Properties

Label 585.2.bs.a
Level $585$
Weight $2$
Character orbit 585.bs
Analytic conductor $4.671$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( -\beta_{2} - \beta_{6} - \beta_{10} ) q^{4} + ( 1 + \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( -\beta_{2} - \beta_{6} - \beta_{10} ) q^{4} + ( 1 + \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( 1 + \beta_{6} - \beta_{8} + \beta_{10} ) q^{10} + ( -\beta_{5} + \beta_{8} - \beta_{9} ) q^{11} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{13} + ( 4 - \beta_{2} ) q^{14} + ( -3 - \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{16} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{17} + ( \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{7} - \beta_{10} ) q^{20} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{22} + ( -\beta_{4} + \beta_{11} ) q^{23} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{25} + ( -2 + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{10} ) q^{26} + ( -1 - 4 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{28} + ( -3 - 3 \beta_{6} ) q^{29} + ( -2 + 2 \beta_{2} ) q^{31} + ( -2 \beta_{1} - \beta_{3} ) q^{32} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{34} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{35} + ( -1 + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{37} + ( 1 - 3 \beta_{1} - 3 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{38} + ( 7 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{40} + ( -3 - 3 \beta_{6} - 2 \beta_{10} ) q^{41} + ( -\beta_{1} + 3 \beta_{3} ) q^{43} + ( \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{44} + ( -\beta_{2} - 2 \beta_{6} - \beta_{10} ) q^{46} + ( -2 + 6 \beta_{1} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{47} + ( 1 + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{49} + ( 6 + 3 \beta_{4} + 6 \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{50} + ( 1 + 4 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{52} + ( 1 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 4 \beta_{11} ) q^{53} + ( -3 - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{55} + ( -4 \beta_{2} - \beta_{5} - 5 \beta_{6} + \beta_{7} - 4 \beta_{10} ) q^{56} -3 \beta_{1} q^{58} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{59} + ( -2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{61} + ( 2 + 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{62} + ( -1 - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{64} + ( -5 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 6 \beta_{6} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{65} + ( -3 \beta_{4} - \beta_{11} ) q^{67} + ( 2 \beta_{4} + \beta_{11} ) q^{68} + ( 4 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{11} ) q^{70} + ( -\beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{71} + ( -1 + 6 \beta_{1} - 2 \beta_{3} + 6 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{73} + ( -\beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{74} + ( -2 - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{76} + ( -2 + 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} ) q^{77} + ( -8 - 2 \beta_{2} ) q^{79} + ( -6 - 5 \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{9} ) q^{80} + ( -7 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{82} + ( 2 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{11} ) q^{83} + ( -3 \beta_{1} + \beta_{3} - 2 \beta_{6} - 3 \beta_{7} ) q^{85} + \beta_{2} q^{86} + ( -2 + 5 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{88} + ( -2 + \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{9} + 4 \beta_{10} ) q^{89} + ( -4 + \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{91} + ( -1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{92} + ( 10 - 2 \beta_{5} + 10 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{94} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{10} ) q^{95} + ( 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( -\beta_{5} - \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} + 6 q^{5} + O(q^{10}) \) \( 12 q + 4 q^{4} + 6 q^{5} + 7 q^{10} + 44 q^{14} - 16 q^{16} + 12 q^{19} + q^{20} - 2 q^{25} - 24 q^{26} - 18 q^{29} - 16 q^{31} + 16 q^{34} - 10 q^{35} + 70 q^{40} - 14 q^{41} + 4 q^{44} + 10 q^{46} + 6 q^{49} + 31 q^{50} - 26 q^{55} + 16 q^{56} + 4 q^{59} + 6 q^{61} - 12 q^{64} - 23 q^{65} + 20 q^{70} + 12 q^{71} - 8 q^{74} - 10 q^{76} - 104 q^{79} - 33 q^{80} + 21 q^{85} + 4 q^{86} - 20 q^{89} - 44 q^{91} + 56 q^{94} - 20 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 16 \nu^{10} - 108 \nu^{8} + 729 \nu^{6} - 184 \nu^{4} + 20 \nu^{2} + 3531 \)\()/1222\)
\(\beta_{3}\)\(=\)\((\)\( 108 \nu^{11} - 729 \nu^{9} + 4768 \nu^{7} - 1242 \nu^{5} + 135 \nu^{3} + 11156 \nu \)\()/1222\)
\(\beta_{4}\)\(=\)\((\)\( 135 \nu^{11} - 1064 \nu^{9} + 7182 \nu^{7} - 9801 \nu^{5} + 12236 \nu^{3} - 1330 \nu \)\()/1222\)
\(\beta_{5}\)\(=\)\((\)\( -92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045 \)\()/2444\)
\(\beta_{6}\)\(=\)\((\)\( 135 \nu^{10} - 1064 \nu^{8} + 7182 \nu^{6} - 9801 \nu^{4} + 12236 \nu^{2} - 1330 \)\()/1222\)
\(\beta_{7}\)\(=\)\((\)\( -92 \nu^{11} - 293 \nu^{10} + 621 \nu^{9} + 2436 \nu^{8} - 4039 \nu^{7} - 16443 \nu^{6} + 1058 \nu^{5} + 26893 \nu^{4} - 115 \nu^{3} - 28014 \nu^{2} - 5181 \nu + 3045 \)\()/2444\)
\(\beta_{8}\)\(=\)\((\)\( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524 \)\()/2444\)
\(\beta_{9}\)\(=\)\((\)\( 655 \nu^{11} + 92 \nu^{10} - 5185 \nu^{9} - 621 \nu^{8} + 34846 \nu^{7} + 4039 \nu^{6} - 47553 \nu^{5} - 1058 \nu^{4} + 52601 \nu^{3} + 115 \nu^{2} - 524 \nu + 2737 \)\()/2444\)
\(\beta_{10}\)\(=\)\((\)\( -405 \nu^{10} + 3192 \nu^{8} - 21546 \nu^{6} + 29403 \nu^{4} - 35486 \nu^{2} + 324 \)\()/1222\)
\(\beta_{11}\)\(=\)\( \nu^{11} - 8 \nu^{9} + 54 \nu^{7} - 78 \nu^{5} + 92 \nu^{3} - 10 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + 3 \beta_{6} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 5 \beta_{4} - \beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{10} - \beta_{7} + 18 \beta_{6} + \beta_{5} + 7 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-7 \beta_{11} + 8 \beta_{9} + 8 \beta_{8} + 8 \beta_{6} + 8 \beta_{5} + 30 \beta_{4} + 8\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + 46 \beta_{2} - 107\)
