Properties

Label 21.30.a.a
Level $21$
Weight $30$
Character orbit 21.a
Self dual yes
Analytic conductor $111.884$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(111.883889004\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 110200839 x^{4} - 84515300136 x^{3} + 2909822349515976 x^{2} + 3144706297269234912 x - 12496546686906381337200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{2}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 744 + \beta_{1} ) q^{2} + 4782969 q^{3} + ( 51420432 + 6090 \beta_{1} + \beta_{2} ) q^{4} + ( -5676166830 - 591 \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{5} + ( 3558528936 + 4782969 \beta_{1} ) q^{6} + 678223072849 q^{7} + ( 3217868305536 - 143544512 \beta_{1} + 8648 \beta_{2} - 344 \beta_{3} + 1120 \beta_{4} + 232 \beta_{5} ) q^{8} + 22876792454961 q^{9} +O(q^{10})\) \( q +(744 + \beta_{1}) q^{2} +4782969 q^{3} +(51420432 + 6090 \beta_{1} + \beta_{2}) q^{4} +(-5676166830 - 591 \beta_{1} - 3 \beta_{2} - \beta_{5}) q^{5} +(3558528936 + 4782969 \beta_{1}) q^{6} +678223072849 q^{7} +(3217868305536 - 143544512 \beta_{1} + 8648 \beta_{2} - 344 \beta_{3} + 1120 \beta_{4} + 232 \beta_{5}) q^{8} +22876792454961 q^{9} +(-4569402235840 - 6485585898 \beta_{1} - 284767 \beta_{2} - 9071 \beta_{3} - 45478 \beta_{4} - 5923 \beta_{5}) q^{10} +(-323943500004000 + 17057486581 \beta_{1} - 444167 \beta_{2} - 48907 \beta_{3} + 132623 \beta_{4} + 5436 \beta_{5}) q^{11} +(245942332222608 + 29128281210 \beta_{1} + 4782969 \beta_{2}) q^{12} +(-2241791475679002 - 93985491756 \beta_{1} - 45344 \beta_{2} + 1588925 \beta_{3} - 240461 \beta_{4} + 250237 \beta_{5}) q^{13} +(504597966199656 + 678223072849 \beta_{1}) q^{14} +(-27148929986718270 - 2826734679 \beta_{1} - 14348907 \beta_{2} - 4782969 \beta_{5}) q^{15} +(-109580771422981376 + 1898348021472 \beta_{1} - 273109456 \beta_{2} + 5713280 \beta_{3} + 10278400 \beta_{4} + 11014528 \beta_{5}) q^{16} +(-148622481256200234 - 5821791176615 \beta_{1} + 1062930513 \beta_{2} - 2068365 \beta_{3} - 53137851 \beta_{4} + 1378502 \beta_{5}) q^{17} +(17020333586490984 + 22876792454961 \beta_{1}) q^{18} +(-1131202401949959836 - 61208271185958 \beta_{1} - 4109059910 \beta_{2} - 111731073 \beta_{3} - 102587859 \beta_{4} - 78001335 \beta_{5}) q^{19} +(-767786632537895040 - 127190346102928 \beta_{1} - 14044561096 \beta_{2} - 158692064 \beta_{3} - 302461952 \beta_{4} + 14260512 \beta_{5}) q^{20} +3243919932521508681 q^{21} +(9784470097063426000 - 361783545788484 \beta_{1} + 57650146979 \beta_{2} + 980530971 \beta_{3} - 1393271586 \beta_{4} + 809780319 \beta_{5}) q^{22} +(690178141643620884 - 592269303269409 \beta_{1} + 26620167759 \beta_{2} + 3961127877 \beta_{3} + 4034571435 \beta_{4} + 2895459626 \beta_{5}) q^{23} +(15390964351461216384 - 686568951016128 \beta_{1} + 41363115912 \beta_{2} - 1645341336 \beta_{3} + 5356925280 \beta_{4} + 1109648808 \beta_{5}) q^{24} +(45356117788687205175 + 209514468299760 \beta_{1} - 93331637560 \beta_{2} - 14531203055 \beta_{3} + 2702799635 \beta_{4} + 5638961885 \beta_{5}) q^{25} +(-56906696082014935728 - 2933017031652162 \beta_{1} - 503677350878 \beta_{2} - 2816628022 \beta_{3} + 19480177028 \beta_{4} + 6794612162 \beta_{5}) q^{26} +\)\(10\!