LMFDB - Newform orbit 1575.2.m.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1268,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1268");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.m (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:	$12.5764383184$
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} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
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_{12} + \beta_{5} + \beta_{2} + 1) q^{2} + (2 \beta_{12} + \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + (2 \beta_{12} + \beta_{10} + \beta_{7} + \cdots - 1) q^{8}+O(q^{10}) $ q + (b12 + b5 + b2 + 1) * q^2 + (2b12 + b10 - b8 + b5 - b3 + b2) q^4 - b14 * q^7 + (2b12 + b10 + b7 - b3 - 1) q^8	$ q + (\beta_{12} + \beta_{5} + \beta_{2} + 1) q^{2} + (2 \beta_{12} + \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + ( - \beta_{12} - \beta_{10} + \beta_{8} + 1) q^{98}+O(q^{100}) $ q + (b12 + b5 + b2 + 1) * q^2 + (2b12 + b10 - b8 + b5 - b3 + b2) q^4 - b14 * q^7 + (2b12 + b10 + b7 - b3 - 1) q^8 + (-2b9 + b6) q^11 + 2b13 q^13 + (b11 - b1) * q^14 + (b10 - b7 - b5 - 2) * q^16 + (2b12 - 2b7 + 2b5 - 2b3 + 2b2) q^17 + (4b12 + 2b10 - 2b8 + 2b5 + 2b3 + 2b2) * q^19 + (-b14 + b4) * q^22 + (4b12 + b10 - 2b8 - 4) * q^23 + 2b6 q^26 + (-b15 + b14 + b13 + b11 + b9 + b6 - b1) * q^28 + (2b13 + b11 - 2b4) * q^29 + (2b10 + 2b8 - 2b5 + 2b2 - 2) * q^31 + (2b12 - b7 + 2b5 - b3 - b2 + 1) * q^32 + (8b12 + 4b10 - 2b8 + 4b5 - 2b3 + 2b2) * q^34 + (-3b14 - 4b4) * q^37 + (8b12 + 2b10 - 4b8 + 2b7 - 2b3 - 6) q^38 + (-2b13 + 4b6 - 2b4) q^41 + (-b15 + 2b14 + 2b13 + 2b11 + 4b9 + 2b6 - 4b1) * q^43 + (b15 + b14 + 2b11 + 3b1) * q^44 + (4b10 - 4b8 + b7 - 4b5 - 4b2 - 10) * q^46 + (-4b12 - 4b5 + 2b2 - 4) q^47 - b12 * q^49 + (2b14 - 2b4) * q^52 + (-b12 - 4b10 + 4b8 + 1) * q^53 + (-2b15 + 2b14 + b13 + b6 + b4) * q^56 + (3b14 + b13 + 3b11 + b9 + 3b6 - b1) q^58 + (2b15 + 2b14 - 4b13 - 4b11 + 4b4 + 6b1) * q^59 + (2b8 + 2b2 + 8) * q^61 + (-2b12 + 2b7 - 2b5 + 2b3 - 2b2) q^62 + (7b12 + 5b10 - 2b8 + 5b5 + 2b2) q^64 + (b15 - 10b14 + b11 - 2b9 - b6 - 2b4 - 2b1) * q^67 + (10b12 + 6b10 - 4b8 - 10) q^68 + (-5b15 + 5b14 + 2b13 + 2b9 + 9b6 + 2b4) * q^71 + (8b15 - 2b14 - 2b13 - 2b11 - 6b9 - 2b6 + 6b1) q^73 + (4b15 + 4b14 + 7b11 - 3b1) * q^74 + (6b10 - 4b8 + 6b7 - 6b5 - 4b2 - 8) q^76 + (b5 - 2b2) q^77 + (-9b12 - 2b10 + 4b8 - 2b5 + 2b3 - 4b2) * q^79 + (2b15 + 6b14 + 2b11 - 2b6 + 4b4) q^82 + (-4b12 - 6b10 + 8b8 + 4) q^83 + (-9b15 + 9b14 + 2b13 + 3b9 + 5b6 + 2b4) * q^86 + (2b15 + b14 + 4b13 + b11 + b9 + b6 - b1) * q^88 + (-4b11 + 6b1) * q^89 + (-2b7 - 2) q^91 + (-11b12 + 4b7 - 10b5 + 4b3 - 7b2 - 7) q^92 + (-10b12 - 4b10 + 4b8 - 4b5 + 4b3 - 4b2) * q^94 + (4b15 - 6b14 + 4b11 - 4b9 - 4b6 - 4b1) * q^97 + (-b12 - b10 + b8 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} - 16 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 - 16 * q^8	$ 16 q + 4 q^{2} - 16 q^{8} - 8 q^{16} - 8 q^{17} - 48 q^{23} - 16 q^{31} + 12 q^{32} - 80 q^{38} - 72 q^{46} - 40 q^{47} - 32 q^{53} + 112 q^{61} + 8 q^{62} - 96 q^{68} - 48 q^{76} - 16 q^{83} - 16 q^{91} - 36 q^{92} + 4 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 - 16 * q^8 - 8 * q^16 - 8 * q^17 - 48 * q^23 - 16 * q^31 + 12 * q^32 - 80 * q^38 - 72 * q^46 - 40 * q^47 - 32 * q^53 + 112 * q^61 + 8 * q^62 - 96 * q^68 - 48 * q^76 - 16 * q^83 - 16 * q^91 - 36 * q^92 + 4 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8*x^14 + 49*x^12 - 104*x^10 + 160*x^8 - 104*x^6 + 49*x^4 - 8*x^2 + 1 :

