Properties

Label 8001.2.a.u
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 18
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} - 5421 x^{10} - 7882 x^{9} + 13376 x^{8} + 7948 x^{7} - 15795 x^{6} - 3858 x^{5} + 8199 x^{4} + 1453 x^{3} - 1610 x^{2} - 380 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-9056232 \nu^{17} + 56952713 \nu^{16} + 63697503 \nu^{15} - 1083863790 \nu^{14} + 1061485425 \nu^{13} + 7576909534 \nu^{12} - 14966699558 \nu^{11} - 22572883200 \nu^{10} + 73124433062 \nu^{9} + 17899620183 \nu^{8} - 170430820251 \nu^{7} + 41863070067 \nu^{6} + 192141347449 \nu^{5} - 79413530246 \nu^{4} - 95483657380 \nu^{3} + 30790742789 \nu^{2} + 19199072948 \nu - 228715354\)\()/ 365415070 \)
\(\beta_{4}\)\(=\)\((\)\(12516449 \nu^{17} - 35576866 \nu^{16} - 322120881 \nu^{15} + 878461675 \nu^{14} + 3459985365 \nu^{13} - 8859247938 \nu^{12} - 20043398304 \nu^{11} + 46965576220 \nu^{10} + 67169495191 \nu^{9} - 139958996266 \nu^{8} - 129295866318 \nu^{7} + 231201547396 \nu^{6} + 134129812547 \nu^{5} - 191805387798 \nu^{4} - 67208803935 \nu^{3} + 60307885677 \nu^{2} + 15314260994 \nu - 1868332582\)\()/ 365415070 \)
\(\beta_{5}\)\(=\)\((\)\(-25066381 \nu^{17} + 165457944 \nu^{16} + 197605069 \nu^{15} - 3294331535 \nu^{14} + 2503016525 \nu^{13} + 24999769422 \nu^{12} - 37913753504 \nu^{11} - 89306242820 \nu^{10} + 187798500441 \nu^{9} + 143893003544 \nu^{8} - 436801227938 \nu^{7} - 57150890634 \nu^{6} + 479794667297 \nu^{5} - 71979148918 \nu^{4} - 217408254535 \nu^{3} + 44636251777 \nu^{2} + 35680796674 \nu - 2904833932\)\()/ 730830140 \)
\(\beta_{6}\)\(=\)\((\)\(-32625911 \nu^{17} + 257402314 \nu^{16} + 38720489 \nu^{15} - 4899416105 \nu^{14} + 7765546185 \nu^{13} + 34400341342 \nu^{12} - 85621658304 \nu^{11} - 104551063080 \nu^{10} + 388747945111 \nu^{9} + 96698100714 \nu^{8} - 864214618628 \nu^{7} + 143202903956 \nu^{6} + 917584185897 \nu^{5} - 295691214778 \nu^{4} - 402286545165 \nu^{3} + 104139037497 \nu^{2} + 68310107754 \nu - 523550232\)\()/ 730830140 \)
\(\beta_{7}\)\(=\)\((\)\(-40185441 \nu^{17} + 349346684 \nu^{16} - 120164091 \nu^{15} - 6504500675 \nu^{14} + 13028075845 \nu^{13} + 43800913262 \nu^{12} - 133329563104 \nu^{11} - 119795883340 \nu^{10} + 589697389781 \nu^{9} + 49503197884 \nu^{8} - 1291628009318 \nu^{7} + 343556698546 \nu^{6} + 1354642874357 \nu^{5} - 520134110778 \nu^{4} - 579856534395 \nu^{3} + 168757634197 \nu^{2} + 87053646174 \nu - 2527247372\)\()/ 730830140 \)
\(\beta_{8}\)\(=\)\((\)\(-20878693 \nu^{17} + 77409897 \nu^{16} + 451307397 \nu^{15} - 1771179645 \nu^{14} - 3907179050 \nu^{13} + 16305558241 \nu^{12} + 17519717313 \nu^{11} - 77527845145 \nu^{10} - 44085517162 \nu^{9} + 202940243242 \nu^{8} + 63666071141 \nu^{7} - 287556922462 \nu^{6} - 53922064354 \nu^{5} + 199447839821 \nu^{4} + 29523656040 \nu^{3} - 50247230349 \nu^{2} - 11117097738 \nu + 723097034\)\()/ 365415070 \)
