Properties

Label 8022.2.a.z
Level 8022
Weight 2
Character orbit 8022.a
Self dual Yes
Analytic conductor 64.056
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8022.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(53\) \(x^{13}\mathstrut -\mathstrut \) \(x^{12}\mathstrut +\mathstrut \) \(1068\) \(x^{11}\mathstrut +\mathstrut \) \(45\) \(x^{10}\mathstrut -\mathstrut \) \(10139\) \(x^{9}\mathstrut -\mathstrut \) \(615\) \(x^{8}\mathstrut +\mathstrut \) \(45390\) \(x^{7}\mathstrut +\mathstrut \) \(2130\) \(x^{6}\mathstrut -\mathstrut \) \(84842\) \(x^{5}\mathstrut +\mathstrut \) \(7822\) \(x^{4}\mathstrut +\mathstrut \) \(62828\) \(x^{3}\mathstrut -\mathstrut \) \(16144\) \(x^{2}\mathstrut -\mathstrut \) \(13616\) \(x\mathstrut +\mathstrut \) \(4704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 7 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47844143301035099\) \(\nu^{14}\mathstrut +\mathstrut \) \(49332284762810190\) \(\nu^{13}\mathstrut +\mathstrut \) \(2589151861807955487\) \(\nu^{12}\mathstrut -\mathstrut \) \(2513103468751495587\) \(\nu^{11}\mathstrut -\mathstrut \) \(53924977414688377018\) \(\nu^{10}\mathstrut +\mathstrut \) \(47291815758556009753\) \(\nu^{9}\mathstrut +\mathstrut \) \(542030391828967189447\) \(\nu^{8}\mathstrut -\mathstrut \) \(397820650922556789157\) \(\nu^{7}\mathstrut -\mathstrut \) \(2701455319595076197340\) \(\nu^{6}\mathstrut +\mathstrut \) \(1401466170449747084734\) \(\nu^{5}\mathstrut +\mathstrut \) \(6192320267897190525666\) \(\nu^{4}\mathstrut -\mathstrut \) \(1454391463816287278438\) \(\nu^{3}\mathstrut -\mathstrut \) \(5379972760527430385824\) \(\nu^{2}\mathstrut +\mathstrut \) \(216349037206284972520\) \(\nu\mathstrut +\mathstrut \) \(1152392517613515277584\)\()/\)\(71344467406408743984\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(145674275134685923\) \(\nu^{14}\mathstrut -\mathstrut \) \(95336249374922946\) \(\nu^{13}\mathstrut +\mathstrut \) \(7649328658641877611\) \(\nu^{12}\mathstrut +\mathstrut \) \(5190919725719812869\) \(\nu^{11}\mathstrut -\mathstrut \) \(151656890480087505806\) \(\nu^{10}\mathstrut -\mathstrut \) \(107643012000915815299\) \(\nu^{9}\mathstrut +\mathstrut \) \(1395329695881849133439\) \(\nu^{8}\mathstrut +\mathstrut \) \(1033349294454819909079\) \(\nu^{7}\mathstrut -\mathstrut \) \(5832120207138219564504\) \(\nu^{6}\mathstrut -\mathstrut \) \(4340534606751413151214\) \(\nu^{5}\mathstrut +\mathstrut \) \(9152779826696919384090\) \(\nu^{4}\mathstrut +\mathstrut \) \(5432623156253615496098\) \(\nu^{3}\mathstrut -\mathstrut \) \(5416320835690325800040\) \(\nu^{2}\mathstrut -\mathstrut \) \(1863587837228183355760\) \(\nu\mathstrut +\mathstrut \) \(972715425559024637664\)\()/\)\(71344467406408743984\)