\cdots\!09\)\( q^{27} +(34874523398263050768 + 4130378513650410 \beta_{1} + 678223072849 \beta_{2}) q^{28} +(\)\(43\!\cdots\!02\)\( + 982250529550010 \beta_{1} - 1691318299566 \beta_{2} + 65444977359 \beta_{3} - 176471127387 \beta_{4} + 82907066345 \beta_{5}) q^{29} +(-21855309242553408960 - 31020356296971162 \beta_{1} - 1362031733223 \beta_{2} - 43386311799 \beta_{3} - 217519864182 \beta_{4} - 28329525387 \beta_{5}) q^{30} +(-\)\(20\!\cdots\!84\)\( + 67465494298454502 \beta_{1} - 6119678429298 \beta_{2} + 7813673925 \beta_{3} + 218987040039 \beta_{4} + 107999200059 \beta_{5}) q^{31} +(-\)\(69\!\cdots\!44\)\( - 108304780259183104 \beta_{1} + 149123467904 \beta_{2} + 424727030656 \beta_{3} - 486398107136 \beta_{4} - 16819095680 \beta_{5}) q^{32} +(-\)\(15\!\cdots\!00\)\( + 81585429534838989 \beta_{1} - 2124436991823 \beta_{2} - 233920664883 \beta_{3} + 634331697687 \beta_{4} + 26000219484 \beta_{5}) q^{33} +(-\)\(35\!\cdots\!96\)\( + 147129343319431218 \beta_{1} - 12978565070661 \beta_{2} - 604193804285 \beta_{3} + 1598806645358 \beta_{4} - 203086140889 \beta_{5}) q^{34} +(-\)\(38\!\cdots\!70\)\( - 400829836053759 \beta_{1} - 2034669218547 \beta_{2} - 678223072849 \beta_{5}) q^{35} +(\)\(11\!\cdots\!52\)\( + 139319666050712490 \beta_{1} + 22876792454961 \beta_{2}) q^{36} +(-\)\(22\!\cdots\!18\)\( - 98860070369914356 \beta_{1} - 66774050316036 \beta_{2} - 3031967927097 \beta_{3} + 3811577568333 \beta_{4} + 373292788623 \beta_{5}) q^{37} +(-\)\(36\!\cdots\!24\)\( - 2714294163929608220 \beta_{1} - 83944568059564 \beta_{2} + 422365955596 \beta_{3} - 7710444137864 \beta_{4} - 2753012291300 \beta_{5}) q^{38} +(-\)\(10\!\cdots\!38\)\( - 449529693518703564 \beta_{1} - 216878946336 \beta_{2} + 7599779018325 \beta_{3} - 1150117508709 \beta_{4} + 1196875813653 \beta_{5}) q^{39} +(-\)\(72\!\cdots\!60\)\( - 2309798106831787392 \beta_{1} - 24929704244736 \beta_{2} + 8795997054400 \beta_{3} + 10599424467200 \beta_{4} - 3299916469312 \beta_{5}) q^{40} +(-\)\(43\!\cdots\!74\)\( - 2476700058856846621 \beta_{1} - 259786341547973 \beta_{2} - 14758502220553 \beta_{3} - 31864421988631 \beta_{4} - 7524217233120 \beta_{5}) q^{41} +(\)\(24\!\cdots\!64\)\( + 3243919932521508681 \beta_{1}) q^{42} +(\)\(23\!\cdots\!68\)\( - 1339779499421488200 \beta_{1} + 1341578937492396 \beta_{2} - 8533588407200 \beta_{3} - 11756772988420 \beta_{4} - 3203699624176 \beta_{5}) q^{43} +(-\)\(31\!\cdots\!04\)\( + 16293383396990983508 \beta_{1} - 300797687063614 \beta_{2} + 5255744184352 \beta_{3} + 41452076829952 \beta_{4} + 7606991510048 \beta_{5}) q^{44} +(-\)\(12\!\cdots\!30\)\( - 13520184340881951 \beta_{1} - 68630377364883 \beta_{2} - 22876792454961 \beta_{5}) q^{45} +(-\)\(34\!\cdots\!80\)\( + 5216542458030418776 \beta_{1} + 26153915239085 \beta_{2} + 29225884170469 \beta_{3} + 142799305242146 \beta_{4} + 74005348681121 \beta_{5}) q^{46} +(-\)\(42\!\cdots\!64\)\( - 21635389244817491114 \beta_{1} + 2010953326187694 \beta_{2} - 50265114003000 \beta_{3} - 118233000268968 \beta_{4} + 63927372370386 \beta_{5}) q^{47} +(-\)\(52\!\cdots\!44\)\( + 9079739737911910368 \beta_{1} - 1306274061654864 \beta_{2} + 27326441128320 \beta_{3} + 49161268569600 \beta_{4} + 52682145973632 \beta_{5}) q^{48} +\)\(45\!