LMFDB - Newform orbit 1050.3.h.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,3,Mod(349,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1050.349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.h (of order $2$, degree $1$, 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:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 210)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{2} - \beta_{3} q^{3} - 2 q^{4} + \beta_{11} q^{6} + (\beta_{12} + 2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{8} + 3 q^{9}+O(q^{10}) $ q + b2 * q^2 - b3 * q^3 - 2 * q^4 + b11 * q^6 + (b12 + 2b2 - b1) q^7 - 2b2 q^8 + 3 * q^9	$ q + \beta_{2} q^{2} - \beta_{3} q^{3} - 2 q^{4} + \beta_{11} q^{6} + (\beta_{12} + 2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{8} + 3 q^{9} + ( - \beta_{14} - 3 \beta_{9} + \cdots - 2) q^{11}+ \cdots + ( - 3 \beta_{14} - 9 \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) $ q + b2 * q^2 - b3 * q^3 - 2 * q^4 + b11 * q^6 + (b12 + 2b2 - b1) q^7 - 2b2 q^8 + 3 * q^9 + (-b14 - 3b9 + 3b8 - 2) * q^11 + 2b3 q^12 + (-b13 - b12 + b5 - 2b3 - b1) q^13 + (b14 + b7 - b4 - 4) * q^14 + 4 * q^16 + (2b5 - 3b1) * q^17 + 3b2 q^18 + (-7b11 - b9 - b8 - b7 + 3b4) * q^19 + (2b11 + b10 + 3b9) * q^21 + (-b13 + b12 + 6b6 - 2b2) * q^22 + (2b15 + b13 - b12 - 8b2) * q^23 - 2b11 q^24 + (2b11 - b9 - b8 + b7 + 2b4) * q^26 - 3b3 q^27 + (-2b12 - 4b2 + 2b1) q^28 + (2b10 + 7b9 - 7b8 + 18) q^29 + (b11 - b9 - b8 + b7 - 7b4) q^31 + 4b2 q^32 + (-3b13 - 3b12 - 3b5 + 2b3) * q^33 + (-2b9 - 2b8 + 3b7) q^34 - 6 * q^36 + (-2b15 - 4b13 + 4b12 - 3b6 - 10b2) q^37 + (3b13 + 3b12 - 2b5 - 14b3 - 2b1) q^38 + (b14 + b10 - 3b9 + 3b8 + 6) * q^39 + (-4b11 - 4b9 - 4b8 + b7 - 2b4) * q^41 + (b15 - 3b6 + 3b5 + 4b3) q^42 + (4b15 - 2b13 + 2b12 + 4b6 + 4b2) q^43 + (2b14 + 6b9 - 6b8 + 4) q^44 + (-2b14 - 4b10 + 16) * q^46 + (-7b13 - 7b12 + 8b5 - 6b3 - 2b1) q^47 - 4b3 q^48 + (8b14 + b7 + 6b4 + 3) * q^49 + (2b14 + 3b10) * q^51 + (2b13 + 2b12 - 2b5 + 4b3 + 2b1) q^52 + (-b15 - 7b13 + 7b12 - 8b6 - 11b2) * q^53 + 3b11 q^54 + (-2b14 - 2b7 + 2b4 + 8) q^56 + (-b15 + b13 - b12 + 9b6 - 21b2) * q^57 + (2b15 - 14b6 + 18b2) q^58 + (14b11 + 4b7 + 12b4) q^59 + (-16b11 + 8b9 + 8b8 + 5b7 + 14b4) q^61 + (-7b13 - 7b12 - 2b5 + 2b3 + 2b1) q^62 + (3b12 + 6b2 - 3b1) q^63 - 8 * q^64 + (-2b11 + 3b9 + 3b8 + 6b4) * q^66 + (2b15 + 3b13 - 3b12 - 12b6 - 36b2) q^67 + (-4b5 + 6b1) * q^68 + (-8b11 - 3b9 - 3b8 + 6b7) * q^69 + (15b14 - 2b10 + 7b9 - 7b8 + 22) * q^71 - 6b2 q^72 + (b13 + b12 - 9b5 - 38b3 - 7b1) q^73 + (8b14 + 4b10 - 3b9 + 3b8 + 20) * q^74 + (14b11 + 2b9 + 2b8 + 2b7 - 6b4) q^76 + (3b15 - 7b13 - 5b12 + 12b6 - 12b5 + 12b3 + 11b2 + 5b1) * q^77 + (b15 + b13 - b12 + 6b6 + 6b2) * q^78 + (8b14 - 8b9 + 8b8 + 36) q^79 + 9 * q^81 + (-2b13 - 2b12 - 8b5 - 8b3 + 2b1) q^82 + (3b13 + 3b12 - 20b5 - 2b3 - 2b1) q^83 + (-4b11 - 2b10 - 6b9) q^84 + (4b14 - 8b10 + 4b9 - 4b8 - 8) * q^86 + (7b13 + 7b12 - 18b3 - 6b1) * q^87 + (2b13 - 2b12 - 12b6 + 4b2) * q^88 + (4b11 - 5b9 - 5b8 + 16b7) * q^89 + (-2b14 - b11 + 3b10 + 9b9 - 7b8 + 5b7 + 9b4 + 8) * q^91 + (-4b15 - 2b13 + 2b12 + 16b2) * q^92 + (b15 + b13 - b12 - 21b6 + 3b2) * q^93 + (6b11 - 8b9 - 8b8 + 2b7 + 14b4) q^94 + 4b11 q^96 + (3b13 + 3b12 - 3b5 + 30b3 - 17b1) q^97 + (14b13 - 2b12 + 3b2 + 2b1) * q^98 + (-3b14 - 9b9 + 9b8 - 