Properties

Label 8001.2.a.w
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} - 56 x^{11} + 39579 x^{10} - 17664 x^{9} - 52271 x^{8} + 35701 x^{7} + 32493 x^{6} - 25504 x^{5} - 8607 x^{4} + 6812 x^{3} + 609 x^{2} - 425 x + 31\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} - 56 x^{11} + 39579 x^{10} - 17664 x^{9} - 52271 x^{8} + 35701 x^{7} + 32493 x^{6} - 25504 x^{5} - 8607 x^{4} + 6812 x^{3} + 609 x^{2} - 425 x + 31\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\(-80574 \nu^{19} + 88265 \nu^{18} + 4448116 \nu^{17} - 11956345 \nu^{16} - 63248616 \nu^{15} + 232480171 \nu^{14} + 345773311 \nu^{13} - 1910206802 \nu^{12} - 405883602 \nu^{11} + 7970416630 \nu^{10} - 3293131310 \nu^{9} - 17070805117 \nu^{8} + 13762066276 \nu^{7} + 16264479010 \nu^{6} - 19092142347 \nu^{5} - 3272911391 \nu^{4} + 8444727149 \nu^{3} - 1113016448 \nu^{2} - 888775901 \nu + 123446515\)\()/4334246\)
\(\beta_{5}\)\(=\)\((\)\(112041 \nu^{19} - 1197675 \nu^{18} + 2264887 \nu^{17} + 17783216 \nu^{16} - 73262421 \nu^{15} - 57671117 \nu^{14} + 647447518 \nu^{13} - 364390775 \nu^{12} - 2623374358 \nu^{11} + 3210092650 \nu^{10} + 5054834351 \nu^{9} - 9103752501 \nu^{8} - 3527560932 \nu^{7} + 10907649962 \nu^{6} - 969053095 \nu^{5} - 4410403111 \nu^{4} + 1105027913 \nu^{3} + 377557653 \nu^{2} - 99590538 \nu + 2196193\)\()/2167123\)
\(\beta_{6}\)\(=\)\((\)\(16028 \nu^{19} - 143864 \nu^{18} + 122280 \nu^{17} + 2470224 \nu^{16} - 6859382 \nu^{15} - 12972651 \nu^{14} + 67984425 \nu^{13} - 1910609 \nu^{12} - 295889912 \nu^{11} + 245045373 \nu^{10} + 599709512 \nu^{9} - 851828022 \nu^{8} - 418277501 \nu^{7} + 1092553461 \nu^{6} - 169615722 \nu^{5} - 415699503 \nu^{4} + 165471717 \nu^{3} + 18340672 \nu^{2} - 13862243 \nu + 1383690\)\()/309589\)
\(\beta_{7}\)\(=\)\((\)\(-248879 \nu^{19} + 3978132 \nu^{18} - 13988931 \nu^{17} - 48139158 \nu^{16} + 338768723 \nu^{15} - 38956001 \nu^{14} - 2759046818 \nu^{13} + 3143549108 \nu^{12} + 10402831226 \nu^{11} - 18846899174 \nu^{10} - 17448853443 \nu^{9} + 48415589078 \nu^{8} + 5425063462 \nu^{7} - 55791961163 \nu^{6} + 14568997447 \nu^{5} + 22437701529 \nu^{4} - 8633213124 \nu^{3} - 1967839611 \nu^{2} + 676572227 \nu - 44644594\)\()/4334246\)
\(\beta_{8}\)\(=\)\((\)\(130444 \nu^{19} - 1062848 \nu^{18} + 203009 \nu^{17} + 19282840 \nu^{16} - 38105845 \nu^{15} - 122444455 \nu^{14} + 401582731 \nu^{13} + 251551762 \nu^{12} - 1805118745 \nu^{11} + 509262882 \nu^{10} + 3897308205 \nu^{9} - 3070652687 \nu^{8} - 3512303592 \nu^{7} + 4434806592 \nu^{6} + 364613036 \nu^{5} - 1821783462 \nu^{4} + 433924998 \nu^{3} + 117630811 \nu^{2} - 52984391 \nu + 4552939\)\()/2167123\)
\(\beta_{9}\)\(=\)\((\)\(-146869 \nu^{19} + 1305396 \nu^{18} - 1062848 \nu^{17} - 22121079 \nu^{16} + 59524946 \nu^{15} + 117722164 \nu^{14} - 581410080 \nu^{13} - 35646282 \nu^{12} + 2524202668 \nu^{11} - 1796894081 \nu^{10} - 5303665269 \nu^{9} + 6491602221 \nu^{8} + 4606336812 \nu^{7} - 8755673761 \nu^{6} - 337407825 \nu^{5} + 4110360012 \nu^{4} - 557681979 \nu^{3} - 566546630 \nu^{2} + 28187590 \nu + 9434934\)\()/2167123\)
