LMFDB - Newform orbit 2019.1.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2019,1,Mod(95,2019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2019, base_ring=CyclotomicField(56))

chi = DirichletCharacter(H, H._module([28, 33]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2019.95");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2019 = 3 \cdot 673 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2019.bg (of order $56$, degree $24$, 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:	$1.00761226051$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{56})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{56}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{56} - \cdots)$

$q$-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_{56}^{6} q^{3} - \zeta_{56}^{14} q^{4} + (\zeta_{56}^{17} - \zeta_{56}^{13}) q^{7} + \zeta_{56}^{12} q^{9} +O(q^{10}) $ q - z^6 * q^3 - z^14 * q^4 + (z^17 - z^13) * q^7 + z^12 * q^9	$ q - \zeta_{56}^{6} q^{3} - \zeta_{56}^{14} q^{4} + (\zeta_{56}^{17} - \zeta_{56}^{13}) q^{7} + \zeta_{56}^{12} q^{9} + \zeta_{56}^{20} q^{12} + ( - \zeta_{56}^{26} - \zeta_{56}^{22}) q^{13} - q^{16} + ( - \zeta_{56}^{24} - \zeta_{56}^{13}) q^{19} + ( - \zeta_{56}^{23} + \zeta_{56}^{19}) q^{21} - \zeta_{56}^{3} q^{25} - \zeta_{56}^{18} q^{27} + (\zeta_{56}^{27} + \zeta_{56}^{3}) q^{28} + (\zeta_{56}^{23} - \zeta_{56}^{10}) q^{31} - \zeta_{56}^{26} q^{36} + ( - \zeta_{56}^{19} + \zeta_{56}) q^{37} + ( - \zeta_{56}^{4} - 1) q^{39} + (\zeta_{56}^{22} + \zeta_{56}^{21}) q^{43} + \zeta_{56}^{6} q^{48} + (\zeta_{56}^{26} + \cdots + \zeta_{56}^{2}) q^{49} + \cdots + ( - \zeta_{56}^{14} + \zeta_{56}^{10}) q^{97} +O(q^{100}) $ q - z^6 * q^3 - z^14 * q^4 + (z^17 - z^13) * q^7 + z^12 * q^9 + z^20 * q^12 + (-z^26 - z^22) * q^13 - q^16 + (-z^24 - z^13) * q^19 + (-z^23 + z^19) * q^21 - z^3 * q^25 - z^18 * q^27 + (z^27 + z^3) * q^28 + (z^23 - z^10) * q^31 - z^26 * q^36 + (-z^19 + z) * q^37 + (-z^4 - 1) * q^39 + (z^22 + z^21) * q^43 + z^6 * q^48 + (z^26 - z^6 + z^2) * q^49 + (-z^12 - z^8) * q^52 + (z^19 - z^2) * q^57 + (-z^19 + z^18) * q^61 + (-z^25 - z) * q^63 + z^14 * q^64 + (z^21 - z^2) * q^67 + (z^3 - z) * q^73 + z^9 * q^75 + (z^27 - z^10) * q^76 + (-z^16 + z^11) * q^79 + z^24 * q^81 + (-z^9 + z^5) * q^84 + (z^15 - z^7) * q^91 + (z^16 + z) * q^93 + (-z^14 + z^10) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} + 4 q^{12} - 24 q^{16} + 4 q^{19} - 28 q^{39} + 4 q^{79} - 4 q^{81} - 4 q^{93}+O(q^{100}) $ 24 * q + 4 * q^9 + 4 * q^12 - 24 * q^16 + 4 * q^19 - 28 * q^39 + 4 * q^79 - 4 * q^81 - 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2019\mathbb{Z}\right)^\times$.

$n$	$674$	$1351$
$\chi(n)$	$-1$	$-\zeta_{56}^{3}$

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} $

95.1

−0.330279	−	0.943883i
−0.993712	−	0.111964i
0.993712	+	0.111964i
0.330279	+	0.943883i
0.993712	−	0.111964i
0.532032	+	0.846724i
0.330279	−	0.943883i
0.111964	−	0.993712i
−0.532032	+	0.846724i
−0.846724	+	0.532032i
0.943883	+	0.330279i
0.111964	+	0.993712i
0.943883	−	0.330279i
0.846724	+	0.532032i
−0.846724	−	0.532032i
−0.943883	+	0.330279i
−0.111964	−	0.993712i
−0.943883	−	0.330279i
0.846724	−	0.532032i
0.532032	−	0.846724i