$\beta_{1}$	$=$	$ ( 23 \nu^{15} + 112 \nu^{13} - 1152 \nu^{11} + 11432 \nu^{9} - 22976 \nu^{7} + 36864 \nu^{5} + \cdots + 4720 \nu ) / 3168 $ (23v^15 + 112v^13 - 1152v^11 + 11432v^9 - 22976v^7 + 36864v^5 - 16729v^3 + 4720v) / 3168
$\beta_{2}$	$=$	$ ( 83\nu^{14} - 626\nu^{12} + 3648\nu^{10} - 5944\nu^{8} + 4528\nu^{6} + 4704\nu^{4} - 6973\nu^{2} + 1678 ) / 3168 $ (83v^14 - 626v^12 + 3648v^10 - 5944v^8 + 4528v^6 + 4704v^4 - 6973*v^2 + 1678) / 3168
$\beta_{3}$	$=$	$ ( 353 \nu^{14} - 3056 \nu^{12} + 18912 \nu^{10} - 46120 \nu^{8} + 68896 \nu^{6} - 49728 \nu^{4} + \cdots - 1616 ) / 3168 $ (353v^14 - 3056v^12 + 18912v^10 - 46120v^8 + 68896v^6 - 49728v^4 + 13169*v^2 - 1616) / 3168
$\beta_{4}$	$=$	$ ( 356 \nu^{15} - 3017 \nu^{13} + 18624 \nu^{11} - 44032 \nu^{9} + 66760 \nu^{7} - 50016 \nu^{5} + \cdots + 103 \nu ) / 3168 $ (356v^15 - 3017v^13 + 18624v^11 - 44032v^9 + 66760v^7 - 50016v^5 + 11492v^3 + 103v) / 3168
$\beta_{5}$	$=$	$ ( 379\nu^{14} - 2905\nu^{12} + 17472\nu^{10} - 32600\nu^{8} + 43784\nu^{6} - 13152\nu^{4} - 5237\nu^{2} + 1655 ) / 3168 $ (379v^14 - 2905v^12 + 17472v^10 - 32600v^8 + 43784v^6 - 13152v^4 - 5237*v^2 + 1655) / 3168
$\beta_{6}$	$=$	$ ( 127\nu^{15} - 945\nu^{13} + 5760\nu^{11} - 10520\nu^{9} + 17640\nu^{7} - 10080\nu^{5} + 8911\nu^{3} - 225\nu ) / 1056 $ (127v^15 - 945v^13 + 5760v^11 - 10520v^9 + 17640v^7 - 10080v^5 + 8911v^3 - 225v) / 1056
$\beta_{7}$	$=$	$ ( 140\nu^{14} - 1051\nu^{12} + 6272\nu^{10} - 10976\nu^{8} + 14104\nu^{6} - 3584\nu^{4} + 588\nu^{2} - 1323 ) / 1056 $ (140v^14 - 1051v^12 + 6272v^10 - 10976v^8 + 14104v^6 - 3584v^4 + 588*v^2 - 1323) / 1056
$\beta_{8}$	$=$	$ ( 442\nu^{14} - 3307\nu^{12} + 19872\nu^{10} - 35216\nu^{8} + 49880\nu^{6} - 18144\nu^{4} + 9178\nu^{2} + 101 ) / 3168 $ (442v^14 - 3307v^12 + 19872v^10 - 35216v^8 + 49880v^6 - 18144v^4 + 9178*v^2 + 101) / 3168
$\beta_{9}$	$=$	$ ( 196 \nu^{15} - 1533 \nu^{13} + 9344 \nu^{11} - 18816 \nu^{9} + 28616 \nu^{7} - 16352 \nu^{5} + \cdots - 365 \nu ) / 1056 $ (196v^15 - 1533v^13 + 9344v^11 - 18816v^9 + 28616v^7 - 16352v^5 + 8708v^3 - 365v) / 1056
$\beta_{10}$	$=$	$ ( -611\nu^{14} + 4487\nu^{12} - 26880\nu^{10} + 45016\nu^{8} - 62872\nu^{6} + 12192\nu^{4} - 9395\nu^{2} - 457 ) / 3168 $ (-611v^14 + 4487v^12 - 26880v^10 + 45016v^8 - 62872v^6 + 12192v^4 - 9395*v^2 - 457) / 3168
$\beta_{11}$	$=$	$ ( 53\nu^{15} - 485\nu^{13} + 3072\nu^{11} - 8392\nu^{9} + 14152\nu^{7} - 13728\nu^{5} + 6629\nu^{3} - 1877\nu ) / 288 $ (53v^15 - 485v^13 + 3072v^11 - 8392v^9 + 14152v^7 - 13728v^5 + 6629v^3 - 1877v) / 288
$\beta_{12}$	$=$	$ ( -59\nu^{14} + 476\nu^{12} - 2916\nu^{10} + 6280\nu^{8} - 9538\nu^{6} + 6192\nu^{4} - 2351\nu^{2} + 260 ) / 198 $ (-59v^14 + 476v^12 - 2916v^10 + 6280v^8 - 9538v^6 + 6192v^4 - 2351*v^2 + 260) / 198
$\beta_{13}$	$=$	$ ( 967 \nu^{15} - 7504 \nu^{13} + 45504 \nu^{11} - 89048 \nu^{9} + 129632 \nu^{7} - 62208 \nu^{5} + \cdots - 2608 \nu ) / 3168 $ (967v^15 - 7504v^13 + 45504v^11 - 89048v^9 + 129632v^7 - 62208v^5 + 20887v^3 - 2608v) / 3168
$\beta_{14}$	$=$	$ ( 598 \nu^{15} - 4711 \nu^{13} + 28656 \nu^{11} - 58112 \nu^{9} + 85064 \nu^{7} - 44208 \nu^{5} + \cdots + 2105 \nu ) / 1584 $ (598v^15 - 4711v^13 + 28656v^11 - 58112v^9 + 85064v^7 - 44208v^5 + 13846v^3 + 2105v) / 1584
$\beta_{15}$	$=$	$ ( - 1049 \nu^{15} + 8330 \nu^{13} - 50832 \nu^{11} + 105544 \nu^{9} - 158368 \nu^{7} + 94896 \nu^{5} + \cdots + 5210 \nu ) / 1584 $ (-1049v^15 + 8330v^13 - 50832v^11 + 105544v^9 - 158368v^7 + 94896v^5 - 39641v^3 + 5210v) / 1584

