LMFDB - Newform orbit 1600.2.s.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 1]))

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.s (of order $4$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 $ x^16 - 2x^14 + 6x^12 - 12x^10 + 36x^8 - 48x^6 + 96x^4 - 128*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{3} - \beta_{7} q^{7} + ( - \beta_{12} + 1) q^{9} + (\beta_{10} - 1) q^{11} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{13} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{4}) q^{17} + (\beta_{14} + \beta_{12} + \cdots - \beta_1) q^{19}+ \cdots + ( - \beta_{14} - \beta_{12} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q - b4 * q^3 - b7 * q^7 + (-b12 + 1) * q^9 + (b10 - 1) * q^11 + (b7 - b5 - b3) * q^13 + (-b7 - b6 + b5 - b4) * q^17 + (b14 + b12 + b9 - b1) * q^19 + (b15 - b11 - 1) * q^21 + (b3 - b2) * q^23 + (b13 - 2b8 - 2b7 - b6 - 2b4 + 2b2) * q^27 + (b15 - b14 + b12 + b11 - 1) * q^29 + (b15 - b9 + b1 - 1) * q^31 + (b13 - 2b8 - b3 + b2) q^33 + (b13 + b8 + b7 + b6 - 2b3) q^37 + (-2b14 + b11 + b10 - 3b9) * q^39 + (b15 - b14 + b11 + b10 + b1 - 1) * q^41 + (b13 + b6 - b5) * q^43 + (-2b13 + b8) q^47 + (-b15 + b14 + b11 + b10 + b9 - b1 + 1) * q^49 + (b15 + b14 - b12 - b11 + 3b9 + 2) q^51 + (-b13 + b6 + 3b4 - b2) q^53 + (-b13 - b8 + 3b5 + 3b4 + 2b3 - 2b2) * q^57 + (b15 + b14 - b12 - 1) * q^59 + (b14 + b12 + 2b10 - 2) q^61 + (-b8 - 3b7 - 2b6 + 2b5 - 2b4 + b3 + b2) * q^63 + (b13 - 2b8 + b6 + b5 + 2b3) * q^67 + (b14 + b12 - b10 - b1 + 2) * q^69 + (b15 + 2b12 - b11 + b10 - b1 + 1) q^71 + (-b8 + b7 - b6 + b3 + b2) * q^73 + (-b13 + 2b8 + 2b7 + b6 - 3b4 + b2) q^77 + (-2b12 + b11 - b10 + b9 + 6) q^79 + (-b12 - 2b11 + 2b10 - 2b9 - 1) q^81 + (-b13 + 2b8 + 2b7 + b6 + b4) * q^83 + (3b8 + 4b7 + 2b6 - 4b5 + 4b4 - 3b3 - 3b2) q^87 + (b15 - b11 + b10 - b1 - 1) * q^89 + (-b14 - b12 + b10 - 5b9 + b1 + 3) q^91 + (-b13 + 3b8 - 3b7 - b6 + 2b5) q^93 + (-b8 - b7 - 3b6 - 2b5 + 2b4 + b3 + b2) q^97 + (-b14 - b12 + b9 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9} - 8 q^{11} - 8 q^{19} - 16 q^{29} + 48 q^{51} - 8 q^{59} - 16 q^{61} + 16 q^{69} + 32 q^{71} + 80 q^{79} + 16 q^{81} + 64 q^{91} - 8 q^{99}+O(q^{100}) $ 16 * q + 16 * q^9 - 8 * q^11 - 8 * q^19 - 16 * q^29 + 48 * q^51 - 8 * q^59 - 16 * q^61 + 16 * q^69 + 32 * q^71 + 80 * q^79 + 16 * q^81 + 64 * q^91 - 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 $ x^16 - 2*x^14 + 6*x^12 - 12*x^10 + 36*x^8 - 48*x^6 + 96*x^4 - 128*x^2 + 256 :

