Properties

Label 6036.2.a.g
Level 6036
Weight 2
Character orbit 6036.a
Self dual Yes
Analytic conductor 48.198
Analytic rank 1
Dimension 15
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.19770266\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( -\beta_{8} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( -\beta_{8} q^{7} \) \(+ q^{9}\) \( + ( \beta_{4} + \beta_{10} ) q^{11} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{13} \) \( + ( -1 + \beta_{1} ) q^{15} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{14} ) q^{17} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{19} \) \( -\beta_{8} q^{21} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{23} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{25} \) \(+ q^{27}\) \( + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{29} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{31} \) \( + ( \beta_{4} + \beta_{10} ) q^{33} \) \( + ( -\beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{35} \) \( + ( 1 - \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{37} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{39} \) \( + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{43} \) \( + ( -1 + \beta_{1} ) q^{45} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{47} \) \( + ( -1 - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{14} ) q^{49} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{14} ) q^{51} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{53} \) \( + ( -3 - \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 3 \beta_{10} - 2 \beta_{12} + \beta_{14} ) q^{55} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{57} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{59} \) \( + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{61} \) \( -\beta_{8} q^{63} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{65} \) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{67} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{69} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{71} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{73} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{75} \) \( + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{77} \) \( + ( -4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{13} + \beta_{14} ) q^{79} \) \(+ q^{81}\) \( + ( -4 - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{12} ) q^{83} \) \( + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{85} \) \( + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{87} \) \( + ( -1 + 3 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{89} \) \( + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{13} ) q^{91} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{93} \) \( + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + 4 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{95} \) \( + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{97} \) \( + ( \beta_{4} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(15q \) \(\mathstrut +\mathstrut 15q^{3} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut +\mathstrut 15q^{3} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut -\mathstrut 11q^{15} \) \(\mathstrut -\mathstrut 5q^{17} \) \(\mathstrut +\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 32q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 15q^{27} \) \(\mathstrut -\mathstrut 23q^{29} \) \(\mathstrut -\mathstrut 13q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 14q^{39} \) \(\mathstrut -\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut 20q^{47} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 30q^{53} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut +\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 14q^{59} \) \(\mathstrut -\mathstrut 38q^{61} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 32q^{69} \) \(\mathstrut -\mathstrut 41q^{71} \) \(\mathstrut -\mathstrut 19q^{73} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 39q^{77} \) \(\mathstrut -\mathstrut 27q^{79} \) \(\mathstrut +\mathstrut 15q^{81} \) \(\mathstrut -\mathstrut 17q^{83} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut -\mathstrut 23q^{87} \) \(\mathstrut -\mathstrut 23q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 13q^{93} \) \(\mathstrut -\mathstrut 30q^{95} \) \(\mathstrut -\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(4\) \(x^{14}\mathstrut -\mathstrut \) \(27\) \(x^{13}\mathstrut +\mathstrut \) \(106\) \(x^{12}\mathstrut +\mathstrut \) \(266\) \(x^{11}\mathstrut -\mathstrut \) \(1004\) \(x^{10}\mathstrut -\mathstrut \) \(1105\) \(x^{9}\mathstrut +\mathstrut \) \(4076\) \(x^{8}\mathstrut +\mathstrut \) \(1501\) \(x^{7}\mathstrut -\mathstrut \) \(7100\) \(x^{6}\mathstrut -\mathstrut \) \(134\) \(x^{5}\mathstrut +\mathstrut \) \(5356\) \(x^{4}\mathstrut -\mathstrut \) \(1041\) \(x^{3}\mathstrut -\mathstrut \) \(1381\) \(x^{2}\mathstrut +\mathstrut \) \(543\) \(x\mathstrut -\mathstrut \) \(44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(358340218\) \(\nu^{14}\mathstrut -\mathstrut \) \(3455243067\) \(\nu^{13}\mathstrut +\mathstrut \) \(1639898571\) \(\nu^{12}\mathstrut +\mathstrut \) \(78821293300\) \(\nu^{11}\mathstrut -\mathstrut \) \(197604821838\) \(\nu^{10}\mathstrut -\mathstrut \) \(551873907794\) \(\nu^{9}\mathstrut +\mathstrut \) \(2268619321365\) \(\nu^{8}\mathstrut +\mathstrut \) \(706301147372\) \(\nu^{7}\mathstrut -\mathstrut \) \(9186271236513\) \(\nu^{6}\mathstrut +\mathstrut \) \(4454535017866\) \(\nu^{5}\mathstrut +\mathstrut \) \(11397811685951\) \(\nu^{4}\mathstrut -\mathstrut \) \(8590765226022\) \(\nu^{3}\mathstrut -\mathstrut \) \(3528723902040\) \(\nu^{2}\mathstrut +\mathstrut \) \(4026629840681\) \(\nu\mathstrut -\mathstrut \) \(546137578798\)\()/\)\(58444065093\)
\(\beta_{3}\)\(=\)\((\)\(423712226\) \(\nu^{14}\mathstrut -\mathstrut \) \(628785145\) \(\nu^{13}\mathstrut -\mathstrut \) \(16928063565\) \(\nu^{12}\mathstrut +\mathstrut \) \(21911983318\) \(\nu^{11}\mathstrut +\mathstrut \) \(253795233195\) \(\nu^{10}\mathstrut -\mathstrut \) \(288553642002\) \(\nu^{9}\mathstrut -\mathstrut \) \(1737416359970\) \(\nu^{8}\mathstrut +\mathstrut \) \(1834247254936\) \(\nu^{7}\mathstrut +\mathstrut \) \(5206469459211\) \(\nu^{6}\mathstrut -\mathstrut \) \(5774695912957\) \(\nu^{5}\mathstrut -\mathstrut \) \(5530760533108\) \(\nu^{4}\mathstrut +\mathstrut \) \(6569919538899\) \(\nu^{3}\mathstrut +\mathstrut \) \(1460693112978\) \(\nu^{2}\mathstrut -\mathstrut \) \(2298192379866\) \(\nu\mathstrut +\mathstrut \) \(400272956308\)\()/\)\(19481355031\)