\cdots\!01\)\( q^{49} +(\)\(15\!\cdots\!00\)\( + 18255330662392612575 \beta_{1} + 6010957793858390 \beta_{2} + 140543749535230 \beta_{3} + 47329408520140 \beta_{4} - 38212914071450 \beta_{5}) q^{50} +(-\)\(71\!\cdots\!46\)\( - 27845446722223069935 \beta_{1} + 5083963692833097 \beta_{2} - 9892925675685 \beta_{3} - 254156694059619 \beta_{4} + 6593332332438 \beta_{5}) q^{51} +(-\)\(56\!\cdots\!44\)\( - \)\(17\!\cdots\!08\)\( \beta_{1} + 2408838032441698 \beta_{2} - 509146777830912 \beta_{3} - 293818654661376 \beta_{4} - 65107564953600 \beta_{5}) q^{52} +(-\)\(45\!\cdots\!18\)\( - \)\(10\!\cdots\!92\)\( \beta_{1} - 13467227300841032 \beta_{2} + 86858760895283 \beta_{3} + 18672529352609 \beta_{4} - 57463512778145 \beta_{5}) q^{53} +(\)\(81\!\cdots\!96\)\( + \)\(10\!\cdots\!09\)\( \beta_{1}) q^{54} +(-\)\(16\!\cdots\!60\)\( - \)\(35\!\cdots\!82\)\( \beta_{1} + 4946050374865994 \beta_{2} + 678315913687875 \beta_{3} - 224292762930375 \beta_{4} - 40303275582627 \beta_{5}) q^{55} +(\)\(21\!\cdots\!64\)\( - 97355200019250154688 \beta_{1} + 5865273133998152 \beta_{2} - 233308737060056 \beta_{3} + 759609841590880 \beta_{4} + 157347752900968 \beta_{5}) q^{56} +(-\)\(54\!\cdots\!84\)\( - \)\(29\!\cdots\!02\)\( \beta_{1} - 19653506168672790 \beta_{2} - 534406258495737 \beta_{3} - 490674549373371 \beta_{4} - 373077967263615 \beta_{5}) q^{57} +(\)\(89\!\cdots\!76\)\( - \)\(10\!\cdots\!10\)\( \beta_{1} - 34045096711731116 \beta_{2} + 390123438747340 \beta_{3} + 3095937824551160 \beta_{4} - 1134107597708068 \beta_{5}) q^{58} +(\)\(49\!\cdots\!96\)\( - \)\(87\!\cdots\!42\)\( \beta_{1} - 52387504793002982 \beta_{2} + 2464918912923920 \beta_{3} + 2262352481892152 \beta_{4} - 1433910578527258 \beta_{5}) q^{59} +(-\)\(36\!\cdots\!60\)\( - \)\(60\!\cdots\!32\)\( \beta_{1} - 67174700340774024 \beta_{2} - 759019222658016 \beta_{3} - 1446666140095488 \beta_{4} + 68207586820128 \beta_{5}) q^{60} +(-\)\(15\!\cdots\!86\)\( - \)\(33\!\cdots\!98\)\( \beta_{1} + 49069304696326478 \beta_{2} - 2957703743528869 \beta_{3} - 7194122870045747 \beta_{4} + 639487640501581 \beta_{5}) q^{61} +(\)\(39\!\cdots\!32\)\( - \)\(17\!\cdots\!32\)\( \beta_{1} + 124135362705670692 \beta_{2} + 4238628104735820 \beta_{3} - 3752101202904456 \beta_{4} + 1024285496686428 \beta_{5}) q^{62} +\)\(15\!\cdots\!89\)\( q^{63} +(-\)\(53\!\cdots\!64\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} - 179850502666168064 \beta_{2} - 6770346686664704 \beta_{3} - 797469328113664 \beta_{4} - 8049017098522624 \beta_{5}) q^{64} +(\)\(44\!\cdots\!40\)\( - \)\(12\!\cdots\!82\)\( \beta_{1} - 76767777181071470 \beta_{2} - 18909814010712118 \beta_{3} + 7277275946401526 \beta_{4} + 10599430171737888 \beta_{5}) q^{65} +(\)\(46\!\cdots\!00\)\( - \)\(17\!\cdots\!96\)\( \beta_{1} + 275738865846000651 \beta_{2} + 4689849237832899 \beta_{3} - 6663974804418834 \beta_{4} + 3873154162587111 \beta_{5}) q^{66} +(-\)\(39\!\cdots\!60\)\( + \)\(47\!\cdots\!50\)\( \beta_{1} - 87733283903287850 \beta_{2} + 22346751900343972 \beta_{3} + 31830198679665440 \beta_{4} + 5349541579450730 \beta_{5}) q^{67} +(\)\(16\!\cdots\!64\)\( - \)\(35\!\cdots\!64\)\( \beta_{1} - 23996070280010072 \beta_{2} + 12878367012880544 \beta_{3} - 8188361124434176 \beta_{4} + 5780073857607328 \beta_{5}) q^{68} +(\)\(33\!