LMFDB - Newform orbit 1776.2.e.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1776,2,Mod(815,1776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1776, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1776.815");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1776 = 2^{4} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1776.e (of order $2$, degree $1$, 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:	$14.1814313990$
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} - 5 x^{11} + 10 x^{10} - 11 x^{9} + 15 x^{8} - 42 x^{7} + 92 x^{6} - 126 x^{5} + 135 x^{4} + \cdots + 729 $ x^12 - 5x^11 + 10x^10 - 11x^9 + 15x^8 - 42x^7 + 92x^6 - 126x^5 + 135x^4 - 297x^3 + 810x^2 - 1215*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
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_{8} q^{3} + \beta_{10} q^{5} - \beta_{11} q^{7} + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{9}+O(q^{10}) $ q - b8 * q^3 + b10 * q^5 - b11 * q^7 + (b10 + b9 - b6 - b2) * q^9	$ q - \beta_{8} q^{3} + \beta_{10} q^{5} - \beta_{11} q^{7} + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{9}+ \cdots + (3 \beta_{11} + 3 \beta_{5} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) $ q - b8 * q^3 + b10 * q^5 - b11 * q^7 + (b10 + b9 - b6 - b2) * q^9 + (b8 + b7 + b4 - b1) * q^11 - 2b3 q^13 + (-b11 - b4) * q^15 + (b9 - b2) * q^17 + (b11 - b8 - b1) * q^19 + (2b10 + b3 + b2 - 1) q^21 + (b8 - b7 + 2b4 - b1) q^23 + (b6 + b3 + b2 + 1) * q^25 + (-b11 + b8 - b7 - b5 - b1) * q^27 + (b9 - 2b6 + b2) q^29 + (b11 - b8 - b1) * q^31 + (3b6 - 3) q^33 + (2b8 + b7 - b4 - 2b1) * q^35 + q^37 + (2b5 + 2b4 - 2b1) q^39 + (-b10 + b9 - b6) * q^41 + (-b11 - b8 - 2b5 - b1) q^43 + (b10 + 2b3 + 2b2 - 2) * q^45 + (b8 - b7 - b4 - b1) * q^47 + (3b6 + 2b3 + 3b2 - 1) q^49 + (b11 - b7 - 2b1) q^51 + (-3b10 - b9 + b6) q^53 + (4b11 - 2b8 + 2b5 - 2b1) * q^55 + (-b10 + b9 - b6 - b3 - 2b2 - 2) q^57 + (b8 - b4 - b1) * q^59 + (-2b6 - 2b3 - 2b2 + 4) q^61 + (-2b11 + b8 + b7 - b5 - 2b4) * q^63 + (2b10 + 2b9 - 2b2) q^65 + (b11 - 3b8 + 2b5 - 3b1) q^67 + (b10 - 2b9 - b6 - 3b3 - b2) * q^69 + (b8 + b7 - b4 - b1) * q^71 + (-2b6 + b3 - 2b2 + 6) * q^73 + (b11 - 2b8 + b7 - b4 - b1) q^75 + (-6b10 + 3b6 - 3b2) q^77 + (b11 - b8 + 2b5 - b1) q^79 + (b10 - b9 - 2b6 - 2b3 + 2b2 - 4) q^81 + (b8 - 6b4 - b1) q^83 + (-b6 + b3 - b2 + 2) * q^85 + (-b11 + 2b8 + b7 - 2b5 + 4b4 - 2b1) * q^87 + (b10 - 2b9 + 2b6) * q^89 + (-2b8 - 2b1) * q^91 + (-b10 + b9 - b6 - b3 - 2b2 - 2) q^93 + (-2b8 - b7 - b4 + 2b1) * q^95 + (-2b6 + 2b3 - 2b2 + 8) q^97 + (3b11 + 3b5 - 3b4 - 3b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{9}+O(q^{10}) $ 12 * q - 2 * q^9	$ 12 q - 2 q^{9} + 8 q^{13} - 14 q^{21} + 12 q^{25} - 30 q^{33} + 12 q^{37} - 28 q^{45} - 8 q^{49} - 24 q^{57} + 48 q^{61} + 4 q^{69} + 60 q^{73} - 42 q^{81} + 16 q^{85} - 24 q^{93} + 80 q^{97}+O(q^{100}) $ 12 * q - 2 * q^9 + 8 * q^13 - 14 * q^21 + 12 * q^25 - 30 * q^33 + 12 * q^37 - 28 * q^45 - 8 * q^49 - 24 * q^57 + 48 * q^61 + 4 * q^69 + 60 * q^73 - 42 * q^81 + 16 * q^85 - 24 * q^93 + 80 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 5 x^{11} + 10 x^{10} - 11 x^{9} + 15 x^{8} - 42 x^{7} + 92 x^{6} - 126 x^{5} + 135 x^{4} + \cdots + 729 $ x^12 - 5*x^11 + 10*x^10 - 11*x^9 + 15*x^8 - 42*x^7 + 92*x^6 - 126*x^5 + 135*x^4 - 297*x^3 + 810*x^2 - 1215*x + 729 :

