LMFDB - Newform orbit 525.2.j.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(218,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

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

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

chi := DirichletCharacter("525.218");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.j (of order $4$, degree $2$, 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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.6040479020157644046336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 $ x^16 - 7x^12 - 32x^8 - 567*x^4 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
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_{15} + \beta_{13} - \beta_{10}) q^{2} + \beta_{7} q^{3} + ( - 2 \beta_{14} + \beta_{11} - \beta_{2}) q^{4} + ( - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{6} - \beta_{12} q^{7} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7}) q^{8} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{9}+O(q^{10}) $ q + (-b15 + b13 - b10) * q^2 + b7 * q^3 + (-2b14 + b11 - b2) q^4 + (-2b6 - b3 - b1 + 1) q^6 - b12 * q^7 + (-b9 + 3b8 - b7) q^8 + (-b14 + b11 + b4 - b2) * q^9	$ q + ( - \beta_{15} + \beta_{13} - \beta_{10}) q^{2} + \beta_{7} q^{3} + ( - 2 \beta_{14} + \beta_{11} - \beta_{2}) q^{4} + ( - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{6} - \beta_{12} q^{7} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7}) q^{8} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{9}+ \cdots + (7 \beta_{14} + 4 \beta_{11} + \cdots + 8 \beta_{2}) q^{99}+O(q^{100}) $ q + (-b15 + b13 - b10) * q^2 + b7 * q^3 + (-2b14 + b11 - b2) q^4 + (-2b6 - b3 - b1 + 1) q^6 - b12 * q^7 + (-b9 + 3b8 - b7) q^8 + (-b14 + b11 + b4 - b2) * q^9 + (2b6 + 3b3 + 3b1) q^11 + (-b15 + b13 - 4b12 + b10) q^12 + (b9 - b7) * q^13 + (-b4 - b2) * q^14 + (b3 - b1 - 3) * q^16 + (-2b15 + b13 - b10) q^17 + (-3b9 + 3b8 - b7) * q^18 - 4b11 q^19 - b1 * q^21 + (-2b13 + 8b12 - 2b10) q^22 - 2b8 q^23 + (-2b14 - b11 - 4b4 - 5b2) q^24 + (4b6 + 2b3 + 2b1) q^26 + (-b15 + 2b13 - 4b12 - 2b10) q^27 + (b9 - b7 - b5) * q^28 + (2b4 - b2) q^29 + (2b3 - 2b1) * q^31 + (b15 - 3b13 + 3b10) * q^32 + (b9 - 5b8 + 2b7 - 4b5) q^33 + (-4b14 + 4b11 - 2b2) q^34 + (-2b6 - b3 - 5b1 - 5) * q^36 + (2b13 + 2b10) * q^37 + (4b9 - 4b8 + 4b7) q^38 + (b14 - 4b11 - b4 + b2) q^39 + (-4b6 + 2b3 + 2b1) q^41 + (2b15 - b13 - b12 + b10) q^42 + 8b5 q^43 + (12b4 + 6b2) * q^44 + (-2b3 + 2b1 + 2) * q^46 + (4b15 - b13 + b10) q^47 + (b9 + b8 - 3b7 - 4b5) * q^48 - b11 * q^49 + (-3b6 - 3b3 - 2b1) q^51 + (-2b13 + 8b12 - 2b10) q^52 + (-2b9 - 4b8 - 2b7) q^53 + (-2b14 + 5b11 - 4b4 - 5b2) * q^54 + (3b6 + b3 + b1) q^56 - 4b13 q^57 + (-2b9 + 2b7) * q^58 + (-4b4 + 2b2) * q^59 + (2b3 - 2b1 + 2) * q^61 + (8b15 - 4b13 + 4b10) q^62 + (b9 + b8 - b5) * q^63 + (-2b14 - b11 - b2) q^64 + (2b6 - 2b3 + 8b1 + 8) q^66 + (2b13 + 8b12 + 2b10) q^67 + (-6b9 + 8b8 - 6b7) q^68 + (2b14 + 2b11 + 2b4 + 4b2) * q^69 + (4b6 + 2b3 + 2b1) q^71 + (7b15 - 5b13 - 8b12 + 5b10) * q^72 + (2b9 - 2b7 + 8b5) q^73 + (-8b4 - 4b2) * q^74 + (-4b3 + 4b1 + 4) * q^76 + (-2b15 + 3b13 - 3b10) q^77 + (6b9 - 6b8 + 4b7) q^78 + (6b14 + 8b11 + 3b2) q^79 + (-3b6 - 5b1 + 3) * q^81 + (4b13 + 4b10) * q^82 + (8b9 - 2b8 + 8b7) q^83 + (2b14 - 4b11 - b4 + b2) * q^84 + (-8b6 - 8b3 - 8b1) q^86 + (-b15 + 2b13 - 4b12 - 5b10) q^87 + (-8b9 + 8b7 + 8b5) q^88 + (-8b4 - 4b2) * q^89 + (-b3 + b1) * q^91 + (-6b15 + 6b13 - 6b10) q^92 + (2b9 + 2b8 - 8b5) q^93 + (8b14 - 6b11 + 4b2) q^94 + (4b6 - b3 + b1 - 5) q^96 + (-b13 - b10) * q^97 + (b9 - b8 + b7) * q^98 + (7b14 + 4b11 + 7b4 + 8b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{6}+O(q^{10}) $ 16 * q + 16 * q^6	$ 16 q + 16 q^{6} - 56 q^{16} - 4 q^{21} - 16 q^{31} - 96 q^{36} + 48 q^{46} + 4 q^{51} + 16 q^{61} + 168 q^{66} + 96 q^{76} + 28 q^{81} + 8 q^{91} - 72 q^{96}+O(q^{100}) $ 16 * q + 16 * q^6 - 56 * q^16 - 4 * q^21 - 16 * q^31 - 96 * q^36 + 48 * q^46 + 4 * q^51 + 16 * q^61 + 168 * q^66 + 96 * q^76 + 28 * q^81 + 8 * q^91 - 72 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 $ x^16 - 7*x^12 - 32*x^8 - 567*x^4 + 6561 :

