Properties

Label 8034.2.a.ba
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 14
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-266306853678383 \nu^{13} + 134001969718133 \nu^{12} + 11730896555981242 \nu^{11} - 3739404291058554 \nu^{10} - 191585597830484336 \nu^{9} + 24886922325561149 \nu^{8} + 1414782771710524552 \nu^{7} + 84321828569428448 \nu^{6} - 4492827496115707927 \nu^{5} - 915363310465441330 \nu^{4} + 4255301279762091841 \nu^{3} + 422048253089054251 \nu^{2} - 1047631892432145166 \nu + 11371595281753640\)\()/ 43692742328669662 \)
\(\beta_{3}\)\(=\)\((\)\(-276561632426969 \nu^{13} + 421045515071387 \nu^{12} + 11831212064366610 \nu^{11} - 15979394388187493 \nu^{10} - 186676696579320143 \nu^{9} + 217017383487373922 \nu^{8} + 1324468395061602794 \nu^{7} - 1262761014021014640 \nu^{6} - 4033414366634615670 \nu^{5} + 3051387240952734794 \nu^{4} + 3669499430064327162 \nu^{3} - 2701606611286975207 \nu^{2} - 780973362285615831 \nu + 588284015133280532\)\()/ 21846371164334831 \)
\(\beta_{4}\)\(=\)\((\)\(281008689100883 \nu^{13} - 281957981628135 \nu^{12} - 12211701038963858 \nu^{11} + 10147274836518033 \nu^{10} + 196191030805468054 \nu^{9} - 127532267940051610 \nu^{8} - 1419834757376023830 \nu^{7} + 652528796048380961 \nu^{6} + 4397320407290358181 \nu^{5} - 1303030165431025011 \nu^{4} - 3995402229552693528 \nu^{3} + 1346052708231791049 \nu^{2} + 908907497419418757 \nu - 376805384643590757\)\()/ 21846371164334831 \)
\(\beta_{5}\)\(=\)\((\)\(563276920787713 \nu^{13} - 153049032331183 \nu^{12} - 24789913096510800 \nu^{11} + 2179601153394382 \nu^{10} + 403199471683301528 \nu^{9} + 41321895819192391 \nu^{8} - 2942272434616681480 \nu^{7} - 888965469465066868 \nu^{6} + 9011134325268674427 \nu^{5} + 4368493534082155588 \nu^{4} - 7216360830653724145 \nu^{3} - 3783713813055134269 \nu^{2} + 1120836385553257082 \nu + 727260622566264690\)\()/ 43692742328669662 \)
\(\beta_{6}\)\(=\)\((\)\(587894636532799 \nu^{13} - 401515966880495 \nu^{12} - 25559574153488160 \nu^{11} + 12469830127322176 \nu^{10} + 410367492349937474 \nu^{9} - 114850747220457015 \nu^{8} - 2960764032107237352 \nu^{7} + 169914683015283554 \nu^{6} + 9072106839066945823 \nu^{5} + 1281274510944306808 \nu^{4} - 7818651825966159249 \nu^{3} - 1046735607854569041 \nu^{2} + 1635447715064563388 \nu + 69082076903089780\)\()/ 43692742328669662 \)
\(\beta_{7}\)\(=\)\((\)\(600547475775077 \nu^{13} + 160676916331065 \nu^{12} - 26573482465291666 \nu^{11} - 12031687706537128 \nu^{10} + 434206046573977704 \nu^{9} + 279533615334483141 \nu^{8} - 3175735116060119102 \nu^{7} - 2683158768977673228 \nu^{6} + 9656661921220973037 \nu^{5} + 10041227588618067172 \nu^{4} - 7220568332423014109 \nu^{3} - 8543202845942904737 \nu^{2} + 983162040010572610 \nu + 1584010040979027426\)\()/ 43692742328669662 \)
\(\beta_{8}\)\(=\)\((\)\(323027709062960 \nu^{13} - 172381882120881 \nu^{12} - 14134878235621520 \nu^{11} + 4846129448665246 \nu^{10} + 228696579415282775 \nu^{9} - 32681850003000513 \nu^{8} - 1664502571125875459 \nu^{7} - 111735197101452200 \nu^{6} + 5139130808556126422 \nu^{5} + 1294647822855345817 \nu^{4} - 4402465041131927787 \nu^{3} - 1059281835457961059 \nu^{2} + 828635442318437489 \nu + 87259893240249591\)\()/ 21846371164334831 \)