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 48 q^{9}+O(q^{10}) $ 16 * q - 32 * q^4 + 48 * q^9	$ 16 q - 32 q^{4} + 48 q^{9} - 32 q^{11} - 64 q^{14} + 64 q^{16} + 288 q^{29} - 96 q^{36} + 96 q^{39} + 64 q^{44} + 256 q^{46} + 48 q^{49} + 128 q^{56} - 128 q^{64} + 352 q^{71} + 320 q^{74} + 576 q^{79} + 144 q^{81} - 128 q^{86} + 128 q^{91} - 96 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 + 48 * q^9 - 32 * q^11 - 64 * q^14 + 64 * q^16 + 288 * q^29 - 96 * q^36 + 96 * q^39 + 64 * q^44 + 256 * q^46 + 48 * q^49 + 128 * q^56 - 128 * q^64 + 352 * q^71 + 320 * q^74 + 576 * q^79 + 144 * q^81 - 128 * q^86 + 128 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7*x^12 + 48*x^8 - 7*x^4 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{12} + 161 ) / 36 $ (v^12 + 161) / 36
$\beta_{2}$	$=$	$ ( -\nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} - 305\nu^{3} + 119\nu ) / 72 $ (-v^15 - 17v^13 + 120v^9 - 816v^5 - 305v^3 + 119*v) / 72
$\beta_{3}$	$=$	$ ( 9\nu^{14} - 64\nu^{10} + 440\nu^{6} - 127\nu^{2} ) / 24 $ (9v^14 - 64v^10 + 440v^6 - 127v^2) / 24
$\beta_{4}$	$=$	$ ( 7\nu^{12} - 48\nu^{8} + 336\nu^{4} - 25 ) / 12 $ (7v^12 - 48v^8 + 336*v^4 - 25) / 12
$\beta_{5}$	$=$	$ ( -\nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} - 341\nu^{3} - 133\nu ) / 36 $ (-v^15 + 19v^13 - 132v^9 + 912v^5 - 341v^3 - 133*v) / 36
$\beta_{6}$	$=$	$ ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 9 $ (7v^14 - 48v^10 + 330*v^6 - v^2) / 9
$\beta_{7}$	$=$	$ ( \nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} + 341\nu^{3} - 133\nu ) / 18 $ (v^15 + 19v^13 - 132v^9 + 912v^5 + 341v^3 - 133*v) / 18
$\beta_{8}$	$=$	$ ( - \nu^{15} - 63 \nu^{14} + 17 \nu^{13} + 432 \nu^{10} - 120 \nu^{9} - 2952 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 $ (-v^15 - 63v^14 + 17v^13 + 432v^10 - 120v^9 - 2952v^6 + 816v^5 - 305v^3 + 9v^2 - 119*v) / 72
$\beta_{9}$	$=$	$ ( \nu^{15} - 63 \nu^{14} - 17 \nu^{13} + 432 \nu^{10} + 120 \nu^{9} - 2952 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 $ (v^15 - 63v^14 - 17v^13 + 432v^10 + 120v^9 - 2952v^6 - 816v^5 + 305v^3 + 9v^2 + 119*v) / 72
$\beta_{10}$	$=$	$ ( 5\nu^{14} - 36\nu^{10} + 246\nu^{6} - 71\nu^{2} ) / 3 $ (5v^14 - 36v^10 + 246v^6 - 71v^2) / 3
$\beta_{11}$	$=$	$ ( -15\nu^{15} - 3\nu^{13} + 104\nu^{11} + 20\nu^{9} - 712\nu^{7} - 136\nu^{5} + 53\nu^{3} - 19\nu ) / 12 $ (-15v^15 - 3v^13 + 104v^11 + 20v^9 - 712v^7 - 136v^5 + 53v^3 - 19v) / 12
$\beta_{12}$	$=$	$ ( - 15 \nu^{15} + 3 \nu^{13} - 8 \nu^{12} + 104 \nu^{11} - 20 \nu^{9} + 56 \nu^{8} - 712 \nu^{7} + \cdots + 28 ) / 12 $ (-15v^15 + 3v^13 - 8v^12 + 104v^11 - 20v^9 + 56v^8 - 712v^7 + 136v^5 - 376v^4 + 53v^3 + 19*v + 28) / 12
$\beta_{13}$	$=$	$ ( - 15 \nu^{15} + 3 \nu^{13} + 8 \nu^{12} + 104 \nu^{11} - 20 \nu^{9} - 56 \nu^{8} - 712 \nu^{7} + \cdots - 28 ) / 12 $ (-15v^15 + 3v^13 + 8v^12 + 104v^11 - 20v^9 - 56v^8 - 712v^7 + 136v^5 + 376v^4 + 53v^3 + 19*v - 28) / 12
$\beta_{14}$	$=$	$ ( 67\nu^{15} - 13\nu^{13} - 464\nu^{11} + 88\nu^{9} + 3184\nu^{7} - 608\nu^{5} - 237\nu^{3} - 85\nu ) / 24 $ (67v^15 - 13v^13 - 464v^11 + 88v^9 + 3184v^7 - 608v^5 - 237v^3 - 85v) / 24
$\beta_{15}$	$=$	$ ( 67\nu^{15} + 13\nu^{13} - 464\nu^{11} - 88\nu^{9} + 3184\nu^{7} + 608\nu^{5} - 237\nu^{3} + 85\nu ) / 12 $ (67v^15 + 13v^13 - 464v^11 - 88v^9 + 3184v^7 + 608v^5 - 237v^3 + 85v) / 12

