LMFDB - Newform orbit 33.3.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,3,Mod(5,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("33.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	33.h (of order $10$, degree $4$, 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:	$0.899184872389$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$-\zeta_{20}^{2}$	$-1$

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} $

5.1

−0.951057	+	0.309017i
0.951057	−	0.309017i
−0.587785	−	0.809017i
0.587785	+	0.809017i
−0.951057	−	0.309017i
0.951057	+	0.309017i
−0.587785	+	0.809017i
0.587785	−	0.809017i

−0.951057

0.309017i

0.303706

2.98459i

−2.42705

1.76336i

4.16750

1.35410i

−1.21113

−

2.74466i

1.73607

−

1.26133i

4.11450

−

5.66312i

−8.81553

1.81288i

−4.38197

5.2

0.951057

−

0.309017i

2.93236

−

0.633446i

−2.42705

1.76336i

−4.16750

−

1.35410i

2.59310

−

1.50859i

1.73607

−

1.26133i

−4.11450

5.66312i

8.19749

−

3.71499i

−4.38197

14.1

−0.587785

−

0.809017i

−2.74466

−

1.21113i

0.927051

−

2.85317i

3.88998

−

5.35410i

0.633446

2.93236i

−2.73607

8.42075i

−6.65740

2.16312i

6.06633

6.64828i

−6.61803

14.2

0.587785

0.809017i

1.50859

−

2.59310i

0.927051

−

2.85317i

−3.88998

5.35410i

2.98459

−

0.303706i

−2.73607

8.42075i

6.65740

−

2.16312i

−4.44829

−

7.82385i

−6.61803

20.1

−0.951057

−

0.309017i

0.303706

−

2.98459i

−2.42705

−

1.76336i

4.16750

−

1.35410i

−1.21113

2.74466i

1.73607

1.26133i

4.11450

5.66312i

−8.81553

−

1.81288i

−4.38197

20.2

0.951057

0.309017i

2.93236

0.633446i

−2.42705

−

1.76336i

−4.16750

1.35410i

2.59310

1.50859i

1.73607

1.26133i

−4.11450

−

5.66312i

8.19749

3.71499i

−4.38197

26.1

−0.587785

0.809017i

−2.74466

1.21113i

0.927051

2.85317i

3.88998

5.35410i

0.633446

−

2.93236i

−2.73607

−

8.42075i

−6.65740

−

2.16312i

6.06633

−

6.64828i

−6.61803

26.2

0.587785

−

0.809017i

1.50859

2.59310i

0.927051

2.85317i

−3.88998

−

5.35410i

2.98459

0.303706i

−2.73607

−

8.42075i

6.65740

2.16312i

−4.44829

7.82385i

−6.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.c	even	5	1	inner
33.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	33.3.h.a	✓	8
3.b	odd	2	1	inner	33.3.h.a	✓	8