\(\beta_{9}\)\(=\)\((\)\(-682482407227411 \nu^{13} - 152764996823 \nu^{12} + 30122063309684406 \nu^{11} + 5643504601298036 \nu^{10} - 491662813166917868 \nu^{9} - 187136585825562229 \nu^{8} + 3607458160860328818 \nu^{7} + 2101336739470417196 \nu^{6} - 11175426298123444609 \nu^{5} - 8583170474917069942 \nu^{4} + 9392486941861559115 \nu^{3} + 7853101744503182057 \nu^{2} - 1800196355230650124 \nu - 1698473816153287560\)\()/ 43692742328669662 \)
\(\beta_{10}\)\(=\)\((\)\(352781396910762 \nu^{13} - 166497424213521 \nu^{12} - 15413816930355473 \nu^{11} + 4333005041742547 \nu^{10} + 248673668063896890 \nu^{9} - 20208645124353222 \nu^{8} - 1799297386675220078 \nu^{7} - 231617478407447300 \nu^{6} + 5476590592887655990 \nu^{5} + 1713977797830856848 \nu^{4} - 4440659955237449610 \nu^{3} - 1259504021300032236 \nu^{2} + 849290650220779131 \nu + 91607361939384532\)\()/ 21846371164334831 \)
\(\beta_{11}\)\(=\)\((\)\(756784517059657 \nu^{13} - 144783307200521 \nu^{12} - 33253610912031224 \nu^{11} - 62661878222990 \nu^{10} + 540215049753776070 \nu^{9} + 110071620789847963 \nu^{8} - 3944695232805112230 \nu^{7} - 1643994001829427568 \nu^{6} + 12173804052853690563 \nu^{5} + 7459946733255625440 \nu^{4} - 10254865555632034835 \nu^{3} - 6870285231682939387 \nu^{2} + 2137547859976077336 \nu + 1424281961283445274\)\()/ 43692742328669662 \)
\(\beta_{12}\)\(=\)\((\)\(-461379618788969 \nu^{13} + 101629522210970 \nu^{12} + 20312989919152389 \nu^{11} - 683299742724623 \nu^{10} - 330994820970097058 \nu^{9} - 52645645511792452 \nu^{8} + 2429698868506903576 \nu^{7} + 870810312435771415 \nu^{6} - 7583560590369499978 \nu^{5} - 4027119627133068275 \nu^{4} + 6656963138830352527 \nu^{3} + 3488559562581261272 \nu^{2} - 1371444613064346176 \nu - 641231492639617501\)\()/ 21846371164334831 \)
\(\beta_{13}\)\(=\)\((\)\(1399081209291127 \nu^{13} - 773564887767339 \nu^{12} - 61270437020665254 \nu^{11} + 22466875749627886 \nu^{10} + 992798322812597212 \nu^{9} - 170706210676448595 \nu^{8} - 7249462015732823470 \nu^{7} - 234855379984642544 \nu^{6} + 22595322669123401215 \nu^{5} + 4786003452335976276 \nu^{4} - 20309684730415768281 \nu^{3} - 4247182212949563787 \nu^{2} + 4702972689934548944 \nu + 812680240968073994\)\()/ 43692742328669662 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{13} + \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 7\)
\(\nu^{3}\)\(=\)\(-\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + 12 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-16 \beta_{13} + 2 \beta_{12} + 18 \beta_{11} - 3 \beta_{10} - \beta_{9} + 14 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 17 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} + 82\)
\(\nu^{5}\)\(=\)\(-23 \beta_{13} - 18 \beta_{12} - 10 \beta_{11} + 6 \beta_{10} - 19 \beta_{9} - 4 \beta_{8} - 28 \beta_{7} + 23 \beta_{6} + 21 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 159 \beta_{1} + 25\)
