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Properties

Label	603.2.u.a

Level	$603$
Weight	$2$
Character orbit	603.u
Analytic conductor	$4.815$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [603,2,Mod(64,603)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(603, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("603.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.u (of order \(11\), degree \(10\), 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:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 67)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-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 primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\zeta_{22}^{9} - \zeta_{22}^{6} + \cdots - 1) q^{2}+ \cdots + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots + 1) q^{8}+O(q^{10}) \) q + (z^9 - z^6 - z^4 + z^3 - z^2 + z - 1) * q^2 + (-z^8 - z^7 + z^6 + z^4 + z^2 - z - 1) * q^4 + (-z^5 + 1) * q^5 + (z^9 - z^8 + z^7 - z^6 + z^3 - z^2 + z) * q^7 + (-z^9 + z^7 + 2z^6 + z^5 - z^3 - z + 1) q^8	\( q + (\zeta_{22}^{9} - \zeta_{22}^{6} + \cdots - 1) q^{2}+ \cdots + (5 \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots - 6) q^{98}+O(q^{100}) \) q + (z^9 - z^6 - z^4 + z^3 - z^2 + z - 1) * q^2 + (-z^8 - z^7 + z^6 + z^4 + z^2 - z - 1) * q^4 + (-z^5 + 1) * q^5 + (z^9 - z^8 + z^7 - z^6 + z^3 - z^2 + z) * q^7 + (-z^9 + z^7 + 2z^6 + z^5 - z^3 - z + 1) q^8 + (2z^9 - z^8 + z^7 - 2z^6 + z^5 - z^4 + 2z^3 - z^2 + z - 2) q^10 + (z^9 - z^8 + z^7 - z^5 - z^4 - z^3 + z^2 + z + 1) * q^11 + (3z^9 - z^8 + z^7 - 2z^6 + z^5 - 2z^4 + z^3 - z^2 + 3z) * q^13 + (-z^8 - z^6 + z^5 + z^3 - z^2 - 1) * q^14 + (-2z^9 + 3z^8 - 2z^7 - 3z^5 + z^4 - 3z^3 + 3z^2 - z + 3) * q^16 + (-z^8 - 3z^7 - z^6) q^17 + (-4z^9 - z^7 + 2z^6 - 3z^5 + 4z^4 - 4z^3 + 3z^2 - 2z + 1) q^19 + (-z^9 - z^8 - 2z^7 + 2z^6 + z^5 + z^4 - 2z) q^20 + (5z^9 - 4z^8 + 3z^7 - 7z^6 + 5z^5 - 5z^4 + 7z^3 - 3z^2 + 4z - 5) q^22 + (3z^9 - 4z^8 + 4z^7 - 4z^6 + 4z^5 - 3z^4 - 2z^2 + 3z - 2) * q^23 + (z^9 - z^8 + z^7 - z^6 + 4z^5 - z^4 + z^3 - z^2 + z) q^25 + (3z^9 - 4z^8 - z^7 - 4z^6 + 3z^5 + 5z^3 - 2z^2 + 2z - 5) q^26 + (-z^9 - z^7 + 4z^6 - z^5 - z^3 - 2z + 2) * q^28 + (2z^9 - 3z^8 + z^7 - z^6 + z^5 - z^4 + 3z^3 - 2z^2 + 1) * q^29 + (2z^9 - z^8 - 2z^7 + 3z^6 - 2z^5 - z^4 + 2z^3) q^31 + (z^9 + z^8 + z^7 - 3z^6 + 2z^5 - 2z^4 + 3z^3 - z^2 - z - 1) * q^32 + (z^7 + 3z^6 + 2z^5 + 2z^4 - 2z^3 - 2z^2 - 3z - 1) * q^34 + (z^9 - 2z^8 + 2z^7 - 2z^6 + 2z^3 - 2z^2 + 2z - 1) * q^35 + (2z^9 - 4z^6 + 4z^5 - 2z^2 - 1) * q^37 + (2z^9 + 7z^7 - 4z^6 + 4z^5 - 7z^4 - 2z^2 + 1) * q^38 + (-2z^9 + 2z^8 + 4z^6 - z^5 + z^4 - 3z^3 + z^2 - z + 4) * q^40 + (-2z^9 + 2z^8 - 6z^7 + 2z^6 - 2z^5 - 3z^3 + 3z^2 - 3z + 3) * q^41 + (-3z^9 - 2z^8 - z^7 - 2z^6 - 3z^5 - z^3 + 4z^2 - 4z + 1) * q^43 + (-6z^9 + z^8 - 6z^7 + 6z^6 - z^5 + 6z^4 - z^2 - 6z - 1) q^44 + (-2z^6 + 5z^5 - 4z^4 + 6z^3 - 4z^2 + 5z - 2) * q^46 + (5z^9 - 4z^8 + 3z^7 - 3z^6 + 4z^5 - 5z^4 - 7z^2 + 6z - 7) * q^47 + (-2z^8 + 3z^7 - 3z^6 + z^5 - 6z^4 + z^3 - 3z^2 + 3z - 2) * q^49 + (-3z^9 + 3z^8 - 4z^7 + 3z^6 - 3z^5 - 2z^3 - z^2 + z + 2) * q^50 + (-9z^9 - 2z^7 + 9z^6 - z^5 + 5z^4 - 5z^3 + z^2 - 9z + 2) * q^52 + (-2z^8 + 2z^7 + 3z^6 + z^5 + 3z^4 + z^3 + 3z^2 + 2z - 2) * q^53 + (3z^9 - z^8 + z^7 - 2z^6 - z^5 - 2z^4 + z^3 - z^2 + 3z) * q^55 + (2z^9 + 2z^7 - z^5 - 2z^4 + 2z + 1) * q^56 + (-2z^9 + z^8 - z^7 + 2z^6 - 3z^2) q^58 + (-z^9 + z^8 - 6z^7 - 6z^5 + z^4 - z^3 - 6z + 6) q^59 + (-z^9 + z^8 - 3z^6 - 5z^5 + 3z^4 - 5z^3 + 3z^2 - 5z - 3) * q^61 + (z^9 - 4z^8 + z^7 + 3z^4 - 2z^3 + 2z^2 - 3z) q^62 + (z^9 + z^8 - z^5 - z^4 - 4z^2 + 5z - 4) * q^64 + (4z^9 - z^8 + z^7 - 4z^6 - z^4 + 3z^3 - z^2 + 3z - 1) * q^65 + (2z^9 + 2z^8 - 3z^7 + z^6 - z^5 - 4z^4 - z^3 - 4z^2 + 3) q^67 + (-4z^9 + 5z^8 + 2z^7 + 3z^6 - 3z^5 - 2z^4 - 5z^3 + 4z^2 + 5) * q^68 + (-z^9 - z^8 + z^5 + z^4 - z^2 - z - 1) * q^70 + (4z^7 - z^6 + 4z^5 + 2z^4 + 4z^3 - z^2 + 4z) q^71 + (-3z^9 + 3z^8 + z^6 + z^5 + z^4 + z^2 + z + 1) * q^73 + (-5z^9 + 4z^8 - 4z^7 + 5z^6 + 3z^4 - z^3 - z^2 - z + 3) q^74 + (6z^9 - 6z^6 - 4z^4 + 11z^3 - 7z^2 + 11z - 4) * q^76 + (3z^9 - 4z^8 + 2z^7 - 3z^6 + z^5 - z^4 + 3z^3 - 2z^2 + 4z - 3) q^77 + (-5z^9 + 5z^8 + z^7 - z^5 + z^3 + 5z^2 - 5z) * q^79 + (3z^8 - 2z^7 - 2z^6 - 3z^5 - 2z^4 - 2z^3 + 3z^2) q^80 + (5z^9 + 3z^7 + 2z^5 - z^4 + z^3 - 2z^2 - 3) * q^82 + (-z^9 + 10z^8 - z^7 - 3z^5 + 3z^4 - 7z^3 + 7z^2 - 3z + 3) * q^83 + (-z^8 - 3z^7 - z^6 - z^2 - 3z - 1) * q^85 + (z^9 + 6z^7 - z^6 + 9z^5 - 7z^4 + 7z^3 - 9z^2 + z - 6) q^86 + (-8z^9 + 8z^8 + 11z^6 - 7z^5 + 5z^4 - 11z^3 + 5z^2 - 7z + 11) * q^88 + (-7z^9 + 10z^8 - 4z^7 + 10z^6 - 11z^5 + 11z^4 - 10z^3 + 4z^2 - 10z + 7) q^89 + (3z^9 - 4z^8 + z^7 - 4z^6 + 3z^5 + 3z^3 - 3z^2 + 3z - 3) q^91 + (-4z^9 + 2z^8 - 3z^7 + 8z^6 - 11z^5 + 8z^4 - 3z^3 + 2z^2 - 4z) q^92 + (4z^6 + z^5 + 7z^4 - 2z^3 + 7z^2 + z + 4) * q^94 + (-5z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - 5z^3) * q^95 + (-6z^9 + 6z^8 - 4z^7 + 7z^6 - 7z^5 + 4z^4 - 6z^3 + 6z^2 + 10) * q^97 + (5z^9 + z^8 + z^6 + z^4 + 5z^3 + 6z - 6) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} - 14 q^{4} + 9 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) 10 * q - 4 * q^2 - 14 * q^4 + 9 * q^5 + 7 * q^7 + 7 * q^8	\( 10 q - 4 q^{2} - 14 q^{4} + 9 q^{5} + 7 q^{7} + 7 q^{8} - 8 q^{10} + 12 q^{11} + 15 q^{13} - 5 q^{14} + 12 q^{16} - q^{17} - 13 q^{19} - 6 q^{20} - 7 q^{22} + 7 q^{23} + 12 q^{25} - 28 q^{26} + 10 q^{28} + 24 q^{29} - q^{31} + q^{32} - 15 q^{34} + 3 q^{35} + 2 q^{37} + 36 q^{38} + 25 q^{40} + 7 q^{41} - 2 q^{43} - 41 q^{44} + 6 q^{46} - 33 q^{47} + 2 q^{49} + 4 q^{50} - 21 q^{52} - 21 q^{53} + 13 q^{55} + 17 q^{56} - 3 q^{58} + 38 q^{59} - 50 q^{61} - 4 q^{62} - 31 q^{64} + 8 q^{65} + 32 q^{67} + 30 q^{68} - 10 q^{70} + 16 q^{71} + 3 q^{73} + 8 q^{74} + 5 q^{76} - 7 q^{77} - 19 q^{79} - 9 q^{80} - 16 q^{82} - 5 q^{83} - 13 q^{85} - 19 q^{86} + 48 q^{88} - 7 q^{89} - 6 q^{91} - 45 q^{92} + 22 q^{94} - 15 q^{95} + 54 q^{97} - 47 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 - 14 * q^4 + 9 * q^5 + 7 * q^7 + 7 * q^8 - 8 * q^10 + 12 * q^11 + 15 * q^13 - 5 * q^14 + 12 * q^16 - q^17 - 13 * q^19 - 6 * q^20 - 7 * q^22 + 7 * q^23 + 12 * q^25 - 28 * q^26 + 10 * q^28 + 24 * q^29 - q^31 + q^32 - 15 * q^34 + 3 * q^35 + 2 * q^37 + 36 * q^38 + 25 * q^40 + 7 * q^41 - 2 * q^43 - 41 * q^44 + 6 * q^46 - 33 * q^47 + 2 * q^49 + 4 * q^50 - 21 * q^52 - 21 * q^53 + 13 * q^55 + 17 * q^56 - 3 * q^58 + 38 * q^59 - 50 * q^61 - 4 * q^62 - 31 * q^64 + 8 * q^65 + 32 * q^67 + 30 * q^68 - 10 * q^70 + 16 * q^71 + 3 * q^73 + 8 * q^74 + 5 * q^76 - 7 * q^77 - 19 * q^79 - 9 * q^80 - 16 * q^82 - 5 * q^83 - 13 * q^85 - 19 * q^86 + 48 * q^88 - 7 * q^89 - 6 * q^91 - 45 * q^92 + 22 * q^94 - 15 * q^95 + 54 * q^97 - 47 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(\zeta_{22}^{4}\)	\(1\)