$\beta_{1}$	$=$	$ ( -\nu^{12} + 2\nu^{10} - 6\nu^{8} + 12\nu^{6} - 20\nu^{4} + 48\nu^{2} - 48 ) / 16 $ (-v^12 + 2v^10 - 6v^8 + 12v^6 - 20v^4 + 48*v^2 - 48) / 16
$\beta_{2}$	$=$	$ ( -\nu^{15} + 18\nu^{13} - 42\nu^{11} + 68\nu^{9} - 60\nu^{7} + 224\nu^{5} - 432\nu^{3} + 320\nu ) / 448 $ (-v^15 + 18v^13 - 42v^11 + 68v^9 - 60v^7 + 224v^5 - 432v^3 + 320*v) / 448
$\beta_{3}$	$=$	$ ( 5\nu^{15} - 6\nu^{13} - 42\nu^{11} + 108\nu^{9} + 132\nu^{7} + 480\nu^{3} + 1088\nu ) / 1344 $ (5v^15 - 6v^13 - 42v^11 + 108v^9 + 132v^7 + 480v^3 + 1088*v) / 1344
$\beta_{4}$	$=$	$ ( \nu^{15} - 4\nu^{13} - 12\nu^{9} + 32\nu^{7} - 72\nu^{3} - 96\nu ) / 224 $ (v^15 - 4v^13 - 12v^9 + 32v^7 - 72v^3 - 96*v) / 224
$\beta_{5}$	$=$	$ ( \nu^{15} + 6\nu^{11} - 24\nu^{9} + 36\nu^{7} - 24\nu^{5} + 144\nu^{3} - 128\nu ) / 192 $ (v^15 + 6v^11 - 24v^9 + 36v^7 - 24v^5 + 144v^3 - 128v) / 192
$\beta_{6}$	$=$	$ ( 2\nu^{15} + 6\nu^{13} + 21\nu^{11} + 18\nu^{9} + 78\nu^{7} + 84\nu^{5} + 276\nu^{3} + 32\nu ) / 336 $ (2v^15 + 6v^13 + 21v^11 + 18v^9 + 78v^7 + 84v^5 + 276v^3 + 32v) / 336
$\beta_{7}$	$=$	$ ( -3\nu^{15} - 2\nu^{13} - 14\nu^{11} - 20\nu^{9} + 44\nu^{7} - 224\nu^{5} + 48\nu^{3} - 384\nu ) / 448 $ (-3v^15 - 2v^13 - 14v^11 - 20v^9 + 44v^7 - 224v^5 + 48v^3 - 384v) / 448
$\beta_{8}$	$=$	$ ( -3\nu^{15} + 12\nu^{13} - 42\nu^{11} + 64\nu^{9} - 124\nu^{7} + 280\nu^{5} - 176\nu^{3} + 64\nu ) / 448 $ (-3v^15 + 12v^13 - 42v^11 + 64v^9 - 124v^7 + 280v^5 - 176v^3 + 64v) / 448
$\beta_{9}$	$=$	$ ( 11\nu^{14} - 30\nu^{12} + 42\nu^{10} - 132\nu^{8} + 156\nu^{6} - 336\nu^{4} + 384\nu^{2} - 832 ) / 1344 $ (11v^14 - 30v^12 + 42v^10 - 132v^8 + 156v^6 - 336v^4 + 384*v^2 - 832) / 1344
$\beta_{10}$	$=$	$ ( -5\nu^{14} + 6\nu^{12} - 14\nu^{10} + 116\nu^{8} - 244\nu^{6} + 448\nu^{4} - 256\nu^{2} + 1600 ) / 448 $ (-5v^14 + 6v^12 - 14v^10 + 116v^8 - 244v^6 + 448v^4 - 256*v^2 + 1600) / 448
$\beta_{11}$	$=$	$ ( 4\nu^{14} - 9\nu^{12} + 42\nu^{10} - 6\nu^{8} + 156\nu^{6} - 84\nu^{4} + 48\nu^{2} + 64 ) / 336 $ (4v^14 - 9v^12 + 42v^10 - 6v^8 + 156v^6 - 84v^4 + 48*v^2 + 64) / 336
$\beta_{12}$	$=$	$ ( -\nu^{14} - 2\nu^{12} - 6\nu^{10} + 4\nu^{8} - 4\nu^{6} - 64\nu^{2} ) / 64 $ (-v^14 - 2v^12 - 6v^10 + 4v^8 - 4v^6 - 64*v^2) / 64
$\beta_{13}$	$=$	$ ( -11\nu^{15} + 9\nu^{13} - 42\nu^{11} + 90\nu^{9} - 156\nu^{7} + 420\nu^{5} - 216\nu^{3} + 1504\nu ) / 672 $ (-11v^15 + 9v^13 - 42v^11 + 90v^9 - 156v^7 + 420v^5 - 216v^3 + 1504v) / 672
$\beta_{14}$	$=$	$ ( 23\nu^{14} + 6\nu^{12} + 42\nu^{10} - 108\nu^{8} + 540\nu^{6} + 672\nu^{4} + 1536\nu^{2} - 640 ) / 1344 $ (23v^14 + 6v^12 + 42v^10 - 108v^8 + 540v^6 + 672v^4 + 1536*v^2 - 640) / 1344
$\beta_{15}$	$=$	$ ( 31\nu^{14} + 30\nu^{12} + 42\nu^{10} + 132\nu^{8} + 348\nu^{6} + 2304\nu^{2} + 832 ) / 1344 $ (31v^14 + 30v^12 + 42v^10 + 132v^8 + 348v^6 + 2304v^2 + 832) / 1344

