LMFDB - Newform orbit 1575.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1574,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1574");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.g (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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{2} + ( - \beta_{4} + \beta_1) q^{7} - 2 \beta_{6} q^{8}+O(q^{10}) $ q + b6 * q^2 + (-b4 + b1) * q^7 - 2b6 q^8	\( q + \beta_{6} q^{2} + ( - \beta_{4} + \beta_1) q^{7} - 2 \beta_{6} q^{8} + \beta_{3} q^{11} + (2 \beta_{4} - \beta_1) q^{13} - \beta_{5} q^{14} - 4 q^{16} + ( - 2 \beta_{7} + \beta_{6}) q^{17} + ( - 2 \beta_{2} + 1) q^{19} + 2 \beta_1 q^{22} + 5 \beta_{6} q^{23} + (2 \beta_{5} + \beta_{3}) q^{26} - \beta_{3} q^{29} + ( - 2 \beta_{2} + 1) q^{31} + ( - 4 \beta_{2} + 2) q^{34} - 8 \beta_1 q^{37} + ( - 2 \beta_{7} + \beta_{6}) q^{38} + 5 \beta_1 q^{43} + 10 q^{46} + ( - 2 \beta_{7} + \beta_{6}) q^{47} + ( - \beta_{2} + 7) q^{49} - 4 \beta_{6} q^{53} + 2 \beta_{5} q^{56} - 2 \beta_1 q^{58} + ( - 2 \beta_{5} - \beta_{3}) q^{59} + (2 \beta_{2} - 1) q^{61} + ( - 2 \beta_{7} + \beta_{6}) q^{62} + 8 q^{64} + 7 \beta_1 q^{67} - 2 \beta_{3} q^{71} + (4 \beta_{4} - 2 \beta_1) q^{73} - 8 \beta_{3} q^{74} - \beta_{7} q^{77} - 14 q^{79} + ( - 2 \beta_{7} + \beta_{6}) q^{83} + 5 \beta_{3} q^{86} - 4 \beta_1 q^{88} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{89} + (\beta_{2} - 14) q^{91} + ( - 4 \beta_{2} + 2) q^{94} + (2 \beta_{4} - \beta_1) q^{97} + ( - \beta_{7} + 7 \beta_{6}) q^{98}+O(q^{100}) \) q + b6 * q^2 + (-b4 + b1) * q^7 - 2b6 q^8 + b3 * q^11 + (2b4 - b1) q^13 - b5 * q^14 - 4 * q^16 + (-2b7 + b6) q^17 + (-2b2 + 1) q^19 + 2b1 q^22 + 5b6 q^23 + (2b5 + b3) q^26 - b3 * q^29 + (-2b2 + 1) q^31 + (-4b2 + 2) q^34 - 8b1 q^37 + (-2b7 + b6) q^38 + 5b1 q^43 + 10 * q^46 + (-2b7 + b6) q^47 + (-b2 + 7) * q^49 - 4b6 q^53 + 2b5 q^56 - 2b1 q^58 + (-2b5 - b3) q^59 + (2b2 - 1) q^61 + (-2b7 + b6) q^62 + 8 * q^64 + 7b1 q^67 - 2b3 q^71 + (4b4 - 2b1) * q^73 - 8b3 q^74 - b7 * q^77 - 14 * q^79 + (-2b7 + b6) q^83 + 5b3 q^86 - 4b1 q^88 + (-4b5 - 2b3) * q^89 + (b2 - 14) * q^91 + (-4b2 + 2) q^94 + (2b4 - b1) q^97 + (-b7 + 7b6) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{16} + 80 q^{46} + 52 q^{49} + 64 q^{64} - 112 q^{79} - 108 q^{91}+O(q^{100}) $ 8 * q - 32 * q^16 + 80 * q^46 + 52 * q^49 + 64 * q^64 - 112 * q^79 - 108 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{24}^{6} $ v^6
$\beta_{2}$	$=$	$ 3\zeta_{24}^{4} - 1 $ 3*v^4 - 1
$\beta_{3}$	$=$	$ \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} $ v^5 + v^3 - v
$\beta_{4}$	$=$	$ -\zeta_{24}^{6} + 3\zeta_{24}^{2} $ -v^6 + 3*v^2
$\beta_{5}$	$=$	$ -3\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} $ -3v^7 + v^5 + v^3 + 2v
$\beta_{6}$	$=$	$ -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} $ -v^5 + v^3 + v
$\beta_{7}$	$=$	$ 3\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24} $ 3v^7 + v^5 - v^3 + 2v