\(\beta_{10}\)\(=\)\((\)\(452378 \nu^{19} - 3636757 \nu^{18} + 479452 \nu^{17} + 64942855 \nu^{16} - 120306036 \nu^{15} - 413198975 \nu^{14} + 1229305933 \nu^{13} + 962389260 \nu^{12} - 5289287134 \nu^{11} + 625710296 \nu^{10} + 10741270610 \nu^{9} - 6399002549 \nu^{8} - 8503960458 \nu^{7} + 8810093974 \nu^{6} - 662651141 \nu^{5} - 2565501807 \nu^{4} + 1987510173 \nu^{3} - 333016752 \nu^{2} - 247405745 \nu + 51386023\)\()/4334246\)
\(\beta_{11}\)\(=\)\((\)\(239299 \nu^{19} - 839918 \nu^{18} - 6727183 \nu^{17} + 25057160 \nu^{16} + 75577807 \nu^{15} - 306448870 \nu^{14} - 430391229 \nu^{13} + 1985694746 \nu^{12} + 1274010673 \nu^{11} - 7326712947 \nu^{10} - 1640919567 \nu^{9} + 15286073938 \nu^{8} - 166042331 \nu^{7} - 16629421700 \nu^{6} + 1992882190 \nu^{5} + 7760740247 \nu^{4} - 741296302 \nu^{3} - 1279308585 \nu^{2} - 50188832 \nu + 33626793\)\()/2167123\)
\(\beta_{12}\)\(=\)\((\)\(513169 \nu^{19} - 3359961 \nu^{18} - 4253023 \nu^{17} + 67902593 \nu^{16} - 49716311 \nu^{15} - 524918352 \nu^{14} + 801626921 \nu^{13} + 1873963420 \nu^{12} - 4115132892 \nu^{11} - 2626187800 \nu^{10} + 9547027119 \nu^{9} - 992559901 \nu^{8} - 8895240438 \nu^{7} + 5258752701 \nu^{6} + 511422678 \nu^{5} - 2384056922 \nu^{4} + 1615698465 \nu^{3} + 53574147 \nu^{2} - 261345936 \nu + 14104789\)\()/4334246\)
\(\beta_{13}\)\(=\)\((\)\(600491 \nu^{19} - 5014091 \nu^{18} + 1433657 \nu^{17} + 92851141 \nu^{16} - 196388497 \nu^{15} - 595632982 \nu^{14} + 2117597099 \nu^{13} + 1139674102 \nu^{12} - 9839915902 \nu^{11} + 3476162040 \nu^{10} + 22105769119 \nu^{9} - 18817836025 \nu^{8} - 21121831192 \nu^{7} + 27608913433 \nu^{6} + 3331895996 \nu^{5} - 12057424380 \nu^{4} + 2030109247 \nu^{3} + 1034875291 \nu^{2} - 246014754 \nu + 34267467\)\()/4334246\)
\(\beta_{14}\)\(=\)\((\)\(-410993 \nu^{19} + 2213625 \nu^{18} + 6917810 \nu^{17} - 52563575 \nu^{16} - 29025299 \nu^{15} + 513862951 \nu^{14} - 139289961 \nu^{13} - 2668384327 \nu^{12} + 1733798795 \nu^{11} + 7888472785 \nu^{10} - 6649332298 \nu^{9} - 13110138889 \nu^{8} + 12281615752 \nu^{7} + 11099392631 \nu^{6} - 10831511067 \nu^{5} - 3600113430 \nu^{4} + 3719573150 \nu^{3} + 165170286 \nu^{2} - 303545763 \nu + 31062379\)\()/2167123\)
\(\beta_{15}\)\(=\)\((\)\(-413388 \nu^{19} + 3462562 \nu^{18} - 1511903 \nu^{17} - 60346403 \nu^{16} + 134466805 \nu^{15} + 354990510 \nu^{14} - 1345384350 \nu^{13} - 525261149 \nu^{12} + 5839741304 \nu^{11} - 2589729038 \nu^{10} - 12179494628 \nu^{9} + 11432686680 \nu^{8} + 10435235153 \nu^{7} - 15517251270 \nu^{6} - 583856352 \nu^{5} + 6479767911 \nu^{4} - 1490693552 \nu^{3} - 545381160 \nu^{2} + 147237033 \nu - 3502341\)\()/2167123\)