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

$\nu$	$=$	$ ( -\beta_{13} + \beta_{9} + \beta_{4} + \beta_1 ) / 2 $ (-b13 + b9 + b4 + b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{12} - \beta_{10} - 3\beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 2 ) / 2 $ (-b12 - b10 - 3*b8 + b7 - b5 - b3 + b2 + 2) / 2
$\nu^{3}$	$=$	$ \beta_{15} - \beta_{14} + \beta_{13} + 4\beta_{9} - \beta_{6} + \beta_{4} $ b15 - b14 + b13 + 4*b9 - b6 + b4
$\nu^{4}$	$=$	$ ( -6\beta_{12} - 2\beta_{10} - 5\beta_{8} - 5\beta_{7} - 4\beta_{5} - 7\beta_{3} + 17\beta_{2} - 9 ) / 2 $ (-6b12 - 2b10 - 5b8 - 5b7 - 4b5 - 7b3 + 17*b2 - 9) / 2
$\nu^{5}$	$=$	$ ( 10\beta_{15} - 2\beta_{14} + 21\beta_{13} - 7\beta_{11} + 21\beta_{9} - 5\beta_{6} - 7\beta_{4} - 35\beta_1 ) / 2 $ (10b15 - 2b14 + 21b13 - 7b11 + 21b9 - 5b6 - 7b4 - 35b1) / 2
$\nu^{6}$	$=$	$ 7\beta_{10} + 34\beta_{8} - 26\beta_{7} - 7\beta_{5} + 34\beta_{2} - 47 $ 7b10 + 34b8 - 26b7 - 7b5 + 34*b2 - 47
$\nu^{7}$	$=$	$ ( - 16 \beta_{15} + 52 \beta_{14} + 33 \beta_{13} - 42 \beta_{11} - 115 \beta_{9} + 26 \beta_{6} + \cdots - 197 \beta_1 ) / 2 $ (-16b15 + 52b14 + 33b13 - 42b11 - 115b9 + 26b6 - 115b4 - 197b1) / 2
$\nu^{8}$	$=$	$ ( 197\beta_{12} + 115\beta_{10} + 521\beta_{8} - 141\beta_{7} + 33\beta_{5} + 239\beta_{3} - 141\beta_{2} - 256 ) / 2 $ (197b12 + 115b10 + 521b8 - 141b7 + 33b5 + 239b3 - 141*b2 - 256) / 2
$\nu^{9}$	$=$	$ -190\beta_{15} + 190\beta_{14} - 231\beta_{13} - 636\beta_{9} + 141\beta_{6} - 231\beta_{4} $ -190b15 + 190b14 - 231b13 - 636b9 + 141b6 - 231b4
$\nu^{10}$	$=$	$ ( 1098 \beta_{12} + 174 \beta_{10} + 777 \beta_{8} + 777 \beta_{7} + 636 \beta_{5} + 1337 \beta_{3} + \cdots + 1413 ) / 2 $ (1098b12 + 174b10 + 777b8 + 777b7 + 636b5 + 1337b3 - 2891*b2 + 1413) / 2
$\nu^{11}$	$=$	$ ( - 1554 \beta_{15} + 560 \beta_{14} - 3527 \beta_{13} + 1337 \beta_{11} - 3527 \beta_{9} + \cdots + 6103 \beta_1 ) / 2 $ (-1554b15 + 560b14 - 3527b13 + 1337b11 - 3527b9 + 777b6 + 951b4 + 6103b1) / 2
$\nu^{12}$	$=$	$ -1288\beta_{10} - 5872\beta_{8} + 4304\beta_{7} + 1288\beta_{5} - 5872\beta_{2} + 7831 $ -1288b10 - 5872b8 + 4304b7 + 1288b5 - 5872*b2 + 7831
$\nu^{13}$	$=$	$ ( 3136 \beta_{15} - 8608 \beta_{14} - 5255 \beta_{13} + 7440 \beta_{11} + 19575 \beta_{9} + \cdots + 33895 \beta_1 ) / 2 $ (3136b15 - 8608b14 - 5255b13 + 7440b11 + 19575b9 - 4304b6 + 19575b4 + 33895b1) / 2
$\nu^{14}$	$=$	$ ( - 33895 \beta_{12} - 19575 \beta_{10} - 89093 \beta_{8} + 23879 \beta_{7} - 5255 \beta_{5} + \cdots + 43454 ) / 2 $ (-33895b12 - 19575b10 - 89093b8 + 23879b7 - 5255b5 - 41335b3 + 23879*b2 + 43454) / 2
$\nu^{15}$	$=$	$ 32607\beta_{15} - 32607\beta_{14} + 39767\beta_{13} + 108668\beta_{9} - 23879\beta_{6} + 39767\beta_{4} $ 32607b15 - 32607b14 + 39767b13 + 108668b9 - 23879b6 + 39767b4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-\beta_{12}$	$1$	$-1$

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} $

1268.1

0.670418	+	0.387066i
−0.670418	−	0.387066i
−2.04058	+	1.17813i
2.04058	−	1.17813i
−1.11871	−	0.645885i
1.11871	+	0.645885i
0.367543	−	0.212201i
−0.367543	+	0.212201i
0.670418	−	0.387066i
−0.670418	+	0.387066i
−2.04058	−	1.17813i
2.04058	+	1.17813i
−1.11871	+	0.645885i
1.11871	−	0.645885i
0.367543	+	0.212201i
−0.367543	−	0.212201i

