Properties

Label 21.30.a.c
Level $21$
Weight $30$
Character orbit 21.a
Self dual yes
Analytic conductor $111.884$
Analytic rank $0$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 678740466 x^{5} - 2954969748680 x^{4} + 92148363808982720 x^{3} + 418126965823508980224 x^{2} - 2978683903673490966301184 x - 14143029174281721602229028864\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{12}\cdot 5^{3}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + ( 8696 - \beta_{1} ) q^{2} -4782969 q^{3} + ( 314449162 - 4328 \beta_{1} + \beta_{2} ) q^{4} + ( -408826210 + 82488 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( -41592698424 + 4782969 \beta_{1} ) q^{6} + 678223072849 q^{7} + ( 1423602342714 - 441199168 \beta_{1} + 3544 \beta_{2} - 352 \beta_{3} - 2 \beta_{4} ) q^{8} + 22876792454961 q^{9} +O(q^{10})\) \( q +(8696 - \beta_{1}) q^{2} -4782969 q^{3} +(314449162 - 4328 \beta_{1} + \beta_{2}) q^{4} +(-408826210 + 82488 \beta_{1} + 3 \beta_{2} + \beta_{3}) q^{5} +(-41592698424 + 4782969 \beta_{1}) q^{6} +678223072849 q^{7} +(1423602342714 - 441199168 \beta_{1} + 3544 \beta_{2} - 352 \beta_{3} - 2 \beta_{4}) q^{8} +22876792454961 q^{9} +(-67539118043385 - 1972077819 \beta_{1} - 168321 \beta_{2} + 20667 \beta_{3} - 17 \beta_{4} + \beta_{5} + 6 \beta_{6}) q^{10} +(19409214331628 + 7586402528 \beta_{1} - 618820 \beta_{2} + 9849 \beta_{3} - 13 \beta_{4} - 51 \beta_{5} + 75 \beta_{6}) q^{11} +(-1504000593921978 + 20700689832 \beta_{1} - 4782969 \beta_{2}) q^{12} +(1734165931996162 - 83288086157 \beta_{1} - 1192377 \beta_{2} + 825395 \beta_{3} - 1523 \beta_{4} - 140 \beta_{5} + 139 \beta_{6}) q^{13} +(5897827841494904 - 678223072849 \beta_{1}) q^{14} +(1955403088817490 - 394537546872 \beta_{1} - 14348907 \beta_{2} - 4782969 \beta_{3}) q^{15} +(185800942862154156 + 398556728528 \beta_{1} + 393252664 \beta_{2} + 3705296 \beta_{3} - 30092 \beta_{4} + 3376 \beta_{5} + 9760 \beta_{6}) q^{16} +(-55120730342525114 - 12282200165263 \beta_{1} - 107556564 \beta_{2} - 7768580 \beta_{3} - 29775 \beta_{4} + 7602 \beta_{5} - 20133 \beta_{6}) q^{17} +(198936587188340856 - 22876792454961 \beta_{1}) q^{18} +(103719039700182460 + 21180389119784 \beta_{1} - 1161716579 \beta_{2} - 79809886 \beta_{3} - 73531 \beta_{4} - 40837 \beta_{5} - 289907 \beta_{6}) q^{19} +(1161819538426439440 + 147308156546104 \beta_{1} + 3162334258 \beta_{2} + 131514888 \beta_{3} - 188876 \beta_{4} + 28248 \beta_{5} + 386448 \beta_{6}) q^{20} -3243919932521508681 q^{21} +(-5716327147196042129 + 370086257567011 \beta_{1} + 1207377479 \beta_{2} + 809715943 \beta_{3} + 1511479 \beta_{4} + 123205 \beta_{5} + 2685662 \beta_{6}) q^{22} +(11660057307345038204 + 1416390009995205 \beta_{1} + 16916944800 \beta_{2} + 1021602694 \beta_{3} - 199929 \beta_{4} - 1140648 \beta_{5} - 3822039 \beta_{6}) q^{23} +(-6809045873528437866 + 2110241943369792 \beta_{1} - 16950842136 \beta_{2} + 1683605088 \beta_{3} + 9565938 \beta_{4}) q^{24} +(-49240739727133615041 - 1377390714187940 \beta_{1} - 32102962713 \beta_{2} + 5498016764 \beta_{3} + 6368829 \beta_{4} + 1541775 \beta_{5} - 8668499 \beta_{6}) q^{25} +(79686423331505066792 - 534414853863522 \beta_{1} + 378954814184 \beta_{2} + 28229495716 \beta_{3} - 25105432 \beta_{4} + 4109100 \beta_{5} + 19097544 \beta_{6}) q^{26} -\)\(10\!\cdots\!09\)\( q^{27} +(\)\(21\!\cdots\!38\)\( - 2935349459290472 \beta_{1} + 678223072849 \beta_{2}) q^{28} +(\)\(36\!\cdots\!50\)\( + 7703944713097870 \beta_{1} + 1091084490525 \beta_{2} + 9092610088 \beta_{3} - 82500027 \beta_{4} + 11267085 \beta_{5} - 69656295 \beta_{6}) q^{29} +(\)\(32\!\cdots\!65\)\( + 9432387073864611 \beta_{1} + 805074125049 \beta_{2} - 98849620323 \beta_{3} + 81310473 \beta_{4} - 4782969 \beta_{5} - 28697814 \beta_{6}) q^{30} +(\)\(44\!\cdots\!72\)\( - 115755682829944826 \beta_{1} + 1382004620241 \beta_{2} - 58233577144 \beta_{3} + 36655001 \beta_{4} - 150916063 \beta_{5} + 106931677 \beta_{6}) q^{31} +(\)\(54\!\cdots\!52\)\( - 218980913536551200 \beta_{1} + 4783971216016 \beta_{2} + 417105802208 \beta_{3} - 310688408 \beta_{4} + 9826080 \beta_{5} + 449012928 \beta_{6}) q^{32} +(-92833670462532443532 - 36285528112945632 \beta_{1} + 2959796876580 \beta_{2} - 47107461681 \beta_{3} + 62178597 \beta_{4} + 243931419 \beta_{5} - 358722675 \beta_{6}) q^{33} +(\)\(90\!\cdots\!25\)\( + 181963995337575355 \beta_{1} + 18284032151701 \beta_{2} - 58044027723 \beta_{3} - 112064027 \beta_{4} + 61245391 \beta_{5} - 503389542 \beta_{6}) q^{34} +(-\)\(27\!\cdots\!90\)\( + 55945264833168312 \beta_{1} + 2034669218547 \beta_{2} + 678223072849 \beta_{3}) q^{35} +(\)\(71\!\cdots\!82\)\( - 99010757745071208 \beta_{1} + 22876792454961 \beta_{2}) q^{36} +(\)\(17\!\cdots\!62\)\( + 622602088720980530 \beta_{1} + 20797431523767 \beta_{2} - 1787552672474 \beta_{3} + 1282289689 \beta_{4} - 644896139 \beta_{5} + 3782121077 \beta_{6}) q^{37} +(-\)\(15\!\cdots\!86\)\( + 594449110917444422 \beta_{1} - 19091297625218 \beta_{2} - 3383629703854 \beta_{3} + 5414380126 \beta_{4} + 1176050262 \beta_{5} - 7909817148 \beta_{6}) q^{38} +(-\)\(82\!