LMFDB - Newform orbit 896.2.m.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(225,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("896.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.m (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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.20138089353117696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 $ x^12 - 3x^10 - 2x^9 + 2x^8 + 4x^7 + 2x^6 + 8x^5 + 8x^4 - 16x^3 - 48*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 112)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + \beta_1) q^{9}+O(q^{10}) $ q + b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 + b3 * q^7 + (-b4 - 2b3 - b2 + b1) q^9	$ q + \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + \beta_1) q^{9}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) $ q + b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 + b3 * q^7 + (-b4 - 2b3 - b2 + b1) q^9 + b8 * q^11 + (b11 + b9 - b6 + b1) * q^13 + (-b11 - b6 - b5 - 3) * q^15 + (b10 - 1) * q^17 + (-b9 - b6 - b4) * q^19 + b5 * q^21 + (b9 - b8 + b7 - 2b3 + b1) q^23 + (-b9 + b8 - b7 - b6 + b5 + 3b3 - b1) q^25 + (b10 - b8 + b7 - 2b5 - b3 + b2 - 1) q^27 + (-b11 - b1) * q^29 + (b6 + b5) * q^31 + (b10 - b9 - b8 - b6 - b5 - 1) * q^33 + (b6 - b4 - b3 + 1) * q^35 + (b3 - 2b2 + 1) q^37 + (b7 + 2b6 - 2b5 - b1) * q^39 + (b6 - b5 - b4 - 4b3 - b2) q^41 + (-b11 + b10 - b8 + b7 - 2b3 + b1 - 2) q^43 + (-b11 + b10 - b9 - b7 - 3b6 + 5b3 - b1 - 5) * q^45 + (2b11 + b9 + b8 - b6 - b5 - b4 + b2 + 2) q^47 - q^49 + (-b11 - b3 - b1 + 1) * q^51 + (b10 - 2b8 + b7 + 2b2) * q^53 + (-b9 + b8 - b4 - 2b3 - b2 - 2b1) * q^55 + (b9 - b8 + b7 + b6 - b5 + 2b4 + 6b3 + 2b2 - b1) q^57 + (-b10 + b8 - b7 - b5 + b3 - b2 + 1) * q^59 + (b10 - b7 + b6 - b4 - 2b3 + 2) q^61 + (b11 - b4 + b2 + 2) * q^63 + (b11 + 2b9 + 2b8 - 2b6 - 2b5 + b4 - b2 + 1) * q^65 + (b11 + b9 - 2b4 + b3 + b1 - 1) q^67 + (-b11 + b8 - 2b5 - b3 - b2 + b1 - 1) q^69 + (b7 + b6 - b5 - b3 + 2b1) q^71 + (-b7 - b6 + b5 - b4 + b3 - b2 - 2b1) q^73 + (2b11 - b8 + 3b5 + 6b3 + 3b2 - 2b1 + 6) q^75 - b9 * q^77 + (b10 - b9 - b8 - b6 - b5 + 1) * q^79 + (-b11 - b10 + b9 + b8 + b6 + b5 - 3) * q^81 + (2b11 - b10 + 2b9 + b7 - b6 - 2b4 - 3b3 + 2b1 + 3) q^83 + (b11 - b10 + 2b8 - b7 + 2b3 + 2b2 - b1 + 2) q^85 + (-b9 + b8 - 2b7 - 3b6 + 3b5 + b4 + 6b3 + b2) * q^87 + (b9 - b8 + b7 - 3b6 + 3b5 + 3b3 + 2b1) * q^89 + (b11 + b8 - b5 - b1) * q^91 + (b11 - 2b4 - 5b3 + b1 + 5) * q^93 + (-b11 + b10 - 2b9 - 2b8 + 2b6 + 2b5 - 2b4 + 2b2 + 2) * q^95 + (-b10 - b9 - b8 + 2b6 + 2b5 + b4 - b2 + 5) * q^97 + (-2b11 - b10 - b9 + b7 + 2b6 + 2b4 + 3b3 - 2b1 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 4 q^{5}+O(q^{10}) $ 12 * q - 4 * q^3 - 4 * q^5	$ 12 q - 4 q^{3} - 4 q^{5} - 24 q^{15} - 8 q^{17} - 4 q^{21} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{35} + 20 q^{37} - 16 q^{43} - 40 q^{45} + 16 q^{47} - 12 q^{49} + 16 q^{51} - 4 q^{53} + 16 q^{59} + 20 q^{61} + 12 q^{63} + 32 q^{65} - 24 q^{67} + 4 q^{69} + 40 q^{75} + 24 q^{79} - 44 q^{81} + 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 - 4 * q^5 - 24 * q^15 - 8 * q^17 - 4 * q^21 - 4 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^35 + 20 * q^37 - 16 * q^43 - 40 * q^45 + 16 * q^47 - 12 * q^49 + 16 * q^51 - 4 * q^53 + 16 * q^59 + 20 * q^61 + 12 * q^63 + 32 * q^65 - 24 * q^67 + 4 * q^69 + 40 * q^75 + 24 * q^79 - 44 * q^81 + 20 * q^83 + 8 * q^85 + 48 * q^93 + 48 * q^97 - 32 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 $ x^12 - 3*x^10 - 2*x^9 + 2*x^8 + 4*x^7 + 2*x^6 + 8*x^5 + 8*x^4 - 16*x^3 - 48*x^2 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{11} + 14 \nu^{10} + \nu^{9} - 20 \nu^{8} - 54 \nu^{7} + 16 \nu^{6} + 34 \nu^{5} + 36 \nu^{4} + \cdots - 608 ) / 128 $ (v^11 + 14v^10 + v^9 - 20v^8 - 54v^7 + 16v^6 + 34v^5 + 36v^4 + 192v^3 + 240v^2 - 144*v - 608) / 128
$\beta_{2}$	$=$	$ ( 3 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 28 \nu^{8} - 34 \nu^{7} - 16 \nu^{6} - 26 \nu^{5} + 44 \nu^{4} + \cdots - 416 ) / 128 $ (3v^11 + 10v^10 + 3v^9 - 28v^8 - 34v^7 - 16v^6 - 26v^5 + 44v^4 + 192v^3 + 208v^2 - 176*v - 416) / 128
$\beta_{3}$	$=$	$ ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + \cdots + 32 ) / 128 $ (5v^11 + 6v^10 + 5v^9 - 4v^8 - 14v^7 - 16v^6 - 22v^5 - 12v^4 + 64v^3 + 112v^2 + 48*v + 32) / 128
$\beta_{4}$	$=$	$ ( - 11 \nu^{11} - 10 \nu^{10} + 21 \nu^{9} + 44 \nu^{8} + 18 \nu^{7} - 16 \nu^{6} + 10 \nu^{5} + \cdots + 416 ) / 128 $ (-11v^11 - 10v^10 + 21v^9 + 44v^8 + 18v^7 - 16v^6 + 10v^5 - 108v^4 - 256v^3 - 80v^2 + 304*v + 416) / 128
$\beta_{5}$	$=$	$ ( - 5 \nu^{11} - 14 \nu^{10} + 3 \nu^{9} + 28 \nu^{8} + 38 \nu^{7} + 16 \nu^{6} + 6 \nu^{5} - 36 \nu^{4} + \cdots + 352 ) / 64 $ (-5v^11 - 14v^10 + 3v^9 + 28v^8 + 38v^7 + 16v^6 + 6v^5 - 36v^4 - 176v^3 - 176v^2 + 80*v + 352) / 64
$\beta_{6}$	$=$	$ ( -\nu^{11} - \nu^{10} + 2\nu^{9} + 5\nu^{8} + 3\nu^{7} - 4\nu^{5} - 14\nu^{4} - 26\nu^{3} - 8\nu^{2} + 40\nu + 48 ) / 8 $ (-v^11 - v^10 + 2v^9 + 5v^8 + 3v^7 - 4v^5 - 14v^4 - 26v^3 - 8v^2 + 40v + 48) / 8
$\beta_{7}$	$=$	$ ( 21 \nu^{11} + 54 \nu^{10} + 21 \nu^{9} - 52 \nu^{8} - 78 \nu^{7} - 112 \nu^{6} - 118 \nu^{5} + \cdots - 480 ) / 128 $ (21v^11 + 54v^10 + 21v^9 - 52v^8 - 78v^7 - 112v^6 - 118v^5 + 84v^4 + 448v^3 + 752v^2 + 176*v - 480) / 128
$\beta_{8}$	$=$	$ ( - 23 \nu^{11} - 18 \nu^{10} + 41 \nu^{9} + 60 \nu^{8} + 58 \nu^{7} + 48 \nu^{6} - 14 \nu^{5} + \cdots + 544 ) / 128 $ (-23v^11 - 18v^10 + 41v^9 + 60v^8 + 58v^7 + 48v^6 - 14v^5 - 220v^4 - 448v^3 - 144v^2 + 624*v + 544) / 128
$\beta_{9}$	$=$	$ ( - 23 \nu^{11} - 34 \nu^{10} + 41 \nu^{9} + 108 \nu^{8} + 90 \nu^{7} + 16 \nu^{6} - 78 \nu^{5} + \cdots + 1312 ) / 128 $ (-23v^11 - 34v^10 + 41v^9 + 108v^8 + 90v^7 + 16v^6 - 78v^5 - 252v^4 - 576v^3 - 528v^2 + 880*v + 1312) / 128
$\beta_{10}$	$=$	$ ( - 17 \nu^{11} - 22 \nu^{10} + 31 \nu^{9} + 76 \nu^{8} + 54 \nu^{7} - 34 \nu^{5} - 180 \nu^{4} + \cdots + 800 ) / 64 $ (-17v^11 - 22v^10 + 31v^9 + 76v^8 + 54v^7 - 34v^5 - 180v^4 - 416v^3 - 176v^2 + 720v + 800) / 64
$\beta_{11}$	$=$	$ ( 19 \nu^{11} + 18 \nu^{10} - 29 \nu^{9} - 68 \nu^{8} - 66 \nu^{7} - 16 \nu^{6} + 38 \nu^{5} + 220 \nu^{4} + \cdots - 736 ) / 64 $ (19v^11 + 18v^10 - 29v^9 - 68v^8 - 66v^7 - 16v^6 + 38v^5 + 220v^4 + 352v^3 + 240v^2 - 560*v - 736) / 64