\(\nu^{6}\)\(=\)\(-252 \beta_{13} + 35 \beta_{12} + 292 \beta_{11} - 69 \beta_{10} - 15 \beta_{9} + 186 \beta_{8} - 181 \beta_{7} + 190 \beta_{6} + 12 \beta_{5} + 268 \beta_{4} + 178 \beta_{3} - 76 \beta_{2} + 90 \beta_{1} + 1070\)
\(\nu^{7}\)\(=\)\(-454 \beta_{13} - 291 \beta_{12} - 23 \beta_{11} + 114 \beta_{10} - 290 \beta_{9} - 91 \beta_{8} - 537 \beta_{7} + 413 \beta_{6} + 379 \beta_{5} + 116 \beta_{4} + 123 \beta_{3} - 126 \beta_{2} + 2248 \beta_{1} + 493\)
\(\nu^{8}\)\(=\)\(-3992 \beta_{13} + 446 \beta_{12} + 4637 \beta_{11} - 1265 \beta_{10} - 143 \beta_{9} + 2477 \beta_{8} - 2713 \beta_{7} + 3001 \beta_{6} + 451 \beta_{5} + 4163 \beta_{4} + 2583 \beta_{3} - 1435 \beta_{2} + 2047 \beta_{1} + 14763\)
\(\nu^{9}\)\(=\)\(-8449 \beta_{13} - 4639 \beta_{12} + 1814 \beta_{11} + 1546 \beta_{10} - 4099 \beta_{9} - 1550 \beta_{8} - 9328 \beta_{7} + 7012 \beta_{6} + 6459 \beta_{5} + 3029 \beta_{4} + 2427 \beta_{3} - 2861 \beta_{2} + 33152 \beta_{1} + 9432\)
\(\nu^{10}\)\(=\)\(-63639 \beta_{13} + 4544 \beta_{12} + 73258 \beta_{11} - 21545 \beta_{10} - 620 \beta_{9} + 33285 \beta_{8} - 42376 \beta_{7} + 48823 \beta_{6} + 11248 \beta_{5} + 64765 \beta_{4} + 38846 \beta_{3} - 24636 \beta_{2} + 41751 \beta_{1} + 211587\)
\(\nu^{11}\)\(=\)\(-152268 \beta_{13} - 73921 \beta_{12} + 60922 \beta_{11} + 17182 \beta_{10} - 56254 \beta_{9} - 23521 \beta_{8} - 156735 \beta_{7} + 117792 \beta_{6} + 107248 \beta_{5} + 67121 \beta_{4} + 45133 \beta_{3} - 56731 \beta_{2} + 502445 \beta_{1} + 179584\)
\(\nu^{12}\)\(=\)\(-1020459 \beta_{13} + 30075 \beta_{12} + 1158028 \beta_{11} - 355420 \beta_{10} + 12256 \beta_{9} + 452268 \beta_{8} - 676595 \beta_{7} + 800959 \beta_{6} + 236684 \beta_{5} + 1013938 \beta_{4} + 596937 \beta_{3} - 407361 \beta_{2} + 802077 \beta_{1} + 3117311\)
\(\nu^{13}\)\(=\)\(-2688292 \beta_{13} - 1179924 \beta_{12} + 1428695 \beta_{11} + 142999 \beta_{10} - 765007 \beta_{9} - 332940 \beta_{8} - 2603136 \beta_{7} + 1981261 \beta_{6} + 1759863 \beta_{5} + 1361075 \beta_{4} + 820748 \beta_{3} - 1049738 \beta_{2} + 7757649 \beta_{1} + 3383769\)

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
4.07159
3.43788
3.09495
2.71343
0.924862
0.891518
0.503883
−0.522326
−0.643572
−1.05309
−2.46518
−2.56958
−3.59906
−3.78531
−1.00000 −1.00000 1.00000 −4.07159 1.00000 2.58120 −1.00000 1.00000 4.07159
1.2 −1.00000 −1.00000 1.00000 −3.43788 1.00000 −0.875663 −1.00000 1.00000 3.43788
1.3 −1.00000 −1.00000 1.00000 −3.09495 1.00000 3.97310 −1.00000 1.00000 3.09495
1.4 −1.00000 −1.00000 1.00000 −2.71343 1.00000 −2.34175 −1.00000 1.00000 2.71343
1.5 −1.00000 −1.00000 1.00000 −0.924862 1.00000 −4.24556 −1.00000 1.00000 0.924862
1.6 −1.00000 −1.00000 1.00000 −0.891518 1.00000 4.07845 −1.00000 1.00000 0.891518
1.7 −1.00000 −1.00000 1.00000 −0.503883 1.00000 2.75905 −1.00000 1.00000 0.503883
1.8 −1.00000 −1.00000 1.00000 0.522326 1.00000 −2.74868 −1.00000 1.00000 −0.522326
1.9 −1.00000 −1.00000 1.00000 0.643572 1.00000 1.17705 −1.00000 1.00000 −0.643572
