Properties

Label 21.30.a.b
Level $21$
Weight $30$
Character orbit 21.a
Self dual yes
Analytic conductor $111.884$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 2752306353 x^{5} - 358739735184 x^{4} + 2112044285875555887 x^{3} - 87015327743334724410 x^{2} - 372546762120092709645386335 x - 1910221430295883859730026686804\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{12}\cdot 5^{2}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 597 - \beta_{1} ) q^{2} -4782969 q^{3} + ( 249858685 - 996 \beta_{1} + \beta_{2} ) q^{4} + ( -245259623 + 12344 \beta_{1} + 11 \beta_{2} + \beta_{3} ) q^{5} + ( -2855432493 + 4782969 \beta_{1} ) q^{6} -678223072849 q^{7} + ( 611826440756 - 144534335 \beta_{1} + 489 \beta_{2} + 199 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{8} + 22876792454961 q^{9} +O(q^{10})\) \( q +(597 - \beta_{1}) q^{2} -4782969 q^{3} +(249858685 - 996 \beta_{1} + \beta_{2}) q^{4} +(-245259623 + 12344 \beta_{1} + 11 \beta_{2} + \beta_{3}) q^{5} +(-2855432493 + 4782969 \beta_{1}) q^{6} -678223072849 q^{7} +(611826440756 - 144534335 \beta_{1} + 489 \beta_{2} + 199 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6}) q^{8} +22876792454961 q^{9} +(-9854364286348 - 4358460806 \beta_{1} + 76611 \beta_{2} + 12876 \beta_{3} - 25 \beta_{4} - 75 \beta_{5} - 25 \beta_{6}) q^{10} +(362747308383 + 3131926725 \beta_{1} + 1158441 \beta_{2} - 42882 \beta_{3} - 60 \beta_{4} + 101 \beta_{5} + 155 \beta_{6}) q^{11} +(-1195066344735765 + 4763837124 \beta_{1} - 4782969 \beta_{2}) q^{12} +(2130535773608112 + 124404788955 \beta_{1} - 2371840 \beta_{2} - 53967 \beta_{3} + 1384 \beta_{4} + 4031 \beta_{5} - 431 \beta_{6}) q^{13} +(-404899174490853 + 678223072849 \beta_{1}) q^{14} +(1173069173760687 - 59040969336 \beta_{1} - 52612659 \beta_{2} - 4782969 \beta_{3}) q^{15} +(-20118769287547100 - 318691054319 \beta_{1} - 16845327 \beta_{2} + 654327 \beta_{3} - 41942 \beta_{4} + 40774 \beta_{5} - 2345 \beta_{6}) q^{16} +(-105081299501944529 + 2878840625165 \beta_{1} + 249328031 \beta_{2} + 2366404 \beta_{3} - 5952 \beta_{4} + 100393 \beta_{5} + 145015 \beta_{6}) q^{17} +(13657445095611717 - 22876792454961 \beta_{1}) q^{18} +(461696187157330020 - 18966601538385 \beta_{1} - 622632582 \beta_{2} - 95747217 \beta_{3} - 183508 \beta_{4} - 763289 \beta_{5} + 86633 \beta_{6}) q^{19} +(3553157731740329028 - 29822749024734 \beta_{1} + 2839514654 \beta_{2} + 23137214 \beta_{3} - 1504300 \beta_{4} + 65100 \beta_{5} - 1335650 \beta_{6}) q^{20} +3243919932521508681 q^{21} +(-2462695392651852750 - 504330063561564 \beta_{1} - 28676788677 \beta_{2} - 413261004 \beta_{3} + 7750447 \beta_{4} - 880243 \beta_{5} + 1302535 \beta_{6}) q^{22} +(-6026165256147493035 - 635752663945523 \beta_{1} - 34348447609 \beta_{2} - 35550316 \beta_{3} + 8831288 \beta_{4} + 3410673 \beta_{5} + 1454527 \beta_{6}) q^{23} +(-2926346899516284564 + 691303243740615 \beta_{1} - 2338871841 \beta_{2} - 951810831 \beta_{3} - 9565938 \beta_{4} + 9565938 \beta_{5} + 4782969 \beta_{6}) q^{24} +(6687093438331474237 - 2661674416559161 \beta_{1} - 113873665584 \beta_{2} - 7254730119 \beta_{3} - 26129700 \beta_{4} + 5781775 \beta_{5} - 34035375 \beta_{6}) q^{25} +(-96556512046221389110 - 1065151810758862 \beta_{1} - 182807255998 \beta_{2} - 21930118656 \beta_{3} + 25090474 \beta_{4} - 62499202 \beta_{5} + 38134354 \beta_{6}) q^{26} -\)\(10\!\cdots\!09\)\( q^{27} +(-\)\(16\!\cdots\!65\)\( + 675510180557604 \beta_{1} - 678223072849 \beta_{2}) q^{28} +(-\)\(34\!