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 1 ) / 4 $ (b10 - b6 - b5 - b4 - b2 - 1) / 4
$\nu^{2}$	$=$	$ ( \beta_{10} - 2\beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 3 ) / 4 $ (b10 - 2*b9 + b6 + b5 - b4 + b2 + 3) / 4
$\nu^{3}$	$=$	$ ( -2\beta_{11} + \beta_{10} - 2\beta_{8} - 3\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4 $ (-2b11 + b10 - 2b8 - 3b6 + b5 - b4 + 2b3 + b2 + 2*b1 + 1) / 4
$\nu^{4}$	$=$	$ ( 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - 8 \beta_{3} + \cdots + 5 ) / 4 $ (2b11 + b10 + 2b9 + 2b7 - b6 + 3b5 - b4 - 8b3 + b2 + 2b1 + 5) / 4
$\nu^{5}$	$=$	$ ( -2\beta_{11} + \beta_{10} - 2\beta_{9} - 7\beta_{6} + \beta_{5} + 5\beta_{4} + 6\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 4 $ (-2b11 + b10 - 2b9 - 7b6 + b5 + 5b4 + 6b3 - 5b2 + 2*b1 - 3) / 4
$\nu^{6}$	$=$	$ ( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + \cdots + 11 ) / 4 $ (2b11 - b10 + 2b9 + 4b8 + 3b6 + 3b5 - 9b4 + 2b3 - 3b2 + 6*b1 + 11) / 4
$\nu^{7}$	$=$	$ ( - 6 \beta_{11} + \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} + \cdots - 15 ) / 4 $ (-6b11 + b10 - 6b9 + 4b7 - 7b6 + 9b5 - 3b4 - 2b3 - b2 - 2b1 - 15) / 4
$\nu^{8}$	$=$	$ ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 12 \beta_{8} + 4 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} + \cdots - 9 ) / 4 $ (-2b11 - b10 + 2b9 - 12b8 + 4b7 + 7b6 + 15b5 - b4 - 6b3 - 3b2 + 18*b1 - 9) / 4
$\nu^{9}$	$=$	$ ( 2 \beta_{11} - 7 \beta_{10} + 14 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 23 \beta_{6} + 17 \beta_{5} + \cdots - 23 ) / 4 $ (2b11 - 7b10 + 14b9 + 12b8 + 4b7 - 23b6 + 17b5 + 17b4 + 30b3 + 11b2 + 6*b1 - 23) / 4
$\nu^{10}$	$=$	$ ( - 2 \beta_{11} - 21 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} + 16 \beta_{7} + 15 \beta_{6} - \beta_{5} + \cdots + 19 ) / 4 $ (-2b11 - 21b10 + 10b9 + 4b8 + 16b7 + 15b6 - b5 + 7b4 - 50b3 - 35b2 + 10b1 + 19) / 4
$\nu^{11}$	$=$	$ ( 10 \beta_{11} - 7 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 17 \beta_{5} + \cdots - 111 ) / 4 $ (10b11 - 7b10 + 6b9 + 12b8 - 4b7 - 7b6 + 17b5 + b4 + 110b3 - 29b2 - 2b1 - 111) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$1$	$\beta_{3}$