\cdots\!96\)\( - \)\(28\!\cdots\!21\)\( \beta_{1} + 127323437166096471 \beta_{2} + 18945951840726813 \beta_{3} + 19297230101890515 \beta_{4} + 13848893631909594 \beta_{5}) q^{69} +(-\)\(30\!\cdots\!60\)\( - \)\(43\!\cdots\!02\)\( \beta_{1} - 193135549785991183 \beta_{2} - 6152161493813279 \beta_{3} - 30844228907026822 \beta_{4} - 4017115260484627 \beta_{5}) q^{70} +(-\)\(19\!\cdots\!12\)\( - \)\(19\!\cdots\!27\)\( \beta_{1} + 368036433772995001 \beta_{2} - 48287148454711525 \beta_{3} - 10616739647181667 \beta_{4} - 13671169008695722 \beta_{5}) q^{71} +(\)\(73\!\cdots\!96\)\( - \)\(32\!\cdots\!32\)\( \beta_{1} + 197838501150502728 \beta_{2} - 7869616604506584 \beta_{3} + 25622007549556320 \beta_{4} + 5307415849550952 \beta_{5}) q^{72} +(\)\(76\!\cdots\!70\)\( + \)\(35\!\cdots\!10\)\( \beta_{1} + 1713410343420920318 \beta_{2} + 22108504357900840 \beta_{3} - 6747220750339696 \beta_{4} - 26589577512076438 \beta_{5}) q^{73} +(-\)\(74\!\cdots\!64\)\( - \)\(42\!\cdots\!50\)\( \beta_{1} + 1429587615864967254 \beta_{2} + 53244411228471918 \beta_{3} - 98926733349655380 \beta_{4} + 4179931193815830 \beta_{5}) q^{74} +(\)\(21\!\cdots\!75\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} - 446402329168715640 \beta_{2} - 69502293744770295 \beta_{3} + 12927406867416315 \beta_{4} + 26970979888136565 \beta_{5}) q^{75} +(-\)\(10\!\cdots\!28\)\( - \)\(44\!\cdots\!64\)\( \beta_{1} - 3167948992937710228 \beta_{2} + 22507291691724992 \beta_{3} - 100720478950380800 \beta_{4} - 54865106085928256 \beta_{5}) q^{76} +(-\)\(21\!\cdots\!00\)\( + \)\(11\!\cdots\!69\)\( \beta_{1} - 301244307598121783 \beta_{2} - 33169855823826043 \beta_{3} + 89947978590452927 \beta_{4} + 3686820624007164 \beta_{5}) q^{77} +(-\)\(27\!\cdots\!32\)\( - \)\(14\!\cdots\!78\)\( \beta_{1} - 2409073155251596782 \beta_{2} - 13471844513757318 \beta_{3} + 93173082839436132 \beta_{4} + 32498419337868978 \beta_{5}) q^{78} +(-\)\(16\!\cdots\!08\)\( - \)\(14\!\cdots\!02\)\( \beta_{1} + 4701544694187407806 \beta_{2} - 54586473633965084 \beta_{3} + 189637762376125820 \beta_{4} - 142236937723167826 \beta_{5}) q^{79} +(-\)\(99\!\cdots\!00\)\( - \)\(24\!\cdots\!40\)\( \beta_{1} + 4032264615172395648 \beta_{2} + 88670885434799616 \beta_{3} - 35542778734835712 \beta_{4} + 100126602672596480 \beta_{5}) q^{80} +\)\(52\!\cdots\!21\)\( q^{81} +(-\)\(14\!\cdots\!60\)\( - \)\(96\!\cdots\!46\)\( \beta_{1} - 7715502751720710339 \beta_{2} - 102372882186484155 \beta_{3} - 460470717541311390 \beta_{4} - 437558752338866559 \beta_{5}) q^{82} +(-\)\(32\!\cdots\!36\)\( + \)\(10\!\cdots\!04\)\( \beta_{1} - 5099668624202352748 \beta_{2} + 357639493269765982 \beta_{3} + 504083039018852194 \beta_{4} + 179853988202903658 \beta_{5}) q^{83} +(\)\(16\!\cdots\!92\)\( + \)\(19\!\cdots\!90\)\( \beta_{1} + 3243919932521508681 \beta_{2}) q^{84} +(-\)\(69\!\cdots\!80\)\( + \)\(92\!\cdots\!24\)\( \beta_{1} - 9618784605150275504 \beta_{2} + 882486258359123 \beta_{3} - 55247926959039311 \beta_{4} + 265892088924770599 \beta_{5}) q^{85} +(-\)\(77\!\cdots\!56\)\( + \)\(43\!\cdots\!48\)\( \beta_{1} + 1121432900346975736 \beta_{2} - 528417368079624232 \beta_{3} + 1407707207219054192 \beta_{4} + 155189613706824440 \beta_{5}) q^{86} +(\)\(20\!