\(\beta_{16}\)\(=\)\((\)\(-1008515 \nu^{19} + 6530918 \nu^{18} + 8850087 \nu^{17} - 132965184 \nu^{16} + 86306181 \nu^{15} + 1050649075 \nu^{14} - 1488281912 \nu^{13} - 3982849598 \nu^{12} + 7914662926 \nu^{11} + 6906341856 \nu^{10} - 19645694201 \nu^{9} - 2787853840 \nu^{8} + 22331737484 \nu^{7} - 4642008039 \nu^{6} - 8848072497 \nu^{5} + 2556273551 \nu^{4} + 817071694 \nu^{3} - 21377165 \nu^{2} - 16996363 \nu - 10476190\)\()/4334246\)
\(\beta_{17}\)\(=\)\((\)\(-579954 \nu^{19} + 3614942 \nu^{18} + 6341910 \nu^{17} - 77052369 \nu^{16} + 25536710 \nu^{15} + 659355662 \nu^{14} - 695262384 \nu^{13} - 2896884557 \nu^{12} + 4231996085 \nu^{11} + 6882461020 \nu^{10} - 12292561414 \nu^{9} - 8381823921 \nu^{8} + 18384933518 \nu^{7} + 4109230177 \nu^{6} - 13340039557 \nu^{5} + 136389714 \nu^{4} + 4022677293 \nu^{3} - 483654127 \nu^{2} - 315155203 \nu + 42358963\)\()/2167123\)
\(\beta_{18}\)\(=\)\((\)\(170912 \nu^{19} - 1102231 \nu^{18} - 1613038 \nu^{17} + 22982443 \nu^{16} - 12939568 \nu^{15} - 189215421 \nu^{14} + 248577527 \nu^{13} + 774985956 \nu^{12} - 1408256152 \nu^{11} - 1603313970 \nu^{10} + 3845197606 \nu^{9} + 1393415281 \nu^{8} - 5275677336 \nu^{7} + 3660472 \nu^{6} + 3356197997 \nu^{5} - 455638987 \nu^{4} - 921371945 \nu^{3} + 151871254 \nu^{2} + 82480027 \nu - 10261733\)\()/619178\)
\(\beta_{19}\)\(=\)\((\)\(1492257 \nu^{19} - 9628117 \nu^{18} - 13332635 \nu^{17} + 194372481 \nu^{16} - 107507193 \nu^{15} - 1560069588 \nu^{14} + 1932311183 \nu^{13} + 6369622840 \nu^{12} - 10456617166 \nu^{11} - 13947586220 \nu^{10} + 27726528907 \nu^{9} + 15505936931 \nu^{8} - 38183605866 \nu^{7} - 6618531811 \nu^{6} + 25857674064 \nu^{5} - 907539212 \nu^{4} - 7654017971 \nu^{3} + 1064221961 \nu^{2} + 698367060 \nu - 78687283\)\()/4334246\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{18} + \beta_{15} + \beta_{14} + \beta_{10} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{19} + 2 \beta_{18} + \beta_{17} + 2 \beta_{16} + \beta_{15} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} + 27 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(12 \beta_{18} + \beta_{17} + \beta_{16} + 11 \beta_{15} + 9 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 12 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{4} + 12 \beta_{3} + 58 \beta_{2} + 13 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(11 \beta_{19} + 25 \beta_{18} + 12 \beta_{17} + 23 \beta_{16} + 14 \beta_{15} - \beta_{14} - \beta_{13} - 14 \beta_{11} + 15 \beta_{10} - 15 \beta_{9} + 14 \beta_{8} + 13 \beta_{7} + 8 \beta_{6} - 11 \beta_{5} + 26 \beta_{4} + 70 \beta_{3} + 95 \beta_{2} + 159 \beta_{1} + 69\)
\(\nu^{8}\)\(=\)\(3 \beta_{19} + 107 \beta_{18} + 16 \beta_{17} + 17 \beta_{16} + 95 \beta_{15} + 61 \beta_{14} - 12 \beta_{13} + 12 \beta_{12} - 31 \beta_{11} + 108 \beta_{10} - 18 \beta_{9} + 16 \beta_{8} + 16 \beta_{7} - 15 \beta_{6} - 2 \beta_{5} + 34 \beta_{4} + 110 \beta_{3} + 415 \beta_{2} + 122 \beta_{1} + 427\)