1.10 −1.00000 −1.00000 1.00000 1.05309 1.00000 −0.189425 −1.00000 1.00000 −1.05309
1.11 −1.00000 −1.00000 1.00000 2.46518 1.00000 −0.952790 −1.00000 1.00000 −2.46518
1.12 −1.00000 −1.00000 1.00000 2.56958 1.00000 3.96287 −1.00000 1.00000 −2.56958
1.13 −1.00000 −1.00000 1.00000 3.59906 1.00000 0.919823 −1.00000 1.00000 −3.59906
1.14 −1.00000 −1.00000 1.00000 3.78531 1.00000 −3.09768 −1.00000 1.00000 −3.78531
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 8034.2.a.ba 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.ba 14 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(-1\)

Hecke kernels

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

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{14} \)
$3$ \( ( 1 + T )^{14} \)
$5$ \( 1 + T + 26 T^{2} + 29 T^{3} + 357 T^{4} + 421 T^{5} + 3562 T^{6} + 4277 T^{7} + 28796 T^{8} + 35021 T^{9} + 197621 T^{10} + 241828 T^{11} + 1189074 T^{12} + 1418301 T^{13} + 6329698 T^{14} + 7091505 T^{15} + 29726850 T^{16} + 30228500 T^{17} + 123513125 T^{18} + 109440625 T^{19} + 449937500 T^{20} + 334140625 T^{21} + 1391406250 T^{22} + 822265625 T^{23} + 3486328125 T^{24} + 1416015625 T^{25} + 6347656250 T^{26} + 1220703125 T^{27} + 6103515625 T^{28} \)
$7$ \( 1 - 5 T + 57 T^{2} - 239 T^{3} + 1611 T^{4} - 5855 T^{5} + 30121 T^{6} - 97176 T^{7} + 418908 T^{8} - 1219306 T^{9} + 4614344 T^{10} - 12255191 T^{11} + 41803831 T^{12} - 102002004 T^{13} + 318158398 T^{14} - 714014028 T^{15} + 2048387719 T^{16} - 4203530513 T^{17} + 11079039944 T^{18} - 20492875942 T^{19} + 49284107292 T^{20} - 80028614568 T^{21} + 173641570921 T^{22} - 236270368985 T^{23} + 455067626139 T^{24} - 472581091577 T^{25} + 788953370457 T^{26} - 484445052035 T^{27} + 678223072849 T^{28} \)
$11$ \( 1 + T + 47 T^{2} - 9 T^{3} + 1093 T^{4} - 1167 T^{5} + 18072 T^{6} - 28964 T^{7} + 240068 T^{8} - 401230 T^{9} + 2852393 T^{10} - 3329744 T^{11} + 32846446 T^{12} - 18794541 T^{13} + 367939200 T^{14} - 206739951 T^{15} + 3974419966 T^{16} - 4431889264 T^{17} + 41761885913 T^{18} - 64618492730 T^{19} + 425295106148 T^{20} - 564426420844 T^{21} + 3873893697432 T^{22} - 2751724955397 T^{23} + 28349605088893 T^{24} - 2567805035499 T^{25} + 147506133705887 T^{26} + 34522712143931 T^{27} + 379749833583241 T^{28} \)
$13$ \( ( 1 + T )^{14} \)
$17$ \( 1 + 12 T + 146 T^{2} + 1179 T^{3} + 9205 T^{4} + 57844 T^{5} + 354707 T^{6} + 1891121 T^{7} + 10018184 T^{8} + 48084616 T^{9} + 233639956 T^{10} + 1050210241 T^{11} + 4794849506 T^{12} + 20320486587 T^{13} + 87060872590 T^{14} + 345448271979 T^{15} + 1385711507234 T^{16} + 5159682914033 T^{17} + 19513842765076 T^{18} + 68273278619912 T^{19} + 241814607554696 T^{20} + 776000081622433 T^{21} + 2474349994624787 T^{22} + 6859597128092468 T^{23} + 18557223853633045 T^{24} + 40406565746699307 T^{25} + 85062846635545106 T^{26} + 118854936394871244 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 10 T + 120 T^{2} - 786 T^{3} + 6802 T^{4} - 39806 T^{5} + 293600 T^{6} - 1538069 T^{7} + 9895152 