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

$\nu$	$=$	$ ( \beta_{13} - 2\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 4 $ (b13 - 2*b8 - b7 - b6 + b5 - b4 + b3 + b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{11} + \beta_{10} + \beta _1 - 1 ) / 4 $ (b15 - b11 + b10 + b1 - 1) / 4
$\nu^{3}$	$=$	$ ( \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 2 $ (b8 + b5 - b4 + b3 - b2) / 2
$\nu^{4}$	$=$	$ ( -\beta_{15} + 2\beta_{14} + \beta_{10} - 2 ) / 2 $ (-b15 + 2*b14 + b10 - 2) / 2
$\nu^{5}$	$=$	$ ( \beta_{13} + 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} ) / 2 $ (b13 + 2b8 - b7 + b6 + b5 + 3b4 - b3 - b2) / 2
$\nu^{6}$	$=$	$ \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 3\beta_{9} + 2 $ b14 + b12 + b11 - b10 - 3*b9 + 2
$\nu^{7}$	$=$	$ \beta_{13} - \beta_{8} + 2\beta_{7} + 2\beta_{6} + 3\beta_{4} + \beta_{2} $ b13 - b8 + 2b7 + 2b6 + 3*b4 + b2
$\nu^{8}$	$=$	$ 3\beta_{15} - 2\beta_{14} + 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} - 8 $ 3b15 - 2b14 + 2b12 + 2b11 + b10 - 2*b9 - 8
$\nu^{9}$	$=$	$ -2\beta_{13} + 2\beta_{8} + 2\beta_{7} + 4\beta_{6} - 8\beta_{5} + 2\beta_{3} - 2\beta_{2} $ -2b13 + 2b8 + 2b7 + 4b6 - 8b5 + 2b3 - 2*b2
$\nu^{10}$	$=$	$ -3\beta_{15} - 2\beta_{14} - 6\beta_{12} + 5\beta_{11} + 3\beta_{10} - 2\beta_{9} + \beta _1 - 9 $ -3b15 - 2b14 - 6b12 + 5b11 + 3b10 - 2b9 + b1 - 9
$\nu^{11}$	$=$	$ -4\beta_{8} - 2\beta_{7} + 6\beta_{6} - 8\beta_{5} - 2\beta_{4} - 8\beta_{3} - 2\beta_{2} $ -4b8 - 2b7 + 6b6 - 8b5 - 2b4 - 8b3 - 2*b2
$\nu^{12}$	$=$	$ -2\beta_{15} - 12\beta_{12} - 2\beta_{11} - 10\beta_{10} - 28\beta_{9} - 2\beta _1 + 14 $ -2b15 - 12b12 - 2b11 - 10b10 - 28b9 - 2b1 + 14
$\nu^{13}$	$=$	$ -2\beta_{13} - 12\beta_{8} + 2\beta_{7} + 2\beta_{6} + 14\beta_{5} - 22\beta_{4} - 10\beta_{3} + 22\beta_{2} $ -2b13 - 12b8 + 2b7 + 2b6 + 14b5 - 22b4 - 10b3 + 22b2
$\nu^{14}$	$=$	$ 18\beta_{15} - 6\beta_{11} - 6\beta_{10} + 72\beta_{9} - 18\beta _1 + 2 $ 18b15 - 6b11 - 6b10 + 72b9 - 18*b1 + 2
$\nu^{15}$	$=$	$ -40\beta_{13} - 4\beta_{8} - 56\beta_{7} - 32\beta_{6} + 20\beta_{5} - 20\beta_{4} + 44\beta_{3} + 20\beta_{2} $ -40b13 - 4b8 - 56b7 - 32b6 + 20b5 - 20b4 + 44b3 + 20b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$\beta_{9}$	$\beta_{9}$	$-1$

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} $

207.1

−1.24570	−	0.669507i
0.601202	−	1.28006i
0.859408	+	1.12313i
1.35949	−	0.389597i
−1.35949	+	0.389597i
−0.859408	−	1.12313i
−0.601202	+	1.28006i
1.24570	+	0.669507i
−1.24570	+	0.669507i
0.601202	+	1.28006i
0.859408	−	1.12313i
1.35949	+	0.389597i
−1.35949	−	0.389597i
−0.859408	+	1.12313i
−0.601202	−	1.28006i
1.24570	−	0.669507i

