Properties

Label 4012.2.a.i
Level 4012
Weight 2
Character orbit 4012.a
Self dual yes
Analytic conductor 32.036
Analytic rank 0
Dimension 18
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{9} q^{5} -\beta_{8} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{9} q^{5} -\beta_{8} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{4} ) q^{11} + \beta_{7} q^{13} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{9} + \beta_{12} + \beta_{13} - \beta_{17} ) q^{15} - q^{17} + ( -\beta_{6} + \beta_{9} + \beta_{17} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - 2 \beta_{15} - \beta_{17} ) q^{21} + ( 2 + \beta_{11} - \beta_{13} - \beta_{14} ) q^{23} + ( \beta_{1} + \beta_{2} + \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{25} + ( 1 + 2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} - \beta_{17} ) q^{27} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{17} ) q^{29} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{31} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{13} - \beta_{17} ) q^{33} + ( 3 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{35} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{9} + \beta_{12} - \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{37} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{16} ) q^{39} + ( 3 - \beta_{1} - \beta_{2} + \beta_{11} + \beta_{15} + \beta_{16} ) q^{41} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{14} - 3 \beta_{15} - \beta_{16} ) q^{43} + ( 5 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{11} + 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} - 2 \beta_{17} ) q^{45} + ( 3 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{47} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{17} ) q^{49} -\beta_{1} q^{51} + ( -3 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{55} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{9} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{17} ) q^{57} - q^{59} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{61} + ( 7 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{63} + ( -3 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{65} + ( 5 - \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{17} ) q^{67} + ( -1 + 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{69} + ( -3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{71} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{73} + ( 1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{9} + \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{75} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} + 2 \beta_{13} + \beta_{15} + 2 \beta_{16} ) q^{77} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{79} + ( 3 - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{81} + ( 1 + 3 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{83} + \beta_{9} q^{85} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + 3 \beta_{9} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{87} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 