\cdots\!38\)\( + \)\(46\!\cdots\!90\)\( \beta_{1} - 8089522995956891454 \beta_{2} + 313021297913798871 \beta_{3} - 844055931687072003 \beta_{4} + 396541928209078305 \beta_{5}) q^{87} +(\)\(42\!\cdots\!56\)\( + \)\(15\!\cdots\!56\)\( \beta_{1} - 6307391017821233968 \beta_{2} - 158752245039035312 \beta_{3} + 475146500206312640 \beta_{4} - 91117605485164336 \beta_{5}) q^{88} +(\)\(48\!\cdots\!82\)\( - \)\(24\!\cdots\!73\)\( \beta_{1} + 10344521586371949551 \beta_{2} - 59447691099477215 \beta_{3} - 3557717100211232129 \beta_{4} - 19573140354839578 \beta_{5}) q^{89} +(-\)\(10\!\cdots\!40\)\( - \)\(14\!\cdots\!78\)\( \beta_{1} - 6514555557021879087 \beta_{2} - 207515384358951231 \beta_{3} - 1040390767266716358 \beta_{4} - 135499241710734003 \beta_{5}) q^{90} +(-\)\(15\!\cdots\!98\)\( - \)\(63\!\cdots\!44\)\( \beta_{1} - 30753347015265056 \beta_{2} + 1077645596026597325 \beta_{3} - 163086198320343389 \beta_{4} + 169716507080515213 \beta_{5}) q^{91} +(\)\(24\!\cdots\!96\)\( + 93253367406405341492 \beta_{1} + 32615953204178208434 \beta_{2} - 771914525670984608 \beta_{3} + 148601467127771392 \beta_{4} + 178245079908134496 \beta_{5}) q^{92} +(-\)\(96\!\cdots\!96\)\( + \)\(32\!\cdots\!38\)\( \beta_{1} - 29270232217301025762 \beta_{2} + 37372560159383325 \beta_{3} + 1047408223908295791 \beta_{4} + 516556825906995171 \beta_{5}) q^{93} +(-\)\(13\!\cdots\!28\)\( + \)\(68\!\cdots\!88\)\( \beta_{1} - 9284908469070882362 \beta_{2} - 487163048602210650 \beta_{3} + 5512506134357684124 \beta_{4} - 432027701783743122 \beta_{5}) q^{94} +(\)\(16\!\cdots\!20\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} + 71447212765198463228 \beta_{2} + 1489111668559021252 \beta_{3} + 2198155934531901436 \beta_{4} + 497731415270908184 \beta_{5}) q^{95} +(-\)\(33\!\cdots\!36\)\( - \)\(51\!\cdots\!76\)\( \beta_{1} + 713252924157326976 \beta_{2} + 2031456221089697664 \beta_{3} - 2326427068090166784 \beta_{4} - 80445213245473920 \beta_{5}) q^{96} +(-\)\(13\!\cdots\!98\)\( + \)\(21\!\cdots\!58\)\( \beta_{1} + 15858072435546818426 \beta_{2} - 5049324954526995624 \beta_{3} - 487741975566638664 \beta_{4} - 134482427457456954 \beta_{5}) q^{97} +(\)\(34\!\cdots\!44\)\( + \)\(45\!\cdots\!01\)\( \beta_{1}) q^{98} +(-\)\(74\!\cdots\!00\)\( + \)\(39\!\cdots\!41\)\( \beta_{1} - 10161116274342662487 \beta_{2} - 1118835288594777627 \beta_{3} + 3033988845754292703 \beta_{4} + 124358243785167996 \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4464q^{2} + 28697814q^{3} + 308522592q^{4} - 34057000980q^{5} + 21351173616q^{6} + 4069338437094q^{7} + 19307209833216q^{8} + 137260754729766q^{9} + O(q^{10}) \) \( 6q + 4464q^{2} + 28697814q^{3} + 308522592q^{4} - 34057000980q^{5} + 21351173616q^{6} + 4069338437094q^{7} + 19307209833216q^{8} + 137260754729766q^{9} - 27416413415040q^{10} - 1943661000024000q^{11} + 1475653993335648q^{12} - 13450748854074012q^{13} + 3027587797197936q^{14} - 162893579920309620q^{15} - 657484628537888256q^{16} - 891734887537201404q^{17} + 102122001518945904q^{18} - 6787214411699759016q^{19} - 4606719795227370240q^{20} + 19463519595129052086q^{21} + 58706820582380556000q^{22} + 4141068849861725304q^{23} + 92345786108767298304q^{24} + \)\(27\!