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

$\nu$	$=$	$ ( \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{9} + \beta_{8} + \cdots - 2 \beta_{2} ) / 16 $ (b15 - 2b14 - b13 - b12 + 2b11 - b9 + b8 - b7 - 2b5 - 2b2) / 16
$\nu^{2}$	$=$	$ ( \beta_{10} + \beta_{9} + \beta_{8} + 3\beta_{6} - 6\beta_{3} ) / 8 $ (b10 + b9 + b8 + 3b6 - 6b3) / 8
$\nu^{3}$	$=$	$ ( -2\beta_{9} + 2\beta_{8} + \beta_{7} - 2\beta_{5} + 4\beta_{2} ) / 4 $ (-2b9 + 2b8 + b7 - 2b5 + 4b2) / 4
$\nu^{4}$	$=$	$ ( -3\beta_{13} + 3\beta_{12} + 7\beta_{4} - 3\beta _1 + 14 ) / 8 $ (-3b13 + 3b12 + 7b4 - 3b1 + 14) / 8
$\nu^{5}$	$=$	$ ( 5 \beta_{15} - 10 \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 22 \beta_{11} + 11 \beta_{9} + \cdots + 22 \beta_{2} ) / 16 $ (5b15 - 10b14 - 11b13 - 11b12 + 22b11 + 11b9 - 11b8 + 5b7 + 10b5 + 22b2) / 16
$\nu^{6}$	$=$	$ ( 4\beta_{9} + 4\beta_{8} + 9\beta_{6} ) / 2 $ (4b9 + 4b8 + 9*b6) / 2
$\nu^{7}$	$=$	$ ( 13 \beta_{15} + 26 \beta_{14} + 29 \beta_{13} + 29 \beta_{12} + 58 \beta_{11} - 29 \beta_{9} + \cdots + 58 \beta_{2} ) / 16 $ (13b15 + 26b14 + 29b13 + 29b12 + 58b11 - 29b9 + 29b8 + 13b7 - 26b5 + 58b2) / 16
$\nu^{8}$	$=$	$ ( -21\beta_{13} + 21\beta_{12} + 47\beta_{4} + 21\beta _1 - 94 ) / 8 $ (-21b13 + 21b12 + 47b4 + 21b1 - 94) / 8
$\nu^{9}$	$=$	$ ( 38\beta_{9} - 38\beta_{8} + 17\beta_{7} + 34\beta_{5} + 76\beta_{2} ) / 4 $ (38b9 - 38b8 + 17b7 + 34b5 + 76*b2) / 4
$\nu^{10}$	$=$	$ ( -55\beta_{10} + 55\beta_{9} + 55\beta_{8} + 123\beta_{6} + 246\beta_{3} ) / 8 $ (-55b10 + 55b9 + 55b8 + 123b6 + 246*b3) / 8
$\nu^{11}$	$=$	$ ( 89 \beta_{15} + 178 \beta_{14} + 199 \beta_{13} + 199 \beta_{12} + 398 \beta_{11} + \cdots - 398 \beta_{2} ) / 16 $ (89b15 + 178b14 + 199b13 + 199b12 + 398b11 + 199b9 - 199b8 - 89b7 + 178b5 - 398b2) / 16
$\nu^{12}$	$=$	$ 36\beta _1 - 161 $ 36*b1 - 161
$\nu^{13}$	$=$	$ ( - 233 \beta_{15} + 466 \beta_{14} + 521 \beta_{13} + 521 \beta_{12} - 1042 \beta_{11} + \cdots + 1042 \beta_{2} ) / 16 $ (-233b15 + 466b14 + 521b13 + 521b12 - 1042b11 + 521b9 - 521b8 + 233b7 + 466b5 + 1042b2) / 16
$\nu^{14}$	$=$	$ ( -377\beta_{10} - 377\beta_{9} - 377\beta_{8} - 843\beta_{6} + 1686\beta_{3} ) / 8 $ (-377b10 - 377b9 - 377b8 - 843b6 + 1686*b3) / 8
$\nu^{15}$	$=$	$ ( 682\beta_{9} - 682\beta_{8} - 305\beta_{7} + 610\beta_{5} - 1364\beta_{2} ) / 4 $ (682b9 - 682b8 - 305b7 + 610b5 - 1364*b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$-1$	$1$

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} $

349.1

−0.159959	−	0.596975i
0.159959	−	0.596975i
0.418778	+	1.56290i
−0.418778	+	1.56290i
−0.596975	+	0.159959i
0.596975	+	0.159959i
1.56290	−	0.418778i
−1.56290	−	0.418778i
−0.159959	+	0.596975i
0.159959	+	0.596975i
0.418778	−	1.56290i
−0.418778	−	1.56290i
−0.596975	−	0.159959i
0.596975	−	0.159959i
1.56290	+	0.418778i
−1.56290	+	0.418778i