\(\beta_{5}\)\(=\)\((\)\(253373654766014459\) \(\nu^{14}\mathstrut +\mathstrut \) \(163121527367248846\) \(\nu^{13}\mathstrut -\mathstrut \) \(13320607269945514819\) \(\nu^{12}\mathstrut -\mathstrut \) \(8822558076321033065\) \(\nu^{11}\mathstrut +\mathstrut \) \(264823325479108909458\) \(\nu^{10}\mathstrut +\mathstrut \) \(181506316507140562235\) \(\nu^{9}\mathstrut -\mathstrut \) \(2451587671951550009307\) \(\nu^{8}\mathstrut -\mathstrut \) \(1726900772962635860003\) \(\nu^{7}\mathstrut +\mathstrut \) \(10398114673415142018212\) \(\nu^{6}\mathstrut +\mathstrut \) \(7188181795410538835814\) \(\nu^{5}\mathstrut -\mathstrut \) \(16967789826021060630834\) \(\nu^{4}\mathstrut -\mathstrut \) \(8917630560881720595370\) \(\nu^{3}\mathstrut +\mathstrut \) \(10486974637156346458736\) \(\nu^{2}\mathstrut +\mathstrut \) \(2868338026195500538272\) \(\nu\mathstrut -\mathstrut \) \(2002669376503714503696\)\()/\)\(71344467406408743984\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(143980564676940343\) \(\nu^{14}\mathstrut -\mathstrut \) \(67158442937616560\) \(\nu^{13}\mathstrut +\mathstrut \) \(7596232450925625803\) \(\nu^{12}\mathstrut +\mathstrut \) \(3677196111640734007\) \(\nu^{11}\mathstrut -\mathstrut \) \(151809858709144529196\) \(\nu^{10}\mathstrut -\mathstrut \) \(76830397287487824811\) \(\nu^{9}\mathstrut +\mathstrut \) \(1417576896961580856189\) \(\nu^{8}\mathstrut +\mathstrut \) \(742126355596970318929\) \(\nu^{7}\mathstrut -\mathstrut \) \(6113609103065956762018\) \(\nu^{6}\mathstrut -\mathstrut \) \(3104114478921165345438\) \(\nu^{5}\mathstrut +\mathstrut \) \(10365451786874507600046\) \(\nu^{4}\mathstrut +\mathstrut \) \(3578981562674026030742\) \(\nu^{3}\mathstrut -\mathstrut \) \(6578574159891447888460\) \(\nu^{2}\mathstrut -\mathstrut \) \(834595104935603406168\) \(\nu\mathstrut +\mathstrut \) \(1206422294142049350504\)\()/\)\(35672233703204371992\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(435077682637100285\) \(\nu^{14}\mathstrut -\mathstrut \) \(225793991413084982\) \(\nu^{13}\mathstrut +\mathstrut \) \(22992219471492275225\) \(\nu^{12}\mathstrut +\mathstrut \) \(12311307959306123203\) \(\nu^{11}\mathstrut -\mathstrut \) \(460925788792527955646\) \(\nu^{10}\mathstrut -\mathstrut \) \(256299807939580952225\) \(\nu^{9}\mathstrut +\mathstrut \) \(4330733628662455297817\) \(\nu^{8}\mathstrut +\mathstrut \) \(2476016431170807843125\) \(\nu^{7}\mathstrut -\mathstrut \) \(18940119918113541544108\) \(\nu^{6}\mathstrut -\mathstrut \) \(10495753003077381214318\) \(\nu^{5}\mathstrut +\mathstrut \) \(33373539998586804849870\) \(\nu^{4}\mathstrut +\mathstrut \) \(13200028444408522075990\) \(\nu^{3}\mathstrut -\mathstrut \) \(22952407479971842707792\) \(\nu^{2}\mathstrut -\mathstrut \) \(3899901115490818892152\) \(\nu\mathstrut +\mathstrut \) \(4542537945343916384832\)\()/\)\(71344467406408743984\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(112790344931326241\) \(\nu^{14}\mathstrut -\mathstrut \) \(87742709231470043\) \(\nu^{13}\mathstrut +\mathstrut \) \(5922298344475101917\) \(\nu^{12}\mathstrut +\mathstrut \) \(4708706949542762884\) \(\nu^{11}\mathstrut -\mathstrut \) \(117429187152098562389\) \(\nu^{10}\mathstrut -\mathstrut \) \(95940771488732908961\) \(\nu^{9}\mathstrut +\mathstrut \) \(1080789454132306352144\) \(\nu^{8}\mathstrut +\mathstrut \) \(902828136570275154188\) \(\nu^{7}\mathstrut -\mathstrut \) \(4519906876304426386669\) \(\nu^{6}\mathstrut -\mathstrut \) \(3712164418355826904708\) \(\nu^{5}\mathstrut +\mathstrut \) \(7087919805467433423612\) \(\nu^{4}\mathstrut +\mathstrut \) \(4534350434211526941760\) \(\nu^{3}\mathstrut -\mathstrut \) \(4170727637139449111910\) \(\nu^{2}\mathstrut -\mathstrut \) \(1280281318799657216572\) \(\nu\mathstrut +\mathstrut \) \(805397881609767914556\)\()/\)\(17836116851602185996\)
\(\beta_{9}\)\(=\)\((\)\(128794562939054244\) \(\nu^{14}\mathstrut +\mathstrut \) \(77237585175435763\) \(\nu^{13}\mathstrut -\mathstrut \) \(6773705946064420810\) \(\nu^{12}\mathstrut -\mathstrut \) \(4200761904910892459\) \(\nu^{11}\mathstrut +\mathstrut \) \(134696803396025775311\) \(\nu^{10}\mathstrut +\mathstrut \) \(87023924027426484918\) \(\nu^{9}\mathstrut -\mathstrut \) \(1246618106742559952105\) \(\nu^{8}\mathstrut -\mathstrut \) \(834512276572118834799\) \(\nu^{7}\mathstrut +\mathstrut \) \(5277326861489579230277\) \(\nu^{6}\mathstrut +\mathstrut \) \(3499047092328247027912\) \(\nu^{5}\mathstrut -\mathstrut \) \(8534637097252352622594\) \(\nu^{4}\mathstrut -\mathstrut \) \(4313146196078542278522\) \(\nu^{3}\mathstrut +\mathstrut \) \(5076173451478455403366\) \(\nu^{2}\mathstrut +\mathstrut \) \(1190538423485802479440\) \(\nu\mathstrut -\mathstrut \) \(895775762206901213472\)\()/\)\(17836116851602185996\)
\(\beta_{10}\)\(=\)\((\)\(279026151355819425\) \(\nu^{14}\mathstrut +\mathstrut \) \(207687442650093487\) \(\nu^{13}\mathstrut -\mathstrut \) \(14667014841434591587\) \(\nu^{12}\mathstrut -\mathstrut \) \(11207020184843952140\) \(\nu^{11}\mathstrut +\mathstrut \) \(291401389817549796611\) \(\nu^{10}\mathstrut +\mathstrut \) \(229919971336010242191\) \(\nu^{9}\mathstrut -\mathstrut \) \(2692389644334917119448\) \(\nu^{8}\mathstrut -\mathstrut \) \(2182757191771485703410\) \(\nu^{7}\mathstrut +\mathstrut \) \(11355123931290340017479\) \(\nu^{6}\mathstrut +\mathstrut \) \(9092146572248850432838\) \(\nu^{5}\mathstrut -\mathstrut \) \(18187811553138106782456\) \(\nu^{4}\mathstrut -\mathstrut \) \(11445320208127070346804\) \(\nu^{3}\mathstrut +\mathstrut \) \(10809145508320427944858\) \(\nu^{2}\mathstrut +\mathstrut \) \(3343798898748498202936\) \(\nu\mathstrut -\mathstrut \) \(1957215294219116942160\)\()/\)\(35672233703204371992\)