\(\nu^{9}\)\(=\)\(94 \beta_{19} + 233 \beta_{18} + 112 \beta_{17} + 200 \beta_{16} + 143 \beta_{15} - 18 \beta_{14} - 14 \beta_{13} + \beta_{12} - 142 \beta_{11} + 159 \beta_{10} - 157 \beta_{9} + 137 \beta_{8} + 125 \beta_{7} + 42 \beta_{6} - 90 \beta_{5} + 249 \beta_{4} + 522 \beta_{3} + 760 \beta_{2} + 998 \beta_{1} + 508\)
\(\nu^{10}\)\(=\)\(55 \beta_{19} + 861 \beta_{18} + 181 \beta_{17} + 195 \beta_{16} + 758 \beta_{15} + 368 \beta_{14} - 104 \beta_{13} + 107 \beta_{12} - 335 \beta_{11} + 881 \beta_{10} - 223 \beta_{9} + 178 \beta_{8} + 183 \beta_{7} - 159 \beta_{6} - 29 \beta_{5} + 388 \beta_{4} + 915 \beta_{3} + 2969 \beta_{2} + 1029 \beta_{1} + 2627\)
\(\nu^{11}\)\(=\)\(749 \beta_{19} + 1951 \beta_{18} + 964 \beta_{17} + 1587 \beta_{16} + 1289 \beta_{15} - 215 \beta_{14} - 130 \beta_{13} + 28 \beta_{12} - 1272 \beta_{11} + 1465 \beta_{10} - 1424 \beta_{9} + 1164 \beta_{8} + 1082 \beta_{7} + 139 \beta_{6} - 655 \beta_{5} + 2129 \beta_{4} + 3829 \beta_{3} + 5886 \beta_{2} + 6543 \beta_{1} + 3700\)
\(\nu^{12}\)\(=\)\(682 \beta_{19} + 6626 \beta_{18} + 1783 \beta_{17} + 1903 \beta_{16} + 5847 \beta_{15} + 2054 \beta_{14} - 784 \beta_{13} + 866 \beta_{12} - 3132 \beta_{11} + 6881 \beta_{10} - 2321 \beta_{9} + 1695 \beta_{8} + 1823 \beta_{7} - 1465 \beta_{6} - 274 \beta_{5} + 3757 \beta_{4} + 7272 \beta_{3} + 21318 \beta_{2} + 8274 \beta_{1} + 16903\)
\(\nu^{13}\)\(=\)\(5849 \beta_{19} + 15528 \beta_{18} + 8007 \beta_{17} + 12128 \beta_{16} + 10902 \beta_{15} - 2161 \beta_{14} - 997 \beta_{13} + 443 \beta_{12} - 10725 \beta_{11} + 12567 \beta_{10} - 12026 \beta_{9} + 9231 \beta_{8} + 8945 \beta_{7} - 237 \beta_{6} - 4470 \beta_{5} + 17254 \beta_{4} + 27907 \beta_{3} + 44880 \beta_{2} + 44173 \beta_{1} + 26897\)
\(\nu^{14}\)\(=\)\(7165 \beta_{19} + 49927 \beta_{18} + 16332 \beta_{17} + 17066 \beta_{16} + 44367 \beta_{15} + 10564 \beta_{14} - 5421 \beta_{13} + 6771 \beta_{12} - 27216 \beta_{11} + 52601 \beta_{10} - 21850 \beta_{9} + 14829 \beta_{8} + 16853 \beta_{7} - 12569 \beta_{6} - 2112 \beta_{5} + 33413 \beta_{4} + 56412 \beta_{3} + 153805 \beta_{2} + 64837 \beta_{1} + 112151\)
\(\nu^{15}\)\(=\)\(45504 \beta_{19} + 120296 \beta_{18} + 65244 \beta_{17} + 91153 \beta_{16} + 88859 \beta_{15} - 19819 \beta_{14} - 6721 \beta_{13} + 5434 \beta_{12} - 87508 \beta_{11} + 103472 \beta_{10} - 97615 \beta_{9} + 70564 \beta_{8} + 72322 \beta_{7} - 10263 \beta_{6} - 29212 \beta_{5} + 135979 \beta_{4} + 203042 \beta_{3} + 339458 \beta_{2} + 304274 \beta_{1} + 195625\)
\(\nu^{16}\)\(=\)\(68741 \beta_{19} + 372420 \beta_{18} + 143070 \beta_{17} + 145540 \beta_{16} + 333767 \beta_{15} + 47771 \beta_{14} - 34977 \beta_{13} + 52411 \beta_{12} - 226950 \beta_{11} + 397689 \beta_{10} - 193095 \beta_{9} + 123205 \beta_{8} + 148769 \beta_{7} - 103573 \beta_{6} - 14172 \beta_{5} + 282868 \beta_{4} + 431591 \beta_{3} + 1115180 \beta_{2} + 500189 \beta_{1} + 760028\)
