LMFDB - Newform orbit 1764.2.e.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1079,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1764.1079");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 252)
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_{6} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{9} + \beta_{6}) q^{8}+O(q^{10}) $ q + b6 * q^2 + (b10 + 1) * q^4 - b4 * q^5 + (-b9 + b6) * q^8	$ q + \beta_{6} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{9} + \beta_{6}) q^{8} + ( - \beta_{10} + \beta_{8} - \beta_{5} + \cdots + 1) q^{10}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{5} - 1) q^{97}+O(q^{100}) $ q + b6 * q^2 + (b10 + 1) * q^4 - b4 * q^5 + (-b9 + b6) * q^8 + (-b10 + b8 - b5 - b2 + 1) * q^10 + (b7 + b6 + b3) * q^11 + (b11 + b10 + b8 + b2) * q^13 + (b10 + b8 - 1) * q^16 + (-b4 + 2b1) q^17 + (2b10 - b8 + b2) q^19 + (-2b3 - 2b1) * q^20 + (b11 + b10 + b8 + 2) * q^22 + (b7 - 3b6 + 2b4 - 3b3 - 2b1) * q^23 + (-b11 - b10 - b8 - b2 - 1) * q^25 + (2b7 + 2b3 - 2b1) q^26 + b1 * q^29 + (-b11 - b8 + b5 + 3b2 - 1) q^31 + (-b9 + b7 - b6 - b4) * q^32 + (-b10 + b8 - b5 + b2 + 1) * q^34 + (b11 - b10 + b8 - 2b5 + b2) q^37 + (-2b9 + 2b4) * q^38 + (-2b5 - 2b2) * q^40 + (-2b9 + 2b6 + b4 - 2b3 - 2b1) * q^41 + (-b11 - 2b10 + b5 + 2b2 - 1) * q^43 + (b7 + 2b6 - b4 + 2b3 - 2b1) q^44 + (b11 - b10 - b8 - 2b5 - 4) q^46 + (4b7 + 2b4 - 2b1) q^47 + (-2b7 - b6 - 2b3 + 2b1) q^50 + (2b11 + 2b8 - 2b2 + 2) q^52 + (2b9 - 2b6 - 2b4 + 2b3 - b1) * q^53 + (3b8 - 2b5 - 3b2 + 2) q^55 + b2 * q^58 + (2b9 - 2b7 + 4b6 - 2b4 + 2b1) q^59 + (-b11 - 2b10 - b8 - b5 - b2 - 5) q^61 + (2b7 + 2b4 + 4b1) q^62 + (b11 - 2b10 + 3b8 - 2b5 - b2 - 1) q^64 + (-2b9 + 2b6 - 2b4 - 2b3 - 2b1) q^65 + (-b11 + 3b10 - b8 + 3b2) * q^67 + (2b7 + 2b4 - 2b3 - 2b1) * q^68 + (2b9 - 3b7 + 3b6 - 2b4 - b3 + 2b1) q^71 + (-2b11 + b10 - 2b8 + 3b5 - 2b2 + 1) * q^73 + (2b7 - 2b6 + 4b4 - 2b3 - 6b1) q^74 + (2b10 + 2b5 + 2b2 - 6) q^76 + (b11 + 3b10 + b8 - 2b5 - 3b2 + 2) q^79 + (-2b9 - 2b7 - 2b6 + 2b4 - 4b3 - 4b1) * q^80 + (3b10 + b8 - b5 - b2 - 3) q^82 + (2b7 - 2b6 + 2b4 - 2b3 - 2b1) q^83 + (-b11 + b10 - b8 + 2b5 - b2 - 4) q^85 + (2b9 + 2b7 + 4b1) q^86 + (b11 + b10 + 2b8 - 3b2 + 4) * q^88 + (2b9 - 2b6 - 3b4 + 2b3 - 2b1) q^89 + (-b7 - 6b6 + 5b4 - 2b3 - 6b1) * q^92 + (4b11 + 2b10 + 2b8 - 2b5 + 2) * q^94 + (2b9 + 2b7 + 4b6) q^95 + (-3b10 - 3b5 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4}+O(q^{10}) $ 12 * q + 8 * q^4	$ 12 q + 8 q^{4} + 8 q^{10} - 20 q^{16} + 20 q^{22} - 12 q^{25} + 16 q^{34} + 8 q^{37} - 8 q^{40} - 36 q^{46} + 16 q^{52} + 4 q^{58} - 56 q^{61} - 16 q^{64} - 72 q^{76} - 56 q^{82} - 56 q^{85} + 28 q^{88} + 24 q^{94}+O(q^{100}) $ 12 * q + 8 * q^4 + 8 * q^10 - 20 * q^16 + 20 * q^22 - 12 * q^25 + 16 * q^34 + 8 * q^37 - 8 * q^40 - 36 * q^46 + 16 * q^52 + 4 * q^58 - 56 * q^61 - 16 * q^64 - 72 * q^76 - 56 * q^82 - 56 * q^85 + 28 * q^88 + 24 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4*x^10 + 13*x^8 - 28*x^6 + 52*x^4 - 64*x^2 + 64 :