−0.913419

−

0.913419i

−

0.331331i

−0.707107

0.707107i

−2.12948

2.12948i

1268.2

−0.913419

−

0.913419i

−

0.331331i

0.707107

−

0.707107i

−2.12948

2.12948i

1268.3

−0.300098

−

0.300098i

−

1.81988i

−0.707107

0.707107i

−1.14634

1.14634i

1268.4

−0.300098

−

0.300098i

−

1.81988i

0.707107

−

0.707107i

−1.14634

1.14634i

1268.5

0.547394

0.547394i

−

1.40072i

−0.707107

0.707107i

1.86153

−

1.86153i

1268.6

0.547394

0.547394i

−

1.40072i

0.707107

−

0.707107i

1.86153

−

1.86153i

1268.7

1.66612

1.66612i

3.55193i

−0.707107

0.707107i

−2.58571

2.58571i

1268.8

1.66612

1.66612i

3.55193i

0.707107

−

0.707107i

−2.58571

2.58571i

1457.1

−0.913419

0.913419i

0.331331i

−0.707107

−

0.707107i

−2.12948

−

2.12948i

1457.2

−0.913419

0.913419i

0.331331i

0.707107

0.707107i

−2.12948

−

2.12948i

1457.3

−0.300098

0.300098i

1.81988i

−0.707107

−

0.707107i

−1.14634

−

1.14634i

1457.4

−0.300098

0.300098i

1.81988i

0.707107

0.707107i

−1.14634

−

1.14634i

1457.5

0.547394

−

0.547394i

1.40072i

−0.707107

−

0.707107i

1.86153

1.86153i

1457.6

0.547394

−

0.547394i

1.40072i

0.707107

0.707107i

1.86153

1.86153i

1457.7

1.66612

−

1.66612i

−

3.55193i

−0.707107

−

0.707107i

−2.58571

−

2.58571i

1457.8

1.66612

−

1.66612i

−

3.55193i

0.707107

0.707107i

−2.58571

−

2.58571i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.2.m.f	yes	16
3.b	odd	2	1		1575.2.m.e	✓	16
5.b	even	2	1		1575.2.m.e	✓	16
5.c	odd	4	1		1575.2.m.e	✓	16
5.c	odd	4	1	inner	1575.2.m.f	yes	16
15.d	odd	2	1	inner	1575.2.m.f	yes	16
15.e	even	4	1		1575.2.m.e	✓	16
15.e	even	4	1	inner	1575.2.m.f	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1575.2.m.e	✓	16	3.b	odd	2	1
1575.2.m.e	✓	16	5.b	even	2	1
1575.2.m.e	✓	16	5.c	odd	4	1
1575.2.m.e	✓	16	15.e	even	4	1
1575.2.m.f	yes	16	1.a	even	1	1	trivial
1575.2.m.f	yes	16	5.c	odd	4	1	inner
1575.2.m.f	yes	16	15.d	odd	2	1	inner
1575.2.m.f	yes	16	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 2T_{2}^{7} + 2T_{2}^{6} + 4T_{2}^{5} + 7T_{2}^{4} - 4T_{2}^{3} + 2T_{2}^{2} + 2T_{2} + 1 $ T2^8 - 2*T2^7 + 2*T2^6 + 4*T2^5 + 7*T2^4 - 4*T2^3 + 2*T2^2 + 2*T2 + 1 acting on $S_{2}^{\mathrm{new}}(1575, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} $ (T^8 - 2T^7 + 2T^6 + 4T^5 + 7T^4 - 4T^3 + 2T^2 + 2*T + 1)^2
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$11$	$ (T^{8} + 48 T^{6} + 570 T^{4} + \cdots + 9)^{2} $ (T^8 + 48T^6 + 570T^4 + 144*T^2 + 9)^2
$13$	$ T^{16} + 736 T^{12} + \cdots + 65536 $ T^16 + 736T^12 + 74496T^8 + 1761280*T^4 + 65536
$17$	$ (T^{8} + 4 T^{7} + \cdots + 43264)^{2} $ (T^8 + 4T^7 + 8T^6 - 32T^5 + 880T^4 + 3200T^3 + 6272T^2 - 23296*T + 43264)^2
$19$	$ (T^{8} + 136 T^{6} + \cdots + 135424)^{2} $ (T^8 + 136T^6 + 6192T^4 + 97408*T^2 + 135424)^2
$23$	$ (T^{8} + 24 T^{7} + \cdots + 19881)^{2} $ (T^8 + 24T^7 + 288T^6 + 2016T^5 + 8934T^4 + 24264T^3 + 41472T^2 + 40608*T + 19881)^2
$29$	$ (T^{8} - 80 T^{6} + \cdots + 11881)^{2} $ (T^8 - 80T^6 + 1434T^4 - 8048*T^2 + 11881)^2
$31$	$ (T^{4} + 4 T^{3} + \cdots + 208)^{4} $ (T^4 + 4T^3 - 72T^2 - 80*T + 208)^4
$37$	$ T^{16} + \cdots + 43617904801 $ T^16 + 9124T^12 + 17592294T^8 + 3718755556*T^4 + 43617904801
$41$	$ (T^{8} + 176 T^{6} + \cdots + 4096)^{2} $ (T^8 + 176T^6 + 4800T^4 + 11264*T^2 + 4096)^2
$43$	$ T^{16} + 19396 T^{12} + \cdots + 4879681 $ T^16 + 19396T^12 + 50703558T^8 + 10235889220*T^4 + 4879681
$47$	$ (T^{8} + 20 T^{7} + \cdots + 43264)^{2} $ (T^8 + 20T^7 + 200T^6 + 320T^5 - 272T^4 + 2560T^3 + 156800T^2 - 116480*T + 43264)^2
$53$	$ (T^{8} + 16 T^{7} + \cdots + 110224)^{2} $ (T^8 + 16T^7 + 128T^6 + 256T^5 + 808T^4 + 10688T^3 + 100352T^2 + 148736*T + 110224)^2
$59$	$ (T^{8} - 368 T^{6} + \cdots + 14868736)^{2} $ (T^8 - 368T^6 + 39792T^4 - 1391360*T^2 + 14868736)^2
$61$	$ (T^{4} - 28 T^{3} + \cdots + 832)^{4} $ (T^4 - 28T^3 + 264T^2 - 928*T + 832)^4