\(\beta_{11}\)\(=\)\((\)\(173119505841442118\) \(\nu^{14}\mathstrut +\mathstrut \) \(120176375325262075\) \(\nu^{13}\mathstrut -\mathstrut \) \(9100209681746782816\) \(\nu^{12}\mathstrut -\mathstrut \) \(6483381202731471853\) \(\nu^{11}\mathstrut +\mathstrut \) \(180824512333209673471\) \(\nu^{10}\mathstrut +\mathstrut \) \(133039038295354289916\) \(\nu^{9}\mathstrut -\mathstrut \) \(1671375884765616054283\) \(\nu^{8}\mathstrut -\mathstrut \) \(1263088163896430650861\) \(\nu^{7}\mathstrut +\mathstrut \) \(7056391543742806488717\) \(\nu^{6}\mathstrut +\mathstrut \) \(5250954159974349563444\) \(\nu^{5}\mathstrut -\mathstrut \) \(11331099879512724663330\) \(\nu^{4}\mathstrut -\mathstrut \) \(6502986235248961530102\) \(\nu^{3}\mathstrut +\mathstrut \) \(6702141628972424751542\) \(\nu^{2}\mathstrut +\mathstrut \) \(1848534323084673928088\) \(\nu\mathstrut -\mathstrut \) \(1195388994843941406456\)\()/\)\(11890744567734790664\)
\(\beta_{12}\)\(=\)\((\)\(1317384423576065221\) \(\nu^{14}\mathstrut +\mathstrut \) \(850209309626005346\) \(\nu^{13}\mathstrut -\mathstrut \) \(69211945835863404749\) \(\nu^{12}\mathstrut -\mathstrut \) \(45976513730063865319\) \(\nu^{11}\mathstrut +\mathstrut \) \(1374118258779025758558\) \(\nu^{10}\mathstrut +\mathstrut \) \(945479263628513599765\) \(\nu^{9}\mathstrut -\mathstrut \) \(12684048256174737262341\) \(\nu^{8}\mathstrut -\mathstrut \) \(8980266604861904662237\) \(\nu^{7}\mathstrut +\mathstrut \) \(53424516372419043995356\) \(\nu^{6}\mathstrut +\mathstrut \) \(37115891207988086050122\) \(\nu^{5}\mathstrut -\mathstrut \) \(85392155189455265874222\) \(\nu^{4}\mathstrut -\mathstrut \) \(44167667913375731550710\) \(\nu^{3}\mathstrut +\mathstrut \) \(50352317577861990590464\) \(\nu^{2}\mathstrut +\mathstrut \) \(11318351887192471778592\) \(\nu\mathstrut -\mathstrut \) \(8797899846853478728656\)\()/\)\(71344467406408743984\)
\(\beta_{13}\)\(=\)\((\)\(698489089123427251\) \(\nu^{14}\mathstrut +\mathstrut \) \(483435293977683721\) \(\nu^{13}\mathstrut -\mathstrut \) \(36727261839951130129\) \(\nu^{12}\mathstrut -\mathstrut \) \(26104614174926647544\) \(\nu^{11}\mathstrut +\mathstrut \) \(730099435097944633141\) \(\nu^{10}\mathstrut +\mathstrut \) \(536023255394080824133\) \(\nu^{9}\mathstrut -\mathstrut \) \(6753625554358573827100\) \(\nu^{8}\mathstrut -\mathstrut \) \(5089691294902940317042\) \(\nu^{7}\mathstrut +\mathstrut \) \(28565379640083352955801\) \(\nu^{6}\mathstrut +\mathstrut \) \(21133358892345838770122\) \(\nu^{5}\mathstrut -\mathstrut \) \(46170514381419896585760\) \(\nu^{4}\mathstrut -\mathstrut \) \(25997785502318239929068\) \(\nu^{3}\mathstrut +\mathstrut \) \(28033124759210837863734\) \(\nu^{2}\mathstrut +\mathstrut \) \(7313700939316505024480\) \(\nu\mathstrut -\mathstrut \) \(5217773626313525601528\)\()/\)\(35672233703204371992\)