−0.433884

−

0.900969i

1.00000i

−0.411851

−

1.17700i

−0.623490

0.781831i

206.1

−0.781831

−

0.623490i

−

1.00000i

0.442244

0.0498289i

0.222521

0.974928i

467.1

−0.781831

−

0.623490i

−

1.00000i

−0.442244

−

0.0498289i

0.222521

0.974928i

578.1

−0.433884

−

0.900969i

1.00000i

0.411851

1.17700i

−0.623490

0.781831i

722.1

−0.781831

0.623490i

1.00000i

−0.442244

0.0498289i

0.222521

−

0.974928i

725.1

−0.974928

0.222521i

−

1.00000i

−0.958689

−

1.52574i

0.900969

−

0.433884i

758.1

−0.433884

0.900969i

−

1.00000i

0.411851

−

1.17700i

−0.623490

−

0.781831i

839.1

0.781831

0.623490i

1.00000i

−0.0498289

0.442244i

0.222521

0.974928i

893.1

−0.974928

−

0.222521i

1.00000i

0.958689

−

1.52574i

0.900969

0.433884i

1022.1

0.974928

−

0.222521i

1.00000i

1.52574

−

0.958689i

0.900969

−

0.433884i

1127.1

0.433884

−

0.900969i

1.00000i

1.17700

0.411851i

−0.623490

−

0.781831i

1196.1

0.781831

−

0.623490i

−

1.00000i

−0.0498289

−

0.442244i

0.222521

−

0.974928i

1220.1

0.433884

0.900969i

−

1.00000i

1.17700

−

0.411851i

−0.623490

0.781831i

1319.1

0.974928

0.222521i

−

1.00000i

−1.52574

−

0.958689i

0.900969

0.433884i

1373.1

0.974928

0.222521i

−

1.00000i

1.52574

0.958689i

0.900969

0.433884i

1472.1

0.433884

0.900969i

−

1.00000i

−1.17700

0.411851i

−0.623490

0.781831i

1496.1

0.781831

−

0.623490i

−

1.00000i

0.0498289

0.442244i

0.222521

−

0.974928i

1565.1

0.433884

−

0.900969i

1.00000i

−1.17700

−

0.411851i

−0.623490

−

0.781831i

1670.1

0.974928

−

0.222521i

1.00000i

−1.52574

0.958689i

0.900969

−

0.433884i

1799.1

−0.974928

−

0.222521i

1.00000i

−0.958689

1.52574i

0.900969

0.433884i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
673.q	even	56	1	inner
2019.bg	odd	56	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2019.1.bg.a	✓	24
3.b	odd	2	1	CM	2019.1.bg.a	✓	24
673.q	even	56	1	inner	2019.1.bg.a	✓	24
2019.bg	odd	56	1	inner	2019.1.bg.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2019.1.bg.a	✓	24	1.a	even	1	1	trivial
2019.1.bg.a	✓	24	3.b	odd	2	1	CM
2019.1.bg.a	✓	24	673.q	even	56	1	inner
2019.1.bg.a	✓	24	2019.bg	odd	56	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2019, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} $ T^24
$3$	$ (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} $ (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$5$	$ T^{24} $ T^24
$7$	$ T^{24} + 12 T^{20} + \cdots + 1 $ T^24 + 12T^20 + 102T^16 - 50T^12 + 653T^8 - 46*T^4 + 1
$11$	$ T^{24} $ T^24
$13$	$ (T^{12} + 35 T^{6} + \cdots + 49)^{2} $ (T^12 + 35T^6 + 98T^4 + 49*T^2 + 49)^2
$17$	$ T^{24} $ T^24
$19$	$ T^{24} - 4 T^{23} + \cdots + 1 $ T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 48T^19 + 48T^18 + 4T^17 - 180T^16 + 600T^15 - 844T^14 + 488T^13 + 1072T^12 - 2268T^11 + 2382T^10 - 1184T^9 + 557T^8 - 1684T^7 + 3366T^6 - 1792T^5 + 196T^4 - 124T^3 + 50T^2 + 12*T + 1
$23$	$ T^{24} $ T^24
$29$	$ T^{24} $ T^24
$31$	$ T^{24} - 2 T^{22} + \cdots + 1 $ T^24 - 2T^22 + 24T^21 + 2T^20 - 40T^19 + 226T^18 + 24T^17 - 296T^16 + 1088T^15 + 44T^14 - 1040T^13 + 2935T^12 - 272T^11 - 1796T^10 + 4212T^9 - 1190T^8 - 864T^7 + 3138T^6 - 1936T^5 + 715T^4 - 88T^3 + 38T^2 + 12T + 1