\cdots\!78\)\( + 398364334158260133 \beta_{1} + 5703102227313 \beta_{2} - 3947838697755 \beta_{3} + 7284461787 \beta_{4} + 669615660 \beta_{5} - 664832691 \beta_{6}) q^{39} +(-\)\(67\!\cdots\!80\)\( - 2901283621254493600 \beta_{1} + 3360487291072 \beta_{2} - 4951589471200 \beta_{3} - 4860642828 \beta_{4} - 1383626976 \beta_{5} + 9395049664 \beta_{6}) q^{40} +(-\)\(37\!\cdots\!06\)\( - 1234491220558016177 \beta_{1} + 56810563514174 \beta_{2} + 1686997514628 \beta_{3} - 9470959939 \beta_{4} - 2993127504 \beta_{5} + 2530661667 \beta_{6}) q^{41} +(-\)\(28\!\cdots\!76\)\( + 3243919932521508681 \beta_{1}) q^{42} +(-\)\(27\!\cdots\!64\)\( - 2657849309208797385 \beta_{1} + 115289980867858 \beta_{2} + 1744442655211 \beta_{3} - 64145266270 \beta_{4} - 9057405469 \beta_{5} - 19878749056 \beta_{6}) q^{43} +(-\)\(34\!\cdots\!44\)\( - 774696485799868088 \beta_{1} - 246243629265056 \beta_{2} + 24242228133064 \beta_{3} - 7947013484 \beta_{4} + 16576961688 \beta_{5} + 30633910032 \beta_{6}) q^{44} +(-\)\(93\!\cdots\!10\)\( + 1887060856024822968 \beta_{1} + 68630377364883 \beta_{2} + 22876792454961 \beta_{3}) q^{45} +(-\)\(99\!\cdots\!81\)\( - 29366360849931030465 \beta_{1} - 1762083037547457 \beta_{2} - 19239798737457 \beta_{3} + 7993905967 \beta_{4} + 19847477629 \beta_{5} - 91376406738 \beta_{6}) q^{46} +(-\)\(16\!\cdots\!24\)\( + 3115660145563943756 \beta_{1} - 649360603258374 \beta_{2} + 55576184939574 \beta_{3} + 127768803108 \beta_{4} - 7584386064 \beta_{5} + 125734866204 \beta_{6}) q^{47} +(-\)\(88\!\cdots\!64\)\( - 1906284477290839632 \beta_{1} - 1880915301079416 \beta_{2} - 17722315903824 \beta_{3} + 143929103148 \beta_{4} - 16147303344 \beta_{5} - 46681777440 \beta_{6}) q^{48} +\)\(45\!\cdots\!01\)\( q^{49} +(\)\(64\!\cdots\!76\)\( + 77315890798764350325 \beta_{1} - 1214751730634552 \beta_{2} + 31195329961316 \beta_{3} + 122873845576 \beta_{4} + 10090962540 \beta_{5} - 263130970296 \beta_{6}) q^{50} +(\)\(26\!\cdots\!66\)\( + 58745382642247805847 \beta_{1} + 514439711358516 \beta_{2} + 37156877314020 \beta_{3} + 142412901975 \beta_{4} - 36360130338 \beta_{5} + 96295514877 \beta_{6}) q^{51} +(\)\(17\!\cdots\!32\)\( - \)\(29\!\cdots\!60\)\( \beta_{1} + 5798015904535462 \beta_{2} + 369217003303872 \beta_{3} - 830943864096 \beta_{4} + 67762143552 \beta_{5} + 750194637696 \beta_{6}) q^{52} +(-\)\(95\!\cdots\!90\)\( - \)\(13\!\cdots\!58\)\( \beta_{1} - 7598559448498705 \beta_{2} - 243309289732486 \beta_{3} + 475588433237 \beta_{4} - 286491234447 \beta_{5} - 327235004031 \beta_{6}) q^{53} +(-\)\(95\!\cdots\!64\)\( + \)\(10\!\cdots\!09\)\( \beta_{1}) q^{54} +(\)\(75\!\cdots\!20\)\( - \)\(15\!