$\beta_{1}$	$=$	$ ( 7\nu^{12} + 32\nu^{8} + 2368\nu^{4} - 6561 ) / 12960 $ (7v^12 + 32v^8 + 2368*v^4 - 6561) / 12960
$\beta_{2}$	$=$	$ ( -5\nu^{14} - 208\nu^{10} + 4048\nu^{6} - 1053\nu^{2} ) / 58320 $ (-5v^14 - 208v^10 + 4048v^6 - 1053v^2) / 58320
$\beta_{3}$	$=$	$ ( 13\nu^{12} - 64\nu^{8} - 416\nu^{4} - 7371 ) / 4320 $ (13v^12 - 64v^8 - 416*v^4 - 7371) / 4320
$\beta_{4}$	$=$	$ ( -25\nu^{14} + 256\nu^{10} - 496\nu^{6} + 34911\nu^{2} ) / 58320 $ (-25v^14 + 256v^10 - 496v^6 + 34911v^2) / 58320
$\beta_{5}$	$=$	$ ( 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} - 20979\nu ) / 19440 $ (37v^13 - 16v^9 - 1184v^5 - 20979v) / 19440
$\beta_{6}$	$=$	$ ( 7\nu^{12} + 32\nu^{8} - 224\nu^{4} - 5265 ) / 1296 $ (7v^12 + 32v^8 - 224*v^4 - 5265) / 1296
$\beta_{7}$	$=$	$ ( \nu^{13} - 1079\nu ) / 480 $ (v^13 - 1079*v) / 480
$\beta_{8}$	$=$	$ ( 11\nu^{13} + 4\nu^{9} + 296\nu^{5} - 6885\nu ) / 4860 $ (11v^13 + 4v^9 + 296v^5 - 6885v) / 4860
$\beta_{9}$	$=$	$ ( -7\nu^{13} - 32\nu^{9} + 224\nu^{5} + 6561\nu ) / 2592 $ (-7v^13 - 32v^9 + 224v^5 + 6561v) / 2592
$\beta_{10}$	$=$	$ ( 31\nu^{15} + 512\nu^{11} - 992\nu^{7} - 17577\nu^{3} ) / 116640 $ (31v^15 + 512v^11 - 992v^7 - 17577v^3) / 116640
$\beta_{11}$	$=$	$ ( -7\nu^{14} - 32\nu^{10} + 62\nu^{6} + 6561\nu^{2} ) / 7290 $ (-7v^14 - 32v^10 + 62v^6 + 6561v^2) / 7290
$\beta_{12}$	$=$	$ ( -\nu^{15} + 359\nu^{3} ) / 2160 $ (-v^15 + 359*v^3) / 2160
$\beta_{13}$	$=$	$ ( 49\nu^{15} + 224\nu^{11} + 1024\nu^{7} - 45927\nu^{3} ) / 69984 $ (49v^15 + 224v^11 + 1024v^7 - 45927v^3) / 69984
$\beta_{14}$	$=$	$ ( 253\nu^{14} + 416\nu^{10} - 8096\nu^{6} - 143451\nu^{2} ) / 116640 $ (253v^14 + 416v^10 - 8096v^6 - 143451v^2) / 116640
$\beta_{15}$	$=$	$ ( -43\nu^{15} - 104\nu^{11} + 2024\nu^{7} + 28593\nu^{3} ) / 43740 $ (-43v^15 - 104v^11 + 2024v^7 + 28593v^3) / 43740

