Properties

Label 8034.2.a.u
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 4 x^{10} - 18 x^{9} + 64 x^{8} + 85 x^{7} - 249 x^{6} - 109 x^{5} + 230 x^{4} + 97 x^{3} - 53 x^{2} - 32 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 18 x^{9} + 64 x^{8} + 85 x^{7} - 249 x^{6} - 109 x^{5} + 230 x^{4} + 97 x^{3} - 53 x^{2} - 32 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25115 \nu^{10} + 123880 \nu^{9} + 347054 \nu^{8} - 1973096 \nu^{7} - 475719 \nu^{6} + 7344639 \nu^{5} - 3375623 \nu^{4} - 4857162 \nu^{3} + 1771243 \nu^{2} + 840961 \nu - 57834 \)\()/57098\)
\(\beta_{3}\)\(=\)\((\)\( -43620 \nu^{10} + 194155 \nu^{9} + 703885 \nu^{8} - 3133089 \nu^{7} - 2408751 \nu^{6} + 12312604 \nu^{5} - 266083 \nu^{4} - 11137378 \nu^{3} + 274806 \nu^{2} + 2744613 \nu + 326464 \)\()/57098\)
\(\beta_{4}\)\(=\)\((\)\(115421 \nu^{10} - 551264 \nu^{9} - 1667268 \nu^{8} + 8763870 \nu^{7} + 3266695 \nu^{6} - 32593715 \nu^{5} + 12149715 \nu^{4} + 22005140 \nu^{3} - 7530379 \nu^{2} - 3554529 \nu + 355594\)\()/114196\)
\(\beta_{5}\)\(=\)\((\)\(-189461 \nu^{10} + 831576 \nu^{9} + 3083592 \nu^{8} - 13311542 \nu^{7} - 10877367 \nu^{6} + 51186895 \nu^{5} + 637757 \nu^{4} - 43024636 \nu^{3} - 2052417 \nu^{2} + 10501909 \nu + 2166734\)\()/114196\)
\(\beta_{6}\)\(=\)\((\)\(-140379 \nu^{10} + 618522 \nu^{9} + 2271966 \nu^{8} - 9891326 \nu^{7} - 7853363 \nu^{6} + 37899765 \nu^{5} - 344375 \nu^{4} - 31274154 \nu^{3} - 665823 \nu^{2} + 7257953 \nu + 1342942\)\()/57098\)
\(\beta_{7}\)\(=\)\((\)\(-366281 \nu^{10} + 1630196 \nu^{9} + 5868288 \nu^{8} - 26121810 \nu^{7} - 19548755 \nu^{6} + 100542707 \nu^{5} - 4436099 \nu^{4} - 84025608 \nu^{3} + 1059979 \nu^{2} + 20040201 \nu + 3395542\)\()/114196\)
\(\beta_{8}\)\(=\)\((\)\(510619 \nu^{10} - 2246898 \nu^{9} - 8303822 \nu^{8} + 36042260 \nu^{7} + 29219587 \nu^{6} - 139385317 \nu^{5} - 1255417 \nu^{4} + 119432904 \nu^{3} + 3928915 \nu^{2} - 28669817 \nu - 5761410\)\()/114196\)
\(\beta_{9}\)\(=\)\((\)\(-671471 \nu^{10} + 2966642 \nu^{9} + 10849434 \nu^{8} - 47518076 \nu^{7} - 37292383 \nu^{6} + 182903005 \nu^{5} - 2609191 \nu^{4} - 153749580 \nu^{3} - 2584379 \nu^{2} + 36919609 \nu + 6971166\)\()/114196\)
\(\beta_{10}\)\(=\)\((\)\(-460175 \nu^{10} + 2024699 \nu^{9} + 7470473 \nu^{8} - 32427051 \nu^{7} - 26090460 \nu^{6} + 124845689 \nu^{5} - 53294 \nu^{4} - 105229792 \nu^{3} - 2225443 \nu^{2} + 24837390 \nu + 4665936\)\()/57098\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 5 \beta_{2} + 11 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(21 \beta_{10} - 19 \beta_{9} + 16 \beta_{8} + \beta_{7} - 8 \beta_{6} + 15 \beta_{5} + 19 \beta_{4} + 10 \beta_{3} + 39 \beta_{2} + 23 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\(81 \beta_{10} - 58 \beta_{9} + 42 \beta_{8} - 5 \beta_{7} - 49 \beta_{6} + 9 \beta_{5} + 66 \beta_{4} + 3 \beta_{3} + 141 \beta_{2} + 173 \beta_{1} + 91\)
\(\nu^{6}\)\(=\)\(441 \beta_{10} - 383 \beta_{9} + 300 \beta_{8} + 49 \beta_{7} - 213 \beta_{6} + 222 \beta_{5} + 394 \beta_{4} + 112 \beta_{3} + 796 \beta_{2} + 525 \beta_{1} + 758\)