\cdots\!14\)\( + 8469787104159457 \beta_{1} - 1035253218530 \beta_{2} - 23864493135 \beta_{3} - 2814148 \beta_{4} - 121456175 \beta_{5} + 52367039 \beta_{6}) q^{29} +(47133118896309607212 + 20846382922813014 \beta_{1} - 366428038059 \beta_{2} - 61585508844 \beta_{3} + 119574225 \beta_{4} + 358722675 \beta_{5} + 119574225 \beta_{6}) q^{30} +(\)\(31\!\cdots\!08\)\( + 22621080589479157 \beta_{1} - 4362896665418 \beta_{2} + 35876514261 \beta_{3} - 555014628 \beta_{4} + 594134597 \beta_{5} - 529513125 \beta_{6}) q^{31} +(-89871943196956239756 + 104933950317937441 \beta_{1} - 5797413725039 \beta_{2} - 50368972505 \beta_{3} - 828679862 \beta_{4} - 291804890 \beta_{5} + 209775111 \beta_{6}) q^{32} +(-1735009130829329127 - 14979908435946525 \beta_{1} - 5540787391329 \beta_{2} + 205103276658 \beta_{3} + 286978140 \beta_{4} - 483079869 \beta_{5} - 741360195 \beta_{6}) q^{33} +(-\)\(23\!\cdots\!80\)\( - 952555595207330 \beta_{1} - 17945557810851 \beta_{2} - 358109002692 \beta_{3} + 3635559913 \beta_{4} - 915102149 \beta_{5} + 2133872065 \beta_{6}) q^{34} +(\)\(16\!\cdots\!27\)\( - 8371985611248056 \beta_{1} - 7460453801339 \beta_{2} - 678223072849 \beta_{3}) q^{35} +(\)\(57\!\cdots\!85\)\( - 22785285285141156 \beta_{1} + 22876792454961 \beta_{2}) q^{36} +(-\)\(70\!\cdots\!44\)\( + 352675973581808849 \beta_{1} - 13792246038796 \beta_{2} + 4578988777587 \beta_{3} - 1708766020 \beta_{4} - 5192458351 \beta_{5} - 37339585 \beta_{6}) q^{37} +(\)\(15\!\cdots\!96\)\( - 213703210104951092 \beta_{1} + 27591515376856 \beta_{2} + 2430753639992 \beta_{3} - 1831538808 \beta_{4} + 20612814744 \beta_{5} - 3594336432 \beta_{6}) q^{38} +(-\)\(10\!\cdots\!28\)\( - 595024249023307395 \beta_{1} + 11344437192960 \beta_{2} + 258122488023 \beta_{3} - 6619629096 \beta_{4} - 19280148039 \beta_{5} + 2061459639 \beta_{6}) q^{39} +(\)\(30\!\cdots\!04\)\( - 2446449758692660962 \beta_{1} - 69060434996578 \beta_{2} + 1698586738002 \beta_{3} - 3449973300 \beta_{4} - 2954193900 \beta_{5} - 26290236750 \beta_{6}) q^{40} +(-\)\(20\!\cdots\!67\)\( - 590747582643519263 \beta_{1} - 172110398048363 \beta_{2} - 550023154534 \beta_{3} + 45348964200 \beta_{4} - 9583315803 \beta_{5} + 32484570939 \beta_{6}) q^{41} +(\)\(19\!\cdots\!57\)\( - 3243919932521508681 \beta_{1}) q^{42} +(\)\(13\!\cdots\!84\)\( + 4477940652776980952 \beta_{1} + 46024975985260 \beta_{2} + 3473077563792 \beta_{3} - 110203980412 \beta_{4} - 41248789876 \beta_{5} - 36842773996 \beta_{6}) q^{43} +(\)\(39\!\cdots\!66\)\( + 12889665365052421946 \beta_{1} + 546776319713420 \beta_{2} - 21557720512658 \beta_{3} + 8222438580 \beta_{4} + 153421598444 \beta_{5} + 62599345166 \beta_{6}) q^{44} +(-\)\(56\!\cdots\!03\)\( + 282391126064038584 \beta_{1} + 251644717004571 \beta_{2} + 22876792454961 \beta_{3}) q^{45} +(\)\(49\!\cdots\!54\)\( + 20578387657999949816 \beta_{1} + 1204631361851869 \beta_{2} - 62448733095108 \beta_{3} - 13955323799 \beta_{4} + 127206739899 \beta_{5} + 186527397697 \beta_{6}) q^{46} +(\)\(10\!\cdots\!90\)\( - 133477166144989512 \beta_{1} + 247340989862758 \beta_{2} - 52493957826550 \beta_{3} + 170700378336 \beta_{4} - 110217668840 \beta_{5} - 185574833768 \beta_{6}) q^{47} +(\)\(96\!\cdots\!00\)\( + 1524289433385093111 \beta_{1} + 80570676835863 \beta_{2} - 3129625756863 \beta_{3} + 200607285798 \beta_{4} - 195020778006 \beta_{5} + 11216062305 \beta_{6}) q^{48} +\)\(45\!\cdots\!01\)\( q^{49} +(\)\(20\!\cdots\!87\)\( + 40372923243845988489 \beta_{1} + 1469852113224766 \beta_{2} + 25710711459856 \beta_{3} - 398552693450 \beta_{4} - 248057056350 \beta_{5} - 536274957250 \beta_{6}) q^{50} +(\)\(50\!\cdots\!01\)\( - 13769405466104814885 \beta_{1} - 1192528243104039 \beta_{2} - 11318436973476 \beta_{3} + 28468231488 \beta_{4} - 480176606817 \beta_{5} - 693602249535 \beta_{6}) q^{51} +(-\)\(36\!