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

$\nu$	$=$	$ ( \beta_{8} - 2\beta_{7} + \beta_{5} ) / 2 $ (b8 - 2*b7 + b5) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{14} + 3\beta_{11} + 3\beta_{4} + 2\beta_{2} ) / 2 $ (2b14 + 3b11 + 3b4 + 2b2) / 2
$\nu^{3}$	$=$	$ \beta_{15} - 2\beta_{13} - 4\beta_{12} + 2\beta_{10} $ b15 - 2b13 - 4b12 + 2*b10
$\nu^{4}$	$=$	$ ( -\beta_{6} + 10\beta _1 + 1 ) / 2 $ (-b6 + 10*b1 + 1) / 2
$\nu^{5}$	$=$	$ ( 2\beta_{9} + 15\beta_{8} - 15\beta_{5} ) / 2 $ (2b9 + 15b8 - 15*b5) / 2
$\nu^{6}$	$=$	$ -5\beta_{11} + 8\beta_{4} + 16\beta_{2} $ -5b11 + 8b4 + 16*b2
$\nu^{7}$	$=$	$ ( 35\beta_{15} + 26\beta_{13} - 35\beta_{12} ) / 2 $ (35b15 + 26b13 - 35*b12) / 2
$\nu^{8}$	$=$	$ ( 39\beta_{6} - 70\beta_{3} + 39 ) / 2 $ (39b6 - 70b3 + 39) / 2
$\nu^{9}$	$=$	$ -74\beta_{9} + 37\beta_{8} - 74\beta_{7} - 68\beta_{5} $ -74b9 + 37b8 - 74b7 - 68b5
$\nu^{10}$	$=$	$ ( -62\beta_{14} - 253\beta_{11} + 253\beta_{4} ) / 2 $ (-62b14 - 253b11 + 253*b4) / 2
$\nu^{11}$	$=$	$ ( 93\beta_{15} + 93\beta_{12} + 506\beta_{10} ) / 2 $ (93b15 + 93b12 + 506*b10) / 2
$\nu^{12}$	$=$	$ 80\beta_{6} + 160\beta_{3} + 160\beta _1 + 679 $ 80b6 + 160b3 + 160*b1 + 679
$\nu^{13}$	$=$	$ ( 1079\beta_{8} - 1198\beta_{7} + 1079\beta_{5} ) / 2 $ (1079b8 - 1198b7 + 1079*b5) / 2
$\nu^{14}$	$=$	$ ( 2158\beta_{14} + 1797\beta_{11} + 1797\beta_{4} + 2158\beta_{2} ) / 2 $ (2158b14 + 1797b11 + 1797b4 + 2158b2) / 2
$\nu^{15}$	$=$	$ 359\beta_{15} - 718\beta_{13} - 3596\beta_{12} + 718\beta_{10} $ 359b15 - 718b13 - 3596b12 + 718b10

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-\beta_{11}$	$-1$	$1$

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} $

218.1

1.73122	−	0.0537601i
0.0537601	−	1.73122i
1.47240	+	0.912166i
−0.912166	−	1.47240i
0.912166	+	1.47240i
−1.47240	−	0.912166i
−0.0537601	+	1.73122i
−1.73122	+	0.0537601i
1.73122	+	0.0537601i
0.0537601	+	1.73122i
1.47240	−	0.912166i
−0.912166	+	1.47240i
0.912166	−	1.47240i
−1.47240	+	0.912166i
−0.0537601	−	1.73122i
−1.73122	−	0.0537601i