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} $

225.1

1.35309	−	0.411286i
−0.605558	−	1.27801i
0.402577	+	1.35570i
−1.12465	+	0.857418i
1.37925	+	0.312504i
−1.40471	+	0.163666i
1.35309	+	0.411286i
−0.605558	+	1.27801i
0.402577	−	1.35570i
−1.12465	−	0.857418i
1.37925	−	0.312504i
−1.40471	−	0.163666i

−2.21570

2.21570i

0.393125

0.393125i

1.00000i

−

6.81864i

225.2

−1.39123

1.39123i

−2.16478

−

2.16478i

1.00000i

−

0.871066i

225.3

−0.631188

0.631188i

2.34259

2.34259i

1.00000i

2.20320i

225.4

−0.416854

0.416854i

1.13169

1.13169i

1.00000i

2.65247i

225.5

0.599978

−

0.599978i

−0.974969

−

0.974969i

1.00000i

2.28005i

225.6

2.05500

−

2.05500i

−2.72766

−

2.72766i

1.00000i

−

5.44602i

673.1

−2.21570

−

2.21570i

0.393125

−

0.393125i

−

1.00000i

6.81864i

673.2

−1.39123

−

1.39123i

−2.16478

2.16478i

−

1.00000i

0.871066i

673.3

−0.631188

−

0.631188i

2.34259

−

2.34259i

−

1.00000i

−

2.20320i

673.4

−0.416854

−

0.416854i

1.13169

−

1.13169i

−

1.00000i

−

2.65247i

673.5

0.599978

0.599978i

−0.974969

0.974969i

−

1.00000i

−

2.28005i

673.6

2.05500

2.05500i

−2.72766

2.72766i

−

1.00000i

5.44602i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	896.2.m.g		12
4.b	odd	2	1		896.2.m.h		12
8.b	even	2	1		112.2.m.d	✓	12
8.d	odd	2	1		448.2.m.d		12
16.e	even	4	1		112.2.m.d	✓	12
16.e	even	4	1	inner	896.2.m.g		12
16.f	odd	4	1		448.2.m.d		12
16.f	odd	4	1		896.2.m.h		12
32.g	even	8	2		7168.2.a.bj		12
32.h	odd	8	2		7168.2.a.bi		12
56.h	odd	2	1		784.2.m.h		12
56.j	odd	6	2		784.2.x.m		24
56.p	even	6	2		784.2.x.l		24
112.l	odd	4	1		784.2.m.h		12
112.w	even	12	2		784.2.x.l		24
112.x	odd	12	2		784.2.x.m		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.m.d	✓	12	8.b	even	2	1
112.2.m.d	✓	12	16.e	even	4	1
448.2.m.d		12	8.d	odd	2	1
448.2.m.d		12	16.f	odd	4	1
784.2.m.h		12	56.h	odd	2	1
784.2.m.h		12	112.l	odd	4	1
784.2.x.l		24	56.p	even	6	2
784.2.x.l		24	112.w	even	12	2
784.2.x.m		24	56.j	odd	6	2
784.2.x.m		24	112.x	odd	12	2
896.2.m.g		12	1.a	even	1	1	trivial
896.2.m.g		12	16.e	even	4	1	inner
896.2.m.h		12	4.b	odd	2	1
896.2.m.h		12	16.f	odd	4	1
7168.2.a.bi		12	32.h	odd	8	2
7168.2.a.bj		12	32.g	even	8	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(896, [\chi])$:

$ T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 76 T_{3}^{8} + 288 T_{3}^{7} + 552 T_{3}^{6} + \cdots + 64 $