\cdots\!50\)\(q^{25} - \)\(34\!\cdots\!68\)\(q^{26} + \)\(65\!\cdots\!54\)\(q^{27} + \)\(20\!\cdots\!08\)\(q^{28} + \)\(25\!\cdots\!12\)\(q^{29} - \)\(13\!\cdots\!60\)\(q^{30} - \)\(12\!\cdots\!04\)\(q^{31} - \)\(41\!\cdots\!64\)\(q^{32} - \)\(92\!\cdots\!00\)\(q^{33} - \)\(21\!\cdots\!76\)\(q^{34} - \)\(23\!\cdots\!20\)\(q^{35} + \)\(70\!\cdots\!12\)\(q^{36} - \)\(13\!\cdots\!08\)\(q^{37} - \)\(22\!\cdots\!44\)\(q^{38} - \)\(64\!\cdots\!28\)\(q^{39} - \)\(43\!\cdots\!60\)\(q^{40} - \)\(26\!\cdots\!44\)\(q^{41} + \)\(14\!\cdots\!84\)\(q^{42} + \)\(14\!\cdots\!08\)\(q^{43} - \)\(18\!\cdots\!24\)\(q^{44} - \)\(77\!\cdots\!80\)\(q^{45} - \)\(20\!\cdots\!80\)\(q^{46} - \)\(25\!\cdots\!84\)\(q^{47} - \)\(31\!\cdots\!64\)\(q^{48} + \)\(27\!\cdots\!06\)\(q^{49} + \)\(94\!\cdots\!00\)\(q^{50} - \)\(42\!\cdots\!76\)\(q^{51} - \)\(33\!\cdots\!64\)\(q^{52} - \)\(27\!\cdots\!08\)\(q^{53} + \)\(48\!\cdots\!76\)\(q^{54} - \)\(96\!\cdots\!60\)\(q^{55} + \)\(13\!\cdots\!84\)\(q^{56} - \)\(32\!\cdots\!04\)\(q^{57} + \)\(53\!\cdots\!56\)\(q^{58} + \)\(29\!\cdots\!76\)\(q^{59} - \)\(22\!\cdots\!60\)\(q^{60} - \)\(95\!\cdots\!16\)\(q^{61} + \)\(23\!\cdots\!92\)\(q^{62} + \)\(93\!\cdots\!34\)\(q^{63} - \)\(32\!\cdots\!84\)\(q^{64} + \)\(26\!\cdots\!40\)\(q^{65} + \)\(28\!\cdots\!00\)\(q^{66} - \)\(23\!\cdots\!60\)\(q^{67} + \)\(98\!\cdots\!84\)\(q^{68} + \)\(19\!\cdots\!76\)\(q^{69} - \)\(18\!\cdots\!60\)\(q^{70} - \)\(11\!\cdots\!72\)\(q^{71} + \)\(44\!\cdots\!76\)\(q^{72} + \)\(45\!\cdots\!20\)\(q^{73} - \)\(44\!\cdots\!84\)\(q^{74} + \)\(13\!\cdots\!50\)\(q^{75} - \)\(60\!\cdots\!68\)\(q^{76} - \)\(13\!\cdots\!00\)\(q^{77} - \)\(16\!\cdots\!92\)\(q^{78} - \)\(10\!\cdots\!48\)\(q^{79} - \)\(59\!\cdots\!00\)\(q^{80} + \)\(31\!\cdots\!26\)\(q^{81} - \)\(87\!\cdots\!60\)\(q^{82} - \)\(19\!\cdots\!16\)\(q^{83} + \)\(10\!\cdots\!52\)\(q^{84} - \)\(41\!\cdots\!80\)\(q^{85} - \)\(46\!\cdots\!36\)\(q^{86} + \)\(12\!\cdots\!28\)\(q^{87} + \)\(25\!\cdots\!36\)\(q^{88} + \)\(29\!\cdots\!92\)\(q^{89} - \)\(62\!\cdots\!40\)\(q^{90} - \)\(91\!\cdots\!88\)\(q^{91} + \)\(14\!\cdots\!76\)\(q^{92} - \)\(57\!\cdots\!76\)\(q^{93} - \)\(78\!\cdots\!68\)\(q^{94} + \)\(99\!\cdots\!20\)\(q^{95} - \)\(19\!\cdots\!16\)\(q^{96} - \)\(78\!\cdots\!88\)\(q^{97} + \)\(20\!\cdots\!64\)\(q^{98} - \)\(44\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 110200839 x^{4} - 84515300136 x^{3} + 2909822349515976 x^{2} + 3144706297269234912 x - 12496546686906381337200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( 16 \nu^{2} - 18408 \nu - 587737808 \)
\(\beta_{3}\)\(=\)\((\)\(404935 \nu^{5} - 579698838 \nu^{4} - 41735794063989 \nu^{3} + 27348147783369450 \nu^{2} + 885061390382623616628 \nu - 100509133526153140458024\)\()/ 75744294807552 \)
\(\beta_{4}\)\(=\)\((\)\(312163 \nu^{5} - 1124954862 \nu^{4} - 24630414484425 \nu^{3} + 64715078296052370 \nu^{2} + 294055251053591202372 \nu - 569537910745315964261064\)\()/ 113616442211328 \)