−

1.41421i

−1.73205

−2.00000

2.44949i

−6.92163

1.04456i

2.82843i

3.00000

349.2

−

1.41421i

−1.73205

−2.00000

2.44949i

−2.02265

−

6.70141i

2.82843i

3.00000

349.3

−

1.41421i

−1.73205

−2.00000

2.44949i

2.02265

−

6.70141i

2.82843i

3.00000

349.4

−

1.41421i

−1.73205

−2.00000

2.44949i

6.92163

1.04456i

2.82843i

3.00000

349.5

−

1.41421i

1.73205

−2.00000

−

2.44949i

−6.92163

1.04456i

2.82843i

3.00000

349.6

−

1.41421i

1.73205

−2.00000

−

2.44949i

−2.02265

−

6.70141i

2.82843i

3.00000

349.7

−

1.41421i

1.73205

−2.00000

−

2.44949i

2.02265

−

6.70141i

2.82843i

3.00000

349.8

−

1.41421i

1.73205

−2.00000

−

2.44949i

6.92163

1.04456i

2.82843i

3.00000

349.9

1.41421i

−1.73205

−2.00000

−

2.44949i

−6.92163

−

1.04456i

−

2.82843i

3.00000

349.10

1.41421i

−1.73205

−2.00000

−

2.44949i

−2.02265

6.70141i

−

2.82843i

3.00000

349.11

1.41421i

−1.73205

−2.00000

−

2.44949i

2.02265

6.70141i

−

2.82843i

3.00000

349.12

1.41421i

−1.73205

−2.00000

−

2.44949i

6.92163

−

1.04456i

−

2.82843i

3.00000

349.13

1.41421i

1.73205

−2.00000

2.44949i

−6.92163

−

1.04456i

−

2.82843i

3.00000

349.14

1.41421i

1.73205

−2.00000

2.44949i

−2.02265

6.70141i

−

2.82843i

3.00000

349.15

1.41421i

1.73205

−2.00000

2.44949i

2.02265

6.70141i

−

2.82843i

3.00000

349.16

1.41421i

1.73205

−2.00000

2.44949i

6.92163

−

1.04456i

−

2.82843i

3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1050.3.h.b		16
5.b	even	2	1	inner	1050.3.h.b		16
5.c	odd	4	1		210.3.f.a	✓	8
5.c	odd	4	1		1050.3.f.b		8
7.b	odd	2	1	inner	1050.3.h.b		16
15.e	even	4	1		630.3.f.c		8
20.e	even	4	1		1680.3.s.a		8
35.c	odd	2	1	inner	1050.3.h.b		16
35.f	even	4	1		210.3.f.a	✓	8
35.f	even	4	1		1050.3.f.b		8
105.k	odd	4	1		630.3.f.c		8
140.j	odd	4	1		1680.3.s.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.3.f.a	✓	8	5.c	odd	4	1
210.3.f.a	✓	8	35.f	even	4	1
630.3.f.c		8	15.e	even	4	1
630.3.f.c		8	105.k	odd	4	1
1050.3.f.b		8	5.c	odd	4	1
1050.3.f.b		8	35.f	even	4	1
1050.3.h.b		16	1.a	even	1	1	trivial
1050.3.h.b		16	5.b	even	2	1	inner
1050.3.h.b		16	7.b	odd	2	1	inner
1050.3.h.b		16	35.c	odd	2	1	inner
1680.3.s.a		8	20.e	even	4	1
1680.3.s.a		8	140.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{4} + 8T_{11}^{3} - 180T_{11}^{2} - 784T_{11} + 964 $ T11^4 + 8*T11^3 - 180*T11^2 - 784*T11 + 964 acting on $S_{3}^{\mathrm{new}}(1050, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$3$	$ (T^{2} - 3)^{8} $ (T^2 - 3)^8
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} - 12 T^{6} + \cdots + 5764801)^{2} $ (T^8 - 12T^6 - 2842T^4 - 28812*T^2 + 5764801)^2
$11$	$ (T^{4} + 8 T^{3} + \cdots + 964)^{4} $ (T^4 + 8T^3 - 180T^2 - 784*T + 964)^4
$13$	$ (T^{8} - 264 T^{6} + \cdots + 14714896)^{2} $ (T^8 - 264T^6 + 25112T^4 - 1014816*T^2 + 14714896)^2