\(\beta_{4}\)\(=\)\((\)\(3567036137\) \(\nu^{14}\mathstrut -\mathstrut \) \(21649170771\) \(\nu^{13}\mathstrut -\mathstrut \) \(62131751904\) \(\nu^{12}\mathstrut +\mathstrut \) \(551620921145\) \(\nu^{11}\mathstrut +\mathstrut \) \(74099057337\) \(\nu^{10}\mathstrut -\mathstrut \) \(4887018761755\) \(\nu^{9}\mathstrut +\mathstrut \) \(3855492281382\) \(\nu^{8}\mathstrut +\mathstrut \) \(16869229948300\) \(\nu^{7}\mathstrut -\mathstrut \) \(22290712534461\) \(\nu^{6}\mathstrut -\mathstrut \) \(16261330615534\) \(\nu^{5}\mathstrut +\mathstrut \) \(32094202063099\) \(\nu^{4}\mathstrut -\mathstrut \) \(234681574773\) \(\nu^{3}\mathstrut -\mathstrut \) \(13294732750773\) \(\nu^{2}\mathstrut +\mathstrut \) \(3937809774856\) \(\nu\mathstrut -\mathstrut \) \(198959454251\)\()/\)\(58444065093\)
\(\beta_{5}\)\(=\)\((\)\(1533791708\) \(\nu^{14}\mathstrut -\mathstrut \) \(8034154554\) \(\nu^{13}\mathstrut -\mathstrut \) \(33188007874\) \(\nu^{12}\mathstrut +\mathstrut \) \(208529518638\) \(\nu^{11}\mathstrut +\mathstrut \) \(200675102334\) \(\nu^{10}\mathstrut -\mathstrut \) \(1906796199932\) \(\nu^{9}\mathstrut +\mathstrut \) \(103512865275\) \(\nu^{8}\mathstrut +\mathstrut \) \(7086723160006\) \(\nu^{7}\mathstrut -\mathstrut \) \(3740738899709\) \(\nu^{6}\mathstrut -\mathstrut \) \(9166061389940\) \(\nu^{5}\mathstrut +\mathstrut \) \(6053346531801\) \(\nu^{4}\mathstrut +\mathstrut \) \(3759925861648\) \(\nu^{3}\mathstrut -\mathstrut \) \(2545481553708\) \(\nu^{2}\mathstrut +\mathstrut \) \(92994480428\) \(\nu\mathstrut +\mathstrut \) \(31492118482\)\()/\)\(19481355031\)
\(\beta_{6}\)\(=\)\((\)\(1731551174\) \(\nu^{14}\mathstrut -\mathstrut \) \(8250353036\) \(\nu^{13}\mathstrut -\mathstrut \) \(38548196723\) \(\nu^{12}\mathstrut +\mathstrut \) \(207446618172\) \(\nu^{11}\mathstrut +\mathstrut \) \(247221177892\) \(\nu^{10}\mathstrut -\mathstrut \) \(1793151763368\) \(\nu^{9}\mathstrut +\mathstrut \) \(43750977609\) \(\nu^{8}\mathstrut +\mathstrut \) \(5955137864209\) \(\nu^{7}\mathstrut -\mathstrut \) \(4622078754486\) \(\nu^{6}\mathstrut -\mathstrut \) \(5686955517688\) \(\nu^{5}\mathstrut +\mathstrut \) \(8186083588136\) \(\nu^{4}\mathstrut +\mathstrut \) \(53788357764\) \(\nu^{3}\mathstrut -\mathstrut \) \(3637991875353\) \(\nu^{2}\mathstrut +\mathstrut \) \(1272169711354\) \(\nu\mathstrut -\mathstrut \) \(117449952898\)\()/\)\(19481355031\)
\(\beta_{7}\)\(=\)\((\)\(5568687106\) \(\nu^{14}\mathstrut -\mathstrut \) \(19093754208\) \(\nu^{13}\mathstrut -\mathstrut \) \(162031083048\) \(\nu^{12}\mathstrut +\mathstrut \) \(502149347719\) \(\nu^{11}\mathstrut +\mathstrut \) \(1780883206311\) \(\nu^{10}\mathstrut -\mathstrut \) \(4681899694094\) \(\nu^{9}\mathstrut -\mathstrut \) \(8834705170920\) \(\nu^{8}\mathstrut +\mathstrut \) \(18554016837980\) \(\nu^{7}\mathstrut +\mathstrut \) \(18163440635214\) \(\nu^{6}\mathstrut -\mathstrut \) \(31919893118456\) \(\nu^{5}\mathstrut -\mathstrut \) \(15219609456415\) \(\nu^{4}\mathstrut +\mathstrut \) \(22577656847343\) \(\nu^{3}\mathstrut +\mathstrut \) \(4020848205666\) \(\nu^{2}\mathstrut -\mathstrut \) \(5018365640773\) \(\nu\mathstrut +\mathstrut \) \(319535662097\)\()/\)\(58444065093\)