2 \beta_{14} - 3 \beta_{15} - \beta_{16} ) q^{89} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{15} - \beta_{17} ) q^{91} + ( -3 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{17} ) q^{93} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{97} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + \beta_{15} + \beta_{16} - 3 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 8q^{3} + 4q^{5} + 2q^{7} + 18q^{9} + O(q^{10}) \) \( 18q + 8q^{3} + 4q^{5} + 2q^{7} + 18q^{9} + 12q^{11} + 2q^{13} - 18q^{17} + 5q^{19} - 3q^{21} + 21q^{23} + 16q^{25} + 26q^{27} + 14q^{29} + 15q^{31} + 19q^{33} + 20q^{35} + 2q^{37} - 14q^{39} + 34q^{41} + 21q^{43} + 49q^{45} + 69q^{47} + 28q^{49} - 8q^{51} - 4q^{53} + 18q^{55} + 5q^{57} - 18q^{59} + 11q^{61} + 35q^{63} + 27q^{65} + 34q^{67} - 4q^{69} + 37q^{71} + 18q^{73} + 72q^{75} + 11q^{77} + 11q^{79} + 30q^{81} + 28q^{83} - 4q^{85} + 7q^{87} + 44q^{89} - 23q^{91} - 3q^{93} - 11q^{95} + 11q^{97} + 56q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} - 17374 x^{10} - 893 x^{9} + 38112 x^{8} - 18700 x^{7} - 32137 x^{6} + 26381 x^{5} + 3964 x^{4} - 5788 x^{3} - 108 x^{2} + 232 x - 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(721980238514 \nu^{17} - 8792294540969 \nu^{16} + 19922000071949 \nu^{15} + 155380660037965 \nu^{14} - 735707335127267 \nu^{13} - 568622846364540 \nu^{12} + 7445947298023033 \nu^{11} - 4273813195078958 \nu^{10} - 33371400511887937 \nu^{9} + 38339043861291599 \nu^{8} + 68469249736557868 \nu^{7} - 103056900210181944 \nu^{6} - 52510563287426852 \nu^{5} + 102057820349258193 \nu^{4} + 3198625341800514 \nu^{3} - 23238733137161126 \nu^{2} - 2215838727764722 \nu + 926602245958234\)\()/ 221648864308394 \)
\(\beta_{4}\)\(=\)\((\)\(3830861910649 \nu^{17} + 47591230451132 \nu^{16} - 573472883791908 \nu^{15} - 137289585232790 \nu^{14} + 12325036003989759 \nu^{13} - 14148964215534769 \nu^{12} - 107106555866144333 \nu^{11} + 184409875945857643 \nu^{10} + 442914820232424934 \nu^{9} - 930141316065082005 \nu^{8} - 849682139411626348 \nu^{7} + 2136821502467077292 \nu^{6} + 583790399310322567 \nu^{5} - 2048651270396240855 \nu^{4} + 4680051383184224 \nu^{3} + 525804686675066276 \nu^{2} + 49267910585755140 \nu - 14341710559057800\)\()/ 886595457233576 \)
\(\beta_{5}\)\(=\)\((\)\(3229865338098 \nu^{17} - 28341808006461 \nu^{16} + 7224068968562 \nu^{15} + 591300911310204 \nu^{14} - 1336425942773296 \nu^{13} - 4004054182384929 \nu^{12} + 15632584348897933 \nu^{11} + 5978431644745797 \nu^{10} - 74099469131188689 \nu^{9} + 38791752370189050 \nu^{8} + 156577499294520397 \nu^{7} - 158561937111261718 \nu^{6} - 122359738200449602 \nu^{5} + 178581023652807271 \nu^{4} + 6230795980002505 \nu^{3} - 37354205935957182 \nu^{2} - 2171567585544804 \nu - 773148687785764\)\()/ 443297728616788 \)