\(\nu^{17}\)\(=\)\(354430 \beta_{19} + 917916 \beta_{18} + 525239 \beta_{17} + 680149 \beta_{16} + 707465 \beta_{15} - 172044 \beta_{14} - 40089 \beta_{13} + 57847 \beta_{12} - 700514 \beta_{11} + 830610 \beta_{10} - 773876 \beta_{9} + 528454 \beta_{8} + 578015 \beta_{7} - 131379 \beta_{6} - 184387 \beta_{5} + 1055444 \beta_{4} + 1478128 \beta_{3} + 2556417 \beta_{2} + 2126070 \beta_{1} + 1424146\)
\(\nu^{18}\)\(=\)\(623089 \beta_{19} + 2765411 \beta_{18} + 1215932 \beta_{17} + 1201403 \beta_{16} + 2498983 \beta_{15} + 158548 \beta_{14} - 210305 \beta_{13} + 405455 \beta_{12} - 1846013 \beta_{11} + 2990011 \beta_{10} - 1636228 \beta_{9} + 989690 \beta_{8} + 1273224 \beta_{7} - 832969 \beta_{6} - 83371 \beta_{5} + 2322373 \beta_{4} + 3274565 \beta_{3} + 8123538 \beta_{2} + 3819421 \beta_{1} + 5228188\)
\(\nu^{19}\)\(=\)\(2765954 \beta_{19} + 6943503 \beta_{18} + 4191957 \beta_{17} + 5061055 \beta_{16} + 5543561 \beta_{15} - 1440901 \beta_{14} - 201460 \beta_{13} + 563630 \beta_{12} - 5542441 \beta_{11} + 6557790 \beta_{10} - 6045266 \beta_{9} + 3911178 \beta_{8} + 4589665 \beta_{7} - 1330431 \beta_{6} - 1126086 \beta_{5} + 8123077 \beta_{4} + 10779432 \beta_{3} + 19206088 \beta_{2} + 15014219 \beta_{1} + 10376694\)

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
2.75954
2.70334
2.58654
2.24329
2.11976
2.02020
1.78791
1.72132
0.831296
0.697877
0.170762
0.102309
−0.343534
−0.723268
−0.731838
−1.60447
−1.86440
−1.86718
−2.12365
−2.48579
−2.75954 0 5.61505 −0.308409 0 1.00000 −9.97588 0 0.851066
1.2 −2.70334 0 5.30805 3.37987 0 1.00000 −8.94278 0 −9.13693
1.3 −2.58654 0 4.69021 −1.18795 0 1.00000 −6.95835 0 3.07267
1.4 −2.24329 0 3.03233 1.67114 0 1.00000 −2.31581 0 −3.74884
1.5 −2.11976 0 2.49337 −3.20666 0 1.00000 −1.04583 0 6.79735
1.6 −2.02020 0 2.08119 3.61502 0 1.00000 −0.164016 0 −7.30304
1.7 −1.78791 0 1.19664 −2.61034 0 1.00000 1.43634 0 4.66707
1.8 −1.72132 0 0.962958 −2.01516 0 1.00000 1.78509 0 3.46875
1.9 −0.831296 0 −1.30895 1.71517 0 1.00000 2.75071 0 −1.42582
1.10 −0.697877 0 −1.51297 0.682552 0 1.00000 2.45162 0 −0.476337
1.11 −0.170762 0 −1.97084 −2.09248 0 1.00000 0.678071 0 0.357317
1.12 −0.102309 0 −1.98953 −2.62692 0 1.00000 0.408164 0 0.268756
1.13 0.343534 0 −1.88198 4.21457 0 1.00000 −1.33360 0 1.44785
1.14 0.723268 0 −1.47688 1.71672 0 1.00000 −2.51472 0 1.24165
1.15 0.731838 0 −1.46441 −3.93264 0 1.00000 −2.53539 0 −2.87806
1.16 1.60447 0 0.574339 −0.248535 0 1.00000 −2.28744 0 −0.398768
1.17 1.86440 0 1.47599 1.88221 0 1.00000 −0.976959 0 3.50920
1.18 1.86718 0 1.48637 −2.52504 0 1.00000 −0.959047 0 −4.71472
1.19 2.12365 0 2.50989 2.22935 0 1.00000 1.08284 0 4.73436
1.20 2.48579 0 4.17918 −3.35246 0 1.00000 5.41698 0 −8.33354
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

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 8001.2.a.w 20
3.b odd 2 1 889.2.a.d 20
21.c even 2 1 6223.2.a.l 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.d 20 3.b odd 2 1
6223.2.a.l 20 21.c even 2 1
