LMFDB - Newform orbit 725.2.l.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(226,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 10]))

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

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

chi := DirichletCharacter("725.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.l (of order $7$, degree $6$, 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:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$.

$n$	$176$	$552$
$\chi(n)$	$-\zeta_{14}$	$1$

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} $

226.1

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

1.12349

−

1.40881i

0.0990311

−

0.433884i

−0.277479

−

1.21572i

−0.500000

−

0.626980i

−0.900969

3.94740i

1.22252

0.588735i

2.52446

1.21572i

326.1

0.277479

1.21572i

1.62349

0.781831i

0.400969

−

0.193096i

−0.500000

2.19064i

0.623490

0.300257i

1.90097

2.38374i

0.153989

0.193096i

401.1

1.12349

1.40881i

0.0990311

0.433884i

−0.277479

1.21572i

−0.500000

0.626980i

−0.900969

−

3.94740i

1.22252

−

0.588735i

2.52446

−

1.21572i

426.1

−0.400969

0.193096i

0.777479

0.974928i

−1.12349

1.40881i

−0.500000

−

0.240787i

−0.222521

−

0.279032i

0.376510

−

1.64960i

0.321552

−

1.40881i

451.1

−0.400969

−

0.193096i

0.777479

−

0.974928i

−1.12349

−

1.40881i

−0.500000

0.240787i

−0.222521

0.279032i

0.376510

1.64960i

0.321552

1.40881i

576.1

0.277479

−

1.21572i

1.62349

−

0.781831i

0.400969

0.193096i

−0.500000

−

2.19064i

0.623490

−

0.300257i

1.90097

−

2.38374i

0.153989

−

0.193096i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	725.2.l.b		6
5.b	even	2	1		29.2.d.a	✓	6
5.c	odd	4	2		725.2.r.b		12
15.d	odd	2	1		261.2.k.a		6
20.d	odd	2	1		464.2.u.f		6
29.d	even	7	1	inner	725.2.l.b		6
145.d	even	2	1		841.2.d.d		6
145.f	odd	4	2		841.2.e.d		12
145.l	even	14	1		841.2.a.f		3
145.l	even	14	2		841.2.d.a		6
145.l	even	14	2		841.2.d.c		6
145.l	even	14	1		841.2.d.d		6
145.n	even	14	1		29.2.d.a	✓	6
145.n	even	14	1		841.2.a.e		3
145.n	even	14	2		841.2.d.b		6
145.n	even	14	2		841.2.d.e		6
145.p	odd	28	2		725.2.r.b		12
145.s	odd	28	2		841.2.b.c		6
145.s	odd	28	4		841.2.e.b		12
145.s	odd	28	4		841.2.e.c		12
145.s	odd	28	2		841.2.e.d		12
435.w	odd	14	1		261.2.k.a		6
435.w	odd	14	1		7569.2.a.r		3
435.bb	odd	14	1		7569.2.a.p		3
580.v	odd	14	1		464.2.u.f		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.2.d.a	✓	6	5.b	even	2	1
29.2.d.a	✓	6	145.n	even	14	1
261.2.k.a		6	15.d	odd	2	1
261.2.k.a		6	435.w	odd	14	1
464.2.u.f		6	20.d	odd	2	1
464.2.u.f		6	580.v	odd	14	1
725.2.l.b		6	1.a	even	1	1	trivial
725.2.l.b		6	29.d	even	7	1	inner
725.2.r.b		12	5.c	odd	4	2
725.2.r.b		12	145.p	odd	28	2
