 2b6 + 2b3 - 2b1 + 5) q^59 + (-2b11 - 2b10 - 2b7 + b5 - b4 - 2b2) * q^61 + (2b9 - b8 - b6 - b3 + 3) q^65 + (b9 - 2b8 - 4b6 - 2b3 - 2b1 + 1) * q^67 + (2b11 + 2b7 + 2b4) q^71 + (-b6 - 2b3 + 2b1 + 2) * q^73 + (-b11 + 2b7 - 2b5 - b4) * q^77 + (b10 + b7 + 2b5 + 4b4 - b2) * q^79 + (-2b9 + 2b8 - 8b6 - 4b3 - 2b1 + 2) q^83 + (-b11 + 4b10 + 3b7 + b4) * q^85 + (-b6 - 1) * q^89 + (b9 - 3b8 + 2b3 - b1 + 8) * q^91 + (2b11 - b7 - 2b5 + 3b4 - 5b2) * q^95 + (7b9 - 7b8 + 2b6 + 2b3 + b1 + 6) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 6 q^{11} + 6 q^{17} + 6 q^{19} + 18 q^{35} - 12 q^{49} + 42 q^{59} + 12 q^{65} - 30 q^{67} + 18 q^{73} - 18 q^{89} + 72 q^{91}+O(q^{100}) $ 12 * q - 6 * q^11 + 6 * q^17 + 6 * q^19 + 18 * q^35 - 12 * q^49 + 42 * q^59 + 12 * q^65 - 30 * q^67 + 18 * q^73 - 18 * q^89 + 72 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 $ x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4 :

$\beta_{1}$	$=$	$ ( - 6789 \nu^{11} + 407977 \nu^{10} - 701704 \nu^{9} - 1014429 \nu^{8} - 865016 \nu^{7} + \cdots - 9570816 ) / 5500348 $ (-6789v^11 + 407977v^10 - 701704v^9 - 1014429v^8 - 865016v^7 + 19540967v^6 - 53823631v^5 + 103674982v^4 - 118229724v^3 + 58499210v^2 - 2665028*v - 9570816) / 5500348
$\beta_{2}$	$=$	$ ( - 76837 \nu^{11} - 171179 \nu^{10} + 509986 \nu^{9} + 1428477 \nu^{8} - 2520360 \nu^{7} + \cdots - 7508164 ) / 5500348 $ (-76837v^11 - 171179v^10 + 509986v^9 + 1428477v^8 - 2520360v^7 - 4710591v^6 + 8507441v^5 - 20646084v^4 + 15569686v^3 + 23756054v^2 - 19439928*v - 7508164) / 5500348
$\beta_{3}$	$=$	$ ( 54041 \nu^{11} - 302383 \nu^{10} + 439891 \nu^{9} + 184300 \nu^{8} + 2261772 \nu^{7} + \cdots - 178398 ) / 2750174 $ (54041v^11 - 302383v^10 + 439891v^9 + 184300v^8 + 2261772v^7 - 17461376v^6 + 49851815v^5 - 91525793v^4 + 115305627v^3 - 80402588v^2 + 26100310*v - 178398) / 2750174
$\beta_{4}$	$=$	$ ( - 127195 \nu^{11} + 82977 \nu^{10} + 563109 \nu^{9} + 592838 \nu^{8} - 5748844 \nu^{7} + \cdots - 2572044 ) / 2750174 $ (-127195v^11 + 82977v^10 + 563109v^9 + 592838v^8 - 5748844v^7 + 9850368v^6 - 14441625v^5 + 3585319v^4 + 16374455v^3 - 12968760v^2 + 70472*v - 2572044) / 2750174
$\beta_{5}$	$=$	$ ( 295642 \nu^{11} - 410251 \nu^{10} - 714509 \nu^{9} - 877952 \nu^{8} + 12614519 \nu^{7} + \cdots + 15787424 ) / 5500348 $ (295642v^11 - 410251v^10 - 714509v^9 - 877952v^8 + 12614519v^7 - 34529048v^6 + 71188533v^5 - 75428613v^4 + 41121960v^3 + 4078592v^2 - 31162314*v + 15787424) / 5500348
$\beta_{6}$	$=$	$ ( 2236 \nu^{11} - 4785 \nu^{10} - 6307 \nu^{9} + 1316 \nu^{8} + 113135 \nu^{7} - 323728 \nu^{6} + \cdots + 27316 ) / 41356 $ (2236v^11 - 4785v^10 - 6307v^9 + 1316v^8 + 113135v^7 - 323728v^6 + 580379v^5 - 648191v^4 + 212358v^3 + 208368v^2 - 142154*v + 27316) / 41356
$\beta_{7}$	$=$	$ ( - 588552 \nu^{11} + 1471277 \nu^{10} + 959613 \nu^{9} - 594282 \nu^{8} - 29122539 \nu^{7} + \cdots + 3770204 ) / 5500348 $ (-588552v^11 + 1471277v^10 + 959613v^9 - 594282v^8 - 29122539v^7 + 96716674v^6 - 193796667v^5 + 255549677v^4 - 178977012v^3 + 33607448v^2 + 7943010*v + 3770204) / 5500348
$\beta_{8}$	$=$	$ ( 658600 \nu^{11} - 892121 \nu^{10} - 2171303 \nu^{9} - 1848624 \nu^{8} + 30777883 \nu^{7} + \cdots - 5832856 ) / 5500348 $ (658600v^11 - 892121v^10 - 2171303v^9 - 1848624v^8 + 30777883v^7 - 72465116v^6 + 131465595v^5 - 131228611v^4 + 45177602v^3 + 1135708v^2 + 19832586*v - 5832856) / 5500348
$\beta_{9}$	$=$	$ ( - 721311 \nu^{11} + 2509218 \nu^{10} - 349445 \nu^{9} - 2200105 \nu^{8} - 35847041 \nu^{7} + \cdots - 2329388 ) / 5500348 $ (-721311v^11 + 2509218v^10 - 349445v^9 - 2200105v^8 - 35847041v^7 + 152603387v^6 - 342514582v^5 + 520339833v^4 - 478822416v^3 + 211570982v^2 - 29719866*v - 2329388) / 5500348
$\beta_{10}$	$=$	$ ( 944653 \nu^{11} - 1864021 \nu^{10} - 2371448 \nu^{9} - 304551 \nu^{8} + 45642708 \nu^{7} + \cdots + 938824 ) / 5500348 $ (944653v^11 - 1864021v^10 - 2371448v^9 - 304551v^8 + 45642708v^7 - 132130403v^6 + 248926171v^5 - 285355566v^4 + 136789536v^3 + 36475726v^2 - 33967380*v + 938824) / 5500348
$\beta_{11}$	$=$	$ ( - 4510 \nu^{11} + 15163 \nu^{10} - 1205 \nu^{9} - 9766 \nu^{8} - 224935 \nu^{7} + 920210 \nu^{6} + \cdots + 76880 ) / 26068 $ (-4510v^11 + 15163v^10 - 1205v^9 - 9766v^8 - 224935v^7 + 920210v^6 - 2085585v^5 + 3227207v^4 - 3042430v^3 + 1622152v^2 - 452518*v + 76880) / 26068

