Properties

Label 4005.2.a.x
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 17
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} - 2456 x^{9} - 7002 x^{8} + 6279 x^{7} + 7299 x^{6} - 7119 x^{5} - 3066 x^{4} + 3184 x^{3} + 99 x^{2} - 231 x + 24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(685 \nu^{16} - 1978 \nu^{15} - 23375 \nu^{14} + 80742 \nu^{13} + 247035 \nu^{12} - 1063752 \nu^{11} - 927300 \nu^{10} + 6222061 \nu^{9} + 79227 \nu^{8} - 17370991 \nu^{7} + 6231634 \nu^{6} + 22769593 \nu^{5} - 11572552 \nu^{4} - 11774626 \nu^{3} + 6374744 \nu^{2} + 982243 \nu - 368684\)\()/31022\)
\(\beta_{5}\)\(=\)\((\)\(-1070 \nu^{16} + 1165 \nu^{15} + 40249 \nu^{14} - 75174 \nu^{13} - 484380 \nu^{12} + 1137763 \nu^{11} + 2444697 \nu^{10} - 7082375 \nu^{9} - 4941450 \nu^{8} + 20342129 \nu^{7} + 1670236 \nu^{6} - 26893622 \nu^{5} + 5596139 \nu^{4} + 13977281 \nu^{3} - 4729743 \nu^{2} - 1384065 \nu + 300552\)\()/31022\)
\(\beta_{6}\)\(=\)\((\)\(6613 \nu^{16} - 17420 \nu^{15} - 180375 \nu^{14} + 520530 \nu^{13} + 1766251 \nu^{12} - 5814854 \nu^{11} - 7314342 \nu^{10} + 30806921 \nu^{9} + 9574065 \nu^{8} - 81028001 \nu^{7} + 12506146 \nu^{6} + 103349077 \nu^{5} - 39644890 \nu^{4} - 55867828 \nu^{3} + 25227292 \nu^{2} + 7528791 \nu - 1639268\)\()/62044\)
\(\beta_{7}\)\(=\)\((\)\(-7257 \nu^{16} + 30443 \nu^{15} + 137250 \nu^{14} - 693400 \nu^{13} - 837779 \nu^{12} + 6167935 \nu^{11} + 943861 \nu^{10} - 27194200 \nu^{9} + 8863387 \nu^{8} + 62567552 \nu^{7} - 34862246 \nu^{6} - 72621151 \nu^{5} + 46524231 \nu^{4} + 36780711 \nu^{3} - 21561069 \nu^{2} - 4918520 \nu + 1063920\)\()/62044\)
\(\beta_{8}\)\(=\)\((\)\(6850 \nu^{16} - 35291 \nu^{15} - 94151 \nu^{14} + 714354 \nu^{13} + 174722 \nu^{12} - 5503379 \nu^{11} + 2779047 \nu^{10} + 20604597 \nu^{9} - 16766182 \nu^{8} - 40485931 \nu^{7} + 37622828 \nu^{6} + 41160644 \nu^{5} - 37720701 \nu^{4} - 18925679 \nu^{3} + 14593081 \nu^{2} + 2423683 \nu - 739750\)\()/31022\)
\(\beta_{9}\)\(=\)\((\)\(-14426 \nu^{16} + 61515 \nu^{15} + 278177 \nu^{14} - 1417322 \nu^{13} - 1771986 \nu^{12} + 12781181 \nu^{11} + 2666987 \nu^{10} - 57266037 \nu^{9} + 14936458 \nu^{8} + 134174293 \nu^{7} - 65290780 \nu^{6} - 158712488 \nu^{5} + 91148769 \nu^{4} + 81480431 \nu^{3} - 44547125 \nu^{2} - 10787091 \nu + 2544040\)\()/62044\)
\(\beta_{10}\)\(=\)\((\)\(7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} - 6567131 \nu^{11} + 1851747 \nu^{10} + 26826658 \nu^{9} - 16686955 \nu^{8} - 57856922 \nu^{7} + 43854462 \nu^{6} + 63961259 \nu^{5} - 49324275 \nu^{4} - 30979503 \nu^{3} + 21184979 \nu^{2} + 3902278 \nu - 1263544\)\()/31022\)
\(\beta_{11}\)\(=\)\((\)\(7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} - 6567131 \nu^{11} + 1851747 \nu^{10} + 26826658 \nu^{9} - 16686955 \nu^{8} - 57856922 \nu^{7} + 43854462 \nu^{6} + 63961259 \nu^{5} - 49293253 \nu^{4} - 30979503 \nu^{3} + 20936803 \nu^{2} + 3871256 \nu - 1015368\)\()/31022\)
\(\beta_{12}\)\(=\)\((\)\(15238 \nu^{16} - 70517 \nu^{15} - 270335 \nu^{14} + 1582686 \nu^{13} + 1432154 \nu^{12} - 13900087 \nu^{11} + 185359 \nu^{10} + 60904047 \nu^{9} - 25130162 \nu^{8} - 141089899 \nu^{7} + 83217368 \nu^{6} + 167347268 \nu^{5} - 106903155 \nu^{4} - 87031309 \nu^{3} + 50859751 \nu^{2} + 11368909 \nu - 3029808\)\()/62044\)