$17$	$ (T^{4} - 440 T^{2} + 19600)^{4} $ (T^4 - 440*T^2 + 19600)^4
$19$	$ (T^{8} + 1848 T^{6} + \cdots + 2869102096)^{2} $ (T^8 + 1848T^6 + 1069208T^4 + 192546528*T^2 + 2869102096)^2
$23$	$ (T^{8} + 2672 T^{6} + \cdots + 2975047936)^{2} $ (T^8 + 2672T^6 + 1893984T^4 + 381671168*T^2 + 2975047936)^2
$29$	$ (T^{4} - 72 T^{3} + \cdots - 281456)^{4} $ (T^4 - 72T^3 + 680T^2 + 22176*T - 281456)^4
$31$	$ (T^{8} + 2616 T^{6} + \cdots + 79120688656)^{2} $ (T^8 + 2616T^6 + 2165912T^4 + 709481184*T^2 + 79120688656)^2
$37$	$ (T^{8} + 6704 T^{6} + \cdots + 780013977856)^{2} $ (T^8 + 6704T^6 + 13155936T^4 + 7881016064*T^2 + 780013977856)^2
$41$	$ (T^{8} + 2016 T^{6} + \cdots + 7126061056)^{2} $ (T^8 + 2016T^6 + 1338752T^4 + 301676544*T^2 + 7126061056)^2
$43$	$ (T^{8} + \cdots + 7046179127296)^{2} $ (T^8 + 9024T^6 + 23700992T^4 + 22632677376*T^2 + 7046179127296)^2
$47$	$ (T^{8} + \cdots + 1081300500736)^{2} $ (T^8 - 8016T^6 + 19145312T^4 - 12067017984*T^2 + 1081300500736)^2
$53$	$ (T^{8} + \cdots + 13854147517456)^{2} $ (T^8 + 14232T^6 + 67543448T^4 + 110015218272*T^2 + 13854147517456)^2
$59$	$ (T^{8} + \cdots + 9016903929856)^{2} $ (T^8 + 14176T^6 + 44234112T^4 + 40668043264*T^2 + 9016903929856)^2
$61$	$ (T^{8} + \cdots + 11\!\cdots\!76)^{2} $ (T^8 + 24672T^6 + 221426048T^4 + 851048896512*T^2 + 1168834660274176)^2
$67$	$ (T^{8} + \cdots + 54339071222016)^{2} $ (T^8 + 16752T^6 + 95227488T^4 + 193265747712*T^2 + 54339071222016)^2
$71$	$ (T^{4} - 88 T^{3} + \cdots + 25706884)^{4} $ (T^4 - 88T^3 - 11860T^2 + 808624*T + 25706884)^4
$73$	$ (T^{8} - 24584 T^{6} + \cdots + 483086161936)^{2} $ (T^8 - 24584T^6 + 170869272T^4 - 363296139296*T^2 + 483086161936)^2
$79$	$ (T^{4} - 144 T^{3} + \cdots - 2641664)^{4} $ (T^4 - 144T^3 + 2912T^2 + 163584*T - 2641664)^4
$83$	$ (T^{8} + \cdots + 177346111242496)^{2} $ (T^8 - 17232T^6 + 102712928T^4 - 242767656192*T^2 + 177346111242496)^2
$89$	$ (T^{8} + \cdots + 86\!\cdots\!96)^{2} $ (T^8 + 43344T^6 + 655305056T^4 + 4054419197184*T^2 + 8614575218749696)^2
$97$	$ (T^{8} + \cdots + 76019496215056)^{2} $ (T^8 - 35144T^6 + 290961816T^4 - 312878921504*T^2 + 76019496215056)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} - \beta_{3} q^{3} - 2 q^{4} + \beta_{11} q^{6} + (\beta_{12} + 2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{8} + 3 q^{9}+O(q^{10}) \) q + b2 * q^2 - b3 * q^3 - 2 * q^4 + b11 * q^6 + (b12 + 2b2 - b1) q^7 - 2b2 q^8 + 3 * q^9	\( q + \beta_{2} q^{2} - \beta_{3} q^{3} - 2 q^{4} + \beta_{11} q^{6} + (\beta_{12} + 2 \beta_{2} - \beta_1) q^{7} - 2 \beta_{2} q^{8} + 3 q^{9} + ( - \beta_{14} - 3 \beta_{9} + \cdots - 2) q^{11}+ \cdots + ( - 3 \beta_{14} - 9 \beta_{9} + \cdots - 6) q^{99}+O(q^{100}) \) q + b2 * q^2 - b3 * q^3 - 2 * q^4 + b11 * q^6 + (b12 + 2b2 - b1) q^7 - 2b2 q^8 + 3 * q^9 + (-b14 - 3b9 + 3b8 - 2) * q^11 + 2b3 q^12 + (-b13 - b12 + b5 - 2b3 - b1) q^13 + (b14 + b7 - b4 - 4) * q^14 + 4 * q^16 + (2b5 - 3b1) * q^17 + 3b2 q^18 + (-7b11 - b9 - b8 - b7 + 3b4) * q^19 + (2b11 + b10 + 3b9) * q^21 + (-b13 + b12 + 6b6 - 2b2) * q^22 + (2b15 + b13 - b12 - 8b2) * q^23 - 2b11 q^24 + (2b11 - b9 - b8 + b7 + 2b4) * q^26 - 3b3 q^27 + (-2b12 - 4b2 + 2b1) q^28 + (2b10 + 7b9 - 7b8 + 18) q^29 + (b11 - b9 - b8 + b7 - 7b4) q^31 + 4b2 q^32 + (-3b13 - 3b12 - 3b5 + 2b3) * q^33 + (-2b9 - 2b8 + 3b7) q^34 - 6 * q^36 + (-2b15 - 4b13 + 4b12 - 3b6 - 10b2) q^37 + (3b13 + 3b12 - 2b5 - 14b3 - 2b1) q^38 + (b14 + b10 - 3b9 + 3b8 + 6) * q^39 + (-4b11 - 4b9 - 4b8 + b7 - 2b4) * q^41 + (b15 - 3b6 + 3b5 + 4b3) q^42 + (4b15 - 2b13 + 2b12 + 4b6 + 4b2) q^43 + (2b14 + 6b9 - 6b8 + 4) q^44 + (-2b14 - 4b10 + 16) * q^46 + (-7b13 - 7b12 + 8b5 - 6b3 - 2b1) q^47 - 4b3 q^48 + (8b14 + b7 + 6b4 + 3) * q^49 + (2b14 + 3b10) * q^51 + (2b13 + 2b12 - 2b5 + 4b3 + 2b1) q^52 + (-b15 - 7b13 + 7b12 - 8b6 - 11b2) * q^53 + 3b11 q^54 + (-2b14 - 2b7 + 2b4 + 8) q^56 + (-b15 + b13 - b12 + 9b6 - 21b2) * q^57 + (2b15 - 14b6 + 18b2) q^58 + (14b11 + 4b7 + 12b4) q^59 + (-16b11 + 8b9 + 8b8 + 5b7 + 14b4) q^61 + (-7b13 - 7b12 - 2b5 + 2b3 + 2b1) q^62 + (3b12 + 6b2 - 3b1) q^63 - 8 * q^64 + (-2b11 + 3b9 + 3b8 + 6b4) * q^66 + (2b15 + 3b13 - 3b12 - 12b6 - 36b2) q^67 + (-4b5 + 6b1) * q^68 + (-8b11 - 3b9 - 3b8 + 6b7) * q^69 + (15b14 - 2b10 + 7b9 - 7b8 + 22) * q^71 - 6b2 q^72 + (b13 + b12 - 9b5 - 38b3 - 7b1) q^73 + (8b14 + 4b10 - 3b9 + 3b8 + 20) * q^74 + (14b11 + 2b9 + 2b8 + 2b7 - 6b4) q^76 + (3b15 - 7b13 - 5b12 + 12b6 - 12b5 + 12b3 + 11b2 + 5b1) * q^77 + (b15 + b13 - b12 + 6b6 + 6b2) * q^78 + (8b14 - 8b9 + 8b8 + 36) q^79 + 9 * q^81 + (-2b13 - 2b12 - 8b5 - 8b3 + 2b1) q^82 + (3b13 + 3b12 - 20b5 - 2b3 - 2b1) q^83 + (-4b11 - 2b10 - 6b9) q^84 + (4b14 - 8b10 + 4b9 - 4b8 - 8) * q^86 + (7b13 + 7b12 - 18b3 - 6b1) * q^87 + (2b13 - 2b12 - 12b6 + 4b2) * q^88 + (4b11 - 5b9 - 5b8 + 16b7) * q^89 + (-2b14 - b11 + 3b10 + 9b9 - 7b8 + 5b7 + 9b4 + 8) * q^91 + (-4b15 - 2b13 + 2b12 + 16b2) * q^92 + (b15 + b13 - b12 - 21b6 + 3b2) * q^93 + (6b11 - 8b9 - 8b8 + 2b7 + 14b4) q^94 + 4b11 q^96 + (3b13 + 3b12 - 3b5 + 30b3 - 17b1) q^97 + (14b13 - 2b12 + 3b2 + 2b1) * q^98 + (-3b14 - 9b9 + 9b8 - 6) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 48 q^{9}+O(q^{10}) \) 16 * q - 32 * q^4 + 48 * q^9	\( 16 q - 32 q^{4} + 48 q^{9} - 32 q^{11} - 64 q^{14} + 64 q^{16} + 288 q^{29} - 96 q^{36} + 96 q^{39} + 64 q^{44} + 256 q^{46} + 48 q^{49} + 128 q^{56} - 128 q^{64} + 352 q^{71} + 320 q^{74} + 576 q^{79} + 144 q^{81} - 128 q^{86} + 128 q^{91} - 96 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 + 48 * q^9 - 32 * q^11 - 64 * q^14 + 64 * q^16 + 288 * q^29 - 96 * q^36 + 96 * q^39 + 64 * q^44 + 256 * q^46 + 48 * q^49 + 128 * q^56 - 128 * q^64 + 352 * q^71 + 320 * q^74 + 576 * q^79 + 144 * q^81 - 128 * q^86 + 128 * q^91 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1050.3.h.b