$67$	$ T^{16} + \cdots + 23811286661761 $ T^16 + 41092T^12 + 307641222T^8 + 513652626820*T^4 + 23811286661761
$71$	$ (T^{8} + 464 T^{6} + \cdots + 28270489)^{2} $ (T^8 + 464T^6 + 66762T^4 + 2997872*T^2 + 28270489)^2
$73$	$ T^{16} + \cdots + 8590432731136 $ T^16 + 84448T^12 + 1716953856T^8 + 10192330350592*T^4 + 8590432731136
$79$	$ (T^{8} + 244 T^{6} + \cdots + 786769)^{2} $ (T^8 + 244T^6 + 13110T^4 + 225940*T^2 + 786769)^2
$83$	$ (T^{8} + 8 T^{7} + \cdots + 42823936)^{2} $ (T^8 + 8T^7 + 32T^6 - 352T^5 + 23776T^4 + 174976T^3 + 700928T^2 - 7748096*T + 42823936)^2
$89$	$ (T^{8} - 224 T^{6} + \cdots + 6310144)^{2} $ (T^8 - 224T^6 + 17136T^4 - 548864*T^2 + 6310144)^2
$97$	$ T^{16} + \cdots + 1871773696 $ T^16 + 57664T^12 + 136558080T^8 + 72550137856*T^4 + 1871773696
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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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{12} + \beta_{5} + \beta_{2} + 1) q^{2} + (2 \beta_{12} + \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + (2 \beta_{12} + \beta_{10} + \beta_{7} + \cdots - 1) q^{8}+O(q^{10}) \) q + (b12 + b5 + b2 + 1) * q^2 + (2b12 + b10 - b8 + b5 - b3 + b2) q^4 - b14 * q^7 + (2b12 + b10 + b7 - b3 - 1) q^8	\( q + (\beta_{12} + \beta_{5} + \beta_{2} + 1) q^{2} + (2 \beta_{12} + \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + ( - \beta_{12} - \beta_{10} + \beta_{8} + 1) q^{98}+O(q^{100}) \) q + (b12 + b5 + b2 + 1) * q^2 + (2b12 + b10 - b8 + b5 - b3 + b2) q^4 - b14 * q^7 + (2b12 + b10 + b7 - b3 - 1) q^8 + (-2b9 + b6) q^11 + 2b13 q^13 + (b11 - b1) * q^14 + (b10 - b7 - b5 - 2) * q^16 + (2b12 - 2b7 + 2b5 - 2b3 + 2b2) q^17 + (4b12 + 2b10 - 2b8 + 2b5 + 2b3 + 2b2) * q^19 + (-b14 + b4) * q^22 + (4b12 + b10 - 2b8 - 4) * q^23 + 2b6 q^26 + (-b15 + b14 + b13 + b11 + b9 + b6 - b1) * q^28 + (2b13 + b11 - 2b4) * q^29 + (2b10 + 2b8 - 2b5 + 2b2 - 2) * q^31 + (2b12 - b7 + 2b5 - b3 - b2 + 1) * q^32 + (8b12 + 4b10 - 2b8 + 4b5 - 2b3 + 2b2) * q^34 + (-3b14 - 4b4) * q^37 + (8b12 + 2b10 - 4b8 + 2b7 - 2b3 - 6) q^38 + (-2b13 + 4b6 - 2b4) q^41 + (-b15 + 2b14 + 2b13 + 2b11 + 4b9 + 2b6 - 4b1) * q^43 + (b15 + b14 + 2b11 + 3b1) * q^44 + (4b10 - 4b8 + b7 - 4b5 - 4b2 - 10) * q^46 + (-4b12 - 4b5 + 2b2 - 4) q^47 - b12 * q^49 + (2b14 - 2b4) * q^52 + (-b12 - 4b10 + 4b8 + 1) * q^53 + (-2b15 + 2b14 + b13 + b6 + b4) * q^56 + (3b14 + b13 + 3b11 + b9 + 3b6 - b1) q^58 + (2b15 + 2b14 - 4b13 - 4b11 + 4b4 + 6b1) * q^59 + (2b8 + 2b2 + 8) * q^61 + (-2b12 + 2b7 - 2b5 + 2b3 - 2b2) q^62 + (7b12 + 5b10 - 2b8 + 5b5 + 2b2) q^64 + (b15 - 10b14 + b11 - 2b9 - b6 - 2b4 - 2b1) * q^67 + (10b12 + 6b10 - 4b8 - 10) q^68 + (-5b15 + 5b14 + 2b13 + 2b9 + 9b6 + 2b4) * q^71 + (8b15 - 2b14 - 2b13 - 2b11 - 6b9 - 2b6 + 6b1) q^73 + (4b15 + 4b14 + 7b11 - 3b1) * q^74 + (6b10 - 4b8 + 6b7 - 6b5 - 4b2 - 8) q^76 + (b5 - 2b2) q^77 + (-9b12 - 2b10 + 4b8 - 2b5 + 2b3 - 4b2) * q^79 + (2b15 + 6b14 + 2b11 - 2b6 + 4b4) q^82 + (-4b12 - 6b10 + 8b8 + 4) q^83 + (-9b15 + 9b14 + 2b13 + 3b9 + 5b6 + 2b4) * q^86 + (2b15 + b14 + 4b13 + b11 + b9 + b6 - b1) * q^88 + (-4b11 + 6b1) * q^89 + (-2b7 - 2) q^91 + (-11b12 + 4b7 - 10b5 + 4b3 - 7b2 - 7) q^92 + (-10b12 - 4b10 + 4b8 - 4b5 + 4b3 - 4b2) * q^94 + (4b15 - 6b14 + 4b11 - 4b9 - 4b6 - 4b1) * q^97 + (-b12 - b10 + b8 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 16 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 - 16 * q^8	\( 16 q + 4 q^{2} - 16 q^{8} - 8 q^{16} - 8 q^{17} - 48 q^{23} - 16 q^{31} + 12 q^{32} - 80 q^{38} - 72 q^{46} - 40 q^{47} - 32 q^{53} + 112 q^{61} + 8 q^{62} - 96 q^{68} - 48 q^{76} - 16 q^{83} - 16 q^{91} - 36 q^{92} + 4 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 - 16 * q^8 - 8 * q^16 - 8 * q^17 - 48 * q^23 - 16 * q^31 + 12 * q^32 - 80 * q^38 - 72 * q^46 - 40 * q^47 - 32 * q^53 + 112 * q^61 + 8 * q^62 - 96 * q^68 - 48 * q^76 - 16 * q^83 - 16 * q^91 - 36 * q^92 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1575.2.m.f