T^{8} - 47448685 T^{9} + 278127368 T^{10} - 1237611429 T^{11} + 6641904965 T^{12} - 27317844823 T^{13} + 135542121088 T^{14} - 519039051637 T^{15} + 2397727692365 T^{16} - 8488776791511 T^{17} + 36245836725128 T^{18} - 117487641479815 T^{19} + 465526143468912 T^{20} - 1374836411731991 T^{21} + 4986374108837600 T^{22} - 12844906497790874 T^{23} + 41703512685562402 T^{24} - 91561343494000134 T^{25} + 265597790287939320 T^{26} - 420529834622570590 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 + T + 146 T^{2} + 24 T^{3} + 10990 T^{4} - 4388 T^{5} + 580134 T^{6} - 461554 T^{7} + 23923309 T^{8} - 25743097 T^{9} + 811618435 T^{10} - 1003391134 T^{11} + 23330385884 T^{12} - 29594442892 T^{13} + 577062042746 T^{14} - 680672186516 T^{15} + 12341774132636 T^{16} - 12208259927378 T^{17} + 227124114468835 T^{18} - 165691402174271 T^{19} + 3541508315636701 T^{20} - 1571510804364638 T^{21} + 45430865135007654 T^{22} - 7903457878499644 T^{23} + 455277358238002510 T^{24} + 22867434189934248 T^{25} + 3199535167074966866 T^{26} + 504036361936467383 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 - 6 T + 221 T^{2} - 1312 T^{3} + 25500 T^{4} - 145405 T^{5} + 1992382 T^{6} - 10821112 T^{7} + 117210411 T^{8} - 601700609 T^{9} + 5479693190 T^{10} - 26359011388 T^{11} + 209605421418 T^{12} - 934182931232 T^{13} + 6656663786778 T^{14} - 27091305005728 T^{15} + 176278159412538 T^{16} - 642869928741932 T^{17} + 3875682879116390 T^{18} - 12341570844589741 T^{19} + 69719485926794931 T^{20} - 186662843525835608 T^{21} + 996681948748063102 T^{22} - 2109411560621231945 T^{23} + 10728034449155125500 T^{24} - 16007068812606047648 T^{25} + 78193067088408658061 T^{26} - 61563772277751613134 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 20 T + 460 T^{2} - 6228 T^{3} + 86212 T^{4} - 910676 T^{5} + 9578480 T^{6} - 84341457 T^{7} + 733512374 T^{8} - 5576136615 T^{9} + 41749585244 T^{10} - 279324702071 T^{11} + 1838783091861 T^{12} - 10933646049901 T^{13} + 63949041245744 T^{14} - 338943027546931 T^{15} + 1767070551278421 T^{16} - 8321362199397161 T^{17} + 38556618714124124 T^{18} - 159640057147463865 T^{19} + 650994931984048694 T^{20} - 2320453960000499727 T^{21} + 8169399744307869680 T^{22} - 24077929350791223596 T^{23} + 70661793877188815812 T^{24} - \)\(15\!\cdots\!68\)\( T^{25} + \)\(36\!\cdots\!60\)\( T^{26} - \)\(48\!\cdots\!20\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 + 3 T + 321 T^{2} + 1045 T^{3} + 50594 T^{4} + 168589 T^{5} + 5241883 T^{6} + 17060148 T^{7} + 401317130 T^{8} + 1236128166 T^{9} + 24116562479 T^{10} + 69122313326 T^{11} + 1177989745387 T^{12} + 3113686396783 T^{13} + 47707103466890 T^{14} + 115206396680971 T^{15} + 1612667961434803 T^{16} + 3501252536901878 T^{17} + 45198320852205119 T^{18} + 85718018389592862 T^{19} + 1029669958825086170 T^{20} + 1619551873806795684 T^{21} + 18412006337357773243 T^{22} + 21910119750312236353 T^{23} + \)\(24\!\cdots\!06\)\( T^{24} + \)\(18\!\cdots\!85\)\( T^{25} + \)\(21\!\cdots\!01\)\( T^{26} + \)\(73\!