$\beta_{1}$	$=$	$ ( \nu^{11} - 2 \nu^{10} + 4 \nu^{9} + \nu^{8} + 18 \nu^{7} + 12 \nu^{6} + 128 \nu^{5} + 258 \nu^{4} + \cdots - 243 ) / 3888 $ (v^11 - 2v^10 + 4v^9 + v^8 + 18v^7 + 12v^6 + 128v^5 + 258v^4 + 909v^3 - 1458v^2 + 324*v - 243) / 3888
$\beta_{2}$	$=$	$ ( - 17 \nu^{11} + 40 \nu^{10} - 62 \nu^{9} + 25 \nu^{8} - 174 \nu^{7} + 300 \nu^{6} - 592 \nu^{5} + \cdots + 8505 ) / 3888 $ (-17v^11 + 40v^10 - 62v^9 + 25v^8 - 174v^7 + 300v^6 - 592v^5 + 1134v^4 - 1233v^3 + 2916v^2 - 9882*v + 8505) / 3888
$\beta_{3}$	$=$	$ ( 7 \nu^{11} - 19 \nu^{10} + 17 \nu^{9} - 16 \nu^{8} + 46 \nu^{7} - 126 \nu^{6} + 170 \nu^{5} + \cdots - 1620 ) / 648 $ (7v^11 - 19v^10 + 17v^9 - 16v^8 + 46v^7 - 126v^6 + 170v^5 - 220v^4 + 387v^3 - 1143v^2 + 2349*v - 1620) / 648
$\beta_{4}$	$=$	$ ( - 19 \nu^{11} + 83 \nu^{10} - 121 \nu^{9} + 98 \nu^{8} - 252 \nu^{7} + 654 \nu^{6} - 1136 \nu^{5} + \cdots + 12150 ) / 1944 $ (-19v^11 + 83v^10 - 121v^9 + 98v^8 - 252v^7 + 654v^6 - 1136v^5 + 1560v^4 - 1521v^3 + 4455v^2 - 10935*v + 12150) / 1944
$\beta_{5}$	$=$	$ ( 41 \nu^{11} - 163 \nu^{10} + 227 \nu^{9} - 220 \nu^{8} + 450 \nu^{7} - 1200 \nu^{6} + 2278 \nu^{5} + \cdots - 22842 ) / 1944 $ (41v^11 - 163v^10 + 227v^9 - 220v^8 + 450v^7 - 1200v^6 + 2278v^5 - 2274v^4 + 2565v^3 - 9045v^2 + 23895*v - 22842) / 1944
$\beta_{6}$	$=$	$ ( 103 \nu^{11} - 326 \nu^{10} + 472 \nu^{9} - 449 \nu^{8} + 906 \nu^{7} - 2832 \nu^{6} + 5156 \nu^{5} + \cdots - 47385 ) / 3888 $ (103v^11 - 326v^10 + 472v^9 - 449v^8 + 906v^7 - 2832v^6 + 5156v^5 - 5526v^4 + 6255v^3 - 21762v^2 + 48924*v - 47385) / 3888
$\beta_{7}$	$=$	$ ( 17 \nu^{11} - 73 \nu^{10} + 107 \nu^{9} - 70 \nu^{8} + 156 \nu^{7} - 546 \nu^{6} + 1024 \nu^{5} + \cdots - 10530 ) / 648 $ (17v^11 - 73v^10 + 107v^9 - 70v^8 + 156v^7 - 546v^6 + 1024v^5 - 1128v^4 + 939v^3 - 4221v^2 + 10557*v - 10530) / 648
$\beta_{8}$	$=$	$ ( 131 \nu^{11} - 460 \nu^{10} + 578 \nu^{9} - 463 \nu^{8} + 1278 \nu^{7} - 3792 \nu^{6} + 6292 \nu^{5} + \cdots - 56619 ) / 3888 $ (131v^11 - 460v^10 + 578v^9 - 463v^8 + 1278v^7 - 3792v^6 + 6292v^5 - 6342v^4 + 7695v^3 - 28620v^2 + 64962*v - 56619) / 3888
$\beta_{9}$	$=$	$ ( 127 \nu^{11} - 476 \nu^{10} + 646 \nu^{9} - 527 \nu^{8} + 1110 \nu^{7} - 3804 \nu^{6} + 7220 \nu^{5} + \cdots - 65367 ) / 3888 $ (127v^11 - 476v^10 + 646v^9 - 527v^8 + 1110v^7 - 3804v^6 + 7220v^5 - 7098v^4 + 7659v^3 - 26352v^2 + 65934*v - 65367) / 3888
$\beta_{10}$	$=$	$ ( - 43 \nu^{11} + 146 \nu^{10} - 202 \nu^{9} + 152 \nu^{8} - 381 \nu^{7} + 1194 \nu^{6} - 2057 \nu^{5} + \cdots + 19440 ) / 972 $ (-43v^11 + 146v^10 - 202v^9 + 152v^8 - 381v^7 + 1194v^6 - 2057v^5 + 2040v^4 - 2448v^3 + 8532v^2 - 20493*v + 19440) / 972
$\beta_{11}$	$=$	$ ( 140 \nu^{11} - 433 \nu^{10} + 533 \nu^{9} - 427 \nu^{8} + 1224 \nu^{7} - 3486 \nu^{6} + 5986 \nu^{5} + \cdots - 49815 ) / 1944 $ (140v^11 - 433v^10 + 533v^9 - 427v^8 + 1224v^7 - 3486v^6 + 5986v^5 - 5766v^4 + 7092v^3 - 27945v^2 + 59049*v - 49815) / 1944