\(\beta_{5}\)\(=\)\((\)\(-1212725 \nu^{5} + 8282972802 \nu^{4} + 114843506785023 \nu^{3} - 603729766612121598 \nu^{2} - 2384064470084661333468 \nu + 6096417507051005377561464\)\()/ 227232884422656 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 4602 \beta_{1} + 587737808\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(29 \beta_{5} + 140 \beta_{4} - 43 \beta_{3} + 802 \beta_{2} + 114783130 \beta_{1} + 338061200544\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(645256 \beta_{5} + 434080 \beta_{4} + 421064 \beta_{3} + 82193003 \beta_{2} + 559830704094 \beta_{1} + 33731094492117168\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(3450838047 \beta_{5} + 14740215108 \beta_{4} - 2634100857 \beta_{3} + 107725010466 \beta_{2} + 7704399005285550 \beta_{1} + 41126337907031861280\)\()/8\)

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
−7908.19
−5362.48
−3286.24
1705.15
5827.19
9024.57
−30888.8 4.78297e6 4.17246e8 −8.65686e9 −1.47740e11 6.78223e11 3.69508e12 2.28768e13 2.67400e14
1.2 −20705.9 4.78297e6 −1.08136e8 1.18612e10 −9.90357e10 6.78223e11 1.33555e13 2.28768e13 −2.45596e14
1.3 −12401.0 4.78297e6 −3.83087e8 −2.55800e10 −5.93134e10 6.78223e11 1.14084e13 2.28768e13 3.17217e14
1.4 7564.59 4.78297e6 −4.79648e8 −7.91331e9 3.61812e10 6.78223e11 −7.68955e12 2.28768e13 −5.98610e13
1.5 24052.8 4.78297e6 4.16645e7 1.31167e10 1.15044e11 6.78223e11 −1.19111e13 2.28768e13 3.15492e14
1.6 36842.3 4.78297e6 8.20484e8 −1.68846e10 1.76216e11 6.78223e11 1.04489e13 2.28768e13 −6.22068e14
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

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 21.30.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.a.a 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 4464 T_{2}^{5} - 1754910384 T_{2}^{4} - 169892674560 T_{2}^{3} + \)\(75\!\cdots\!40\)\( T_{2}^{2} + \)\(21\!\cdots\!48\)\( T_{2} - \)\(53\!\cdots\!68\)\( \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(53\!\cdots\!68\)\( + \)\(21\!\cdots\!48\)\( T + 751135946466263040 T^{2} - 169892674560 T^{3} - 1754910384 T^{4} - 4464 T^{5} + T^{6} \)
$3$ \( ( -4782969 + T )^{6} \)
$5$ \( \)\(46\!\cdots\!00\)\( + \)\(82\!\cdots\!00\)\( T - \)\(34\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} - \)\(11\!\cdots\!00\)\( T^{4} + 34057000980 T^{5} + T^{6} \)
$7$ \( ( -678223072849 + T )^{6} \)
$11$ \( -\)\(20\!\cdots\!40\)\( + \)\(46\!\cdots\!08\)\( T + \)\(99\!\cdots\!00\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} - \)\(16\!\cdots\!80\)\( T^{4} + 1943661000024000 T^{5} + T^{6} \)
$13$ \( \)\(37\!\cdots\!84\)\( + \)\(12\!\cdots\!84\)\( T + \)\(64\!\cdots\!60\)\( T^{2} - \)\(10\!\cdots\!24\)\( T^{3} - \)\(74\!\cdots\!92\)\( T^{4} + 13450748854074012 T^{5} + T^{6} \)
$17$ \( \)\(34\!\cdots\!44\)\( - \)\(12\!\cdots\!88\)\( T - \)\(21\!\cdots\!56\)\( T^{2} - \)\(82\!\cdots\!04\)\( T^{3} - \)\(65\!\cdots\!84\)\( T^{4} + 891734887537201404 T^{5} + T^{6} \)
$19$ \( -\)\(55\!\cdots\!40\)\( + \)\(19\!\cdots\!48\)\( T - \)\(27\!\cdots\!60\)\( T^{2} - \)\(27\!\cdots\!88\)\( T^{3} - \)\(24\!\cdots\!28\)\( T^{4} + 6787214411699759016 T^{5} + T^{6} \)
$23$ \( \)\(16\!\cdots\!72\)\( - \)\(21\!\cdots\!16\)\( T + \)\(17\!\cdots\!32\)\( T^{2} + \)\(36\!\cdots\!08\)\( T^{3} - \)\(90\!\cdots\!48\)\( T^{4} - 4141068849861725304 T^{5} + T^{6} \)
$29$ \( \)\(11\!\cdots\!80\)\( - \)\(82\!\cdots\!72\)\( T - \)\(56\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!88\)\( T^{3} - \)\(69\!\cdots\!28\)\( T^{4} - \)\(25\!\cdots\!12\)\( T^{5} + T^{6} \)