−3.07455

−1.47763

−

1.47763i

6.45288

207.2

−2.02856

3.26957

3.26957i

1.11507

207.3

−1.54564

−1.17442

−

1.17442i

−0.611000

207.4

−0.207468

−1.32185

−

1.32185i

−2.95696

207.5

0.207468

1.32185

1.32185i

−2.95696

207.6

1.54564

1.17442

1.17442i

−0.611000

207.7

2.02856

−3.26957

−

3.26957i

1.11507

207.8

3.07455

1.47763

1.47763i

6.45288

943.1

−3.07455

−1.47763

1.47763i

6.45288

943.2

−2.02856

3.26957

−

3.26957i

1.11507

943.3

−1.54564

−1.17442

1.17442i

−0.611000

943.4

−0.207468

−1.32185

1.32185i

−2.95696

943.5

0.207468

1.32185

−

1.32185i

−2.95696

943.6

1.54564

1.17442

−

1.17442i

−0.611000

943.7

2.02856

−3.26957

3.26957i

1.11507

943.8

3.07455

1.47763

−

1.47763i

6.45288

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
80.j	even	4	1	inner
80.s	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.2.s.c		16
4.b	odd	2	1		400.2.s.c	yes	16
5.b	even	2	1	inner	1600.2.s.c		16
5.c	odd	4	2		1600.2.j.c		16
16.e	even	4	1		400.2.j.c	✓	16
16.f	odd	4	1		1600.2.j.c		16
20.d	odd	2	1		400.2.s.c	yes	16
20.e	even	4	2		400.2.j.c	✓	16
80.i	odd	4	1		400.2.s.c	yes	16
80.j	even	4	1	inner	1600.2.s.c		16
80.k	odd	4	1		1600.2.j.c		16
80.q	even	4	1		400.2.j.c	✓	16
80.s	even	4	1	inner	1600.2.s.c		16
80.t	odd	4	1		400.2.s.c	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.2.j.c	✓	16	16.e	even	4	1
400.2.j.c	✓	16	20.e	even	4	2
400.2.j.c	✓	16	80.q	even	4	1
400.2.s.c	yes	16	4.b	odd	2	1
400.2.s.c	yes	16	20.d	odd	2	1
400.2.s.c	yes	16	80.i	odd	4	1
400.2.s.c	yes	16	80.t	odd	4	1
1600.2.j.c		16	5.c	odd	4	2
1600.2.j.c		16	16.f	odd	4	1
1600.2.j.c		16	80.k	odd	4	1
1600.2.s.c		16	1.a	even	1	1	trivial
1600.2.s.c		16	5.b	even	2	1	inner
1600.2.s.c		16	80.j	even	4	1	inner
1600.2.s.c		16	80.s	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 16T_{3}^{6} + 72T_{3}^{4} - 96T_{3}^{2} + 4 $ T3^8 - 16*T3^6 + 72*T3^4 - 96*T3^2 + 4 acting on $S_{2}^{\mathrm{new}}(1600, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 16 T^{6} + 72 T^{4} + \cdots + 4)^{2} $ (T^8 - 16T^6 + 72T^4 - 96*T^2 + 4)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} + 496 T^{12} + \cdots + 810000 $ T^16 + 496T^12 + 18248T^8 + 217024*T^4 + 810000
$11$	$ (T^{8} + 4 T^{7} + \cdots + 7056)^{2} $ (T^8 + 4T^7 + 8T^6 - 40T^5 + 232T^4 + 464T^3 + 800T^2 - 3360*T + 7056)^2
$13$	$ (T^{8} + 56 T^{6} + \cdots + 5184)^{2} $ (T^8 + 56T^6 + 896T^4 + 4160*T^2 + 5184)^2
$17$	$ T^{16} + \cdots + 1600000000 $ T^16 + 1856T^12 + 793728T^8 + 70569984*T^4 + 1600000000
$19$	$ (T^{8} + 4 T^{7} + \cdots + 38416)^{2} $ (T^8 + 4T^7 + 8T^6 - 8T^5 + 1544T^4 + 6608T^3 + 14112T^2 - 32928*T + 38416)^2
$23$	$ T^{16} + 2448 T^{12} + \cdots + 10000 $ T^16 + 2448T^12 + 470280T^8 + 2498624*T^4 + 10000
$29$	$ (T^{8} + 8 T^{7} + \cdots + 242064)^{2} $ (T^8 + 8T^7 + 32T^6 + 80T^5 + 3688T^4 + 29728T^3 + 123008T^2 + 244032*T + 242064)^2
$31$	$ (T^{8} + 160 T^{6} + \cdots + 1849600)^{2} $ (T^8 + 160T^6 + 9056T^4 + 216576*T^2 + 1849600)^2
$37$	$ (T^{8} + 144 T^{6} + \cdots + 9216)^{2} $ (T^8 + 144T^6 + 2816T^4 + 9728*T^2 + 9216)^2
$41$	$ (T^{8} + 192 T^{6} + \cdots + 90000)^{2} $ (T^8 + 192T^6 + 8616T^4 + 77824*T^2 + 90000)^2
$43$	$ (T^{8} + 96 T^{6} + \cdots + 36)^{2} $ (T^8 + 96T^6 + 632T^4 + 608*T^2 + 36)^2
$47$	$ T^{16} + 20464 T^{12} + \cdots + 38416 $ T^16 + 20464T^12 + 25159112T^8 + 47051712*T^4 + 38416
$53$	$ (T^{8} - 200 T^{6} + \cdots + 1915456)^{2} $ (T^8 - 200T^6 + 13760T^4 - 353472*T^2 + 1915456)^2
$59$	$ (T^{8} + 4 T^{7} + \cdots + 1192464)^{2} $ (T^8 + 4T^7 + 8T^6 + 88T^5 + 13192T^4 + 68048T^3 + 170528T^2 - 637728*T + 1192464)^2
$61$	$ (T^{8} + 8 T^{7} + \cdots + 3625216)^{2} $ (T^8 + 8T^7 + 32T^6 + 416T^5 + 20192T^4 + 199552T^3 + 1036800T^2 + 2741760*T + 3625216)^2
$67$	$ (T^{8} + 336 T^{6} + \cdots + 9771876)^{2} $ (T^8 + 336T^6 + 36792T^4 + 1428800*T^2 + 9771876)^2
$71$	$ (T^{4} - 8 T^{3} + \cdots - 192)^{4} $ (T^4 - 8T^3 - 128T^2 + 1088*T - 192)^4
$73$	$ T^{16} + \cdots + 14281868906496 $ T^16 + 22208T^12 + 116635776T^8 + 163305631744*T^4 + 14281868906496
$79$	$ (T^{4} - 20 T^{3} + \cdots - 6000)^{4} $ (T^4 - 20T^3 + 8T^2 + 1552*T - 6000)^4
$83$	$ (T^{8} - 256 T^{6} + \cdots + 4235364)^{2} $ (T^8 - 256T^6 + 19416T^4 - 509600*T^2 + 4235364)^2
$89$	$ (T^{4} - 136 T^{2} + \cdots - 240)^{4} $ (T^4 - 136T^2 + 512T - 240)^4
$97$	$ T^{16} + \cdots + 77\!\cdots\!00 $ T^16 + 136640T^12 + 4644204672T^8 + 35550049398784*T^4 + 77407703861760000
show more
show less