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

$\nu$	$=$	$ ( \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 ) / 2 $ (b8 + b7 + b2 - b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{11} - 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - 2\beta_{7} - 4\beta_{6} + \beta_{4} - 3\beta_{3} - 2\beta _1 + 1 ) / 2 $ (b11 - 2b10 - 2b9 + 3b8 - 2b7 - 4b6 + b4 - 3b3 - 2*b1 + 1) / 2
$\nu^{3}$	$=$	$ ( \beta_{11} + 2\beta_{10} + 7\beta_{7} + 14\beta_{6} + 7\beta_{4} + 7\beta_{3} - \beta_{2} + 5\beta _1 - 3 ) / 2 $ (b11 + 2b10 + 7b7 + 14b6 + 7b4 + 7b3 - b2 + 5b1 - 3) / 2
$\nu^{4}$	$=$	$ ( 4 \beta_{11} - 20 \beta_{10} - 8 \beta_{9} + 16 \beta_{8} - 12 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + \cdots - 25 ) / 2 $ (4b11 - 20b10 - 8b9 + 16b8 - 12b7 - 15b6 + 3b5 - 21b4 - 14b3 + 14b2 - 22*b1 - 25) / 2
$\nu^{5}$	$=$	$ ( 14 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} - 3 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 13 \beta_{5} + \cdots + 56 ) / 2 $ (14b11 + 13b10 - 37b9 - 3b8 + 8b7 - 5b6 - 13b5 + 62b4 - 13b3 - 51b2 + 55*b1 + 56) / 2
$\nu^{6}$	$=$	$ ( - 11 \beta_{11} + 24 \beta_{10} + 54 \beta_{9} - 9 \beta_{8} + 136 \beta_{7} + 217 \beta_{6} + \cdots - 278 ) / 2 $ (-11b11 + 24b10 + 54b9 - 9b8 + 136b7 + 217b6 + 9b5 - 68b4 + 119b3 + 78b2 - 62*b1 - 278) / 2
$\nu^{7}$	$=$	$ ( 114 \beta_{11} - 291 \beta_{10} - 326 \beta_{9} + 194 \beta_{8} - 542 \beta_{7} - 901 \beta_{6} + \cdots + 247 ) / 2 $ (114b11 - 291b10 - 326b9 + 194b8 - 542b7 - 901b6 - 55b5 - 141b4 - 580b3 - 148b2 - 165*b1 + 247) / 2
$\nu^{8}$	$=$	$ ( - 113 \beta_{11} + 1276 \beta_{10} + 404 \beta_{9} - 940 \beta_{8} + 1655 \beta_{7} + 2527 \beta_{6} + \cdots + 96 ) / 2 $ (-113b11 + 1276b10 + 404b9 - 940b8 + 1655b7 + 2527b6 - 195b5 + 1312b4 + 1588b3 - 652b2 + 1500*b1 + 96) / 2
$\nu^{9}$	$=$	$ ( - 25 \beta_{11} - 2766 \beta_{10} + 72 \beta_{9} + 1773 \beta_{8} - 2716 \beta_{7} - 3969 \beta_{6} + \cdots - 4406 ) / 2 $ (-25b11 - 2766b10 + 72b9 + 1773b8 - 2716b7 - 3969b6 + 525b5 - 5278b4 - 2352b3 + 2895b2 - 5001*b1 - 4406) / 2
$\nu^{10}$	$=$	$ ( 1301 \beta_{11} + 4480 \beta_{10} - 4337 \beta_{9} - 3460 \beta_{8} - 1298 \beta_{7} - 4210 \beta_{6} + \cdots + 18488 ) / 2 $ (1301b11 + 4480b10 - 4337b9 - 3460b8 - 1298b7 - 4210b6 - 2532b5 + 12294b4 - 2171b3 - 10904b2 + 12175*b1 + 18488) / 2
$\nu^{11}$	$=$	$ ( - 7243 \beta_{11} + 10057 \beta_{10} + 22468 \beta_{9} - 7768 \beta_{8} + 29731 \beta_{7} + \cdots - 47282 ) / 2 $ (-7243b11 + 10057b10 + 22468b9 - 7768b8 + 29731b7 + 51343b6 + 4507b5 - 15000b4 + 33291b3 + 21451b2 - 10568*b1 - 47282) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1 - \beta_{6}$	$-1$	$1$