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

$\nu$	$=$	$ ( -\beta_{11} + 2\beta_{8} + \beta_{5} + 3\beta_{4} - 3\beta_{2} - \beta _1 + 3 ) / 6 $ (-b11 + 2b8 + b5 + 3b4 - 3*b2 - b1 + 3) / 6
$\nu^{2}$	$=$	$ ( -\beta_{11} - 3\beta_{10} + 3\beta_{9} - \beta_{8} - 3\beta_{7} - 3\beta_{6} + \beta_{5} - 3\beta_{2} - 4\beta _1 + 3 ) / 6 $ (-b11 - 3b10 + 3b9 - b8 - 3b7 - 3b6 + b5 - 3b2 - 4b1 + 3) / 6
$\nu^{3}$	$=$	$ ( - 4 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + \cdots + 6 ) / 6 $ (-4b11 - 3b10 + 3b9 + 2b8 - 3b7 - 2b5 - 3b4 + 6b3 - 3b2 + 14b1 + 6) / 6
$\nu^{4}$	$=$	$ ( - \beta_{11} - 9 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} + 7 \beta_{5} + \cdots - 9 ) / 6 $ (-b11 - 9b10 - 3b9 + 2b8 - 3b7 - 6b6 + 7b5 + 12b4 - 6b3 + 6b2 + 11b1 - 9) / 6
$\nu^{5}$	$=$	$ ( 11 \beta_{11} + 12 \beta_{10} + 24 \beta_{9} - 13 \beta_{8} - 6 \beta_{7} - 9 \beta_{6} + \beta_{5} + \cdots + 9 ) / 6 $ (11b11 + 12b10 + 24b9 - 13b8 - 6b7 - 9b6 + b5 + 3b4 - 24b3 - 12b2 + 14b1 + 9) / 6
$\nu^{6}$	$=$	$ ( 13 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} - 29 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 14 \beta_{5} + \cdots + 3 ) / 3 $ (13b11 + 12b10 + 6b9 - 29b8 + 6b7 - 9b6 + 14b5 - 6b4 + 3b3 - 15b2 + 19*b1 + 3) / 3
$\nu^{7}$	$=$	$ ( 35 \beta_{11} - 24 \beta_{9} - 10 \beta_{8} - 30 \beta_{7} - 48 \beta_{6} + 37 \beta_{5} - 63 \beta_{4} + \cdots - 21 ) / 6 $ (35b11 - 24b9 - 10b8 - 30b7 - 48b6 + 37b5 - 63b4 - 84b3 - 39*b2 - b1 - 21) / 6
$\nu^{8}$	$=$	$ ( 29 \beta_{11} - 39 \beta_{10} + 27 \beta_{9} - 7 \beta_{8} + 21 \beta_{7} - 117 \beta_{6} - 95 \beta_{5} + \cdots - 9 ) / 6 $ (29b11 - 39b10 + 27b9 - 7b8 + 21b7 - 117b6 - 95b5 - 36b4 - 78b3 - 129b2 + 74*b1 - 9) / 6
$\nu^{9}$	$=$	$ ( 8 \beta_{11} - 267 \beta_{10} - 201 \beta_{9} - 226 \beta_{8} + 75 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} + \cdots + 354 ) / 6 $ (8b11 - 267b10 - 201b9 - 226b8 + 75b7 + 12b6 - 2b5 - 81b4 - 90b3 + 69b2 + 122*b1 + 354) / 6
$\nu^{10}$	$=$	$ ( - 43 \beta_{11} - 189 \beta_{10} - 219 \beta_{9} + 32 \beta_{8} - 51 \beta_{7} + 336 \beta_{6} + \cdots + 99 ) / 6 $ (-43b11 - 189b10 - 219b9 + 32b8 - 51b7 + 336b6 - 89b5 + 192b4 - 384b3 - 264b2 - 19*b1 + 99) / 6
$\nu^{11}$	$=$	$ ( 263 \beta_{11} - 420 \beta_{10} - 24 \beta_{9} - 925 \beta_{8} - 132 \beta_{7} + 147 \beta_{6} + \cdots - 327 ) / 6 $ (263b11 - 420b10 - 24b9 - 925b8 - 132b7 + 147b6 - 59b5 + 297b4 + 276b3 - 300b2 - 574*b1 - 327) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times$.

