Properties

Label 8001.2.a.s
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(4\) \(x^{15}\mathstrut -\mathstrut \) \(18\) \(x^{14}\mathstrut +\mathstrut \) \(83\) \(x^{13}\mathstrut +\mathstrut \) \(112\) \(x^{12}\mathstrut -\mathstrut \) \(668\) \(x^{11}\mathstrut -\mathstrut \) \(235\) \(x^{10}\mathstrut +\mathstrut \) \(2648\) \(x^{9}\mathstrut -\mathstrut \) \(298\) \(x^{8}\mathstrut -\mathstrut \) \(5422\) \(x^{7}\mathstrut +\mathstrut \) \(2075\) \(x^{6}\mathstrut +\mathstrut \) \(5385\) \(x^{5}\mathstrut -\mathstrut \) \(3163\) \(x^{4}\mathstrut -\mathstrut \) \(1882\) \(x^{3}\mathstrut +\mathstrut \) \(1614\) \(x^{2}\mathstrut -\mathstrut \) \(200\) \(x\mathstrut -\mathstrut \) \(20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 7 \nu^{2} - \nu + 6 \)
\(\beta_{5}\)\(=\)\((\)\(16172\) \(\nu^{15}\mathstrut -\mathstrut \) \(233047\) \(\nu^{14}\mathstrut +\mathstrut \) \(545740\) \(\nu^{13}\mathstrut +\mathstrut \) \(3569234\) \(\nu^{12}\mathstrut -\mathstrut \) \(13969268\) \(\nu^{11}\mathstrut -\mathstrut \) \(15371453\) \(\nu^{10}\mathstrut +\mathstrut \) \(106160280\) \(\nu^{9}\mathstrut -\mathstrut \) \(7403894\) \(\nu^{8}\mathstrut -\mathstrut \) \(351255852\) \(\nu^{7}\mathstrut +\mathstrut \) \(189480617\) \(\nu^{6}\mathstrut +\mathstrut \) \(508824053\) \(\nu^{5}\mathstrut -\mathstrut \) \(399632645\) \(\nu^{4}\mathstrut -\mathstrut \) \(240231533\) \(\nu^{3}\mathstrut +\mathstrut \) \(249231930\) \(\nu^{2}\mathstrut -\mathstrut \) \(32996882\) \(\nu\mathstrut -\mathstrut \) \(1833530\)\()/739814\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(55506\) \(\nu^{15}\mathstrut +\mathstrut \) \(171907\) \(\nu^{14}\mathstrut +\mathstrut \) \(1127781\) \(\nu^{13}\mathstrut -\mathstrut \) \(3471014\) \(\nu^{12}\mathstrut -\mathstrut \) \(8968081\) \(\nu^{11}\mathstrut +\mathstrut \) \(26857942\) \(\nu^{10}\mathstrut +\mathstrut \) \(35796817\) \(\nu^{9}\mathstrut -\mathstrut \) \(100836497\) \(\nu^{8}\mathstrut -\mathstrut \) \(75361440\) \(\nu^{7}\mathstrut +\mathstrut \) \(193857030\) \(\nu^{6}\mathstrut +\mathstrut \) \(74881691\) \(\nu^{5}\mathstrut -\mathstrut \) \(187228846\) \(\nu^{4}\mathstrut -\mathstrut \) \(20078726\) \(\nu^{3}\mathstrut +\mathstrut \) \(77060622\) \(\nu^{2}\mathstrut -\mathstrut \) \(7448006\) \(\nu\mathstrut -\mathstrut \) \(3035278\)\()/739814\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(79649\) \(\nu^{15}\mathstrut +\mathstrut \) \(179031\) \(\nu^{14}\mathstrut +\mathstrut \) \(1937788\) \(\nu^{13}\mathstrut -\mathstrut \) \(4128253\) \(\nu^{12}\mathstrut -\mathstrut \) \(18731714\) \(\nu^{11}\mathstrut +\mathstrut \) \(37509563\) \(\nu^{10}\mathstrut +\mathstrut \) \(90669787\) \(\nu^{9}\mathstrut -\mathstrut \) \(170332316\) \(\nu^{8}\mathstrut -\mathstrut \) \(224913536\) \(\nu^{7}\mathstrut +\mathstrut \) \(404529319\) \(\nu^{6}\mathstrut +\mathstrut \) \(250701988\) \(\nu^{5}\mathstrut -\mathstrut \) \(476307526\) \(\nu^{4}\mathstrut -\mathstrut \) \(57970538\) \(\nu^{3}\mathstrut +\mathstrut \) \(219774482\) \(\nu^{2}\mathstrut -\mathstrut \) \(52297458\) \(\nu\mathstrut -\mathstrut \) \(1805692\)\()/739814\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(49011\) \(\nu^{15}\mathstrut +\mathstrut \) \(202993\) \(\nu^{14}\mathstrut +\mathstrut \) \(777233\) \(\nu^{13}\mathstrut -\mathstrut \) \(3856011\) \(\nu^{12}\mathstrut -\mathstrut \) \(3798508\) \(\nu^{11}\mathstrut +\mathstrut \) \(27243628\) \(\nu^{10}\mathstrut +\mathstrut \) \(3013336\) \(\nu^{9}\mathstrut -\mathstrut \) \(88425697\) \(\nu^{8}\mathstrut +\mathstrut \) \(22495878\) \(\nu^{7}\mathstrut +\mathstrut \) \(131065711\) \(\nu^{6}\mathstrut -\mathstrut \) \(53417264\) \(\nu^{5}\mathstrut -\mathstrut \) \(71773574\) \(\nu^{4}\mathstrut +\mathstrut \) \(32187666\) \(\nu^{3}\mathstrut +\mathstrut \) \(1013322\) \(\nu^{2}\mathstrut +\mathstrut \) \(932298\) \(\nu\mathstrut +\mathstrut \) \(482636\)\()/369907\)
\(\beta_{9}\)\(=\)\((\)\(123393\) \(\nu^{15}\mathstrut -\mathstrut \) \(271059\) \(\nu^{14}\mathstrut -\mathstrut \) \(3028354\) \(\nu^{13}\mathstrut +\mathstrut \) \(6241792\) \(\nu^{12}\mathstrut +\mathstrut \) \(29665237\) \(\nu^{11}\mathstrut -\mathstrut \) \(56580819\) \(\nu^{10}\mathstrut -\mathstrut \) \(146738516\) \(\nu^{9}\mathstrut +\mathstrut \) \(256052832\) \(\nu^{8}\mathstrut +\mathstrut \) \(378583411\) \(\nu^{7}\mathstrut -\mathstrut \) \(605454280\) \(\nu^{6}\mathstrut -\mathstrut \) \(461426392\) \(\nu^{5}\mathstrut +\mathstrut \) \(709532970\) \(\nu^{4}\mathstrut +\mathstrut \) \(170356469\) \(\nu^{3}\mathstrut -\mathstrut \) \(325487958\) \(\nu^{2}\mathstrut +\mathstrut \) \(50732374\) \(\nu\mathstrut +\mathstrut \) \(3426510\)\()/739814\)
\(\beta_{10}\)\(=\)\((\)\(125049\) \(\nu^{15}\mathstrut -\mathstrut \) \(467479\) \(\nu^{14}\mathstrut -\mathstrut \) \(2288559\) \(\nu^{13}\mathstrut +\mathstrut \) \(9361224\) \(\nu^{12}\mathstrut +\mathstrut \) \(15315132\) \(\nu^{11}\mathstrut -\mathstrut \) \(71647986\) \(\nu^{10}\mathstrut -\mathstrut \) \(45326027\) \(\nu^{9}\mathstrut +\mathstrut \) \(264664955\) \(\nu^{8}\mathstrut +\mathstrut \) \(52335901\) \(\nu^{7}\mathstrut -\mathstrut \) \(493686395\) \(\nu^{6}\mathstrut +\mathstrut \) \(12589986\) \(\nu^{5}\mathstrut +\mathstrut \) \(444594657\) \(\nu^{4}\mathstrut -\mathstrut \) \(76420466\) \(\nu^{3}\mathstrut -\mathstrut \) \(158630492\) \(\nu^{2}\mathstrut +\mathstrut \) \(46767048\) \(\nu\mathstrut +\mathstrut \) \(4758460\)\()/739814\)
