Properties

Label 8034.2.a.bc
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(x^{14}\mathstrut -\mathstrut \) \(48\) \(x^{13}\mathstrut +\mathstrut \) \(44\) \(x^{12}\mathstrut +\mathstrut \) \(872\) \(x^{11}\mathstrut -\mathstrut \) \(707\) \(x^{10}\mathstrut -\mathstrut \) \(7580\) \(x^{9}\mathstrut +\mathstrut \) \(5112\) \(x^{8}\mathstrut +\mathstrut \) \(33191\) \(x^{7}\mathstrut -\mathstrut \) \(16428\) \(x^{6}\mathstrut -\mathstrut \) \(71361\) \(x^{5}\mathstrut +\mathstrut \) \(21747\) \(x^{4}\mathstrut +\mathstrut \) \(65434\) \(x^{3}\mathstrut -\mathstrut \) \(11840\) \(x^{2}\mathstrut -\mathstrut \) \(17600\) \(x\mathstrut +\mathstrut \) \(2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(2537687267978126305\) \(\nu^{14}\mathstrut -\mathstrut \) \(4451260489557971801\) \(\nu^{13}\mathstrut -\mathstrut \) \(118607585529944247160\) \(\nu^{12}\mathstrut +\mathstrut \) \(201894771958174626540\) \(\nu^{11}\mathstrut +\mathstrut \) \(2065774167255360616712\) \(\nu^{10}\mathstrut -\mathstrut \) \(3389041191171673838947\) \(\nu^{9}\mathstrut -\mathstrut \) \(16731997871850458694196\) \(\nu^{8}\mathstrut +\mathstrut \) \(26184651556522014754456\) \(\nu^{7}\mathstrut +\mathstrut \) \(64612006487857426157383\) \(\nu^{6}\mathstrut -\mathstrut \) \(94142583432419588002804\) \(\nu^{5}\mathstrut -\mathstrut \) \(109525350370822082766017\) \(\nu^{4}\mathstrut +\mathstrut \) \(145425424067694665372939\) \(\nu^{3}\mathstrut +\mathstrut \) \(53532513774968079841586\) \(\nu^{2}\mathstrut -\mathstrut \) \(75481126501892671808752\) \(\nu\mathstrut +\mathstrut \) \(10315145682746966682944\)\()/\)\(44\!\cdots\!36\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(3699895265701566353\) \(\nu^{14}\mathstrut +\mathstrut \) \(9148574385842311585\) \(\nu^{13}\mathstrut +\mathstrut \) \(173370522950640285632\) \(\nu^{12}\mathstrut -\mathstrut \) \(401147448883605874380\) \(\nu^{11}\mathstrut -\mathstrut \) \(3040455429814694571304\) \(\nu^{10}\mathstrut +\mathstrut \) \(6347593527480726095411\) \(\nu^{9}\mathstrut +\mathstrut \) \(25017731332666579088492\) \(\nu^{8}\mathstrut -\mathstrut \) \(44104795536205132068824\) \(\nu^{7}\mathstrut -\mathstrut \) \(100220807092977766199447\) \(\nu^{6}\mathstrut +\mathstrut \) \(130523696906252529491036\) \(\nu^{5}\mathstrut +\mathstrut \) \(189558836334691265479921\) \(\nu^{4}\mathstrut -\mathstrut \) \(141476191243349227852339\) \(\nu^{3}\mathstrut -\mathstrut \) \(143327565524240360562922\) \(\nu^{2}\mathstrut +\mathstrut \) \(51082916732053674000704\) \(\nu\mathstrut +\mathstrut \) \(31232388687353964064640\)\()/\)\(44\!\cdots\!36\)