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.s (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} - \beta_{7} q^{7} + ( - \beta_{12} + 1) q^{9} + (\beta_{10} - 1) q^{11} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{13} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{4}) q^{17} + (\beta_{14} + \beta_{12} + \cdots - \beta_1) q^{19}+ \cdots + ( - \beta_{14} - \beta_{12} + \cdots - \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 - b7 * q^7 + (-b12 + 1) * q^9 + (b10 - 1) * q^11 + (b7 - b5 - b3) * q^13 + (-b7 - b6 + b5 - b4) * q^17 + (b14 + b12 + b9 - b1) * q^19 + (b15 - b11 - 1) * q^21 + (b3 - b2) * q^23 + (b13 - 2b8 - 2b7 - b6 - 2b4 + 2b2) * q^27 + (b15 - b14 + b12 + b11 - 1) * q^29 + (b15 - b9 + b1 - 1) * q^31 + (b13 - 2b8 - b3 + b2) q^33 + (b13 + b8 + b7 + b6 - 2b3) q^37 + (-2b14 + b11 + b10 - 3b9) * q^39 + (b15 - b14 + b11 + b10 + b1 - 1) * q^41 + (b13 + b6 - b5) * q^43 + (-2b13 + b8) q^47 + (-b15 + b14 + b11 + b10 + b9 - b1 + 1) * q^49 + (b15 + b14 - b12 - b11 + 3b9 + 2) q^51 + (-b13 + b6 + 3b4 - b2) q^53 + (-b13 - b8 + 3b5 + 3b4 + 2b3 - 2b2) * q^57 + (b15 + b14 - b12 - 1) * q^59 + (b14 + b12 + 2b10 - 2) q^61 + (-b8 - 3b7 - 2b6 + 2b5 - 2b4 + b3 + b2) * q^63 + (b13 - 2b8 + b6 + b5 + 2b3) * q^67 + (b14 + b12 - b10 - b1 + 2) * q^69 + (b15 + 2b12 - b11 + b10 - b1 + 1) q^71 + (-b8 + b7 - b6 + b3 + b2) * q^73 + (-b13 + 2b8 + 2b7 + b6 - 3b4 + b2) q^77 + (-2b12 + b11 - b10 + b9 + 6) q^79 + (-b12 - 2b11 + 2b10 - 2b9 - 1) q^81 + (-b13 + 2b8 + 2b7 + b6 + b4) * q^83 + (3b8 + 4b7 + 2b6 - 4b5 + 4b4 - 3b3 - 3b2) q^87 + (b15 - b11 + b10 - b1 - 1) * q^89 + (-b14 - b12 + b10 - 5b9 + b1 + 3) q^91 + (-b13 + 3b8 - 3b7 - b6 + 2b5) q^93 + (-b8 - b7 - 3b6 - 2b5 + 2b4 + b3 + b2) q^97 + (-b14 - b12 + b9 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9} - 8 q^{11} - 8 q^{19} - 16 q^{29} + 48 q^{51} - 8 q^{59} - 16 q^{61} + 16 q^{69} + 32 q^{71} + 80 q^{79} + 16 q^{81} + 64 q^{91} - 8 q^{99}+O(q^{100}) \) 16 * q + 16 * q^9 - 8 * q^11 - 8 * q^19 - 16 * q^29 + 48 * q^51 - 8 * q^59 - 16 * q^61 + 16 * q^69 + 32 * q^71 + 80 * q^79 + 16 * q^81 + 64 * q^91 - 8 * q^99