\(\beta_{8}\)\(=\)\((\)\(1924356769\) \(\nu^{14}\mathstrut -\mathstrut \) \(11454090727\) \(\nu^{13}\mathstrut -\mathstrut \) \(34583644911\) \(\nu^{12}\mathstrut +\mathstrut \) \(292039374175\) \(\nu^{11}\mathstrut +\mathstrut \) \(68286852293\) \(\nu^{10}\mathstrut -\mathstrut \) \(2588425841213\) \(\nu^{9}\mathstrut +\mathstrut \) \(1813047829279\) \(\nu^{8}\mathstrut +\mathstrut \) \(8924962826124\) \(\nu^{7}\mathstrut -\mathstrut \) \(10999683311240\) \(\nu^{6}\mathstrut -\mathstrut \) \(8457780235689\) \(\nu^{5}\mathstrut +\mathstrut \) \(15997332900914\) \(\nu^{4}\mathstrut -\mathstrut \) \(665985805523\) \(\nu^{3}\mathstrut -\mathstrut \) \(6663639693128\) \(\nu^{2}\mathstrut +\mathstrut \) \(2460004392646\) \(\nu\mathstrut -\mathstrut \) \(142936166389\)\()/\)\(19481355031\)
\(\beta_{9}\)\(=\)\((\)\(6083709676\) \(\nu^{14}\mathstrut -\mathstrut \) \(20946481836\) \(\nu^{13}\mathstrut -\mathstrut \) \(170476323909\) \(\nu^{12}\mathstrut +\mathstrut \) \(529474125973\) \(\nu^{11}\mathstrut +\mathstrut \) \(1768601832687\) \(\nu^{10}\mathstrut -\mathstrut \) \(4611036393497\) \(\nu^{9}\mathstrut -\mathstrut \) \(7918515647748\) \(\nu^{8}\mathstrut +\mathstrut \) \(16007511236417\) \(\nu^{7}\mathstrut +\mathstrut \) \(12916624866327\) \(\nu^{6}\mathstrut -\mathstrut \) \(21508962230990\) \(\nu^{5}\mathstrut -\mathstrut \) \(8733916171558\) \(\nu^{4}\mathstrut +\mathstrut \) \(11347474643385\) \(\nu^{3}\mathstrut +\mathstrut \) \(2349937306503\) \(\nu^{2}\mathstrut -\mathstrut \) \(1781773064236\) \(\nu\mathstrut -\mathstrut \) \(157050312484\)\()/\)\(58444065093\)
\(\beta_{10}\)\(=\)\((\)\(7260291644\) \(\nu^{14}\mathstrut -\mathstrut \) \(32010037530\) \(\nu^{13}\mathstrut -\mathstrut \) \(175831046238\) \(\nu^{12}\mathstrut +\mathstrut \) \(814474288403\) \(\nu^{11}\mathstrut +\mathstrut \) \(1410838550580\) \(\nu^{10}\mathstrut -\mathstrut \) \(7184392173223\) \(\nu^{9}\mathstrut -\mathstrut \) \(3323137051056\) \(\nu^{8}\mathstrut +\mathstrut \) \(25024944501244\) \(\nu^{7}\mathstrut -\mathstrut \) \(5863573811574\) \(\nu^{6}\mathstrut -\mathstrut \) \(28904743145578\) \(\nu^{5}\mathstrut +\mathstrut \) \(16205144774653\) \(\nu^{4}\mathstrut +\mathstrut \) \(8297232990042\) \(\nu^{3}\mathstrut -\mathstrut \) \(8214201190113\) \(\nu^{2}\mathstrut +\mathstrut \) \(1781074947118\) \(\nu\mathstrut -\mathstrut \) \(54571509779\)\()/\)\(58444065093\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(7262151155\) \(\nu^{14}\mathstrut +\mathstrut \) \(34249747695\) \(\nu^{13}\mathstrut +\mathstrut \) \(171462500097\) \(\nu^{12}\mathstrut -\mathstrut \) \(888312598286\) \(\nu^{11}\mathstrut -\mathstrut \) \(1307054951988\) \(\nu^{10}\mathstrut +\mathstrut \) \(8109490478674\) \(\nu^{9}\mathstrut +\mathstrut \) \(2542527461046\) \(\nu^{8}\mathstrut -\mathstrut \) \(30237803769097\) \(\nu^{7}\mathstrut +\mathstrut \) \(7854912941457\) \(\nu^{6}\mathstrut +\mathstrut \) \(40974759929053\) \(\nu^{5}\mathstrut -\mathstrut \) \(18546626159971\) \(\nu^{4}\mathstrut -\mathstrut \) \(19020179732652\) \(\nu^{3}\mathstrut +\mathstrut \) \(9683158812840\) \(\nu^{2}\mathstrut +\mathstrut \) \(965571711005\) \(\nu\mathstrut -\mathstrut \) \(461279859325\)\()/\)\(58444065093\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(8881419661\) \(\nu^{14}\mathstrut +\mathstrut \) \(37650120882\) \(\nu^{13}\mathstrut +\mathstrut \) \(223317906615\) \(\nu^{12}\mathstrut -\mathstrut \) \(963258026887\) \(\nu^{11}\mathstrut -\mathstrut \) \(1943112254400\) \(\nu^{10}\mathstrut +\mathstrut \) \(8575689751328\) \(\nu^{9}\mathstrut +\mathstrut \) \(6103608989484\) \(\nu^{8}\mathstrut -\mathstrut \) \(30533855283224\) \(\nu^{7}\mathstrut -\mathstrut \) \(737955429000\) \(\nu^{6}\mathstrut +\mathstrut \) \(38258309514041\) \(\nu^{5}\mathstrut -\mathstrut \) \(8610542857280\) \(\nu^{4}\mathstrut -\mathstrut \) \(15180056486778\) \(\nu^{3}\mathstrut +\mathstrut \) \(4844271601440\) \(\nu^{2}\mathstrut -\mathstrut \) \(98458986287\) \(\nu\mathstrut +\mathstrut \) \(92979418225\)\()/\)\(58444065093\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(9385042712\) \(\nu^{14}\mathstrut +\mathstrut \) \(47363234112\) \(\nu^{13}\mathstrut +\mathstrut \) \(204278546022\) \(\nu^{12}\mathstrut -\mathstrut \) \(1208912683499\) \(\nu^{11}\mathstrut -\mathstrut \) \(1246495383216\) \(\nu^{10}\mathstrut +\mathstrut \) \(10729352393770\) \(\nu^{9}\mathstrut -\mathstrut \) \(669825226899\) \(\nu^{8}\mathstrut -\mathstrut \) \(37488998318524\) \(\nu^{7}\mathstrut +\mathstrut \) \(24162681101964\) \(\nu^{6}\mathstrut +\mathstrut \) \(40592532372535\) \(\nu^{5}\mathstrut -\mathstrut \) \(39366359011501\) \(\nu^{4}\mathstrut -\mathstrut \) \(6976084915473\) \(\nu^{3}\mathstrut +\mathstrut \) \(15978620346393\) \(\nu^{2}\mathstrut -\mathstrut \) \(5480104040764\) \(\nu\mathstrut +\mathstrut \) \(716312928815\)\()/\)\(58444065093\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(4057390393\) \(\nu^{14}\mathstrut +\mathstrut \) \(20201794850\) \(\nu^{13}\mathstrut +\mathstrut \) \(89153456075\) \(\nu^{12}\mathstrut -\mathstrut \) \(515653560454\) \(\nu^{11}\mathstrut -\mathstrut \) \(556047690094\) \(\nu^{10}\mathstrut +\mathstrut \) \(4576721419850\) \(\nu^{9}\mathstrut -\mathstrut \) \(202805385751\) \(\nu^{8}\mathstrut -\mathstrut \) \(16011768064319\) \(\nu^{7}\mathstrut +\mathstrut \) \(10599496006485\) \(\nu^{6}\mathstrut +\mathstrut \) \(17503280621078\) \(\nu^{5}\mathstrut -\mathstrut \) \(18525088631636\) \(\nu^{4}\mathstrut -\mathstrut \) \(2788505274183\) \(\nu^{3}\mathstrut +\mathstrut \) \(8383557145413\) \(\nu^{2}\mathstrut -\mathstrut \) \(2704514891878\) \(\nu\mathstrut +\mathstrut \) \(164727575441\)\()/\)\(19481355031\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(11\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{14}\mathstrut -\mathstrut \) \(13\) \(\beta_{13}\mathstrut +\mathstrut \) \(9\) \(\beta_{12}\mathstrut -\mathstrut \) \(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(107\) \(\beta_{1}\mathstrut +\mathstrut \) \(57\)
\(\nu^{6}\)\(=\)\(7\) \(\beta_{14}\mathstrut -\mathstrut \) \(9\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{12}\mathstrut -\mathstrut \) \(123\) \(\beta_{11}\mathstrut -\mathstrut \) \(122\) \(\beta_{10}\mathstrut +\mathstrut \) \(43\) \(\beta_{9}\mathstrut +\mathstrut \) \(33\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(75\) \(\beta_{6}\mathstrut -\mathstrut \) \(30\) \(\beta_{5}\mathstrut -\mathstrut \) \(98\) \(\beta_{4}\mathstrut -\mathstrut \) \(102\) \(\beta_{3}\mathstrut -\mathstrut \) \(118\) \(\beta_{2}\mathstrut +\mathstrut \) \(134\) \(\beta_{1}\mathstrut +\mathstrut \) \(515\)