$ T_{11}^{12} - 32 T_{11}^{9} + 740 T_{11}^{8} - 896 T_{11}^{7} + 512 T_{11}^{6} - 896 T_{11}^{5} + \cdots + 541696 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 4 T^{11} + \cdots + 64 $ T^12 + 4T^11 + 8T^10 + 4T^9 + 76T^8 + 288T^7 + 552T^6 + 376T^5 + 164T^4 + 144T^3 + 288T^2 + 192*T + 64
$5$	$ T^{12} + 4 T^{11} + \cdots + 2304 $ T^12 + 4T^11 + 8T^10 - 12T^9 + 140T^8 + 504T^7 + 968T^6 - 1288T^5 + 1828T^4 + 3072T^3 + 4608T^2 - 4608*T + 2304
$7$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$11$	$ T^{12} - 32 T^{9} + \cdots + 541696 $ T^12 - 32T^9 + 740T^8 - 896T^7 + 512T^6 - 896T^5 + 67968T^4 - 89088T^3 + 51200T^2 + 235520*T + 541696
$13$	$ T^{12} - 20 T^{9} + \cdots + 3211264 $ T^12 - 20T^9 + 1724T^8 - 1376T^7 + 200T^6 + 28456T^5 + 297732T^4 + 216576T^3 + 32768T^2 - 458752*T + 3211264
$17$	$ (T^{6} + 4 T^{5} - 44 T^{4} + \cdots - 96)^{2} $ (T^6 + 4T^5 - 44T^4 - 200T^3 + 40T^2 + 288*T - 96)^2
$19$	$ T^{12} + 76 T^{9} + \cdots + 2849344 $ T^12 + 76T^9 + 2444T^8 + 4024T^7 + 2888T^6 - 9016T^5 + 466212T^4 + 565552T^3 + 352800T^2 - 1417920*T + 2849344
$23$	$ T^{12} + 136 T^{10} + \cdots + 10137856 $ T^12 + 136T^10 + 7096T^8 + 175744T^6 + 2044688T^4 + 9219456*T^2 + 10137856
$29$	$ T^{12} - 4 T^{11} + \cdots + 8620096 $ T^12 - 4T^11 + 8T^10 + 216T^9 + 3084T^8 - 928T^7 + 2368T^6 + 68544T^5 + 995888T^4 - 13888T^3 + 21632T^2 + 610688*T + 8620096
$31$	$ (T^{6} + 4 T^{5} - 16 T^{4} + \cdots + 64)^{2} $ (T^6 + 4T^5 - 16T^4 - 72T^3 - 24T^2 + 96*T + 64)^2
$37$	$ T^{12} - 20 T^{11} + \cdots + 5053504 $ T^12 - 20T^11 + 200T^10 - 936T^9 + 3532T^8 - 24992T^7 + 231488T^6 - 1060672T^5 + 2719280T^4 - 2835776T^3 + 332928T^2 + 1834368*T + 5053504
$41$	$ T^{12} + 120 T^{10} + \cdots + 25600 $ T^12 + 120T^10 + 4704T^8 + 68544T^6 + 345920T^4 + 392704*T^2 + 25600
$43$	$ T^{12} + 16 T^{11} + \cdots + 23040000 $ T^12 + 16T^11 + 128T^10 + 768T^9 + 11780T^8 + 158208T^7 + 1318400T^6 + 6805376T^5 + 22906624T^4 + 49373184T^3 + 68585472T^2 + 56217600*T + 23040000
$47$	$ (T^{6} - 8 T^{5} + \cdots + 19776)^{2} $ (T^6 - 8T^5 - 104T^4 + 1032T^3 - 1208T^2 - 8928*T + 19776)^2
$53$	$ T^{12} + 4 T^{11} + \cdots + 78400 $ T^12 + 4T^11 + 8T^10 - 248T^9 + 19372T^8 + 59808T^7 + 115008T^6 - 2979008T^5 + 22600496T^4 - 25960896T^3 + 14709888T^2 + 1518720*T + 78400
$59$	$ T^{12} + \cdots + 119596096 $ T^12 - 16T^11 + 128T^10 - 124T^9 + 1996T^8 - 36840T^7 + 341640T^6 - 218536T^5 - 899932T^4 + 4793520T^3 + 62272800T^2 + 122045760*T + 119596096
$61$	$ T^{12} - 20 T^{11} + \cdots + 256 $ T^12 - 20T^11 + 200T^10 - 172T^9 + 2188T^8 - 34632T^7 + 269832T^6 - 568008T^5 + 339748T^4 + 1728352T^3 + 4476032T^2 - 47872*T + 256
$67$	$ T^{12} + 24 T^{11} + \cdots + 3686400 $ T^12 + 24T^11 + 288T^10 + 1072T^9 + 1124T^8 - 9344T^7 + 26624T^6 + 66560T^5 + 150016T^4 - 1548288T^3 + 4718592T^2 - 5898240*T + 3686400
$71$	$ T^{12} + 432 T^{10} + \cdots + 6553600 $ T^12 + 432T^10 + 60960T^8 + 2931456T^6 + 15382784T^4 + 18743296*T^2 + 6553600
$73$	$ T^{12} + \cdots + 3050573824 $ T^12 + 616T^10 + 134384T^8 + 12430080T^6 + 473632512T^4 + 6278768640*T^2 + 3050573824
$79$	$ (T^{6} - 12 T^{5} + \cdots - 2240)^{2} $ (T^6 - 12T^5 - 44T^4 + 576T^3 - 112T^2 - 3392*T - 2240)^2
$83$	$ T^{12} + \cdots + 320093429824 $ T^12 - 20T^11 + 200T^10 - 1116T^9 + 34732T^8 - 656608T^7 + 6808488T^6 - 33954632T^5 + 208394468T^4 - 2610661520T^3 + 27522333728T^2 - 132738225088*T + 320093429824