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	1600.2.s.c

Level	$1600$
Weight	$2$
Character orbit	1600.s
Analytic conductor	$12.776$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(12.7760643234\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 \) x^16 - 2x^14 + 6x^12 - 12x^10 + 36x^8 - 48x^6 + 96x^4 - 128*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{12} + 2\nu^{10} - 6\nu^{8} + 12\nu^{6} - 20\nu^{4} + 48\nu^{2} - 48 ) / 16 \) (-v^12 + 2v^10 - 6v^8 + 12v^6 - 20v^4 + 48*v^2 - 48) / 16
\(\beta_{2}\)	\(=\)	\( ( -\nu^{15} + 18\nu^{13} - 42\nu^{11} + 68\nu^{9} - 60\nu^{7} + 224\nu^{5} - 432\nu^{3} + 320\nu ) / 448 \) (-v^15 + 18v^13 - 42v^11 + 68v^9 - 60v^7 + 224v^5 - 432v^3 + 320*v) / 448
\(\beta_{3}\)	\(=\)	\( ( 5\nu^{15} - 6\nu^{13} - 42\nu^{11} + 108\nu^{9} + 132\nu^{7} + 480\nu^{3} + 1088\nu ) / 1344 \) (5v^15 - 6v^13 - 42v^11 + 108v^9 + 132v^7 + 480v^3 + 1088*v) / 1344
\(\beta_{4}\)	\(=\)	\( ( \nu^{15} - 4\nu^{13} - 12\nu^{9} + 32\nu^{7} - 72\nu^{3} - 96\nu ) / 224 \) (v^15 - 4v^13 - 12v^9 + 32v^7 - 72v^3 - 96*v) / 224
\(\beta_{5}\)	\(=\)	\( ( \nu^{15} + 6\nu^{11} - 24\nu^{9} + 36\nu^{7} - 24\nu^{5} + 144\nu^{3} - 128\nu ) / 192 \) (v^15 + 6v^11 - 24v^9 + 36v^7 - 24v^5 + 144v^3 - 128v) / 192
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{15} + 6\nu^{13} + 21\nu^{11} + 18\nu^{9} + 78\nu^{7} + 84\nu^{5} + 276\nu^{3} + 32\nu ) / 336 \) (2v^15 + 6v^13 + 21v^11 + 18v^9 + 78v^7 + 84v^5 + 276v^3 + 32v) / 336
\(\beta_{7}\)	\(=\)	\( ( -3\nu^{15} - 2\nu^{13} - 14\nu^{11} - 20\nu^{9} + 44\nu^{7} - 224\nu^{5} + 48\nu^{3} - 384\nu ) / 448 \) (-3v^15 - 2v^13 - 14v^11 - 20v^9 + 44v^7 - 224v^5 + 48v^3 - 384v) / 448
\(\beta_{8}\)	\(=\)	\( ( -3\nu^{15} + 12\nu^{13} - 42\nu^{11} + 64\nu^{9} - 124\nu^{7} + 280\nu^{5} - 176\nu^{3} + 64\nu ) / 448 \) (-3v^15 + 12v^13 - 42v^11 + 64v^9 - 124v^7 + 280v^5 - 176v^3 + 64v) / 448
\(\beta_{9}\)	\(=\)	\( ( 11\nu^{14} - 30\nu^{12} + 42\nu^{10} - 132\nu^{8} + 156\nu^{6} - 336\nu^{4} + 384\nu^{2} - 832 ) / 1344 \) (11v^14 - 30v^12 + 42v^10 - 132v^8 + 156v^6 - 336v^4 + 384*v^2 - 832) / 1344
\(\beta_{10}\)	\(=\)	\( ( -5\nu^{14} + 6\nu^{12} - 14\nu^{10} + 116\nu^{8} - 244\nu^{6} + 448\nu^{4} - 256\nu^{2} + 1600 ) / 448 \) (-5v^14 + 6v^12 - 14v^10 + 116v^8 - 244v^6 + 448v^4 - 256*v^2 + 1600) / 448
\(\beta_{11}\)	\(=\)	\( ( 4\nu^{14} - 9\nu^{12} + 42\nu^{10} - 6\nu^{8} + 156\nu^{6} - 84\nu^{4} + 48\nu^{2} + 64 ) / 336 \) (4v^14 - 9v^12 + 42v^10 - 6v^8 + 156v^6 - 84v^4 + 48*v^2 + 64) / 336
\(\beta_{12}\)	\(=\)	\( ( -\nu^{14} - 2\nu^{12} - 6\nu^{10} + 4\nu^{8} - 4\nu^{6} - 64\nu^{2} ) / 64 \) (-v^14 - 2v^12 - 6v^10 + 4v^8 - 4v^6 - 64*v^2) / 64
\(\beta_{13}\)	\(=\)	\( ( -11\nu^{15} + 9\nu^{13} - 42\nu^{11} + 90\nu^{9} - 156\nu^{7} + 420\nu^{5} - 216\nu^{3} + 1504\nu ) / 672 \) (-11v^15 + 9v^13 - 42v^11 + 90v^9 - 156v^7 + 420v^5 - 216v^3 + 1504v) / 672
\(\beta_{14}\)	\(=\)	\( ( 23\nu^{14} + 6\nu^{12} + 42\nu^{10} - 108\nu^{8} + 540\nu^{6} + 672\nu^{4} + 1536\nu^{2} - 640 ) / 1344 \) (23v^14 + 6v^12 + 42v^10 - 108v^8 + 540v^6 + 672v^4 + 1536*v^2 - 640) / 1344
\(\beta_{15}\)	\(=\)	\( ( 31\nu^{14} + 30\nu^{12} + 42\nu^{10} + 132\nu^{8} + 348\nu^{6} + 2304\nu^{2} + 832 ) / 1344 \) (31v^14 + 30v^12 + 42v^10 + 132v^8 + 348v^6 + 2304v^2 + 832) / 1344