\(\beta_{9}\)\(=\)\((\)\(8395741 \nu^{17} - 64482942 \nu^{16} - 11290113 \nu^{15} + 1199594231 \nu^{14} - 1945179675 \nu^{13} - 8038582192 \nu^{12} + 21352465070 \nu^{11} + 21465866534 \nu^{10} - 96183962599 \nu^{9} - 5111788540 \nu^{8} + 212368884076 \nu^{7} - 75583142318 \nu^{6} - 226469839587 \nu^{5} + 112899717876 \nu^{4} + 104809072213 \nu^{3} - 38853986253 \nu^{2} - 20711179258 \nu - 76030588\)\()/ 146166028 \)
\(\beta_{10}\)\(=\)\((\)\(50383667 \nu^{17} - 161191148 \nu^{16} - 1204101853 \nu^{15} + 3787887405 \nu^{14} + 11956797505 \nu^{13} - 35940666424 \nu^{12} - 64188753322 \nu^{11} + 176493100810 \nu^{10} + 201153933563 \nu^{9} - 476359522038 \nu^{8} - 365447236374 \nu^{7} + 688564895308 \nu^{6} + 355164023581 \nu^{5} - 472380557164 \nu^{4} - 157197227985 \nu^{3} + 110627356231 \nu^{2} + 31151563462 \nu - 2009629216\)\()/ 730830140 \)
\(\beta_{11}\)\(=\)\((\)\(80549203 \nu^{17} - 492648012 \nu^{16} - 821028837 \nu^{15} + 9925282045 \nu^{14} - 3917777155 \nu^{13} - 77138965156 \nu^{12} + 85375182182 \nu^{11} + 290745846970 \nu^{10} - 439435499053 \nu^{9} - 542716950162 \nu^{8} + 1006286981314 \nu^{7} + 442575263692 \nu^{6} - 1037561853751 \nu^{5} - 95993813136 \nu^{4} + 391615540795 \nu^{3} - 4720823941 \nu^{2} - 38822492922 \nu + 3906261256\)\()/ 730830140 \)
\(\beta_{12}\)\(=\)\((\)\(100575831 \nu^{17} - 597870654 \nu^{16} - 1070376159 \nu^{15} + 11961026425 \nu^{14} - 3958878145 \nu^{13} - 91593180032 \nu^{12} + 99303778874 \nu^{11} + 333681577930 \nu^{10} - 522661745161 \nu^{9} - 568242407784 \nu^{8} + 1220905295328 \nu^{7} + 318287907994 \nu^{6} - 1310588271257 \nu^{5} + 131621699248 \nu^{4} + 559009857575 \nu^{3} - 98821283147 \nu^{2} - 79152820674 \nu + 4483050272\)\()/ 730830140 \)
\(\beta_{13}\)\(=\)\((\)\(-101911933 \nu^{17} + 591076942 \nu^{16} + 1177422127 \nu^{15} - 11988946255 \nu^{14} + 2088222575 \nu^{13} + 94092714686 \nu^{12} - 85031622852 \nu^{11} - 359907129280 \nu^{10} + 466570180813 \nu^{9} + 687964139822 \nu^{8} - 1102540718984 \nu^{7} - 586410163532 \nu^{6} + 1177689206511 \nu^{5} + 145204756046 \nu^{4} - 478582900535 \nu^{3} - 1332536689 \nu^{2} + 52828018842 \nu + 1025152864\)\()/ 730830140 \)
\(\beta_{14}\)\(=\)\((\)\(-10957014 \nu^{17} + 55541023 \nu^{16} + 166873740 \nu^{15} - 1164950524 \nu^{14} - 620946577 \nu^{13} + 9612588205 \nu^{12} - 2152998819 \nu^{11} - 39807766547 \nu^{10} + 21114688373 \nu^{9} + 87382928254 \nu^{8} - 54458600727 \nu^{7} - 98803608930 \nu^{6} + 54227047741 \nu^{5} + 52720255893 \nu^{4} - 14672012293 \nu^{3} - 12177807586 \nu^{2} - 852708688 \nu + 267274566\)\()/73083014\)
\(\beta_{15}\)\(=\)\((\)\(33634461 \nu^{17} - 212119814 \nu^{16} - 304035654 \nu^{15} + 4222090590 \nu^{14} - 2463442190 \nu^{13} - 32123264007 \nu^{12} + 42663320289 \nu^{11} + 116114548090 \nu^{10} - 213829531481 \nu^{9} - 196165560134 \nu^{8} + 491688834403 \nu^{7} + 110801814894 \nu^{6} - 523818348102 \nu^{5} + 37804322218 \nu^{4} + 220034653890 \nu^{3} - 26980577947 \nu^{2} - 29553701509 \nu + 1153552087\)\()/ 182707535 \)