8001.2.a.w 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 31 + 425 T + 609 T^{2} - 6812 T^{3} - 8607 T^{4} + 25504 T^{5} + 32493 T^{6} - 35701 T^{7} - 52271 T^{8} + 17664 T^{9} + 39579 T^{10} + 56 T^{11} - 15474 T^{12} - 2977 T^{13} + 3125 T^{14} + 1061 T^{15} - 274 T^{16} - 152 T^{17} + 8 T^{19} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( -203968 - 1241664 T - 467616 T^{2} + 5947520 T^{3} + 2267944 T^{4} - 8263246 T^{5} - 2710324 T^{6} + 5458748 T^{7} + 1651483 T^{8} - 2020391 T^{9} - 595740 T^{10} + 445364 T^{11} + 132200 T^{12} - 58925 T^{13} - 17889 T^{14} + 4519 T^{15} + 1412 T^{16} - 183 T^{17} - 59 T^{18} + 3 T^{19} + T^{20} \)
$7$ \( ( -1 + T )^{20} \)
$11$ \( -11392 + 743008 T + 5850048 T^{2} - 27027632 T^{3} - 49644343 T^{4} + 44572044 T^{5} + 139103627 T^{6} + 86640991 T^{7} - 15309410 T^{8} - 40435362 T^{9} - 14554935 T^{10} + 2269822 T^{11} + 2963926 T^{12} + 654604 T^{13} - 69539 T^{14} - 56149 T^{15} - 8338 T^{16} + 311 T^{17} + 231 T^{18} + 26 T^{19} + T^{20} \)
$13$ \( -11509712 + 123229184 T - 267988376 T^{2} - 8051824 T^{3} + 412526113 T^{4} - 121262151 T^{5} - 260063341 T^{6} + 81927040 T^{7} + 87825864 T^{8} - 22941885 T^{9} - 17003830 T^{10} + 3422988 T^{11} + 1948454 T^{12} - 293467 T^{13} - 133133 T^{14} + 14430 T^{15} + 5297 T^{16} - 376 T^{17} - 113 T^{18} + 4 T^{19} + T^{20} \)
$17$ \( 7648759472 - 8595851872 T - 9474026928 T^{2} + 10948112988 T^{3} + 5371681587 T^{4} - 5708369688 T^{5} - 1878041878 T^{6} + 1575342985 T^{7} + 431250253 T^{8} - 249625165 T^{9} - 63456729 T^{10} + 23322430 T^{11} + 5772535 T^{12} - 1280814 T^{13} - 314482 T^{14} + 39812 T^{15} + 9778 T^{16} - 635 T^{17} - 157 T^{18} + 4 T^{19} + T^{20} \)
$19$ \( -3061190104 + 30025002182 T + 69148371798 T^{2} - 107845712030 T^{3} - 18225293377 T^{4} + 59314435216 T^{5} - 6131489686 T^{6} - 12450389728 T^{7} + 2517632078 T^{8} + 1285696564 T^{9} - 334962411 T^{10} - 72546101 T^{11} + 22606963 T^{12} + 2304155 T^{13} - 858326 T^{14} - 39994 T^{15} + 18521 T^{16} + 339 T^{17} - 212 T^{18} - T^{19} + T^{20} \)
$23$ \( -37130752 + 366411840 T + 2649502560 T^{2} - 19929353552 T^{3} - 43254838200 T^{4} - 14799400580 T^{5} + 26717847572 T^{6} + 28596400746 T^{7} + 9447308331 T^{8} - 692757974 T^{9} - 1298779482 T^{10} - 289807035 T^{11} + 12801829 T^{12} + 14642873 T^{13} + 1746801 T^{14} - 133360 T^{15} - 45310 T^{16} - 2394 T^{17} + 232 T^{18} + 31 T^{19} + T^{20} \)
$29$ \( -847113116416 + 271304454400 T + 1828350979072 T^{2} + 11030718624 T^{3} - 1189791340640 T^{4} - 120001447520 T^{5} + 345347477608 T^{6} + 49675716454 T^{7} - 48840619699 T^{8} - 8766692964 T^{9} + 3285030631 T^{10} + 742257689 T^{11} - 90747674 T^{12} - 28533905 T^{13} + 678071 T^{14} + 532441 T^{15} + 11912 T^{16} - 4726 T^{17} - 224 T^{18} + 16 T^{19} + T^{20} \)
$31$ \( 767665605056 + 1467138502526 T - 801345199498 T^{2} - 2118789884822 T^{3} + 124926648405 T^{4} + 863070683686 T^{5} - 16800684379 T^{6} - 152097883632 T^{7} + 7125020075 T^{8} + 13604536595 T^{9} - 1056351736 T^{10} - 660684814 T^{11} + 67187664 T^{12} + 17864806 T^{13} - 2142373 T^{14} - 265951 T^{15} + 35930 T^{16} + 2016 T^{17} - 302 T^{18} - 6 T^{19} + T^{20} \)