$89$	$ T^{12} + \cdots + 35476475904 $ T^12 + 552T^10 + 101680T^8 + 8448256T^6 + 338121472T^4 + 6091352064*T^2 + 35476475904
$97$	$ (T^{6} - 24 T^{5} + \cdots - 39008)^{2} $ (T^6 - 24T^5 - 60T^4 + 2568T^3 + 8168T^2 - 11104*T - 39008)^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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + \beta_1) q^{9}+O(q^{10}) \) q + b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 + b3 * q^7 + (-b4 - 2b3 - b2 + b1) q^9	\( q + \beta_{6} q^{3} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{5} + \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} + \cdots + \beta_1) q^{9}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) q + b6 * q^3 + (-b5 - b3 - b2 - 1) * q^5 + b3 * q^7 + (-b4 - 2b3 - b2 + b1) q^9 + b8 * q^11 + (b11 + b9 - b6 + b1) * q^13 + (-b11 - b6 - b5 - 3) * q^15 + (b10 - 1) * q^17 + (-b9 - b6 - b4) * q^19 + b5 * q^21 + (b9 - b8 + b7 - 2b3 + b1) q^23 + (-b9 + b8 - b7 - b6 + b5 + 3b3 - b1) q^25 + (b10 - b8 + b7 - 2b5 - b3 + b2 - 1) q^27 + (-b11 - b1) * q^29 + (b6 + b5) * q^31 + (b10 - b9 - b8 - b6 - b5 - 1) * q^33 + (b6 - b4 - b3 + 1) * q^35 + (b3 - 2b2 + 1) q^37 + (b7 + 2b6 - 2b5 - b1) * q^39 + (b6 - b5 - b4 - 4b3 - b2) q^41 + (-b11 + b10 - b8 + b7 - 2b3 + b1 - 2) q^43 + (-b11 + b10 - b9 - b7 - 3b6 + 5b3 - b1 - 5) * q^45 + (2b11 + b9 + b8 - b6 - b5 - b4 + b2 + 2) q^47 - q^49 + (-b11 - b3 - b1 + 1) * q^51 + (b10 - 2b8 + b7 + 2b2) * q^53 + (-b9 + b8 - b4 - 2b3 - b2 - 2b1) * q^55 + (b9 - b8 + b7 + b6 - b5 + 2b4 + 6b3 + 2b2 - b1) q^57 + (-b10 + b8 - b7 - b5 + b3 - b2 + 1) * q^59 + (b10 - b7 + b6 - b4 - 2b3 + 2) q^61 + (b11 - b4 + b2 + 2) * q^63 + (b11 + 2b9 + 2b8 - 2b6 - 2b5 + b4 - b2 + 1) * q^65 + (b11 + b9 - 2b4 + b3 + b1 - 1) q^67 + (-b11 + b8 - 2b5 - b3 - b2 + b1 - 1) q^69 + (b7 + b6 - b5 - b3 + 2b1) q^71 + (-b7 - b6 + b5 - b4 + b3 - b2 - 2b1) q^73 + (2b11 - b8 + 3b5 + 6b3 + 3b2 - 2b1 + 6) q^75 - b9 * q^77 + (b10 - b9 - b8 - b6 - b5 + 1) * q^79 + (-b11 - b10 + b9 + b8 + b6 + b5 - 3) * q^81 + (2b11 - b10 + 2b9 + b7 - b6 - 2b4 - 3b3 + 2b1 + 3) q^83 + (b11 - b10 + 2b8 - b7 + 2b3 + 2b2 - b1 + 2) q^85 + (-b9 + b8 - 2b7 - 3b6 + 3b5 + b4 + 6b3 + b2) * q^87 + (b9 - b8 + b7 - 3b6 + 3b5 + 3b3 + 2b1) * q^89 + (b11 + b8 - b5 - b1) * q^91 + (b11 - 2b4 - 5b3 + b1 + 5) * q^93 + (-b11 + b10 - 2b9 - 2b8 + 2b6 + 2b5 - 2b4 + 2b2 + 2) * q^95 + (-b10 - b9 - b8 + 2b6 + 2b5 + b4 - b2 + 5) * q^97 + (-2b11 - b10 - b9 + b7 + 2b6 + 2b4 + 3b3 - 2b1 - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) 12 * q - 4 * q^3 - 4 * q^5	\( 12 q - 4 q^{3} - 4 q^{5} - 24 q^{15} - 8 q^{17} - 4 q^{21} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{35} + 20 q^{37} - 16 q^{43} - 40 q^{45} + 16 q^{47} - 12 q^{49} + 16 q^{51} - 4 q^{53} + 16 q^{59} + 20 q^{61} + 12 q^{63} + 32 q^{65} - 24 q^{67} + 4 q^{69} + 40 q^{75} + 24 q^{79} - 44 q^{81} + 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 - 4 * q^5 - 24 * q^15 - 8 * q^17 - 4 * q^21 - 4 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^35 + 20 * q^37 - 16 * q^43 - 40 * q^45 + 16 * q^47 - 12 * q^49 + 16 * q^51 - 4 * q^53 + 16 * q^59 + 20 * q^61 + 12 * q^63 + 32 * q^65 - 24 * q^67 + 4 * q^69 + 40 * q^75 + 24 * q^79 - 44 * q^81 + 20 * q^83 + 8 * q^85 + 48 * q^93 + 48 * q^97 - 32 * 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	896.2.m.g