841.2.a.e		3	145.n	even	14	1
841.2.a.f		3	145.l	even	14	1
841.2.b.c		6	145.s	odd	28	2
841.2.d.a		6	145.l	even	14	2
841.2.d.b		6	145.n	even	14	2
841.2.d.c		6	145.l	even	14	2
841.2.d.d		6	145.d	even	2	1
841.2.d.d		6	145.l	even	14	1
841.2.d.e		6	145.n	even	14	2
841.2.e.b		12	145.s	odd	28	4
841.2.e.c		12	145.s	odd	28	4
841.2.e.d		12	145.f	odd	4	2
841.2.e.d		12	145.s	odd	28	2
7569.2.a.p		3	435.bb	odd	14	1
7569.2.a.r		3	435.w	odd	14	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 2T_{2}^{5} + 4T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + 3T_{2} + 1 $ T2^6 - 2*T2^5 + 4*T2^4 - T2^3 + 2*T2^2 + 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(725, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - T^3 + 2T^2 + 3T + 1
$3$	$ T^{6} - 5 T^{5} + \cdots + 1 $ T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + T^{5} + 15 T^{4} + \cdots + 1 $ T^6 + T^5 + 15T^4 - 13T^3 + T^2 + T + 1
$11$	$ T^{6} + 11 T^{5} + \cdots + 1681 $ T^6 + 11T^5 + 79T^4 + 365T^3 + 1089T^2 + 1927*T + 1681
$13$	$ T^{6} - 5 T^{5} + \cdots + 1849 $ T^6 - 5T^5 + 25T^4 - 181T^3 + 513T^2 + 1075*T + 1849
$17$	$ (T^{3} + 4 T^{2} - 4 T - 8)^{2} $ (T^3 + 4T^2 - 4T - 8)^2
$19$	$ T^{6} - T^{5} + \cdots + 169 $ T^6 - T^5 + T^4 - 15T^3 + 29T^2 + 13*T + 169
$23$	$ T^{6} - 7 T^{5} + \cdots + 2401 $ T^6 - 7T^5 + 49T^4 - 245T^3 + 1029T^2 - 2401*T + 2401
$29$	$ T^{6} - 6 T^{5} + \cdots + 24389 $ T^6 - 6T^5 - 13T^4 + 316T^3 - 377T^2 - 5046*T + 24389
$31$	$ T^{6} - 5 T^{5} + \cdots + 6889 $ T^6 - 5T^5 - 3T^4 - 139T^3 + 1885T^2 - 2075*T + 6889
$37$	$ T^{6} + 11 T^{5} + \cdots + 1681 $ T^6 + 11T^5 + 79T^4 + 365T^3 + 1089T^2 + 1927*T + 1681
$41$	$ (T^{3} - 10 T^{2} + 24 T - 8)^{2} $ (T^3 - 10T^2 + 24T - 8)^2
$43$	$ T^{6} + 13 T^{5} + \cdots + 169 $ T^6 + 13T^5 + 85T^4 + 265T^3 + 337T^2 + 13*T + 169
$47$	$ T^{6} + 11 T^{5} + \cdots + 57121 $ T^6 + 11T^5 + 65T^4 + 295T^3 + 1257T^2 + 2151*T + 57121
$53$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 - 5T^4 + 41T^3 + 417T^2 - 9*T + 1
$59$	$ (T^{3} + 28 T^{2} + \cdots + 728)^{2} $ (T^3 + 28T^2 + 252T + 728)^2
$61$	$ T^{6} - 3 T^{5} + \cdots + 169 $ T^6 - 3T^5 + 37T^4 - 13T^3 - 3T^2 - 117*T + 169
$67$	$ T^{6} + 19 T^{5} + \cdots + 169 $ T^6 + 19T^5 + 235T^4 + 167T^3 + 1017T^2 - 767*T + 169
$71$	$ T^{6} - 21 T^{5} + \cdots + 35721 $ T^6 - 21T^5 + 189T^4 - 945T^3 + 3969T^2 + 11907*T + 35721
$73$	$ T^{6} - 25 T^{5} + \cdots + 175561 $ T^6 - 25T^5 + 373T^4 - 3473T^3 + 22425T^2 - 90085*T + 175561
$79$	$ T^{6} + 9 T^{5} + \cdots + 729 $ T^6 + 9T^5 + 81T^4 - 27T^3 + 2025T^2 - 2187*T + 729
$83$	$ T^{6} + 17 T^{5} + \cdots + 28561 $ T^6 + 17T^5 + 219T^4 + 1693T^3 + 8089T^2 + 22139*T + 28561
$89$	$ T^{6} - 7 T^{5} + \cdots + 8281 $ T^6 - 7T^5 + 119T^4 - 1337T^3 + 6321T^2 - 9555*T + 8281
