Properties

Label 8001.2.a.r
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} - 2747 x^{7} + 5821 x^{6} - 158 x^{5} - 3341 x^{4} + 1002 x^{3} + 416 x^{2} - 148 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} - 2747 x^{7} + 5821 x^{6} - 158 x^{5} - 3341 x^{4} + 1002 x^{3} + 416 x^{2} - 148 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\(34171 \nu^{15} - 45717 \nu^{14} - 952649 \nu^{13} + 996036 \nu^{12} + 10901947 \nu^{11} - 8612786 \nu^{10} - 64480493 \nu^{9} + 37846618 \nu^{8} + 203131982 \nu^{7} - 90034741 \nu^{6} - 313228957 \nu^{5} + 112849336 \nu^{4} + 182483997 \nu^{3} - 61371742 \nu^{2} - 23762490 \nu + 10476522\)\()/3260350\)
\(\beta_{5}\)\(=\)\((\)\(-95749 \nu^{15} + 241013 \nu^{14} + 1943846 \nu^{13} - 4685549 \nu^{12} - 14893598 \nu^{11} + 33446774 \nu^{10} + 52754722 \nu^{9} - 102849072 \nu^{8} - 85522528 \nu^{7} + 106565099 \nu^{6} + 65698528 \nu^{5} + 47018671 \nu^{4} - 57366413 \nu^{3} - 63764457 \nu^{2} + 11672715 \nu + 5521822\)\()/1630175\)
\(\beta_{6}\)\(=\)\((\)\(138556 \nu^{15} - 479387 \nu^{14} - 2447289 \nu^{13} + 9459921 \nu^{12} + 14196617 \nu^{11} - 69724171 \nu^{10} - 19774673 \nu^{9} + 231708473 \nu^{8} - 83578798 \nu^{7} - 316169501 \nu^{6} + 271416573 \nu^{5} + 75977871 \nu^{4} - 188741583 \nu^{3} + 54036313 \nu^{2} + 31200060 \nu - 10011933\)\()/1630175\)
\(\beta_{7}\)\(=\)\((\)\(312927 \nu^{15} - 1689159 \nu^{14} - 4050143 \nu^{13} + 33947762 \nu^{12} + 2433199 \nu^{11} - 256582162 \nu^{10} + 180181899 \nu^{9} + 886177826 \nu^{8} - 982488526 \nu^{7} - 1312231707 \nu^{6} + 1885163751 \nu^{5} + 532378722 \nu^{4} - 1161023271 \nu^{3} + 13012206 \nu^{2} + 194829910 \nu - 7559716\)\()/3260350\)
\(\beta_{8}\)\(=\)\((\)\(174111 \nu^{15} - 416897 \nu^{14} - 3848034 \nu^{13} + 8756876 \nu^{12} + 33610177 \nu^{11} - 71442151 \nu^{10} - 147009638 \nu^{9} + 284665963 \nu^{8} + 336839837 \nu^{7} - 569728931 \nu^{6} - 387906862 \nu^{5} + 524853251 \nu^{4} + 199086302 \nu^{3} - 178089422 \nu^{2} - 29637740 \nu + 12678877\)\()/1630175\)
\(\beta_{9}\)\(=\)\((\)\(-62553 \nu^{15} + 185357 \nu^{14} + 1244303 \nu^{13} - 3782409 \nu^{12} - 9231198 \nu^{11} + 29429019 \nu^{10} + 30643366 \nu^{9} - 108077121 \nu^{8} - 40734444 \nu^{7} + 186611071 \nu^{6} + 9095064 \nu^{5} - 131435121 \nu^{4} - 543294 \nu^{3} + 37570814 \nu^{2} + 3158402 \nu - 3490290\)\()/326035\)
\(\beta_{10}\)\(=\)\((\)\(322256 \nu^{15} - 1375567 \nu^{14} - 5035219 \nu^{13} + 27365426 \nu^{12} + 19868527 \nu^{11} - 204238826 \nu^{10} + 53766867 \nu^{9} + 693632208 \nu^{8} - 546168183 \nu^{7} - 997385891 \nu^{6} + 1188437858 \nu^{5} + 353563286 \nu^{4} - 736318918 \nu^{3} + 56736448 \nu^{2} + 98969000 \nu - 11457663\)\()/1630175\)