Level	$896$
Weight	$2$
Character orbit	896.m
Analytic conductor	$7.155$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	12.0.20138089353117696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) x^12 - 3x^10 - 2x^9 + 2x^8 + 4x^7 + 2x^6 + 8x^5 + 8x^4 - 16x^3 - 48*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{9} \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} + 14 \nu^{10} + \nu^{9} - 20 \nu^{8} - 54 \nu^{7} + 16 \nu^{6} + 34 \nu^{5} + 36 \nu^{4} + \cdots - 608 ) / 128 \) (v^11 + 14v^10 + v^9 - 20v^8 - 54v^7 + 16v^6 + 34v^5 + 36v^4 + 192v^3 + 240v^2 - 144*v - 608) / 128
\(\beta_{2}\)	\(=\)	\( ( 3 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 28 \nu^{8} - 34 \nu^{7} - 16 \nu^{6} - 26 \nu^{5} + 44 \nu^{4} + \cdots - 416 ) / 128 \) (3v^11 + 10v^10 + 3v^9 - 28v^8 - 34v^7 - 16v^6 - 26v^5 + 44v^4 + 192v^3 + 208v^2 - 176*v - 416) / 128
\(\beta_{3}\)	\(=\)	\( ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + \cdots + 32 ) / 128 \) (5v^11 + 6v^10 + 5v^9 - 4v^8 - 14v^7 - 16v^6 - 22v^5 - 12v^4 + 64v^3 + 112v^2 + 48*v + 32) / 128
\(\beta_{4}\)	\(=\)	\( ( - 11 \nu^{11} - 10 \nu^{10} + 21 \nu^{9} + 44 \nu^{8} + 18 \nu^{7} - 16 \nu^{6} + 10 \nu^{5} + \cdots + 416 ) / 128 \) (-11v^11 - 10v^10 + 21v^9 + 44v^8 + 18v^7 - 16v^6 + 10v^5 - 108v^4 - 256v^3 - 80v^2 + 304*v + 416) / 128
\(\beta_{5}\)	\(=\)	\( ( - 5 \nu^{11} - 14 \nu^{10} + 3 \nu^{9} + 28 \nu^{8} + 38 \nu^{7} + 16 \nu^{6} + 6 \nu^{5} - 36 \nu^{4} + \cdots + 352 ) / 64 \) (-5v^11 - 14v^10 + 3v^9 + 28v^8 + 38v^7 + 16v^6 + 6v^5 - 36v^4 - 176v^3 - 176v^2 + 80*v + 352) / 64
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} - \nu^{10} + 2\nu^{9} + 5\nu^{8} + 3\nu^{7} - 4\nu^{5} - 14\nu^{4} - 26\nu^{3} - 8\nu^{2} + 40\nu + 48 ) / 8 \) (-v^11 - v^10 + 2v^9 + 5v^8 + 3v^7 - 4v^5 - 14v^4 - 26v^3 - 8v^2 + 40v + 48) / 8
\(\beta_{7}\)	\(=\)	\( ( 21 \nu^{11} + 54 \nu^{10} + 21 \nu^{9} - 52 \nu^{8} - 78 \nu^{7} - 112 \nu^{6} - 118 \nu^{5} + \cdots - 480 ) / 128 \) (21v^11 + 54v^10 + 21v^9 - 52v^8 - 78v^7 - 112v^6 - 118v^5 + 84v^4 + 448v^3 + 752v^2 + 176*v - 480) / 128
\(\beta_{8}\)	\(=\)	\( ( - 23 \nu^{11} - 18 \nu^{10} + 41 \nu^{9} + 60 \nu^{8} + 58 \nu^{7} + 48 \nu^{6} - 14 \nu^{5} + \cdots + 544 ) / 128 \) (-23v^11 - 18v^10 + 41v^9 + 60v^8 + 58v^7 + 48v^6 - 14v^5 - 220v^4 - 448v^3 - 144v^2 + 624*v + 544) / 128
\(\beta_{9}\)	\(=\)	\( ( - 23 \nu^{11} - 34 \nu^{10} + 41 \nu^{9} + 108 \nu^{8} + 90 \nu^{7} + 16 \nu^{6} - 78 \nu^{5} + \cdots + 1312 ) / 128 \) (-23v^11 - 34v^10 + 41v^9 + 108v^8 + 90v^7 + 16v^6 - 78v^5 - 252v^4 - 576v^3 - 528v^2 + 880*v + 1312) / 128
\(\beta_{10}\)	\(=\)	\( ( - 17 \nu^{11} - 22 \nu^{10} + 31 \nu^{9} + 76 \nu^{8} + 54 \nu^{7} - 34 \nu^{5} - 180 \nu^{4} + \cdots + 800 ) / 64 \) (-17v^11 - 22v^10 + 31v^9 + 76v^8 + 54v^7 - 34v^5 - 180v^4 - 416v^3 - 176v^2 + 720v + 800) / 64
\(\beta_{11}\)	\(=\)	\( ( 19 \nu^{11} + 18 \nu^{10} - 29 \nu^{9} - 68 \nu^{8} - 66 \nu^{7} - 16 \nu^{6} + 38 \nu^{5} + 220 \nu^{4} + \cdots - 736 ) / 64 \) (19v^11 + 18v^10 - 29v^9 - 68v^8 - 66v^7 - 16v^6 + 38v^5 + 220v^4 + 352v^3 + 240v^2 - 560*v - 736) / 64