\(\beta_{4}\)\(=\)\((\)\(3773756680062282389\) \(\nu^{14}\mathstrut -\mathstrut \) \(6621368153969086837\) \(\nu^{13}\mathstrut -\mathstrut \) \(173187246246377101328\) \(\nu^{12}\mathstrut +\mathstrut \) \(299029290976494333660\) \(\nu^{11}\mathstrut +\mathstrut \) \(2937458974583126918344\) \(\nu^{10}\mathstrut -\mathstrut \) \(4977240979565848871999\) \(\nu^{9}\mathstrut -\mathstrut \) \(22880252731934338749548\) \(\nu^{8}\mathstrut +\mathstrut \) \(37839894450753416967704\) \(\nu^{7}\mathstrut +\mathstrut \) \(83503603148532481698995\) \(\nu^{6}\mathstrut -\mathstrut \) \(132396296199577483382588\) \(\nu^{5}\mathstrut -\mathstrut \) \(131442940707426684242005\) \(\nu^{4}\mathstrut +\mathstrut \) \(201052630181593672596943\) \(\nu^{3}\mathstrut +\mathstrut \) \(56590561866592878591298\) \(\nu^{2}\mathstrut -\mathstrut \) \(105730885302223976198336\) \(\nu\mathstrut +\mathstrut \) \(15999879528543583420672\)\()/\)\(44\!\cdots\!36\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(480421702545662123\) \(\nu^{14}\mathstrut +\mathstrut \) \(43110065350197505\) \(\nu^{13}\mathstrut +\mathstrut \) \(23213090564296575506\) \(\nu^{12}\mathstrut -\mathstrut \) \(740490576502024596\) \(\nu^{11}\mathstrut -\mathstrut \) \(423613518372852698680\) \(\nu^{10}\mathstrut -\mathstrut \) \(14892506323265156095\) \(\nu^{9}\mathstrut +\mathstrut \) \(3671032048907900349578\) \(\nu^{8}\mathstrut +\mathstrut \) \(411319014825229416136\) \(\nu^{7}\mathstrut -\mathstrut \) \(15679149848322058779965\) \(\nu^{6}\mathstrut -\mathstrut \) \(3172708855946150385058\) \(\nu^{5}\mathstrut +\mathstrut \) \(30948944380910040502771\) \(\nu^{4}\mathstrut +\mathstrut \) \(8259757592004157683545\) \(\nu^{3}\mathstrut -\mathstrut \) \(22217767129638255474988\) \(\nu^{2}\mathstrut -\mathstrut \) \(4580081011050082891228\) \(\nu\mathstrut +\mathstrut \) \(2332369560757484941472\)\()/\)\(55\!\cdots\!92\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(3904290949793122853\) \(\nu^{14}\mathstrut +\mathstrut \) \(14888092941463260925\) \(\nu^{13}\mathstrut +\mathstrut \) \(169062781332764512088\) \(\nu^{12}\mathstrut -\mathstrut \) \(668437526636551236252\) \(\nu^{11}\mathstrut -\mathstrut \) \(2592166138155870798184\) \(\nu^{10}\mathstrut +\mathstrut \) \(10967030382881649513455\) \(\nu^{9}\mathstrut +\mathstrut \) \(16479152638452021523460\) \(\nu^{8}\mathstrut -\mathstrut \) \(80762745305159870682104\) \(\nu^{7}\mathstrut -\mathstrut \) \(35045824171083536765987\) \(\nu^{6}\mathstrut +\mathstrut \) \(263743776775326664831940\) \(\nu^{5}\mathstrut -\mathstrut \) \(17153063243581124000891\) \(\nu^{4}\mathstrut -\mathstrut \) \(329708970008761859686711\) \(\nu^{3}\mathstrut +\mathstrut \) \(86096965550968147850822\) \(\nu^{2}\mathstrut +\mathstrut \) \(85313542688545755485744\) \(\nu\mathstrut -\mathstrut \) \(17231425109937506795776\)\()/\)\(44\!\cdots\!36\)