\(\beta_{11}\)\(=\)\((\)\(-387602 \nu^{15} + 1093459 \nu^{14} + 7892793 \nu^{13} - 22401737 \nu^{12} - 60887074 \nu^{11} + 175479737 \nu^{10} + 217918476 \nu^{9} - 652127926 \nu^{8} - 350064674 \nu^{7} + 1148743407 \nu^{6} + 201090249 \nu^{5} - 822768247 \nu^{4} - 52865479 \nu^{3} + 185190694 \nu^{2} + 5524390 \nu + 2433991\)\()/1630175\)
\(\beta_{12}\)\(=\)\((\)\(397191 \nu^{15} - 1572742 \nu^{14} - 6632689 \nu^{13} + 31550366 \nu^{12} + 32948807 \nu^{11} - 238552241 \nu^{10} + 2632702 \nu^{9} + 829385698 \nu^{8} - 449364548 \nu^{7} - 1260230866 \nu^{6} + 1103837898 \nu^{5} + 577130586 \nu^{4} - 695428333 \nu^{3} - 15812362 \nu^{2} + 100033165 \nu - 6777548\)\()/1630175\)
\(\beta_{13}\)\(=\)\((\)\(443728 \nu^{15} - 1561421 \nu^{14} - 8006447 \nu^{13} + 31434138 \nu^{12} + 49072551 \nu^{11} - 239584888 \nu^{10} - 92916154 \nu^{9} + 848762304 \nu^{8} - 151345429 \nu^{7} - 1357041408 \nu^{6} + 646884804 \nu^{5} + 763592393 \nu^{4} - 420987859 \nu^{3} - 120963051 \nu^{2} + 53266175 \nu + 3556906\)\()/1630175\)
\(\beta_{14}\)\(=\)\((\)\(509784 \nu^{15} - 2052553 \nu^{14} - 8419656 \nu^{13} + 41075579 \nu^{12} + 40710358 \nu^{11} - 309487504 \nu^{10} + 12346983 \nu^{9} + 1070084317 \nu^{8} - 592856117 \nu^{7} - 1609704994 \nu^{6} + 1400337842 \nu^{5} + 719985124 \nu^{4} - 824771807 \nu^{3} - 26429048 \nu^{2} + 92347345 \nu - 4486597\)\()/1630175\)
\(\beta_{15}\)\(=\)\((\)\(540201 \nu^{15} - 2114267 \nu^{14} - 9125559 \nu^{13} + 42440381 \nu^{12} + 47185962 \nu^{11} - 321313406 \nu^{10} - 17132488 \nu^{9} + 1120057338 \nu^{8} - 525465538 \nu^{7} - 1711378566 \nu^{6} + 1336320913 \nu^{5} + 794016061 \nu^{4} - 824333673 \nu^{3} - 8611797 \nu^{2} + 109330955 \nu - 15755608\)\()/1630175\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{15} + \beta_{13} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + 9 \beta_{3} + 10 \beta_{2} + 28 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(\beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 11 \beta_{11} - 10 \beta_{10} + 13 \beta_{8} + 12 \beta_{6} - 2 \beta_{4} + 9 \beta_{3} + 57 \beta_{2} + 2 \beta_{1} + 85\)
\(\nu^{7}\)\(=\)\(-11 \beta_{15} + \beta_{14} + 8 \beta_{13} + 3 \beta_{11} + \beta_{10} + 10 \beta_{9} + 17 \beta_{8} + 11 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} - 15 \beta_{4} + 65 \beta_{3} + 84 \beta_{2} + 167 \beta_{1} + 40\)