\(\nu^{7}\)\(=\)\(1863 \beta_{10} - 1428 \beta_{9} + 1067 \beta_{8} + 109 \beta_{7} - 1067 \beta_{6} + 330 \beta_{5} + 1612 \beta_{4} + 82 \beta_{3} + 3287 \beta_{2} + 3131 \beta_{1} + 2199\)
\(\nu^{8}\)\(=\)\(9266 \beta_{10} - 7869 \beta_{9} + 5979 \beta_{8} + 1309 \beta_{7} - 4789 \beta_{6} + 3571 \beta_{5} + 8308 \beta_{4} + 1410 \beta_{3} + 16589 \beta_{2} + 11579 \beta_{1} + 13651\)
\(\nu^{9}\)\(=\)\(40813 \beta_{10} - 32453 \beta_{9} + 24224 \beta_{8} + 4636 \beta_{7} - 22775 \beta_{6} + 8756 \beta_{5} + 36143 \beta_{4} + 1962 \beta_{3} + 72387 \beta_{2} + 60981 \beta_{1} + 49344\)
\(\nu^{10}\)\(=\)\(194994 \beta_{10} - 163463 \beta_{9} + 122607 \beta_{8} + 30061 \beta_{7} - 103522 \beta_{6} + 63091 \beta_{5} + 175406 \beta_{4} + 20202 \beta_{3} + 348138 \beta_{2} + 250349 \beta_{1} + 264771\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.740393
−0.403660
1.71784
1.09038
3.12575
−3.29880
4.59586
−2.14636
−0.444701
−0.218590
0.722678
1.00000 −1.00000 1.00000 −3.99294 −1.00000 −2.56934 1.00000 1.00000 −3.99294
1.2 1.00000 −1.00000 1.00000 −2.98995 −1.00000 3.36753 1.00000 1.00000 −2.98995
1.3 1.00000 −1.00000 1.00000 −2.58633 −1.00000 3.26383 1.00000 1.00000 −2.58633
1.4 1.00000 −1.00000 1.00000 −1.28102 −1.00000 −0.150820 1.00000 1.00000 −1.28102
1.5 1.00000 −1.00000 1.00000 −0.539721 −1.00000 0.708670 1.00000 1.00000 −0.539721
1.6 1.00000 −1.00000 1.00000 0.113923 −1.00000 2.65133 1.00000 1.00000 0.113923
1.7 1.00000 −1.00000 1.00000 0.819538 −1.00000 −3.89413 1.00000 1.00000 0.819538
1.8 1.00000 −1.00000 1.00000 1.41909 −1.00000 −0.663444 1.00000 1.00000 1.41909
1.9 1.00000 −1.00000 1.00000 1.48470 −1.00000 −3.17363 1.00000 1.00000 1.48470
1.10 1.00000 −1.00000 1.00000 2.47956 −1.00000 −2.89002 1.00000 1.00000 2.47956
1.11 1.00000 −1.00000 1.00000 3.07315 −1.00000 1.35002 1.00000 1.00000 3.07315
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8034.2.a.u 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.u 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{11} + \cdots\)
\(T_{7}^{11} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{11} \)
$3$ \( ( 1 + T )^{11} \)
$5$ \( 32 - 272 T - 190 T^{2} + 998 T^{3} - 45 T^{4} - 805 T^{5} + 141 T^{6} + 221 T^{7} - 33 T^{8} - 25 T^{9} + 2 T^{10} + T^{11} \)
$7$ \( -256 - 1616 T + 1240 T^{2} + 4397 T^{3} - 1977 T^{4} - 2192 T^{5} + 566 T^{6} + 417 T^{7} - 58 T^{8} - 34 T^{9} + 2 T^{10} + T^{11} \)
$11$ \( -13030 + 103624 T + 105565 T^{2} - 45170 T^{3} - 42139 T^{4} + 6189 T^{5} + 6317 T^{6} - 172 T^{7} - 415 T^{8} - 20 T^{9} + 10 T^{10} + T^{11} \)
$13$ \( ( 1 + T )^{11} \)
$17$ \( -5632 + 17856 T - 10022 T^{2} - 13871 T^{3} + 11555 T^{4} + 2041 T^{5} - 3081 T^{6} + 206 T^{7} + 247 T^{8} - 34 T^{9} - 6 T^{10} + T^{11} \)
$19$ \( 704 + 168 T - 5360 T^{2} + 4555 T^{3} + 4489 T^{4} - 5180 T^{5} - 381 T^{6} + 1125 T^{7} - 76 T^{8} - 62 T^{9} + 3 T^{10} + T^{11} \)