\cdots\!84\)\( \beta_{1} + 3756129087506639 \beta_{2} - 843376721564758 \beta_{3} - 247602128377 \beta_{4} + 290902068821 \beta_{5} - 828091299929 \beta_{6}) q^{55} +(\)\(96\!\cdots\!86\)\( - \)\(29\!\cdots\!32\)\( \beta_{1} + 2403622570176856 \beta_{2} - 238734521642848 \beta_{3} - 1356446145698 \beta_{4}) q^{56} +(-\)\(49\!\cdots\!40\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} + 5556454384143051 \beta_{2} + 381728210631534 \beta_{3} + 351696493539 \beta_{4} + 195322105053 \beta_{5} + 1386616193883 \beta_{6}) q^{57} +(-\)\(28\!\cdots\!78\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} + 4244469819706334 \beta_{2} - 150767110577166 \beta_{3} - 3095894655554 \beta_{4} + 302072645686 \beta_{5} - 1522052675580 \beta_{6}) q^{58} +(\)\(36\!\cdots\!92\)\( - \)\(44\!\cdots\!74\)\( \beta_{1} + 40998299507707766 \beta_{2} - 881125399334492 \beta_{3} + 2568211458728 \beta_{4} + 384523615974 \beta_{5} + 1331923922364 \beta_{6}) q^{59} +(-\)\(55\!\cdots\!60\)\( - \)\(70\!\cdots\!76\)\( \beta_{1} - 15125346723652002 \beta_{2} - 629031632342472 \beta_{3} + 903388052844 \beta_{4} - 135109308312 \beta_{5} - 1848368804112 \beta_{6}) q^{60} +(\)\(40\!\cdots\!54\)\( - \)\(31\!\cdots\!33\)\( \beta_{1} + 1480418592134191 \beta_{2} - 1513027631470767 \beta_{3} + 5495378922983 \beta_{4} - 1800062708182 \beta_{5} - 2119370189547 \beta_{6}) q^{61} +(\)\(93\!\cdots\!74\)\( - \)\(88\!\cdots\!02\)\( \beta_{1} + 120187681271457942 \beta_{2} - 2013040701010950 \beta_{3} + 922415587830 \beta_{4} + 481377863118 \beta_{5} + 3720827079828 \beta_{6}) q^{62} +\)\(15\!\cdots\!89\)\( q^{63} +(\)\(74\!\cdots\!60\)\( - \)\(30\!\cdots\!80\)\( \beta_{1} + 76190991121317920 \beta_{2} + 10941719478130624 \beta_{3} - 5931753703024 \beta_{4} - 2220563463616 \beta_{5} + 11738325377408 \beta_{6}) q^{64} +(\)\(96\!\cdots\!80\)\( - \)\(42\!\cdots\!94\)\( \beta_{1} - 13750555358579260 \beta_{2} + 1130373720288072 \beta_{3} + 5614974844934 \beta_{4} + 1161465625008 \beta_{5} - 6750274295622 \beta_{6}) q^{65} +(\)\(27\!\cdots\!01\)\( - \)\(17\!\cdots\!59\)\( \beta_{1} - 5774849053355151 \beta_{2} - 3872846254174767 \beta_{3} - 7229357201151 \beta_{4} - 589285695645 \beta_{5} - 12845438090478 \beta_{6}) q^{66} +(-\)\(31\!\cdots\!48\)\( - \)\(40\!\cdots\!19\)\( \beta_{1} - 221271116084163472 \beta_{2} + 10471303033851419 \beta_{3} + 28029315710634 \beta_{4} + 3303731577189 \beta_{5} + 1979775975844 \beta_{6}) q^{67} +(-\)\(32\!\cdots\!12\)\( - \)\(15\!\cdots\!44\)\( \beta_{1} - 147506023477843762 \beta_{2} - 6434359735755688 \beta_{3} - 18864816685636 \beta_{4} - 2800402364472 \beta_{5} - 3638600537808 \beta_{6}) q^{68} +(-\)\(55\!