LMFDB - Newform orbit 2900.1.cg.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,1,Mod(43,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.43"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 21, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2900.cg (of order $28$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.44728853664$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{28}^{6} q^{2} + ( - \zeta_{28}^{3} + \zeta_{28}) q^{3} + \zeta_{28}^{12} q^{4} + ( - \zeta_{28}^{9} + \zeta_{28}^{7}) q^{6} + ( - \zeta_{28}^{7} + \zeta_{28}^{4}) q^{7} - \zeta_{28}^{4} q^{8} + \cdots + ( - \zeta_{28}^{6} + \zeta_{28}^{3} - 1) q^{98} +O(q^{100}) $ q + z^6 * q^2 + (-z^3 + z) * q^3 + z^12 * q^4 + (-z^9 + z^7) * q^6 + (-z^7 + z^4) * q^7 - z^4 * q^8 + (z^6 - z^4 + z^2) * q^9 + (z^13 + z) * q^12 + (-z^13 + z^10) * q^14 - z^10 * q^16 + (z^12 - z^10 + z^8) * q^18 + (z^10 - z^8 - z^7 + z^5) * q^21 + (-z^7 + z^2) * q^23 + (z^7 - z^5) * q^24 + (-z^9 + z^7 - z^5 + z^3) * q^27 + (z^5 - z^2) * q^28 + z^9 * q^29 + z^2 * q^32 + (-z^4 + z^2 - 1) * q^36 + (-z^4 + z^3) * q^41 + (-z^13 + z^11 - z^2 + 1) * q^42 + (-z^12 + z^4) * q^43 + (-z^13 + z^8) * q^46 + (z^12 - z^8) * q^47 + (z^13 - z^11) * q^48 + (-z^11 + z^8 - 1) * q^49 + (z^13 - z^11 + z^9 + z) * q^54 + (z^11 - z^8) * q^56 - z * q^58 + (z^9 + z^2) * q^61 + (-z^13 + z^11 + z^10 - z^9 - z^8 + z^6) * q^63 + z^8 * q^64 + (-z^11 - z^4) * q^67 + (z^10 - z^8 - z^5 + z^3) * q^69 + (-z^10 + z^8 - z^6) * q^72 + (z^12 - z^10 + z^8 - z^6 + z^4) * q^81 + (-z^10 + z^9) * q^82 + (z^9 - z^8) * q^83 + (-z^8 + z^6 + z^5 - z^3) * q^84 + (z^10 + z^4) * q^86 + (-z^12 + z^10) * q^87 + (z^4 + z) * q^89 + (z^5 - 1) * q^92 + (-z^4 + 1) * q^94 + (-z^5 + z^3) * q^96 + (-z^6 + z^3 - 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 6 q^{9} + 2 q^{14} - 2 q^{16} - 6 q^{18} + 4 q^{21} + 2 q^{23} - 2 q^{28} + 2 q^{32} - 8 q^{36} + 2 q^{41} + 10 q^{42} - 2 q^{46} - 14 q^{49} + 2 q^{56} + 2 q^{61}+ \cdots - 14 q^{98}+O(q^{100}) $ 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 6 * q^9 + 2 * q^14 - 2 * q^16 - 6 * q^18 + 4 * q^21 + 2 * q^23 - 2 * q^28 + 2 * q^32 - 8 * q^36 + 2 * q^41 + 10 * q^42 - 2 * q^46 - 14 * q^49 + 2 * q^56 + 2 * q^61 + 6 * q^63 - 2 * q^64 + 2 * q^67 + 4 * q^69 - 6 * q^72 - 10 * q^81 - 2 * q^82 + 2 * q^83 + 4 * q^84 + 4 * q^87 - 2 * q^89 - 12 * q^92 + 14 * q^94 - 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$-\zeta_{28}^{5}$	$\zeta_{28}^{7}$	$-1$

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} $

43.1

−0.781831	+	0.623490i
−0.433884	−	0.900969i
0.433884	−	0.900969i
−0.781831	−	0.623490i
0.781831	+	0.623490i
−0.433884	+	0.900969i
0.433884	+	0.900969i
0.781831	−	0.623490i
0.974928	−	0.222521i
−0.974928	−	0.222521i
0.974928	+	0.222521i
−0.974928	+	0.222521i