\(\beta_{14}\)\(=\)\((\)\(1413151433182310851\) \(\nu^{14}\mathstrut +\mathstrut \) \(860002258497780510\) \(\nu^{13}\mathstrut -\mathstrut \) \(74317793613973453179\) \(\nu^{12}\mathstrut -\mathstrut \) \(46651004556455924913\) \(\nu^{11}\mathstrut +\mathstrut \) \(1477960742755837792178\) \(\nu^{10}\mathstrut +\mathstrut \) \(963630109770286975987\) \(\nu^{9}\mathstrut -\mathstrut \) \(13685037087157198369715\) \(\nu^{8}\mathstrut -\mathstrut \) \(9209169621689696787019\) \(\nu^{7}\mathstrut +\mathstrut \) \(58031340399936060374532\) \(\nu^{6}\mathstrut +\mathstrut \) \(38424449791032430708966\) \(\nu^{5}\mathstrut -\mathstrut \) \(94498052084908879726242\) \(\nu^{4}\mathstrut -\mathstrut \) \(46806338252758402728074\) \(\nu^{3}\mathstrut +\mathstrut \) \(57679223018953714068464\) \(\nu^{2}\mathstrut +\mathstrut \) \(12890322571316985517456\) \(\nu\mathstrut -\mathstrut \) \(10648217680809741481920\)\()/\)\(71344467406408743984\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(87\)
\(\nu^{5}\)\(=\)\(24\) \(\beta_{14}\mathstrut -\mathstrut \) \(27\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(49\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(37\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(188\) \(\beta_{1}\mathstrut +\mathstrut \) \(37\)
\(\nu^{6}\)\(=\)\(-\)\(49\) \(\beta_{14}\mathstrut -\mathstrut \) \(44\) \(\beta_{13}\mathstrut +\mathstrut \) \(48\) \(\beta_{12}\mathstrut +\mathstrut \) \(22\) \(\beta_{11}\mathstrut +\mathstrut \) \(8\) \(\beta_{10}\mathstrut +\mathstrut \) \(36\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(39\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(54\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(225\) \(\beta_{2}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1201\)
\(\nu^{7}\)\(=\)\(478\) \(\beta_{14}\mathstrut -\mathstrut \) \(541\) \(\beta_{13}\mathstrut -\mathstrut \) \(57\) \(\beta_{12}\mathstrut -\mathstrut \) \(355\) \(\beta_{11}\mathstrut +\mathstrut \) \(968\) \(\beta_{10}\mathstrut -\mathstrut \) \(233\) \(\beta_{9}\mathstrut -\mathstrut \) \(396\) \(\beta_{8}\mathstrut +\mathstrut \) \(41\) \(\beta_{7}\mathstrut -\mathstrut \) \(241\) \(\beta_{6}\mathstrut -\mathstrut \) \(126\) \(\beta_{5}\mathstrut +\mathstrut \) \(621\) \(\beta_{4}\mathstrut +\mathstrut \) \(81\) \(\beta_{3}\mathstrut -\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(2820\) \(\beta_{1}\mathstrut +\mathstrut \) \(591\)
\(\nu^{8}\)\(=\)\(-\)\(968\) \(\beta_{14}\mathstrut -\mathstrut \) \(753\) \(\beta_{13}\mathstrut +\mathstrut \) \(891\) \(\beta_{12}\mathstrut +\mathstrut \) \(457\) \(\beta_{11}\mathstrut -\mathstrut \) \(170\) \(\beta_{10}\mathstrut +\mathstrut \) \(902\) \(\beta_{9}\mathstrut -\mathstrut \) \(70\) \(\beta_{8}\mathstrut -\mathstrut \) \(1252\) \(\beta_{7}\mathstrut +\mathstrut \) \(938\) \(\beta_{6}\mathstrut -\mathstrut \) \(225\) \(\beta_{5}\mathstrut -\mathstrut \) \(1132\) \(\beta_{4}\mathstrut +\mathstrut \) \(262\) \(\beta_{3}\mathstrut +\mathstrut \) \(3441\) \(\beta_{2}\mathstrut -\mathstrut \) \(224\) \(\beta_{1}\mathstrut +\mathstrut \) \(17301\)