Level	$1575$
Weight	$2$
Character orbit	1575.m
Analytic conductor	$12.576$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(12.5764383184\)
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:	\( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 23 \nu^{15} + 112 \nu^{13} - 1152 \nu^{11} + 11432 \nu^{9} - 22976 \nu^{7} + 36864 \nu^{5} + \cdots + 4720 \nu ) / 3168 \) (23v^15 + 112v^13 - 1152v^11 + 11432v^9 - 22976v^7 + 36864v^5 - 16729v^3 + 4720v) / 3168
\(\beta_{2}\)	\(=\)	\( ( 83\nu^{14} - 626\nu^{12} + 3648\nu^{10} - 5944\nu^{8} + 4528\nu^{6} + 4704\nu^{4} - 6973\nu^{2} + 1678 ) / 3168 \) (83v^14 - 626v^12 + 3648v^10 - 5944v^8 + 4528v^6 + 4704v^4 - 6973*v^2 + 1678) / 3168
\(\beta_{3}\)	\(=\)	\( ( 353 \nu^{14} - 3056 \nu^{12} + 18912 \nu^{10} - 46120 \nu^{8} + 68896 \nu^{6} - 49728 \nu^{4} + \cdots - 1616 ) / 3168 \) (353v^14 - 3056v^12 + 18912v^10 - 46120v^8 + 68896v^6 - 49728v^4 + 13169*v^2 - 1616) / 3168
\(\beta_{4}\)	\(=\)	\( ( 356 \nu^{15} - 3017 \nu^{13} + 18624 \nu^{11} - 44032 \nu^{9} + 66760 \nu^{7} - 50016 \nu^{5} + \cdots + 103 \nu ) / 3168 \) (356v^15 - 3017v^13 + 18624v^11 - 44032v^9 + 66760v^7 - 50016v^5 + 11492v^3 + 103v) / 3168
\(\beta_{5}\)	\(=\)	\( ( 379\nu^{14} - 2905\nu^{12} + 17472\nu^{10} - 32600\nu^{8} + 43784\nu^{6} - 13152\nu^{4} - 5237\nu^{2} + 1655 ) / 3168 \) (379v^14 - 2905v^12 + 17472v^10 - 32600v^8 + 43784v^6 - 13152v^4 - 5237*v^2 + 1655) / 3168
\(\beta_{6}\)	\(=\)	\( ( 127\nu^{15} - 945\nu^{13} + 5760\nu^{11} - 10520\nu^{9} + 17640\nu^{7} - 10080\nu^{5} + 8911\nu^{3} - 225\nu ) / 1056 \) (127v^15 - 945v^13 + 5760v^11 - 10520v^9 + 17640v^7 - 10080v^5 + 8911v^3 - 225v) / 1056
\(\beta_{7}\)	\(=\)	\( ( 140\nu^{14} - 1051\nu^{12} + 6272\nu^{10} - 10976\nu^{8} + 14104\nu^{6} - 3584\nu^{4} + 588\nu^{2} - 1323 ) / 1056 \) (140v^14 - 1051v^12 + 6272v^10 - 10976v^8 + 14104v^6 - 3584v^4 + 588*v^2 - 1323) / 1056
\(\beta_{8}\)	\(=\)	\( ( 442\nu^{14} - 3307\nu^{12} + 19872\nu^{10} - 35216\nu^{8} + 49880\nu^{6} - 18144\nu^{4} + 9178\nu^{2} + 101 ) / 3168 \) (442v^14 - 3307v^12 + 19872v^10 - 35216v^8 + 49880v^6 - 18144v^4 + 9178*v^2 + 101) / 3168
\(\beta_{9}\)	\(=\)	\( ( 196 \nu^{15} - 1533 \nu^{13} + 9344 \nu^{11} - 18816 \nu^{9} + 28616 \nu^{7} - 16352 \nu^{5} + \cdots - 365 \nu ) / 1056 \) (196v^15 - 1533v^13 + 9344v^11 - 18816v^9 + 28616v^7 - 16352v^5 + 8708v^3 - 365v) / 1056
\(\beta_{10}\)	\(=\)	\( ( -611\nu^{14} + 4487\nu^{12} - 26880\nu^{10} + 45016\nu^{8} - 62872\nu^{6} + 12192\nu^{4} - 9395\nu^{2} - 457 ) / 3168 \) (-611v^14 + 4487v^12 - 26880v^10 + 45016v^8 - 62872v^6 + 12192v^4 - 9395*v^2 - 457) / 3168
\(\beta_{11}\)	\(=\)	\( ( 53\nu^{15} - 485\nu^{13} + 3072\nu^{11} - 8392\nu^{9} + 14152\nu^{7} - 13728\nu^{5} + 6629\nu^{3} - 1877\nu ) / 288 \) (53v^15 - 485v^13 + 3072v^11 - 8392v^9 + 14152v^7 - 13728v^5 + 6629v^3 - 1877v) / 288
\(\beta_{12}\)	\(=\)	\( ( -59\nu^{14} + 476\nu^{12} - 2916\nu^{10} + 6280\nu^{8} - 9538\nu^{6} + 6192\nu^{4} - 2351\nu^{2} + 260 ) / 198 \) (-59v^14 + 476v^12 - 2916v^10 + 6280v^8 - 9538v^6 + 6192v^4 - 2351*v^2 + 260) / 198
\(\beta_{13}\)	\(=\)	\( ( 967 \nu^{15} - 7504 \nu^{13} + 45504 \nu^{11} - 89048 \nu^{9} + 129632 \nu^{7} - 62208 \nu^{5} + \cdots - 2608 \nu ) / 3168 \) (967v^15 - 7504v^13 + 45504v^11 - 89048v^9 + 129632v^7 - 62208v^5 + 20887v^3 - 2608v) / 3168
\(\beta_{14}\)	\(=\)	\( ( 598 \nu^{15} - 4711 \nu^{13} + 28656 \nu^{11} - 58112 \nu^{9} + 85064 \nu^{7} - 44208 \nu^{5} + \cdots + 2105 \nu ) / 1584 \) (598v^15 - 4711v^13 + 28656v^11 - 58112v^9 + 85064v^7 - 44208v^5 + 13846v^3 + 2105v) / 1584
\(\beta_{15}\)	\(=\)	\( ( - 1049 \nu^{15} + 8330 \nu^{13} - 50832 \nu^{11} + 105544 \nu^{9} - 158368 \nu^{7} + 94896 \nu^{5} + \cdots + 5210 \nu ) / 1584 \) (-1049v^15 + 8330v^13 - 50832v^11 + 105544v^9 - 158368v^7 + 94896v^5 - 39641v^3 + 5210v) / 1584