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 1 ) / 4 \) (b10 - b6 - b5 - b4 - b2 - 1) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} - 2\beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 3 ) / 4 \) (b10 - 2*b9 + b6 + b5 - b4 + b2 + 3) / 4
\(\nu^{3}\)	\(=\)	\( ( -2\beta_{11} + \beta_{10} - 2\beta_{8} - 3\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4 \) (-2b11 + b10 - 2b8 - 3b6 + b5 - b4 + 2b3 + b2 + 2*b1 + 1) / 4
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - 8 \beta_{3} + \cdots + 5 ) / 4 \) (2b11 + b10 + 2b9 + 2b7 - b6 + 3b5 - b4 - 8b3 + b2 + 2b1 + 5) / 4
\(\nu^{5}\)	\(=\)	\( ( -2\beta_{11} + \beta_{10} - 2\beta_{9} - 7\beta_{6} + \beta_{5} + 5\beta_{4} + 6\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 4 \) (-2b11 + b10 - 2b9 - 7b6 + b5 + 5b4 + 6b3 - 5b2 + 2*b1 - 3) / 4
\(\nu^{6}\)	\(=\)	\( ( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + \cdots + 11 ) / 4 \) (2b11 - b10 + 2b9 + 4b8 + 3b6 + 3b5 - 9b4 + 2b3 - 3b2 + 6*b1 + 11) / 4
\(\nu^{7}\)	\(=\)	\( ( - 6 \beta_{11} + \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} + \cdots - 15 ) / 4 \) (-6b11 + b10 - 6b9 + 4b7 - 7b6 + 9b5 - 3b4 - 2b3 - b2 - 2b1 - 15) / 4
\(\nu^{8}\)	\(=\)	\( ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 12 \beta_{8} + 4 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} + \cdots - 9 ) / 4 \) (-2b11 - b10 + 2b9 - 12b8 + 4b7 + 7b6 + 15b5 - b4 - 6b3 - 3b2 + 18*b1 - 9) / 4
\(\nu^{9}\)	\(=\)	\( ( 2 \beta_{11} - 7 \beta_{10} + 14 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 23 \beta_{6} + 17 \beta_{5} + \cdots - 23 ) / 4 \) (2b11 - 7b10 + 14b9 + 12b8 + 4b7 - 23b6 + 17b5 + 17b4 + 30b3 + 11b2 + 6*b1 - 23) / 4
\(\nu^{10}\)	\(=\)	\( ( - 2 \beta_{11} - 21 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} + 16 \beta_{7} + 15 \beta_{6} - \beta_{5} + \cdots + 19 ) / 4 \) (-2b11 - 21b10 + 10b9 + 4b8 + 16b7 + 15b6 - b5 + 7b4 - 50b3 - 35b2 + 10b1 + 19) / 4
\(\nu^{11}\)	\(=\)	\( ( 10 \beta_{11} - 7 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 17 \beta_{5} + \cdots - 111 ) / 4 \) (10b11 - 7b10 + 6b9 + 12b8 - 4b7 - 7b6 + 17b5 + b4 + 110b3 - 29b2 - 2b1 - 111) / 4