$n$	$223$	$593$	$1297$	$1333$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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} $

815.1

1.22118	−	1.22830i
1.22118	+	1.22830i
1.56305	+	0.746251i
1.56305	−	0.746251i
−1.50850	+	0.851135i
−1.50850	−	0.851135i
0.430781	−	1.67763i
0.430781	+	1.67763i
−0.933602	+	1.45890i
−0.933602	−	1.45890i
1.72710	+	0.130928i
1.72710	−	0.130928i

−1.57294

−

0.725164i

2.86674i

−

4.50920i

1.94827

2.28128i

815.2

−1.57294

0.725164i

−

2.86674i

4.50920i

1.94827

−

2.28128i

815.3

−1.14661

−

1.29818i

−

1.03994i

1.19241i

−0.370556

2.97703i

815.4

−1.14661

1.29818i

1.03994i

−

1.19241i

−0.370556

−

2.97703i

815.5

−0.679074

−

1.59338i

1.64327i

−

1.11590i

−2.07772

2.16404i

815.6

−0.679074

1.59338i

−

1.64327i

1.11590i

−2.07772

−

2.16404i

815.7

0.679074

−

1.59338i

−

1.64327i

−

1.11590i

−2.07772

−

2.16404i

815.8

0.679074

1.59338i

1.64327i

1.11590i

−2.07772

2.16404i

815.9

1.14661

−

1.29818i

1.03994i

1.19241i

−0.370556

−

2.97703i

815.10

1.14661

1.29818i

−

1.03994i

−

1.19241i

−0.370556

2.97703i

815.11

1.57294

−

0.725164i

−

2.86674i

−

4.50920i

1.94827

−

2.28128i

815.12

1.57294

0.725164i

2.86674i

4.50920i

1.94827

2.28128i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1776.2.e.g	✓	12
3.b	odd	2	1	inner	1776.2.e.g	✓	12
4.b	odd	2	1	inner	1776.2.e.g	✓	12
12.b	even	2	1	inner	1776.2.e.g	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1776.2.e.g	✓	12	1.a	even	1	1	trivial
1776.2.e.g	✓	12	3.b	odd	2	1	inner
1776.2.e.g	✓	12	4.b	odd	2	1	inner
1776.2.e.g	✓	12	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1776, [\chi])$:

$ T_{5}^{6} + 12T_{5}^{4} + 34T_{5}^{2} + 24 $

$ T_{7}^{6} + 23T_{7}^{4} + 56T_{7}^{2} + 36 $

$ T_{11}^{6} - 57T_{11}^{4} + 864T_{11}^{2} - 1944 $

$ T_{23}^{6} - 80T_{23}^{4} + 1526T_{23}^{2} - 216 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + T^{10} + \cdots + 729 $ T^12 + T^10 + 11T^8 + 6T^6 + 99T^4 + 81T^2 + 729
$5$	$ (T^{6} + 12 T^{4} + \cdots + 24)^{2} $ (T^6 + 12T^4 + 34T^2 + 24)^2
$7$	$ (T^{6} + 23 T^{4} + \cdots + 36)^{2} $ (T^6 + 23T^4 + 56T^2 + 36)^2
$11$	$ (T^{6} - 57 T^{4} + \cdots - 1944)^{2} $ (T^6 - 57T^4 + 864T^2 - 1944)^2
$13$	$ (T^{3} - 2 T^{2} - 24 T + 16)^{4} $ (T^3 - 2T^2 - 24T + 16)^4
$17$	$ (T^{6} + 24 T^{4} + \cdots + 384)^{2} $ (T^6 + 24T^4 + 178T^2 + 384)^2
$19$	$ (T^{6} + 28 T^{4} + \cdots + 576)^{2} $ (T^6 + 28T^4 + 236T^2 + 576)^2
$23$	$ (T^{6} - 80 T^{4} + \cdots - 216)^{2} $ (T^6 - 80T^4 + 1526T^2 - 216)^2
$29$	$ (T^{6} + 100 T^{4} + \cdots + 23064)^{2} $ (T^6 + 100T^4 + 2962T^2 + 23064)^2
$31$	$ (T^{6} + 28 T^{4} + \cdots + 576)^{2} $ (T^6 + 28T^4 + 236T^2 + 576)^2
$37$	$ (T - 1)^{12} $ (T - 1)^12
$41$	$ (T^{6} + 51 T^{4} + \cdots + 1944)^{2} $ (T^6 + 51T^4 + 648T^2 + 1944)^2
$43$	$ (T^{6} + 144 T^{4} + \cdots + 11664)^{2} $ (T^6 + 144T^4 + 4860T^2 + 11664)^2
$47$	$ (T^{6} - 65 T^{4} + \cdots - 216)^{2} $ (T^6 - 65T^4 + 512T^2 - 216)^2
$53$	$ (T^{6} + 99 T^{4} + \cdots + 1944)^{2} $ (T^6 + 99T^4 + 1000T^2 + 1944)^2
$59$	$ (T^{6} - 44 T^{4} + \cdots - 384)^{2} $ (T^6 - 44T^4 + 470T^2 - 384)^2
$61$	$ (T^{3} - 12 T^{2} + \cdots + 32)^{4} $ (T^3 - 12T^2 - 8T + 32)^4
$67$	$ (T^{6} + 208 T^{4} + \cdots + 144)^{2} $ (T^6 + 208T^4 + 380T^2 + 144)^2
$71$	$ (T^{6} - 97 T^{4} + \cdots - 24)^{2} $ (T^6 - 97T^4 + 128T^2 - 24)^2
$73$	$ (T^{3} - 15 T^{2} + \cdots + 466)^{4} $ (T^3 - 15T^2 - 14T + 466)^4
$79$	$ (T^{6} + 100 T^{4} + \cdots + 20736)^{2} $ (T^6 + 100T^4 + 2636T^2 + 20736)^2
$83$	$ (T^{6} - 569 T^{4} + \cdots - 2396544)^{2} $ (T^6 - 569T^4 + 89120T^2 - 2396544)^2
$89$	$ (T^{6} + 144 T^{4} + \cdots + 6144)^{2} $ (T^6 + 144T^4 + 3778T^2 + 6144)^2
$97$	$ (T^{3} - 20 T^{2} + \cdots + 992)^{4} $ (T^3 - 20T^2 + 8T + 992)^4
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1776.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{8} q^{3} + \beta_{10} q^{5} - \beta_{11} q^{7} + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) q - b8 * q^3 + b10 * q^5 - b11 * q^7 + (b10 + b9 - b6 - b2) * q^9	\( q - \beta_{8} q^{3} + \beta_{10} q^{5} - \beta_{11} q^{7} + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{9}+ \cdots + (3 \beta_{11} + 3 \beta_{5} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) q - b8 * q^3 + b10 * q^5 - b11 * q^7 + (b10 + b9 - b6 - b2) * q^9 + (b8 + b7 + b4 - b1) * q^11 - 2b3 q^13 + (-b11 - b4) * q^15 + (b9 - b2) * q^17 + (b11 - b8 - b1) * q^19 + (2b10 + b3 + b2 - 1) q^21 + (b8 - b7 + 2b4 - b1) q^23 + (b6 + b3 + b2 + 1) * q^25 + (-b11 + b8 - b7 - b5 - b1) * q^27 + (b9 - 2b6 + b2) q^29 + (b11 - b8 - b1) * q^31 + (3b6 - 3) q^33 + (2b8 + b7 - b4 - 2b1) * q^35 + q^37 + (2b5 + 2b4 - 2b1) q^39 + (-b10 + b9 - b6) * q^41 + (-b11 - b8 - 2b5 - b1) q^43 + (b10 + 2b3 + 2b2 - 2) * q^45 + (b8 - b7 - b4 - b1) * q^47 + (3b6 + 2b3 + 3b2 - 1) q^49 + (b11 - b7 - 2b1) q^51 + (-3b10 - b9 + b6) q^53 + (4b11 - 2b8 + 2b5 - 2b1) * q^55 + (-b10 + b9 - b6 - b3 - 2b2 - 2) q^57 + (b8 - b4 - b1) * q^59 + (-2b6 - 2b3 - 2b2 + 4) q^61 + (-2b11 + b8 + b7 - b5 - 2b4) * q^63 + (2b10 + 2b9 - 2b2) q^65 + (b11 - 3b8 + 2b5 - 3b1) q^67 + (b10 - 2b9 - b6 - 3b3 - b2) * q^69 + (b8 + b7 - b4 - b1) * q^71 + (-2b6 + b3 - 2b2 + 6) * q^73 + (b11 - 2b8 + b7 - b4 - b1) q^75 + (-6b10 + 3b6 - 3b2) q^77 + (b11 - b8 + 2b5 - b1) q^79 + (b10 - b9 - 2b6 - 2b3 + 2b2 - 4) q^81 + (b8 - 6b4 - b1) q^83 + (-b6 + b3 - b2 + 2) * q^85 + (-b11 + 2b8 + b7 - 2b5 + 4b4 - 2b1) * q^87 + (b10 - 2b9 + 2b6) * q^89 + (-2b8 - 2b1) * q^91 + (-b10 + b9 - b6 - b3 - 2b2 - 2) q^93 + (-2b8 - b7 - b4 + 2b1) * q^95 + (-2b6 + 2b3 - 2b2 + 8) q^97 + (3b11 + 3b5 - 3b4 - 3b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{9}+O(q^{10}) \) 12 * q - 2 * q^9	\( 12 q - 2 q^{9} + 8 q^{13} - 14 q^{21} + 12 q^{25} - 30 q^{33} + 12 q^{37} - 28 q^{45} - 8 q^{49} - 24 q^{57} + 48 q^{61} + 4 q^{69} + 60 q^{73} - 42 q^{81} + 16 q^{85} - 24 q^{93} + 80 q^{97}+O(q^{100}) \) 12 * q - 2 * q^9 + 8 * q^13 - 14 * q^21 + 12 * q^25 - 30 * q^33 + 12 * q^37 - 28 * q^45 - 8 * q^49 - 24 * q^57 + 48 * q^61 + 4 * q^69 + 60 * q^73 - 42 * q^81 + 16 * q^85 - 24 * q^93 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	1776.2.e.g