\(\nu\)	\(=\)	\( ( -\beta_{13} + \beta_{9} + \beta_{4} + \beta_1 ) / 2 \) (-b13 + b9 + b4 + b1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{12} - \beta_{10} - 3\beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 2 ) / 2 \) (-b12 - b10 - 3*b8 + b7 - b5 - b3 + b2 + 2) / 2
\(\nu^{3}\)	\(=\)	\( \beta_{15} - \beta_{14} + \beta_{13} + 4\beta_{9} - \beta_{6} + \beta_{4} \) b15 - b14 + b13 + 4*b9 - b6 + b4
\(\nu^{4}\)	\(=\)	\( ( -6\beta_{12} - 2\beta_{10} - 5\beta_{8} - 5\beta_{7} - 4\beta_{5} - 7\beta_{3} + 17\beta_{2} - 9 ) / 2 \) (-6b12 - 2b10 - 5b8 - 5b7 - 4b5 - 7b3 + 17*b2 - 9) / 2
\(\nu^{5}\)	\(=\)	\( ( 10\beta_{15} - 2\beta_{14} + 21\beta_{13} - 7\beta_{11} + 21\beta_{9} - 5\beta_{6} - 7\beta_{4} - 35\beta_1 ) / 2 \) (10b15 - 2b14 + 21b13 - 7b11 + 21b9 - 5b6 - 7b4 - 35b1) / 2
\(\nu^{6}\)	\(=\)	\( 7\beta_{10} + 34\beta_{8} - 26\beta_{7} - 7\beta_{5} + 34\beta_{2} - 47 \) 7b10 + 34b8 - 26b7 - 7b5 + 34*b2 - 47
\(\nu^{7}\)	\(=\)	\( ( - 16 \beta_{15} + 52 \beta_{14} + 33 \beta_{13} - 42 \beta_{11} - 115 \beta_{9} + 26 \beta_{6} + \cdots - 197 \beta_1 ) / 2 \) (-16b15 + 52b14 + 33b13 - 42b11 - 115b9 + 26b6 - 115b4 - 197b1) / 2
\(\nu^{8}\)	\(=\)	\( ( 197\beta_{12} + 115\beta_{10} + 521\beta_{8} - 141\beta_{7} + 33\beta_{5} + 239\beta_{3} - 141\beta_{2} - 256 ) / 2 \) (197b12 + 115b10 + 521b8 - 141b7 + 33b5 + 239b3 - 141*b2 - 256) / 2
\(\nu^{9}\)	\(=\)	\( -190\beta_{15} + 190\beta_{14} - 231\beta_{13} - 636\beta_{9} + 141\beta_{6} - 231\beta_{4} \) -190b15 + 190b14 - 231b13 - 636b9 + 141b6 - 231b4
\(\nu^{10}\)	\(=\)	\( ( 1098 \beta_{12} + 174 \beta_{10} + 777 \beta_{8} + 777 \beta_{7} + 636 \beta_{5} + 1337 \beta_{3} + \cdots + 1413 ) / 2 \) (1098b12 + 174b10 + 777b8 + 777b7 + 636b5 + 1337b3 - 2891*b2 + 1413) / 2
\(\nu^{11}\)	\(=\)	\( ( - 1554 \beta_{15} + 560 \beta_{14} - 3527 \beta_{13} + 1337 \beta_{11} - 3527 \beta_{9} + \cdots + 6103 \beta_1 ) / 2 \) (-1554b15 + 560b14 - 3527b13 + 1337b11 - 3527b9 + 777b6 + 951b4 + 6103b1) / 2
\(\nu^{12}\)	\(=\)	\( -1288\beta_{10} - 5872\beta_{8} + 4304\beta_{7} + 1288\beta_{5} - 5872\beta_{2} + 7831 \) -1288b10 - 5872b8 + 4304b7 + 1288b5 - 5872*b2 + 7831
\(\nu^{13}\)	\(=\)	\( ( 3136 \beta_{15} - 8608 \beta_{14} - 5255 \beta_{13} + 7440 \beta_{11} + 19575 \beta_{9} + \cdots + 33895 \beta_1 ) / 2 \) (3136b15 - 8608b14 - 5255b13 + 7440b11 + 19575b9 - 4304b6 + 19575b4 + 33895b1) / 2
\(\nu^{14}\)	\(=\)	\( ( - 33895 \beta_{12} - 19575 \beta_{10} - 89093 \beta_{8} + 23879 \beta_{7} - 5255 \beta_{5} + \cdots + 43454 ) / 2 \) (-33895b12 - 19575b10 - 89093b8 + 23879b7 - 5255b5 - 41335b3 + 23879*b2 + 43454) / 2
\(\nu^{15}\)	\(=\)	\( 32607\beta_{15} - 32607\beta_{14} + 39767\beta_{13} + 108668\beta_{9} - 23879\beta_{6} + 39767\beta_{4} \) 32607b15 - 32607b14 + 39767b13 + 108668b9 - 23879b6 + 39767b4