\(\nu^{7}\)\(=\)\(-\)\(4\) \(\beta_{14}\mathstrut -\mathstrut \) \(162\) \(\beta_{13}\mathstrut +\mathstrut \) \(65\) \(\beta_{12}\mathstrut -\mathstrut \) \(182\) \(\beta_{11}\mathstrut -\mathstrut \) \(217\) \(\beta_{10}\mathstrut +\mathstrut \) \(126\) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(33\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(239\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(88\) \(\beta_{3}\mathstrut -\mathstrut \) \(221\) \(\beta_{2}\mathstrut +\mathstrut \) \(1207\) \(\beta_{1}\mathstrut +\mathstrut \) \(864\)
\(\nu^{8}\)\(=\)\(165\) \(\beta_{14}\mathstrut -\mathstrut \) \(229\) \(\beta_{13}\mathstrut +\mathstrut \) \(140\) \(\beta_{12}\mathstrut -\mathstrut \) \(1450\) \(\beta_{11}\mathstrut -\mathstrut \) \(1449\) \(\beta_{10}\mathstrut +\mathstrut \) \(508\) \(\beta_{9}\mathstrut +\mathstrut \) \(447\) \(\beta_{8}\mathstrut -\mathstrut \) \(93\) \(\beta_{7}\mathstrut +\mathstrut \) \(785\) \(\beta_{6}\mathstrut -\mathstrut \) \(610\) \(\beta_{5}\mathstrut -\mathstrut \) \(986\) \(\beta_{4}\mathstrut -\mathstrut \) \(1064\) \(\beta_{3}\mathstrut -\mathstrut \) \(1289\) \(\beta_{2}\mathstrut +\mathstrut \) \(2214\) \(\beta_{1}\mathstrut +\mathstrut \) \(5723\)
\(\nu^{9}\)\(=\)\(172\) \(\beta_{14}\mathstrut -\mathstrut \) \(2064\) \(\beta_{13}\mathstrut +\mathstrut \) \(322\) \(\beta_{12}\mathstrut -\mathstrut \) \(3074\) \(\beta_{11}\mathstrut -\mathstrut \) \(3546\) \(\beta_{10}\mathstrut +\mathstrut \) \(1307\) \(\beta_{9}\mathstrut +\mathstrut \) \(60\) \(\beta_{8}\mathstrut -\mathstrut \) \(415\) \(\beta_{7}\mathstrut +\mathstrut \) \(491\) \(\beta_{6}\mathstrut -\mathstrut \) \(3527\) \(\beta_{5}\mathstrut -\mathstrut \) \(148\) \(\beta_{4}\mathstrut -\mathstrut \) \(1382\) \(\beta_{3}\mathstrut -\mathstrut \) \(2846\) \(\beta_{2}\mathstrut +\mathstrut \) \(14165\) \(\beta_{1}\mathstrut +\mathstrut \) \(11937\)
\(\nu^{10}\)\(=\)\(2910\) \(\beta_{14}\mathstrut -\mathstrut \) \(4243\) \(\beta_{13}\mathstrut +\mathstrut \) \(685\) \(\beta_{12}\mathstrut -\mathstrut \) \(17859\) \(\beta_{11}\mathstrut -\mathstrut \) \(18104\) \(\beta_{10}\mathstrut +\mathstrut \) \(5564\) \(\beta_{9}\mathstrut +\mathstrut \) \(5696\) \(\beta_{8}\mathstrut -\mathstrut \) \(892\) \(\beta_{7}\mathstrut +\mathstrut \) \(8897\) \(\beta_{6}\mathstrut -\mathstrut \) \(10620\) \(\beta_{5}\mathstrut -\mathstrut \) \(10158\) \(\beta_{4}\mathstrut -\mathstrut \) \(11285\) \(\beta_{3}\mathstrut -\mathstrut \) \(14254\) \(\beta_{2}\mathstrut +\mathstrut \) \(32736\) \(\beta_{1}\mathstrut +\mathstrut \) \(64963\)
\(\nu^{11}\)\(=\)\(5604\) \(\beta_{14}\mathstrut -\mathstrut \) \(26986\) \(\beta_{13}\mathstrut -\mathstrut \) \(1347\) \(\beta_{12}\mathstrut -\mathstrut \) \(46638\) \(\beta_{11}\mathstrut -\mathstrut \) \(52501\) \(\beta_{10}\mathstrut +\mathstrut \) \(13310\) \(\beta_{9}\mathstrut +\mathstrut \) \(3562\) \(\beta_{8}\mathstrut -\mathstrut \) \(4672\) \(\beta_{7}\mathstrut +\mathstrut \) \(12536\) \(\beta_{6}\mathstrut -\mathstrut \) \(51652\) \(\beta_{5}\mathstrut -\mathstrut \) \(5829\) \(\beta_{4}\mathstrut -\mathstrut \) \(18983\) \(\beta_{3}\mathstrut -\mathstrut \) \(35116\) \(\beta_{2}\mathstrut +\mathstrut \) \(171297\) \(\beta_{1}\mathstrut +\mathstrut \) \(156951\)