\(\nu^{8}\)\(=\)\(-3 \beta_{15} + 15 \beta_{14} - 28 \beta_{13} + 76 \beta_{12} + 94 \beta_{11} - 76 \beta_{10} - \beta_{9} + 125 \beta_{8} + 3 \beta_{7} + 108 \beta_{6} - 2 \beta_{5} - 33 \beta_{4} + 65 \beta_{3} + 398 \beta_{2} + 33 \beta_{1} + 514\)
\(\nu^{9}\)\(=\)\(-90 \beta_{15} + 13 \beta_{14} + 41 \beta_{13} + \beta_{12} + 56 \beta_{11} + 16 \beta_{10} + 72 \beta_{9} + 198 \beta_{8} + 94 \beta_{7} + 41 \beta_{6} - 108 \beta_{5} - 160 \beta_{4} + 439 \beta_{3} + 671 \beta_{2} + 1040 \beta_{1} + 386\)
\(\nu^{10}\)\(=\)\(-47 \beta_{15} + 152 \beta_{14} - 283 \beta_{13} + 524 \beta_{12} + 745 \beta_{11} - 524 \beta_{10} - 21 \beta_{9} + 1081 \beta_{8} + 56 \beta_{7} + 880 \beta_{6} - 41 \beta_{5} - 376 \beta_{4} + 446 \beta_{3} + 2789 \beta_{2} + 373 \beta_{1} + 3243\)
\(\nu^{11}\)\(=\)\(-659 \beta_{15} + 124 \beta_{14} + 113 \beta_{13} + 11 \beta_{12} + 693 \beta_{11} + 168 \beta_{10} + 447 \beta_{9} + 1967 \beta_{8} + 745 \beta_{7} + 540 \beta_{6} - 880 \beta_{5} - 1497 \beta_{4} + 2892 \beta_{3} + 5242 \beta_{2} + 6685 \beta_{1} + 3289\)
\(\nu^{12}\)\(=\)\(-488 \beta_{15} + 1308 \beta_{14} - 2518 \beta_{13} + 3447 \beta_{12} + 5755 \beta_{11} - 3453 \beta_{10} - 287 \beta_{9} + 8905 \beta_{8} + 693 \beta_{7} + 6864 \beta_{6} - 540 \beta_{5} - 3687 \beta_{4} + 3041 \beta_{3} + 19751 \beta_{2} + 3616 \beta_{1} + 21084\)
\(\nu^{13}\)\(=\)\(-4564 \beta_{15} + 1055 \beta_{14} - 621 \beta_{13} + 40 \beta_{12} + 7185 \beta_{11} + 1480 \beta_{10} + 2467 \beta_{9} + 17932 \beta_{8} + 5755 \beta_{7} + 5856 \beta_{6} - 6864 \beta_{5} - 13129 \beta_{4} + 18896 \beta_{3} + 40463 \beta_{2} + 44042 \beta_{1} + 26378\)
\(\nu^{14}\)\(=\)\(-4226 \beta_{15} + 10308 \beta_{14} - 20979 \beta_{13} + 22018 \beta_{12} + 44116 \beta_{11} - 22192 \beta_{10} - 3248 \beta_{9} + 71487 \beta_{8} + 7185 \beta_{7} + 52445 \beta_{6} - 5856 \beta_{5} - 33443 \beta_{4} + 20897 \beta_{3} + 141527 \beta_{2} + 32386 \beta_{1} + 140173\)
\(\nu^{15}\)\(=\)\(-30550 \beta_{15} + 8435 \beta_{14} - 14647 \beta_{13} - 580 \beta_{12} + 67671 \beta_{11} + 11924 \beta_{10} + 11585 \beta_{9} + 155148 \beta_{8} + 44116 \beta_{7} + 57046 \beta_{6} - 52445 \beta_{5} - 111002 \beta_{4} + 123435 \beta_{3} + 310155 \beta_{2} + 296102 \beta_{1} + 204740\)

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
2.76723
2.62936
2.41798
1.84800
1.68977
1.55853
1.09349
0.639731
0.298113
0.0296587
−0.415886
−0.823041
−1.82937
−2.04658
−2.36717
−2.48981
−2.76723 0 5.65757 3.65468 0 −1.00000 −10.1213 0 −10.1133
1.2 −2.62936 0 4.91351 0.281520 0 −1.00000 −7.66067 0 −0.740216