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} $

271.1

0.186445	−	1.54034i
0.609850	+	0.457915i
−0.0263223	−	0.217464i
1.09935	−	0.468876i
2.00233	+	0.854000i
−2.37165	+	1.78079i
0.186445	+	1.54034i
0.609850	−	0.457915i
−0.0263223	+	0.217464i
1.09935	+	0.468876i
2.00233	−	0.854000i
−2.37165	−	1.78079i

−1.59713

2.76632i

−0.694153

−

2.55307i

271.2

−1.03926

1.80005i

−1.25203

2.33076i

271.3

−0.345107

0.597743i

2.63639

−

0.222310i

271.4

0.345107

−

0.597743i

−2.63639

0.222310i

271.5

1.03926

−

1.80005i

1.25203

−

2.33076i

271.6

1.59713

−

2.76632i

0.694153

2.55307i

1711.1

−1.59713

−

2.76632i

−0.694153

2.55307i

1711.2

−1.03926

−

1.80005i

−1.25203

−

2.33076i

1711.3

−0.345107

−

0.597743i

2.63639

0.222310i

1711.4

0.345107

0.597743i

−2.63639

−

0.222310i

1711.5

1.03926

1.80005i

1.25203

2.33076i

1711.6

1.59713

2.76632i

0.694153

−

2.55307i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
8.d	odd	2	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2016.2.bs.a		12
3.b	odd	2	1		224.2.q.a		12
4.b	odd	2	1		504.2.bk.a		12
7.d	odd	6	1	inner	2016.2.bs.a		12
8.b	even	2	1		504.2.bk.a		12
8.d	odd	2	1	inner	2016.2.bs.a		12
12.b	even	2	1		56.2.m.a	✓	12
21.c	even	2	1		1568.2.q.g		12
21.g	even	6	1		224.2.q.a		12
21.g	even	6	1		1568.2.e.e		12
21.h	odd	6	1		1568.2.e.e		12
21.h	odd	6	1		1568.2.q.g		12
24.f	even	2	1		224.2.q.a		12
24.h	odd	2	1		56.2.m.a	✓	12
28.f	even	6	1		504.2.bk.a		12
56.j	odd	6	1		504.2.bk.a		12
56.m	even	6	1	inner	2016.2.bs.a		12
84.h	odd	2	1		392.2.m.g		12
84.j	odd	6	1		56.2.m.a	✓	12
84.j	odd	6	1		392.2.e.e		12
84.n	even	6	1		392.2.e.e		12
84.n	even	6	1		392.2.m.g		12
168.e	odd	2	1		1568.2.q.g		12
168.i	even	2	1		392.2.m.g		12
168.s	odd	6	1		392.2.e.e		12
168.s	odd	6	1		392.2.m.g		12
168.v	even	6	1		1568.2.e.e		12
168.v	even	6	1		1568.2.q.g		12
168.ba	even	6	1		56.2.m.a	✓	12
168.ba	even	6	1		392.2.e.e		12
168.be	odd	6	1		224.2.q.a		12
168.be	odd	6	1		1568.2.e.e		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.m.a	✓	12	12.b	even	2	1
56.2.m.a	✓	12	24.h	odd	2	1
56.2.m.a	✓	12	84.j	odd	6	1
56.2.m.a	✓	12	168.ba	even	6	1
224.2.q.a		12	3.b	odd	2	1
224.2.q.a		12	21.g	even	6	1
224.2.q.a		12	24.f	even	2	1
224.2.q.a		12	168.be	odd	6	1
392.2.e.e		12	84.j	odd	6	1
392.2.e.e		12	84.n	even	6	1
392.2.e.e		12	168.s	odd	6	1
392.2.e.e		12	168.ba	even	6	1
392.2.m.g		12	84.h	odd	2	1
392.2.m.g		12	84.n	even	6	1
392.2.m.g		12	168.i	even	2	1