$\beta_{1}$	$=$	$ ( -\nu^{11} + 4\nu^{9} - 5\nu^{7} + 12\nu^{5} - 12\nu^{3} + 16\nu ) / 32 $ (-v^11 + 4v^9 - 5v^7 + 12v^5 - 12v^3 + 16*v) / 32
$\beta_{2}$	$=$	$ ( -\nu^{10} + 4\nu^{8} - 5\nu^{6} + 12\nu^{4} - 12\nu^{2} + 16 ) / 16 $ (-v^10 + 4v^8 - 5v^6 + 12v^4 - 12v^2 + 16) / 16
$\beta_{3}$	$=$	$ ( -\nu^{11} + 4\nu^{9} - 13\nu^{7} + 28\nu^{5} - 20\nu^{3} + 32\nu ) / 32 $ (-v^11 + 4v^9 - 13v^7 + 28v^5 - 20v^3 + 32*v) / 32
$\beta_{4}$	$=$	$ ( \nu^{9} - 2\nu^{7} + 9\nu^{5} - 10\nu^{3} + 16\nu ) / 8 $ (v^9 - 2v^7 + 9v^5 - 10v^3 + 16v) / 8
$\beta_{5}$	$=$	$ ( -\nu^{10} + 4\nu^{8} - 13\nu^{6} + 28\nu^{4} - 20\nu^{2} + 32 ) / 16 $ (-v^10 + 4v^8 - 13v^6 + 28v^4 - 20v^2 + 32) / 16
$\beta_{6}$	$=$	$ ( -\nu^{11} + 4\nu^{9} - 13\nu^{7} + 28\nu^{5} - 52\nu^{3} + 64\nu ) / 32 $ (-v^11 + 4v^9 - 13v^7 + 28v^5 - 52v^3 + 64*v) / 32
$\beta_{7}$	$=$	$ ( -\nu^{11} - 4\nu^{9} + 11\nu^{7} - 28\nu^{5} + 36\nu^{3} - 48\nu ) / 32 $ (-v^11 - 4v^9 + 11v^7 - 28v^5 + 36v^3 - 48*v) / 32
$\beta_{8}$	$=$	$ ( -3\nu^{10} + 8\nu^{8} - 23\nu^{6} + 32\nu^{4} - 44\nu^{2} + 16 ) / 16 $ (-3v^10 + 8v^8 - 23v^6 + 32v^4 - 44*v^2 + 16) / 16
$\beta_{9}$	$=$	$ ( 3\nu^{11} - 8\nu^{9} + 23\nu^{7} - 32\nu^{5} + 44\nu^{3} + 16\nu ) / 32 $ (3v^11 - 8v^9 + 23v^7 - 32v^5 + 44v^3 + 16v) / 32
$\beta_{10}$	$=$	$ ( -\nu^{10} + 4\nu^{8} - 13\nu^{6} + 28\nu^{4} - 52\nu^{2} + 48 ) / 16 $ (-v^10 + 4v^8 - 13v^6 + 28v^4 - 52v^2 + 48) / 16
$\beta_{11}$	$=$	$ ( \nu^{10} - 8\nu^{8} + 21\nu^{6} - 32\nu^{4} + 60\nu^{2} - 48 ) / 16 $ (v^10 - 8v^8 + 21v^6 - 32v^4 + 60v^2 - 48) / 16