1.3 −2.41798 0 3.84663 −2.86533 0 −1.00000 −4.46511 0 6.92831
1.4 −1.84800 0 1.41512 3.49192 0 −1.00000 1.08086 0 −6.45308
1.5 −1.68977 0 0.855315 −2.75353 0 −1.00000 1.93425 0 4.65282
1.6 −1.55853 0 0.429028 1.78044 0 −1.00000 2.44841 0 −2.77487
1.7 −1.09349 0 −0.804288 −1.42245 0 −1.00000 3.06645 0 1.55543
1.8 −0.639731 0 −1.59074 2.86363 0 −1.00000 2.29711 0 −1.83195
1.9 −0.298113 0 −1.91113 −2.54722 0 −1.00000 1.16596 0 0.759360
1.10 −0.0296587 0 −1.99912 −3.07938 0 −1.00000 0.118609 0 0.0913304
1.11 0.415886 0 −1.82704 0.233776 0 −1.00000 −1.59161 0 0.0972242
1.12 0.823041 0 −1.32260 3.74037 0 −1.00000 −2.73464 0 3.07848
1.13 1.82937 0 1.34661 1.45514 0 −1.00000 −1.19530 0 2.66200
1.14 2.04658 0 2.18847 2.05615 0 −1.00000 0.385720 0 4.20807
1.15 2.36717 0 3.60352 −4.44142 0 −1.00000 3.79580 0 −10.5136
1.16 2.48981 0 4.19916 −1.44828 0 −1.00000 5.47548 0 −3.60594
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.r 16
3.b odd 2 1 2667.2.a.o 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.o 16 3.b odd 2 1
8001.2.a.r 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 148 T + 416 T^{2} - 1002 T^{3} - 3341 T^{4} + 158 T^{5} + 5821 T^{6} + 2747 T^{7} - 3082 T^{8} - 2282 T^{9} + 565 T^{10} + 712 T^{11} + 9 T^{12} - 98 T^{13} - 13 T^{14} + 5 T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( -27136 + 208768 T - 344128 T^{2} - 359328 T^{3} + 542544 T^{4} + 232104 T^{5} - 316420 T^{6} - 73336 T^{7} + 91364 T^{8} + 12600 T^{9} - 14371 T^{10} - 1185 T^{11} + 1249 T^{12} + 56 T^{13} - 56 T^{14} - T^{15} + T^{16} \)
$7$ \( ( 1 + T )^{16} \)
$11$ \( -1179648 - 8364032 T - 22375424 T^{2} - 26521856 T^{3} - 9532800 T^{4} + 7245280 T^{5} + 6803293 T^{6} + 563202 T^{7} - 1042308 T^{8} - 269445 T^{9} + 56472 T^{10} + 24672 T^{11} - 428 T^{12} - 888 T^{13} - 49 T^{14} + 11 T^{15} + T^{16} \)
$13$ \( -61239104 + 348343936 T - 665531472 T^{2} + 451956448 T^{3} + 14462820 T^{4} - 139088560 T^{5} + 38601569 T^{6} + 11317384 T^{7} - 6109748 T^{8} + 24926 T^{9} + 352702 T^{10} - 39214 T^{11} - 7668 T^{12} + 1562 T^{13} + 11 T^{14} - 18 T^{15} + T^{16} \)
$17$ \( 939096 - 10373798 T - 26553199 T^{2} + 87531428 T^{3} - 57680354 T^{4} - 9307050 T^{5} + 21989368 T^{6} - 5624087 T^{7} - 1463057 T^{8} + 802361 T^{9} - 19265 T^{10} - 37776 T^{11} + 4040 T^{12} + 735 T^{13} - 114 T^{14} - 5 T^{15} + T^{16} \)
$19$ \( -18205664 + 15264728 T + 123230100 T^{2} - 60135700 T^{3} - 207031320 T^{4} - 17673676 T^{5} + 69330555 T^{6} + 19474268 T^{7} - 4946635 T^{8} - 2164370 T^{9} + 67805 T^{10} + 91651 T^{11} + 3733 T^{12} - 1673 T^{13} - 123 T^{14} + 11 T^{15} + T^{16} \)
