LMFDB - Newform orbit 450.2.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,2,Mod(17,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 13]))

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

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

chi := DirichletCharacter("450.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	450.s (of order $20$, degree $8$, 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:	$3.59326809096$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$\zeta_{40}^{2}$

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} $

17.1

−0.453990	+	0.891007i
0.453990	−	0.891007i
−0.453990	−	0.891007i
0.453990	+	0.891007i
−0.891007	+	0.453990i
0.891007	−	0.453990i
−0.891007	−	0.453990i
0.891007	+	0.453990i
0.156434	+	0.987688i
−0.156434	−	0.987688i
0.156434	−	0.987688i
−0.156434	+	0.987688i
0.987688	+	0.156434i
−0.987688	−	0.156434i
0.987688	−	0.156434i
−0.987688	+	0.156434i

−0.891007

−

0.453990i

0.587785

0.809017i

−1.16110

−

1.91098i

−1.83927

−

1.83927i

−0.156434

−

0.987688i

0.166977

2.22982i

17.2

0.891007

0.453990i

0.587785

0.809017i

1.16110

1.91098i

−1.83927

−

1.83927i

0.156434

0.987688i

0.166977

2.22982i

53.1

−0.891007

0.453990i

0.587785

−

0.809017i

−1.16110

1.91098i

−1.83927

1.83927i

−0.156434

0.987688i

0.166977

−

2.22982i

53.2

0.891007

−

0.453990i

0.587785

−

0.809017i

1.16110

−

1.91098i

−1.83927

1.83927i

0.156434

−

0.987688i

0.166977

−

2.22982i

197.1

−0.453990

−

0.891007i

−0.587785

0.809017i

−0.183900

2.22849i

−3.01484

−

3.01484i

0.987688

0.156434i

2.06909

−

0.847859i

197.2

0.453990

0.891007i

−0.587785

0.809017i

0.183900

−

2.22849i

−3.01484

−

3.01484i

−0.987688

−

0.156434i

2.06909

−

0.847859i

233.1

−0.453990

0.891007i

−0.587785

−

0.809017i

−0.183900

−

2.22849i

−3.01484

3.01484i

0.987688

−

0.156434i

2.06909

0.847859i

233.2

0.453990

−

0.891007i

−0.587785

−

0.809017i

0.183900

2.22849i

−3.01484

3.01484i

−0.987688

0.156434i

2.06909

0.847859i

287.1

−0.987688

0.156434i

0.951057

−

0.309017i

0.863541

−

2.06259i

1.87811

1.87811i

−0.891007

0.453990i

−0.530249

2.17229i

287.2

0.987688

−

0.156434i

0.951057

−

0.309017i

−0.863541

2.06259i

1.87811

1.87811i

0.891007

−

0.453990i

−0.530249

2.17229i

323.1

−0.987688