\(\beta_{13}\)\(=\)\((\)\(-5003 \nu^{16} + 23640 \nu^{15} + 78789 \nu^{14} - 491641 \nu^{13} - 318258 \nu^{12} + 3931692 \nu^{11} - 700684 \nu^{10} - 15463187 \nu^{9} + 8035982 \nu^{8} + 32135471 \nu^{7} - 21250233 \nu^{6} - 34474069 \nu^{5} + 23881305 \nu^{4} + 16421556 \nu^{3} - 10374066 \nu^{2} - 1994666 \nu + 580367\)\()/15511\)
\(\beta_{14}\)\(=\)\((\)\(-12523 \nu^{16} + 61545 \nu^{15} + 189010 \nu^{14} - 1274666 \nu^{13} - 638709 \nu^{12} + 10147647 \nu^{11} - 2869597 \nu^{10} - 39770336 \nu^{9} + 23674113 \nu^{8} + 82744596 \nu^{7} - 58289788 \nu^{6} - 89735141 \nu^{5} + 62257615 \nu^{4} + 43991657 \nu^{3} - 25895951 \nu^{2} - 5969520 \nu + 1508748\)\()/31022\)
\(\beta_{15}\)\(=\)\((\)\(-16906 \nu^{16} + 80451 \nu^{15} + 276079 \nu^{14} - 1727532 \nu^{13} - 1200628 \nu^{12} + 14402921 \nu^{11} - 2192535 \nu^{10} - 59629455 \nu^{9} + 30378002 \nu^{8} + 130973989 \nu^{7} - 86449694 \nu^{6} - 148302258 \nu^{5} + 102577437 \nu^{4} + 74063549 \nu^{3} - 46121451 \nu^{2} - 9366361 \nu + 2614408\)\()/31022\)
\(\beta_{16}\)\(=\)\((\)\(-38615 \nu^{16} + 177511 \nu^{15} + 642124 \nu^{14} - 3780588 \nu^{13} - 3038785 \nu^{12} + 31183239 \nu^{11} - 2039975 \nu^{10} - 127378426 \nu^{9} + 54823177 \nu^{8} + 275564722 \nu^{7} - 161135054 \nu^{6} - 306983937 \nu^{5} + 190626927 \nu^{4} + 150613483 \nu^{3} - 84980589 \nu^{2} - 18623890 \nu + 4923384\)\()/62044\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{10} + 8 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{11} - \beta_{8} - \beta_{4} + 9 \beta_{3} + \beta_{2} + 30 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(11 \beta_{11} - 9 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + 57 \beta_{2} + 11 \beta_{1} + 99\)
\(\nu^{7}\)\(=\)\(-\beta_{16} + \beta_{14} + \beta_{13} + 14 \beta_{11} - \beta_{10} + \beta_{9} - 11 \beta_{8} + 2 \beta_{7} + \beta_{6} - 12 \beta_{4} + 69 \beta_{3} + 15 \beta_{2} + 195 \beta_{1} + 71\)
\(\nu^{8}\)\(=\)\(-2 \beta_{16} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 95 \beta_{11} - 67 \beta_{10} + 12 \beta_{9} - 13 \beta_{8} + 16 \beta_{7} + 14 \beta_{6} + \beta_{5} - 15 \beta_{4} + 33 \beta_{3} + 396 \beta_{2} + 100 \beta_{1} + 654\)
\(\nu^{9}\)\(=\)\(-16 \beta_{16} - \beta_{15} + 15 \beta_{14} + 20 \beta_{13} - \beta_{12} + 141 \beta_{11} - 17 \beta_{10} + 16 \beta_{9} - 89 \beta_{8} + 35 \beta_{7} + 19 \beta_{6} - 110 \beta_{4} + 510 \beta_{3} + 160 \beta_{2} + 1317 \beta_{1} + 556\)
\(\nu^{10}\)\(=\)\(-34 \beta_{16} - 21 \beta_{15} + 18 \beta_{14} + 73 \beta_{13} - 38 \beta_{12} + 758 \beta_{11} - 475 \beta_{10} + 108 \beta_{9} - 119 \beta_{8} + 182 \beta_{7} + 146 \beta_{6} + 16 \beta_{5} - 160 \beta_{4} + 374 \beta_{3} + 2735 \beta_{2} + 863 \beta_{1} + 4461\)
\(\nu^{11}\)\(=\)\(-175 \beta_{16} - 25 \beta_{15} + 157 \beta_{14} + 258 \beta_{13} - 29 \beta_{12} + 1254 \beta_{11} - 192 \beta_{10} + 177 \beta_{9} - 643 \beta_{8} + 419 \beta_{7} + 240 \beta_{6} - 3 \beta_{5} - 920 \beta_{4} + 3738 \beta_{3} + 1489 \beta_{2} + 9078 \beta_{1} + 4378\)