\(\beta_{11}\)\(=\)\((\)\(129018\) \(\nu^{15}\mathstrut -\mathstrut \) \(383386\) \(\nu^{14}\mathstrut -\mathstrut \) \(2688667\) \(\nu^{13}\mathstrut +\mathstrut \) \(7911672\) \(\nu^{12}\mathstrut +\mathstrut \) \(21971299\) \(\nu^{11}\mathstrut -\mathstrut \) \(62975173\) \(\nu^{10}\mathstrut -\mathstrut \) \(89947843\) \(\nu^{9}\mathstrut +\mathstrut \) \(244582021\) \(\nu^{8}\mathstrut +\mathstrut \) \(193199050\) \(\nu^{7}\mathstrut -\mathstrut \) \(484210085\) \(\nu^{6}\mathstrut -\mathstrut \) \(196229912\) \(\nu^{5}\mathstrut +\mathstrut \) \(461790873\) \(\nu^{4}\mathstrut +\mathstrut \) \(51084111\) \(\nu^{3}\mathstrut -\mathstrut \) \(166221504\) \(\nu^{2}\mathstrut +\mathstrut \) \(26121758\) \(\nu\mathstrut +\mathstrut \) \(725548\)\()/739814\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(137239\) \(\nu^{15}\mathstrut +\mathstrut \) \(515922\) \(\nu^{14}\mathstrut +\mathstrut \) \(2401726\) \(\nu^{13}\mathstrut -\mathstrut \) \(10047088\) \(\nu^{12}\mathstrut -\mathstrut \) \(14807929\) \(\nu^{11}\mathstrut +\mathstrut \) \(73837244\) \(\nu^{10}\mathstrut +\mathstrut \) \(36812718\) \(\nu^{9}\mathstrut -\mathstrut \) \(256493020\) \(\nu^{8}\mathstrut -\mathstrut \) \(24053391\) \(\nu^{7}\mathstrut +\mathstrut \) \(434181743\) \(\nu^{6}\mathstrut -\mathstrut \) \(30567173\) \(\nu^{5}\mathstrut -\mathstrut \) \(332083099\) \(\nu^{4}\mathstrut +\mathstrut \) \(31220698\) \(\nu^{3}\mathstrut +\mathstrut \) \(88655714\) \(\nu^{2}\mathstrut +\mathstrut \) \(2540988\) \(\nu\mathstrut -\mathstrut \) \(4842758\)\()/739814\)
\(\beta_{13}\)\(=\)\((\)\(145145\) \(\nu^{15}\mathstrut -\mathstrut \) \(426142\) \(\nu^{14}\mathstrut -\mathstrut \) \(3134036\) \(\nu^{13}\mathstrut +\mathstrut \) \(9144531\) \(\nu^{12}\mathstrut +\mathstrut \) \(26680262\) \(\nu^{11}\mathstrut -\mathstrut \) \(76732984\) \(\nu^{10}\mathstrut -\mathstrut \) \(113889853\) \(\nu^{9}\mathstrut +\mathstrut \) \(320134980\) \(\nu^{8}\mathstrut +\mathstrut \) \(252475352\) \(\nu^{7}\mathstrut -\mathstrut \) \(697758738\) \(\nu^{6}\mathstrut -\mathstrut \) \(255070845\) \(\nu^{5}\mathstrut +\mathstrut \) \(754990315\) \(\nu^{4}\mathstrut +\mathstrut \) \(44791941\) \(\nu^{3}\mathstrut -\mathstrut \) \(321397232\) \(\nu^{2}\mathstrut +\mathstrut \) \(60680346\) \(\nu\mathstrut +\mathstrut \) \(5228678\)\()/739814\)