−

0.156434i

0.951057

0.309017i

0.863541

2.06259i

1.87811

−

1.87811i

−0.891007

−

0.453990i

−0.530249

−

2.17229i

323.2

0.987688

0.156434i

0.951057

0.309017i

−0.863541

−

2.06259i

1.87811

−

1.87811i

0.891007

0.453990i

−0.530249

−

2.17229i

377.1

−0.156434

0.987688i

−0.951057

−

0.309017i

1.69480

1.45865i

−0.0240055

−

0.0240055i

0.453990

−

0.891007i

−1.70582

1.44575i

377.2

0.156434

−

0.987688i

−0.951057

−

0.309017i

−1.69480

−

1.45865i

−0.0240055

−

0.0240055i

−0.453990

0.891007i

−1.70582

1.44575i

413.1

−0.156434

−

0.987688i

−0.951057

0.309017i

1.69480

−

1.45865i

−0.0240055

0.0240055i

0.453990

0.891007i

−1.70582

−

1.44575i

413.2

0.156434

0.987688i

−0.951057

0.309017i

−1.69480

1.45865i

−0.0240055

0.0240055i

−0.453990

−

0.891007i

−1.70582

−

1.44575i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
25.f	odd	20	1	inner
75.l	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.2.s.a	✓	16
3.b	odd	2	1	inner	450.2.s.a	✓	16
25.f	odd	20	1	inner	450.2.s.a	✓	16
75.l	even	20	1	inner	450.2.s.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.2.s.a	✓	16	1.a	even	1	1	trivial
450.2.s.a	✓	16	3.b	odd	2	1	inner
450.2.s.a	✓	16	25.f	odd	20	1	inner
450.2.s.a	✓	16	75.l	even	20	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} + 6T_{7}^{7} + 18T_{7}^{6} + 51T_{7}^{4} + 300T_{7}^{3} + 882T_{7}^{2} + 42T_{7} + 1 $ T7^8 + 6*T7^7 + 18*T7^6 + 51*T7^4 + 300*T7^3 + 882*T7^2 + 42*T7 + 1 acting on $S_{2}^{\mathrm{new}}(450, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{12} + \cdots + 1 $ T^16 - T^12 + T^8 - T^4 + 1
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 20 T^{14} + \cdots + 390625 $ T^16 + 20T^14 + 215T^12 + 1600T^10 + 9025T^8 + 40000T^6 + 134375T^4 + 312500*T^2 + 390625
$7$	$ (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 1)^{2} $ (T^8 + 6T^7 + 18T^6 + 51T^4 + 300T^3 + 882T^2 + 42T + 1)^2
$11$	$ T^{16} - 56 T^{14} + \cdots + 38950081 $ T^16 - 56T^14 + 1257T^12 - 3568T^10 + 89030T^8 - 416872T^6 + 1921212T^4 - 7014884*T^2 + 38950081
$13$	$ (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 16)^{2} $ (T^8 + 6T^7 + 18T^6 + 200T^5 + 1436T^4 + 1040T^3 + 392T^2 + 112*T + 16)^2
$17$	$ T^{16} - 256 T^{12} + \cdots + 16777216 $ T^16 - 256T^12 + 25600T^10 + 352256T^8 - 6553600T^6 + 21233664T^4 + 20971520T^2 + 16777216
