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 $1$

Related objects

Downloads

Learn more about

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:

Atkin-Lehner signs

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

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

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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{14} \)
$3$ \( ( 1 + T )^{14} \)
$5$ \( -1492 + 506 T + 10504 T^{2} - 2997 T^{3} - 22759 T^{4} + 4126 T^{5} + 18441 T^{6} - 2268 T^{7} - 5438 T^{8} + 451 T^{9} + 722 T^{10} - 36 T^{11} - 44 T^{12} + T^{13} + T^{14} \)
$7$ \( -6624 - 35664 T + 13691 T^{2} + 91773 T^{3} - 14049 T^{4} - 76528 T^{5} + 11179 T^{6} + 23945 T^{7} - 3857 T^{8} - 3377 T^{9} + 596 T^{10} + 216 T^{11} - 41 T^{12} - 5 T^{13} + T^{14} \)
$11$ \( 1214688 + 3400368 T - 1712584 T^{2} - 5631736 T^{3} - 1309666 T^{4} + 1603454 T^{5} + 576657 T^{6} - 168983 T^{7} - 74570 T^{8} + 7787 T^{9} + 4206 T^{10} - 152 T^{11} - 107 T^{12} + T^{13} + T^{14} \)
$13$ \( ( 1 + T )^{14} \)
$17$ \( -19899904 + 36525952 T + 22063840 T^{2} - 45698312 T^{3} + 2868356 T^{4} + 10790934 T^{5} - 895595 T^{6} - 1163983 T^{7} + 36603 T^{8} + 62791 T^{9} + 1674 T^{10} - 1473 T^{11} - 92 T^{12} + 12 T^{13} + T^{14} \)
$19$ \( 558905344 + 32804992 T - 797678016 T^{2} + 306966800 T^{3} + 93855336 T^{4} - 52938412 T^{5} - 2659364 T^{6} + 3483973 T^{7} - 99890 T^{8} - 110182 T^{9} + 7239 T^{10} + 1684 T^{11} - 146 T^{12} - 10 T^{13} + T^{14} \)
$23$ \( 43362048 + 32937648 T - 64827532 T^{2} - 75857116 T^{3} - 9307685 T^{4} + 14339739 T^{5} + 3976973 T^{6} - 892666 T^{7} - 332000 T^{8} + 23925 T^{9} + 11427 T^{10} - 275 T^{11} - 176 T^{12} + T^{13} + T^{14} \)
$29$ \( -204714696 + 353641740 T + 502617454 T^{2} + 29970973 T^{3} - 106757683 T^{4} - 15202724 T^{5} + 9834373 T^{6} + 1408449 T^{7} - 487814 T^{8} - 54867 T^{9} + 13349 T^{10} + 950 T^{11} - 185 T^{12} - 6 T^{13} + T^{14} \)
$31$ \( 38404096 - 151546080 T + 196832560 T^{2} - 92856264 T^{3} - 7664616 T^{4} + 21485738 T^{5} - 5527192 T^{6} - 659685 T^{7} + 467890 T^{8} - 36228 T^{9} - 10911 T^{10} + 1832 T^{11} + 26 T^{12} - 20 T^{13} + T^{14} \)
$37$ \( -1146304 - 5918944 T + 7631440 T^{2} + 13743888 T^{3} - 12797008 T^{4} - 3396698 T^{5} + 3857950 T^{6} + 161027 T^{7} - 384781 T^{8} + 10229 T^{9} + 13483 T^{10} - 398 T^{11} - 197 T^{12} + 3 T^{13} + T^{14} \)
$41$ \( -2067248 - 14150352 T - 34689472 T^{2} - 35059612 T^{3} - 9166917 T^{4} + 6811232 T^{5} + 3547198 T^{6} - 283773 T^{7} - 312044 T^{8} + 308 T^{9} + 11536 T^{10} + 102 T^{11} - 182 T^{12} - T^{13} + T^{14} \)
$43$ \( 52664932864 - 20059452672 T - 9504277600 T^{2} + 4563979232 T^{3} + 445978992 T^{4} - 379743116 T^{5} + 7739706 T^{6} + 14439259 T^{7} - 1082707 T^{8} - 261278 T^{9} + 28056 T^{10} + 2120 T^{11} - 284 T^{12} - 6 T^{13} + T^{14} \)