\(\beta_{6}\)\(=\)\((\)\(-3998990215870 \nu^{17} + 48822273723019 \nu^{16} - 95914145833494 \nu^{15} - 926486305202038 \nu^{14} + 3758734853696590 \nu^{13} + 4674117759559001 \nu^{12} - 38722068240010909 \nu^{11} + 10159045369925169 \nu^{10} + 176233209704042289 \nu^{9} - 156608643808120130 \nu^{8} - 369918339657289877 \nu^{7} + 465163701156950644 \nu^{6} + 302735513077342338 \nu^{5} - 488011641581764889 \nu^{4} - 42529357188719131 \nu^{3} + 117684572825314984 \nu^{2} + 15555876709359224 \nu - 2577127790916876\)\()/ 443297728616788 \)
\(\beta_{7}\)\(=\)\((\)\(-13022680202683 \nu^{17} + 118335650003618 \nu^{16} - 44771164178744 \nu^{15} - 2504859808443810 \nu^{14} + 5858001428716411 \nu^{13} + 17617430837766009 \nu^{12} - 68767243551113023 \nu^{11} - 33470024699481755 \nu^{10} + 335301431568089064 \nu^{9} - 124869146258752505 \nu^{8} - 759407407344539158 \nu^{7} + 593603282669244732 \nu^{6} + 729456897339086163 \nu^{5} - 727242628798399153 \nu^{4} - 213089357105068366 \nu^{3} + 210304169983562996 \nu^{2} + 37001723229781356 \nu - 5195292962197536\)\()/ 886595457233576 \)
\(\beta_{8}\)\(=\)\((\)\(-10740292040922 \nu^{17} + 79117903472021 \nu^{16} + 92870769217390 \nu^{15} - 1850035038846512 \nu^{14} + 1667867302396844 \nu^{13} + 16086137984186553 \nu^{12} - 27196147917604641 \nu^{11} - 62934754460213401 \nu^{10} + 144258069159704345 \nu^{9} + 100056788353315650 \nu^{8} - 334405675482316269 \nu^{7} - 12326711344075138 \nu^{6} + 312468213116224454 \nu^{5} - 79163093196098791 \nu^{4} - 71491791608438813 \nu^{3} + 13826534439622850 \nu^{2} + 5554471852155572 \nu - 813298060986948\)\()/ 443297728616788 \)
\(\beta_{9}\)\(=\)\((\)\(33959676594129 \nu^{17} - 244226734403362 \nu^{16} - 339891834715816 \nu^{15} + 5819918185918742 \nu^{14} - 4245767604975569 \nu^{13} - 52276748204292455 \nu^{12} + 77888712930711393 \nu^{11} + 218104344572336437 \nu^{10} - 431291327425516436 \nu^{9} - 410563817675315645 \nu^{8} + 1050176863097033118 \nu^{7} + 240664778693316116 \nu^{6} - 1081581921979083169 \nu^{5} + 90985224423031575 \nu^{4} + 340304923142133006 \nu^{3} - 34519210394347964 \nu^{2} - 19663175760623644 \nu + 3721085582082608\)\()/ 886595457233576 \)
\(\beta_{10}\)\(=\)\((\)\(-25214595273041 \nu^{17} + 189190565282336 \nu^{16} + 190733503488708 \nu^{15} - 4362707258208572 \nu^{14} + 4551293117275079 \nu^{13} + 36959384933538211 \nu^{12} - 69373780767620221 \nu^{11} - 136233524368167603 \nu^{10} + 360617565778697220 \nu^{9} + 174131370845942773 \nu^{8} - 823035531340428794 \nu^{7} + 117468256302145770 \nu^{6} + 750442873636541309 \nu^{5} - 335379413156274269 \nu^{4} - 153579131812724200 \nu^{3} + 74616740535521018 \nu^{2} + 7869748119362996 \nu - 2555424121174580\)\()/ 443297728616788 \)
\(\beta_{11}\)\(=\)\((\)\(60700263486941 \nu^{17} - 478320509610002 \nu^{16} - 288622338255540 \nu^{15} + 10692032400216810 \nu^{14} - 14938440419646921 \nu^{13} - 85238794749149239 \nu^{12} + 201539076508820437 \nu^{11} + 268704783294678477 \nu^{10} - 1002553424780760044 \nu^{9} - 109607014231966189 \nu^{8} + 2192698808647176082 \nu^{7} - 981512517810791100 \nu^{6} - 1831423361688449061 \nu^{5} + 1411410964749415411 \nu^{4} + 220519163542968282 \nu^{3} - 260745438390303108 \nu^{2} - 23357849237344556 \nu + 3826534657429544\)\()/ 886595457233576 \)
