Properties

Label 770.2.i.m
Level $770$
Weight $2$
Character orbit 770.i
Analytic conductor $6.148$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 12 x^{8} - 6 x^{7} + 113 x^{6} - 43 x^{5} + 381 x^{4} - 75 x^{3} + 982 x^{2} - 217 x + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 12 x^{8} - 6 x^{7} + 113 x^{6} - 43 x^{5} + 381 x^{4} - 75 x^{3} + 982 x^{2} - 217 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 98501 \nu^{9} + 122729 \nu^{8} - 114811 \nu^{7} + 622151 \nu^{6} - 1745919 \nu^{5} + 14212810 \nu^{4} - 112651589 \nu^{3} + 46458865 \nu^{2} - 10309236 \nu + 87558415 \)\()/ 383873037 \)
\(\beta_{3}\)\(=\)\((\)\(-122729 \nu^{9} + 1296823 \nu^{8} - 1213157 \nu^{7} + 12876532 \nu^{6} - 18448353 \nu^{5} + 150180470 \nu^{4} - 53846440 \nu^{3} + 490910255 \nu^{2} - 108933132 \nu + 1924191734\)\()/ 383873037 \)
\(\beta_{4}\)\(=\)\((\)\(-12763717 \nu^{9} - 14388479 \nu^{8} - 180540162 \nu^{7} - 215638669 \nu^{6} - 1995358828 \nu^{5} - 1979981394 \nu^{4} - 7975455071 \nu^{3} - 8951124856 \nu^{2} - 27837760897 \nu - 10157690340\)\()/ 5374222518 \)
\(\beta_{5}\)\(=\)\((\)\(-21436535 \nu^{9} + 39922932 \nu^{8} - 231347611 \nu^{7} + 566697487 \nu^{6} - 2767982449 \nu^{5} + 4309630396 \nu^{4} - 10138708261 \nu^{3} + 11608372179 \nu^{2} - 32399919421 \nu + 16808160187\)\()/ 5374222518 \)
\(\beta_{6}\)\(=\)\((\)\(-12508345 \nu^{9} + 689507 \nu^{8} - 149241037 \nu^{7} + 74246393 \nu^{6} - 1409087928 \nu^{5} + 525637402 \nu^{4} - 4666189775 \nu^{3} + 149564752 \nu^{2} - 11957982735 \nu + 2642146213\)\()/ 2687111259 \)
\(\beta_{7}\)\(=\)\((\)\(-57287134 \nu^{9} - 43187207 \nu^{8} - 636536463 \nu^{7} - 21732244 \nu^{6} - 5461553071 \nu^{5} - 1179137496 \nu^{4} - 15873003278 \nu^{3} - 3420594307 \nu^{2} - 41925208336 \nu - 7072023903\)\()/ 5374222518 \)
\(\beta_{8}\)\(=\)\((\)\(-63494973 \nu^{9} - 29110928 \nu^{8} - 843704947 \nu^{7} - 4838781 \nu^{6} - 7861846037 \nu^{5} + 137283730 \nu^{4} - 28690819467 \nu^{3} - 1596430933 \nu^{2} - 72154008905 \nu + 6315431857\)\()/ 5374222518 \)
\(\beta_{9}\)\(=\)\((\)\(-62541725 \nu^{9} + 3447535 \nu^{8} - 746205185 \nu^{7} + 371231965 \nu^{6} - 7045439640 \nu^{5} + 2628187010 \nu^{4} - 23330948875 \nu^{3} + 3434935019 \nu^{2} - 59789913675 \nu + 13210731065\)\()/ 2687111259 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - 5 \beta_{6}\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - 6 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-10 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 36 \beta_{6} + \beta_{5} + \beta_{4} + 10 \beta_{3} + 2 \beta_{1} - 36\)
\(\nu^{5}\)\(=\)\(15 \beta_{9} - 12 \beta_{8} - 32 \beta_{6} - 12 \beta_{5} + 41 \beta_{2} - 41 \beta_{1}\)
\(\nu^{6}\)\(=\)\(15 \beta_{8} - 15 \beta_{7} + 24 \beta_{5} - 39 \beta_{4} - 92 \beta_{3} - 35 \beta_{2} + 283\)