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

$\nu$	$=$	$ ( \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_1 ) / 2 $ (b9 + b6 - b4 + b3 + b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{10} + \beta_{5} + 1 ) / 2 $ (-b10 + b5 + 1) / 2
$\nu^{3}$	$=$	$ ( \beta_{9} - \beta_{6} - \beta_{4} + 3\beta_{3} + \beta_1 ) / 2 $ (b9 - b6 - b4 + 3*b3 + b1) / 2
$\nu^{4}$	$=$	$ ( \beta_{11} + \beta_{10} - \beta_{8} + 2\beta_{5} + \beta_{2} - 4 ) / 2 $ (b11 + b10 - b8 + 2*b5 + b2 - 4) / 2
$\nu^{5}$	$=$	$ ( -\beta_{9} + 2\beta_{7} - 3\beta_{6} + 5\beta_{4} + \beta_{3} - 3\beta_1 ) / 2 $ (-b9 + 2b7 - 3b6 + 5b4 + b3 - 3b1) / 2
$\nu^{6}$	$=$	$ ( 2\beta_{11} + 3\beta_{10} - 2\beta_{8} - \beta_{5} + 6\beta_{2} - 5 ) / 2 $ (2b11 + 3b10 - 2b8 - b5 + 6b2 - 5) / 2
$\nu^{7}$	$=$	$ ( -\beta_{9} + 4\beta_{7} - 3\beta_{6} + 9\beta_{4} - 7\beta_{3} + 3\beta_1 ) / 2 $ (-b9 + 4b7 - 3b6 + 9b4 - 7b3 + 3*b1) / 2
$\nu^{8}$	$=$	$ ( -5\beta_{11} - 5\beta_{10} - 3\beta_{8} - 2\beta_{5} + 11\beta_{2} - 4 ) / 2 $ (-5b11 - 5b10 - 3b8 - 2b5 + 11*b2 - 4) / 2
$\nu^{9}$	$=$	$ ( \beta_{9} - 10\beta_{7} - 5\beta_{6} - 5\beta_{4} - 9\beta_{3} + 27\beta_1 ) / 2 $ (b9 - 10b7 - 5b6 - 5b4 - 9b3 + 27*b1) / 2
$\nu^{10}$	$=$	$ ( -18\beta_{11} - 11\beta_{10} - 14\beta_{8} + 9\beta_{5} - 6\beta_{2} - 19 ) / 2 $ (-18b11 - 11b10 - 14b8 + 9b5 - 6*b2 - 19) / 2
$\nu^{11}$	$=$	$ ( \beta_{9} - 36\beta_{7} - 13\beta_{6} - 9\beta_{4} - 9\beta_{3} - 3\beta_1 ) / 2 $ (b9 - 36b7 - 13b6 - 9b4 - 9b3 - 3*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$-1$	$1$

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} $

1079.1

−1.35489	+	0.405301i
−1.35489	−	0.405301i
−1.16947	+	0.795191i
−1.16947	−	0.795191i
−0.892524	+	1.09700i
−0.892524	−	1.09700i
0.892524	+	1.09700i
0.892524	−	1.09700i
1.16947	+	0.795191i
1.16947	−	0.795191i
1.35489	+	0.405301i
1.35489	−	0.405301i