\cdots\!54\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} + 3542882861085830 \beta_{2} - 160002386498220 \beta_{3} + 17300626040 \beta_{4} + 1410185097160 \beta_{5} + 1190153397716 \beta_{6}) q^{52} +(-\)\(14\!\cdots\!00\)\( + 4085953379232056293 \beta_{1} - 3652123540608320 \beta_{2} + 536992026991123 \beta_{3} + 1018514283292 \beta_{4} - 2306413482619 \beta_{5} + 207290113723 \beta_{6}) q^{53} +(-\)\(65\!\cdots\!73\)\( + \)\(10\!\cdots\!09\)\( \beta_{1}) q^{54} +(-\)\(23\!\cdots\!60\)\( + \)\(21\!\cdots\!55\)\( \beta_{1} - 1734728087267630 \beta_{2} - 437100101404005 \beta_{3} - 2154219888100 \beta_{4} + 590873457075 \beta_{5} + 189028374125 \beta_{6}) q^{55} +(-\)\(41\!\cdots\!44\)\( + 98026520815886770415 \beta_{1} - 331651082623161 \beta_{2} - 134966391496951 \beta_{3} - 1356446145698 \beta_{4} + 1356446145698 \beta_{5} + 678223072849 \beta_{6}) q^{56} +(-\)\(22\!\cdots\!80\)\( + 90716667193447765065 \beta_{1} + 2978032338095958 \beta_{2} + 457955970747273 \beta_{3} + 877713075252 \beta_{4} + 3650787625041 \beta_{5} - 414362953377 \beta_{6}) q^{57} +(-\)\(68\!\cdots\!50\)\( + \)\(79\!\cdots\!18\)\( \beta_{1} - 7447099409856024 \beta_{2} - 42855450755640 \beta_{3} - 1122136621000 \beta_{4} + 6412921787560 \beta_{5} + 1333218636656 \beta_{6}) q^{58} +(-\)\(73\!\cdots\!70\)\( + \)\(22\!\cdots\!88\)\( \beta_{1} + 8976828729417650 \beta_{2} + 1956443877997814 \beta_{3} + 1206118856600 \beta_{4} - 643689807872 \beta_{5} - 691405276528 \beta_{6}) q^{59} +(-\)\(16\!\cdots\!32\)\( + \)\(14\!\cdots\!46\)\( \beta_{1} - 13581310565127726 \beta_{2} - 110664577308366 \beta_{3} + 7195020266700 \beta_{4} - 311371281900 \beta_{5} + 6388372544850 \beta_{6}) q^{60} +(-\)\(16\!\cdots\!06\)\( + \)\(10\!\cdots\!25\)\( \beta_{1} - 42043969133250734 \beta_{2} - 192530903588739 \beta_{3} + 1999371278864 \beta_{4} - 8692734582823 \beta_{5} - 9437378375785 \beta_{6}) q^{61} +(-\)\(17\!\cdots\!12\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - 66214895212289992 \beta_{2} + 351701605837224 \beta_{3} - 14069933455352 \beta_{4} - 18567440619752 \beta_{5} - 6855882083808 \beta_{6}) q^{62} -\)\(15\!\cdots\!89\)\( q^{63} +(-\)\(71\!\cdots\!92\)\( + \)\(27\!\cdots\!09\)\( \beta_{1} - 173530855779112527 \beta_{2} + 2804415307038855 \beta_{3} + 14260575352586 \beta_{4} - 9182763807130 \beta_{5} - 1727141429017 \beta_{6}) q^{64} +(-\)\(16\!\cdots\!22\)\( + \)\(50\!\cdots\!66\)\( \beta_{1} - 6390028630004346 \beta_{2} + 2818083414067564 \beta_{3} + 23728709234000 \beta_{4} - 2102321398250 \beta_{5} + 9035538897050 \beta_{6}) q^{65} +(\)\(11\!\cdots\!50\)\( + \)\(24\!\cdots\!16\)\( \beta_{1} + 137160191261642013 \beta_{2} + 1976614571040876 \beta_{3} - 37070147737143 \beta_{4} + 4210174981467 \beta_{5} - 6229984526415 \beta_{6}) q^{66} +(-\)\(45\!\cdots\!50\)\( + \)\(57\!\cdots\!88\)\( \beta_{1} + 65441293716090358 \beta_{2} - 12419696698348326 \beta_{3} - 1398021596772 \beta_{4} - 25089682572112 \beta_{5} - 3543630528144 \beta_{6}) q^{67} +(\)\(55\!\cdots\!68\)\( + \)\(84\!\cdots\!78\)\( \beta_{1} + 93940994220715746 \beta_{2} - 24035970260715902 \beta_{3} + 11498661029228 \beta_{4} + 90392282376436 \beta_{5} + 14627722977890 \beta_{6}) q^{68} +(\)\(28\!\cdots\!15\)\( + \)\(30\!\cdots\!87\)\( \beta_{1} + 164287560111971121 \beta_{2} + 170036059368204 \beta_{3} - 42239776734072 \beta_{4} - 16313143228137 \beta_{5} - 6956957550663 \beta_{6}) q^{69} +(\)\(66\!\cdots\!52\)\( + \)\(29\!