Level	$1776$
Weight	$2$
Character orbit	1776.e
Analytic conductor	$14.181$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(14.1814313990\)
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:	\( x^{12} - 5 x^{11} + 10 x^{10} - 11 x^{9} + 15 x^{8} - 42 x^{7} + 92 x^{6} - 126 x^{5} + 135 x^{4} + \cdots + 729 \) x^12 - 5x^11 + 10x^10 - 11x^9 + 15x^8 - 42x^7 + 92x^6 - 126x^5 + 135x^4 - 297x^3 + 810x^2 - 1215*x + 729
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} - 2 \nu^{10} + 4 \nu^{9} + \nu^{8} + 18 \nu^{7} + 12 \nu^{6} + 128 \nu^{5} + 258 \nu^{4} + \cdots - 243 ) / 3888 \) (v^11 - 2v^10 + 4v^9 + v^8 + 18v^7 + 12v^6 + 128v^5 + 258v^4 + 909v^3 - 1458v^2 + 324*v - 243) / 3888
\(\beta_{2}\)	\(=\)	\( ( - 17 \nu^{11} + 40 \nu^{10} - 62 \nu^{9} + 25 \nu^{8} - 174 \nu^{7} + 300 \nu^{6} - 592 \nu^{5} + \cdots + 8505 ) / 3888 \) (-17v^11 + 40v^10 - 62v^9 + 25v^8 - 174v^7 + 300v^6 - 592v^5 + 1134v^4 - 1233v^3 + 2916v^2 - 9882*v + 8505) / 3888
\(\beta_{3}\)	\(=\)	\( ( 7 \nu^{11} - 19 \nu^{10} + 17 \nu^{9} - 16 \nu^{8} + 46 \nu^{7} - 126 \nu^{6} + 170 \nu^{5} + \cdots - 1620 ) / 648 \) (7v^11 - 19v^10 + 17v^9 - 16v^8 + 46v^7 - 126v^6 + 170v^5 - 220v^4 + 387v^3 - 1143v^2 + 2349*v - 1620) / 648
\(\beta_{4}\)	\(=\)	\( ( - 19 \nu^{11} + 83 \nu^{10} - 121 \nu^{9} + 98 \nu^{8} - 252 \nu^{7} + 654 \nu^{6} - 1136 \nu^{5} + \cdots + 12150 ) / 1944 \) (-19v^11 + 83v^10 - 121v^9 + 98v^8 - 252v^7 + 654v^6 - 1136v^5 + 1560v^4 - 1521v^3 + 4455v^2 - 10935*v + 12150) / 1944
\(\beta_{5}\)	\(=\)	\( ( 41 \nu^{11} - 163 \nu^{10} + 227 \nu^{9} - 220 \nu^{8} + 450 \nu^{7} - 1200 \nu^{6} + 2278 \nu^{5} + \cdots - 22842 ) / 1944 \) (41v^11 - 163v^10 + 227v^9 - 220v^8 + 450v^7 - 1200v^6 + 2278v^5 - 2274v^4 + 2565v^3 - 9045v^2 + 23895*v - 22842) / 1944
\(\beta_{6}\)	\(=\)	\( ( 103 \nu^{11} - 326 \nu^{10} + 472 \nu^{9} - 449 \nu^{8} + 906 \nu^{7} - 2832 \nu^{6} + 5156 \nu^{5} + \cdots - 47385 ) / 3888 \) (103v^11 - 326v^10 + 472v^9 - 449v^8 + 906v^7 - 2832v^6 + 5156v^5 - 5526v^4 + 6255v^3 - 21762v^2 + 48924*v - 47385) / 3888
\(\beta_{7}\)	\(=\)	\( ( 17 \nu^{11} - 73 \nu^{10} + 107 \nu^{9} - 70 \nu^{8} + 156 \nu^{7} - 546 \nu^{6} + 1024 \nu^{5} + \cdots - 10530 ) / 648 \) (17v^11 - 73v^10 + 107v^9 - 70v^8 + 156v^7 - 546v^6 + 1024v^5 - 1128v^4 + 939v^3 - 4221v^2 + 10557*v - 10530) / 648
\(\beta_{8}\)	\(=\)	\( ( 131 \nu^{11} - 460 \nu^{10} + 578 \nu^{9} - 463 \nu^{8} + 1278 \nu^{7} - 3792 \nu^{6} + 6292 \nu^{5} + \cdots - 56619 ) / 3888 \) (131v^11 - 460v^10 + 578v^9 - 463v^8 + 1278v^7 - 3792v^6 + 6292v^5 - 6342v^4 + 7695v^3 - 28620v^2 + 64962*v - 56619) / 3888
\(\beta_{9}\)	\(=\)	\( ( 127 \nu^{11} - 476 \nu^{10} + 646 \nu^{9} - 527 \nu^{8} + 1110 \nu^{7} - 3804 \nu^{6} + 7220 \nu^{5} + \cdots - 65367 ) / 3888 \) (127v^11 - 476v^10 + 646v^9 - 527v^8 + 1110v^7 - 3804v^6 + 7220v^5 - 7098v^4 + 7659v^3 - 26352v^2 + 65934*v - 65367) / 3888
\(\beta_{10}\)	\(=\)	\( ( - 43 \nu^{11} + 146 \nu^{10} - 202 \nu^{9} + 152 \nu^{8} - 381 \nu^{7} + 1194 \nu^{6} - 2057 \nu^{5} + \cdots + 19440 ) / 972 \) (-43v^11 + 146v^10 - 202v^9 + 152v^8 - 381v^7 + 1194v^6 - 2057v^5 + 2040v^4 - 2448v^3 + 8532v^2 - 20493*v + 19440) / 972
\(\beta_{11}\)	\(=\)	\( ( 140 \nu^{11} - 433 \nu^{10} + 533 \nu^{9} - 427 \nu^{8} + 1224 \nu^{7} - 3486 \nu^{6} + 5986 \nu^{5} + \cdots - 49815 ) / 1944 \) (140v^11 - 433v^10 + 533v^9 - 427v^8 + 1224v^7 - 3486v^6 + 5986v^5 - 5766v^4 + 7092v^3 - 27945v^2 + 59049*v - 49815) / 1944