−1.35489

−

0.405301i

1.67146

1.09828i

−

3.31339i

−1.81951

−

2.16549i

−1.34292

4.48929i

1079.2

−1.35489

0.405301i

1.67146

−

1.09828i

3.31339i

−1.81951

2.16549i

−1.34292

−

4.48929i

1079.3

−1.16947

−

0.795191i

0.735342

1.85991i

0.665647i

0.619022

−

2.75986i

0.529317

−

0.778457i

1079.4

−1.16947

0.795191i

0.735342

−

1.85991i

−

0.665647i

0.619022

2.75986i

0.529317

0.778457i

1079.5

−0.892524

−

1.09700i

−0.406803

1.95819i

2.56483i

2.51121

−

1.30147i

2.81361

−

2.28917i

1079.6

−0.892524

1.09700i

−0.406803

−

1.95819i

−

2.56483i

2.51121

1.30147i

2.81361

2.28917i

1079.7

0.892524

−

1.09700i

−0.406803

−

1.95819i

2.56483i

−2.51121

−

1.30147i

2.81361

2.28917i

1079.8

0.892524

1.09700i

−0.406803

1.95819i

−

2.56483i

−2.51121

1.30147i

2.81361

−

2.28917i

1079.9

1.16947

−

0.795191i

0.735342

−

1.85991i

0.665647i

−0.619022

−

2.75986i

0.529317

0.778457i

1079.10

1.16947

0.795191i

0.735342

1.85991i

−

0.665647i

−0.619022

2.75986i

0.529317

−

0.778457i

1079.11

1.35489

−

0.405301i

1.67146

−

1.09828i

−

3.31339i

1.81951

−

2.16549i

−1.34292

−

4.48929i

1079.12

1.35489

0.405301i

1.67146

1.09828i

3.31339i

1.81951

2.16549i

−1.34292

4.48929i

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	1764.2.e.g		12
3.b	odd	2	1	inner	1764.2.e.g		12
4.b	odd	2	1	inner	1764.2.e.g		12
7.b	odd	2	1		252.2.e.a	✓	12
12.b	even	2	1	inner	1764.2.e.g		12
21.c	even	2	1		252.2.e.a	✓	12
28.d	even	2	1		252.2.e.a	✓	12
56.e	even	2	1		4032.2.h.h		12
56.h	odd	2	1		4032.2.h.h		12
84.h	odd	2	1		252.2.e.a	✓	12
168.e	odd	2	1		4032.2.h.h		12
168.i	even	2	1		4032.2.h.h		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.e.a	✓	12	7.b	odd	2	1
252.2.e.a	✓	12	21.c	even	2	1
252.2.e.a	✓	12	28.d	even	2	1
252.2.e.a	✓	12	84.h	odd	2	1
1764.2.e.g		12	1.a	even	1	1	trivial
1764.2.e.g		12	3.b	odd	2	1	inner
1764.2.e.g		12	4.b	odd	2	1	inner
1764.2.e.g		12	12.b	even	2	1	inner
4032.2.h.h		12	56.e	even	2	1
4032.2.h.h		12	56.h	odd	2	1
4032.2.h.h		12	168.e	odd	2	1
4032.2.h.h		12	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1764, [\chi])$:

$ T_{5}^{6} + 18T_{5}^{4} + 80T_{5}^{2} + 32 $

$ T_{13}^{3} - 28T_{13} + 16 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 4 T^{10} + \cdots + 64 $ T^12 - 4T^10 + 13T^8 - 28T^6 + 52T^4 - 64*T^2 + 64
$3$	$ T^{12} $ T^12
$5$	$ (T^{6} + 18 T^{4} + \cdots + 32)^{2} $ (T^6 + 18T^4 + 80T^2 + 32)^2
$7$	$ T^{12} $ T^12
$11$	$ (T^{6} - 28 T^{4} + \cdots - 128)^{2} $ (T^6 - 28T^4 + 132T^2 - 128)^2
$13$	$ (T^{3} - 28 T + 16)^{4} $ (T^3 - 28*T + 16)^4
$17$	$ (T^{6} + 34 T^{4} + \cdots + 32)^{2} $ (T^6 + 34T^4 + 144T^2 + 32)^2
$19$	$ (T^{6} + 64 T^{4} + \cdots + 4096)^{2} $ (T^6 + 64T^4 + 1088T^2 + 4096)^2
$23$	$ (T^{6} - 140 T^{4} + \cdots - 67712)^{2} $ (T^6 - 140T^4 + 5924T^2 - 67712)^2
$29$	$ (T^{2} + 2)^{6} $ (T^2 + 2)^6
$31$	$ (T^{6} + 128 T^{4} + \cdots + 65536)^{2} $ (T^6 + 128T^4 + 5184T^2 + 65536)^2
$37$	$ (T^{3} - 2 T^{2} + \cdots - 128)^{4} $ (T^3 - 2T^2 - 64T - 128)^4
$41$	$ (T^{6} + 114 T^{4} + \cdots + 53792)^{2} $ (T^6 + 114T^4 + 4304T^2 + 53792)^2
$43$	$ (T^{6} + 160 T^{4} + \cdots + 16384)^{2} $ (T^6 + 160T^4 + 4352T^2 + 16384)^2
$47$	$ (T^{6} - 272 T^{4} + \cdots - 524288)^{2} $ (T^6 - 272T^4 + 22592T^2 - 524288)^2
$53$	$ (T^{6} + 134 T^{4} + \cdots + 2312)^{2} $ (T^6 + 134T^4 + 2444T^2 + 2312)^2
$59$	$ (T^{6} - 208 T^{4} + \cdots - 8192)^{2} $ (T^6 - 208T^4 + 4672T^2 - 8192)^2
$61$	$ (T^{3} + 14 T^{2} + 4 T - 8)^{4} $ (T^3 + 14T^2 + 4T - 8)^4
$67$	$ (T^{6} + 216 T^{4} + \cdots + 16384)^{2} $ (T^6 + 216T^4 + 3728T^2 + 16384)^2
$71$	$ (T^{6} - 268 T^{4} + \cdots - 36992)^{2} $ (T^6 - 268T^4 + 6180T^2 - 36992)^2
$73$	$ (T^{3} - 172 T + 352)^{4} $ (T^3 - 172*T + 352)^4
$79$	$ (T^{6} + 312 T^{4} + \cdots + 16384)^{2} $ (T^6 + 312T^4 + 24336T^2 + 16384)^2
$83$	$ (T^{6} - 128 T^{4} + \cdots - 32768)^{2} $ (T^6 - 128T^4 + 4352T^2 - 32768)^2
$89$	$ (T^{6} + 226 T^{4} + \cdots + 170528)^{2} $ (T^6 + 226T^4 + 11536T^2 + 170528)^2
$97$	$ (T^{3} - 156 T - 128)^{4} $ (T^3 - 156*T - 128)^4
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{9} + \beta_{6}) q^{8}+O(q^{10}) \) q + b6 * q^2 + (b10 + 1) * q^4 - b4 * q^5 + (-b9 + b6) * q^8	\( q + \beta_{6} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{9} + \beta_{6}) q^{8} + ( - \beta_{10} + \beta_{8} - \beta_{5} + \cdots + 1) q^{10}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{5} - 