11.b	odd	2	1		363.3.h.g		8
11.c	even	5	1	inner	33.3.h.a	✓	8
11.c	even	5	1		363.3.b.f		4
11.c	even	5	2		363.3.h.h		8
11.d	odd	10	1		363.3.b.g		4
11.d	odd	10	1		363.3.h.g		8
11.d	odd	10	2		363.3.h.i		8
33.d	even	2	1		363.3.h.g		8
33.f	even	10	1		363.3.b.g		4
33.f	even	10	1		363.3.h.g		8
33.f	even	10	2		363.3.h.i		8
33.h	odd	10	1	inner	33.3.h.a	✓	8
33.h	odd	10	1		363.3.b.f		4
33.h	odd	10	2		363.3.h.h		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.3.h.a	✓	8	1.a	even	1	1	trivial
33.3.h.a	✓	8	3.b	odd	2	1	inner
33.3.h.a	✓	8	11.c	even	5	1	inner
33.3.h.a	✓	8	33.h	odd	10	1	inner
363.3.b.f		4	11.c	even	5	1
363.3.b.f		4	33.h	odd	10	1
363.3.b.g		4	11.d	odd	10	1
363.3.b.g		4	33.f	even	10	1
363.3.h.g		8	11.b	odd	2	1
363.3.h.g		8	11.d	odd	10	1
363.3.h.g		8	33.d	even	2	1
363.3.h.g		8	33.f	even	10	1
363.3.h.h		8	11.c	even	5	2
363.3.h.h		8	33.h	odd	10	2
363.3.h.i		8	11.d	odd	10	2
363.3.h.i		8	33.f	even	10	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 $ T2^8 - T2^6 + T2^4 - T2^2 + 1 acting on $S_{3}^{\mathrm{new}}(33, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{8} - 4 T^{7} + \cdots + 6561 $ T^8 - 4T^7 + 7T^6 + 8T^5 - 95T^4 + 72T^3 + 567T^2 - 2916*T + 6561
$5$	$ T^{8} - 4 T^{6} + \cdots + 707281 $ T^8 - 4T^6 + 1446T^4 - 49619*T^2 + 707281
$7$	$ (T^{4} + 2 T^{3} + \cdots + 361)^{2} $ (T^4 + 2T^3 + 64T^2 - 247*T + 361)^2
$11$	$ T^{8} - 316 T^{6} + \cdots + 214358881 $ T^8 - 316T^6 + 49126T^4 - 4626556*T^2 + 214358881
$13$	$ (T^{4} - 16 T^{3} + \cdots + 13456)^{2} $ (T^4 - 16T^3 + 136T^2 - 696*T + 13456)^2
$17$	$ T^{8} - 36 T^{6} + \cdots + 1679616 $ T^8 - 36T^6 + 1296T^4 - 46656*T^2 + 1679616
$19$	$ (T^{4} - 56 T^{3} + \cdots + 80656)^{2} $ (T^4 - 56T^3 + 1336T^2 - 9656*T + 80656)^2
$23$	$ (T^{4} + 1008 T^{2} + 215296)^{2} $ (T^4 + 1008*T^2 + 215296)^2
$29$	$ T^{8} + \cdots + 2449228130001 $ T^8 + 261T^6 + 3966246T^4 - 5014263204*T^2 + 2449228130001
$31$	$ (T^{4} + 9 T^{3} + \cdots + 36481)^{2} $ (T^4 + 9T^3 + 796T^2 - 8786*T + 36481)^2
$37$	$ (T^{4} + 60 T^{3} + \cdots + 336400)^{2} $ (T^4 + 60T^3 + 1840T^2 + 29000*T + 336400)^2
$41$	$ T^{8} + \cdots + 3544535296 $ T^8 + 1124T^6 + 3554016T^4 - 181942016*T^2 + 3544535296
$43$	$ (T^{2} + 66 T + 1044)^{4} $ (T^2 + 66*T + 1044)^4
$47$	$ T^{8} - 496 T^{6} + \cdots + 33362176 $ T^8 - 496T^6 + 92896T^4 + 947264*T^2 + 33362176