\(\nu^{7}\)\(=\)\(54 \beta_{7} + 54 \beta_{5} + 46 \beta_{3} - 191 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-299 \beta_{10} - 54 \beta_{9} + 54 \beta_{8} - 689 \beta_{6} - 54 \beta_{5} - 689\)
\(\nu^{9}\)\(=\)\(299 \beta_{11} - 353 \beta_{9} - 353 \beta_{8} + 353 \beta_{7} - 353 \beta_{6} - 1233 \beta_{4} + 299 \beta_{3} - 1233 \beta_{1} - 353\)
\(\nu^{10}\)\(=\)\(-1939 \beta_{10} + 353 \beta_{7} - 4812 \beta_{6} - 353 \beta_{5} - 1939 \beta_{2}\)
\(\nu^{11}\)\(=\)\(1939 \beta_{11} - 2292 \beta_{9} - 2292 \beta_{8} - 2292 \beta_{6} - 2292 \beta_{5} - 7984 \beta_{4} - 2292\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−2.20467 1.27287i
−1.02826 0.593667i
−0.286513 0.165418i
0.286513 + 0.165418i
1.02826 + 0.593667i
2.20467 + 1.27287i
−2.20467 + 1.27287i
−1.02826 + 0.593667i
−0.286513 + 0.165418i
0.286513 0.165418i
1.02826 0.593667i
2.20467 1.27287i
−2.20467 + 1.27287i 0 2.24039 3.88048i 0.817544 2.08125i 0 −2.54486 1.46928i 6.31544i 0 0.846746 + 5.62912i
289.2 −1.02826 + 0.593667i 0 −0.295120 + 0.511162i −1.44045 + 1.71029i 0 −1.75765 1.01478i 3.07548i 0 0.465813 2.61378i
289.3 −0.286513 + 0.165418i 0 −0.945274 + 1.63726i 2.12291 0.702335i 0 −2.90420 1.67674i 1.28714i 0 −0.492061 + 0.552395i
289.4 0.286513 0.165418i 0 −0.945274 + 1.63726i 2.12291 + 0.702335i 0 2.90420 + 1.67674i 1.28714i 0 0.724419 0.149939i
289.5 1.02826 0.593667i 0 −0.295120 + 0.511162i −1.44045 1.71029i 0 1.75765 + 1.01478i 3.07548i 0 −2.49650 0.903481i
289.6 2.20467 1.27287i 0 2.24039 3.88048i 0.817544 + 2.08125i 0 2.54486 + 1.46928i 6.31544i 0 4.45158 + 3.54786i
334.1 −2.20467 1.27287i 0 2.24039 + 3.88048i 0.817544 + 2.08125i 0 −2.54486 + 1.46928i 6.31544i 0 0.846746 5.62912i
334.2 −1.02826 0.593667i 0 −0.295120 0.511162i −1.44045 1.71029i 0 −1.75765 + 1.01478i 3.07548i 0 0.465813 + 2.61378i
334.3 −0.286513 0.165418i 0 −0.945274 1.63726i 2.12291 + 0.702335i 0 −2.90420 + 1.67674i 1.28714i 0 −0.492061 0.552395i
334.4 0.286513 + 0.165418i 0 −0.945274 1.63726i 2.12291 0.702335i 0 2.90420 1.67674i 1.28714i 0 0.724419 + 0.149939i
334.5 1.02826 + 0.593667i 0 −0.295120 0.511162i −1.44045 + 1.71029i 0 1.75765 1.01478i 3.07548i 0 −2.49650 + 0.903481i
334.6 2.20467 + 1.27287i 0 2.24039 + 3.88048i 0.817544 2.08125i 0 2.54486 1.46928i 6.31544i 0 4.45158 3.54786i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 334.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bs.a 12
3.b odd 2 1 65.2.n.a 12
5.b even 2 1 inner 585.2.bs.a 12
12.b even 2 1 1040.2.dh.a 12
13.c even 3 1 inner 585.2.bs.a 12
15.d odd 2 1 65.2.n.a 12
15.e even 4 2 325.2.e.e 12
39.d odd 2 1 845.2.n.e 12
39.f even 4 2 845.2.l.f 24
39.h odd 6 1 845.2.b.e 6
39.h odd 6 1 845.2.n.e 12
39.i odd 6 1 65.2.n.a 12
39.i odd 6 1 845.2.b.d 6
39.k even 12 2 845.2.d.d 12
39.k even 12 2 845.2.l.f 24
60.h even 2 1 1040.2.dh.a 12
65.n even 6 1 inner 585.2.bs.a 12
156.p even 6 1 1040.2.dh.a 12
195.e odd 2 1 845.2.n.e 12
195.n even 4 2 845.2.l.f 24
195.x odd 6 1 65.2.n.a 12
195.x odd 6 1 845.2.b.d 6
195.y odd 6 1 845.2.b.e 6
195.y odd 6 1 845.2.n.e 12
195.bf even 12 2 4225.2.a.bq 6
195.bh even 12 2 845.2.d.d 12
195.bh even 12 2 845.2.l.f 24
195.bl even 12 2 325.2.e.e 12