\(\beta_{7}\)\(=\)\((\)\(1437348996203833439\) \(\nu^{14}\mathstrut -\mathstrut \) \(3946760310112449751\) \(\nu^{13}\mathstrut -\mathstrut \) \(62994684520973032856\) \(\nu^{12}\mathstrut +\mathstrut \) \(176671191969983900820\) \(\nu^{11}\mathstrut +\mathstrut \) \(984697502339295110968\) \(\nu^{10}\mathstrut -\mathstrut \) \(2894763636328689608669\) \(\nu^{9}\mathstrut -\mathstrut \) \(6497133196312926368636\) \(\nu^{8}\mathstrut +\mathstrut \) \(21396335118373983609512\) \(\nu^{7}\mathstrut +\mathstrut \) \(15463140566778105369785\) \(\nu^{6}\mathstrut -\mathstrut \) \(71147018611462724250524\) \(\nu^{5}\mathstrut +\mathstrut \) \(668266301521186841729\) \(\nu^{4}\mathstrut +\mathstrut \) \(96396192676307847030949\) \(\nu^{3}\mathstrut -\mathstrut \) \(26343781779899371656962\) \(\nu^{2}\mathstrut -\mathstrut \) \(36576815788443985896752\) \(\nu\mathstrut +\mathstrut \) \(4137311857191453969280\)\()/\)\(14\!\cdots\!12\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(216776146664850899\) \(\nu^{14}\mathstrut +\mathstrut \) \(833519197187931691\) \(\nu^{13}\mathstrut +\mathstrut \) \(9225298037107150072\) \(\nu^{12}\mathstrut -\mathstrut \) \(37446683496667123780\) \(\nu^{11}\mathstrut -\mathstrut \) \(136909874293211051608\) \(\nu^{10}\mathstrut +\mathstrut \) \(615042765945550203385\) \(\nu^{9}\mathstrut +\mathstrut \) \(807482146801725193676\) \(\nu^{8}\mathstrut -\mathstrut \) \(4539938442156400956040\) \(\nu^{7}\mathstrut -\mathstrut \) \(1259609430605682009317\) \(\nu^{6}\mathstrut +\mathstrut \) \(14947524072875232440620\) \(\nu^{5}\mathstrut -\mathstrut \) \(2596195015386327990477\) \(\nu^{4}\mathstrut -\mathstrut \) \(19543885760794156101025\) \(\nu^{3}\mathstrut +\mathstrut \) \(5767233907927992512410\) \(\nu^{2}\mathstrut +\mathstrut \) \(6876707545146179250864\) \(\nu\mathstrut -\mathstrut \) \(1099608980575046137664\)\()/\)\(16\!\cdots\!68\)
\(\beta_{9}\)\(=\)\((\)\(2262060993304819351\) \(\nu^{14}\mathstrut -\mathstrut \) \(3898546964586732887\) \(\nu^{13}\mathstrut -\mathstrut \) \(102677561674421271760\) \(\nu^{12}\mathstrut +\mathstrut \) \(172180740620193204084\) \(\nu^{11}\mathstrut +\mathstrut \) \(1709275636723449924632\) \(\nu^{10}\mathstrut -\mathstrut \) \(2762375590140558319621\) \(\nu^{9}\mathstrut -\mathstrut \) \(12869509104005418846724\) \(\nu^{8}\mathstrut +\mathstrut \) \(19631778341601514731400\) \(\nu^{7}\mathstrut +\mathstrut \) \(44047896643808091309313\) \(\nu^{6}\mathstrut -\mathstrut \) \(59268935086500033928756\) \(\nu^{5}\mathstrut -\mathstrut \) \(62006168022988305824471\) \(\nu^{4}\mathstrut +\mathstrut \) \(60676860335814627002837\) \(\nu^{3}\mathstrut +\mathstrut \) \(22877990863326377997014\) \(\nu^{2}\mathstrut -\mathstrut \) \(5496566174053900711552\) \(\nu\mathstrut +\mathstrut \) \(5241129959594824984640\)\()/\)\(14\!\cdots\!12\)