1) q^{97}+O(q^{100}) \) q + b6 * q^2 + (b10 + 1) * q^4 - b4 * q^5 + (-b9 + b6) * q^8 + (-b10 + b8 - b5 - b2 + 1) * q^10 + (b7 + b6 + b3) * q^11 + (b11 + b10 + b8 + b2) * q^13 + (b10 + b8 - 1) * q^16 + (-b4 + 2b1) q^17 + (2b10 - b8 + b2) q^19 + (-2b3 - 2b1) * q^20 + (b11 + b10 + b8 + 2) * q^22 + (b7 - 3b6 + 2b4 - 3b3 - 2b1) * q^23 + (-b11 - b10 - b8 - b2 - 1) * q^25 + (2b7 + 2b3 - 2b1) q^26 + b1 * q^29 + (-b11 - b8 + b5 + 3b2 - 1) q^31 + (-b9 + b7 - b6 - b4) * q^32 + (-b10 + b8 - b5 + b2 + 1) * q^34 + (b11 - b10 + b8 - 2b5 + b2) q^37 + (-2b9 + 2b4) * q^38 + (-2b5 - 2b2) * q^40 + (-2b9 + 2b6 + b4 - 2b3 - 2b1) * q^41 + (-b11 - 2b10 + b5 + 2b2 - 1) * q^43 + (b7 + 2b6 - b4 + 2b3 - 2b1) q^44 + (b11 - b10 - b8 - 2b5 - 4) q^46 + (4b7 + 2b4 - 2b1) q^47 + (-2b7 - b6 - 2b3 + 2b1) q^50 + (2b11 + 2b8 - 2b2 + 2) q^52 + (2b9 - 2b6 - 2b4 + 2b3 - b1) * q^53 + (3b8 - 2b5 - 3b2 + 2) q^55 + b2 * q^58 + (2b9 - 2b7 + 4b6 - 2b4 + 2b1) q^59 + (-b11 - 2b10 - b8 - b5 - b2 - 5) q^61 + (2b7 + 2b4 + 4b1) q^62 + (b11 - 2b10 + 3b8 - 2b5 - b2 - 1) q^64 + (-2b9 + 2b6 - 2b4 - 2b3 - 2b1) q^65 + (-b11 + 3b10 - b8 + 3b2) * q^67 + (2b7 + 2b4 - 2b3 - 2b1) * q^68 + (2b9 - 3b7 + 3b6 - 2b4 - b3 + 2b1) q^71 + (-2b11 + b10 - 2b8 + 3b5 - 2b2 + 1) * q^73 + (2b7 - 2b6 + 4b4 - 2b3 - 6b1) q^74 + (2b10 + 2b5 + 2b2 - 6) q^76 + (b11 + 3b10 + b8 - 2b5 - 3b2 + 2) q^79 + (-2b9 - 2b7 - 2b6 + 2b4 - 4b3 - 4b1) * q^80 + (3b10 + b8 - b5 - b2 - 3) q^82 + (2b7 - 2b6 + 2b4 - 2b3 - 2b1) q^83 + (-b11 + b10 - b8 + 2b5 - b2 - 4) q^85 + (2b9 + 2b7 + 4b1) q^86 + (b11 + b10 + 2b8 - 3b2 + 4) * q^88 + (2b9 - 2b6 - 3b4 + 2b3 - 2b1) q^89 + (-b7 - 6b6 + 5b4 - 2b3 - 6b1) * q^92 + (4b11 + 2b10 + 2b8 - 2b5 + 2) * q^94 + (2b9 + 2b7 + 4b6) q^95 + (-3b10 - 3b5 - 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4}+O(q^{10}) \) 12 * q + 8 * q^4	\( 12 q + 8 q^{4} + 8 q^{10} - 20 q^{16} + 20 q^{22} - 12 q^{25} + 16 q^{34} + 8 q^{37} - 8 q^{40} - 36 q^{46} + 16 q^{52} + 4 q^{58} - 56 q^{61} - 16 q^{64} - 72 q^{76} - 56 q^{82} - 56 q^{85} + 28 q^{88} + 24 q^{94}+O(q^{100}) \) 12 * q + 8 * q^4 + 8 * q^10 - 20 * q^16 + 20 * q^22 - 12 * q^25 + 16 * q^34 + 8 * q^37 - 8 * q^40 - 36 * q^46 + 16 * q^52 + 4 * q^58 - 56 * q^61 - 16 * q^64 - 72 * q^76 - 56 * q^82 - 56 * q^85 + 28 * q^88 + 24 * q^94