$37$	$ T^{24} + 4 T^{22} + \cdots + 1 $ T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} - 2 T^{22} + \cdots + 1 $ T^24 - 2T^22 + 4T^21 + 9T^20 - 16T^19 + 2T^18 + 60T^17 - 65T^16 + 32T^15 + 366T^14 - 444T^13 - 439T^12 - 624T^11 - 1628T^10 + 380T^9 + 1673T^8 + 2488T^7 + 2956T^6 + 956T^5 + 715T^4 + 424T^3 + 136T^2 + 16T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} - 2 T^{22} + \cdots + 1 $ T^24 - 2T^22 - 24T^21 + 2T^20 + 40T^19 + 226T^18 - 24T^17 - 296T^16 - 1088T^15 + 44T^14 + 1040T^13 + 2935T^12 + 272T^11 - 1796T^10 - 4212T^9 - 1190T^8 + 864T^7 + 3138T^6 + 1936T^5 + 715T^4 + 88T^3 + 38T^2 - 12T + 1
$67$	$ T^{24} - 2 T^{22} + \cdots + 1 $ T^24 - 2T^22 + 4T^21 + 9T^20 - 16T^19 + 2T^18 + 60T^17 - 65T^16 + 32T^15 + 366T^14 - 444T^13 - 439T^12 - 624T^11 - 1628T^10 + 380T^9 + 1673T^8 + 2488T^7 + 2956T^6 + 956T^5 + 715T^4 + 424T^3 + 136T^2 + 16T + 1
$71$	$ T^{24} $ T^24
$73$	$ T^{24} + 4 T^{22} + \cdots + 1 $ T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$79$	$ T^{24} - 4 T^{23} + \cdots + 1 $ T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 48T^19 + 48T^18 + 4T^17 - 180T^16 + 600T^15 - 844T^14 + 488T^13 + 1072T^12 - 2268T^11 + 2382T^10 - 1184T^9 + 557T^8 - 1684T^7 + 3366T^6 - 1792T^5 + 196T^4 - 124T^3 + 50T^2 + 12*T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ (T^{12} + 3 T^{10} + \cdots + 1)^{2} $ (T^12 + 3T^10 + 9T^8 - T^6 + 25T^4 - 9T^2 + 1)^2
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2019 = 3 \cdot 673 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2019.bg (of order \(56\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{56}^{6} q^{3} - \zeta_{56}^{14} q^{4} + (\zeta_{56}^{17} - \zeta_{56}^{13}) q^{7} + \zeta_{56}^{12} q^{9} +O(q^{10}) \) q - z^6 * q^3 - z^14 * q^4 + (z^17 - z^13) * q^7 + z^12 * q^9	\( q - \zeta_{56}^{6} q^{3} - \zeta_{56}^{14} q^{4} + (\zeta_{56}^{17} - \zeta_{56}^{13}) q^{7} + \zeta_{56}^{12} q^{9} + \zeta_{56}^{20} q^{12} + ( - \zeta_{56}^{26} - \zeta_{56}^{22}) q^{13} - q^{16} + ( - \zeta_{56}^{24} - \zeta_{56}^{13}) q^{19} + ( - \zeta_{56}^{23} + \zeta_{56}^{19}) q^{21} - \zeta_{56}^{3} q^{25} - \zeta_{56}^{18} q^{27} + (\zeta_{56}^{27} + \zeta_{56}^{3}) q^{28} + (\zeta_{56}^{23} - \zeta_{56}^{10}) q^{31} - \zeta_{56}^{26} q^{36} + ( - \zeta_{56}^{19} + \zeta_{56}) q^{37} + ( - \zeta_{56}^{4} - 1) q^{39} + (\zeta_{56}^{22} + \zeta_{56}^{21}) q^{43} + \zeta_{56}^{6} q^{48} + (\zeta_{56}^{26} + \cdots + \zeta_{56}^{2}) q^{49} + \cdots + ( - \zeta_{56}^{14} + \zeta_{56}^{10}) q^{97} +O(q^{100}) \) q - z^6 * q^3 - z^14 * q^4 + (z^17 - z^13) * q^7 + z^12 * q^9 + z^20 * q^12 + (-z^26 - z^22) * q^13 - q^16 + (-z^24 - z^13) * q^19 + (-z^23 + z^19) * q^21 - z^3 * q^25 - z^18 * q^27 + (z^27 + z^3) * q^28 + (z^23 - z^10) * q^31 - z^26 * q^36 + (-z^19 + z) * q^37 + (-z^4 - 1) * q^39 + (z^22 + z^21) * q^43 + z^6 * q^48 + (z^26 - z^6 + z^2) * q^49 + (-z^12 - z^8) * q^52 + (z^19 - z^2) * q^57 + (-z^19 + z^18) * q^61 + (-z^25 - z) * q^63 + z^14 * q^64 + (z^21 - z^2) * q^67 + (z^3 - z) * q^73 + z^9 * q^75 + (z^27 - z^10) * q^76 + (-z^16 + z^11) * q^79 + z^24 * q^81 + (-z^9 + z^5) * q^84 + (z^15 - z^7) * q^91 + (z^16 + z) * q^93 + (-z^14 + z^10) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} + 4 q^{12} - 24 q^{16} + 4 q^{19} - 28 q^{39} + 4 q^{79} - 4 q^{81} - 4 q^{93}+O(q^{100}) \) 24 * q + 4 * q^9 + 4 * q^12 - 24 * q^16 + 4 * q^19 - 28 * q^39 + 4 * q^79 - 4 * q^81 - 4 * q^93