195.bl even 12 2 4225.2.a.br 6
780.br even 6 1 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 3.b odd 2 1
65.2.n.a 12 15.d odd 2 1
65.2.n.a 12 39.i odd 6 1
65.2.n.a 12 195.x odd 6 1
325.2.e.e 12 15.e even 4 2
325.2.e.e 12 195.bl even 12 2
585.2.bs.a 12 1.a even 1 1 trivial
585.2.bs.a 12 5.b even 2 1 inner
585.2.bs.a 12 13.c even 3 1 inner
585.2.bs.a 12 65.n even 6 1 inner
845.2.b.d 6 39.i odd 6 1
845.2.b.d 6 195.x odd 6 1
845.2.b.e 6 39.h odd 6 1
845.2.b.e 6 195.y odd 6 1
845.2.d.d 12 39.k even 12 2
845.2.d.d 12 195.bh even 12 2
845.2.l.f 24 39.f even 4 2
845.2.l.f 24 39.k even 12 2
845.2.l.f 24 195.n even 4 2
845.2.l.f 24 195.bh even 12 2
845.2.n.e 12 39.d odd 2 1
845.2.n.e 12 39.h odd 6 1
845.2.n.e 12 195.e odd 2 1
845.2.n.e 12 195.y odd 6 1
1040.2.dh.a 12 12.b even 2 1
1040.2.dh.a 12 60.h even 2 1
1040.2.dh.a 12 156.p even 6 1
1040.2.dh.a 12 780.br even 6 1
4225.2.a.bq 6 195.bf even 12 2
4225.2.a.br 6 195.bl even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 8 T_{2}^{10} + 54 T_{2}^{8} - 78 T_{2}^{6} + 92 T_{2}^{4} - 10 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T^{2} + 92 T^{4} - 78 T^{6} + 54 T^{8} - 8 T^{10} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 125 - 75 T + 25 T^{2} - 10 T^{3} + 5 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$7$ \( 160000 - 71600 T^{2} + 22441 T^{4} - 3496 T^{6} + 397 T^{8} - 24 T^{10} + T^{12} \)
$11$ \( ( 64 + 104 T + 169 T^{2} + 16 T^{3} + 13 T^{4} + T^{6} )^{2} \)
$13$ \( 4826809 - 428415 T^{2} + 6591 T^{4} - 322 T^{6} + 39 T^{8} - 15 T^{10} + T^{12} \)
$17$ \( 28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12} \)
$19$ \( ( 100 - 10 T + 61 T^{2} - 14 T^{3} + 37 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$23$ \( 16 - 140 T^{2} + 1177 T^{4} - 412 T^{6} + 109 T^{8} - 12 T^{10} + T^{12} \)
$29$ \( ( 9 + 3 T + T^{2} )^{6} \)
$31$ \( ( 40 - 40 T + 4 T^{2} + T^{3} )^{4} \)
$37$ \( 28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12} \)
$41$ \( ( 25 - 145 T + 806 T^{2} - 213 T^{3} + 78 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$43$ \( 65536 - 72448 T^{2} + 59609 T^{4} - 22128 T^{6} + 6117 T^{8} - 80 T^{10} + T^{12} \)
$47$ \( ( 270400 + 14640 T^{2} + 236 T^{4} + T^{6} )^{2} \)
$53$ \( ( 400 + 1040 T^{2} + 171 T^{4} + T^{6} )^{2} \)
$59$ \( ( 18496 - 7480 T + 3297 T^{2} - 162 T^{3} + 59 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$61$ \( ( 13225 - 5635 T + 2746 T^{2} - 83 T^{3} + 58 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$67$ \( 406586896 - 52486892 T^{2} + 4759209 T^{4} - 219972 T^{6} + 7397 T^{8} - 100 T^{10} + T^{12} \)
$71$ \( ( 676 - 26 T + 157 T^{2} - 46 T^{3} + 37 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$73$ \( ( 250000 + 13900 T^{2} + 215 T^{4} + T^{6} )^{2} \)
$79$ \( ( 160 + 180 T + 26 T^{2} + T^{3} )^{4} \)
$83$ \( ( 640000 + 23600 T^{2} + 276 T^{4} + T^{6} )^{2} \)
$89$ \( ( 2515396 + 249002 T + 40509 T^{2} + 1602 T^{3} + 257 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$97$ \( 41740124416 - 2934418352 T^{2} + 149090649 T^{4} - 3613032 T^{6} + 64037 T^{8} - 280 T^{10} + T^{12} \)
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