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)

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

0.142315	−	0.989821i
0.959493	−	0.281733i
−0.841254	+	0.540641i
0.654861	−	0.755750i
−0.415415	−	0.909632i
0.142315	+	0.989821i
−0.415415	+	0.909632i
0.959493	+	0.281733i
−0.841254	−	0.540641i
0.654861	+	0.755750i

0.459493

+

0.134919i

0

−1.48958

−

0.957293i

0.345139

+

0.755750i

0

1.04408

+

0.306569i

−1.18251

−

1.36469i

0

0.0566239

+

0.393828i

−1.34125

+

0.861971i

0

0.225136

−

0.492980i

0.857685

+

0.989821i

0

0.313607

−

0.201543i

−0.330830

−

2.30097i

0

−2.00357

−

0.588302i

−0.915415

+

2.00448i

0

−1.87023

−

2.15836i

0.0405070

−

0.281733i

0

0.226900

−

0.496841i

1.80972

−

0.531382i

0

0.527646

+

0.339098i

−0.357685

−

2.48775i

0

−4.14200

+

1.21620i

1.41542

−

0.909632i

0

−0.198939

−

1.38365i

2.41899

+

5.29684i

0

−2.76921

−

3.19584i

0.154861

−

0.178719i

0

0.276671

+

1.92429i

1.84125

−

0.540641i

0

2.11435

−

2.44009i

0.784630

+

0.504251i

0

0.188515

−

0.412791i

0.459493

−

0.134919i

0

−1.48958

+

0.957293i

0.345139

−

0.755750i

0

1.04408

−

0.306569i

−1.18251

+

1.36469i

0

0.0566239

−

0.393828i

0.154861

+

0.178719i

0

0.276671

−

1.92429i

1.84125

+

0.540641i

0

2.11435

+

2.44009i

0.784630

−

0.504251i

0

0.188515

+

0.412791i

−1.34125

−

0.861971i

0

0.225136

+

0.492980i

0.857685

−

0.989821i

0

0.313607

+

0.201543i

−0.330830

+

2.30097i

0

−2.00357

+

0.588302i

−0.915415

−

2.00448i

0

−1.87023

+

2.15836i

0.0405070

+

0.281733i

0

0.226900

+

0.496841i

1.80972

+

0.531382i

0

0.527646

−

0.339098i

−0.357685

+

2.48775i

0

−4.14200

−

1.21620i

1.41542

+

0.909632i

0

−0.198939

+

1.38365i

2.41899

−

5.29684i

0

−2.76921

+

3.19584i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
67.e	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	603.2.u.a		10
3.b	odd	2	1		67.2.e.b	✓	10
67.e	even	11	1	inner	603.2.u.a		10
201.j	even	22	1		4489.2.a.h		5
201.k	odd	22	1		67.2.e.b	✓	10
201.k	odd	22	1		4489.2.a.i		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
67.2.e.b	✓	10	3.b	odd	2	1
67.2.e.b	✓	10	201.k	odd	22	1
603.2.u.a		10	1.a	even	1	1	trivial
603.2.u.a		10	67.e	even	11	1	inner
4489.2.a.h		5	201.j	even	22	1
4489.2.a.i		5	201.k	odd	22	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 4T_{2}^{9} + 16T_{2}^{8} + 31T_{2}^{7} + 47T_{2}^{6} + 23T_{2}^{5} - 18T_{2}^{4} - 39T_{2}^{3} + 31T_{2}^{2} - 8T_{2} + 1 \) T2^10 + 4*T2^9 + 16*T2^8 + 31*T2^7 + 47*T2^6 + 23*T2^5 - 18*T2^4 - 39*T2^3 + 31*T2^2 - 8*T2 + 1 acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{10} + 4 T^{9} + \cdots + 1 \) T^10 + 4T^9 + 16T^8 + 31T^7 + 47T^6 + 23T^5 - 18T^4 - 39T^3 + 31T^2 - 8*T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} - 9 T^{9} + \cdots + 1 \) T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$7$	\( T^{10} - 7 T^{9} + \cdots + 1 \) T^10 - 7T^9 + 27T^8 - 57T^7 + 91T^6 - 120T^5 + 114T^4 - 72T^3 + 31T^2 - 8*T + 1