\(\beta_{14}\)\(=\)\((\)\(158942\) \(\nu^{15}\mathstrut -\mathstrut \) \(398038\) \(\nu^{14}\mathstrut -\mathstrut \) \(3767459\) \(\nu^{13}\mathstrut +\mathstrut \) \(8967247\) \(\nu^{12}\mathstrut +\mathstrut \) \(35638598\) \(\nu^{11}\mathstrut -\mathstrut \) \(79277367\) \(\nu^{10}\mathstrut -\mathstrut \) \(170414762\) \(\nu^{9}\mathstrut +\mathstrut \) \(348630065\) \(\nu^{8}\mathstrut +\mathstrut \) \(425502097\) \(\nu^{7}\mathstrut -\mathstrut \) \(797772424\) \(\nu^{6}\mathstrut -\mathstrut \) \(500952602\) \(\nu^{5}\mathstrut +\mathstrut \) \(901431563\) \(\nu^{4}\mathstrut +\mathstrut \) \(177265028\) \(\nu^{3}\mathstrut -\mathstrut \) \(399924308\) \(\nu^{2}\mathstrut +\mathstrut \) \(51868034\) \(\nu\mathstrut +\mathstrut \) \(7379728\)\()/739814\)
\(\beta_{15}\)\(=\)\((\)\(322032\) \(\nu^{15}\mathstrut -\mathstrut \) \(1082571\) \(\nu^{14}\mathstrut -\mathstrut \) \(6442221\) \(\nu^{13}\mathstrut +\mathstrut \) \(22367158\) \(\nu^{12}\mathstrut +\mathstrut \) \(49725923\) \(\nu^{11}\mathstrut -\mathstrut \) \(178585472\) \(\nu^{10}\mathstrut -\mathstrut \) \(187902897\) \(\nu^{9}\mathstrut +\mathstrut \) \(698572039\) \(\nu^{8}\mathstrut +\mathstrut \) \(358046232\) \(\nu^{7}\mathstrut -\mathstrut \) \(1405569986\) \(\nu^{6}\mathstrut -\mathstrut \) \(279881591\) \(\nu^{5}\mathstrut +\mathstrut \) \(1389436646\) \(\nu^{4}\mathstrut -\mathstrut \) \(43714386\) \(\nu^{3}\mathstrut -\mathstrut \) \(540456742\) \(\nu^{2}\mathstrut +\mathstrut \) \(124105592\) \(\nu\mathstrut +\mathstrut \) \(7468534\)\()/739814\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{7}\)\(=\)\(15\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(13\) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{12}\mathstrut -\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(55\) \(\beta_{3}\mathstrut +\mathstrut \) \(69\) \(\beta_{2}\mathstrut +\mathstrut \) \(209\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)
\(\nu^{8}\)\(=\)\(33\) \(\beta_{15}\mathstrut -\mathstrut \) \(19\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{12}\mathstrut -\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(61\) \(\beta_{10}\mathstrut +\mathstrut \) \(34\) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(99\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(292\) \(\beta_{2}\mathstrut +\mathstrut \) \(175\) \(\beta_{1}\mathstrut +\mathstrut \) \(520\)
\(\nu^{9}\)\(=\)\(162\) \(\beta_{15}\mathstrut -\mathstrut \) \(146\) \(\beta_{14}\mathstrut -\mathstrut \) \(125\) \(\beta_{13}\mathstrut -\mathstrut \) \(139\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(196\) \(\beta_{10}\mathstrut +\mathstrut \) \(127\) \(\beta_{9}\mathstrut +\mathstrut \) \(142\) \(\beta_{8}\mathstrut +\mathstrut \) \(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(106\) \(\beta_{6}\mathstrut +\mathstrut \) \(163\) \(\beta_{4}\mathstrut +\mathstrut \) \(370\) \(\beta_{3}\mathstrut +\mathstrut \) \(510\) \(\beta_{2}\mathstrut +\mathstrut \) \(1466\) \(\beta_{1}\mathstrut +\mathstrut \) \(203\)