\cdots\!76\)\( - \)\(67\!\cdots\!45\)\( \beta_{1} - 80913222553111200 \beta_{2} - 4886294015718486 \beta_{3} + 956254209201 \beta_{4} + 5455684023912 \beta_{5} + 18280694053791 \beta_{6}) q^{69} +(-\)\(45\!\cdots\!65\)\( - \)\(13\!\cdots\!31\)\( \beta_{1} - 114159185845016529 \beta_{2} + 14016836246570283 \beta_{3} - 11529792238433 \beta_{4} + 678223072849 \beta_{5} + 4069338437094 \beta_{6}) q^{70} +(\)\(33\!\cdots\!48\)\( - \)\(93\!\cdots\!71\)\( \beta_{1} - 222653952173944324 \beta_{2} + 14124493367714212 \beta_{3} - 34421360637715 \beta_{4} + 14371112933010 \beta_{5} - 49117089642369 \beta_{6}) q^{71} +(\)\(32\!\cdots\!54\)\( - \)\(10\!\cdots\!48\)\( \beta_{1} + 81075352460381784 \beta_{2} - 8052630944146272 \beta_{3} - 45753584909922 \beta_{4}) q^{72} +(\)\(20\!\cdots\!38\)\( - \)\(30\!\cdots\!86\)\( \beta_{1} + 40245142714031538 \beta_{2} + 48916511830915332 \beta_{3} + 14405148969460 \beta_{4} - 24790856833694 \beta_{5} + 13923255987192 \beta_{6}) q^{73} +(-\)\(46\!\cdots\!08\)\( - \)\(18\!\cdots\!74\)\( \beta_{1} - 486874245518509068 \beta_{2} - 28144764708791856 \beta_{3} - 19968785609676 \beta_{4} - 11532162677520 \beta_{5} + 94134871306080 \beta_{6}) q^{74} +(\)\(23\!\cdots\!29\)\( + \)\(65\!\cdots\!60\)\( \beta_{1} + 153547475464434897 \beta_{2} - 26296843743692316 \beta_{3} - 30461911673301 \beta_{4} - 7374262029975 \beta_{5} + 41461161993531 \beta_{6}) q^{75} +(-\)\(65\!\cdots\!48\)\( + \)\(14\!\cdots\!72\)\( \beta_{1} - 1549811552319703292 \beta_{2} - 113596319312138480 \beta_{3} + 223454004818280 \beta_{4} + 24890712837552 \beta_{5} - 130577485582048 \beta_{6}) q^{76} +(\)\(13\!\cdots\!72\)\( + \)\(51\!\cdots\!72\)\( \beta_{1} - 419698001940418180 \beta_{2} + 6679819044489801 \beta_{3} - 8816899947037 \beta_{4} - 34589376715299 \beta_{5} + 50866730463675 \beta_{6}) q^{77} +(-\)\(38\!\cdots\!48\)\( + \)\(25\!\cdots\!18\)\( \beta_{1} - 1812529128642832296 \beta_{2} - 135020802895260804 \beta_{3} + 120078502987608 \beta_{4} - 19653697917900 \beta_{5} - 91342960928136 \beta_{6}) q^{78} +(\)\(63\!\cdots\!28\)\( - \)\(49\!\cdots\!60\)\( \beta_{1} + 854632899422858882 \beta_{2} + 204865068336022138 \beta_{3} - 44976876040612 \beta_{4} + 86028083293124 \beta_{5} - 58095701214916 \beta_{6}) q^{79} +(\)\(10\!\cdots\!00\)\( - \)\(93\!\cdots\!36\)\( \beta_{1} + 3480009768728745168 \beta_{2} - 82671745120300832 \beta_{3} + 40991451727704 \beta_{4} - 30668342130912 \beta_{5} + 69235270235328 \beta_{6}) q^{80} +\)\(52\!\cdots\!21\)\( q^{81} +(\)\(63\!