−1.78498

−

1.78498i

−1.47240

−

0.912166i

4.37228i

1.00000

4.25639i

0.707107

−

0.707107i

4.23447

−

4.23447i

1.33591

2.68614i

218.2

−1.78498

−

1.78498i

0.912166

1.47240i

4.37228i

1.00000

−

4.25639i

−0.707107

0.707107i

4.23447

−

4.23447i

−1.33591

2.68614i

218.3

−0.560232

−

0.560232i

−1.73122

0.0537601i

−

1.37228i

1.00000

0.939764i

−0.707107

0.707107i

−1.88926

1.88926i

2.99422

−

0.186141i

218.4

−0.560232

−

0.560232i

−0.0537601

1.73122i

−

1.37228i

1.00000

−

0.939764i

0.707107

−

0.707107i

−1.88926

1.88926i

−2.99422

−

0.186141i

218.5

0.560232

0.560232i

0.0537601

−

1.73122i

−

1.37228i

1.00000

−

0.939764i

−0.707107

0.707107i

1.88926

−

1.88926i

−2.99422

−

0.186141i

218.6

0.560232

0.560232i

1.73122

−

0.0537601i

−

1.37228i

1.00000

0.939764i

0.707107

−

0.707107i

1.88926

−

1.88926i

2.99422

−

0.186141i

218.7

1.78498

1.78498i

−0.912166

−

1.47240i

4.37228i

1.00000

−

4.25639i

0.707107

−

0.707107i

−4.23447

4.23447i

−1.33591

2.68614i

218.8

1.78498

1.78498i

1.47240

0.912166i

4.37228i

1.00000

4.25639i

−0.707107

0.707107i

−4.23447

4.23447i

1.33591

2.68614i

407.1

−1.78498

1.78498i

−1.47240

0.912166i

−

4.37228i

1.00000

−

4.25639i

0.707107

0.707107i

4.23447

4.23447i

1.33591

−

2.68614i

407.2

−1.78498

1.78498i

0.912166

−

1.47240i

−

4.37228i

1.00000

4.25639i

−0.707107

−

0.707107i

4.23447

4.23447i

−1.33591

−

2.68614i

407.3

−0.560232

0.560232i

−1.73122

−

0.0537601i

1.37228i

1.00000

−

0.939764i

−0.707107

−

0.707107i

−1.88926

−

1.88926i

2.99422

0.186141i

407.4

−0.560232

0.560232i

−0.0537601

−

1.73122i

1.37228i

1.00000

0.939764i

0.707107

0.707107i

−1.88926

−

1.88926i

−2.99422

0.186141i

407.5

0.560232

−

0.560232i

0.0537601

1.73122i

1.37228i

1.00000

0.939764i

−0.707107

−

0.707107i

1.88926

1.88926i

−2.99422

0.186141i

407.6

0.560232

−

0.560232i

1.73122

0.0537601i

1.37228i

1.00000

−

0.939764i

0.707107

0.707107i

1.88926

1.88926i

2.99422

0.186141i

407.7

1.78498

−

1.78498i

−0.912166

1.47240i

−

4.37228i

1.00000

4.25639i

0.707107

0.707107i

−4.23447

−

4.23447i

−1.33591

−

2.68614i

407.8

1.78498

−

1.78498i

1.47240

−

0.912166i

−

4.37228i

1.00000

−

4.25639i

−0.707107

−

0.707107i

−4.23447

−

4.23447i

1.33591

−

2.68614i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.j.a	✓	16
3.b	odd	2	1	inner	525.2.j.a	✓	16
5.b	even	2	1	inner	525.2.j.a	✓	16
5.c	odd	4	2	inner	525.2.j.a	✓	16
15.d	odd	2	1	inner	525.2.j.a	✓	16