$37$ \( -1123613199576 + 486820420380 T + 1791163175704 T^{2} - 1143243043428 T^{3} - 636548286199 T^{4} + 594809870831 T^{5} + 15331673692 T^{6} - 100902301895 T^{7} + 11478563508 T^{8} + 8066558277 T^{9} - 1459687483 T^{10} - 347204956 T^{11} + 79424120 T^{12} + 8405024 T^{13} - 2303871 T^{14} - 112938 T^{15} + 36838 T^{16} + 770 T^{17} - 304 T^{18} - 2 T^{19} + T^{20} \)
$41$ \( -9083203177744 + 28703459923744 T + 60521554648536 T^{2} - 41912608109288 T^{3} - 36533697788801 T^{4} + 4885855365900 T^{5} + 5996235447145 T^{6} + 18606491766 T^{7} - 448630618929 T^{8} - 29128469024 T^{9} + 17930454439 T^{10} + 1866581889 T^{11} - 398100146 T^{12} - 55613963 T^{13} + 4580995 T^{14} + 889048 T^{15} - 17271 T^{16} - 7377 T^{17} - 124 T^{18} + 25 T^{19} + T^{20} \)
$43$ \( -2352002970368 - 17920022981184 T + 2445634297120 T^{2} + 28393514163728 T^{3} + 10223298919768 T^{4} - 6502983248588 T^{5} - 3272409325924 T^{6} + 430123209302 T^{7} + 361607146459 T^{8} + 2131479486 T^{9} - 18176037955 T^{10} - 1039281936 T^{11} + 483180882 T^{12} + 37001902 T^{13} - 7464753 T^{14} - 576573 T^{15} + 70198 T^{16} + 4338 T^{17} - 389 T^{18} - 13 T^{19} + T^{20} \)
$47$ \( 7160930336 + 71173662942 T + 253604313244 T^{2} + 395642260632 T^{3} + 240668312075 T^{4} - 45188842833 T^{5} - 134888069040 T^{6} - 58172992896 T^{7} + 2758431947 T^{8} + 9320545912 T^{9} + 2380670340 T^{10} - 105545010 T^{11} - 132366135 T^{12} - 13062223 T^{13} + 2283441 T^{14} + 445588 T^{15} - 5998 T^{16} - 5095 T^{17} - 166 T^{18} + 19 T^{19} + T^{20} \)
$53$ \( -4131688997632 + 13914455818880 T + 75866371451520 T^{2} - 56827390892416 T^{3} - 65681183988704 T^{4} + 31415556710904 T^{5} + 17717190699976 T^{6} - 2840451729316 T^{7} - 1939999484851 T^{8} - 32373350671 T^{9} + 75842140008 T^{10} + 6304416954 T^{11} - 1303840520 T^{12} - 173274592 T^{13} + 9148718 T^{14} + 2067407 T^{15} + 1749 T^{16} - 11454 T^{17} - 308 T^{18} + 24 T^{19} + T^{20} \)
$59$ \( 11947401797632 + 47825973970944 T + 23982240732672 T^{2} - 47790414039616 T^{3} - 25234668196272 T^{4} + 13243381314504 T^{5} + 8041641764388 T^{6} - 991295509906 T^{7} - 1022181339963 T^{8} - 51033833400 T^{9} + 47148653357 T^{10} + 5687967717 T^{11} - 883337144 T^{12} - 161364760 T^{13} + 5534490 T^{14} + 1972023 T^{15} + 19559 T^{16} - 10997 T^{17} - 345 T^{18} + 23 T^{19} + T^{20} \)
$61$ \( 1381335112304 - 8874153165696 T + 13877656435424 T^{2} + 2111974119260 T^{3} - 15236508853913 T^{4} + 1892451432134 T^{5} + 6341906048523 T^{6} - 326214780229 T^{7} - 1156834225343 T^{8} - 105942000111 T^{9} + 62186817476 T^{10} + 8661306228 T^{11} - 1315022877 T^{12} - 232374376 T^{13} + 10107723 T^{14} + 2724373 T^{15} + 3814 T^{16} - 14260 T^{17} - 345 T^{18} + 27 T^{19} + T^{20} \)