$23$ \( 885006336 - 1962362368 T - 1118563072 T^{2} + 2777301760 T^{3} + 313746112 T^{4} - 913579808 T^{5} - 6053312 T^{6} + 107080624 T^{7} + 606292 T^{8} - 6061328 T^{9} - 157913 T^{10} + 176098 T^{11} + 8415 T^{12} - 2467 T^{13} - 160 T^{14} + 13 T^{15} + T^{16} \)
$29$ \( 24336668672 + 30142906848 T - 1675831296 T^{2} - 14148060280 T^{3} - 3258186392 T^{4} + 2040741606 T^{5} + 777392167 T^{6} - 76283362 T^{7} - 59428567 T^{8} - 2286180 T^{9} + 1852743 T^{10} + 193194 T^{11} - 21848 T^{12} - 3818 T^{13} + 8 T^{14} + 24 T^{15} + T^{16} \)
$31$ \( 63774720 - 1312400896 T - 7038464 T^{2} + 1855853920 T^{3} + 368632800 T^{4} - 719197446 T^{5} - 283400427 T^{6} + 62123044 T^{7} + 51515713 T^{8} + 7050827 T^{9} - 1320819 T^{10} - 489813 T^{11} - 44073 T^{12} + 1767 T^{13} + 607 T^{14} + 42 T^{15} + T^{16} \)
$37$ \( 39421737428 + 16074547284 T - 29627325635 T^{2} - 3650308220 T^{3} + 8468827733 T^{4} - 594178683 T^{5} - 1007814474 T^{6} + 201275717 T^{7} + 42779044 T^{8} - 16317356 T^{9} + 420032 T^{10} + 422797 T^{11} - 58414 T^{12} + 285 T^{13} + 501 T^{14} - 40 T^{15} + T^{16} \)
$41$ \( 16289945096 - 109425368814 T + 217123192233 T^{2} - 177296209815 T^{3} + 51942179735 T^{4} + 6696670185 T^{5} - 5960829549 T^{6} + 313892087 T^{7} + 240883799 T^{8} - 20351754 T^{9} - 5147630 T^{10} + 389089 T^{11} + 62205 T^{12} - 3119 T^{13} - 394 T^{14} + 9 T^{15} + T^{16} \)
$43$ \( 41489887232 - 53869608960 T - 36774800640 T^{2} + 22134681728 T^{3} + 11927863680 T^{4} - 2601630528 T^{5} - 1537015488 T^{6} + 104851432 T^{7} + 92592520 T^{8} - 70456 T^{9} - 2766767 T^{10} - 77508 T^{11} + 41667 T^{12} + 1377 T^{13} - 316 T^{14} - 7 T^{15} + T^{16} \)
$47$ \( 14109523968 - 30299066368 T - 33986376960 T^{2} + 3164929536 T^{3} + 11794207764 T^{4} + 2863572924 T^{5} - 773849045 T^{6} - 399684257 T^{7} - 19556301 T^{8} + 15292602 T^{9} + 2623371 T^{10} - 75363 T^{11} - 53211 T^{12} - 3574 T^{13} + 189 T^{14} + 31 T^{15} + T^{16} \)
$53$ \( -443170186752 - 198052307872 T + 180696190064 T^{2} + 90189284672 T^{3} - 22818806676 T^{4} - 15575563488 T^{5} + 237147455 T^{6} + 1212752963 T^{7} + 149147211 T^{8} - 33614746 T^{9} - 9659696 T^{10} - 441631 T^{11} + 129116 T^{12} + 23588 T^{13} + 1773 T^{14} + 66 T^{15} + T^{16} \)
$59$ \( -219737088000 - 817016531200 T - 115227370560 T^{2} + 541342008032 T^{3} + 210192647744 T^{4} - 27816997840 T^{5} - 18737338236 T^{6} + 44197692 T^{7} + 636875872 T^{8} + 17854814 T^{9} - 10941509 T^{10} - 358776 T^{11} + 102525 T^{12} + 2687 T^{13} - 502 T^{14} - 7 T^{15} + T^{16} \)