\(\nu^{10}\)\(=\)\(378\) \(\beta_{15}\mathstrut -\mathstrut \) \(239\) \(\beta_{14}\mathstrut -\mathstrut \) \(59\) \(\beta_{13}\mathstrut +\mathstrut \) \(62\) \(\beta_{12}\mathstrut -\mathstrut \) \(129\) \(\beta_{11}\mathstrut -\mathstrut \) \(657\) \(\beta_{10}\mathstrut +\mathstrut \) \(388\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(218\) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut +\mathstrut \) \(154\) \(\beta_{5}\mathstrut +\mathstrut \) \(837\) \(\beta_{4}\mathstrut +\mathstrut \) \(174\) \(\beta_{3}\mathstrut +\mathstrut \) \(1942\) \(\beta_{2}\mathstrut +\mathstrut \) \(1657\) \(\beta_{1}\mathstrut +\mathstrut \) \(3232\)
\(\nu^{11}\)\(=\)\(1538\) \(\beta_{15}\mathstrut -\mathstrut \) \(1365\) \(\beta_{14}\mathstrut -\mathstrut \) \(1076\) \(\beta_{13}\mathstrut -\mathstrut \) \(1217\) \(\beta_{12}\mathstrut +\mathstrut \) \(47\) \(\beta_{11}\mathstrut -\mathstrut \) \(1930\) \(\beta_{10}\mathstrut +\mathstrut \) \(1130\) \(\beta_{9}\mathstrut +\mathstrut \) \(1286\) \(\beta_{8}\mathstrut +\mathstrut \) \(259\) \(\beta_{7}\mathstrut +\mathstrut \) \(840\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(1560\) \(\beta_{4}\mathstrut +\mathstrut \) \(2500\) \(\beta_{3}\mathstrut +\mathstrut \) \(3747\) \(\beta_{2}\mathstrut +\mathstrut \) \(10526\) \(\beta_{1}\mathstrut +\mathstrut \) \(1641\)
\(\nu^{12}\)\(=\)\(3733\) \(\beta_{15}\mathstrut -\mathstrut \) \(2530\) \(\beta_{14}\mathstrut -\mathstrut \) \(752\) \(\beta_{13}\mathstrut +\mathstrut \) \(223\) \(\beta_{12}\mathstrut -\mathstrut \) \(982\) \(\beta_{11}\mathstrut -\mathstrut \) \(6181\) \(\beta_{10}\mathstrut +\mathstrut \) \(3756\) \(\beta_{9}\mathstrut +\mathstrut \) \(537\) \(\beta_{8}\mathstrut +\mathstrut \) \(2236\) \(\beta_{7}\mathstrut +\mathstrut \) \(274\) \(\beta_{6}\mathstrut +\mathstrut \) \(1350\) \(\beta_{5}\mathstrut +\mathstrut \) \(6877\) \(\beta_{4}\mathstrut +\mathstrut \) \(1629\) \(\beta_{3}\mathstrut +\mathstrut \) \(13248\) \(\beta_{2}\mathstrut +\mathstrut \) \(14610\) \(\beta_{1}\mathstrut +\mathstrut \) \(20471\)
\(\nu^{13}\)\(=\)\(13671\) \(\beta_{15}\mathstrut -\mathstrut \) \(12084\) \(\beta_{14}\mathstrut -\mathstrut \) \(8781\) \(\beta_{13}\mathstrut -\mathstrut \) \(10051\) \(\beta_{12}\mathstrut +\mathstrut \) \(703\) \(\beta_{11}\mathstrut -\mathstrut \) \(17520\) \(\beta_{10}\mathstrut +\mathstrut \) \(9659\) \(\beta_{9}\mathstrut +\mathstrut \) \(11081\) \(\beta_{8}\mathstrut +\mathstrut \) \(3015\) \(\beta_{7}\mathstrut +\mathstrut \) \(6340\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(13979\) \(\beta_{4}\mathstrut +\mathstrut \) \(17072\) \(\beta_{3}\mathstrut +\mathstrut \) \(27638\) \(\beta_{2}\mathstrut +\mathstrut \) \(76824\) \(\beta_{1}\mathstrut +\mathstrut \) \(12767\)