−0.623490

0.781831i

−1.21572

−

0.277479i

−0.222521

−

0.974928i

0.974928

−

0.777479i

−0.900969

0.566116i

0.900969

0.433884i

0.500000

0.240787i

243.1

0.900969

0.433884i

−1.40881

−

1.12349i

0.623490

0.781831i

−0.781831

−

1.62349i

−0.222521

0.0250721i

0.222521

0.974928i

0.500000

2.19064i

543.1

0.900969

−

0.433884i

1.40881

−

1.12349i

0.623490

−

0.781831i

0.781831

−

1.62349i

−0.222521

1.97493i

0.222521

−

0.974928i

0.500000

−

2.19064i

607.1

−0.623490

−

0.781831i

−1.21572

0.277479i

−0.222521

0.974928i

0.974928

0.777479i

−0.900969

−

0.566116i

0.900969

−

0.433884i

0.500000

−

0.240787i

843.1

−0.623490

−

0.781831i

1.21572

−

0.277479i

−0.222521

0.974928i

−0.974928

−

0.777479i

−0.900969

1.43388i

0.900969

−

0.433884i

0.500000

−

0.240787i

907.1

0.900969

−

0.433884i

−1.40881

1.12349i

0.623490

−

0.781831i

−0.781831

1.62349i

−0.222521

−

0.0250721i

0.222521

−

0.974928i

0.500000

−

2.19064i

1207.1

0.900969

0.433884i

1.40881

1.12349i

0.623490

0.781831i

0.781831

1.62349i

−0.222521

−

1.97493i

0.222521

0.974928i

0.500000

2.19064i

1407.1

−0.623490

0.781831i

1.21572

0.277479i

−0.222521

−

0.974928i

−0.974928

0.777479i

−0.900969

−

1.43388i

0.900969

0.433884i

0.500000

0.240787i

1643.1

0.222521

−

0.974928i

0.193096

0.400969i

−0.900969

−

0.433884i

0.433884

−

0.0990311i

0.623490

0.218169i

−0.623490

0.781831i

0.500000

−

0.626980i

1743.1

0.222521

0.974928i

−0.193096

0.400969i

−0.900969

0.433884i

−0.433884

−

0.0990311i

0.623490

1.78183i

−0.623490

−

0.781831i

0.500000

0.626980i

2607.1

0.222521

0.974928i

0.193096

−

0.400969i

−0.900969

0.433884i

0.433884

0.0990311i

0.623490

−

0.218169i

−0.623490

−

0.781831i

0.500000

0.626980i

2707.1

0.222521

−

0.974928i

−0.193096

−

0.400969i

−0.900969

−

0.433884i

−0.433884

0.0990311i

0.623490

−

1.78183i

−0.623490

0.781831i

0.500000

−

0.626980i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
145.o	even	28	1	inner
580.bm	odd	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2900.1.cg.c	yes	12
4.b	odd	2	1		2900.1.cg.a	yes	12
5.b	even	2	1		2900.1.cg.a	yes	12
5.c	odd	4	1		2900.1.bx.a	✓	12
5.c	odd	4	1		2900.1.bx.b	yes	12
20.d	odd	2	1	CM	2900.1.cg.c	yes	12
20.e	even	4	1		2900.1.bx.a	✓	12
20.e	even	4	1		2900.1.bx.b	yes	12
29.f	odd	28	1		2900.1.bx.b	yes	12
116.l	even	28	1		2900.1.bx.a	✓	12
145.o	even	28	1	inner	2900.1.cg.c	yes	12
145.s	odd	28	1		2900.1.bx.a	✓	12
145.t	even	28	1		2900.1.cg.a	yes	12
580.bd	odd	28	1		2900.1.cg.a	yes	12
580.be	even	28	1		2900.1.bx.b	yes	12
580.bm	odd	28	1	inner	2900.1.cg.c	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2900.1.bx.a	✓	12	5.c	odd	4	1
2900.1.bx.a	✓	12	20.e	even	4	1
2900.1.bx.a	✓	12	116.l	even	28	1
2900.1.bx.a	✓	12	145.s	odd	28	1
2900.1.bx.b	yes	12	5.c	odd	4	1
2900.1.bx.b	yes	12	20.e	even	4	1
2900.1.bx.b	yes	12	29.f	odd	28	1
2900.1.bx.b	yes	12	580.be	even	28	1
2900.1.cg.a	yes	12	4.b	odd	2	1
2900.1.cg.a	yes	12	5.b	even	2	1
2900.1.cg.a	yes	12	145.t	even	28	1
2900.1.cg.a	yes	12	580.bd	odd	28	1
2900.1.cg.c	yes	12	1.a	even	1	1	trivial
2900.1.cg.c	yes	12	20.d	odd	2	1	CM
2900.1.cg.c	yes	12	145.o	even	28	1	inner
2900.1.cg.c	yes	12	580.bm	odd	28	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(2900, [\chi])$:

$ T_{3}^{12} - 4T_{3}^{10} + 16T_{3}^{8} - 29T_{3}^{6} + 18T_{3}^{4} + 5T_{3}^{2} + 1 $

T3^12 - 4*T3^10 + 16*T3^8 - 29*T3^6 + 18*T3^4 + 5*T3^2 + 1

$ T_{7}^{12} + 2 T_{7}^{11} + 9 T_{7}^{10} + 14 T_{7}^{9} + 31 T_{7}^{8} + 34 T_{7}^{7} + 41 T_{7}^{6} + \cdots + 1 $

T7^12 + 2*T7^11 + 9*T7^10 + 14*T7^9 + 31*T7^8 + 34*T7^7 + 41*T7^6 + 12*T7^5 - 23*T7^4 - 28*T7^3 + 11*T7^2 + 8*T7 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{12} - 4 T^{10} + \cdots + 1 $ T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 9T^10 + 14T^9 + 31T^8 + 34T^7 + 41T^6 + 12T^5 - 23T^4 - 28T^3 + 11T^2 + 8*T + 1
$11$	$ T^{12} $ T^12
$13$	$ T^{12} $ T^12
$17$	$ T^{12} $ T^12
$19$	$ T^{12} $ T^12
$23$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 9T^10 - 14T^9 + 31T^8 - 34T^7 + 41T^6 - 12T^5 - 23T^4 + 28T^3 + 11T^2 - 8*T + 1
$29$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$31$	$ T^{12} $ T^12
$37$	$ T^{12} $ T^12
$41$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 2T^10 + 17T^8 - 34T^7 + 34T^6 + 2T^5 + 26T^4 - 42T^3 + 32T^2 - 8T + 1
$43$	$ (T^{6} - 7 T^{3} + 7 T + 7)^{2} $ (T^6 - 7T^3 + 7T + 7)^2
$47$	$ (T^{6} - 7 T^{3} + 7 T + 7)^{2} $ (T^6 - 7T^3 + 7T + 7)^2
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ T^{12} - 2 T^{11} + \cdots + 64 $ T^12 - 2T^11 + 2T^10 - 4T^8 + 8T^7 - 8T^6 + 16T^5 - 16T^4 + 32T^2 - 64*T + 64
$67$	$ T^{12} - 2 T^{11} + \cdots + 64 $ T^12 - 2T^11 + 2T^10 - 4T^8 + 8T^7 - 8T^6 + 16T^5 - 16T^4 + 32T^2 - 64*T + 64
$71$	$ T^{12} $ T^12
$73$	$ T^{12} $ T^12
$79$	$ T^{12} $ T^12
$83$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 2T^10 - 4T^8 - 20T^7 + 48T^6 - 54T^5 + 61T^4 - 14T^3 + 4T^2 + 6T + 1
$89$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 - 4T^8 + 20T^7 + 48T^6 + 54T^5 + 61T^4 + 14T^3 + 4T^2 - 6T + 1
$97$	$ T^{12} $ T^12
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2900.cg (of order \(28\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{28}^{6} q^{2} + ( - \zeta_{28}^{3} + \zeta_{28}) q^{3} + \zeta_{28}^{12} q^{4} + ( - \zeta_{28}^{9} + \zeta_{28}^{7}) q^{6} + ( - \zeta_{28}^{7} + \zeta_{28}^{4}) q^{7} - \zeta_{28}^{4} q^{8} + \cdots + ( - \zeta_{28}^{6} + \zeta_{28}^{3} - 1) q^{98} +O(q^{100}) \) q + z^6 * q^2 + (-z^3 + z) * q^3 + z^12 * q^4 + (-z^9 + z^7) * q^6 + (-z^7 + z^4) * q^7 - z^4 * q^8 + (z^6 - z^4 + z^2) * q^9 + (z^13 + z) * q^12 + (-z^13 + z^10) * q^14 - z^10 * q^16 + (z^12 - z^10 + z^8) * q^18 + (z^10 - z^8 - z^7 + z^5) * q^21 + (-z^7 + z^2) * q^23 + (z^7 - z^5) * q^24 + (-z^9 + z^7 - z^5 + z^3) * q^27 + (z^5 - z^2) * q^28 + z^9 * q^29 + z^2 * q^32 + (-z^4 + z^2 - 1) * q^36 + (-z^4 + z^3) * q^41 + (-z^13 + z^11 - z^2 + 1) * q^42 + (-z^12 + z^4) * q^43 + (-z^13 + z^8) * q^46 + (z^12 - z^8) * q^47 + (z^13 - z^11) * q^48 + (-z^11 + z^8 - 1) * q^49 + (z^13 - z^11 + z^9 + z) * q^54 + (z^11 - z^8) * q^56 - z * q^58 + (z^9 + z^2) * q^61 + (-z^13 + z^11 + z^10 - z^9 - z^8 + z^6) * q^63 + z^8 * q^64 + (-z^11 - z^4) * q^67 + (z^10 - z^8 - z^5 + z^3) * q^69 + (-z^10 + z^8 - z^6) * q^72 + (z^12 - z^10 + z^8 - z^6 + z^4) * q^81 + (-z^10 + z^9) * q^82 + (z^9 - z^8) * q^83 + (-z^8 + z^6 + z^5 - z^3) * q^84 + (z^10 + z^4) * q^86 + (-z^12 + z^10) * q^87 + (z^4 + z) * q^89 + (z^5 - 1) * q^92 + (-z^4 + 1) * q^94 + (-z^5 + z^3) * q^96 + (-z^6 + z^3 - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 6 q^{9} + 2 q^{14} - 2 q^{16} - 6 q^{18} + 4 q^{21} + 2 q^{23} - 2 q^{28} + 2 q^{32} - 8 q^{36} + 2 q^{41} + 10 q^{42} - 2 q^{46} - 14 q^{49} + 2 q^{56} + 2 q^{61}+ \cdots - 14 q^{98}+O(q^{100}) \) 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 6 * q^9 + 2 * q^14 - 2 * q^16 - 6 * q^18 + 4 * q^21 + 2 * q^23 - 2 * q^28 + 2 * q^32 - 8 * q^36 + 2 * q^41 + 10 * q^42 - 2 * q^46 - 14 * q^49 + 2 * q^56 + 2 * q^61 + 6 * q^63 - 2 * q^64 + 2 * q^67 + 4 * q^69 - 6 * q^72 - 10 * q^81 - 2 * q^82 + 2 * q^83 + 4 * q^84 + 4 * q^87 - 2 * q^89 - 12 * q^92 + 14 * q^94 - 14 * q^98