15.e	even	4	2	inner	525.2.j.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.2.j.a	✓	16	1.a	even	1	1	trivial
525.2.j.a	✓	16	3.b	odd	2	1	inner
525.2.j.a	✓	16	5.b	even	2	1	inner
525.2.j.a	✓	16	5.c	odd	4	2	inner
525.2.j.a	✓	16	15.d	odd	2	1	inner
525.2.j.a	✓	16	15.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 41T_{2}^{4} + 16 $ T2^8 + 41*T2^4 + 16 acting on $S_{2}^{\mathrm{new}}(525, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 41 T^{4} + 16)^{2} $ (T^8 + 41*T^4 + 16)^2
$3$	$ T^{16} - 7 T^{12} + \cdots + 6561 $ T^16 - 7T^12 - 32T^8 - 567*T^4 + 6561
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$11$	$ (T^{4} + 51 T^{2} + 576)^{4} $ (T^4 + 51*T^2 + 576)^4
$13$	$ (T^{8} + 161 T^{4} + 4096)^{2} $ (T^8 + 161*T^4 + 4096)^2
$17$	$ (T^{8} + 329 T^{4} + 256)^{2} $ (T^8 + 329*T^4 + 256)^2
$19$	$ (T^{2} + 16)^{8} $ (T^2 + 16)^8
$23$	$ (T^{4} + 144)^{4} $ (T^4 + 144)^4
$29$	$ (T^{4} - 43 T^{2} + 256)^{4} $ (T^4 - 43*T^2 + 256)^4
$31$	$ (T^{2} + 2 T - 32)^{8} $ (T^2 + 2*T - 32)^8
$37$	$ (T^{8} + 2576 T^{4} + 1048576)^{2} $ (T^8 + 2576*T^4 + 1048576)^2
$41$	$ (T^{4} + 172 T^{2} + 4096)^{4} $ (T^4 + 172*T^2 + 4096)^4
$43$	$ (T^{4} + 4096)^{4} $ (T^4 + 4096)^4
$47$	$ (T^{8} + 3929 T^{4} + 1336336)^{2} $ (T^8 + 3929*T^4 + 1336336)^2
$53$	$ (T^{8} + 21392 T^{4} + 16777216)^{2} $ (T^8 + 21392*T^4 + 16777216)^2
$59$	$ (T^{4} - 172 T^{2} + 4096)^{4} $ (T^4 - 172*T^2 + 4096)^4
$61$	$ (T^{2} - 2 T - 32)^{8} $ (T^2 - 2*T - 32)^8
$67$	$ (T^{8} + 26384 T^{4} + 65536)^{2} $ (T^8 + 26384*T^4 + 65536)^2
$71$	$ (T^{4} + 76 T^{2} + 256)^{4} $ (T^4 + 76*T^2 + 256)^4
$73$	$ (T^{8} + 26384 T^{4} + 65536)^{2} $ (T^8 + 26384*T^4 + 65536)^2
$79$	$ (T^{4} + 233 T^{2} + 1024)^{4} $ (T^4 + 233*T^2 + 1024)^4
$83$	$ (T^{8} + 87584 T^{4} + 723394816)^{2} $ (T^8 + 87584*T^4 + 723394816)^2
$89$	$ (T^{4} - 304 T^{2} + 4096)^{4} $ (T^4 - 304*T^2 + 4096)^4
$97$	$ (T^{8} + 161 T^{4} + 4096)^{2} $ (T^8 + 161*T^4 + 4096)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{15} + \beta_{13} - \beta_{10}) q^{2} + \beta_{7} q^{3} + ( - 2 \beta_{14} + \beta_{11} - \beta_{2}) q^{4} + ( - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{6} - \beta_{12} q^{7} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7}) q^{8} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) q + (-b15 + b13 - b10) * q^2 + b7 * q^3 + (-2b14 + b11 - b2) q^4 + (-2b6 - b3 - b1 + 1) q^6 - b12 * q^7 + (-b9 + 3b8 - b7) q^8 + (-b14 + b11 + b4 - b2) * q^9	\( q + ( - \beta_{15} + \beta_{13} - \beta_{10}) q^{2} + \beta_{7} q^{3} + ( - 2 \beta_{14} + \beta_{11} - \beta_{2}) q^{4} + ( - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{6} - \beta_{12} q^{7} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7}) q^{8} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{2}) q^{9}+ \cdots + (7 \beta_{14} + 4 \beta_{11} + \cdots + 8 \beta_{2}) q^{99}+O(q^{100}) \) q + (-b15 + b13 - b10) * q^2 + b7 * q^3 + (-2b14 + b11 - b2) q^4 + (-2b6 - b3 - b1 + 1) q^6 - b12 * q^7 + (-b9 + 3b8 - b7) q^8 + (-b14 + b11 + b4 - b2) * q^9 + (2b6 + 3b3 + 3b1) q^11 + (-b15 + b13 - 4b12 + b10) q^12 + (b9 - b7) * q^13 + (-b4 - b2) * q^14 + (b3 - b1 - 3) * q^16 + (-2b15 + b13 - b10) q^17 + (-3b9 + 3b8 - b7) * q^18 - 4b11 q^19 - b1 * q^21 + (-2b13 + 8b12 - 2b10) q^22 - 2b8 q^23 + (-2b14 - b11 - 4b4 - 5b2) q^24 + (4b6 + 2b3 + 2b1) q^26 + (-b15 + 2b13 - 4b12 - 2b10) q^27 + (b9 - b7 - b5) * q^28 + (2b4 - b2) q^29 + (2b3 - 2b1) * q^31 + (b15 - 3b13 + 3b10) * q^32 + (b9 - 5b8 + 2b7 - 4b5) q^33 + (-4b14 + 4b11 - 2b2) q^34 + (-2b6 - b3 - 5b1 - 5) * q^36 + (2b13 + 2b10) * q^37 + (4b9 - 4b8 + 4b7) q^38 + (b14 - 4b11 - b4 + b2) q^39 + (-4b6 + 2b3 + 2b1) q^41 + (2b15 - b13 - b12 + b10) q^42 + 8b5 q^43 + (12b4 + 6b2) * q^44 + (-2b3 + 2b1 + 2) * q^46 + (4b15 - b13 + b10) q^47 + (b9 + b8 - 3b7 - 4b5) * q^48 - b11 * q^49 + (-3b6 - 3b3 - 2b1) q^51 + (-2b13 + 8b12 - 2b10) q^52 + (-2b9 - 4b8 - 2b7) q^53 + (-2b14 + 5b11 - 4b4 - 5b2) * q^54 + (3b6 + b3 + b1) q^56 - 4b13 q^57 + (-2b9 + 2b7) * q^58 + (-4b4 + 2b2) * q^59 + (2b3 - 2b1 + 2) * q^61 + (8b15 - 4b13 + 4b10) q^62 + (b9 + b8 - b5) * q^63 + (-2b14 - b11 - b2) q^64 + (2b6 - 2b3 + 8b1 + 8) q^66 + (2b13 + 8b12 + 2b10) q^67 + (-6b9 + 8b8 - 6b7) q^68 + (2b14 + 2b11 + 2b4 + 4b2) * q^69 + (4b6 + 2b3 + 2b1) q^71 + (7b15 - 5b13 - 8b12 + 5b10) * q^72 + (2b9 - 2b7 + 8b5) q^73 + (-8b4 - 4b2) * q^74 + (-4b3 + 4b1 + 4) * q^76 + (-2b15 + 3b13 - 3b10) q^77 + (6b9 - 6b8 + 4b7) q^78 + (6b14 + 8b11 + 3b2) q^79 + (-3b6 - 5b1 + 3) * q^81 + (4b13 + 4b10) * q^82 + (8b9 - 2b8 + 8b7) q^83 + (2b14 - 4b11 - b4 + b2) * q^84 + (-8b6 - 8b3 - 8b1) q^86 + (-b15 + 2b13 - 4b12 - 5b10) q^87 + (-8b9 + 8b7 + 8b5) q^88 + (-8b4 - 4b2) * q^89 + (-b3 + b1) * q^91 + (-6b15 + 6b13 - 6b10) q^92 + (2b9 + 2b8 - 8b5) q^93 + (8b14 - 6b11 + 4b2) q^94 + (4b6 - b3 + b1 - 5) q^96 + (-b13 - b10) * q^97 + (b9 - b8 + b7) * q^98 + (7b14 + 4b11 + 7b4 + 8b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{6}+O(q^{10}) \) 16 * q + 16 * q^6	\( 16 q + 16 q^{6} - 56 q^{16} - 4 q^{21} - 16 q^{31} - 96 q^{36} + 48 q^{46} + 4 q^{51} + 16 q^{61} + 168 q^{66} + 96 q^{76} + 28 q^{81} + 8 q^{91} - 72 q^{96}+O(q^{100}) \) 16 * q + 16 * q^6 - 56 * q^16 - 4 * q^21 - 16 * q^31 - 96 * q^36 + 48 * q^46 + 4 * q^51 + 16 * q^61 + 168 * q^66 + 96 * q^76 + 28 * q^81 + 8 * q^91 - 72 * q^96