Display $\zeta_{24}^j$ in terms of $\beta_i$

$\zeta_{24}$	$=$	$ ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} ) / 6 $ (b7 + b6 + b5 - b3) / 6
$\zeta_{24}^{2}$	$=$	$ ( \beta_{4} + \beta_1 ) / 3 $ (b4 + b1) / 3
$\zeta_{24}^{3}$	$=$	$ ( \beta_{6} + \beta_{3} ) / 2 $ (b6 + b3) / 2
$\zeta_{24}^{4}$	$=$	$ ( \beta_{2} + 1 ) / 3 $ (b2 + 1) / 3
$\zeta_{24}^{5}$	$=$	$ ( \beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{3} ) / 6 $ (b7 - 2b6 + b5 + 2b3) / 6
$\zeta_{24}^{6}$	$=$	$ \beta_1 $ b1
$\zeta_{24}^{7}$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} ) / 6 $ (b7 + b6 - b5 + b3) / 6

Display $\beta_i$ in terms of $\zeta_{24}^j$

Character values

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

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

1574.1

−0.965926	+	0.258819i
−0.965926	−	0.258819i
0.258819	−	0.965926i
0.258819	+	0.965926i
0.965926	−	0.258819i
0.965926	+	0.258819i
−0.258819	+	0.965926i
−0.258819	−	0.965926i

−1.41421

−2.59808

−

0.500000i

2.82843

1574.2

−1.41421

−2.59808

0.500000i

2.82843

1574.3

−1.41421

2.59808

−

0.500000i

2.82843

1574.4

−1.41421

2.59808

0.500000i

2.82843

1574.5

1.41421

−2.59808

−

0.500000i

−2.82843

1574.6

1.41421

−2.59808

0.500000i

−2.82843

1574.7

1.41421

2.59808

−

0.500000i

−2.82843

1574.8

1.41421

2.59808

0.500000i

−2.82843

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
35.c	odd	2	1	inner
105.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.2.g.b		8
3.b	odd	2	1	inner	1575.2.g.b		8
5.b	even	2	1	inner	1575.2.g.b		8
5.c	odd	4	1		1575.2.b.d	✓	4
5.c	odd	4	1		1575.2.b.e	yes	4
7.b	odd	2	1	inner	1575.2.g.b		8
15.d	odd	2	1	inner	1575.2.g.b		8
15.e	even	4	1		1575.2.b.d	✓	4
15.e	even	4	1		1575.2.b.e	yes	4
21.c	even	2	1	inner	1575.2.g.b		8
35.c	odd	2	1	inner	1575.2.g.b		8
35.f	even	4	1		1575.2.b.d	✓	4
35.f	even	4	1		1575.2.b.e	yes	4
105.g	even	2	1	inner	1575.2.g.b		8
105.k	odd	4	1		1575.2.b.d	✓	4
105.k	odd	4	1		1575.2.b.e	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1575.2.b.d	✓	4	5.c	odd	4	1
1575.2.b.d	✓	4	15.e	even	4	1
1575.2.b.d	✓	4	35.f	even	4	1
1575.2.b.d	✓	4	105.k	odd	4	1
1575.2.b.e	yes	4	5.c	odd	4	1
1575.2.b.e	yes	4	15.e	even	4	1
1575.2.b.e	yes	4	35.f	even	4	1
1575.2.b.e	yes	4	105.k	odd	4	1
1575.2.g.b		8	1.a	even	1	1	trivial
1575.2.g.b		8	3.b	odd	2	1	inner
1575.2.g.b		8	5.b	even	2	1	inner
1575.2.g.b		8	7.b	odd	2	1	inner
1575.2.g.b		8	15.d	odd	2	1	inner
1575.2.g.b		8	21.c	even	2	1	inner
1575.2.g.b		8	35.c	odd	2	1	inner
1575.2.g.b		8	105.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1575, [\chi])$:

$ T_{2}^{2} - 2 $

$ T_{59}^{2} - 54 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2)^{4} $ (T^2 - 2)^4
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} - 13 T^{2} + 49)^{2} $ (T^4 - 13*T^2 + 49)^2
$11$	$ (T^{2} + 2)^{4} $ (T^2 + 2)^4
$13$	$ (T^{2} - 27)^{4} $ (T^2 - 27)^4
$17$	$ (T^{2} + 54)^{4} $ (T^2 + 54)^4
$19$	$ (T^{2} + 27)^{4} $ (T^2 + 27)^4
$23$	$ (T^{2} - 50)^{4} $ (T^2 - 50)^4
$29$	$ (T^{2} + 2)^{4} $ (T^2 + 2)^4
$31$	$ (T^{2} + 27)^{4} $ (T^2 + 27)^4
$37$	$ (T^{2} + 64)^{4} $ (T^2 + 64)^4
$41$	$ T^{8} $ T^8
$43$	$ (T^{2} + 25)^{4} $ (T^2 + 25)^4
$47$	$ (T^{2} + 54)^{4} $ (T^2 + 54)^4
$53$	$ (T^{2} - 32)^{4} $ (T^2 - 32)^4
$59$	$ (T^{2} - 54)^{4} $ (T^2 - 54)^4
$61$	$ (T^{2} + 27)^{4} $ (T^2 + 27)^4
$67$	$ (T^{2} + 49)^{4} $ (T^2 + 49)^4
$71$	$ (T^{2} + 8)^{4} $ (T^2 + 8)^4
$73$	$ (T^{2} - 108)^{4} $ (T^2 - 108)^4
$79$	$ (T + 14)^{8} $ (T + 14)^8
$83$	$ (T^{2} + 54)^{4} $ (T^2 + 54)^4
$89$	$ (T^{2} - 216)^{4} $ (T^2 - 216)^4
$97$	$ (T^{2} - 27)^{4} $ (T^2 - 27)^4
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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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + ( - \beta_{4} + \beta_1) q^{7} - 2 \beta_{6} q^{8}+O(q^{10}) \) q + b6 * q^2 + (-b4 + b1) * q^7 - 2b6 q^8	\( q + \beta_{6} q^{2} + ( - \beta_{4} + \beta_1) q^{7} - 2 \beta_{6} q^{8} + \beta_{3} q^{11} + (2 \beta_{4} - \beta_1) q^{13} - \beta_{5} q^{14} - 4 q^{16} + ( - 2 \beta_{7} + \beta_{6}) q^{17} + ( - 2 \beta_{2} + 1) q^{19} + 2 \beta_1 q^{22} + 5 \beta_{6} q^{23} + (2 \beta_{5} + \beta_{3}) q^{26} - \beta_{3} q^{29} + ( - 2 \beta_{2} + 1) q^{31} + ( - 4 \beta_{2} + 2) q^{34} - 8 \beta_1 q^{37} + ( - 2 \beta_{7} + \beta_{6}) q^{38} + 5 \beta_1 q^{43} + 10 q^{46} + ( - 2 \beta_{7} + \beta_{6}) q^{47} + ( - \beta_{2} + 7) q^{49} - 4 \beta_{6} q^{53} + 2 \beta_{5} q^{56} - 2 \beta_1 q^{58} + ( - 2 \beta_{5} - \beta_{3}) q^{59} + (2 \beta_{2} - 1) q^{61} + ( - 2 \beta_{7} + \beta_{6}) q^{62} + 8 q^{64} + 7 \beta_1 q^{67} - 2 \beta_{3} q^{71} + (4 \beta_{4} - 2 \beta_1) q^{73} - 8 \beta_{3} q^{74} - \beta_{7} q^{77} - 14 q^{79} + ( - 2 \beta_{7} + \beta_{6}) q^{83} + 5 \beta_{3} q^{86} - 4 \beta_1 q^{88} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{89} + (\beta_{2} - 14) q^{91} + ( - 4 \beta_{2} + 2) q^{94} + (2 \beta_{4} - \beta_1) q^{97} + ( - \beta_{7} + 7 \beta_{6}) q^{98}+O(q^{100}) \) q + b6 * q^2 + (-b4 + b1) * q^7 - 2b6 q^8 + b3 * q^11 + (2b4 - b1) q^13 - b5 * q^14 - 4 * q^16 + (-2b7 + b6) q^17 + (-2b2 + 1) q^19 + 2b1 q^22 + 5b6 q^23 + (2b5 + b3) q^26 - b3 * q^29 + (-2b2 + 1) q^31 + (-4b2 + 2) q^34 - 8b1 q^37 + (-2b7 + b6) q^38 + 5b1 q^43 + 10 * q^46 + (-2b7 + b6) q^47 + (-b2 + 7) * q^49 - 4b6 q^53 + 2b5 q^56 - 2b1 q^58 + (-2b5 - b3) q^59 + (2b2 - 1) q^61 + (-2b7 + b6) q^62 + 8 * q^64 + 7b1 q^67 - 2b3 q^71 + (4b4 - 2b1) * q^73 - 8b3 q^74 - b7 * q^77 - 14 * q^79 + (-2b7 + b6) q^83 + 5b3 q^86 - 4b1 q^88 + (-4b5 - 2b3) * q^89 + (b2 - 14) * q^91 + (-4b2 + 2) q^94 + (2b4 - b1) q^97 + (-b7 + 7b6) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{16} + 80 q^{46} + 52 q^{49} + 64 q^{64} - 112 q^{79} - 108 q^{91}+O(q^{100}) \) 8 * q - 32 * q^16 + 80 * q^46 + 52 * q^49 + 64 * q^64 - 112 * q^79 - 108 * q^91