\(\beta_{10}\)\(=\)\((\)\(7389528242931052763\) \(\nu^{14}\mathstrut -\mathstrut \) \(17619842819561824147\) \(\nu^{13}\mathstrut -\mathstrut \) \(335494621667721213464\) \(\nu^{12}\mathstrut +\mathstrut \) \(781347790842433004388\) \(\nu^{11}\mathstrut +\mathstrut \) \(5596799573243534456152\) \(\nu^{10}\mathstrut -\mathstrut \) \(12587537843796393862097\) \(\nu^{9}\mathstrut -\mathstrut \) \(42434297297760917373068\) \(\nu^{8}\mathstrut +\mathstrut \) \(89945009312798312263112\) \(\nu^{7}\mathstrut +\mathstrut \) \(148333866313014220649117\) \(\nu^{6}\mathstrut -\mathstrut \) \(276567011587519356139052\) \(\nu^{5}\mathstrut -\mathstrut \) \(225796791826086771929275\) \(\nu^{4}\mathstrut +\mathstrut \) \(305614434437315646882457\) \(\nu^{3}\mathstrut +\mathstrut \) \(122418852737580306892054\) \(\nu^{2}\mathstrut -\mathstrut \) \(64975688162446145854064\) \(\nu\mathstrut -\mathstrut \) \(20760519604905515508992\)\()/\)\(44\!\cdots\!36\)
\(\beta_{11}\)\(=\)\((\)\(8766750062989708633\) \(\nu^{14}\mathstrut -\mathstrut \) \(22529199141324906641\) \(\nu^{13}\mathstrut -\mathstrut \) \(383811818206987097848\) \(\nu^{12}\mathstrut +\mathstrut \) \(995941970920453001676\) \(\nu^{11}\mathstrut +\mathstrut \) \(6001036075358757513992\) \(\nu^{10}\mathstrut -\mathstrut \) \(15968681453075819058955\) \(\nu^{9}\mathstrut -\mathstrut \) \(39886372998134487110740\) \(\nu^{8}\mathstrut +\mathstrut \) \(113304212701430619470680\) \(\nu^{7}\mathstrut +\mathstrut \) \(99684232238791936686799\) \(\nu^{6}\mathstrut -\mathstrut \) \(345811571939604474832852\) \(\nu^{5}\mathstrut -\mathstrut \) \(22639092092040808747577\) \(\nu^{4}\mathstrut +\mathstrut \) \(392231796537565988390195\) \(\nu^{3}\mathstrut -\mathstrut \) \(156334774693565882680990\) \(\nu^{2}\mathstrut -\mathstrut \) \(114807959618689508407408\) \(\nu\mathstrut +\mathstrut \) \(54269842840724729422208\)\()/\)\(44\!\cdots\!36\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(4417538992890062809\) \(\nu^{14}\mathstrut +\mathstrut \) \(10268326608661339349\) \(\nu^{13}\mathstrut +\mathstrut \) \(197311809783773428132\) \(\nu^{12}\mathstrut -\mathstrut \) \(456611053302482919612\) \(\nu^{11}\mathstrut -\mathstrut \) \(3196413318423942416024\) \(\nu^{10}\mathstrut +\mathstrut \) \(7398646108103984300491\) \(\nu^{9}\mathstrut +\mathstrut \) \(22849031189715444748168\) \(\nu^{8}\mathstrut -\mathstrut \) \(53545307775003849319096\) \(\nu^{7}\mathstrut -\mathstrut \) \(69555529379053943147695\) \(\nu^{6}\mathstrut +\mathstrut \) \(169932974581872220205488\) \(\nu^{5}\mathstrut +\mathstrut \) \(70235966360674007667737\) \(\nu^{4}\mathstrut -\mathstrut \) \(207964216939141801609367\) \(\nu^{3}\mathstrut +\mathstrut \) \(8344259772351224143546\) \(\nu^{2}\mathstrut +\mathstrut \) \(71793846729075368713576\) \(\nu\mathstrut -\mathstrut \) \(11807950134708017239328\)\()/\)\(22\!\cdots\!68\)
\(\beta_{13}\)\(=\)\((\)\(7599288743075757563\) \(\nu^{14}\mathstrut -\mathstrut \) \(14675949069150921499\) \(\nu^{13}\mathstrut -\mathstrut \) \(343394734237158555296\) \(\nu^{12}\mathstrut +\mathstrut \) \(648617517135946752852\) \(\nu^{11}\mathstrut +\mathstrut \) \(5675871771319530293080\) \(\nu^{10}\mathstrut -\mathstrut \) \(10416374257422677241905\) \(\nu^{9}\mathstrut -\mathstrut \) \(42209961752351543724788\) \(\nu^{8}\mathstrut +\mathstrut \) \(74291750278280117421560\) \(\nu^{7}\mathstrut +\mathstrut \) \(141106197202116037886717\) \(\nu^{6}\mathstrut -\mathstrut \) \(228640259637922398414884\) \(\nu^{5}\mathstrut -\mathstrut \) \(190386748714222964816683\) \(\nu^{4}\mathstrut +\mathstrut \) \(258839300934634508381089\) \(\nu^{3}\mathstrut +\mathstrut \) \(69730647157650718059982\) \(\nu^{2}\mathstrut -\mathstrut \) \(62443733500550485578896\) \(\nu\mathstrut -\mathstrut \) \(2219445897514458889376\)\()/\)\(22\!\cdots\!68\)
\(\beta_{14}\)\(=\)\((\)\(5118564908078654033\) \(\nu^{14}\mathstrut -\mathstrut \) \(13012426933683220879\) \(\nu^{13}\mathstrut -\mathstrut \) \(226696578225165112178\) \(\nu^{12}\mathstrut +\mathstrut \) \(578099458562396288700\) \(\nu^{11}\mathstrut +\mathstrut \) \(3621424480522273550704\) \(\nu^{10}\mathstrut -\mathstrut \) \(9348618128120933493683\) \(\nu^{9}\mathstrut -\mathstrut \) \(25231217957008947225314\) \(\nu^{8}\mathstrut +\mathstrut \) \(67395099414387524537648\) \(\nu^{7}\mathstrut +\mathstrut \) \(72665931767149837846055\) \(\nu^{6}\mathstrut -\mathstrut \) \(212442650490802176795278\) \(\nu^{5}\mathstrut -\mathstrut \) \(62438278530719538088105\) \(\nu^{4}\mathstrut +\mathstrut \) \(256098528986214242270833\) \(\nu^{3}\mathstrut -\mathstrut \) \(19685540454780823356992\) \(\nu^{2}\mathstrut -\mathstrut \) \(79244364779543304621452\) \(\nu\mathstrut +\mathstrut \) \(9700285797997532462848\)\()/\)\(11\!\cdots\!84\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(34\) \(\beta_{14}\mathstrut +\mathstrut \) \(32\) \(\beta_{13}\mathstrut +\mathstrut \) \(5\) \(\beta_{12}\mathstrut +\mathstrut \) \(19\) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(19\) \(\beta_{2}\mathstrut -\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(85\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{14}\mathstrut +\mathstrut \) \(8\) \(\beta_{13}\mathstrut -\mathstrut \) \(75\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\) \(\beta_{11}\mathstrut -\mathstrut \) \(27\) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(168\) \(\beta_{1}\mathstrut -\mathstrut \) \(15\)