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	2019.1.bg.a

Level	$2019$
Weight	$1$
Character orbit	2019.bg
Analytic conductor	$1.008$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{56}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.00761226051\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{56})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{56}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{56} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 95.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
673.q	even	56	1	inner
2019.bg	odd	56	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} \) T^24
$3$	\( (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} \) (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$5$	\( T^{24} \) T^24
$7$	\( T^{24} + 12 T^{20} + \cdots + 1 \) T^24 + 12T^20 + 102T^16 - 50T^12 + 653T^8 - 46*T^4 + 1
$11$	\( T^{24} \) T^24
$13$	\( (T^{12} + 35 T^{6} + \cdots + 49)^{2} \) (T^12 + 35T^6 + 98T^4 + 49*T^2 + 49)^2
$17$	\( T^{24} \) T^24
$19$	\( T^{24} - 4 T^{23} + \cdots + 1 \) T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 48T^19 + 48T^18 + 4T^17 - 180T^16 + 600T^15 - 844T^14 + 488T^13 + 1072T^12 - 2268T^11 + 2382T^10 - 1184T^9 + 557T^8 - 1684T^7 + 3366T^6 - 1792T^5 + 196T^4 - 124T^3 + 50T^2 + 12*T + 1
$23$	\( T^{24} \) T^24
$29$	\( T^{24} \) T^24
$31$	\( T^{24} - 2 T^{22} + \cdots + 1 \) T^24 - 2T^22 + 24T^21 + 2T^20 - 40T^19 + 226T^18 + 24T^17 - 296T^16 + 1088T^15 + 44T^14 - 1040T^13 + 2935T^12 - 272T^11 - 1796T^10 + 4212T^9 - 1190T^8 - 864T^7 + 3138T^6 - 1936T^5 + 715T^4 - 88T^3 + 38T^2 + 12T + 1
$37$	\( T^{24} + 4 T^{22} + \cdots + 1 \) T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} - 2 T^{22} + \cdots + 1 \) T^24 - 2T^22 + 4T^21 + 9T^20 - 16T^19 + 2T^18 + 60T^17 - 65T^16 + 32T^15 + 366T^14 - 444T^13 - 439T^12 - 624T^11 - 1628T^10 + 380T^9 + 1673T^8 + 2488T^7 + 2956T^6 + 956T^5 + 715T^4 + 424T^3 + 136T^2 + 16T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} - 2 T^{22} + \cdots + 1 \) T^24 - 2T^22 - 24T^21 + 2T^20 + 40T^19 + 226T^18 - 24T^17 - 296T^16 - 1088T^15 + 44T^14 + 1040T^13 + 2935T^12 + 272T^11 - 1796T^10 - 4212T^9 - 1190T^8 + 864T^7 + 3138T^6 + 1936T^5 + 715T^4 + 88T^3 + 38T^2 - 12T + 1
$67$	\( T^{24} - 2 T^{22} + \cdots + 1 \) T^24 - 2T^22 + 4T^21 + 9T^20 - 16T^19 + 2T^18 + 60T^17 - 65T^16 + 32T^15 + 366T^14 - 444T^13 - 439T^12 - 624T^11 - 1628T^10 + 380T^9 + 1673T^8 + 2488T^7 + 2956T^6 + 956T^5 + 715T^4 + 424T^3 + 136T^2 + 16T + 1
$71$	\( T^{24} \) T^24
$73$	\( T^{24} + 4 T^{22} + \cdots + 1 \) T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$79$	\( T^{24} - 4 T^{23} + \cdots + 1 \) T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 48T^19 + 48T^18 + 4T^17 - 180T^16 + 600T^15 - 844T^14 + 488T^13 + 1072T^12 - 2268T^11 + 2382T^10 - 1184T^9 + 557T^8 - 1684T^7 + 3366T^6 - 1792T^5 + 196T^4 - 124T^3 + 50T^2 + 12*T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( (T^{12} + 3 T^{10} + \cdots + 1)^{2} \) (T^12 + 3T^10 + 9T^8 - T^6 + 25T^4 - 9T^2 + 1)^2
show more
show less

\(n\)	\(674\)	\(1351\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{56}^{3}\)