\cdots\!94\)\( \beta_{1} - 51959347834034739 \beta_{2} - 8732800286003724 \beta_{3} + 16955576821225 \beta_{4} + 50866730463675 \beta_{5} + 16955576821225 \beta_{6}) q^{70} +(-\)\(98\!\cdots\!65\)\( + \)\(16\!\cdots\!55\)\( \beta_{1} + 98010655940602193 \beta_{2} - 3892290932966508 \beta_{3} + 54422379298624 \beta_{4} + 169932005567359 \beta_{5} + 58586991733713 \beta_{6}) q^{71} +(\)\(13\!\cdots\!16\)\( - \)\(33\!\cdots\!35\)\( \beta_{1} + 11186751510475929 \beta_{2} + 4552481698537239 \beta_{3} + 45753584909922 \beta_{4} - 45753584909922 \beta_{5} - 22876792454961 \beta_{6}) q^{72} +(\)\(37\!\cdots\!20\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + 1033514020163382114 \beta_{2} - 44630805995505378 \beta_{3} - 45805854156520 \beta_{4} - 155893612359016 \beta_{5} + 15748002151784 \beta_{6}) q^{73} +(-\)\(28\!\cdots\!54\)\( + \)\(13\!\cdots\!46\)\( \beta_{1} + 202434156721155758 \beta_{2} + 73136259245950304 \beta_{3} - 287730355227322 \beta_{4} - 131305359415342 \beta_{5} - 102303202023810 \beta_{6}) q^{74} +(-\)\(31\!\cdots\!53\)\( + \)\(12\!\cdots\!09\)\( \beta_{1} + 544654212404638896 \beta_{2} + 34699149262543311 \beta_{3} + 124977545079300 \beta_{4} - 27654050589975 \beta_{5} + 162790143528375 \beta_{6}) q^{75} +(-\)\(70\!\cdots\!04\)\( - \)\(16\!\cdots\!24\)\( \beta_{1} - 260469254331828804 \beta_{2} + 19293766507016304 \beta_{3} + 156147184837344 \beta_{4} - 258473723764832 \beta_{5} - 61476707404368 \beta_{6}) q^{76} +(-\)\(24\!\cdots\!67\)\( - \)\(21\!\cdots\!25\)\( \beta_{1} - 785681414734268409 \beta_{2} + 29083561809910818 \beta_{3} + 40693384370940 \beta_{4} - 68500530357749 \beta_{5} - 105124576291595 \beta_{6}) q^{77} +(\)\(46\!\cdots\!90\)\( + \)\(50\!\cdots\!78\)\( \beta_{1} + 874361438413498062 \beta_{2} + 104891077697969664 \beta_{3} - 120006959337306 \beta_{4} + 298931745690738 \beta_{5} - 182395433017026 \beta_{6}) q^{78} +(\)\(16\!\cdots\!78\)\( - \)\(14\!\cdots\!44\)\( \beta_{1} - 844338195810020002 \beta_{2} + 11028567855360846 \beta_{3} + 67399242425248 \beta_{4} + 43893783097932 \beta_{5} - 636861626435084 \beta_{6}) q^{79} +(\)\(34\!\cdots\!68\)\( + \)\(14\!\cdots\!46\)\( \beta_{1} + 2640610389154586174 \beta_{2} + 36736903568473234 \beta_{3} + 130954103261100 \beta_{4} - 297662766786700 \beta_{5} + 353383133392850 \beta_{6}) q^{80} +\)\(52\!\cdots\!21\)\( q^{81} +(\)\(45\!\cdots\!52\)\( + \)\(94\!\cdots\!02\)\( \beta_{1} + 3290711346011475843 \beta_{2} - 242395486268239500 \beta_{3} + 135329151792535 \beta_{4} + 1525047168201285 \beta_{5} + 1135133537639119 \beta_{6}) q^{82} +(-\)\(15\!\cdots\!04\)\( + \)\(42\!\cdots\!94\)\( \beta_{1} - 8678916872905786532 \beta_{2} + 3774778183649974 \beta_{3} + 763418404634320 \beta_{4} - 613229047170162 \beta_{5} + 740955444591650 \beta_{6}) q^{83} +(\)\(81\!\cdots\!85\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} + 3243919932521508681 \beta_{2}) q^{84} +(\)\(13\!\cdots\!54\)\( + \)\(15\!\cdots\!13\)\( \beta_{1} - 10435567561004676928 \beta_{2} - 529930720879669773 \beta_{3} - 1168264687649300 \beta_{4} + 710599524392725 \beta_{5} + 139629127342075 \beta_{6}) q^{85} +(-\)\(34\!\cdots\!96\)\( - \)\(15\!\cdots\!56\)\( \beta_{1} - 9963197079875060752 \beta_{2} + 743337397267979584 \beta_{3} - 1030454252388048 \beta_{4} - 1242927207929072 \beta_{5} - 2227366224551504 \beta_{6}) q^{86} +(\)\(16\!\cdots\!66\)\( - \)\(40\!\cdots\!33\)\( \beta_{1} + 4951584051379215570 \beta_{2} + 114143130865417815 \beta_{3} + 13459982645412 \beta_{4} + 580921119883575 \beta_{5} - 250469924158791 \beta_{6}) q^{87} +(-\)\(85\!\cdots\!32\)\( - \)\(35\!