$97$	$ T^{6} + T^{5} + \cdots + 169 $ T^6 + T^5 + 211T^4 + 1849T^3 + 4761T^2 - 1105T + 169
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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:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.l (of order \(7\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{2}+ \cdots + (\zeta_{14}^{5} - \zeta_{14}^{3} + 1) q^{9}+O(q^{10}) \) q + (-z^5 + z^4 - z^3 - z + 1) * q^2 + (-z^5 + 1) * q^3 + (-z^5 + z^4) * q^4 + (-z^5 - z^3 - z) * q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-z^5 - 2z^3 + z^2 - z + 2) q^8 + (z^5 - z^3 + 1) * q^9	\( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{2}+ \cdots + ( - 3 \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots - 4) q^{99}+O(q^{100}) \) q + (-z^5 + z^4 - z^3 - z + 1) * q^2 + (-z^5 + 1) * q^3 + (-z^5 + z^4) * q^4 + (-z^5 - z^3 - z) * q^6 + (z^5 - 2z^4 + 2z - 1) * q^7 + (-z^5 - 2z^3 + z^2 - z + 2) q^8 + (z^5 - z^3 + 1) * q^9 + (-z^4 - 2z^3 + 2z^2 - 2z - 1) q^11 + (-z^5 + z^4 - z^3 + z^2) * q^12 + (3z^4 - 4z^3 + 2z^2 - 4z + 3) * q^13 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^14 + (2z^4 - 3z^3 - 3z + 2) q^16 + (-2z^5 + 2z^4 - 2z^3 + 2z^2) * q^17 + (2z^4 - z^3 - z^2 - z + 2) q^18 + (-2z^5 + 2z^4 + z^2 + 1) * q^19 + (-z^3 + 1) * q^21 + (4z^5 - 4z^4 + 3z^3 - 3z^2 + 4z - 4) q^22 + (z^5 - 3z^4 - z^3 - 3z^2 + z) * q^23 + (-2z^5 - z^4 - 2z^3 - 2z + 2) q^24 + (-2z^5 + 4z^4 + z^3 - z^2 - 4z + 2) q^26 + (3z^5 + 3z^3 - z + 1) * q^27 + (z^5 + z^4 - z^3 - z^2 + 2) * q^28 + (-2z^5 + 6z^4 - 2z^3 + 2z^2 + 3) * q^29 + (2z^5 - 6z^4 + 2z^3 + z - 1) q^31 + (-3z^5 + 2z^4 + z^3 - z^2 - 2z + 3) q^32 + (3z^5 - 3z^4 - z^2 - 2z - 1) q^33 - 2z^4 q^34 + (-z^5 + z^4 + z^3 - z^2 - z + 1) * q^36 + (3z^3 - 2z^2 + 2z - 3) q^37 + (-3z^5 - z^3 + 2z^2 - 2z + 1) q^38 + (z^5 - z^4 + z^2 - 4z + 1) q^39 + (-2z^5 + 2z^2 + 4) * q^41 + (z^4 - z^3 - z^2 - z + 1) * q^42 + (-3z^5 + 3z^4 - z^3 + 3z^2 - 3z) * q^43 + (3z^5 - 5z^4 + 4z^3 - 5z^2 + 3z) q^44 + (4z^5 + z^4 - z^3 - 4z^2 - 1) * q^46 + (-z^4 + 6z^2 - 1) q^47 + (z^5 - z^4 - z^2 - 3z - 1) q^48 + (z^5 + 3z^3 + 4z^2 - 4z - 3) q^49 + (-2z^5 + 2z^4 - 4z^3 + 4z^2 - 2z + 2) q^51 + (-z^5 - 3z^4 + 6z^3 - 3z^2 - z) q^52 + (-2z^5 + 4z^4 - 2z^3 - z + 1) q^53 + (-4z^5 + 4z^4 + 3z^2 - z + 3) q^54 + (-3z^4 + 2z - 2) * q^56 + (-3z^5 + 2z^4 - 2z^3 + 3z^2 + 2) * q^57 + (-7z^5 + 5z^4 - 3z^3 + 4z^2 - 7z + 7) q^58 + (-2z^5 + 2z^4 - 2z^3 + 2z^2 - 8) * q^59 + (-3z^5 + 4z^4 - 2z^3 + 2z^2 - 4z + 3) q^61 + (5z^5 - 5z^4 - 4z^2 + 7z - 4) * q^62 + (2z^5 - 6z^4 + 2z^3 + 5z - 5) * q^63 + (-2z^5 - 3z^4 - z^3 - 3z^2 - 2z) * q^64 + (4z^5 + 3z^3 - 3z^2 + 3z - 3) * q^66 + (8z^5 + 7z^3 - 4z^2 + 4z - 7) * q^67 + (2z^5 - 2z^4 + 2z^2 - 2z + 2) * q^68 + (-2z^4 - z^3 - 5z^2 - z - 2) * q^69 + (3z^4 - 6z^2 + 