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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

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