\cdots\!20\)\( \beta_{1} - 11131511547098859396 \beta_{2} - 536690133163557612 \beta_{3} + 287558653668520 \beta_{4} - 1699970744350664 \beta_{5} + 865207242514180 \beta_{6}) q^{88} +(-\)\(44\!\cdots\!55\)\( - \)\(16\!\cdots\!49\)\( \beta_{1} - 6624983980975369279 \beta_{2} - 247809329507994768 \beta_{3} - 1049025018083240 \beta_{4} + 3494733484820019 \beta_{5} - 1347612455183731 \beta_{6}) q^{89} +(-\)\(22\!\cdots\!28\)\( - \)\(99\!\cdots\!66\)\( \beta_{1} + 1752613946767017171 \beta_{2} + 294561579650077836 \beta_{3} - 571919811374025 \beta_{4} - 1715759434122075 \beta_{5} - 571919811374025 \beta_{6}) q^{90} +(-\)\(14\!\cdots\!88\)\( - \)\(84\!\cdots\!95\)\( \beta_{1} + 1608636613106172160 \beta_{2} + 36601664572441983 \beta_{3} - 938660732823016 \beta_{4} - 2733917206654319 \beta_{5} + 292314144397919 \beta_{6}) q^{91} +(-\)\(12\!\cdots\!22\)\( - \)\(67\!\cdots\!90\)\( \beta_{1} - 28645959850535398072 \beta_{2} - 928691988874060414 \beta_{3} + 4764620394854252 \beta_{4} - 1342606236190988 \beta_{5} + 1905227825008994 \beta_{6}) q^{92} +(-\)\(15\!\cdots\!52\)\( - \)\(10\!\cdots\!33\)\( \beta_{1} + 20867599500897666042 \beta_{2} - 171596255538420909 \beta_{3} + 2654617760270532 \beta_{4} - 2841727359278493 \beta_{5} + 2532644861968125 \beta_{6}) q^{93} +(\)\(70\!\cdots\!64\)\( - \)\(11\!\cdots\!64\)\( \beta_{1} + 28776170669951184550 \beta_{2} - 797505849469864488 \beta_{3} - 1039375558780434 \beta_{4} + 4274833317000394 \beta_{5} - 414387994487250 \beta_{6}) q^{94} +(-\)\(17\!\cdots\!16\)\( - \)\(60\!\cdots\!52\)\( \beta_{1} + 17090705755843531212 \beta_{2} + 907560621440074792 \beta_{3} - 1712267143710400 \beta_{4} + 512766638258300 \beta_{5} + 1458020297693700 \beta_{6}) q^{95} +(\)\(42\!\cdots\!64\)\( - \)\(50\!\cdots\!29\)\( \beta_{1} + 27728850127036060791 \beta_{2} + 240913234053267345 \beta_{3} + 3963550090870278 \beta_{4} + 1395693742918410 \beta_{5} - 1003347852884559 \beta_{6}) q^{96} +(\)\(19\!\cdots\!80\)\( - \)\(92\!\cdots\!32\)\( \beta_{1} + 58610755646114093238 \beta_{2} - 210941776611783774 \beta_{3} - 1571141089360912 \beta_{4} + 9083133726660992 \beta_{5} - 9820884065148640 \beta_{6}) q^{97} +(\)\(27\!\cdots\!97\)\( - \)\(45\!\cdots\!01\)\( \beta_{1}) q^{98} +(\)\(82\!\cdots\!63\)\( + \)\(71\!\cdots\!25\)\( \beta_{1} + 26501414328317475801 \beta_{2} - 981002614053637602 \beta_{3} - 1372607547297660 \beta_{4} + 2310556037951061 \beta_{5} + 3545902830518955 \beta_{6}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4177q^{2} - 33480783q^{3} + 1749008801q^{4} - 1716792694q^{5} - 19978461513q^{6} - 4747561509943q^{7} + 4282496015841q^{8} + 160137547184727q^{9} + O(q^{10}) \) \( 7q + 4177q^{2} - 33480783q^{3} + 1749008801q^{4} - 1716792694q^{5} - 19978461513q^{6} - 4747561509943q^{7} + 4282496015841q^{8} + 160137547184727q^{9} - 68989267066594q^{10} + 2545492652300q^{11} - 8365454875910169q^{12} + 14913999229538746q^{13} - 2832937775290273q^{14} + 8211366234828486q^{15} - 140832022360508287q^{16} - 735563339328645834q^{17} + 95556362084372097q^{18} + 3231835378045104676q^{19} + 24872044471029723934q^{20} + 22707439527650560767q^{21} - 17239876351747271380q^{22} - 42184428229682843136q^{23} - 20483045686391011929q^{24} + 46804330940039088761q^{25} - \)\(67\!\cdots\!94\)\(q^{26} - \)\(76\!\cdots\!63\)\(q^{27} - \)\(11\!\cdots\!49\)\(q^{28} - \)\(24\!\cdots\!10\)\(q^{29} + \)\(32\!\cdots\!86\)\(q^{30} + \)\(21\!\cdots\!80\)\(q^{31} - \)\(62\!\cdots\!91\)\(q^{32} - 12175012445678678700q^{33} - \)\(16\!\cdots\!74\)\(q^{34} + \)\(11\!\cdots\!06\)\(q^{35} + \)\(40\!\cdots\!61\)\(q^{36} - \)\(49\!\cdots\!34\)\(q^{37} + \)\(10\!\cdots\!56\)\(q^{38} - \)\(71\!