\(\nu\)	\(=\)	\( ( -\beta_{11} + 2\beta_{8} + \beta_{5} + 3\beta_{4} - 3\beta_{2} - \beta _1 + 3 ) / 6 \) (-b11 + 2b8 + b5 + 3b4 - 3*b2 - b1 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} - 3\beta_{10} + 3\beta_{9} - \beta_{8} - 3\beta_{7} - 3\beta_{6} + \beta_{5} - 3\beta_{2} - 4\beta _1 + 3 ) / 6 \) (-b11 - 3b10 + 3b9 - b8 - 3b7 - 3b6 + b5 - 3b2 - 4b1 + 3) / 6
\(\nu^{3}\)	\(=\)	\( ( - 4 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + \cdots + 6 ) / 6 \) (-4b11 - 3b10 + 3b9 + 2b8 - 3b7 - 2b5 - 3b4 + 6b3 - 3b2 + 14b1 + 6) / 6
\(\nu^{4}\)	\(=\)	\( ( - \beta_{11} - 9 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - 6 \beta_{6} + 7 \beta_{5} + \cdots - 9 ) / 6 \) (-b11 - 9b10 - 3b9 + 2b8 - 3b7 - 6b6 + 7b5 + 12b4 - 6b3 + 6b2 + 11b1 - 9) / 6
\(\nu^{5}\)	\(=\)	\( ( 11 \beta_{11} + 12 \beta_{10} + 24 \beta_{9} - 13 \beta_{8} - 6 \beta_{7} - 9 \beta_{6} + \beta_{5} + \cdots + 9 ) / 6 \) (11b11 + 12b10 + 24b9 - 13b8 - 6b7 - 9b6 + b5 + 3b4 - 24b3 - 12b2 + 14b1 + 9) / 6
\(\nu^{6}\)	\(=\)	\( ( 13 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} - 29 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 14 \beta_{5} + \cdots + 3 ) / 3 \) (13b11 + 12b10 + 6b9 - 29b8 + 6b7 - 9b6 + 14b5 - 6b4 + 3b3 - 15b2 + 19*b1 + 3) / 3
\(\nu^{7}\)	\(=\)	\( ( 35 \beta_{11} - 24 \beta_{9} - 10 \beta_{8} - 30 \beta_{7} - 48 \beta_{6} + 37 \beta_{5} - 63 \beta_{4} + \cdots - 21 ) / 6 \) (35b11 - 24b9 - 10b8 - 30b7 - 48b6 + 37b5 - 63b4 - 84b3 - 39*b2 - b1 - 21) / 6
\(\nu^{8}\)	\(=\)	\( ( 29 \beta_{11} - 39 \beta_{10} + 27 \beta_{9} - 7 \beta_{8} + 21 \beta_{7} - 117 \beta_{6} - 95 \beta_{5} + \cdots - 9 ) / 6 \) (29b11 - 39b10 + 27b9 - 7b8 + 21b7 - 117b6 - 95b5 - 36b4 - 78b3 - 129b2 + 74*b1 - 9) / 6
\(\nu^{9}\)	\(=\)	\( ( 8 \beta_{11} - 267 \beta_{10} - 201 \beta_{9} - 226 \beta_{8} + 75 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} + \cdots + 354 ) / 6 \) (8b11 - 267b10 - 201b9 - 226b8 + 75b7 + 12b6 - 2b5 - 81b4 - 90b3 + 69b2 + 122*b1 + 354) / 6
\(\nu^{10}\)	\(=\)	\( ( - 43 \beta_{11} - 189 \beta_{10} - 219 \beta_{9} + 32 \beta_{8} - 51 \beta_{7} + 336 \beta_{6} + \cdots + 99 ) / 6 \) (-43b11 - 189b10 - 219b9 + 32b8 - 51b7 + 336b6 - 89b5 + 192b4 - 384b3 - 264b2 - 19*b1 + 99) / 6
\(\nu^{11}\)	\(=\)	\( ( 263 \beta_{11} - 420 \beta_{10} - 24 \beta_{9} - 925 \beta_{8} - 132 \beta_{7} + 147 \beta_{6} + \cdots - 327 ) / 6 \) (263b11 - 420b10 - 24b9 - 925b8 - 132b7 + 147b6 - 59b5 + 297b4 + 276b3 - 300b2 - 574*b1 - 327) / 6