$53$	$ T^{8} + \cdots + 3969126001 $ T^8 + 101T^6 + 199126T^4 - 45612724*T^2 + 3969126001
$59$	$ T^{8} + \cdots + 276281640625 $ T^8 - 100T^6 + 903750T^4 - 775296875*T^2 + 276281640625
$61$	$ (T^{4} - 168 T^{3} + \cdots + 6533136)^{2} $ (T^4 - 168T^3 + 12024T^2 - 260712*T + 6533136)^2
$67$	$ (T^{2} + 6 T - 5436)^{4} $ (T^2 + 6*T - 5436)^4
$71$	$ T^{8} + \cdots + 11\!\cdots\!56 $ T^8 - 9904T^6 + 45544416T^4 - 74244860864*T^2 + 1134774622925056
$73$	$ (T^{4} + 91 T^{3} + \cdots + 866761)^{2} $ (T^4 + 91T^3 + 3136T^2 - 13034*T + 866761)^2
$79$	$ (T^{4} + 230 T^{3} + \cdots + 55428025)^{2} $ (T^4 + 230T^3 + 25360T^2 + 1302875*T + 55428025)^2
$83$	$ T^{8} + \cdots + 22\!\cdots\!01 $ T^8 - 12004T^6 + 150136006T^4 - 1875937425499*T^2 + 22526107869174001
$89$	$ (T^{4} + 9632 T^{2} + 891136)^{2} $ (T^4 + 9632*T^2 + 891136)^2
$97$	$ (T^{4} - 258 T^{3} + \cdots + 147403881)^{2} $ (T^4 - 258T^3 + 36864T^2 - 2877417*T + 147403881)^2
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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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + \cdots + 2) q^{3}+ \cdots + ( - 8 \zeta_{20}^{7} + \cdots + 4 \zeta_{20}) q^{9}+O(q^{10}) \) q + z * q^2 + (-2z^6 + 2z^5 + 2z^4 - z^3 - 2z^2 + 2z + 2) q^3 - 3z^2 q^4 + (5z^7 - 6z^5 + 6z^3 - 5z) * q^5 + (-2z^7 + 2z^6 + 2z^5 - z^4 - 2z^3 + 2z^2 + 2z) * q^6 + (-3z^4 + 7z^2 - 3) * q^7 - 7z^3 q^8 + (-8z^7 + z^6 + 8z^5 + 4z) q^9	\( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + \cdots + 2) q^{3}+ \cdots + ( - 6 \zeta_{20}^{7} + 20 \zeta_{20}^{6} + \cdots - 32) q^{99}+O(q^{100}) \) q + z * q^2 + (-2z^6 + 2z^5 + 2z^4 - z^3 - 2z^2 + 2z + 2) q^3 - 3z^2 q^4 + (5z^7 - 6z^5 + 6z^3 - 5z) * q^5 + (-2z^7 + 2z^6 + 2z^5 - z^4 - 2z^3 + 2z^2 + 2z) * q^6 + (-3z^4 + 7z^2 - 3) * q^7 - 7z^3 q^8 + (-8z^7 + z^6 + 8z^5 + 4z) q^9 + (-z^6 + z^4 - 5) * q^10 + (2z^7 - 9z^5 + 6z^3 - 12z) * q^11 + (-6z^7 + 3z^5 - 6z^3 - 6) q^12 + (10z^6 - 2z^2 + 2) * q^13 + (-3z^5 + 7z^3 - 3z) q^14 + (10z^7 - 11z^4 - 2z^3 + 8z^2 + 2z - 11) q^15 + 5z^4 q^16 + (-6z^7 + 6z^5 - 6z^3 + 6z) * q^17 + (z^7 + 8z^4 - 4z^2 + 8) * q^18 + (-20z^6 + 2z^4 - 2z^2 + 20) q^19 + (3z^7 - 3z^5 + 15z) q^20 + (11z^7 + 6z^6 - 13z^5 - 6z^4 + 11z^3 + 8) q^21 + (-7z^6 + 4z^4 - 10z^2 - 2) q^22 + (4z^7 + 20z^5 + 4z^3) q^23 + (-7z^6 - 14z^2 - 14z + 14) q^24 + (-z^6 + 12z^4 - 12z^2 + 1) * q^25 + (10z^7 - 2z^3 + 2z) q^26 + (-7z^7 - 7z^5 + 22z^4 + 7z^3 + 7z) q^27 + (9z^6 - 21z^4 + 9z^2) q^28 + (-27z^7 - 33z^3 + 33z) q^29 + (10z^6 - 11z^5 - 12z^4 + 8z^3 + 12z^2 - 11z - 10) * q^30 + (24z^6 + 11z^2 - 11) * q^31 + 33z^5 q^32 + (24z^7 - 23z^6 - 20z^5 - z^4 + 6z^3 - 3z^2 - 12z - 5) * q^33 + 6 * q^34 + (-22z^7 + 22z^5 - 23z) q^35 + (-3z^6 - 24z^5 + 3z^4 + 12z^3 - 3z^2 - 24z + 3) * q^36 + (-22z^4 + 6z^2 - 22) * q^37 + (-20z^7 + 2z^5 - 2z^3 + 20z) * q^38 + (6z^7 - 4z^6 + 16z^5 + 24z^4 - 16z^3 - 4z^2 - 6z) q^39 + (-7z^4 + 42z^2 - 7) * q^40 + (-22z^5 - 8z^3 - 22z) q^41 + (6z^7 - 2z^6 - 6z^5 + 11z^2 + 8z - 11) q^42 + (6z^6 - 6z^4 - 36) * q^43 + (21z^7 - 12z^5 + 30z^3 + 6z) * q^44 + (z^7 + 44z^6 - 6z^5 - 44z^4 + z^3 - 12) q^45 + (24z^6 + 4z^2 - 4) * q^46 + (6z^5 - 14z^3 + 6z) q^47 + (5z^7 + 10z^3 + 10z^2 - 10z) * q^48 + (-33z^6 + 9z^4 - 33z^2) q^49 + (-z^7 + 12z^5 - 12z^3 + z) * q^50 + (-12z^7 + 12z^4 - 6z^2 + 12) q^51 + (-30z^6 + 36z^4 - 36z^2 + 30) q^52 + (-15z^7 + 15z^5 + 2z) q^53 + (-14z^6 + 22z^5 + 14z^4 + 7) q^54 + (-7z^6 + 48z^4 - 43z^2 + 64) q^55 + (21z^7 - 49z^5 + 21z^3) q^56 + (-22z^7 - 40z^6 + 22z^5 - 36z^2 + 56z + 36) q^57 + (-27z^6 - 6z^4 + 6z^2 + 27) q^58 + (30z^7 + 5z^3 - 5z) q^59 + (-30z^7 + 33z^6 + 36z^5 - 24z^4 - 36z^3 + 33z^2 + 30z) q^60 + (54z^6 - 60z^4 + 54z^2) q^61 + (24z^7 + 11z^3 - 11z) q^62 + (4z^6 + 44z^5 - 7z^4 - 52z^3 + 7z^2 + 44z - 4) * q^63 + 13z^6 q^64 + (22z^7 - 74z^5 + 22z^3) q^65 + (-23z^7 + 4z^6 - z^5 - 18z^4 - 3z^3 + 12z^2 - 5z - 24) * q^66 + (-66z^6 + 66z^4 + 30) * q^67 - 18z q^68 + (32z^6 + 8z^5 + 12z^4 + 40z^3 - 12z^2 + 8z - 32) * q^69 + (22z^4 - 45z^2 + 22) * q^70 + (82z^7 - 72z^5 + 72z^3 - 82z) * q^71 + (-7z^7 - 56z^6 + 7z^5 + 28z^4 - 7z^3 - 56z^2 + 7z) q^72 + (-21z^4 - 28z^2 - 21) * q^73 + (-22z^5 + 6z^3 - 22z) q^74 + (-13z^7 - 2z^6 + 13z^5 + 22z^2 - 21z - 22) q^75 + (54z^6 - 54z^4 - 60) * q^76 + (-46z^7 + 64z^5 - 61z^3 + z) q^77 + (-4z^7 + 22z^6 + 24z^5 - 22z^4 - 4z^3 - 6) q^78 + (13z^6 + 81z^2 - 81) * q^79 + (5z^5 - 30z^3 + 5z) q^80 + (8z^7 + 16z^3 + 79z^2 - 16z) * q^81 + (-22z^6 - 8z^4 - 22z^2) q^82 + (-49z^7 - 50z^5 + 50z^3 + 49z) * q^83 + (6z^7 + 18z^4 - 33z^3 - 42z^2 + 33z + 18) q^84 + (30z^6 - 36z^4 + 36z^2 - 30) q^85 + (6z^7 - 6z^5 - 36z) q^86 + (-66z^7 - 21z^6 + 12z^5 + 21z^4 - 66z^3 + 93) q^87 + (21z^6 + 21z^4 + 63z^2 - 49) q^88 + (-48z^7 + 68z^5 - 48z^3) q^89 + (44z^7 - 5z^6 - 44z^5 + z^2 - 12z - 1) * q^90 + (46z^6 - 90z^4 + 90z^2 - 46) q^91 + (-72z^7 - 12z^3 + 12z) q^92 + (46z^7 + 22z^6 - 9z^5 + 26z^4 + 9z^3 + 22z^2 - 46z) q^93 + (6z^6 - 14z^4 + 6z^2) q^94 + (82z^7 + 88z^3 - 88z) q^95 + (33z^6 + 33z^4 + 66z^3 - 33z^2 - 33) * q^96 + (-39z^6 - 99z^2 + 99) * q^97 + (-33z^7 + 9z^5 - 33z^3) q^98 + (-6z^7 + 20z^6 - 6z^5 - 112z^4 + 4z^3 + 16z^2 + 3z - 32) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 - 6 * q^4 + 10 * q^6 - 4 * q^7 + 2 * q^9	\( 8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9} - 44 q^{10} - 48 q^{12} + 32 q^{13} - 50 q^{15} - 10 q^{16} + 40 q^{18} + 112 q^{19} + 88 q^{21} - 58 q^{22} + 70 q^{24} - 42 q^{25} - 44 q^{27} + 78 q^{28} - 12 q^{30} - 18 q^{31} - 90 q^{33} + 48 q^{34} + 6 q^{36} - 120 q^{37} - 64 q^{39} + 42 q^{40} - 70 q^{42} - 264 q^{43} + 80 q^{45} + 24 q^{46} + 20 q^{48} - 150 q^{49} + 60 q^{51} + 36 q^{52} + 316 q^{55} + 136 q^{57} + 186 q^{58} + 180 q^{60} + 336 q^{61} + 4 q^{63} + 26 q^{64} - 124 q^{66} - 24 q^{67} - 240 q^{69} + 42 q^{70} - 280 q^{72} - 182 q^{73} - 136 q^{75} - 264 q^{76} + 40 q^{78} - 460 q^{79} + 158 q^{81} - 72 q^{82} + 24 q^{84} - 36 q^{85} + 660 q^{87} - 266 q^{88} - 16 q^{90} + 84 q^{91} + 36 q^{93} + 52 q^{94} - 330 q^{96} + 516 q^{97} + 40 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 6 * q^4 + 10 * q^6 - 4 * q^7 + 2 * q^9 - 44 * q^10 - 48 * q^12 + 32 * q^13 - 50 * q^15 - 10 * q^16 + 40 * q^18 + 112 * q^19 + 88 * q^21 - 58 * q^22 + 70 * q^24 - 42 * q^25 - 44 * q^27 + 78 * q^28 - 12 * q^30 - 18 * q^31 - 90 * q^33 + 48 * q^34 + 6 * q^36 - 120 * q^37 - 64 * q^39 + 42 * q^40 - 70 * q^42 - 264 * q^43 + 80 * q^45 + 24 * q^46 + 20 * q^48 - 150 * q^49 + 60 * q^51 + 36 * q^52 + 316 * q^55 + 136 * q^57 + 186 * q^58 + 180 * q^60 + 336 * q^61 + 4 * q^63 + 26 * q^64 - 124 * q^66 - 24 * q^67 - 240 * q^69 + 42 * q^70 - 280 * q^72 - 182 * q^73 - 136 * q^75 - 264 * q^76 + 40 * q^78 - 460 * q^79 + 158 * q^81 - 72 * q^82 + 24 * q^84 - 36 * q^85 + 660 * q^87 - 266 * q^88 - 16 * q^90 + 84 * q^91 + 36 * q^93 + 52 * q^94 - 330 * q^96 + 516 * q^97 + 40 * 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	33.3.h.a