$47$ \( -7354606408 - 367380330 T + 7486473692 T^{2} - 2177615159 T^{3} - 833674792 T^{4} + 264244213 T^{5} + 42306206 T^{6} - 12379777 T^{7} - 1266335 T^{8} + 281399 T^{9} + 23352 T^{10} - 3090 T^{11} - 239 T^{12} + 13 T^{13} + T^{14} \)
$53$ \( -231076800 + 2232478080 T - 365383648 T^{2} - 1358126520 T^{3} + 405622416 T^{4} + 145956390 T^{5} - 42137041 T^{6} - 8053511 T^{7} + 1605283 T^{8} + 264466 T^{9} - 22732 T^{10} - 4441 T^{11} + 28 T^{12} + 27 T^{13} + T^{14} \)
$59$ \( -3946534656 + 10599423888 T - 9912094744 T^{2} + 3712320467 T^{3} - 136716388 T^{4} - 271638135 T^{5} + 52312535 T^{6} + 5573914 T^{7} - 2100987 T^{8} - 9952 T^{9} + 36333 T^{10} - 787 T^{11} - 301 T^{12} + 6 T^{13} + T^{14} \)
$61$ \( 185165824 + 773156864 T - 1738042880 T^{2} - 1313450624 T^{3} + 446733752 T^{4} + 413638160 T^{5} + 33705510 T^{6} - 18077191 T^{7} - 2431753 T^{8} + 287200 T^{9} + 47548 T^{10} - 1858 T^{11} - 370 T^{12} + 4 T^{13} + T^{14} \)
$67$ \( 796056026 - 1161045096 T - 527849323 T^{2} + 542376432 T^{3} + 103471082 T^{4} - 90330264 T^{5} - 6856305 T^{6} + 6942646 T^{7} + 37142 T^{8} - 251926 T^{9} + 10109 T^{10} + 3779 T^{11} - 258 T^{12} - 13 T^{13} + T^{14} \)
$71$ \( 992054400 + 2472216960 T + 166081486 T^{2} - 3614888566 T^{3} - 2736648833 T^{4} - 372845561 T^{5} + 139904206 T^{6} + 26774785 T^{7} - 2919799 T^{8} - 572696 T^{9} + 36176 T^{10} + 5244 T^{11} - 281 T^{12} - 18 T^{13} + T^{14} \)
$73$ \( 48509712950 - 10621597860 T - 18672539877 T^{2} + 4768337124 T^{3} + 1772596711 T^{4} - 542117132 T^{5} - 56824488 T^{6} + 25221011 T^{7} + 142698 T^{8} - 531368 T^{9} + 22500 T^{10} + 4694 T^{11} - 329 T^{12} - 11 T^{13} + T^{14} \)
$79$ \( -41343457536 - 56280219840 T + 181082689568 T^{2} - 86440842176 T^{3} + 5794989336 T^{4} + 4185462776 T^{5} - 788933940 T^{6} - 30310401 T^{7} + 17557455 T^{8} - 851577 T^{9} - 116629 T^{10} + 11936 T^{11} - 17 T^{12} - 33 T^{13} + T^{14} \)
$83$ \( -246609101456 - 515986047096 T - 296773270322 T^{2} - 14851764363 T^{3} + 26388152020 T^{4} + 3884828875 T^{5} - 712209520 T^{6} - 140823794 T^{7} + 6457482 T^{8} + 2013386 T^{9} + 7934 T^{10} - 12138 T^{11} - 336 T^{12} + 25 T^{13} + T^{14} \)
$89$ \( 244019584 - 868816224 T - 807976592 T^{2} + 2339975064 T^{3} - 442497544 T^{4} - 732271042 T^{5} + 238236086 T^{6} + 20095729 T^{7} - 9663745 T^{8} - 268819 T^{9} + 124162 T^{10} + 1623 T^{11} - 617 T^{12} - 3 T^{13} + T^{14} \)
$97$ \( -175600576384 - 139489340736 T + 60177205760 T^{2} + 55029574032 T^{3} - 6520778120 T^{4} - 5993912140 T^{5} + 519932060 T^{6} + 178736743 T^{7} - 15344419 T^{8} - 1982070 T^{9} + 173973 T^{10} + 8182 T^{11} - 734 T^{12} - 11 T^{13} + T^{14} \)
show more
show less