\(n\)	\(223\)	\(593\)	\(1297\)	\(1333\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 815.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + T^{10} + \cdots + 729 \) T^12 + T^10 + 11T^8 + 6T^6 + 99T^4 + 81T^2 + 729
$5$	\( (T^{6} + 12 T^{4} + \cdots + 24)^{2} \) (T^6 + 12T^4 + 34T^2 + 24)^2
$7$	\( (T^{6} + 23 T^{4} + \cdots + 36)^{2} \) (T^6 + 23T^4 + 56T^2 + 36)^2
$11$	\( (T^{6} - 57 T^{4} + \cdots - 1944)^{2} \) (T^6 - 57T^4 + 864T^2 - 1944)^2
$13$	\( (T^{3} - 2 T^{2} - 24 T + 16)^{4} \) (T^3 - 2T^2 - 24T + 16)^4
$17$	\( (T^{6} + 24 T^{4} + \cdots + 384)^{2} \) (T^6 + 24T^4 + 178T^2 + 384)^2
$19$	\( (T^{6} + 28 T^{4} + \cdots + 576)^{2} \) (T^6 + 28T^4 + 236T^2 + 576)^2
$23$	\( (T^{6} - 80 T^{4} + \cdots - 216)^{2} \) (T^6 - 80T^4 + 1526T^2 - 216)^2
$29$	\( (T^{6} + 100 T^{4} + \cdots + 23064)^{2} \) (T^6 + 100T^4 + 2962T^2 + 23064)^2
$31$	\( (T^{6} + 28 T^{4} + \cdots + 576)^{2} \) (T^6 + 28T^4 + 236T^2 + 576)^2
$37$	\( (T - 1)^{12} \) (T - 1)^12
$41$	\( (T^{6} + 51 T^{4} + \cdots + 1944)^{2} \) (T^6 + 51T^4 + 648T^2 + 1944)^2
$43$	\( (T^{6} + 144 T^{4} + \cdots + 11664)^{2} \) (T^6 + 144T^4 + 4860T^2 + 11664)^2
$47$	\( (T^{6} - 65 T^{4} + \cdots - 216)^{2} \) (T^6 - 65T^4 + 512T^2 - 216)^2
$53$	\( (T^{6} + 99 T^{4} + \cdots + 1944)^{2} \) (T^6 + 99T^4 + 1000T^2 + 1944)^2
$59$	\( (T^{6} - 44 T^{4} + \cdots - 384)^{2} \) (T^6 - 44T^4 + 470T^2 - 384)^2
$61$	\( (T^{3} - 12 T^{2} + \cdots + 32)^{4} \) (T^3 - 12T^2 - 8T + 32)^4
$67$	\( (T^{6} + 208 T^{4} + \cdots + 144)^{2} \) (T^6 + 208T^4 + 380T^2 + 144)^2
$71$	\( (T^{6} - 97 T^{4} + \cdots - 24)^{2} \) (T^6 - 97T^4 + 128T^2 - 24)^2
$73$	\( (T^{3} - 15 T^{2} + \cdots + 466)^{4} \) (T^3 - 15T^2 - 14T + 466)^4
$79$	\( (T^{6} + 100 T^{4} + \cdots + 20736)^{2} \) (T^6 + 100T^4 + 2636T^2 + 20736)^2
$83$	\( (T^{6} - 569 T^{4} + \cdots - 2396544)^{2} \) (T^6 - 569T^4 + 89120T^2 - 2396544)^2
$89$	\( (T^{6} + 144 T^{4} + \cdots + 6144)^{2} \) (T^6 + 144T^4 + 3778T^2 + 6144)^2
$97$	\( (T^{3} - 20 T^{2} + \cdots + 992)^{4} \) (T^3 - 20T^2 + 8T + 992)^4
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