\(\nu^{12}\)\(=\)\(46561\) \(\beta_{14}\mathstrut -\mathstrut \) \(69367\) \(\beta_{13}\mathstrut -\mathstrut \) \(7687\) \(\beta_{12}\mathstrut -\mathstrut \) \(227013\) \(\beta_{11}\mathstrut -\mathstrut \) \(233283\) \(\beta_{10}\mathstrut +\mathstrut \) \(57428\) \(\beta_{9}\mathstrut +\mathstrut \) \(71262\) \(\beta_{8}\mathstrut -\mathstrut \) \(8638\) \(\beta_{7}\mathstrut +\mathstrut \) \(107050\) \(\beta_{6}\mathstrut -\mathstrut \) \(170621\) \(\beta_{5}\mathstrut -\mathstrut \) \(106641\) \(\beta_{4}\mathstrut -\mathstrut \) \(121093\) \(\beta_{3}\mathstrut -\mathstrut \) \(158473\) \(\beta_{2}\mathstrut +\mathstrut \) \(459141\) \(\beta_{1}\mathstrut +\mathstrut \) \(748880\)
\(\nu^{13}\)\(=\)\(117637\) \(\beta_{14}\mathstrut -\mathstrut \) \(360309\) \(\beta_{13}\mathstrut -\mathstrut \) \(80333\) \(\beta_{12}\mathstrut -\mathstrut \) \(672857\) \(\beta_{11}\mathstrut -\mathstrut \) \(743383\) \(\beta_{10}\mathstrut +\mathstrut \) \(128938\) \(\beta_{9}\mathstrut +\mathstrut \) \(81430\) \(\beta_{8}\mathstrut -\mathstrut \) \(48790\) \(\beta_{7}\mathstrut +\mathstrut \) \(220774\) \(\beta_{6}\mathstrut -\mathstrut \) \(750708\) \(\beta_{5}\mathstrut -\mathstrut \) \(103369\) \(\beta_{4}\mathstrut -\mathstrut \) \(242563\) \(\beta_{3}\mathstrut -\mathstrut \) \(420356\) \(\beta_{2}\mathstrut +\mathstrut \) \(2118406\) \(\beta_{1}\mathstrut +\mathstrut \) \(2003516\)
\(\nu^{14}\)\(=\)\(713312\) \(\beta_{14}\mathstrut -\mathstrut \) \(1063396\) \(\beta_{13}\mathstrut -\mathstrut \) \(319108\) \(\beta_{12}\mathstrut -\mathstrut \) \(2948055\) \(\beta_{11}\mathstrut -\mathstrut \) \(3058630\) \(\beta_{10}\mathstrut +\mathstrut \) \(551144\) \(\beta_{9}\mathstrut +\mathstrut \) \(891599\) \(\beta_{8}\mathstrut -\mathstrut \) \(80148\) \(\beta_{7}\mathstrut +\mathstrut \) \(1344725\) \(\beta_{6}\mathstrut -\mathstrut \) \(2613905\) \(\beta_{5}\mathstrut -\mathstrut \) \(1136061\) \(\beta_{4}\mathstrut -\mathstrut \) \(1307964\) \(\beta_{3}\mathstrut -\mathstrut \) \(1760913\) \(\beta_{2}\mathstrut +\mathstrut \) \(6260776\) \(\beta_{1}\mathstrut +\mathstrut \) \(8730595\)

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
−3.01525
−2.82901
−2.61011
−1.04745
−1.01946
−0.927807
0.117275
0.357651
0.546893
1.03500
1.33620
1.40631
3.48442
3.52096
3.64439
0 1.00000 0 −4.01525 0 −3.41124 0 1.00000 0
1.2 0 1.00000 0 −3.82901 0 3.22079 0 1.00000 0
1.3 0 1.00000 0 −3.61011 0 0.982804 0 1.00000 0
1.4 0 1.00000 0 −2.04745 0 3.89716 0 1.00000 0
1.5 0 1.00000 0 −2.01946 0 −2.53250 0 1.00000 0
1.6 0 1.00000 0 −1.92781 0 2.55035 0 1.00000 0
1.7 0 1.00000 0 −0.882725 0 −2.85663 0 1.00000 0
1.8 0 1.00000 0 −0.642349 0 −2.57693 0 1.00000 0
1.9 0 1.00000 0 −0.453107 0 2.17063 0 1.00000 0
1.10 0 1.00000 0 0.0349978 0 0.637505 0 1.00000 0
1.11 0 1.00000 0 0.336200 0 0.568501 0 1.00000 0
1.12 0 1.00000 0 0.406311 0 −3.65793 0 1.00000 0
1.13 0 1.00000 0 2.48442 0 1.05465 0 1.00000 0
1.14 0 1.00000 0 2.52096 0 −1.02518 0 1.00000 0
1.15 0 1.00000 0 2.64439 0 −3.02199 0 1.00000 0
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(503\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

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