$19$	$ (T^{8} + 10 T^{7} + \cdots + 26896)^{2} $ (T^8 + 10T^7 + 26T^6 - 80T^5 - 604T^4 - 880T^3 + 4136T^2 + 16400*T + 26896)^2
$23$	$ T^{16} - 20 T^{14} + \cdots + 256 $ T^16 - 20T^14 + 284T^12 - 160T^10 + 976T^8 - 19840T^6 + 42944T^4 + 6400*T^2 + 256
$29$	$ T^{16} + \cdots + 723394816 $ T^16 - 76T^14 + 13612T^12 - 228448T^10 + 21510480T^8 + 515494528T^6 + 4627945152T^4 + 2935321856*T^2 + 723394816
$31$	$ (T^{8} - 2 T^{7} + \cdots + 2253001)^{2} $ (T^8 - 2T^7 + 13T^6 + 196T^5 + 3330T^4 - 24754T^3 + 293328T^2 + 312208*T + 2253001)^2
$37$	$ (T^{8} - 4 T^{7} + \cdots + 633616)^{2} $ (T^8 - 4T^7 - 72T^6 - 380T^5 + 2136T^4 + 27280T^3 + 112152T^2 + 73232*T + 633616)^2
$41$	$ T^{16} + \cdots + 2186423566336 $ T^16 - 16T^14 + 10432T^12 - 974848T^10 + 41472000T^8 - 741343232T^6 + 11828723712T^4 - 183211393024*T^2 + 2186423566336
$43$	$ (T^{4} - 10 T^{3} + \cdots + 100)^{4} $ (T^4 - 10T^3 + 50T^2 + 100*T + 100)^4
$47$	$ T^{16} + \cdots + 16000000000000 $ T^16 + 2000T^12 + 24000000T^8 - 32000000000*T^4 + 16000000000000
$53$	$ T^{16} + \cdots + 11\!\cdots\!21 $ T^16 + 160T^14 + 12884T^12 - 696980T^10 - 38607874T^8 + 7927288880T^6 - 9813305851T^4 - 34027940361460*T^2 + 1196202532717921
$59$	$ T^{16} + \cdots + 18\!\cdots\!21 $ T^16 - 24T^14 + 41657T^12 + 4362288T^10 + 443328430T^8 + 45787227672T^6 + 4086480564452T^4 + 133845772935084*T^2 + 1875718074012721
$61$	$ (T^{8} - 60 T^{6} + \cdots + 1488400)^{2} $ (T^8 - 60T^6 + 140T^5 + 18960T^4 + 106400T^3 + 753000T^2 + 805200T + 1488400)^2
$67$	$ (T^{8} - 12 T^{7} + \cdots + 430336)^{2} $ (T^8 - 12T^7 + 72T^6 - 160T^5 - 64T^4 + 4160T^3 + 4928T^2 + 146944*T + 430336)^2
$71$	$ T^{16} + \cdots + 26639462656 $ T^16 - 44T^14 + 8332T^12 - 860192T^10 + 43496400T^8 - 927203968T^6 + 8310451392T^4 + 6865517824*T^2 + 26639462656
$73$	$ (T^{8} + 38 T^{7} + \cdots + 4682896)^{2} $ (T^8 + 38T^7 + 822T^6 + 10320T^5 + 75436T^4 + 323760T^3 + 1001208T^2 + 2847824*T + 4682896)^2
$79$	$ (T^{8} + 40 T^{7} + \cdots + 516961)^{2} $ (T^8 + 40T^7 + 596T^6 + 3980T^5 + 14206T^4 + 27680T^3 + 44501T^2 + 186940*T + 516961)^2
$83$	$ T^{16} + \cdots + 838401348535921 $ T^16 - 580T^14 + 124149T^12 - 11433280T^10 + 531736406T^8 - 37981293620T^6 + 20945981381324T^4 + 253365766195080*T^2 + 838401348535921
$89$	$ T^{16} + \cdots + 324415429079296 $ T^16 + 116T^14 + 47252T^12 + 11725728T^10 + 1453925600T^8 + 85325389632T^6 + 2115994810112T^4 - 5990204596736*T^2 + 324415429079296