\(\beta_{16}\)\(=\)\((\)\(100069624 \nu^{17} - 532974031 \nu^{16} - 1376337581 \nu^{15} + 10970946660 \nu^{14} + 2569093715 \nu^{13} - 87933730868 \nu^{12} + 45335398986 \nu^{11} + 347183324810 \nu^{10} - 299845169974 \nu^{9} - 699846001231 \nu^{8} + 735383390107 \nu^{7} + 666522143031 \nu^{6} - 770874664093 \nu^{5} - 239724502658 \nu^{4} + 285055129740 \nu^{3} + 31458483877 \nu^{2} - 25149158196 \nu - 346733612\)\()/ 365415070 \)
\(\beta_{17}\)\(=\)\((\)\(212880147 \nu^{17} - 1116211818 \nu^{16} - 2947927153 \nu^{15} + 22837760465 \nu^{14} + 5940214235 \nu^{13} - 181212148554 \nu^{12} + 92102702768 \nu^{11} + 702455085660 \nu^{10} - 618936385827 \nu^{9} - 1362506463858 \nu^{8} + 1525835050196 \nu^{7} + 1169851753908 \nu^{6} - 1615051615589 \nu^{5} - 258422227194 \nu^{4} + 624723711805 \nu^{3} - 36412665269 \nu^{2} - 71450856478 \nu + 5373162964\)\()/ 730830140 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{16} + \beta_{14} - \beta_{10} + \beta_{7} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(9 \beta_{16} - \beta_{15} + 9 \beta_{14} - \beta_{13} - 9 \beta_{10} + 8 \beta_{7} + \beta_{6} - \beta_{5} + 9 \beta_{2} + 21 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(12 \beta_{16} + 11 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} - 2 \beta_{11} - 13 \beta_{10} + 12 \beta_{7} + 11 \beta_{6} + 59 \beta_{2} + \beta_{1} + 89\)
\(\nu^{7}\)\(=\)\(70 \beta_{16} - 11 \beta_{15} + 69 \beta_{14} - 12 \beta_{13} - \beta_{12} - \beta_{11} - 72 \beta_{10} + 2 \beta_{9} - \beta_{8} + 59 \beta_{7} + 15 \beta_{6} - 14 \beta_{5} + 70 \beta_{2} + 125 \beta_{1} + 86\)
\(\nu^{8}\)\(=\)\(\beta_{17} + 110 \beta_{16} + 92 \beta_{15} + 96 \beta_{14} + 77 \beta_{13} + 2 \beta_{12} - 29 \beta_{11} - 130 \beta_{10} + \beta_{9} - 2 \beta_{8} + 113 \beta_{7} + 97 \beta_{6} - 2 \beta_{5} - \beta_{4} + 427 \beta_{2} + 15 \beta_{1} + 575\)
\(\nu^{9}\)\(=\)\(3 \beta_{17} + 525 \beta_{16} - 88 \beta_{15} + 510 \beta_{14} - 106 \beta_{13} - 14 \beta_{12} - 21 \beta_{11} - 565 \beta_{10} + 30 \beta_{9} - 18 \beta_{8} + 436 \beta_{7} + 158 \beta_{6} - 139 \beta_{5} + \beta_{4} + 4 \beta_{3} + 537 \beta_{2} + 790 \beta_{1} + 698\)
\(\nu^{10}\)\(=\)\(17 \beta_{17} + 926 \beta_{16} + 703 \beta_{15} + 786 \beta_{14} + 543 \beta_{13} + 32 \beta_{12} - 298 \beta_{11} - 1175 \beta_{10} + 24 \beta_{9} - 43 \beta_{8} + 976 \beta_{7} + 799 \beta_{6} - 36 \beta_{5} - 17 \beta_{4} + 4 \beta_{3} + 3077 \beta_{2} + 159 \beta_{1} + 3893\)
\(\nu^{11}\)\(=\)\(56 \beta_{17} + 3904 \beta_{16} - 606 \beta_{15} + 3746 \beta_{14} - 833 \beta_{13} - 137 \beta_{12} - 280 \beta_{11} - 4416 \beta_{10} + 319 \beta_{9} - 224 \beta_{8} + 3268 \beta_{7} + 1458 \beta_{6} - 1195 \beta_{5} + 11 \beta_{4} + 79 \beta_{3} + 4141 \beta_{2} + 5147 \beta_{1} + 5515\)
\(\nu^{12}\)\(=\)\(201 \beta_{17} + 7513 \beta_{16} + 5177 \beta_{15} + 6290 \beta_{14} + 3679 \beta_{13} + 330 \beta_{12} - 2677 \beta_{11} - 10057 \beta_{10} + 348 \beta_{9} - 586 \beta_{8} + 8073 \beta_{7} + 6386 \beta_{6} - 437 \beta_{5} - 205 \beta_{4} + 99 \beta_{3} + 22189 \beta_{2} + 1480 \beta_{1} + 27107\)