392.2.m.g		12	168.s	odd	6	1
504.2.bk.a		12	4.b	odd	2	1
504.2.bk.a		12	8.b	even	2	1
504.2.bk.a		12	28.f	even	6	1
504.2.bk.a		12	56.j	odd	6	1
1568.2.e.e		12	21.g	even	6	1
1568.2.e.e		12	21.h	odd	6	1
1568.2.e.e		12	168.v	even	6	1
1568.2.e.e		12	168.be	odd	6	1
1568.2.q.g		12	21.c	even	2	1
1568.2.q.g		12	21.h	odd	6	1
1568.2.q.g		12	168.e	odd	2	1
1568.2.q.g		12	168.v	even	6	1
2016.2.bs.a		12	1.a	even	1	1	trivial
2016.2.bs.a		12	7.d	odd	6	1	inner
2016.2.bs.a		12	8.d	odd	2	1	inner
2016.2.bs.a		12	56.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} + 15T_{5}^{10} + 174T_{5}^{8} + 723T_{5}^{6} + 2286T_{5}^{4} + 1071T_{5}^{2} + 441 $ T5^12 + 15*T5^10 + 174*T5^8 + 723*T5^6 + 2286*T5^4 + 1071*T5^2 + 441 acting on $S_{2}^{\mathrm{new}}(2016, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 15 T^{10} + \cdots + 441 $ T^12 + 15T^10 + 174T^8 + 723T^6 + 2286T^4 + 1071*T^2 + 441
$7$	$ T^{12} + 6 T^{10} + \cdots + 117649 $ T^12 + 6T^10 - 33T^8 - 700T^6 - 1617T^4 + 14406*T^2 + 117649
$11$	$ (T^{6} + 3 T^{5} + 12 T^{4} + \cdots + 49)^{2} $ (T^6 + 3T^5 + 12T^4 + 5T^3 + 30T^2 + 21*T + 49)^2
$13$	$ (T^{6} - 36 T^{4} + \cdots - 336)^{2} $ (T^6 - 36T^4 + 240T^2 - 336)^2
$17$	$ (T^{6} - 3 T^{5} + \cdots + 1083)^{2} $ (T^6 - 3T^5 - 12T^4 + 45T^3 + 168T^2 - 855*T + 1083)^2
$19$	$ (T^{6} - 3 T^{5} + \cdots + 147)^{2} $ (T^6 - 3T^5 - 18T^4 + 63T^3 + 420T^2 - 441*T + 147)^2
$23$	$ T^{12} - 69 T^{10} + \cdots + 45252529 $ T^12 - 69T^10 + 3450T^8 - 77005T^6 + 1254558T^4 - 8819097*T^2 + 45252529
$29$	$ (T^{6} + 48 T^{4} + \cdots + 112)^{2} $ (T^6 + 48T^4 + 576T^2 + 112)^2
$31$	$ T^{12} + 99 T^{10} + \cdots + 441 $ T^12 + 99T^10 + 8238T^8 + 154695T^6 + 2440890T^4 + 32823*T^2 + 441
$37$	$ T^{12} + \cdots + 3074369809 $ T^12 - 117T^10 + 9222T^8 - 411745T^6 + 13466790T^4 - 247681749*T^2 + 3074369809
$41$	$ (T^{6} + 144 T^{4} + \cdots + 52272)^{2} $ (T^6 + 144T^4 + 5184T^2 + 52272)^2
$43$	$ T^{12} $ T^12
$47$	$ T^{12} + 123 T^{10} + \cdots + 6456681 $ T^12 + 123T^10 + 12990T^8 + 258015T^6 + 4262778T^4 + 5435199*T^2 + 6456681
$53$	$ T^{12} + \cdots + 108651322129 $ T^12 - 213T^10 + 30630T^8 - 2480161T^6 + 147028422T^4 - 4858313397*T^2 + 108651322129
$59$	$ (T^{6} - 21 T^{5} + \cdots + 133563)^{2} $ (T^6 - 21T^5 + 132T^4 + 315T^3 - 4206T^2 - 9495*T + 133563)^2
$61$	$ T^{12} + \cdots + 42362284041 $ T^12 + 207T^10 + 30294T^8 + 2187243T^6 + 115023078T^4 + 2584082655*T^2 + 42362284041