\(\nu^{14}\)\(=\)\(34127\) \(\beta_{15}\mathstrut -\mathstrut \) \(24451\) \(\beta_{14}\mathstrut -\mathstrut \) \(7956\) \(\beta_{13}\mathstrut -\mathstrut \) \(756\) \(\beta_{12}\mathstrut -\mathstrut \) \(6564\) \(\beta_{11}\mathstrut -\mathstrut \) \(54356\) \(\beta_{10}\mathstrut +\mathstrut \) \(33417\) \(\beta_{9}\mathstrut +\mathstrut \) \(8925\) \(\beta_{8}\mathstrut +\mathstrut \) \(20997\) \(\beta_{7}\mathstrut +\mathstrut \) \(2898\) \(\beta_{6}\mathstrut +\mathstrut \) \(10885\) \(\beta_{5}\mathstrut +\mathstrut \) \(55614\) \(\beta_{4}\mathstrut +\mathstrut \) \(14172\) \(\beta_{3}\mathstrut +\mathstrut \) \(92494\) \(\beta_{2}\mathstrut +\mathstrut \) \(123744\) \(\beta_{1}\mathstrut +\mathstrut \) \(131697\)
\(\nu^{15}\)\(=\)\(116930\) \(\beta_{15}\mathstrut -\mathstrut \) \(103619\) \(\beta_{14}\mathstrut -\mathstrut \) \(69592\) \(\beta_{13}\mathstrut -\mathstrut \) \(80533\) \(\beta_{12}\mathstrut +\mathstrut \) \(8580\) \(\beta_{11}\mathstrut -\mathstrut \) \(151698\) \(\beta_{10}\mathstrut +\mathstrut \) \(80862\) \(\beta_{9}\mathstrut +\mathstrut \) \(93120\) \(\beta_{8}\mathstrut +\mathstrut \) \(31789\) \(\beta_{7}\mathstrut +\mathstrut \) \(46689\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(120476\) \(\beta_{4}\mathstrut +\mathstrut \) \(117968\) \(\beta_{3}\mathstrut +\mathstrut \) \(205325\) \(\beta_{2}\mathstrut +\mathstrut \) \(567801\) \(\beta_{1}\mathstrut +\mathstrut \) \(97481\)

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
2.80317
2.63870
2.45059
2.35655
1.50110
1.17466
0.825012
0.716430
0.284398
−0.0643619
−1.14773
−1.35902
−1.40936
−1.84433
−2.35145
−2.57436
−2.80317 0 5.85777 −1.45069 0 −1.00000 −10.8140 0 4.06653
1.2 −2.63870 0 4.96274 −0.422965 0 −1.00000 −7.81779 0 1.11608
1.3 −2.45059 0 4.00539 2.70231 0 −1.00000 −4.91439 0 −6.62224
1.4 −2.35655 0 3.55331 −2.88159 0 −1.00000 −3.66045 0 6.79060
1.5 −1.50110 0 0.253312 2.96917 0 −1.00000 2.62196 0 −4.45703
1.6 −1.17466 0 −0.620175 −3.75278 0 −1.00000 3.07781 0 4.40823
1.7 −0.825012 0 −1.31936 1.66882 0 −1.00000 2.73851 0 −1.37680
1.8 −0.716430 0 −1.48673 −0.617976 0 −1.00000 2.49800 0 0.442737
1.9 −0.284398 0 −1.91912 3.66581 0 −1.00000 1.11459 0 −1.04255
1.10 0.0643619 0 −1.99586 −3.34910 0 −1.00000 −0.257181 0 −0.215554
1.11 1.14773 0 −0.682718 −1.07917 0 −1.00000 −3.07903 0 −1.23860
1.12 1.35902 0 −0.153057 2.54404 0 −1.00000 −2.92605 0 3.45741
1.13 1.40936 0 −0.0136985 −2.06019 0 −1.00000 −2.83803 0 −2.90356
1.14 1.84433 0 1.40154 −2.16633 0 −1.00000 −1.10376 0 −3.99542
1.15 2.35145 0 3.52933 2.01529 0 −1.00000 3.59614 0 4.73886
1.16 2.57436 0 4.62732 −2.78465 0 −1.00000 6.76367 0 −7.16870
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

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