$31$ \( -\)\(77\!\cdots\!28\)\( - \)\(13\!\cdots\!24\)\( T + \)\(13\!\cdots\!52\)\( T^{2} + \)\(32\!\cdots\!04\)\( T^{3} - \)\(38\!\cdots\!96\)\( T^{4} + \)\(12\!\cdots\!04\)\( T^{5} + T^{6} \)
$37$ \( \)\(32\!\cdots\!76\)\( - \)\(25\!\cdots\!20\)\( T - \)\(16\!\cdots\!92\)\( T^{2} - \)\(48\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!96\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{5} + T^{6} \)
$41$ \( -\)\(36\!\cdots\!00\)\( + \)\(81\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} - \)\(58\!\cdots\!20\)\( T^{3} - \)\(22\!\cdots\!92\)\( T^{4} + \)\(26\!\cdots\!44\)\( T^{5} + T^{6} \)
$43$ \( \)\(53\!\cdots\!24\)\( + \)\(19\!\cdots\!24\)\( T + \)\(18\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!36\)\( T^{3} - \)\(10\!\cdots\!52\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{5} + T^{6} \)
$47$ \( -\)\(32\!\cdots\!56\)\( + \)\(26\!\cdots\!88\)\( T + \)\(12\!\cdots\!36\)\( T^{2} - \)\(16\!\cdots\!32\)\( T^{3} - \)\(67\!\cdots\!60\)\( T^{4} + \)\(25\!\cdots\!84\)\( T^{5} + T^{6} \)
$53$ \( -\)\(66\!\cdots\!80\)\( - \)\(37\!\cdots\!36\)\( T + \)\(18\!\cdots\!40\)\( T^{2} + \)\(12\!\cdots\!04\)\( T^{3} - \)\(12\!\cdots\!32\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + T^{6} \)
$59$ \( -\)\(90\!\cdots\!00\)\( + \)\(11\!\cdots\!96\)\( T + \)\(15\!\cdots\!04\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{3} - \)\(78\!\cdots\!24\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + T^{6} \)
$61$ \( \)\(41\!\cdots\!36\)\( + \)\(18\!\cdots\!64\)\( T + \)\(11\!\cdots\!88\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} - \)\(27\!\cdots\!52\)\( T^{4} + \)\(95\!\cdots\!16\)\( T^{5} + T^{6} \)
$67$ \( -\)\(14\!\cdots\!48\)\( + \)\(16\!\cdots\!68\)\( T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(50\!\cdots\!60\)\( T^{3} - \)\(24\!\cdots\!44\)\( T^{4} + \)\(23\!\cdots\!60\)\( T^{5} + T^{6} \)
$71$ \( -\)\(95\!\cdots\!64\)\( - \)\(11\!\cdots\!92\)\( T + \)\(41\!\cdots\!60\)\( T^{2} + \)\(28\!\cdots\!20\)\( T^{3} - \)\(14\!\cdots\!20\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + T^{6} \)
$73$ \( \)\(11\!\cdots\!12\)\( - \)\(59\!\cdots\!72\)\( T + \)\(44\!\cdots\!68\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} - \)\(23\!\cdots\!84\)\( T^{4} - \)\(45\!\cdots\!20\)\( T^{5} + T^{6} \)
$79$ \( -\)\(16\!\cdots\!80\)\( - \)\(41\!\cdots\!48\)\( T - \)\(14\!\cdots\!16\)\( T^{2} - \)\(91\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!44\)\( T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + T^{6} \)
$83$ \( \)\(85\!\cdots\!04\)\( - \)\(49\!\cdots\!24\)\( T - \)\(54\!\cdots\!76\)\( T^{2} - \)\(89\!\cdots\!52\)\( T^{3} + \)\(52\!\cdots\!96\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{5} + T^{6} \)
$89$ \( -\)\(27\!\cdots\!40\)\( - \)\(14\!\cdots\!96\)\( T + \)\(13\!\cdots\!68\)\( T^{2} + \)\(31\!\cdots\!64\)\( T^{3} - \)\(10\!\cdots\!28\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + T^{6} \)
$97$ \( -\)\(16\!\cdots\!36\)\( + \)\(25\!\cdots\!16\)\( T + \)\(94\!\cdots\!04\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} - \)\(17\!\cdots\!64\)\( T^{4} + \)\(78\!\cdots\!88\)\( T^{5} + T^{6} \)
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