$67$	$ (T^{6} + 15 T^{5} + \cdots + 52441)^{2} $ (T^6 + 15T^5 + 198T^4 + 863T^3 + 4164T^2 - 6183*T + 52441)^2
$71$	$ (T^{2} + 28)^{6} $ (T^2 + 28)^6
$73$	$ (T^{6} - 9 T^{5} + \cdots + 1323)^{2} $ (T^6 - 9T^5 - 36T^4 + 567T^3 + 3780T^2 - 3969*T + 1323)^2
$79$	$ T^{12} + \cdots + 31317319089 $ T^12 - 213T^10 + 34074T^8 - 2051901T^6 + 89883054T^4 - 1998842265*T^2 + 31317319089
$83$	$ (T^{6} + 228 T^{4} + \cdots + 192)^{2} $ (T^6 + 228T^4 + 816T^2 + 192)^2
$89$	$ (T^{2} + 3 T + 3)^{6} $ (T^2 + 3*T + 3)^6
$97$	$ (T^{6} + 360 T^{4} + \cdots + 134832)^{2} $ (T^6 + 360T^4 + 19248T^2 + 134832)^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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{7}+O(q^{10}) \) q - b7 * q^5 + (-b11 + b10 + b5) * q^7	\( q - \beta_{7} q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{11} - \beta_{7} + 2 \beta_{5}) q^{13} + (\beta_{9} + \beta_{3} - \beta_1 + 1) q^{17} + ( - 2 \beta_{8} + \beta_{6} + \beta_{3} + \cdots + 1) q^{19}+ \cdots + (7 \beta_{9} - 7 \beta_{8} + 2 \beta_{6} + \cdots + 6) q^{97}+O(q^{100}) \) q - b7 * q^5 + (-b11 + b10 + b5) * q^7 + (-b6 + b3 + b1) * q^11 + (-b11 - b7 + 2b5) q^13 + (b9 + b3 - b1 + 1) * q^17 + (-2b8 + b6 + b3 + 2b1 + 1) * q^19 + (-b10 + b7 + 2b4 + b2) q^23 + (-2b3 - 2b1) * q^25 + (-b11 + 2b10 + b7) q^29 + (-b11 - 2b10 + b7 - b5 + b2) q^31 + (b9 - 3b8 + b6 + b3 + 3b1 + 3) * q^35 + (2b10 + 2b7 - 3b5 + b4 - 2b2) * q^37 + (3b9 - 3b8 + 2b6 - 2b3 - b1 + 2) * q^41 + (-2b11 + b10 + 2b7 + b5 - b4 + b2) * q^47 + (b9 - 3b8 + 2b3 - b1 + 1) * q^49 + (-b11 - b7 + b5 - 4b2) q^53 + (b10 + b4 - 2b2) q^55 + (b9 - 2b6 + 2b3 - 2b1 + 5) q^59 + (-2b11 - 2b10 - 2b7 + b5 - b4 - 2b2) * q^61 + (2b9 - b8 - b6 - b3 + 3) q^65 + (b9 - 2b8 - 4b6 - 2b3 - 2b1 + 1) * q^67 + (2b11 + 2b7 + 2b4) q^71 + (-b6 - 2b3 + 2b1 + 2) * q^73 + (-b11 + 2b7 - 2b5 - b4) * q^77 + (b10 + b7 + 2b5 + 4b4 - b2) * q^79 + (-2b9 + 2b8 - 8b6 - 4b3 - 2b1 + 2) q^83 + (-b11 + 4b10 + 3b7 + b4) * q^85 + (-b6 - 1) * q^89 + (b9 - 3b8 + 2b3 - b1 + 8) * q^91 + (2b11 - b7 - 2b5 + 3b4 - 5b2) * q^95 + (7b9 - 7b8 + 2b6 + 2b3 + b1 + 6) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 6 q^{11} + 6 q^{17} + 6 q^{19} + 18 q^{35} - 12 q^{49} + 42 q^{59} + 12 q^{65} - 30 q^{67} + 18 q^{73} - 18 q^{89} + 72 q^{91}+O(q^{100}) \) 12 * q - 6 * q^11 + 6 * q^17 + 6 * q^19 + 18 * q^35 - 12 * q^49 + 42 * q^59 + 12 * q^65 - 30 * q^67 + 18 * q^73 - 18 * q^89 + 72 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	2016.2.bs.a