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	2900.1.cg.c

Level	$2900$
Weight	$1$
Character orbit	2900.cg
Analytic conductor	$1.447$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{28}$
CM discriminant	-20
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.44728853664\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{28}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

\(n\)	\(901\)	\(1277\)	\(1451\)
\(\chi(n)\)	\(-\zeta_{28}^{5}\)	\(\zeta_{28}^{7}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{12} - 4 T^{10} + \cdots + 1 \) T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$5$	\( T^{12} \) T^12
$7$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 9T^10 + 14T^9 + 31T^8 + 34T^7 + 41T^6 + 12T^5 - 23T^4 - 28T^3 + 11T^2 + 8*T + 1
$11$	\( T^{12} \) T^12
$13$	\( T^{12} \) T^12
$17$	\( T^{12} \) T^12
$19$	\( T^{12} \) T^12
$23$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 9T^10 - 14T^9 + 31T^8 - 34T^7 + 41T^6 - 12T^5 - 23T^4 + 28T^3 + 11T^2 - 8*T + 1
$29$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$31$	\( T^{12} \) T^12
$37$	\( T^{12} \) T^12
$41$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 2T^10 + 17T^8 - 34T^7 + 34T^6 + 2T^5 + 26T^4 - 42T^3 + 32T^2 - 8T + 1
$43$	\( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) (T^6 - 7T^3 + 7T + 7)^2
$47$	\( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) (T^6 - 7T^3 + 7T + 7)^2
$53$	\( T^{12} \) T^12
$59$	\( T^{12} \) T^12
$61$	\( T^{12} - 2 T^{11} + \cdots + 64 \) T^12 - 2T^11 + 2T^10 - 4T^8 + 8T^7 - 8T^6 + 16T^5 - 16T^4 + 32T^2 - 64*T + 64
$67$	\( T^{12} - 2 T^{11} + \cdots + 64 \) T^12 - 2T^11 + 2T^10 - 4T^8 + 8T^7 - 8T^6 + 16T^5 - 16T^4 + 32T^2 - 64*T + 64
$71$	\( T^{12} \) T^12
$73$	\( T^{12} \) T^12
$79$	\( T^{12} \) T^12
$83$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 2T^10 - 4T^8 - 20T^7 + 48T^6 - 54T^5 + 61T^4 - 14T^3 + 4T^2 + 6T + 1
$89$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 - 4T^8 + 20T^7 + 48T^6 + 54T^5 + 61T^4 + 14T^3 + 4T^2 - 6T + 1
$97$	\( T^{12} \) T^12
show more
show less