\(\nu^{12}\)\(=\)\(-387 \beta_{16} - 280 \beta_{15} + 209 \beta_{14} + 891 \beta_{13} - 477 \beta_{12} + 5849 \beta_{11} - 3319 \beta_{10} + 891 \beta_{9} - 946 \beta_{8} + 1795 \beta_{7} + 1366 \beta_{6} + 166 \beta_{5} - 1491 \beta_{4} + 3637 \beta_{3} + 18897 \beta_{2} + 7255 \beta_{1} + 30985\)
\(\nu^{13}\)\(=\)\(-1639 \beta_{16} - 379 \beta_{15} + 1418 \beta_{14} + 2756 \beta_{13} - 465 \beta_{12} + 10502 \beta_{11} - 1842 \beta_{10} + 1700 \beta_{9} - 4398 \beta_{8} + 4260 \beta_{7} + 2539 \beta_{6} - 73 \beta_{5} - 7381 \beta_{4} + 27380 \beta_{3} + 12932 \beta_{2} + 63356 \beta_{1} + 34535\)
\(\nu^{14}\)\(=\)\(-3726 \beta_{16} - 3056 \beta_{15} + 2014 \beta_{14} + 9160 \beta_{13} - 4993 \beta_{12} + 44424 \beta_{11} - 23120 \beta_{10} + 7152 \beta_{9} - 6988 \beta_{8} + 16345 \beta_{7} + 12081 \beta_{6} + 1413 \beta_{5} - 12968 \beta_{4} + 32623 \beta_{3} + 130973 \beta_{2} + 59806 \beta_{1} + 217789\)
\(\nu^{15}\)\(=\)\(-14164 \beta_{16} - 4567 \beta_{15} + 11851 \beta_{14} + 26556 \beta_{13} - 5729 \beta_{12} + 85086 \beta_{11} - 16278 \beta_{10} + 15288 \beta_{9} - 29152 \beta_{8} + 39557 \beta_{7} + 24381 \beta_{6} - 1106 \beta_{5} - 57918 \beta_{4} + 200835 \beta_{3} + 107939 \beta_{2} + 446006 \beta_{1} + 271845\)
\(\nu^{16}\)\(=\)\(-32872 \beta_{16} - 29872 \beta_{15} + 17602 \beta_{14} + 85786 \beta_{13} - 47253 \beta_{12} + 334746 \beta_{11} - 161278 \beta_{10} + 57104 \beta_{9} - 49396 \beta_{8} + 141417 \beta_{7} + 103157 \beta_{6} + 10648 \beta_{5} - 108361 \beta_{4} + 278635 \beta_{3} + 911836 \beta_{2} + 485008 \beta_{1} + 1544374\)

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
−2.56306
−2.53341
−1.59605
−1.58492
−1.30924
−1.02928
−0.320785
0.159847
0.184653
0.829580
1.08305
1.69575
1.89951
2.12059
2.50383
2.71108
2.74886
−2.56306 0 4.56927 1.00000 0 −0.860224 −6.58519 0 −2.56306
1.2 −2.53341 0 4.41816 1.00000 0 3.58669 −6.12619 0 −2.53341
1.3 −1.59605 0 0.547372 1.00000 0 1.65189 2.31847 0 −1.59605
1.4 −1.58492 0 0.511971 1.00000 0 −2.52227 2.35841 0 −1.58492
1.5 −1.30924 0 −0.285886 1.00000 0 4.28141 2.99278 0 −1.30924
1.6 −1.02928 0 −0.940578 1.00000 0 −4.61830 3.02668 0 −1.02928
1.7 −0.320785 0 −1.89710 1.00000 0 4.94917 1.25013 0 −0.320785
1.8 0.159847 0 −1.97445 1.00000 0 −1.46854 −0.635302 0 0.159847
1.9 0.184653 0 −1.96590 1.00000 0 2.68806 −0.732314 0 0.184653
1.10 0.829580 0 −1.31180 1.00000 0 1.06171 −2.74740 0 0.829580
1.11 1.08305 0 −0.826999 1.00000 0 −2.85554 −3.06179 0 1.08305
1.12 1.69575 0 0.875568 1.00000 0 −1.30460 −1.90676 0 1.69575
1.13 1.89951 0 1.60815 1.00000 0 2.75204 −0.744322 0 1.89951
1.14 2.12059 0 2.49688 1.00000 0 4.22149 1.05368 0 2.12059
1.15 2.50383 0 4.26915 1.00000 0 −2.88177 5.68155 0 2.50383
1.16 2.71108 0 5.34994 1.00000 0 −0.946340 9.08194 0 2.71108
1.17 2.74886 0 5.55624 1.00000 0 4.26511 9.77563 0 2.74886
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(89\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{17} - \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} - \cdots\)