Level	$2016$
Weight	$2$
Character orbit	2016.bs
Analytic conductor	$16.098$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.0978410475\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.144054149089536.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) x^12 - 3x^11 + x^9 + 48x^8 - 189x^7 + 431x^6 - 654x^5 + 624x^4 - 340x^3 + 96x^2 - 12*x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 6789 \nu^{11} + 407977 \nu^{10} - 701704 \nu^{9} - 1014429 \nu^{8} - 865016 \nu^{7} + \cdots - 9570816 ) / 5500348 \) (-6789v^11 + 407977v^10 - 701704v^9 - 1014429v^8 - 865016v^7 + 19540967v^6 - 53823631v^5 + 103674982v^4 - 118229724v^3 + 58499210v^2 - 2665028*v - 9570816) / 5500348
\(\beta_{2}\)	\(=\)	\( ( - 76837 \nu^{11} - 171179 \nu^{10} + 509986 \nu^{9} + 1428477 \nu^{8} - 2520360 \nu^{7} + \cdots - 7508164 ) / 5500348 \) (-76837v^11 - 171179v^10 + 509986v^9 + 1428477v^8 - 2520360v^7 - 4710591v^6 + 8507441v^5 - 20646084v^4 + 15569686v^3 + 23756054v^2 - 19439928*v - 7508164) / 5500348
\(\beta_{3}\)	\(=\)	\( ( 54041 \nu^{11} - 302383 \nu^{10} + 439891 \nu^{9} + 184300 \nu^{8} + 2261772 \nu^{7} + \cdots - 178398 ) / 2750174 \) (54041v^11 - 302383v^10 + 439891v^9 + 184300v^8 + 2261772v^7 - 17461376v^6 + 49851815v^5 - 91525793v^4 + 115305627v^3 - 80402588v^2 + 26100310*v - 178398) / 2750174
\(\beta_{4}\)	\(=\)	\( ( - 127195 \nu^{11} + 82977 \nu^{10} + 563109 \nu^{9} + 592838 \nu^{8} - 5748844 \nu^{7} + \cdots - 2572044 ) / 2750174 \) (-127195v^11 + 82977v^10 + 563109v^9 + 592838v^8 - 5748844v^7 + 9850368v^6 - 14441625v^5 + 3585319v^4 + 16374455v^3 - 12968760v^2 + 70472*v - 2572044) / 2750174
\(\beta_{5}\)	\(=\)	\( ( 295642 \nu^{11} - 410251 \nu^{10} - 714509 \nu^{9} - 877952 \nu^{8} + 12614519 \nu^{7} + \cdots + 15787424 ) / 5500348 \) (295642v^11 - 410251v^10 - 714509v^9 - 877952v^8 + 12614519v^7 - 34529048v^6 + 71188533v^5 - 75428613v^4 + 41121960v^3 + 4078592v^2 - 31162314*v + 15787424) / 5500348
\(\beta_{6}\)	\(=\)	\( ( 2236 \nu^{11} - 4785 \nu^{10} - 6307 \nu^{9} + 1316 \nu^{8} + 113135 \nu^{7} - 323728 \nu^{6} + \cdots + 27316 ) / 41356 \) (2236v^11 - 4785v^10 - 6307v^9 + 1316v^8 + 113135v^7 - 323728v^6 + 580379v^5 - 648191v^4 + 212358v^3 + 208368v^2 - 142154*v + 27316) / 41356
\(\beta_{7}\)	\(=\)	\( ( - 588552 \nu^{11} + 1471277 \nu^{10} + 959613 \nu^{9} - 594282 \nu^{8} - 29122539 \nu^{7} + \cdots + 3770204 ) / 5500348 \) (-588552v^11 + 1471277v^10 + 959613v^9 - 594282v^8 - 29122539v^7 + 96716674v^6 - 193796667v^5 + 255549677v^4 - 178977012v^3 + 33607448v^2 + 7943010*v + 3770204) / 5500348
\(\beta_{8}\)	\(=\)	\( ( 658600 \nu^{11} - 892121 \nu^{10} - 2171303 \nu^{9} - 1848624 \nu^{8} + 30777883 \nu^{7} + \cdots - 5832856 ) / 5500348 \) (658600v^11 - 892121v^10 - 2171303v^9 - 1848624v^8 + 30777883v^7 - 72465116v^6 + 131465595v^5 - 131228611v^4 + 45177602v^3 + 1135708v^2 + 19832586*v - 5832856) / 5500348
\(\beta_{9}\)	\(=\)	\( ( - 721311 \nu^{11} + 2509218 \nu^{10} - 349445 \nu^{9} - 2200105 \nu^{8} - 35847041 \nu^{7} + \cdots - 2329388 ) / 5500348 \) (-721311v^11 + 2509218v^10 - 349445v^9 - 2200105v^8 - 35847041v^7 + 152603387v^6 - 342514582v^5 + 520339833v^4 - 478822416v^3 + 211570982v^2 - 29719866*v - 2329388) / 5500348
\(\beta_{10}\)	\(=\)	\( ( 944653 \nu^{11} - 1864021 \nu^{10} - 2371448 \nu^{9} - 304551 \nu^{8} + 45642708 \nu^{7} + \cdots + 938824 ) / 5500348 \) (944653v^11 - 1864021v^10 - 2371448v^9 - 304551v^8 + 45642708v^7 - 132130403v^6 + 248926171v^5 - 285355566v^4 + 136789536v^3 + 36475726v^2 - 33967380*v + 938824) / 5500348
\(\beta_{11}\)	\(=\)	\( ( - 4510 \nu^{11} + 15163 \nu^{10} - 1205 \nu^{9} - 9766 \nu^{8} - 224935 \nu^{7} + 920210 \nu^{6} + \cdots + 76880 ) / 26068 \) (-4510v^11 + 15163v^10 - 1205v^9 - 9766v^8 - 224935v^7 + 920210v^6 - 2085585v^5 + 3227207v^4 - 3042430v^3 + 1622152v^2 - 452518*v + 76880) / 26068