\(\nu^{9}\)\(=\)\(8912\) \(\beta_{14}\mathstrut -\mathstrut \) \(9849\) \(\beta_{13}\mathstrut -\mathstrut \) \(1287\) \(\beta_{12}\mathstrut -\mathstrut \) \(6009\) \(\beta_{11}\mathstrut +\mathstrut \) \(17662\) \(\beta_{10}\mathstrut -\mathstrut \) \(5529\) \(\beta_{9}\mathstrut -\mathstrut \) \(6731\) \(\beta_{8}\mathstrut +\mathstrut \) \(1117\) \(\beta_{7}\mathstrut -\mathstrut \) \(5307\) \(\beta_{6}\mathstrut -\mathstrut \) \(1675\) \(\beta_{5}\mathstrut +\mathstrut \) \(10188\) \(\beta_{4}\mathstrut +\mathstrut \) \(1597\) \(\beta_{3}\mathstrut -\mathstrut \) \(1356\) \(\beta_{2}\mathstrut +\mathstrut \) \(43266\) \(\beta_{1}\mathstrut +\mathstrut \) \(9179\)
\(\nu^{10}\)\(=\)\(-\)\(17802\) \(\beta_{14}\mathstrut -\mathstrut \) \(11609\) \(\beta_{13}\mathstrut +\mathstrut \) \(15244\) \(\beta_{12}\mathstrut +\mathstrut \) \(9225\) \(\beta_{11}\mathstrut -\mathstrut \) \(8674\) \(\beta_{10}\mathstrut +\mathstrut \) \(19286\) \(\beta_{9}\mathstrut -\mathstrut \) \(310\) \(\beta_{8}\mathstrut -\mathstrut \) \(24290\) \(\beta_{7}\mathstrut +\mathstrut \) \(19432\) \(\beta_{6}\mathstrut -\mathstrut \) \(5136\) \(\beta_{5}\mathstrut -\mathstrut \) \(21634\) \(\beta_{4}\mathstrut +\mathstrut \) \(7264\) \(\beta_{3}\mathstrut +\mathstrut \) \(53454\) \(\beta_{2}\mathstrut -\mathstrut \) \(6196\) \(\beta_{1}\mathstrut +\mathstrut \) \(255902\)
\(\nu^{11}\)\(=\)\(160537\) \(\beta_{14}\mathstrut -\mathstrut \) \(171975\) \(\beta_{13}\mathstrut -\mathstrut \) \(26460\) \(\beta_{12}\mathstrut -\mathstrut \) \(99073\) \(\beta_{11}\mathstrut +\mathstrut \) \(309620\) \(\beta_{10}\mathstrut -\mathstrut \) \(114845\) \(\beta_{9}\mathstrut -\mathstrut \) \(112359\) \(\beta_{8}\mathstrut +\mathstrut \) \(25479\) \(\beta_{7}\mathstrut -\mathstrut \) \(104040\) \(\beta_{6}\mathstrut -\mathstrut \) \(25580\) \(\beta_{5}\mathstrut +\mathstrut \) \(166205\) \(\beta_{4}\mathstrut +\mathstrut \) \(28061\) \(\beta_{3}\mathstrut -\mathstrut \) \(27830\) \(\beta_{2}\mathstrut +\mathstrut \) \(674602\) \(\beta_{1}\mathstrut +\mathstrut \) \(140454\)
\(\nu^{12}\)\(=\)\(-\)\(316885\) \(\beta_{14}\mathstrut -\mathstrut \) \(167892\) \(\beta_{13}\mathstrut +\mathstrut \) \(252984\) \(\beta_{12}\mathstrut +\mathstrut \) \(180405\) \(\beta_{11}\mathstrut -\mathstrut \) \(242203\) \(\beta_{10}\mathstrut +\mathstrut \) \(379427\) \(\beta_{9}\mathstrut +\mathstrut \) \(18451\) \(\beta_{8}\mathstrut -\mathstrut \) \(447357\) \(\beta_{7}\mathstrut +\mathstrut \) \(377629\) \(\beta_{6}\mathstrut -\mathstrut \) \(101949\) \(\beta_{5}\mathstrut -\mathstrut \) \(395270\) \(\beta_{4}\mathstrut +\mathstrut \) \(166107\) \(\beta_{3}\mathstrut +\mathstrut \) \(840807\) \(\beta_{2}\mathstrut -\mathstrut \) \(151204\) \(\beta_{1}\mathstrut +\mathstrut \) \(3856280\)