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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	525.2.j.a

Level	$525$
Weight	$2$
Character orbit	525.j
Analytic conductor	$4.192$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	16.0.6040479020157644046336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) x^16 - 7x^12 - 32x^8 - 567*x^4 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 7\nu^{12} + 32\nu^{8} + 2368\nu^{4} - 6561 ) / 12960 \) (7v^12 + 32v^8 + 2368*v^4 - 6561) / 12960
\(\beta_{2}\)	\(=\)	\( ( -5\nu^{14} - 208\nu^{10} + 4048\nu^{6} - 1053\nu^{2} ) / 58320 \) (-5v^14 - 208v^10 + 4048v^6 - 1053v^2) / 58320
\(\beta_{3}\)	\(=\)	\( ( 13\nu^{12} - 64\nu^{8} - 416\nu^{4} - 7371 ) / 4320 \) (13v^12 - 64v^8 - 416*v^4 - 7371) / 4320
\(\beta_{4}\)	\(=\)	\( ( -25\nu^{14} + 256\nu^{10} - 496\nu^{6} + 34911\nu^{2} ) / 58320 \) (-25v^14 + 256v^10 - 496v^6 + 34911v^2) / 58320
\(\beta_{5}\)	\(=\)	\( ( 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} - 20979\nu ) / 19440 \) (37v^13 - 16v^9 - 1184v^5 - 20979v) / 19440
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{12} + 32\nu^{8} - 224\nu^{4} - 5265 ) / 1296 \) (7v^12 + 32v^8 - 224*v^4 - 5265) / 1296
\(\beta_{7}\)	\(=\)	\( ( \nu^{13} - 1079\nu ) / 480 \) (v^13 - 1079*v) / 480
\(\beta_{8}\)	\(=\)	\( ( 11\nu^{13} + 4\nu^{9} + 296\nu^{5} - 6885\nu ) / 4860 \) (11v^13 + 4v^9 + 296v^5 - 6885v) / 4860
\(\beta_{9}\)	\(=\)	\( ( -7\nu^{13} - 32\nu^{9} + 224\nu^{5} + 6561\nu ) / 2592 \) (-7v^13 - 32v^9 + 224v^5 + 6561v) / 2592
\(\beta_{10}\)	\(=\)	\( ( 31\nu^{15} + 512\nu^{11} - 992\nu^{7} - 17577\nu^{3} ) / 116640 \) (31v^15 + 512v^11 - 992v^7 - 17577v^3) / 116640
\(\beta_{11}\)	\(=\)	\( ( -7\nu^{14} - 32\nu^{10} + 62\nu^{6} + 6561\nu^{2} ) / 7290 \) (-7v^14 - 32v^10 + 62v^6 + 6561v^2) / 7290
\(\beta_{12}\)	\(=\)	\( ( -\nu^{15} + 359\nu^{3} ) / 2160 \) (-v^15 + 359*v^3) / 2160
\(\beta_{13}\)	\(=\)	\( ( 49\nu^{15} + 224\nu^{11} + 1024\nu^{7} - 45927\nu^{3} ) / 69984 \) (49v^15 + 224v^11 + 1024v^7 - 45927v^3) / 69984
\(\beta_{14}\)	\(=\)	\( ( 253\nu^{14} + 416\nu^{10} - 8096\nu^{6} - 143451\nu^{2} ) / 116640 \) (253v^14 + 416v^10 - 8096v^6 - 143451v^2) / 116640
\(\beta_{15}\)	\(=\)	\( ( -43\nu^{15} - 104\nu^{11} + 2024\nu^{7} + 28593\nu^{3} ) / 43740 \) (-43v^15 - 104v^11 + 2024v^7 + 28593v^3) / 43740