$23$ \( 13021480 - 5946748 T - 3292898 T^{2} + 1443285 T^{3} + 325761 T^{4} - 133892 T^{5} - 15573 T^{6} + 5926 T^{7} + 354 T^{8} - 125 T^{9} - 3 T^{10} + T^{11} \)
$29$ \( 200864 + 63228 T - 431244 T^{2} - 41417 T^{3} + 199858 T^{4} + 46746 T^{5} - 26504 T^{6} - 11743 T^{7} - 1187 T^{8} + 90 T^{9} + 22 T^{10} + T^{11} \)
$31$ \( 9523904 - 6848350 T - 1766874 T^{2} + 1878575 T^{3} + 48319 T^{4} - 182086 T^{5} + 6901 T^{6} + 7623 T^{7} - 396 T^{8} - 140 T^{9} + 5 T^{10} + T^{11} \)
$37$ \( -22305200 - 59409734 T - 49960944 T^{2} - 11935495 T^{3} + 2339890 T^{4} + 1159084 T^{5} + 48450 T^{6} - 26111 T^{7} - 3025 T^{8} + 82 T^{9} + 26 T^{10} + T^{11} \)
$41$ \( 164675840 - 266119296 T - 16780020 T^{2} + 36555840 T^{3} - 575793 T^{4} - 1687731 T^{5} + 62664 T^{6} + 33269 T^{7} - 1168 T^{8} - 297 T^{9} + 6 T^{10} + T^{11} \)
$43$ \( 93337088 - 82675104 T + 5440672 T^{2} + 11260158 T^{3} - 1698627 T^{4} - 597917 T^{5} + 86122 T^{6} + 16328 T^{7} - 1518 T^{8} - 216 T^{9} + 8 T^{10} + T^{11} \)
$47$ \( -130160 + 2189044 T + 906986 T^{2} - 2767802 T^{3} + 1002175 T^{4} + 74766 T^{5} - 81732 T^{6} + 6875 T^{7} + 1418 T^{8} - 180 T^{9} - 6 T^{10} + T^{11} \)
$53$ \( 48160 + 1090428 T - 2127264 T^{2} - 2625495 T^{3} + 1496183 T^{4} + 984889 T^{5} + 80840 T^{6} - 22568 T^{7} - 3395 T^{8} + 36 T^{9} + 25 T^{10} + T^{11} \)
$59$ \( 279808 + 867808 T - 237072 T^{2} - 1309742 T^{3} + 478071 T^{4} + 100444 T^{5} - 59280 T^{6} + 3576 T^{7} + 1245 T^{8} - 137 T^{9} - 7 T^{10} + T^{11} \)
$61$ \( -7048960 - 15810496 T - 7762152 T^{2} + 4464862 T^{3} + 3991095 T^{4} + 150265 T^{5} - 395644 T^{6} - 86400 T^{7} - 4922 T^{8} + 256 T^{9} + 36 T^{10} + T^{11} \)
$67$ \( -325432148 - 137725074 T + 96003998 T^{2} + 22941671 T^{3} - 9300516 T^{4} - 1077089 T^{5} + 292735 T^{6} + 26011 T^{7} - 3422 T^{8} - 287 T^{9} + 12 T^{10} + T^{11} \)
$71$ \( -137830160 - 81110996 T + 68004974 T^{2} + 22088578 T^{3} - 12012285 T^{4} - 885050 T^{5} + 462983 T^{6} + 29195 T^{7} - 4941 T^{8} - 321 T^{9} + 15 T^{10} + T^{11} \)
$73$ \( 71448320 - 261546922 T + 100879356 T^{2} + 143007911 T^{3} - 37722334 T^{4} - 6124196 T^{5} + 873699 T^{6} + 92359 T^{7} - 5950 T^{8} - 534 T^{9} + 12 T^{10} + T^{11} \)
$79$ \( 17148880 + 66923728 T + 36790336 T^{2} - 3549920 T^{3} - 4580269 T^{4} - 142007 T^{5} + 192375 T^{6} + 11549 T^{7} - 3140 T^{8} - 213 T^{9} + 15 T^{10} + T^{11} \)
$83$ \( 7389723520 + 4839904336 T + 69466568 T^{2} - 480107162 T^{3} - 80486571 T^{4} + 3626930 T^{5} + 1497742 T^{6} + 38933 T^{7} - 8879 T^{8} - 441 T^{9} + 16 T^{10} + T^{11} \)
$89$ \( 887083844 - 208572054 T - 315023446 T^{2} + 62701597 T^{3} + 26366304 T^{4} - 6119200 T^{5} - 290471 T^{6} + 109079 T^{7} - 50 T^{8} - 606 T^{9} + 2 T^{10} + T^{11} \)
$97$ \( -18389296 - 29688228 T + 39505808 T^{2} + 7543567 T^{3} - 6903716 T^{4} - 902309 T^{5} + 331797 T^{6} + 40695 T^{7} - 3736 T^{8} - 399 T^{9} + 10 T^{10} + T^{11} \)
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