\cdots\!03\)\( + \)\(41\!\cdots\!81\)\( \beta_{1} + 3451511524345831707 \beta_{2} + 78936517846388131 \beta_{3} - 212036369196341 \beta_{4} + 30171337493689 \beta_{5} + 162987372383510 \beta_{6}) q^{82} +(\)\(67\!\cdots\!64\)\( - \)\(12\!\cdots\!74\)\( \beta_{1} + 3538946040791289610 \beta_{2} - 178147069832273802 \beta_{3} + 171559463310298 \beta_{4} + 53036744270268 \beta_{5} - 208585851460242 \beta_{6}) q^{83} +(-\)\(10\!\cdots\!22\)\( + \)\(14\!\cdots\!68\)\( \beta_{1} - 3243919932521508681 \beta_{2}) q^{84} +(-\)\(19\!\cdots\!00\)\( - \)\(59\!\cdots\!82\)\( \beta_{1} - 3083794899713437305 \beta_{2} + 161342347658616626 \beta_{3} - 55100601598043 \beta_{4} - 65583917886911 \beta_{5} + 273180499295089 \beta_{6}) q^{85} +(\)\(18\!\cdots\!08\)\( - \)\(39\!\cdots\!20\)\( \beta_{1} + 15739499184228170300 \beta_{2} + 192242766164942308 \beta_{3} - 751483076925316 \beta_{4} + 236502301135980 \beta_{5} - 95984608595448 \beta_{6}) q^{86} +(-\)\(17\!\cdots\!50\)\( - \)\(36\!\cdots\!30\)\( \beta_{1} - 5218623294561868725 \beta_{2} - 43489672179991272 \beta_{3} + 394595071640163 \beta_{4} - 53890118275365 \beta_{5} + 333163899639855 \beta_{6}) q^{87} +(\)\(65\!\cdots\!68\)\( + \)\(32\!\cdots\!24\)\( \beta_{1} + 2446714360506659088 \beta_{2} + 566131584881059296 \beta_{3} - 1306551008107304 \beta_{4} - 220708749591008 \beta_{5} - 520276067873088 \beta_{6}) q^{88} +(\)\(75\!\cdots\!46\)\( + \)\(82\!\cdots\!69\)\( \beta_{1} + 16594499190989417512 \beta_{2} - 881598047170076498 \beta_{3} + 293710629390019 \beta_{4} - 632365299092544 \beta_{5} - 182902686595395 \beta_{6}) q^{89} +(-\)\(15\!\cdots\!85\)\( - \)\(45\!\cdots\!59\)\( \beta_{1} - 3850644582811490481 \beta_{2} + 472794669666678987 \beta_{3} - 388905471734337 \beta_{4} + 22876792454961 \beta_{5} + 137260754729766 \beta_{6}) q^{90} +(\)\(11\!\cdots\!38\)\( - \)\(56\!\cdots\!93\)\( \beta_{1} - 808697592934472073 \beta_{2} + 559801933214200355 \beta_{3} - 1032933739949027 \beta_{4} - 94951230198860 \beta_{5} + 94273007126011 \beta_{6}) q^{91} +(\)\(78\!\cdots\!76\)\( + \)\(15\!\cdots\!28\)\( \beta_{1} + 14173294980388072204 \beta_{2} - 1002032480530885368 \beta_{3} + 4223459780185396 \beta_{4} + 729687737997144 \beta_{5} - 863907073197168 \beta_{6}) q^{92} +(-\)\(21\!\cdots\!68\)\( + \)\(55\!\cdots\!94\)\( \beta_{1} - 6610085256469475529 \beta_{2} + 278529394238860536 \beta_{3} - 175319733477969 \beta_{4} + 721826850931047 \beta_{5} - 511450896209013 \beta_{6}) q^{93} +(-\)\(38\!\cdots\!18\)\( + \)\(59\!