\(n\)	\(127\)	\(451\)	\(1226\)
\(\chi(n)\)	\(-\beta_{12}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1268.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) (T^8 - 2T^7 + 2T^6 + 4T^5 + 7T^4 - 4T^3 + 2T^2 + 2*T + 1)^2
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$11$	\( (T^{8} + 48 T^{6} + 570 T^{4} + \cdots + 9)^{2} \) (T^8 + 48T^6 + 570T^4 + 144*T^2 + 9)^2
$13$	\( T^{16} + 736 T^{12} + \cdots + 65536 \) T^16 + 736T^12 + 74496T^8 + 1761280*T^4 + 65536
$17$	\( (T^{8} + 4 T^{7} + \cdots + 43264)^{2} \) (T^8 + 4T^7 + 8T^6 - 32T^5 + 880T^4 + 3200T^3 + 6272T^2 - 23296*T + 43264)^2
$19$	\( (T^{8} + 136 T^{6} + \cdots + 135424)^{2} \) (T^8 + 136T^6 + 6192T^4 + 97408*T^2 + 135424)^2
$23$	\( (T^{8} + 24 T^{7} + \cdots + 19881)^{2} \) (T^8 + 24T^7 + 288T^6 + 2016T^5 + 8934T^4 + 24264T^3 + 41472T^2 + 40608*T + 19881)^2
$29$	\( (T^{8} - 80 T^{6} + \cdots + 11881)^{2} \) (T^8 - 80T^6 + 1434T^4 - 8048*T^2 + 11881)^2
$31$	\( (T^{4} + 4 T^{3} + \cdots + 208)^{4} \) (T^4 + 4T^3 - 72T^2 - 80*T + 208)^4
$37$	\( T^{16} + \cdots + 43617904801 \) T^16 + 9124T^12 + 17592294T^8 + 3718755556*T^4 + 43617904801
$41$	\( (T^{8} + 176 T^{6} + \cdots + 4096)^{2} \) (T^8 + 176T^6 + 4800T^4 + 11264*T^2 + 4096)^2
$43$	\( T^{16} + 19396 T^{12} + \cdots + 4879681 \) T^16 + 19396T^12 + 50703558T^8 + 10235889220*T^4 + 4879681
$47$	\( (T^{8} + 20 T^{7} + \cdots + 43264)^{2} \) (T^8 + 20T^7 + 200T^6 + 320T^5 - 272T^4 + 2560T^3 + 156800T^2 - 116480*T + 43264)^2
$53$	\( (T^{8} + 16 T^{7} + \cdots + 110224)^{2} \) (T^8 + 16T^7 + 128T^6 + 256T^5 + 808T^4 + 10688T^3 + 100352T^2 + 148736*T + 110224)^2
$59$	\( (T^{8} - 368 T^{6} + \cdots + 14868736)^{2} \) (T^8 - 368T^6 + 39792T^4 - 1391360*T^2 + 14868736)^2
$61$	\( (T^{4} - 28 T^{3} + \cdots + 832)^{4} \) (T^4 - 28T^3 + 264T^2 - 928*T + 832)^4
$67$	\( T^{16} + \cdots + 23811286661761 \) T^16 + 41092T^12 + 307641222T^8 + 513652626820*T^4 + 23811286661761
$71$	\( (T^{8} + 464 T^{6} + \cdots + 28270489)^{2} \) (T^8 + 464T^6 + 66762T^4 + 2997872*T^2 + 28270489)^2
$73$	\( T^{16} + \cdots + 8590432731136 \) T^16 + 84448T^12 + 1716953856T^8 + 10192330350592*T^4 + 8590432731136
$79$	\( (T^{8} + 244 T^{6} + \cdots + 786769)^{2} \) (T^8 + 244T^6 + 13110T^4 + 225940*T^2 + 786769)^2
$83$	\( (T^{8} + 8 T^{7} + \cdots + 42823936)^{2} \) (T^8 + 8T^7 + 32T^6 - 352T^5 + 23776T^4 + 174976T^3 + 700928T^2 - 7748096*T + 42823936)^2
$89$	\( (T^{8} - 224 T^{6} + \cdots + 6310144)^{2} \) (T^8 - 224T^6 + 17136T^4 - 548864*T^2 + 6310144)^2
$97$	\( T^{16} + \cdots + 1871773696 \) T^16 + 57664T^12 + 136558080T^8 + 72550137856*T^4 + 1871773696
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