Level	$1050$
Weight	$3$
Character orbit	1050.h
Analytic conductor	$28.610$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(28.6104277578\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{28} \)
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{12} + 161 ) / 36 \) (v^12 + 161) / 36
\(\beta_{2}\)	\(=\)	\( ( -\nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} - 305\nu^{3} + 119\nu ) / 72 \) (-v^15 - 17v^13 + 120v^9 - 816v^5 - 305v^3 + 119*v) / 72
\(\beta_{3}\)	\(=\)	\( ( 9\nu^{14} - 64\nu^{10} + 440\nu^{6} - 127\nu^{2} ) / 24 \) (9v^14 - 64v^10 + 440v^6 - 127v^2) / 24
\(\beta_{4}\)	\(=\)	\( ( 7\nu^{12} - 48\nu^{8} + 336\nu^{4} - 25 ) / 12 \) (7v^12 - 48v^8 + 336*v^4 - 25) / 12
\(\beta_{5}\)	\(=\)	\( ( -\nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} - 341\nu^{3} - 133\nu ) / 36 \) (-v^15 + 19v^13 - 132v^9 + 912v^5 - 341v^3 - 133*v) / 36
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 9 \) (7v^14 - 48v^10 + 330*v^6 - v^2) / 9
\(\beta_{7}\)	\(=\)	\( ( \nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} + 341\nu^{3} - 133\nu ) / 18 \) (v^15 + 19v^13 - 132v^9 + 912v^5 + 341v^3 - 133*v) / 18
\(\beta_{8}\)	\(=\)	\( ( - \nu^{15} - 63 \nu^{14} + 17 \nu^{13} + 432 \nu^{10} - 120 \nu^{9} - 2952 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 \) (-v^15 - 63v^14 + 17v^13 + 432v^10 - 120v^9 - 2952v^6 + 816v^5 - 305v^3 + 9v^2 - 119*v) / 72
\(\beta_{9}\)	\(=\)	\( ( \nu^{15} - 63 \nu^{14} - 17 \nu^{13} + 432 \nu^{10} + 120 \nu^{9} - 2952 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 \) (v^15 - 63v^14 - 17v^13 + 432v^10 + 120v^9 - 2952v^6 - 816v^5 + 305v^3 + 9v^2 + 119*v) / 72
\(\beta_{10}\)	\(=\)	\( ( 5\nu^{14} - 36\nu^{10} + 246\nu^{6} - 71\nu^{2} ) / 3 \) (5v^14 - 36v^10 + 246v^6 - 71v^2) / 3
\(\beta_{11}\)	\(=\)	\( ( -15\nu^{15} - 3\nu^{13} + 104\nu^{11} + 20\nu^{9} - 712\nu^{7} - 136\nu^{5} + 53\nu^{3} - 19\nu ) / 12 \) (-15v^15 - 3v^13 + 104v^11 + 20v^9 - 712v^7 - 136v^5 + 53v^3 - 19v) / 12
\(\beta_{12}\)	\(=\)	\( ( - 15 \nu^{15} + 3 \nu^{13} - 8 \nu^{12} + 104 \nu^{11} - 20 \nu^{9} + 56 \nu^{8} - 712 \nu^{7} + \cdots + 28 ) / 12 \) (-15v^15 + 3v^13 - 8v^12 + 104v^11 - 20v^9 + 56v^8 - 712v^7 + 136v^5 - 376v^4 + 53v^3 + 19*v + 28) / 12
\(\beta_{13}\)	\(=\)	\( ( - 15 \nu^{15} + 3 \nu^{13} + 8 \nu^{12} + 104 \nu^{11} - 20 \nu^{9} - 56 \nu^{8} - 712 \nu^{7} + \cdots - 28 ) / 12 \) (-15v^15 + 3v^13 + 8v^12 + 104v^11 - 20v^9 - 56v^8 - 712v^7 + 136v^5 + 376v^4 + 53v^3 + 19*v - 28) / 12
\(\beta_{14}\)	\(=\)	\( ( 67\nu^{15} - 13\nu^{13} - 464\nu^{11} + 88\nu^{9} + 3184\nu^{7} - 608\nu^{5} - 237\nu^{3} - 85\nu ) / 24 \) (67v^15 - 13v^13 - 464v^11 + 88v^9 + 3184v^7 - 608v^5 - 237v^3 - 85v) / 24
\(\beta_{15}\)	\(=\)	\( ( 67\nu^{15} + 13\nu^{13} - 464\nu^{11} - 88\nu^{9} + 3184\nu^{7} + 608\nu^{5} - 237\nu^{3} + 85\nu ) / 12 \) (67v^15 + 13v^13 - 464v^11 - 88v^9 + 3184v^7 + 608v^5 - 237v^3 + 85v) / 12