\(\beta_{12}\)\(=\)\((\)\(-64223237008785 \nu^{17} + 539084204709562 \nu^{16} + 81357560610632 \nu^{15} - 11710397156422866 \nu^{14} + 21166954016967573 \nu^{13} + 87837555810305059 \nu^{12} - 261703414914380485 \nu^{11} - 227073674080421397 \nu^{10} + 1264964784404379456 \nu^{9} - 206997387858497275 \nu^{8} - 2711299217705068346 \nu^{7} + 1796551460295109904 \nu^{6} + 2182131551810517545 \nu^{5} - 2191000025769834679 \nu^{4} - 177198874843194846 \nu^{3} + 415845133191300240 \nu^{2} + 23424340673115172 \nu - 9963283153582664\)\()/ 886595457233576 \)
\(\beta_{13}\)\(=\)\((\)\(-37688647737868 \nu^{17} + 267971533358753 \nu^{16} + 388679230699790 \nu^{15} - 6358068531340576 \nu^{14} + 4330758126391390 \nu^{13} + 56693825410091331 \nu^{12} - 81374284776496999 \nu^{11} - 233289544590126007 \nu^{10} + 444417916482658529 \nu^{9} + 425485922166118924 \nu^{8} - 1042263821748804701 \nu^{7} - 220021774917612470 \nu^{6} + 975909258044316960 \nu^{5} - 111684278643633409 \nu^{4} - 207875405300109133 \nu^{3} + 6733463685372634 \nu^{2} + 1177219773180376 \nu - 1739958772516472\)\()/ 443297728616788 \)
\(\beta_{14}\)\(=\)\((\)\(20738042771486 \nu^{17} - 150148962527149 \nu^{16} - 193241956632582 \nu^{15} + 3518231938451895 \nu^{14} - 2860459078067425 \nu^{13} - 30701744481925376 \nu^{12} + 48866970211736716 \nu^{11} + 121031595408829947 \nu^{10} - 260172497518722414 \nu^{9} - 197112789215297770 \nu^{8} + 597183645669358292 \nu^{7} + 41320595955814197 \nu^{6} - 538072198616116060 \nu^{5} + 123850088209191205 \nu^{4} + 96432842921397039 \nu^{3} - 6433339283442403 \nu^{2} - 613174527785290 \nu + 150607842162690\)\()/ 221648864308394 \)
\(\beta_{15}\)\(=\)\((\)\(2295821306687 \nu^{17} - 17156001995758 \nu^{16} - 17887868681760 \nu^{15} + 397008940133822 \nu^{14} - 403834534801947 \nu^{13} - 3384225865140037 \nu^{12} + 6248449737443027 \nu^{11} + 12644424093024211 \nu^{10} - 32751398149840752 \nu^{9} - 17011627017926707 \nu^{8} + 75624975123556534 \nu^{7} - 8189006740723728 \nu^{6} - 70785359758478719 \nu^{5} + 28358086895742889 \nu^{4} + 16444614084075122 \nu^{3} - 6470040163960736 \nu^{2} - 1228312191468324 \nu + 240505445116088\)\()/ 20618499005432 \)
\(\beta_{16}\)\(=\)\((\)\(-83524578661994 \nu^{17} + 599721831065959 \nu^{16} + 816991773268750 \nu^{15} - 14153817184728346 \nu^{14} + 10718678082361394 \nu^{13} + 125010733622999949 \nu^{12} - 191483409046659589 \nu^{11} - 504321692370034015 \nu^{10} + 1040978183727296037 \nu^{9} + 870661205166077602 \nu^{8} - 2463306567788116773 \nu^{7} - 306122927946215072 \nu^{6} + 2390048740897147574 \nu^{5} - 446980437277724669 \nu^{4} - 625847651795991035 \nu^{3} + 90322232452229724 \nu^{2} + 41997132578678184 \nu - 4915948560145472\)\()/ 443297728616788 \)
