LMFDB - Newform orbit 1053.2.e.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1053,2,Mod(352,1053)] 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("1053.352"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,-2,0,-2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (2 \beta_{3} + 2 \beta_1) q^{5} - 2 \beta_1 q^{7} + (\beta_{3} + 3) q^{8} + ( - 2 \beta_{3} - 4) q^{10} + ( - 2 \beta_{2} - 2) q^{11}+ \cdots + (\beta_{3} + 1) q^{98}+O(q^{100}) $ q + (-b2 + b1 - 1) * q^2 + (-2b3 + b2 - 2b1) * q^4 + (2b3 + 2b1) * q^5 - 2b1 q^7 + (b3 + 3) * q^8 + (-2b3 - 4) q^10 + (-2b2 - 2) q^11 - b2 * q^13 + (2b3 - 4b2 + 2b1) q^14 + (-3b2 - 3) q^16 + (4b3 - 2) q^17 + 2b3 q^19 + (8b2 - 2b1 + 8) * q^20 + (-2b3 + 2b2 - 2b1) q^22 + 4b2 q^23 + (-3b2 - 3) q^25 + (-b3 - 1) * q^26 + (-2b3 - 8) q^28 + (2b2 + 2) q^29 + (2b3 - 4b2 + 2b1) q^31 + (-b3 - 3b2 - b1) q^32 + (-6b2 + 2b1 - 6) * q^34 + 8 * q^35 + (4b3 - 2) q^37 + (-4b2 + 2b1 - 4) * q^38 + (6b3 - 4b2 + 6b1) q^40 + (2b3 - 8b2 + 2b1) q^41 + (-4b2 + 4b1 - 4) * q^43 + (4b3 + 2) q^44 + (4b3 + 4) q^46 + (-6b2 - 4b1 - 6) * q^47 + b2 * q^49 + (-3b3 + 3b2 - 3b1) q^50 + (b2 - 2b1 + 1) q^52 + 2 * q^53 - 4b3 q^55 + (4b2 - 6b1 + 4) * q^56 + (2b3 - 2b2 + 2b1) q^58 + (-4b3 - 2b2 - 4b1) q^59 + (-2b2 - 8b1 - 2) * q^61 + (-6b3 - 8) q^62 + (-2b3 - 7) q^64 + 2b1 q^65 + (2b3 + 4b2 + 2b1) q^67 + 14b2 q^68 + (-8b2 + 8b1 - 8) * q^70 - 2 * q^71 + (4b3 + 6) q^73 + (-6b2 + 2b1 - 6) * q^74 + (-2b3 + 8b2 - 2b1) q^76 + (4b3 + 4b1) * q^77 + 8b1 q^79 - 6b3 q^80 + (-10b3 - 12) q^82 + (-2b2 + 4b1 - 2) * q^83 + (-4b3 - 16b2 - 4b1) q^85 + (-8b3 + 12b2 - 8b1) q^86 + (-6b2 + 2b1 - 6) * q^88 + (2b3 - 12) q^89 + 2b3 q^91 + (-4b2 + 8b1 - 4) * q^92 + (-2b3 - 2b2 - 2b1) q^94 - 8b2 q^95 + (2b2 - 4b1 + 2) * q^97 + (b3 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} + 12 q^{8} - 16 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} - 6 q^{16} - 8 q^{17} + 16 q^{20} - 4 q^{22} - 8 q^{23} - 6 q^{25} - 4 q^{26} - 32 q^{28} + 4 q^{29} + 8 q^{31} + 6 q^{32}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 + 12 * q^8 - 16 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 - 6 * q^16 - 8 * q^17 + 16 * q^20 - 4 * q^22 - 8 * q^23 - 6 * q^25 - 4 * q^26 - 32 * q^28 + 4 * q^29 + 8 * q^31 + 6 * q^32 - 12 * q^34 + 32 * q^35 - 8 * q^37 - 8 * q^38 + 8 * q^40 + 16 * q^41 - 8 * q^43 + 8 * q^44 + 16 * q^46 - 12 * q^47 - 2 * q^49 - 6 * q^50 + 2 * q^52 + 8 * q^53 + 8 * q^56 + 4 * q^58 + 4 * q^59 - 4 * q^61 - 32 * q^62 - 28 * q^64 - 8 * q^67 - 28 * q^68 - 16 * q^70 - 8 * q^71 + 24 * q^73 - 12 * q^74 - 16 * q^76 - 48 * q^82 - 4 * q^83 + 32 * q^85 - 24 * q^86 - 12 * q^88 - 48 * q^89 - 8 * q^92 + 4 * q^94 + 16 * q^95 + 4 * q^97 + 4 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$\beta_{2}$	$1$