\(\nu\)	\(=\)	\( ( \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 ) / 2 \) (b8 + b7 + b2 - b1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} - 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - 2\beta_{7} - 4\beta_{6} + \beta_{4} - 3\beta_{3} - 2\beta _1 + 1 ) / 2 \) (b11 - 2b10 - 2b9 + 3b8 - 2b7 - 4b6 + b4 - 3b3 - 2*b1 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + 2\beta_{10} + 7\beta_{7} + 14\beta_{6} + 7\beta_{4} + 7\beta_{3} - \beta_{2} + 5\beta _1 - 3 ) / 2 \) (b11 + 2b10 + 7b7 + 14b6 + 7b4 + 7b3 - b2 + 5b1 - 3) / 2
\(\nu^{4}\)	\(=\)	\( ( 4 \beta_{11} - 20 \beta_{10} - 8 \beta_{9} + 16 \beta_{8} - 12 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + \cdots - 25 ) / 2 \) (4b11 - 20b10 - 8b9 + 16b8 - 12b7 - 15b6 + 3b5 - 21b4 - 14b3 + 14b2 - 22*b1 - 25) / 2
\(\nu^{5}\)	\(=\)	\( ( 14 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} - 3 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 13 \beta_{5} + \cdots + 56 ) / 2 \) (14b11 + 13b10 - 37b9 - 3b8 + 8b7 - 5b6 - 13b5 + 62b4 - 13b3 - 51b2 + 55*b1 + 56) / 2
\(\nu^{6}\)	\(=\)	\( ( - 11 \beta_{11} + 24 \beta_{10} + 54 \beta_{9} - 9 \beta_{8} + 136 \beta_{7} + 217 \beta_{6} + \cdots - 278 ) / 2 \) (-11b11 + 24b10 + 54b9 - 9b8 + 136b7 + 217b6 + 9b5 - 68b4 + 119b3 + 78b2 - 62*b1 - 278) / 2
\(\nu^{7}\)	\(=\)	\( ( 114 \beta_{11} - 291 \beta_{10} - 326 \beta_{9} + 194 \beta_{8} - 542 \beta_{7} - 901 \beta_{6} + \cdots + 247 ) / 2 \) (114b11 - 291b10 - 326b9 + 194b8 - 542b7 - 901b6 - 55b5 - 141b4 - 580b3 - 148b2 - 165*b1 + 247) / 2
\(\nu^{8}\)	\(=\)	\( ( - 113 \beta_{11} + 1276 \beta_{10} + 404 \beta_{9} - 940 \beta_{8} + 1655 \beta_{7} + 2527 \beta_{6} + \cdots + 96 ) / 2 \) (-113b11 + 1276b10 + 404b9 - 940b8 + 1655b7 + 2527b6 - 195b5 + 1312b4 + 1588b3 - 652b2 + 1500*b1 + 96) / 2
\(\nu^{9}\)	\(=\)	\( ( - 25 \beta_{11} - 2766 \beta_{10} + 72 \beta_{9} + 1773 \beta_{8} - 2716 \beta_{7} - 3969 \beta_{6} + \cdots - 4406 ) / 2 \) (-25b11 - 2766b10 + 72b9 + 1773b8 - 2716b7 - 3969b6 + 525b5 - 5278b4 - 2352b3 + 2895b2 - 5001*b1 - 4406) / 2
\(\nu^{10}\)	\(=\)	\( ( 1301 \beta_{11} + 4480 \beta_{10} - 4337 \beta_{9} - 3460 \beta_{8} - 1298 \beta_{7} - 4210 \beta_{6} + \cdots + 18488 ) / 2 \) (1301b11 + 4480b10 - 4337b9 - 3460b8 - 1298b7 - 4210b6 - 2532b5 + 12294b4 - 2171b3 - 10904b2 + 12175*b1 + 18488) / 2
\(\nu^{11}\)	\(=\)	\( ( - 7243 \beta_{11} + 10057 \beta_{10} + 22468 \beta_{9} - 7768 \beta_{8} + 29731 \beta_{7} + \cdots - 47282 ) / 2 \) (-7243b11 + 10057b10 + 22468b9 - 7768b8 + 29731b7 + 51343b6 + 4507b5 - 15000b4 + 33291b3 + 21451b2 - 10568*b1 - 47282) / 2