\(\beta_{17}\)\(=\)\((\)\(-180069207323109 \nu^{17} + 1359427611894492 \nu^{16} + 1313595443605624 \nu^{15} - 31312823898472886 \nu^{14} + 33756602890160281 \nu^{13} + 264706668860550137 \nu^{12} - 507981876728244887 \nu^{11} - 971137358684465475 \nu^{10} + 2636145445743970938 \nu^{9} + 1221007760565960533 \nu^{8} - 6020469856393215568 \nu^{7} + 884958557799693236 \nu^{6} + 5492208908050632229 \nu^{5} - 2390650266780158997 \nu^{4} - 1111520133995208436 \nu^{3} + 481468355205050148 \nu^{2} + 62720763267435044 \nu - 17682230665574032\)\()/ 886595457233576 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{17} + \beta_{16} + 3 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 9 \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(-12 \beta_{17} + 2 \beta_{16} - 7 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} - 15 \beta_{9} + 2 \beta_{8} - 13 \beta_{7} - \beta_{6} - 10 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 63 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(-16 \beta_{17} + 15 \beta_{16} + 41 \beta_{15} - 7 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + 12 \beta_{11} + 13 \beta_{10} - 50 \beta_{9} + 16 \beta_{8} - 22 \beta_{7} - 3 \beta_{6} - \beta_{5} + 30 \beta_{4} + 28 \beta_{3} + 33 \beta_{2} + 80 \beta_{1} + 246\)
\(\nu^{7}\)\(=\)\(-127 \beta_{17} + 37 \beta_{16} - 29 \beta_{15} + 52 \beta_{14} + 57 \beta_{13} + 146 \beta_{12} + 90 \beta_{11} + 90 \beta_{10} - 193 \beta_{9} + 31 \beta_{8} - 148 \beta_{7} - 22 \beta_{6} - 87 \beta_{5} + 43 \beta_{4} + 43 \beta_{3} - 46 \beta_{2} + 508 \beta_{1} + 227\)
\(\nu^{8}\)\(=\)\(-222 \beta_{17} + 185 \beta_{16} + 456 \beta_{15} - 20 \beta_{14} + 238 \beta_{13} + 240 \beta_{12} + 129 \beta_{11} + 144 \beta_{10} - 663 \beta_{9} + 186 \beta_{8} - 326 \beta_{7} - 53 \beta_{6} - 18 \beta_{5} + 362 \beta_{4} + 321 \beta_{3} + 126 \beta_{2} + 719 \beta_{1} + 2115\)
\(\nu^{9}\)\(=\)\(-1349 \beta_{17} + 511 \beta_{16} + 86 \beta_{15} + 637 \beta_{14} + 824 \beta_{13} + 1582 \beta_{12} + 826 \beta_{11} + 814 \beta_{10} - 2344 \beta_{9} + 377 \beta_{8} - 1635 \beta_{7} - 297 \beta_{6} - 716 \beta_{5} + 668 \beta_{4} + 637 \beta_{3} - 741 \beta_{2} + 4180 \beta_{1} + 2618\)
\(\nu^{10}\)\(=\)\(-2892 \beta_{17} + 2146 \beta_{16} + 4803 \beta_{15} + 250 \beta_{14} + 3098 \beta_{13} + 3110 \beta_{12} + 1413 \beta_{11} + 1546 \beta_{10} - 8077 \beta_{9} + 1940 \beta_{8} - 4175 \beta_{7} - 661 \beta_{6} - 185 \beta_{5} + 4110 \beta_{4} + 3477 \beta_{3} - 436 \beta_{2} + 6511 \beta_{1} + 18881\)
\(\nu^{11}\)\(=\)\(-14580 \beta_{17} + 6321 \beta_{16} + 4094 \beta_{15} + 6863 \beta_{14} + 10740 \beta_{13} + 16972 \beta_{12} + 7912 \beta_{11} + 7552 \beta_{10} - 27508 \beta_{9} + 4182 \beta_{8} - 17889 \beta_{7} - 3321 \beta_{6} - 5633 \beta_{5} + 9031 \beta_{4} + 8116 \beta_{3} - 10320 \beta_{2} + 34945 \beta_{1} + 29208\)
\(\nu^{12}\)\(=\)\(-35985 \beta_{17} + 24210 \beta_{16} + 49797 \beta_{15} + 6043 \beta_{14} + 38481 \beta_{13} + 38185 \beta_{12} + 15877 \beta_{11} + 16448 \beta_{10} - 94249 \beta_{9} + 19291 \beta_{8} - 49876 \beta_{7} - 7169 \beta_{6} - 1155 \beta_{5} + 45651 \beta_{4} + 36934 \beta_{3} - 19306 \beta_{2} + 59197 \beta_{1} + 174104\)