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(1\)	\(1\)	\(\beta_{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 225.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 4 T^{11} + \cdots + 64 \) T^12 + 4T^11 + 8T^10 + 4T^9 + 76T^8 + 288T^7 + 552T^6 + 376T^5 + 164T^4 + 144T^3 + 288T^2 + 192*T + 64
$5$	\( T^{12} + 4 T^{11} + \cdots + 2304 \) T^12 + 4T^11 + 8T^10 - 12T^9 + 140T^8 + 504T^7 + 968T^6 - 1288T^5 + 1828T^4 + 3072T^3 + 4608T^2 - 4608*T + 2304
$7$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$11$	\( T^{12} - 32 T^{9} + \cdots + 541696 \) T^12 - 32T^9 + 740T^8 - 896T^7 + 512T^6 - 896T^5 + 67968T^4 - 89088T^3 + 51200T^2 + 235520*T + 541696
$13$	\( T^{12} - 20 T^{9} + \cdots + 3211264 \) T^12 - 20T^9 + 1724T^8 - 1376T^7 + 200T^6 + 28456T^5 + 297732T^4 + 216576T^3 + 32768T^2 - 458752*T + 3211264
$17$	\( (T^{6} + 4 T^{5} - 44 T^{4} + \cdots - 96)^{2} \) (T^6 + 4T^5 - 44T^4 - 200T^3 + 40T^2 + 288*T - 96)^2
$19$	\( T^{12} + 76 T^{9} + \cdots + 2849344 \) T^12 + 76T^9 + 2444T^8 + 4024T^7 + 2888T^6 - 9016T^5 + 466212T^4 + 565552T^3 + 352800T^2 - 1417920*T + 2849344
$23$	\( T^{12} + 136 T^{10} + \cdots + 10137856 \) T^12 + 136T^10 + 7096T^8 + 175744T^6 + 2044688T^4 + 9219456*T^2 + 10137856
$29$	\( T^{12} - 4 T^{11} + \cdots + 8620096 \) T^12 - 4T^11 + 8T^10 + 216T^9 + 3084T^8 - 928T^7 + 2368T^6 + 68544T^5 + 995888T^4 - 13888T^3 + 21632T^2 + 610688*T + 8620096
$31$	\( (T^{6} + 4 T^{5} - 16 T^{4} + \cdots + 64)^{2} \) (T^6 + 4T^5 - 16T^4 - 72T^3 - 24T^2 + 96*T + 64)^2
$37$	\( T^{12} - 20 T^{11} + \cdots + 5053504 \) T^12 - 20T^11 + 200T^10 - 936T^9 + 3532T^8 - 24992T^7 + 231488T^6 - 1060672T^5 + 2719280T^4 - 2835776T^3 + 332928T^2 + 1834368*T + 5053504
$41$	\( T^{12} + 120 T^{10} + \cdots + 25600 \) T^12 + 120T^10 + 4704T^8 + 68544T^6 + 345920T^4 + 392704*T^2 + 25600
$43$	\( T^{12} + 16 T^{11} + \cdots + 23040000 \) T^12 + 16T^11 + 128T^10 + 768T^9 + 11780T^8 + 158208T^7 + 1318400T^6 + 6805376T^5 + 22906624T^4 + 49373184T^3 + 68585472T^2 + 56217600*T + 23040000
$47$	\( (T^{6} - 8 T^{5} + \cdots + 19776)^{2} \) (T^6 - 8T^5 - 104T^4 + 1032T^3 - 1208T^2 - 8928*T + 19776)^2
$53$	\( T^{12} + 4 T^{11} + \cdots + 78400 \) T^12 + 4T^11 + 8T^10 - 248T^9 + 19372T^8 + 59808T^7 + 115008T^6 - 2979008T^5 + 22600496T^4 - 25960896T^3 + 14709888T^2 + 1518720*T + 78400
$59$	\( T^{12} + \cdots + 119596096 \) T^12 - 16T^11 + 128T^10 - 124T^9 + 1996T^8 - 36840T^7 + 341640T^6 - 218536T^5 - 899932T^4 + 4793520T^3 + 62272800T^2 + 122045760*T + 119596096
$61$	\( T^{12} - 20 T^{11} + \cdots + 256 \) T^12 - 20T^11 + 200T^10 - 172T^9 + 2188T^8 - 34632T^7 + 269832T^6 - 568008T^5 + 339748T^4 + 1728352T^3 + 4476032T^2 - 47872*T + 256
$67$	\( T^{12} + 24 T^{11} + \cdots + 3686400 \) T^12 + 24T^11 + 288T^10 + 1072T^9 + 1124T^8 - 9344T^7 + 26624T^6 + 66560T^5 + 150016T^4 - 1548288T^3 + 4718592T^2 - 5898240*T + 3686400
$71$	\( T^{12} + 432 T^{10} + \cdots + 6553600 \) T^12 + 432T^10 + 60960T^8 + 2931456T^6 + 15382784T^4 + 18743296*T^2 + 6553600
$73$	\( T^{12} + \cdots + 3050573824 \) T^12 + 616T^10 + 134384T^8 + 12430080T^6 + 473632512T^4 + 6278768640*T^2 + 3050573824
$79$	\( (T^{6} - 12 T^{5} + \cdots - 2240)^{2} \) (T^6 - 12T^5 - 44T^4 + 576T^3 - 112T^2 - 3392*T - 2240)^2
$83$	\( T^{12} + \cdots + 320093429824 \) T^12 - 20T^11 + 200T^10 - 1116T^9 + 34732T^8 - 656608T^7 + 6808488T^6 - 33954632T^5 + 208394468T^4 - 2610661520T^3 + 27522333728T^2 - 132738225088*T + 320093429824
$89$	\( T^{12} + \cdots + 35476475904 \) T^12 + 552T^10 + 101680T^8 + 8448256T^6 + 338121472T^4 + 6091352064*T^2 + 35476475904
$97$	\( (T^{6} - 24 T^{5} + \cdots - 39008)^{2} \) (T^6 - 24T^5 - 60T^4 + 2568T^3 + 8168T^2 - 11104*T - 39008)^2
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