\(n\)	\(127\)	\(577\)	\(1765\)	\(1793\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{6}\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 271.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 15 T^{10} + \cdots + 441 \) T^12 + 15T^10 + 174T^8 + 723T^6 + 2286T^4 + 1071*T^2 + 441
$7$	\( T^{12} + 6 T^{10} + \cdots + 117649 \) T^12 + 6T^10 - 33T^8 - 700T^6 - 1617T^4 + 14406*T^2 + 117649
$11$	\( (T^{6} + 3 T^{5} + 12 T^{4} + \cdots + 49)^{2} \) (T^6 + 3T^5 + 12T^4 + 5T^3 + 30T^2 + 21*T + 49)^2
$13$	\( (T^{6} - 36 T^{4} + \cdots - 336)^{2} \) (T^6 - 36T^4 + 240T^2 - 336)^2
$17$	\( (T^{6} - 3 T^{5} + \cdots + 1083)^{2} \) (T^6 - 3T^5 - 12T^4 + 45T^3 + 168T^2 - 855*T + 1083)^2
$19$	\( (T^{6} - 3 T^{5} + \cdots + 147)^{2} \) (T^6 - 3T^5 - 18T^4 + 63T^3 + 420T^2 - 441*T + 147)^2
$23$	\( T^{12} - 69 T^{10} + \cdots + 45252529 \) T^12 - 69T^10 + 3450T^8 - 77005T^6 + 1254558T^4 - 8819097*T^2 + 45252529
$29$	\( (T^{6} + 48 T^{4} + \cdots + 112)^{2} \) (T^6 + 48T^4 + 576T^2 + 112)^2
$31$	\( T^{12} + 99 T^{10} + \cdots + 441 \) T^12 + 99T^10 + 8238T^8 + 154695T^6 + 2440890T^4 + 32823*T^2 + 441
$37$	\( T^{12} + \cdots + 3074369809 \) T^12 - 117T^10 + 9222T^8 - 411745T^6 + 13466790T^4 - 247681749*T^2 + 3074369809
$41$	\( (T^{6} + 144 T^{4} + \cdots + 52272)^{2} \) (T^6 + 144T^4 + 5184T^2 + 52272)^2
$43$	\( T^{12} \) T^12
$47$	\( T^{12} + 123 T^{10} + \cdots + 6456681 \) T^12 + 123T^10 + 12990T^8 + 258015T^6 + 4262778T^4 + 5435199*T^2 + 6456681
$53$	\( T^{12} + \cdots + 108651322129 \) T^12 - 213T^10 + 30630T^8 - 2480161T^6 + 147028422T^4 - 4858313397*T^2 + 108651322129
$59$	\( (T^{6} - 21 T^{5} + \cdots + 133563)^{2} \) (T^6 - 21T^5 + 132T^4 + 315T^3 - 4206T^2 - 9495*T + 133563)^2
$61$	\( T^{12} + \cdots + 42362284041 \) T^12 + 207T^10 + 30294T^8 + 2187243T^6 + 115023078T^4 + 2584082655*T^2 + 42362284041
$67$	\( (T^{6} + 15 T^{5} + \cdots + 52441)^{2} \) (T^6 + 15T^5 + 198T^4 + 863T^3 + 4164T^2 - 6183*T + 52441)^2
$71$	\( (T^{2} + 28)^{6} \) (T^2 + 28)^6
$73$	\( (T^{6} - 9 T^{5} + \cdots + 1323)^{2} \) (T^6 - 9T^5 - 36T^4 + 567T^3 + 3780T^2 - 3969*T + 1323)^2
$79$	\( T^{12} + \cdots + 31317319089 \) T^12 - 213T^10 + 34074T^8 - 2051901T^6 + 89883054T^4 - 1998842265*T^2 + 31317319089
$83$	\( (T^{6} + 228 T^{4} + \cdots + 192)^{2} \) (T^6 + 228T^4 + 816T^2 + 192)^2
$89$	\( (T^{2} + 3 T + 3)^{6} \) (T^2 + 3*T + 3)^6
$97$	\( (T^{6} + 360 T^{4} + \cdots + 134832)^{2} \) (T^6 + 360T^4 + 19248T^2 + 134832)^2
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