\(\nu^{13}\)\(=\)\(-159475 \beta_{17} + 74086 \beta_{16} + 68613 \beta_{15} + 69730 \beta_{14} + 132398 \beta_{13} + 181738 \beta_{12} + 78951 \beta_{11} + 71986 \beta_{10} - 315341 \beta_{9} + 44109 \beta_{8} - 194856 \beta_{7} - 33746 \beta_{6} - 41832 \beta_{5} + 113140 \beta_{4} + 95878 \beta_{3} - 132962 \beta_{2} + 295770 \beta_{1} + 320459\)
\(\nu^{14}\)\(=\)\(-432410 \beta_{17} + 268942 \beta_{16} + 514908 \beta_{15} + 87030 \beta_{14} + 461917 \beta_{13} + 451976 \beta_{12} + 180676 \beta_{11} + 174265 \beta_{10} - 1072029 \beta_{9} + 187274 \beta_{8} - 573694 \beta_{7} - 72160 \beta_{6} + 1302 \beta_{5} + 502291 \beta_{4} + 389798 \beta_{3} - 329743 \beta_{2} + 538929 \beta_{1} + 1650712\)
\(\nu^{15}\)\(=\)\(-1753792 \beta_{17} + 842636 \beta_{16} + 926227 \beta_{15} + 688583 \beta_{14} + 1575225 \beta_{13} + 1947310 \beta_{12} + 813383 \beta_{11} + 702648 \beta_{10} - 3554459 \beta_{9} + 450659 \beta_{8} - 2116396 \beta_{7} - 323851 \beta_{6} - 281984 \beta_{5} + 1352851 \beta_{4} + 1086363 \beta_{3} - 1633101 \beta_{2} + 2526944 \beta_{1} + 3479739\)
\(\nu^{16}\)\(=\)\(-5060507 \beta_{17} + 2959032 \beta_{16} + 5336145 \beta_{15} + 1066828 \beta_{14} + 5404657 \beta_{13} + 5213807 \beta_{12} + 2059918 \beta_{11} + 1841870 \beta_{10} - 11992809 \beta_{9} + 1794652 \beta_{8} - 6451975 \beta_{7} - 692272 \beta_{6} + 185229 \beta_{5} + 5498809 \beta_{4} + 4108399 \beta_{3} - 4580523 \beta_{2} + 4902760 \beta_{1} + 16021039\)
\(\nu^{17}\)\(=\)\(-19305219 \beta_{17} + 9411007 \beta_{16} + 11390705 \beta_{15} + 6711080 \beta_{14} + 18284331 \beta_{13} + 20889069 \beta_{12} + 8568643 \beta_{11} + 6992451 \beta_{10} - 39568172 \beta_{9} + 4507699 \beta_{8} - 22935540 \beta_{7} - 2988949 \beta_{6} - 1535353 \beta_{5} + 15690584 \beta_{4} + 12014672 \beta_{3} - 19426758 \beta_{2} + 21736118 \beta_{1} + 37517387\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.80641
−2.57211
−2.15409
−1.74374
−1.35164
−0.397638
−0.267006
0.0880346
0.154866
0.839537
0.877348
1.33821
1.56216
2.54274
2.60297
2.97166
3.05303
3.26206
0 −2.80641 0 3.19831 0 4.58875 0 4.87591 0
1.2 0 −2.57211 0 −0.393688 0 −1.17366 0 3.61576 0
1.3 0 −2.15409 0 −0.228433 0 2.56317 0 1.64008 0
1.4 0 −1.74374 0 0.394783 0 −1.78199 0 0.0406123 0
1.5 0 −1.35164 0 1.29976 0 −4.65863 0 −1.17306 0
1.6 0 −0.397638 0 3.51710 0 2.09256 0 −2.84188 0
1.7 0 −0.267006 0 −2.18930 0 1.68463 0 −2.92871 0
1.8 0 0.0880346 0 −2.20461 0 0.606938 0 −2.99225 0
1.9 0 0.154866 0 −1.21070 0 −0.213015 0 −2.97602 0
1.10 0 0.839537 0 −4.04363 0 −2.98927 0 −2.29518 0
1.11 0 0.877348 0 −1.13091 0 −4.50700 0 −2.23026 0
1.12 0 1.33821 0 3.45044 0 −1.09754 0 −1.20921 0
1.13 0 1.56216 0 −0.592812 0 4.71818 0 −0.559654 0
1.14 0 2.54274 0 3.21926 0 1.46547 0 3.46555 0
1.15 0 2.60297 0 0.183919 0 3.24534 0 3.77545 0
1.16 0 2.97166 0 −4.20066 0 0.274921 0 5.83079 0
1.17 0 3.05303 0 2.19982 0 −4.80385 0 6.32102 0
1.18 0 3.26206 0 2.73134 0 1.98500 0 7.64104 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4012.2.a.i 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.i 18 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\(T_{3}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)