\(\nu\)	\(=\)	\( ( \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{9} + \beta_{8} + \cdots - 2 \beta_{2} ) / 16 \) (b15 - 2b14 - b13 - b12 + 2b11 - b9 + b8 - b7 - 2b5 - 2b2) / 16
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} + \beta_{9} + \beta_{8} + 3\beta_{6} - 6\beta_{3} ) / 8 \) (b10 + b9 + b8 + 3b6 - 6b3) / 8
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{9} + 2\beta_{8} + \beta_{7} - 2\beta_{5} + 4\beta_{2} ) / 4 \) (-2b9 + 2b8 + b7 - 2b5 + 4b2) / 4
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{13} + 3\beta_{12} + 7\beta_{4} - 3\beta _1 + 14 ) / 8 \) (-3b13 + 3b12 + 7b4 - 3b1 + 14) / 8
\(\nu^{5}\)	\(=\)	\( ( 5 \beta_{15} - 10 \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 22 \beta_{11} + 11 \beta_{9} + \cdots + 22 \beta_{2} ) / 16 \) (5b15 - 10b14 - 11b13 - 11b12 + 22b11 + 11b9 - 11b8 + 5b7 + 10b5 + 22b2) / 16
\(\nu^{6}\)	\(=\)	\( ( 4\beta_{9} + 4\beta_{8} + 9\beta_{6} ) / 2 \) (4b9 + 4b8 + 9*b6) / 2
\(\nu^{7}\)	\(=\)	\( ( 13 \beta_{15} + 26 \beta_{14} + 29 \beta_{13} + 29 \beta_{12} + 58 \beta_{11} - 29 \beta_{9} + \cdots + 58 \beta_{2} ) / 16 \) (13b15 + 26b14 + 29b13 + 29b12 + 58b11 - 29b9 + 29b8 + 13b7 - 26b5 + 58b2) / 16
\(\nu^{8}\)	\(=\)	\( ( -21\beta_{13} + 21\beta_{12} + 47\beta_{4} + 21\beta _1 - 94 ) / 8 \) (-21b13 + 21b12 + 47b4 + 21b1 - 94) / 8
\(\nu^{9}\)	\(=\)	\( ( 38\beta_{9} - 38\beta_{8} + 17\beta_{7} + 34\beta_{5} + 76\beta_{2} ) / 4 \) (38b9 - 38b8 + 17b7 + 34b5 + 76*b2) / 4
\(\nu^{10}\)	\(=\)	\( ( -55\beta_{10} + 55\beta_{9} + 55\beta_{8} + 123\beta_{6} + 246\beta_{3} ) / 8 \) (-55b10 + 55b9 + 55b8 + 123b6 + 246*b3) / 8
\(\nu^{11}\)	\(=\)	\( ( 89 \beta_{15} + 178 \beta_{14} + 199 \beta_{13} + 199 \beta_{12} + 398 \beta_{11} + \cdots - 398 \beta_{2} ) / 16 \) (89b15 + 178b14 + 199b13 + 199b12 + 398b11 + 199b9 - 199b8 - 89b7 + 178b5 - 398b2) / 16
\(\nu^{12}\)	\(=\)	\( 36\beta _1 - 161 \) 36*b1 - 161
\(\nu^{13}\)	\(=\)	\( ( - 233 \beta_{15} + 466 \beta_{14} + 521 \beta_{13} + 521 \beta_{12} - 1042 \beta_{11} + \cdots + 1042 \beta_{2} ) / 16 \) (-233b15 + 466b14 + 521b13 + 521b12 - 1042b11 + 521b9 - 521b8 + 233b7 + 466b5 + 1042b2) / 16
\(\nu^{14}\)	\(=\)	\( ( -377\beta_{10} - 377\beta_{9} - 377\beta_{8} - 843\beta_{6} + 1686\beta_{3} ) / 8 \) (-377b10 - 377b9 - 377b8 - 843b6 + 1686*b3) / 8
\(\nu^{15}\)	\(=\)	\( ( 682\beta_{9} - 682\beta_{8} - 305\beta_{7} + 610\beta_{5} - 1364\beta_{2} ) / 4 \) (682b9 - 682b8 - 305b7 + 610b5 - 1364*b2) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 349.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$3$	\( (T^{2} - 3)^{8} \) (T^2 - 3)^8
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} - 12 T^{6} + \cdots + 5764801)^{2} \) (T^8 - 12T^6 - 2842T^4 - 28812*T^2 + 5764801)^2
$11$	\( (T^{4} + 8 T^{3} + \cdots + 964)^{4} \) (T^4 + 8T^3 - 180T^2 - 784*T + 964)^4
$13$	\( (T^{8} - 264 T^{6} + \cdots + 14714896)^{2} \) (T^8 - 264T^6 + 25112T^4 - 1014816*T^2 + 14714896)^2
$17$	\( (T^{4} - 440 T^{2} + 19600)^{4} \) (T^4 - 440*T^2 + 19600)^4
$19$	\( (T^{8} + 1848 T^{6} + \cdots + 2869102096)^{2} \) (T^8 + 1848T^6 + 1069208T^4 + 192546528*T^2 + 2869102096)^2
$23$	\( (T^{8} + 2672 T^{6} + \cdots + 2975047936)^{2} \) (T^8 + 2672T^6 + 1893984T^4 + 381671168*T^2 + 2975047936)^2
$29$	\( (T^{4} - 72 T^{3} + \cdots - 281456)^{4} \) (T^4 - 72T^3 + 680T^2 + 22176*T - 281456)^4
$31$	\( (T^{8} + 2616 T^{6} + \cdots + 79120688656)^{2} \) (T^8 + 2616T^6 + 2165912T^4 + 709481184*T^2 + 79120688656)^2
$37$	\( (T^{8} + 6704 T^{6} + \cdots + 780013977856)^{2} \) (T^8 + 6704T^6 + 13155936T^4 + 7881016064*T^2 + 780013977856)^2
$41$	\( (T^{8} + 2016 T^{6} + \cdots + 7126061056)^{2} \) (T^8 + 2016T^6 + 1338752T^4 + 301676544*T^2 + 7126061056)^2
$43$	\( (T^{8} + \cdots + 7046179127296)^{2} \) (T^8 + 9024T^6 + 23700992T^4 + 22632677376*T^2 + 7046179127296)^2
$47$	\( (T^{8} + \cdots + 1081300500736)^{2} \) (T^8 - 8016T^6 + 19145312T^4 - 12067017984*T^2 + 1081300500736)^2
$53$	\( (T^{8} + \cdots + 13854147517456)^{2} \) (T^8 + 14232T^6 + 67543448T^4 + 110015218272*T^2 + 13854147517456)^2
$59$	\( (T^{8} + \cdots + 9016903929856)^{2} \) (T^8 + 14176T^6 + 44234112T^4 + 40668043264*T^2 + 9016903929856)^2
$61$	\( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) (T^8 + 24672T^6 + 221426048T^4 + 851048896512*T^2 + 1168834660274176)^2
$67$	\( (T^{8} + \cdots + 54339071222016)^{2} \) (T^8 + 16752T^6 + 95227488T^4 + 193265747712*T^2 + 54339071222016)^2
$71$	\( (T^{4} - 88 T^{3} + \cdots + 25706884)^{4} \) (T^4 - 88T^3 - 11860T^2 + 808624*T + 25706884)^4
$73$	\( (T^{8} - 24584 T^{6} + \cdots + 483086161936)^{2} \) (T^8 - 24584T^6 + 170869272T^4 - 363296139296*T^2 + 483086161936)^2
$79$	\( (T^{4} - 144 T^{3} + \cdots - 2641664)^{4} \) (T^4 - 144T^3 + 2912T^2 + 163584*T - 2641664)^4
$83$	\( (T^{8} + \cdots + 177346111242496)^{2} \) (T^8 - 17232T^6 + 102712928T^4 - 242767656192*T^2 + 177346111242496)^2
$89$	\( (T^{8} + \cdots + 86\!\cdots\!96)^{2} \) (T^8 + 43344T^6 + 655305056T^4 + 4054419197184*T^2 + 8614575218749696)^2
$97$	\( (T^{8} + \cdots + 76019496215056)^{2} \) (T^8 - 35144T^6 + 290961816T^4 - 312878921504*T^2 + 76019496215056)^2
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\(n\)	\(127\)	\(451\)	\(701\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)