\cdots\!91\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 - T + 392 T^{2} - 431 T^{3} + 74963 T^{4} - 84808 T^{5} + 9312788 T^{6} - 10451117 T^{7} + 842827407 T^{8} - 916471376 T^{9} + 58956181431 T^{10} - 60864919040 T^{11} + 3293709367305 T^{12} - 3156265964107 T^{13} + 149454356867042 T^{14} - 129406904528387 T^{15} + 5536725446439705 T^{16} - 4194871085155840 T^{17} + 166596078196643991 T^{18} - 106178891948602576 T^{19} + 4003518040421733087 T^{20} - 2035399702580375077 T^{21} + 74361915854655299348 T^{22} - 27764607092083044488 T^{23} + \)\(10\!\cdots\!63\)\( T^{24} - \)\(23\!\cdots\!71\)\( T^{25} + \)\(88\!\cdots\!52\)\( T^{26} - \)\(92\!\cdots\!21\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 - 6 T + 318 T^{2} - 1234 T^{3} + 49771 T^{4} - 123850 T^{5} + 5264265 T^{6} - 7515939 T^{7} + 424310419 T^{8} - 280342639 T^{9} + 27762581682 T^{10} - 4792935707 T^{11} + 1520722018593 T^{12} + 78771451359 T^{13} + 70801844302318 T^{14} + 3387172408437 T^{15} + 2811815012378457 T^{16} - 381071939256449 T^{17} + 94914742017003282 T^{18} - 41212734865901077 T^{19} + 2682220203972307531 T^{20} - 2042972100144934473 T^{21} + 61529783634365228265 T^{22} - 62246094988378005550 T^{23} + \)\(10\!\cdots\!79\)\( T^{24} - \)\(11\!\cdots\!38\)\( T^{25} + \)\(12\!\cdots\!18\)\( T^{26} - \)\(10\!\cdots\!58\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 + 13 T + 419 T^{2} + 4853 T^{3} + 89575 T^{4} + 923795 T^{5} + 12655911 T^{6} + 117246364 T^{7} + 1313017147 T^{8} + 10991750001 T^{9} + 105566329709 T^{10} + 801112719728 T^{11} + 6777279818617 T^{12} + 46613745020802 T^{13} + 352861467105682 T^{14} + 2190846015977694 T^{15} + 14971011119324953 T^{16} + 83173925900320144 T^{17} + 515130013320742829 T^{18} + 2520902980921595007 T^{19} + 14153294558182246363 T^{20} + 59399718792620746532 T^{21} + \)\(30\!\cdots\!71\)\( T^{22} + \)\(10\!\cdots\!65\)\( T^{23} + \)\(47\!\cdots\!75\)\( T^{24} + \)\(11\!\cdots\!59\)\( T^{25} + \)\(48\!\cdots\!79\)\( T^{26} + \)\(70\!\cdots\!51\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 + 27 T + 770 T^{2} + 14162 T^{3} + 250695 T^{4} + 3591117 T^{5} + 48939583 T^{6} + 581612670 T^{7} + 6580518292 T^{8} + 67135844393 T^{9} + 653741569860 T^{10} + 5837714316066 T^{11} + 49860139272540 T^{12} + 393743995952773 T^{13} + 2977411680612566 T^{14} + 20868431785496969 T^{15} + 140057131216564860 T^{16} + 869101394232957882 T^{17} + 5158335435890502660 T^{18} + 28075907543901920749 T^{19} + \)\(14\!\cdots\!68\)\( T^{20} + \)\(68\!\cdots\!90\)\( T^{21} + \)\(30\!\cdots\!63\)\( T^{22} + \)\(11\!\cdots\!61\)\( T^{23} + \)\(43\!\cdots\!55\)\( T^{24} + \)\(13\!\cdots\!14\)\( T^{25} + \)\(37\!\cdots\!70\)\( T^{26} + \)\(70\!\cdots\!71\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 6 T + 525 T^{2} + 3815 T^{3} + 139996 T^{4} + 1108393 T^{5} + 24939893 T^{6} + 202044681 T^{7} + 3281310437 T^{8} + 26097232160 T^{9} + 334899389329 T^{10} + 2532993097652 T^{11} + 27215924996070 T^{12} + 190432727322865 T^{13} + 1784179620381802 T^{14} + 11235530912049035 T^{15} + 94738634911319670 T^{16} + 520223589402670108 T^{17} + 4058096799179040769 T^{18} + 18657545407828255840 T^{19} + \)\(13\!