Introduction

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Newspace parameters

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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	1764.2.e.g

Level	$1764$
Weight	$2$
Character orbit	1764.e
Analytic conductor	$14.086$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index:	\( 2^{9} \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{11} + 4\nu^{9} - 5\nu^{7} + 12\nu^{5} - 12\nu^{3} + 16\nu ) / 32 \) (-v^11 + 4v^9 - 5v^7 + 12v^5 - 12v^3 + 16*v) / 32
\(\beta_{2}\)	\(=\)	\( ( -\nu^{10} + 4\nu^{8} - 5\nu^{6} + 12\nu^{4} - 12\nu^{2} + 16 ) / 16 \) (-v^10 + 4v^8 - 5v^6 + 12v^4 - 12v^2 + 16) / 16
\(\beta_{3}\)	\(=\)	\( ( -\nu^{11} + 4\nu^{9} - 13\nu^{7} + 28\nu^{5} - 20\nu^{3} + 32\nu ) / 32 \) (-v^11 + 4v^9 - 13v^7 + 28v^5 - 20v^3 + 32*v) / 32
\(\beta_{4}\)	\(=\)	\( ( \nu^{9} - 2\nu^{7} + 9\nu^{5} - 10\nu^{3} + 16\nu ) / 8 \) (v^9 - 2v^7 + 9v^5 - 10v^3 + 16v) / 8
\(\beta_{5}\)	\(=\)	\( ( -\nu^{10} + 4\nu^{8} - 13\nu^{6} + 28\nu^{4} - 20\nu^{2} + 32 ) / 16 \) (-v^10 + 4v^8 - 13v^6 + 28v^4 - 20v^2 + 32) / 16
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} + 4\nu^{9} - 13\nu^{7} + 28\nu^{5} - 52\nu^{3} + 64\nu ) / 32 \) (-v^11 + 4v^9 - 13v^7 + 28v^5 - 52v^3 + 64*v) / 32
\(\beta_{7}\)	\(=\)	\( ( -\nu^{11} - 4\nu^{9} + 11\nu^{7} - 28\nu^{5} + 36\nu^{3} - 48\nu ) / 32 \) (-v^11 - 4v^9 + 11v^7 - 28v^5 + 36v^3 - 48*v) / 32
\(\beta_{8}\)	\(=\)	\( ( -3\nu^{10} + 8\nu^{8} - 23\nu^{6} + 32\nu^{4} - 44\nu^{2} + 16 ) / 16 \) (-3v^10 + 8v^8 - 23v^6 + 32v^4 - 44*v^2 + 16) / 16
\(\beta_{9}\)	\(=\)	\( ( 3\nu^{11} - 8\nu^{9} + 23\nu^{7} - 32\nu^{5} + 44\nu^{3} + 16\nu ) / 32 \) (3v^11 - 8v^9 + 23v^7 - 32v^5 + 44v^3 + 16v) / 32
\(\beta_{10}\)	\(=\)	\( ( -\nu^{10} + 4\nu^{8} - 13\nu^{6} + 28\nu^{4} - 52\nu^{2} + 48 ) / 16 \) (-v^10 + 4v^8 - 13v^6 + 28v^4 - 52v^2 + 48) / 16
\(\beta_{11}\)	\(=\)	\( ( \nu^{10} - 8\nu^{8} + 21\nu^{6} - 32\nu^{4} + 60\nu^{2} - 48 ) / 16 \) (v^10 - 8v^8 + 21v^6 - 32v^4 + 60v^2 - 48) / 16