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} $

352.1

−0.707107	−	1.22474i
0.707107	+	1.22474i
−0.707107	+	1.22474i
0.707107	−	1.22474i

−1.20711

−

2.09077i

−1.91421

3.31552i

1.41421

−

2.44949i

1.41421

2.44949i

4.41421

−6.82843

352.2

0.207107

0.358719i

0.914214

−

1.58346i

−1.41421

2.44949i

−1.41421

−

2.44949i

1.58579

−1.17157

703.1

−1.20711

2.09077i

−1.91421

−

3.31552i

1.41421

2.44949i

1.41421

−

2.44949i

4.41421

−6.82843

703.2

0.207107

−

0.358719i

0.914214

1.58346i

−1.41421

−

2.44949i

−1.41421

2.44949i

1.58579

−1.17157

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1053.2.e.e		4
3.b	odd	2	1		1053.2.e.m		4
9.c	even	3	1		117.2.a.c		2
9.c	even	3	1	inner	1053.2.e.e		4
9.d	odd	6	1		39.2.a.b	✓	2
9.d	odd	6	1		1053.2.e.m		4
36.f	odd	6	1		1872.2.a.w		2
36.h	even	6	1		624.2.a.k		2
45.h	odd	6	1		975.2.a.l		2
45.j	even	6	1		2925.2.a.v		2
45.k	odd	12	2		2925.2.c.u		4
45.l	even	12	2		975.2.c.h		4
63.l	odd	6	1		5733.2.a.u		2
63.o	even	6	1		1911.2.a.h		2
72.j	odd	6	1		2496.2.a.bf		2
72.l	even	6	1		2496.2.a.bi		2
72.n	even	6	1		7488.2.a.cl		2
72.p	odd	6	1		7488.2.a.co		2
99.g	even	6	1		4719.2.a.p		2
117.k	odd	6	1		507.2.e.h		4
117.m	odd	6	1		507.2.e.d		4
117.n	odd	6	1		507.2.a.h		2
117.t	even	6	1		1521.2.a.f		2
117.u	odd	6	1		507.2.e.h		4
117.v	odd	6	1		507.2.e.d		4
117.x	even	12	2		507.2.j.f		8
117.y	odd	12	2		1521.2.b.j		4
117.z	even	12	2		507.2.b.e		4
117.bc	even	12	2		507.2.j.f		8
468.x	even	6	1		8112.2.a.bm		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.a.b	✓	2	9.d	odd	6	1
117.2.a.c		2	9.c	even	3	1
507.2.a.h		2	117.n	odd	6	1
507.2.b.e		4	117.z	even	12	2
507.2.e.d		4	117.m	odd	6	1
507.2.e.d		4	117.v	odd	6	1
507.2.e.h		4	117.k	odd	6	1
507.2.e.h		4	117.u	odd	6	1
507.2.j.f		8	117.x	even	12	2
507.2.j.f		8	117.bc	even	12	2
624.2.a.k		2	36.h	even	6	1
975.2.a.l		2	45.h	odd	6	1
975.2.c.h		4	45.l	even	12	2
1053.2.e.e		4	1.a	even	1	1	trivial
1053.2.e.e		4	9.c	even	3	1	inner
1053.2.e.m		4	3.b	odd	2	1
1053.2.e.m		4	9.d	odd	6	1
1521.2.a.f		2	117.t	even	6	1
1521.2.b.j		4	117.y	odd	12	2
1872.2.a.w		2	36.f	odd	6	1
1911.2.a.h		2	63.o	even	6	1
2496.2.a.bf		2	72.j	odd	6	1
2496.2.a.bi		2	72.l	even	6	1
2925.2.a.v		2	45.j	even	6	1
2925.2.c.u		4	45.k	odd	12	2
4719.2.a.p		2	99.g	even	6	1
5733.2.a.u		2	63.l	odd	6	1
7488.2.a.cl		2	72.n	even	6	1
7488.2.a.co		2	72.p	odd	6	1
8112.2.a.bm		2	468.x	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1053, [\chi])$:

$ T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 $

T2^4 + 2*T2^3 + 5*T2^2 - 2*T2 + 1

$ T_{5}^{4} + 8T_{5}^{2} + 64 $

T5^4 + 8*T5^2 + 64

$ T_{7}^{4} + 8T_{7}^{2} + 64 $

T7^4 + 8*T7^2 + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 5T^2 - 2*T + 1
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 8T^{2} + 64 $ T^4 + 8*T^2 + 64
$7$	$ T^{4} + 8T^{2} + 64 $ T^4 + 8*T^2 + 64
$11$	$ (T^{2} + 2 T + 4)^{2} $ (T^2 + 2*T + 4)^2
$13$	$ (T^{2} - T + 1)^{2} $ (T^2 - T + 1)^2
$17$	$ (T^{2} + 4 T - 28)^{2} $ (T^2 + 4*T - 28)^2
$19$	$ (T^{2} - 8)^{2} $ (T^2 - 8)^2
$23$	$ (T^{2} + 4 T + 16)^{2} $ (T^2 + 4*T + 16)^2
$29$	$ (T^{2} - 2 T + 4)^{2} $ (T^2 - 2*T + 4)^2
$31$	$ T^{4} - 8 T^{3} + \cdots + 64 $ T^4 - 8T^3 + 56T^2 - 64*T + 64
$37$	$ (T^{2} + 4 T - 28)^{2} $ (T^2 + 4*T - 28)^2
$41$	$ T^{4} - 16 T^{3} + \cdots + 3136 $ T^4 - 16T^3 + 200T^2 - 896*T + 3136
$43$	$ T^{4} + 8 T^{3} + \cdots + 256 $ T^4 + 8T^3 + 80T^2 - 128*T + 256
$47$	$ T^{4} + 12 T^{3} + \cdots + 16 $ T^4 + 12T^3 + 140T^2 + 48*T + 16
$53$	$ (T - 2)^{4} $ (T - 2)^4
$59$	$ T^{4} - 4 T^{3} + \cdots + 784 $ T^4 - 4T^3 + 44T^2 + 112*T + 784
$61$	$ T^{4} + 4 T^{3} + \cdots + 15376 $ T^4 + 4T^3 + 140T^2 - 496*T + 15376
$67$	$ T^{4} + 8 T^{3} + \cdots + 64 $ T^4 + 8T^3 + 56T^2 + 64*T + 64
$71$	$ (T + 2)^{4} $ (T + 2)^4
$73$	$ (T^{2} - 12 T + 4)^{2} $ (T^2 - 12*T + 4)^2
$79$	$ T^{4} + 128 T^{2} + 16384 $ T^4 + 128*T^2 + 16384
$83$	$ T^{4} + 4 T^{3} + \cdots + 784 $ T^4 + 4T^3 + 44T^2 - 112*T + 784
$89$	$ (T^{2} + 24 T + 136)^{2} $ (T^2 + 24*T + 136)^2
$97$	$ T^{4} - 4 T^{3} + \cdots + 784 $ T^4 - 4T^3 + 44T^2 + 112*T + 784
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Level:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (2 \beta_{3} + 2 \beta_1) q^{5} - 2 \beta_1 q^{7} + (\beta_{3} + 3) q^{8} + ( - 2 \beta_{3} - 4) q^{10} + ( - 2 \beta_{2} - 2) q^{11}+ \cdots + (\beta_{3} + 1) q^{98}+O(q^{100}) \) q + (-b2 + b1 - 1) * q^2 + (-2b3 + b2 - 2b1) * q^4 + (2b3 + 2b1) * q^5 - 2b1 q^7 + (b3 + 3) * q^8 + (-2b3 - 4) q^10 + (-2b2 - 2) q^11 - b2 * q^13 + (2b3 - 4b2 + 2b1) q^14 + (-3b2 - 3) q^16 + (4b3 - 2) q^17 + 2b3 q^19 + (8b2 - 2b1 + 8) * q^20 + (-2b3 + 2b2 - 2b1) q^22 + 4b2 q^23 + (-3b2 - 3) q^25 + (-b3 - 1) * q^26 + (-2b3 - 8) q^28 + (2b2 + 2) q^29 + (2b3 - 4b2 + 2b1) q^31 + (-b3 - 3b2 - b1) q^32 + (-6b2 + 2b1 - 6) * q^34 + 8 * q^35 + (4b3 - 2) q^37 + (-4b2 + 2b1 - 4) * q^38 + (6b3 - 4b2 + 6b1) q^40 + (2b3 - 8b2 + 2b1) q^41 + (-4b2 + 4b1 - 4) * q^43 + (4b3 + 2) q^44 + (4b3 + 4) q^46 + (-6b2 - 4b1 - 6) * q^47 + b2 * q^49 + (-3b3 + 3b2 - 3b1) q^50 + (b2 - 2b1 + 1) q^52 + 2 * q^53 - 4b3 q^55 + (4b2 - 6b1 + 4) * q^56 + (2b3 - 2b2 + 2b1) q^58 + (-4b3 - 2b2 - 4b1) q^59 + (-2b2 - 8b1 - 2) * q^61 + (-6b3 - 8) q^62 + (-2b3 - 7) q^64 + 2b1 q^65 + (2b3 + 4b2 + 2b1) q^67 + 14b2 q^68 + (-8b2 + 8b1 - 8) * q^70 - 2 * q^71 + (4b3 + 6) q^73 + (-6b2 + 2b1 - 6) * q^74 + (-2b3 + 8b2 - 2b1) q^76 + (4b3 + 4b1) * q^77 + 8b1 q^79 - 6b3 q^80 + (-10b3 - 12) q^82 + (-2b2 + 4b1 - 2) * q^83 + (-4b3 - 16b2 - 4b1) q^85 + (-8b3 + 12b2 - 8b1) q^86 + (-6b2 + 2b1 - 6) * q^88 + (2b3 - 12) q^89 + 2b3 q^91 + (-4b2 + 8b1 - 4) * q^92 + (-2b3 - 2b2 - 2b1) q^94 - 8b2 q^95 + (2b2 - 4b1 + 2) * q^97 + (b3 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 12 q^{8} - 16 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} - 6 q^{16} - 8 q^{17} + 16 q^{20} - 4 q^{22} - 8 q^{23} - 6 q^{25} - 4 q^{26} - 32 q^{28} + 4 q^{29} + 8 q^{31} + 6 q^{32}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 + 12 * q^8 - 16 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 - 6 * q^16 - 8 * q^17 + 16 * q^20 - 4 * q^22 - 8 * q^23 - 6 * q^25 - 4 * q^26 - 32 * q^28 + 4 * q^29 + 8 * q^31 + 6 * q^32 - 12 * q^34 + 32 * q^35 - 8 * q^37 - 8 * q^38 + 8 * q^40 + 16 * q^41 - 8 * q^43 + 8 * q^44 + 16 * q^46 - 12 * q^47 - 2 * q^49 - 6 * q^50 + 2 * q^52 + 8 * q^53 + 8 * q^56 + 4 * q^58 + 4 * q^59 - 4 * q^61 - 32 * q^62 - 28 * q^64 - 8 * q^67 - 28 * q^68 - 16 * q^70 - 8 * q^71 + 24 * q^73 - 12 * q^74 - 16 * q^76 - 48 * q^82 - 4 * q^83 + 32 * q^85 - 24 * q^86 - 12 * q^88 - 48 * q^89 - 8 * q^92 + 4 * q^94 + 16 * q^95 + 4 * q^97 + 4 * q^98