\cdots\!17\)\( T^{20} + \)\(50\!\cdots\!39\)\( T^{21} + \)\(36\!\cdots\!53\)\( T^{22} + \)\(96\!\cdots\!27\)\( T^{23} + \)\(71\!\cdots\!96\)\( T^{24} + \)\(11\!\cdots\!85\)\( T^{25} + \)\(93\!\cdots\!25\)\( T^{26} + \)\(62\!\cdots\!74\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 + 4 T + 484 T^{2} + 1314 T^{3} + 115319 T^{4} + 201434 T^{5} + 18326791 T^{6} + 19012883 T^{7} + 2192118347 T^{8} + 1180380893 T^{9} + 209538594860 T^{10} + 46690441853 T^{11} + 16545695495773 T^{12} + 1220607119195 T^{13} + 1098037338250994 T^{14} + 74457034270895 T^{15} + 61566532939771333 T^{16} + 10597843182235793 T^{17} + 2901238067794977260 T^{18} + 996945335998876793 T^{19} + \)\(11\!\cdots\!67\)\( T^{20} + 59752601840355458543 T^{21} + \)\(35\!\cdots\!71\)\( T^{22} + \)\(23\!\cdots\!94\)\( T^{23} + \)\(82\!\cdots\!19\)\( T^{24} + \)\(57\!\cdots\!54\)\( T^{25} + \)\(12\!\cdots\!64\)\( T^{26} + \)\(64\!\cdots\!24\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 - 13 T + 680 T^{2} - 7544 T^{3} + 211176 T^{4} - 2018649 T^{5} + 39849212 T^{6} - 330189361 T^{7} + 5155084593 T^{8} - 37314891084 T^{9} + 496316070080 T^{10} - 3199008307173 T^{11} + 38737076756099 T^{12} - 231279252876556 T^{13} + 2689074296203136 T^{14} - 15495709942729252 T^{15} + 173890737558128411 T^{16} - 962143335490272999 T^{17} + 10001325182426559680 T^{18} - 50379771317478845988 T^{19} + \)\(46\!\cdots\!17\)\( T^{20} - \)\(20\!\cdots\!03\)\( T^{21} + \)\(16\!\cdots\!92\)\( T^{22} - \)\(54\!\cdots\!03\)\( T^{23} + \)\(38\!\cdots\!24\)\( T^{24} - \)\(92\!\cdots\!52\)\( T^{25} + \)\(55\!\cdots\!80\)\( T^{26} - \)\(71\!\cdots\!31\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 - 18 T + 713 T^{2} - 11370 T^{3} + 255495 T^{4} - 3554696 T^{5} + 59554379 T^{6} - 727778567 T^{7} + 9998837755 T^{8} - 108358619322 T^{9} + 1275902745998 T^{10} - 12340757125686 T^{11} + 127482062456591 T^{12} - 1103461769654179 T^{13} + 10126899044108824 T^{14} - 78345785645446709 T^{15} + 642637076843675231 T^{16} - 4416892723611401946 T^{17} + 32422833568325202638 T^{18} - \)\(19\!\cdots\!22\)\( T^{19} + \)\(12\!\cdots\!55\)\( T^{20} - \)\(66\!\cdots\!97\)\( T^{21} + \)\(38\!\cdots\!19\)\( T^{22} - \)\(16\!\cdots\!76\)\( T^{23} + \)\(83\!\cdots\!95\)\( T^{24} - \)\(26\!\cdots\!70\)\( T^{25} + \)\(11\!\cdots\!33\)\( T^{26} - \)\(20\!\cdots\!98\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 11 T + 693 T^{2} - 5745 T^{3} + 219235 T^{4} - 1334368 T^{5} + 42455980 T^{6} - 171947317 T^{7} + 5691712425 T^{8} - 11648551898 T^{9} + 574015416074 T^{10} - 93417125842 T^{11} + 47527580757994 T^{12} + 53350058703113 T^{13} + 3559539737657822 T^{14} + 3894554285327249 T^{15} + 253274477859350026 T^{16} - 36340850043677314 T^{17} + 16301028123384725834 T^{18} - 24148282039330033514 T^{19} + \)\(86\!\cdots\!25\)\( T^{20} - \)\(18\!\cdots\!49\)\( T^{21} + \)\(34\!\cdots\!80\)\( T^{22} - \)\(78\!\cdots\!84\)\( T^{23} + \)\(94\!