\(\nu\)	\(=\)	\( ( \beta_{13} - 2\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 4 \) (b13 - 2*b8 - b7 - b6 + b5 - b4 + b3 + b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{11} + \beta_{10} + \beta _1 - 1 ) / 4 \) (b15 - b11 + b10 + b1 - 1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) (b8 + b5 - b4 + b3 - b2) / 2
\(\nu^{4}\)	\(=\)	\( ( -\beta_{15} + 2\beta_{14} + \beta_{10} - 2 ) / 2 \) (-b15 + 2*b14 + b10 - 2) / 2
\(\nu^{5}\)	\(=\)	\( ( \beta_{13} + 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) (b13 + 2b8 - b7 + b6 + b5 + 3b4 - b3 - b2) / 2
\(\nu^{6}\)	\(=\)	\( \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 3\beta_{9} + 2 \) b14 + b12 + b11 - b10 - 3*b9 + 2
\(\nu^{7}\)	\(=\)	\( \beta_{13} - \beta_{8} + 2\beta_{7} + 2\beta_{6} + 3\beta_{4} + \beta_{2} \) b13 - b8 + 2b7 + 2b6 + 3*b4 + b2
\(\nu^{8}\)	\(=\)	\( 3\beta_{15} - 2\beta_{14} + 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} - 8 \) 3b15 - 2b14 + 2b12 + 2b11 + b10 - 2*b9 - 8
\(\nu^{9}\)	\(=\)	\( -2\beta_{13} + 2\beta_{8} + 2\beta_{7} + 4\beta_{6} - 8\beta_{5} + 2\beta_{3} - 2\beta_{2} \) -2b13 + 2b8 + 2b7 + 4b6 - 8b5 + 2b3 - 2*b2
\(\nu^{10}\)	\(=\)	\( -3\beta_{15} - 2\beta_{14} - 6\beta_{12} + 5\beta_{11} + 3\beta_{10} - 2\beta_{9} + \beta _1 - 9 \) -3b15 - 2b14 - 6b12 + 5b11 + 3b10 - 2b9 + b1 - 9
\(\nu^{11}\)	\(=\)	\( -4\beta_{8} - 2\beta_{7} + 6\beta_{6} - 8\beta_{5} - 2\beta_{4} - 8\beta_{3} - 2\beta_{2} \) -4b8 - 2b7 + 6b6 - 8b5 - 2b4 - 8b3 - 2*b2
\(\nu^{12}\)	\(=\)	\( -2\beta_{15} - 12\beta_{12} - 2\beta_{11} - 10\beta_{10} - 28\beta_{9} - 2\beta _1 + 14 \) -2b15 - 12b12 - 2b11 - 10b10 - 28b9 - 2b1 + 14
\(\nu^{13}\)	\(=\)	\( -2\beta_{13} - 12\beta_{8} + 2\beta_{7} + 2\beta_{6} + 14\beta_{5} - 22\beta_{4} - 10\beta_{3} + 22\beta_{2} \) -2b13 - 12b8 + 2b7 + 2b6 + 14b5 - 22b4 - 10b3 + 22b2
\(\nu^{14}\)	\(=\)	\( 18\beta_{15} - 6\beta_{11} - 6\beta_{10} + 72\beta_{9} - 18\beta _1 + 2 \) 18b15 - 6b11 - 6b10 + 72b9 - 18*b1 + 2
\(\nu^{15}\)	\(=\)	\( -40\beta_{13} - 4\beta_{8} - 56\beta_{7} - 32\beta_{6} + 20\beta_{5} - 20\beta_{4} + 44\beta_{3} + 20\beta_{2} \) -40b13 - 4b8 - 56b7 - 32b6 + 20b5 - 20b4 + 44b3 + 20b2