$97$	$ (T^{8} + 16 T^{7} + \cdots + 47045881)^{2} $ (T^8 + 16T^7 + 3T^6 - 1050T^5 - 7124T^4 + 101460T^3 + 1856262T^2 + 6776692*T + 47045881)^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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.s (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + ( - \zeta_{40}^{15} + \cdots - \zeta_{40}) q^{5} + \cdots - \zeta_{40}^{13} q^{8} +O(q^{10}) \) q + z^11 * q^2 - z^2 * q^4 + (-z^15 + z^13 + z^11 - z^9 - z^7 + 2z^5 + 2z^3 - z) * q^5 + (2z^14 - 2z^12 - 2z^10 + z^8 + 2z^6 - z^2) * q^7 - z^13 * q^8	\( q + \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + ( - \zeta_{40}^{15} + \cdots - \zeta_{40}) q^{5} + \cdots + ( - 3 \zeta_{40}^{15} + \cdots + \zeta_{40}) q^{98} +O(q^{100}) \) q + z^11 * q^2 - z^2 * q^4 + (-z^15 + z^13 + z^11 - z^9 - z^7 + 2z^5 + 2z^3 - z) * q^5 + (2z^14 - 2z^12 - 2z^10 + z^8 + 2z^6 - z^2) * q^7 - z^13 * q^8 + (z^14 + z^12 + z^10 - 2z^8 + z^4 - 1) q^10 + (-2z^15 + 2z^13 + z^11 - 2z^9 + z^7 + 3z^5 - z) * q^11 + (-z^14 - 2z^8 + 2z^6 + 3z^4 - 2z^2 - 2) * q^13 + (z^15 + z^13 - z^11 - 2z^9 + z^7 + z^3) q^14 + z^4 * q^16 + (-2z^15 - 2z^13 + 2z^11 + 4z^9 - 2z^5 + 2z^3 + 2z) q^17 + (3z^14 - z^10 - z^8 + z^6 + 2z^4 - 3z^2 - 2) q^19 + (-z^15 + z^11 - 2z^7 - z^5 + z^3 - z) q^20 + (z^14 + 2z^12 - z^10 - 3z^8 + 3z^6 + z^4 - 2z^2 - 1) * q^22 + (z^11 - 2z^7 + 2z^3 + z) * q^23 + (-2z^14 + 2z^12 + 2z^10 + 4z^4 + z^2 - 4) * q^25 + (z^15 - 2z^9 - 2z^7 + 3z^5 + 2z^3 - 2z) q^26 + (2z^14 - z^10 - z^4 + 2) q^28 + (-4z^15 + 3z^13 + 4z^11 - 3z^9 - z^7 - z^5 + z) * q^29 + (2z^14 + z^12 - 4z^10 + z^8 - 4z^6 + z^4 + 2z^2) * q^31 + z^15 * q^32 + (2z^14 + 2z^8 + 2z^6 - 2z^2 - 2) * q^34 + (-3z^15 + z^13 - 2z^11 + 2z^9 + 3z^7 - 3z^3 - z) q^35 + (2z^14 - 4z^12 + 2z^10 + 5z^8 - z^6 - z^4 - 3z^2 + 3) q^37 + (z^15 - 2z^13 - z^11 - z^9 - z^7 - 2z^5 + z^3) * q^38 + (-z^14 - 2z^12 + 2z^10 + z^8 - z^6 - z^4 + z^2 + 1) * q^40 + (6z^15 + 4z^13 - 2z^11 + 2z^9 + 2z^7 - 2z^3) * q^41 + (3z^14 + 3z^12 - 4z^10 - 3z^8 + 3z^6 + 1) q^43 + (-2z^15 + z^13 + 2z^11 - 3z^9 - 3z^7 + 2z^5 + z^3 - 2z) * q^44 + (z^12 + 2z^10 - 2z^6 + z^2) * q^46 + (4z^13 + 2z^9 + 2z^5 + 4z) * q^47 + (-4z^14 - 3z^12 + 3z^10 + 3z^8 - 4z^6 - 6z^4 + 3) * q^49 + (4z^15 + z^13 - 4z^11 + 2z^5 - 2z^3 - 2z) q^50 + (z^12 + 2z^10 - 3z^8 - 3z^6 + 3z^4 + 2z^2 - 1) q^52 + (2z^15 - 5z^13 - z^11 - 3z^9 + z^7 + 3z^5 - z^3 + 5z) q^53 + (-4z^14 + 6z^12 + 4z^10 - 3z^8 + z^6 + 4z^4 - 2z^2 - 2) * q^55 + (-z^15 + 2z^11 - 2z^5 + z) * q^56 + (-z^14 + z^10 + z^8 + 3z^6 - 4z^4 - 3z^2 + 4) q^58 + (7z^13 - 7z^11 - 7z^9 + z^7 + 2z^5 - 5z^3 - 6z) * q^59 + (-10z^14 - 2z^12 + 3z^10 + 3z^8 - 5z^6 - 3z^4 + 7z^2 + 2) q^61 + (2z^15 - 2z^13 - z^11 + 4z^9 + z^7 - 6z^5 - 2z^3 + 8z) * q^62 - z^6 * q^64 + (-z^15 - 6z^13 + 2z^11 + 8z^9 - 2z^7 - 3z^5 - 2z) * q^65 + (-4z^12 - 4z^10 + 4z^8 - 2z^4 - 2z^2 + 4) q^67 + (2z^15 - 4z^11 - 2z^9 + 2z^7 - 2z^3 - 2z) * q^68 + (-z^12 - 3z^10 + 6z^6 - z^4 - z^2 - 2) * q^70 + (-2z^15 + 3z^13 + 2z^11 - 3z^9 - 5z^7 + 5z^5 - 5z) q^71 + (2z^14 - 3z^12 - 6z^10 - z^8 + 3z^6 + 3z^4 + 2z^2 - 5) * q^73 + (4z^15 - 4z^13 - 2z^11 + z^9 + 5z^7 - 3z^5 - z^3 - z) q^74 + (-2z^12 + z^10 + 2z^8 - 2z^6 + 2z^2 + 3) * q^76 + (-3z^11 + 8z^9 - 8z^5 + 4z) * q^77 + (-z^14 + 2z^12 + z^10 - 4z^8 - 2z^4 + 5z^2 - 6) * q^79 + (z^9 + z^7 + z^3 - z) * q^80 + (-2z^10 - 4z^6 - 4z^4 - 2) q^82 + (2z^15 - 11z^13 - 4z^11 + 6z^7 - z^5 - 3z^3 + z) q^83 + (2z^14 + 4z^12 + 4z^10 + 2z^8 - 2z^6 + 6z^2 + 2) * q^85 + (-3z^15 + 3z^13 + 4z^11 - 3z^9 - 3z^7 + z) q^86 + (-2z^14 + 3z^10 - 2z^8 - z^6 + z^4 + z^2 + 1) q^88 + (-3z^15 - 3z^13 + 2z^11 + z^9 + 5z^7 - 8z^5 + 2z^3 + z) * q^89 + (-2z^14 + 4z^10 + 9z^8 - 3z^6 - 15z^4 + z^2 + 9) q^91 + (-z^13 + 2z^9 - 2z^5 - z^3) * q^92 + (6z^12 - 2z^8 - 2z^4 - 4) q^94 + (z^15 + 2z^13 - 5z^11 + z^9 + z^7 - 2z^5 - 3z^3 - 6z) q^95 + (-z^14 - 8z^12 + 2z^10 + 2z^8 - 2z^6 - 2z^4 + 8z^2 + 1) * q^97 + (-3z^15 - 4z^13 + 4z^9 + 3z^7 + z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{7}+O(q^{10}) \) 16 * q - 12 * q^7	\( 16 q - 12 q^{7} - 12 q^{13} + 4 q^{16} - 20 q^{19} + 8 q^{22} - 40 q^{25} + 28 q^{28} + 4 q^{31} - 40 q^{34} + 8 q^{37} + 40 q^{43} + 4 q^{46} + 12 q^{52} + 20 q^{55} + 44 q^{58} + 24 q^{67} - 40 q^{70} - 76 q^{73} + 32 q^{76} - 80 q^{79} - 48 q^{82} + 40 q^{85} + 28 q^{88} + 48 q^{91} - 40 q^{94} - 32 q^{97}+O(q^{100}) \) 16 * q - 12 * q^7 - 12 * q^13 + 4 * q^16 - 20 * q^19 + 8 * q^22 - 40 * q^25 + 28 * q^28 + 4 * q^31 - 40 * q^34 + 8 * q^37 + 40 * q^43 + 4 * q^46 + 12 * q^52 + 20 * q^55 + 44 * q^58 + 24 * q^67 - 40 * q^70 - 76 * q^73 + 32 * q^76 - 80 * q^79 - 48 * q^82 + 40 * q^85 + 28 * q^88 + 48 * q^91 - 40 * q^94 - 32 * q^97