\cdots\!15\)\( T^{24} - \)\(18\!\cdots\!65\)\( T^{25} + \)\(15\!\cdots\!53\)\( T^{26} - \)\(18\!\cdots\!63\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 33 T + 1089 T^{2} - 21955 T^{3} + 435186 T^{4} - 6543527 T^{5} + 97884339 T^{6} - 1191992050 T^{7} + 14697771336 T^{8} - 151921536064 T^{9} + 1632154963579 T^{10} - 14939958274018 T^{11} + 147514063862501 T^{12} - 1259006663805439 T^{13} + 11989712661479842 T^{14} - 99461526440629681 T^{15} + 920635272565868741 T^{16} - 7365982087463560702 T^{17} + 63572568035954099899 T^{18} - \)\(46\!\cdots\!36\)\( T^{19} + \)\(35\!\cdots\!56\)\( T^{20} - \)\(22\!\cdots\!50\)\( T^{21} + \)\(14\!\cdots\!79\)\( T^{22} - \)\(78\!\cdots\!13\)\( T^{23} + \)\(41\!\cdots\!86\)\( T^{24} - \)\(16\!\cdots\!45\)\( T^{25} + \)\(64\!\cdots\!49\)\( T^{26} - \)\(15\!\cdots\!87\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 + 25 T + 826 T^{2} + 14837 T^{3} + 300177 T^{4} + 4364942 T^{5} + 68402706 T^{6} + 852425088 T^{7} + 11274422479 T^{8} + 124553623990 T^{9} + 1454382814530 T^{10} + 14570091380449 T^{11} + 154285125617853 T^{12} + 1421274293530685 T^{13} + 13852164490995784 T^{14} + 117965766363046855 T^{15} + 1062870230381389317 T^{16} + 8330988840152792363 T^{17} + 69022566468848204130 T^{18} + \)\(49\!\cdots\!70\)\( T^{19} + \)\(36\!\cdots\!51\)\( T^{20} + \)\(23\!\cdots\!76\)\( T^{21} + \)\(15\!\cdots\!46\)\( T^{22} + \)\(81\!\cdots\!26\)\( T^{23} + \)\(46\!\cdots\!73\)\( T^{24} + \)\(19\!\cdots\!79\)\( T^{25} + \)\(88\!\cdots\!86\)\( T^{26} + \)\(22\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 - 3 T + 629 T^{2} - 1848 T^{3} + 186017 T^{4} - 533416 T^{5} + 34890189 T^{6} - 93023627 T^{7} + 4727064917 T^{8} - 10662916229 T^{9} + 503981695563 T^{10} - 855474831760 T^{11} + 46288243965721 T^{12} - 58080317869029 T^{13} + 4082640989018310 T^{14} - 5169148290343581 T^{15} + 366649180452476041 T^{16} - 603083236671015440 T^{17} + 31620941002602376683 T^{18} - 59542358122442897821 T^{19} + \)\(23\!\cdots\!37\)\( T^{20} - \)\(41\!\cdots\!83\)\( T^{21} + \)\(13\!\cdots\!09\)\( T^{22} - \)\(18\!\cdots\!44\)\( T^{23} + \)\(58\!\cdots\!17\)\( T^{24} - \)\(51\!\cdots\!72\)\( T^{25} + \)\(15\!\cdots\!09\)\( T^{26} - \)\(65\!\cdots\!07\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 11 T + 624 T^{2} - 5689 T^{3} + 175816 T^{4} - 1324798 T^{5} + 29812767 T^{6} - 188735535 T^{7} + 3513075222 T^{8} - 20153207198 T^{9} + 333728514676 T^{10} - 2026678003970 T^{11} + 30373819549249 T^{12} - 210293383957295 T^{13} + 2887845488774970 T^{14} - 20398458243857615 T^{15} + 285787268138883841 T^{16} - 1849694293917311810 T^{17} + 29544745453464227956 T^{18} - \)\(17\!\cdots\!86\)\( T^{19} + \)\(29\!\cdots\!38\)\( T^{20} - \)\(15\!\cdots\!55\)\( T^{21} + \)\(23\!\cdots\!87\)\( T^{22} - \)\(10\!\cdots\!66\)\( T^{23} + \)\(12\!\cdots\!84\)\( T^{24} - \)\(40\!\cdots\!17\)\( T^{25} + \)\(43\!\cdots\!84\)\( T^{26} - \)\(74\!\cdots\!47\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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