\(\nu^{13}\)\(=\)\(2832236\) \(\beta_{14}\mathstrut -\mathstrut \) \(2940922\) \(\beta_{13}\mathstrut -\mathstrut \) \(515670\) \(\beta_{12}\mathstrut -\mathstrut \) \(1608888\) \(\beta_{11}\mathstrut +\mathstrut \) \(5308664\) \(\beta_{10}\mathstrut -\mathstrut \) \(2227972\) \(\beta_{9}\mathstrut -\mathstrut \) \(1867397\) \(\beta_{8}\mathstrut +\mathstrut \) \(528420\) \(\beta_{7}\mathstrut -\mathstrut \) \(1927402\) \(\beta_{6}\mathstrut -\mathstrut \) \(427230\) \(\beta_{5}\mathstrut +\mathstrut \) \(2713138\) \(\beta_{4}\mathstrut +\mathstrut \) \(466018\) \(\beta_{3}\mathstrut -\mathstrut \) \(546474\) \(\beta_{2}\mathstrut +\mathstrut \) \(10646798\) \(\beta_{1}\mathstrut +\mathstrut \) \(2111793\)
\(\nu^{14}\)\(=\)\(-\)\(5547762\) \(\beta_{14}\mathstrut -\mathstrut \) \(2303000\) \(\beta_{13}\mathstrut +\mathstrut \) \(4153305\) \(\beta_{12}\mathstrut +\mathstrut \) \(3431803\) \(\beta_{11}\mathstrut -\mathstrut \) \(5584747\) \(\beta_{10}\mathstrut +\mathstrut \) \(7113938\) \(\beta_{9}\mathstrut +\mathstrut \) \(782491\) \(\beta_{8}\mathstrut -\mathstrut \) \(8001130\) \(\beta_{7}\mathstrut +\mathstrut \) \(7089567\) \(\beta_{6}\mathstrut -\mathstrut \) \(1878821\) \(\beta_{5}\mathstrut -\mathstrut \) \(7044180\) \(\beta_{4}\mathstrut +\mathstrut \) \(3429314\) \(\beta_{3}\mathstrut +\mathstrut \) \(13359936\) \(\beta_{2}\mathstrut -\mathstrut \) \(3420114\) \(\beta_{1}\mathstrut +\mathstrut \) \(58932157\)

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
4.01349
3.88066
3.31212
2.90136
1.18405
0.695582
0.638936
0.434267
−0.600780
−1.10784
−1.55328
−2.53934
−3.41684
−3.71340
−4.12898
−1.00000 1.00000 1.00000 −4.01349 −1.00000 1.00000 −1.00000 1.00000 4.01349
1.2 −1.00000 1.00000 1.00000 −3.88066 −1.00000 1.00000 −1.00000 1.00000 3.88066
1.3 −1.00000 1.00000 1.00000 −3.31212 −1.00000 1.00000 −1.00000 1.00000 3.31212
1.4 −1.00000 1.00000 1.00000 −2.90136 −1.00000 1.00000 −1.00000 1.00000 2.90136
1.5 −1.00000 1.00000 1.00000 −1.18405 −1.00000 1.00000 −1.00000 1.00000 1.18405
1.6 −1.00000 1.00000 1.00000 −0.695582 −1.00000 1.00000 −1.00000 1.00000 0.695582
1.7 −1.00000 1.00000 1.00000 −0.638936 −1.00000 1.00000 −1.00000 1.00000 0.638936
1.8 −1.00000 1.00000 1.00000 −0.434267 −1.00000 1.00000 −1.00000 1.00000 0.434267
1.9 −1.00000 1.00000 1.00000 0.600780 −1.00000 1.00000 −1.00000 1.00000 −0.600780
1.10 −1.00000 1.00000 1.00000 1.10784 −1.00000 1.00000 −1.00000 1.00000 −1.10784
1.11 −1.00000 1.00000 1.00000 1.55328 −1.00000 1.00000 −1.00000 1.00000 −1.55328
1.12 −1.00000 1.00000 1.00000 2.53934 −1.00000 1.00000 −1.00000 1.00000 −2.53934
1.13 −1.00000 1.00000 1.00000 3.41684 −1.00000 1.00000 −1.00000 1.00000 −3.41684
1.14 −1.00000 1.00000 1.00000 3.71340 −1.00000 1.00000 −1.00000 1.00000 −3.71340
1.15 −1.00000 1.00000 1.00000 4.12898 −1.00000 1.00000 −1.00000 1.00000 −4.12898
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(191\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\(T_{5}^{15} - \cdots\)
\(T_{11}^{15} + \cdots\)
\(T_{13}^{15} - \cdots\)