Level	$33$
Weight	$3$
Character orbit	33.h
Analytic conductor	$0.899$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{8} - 4 T^{7} + \cdots + 6561 \) T^8 - 4T^7 + 7T^6 + 8T^5 - 95T^4 + 72T^3 + 567T^2 - 2916*T + 6561
$5$	\( T^{8} - 4 T^{6} + \cdots + 707281 \) T^8 - 4T^6 + 1446T^4 - 49619*T^2 + 707281
$7$	\( (T^{4} + 2 T^{3} + \cdots + 361)^{2} \) (T^4 + 2T^3 + 64T^2 - 247*T + 361)^2
$11$	\( T^{8} - 316 T^{6} + \cdots + 214358881 \) T^8 - 316T^6 + 49126T^4 - 4626556*T^2 + 214358881
$13$	\( (T^{4} - 16 T^{3} + \cdots + 13456)^{2} \) (T^4 - 16T^3 + 136T^2 - 696*T + 13456)^2
$17$	\( T^{8} - 36 T^{6} + \cdots + 1679616 \) T^8 - 36T^6 + 1296T^4 - 46656*T^2 + 1679616
$19$	\( (T^{4} - 56 T^{3} + \cdots + 80656)^{2} \) (T^4 - 56T^3 + 1336T^2 - 9656*T + 80656)^2
$23$	\( (T^{4} + 1008 T^{2} + 215296)^{2} \) (T^4 + 1008*T^2 + 215296)^2
$29$	\( T^{8} + \cdots + 2449228130001 \) T^8 + 261T^6 + 3966246T^4 - 5014263204*T^2 + 2449228130001
$31$	\( (T^{4} + 9 T^{3} + \cdots + 36481)^{2} \) (T^4 + 9T^3 + 796T^2 - 8786*T + 36481)^2
$37$	\( (T^{4} + 60 T^{3} + \cdots + 336400)^{2} \) (T^4 + 60T^3 + 1840T^2 + 29000*T + 336400)^2
$41$	\( T^{8} + \cdots + 3544535296 \) T^8 + 1124T^6 + 3554016T^4 - 181942016*T^2 + 3544535296
$43$	\( (T^{2} + 66 T + 1044)^{4} \) (T^2 + 66*T + 1044)^4
$47$	\( T^{8} - 496 T^{6} + \cdots + 33362176 \) T^8 - 496T^6 + 92896T^4 + 947264*T^2 + 33362176
$53$	\( T^{8} + \cdots + 3969126001 \) T^8 + 101T^6 + 199126T^4 - 45612724*T^2 + 3969126001
$59$	\( T^{8} + \cdots + 276281640625 \) T^8 - 100T^6 + 903750T^4 - 775296875*T^2 + 276281640625
$61$	\( (T^{4} - 168 T^{3} + \cdots + 6533136)^{2} \) (T^4 - 168T^3 + 12024T^2 - 260712*T + 6533136)^2
$67$	\( (T^{2} + 6 T - 5436)^{4} \) (T^2 + 6*T - 5436)^4
$71$	\( T^{8} + \cdots + 11\!\cdots\!56 \) T^8 - 9904T^6 + 45544416T^4 - 74244860864*T^2 + 1134774622925056
$73$	\( (T^{4} + 91 T^{3} + \cdots + 866761)^{2} \) (T^4 + 91T^3 + 3136T^2 - 13034*T + 866761)^2
$79$	\( (T^{4} + 230 T^{3} + \cdots + 55428025)^{2} \) (T^4 + 230T^3 + 25360T^2 + 1302875*T + 55428025)^2
$83$	\( T^{8} + \cdots + 22\!\cdots\!01 \) T^8 - 12004T^6 + 150136006T^4 - 1875937425499*T^2 + 22526107869174001
$89$	\( (T^{4} + 9632 T^{2} + 891136)^{2} \) (T^4 + 9632*T^2 + 891136)^2
$97$	\( (T^{4} - 258 T^{3} + \cdots + 147403881)^{2} \) (T^4 - 258T^3 + 36864T^2 - 2877417*T + 147403881)^2
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\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(-\zeta_{20}^{2}\)	\(-1\)