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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	450.2.s.a

Level	$450$
Weight	$2$
Character orbit	450.s
Analytic conductor	$3.593$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{20})\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 17.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - T^{12} + \cdots + 1 \) T^16 - T^12 + T^8 - T^4 + 1
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 20 T^{14} + \cdots + 390625 \) T^16 + 20T^14 + 215T^12 + 1600T^10 + 9025T^8 + 40000T^6 + 134375T^4 + 312500*T^2 + 390625
$7$	\( (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 1)^{2} \) (T^8 + 6T^7 + 18T^6 + 51T^4 + 300T^3 + 882T^2 + 42T + 1)^2
$11$	\( T^{16} - 56 T^{14} + \cdots + 38950081 \) T^16 - 56T^14 + 1257T^12 - 3568T^10 + 89030T^8 - 416872T^6 + 1921212T^4 - 7014884*T^2 + 38950081
$13$	\( (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 16)^{2} \) (T^8 + 6T^7 + 18T^6 + 200T^5 + 1436T^4 + 1040T^3 + 392T^2 + 112*T + 16)^2
$17$	\( T^{16} - 256 T^{12} + \cdots + 16777216 \) T^16 - 256T^12 + 25600T^10 + 352256T^8 - 6553600T^6 + 21233664T^4 + 20971520T^2 + 16777216
$19$	\( (T^{8} + 10 T^{7} + \cdots + 26896)^{2} \) (T^8 + 10T^7 + 26T^6 - 80T^5 - 604T^4 - 880T^3 + 4136T^2 + 16400*T + 26896)^2
$23$	\( T^{16} - 20 T^{14} + \cdots + 256 \) T^16 - 20T^14 + 284T^12 - 160T^10 + 976T^8 - 19840T^6 + 42944T^4 + 6400*T^2 + 256
$29$	\( T^{16} + \cdots + 723394816 \) T^16 - 76T^14 + 13612T^12 - 228448T^10 + 21510480T^8 + 515494528T^6 + 4627945152T^4 + 2935321856*T^2 + 723394816
$31$	\( (T^{8} - 2 T^{7} + \cdots + 2253001)^{2} \) (T^8 - 2T^7 + 13T^6 + 196T^5 + 3330T^4 - 24754T^3 + 293328T^2 + 312208*T + 2253001)^2
$37$	\( (T^{8} - 4 T^{7} + \cdots + 633616)^{2} \) (T^8 - 4T^7 - 72T^6 - 380T^5 + 2136T^4 + 27280T^3 + 112152T^2 + 73232*T + 633616)^2
$41$	\( T^{16} + \cdots + 2186423566336 \) T^16 - 16T^14 + 10432T^12 - 974848T^10 + 41472000T^8 - 741343232T^6 + 11828723712T^4 - 183211393024*T^2 + 2186423566336
$43$	\( (T^{4} - 10 T^{3} + \cdots + 100)^{4} \) (T^4 - 10T^3 + 50T^2 + 100*T + 100)^4
$47$	\( T^{16} + \cdots + 16000000000000 \) T^16 + 2000T^12 + 24000000T^8 - 32000000000*T^4 + 16000000000000
$53$	\( T^{16} + \cdots + 11\!\cdots\!21 \) T^16 + 160T^14 + 12884T^12 - 696980T^10 - 38607874T^8 + 7927288880T^6 - 9813305851T^4 - 34027940361460*T^2 + 1196202532717921
$59$	\( T^{16} + \cdots + 18\!\cdots\!21 \) T^16 - 24T^14 + 41657T^12 + 4362288T^10 + 443328430T^8 + 45787227672T^6 + 4086480564452T^4 + 133845772935084*T^2 + 1875718074012721
$61$	\( (T^{8} - 60 T^{6} + \cdots + 1488400)^{2} \) (T^8 - 60T^6 + 140T^5 + 18960T^4 + 106400T^3 + 753000T^2 + 805200T + 1488400)^2
$67$	\( (T^{8} - 12 T^{7} + \cdots + 430336)^{2} \) (T^8 - 12T^7 + 72T^6 - 160T^5 - 64T^4 + 4160T^3 + 4928T^2 + 146944*T + 430336)^2
$71$	\( T^{16} + \cdots + 26639462656 \) T^16 - 44T^14 + 8332T^12 - 860192T^10 + 43496400T^8 - 927203968T^6 + 8310451392T^4 + 6865517824*T^2 + 26639462656
$73$	\( (T^{8} + 38 T^{7} + \cdots + 4682896)^{2} \) (T^8 + 38T^7 + 822T^6 + 10320T^5 + 75436T^4 + 323760T^3 + 1001208T^2 + 2847824*T + 4682896)^2
$79$	\( (T^{8} + 40 T^{7} + \cdots + 516961)^{2} \) (T^8 + 40T^7 + 596T^6 + 3980T^5 + 14206T^4 + 27680T^3 + 44501T^2 + 186940*T + 516961)^2
$83$	\( T^{16} + \cdots + 838401348535921 \) T^16 - 580T^14 + 124149T^12 - 11433280T^10 + 531736406T^8 - 37981293620T^6 + 20945981381324T^4 + 253365766195080*T^2 + 838401348535921
$89$	\( T^{16} + \cdots + 324415429079296 \) T^16 + 116T^14 + 47252T^12 + 11725728T^10 + 1453925600T^8 + 85325389632T^6 + 2115994810112T^4 - 5990204596736*T^2 + 324415429079296
$97$	\( (T^{8} + 16 T^{7} + \cdots + 47045881)^{2} \) (T^8 + 16T^7 + 3T^6 - 1050T^5 - 7124T^4 + 101460T^3 + 1856262T^2 + 6776692*T + 47045881)^2
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\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(\zeta_{40}^{2}\)