3) * q^71 + (z^5 - 2z^3 + z) q^72 + (7z^5 - 3z^4 + 5z^3 - 3z^2 + 7z) q^73 + (-z^4 + z^3 + 3z^2 + z - 1) q^74 + (-z^3 + 3z^2 - z) q^76 + (-5z^5 + 5z^4 + z^2 - 6z + 1) q^77 + (3z^3 - z^2 + z - 3) q^78 + (6z^5 + 3z^3 - 3) * q^79 + (6z^5 - z^4 + 4z^3 - z^2 + 6z) q^81 + (-4z^5 - 4z^3 - 2z + 2) q^82 + (-4z^5 + 4z^4 - z^2 - 4z - 1) q^83 + (-z^5 + z^4 - z + 1) * q^84 + (-2z^5 - 3z^4 + 3z^3 + 2z^2 - 3) * q^86 + (-5z^5 + 6z^4 - 4z^3 + 8z^2 - 2z + 5) q^87 + (z^5 + z^4 - z^3 - z^2 - 7) * q^88 + (6z^5 + 6z^3 + z - 1) * q^89 + (9z^5 - 9z^4 - 5z^2 + 12z - 5) * q^91 + (-3z^5 + 4z^4 - 3z^3 - 2z + 2) * q^92 + (2z^5 - 5z^4 + 3z^3 - 5z^2 + 2z) q^93 + (2z^5 - 8z^4 + z^3 - z^2 + 8z - 2) q^94 + (-4z^5 - z^2 + z) q^96 + (8z^5 - 8z^4 - 5z^2 + 8z - 5) * q^97 + (-6z^4 + 7z^3 + 3z^2 + 7z - 6) * q^98 + (-3z^5 + z^4 - z^3 + 3z^2 - 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 5 q^{3} - 2 q^{4} - 3 q^{6} - q^{7} + 7 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 5 * q^3 - 2 * q^4 - 3 * q^6 - q^7 + 7 * q^8 + 6 * q^9	\( 6 q + 2 q^{2} + 5 q^{3} - 2 q^{4} - 3 q^{6} - q^{7} + 7 q^{8} + 6 q^{9} - 11 q^{11} - 4 q^{12} + 5 q^{13} + 9 q^{14} + 4 q^{16} - 8 q^{17} + 9 q^{18} + q^{19} + 5 q^{21} - 6 q^{22} + 7 q^{23} + 7 q^{24} + 4 q^{26} + 11 q^{27} + 12 q^{28} + 6 q^{29} + 5 q^{31} + 13 q^{32} - q^{33} + 2 q^{34} + 5 q^{36} - 11 q^{37} - 2 q^{38} + 3 q^{39} + 20 q^{41} + 4 q^{42} - 13 q^{43} + 20 q^{44} - 11 q^{47} - 6 q^{48} - 22 q^{49} - 2 q^{51} + 10 q^{52} - 3 q^{53} + 6 q^{54} - 7 q^{56} + 2 q^{57} + 16 q^{58} - 56 q^{59} + 3 q^{61} - 3 q^{62} - 15 q^{63} + q^{64} - 5 q^{66} - 19 q^{67} + 12 q^{68} - 7 q^{69} + 21 q^{71} + 25 q^{73} - 6 q^{74} - 5 q^{76} - 11 q^{77} - 13 q^{78} - 9 q^{79} + 18 q^{81} + 2 q^{82} - 17 q^{83} + 3 q^{84} - 16 q^{86} + 5 q^{87} - 42 q^{88} + 7 q^{89} + 5 q^{91} + 17 q^{93} + 8 q^{94} - 2 q^{96} - q^{97} - 19 q^{98} - 32 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 5 * q^3 - 2 * q^4 - 3 * q^6 - q^7 + 7 * q^8 + 6 * q^9 - 11 * q^11 - 4 * q^12 + 5 * q^13 + 9 * q^14 + 4 * q^16 - 8 * q^17 + 9 * q^18 + q^19 + 5 * q^21 - 6 * q^22 + 7 * q^23 + 7 * q^24 + 4 * q^26 + 11 * q^27 + 12 * q^28 + 6 * q^29 + 5 * q^31 + 13 * q^32 - q^33 + 2 * q^34 + 5 * q^36 - 11 * q^37 - 2 * q^38 + 3 * q^39 + 20 * q^41 + 4 * q^42 - 13 * q^43 + 20 * q^44 - 11 * q^47 - 6 * q^48 - 22 * q^49 - 2 * q^51 + 10 * q^52 - 3 * q^53 + 6 * q^54 - 7 * q^56 + 2 * q^57 + 16 * q^58 - 56 * q^59 + 3 * q^61 - 3 * q^62 - 15 * q^63 + q^64 - 5 * q^66 - 19 * q^67 + 12 * q^68 - 7 * q^69 + 21 * q^71 + 25 * q^73 - 6 * q^74 - 5 * q^76 - 11 * q^77 - 13 * q^78 - 9 * q^79 + 18 * q^81 + 2 * q^82 - 17 * q^83 + 3 * q^84 - 16 * q^86 + 5 * q^87 - 42 * q^88 + 7 * q^89 + 5 * q^91 + 17 * q^93 + 8 * q^94 - 2 * q^96 - q^97 - 19 * q^98 - 32 * q^99