Introduction

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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	1053.2.e.e

Level	$1053$
Weight	$2$
Character orbit	1053.e
Analytic conductor	$8.408$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.40824733284\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \) x^4 + 2*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{3} \) 2*b3

$p$	$F_p(T)$
$2$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 5T^2 - 2*T + 1
$3$	\( T^{4} \) T^4
$5$	\( T^{4} + 8T^{2} + 64 \) T^4 + 8*T^2 + 64
$7$	\( T^{4} + 8T^{2} + 64 \) T^4 + 8*T^2 + 64
$11$	\( (T^{2} + 2 T + 4)^{2} \) (T^2 + 2*T + 4)^2
$13$	\( (T^{2} - T + 1)^{2} \) (T^2 - T + 1)^2
$17$	\( (T^{2} + 4 T - 28)^{2} \) (T^2 + 4*T - 28)^2
$19$	\( (T^{2} - 8)^{2} \) (T^2 - 8)^2
$23$	\( (T^{2} + 4 T + 16)^{2} \) (T^2 + 4*T + 16)^2
$29$	\( (T^{2} - 2 T + 4)^{2} \) (T^2 - 2*T + 4)^2
$31$	\( T^{4} - 8 T^{3} + \cdots + 64 \) T^4 - 8T^3 + 56T^2 - 64*T + 64
$37$	\( (T^{2} + 4 T - 28)^{2} \) (T^2 + 4*T - 28)^2
$41$	\( T^{4} - 16 T^{3} + \cdots + 3136 \) T^4 - 16T^3 + 200T^2 - 896*T + 3136
$43$	\( T^{4} + 8 T^{3} + \cdots + 256 \) T^4 + 8T^3 + 80T^2 - 128*T + 256
$47$	\( T^{4} + 12 T^{3} + \cdots + 16 \) T^4 + 12T^3 + 140T^2 + 48*T + 16
$53$	\( (T - 2)^{4} \) (T - 2)^4
$59$	\( T^{4} - 4 T^{3} + \cdots + 784 \) T^4 - 4T^3 + 44T^2 + 112*T + 784
$61$	\( T^{4} + 4 T^{3} + \cdots + 15376 \) T^4 + 4T^3 + 140T^2 - 496*T + 15376
$67$	\( T^{4} + 8 T^{3} + \cdots + 64 \) T^4 + 8T^3 + 56T^2 + 64*T + 64
$71$	\( (T + 2)^{4} \) (T + 2)^4
$73$	\( (T^{2} - 12 T + 4)^{2} \) (T^2 - 12*T + 4)^2
$79$	\( T^{4} + 128 T^{2} + 16384 \) T^4 + 128*T^2 + 16384
$83$	\( T^{4} + 4 T^{3} + \cdots + 784 \) T^4 + 4T^3 + 44T^2 - 112*T + 784
$89$	\( (T^{2} + 24 T + 136)^{2} \) (T^2 + 24*T + 136)^2
$97$	\( T^{4} - 4 T^{3} + \cdots + 784 \) T^4 - 4T^3 + 44T^2 + 112*T + 784
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\(n\)	\(326\)	\(730\)
\(\chi(n)\)	\(\beta_{2}\)	\(1\)

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