$61$ \( 15419695152 - 212813119120 T - 2157107564908 T^{2} - 1220842778620 T^{3} + 395438644656 T^{4} + 157308999934 T^{5} - 33128608399 T^{6} - 6127236110 T^{7} + 1167671429 T^{8} + 105369281 T^{9} - 19556025 T^{10} - 890269 T^{11} + 163141 T^{12} + 3687 T^{13} - 653 T^{14} - 6 T^{15} + T^{16} \)
$67$ \( -50604450816 + 193133407232 T - 247299779008 T^{2} + 110799151040 T^{3} + 14638741344 T^{4} - 28178676856 T^{5} + 7638415596 T^{6} + 108089952 T^{7} - 357518480 T^{8} + 43117500 T^{9} + 3374847 T^{10} - 907845 T^{11} + 14546 T^{12} + 6558 T^{13} - 306 T^{14} - 16 T^{15} + T^{16} \)
$71$ \( -2604300724224 - 7249584983600 T - 3969393459372 T^{2} + 112042350264 T^{3} + 534372442048 T^{4} + 93373365606 T^{5} - 15672151733 T^{6} - 5714371015 T^{7} - 136758927 T^{8} + 116652514 T^{9} + 11885530 T^{10} - 654560 T^{11} - 160392 T^{12} - 4937 T^{13} + 560 T^{14} + 46 T^{15} + T^{16} \)
$73$ \( 46641155536 + 367052382272 T + 779015258676 T^{2} + 290292803332 T^{3} - 115683460368 T^{4} - 43532024354 T^{5} + 10695446287 T^{6} + 1833399094 T^{7} - 497705244 T^{8} - 22264297 T^{9} + 11027730 T^{10} - 243078 T^{11} - 107646 T^{12} + 7072 T^{13} + 281 T^{14} - 39 T^{15} + T^{16} \)
$79$ \( 39713124416832 + 11089903363052 T - 15081977537019 T^{2} - 1657557199771 T^{3} + 1706545980887 T^{4} + 95697788325 T^{5} - 85213170864 T^{6} - 2868804422 T^{7} + 2161910264 T^{8} + 49041167 T^{9} - 29285144 T^{10} - 466929 T^{11} + 210272 T^{12} + 2238 T^{13} - 741 T^{14} - 4 T^{15} + T^{16} \)
$83$ \( -18683904000 + 38940646400 T + 75696966400 T^{2} - 204808792064 T^{3} + 116589277056 T^{4} - 3303822368 T^{5} - 11940754880 T^{6} + 1586639184 T^{7} + 446546488 T^{8} - 68554214 T^{9} - 8306099 T^{10} + 1088798 T^{11} + 89974 T^{12} - 6910 T^{13} - 490 T^{14} + 15 T^{15} + T^{16} \)
$89$ \( 105203797590016 + 20680234328064 T - 43079673319424 T^{2} - 9039760123904 T^{3} + 4118312159488 T^{4} + 639431130144 T^{5} - 185680676064 T^{6} - 16700652224 T^{7} + 4241871120 T^{8} + 189800364 T^{9} - 49704225 T^{10} - 971602 T^{11} + 299739 T^{12} + 2006 T^{13} - 881 T^{14} - T^{15} + T^{16} \)
$97$ \( -14259174215040 - 98902237775904 T + 60806973383424 T^{2} + 5580530925120 T^{3} - 6759049954608 T^{4} + 91741479912 T^{5} + 271376308011 T^{6} - 12990835198 T^{7} - 5008586671 T^{8} + 345604532 T^{9} + 44081956 T^{10} - 3991473 T^{11} - 154480 T^{12} + 21068 T^{13} - 53 T^{14} - 41 T^{15} + T^{16} \)
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