\(\nu\)	\(=\)	\( ( \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_1 ) / 2 \) (b9 + b6 - b4 + b3 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{10} + \beta_{5} + 1 ) / 2 \) (-b10 + b5 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{9} - \beta_{6} - \beta_{4} + 3\beta_{3} + \beta_1 ) / 2 \) (b9 - b6 - b4 + 3*b3 + b1) / 2
\(\nu^{4}\)	\(=\)	\( ( \beta_{11} + \beta_{10} - \beta_{8} + 2\beta_{5} + \beta_{2} - 4 ) / 2 \) (b11 + b10 - b8 + 2*b5 + b2 - 4) / 2
\(\nu^{5}\)	\(=\)	\( ( -\beta_{9} + 2\beta_{7} - 3\beta_{6} + 5\beta_{4} + \beta_{3} - 3\beta_1 ) / 2 \) (-b9 + 2b7 - 3b6 + 5b4 + b3 - 3b1) / 2
\(\nu^{6}\)	\(=\)	\( ( 2\beta_{11} + 3\beta_{10} - 2\beta_{8} - \beta_{5} + 6\beta_{2} - 5 ) / 2 \) (2b11 + 3b10 - 2b8 - b5 + 6b2 - 5) / 2
\(\nu^{7}\)	\(=\)	\( ( -\beta_{9} + 4\beta_{7} - 3\beta_{6} + 9\beta_{4} - 7\beta_{3} + 3\beta_1 ) / 2 \) (-b9 + 4b7 - 3b6 + 9b4 - 7b3 + 3*b1) / 2
\(\nu^{8}\)	\(=\)	\( ( -5\beta_{11} - 5\beta_{10} - 3\beta_{8} - 2\beta_{5} + 11\beta_{2} - 4 ) / 2 \) (-5b11 - 5b10 - 3b8 - 2b5 + 11*b2 - 4) / 2
\(\nu^{9}\)	\(=\)	\( ( \beta_{9} - 10\beta_{7} - 5\beta_{6} - 5\beta_{4} - 9\beta_{3} + 27\beta_1 ) / 2 \) (b9 - 10b7 - 5b6 - 5b4 - 9b3 + 27*b1) / 2
\(\nu^{10}\)	\(=\)	\( ( -18\beta_{11} - 11\beta_{10} - 14\beta_{8} + 9\beta_{5} - 6\beta_{2} - 19 ) / 2 \) (-18b11 - 11b10 - 14b8 + 9b5 - 6*b2 - 19) / 2
\(\nu^{11}\)	\(=\)	\( ( \beta_{9} - 36\beta_{7} - 13\beta_{6} - 9\beta_{4} - 9\beta_{3} - 3\beta_1 ) / 2 \) (b9 - 36b7 - 13b6 - 9b4 - 9b3 - 3*b1) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1079.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - 4 T^{10} + \cdots + 64 \) T^12 - 4T^10 + 13T^8 - 28T^6 + 52T^4 - 64*T^2 + 64
$3$	\( T^{12} \) T^12
$5$	\( (T^{6} + 18 T^{4} + \cdots + 32)^{2} \) (T^6 + 18T^4 + 80T^2 + 32)^2
$7$	\( T^{12} \) T^12
$11$	\( (T^{6} - 28 T^{4} + \cdots - 128)^{2} \) (T^6 - 28T^4 + 132T^2 - 128)^2
$13$	\( (T^{3} - 28 T + 16)^{4} \) (T^3 - 28*T + 16)^4
$17$	\( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) (T^6 + 34T^4 + 144T^2 + 32)^2
$19$	\( (T^{6} + 64 T^{4} + \cdots + 4096)^{2} \) (T^6 + 64T^4 + 1088T^2 + 4096)^2
$23$	\( (T^{6} - 140 T^{4} + \cdots - 67712)^{2} \) (T^6 - 140T^4 + 5924T^2 - 67712)^2
$29$	\( (T^{2} + 2)^{6} \) (T^2 + 2)^6
$31$	\( (T^{6} + 128 T^{4} + \cdots + 65536)^{2} \) (T^6 + 128T^4 + 5184T^2 + 65536)^2
$37$	\( (T^{3} - 2 T^{2} + \cdots - 128)^{4} \) (T^3 - 2T^2 - 64T - 128)^4
$41$	\( (T^{6} + 114 T^{4} + \cdots + 53792)^{2} \) (T^6 + 114T^4 + 4304T^2 + 53792)^2
$43$	\( (T^{6} + 160 T^{4} + \cdots + 16384)^{2} \) (T^6 + 160T^4 + 4352T^2 + 16384)^2
$47$	\( (T^{6} - 272 T^{4} + \cdots - 524288)^{2} \) (T^6 - 272T^4 + 22592T^2 - 524288)^2
$53$	\( (T^{6} + 134 T^{4} + \cdots + 2312)^{2} \) (T^6 + 134T^4 + 2444T^2 + 2312)^2
$59$	\( (T^{6} - 208 T^{4} + \cdots - 8192)^{2} \) (T^6 - 208T^4 + 4672T^2 - 8192)^2
$61$	\( (T^{3} + 14 T^{2} + 4 T - 8)^{4} \) (T^3 + 14T^2 + 4T - 8)^4
$67$	\( (T^{6} + 216 T^{4} + \cdots + 16384)^{2} \) (T^6 + 216T^4 + 3728T^2 + 16384)^2
$71$	\( (T^{6} - 268 T^{4} + \cdots - 36992)^{2} \) (T^6 - 268T^4 + 6180T^2 - 36992)^2
$73$	\( (T^{3} - 172 T + 352)^{4} \) (T^3 - 172*T + 352)^4
$79$	\( (T^{6} + 312 T^{4} + \cdots + 16384)^{2} \) (T^6 + 312T^4 + 24336T^2 + 16384)^2
$83$	\( (T^{6} - 128 T^{4} + \cdots - 32768)^{2} \) (T^6 - 128T^4 + 4352T^2 - 32768)^2
$89$	\( (T^{6} + 226 T^{4} + \cdots + 170528)^{2} \) (T^6 + 226T^4 + 11536T^2 + 170528)^2
$97$	\( (T^{3} - 156 T - 128)^{4} \) (T^3 - 156*T - 128)^4
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\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)