\(\nu\)	\(=\)	\( ( \beta_{8} - 2\beta_{7} + \beta_{5} ) / 2 \) (b8 - 2*b7 + b5) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{14} + 3\beta_{11} + 3\beta_{4} + 2\beta_{2} ) / 2 \) (2b14 + 3b11 + 3b4 + 2b2) / 2
\(\nu^{3}\)	\(=\)	\( \beta_{15} - 2\beta_{13} - 4\beta_{12} + 2\beta_{10} \) b15 - 2b13 - 4b12 + 2*b10
\(\nu^{4}\)	\(=\)	\( ( -\beta_{6} + 10\beta _1 + 1 ) / 2 \) (-b6 + 10*b1 + 1) / 2
\(\nu^{5}\)	\(=\)	\( ( 2\beta_{9} + 15\beta_{8} - 15\beta_{5} ) / 2 \) (2b9 + 15b8 - 15*b5) / 2
\(\nu^{6}\)	\(=\)	\( -5\beta_{11} + 8\beta_{4} + 16\beta_{2} \) -5b11 + 8b4 + 16*b2
\(\nu^{7}\)	\(=\)	\( ( 35\beta_{15} + 26\beta_{13} - 35\beta_{12} ) / 2 \) (35b15 + 26b13 - 35*b12) / 2
\(\nu^{8}\)	\(=\)	\( ( 39\beta_{6} - 70\beta_{3} + 39 ) / 2 \) (39b6 - 70b3 + 39) / 2
\(\nu^{9}\)	\(=\)	\( -74\beta_{9} + 37\beta_{8} - 74\beta_{7} - 68\beta_{5} \) -74b9 + 37b8 - 74b7 - 68b5
\(\nu^{10}\)	\(=\)	\( ( -62\beta_{14} - 253\beta_{11} + 253\beta_{4} ) / 2 \) (-62b14 - 253b11 + 253*b4) / 2
\(\nu^{11}\)	\(=\)	\( ( 93\beta_{15} + 93\beta_{12} + 506\beta_{10} ) / 2 \) (93b15 + 93b12 + 506*b10) / 2
\(\nu^{12}\)	\(=\)	\( 80\beta_{6} + 160\beta_{3} + 160\beta _1 + 679 \) 80b6 + 160b3 + 160*b1 + 679
\(\nu^{13}\)	\(=\)	\( ( 1079\beta_{8} - 1198\beta_{7} + 1079\beta_{5} ) / 2 \) (1079b8 - 1198b7 + 1079*b5) / 2
\(\nu^{14}\)	\(=\)	\( ( 2158\beta_{14} + 1797\beta_{11} + 1797\beta_{4} + 2158\beta_{2} ) / 2 \) (2158b14 + 1797b11 + 1797b4 + 2158b2) / 2
\(\nu^{15}\)	\(=\)	\( 359\beta_{15} - 718\beta_{13} - 3596\beta_{12} + 718\beta_{10} \) 359b15 - 718b13 - 3596b12 + 718b10

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(-\beta_{11}\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 218.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 41 T^{4} + 16)^{2} \) (T^8 + 41*T^4 + 16)^2
$3$	\( T^{16} - 7 T^{12} + \cdots + 6561 \) T^16 - 7T^12 - 32T^8 - 567*T^4 + 6561
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$11$	\( (T^{4} + 51 T^{2} + 576)^{4} \) (T^4 + 51*T^2 + 576)^4
$13$	\( (T^{8} + 161 T^{4} + 4096)^{2} \) (T^8 + 161*T^4 + 4096)^2
$17$	\( (T^{8} + 329 T^{4} + 256)^{2} \) (T^8 + 329*T^4 + 256)^2
$19$	\( (T^{2} + 16)^{8} \) (T^2 + 16)^8
$23$	\( (T^{4} + 144)^{4} \) (T^4 + 144)^4
$29$	\( (T^{4} - 43 T^{2} + 256)^{4} \) (T^4 - 43*T^2 + 256)^4
$31$	\( (T^{2} + 2 T - 32)^{8} \) (T^2 + 2*T - 32)^8
$37$	\( (T^{8} + 2576 T^{4} + 1048576)^{2} \) (T^8 + 2576*T^4 + 1048576)^2
$41$	\( (T^{4} + 172 T^{2} + 4096)^{4} \) (T^4 + 172*T^2 + 4096)^4
$43$	\( (T^{4} + 4096)^{4} \) (T^4 + 4096)^4
$47$	\( (T^{8} + 3929 T^{4} + 1336336)^{2} \) (T^8 + 3929*T^4 + 1336336)^2
$53$	\( (T^{8} + 21392 T^{4} + 16777216)^{2} \) (T^8 + 21392*T^4 + 16777216)^2
$59$	\( (T^{4} - 172 T^{2} + 4096)^{4} \) (T^4 - 172*T^2 + 4096)^4
$61$	\( (T^{2} - 2 T - 32)^{8} \) (T^2 - 2*T - 32)^8
$67$	\( (T^{8} + 26384 T^{4} + 65536)^{2} \) (T^8 + 26384*T^4 + 65536)^2
$71$	\( (T^{4} + 76 T^{2} + 256)^{4} \) (T^4 + 76*T^2 + 256)^4
$73$	\( (T^{8} + 26384 T^{4} + 65536)^{2} \) (T^8 + 26384*T^4 + 65536)^2
$79$	\( (T^{4} + 233 T^{2} + 1024)^{4} \) (T^4 + 233*T^2 + 1024)^4
$83$	\( (T^{8} + 87584 T^{4} + 723394816)^{2} \) (T^8 + 87584*T^4 + 723394816)^2
$89$	\( (T^{4} - 304 T^{2} + 4096)^{4} \) (T^4 - 304*T^2 + 4096)^4
$97$	\( (T^{8} + 161 T^{4} + 4096)^{2} \) (T^8 + 161*T^4 + 4096)^2
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