\(\nu^{6}\)\(=\)\(-\)\(556\) \(\beta_{14}\mathstrut +\mathstrut \) \(502\) \(\beta_{13}\mathstrut +\mathstrut \) \(115\) \(\beta_{12}\mathstrut +\mathstrut \) \(317\) \(\beta_{11}\mathstrut +\mathstrut \) \(229\) \(\beta_{10}\mathstrut -\mathstrut \) \(96\) \(\beta_{9}\mathstrut -\mathstrut \) \(333\) \(\beta_{8}\mathstrut +\mathstrut \) \(313\) \(\beta_{7}\mathstrut +\mathstrut \) \(337\) \(\beta_{6}\mathstrut +\mathstrut \) \(302\) \(\beta_{5}\mathstrut -\mathstrut \) \(252\) \(\beta_{4}\mathstrut -\mathstrut \) \(79\) \(\beta_{3}\mathstrut +\mathstrut \) \(310\) \(\beta_{2}\mathstrut -\mathstrut \) \(247\) \(\beta_{1}\mathstrut +\mathstrut \) \(1198\)
\(\nu^{7}\)\(=\)\(-\)\(31\) \(\beta_{14}\mathstrut +\mathstrut \) \(239\) \(\beta_{13}\mathstrut -\mathstrut \) \(1254\) \(\beta_{12}\mathstrut -\mathstrut \) \(346\) \(\beta_{11}\mathstrut -\mathstrut \) \(562\) \(\beta_{10}\mathstrut -\mathstrut \) \(432\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(849\) \(\beta_{7}\mathstrut -\mathstrut \) \(229\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(414\) \(\beta_{4}\mathstrut -\mathstrut \) \(35\) \(\beta_{3}\mathstrut +\mathstrut \) \(99\) \(\beta_{2}\mathstrut +\mathstrut \) \(2518\) \(\beta_{1}\mathstrut -\mathstrut \) \(180\)
\(\nu^{8}\)\(=\)\(-\)\(9097\) \(\beta_{14}\mathstrut +\mathstrut \) \(7967\) \(\beta_{13}\mathstrut +\mathstrut \) \(2088\) \(\beta_{12}\mathstrut +\mathstrut \) \(5193\) \(\beta_{11}\mathstrut +\mathstrut \) \(3576\) \(\beta_{10}\mathstrut -\mathstrut \) \(888\) \(\beta_{9}\mathstrut -\mathstrut \) \(5611\) \(\beta_{8}\mathstrut +\mathstrut \) \(5351\) \(\beta_{7}\mathstrut +\mathstrut \) \(5729\) \(\beta_{6}\mathstrut +\mathstrut \) \(5021\) \(\beta_{5}\mathstrut -\mathstrut \) \(3952\) \(\beta_{4}\mathstrut -\mathstrut \) \(1609\) \(\beta_{3}\mathstrut +\mathstrut \) \(4887\) \(\beta_{2}\mathstrut -\mathstrut \) \(3873\) \(\beta_{1}\mathstrut +\mathstrut \) \(17955\)
\(\nu^{9}\)\(=\)\(-\)\(619\) \(\beta_{14}\mathstrut +\mathstrut \) \(5095\) \(\beta_{13}\mathstrut -\mathstrut \) \(20805\) \(\beta_{12}\mathstrut -\mathstrut \) \(5891\) \(\beta_{11}\mathstrut -\mathstrut \) \(10625\) \(\beta_{10}\mathstrut -\mathstrut \) \(8055\) \(\beta_{9}\mathstrut -\mathstrut \) \(308\) \(\beta_{8}\mathstrut -\mathstrut \) \(15316\) \(\beta_{7}\mathstrut -\mathstrut \) \(5067\) \(\beta_{6}\mathstrut +\mathstrut \) \(61\) \(\beta_{5}\mathstrut -\mathstrut \) \(6877\) \(\beta_{4}\mathstrut -\mathstrut \) \(750\) \(\beta_{3}\mathstrut +\mathstrut \) \(1621\) \(\beta_{2}\mathstrut +\mathstrut \) \(39150\) \(\beta_{1}\mathstrut -\mathstrut \) \(2196\)