$11$	\( T^{10} - 12 T^{9} + \cdots + 17161 \) T^10 - 12T^9 + 78T^8 - 353T^7 + 1332T^6 - 3862T^5 + 10055T^4 - 22639T^3 + 36400T^2 - 35894*T + 17161
$13$	\( T^{10} - 15 T^{9} + \cdots + 1 \) T^10 - 15T^9 + 137T^8 - 746T^7 + 2676T^6 - 5754T^5 + 7209T^4 - 4856T^3 + 1384T^2 - 14*T + 1
$17$	\( T^{10} + T^{9} + \cdots + 39601 \) T^10 + T^9 + T^8 + 111T^7 + 111T^6 - 615T^5 + 2190T^4 + 2465T^3 + 24740T^2 + 35223*T + 39601
$19$	\( T^{10} + 13 T^{9} + \cdots + 6405961 \) T^10 + 13T^9 + 125T^8 + 580T^7 + 1710T^6 + 1990T^5 + 11471T^4 + 152786T^3 + 1034586T^2 + 3912926*T + 6405961
$23$	\( T^{10} - 7 T^{9} + \cdots + 351649 \) T^10 - 7T^9 + 49T^8 - 167T^7 + 861T^6 + 3653T^5 + 35886T^4 + 146173T^3 + 71278T^2 - 197469*T + 351649
$29$	\( (T^{5} - 12 T^{4} + \cdots + 131)^{2} \) (T^5 - 12T^4 + 29T^3 + 47T^2 - 195T + 131)^2
$31$	\( T^{10} + T^{9} + \cdots + 1849 \) T^10 + T^9 - 65T^8 + 100T^7 + 5050T^6 + 29525T^5 + 99507T^4 + 180126T^3 + 194745T^2 + 7052T + 1849
$37$	\( (T^{5} - T^{4} - 70 T^{3} + \cdots + 2507)^{2} \) (T^5 - T^4 - 70T^3 - 74T^2 + 1125*T + 2507)^2
$41$	\( T^{10} - 7 T^{9} + \cdots + 11881 \) T^10 - 7T^9 + 137T^8 - 827T^7 + 6207T^6 - 17170T^5 + 16570T^4 + 8992T^3 + 45197T^2 - 47088*T + 11881
$43$	\( T^{10} + 2 T^{9} + \cdots + 25816561 \) T^10 + 2T^9 + 37T^8 + 1009T^7 + 7221T^6 + 16620T^5 + 415930T^4 + 3562566T^3 + 10595676T^2 + 15395430*T + 25816561
$47$	\( T^{10} + 33 T^{9} + \cdots + 64009 \) T^10 + 33T^9 + 561T^8 + 7403T^7 + 77858T^6 + 579568T^5 + 3025968T^4 + 9899736T^3 + 16385094T^2 - 1335840*T + 64009
$53$	\( T^{10} + 21 T^{9} + \cdots + 35557369 \) T^10 + 21T^9 + 276T^8 + 3332T^7 + 38094T^6 + 325555T^5 + 1899437T^4 + 7227736T^3 + 18389680T^2 + 33118502*T + 35557369
$59$	\( T^{10} - 38 T^{9} + \cdots + 4363921 \) T^10 - 38T^9 + 839T^8 - 12170T^7 + 125365T^6 - 821789T^5 + 3110134T^4 - 2193392T^3 + 4578171T^2 - 8391513*T + 4363921
$61$	\( T^{10} + \cdots + 2766865201 \) T^10 + 50T^9 + 1422T^8 + 27738T^7 + 400376T^6 + 4359288T^5 + 35654756T^4 + 214123071T^3 + 891983085T^2 + 2281357971*T + 2766865201
$67$	\( T^{10} + \cdots + 1350125107 \) T^10 - 32T^9 + 474T^8 - 5103T^7 + 51404T^6 - 463759T^5 + 3444068T^4 - 22907367T^3 + 142561662T^2 - 644835872*T + 1350125107
$71$	\( T^{10} - 16 T^{9} + \cdots + 380689 \) T^10 - 16T^9 + 300T^8 - 2028T^7 + 17950T^6 - 84822T^5 + 274488T^4 - 187839T^3 - 309305T^2 + 428815*T + 380689
$73$	\( T^{10} - 3 T^{9} + \cdots + 528529 \) T^10 - 3T^9 + 9T^8 + 193T^7 - 502T^6 + 1979T^5 + 18648T^4 - 85149T^3 + 147966T^2 + 61068*T + 528529
$79$	\( T^{10} + 19 T^{9} + \cdots + 109561 \) T^10 + 19T^9 + 141T^8 - 1952T^7 - 16804T^6 + 35408T^5 + 882489T^4 - 710752T^3 + 10545518T^2 - 897010*T + 109561
$83$	\( T^{10} + \cdots + 2943171001 \) T^10 + 5T^9 - 195T^8 + 1379T^7 + 28961T^6 - 137323T^5 + 3714298T^4 + 4842247T^3 + 125011330T^2 + 392180479*T + 2943171001
$89$	\( T^{10} + 7 T^{9} + \cdots + 71757841 \) T^10 + 7T^9 + 181T^8 + 3874T^7 + 55575T^6 + 323355T^5 + 13826091T^4 + 76350599T^3 + 102411692T^2 - 42888673*T + 71757841
$97$	\( (T^{5} - 27 T^{4} + \cdots + 1583)^{2} \) (T^5 - 27T^4 + 120T^3 + 1035T^2 - 2701T + 1583)^2
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