Introduction

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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	1575.2.g.b

Level	$1575$
Weight	$2$
Character orbit	1575.g
Analytic conductor	$12.576$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(12.5764383184\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{6} \) v^6
\(\beta_{2}\)	\(=\)	\( 3\zeta_{24}^{4} - 1 \) 3*v^4 - 1
\(\beta_{3}\)	\(=\)	\( \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \) v^5 + v^3 - v
\(\beta_{4}\)	\(=\)	\( -\zeta_{24}^{6} + 3\zeta_{24}^{2} \) -v^6 + 3*v^2
\(\beta_{5}\)	\(=\)	\( -3\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) -3v^7 + v^5 + v^3 + 2v
\(\beta_{6}\)	\(=\)	\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) -v^5 + v^3 + v
\(\beta_{7}\)	\(=\)	\( 3\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24} \) 3v^7 + v^5 - v^3 + 2v

\(\zeta_{24}\)	\(=\)	\( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} ) / 6 \) (b7 + b6 + b5 - b3) / 6
\(\zeta_{24}^{2}\)	\(=\)	\( ( \beta_{4} + \beta_1 ) / 3 \) (b4 + b1) / 3
\(\zeta_{24}^{3}\)	\(=\)	\( ( \beta_{6} + \beta_{3} ) / 2 \) (b6 + b3) / 2
\(\zeta_{24}^{4}\)	\(=\)	\( ( \beta_{2} + 1 ) / 3 \) (b2 + 1) / 3
\(\zeta_{24}^{5}\)	\(=\)	\( ( \beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{3} ) / 6 \) (b7 - 2b6 + b5 + 2b3) / 6
\(\zeta_{24}^{6}\)	\(=\)	\( \beta_1 \) b1
\(\zeta_{24}^{7}\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} ) / 6 \) (b7 + b6 - b5 + b3) / 6

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1574.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{4} \) (T^2 - 2)^4
$3$	\( T^{8} \) T^8
$5$	\( T^{8} \) T^8
$7$	\( (T^{4} - 13 T^{2} + 49)^{2} \) (T^4 - 13*T^2 + 49)^2
$11$	\( (T^{2} + 2)^{4} \) (T^2 + 2)^4
$13$	\( (T^{2} - 27)^{4} \) (T^2 - 27)^4
$17$	\( (T^{2} + 54)^{4} \) (T^2 + 54)^4
$19$	\( (T^{2} + 27)^{4} \) (T^2 + 27)^4
$23$	\( (T^{2} - 50)^{4} \) (T^2 - 50)^4
$29$	\( (T^{2} + 2)^{4} \) (T^2 + 2)^4
$31$	\( (T^{2} + 27)^{4} \) (T^2 + 27)^4
$37$	\( (T^{2} + 64)^{4} \) (T^2 + 64)^4
$41$	\( T^{8} \) T^8
$43$	\( (T^{2} + 25)^{4} \) (T^2 + 25)^4
$47$	\( (T^{2} + 54)^{4} \) (T^2 + 54)^4
$53$	\( (T^{2} - 32)^{4} \) (T^2 - 32)^4
$59$	\( (T^{2} - 54)^{4} \) (T^2 - 54)^4
$61$	\( (T^{2} + 27)^{4} \) (T^2 + 27)^4
$67$	\( (T^{2} + 49)^{4} \) (T^2 + 49)^4
$71$	\( (T^{2} + 8)^{4} \) (T^2 + 8)^4
$73$	\( (T^{2} - 108)^{4} \) (T^2 - 108)^4
$79$	\( (T + 14)^{8} \) (T + 14)^8
$83$	\( (T^{2} + 54)^{4} \) (T^2 + 54)^4
$89$	\( (T^{2} - 216)^{4} \) (T^2 - 216)^4
$97$	\( (T^{2} - 27)^{4} \) (T^2 - 27)^4
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\(n\)	\(127\)	\(451\)	\(1226\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)