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Newspace parameters

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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	725.2.l.b

Level	$725$
Weight	$2$
Character orbit	725.l
Analytic conductor	$5.789$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.78915414654\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 226.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - T^3 + 2T^2 + 3T + 1
$3$	\( T^{6} - 5 T^{5} + \cdots + 1 \) T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$5$	\( T^{6} \) T^6
$7$	\( T^{6} + T^{5} + 15 T^{4} + \cdots + 1 \) T^6 + T^5 + 15T^4 - 13T^3 + T^2 + T + 1
$11$	\( T^{6} + 11 T^{5} + \cdots + 1681 \) T^6 + 11T^5 + 79T^4 + 365T^3 + 1089T^2 + 1927*T + 1681
$13$	\( T^{6} - 5 T^{5} + \cdots + 1849 \) T^6 - 5T^5 + 25T^4 - 181T^3 + 513T^2 + 1075*T + 1849
$17$	\( (T^{3} + 4 T^{2} - 4 T - 8)^{2} \) (T^3 + 4T^2 - 4T - 8)^2
$19$	\( T^{6} - T^{5} + \cdots + 169 \) T^6 - T^5 + T^4 - 15T^3 + 29T^2 + 13*T + 169
$23$	\( T^{6} - 7 T^{5} + \cdots + 2401 \) T^6 - 7T^5 + 49T^4 - 245T^3 + 1029T^2 - 2401*T + 2401
$29$	\( T^{6} - 6 T^{5} + \cdots + 24389 \) T^6 - 6T^5 - 13T^4 + 316T^3 - 377T^2 - 5046*T + 24389
$31$	\( T^{6} - 5 T^{5} + \cdots + 6889 \) T^6 - 5T^5 - 3T^4 - 139T^3 + 1885T^2 - 2075*T + 6889
$37$	\( T^{6} + 11 T^{5} + \cdots + 1681 \) T^6 + 11T^5 + 79T^4 + 365T^3 + 1089T^2 + 1927*T + 1681
$41$	\( (T^{3} - 10 T^{2} + 24 T - 8)^{2} \) (T^3 - 10T^2 + 24T - 8)^2
$43$	\( T^{6} + 13 T^{5} + \cdots + 169 \) T^6 + 13T^5 + 85T^4 + 265T^3 + 337T^2 + 13*T + 169
$47$	\( T^{6} + 11 T^{5} + \cdots + 57121 \) T^6 + 11T^5 + 65T^4 + 295T^3 + 1257T^2 + 2151*T + 57121
$53$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 - 5T^4 + 41T^3 + 417T^2 - 9*T + 1
$59$	\( (T^{3} + 28 T^{2} + \cdots + 728)^{2} \) (T^3 + 28T^2 + 252T + 728)^2
$61$	\( T^{6} - 3 T^{5} + \cdots + 169 \) T^6 - 3T^5 + 37T^4 - 13T^3 - 3T^2 - 117*T + 169
$67$	\( T^{6} + 19 T^{5} + \cdots + 169 \) T^6 + 19T^5 + 235T^4 + 167T^3 + 1017T^2 - 767*T + 169
$71$	\( T^{6} - 21 T^{5} + \cdots + 35721 \) T^6 - 21T^5 + 189T^4 - 945T^3 + 3969T^2 + 11907*T + 35721
$73$	\( T^{6} - 25 T^{5} + \cdots + 175561 \) T^6 - 25T^5 + 373T^4 - 3473T^3 + 22425T^2 - 90085*T + 175561
$79$	\( T^{6} + 9 T^{5} + \cdots + 729 \) T^6 + 9T^5 + 81T^4 - 27T^3 + 2025T^2 - 2187*T + 729
$83$	\( T^{6} + 17 T^{5} + \cdots + 28561 \) T^6 + 17T^5 + 219T^4 + 1693T^3 + 8089T^2 + 22139*T + 28561
$89$	\( T^{6} - 7 T^{5} + \cdots + 8281 \) T^6 - 7T^5 + 119T^4 - 1337T^3 + 6321T^2 - 9555*T + 8281
$97$	\( T^{6} + T^{5} + \cdots + 169 \) T^6 + T^5 + 211T^4 + 1849T^3 + 4761T^2 - 1105T + 169
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\(n\)	\(176\)	\(552\)
\(\chi(n)\)	\(-\zeta_{14}\)	\(1\)