\cdots\!74\)\(q^{39} + \)\(21\!\cdots\!62\)\(q^{40} - \)\(14\!\cdots\!90\)\(q^{41} + \)\(13\!\cdots\!37\)\(q^{42} + \)\(94\!\cdots\!16\)\(q^{43} + \)\(27\!\cdots\!36\)\(q^{44} - \)\(39\!\cdots\!34\)\(q^{45} + \)\(34\!\cdots\!68\)\(q^{46} + \)\(70\!\cdots\!92\)\(q^{47} + \)\(67\!\cdots\!03\)\(q^{48} + \)\(32\!\cdots\!07\)\(q^{49} + \)\(14\!\cdots\!11\)\(q^{50} + \)\(35\!\cdots\!46\)\(q^{51} - \)\(25\!\cdots\!10\)\(q^{52} - \)\(10\!\cdots\!62\)\(q^{53} - \)\(45\!\cdots\!93\)\(q^{54} - \)\(16\!\cdots\!80\)\(q^{55} - \)\(29\!\cdots\!09\)\(q^{56} - \)\(15\!\cdots\!44\)\(q^{57} - \)\(48\!\cdots\!38\)\(q^{58} - \)\(51\!\cdots\!60\)\(q^{59} - \)\(11\!\cdots\!46\)\(q^{60} - \)\(11\!\cdots\!98\)\(q^{61} - \)\(12\!\cdots\!68\)\(q^{62} - \)\(10\!\cdots\!23\)\(q^{63} - \)\(50\!\cdots\!87\)\(q^{64} - \)\(11\!\cdots\!16\)\(q^{65} + \)\(82\!\cdots\!20\)\(q^{66} - \)\(32\!\cdots\!36\)\(q^{67} + \)\(39\!\cdots\!94\)\(q^{68} + \)\(20\!\cdots\!84\)\(q^{69} + \)\(46\!\cdots\!06\)\(q^{70} - \)\(68\!\cdots\!64\)\(q^{71} + \)\(97\!\cdots\!01\)\(q^{72} + \)\(26\!\cdots\!58\)\(q^{73} - \)\(19\!\cdots\!26\)\(q^{74} - \)\(22\!\cdots\!09\)\(q^{75} - \)\(49\!\cdots\!80\)\(q^{76} - \)\(17\!\cdots\!00\)\(q^{77} + \)\(32\!\cdots\!86\)\(q^{78} + \)\(11\!\cdots\!20\)\(q^{79} + \)\(24\!\cdots\!54\)\(q^{80} + \)\(36\!\cdots\!47\)\(q^{81} + \)\(31\!\cdots\!34\)\(q^{82} - \)\(11\!\cdots\!68\)\(q^{83} + \)\(56\!\cdots\!81\)\(q^{84} + \)\(92\!\cdots\!12\)\(q^{85} - \)\(24\!\cdots\!52\)\(q^{86} + \)\(11\!\cdots\!90\)\(q^{87} - \)\(60\!\cdots\!16\)\(q^{88} - \)\(31\!\cdots\!14\)\(q^{89} - \)\(15\!\cdots\!34\)\(q^{90} - \)\(10\!\cdots\!54\)\(q^{91} - \)\(88\!\cdots\!04\)\(q^{92} - \)\(10\!\cdots\!20\)\(q^{93} + \)\(49\!\cdots\!80\)\(q^{94} - \)\(12\!\cdots\!48\)\(q^{95} + \)\(30\!\cdots\!79\)\(q^{96} + \)\(13\!\cdots\!90\)\(q^{97} + \)\(19\!\cdots\!77\)\(q^{98} + \)\(58\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 2752306353 x^{5} - 358739735184 x^{4} + 2112044285875555887 x^{3} - 87015327743334724410 x^{2} - 372546762120092709645386335 x - 1910221430295883859730026686804\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 198 \nu - 786373188 \)
\(\beta_{3}\)\(=\)\((\)\(6319895207 \nu^{6} - 50001970875867 \nu^{5} - 15624907420453298478 \nu^{4} + 104273977365790594864074 \nu^{3} + 10084030417921651771848590523 \nu^{2} - 64476436637980726131561777143487 \nu - 1123455148970232894396060252717659492\)\()/ \)\(15\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-94219193 \nu^{6} + 41494859698437 \nu^{5} - 526283502760867950 \nu^{4} - 80342387160656002219638 \nu^{3} + 775684448990784494645866971 \nu^{2} + 30413746056769337643484075996833 \nu - 3821779478729192774906555356483300\)\()/ \)\(46\!\cdots\!84\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-7307796347 \nu^{6} + 1808037814332447 \nu^{5} + 28198620533081604438 \nu^{4} - 3506411745985621561767234 \nu^{3} - 40044687795082533778597669503 \nu^{2} + 1313192845601920107415934347080627 \nu + 13090879389143981276382696445464786452\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(262291405687 \nu^{6} - 2215488247132779 \nu^{5} - 782628890409858718926 \nu^{4} + 7713796520117824606466730 \nu^{3} + 629381138765951047845322507083 \nu^{2} - 6477601042981923773625553582884879 \nu - 84417473953748051270437235723414049316\)\()/ \)\(31\!\cdots\!28\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 198 \beta_{1} + 786373188\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 199 \beta_{3} + 1302 \beta_{2} + 1217561550 \beta_{1} + 155756846197\)