$67$ \( -1782036672242688 - 5409327530249472 T - 4397855547200128 T^{2} - 75908857441472 T^{3} + 1065677077709040 T^{4} + 191388964282192 T^{5} - 96532279791992 T^{6} - 19993987843882 T^{7} + 4820155273225 T^{8} + 899971341674 T^{9} - 148460702907 T^{10} - 21305891150 T^{11} + 2861905494 T^{12} + 277154022 T^{13} - 33403855 T^{14} - 1935271 T^{15} + 221139 T^{16} + 6705 T^{17} - 749 T^{18} - 9 T^{19} + T^{20} \)
$71$ \( -670552932030659456 - 227771636177710384 T + 159147839209727768 T^{2} + 90590992661311436 T^{3} + 4735815273076249 T^{4} - 5608707963262716 T^{5} - 1118033515157538 T^{6} + 76700771001233 T^{7} + 43692447791986 T^{8} + 2511216641268 T^{9} - 641553805806 T^{10} - 91897105295 T^{11} + 1611158905 T^{12} + 1068391298 T^{13} + 51361488 T^{14} - 4285706 T^{15} - 492012 T^{16} - 6427 T^{17} + 1176 T^{18} + 63 T^{19} + T^{20} \)
$73$ \( -1196197845840464 + 1063830148460624 T + 1004284796308120 T^{2} - 628443368312032 T^{3} - 417451935376003 T^{4} + 98722047536720 T^{5} + 80668181211792 T^{6} + 1143383365712 T^{7} - 5354488356782 T^{8} - 544439578624 T^{9} + 146974272494 T^{10} + 23614439960 T^{11} - 1575636113 T^{12} - 414691327 T^{13} + 1789048 T^{14} + 3503002 T^{15} + 83044 T^{16} - 14030 T^{17} - 544 T^{18} + 21 T^{19} + T^{20} \)
$79$ \( 443538923776 - 2282264984528 T - 1500061480248 T^{2} + 9459300561532 T^{3} + 11341801535081 T^{4} + 1063676596043 T^{5} - 3245134831538 T^{6} - 1031862570296 T^{7} + 215187055853 T^{8} + 101670966567 T^{9} - 6875544491 T^{10} - 4431631873 T^{11} + 166688480 T^{12} + 102812491 T^{13} - 3708948 T^{14} - 1285186 T^{15} + 54940 T^{16} + 7899 T^{17} - 396 T^{18} - 18 T^{19} + T^{20} \)
$83$ \( -952366014608384 + 1733615114496256 T + 267566560683904 T^{2} - 952578245621248 T^{3} + 73170830787824 T^{4} + 176081556788352 T^{5} - 28400139200872 T^{6} - 13937308647310 T^{7} + 2950039467703 T^{8} + 504599632720 T^{9} - 131318026661 T^{10} - 8837887756 T^{11} + 2929766172 T^{12} + 78577909 T^{13} - 35438663 T^{14} - 360786 T^{15} + 233395 T^{16} + 881 T^{17} - 773 T^{18} - T^{19} + T^{20} \)
$89$ \( 1207053990976 - 14739420486848 T - 69074922086480 T^{2} + 55147198947728 T^{3} + 187905283308780 T^{4} - 222247556774474 T^{5} + 47417847960584 T^{6} + 22197631751920 T^{7} - 8667265327421 T^{8} + 6451087621 T^{9} + 332077319014 T^{10} - 32568526885 T^{11} - 3569545090 T^{12} + 661572082 T^{13} + 932957 T^{14} - 4815653 T^{15} + 153514 T^{16} + 14743 T^{17} - 726 T^{18} - 16 T^{19} + T^{20} \)
$97$ \( -52934605786048 - 178784847691072 T - 25042755502976 T^{2} + 571961697985808 T^{3} + 861062150607752 T^{4} + 451184359391646 T^{5} + 16543022727094 T^{6} - 57809515978290 T^{7} - 12190409552833 T^{8} + 1477696088958 T^{9} + 610730964389 T^{10} + 19526001190 T^{11} - 8805385211 T^{12} - 719024266 T^{13} + 46285061 T^{14} + 6437176 T^{15} - 30278 T^{16} - 23801 T^{17} - 449 T^{18} + 32 T^{19} + T^{20} \)
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