\(n\)	\(577\)	\(901\)	\(1151\)
\(\chi(n)\)	\(\beta_{9}\)	\(\beta_{9}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 207.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 16 T^{6} + 72 T^{4} + \cdots + 4)^{2} \) (T^8 - 16T^6 + 72T^4 - 96*T^2 + 4)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} + 496 T^{12} + \cdots + 810000 \) T^16 + 496T^12 + 18248T^8 + 217024*T^4 + 810000
$11$	\( (T^{8} + 4 T^{7} + \cdots + 7056)^{2} \) (T^8 + 4T^7 + 8T^6 - 40T^5 + 232T^4 + 464T^3 + 800T^2 - 3360*T + 7056)^2
$13$	\( (T^{8} + 56 T^{6} + \cdots + 5184)^{2} \) (T^8 + 56T^6 + 896T^4 + 4160*T^2 + 5184)^2
$17$	\( T^{16} + \cdots + 1600000000 \) T^16 + 1856T^12 + 793728T^8 + 70569984*T^4 + 1600000000
$19$	\( (T^{8} + 4 T^{7} + \cdots + 38416)^{2} \) (T^8 + 4T^7 + 8T^6 - 8T^5 + 1544T^4 + 6608T^3 + 14112T^2 - 32928*T + 38416)^2
$23$	\( T^{16} + 2448 T^{12} + \cdots + 10000 \) T^16 + 2448T^12 + 470280T^8 + 2498624*T^4 + 10000
$29$	\( (T^{8} + 8 T^{7} + \cdots + 242064)^{2} \) (T^8 + 8T^7 + 32T^6 + 80T^5 + 3688T^4 + 29728T^3 + 123008T^2 + 244032*T + 242064)^2
$31$	\( (T^{8} + 160 T^{6} + \cdots + 1849600)^{2} \) (T^8 + 160T^6 + 9056T^4 + 216576*T^2 + 1849600)^2
$37$	\( (T^{8} + 144 T^{6} + \cdots + 9216)^{2} \) (T^8 + 144T^6 + 2816T^4 + 9728*T^2 + 9216)^2
$41$	\( (T^{8} + 192 T^{6} + \cdots + 90000)^{2} \) (T^8 + 192T^6 + 8616T^4 + 77824*T^2 + 90000)^2
$43$	\( (T^{8} + 96 T^{6} + \cdots + 36)^{2} \) (T^8 + 96T^6 + 632T^4 + 608*T^2 + 36)^2
$47$	\( T^{16} + 20464 T^{12} + \cdots + 38416 \) T^16 + 20464T^12 + 25159112T^8 + 47051712*T^4 + 38416
$53$	\( (T^{8} - 200 T^{6} + \cdots + 1915456)^{2} \) (T^8 - 200T^6 + 13760T^4 - 353472*T^2 + 1915456)^2
$59$	\( (T^{8} + 4 T^{7} + \cdots + 1192464)^{2} \) (T^8 + 4T^7 + 8T^6 + 88T^5 + 13192T^4 + 68048T^3 + 170528T^2 - 637728*T + 1192464)^2
$61$	\( (T^{8} + 8 T^{7} + \cdots + 3625216)^{2} \) (T^8 + 8T^7 + 32T^6 + 416T^5 + 20192T^4 + 199552T^3 + 1036800T^2 + 2741760*T + 3625216)^2
$67$	\( (T^{8} + 336 T^{6} + \cdots + 9771876)^{2} \) (T^8 + 336T^6 + 36792T^4 + 1428800*T^2 + 9771876)^2
$71$	\( (T^{4} - 8 T^{3} + \cdots - 192)^{4} \) (T^4 - 8T^3 - 128T^2 + 1088*T - 192)^4
$73$	\( T^{16} + \cdots + 14281868906496 \) T^16 + 22208T^12 + 116635776T^8 + 163305631744*T^4 + 14281868906496
$79$	\( (T^{4} - 20 T^{3} + \cdots - 6000)^{4} \) (T^4 - 20T^3 + 8T^2 + 1552*T - 6000)^4
$83$	\( (T^{8} - 256 T^{6} + \cdots + 4235364)^{2} \) (T^8 - 256T^6 + 19416T^4 - 509600*T^2 + 4235364)^2
$89$	\( (T^{4} - 136 T^{2} + \cdots - 240)^{4} \) (T^4 - 136T^2 + 512T - 240)^4
$97$	\( T^{16} + \cdots + 77\!\cdots\!00 \) T^16 + 136640T^12 + 4644204672T^8 + 35550049398784*T^4 + 77407703861760000
show more
show less