\(\nu^{7}\)\(=\)\(-172 \beta_{9} + 6 \beta_{8} + 122 \beta_{7} + 395 \beta_{6} + 116 \beta_{5} + 116 \beta_{4} + 172 \beta_{3} + 312 \beta_{1} - 395\)
\(\nu^{8}\)\(=\)\(832 \beta_{9} - 404 \beta_{8} - 226 \beta_{7} - 2362 \beta_{6} - 404 \beta_{5} + 226 \beta_{4} + 439 \beta_{2} - 439 \beta_{1}\)
\(\nu^{9}\)\(=\)\(1058 \beta_{8} - 1058 \beta_{7} + 130 \beta_{5} - 1188 \beta_{4} - 1805 \beta_{3} - 2564 \beta_{2} + 4345\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.26598 + 2.19274i
1.08142 + 1.87308i
0.112651 + 0.195118i
−0.922461 1.59775i
−1.53759 2.66319i
1.26598 2.19274i
1.08142 1.87308i
0.112651 0.195118i
−0.922461 + 1.59775i
−1.53759 + 2.66319i
−0.500000 0.866025i −1.26598 + 2.19274i −0.500000 + 0.866025i −0.500000 0.866025i 2.53196 2.22551 1.43078i 1.00000 −1.70541 2.95386i −0.500000 + 0.866025i
221.2 −0.500000 0.866025i −1.08142 + 1.87308i −0.500000 + 0.866025i −0.500000 0.866025i 2.16285 −0.590793 + 2.57895i 1.00000 −0.838952 1.45311i −0.500000 + 0.866025i
221.3 −0.500000 0.866025i −0.112651 + 0.195118i −0.500000 + 0.866025i −0.500000 0.866025i 0.225302 −2.14481 1.54913i 1.00000 1.47462 + 2.55412i −0.500000 + 0.866025i
221.4 −0.500000 0.866025i 0.922461 1.59775i −0.500000 + 0.866025i −0.500000 0.866025i −1.84492 2.59682 + 0.506486i 1.00000 −0.201870 0.349648i −0.500000 + 0.866025i
221.5 −0.500000 0.866025i 1.53759 2.66319i −0.500000 + 0.866025i −0.500000 0.866025i −3.07519 −2.08672 + 1.62653i 1.00000 −3.22839 5.59173i −0.500000 + 0.866025i
331.1 −0.500000 + 0.866025i −1.26598 2.19274i −0.500000 0.866025i −0.500000 + 0.866025i 2.53196 2.22551 + 1.43078i 1.00000 −1.70541 + 2.95386i −0.500000 0.866025i
331.2 −0.500000 + 0.866025i −1.08142 1.87308i −0.500000 0.866025i −0.500000 + 0.866025i 2.16285 −0.590793 2.57895i 1.00000 −0.838952 + 1.45311i −0.500000 0.866025i
331.3 −0.500000 + 0.866025i −0.112651 0.195118i −0.500000 0.866025i −0.500000 + 0.866025i 0.225302 −2.14481 + 1.54913i 1.00000 1.47462 2.55412i −0.500000 0.866025i
331.4 −0.500000 + 0.866025i 0.922461 + 1.59775i −0.500000 0.866025i −0.500000 + 0.866025i −1.84492 2.59682 0.506486i 1.00000 −0.201870 + 0.349648i −0.500000 0.866025i
331.5 −0.500000 + 0.866025i 1.53759 + 2.66319i −0.500000 0.866025i −0.500000 + 0.866025i −3.07519 −2.08672 1.62653i 1.00000 −3.22839 + 5.59173i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.i.m 10
7.c even 3 1 inner 770.2.i.m 10
7.c even 3 1 5390.2.a.ci 5
7.d odd 6 1 5390.2.a.ch 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.m 10 1.a even 1 1 trivial
770.2.i.m 10 7.c even 3 1 inner
5390.2.a.ch 5 7.d odd 6 1
5390.2.a.ci 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\(T_{3}^{10} + \cdots\)
\( T_{13}^{5} + T_{13}^{4} - 28 T_{13}^{3} - 20 T_{13}^{2} + 168 T_{13} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{5} \)