\(\nu^{10}\)\(=\)\(-\)\(149357\) \(\beta_{14}\mathstrut +\mathstrut \) \(127959\) \(\beta_{13}\mathstrut +\mathstrut \) \(35595\) \(\beta_{12}\mathstrut +\mathstrut \) \(84969\) \(\beta_{11}\mathstrut +\mathstrut \) \(56709\) \(\beta_{10}\mathstrut -\mathstrut \) \(7087\) \(\beta_{9}\mathstrut -\mathstrut \) \(93089\) \(\beta_{8}\mathstrut +\mathstrut \) \(90567\) \(\beta_{7}\mathstrut +\mathstrut \) \(95731\) \(\beta_{6}\mathstrut +\mathstrut \) \(83612\) \(\beta_{5}\mathstrut -\mathstrut \) \(62019\) \(\beta_{4}\mathstrut -\mathstrut \) \(30076\) \(\beta_{3}\mathstrut +\mathstrut \) \(76569\) \(\beta_{2}\mathstrut -\mathstrut \) \(62051\) \(\beta_{1}\mathstrut +\mathstrut \) \(277897\)
\(\nu^{11}\)\(=\)\(-\)\(10176\) \(\beta_{14}\mathstrut +\mathstrut \) \(95830\) \(\beta_{13}\mathstrut -\mathstrut \) \(346912\) \(\beta_{12}\mathstrut -\mathstrut \) \(100736\) \(\beta_{11}\mathstrut -\mathstrut \) \(192217\) \(\beta_{10}\mathstrut -\mathstrut \) \(145094\) \(\beta_{9}\mathstrut -\mathstrut \) \(6660\) \(\beta_{8}\mathstrut -\mathstrut \) \(270979\) \(\beta_{7}\mathstrut -\mathstrut \) \(101706\) \(\beta_{6}\mathstrut -\mathstrut \) \(5067\) \(\beta_{5}\mathstrut -\mathstrut \) \(110424\) \(\beta_{4}\mathstrut -\mathstrut \) \(13520\) \(\beta_{3}\mathstrut +\mathstrut \) \(21534\) \(\beta_{2}\mathstrut +\mathstrut \) \(622037\) \(\beta_{1}\mathstrut -\mathstrut \) \(31788\)
\(\nu^{12}\)\(=\)\(-\)\(2460611\) \(\beta_{14}\mathstrut +\mathstrut \) \(2074863\) \(\beta_{13}\mathstrut +\mathstrut \) \(595649\) \(\beta_{12}\mathstrut +\mathstrut \) \(1393860\) \(\beta_{11}\mathstrut +\mathstrut \) \(908995\) \(\beta_{10}\mathstrut -\mathstrut \) \(28519\) \(\beta_{9}\mathstrut -\mathstrut \) \(1536904\) \(\beta_{8}\mathstrut +\mathstrut \) \(1524931\) \(\beta_{7}\mathstrut +\mathstrut \) \(1589921\) \(\beta_{6}\mathstrut +\mathstrut \) \(1395762\) \(\beta_{5}\mathstrut -\mathstrut \) \(977347\) \(\beta_{4}\mathstrut -\mathstrut \) \(540506\) \(\beta_{3}\mathstrut +\mathstrut \) \(1203101\) \(\beta_{2}\mathstrut -\mathstrut \) \(1013593\) \(\beta_{1}\mathstrut +\mathstrut \) \(4386294\)