\(\nu^{4}\)\(=\)\(43 \beta_{6} + 45550 \beta_{5} - 46718 \beta_{4} + 179115 \beta_{3} + 1594738131 \beta_{2} + 985103332825 \beta_{1} + 957457760709060351\)
\(\nu^{5}\)\(=\)\(1934272802 \beta_{6} + 4715610756 \beta_{5} - 3598612484 \beta_{4} - 375736361262 \beta_{3} + 9505074858752 \beta_{2} + 1645203872091364619 \beta_{1} + 774746568355396346030\)
\(\nu^{6}\)\(=\)\(-1089362055798 \beta_{6} + 116925421347972 \beta_{5} - 110975603832612 \beta_{4} + 3213168905040426 \beta_{3} + 2400848827326143733 \beta_{2} + 5249358809696158004808 \beta_{1} + 1293744607007053885587082654\)

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
40216.5
27718.7
20218.4
−6861.66
−9633.19
−34621.1
−37035.6
−39619.5 −4.78297e6 1.03283e9 8.07585e9 1.89499e11 −6.78223e11 −1.96498e13 2.28768e13 −3.19961e14
1.2 −27121.7 −4.78297e6 1.98714e8 −7.72707e9 1.29722e11 −6.78223e11 9.17138e12 2.28768e13 2.09571e14
1.3 −19621.4 −4.78297e6 −1.51871e8 9.23391e9 9.38486e10 −6.78223e11 1.35141e13 2.28768e13 −1.81182e14
1.4 7458.66 −4.78297e6 −4.81239e8 −2.60193e10 −3.56746e10 −6.78223e11 −7.59374e12 2.28768e13 −1.94069e14
1.5 10230.2 −4.78297e6 −4.32214e8 5.84968e9 −4.89307e10 −6.78223e11 −9.91393e12 2.28768e13 5.98434e13
1.6 35218.1 −4.78297e6 7.03444e8 −9.52746e9 −1.68447e11 −6.78223e11 5.86640e12 2.28768e13 −3.35539e14
1.7 37632.6 −4.78297e6 8.79342e8 1.83976e10 −1.79996e11 −6.78223e11 1.28881e13 2.28768e13 6.92348e14
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.a.b 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4177 T_{2}^{6} - 2744828928 T_{2}^{5} + \)\(85\!\cdots\!04\)\( T_{2}^{4} + \)\(21\!\cdots\!40\)\( T_{2}^{3} - \)\(36\!\cdots\!48\)\( T_{2}^{2} - \)\(37\!\cdots\!00\)\( T_{2} + \)\(21\!\cdots\!08\)\( \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(21\!\cdots\!08\)\( - \)\(37\!\cdots\!00\)\( T - \)\(36\!\cdots\!48\)\( T^{2} + 2101382585285360640 T^{3} + 8566937725104 T^{4} - 2744828928 T^{5} - 4177 T^{6} + T^{7} \)
$3$ \( ( 4782969 + T )^{7} \)
$5$ \( \)\(15\!\cdots\!00\)\( - \)\(28\!\cdots\!00\)\( T - \)\(41\!\cdots\!00\)\( T^{2} + \)\(84\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!00\)\( T^{4} - \)\(67\!\cdots\!00\)\( T^{5} + 1716792694 T^{6} + T^{7} \)
$7$ \( ( 678223072849 + T )^{7} \)
$11$ \( \)\(14\!\cdots\!00\)\( + \)\(37\!\cdots\!40\)\( T + \)\(26\!\cdots\!48\)\( T^{2} + \)\(47\!\cdots\!20\)\( T^{3} - \)\(79\!\cdots\!60\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{5} - 2545492652300 T^{6} + T^{7} \)
$13$ \( \)\(45\!\cdots\!44\)\( - \)\(31\!\cdots\!40\)\( T - \)\(13\!\cdots\!64\)\( T^{2} + \)\(11\!\cdots\!44\)\( T^{3} + \)\(86\!\cdots\!52\)\( T^{4} - \)\(70\!\cdots\!44\)\( T^{5} - 14913999229538746 T^{6} + T^{7} \)
$17$ \( -\)\(76\!\cdots\!28\)\( - \)\(11\!\cdots\!72\)\( T + \)\(52\!\cdots\!60\)\( T^{2} + \)\(76\!\cdots\!96\)\( T^{3} - \)\(11\!\cdots\!08\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{5} + 735563339328645834 T^{6} + T^{7} \)
$19$ \( -\)\(44\!\cdots\!80\)\( + \)\(20\!\cdots\!24\)\( T - \)\(26\!\cdots\!32\)\( T^{2} + \)\(67\!\cdots\!76\)\( T^{3} + \)\(64\!\cdots\!68\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} - 3231835378045104676 T^{6} + T^{7} \)
$23$ \( -\)\(57\!\cdots\!76\)\( + \)\(42\!\cdots\!04\)\( T + \)\(32\!\cdots\!00\)\( T^{2} + \)\(94\!\cdots\!16\)\( T^{3} - \)\(21\!\cdots\!04\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{5} + 42184428229682843136 T^{6} + T^{7} \)
$29$ \( -\)\(10\!\cdots\!80\)\( + \)\(25\!\cdots\!96\)\( T + \)\(88\!\cdots\!68\)\( T^{2} - \)\(11\!\cdots\!04\)\( T^{3} - \)\(31\!\cdots\!68\)\( T^{4} + \)\(80\!\cdots\!08\)\( T^{5} + \)\(24\!\cdots\!10\)\( T^{6} + T^{7} \)