$3$ \( 49 + 217 T + 982 T^{2} + 75 T^{3} + 381 T^{4} + 43 T^{5} + 113 T^{6} + 6 T^{7} + 12 T^{8} + T^{10} \)
$5$ \( ( 1 + T + T^{2} )^{5} \)
$7$ \( 16807 - 2401 T^{2} - 1225 T^{3} + 343 T^{4} + 139 T^{5} + 49 T^{6} - 25 T^{7} - 7 T^{8} + T^{10} \)
$11$ \( ( 1 + T + T^{2} )^{5} \)
$13$ \( ( 144 + 168 T - 20 T^{2} - 28 T^{3} + T^{4} + T^{5} )^{2} \)
$17$ \( 50176 + 38528 T + 46384 T^{2} + 6364 T^{3} + 13021 T^{4} + 3449 T^{5} + 1677 T^{6} + 150 T^{7} + 43 T^{8} + T^{10} \)
$19$ \( 92416 + 27360 T + 30900 T^{2} + 6018 T^{3} + 6299 T^{4} + 1159 T^{5} + 651 T^{6} + 66 T^{7} + 37 T^{8} + 4 T^{9} + T^{10} \)
$23$ \( 2304 + 14976 T + 88320 T^{2} + 57120 T^{3} + 30688 T^{4} + 7328 T^{5} + 1884 T^{6} + 264 T^{7} + 65 T^{8} + 7 T^{9} + T^{10} \)
$29$ \( ( -317 + 1247 T + 199 T^{2} - 74 T^{3} - 4 T^{4} + T^{5} )^{2} \)
$31$ \( 576 - 2160 T + 6492 T^{2} - 6654 T^{3} + 5803 T^{4} - 233 T^{5} + 481 T^{6} + 56 T^{7} + 49 T^{8} + 6 T^{9} + T^{10} \)
$37$ \( 107495424 - 8024832 T + 9131940 T^{2} + 740682 T^{3} + 507571 T^{4} + 39251 T^{5} + 13967 T^{6} + 1566 T^{7} + 261 T^{8} + 16 T^{9} + T^{10} \)
$41$ \( ( -3584 + 3320 T - 372 T^{2} - 160 T^{3} + T^{4} + T^{5} )^{2} \)
$43$ \( ( -16 - 88 T + 36 T^{2} + 32 T^{3} - 13 T^{4} + T^{5} )^{2} \)
$47$ \( 94633984 + 14008320 T + 7910400 T^{2} + 770304 T^{3} + 403136 T^{4} + 37088 T^{5} + 10416 T^{6} + 528 T^{7} + 148 T^{8} + 8 T^{9} + T^{10} \)
$53$ \( 1387115536 + 274711744 T + 83939868 T^{2} + 7037256 T^{3} + 1755921 T^{4} + 115425 T^{5} + 25725 T^{6} + 894 T^{7} + 189 T^{8} + 4 T^{9} + T^{10} \)
$59$ \( 289816576 + 22335488 T + 18881536 T^{2} + 2286592 T^{3} + 1001920 T^{4} + 100256 T^{5} + 18996 T^{6} + 1062 T^{7} + 187 T^{8} + 9 T^{9} + T^{10} \)
$61$ \( 25331089 + 13272021 T + 7754016 T^{2} + 969825 T^{3} + 379121 T^{4} + 16427 T^{5} + 16725 T^{6} + 42 T^{7} + 142 T^{8} + 2 T^{9} + T^{10} \)
$67$ \( 421201 + 200541 T + 198023 T^{2} + 502 T^{3} + 33461 T^{4} + 3563 T^{5} + 1925 T^{6} + 126 T^{7} + 63 T^{8} + 5 T^{9} + T^{10} \)
$71$ \( ( 12544 - 6732 T + 499 T^{2} + 187 T^{3} - 28 T^{4} + T^{5} )^{2} \)
$73$ \( 29333056 - 19768400 T + 17075788 T^{2} - 406022 T^{3} + 1463983 T^{4} - 185919 T^{5} + 70484 T^{6} - 1115 T^{7} + 272 T^{8} - T^{9} + T^{10} \)
$79$ \( 6392322304 - 307015680 T + 217184064 T^{2} + 10682304 T^{3} + 4549088 T^{4} + 271784 T^{5} + 62112 T^{6} + 4926 T^{7} + 535 T^{8} + 23 T^{9} + T^{10} \)
$83$ \( ( 12672 - 3840 T - 872 T^{2} + 94 T^{3} + 23 T^{4} + T^{5} )^{2} \)
$89$ \( 176943204 + 14406066 T + 20859849 T^{2} - 5141172 T^{3} + 1926643 T^{4} - 229636 T^{5} + 32092 T^{6} - 1763 T^{7} + 214 T^{8} - 9 T^{9} + T^{10} \)
$97$ \( ( 120192 + 6168 T - 3320 T^{2} - 238 T^{3} + 13 T^{4} + T^{5} )^{2} \)
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