\(\nu^{13}\)\(=\)\(-\)\(143953\) \(\beta_{14}\mathstrut +\mathstrut \) \(1697102\) \(\beta_{13}\mathstrut -\mathstrut \) \(5809478\) \(\beta_{12}\mathstrut -\mathstrut \) \(1731649\) \(\beta_{11}\mathstrut -\mathstrut \) \(3399748\) \(\beta_{10}\mathstrut -\mathstrut \) \(2552103\) \(\beta_{9}\mathstrut -\mathstrut \) \(122438\) \(\beta_{8}\mathstrut -\mathstrut \) \(4738578\) \(\beta_{7}\mathstrut -\mathstrut \) \(1938971\) \(\beta_{6}\mathstrut -\mathstrut \) \(196689\) \(\beta_{5}\mathstrut -\mathstrut \) \(1745788\) \(\beta_{4}\mathstrut -\mathstrut \) \(226830\) \(\beta_{3}\mathstrut +\mathstrut \) \(231759\) \(\beta_{2}\mathstrut +\mathstrut \) \(10022933\) \(\beta_{1}\mathstrut -\mathstrut \) \(564758\)
\(\nu^{14}\)\(=\)\(-\)\(40658804\) \(\beta_{14}\mathstrut +\mathstrut \) \(33887265\) \(\beta_{13}\mathstrut +\mathstrut \) \(9929787\) \(\beta_{12}\mathstrut +\mathstrut \) \(22941355\) \(\beta_{11}\mathstrut +\mathstrut \) \(14689662\) \(\beta_{10}\mathstrut +\mathstrut \) \(623231\) \(\beta_{9}\mathstrut -\mathstrut \) \(25363596\) \(\beta_{8}\mathstrut +\mathstrut \) \(25613853\) \(\beta_{7}\mathstrut +\mathstrut \) \(26363690\) \(\beta_{6}\mathstrut +\mathstrut \) \(23335400\) \(\beta_{5}\mathstrut -\mathstrut \) \(15489614\) \(\beta_{4}\mathstrut -\mathstrut \) \(9506066\) \(\beta_{3}\mathstrut +\mathstrut \) \(19015737\) \(\beta_{2}\mathstrut -\mathstrut \) \(16809082\) \(\beta_{1}\mathstrut +\mathstrut \) \(70152806\)

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
4.06882
3.77963
2.75355
2.43337
2.05101
1.17201
0.676512
0.113262
−0.656329
−1.34829
−1.40985
−1.97610
−3.01123
−3.53778
−4.10857
1.00000 −1.00000 1.00000 −4.06882 −1.00000 4.31915 1.00000 1.00000 −4.06882
1.2 1.00000 −1.00000 1.00000 −3.77963 −1.00000 2.13919 1.00000 1.00000 −3.77963
1.3 1.00000 −1.00000 1.00000 −2.75355 −1.00000 0.824838 1.00000 1.00000 −2.75355
1.4 1.00000 −1.00000 1.00000 −2.43337 −1.00000 −2.53842 1.00000 1.00000 −2.43337
1.5 1.00000 −1.00000 1.00000 −2.05101 −1.00000 −1.63593 1.00000 1.00000 −2.05101
1.6 1.00000 −1.00000 1.00000 −1.17201 −1.00000 −4.40989 1.00000 1.00000 −1.17201
1.7 1.00000 −1.00000 1.00000 −0.676512 −1.00000 −2.72085 1.00000 1.00000 −0.676512
1.8 1.00000 −1.00000 1.00000 −0.113262 −1.00000 1.75034 1.00000 1.00000 −0.113262
1.9 1.00000 −1.00000 1.00000 0.656329 −1.00000 2.89208 1.00000 1.00000 0.656329
1.10 1.00000 −1.00000 1.00000 1.34829 −1.00000 −1.25683 1.00000 1.00000 1.34829
1.11 1.00000 −1.00000 1.00000 1.40985 −1.00000 4.86981 1.00000 1.00000 1.40985
1.12 1.00000 −1.00000 1.00000 1.97610 −1.00000 4.65524 1.00000 1.00000 1.97610
1.13 1.00000 −1.00000 1.00000 3.01123 −1.00000 −0.871929 1.00000 1.00000 3.01123
1.14 1.00000 −1.00000 1.00000 3.53778 −1.00000 −4.09818 1.00000 1.00000 3.53778
1.15 1.00000 −1.00000 1.00000 4.10857 −1.00000 1.08136 1.00000 1.00000 4.10857
\(n\): 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
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{15} + \cdots\)
\(T_{7}^{15} - \cdots\)