\(\nu^{13}\)\(=\)\(688 \beta_{17} + 29038 \beta_{16} - 3737 \beta_{15} + 27583 \beta_{14} - 6182 \beta_{13} - 1180 \beta_{12} - 3051 \beta_{11} - 34447 \beta_{10} + 2981 \beta_{9} - 2384 \beta_{8} + 24810 \beta_{7} + 12611 \beta_{6} - 9514 \beta_{5} + 26 \beta_{4} + 1019 \beta_{3} + 32081 \beta_{2} + 34132 \beta_{1} + 43005\)
\(\nu^{14}\)\(=\)\(2053 \beta_{17} + 59783 \beta_{16} + 37549 \beta_{15} + 49823 \beta_{14} + 24374 \beta_{13} + 2749 \beta_{12} - 22488 \beta_{11} - 83230 \beta_{10} + 4044 \beta_{9} - 6536 \beta_{8} + 65123 \beta_{7} + 50205 \beta_{6} - 4458 \beta_{5} - 2173 \beta_{4} + 1509 \beta_{3} + 160445 \beta_{2} + 12948 \beta_{1} + 192283\)
\(\nu^{15}\)\(=\)\(7080 \beta_{17} + 216596 \beta_{16} - 20422 \beta_{15} + 204059 \beta_{14} - 44489 \beta_{13} - 9693 \beta_{12} - 29755 \beta_{11} - 268136 \beta_{10} + 26170 \beta_{9} - 23273 \beta_{8} + 190018 \beta_{7} + 105128 \beta_{6} - 72351 \beta_{5} - 933 \beta_{4} + 10914 \beta_{3} + 248954 \beta_{2} + 229108 \beta_{1} + 333140\)
\(\nu^{16}\)\(=\)\(19439 \beta_{17} + 470153 \beta_{16} + 271056 \beta_{15} + 392329 \beta_{14} + 159039 \beta_{13} + 19758 \beta_{12} - 181934 \beta_{11} - 673701 \beta_{10} + 41602 \beta_{9} - 65334 \beta_{8} + 517172 \beta_{7} + 390689 \beta_{6} - 41142 \beta_{5} - 21540 \beta_{4} + 18347 \beta_{3} + 1164251 \beta_{2} + 109440 \beta_{1} + 1382442\)
\(\nu^{17}\)\(=\)\(66426 \beta_{17} + 1620828 \beta_{16} - 91207 \beta_{15} + 1516971 \beta_{14} - 314852 \beta_{13} - 78760 \beta_{12} - 271151 \beta_{11} - 2082370 \beta_{10} + 221821 \beta_{9} - 215137 \beta_{8} + 1462990 \beta_{7} + 856189 \beta_{6} - 534253 \beta_{5} - 20129 \beta_{4} + 105540 \beta_{3} + 1931064 \beta_{2} + 1552681 \beta_{1} + 2572050\)

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
−2.55461
−2.24824
−1.90615
−1.70725
−1.33766
−0.665348
−0.432278
−0.375443
0.0366427
0.803961
0.981962
1.52505
1.60620
1.77085
2.40246
2.59597
2.73355
2.77035
−2.55461 0 4.52602 2.12871 0 1.00000 −6.45298 0 −5.43801
1.2 −2.24824 0 3.05458 3.31434 0 1.00000 −2.37095 0 −7.45143
1.3 −1.90615 0 1.63341 −1.97979 0 1.00000 0.698768 0 3.77379
1.4 −1.70725 0 0.914706 −1.08728 0 1.00000 1.85287 0 1.85625
1.5 −1.33766 0 −0.210663 0.660635 0 1.00000 2.95712 0 −0.883705
1.6 −0.665348 0 −1.55731 3.70415 0 1.00000 2.36685 0 −2.46455
1.7 −0.432278 0 −1.81314 4.10000 0 1.00000 1.64833 0 −1.77234
1.8 −0.375443 0 −1.85904 −2.49678 0 1.00000 1.44885 0 0.937397
1.9 0.0366427 0 −1.99866 −1.02187 0 1.00000 −0.146522 0 −0.0374440
1.10 0.803961 0 −1.35365 −0.343335 0 1.00000 −2.69620 0 −0.276028
1.11 0.981962 0 −1.03575 −2.59441 0 1.00000 −2.98099 0 −2.54761
1.12 1.52505 0 0.325779 1.71268 0 1.00000 −2.55327 0 2.61192
1.13 1.60620 0 0.579871 3.54808 0 1.00000 −2.28101 0 5.69892
1.14 1.77085 0 1.13590 −1.38376 0 1.00000 −1.53019 0 −2.45042
1.15 2.40246 0 3.77179 0.666645 0 1.00000 4.25666 0 1.60159
1.16 2.59597 0 4.73905 1.63234 0 1.00000 7.11048 0 4.23750
1.17 2.73355 0 5.47227 −4.25416 0 1.00000 9.49162 0 −11.6289
1.18 2.77035 0 5.67483 3.69380 0 1.00000 10.1806 0 10.2331
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)