$31$ \( \)\(14\!\cdots\!52\)\( + \)\(50\!\cdots\!88\)\( T - \)\(16\!\cdots\!52\)\( T^{2} + \)\(81\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!48\)\( T^{4} - \)\(68\!\cdots\!52\)\( T^{5} - \)\(21\!\cdots\!80\)\( T^{6} + T^{7} \)
$37$ \( \)\(32\!\cdots\!72\)\( - \)\(22\!\cdots\!64\)\( T - \)\(31\!\cdots\!84\)\( T^{2} + \)\(43\!\cdots\!84\)\( T^{3} - \)\(23\!\cdots\!00\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{5} + \)\(49\!\cdots\!34\)\( T^{6} + T^{7} \)
$41$ \( \)\(21\!\cdots\!00\)\( - \)\(26\!\cdots\!00\)\( T - \)\(36\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} - \)\(10\!\cdots\!72\)\( T^{4} - \)\(19\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!90\)\( T^{6} + T^{7} \)
$43$ \( -\)\(33\!\cdots\!76\)\( + \)\(32\!\cdots\!80\)\( T - \)\(72\!\cdots\!84\)\( T^{2} - \)\(14\!\cdots\!36\)\( T^{3} + \)\(71\!\cdots\!12\)\( T^{4} - \)\(44\!\cdots\!84\)\( T^{5} - \)\(94\!\cdots\!16\)\( T^{6} + T^{7} \)
$47$ \( \)\(21\!\cdots\!48\)\( - \)\(33\!\cdots\!32\)\( T + \)\(73\!\cdots\!64\)\( T^{2} - \)\(53\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!12\)\( T^{4} + \)\(14\!\cdots\!92\)\( T^{5} - \)\(70\!\cdots\!92\)\( T^{6} + T^{7} \)
$53$ \( \)\(55\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( T + \)\(15\!\cdots\!56\)\( T^{2} + \)\(48\!\cdots\!80\)\( T^{3} - \)\(18\!\cdots\!84\)\( T^{4} - \)\(36\!\cdots\!72\)\( T^{5} + \)\(10\!\cdots\!62\)\( T^{6} + T^{7} \)
$59$ \( -\)\(50\!\cdots\!60\)\( - \)\(17\!\cdots\!64\)\( T + \)\(93\!\cdots\!80\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} - \)\(60\!\cdots\!88\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(51\!\cdots\!60\)\( T^{6} + T^{7} \)
$61$ \( -\)\(12\!\cdots\!68\)\( - \)\(21\!\cdots\!56\)\( T + \)\(21\!\cdots\!40\)\( T^{2} + \)\(21\!\cdots\!88\)\( T^{3} - \)\(91\!\cdots\!24\)\( T^{4} - \)\(78\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!98\)\( T^{6} + T^{7} \)
$67$ \( -\)\(67\!\cdots\!88\)\( - \)\(82\!\cdots\!16\)\( T + \)\(85\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!68\)\( T^{3} - \)\(17\!\cdots\!44\)\( T^{4} - \)\(20\!\cdots\!04\)\( T^{5} + \)\(32\!\cdots\!36\)\( T^{6} + T^{7} \)
$71$ \( -\)\(21\!\cdots\!08\)\( - \)\(46\!\cdots\!48\)\( T + \)\(25\!\cdots\!68\)\( T^{2} + \)\(76\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!40\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!64\)\( T^{6} + T^{7} \)
$73$ \( -\)\(56\!\cdots\!16\)\( + \)\(70\!\cdots\!76\)\( T - \)\(18\!\cdots\!52\)\( T^{2} - \)\(70\!\cdots\!72\)\( T^{3} + \)\(50\!\cdots\!52\)\( T^{4} - \)\(95\!\cdots\!44\)\( T^{5} - \)\(26\!\cdots\!58\)\( T^{6} + T^{7} \)
$79$ \( \)\(57\!\cdots\!80\)\( + \)\(76\!\cdots\!24\)\( T - \)\(15\!\cdots\!60\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(88\!\cdots\!36\)\( T^{4} + \)\(26\!\cdots\!80\)\( T^{5} - \)\(11\!\cdots\!20\)\( T^{6} + T^{7} \)
$83$ \( -\)\(46\!\cdots\!84\)\( - \)\(31\!\cdots\!40\)\( T + \)\(11\!\cdots\!40\)\( T^{2} + \)\(52\!\cdots\!72\)\( T^{3} - \)\(93\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + T^{7} \)
$89$ \( -\)\(20\!\cdots\!60\)\( - \)\(43\!\cdots\!76\)\( T - \)\(30\!\cdots\!08\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} - \)\(12\!\cdots\!24\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{5} + \)\(31\!\cdots\!14\)\( T^{6} + T^{7} \)
$97$ \( -\)\(34\!\cdots\!68\)\( + \)\(25\!\cdots\!36\)\( T - \)\(52\!\cdots\!44\)\( T^{2} - \)\(23\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!24\)\( T^{4} - \)\(90\!\cdots\!48\)\( T^{5} - \)\(13\!\cdots\!90\)\( T^{6} + T^{7} \)
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