\cdots\!66\)\( \beta_{1} - 26903298971640396794 \beta_{2} + 1578350474515323726 \beta_{3} + 1585055680004454 \beta_{4} - 512272928694774 \beta_{5} + 3198408119722812 \beta_{6}) q^{94} +(-\)\(10\!\cdots\!80\)\( + \)\(41\!\cdots\!24\)\( \beta_{1} - 20537779777485400712 \beta_{2} + 1770329623043532808 \beta_{3} + 997783422654424 \beta_{4} + 132202871790288 \beta_{5} + 2290086743060808 \beta_{6}) q^{95} +(-\)\(25\!\cdots\!88\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} - 22881586023096831504 \beta_{2} - 1995004121680995552 \beta_{3} + 1486013024123352 \beta_{4} - 46997836031520 \beta_{5} - 2147614915223232 \beta_{6}) q^{96} +(\)\(10\!\cdots\!02\)\( + \)\(14\!\cdots\!40\)\( \beta_{1} - 34323140253829508146 \beta_{2} - 851510961894610810 \beta_{3} - 3891601190677588 \beta_{4} + 839141035397756 \beta_{5} - 3803840889788420 \beta_{6}) q^{97} +(\)\(40\!\cdots\!96\)\( - \)\(45\!\cdots\!01\)\( \beta_{1}) q^{98} +(\)\(44\!\cdots\!08\)\( + \)\(17\!\cdots\!08\)\( \beta_{1} - 14156616706978966020 \beta_{2} + 225313528888910889 \beta_{3} - 297398301914493 \beta_{4} - 1166716415203011 \beta_{5} + 1715759434122075 \beta_{6}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 60870q^{2} - 33480783q^{3} + 2201135476q^{4} - 2861618502q^{5} - 291139323030q^{6} + 4747561509943q^{7} + 9964333994280q^{8} + 160137547184727q^{9} + O(q^{10}) \) \( 7q + 60870q^{2} - 33480783q^{3} + 2201135476q^{4} - 2861618502q^{5} - 291139323030q^{6} + 4747561509943q^{7} + 9964333994280q^{8} + 160137547184727q^{9} - 472777770164028q^{10} + 135879674344284q^{11} - 10527962746508244q^{12} + 12138994948536170q^{13} + 41283438444318630q^{14} + 13687032584892438q^{15} + 1300607396354614288q^{16} - 385869676567281066q^{17} + 1392510356733476070q^{18} + 726075641163594644q^{19} + 8133031378709443032q^{20} - 22707439527650560767q^{21} - 40013549861901162600q^{22} + 81623233895572110192q^{23} - 47659100600287417320q^{24} - \)\(34\!\cdots\!51\)\(q^{25} + \)\(55\!\cdots\!00\)\(q^{26} - \)\(76\!\cdots\!63\)\(q^{27} + \)\(14\!\cdots\!24\)\(q^{28} + \)\(25\!\cdots\!06\)\(q^{29} + \)\(22\!\cdots\!32\)\(q^{30} + \)\(30\!\cdots\!52\)\(q^{31} + \)\(37\!\cdots\!80\)\(q^{32} - \)\(64\!\cdots\!96\)\(q^{33} + \)\(63\!\cdots\!00\)\(q^{34} - \)\(19\!\cdots\!98\)\(q^{35} + \)\(50\!\cdots\!36\)\(q^{36} + \)\(12\!\cdots\!66\)\(q^{37} - \)\(10\!\cdots\!20\)\(q^{38} - \)\(58\!\cdots\!30\)\(q^{39} - \)\(47\!\cdots\!16\)\(q^{40} - \)\(26\!\cdots\!54\)\(q^{41} - \)\(19\!\cdots\!70\)\(q^{42} - \)\(19\!\cdots\!80\)\(q^{43} - \)\(24\!\cdots\!88\)\(q^{44} - \)\(65\!\cdots\!22\)\(q^{45} - \)\(69\!\cdots\!80\)\(q^{46} - \)\(11\!\cdots\!48\)\(q^{47} - \)\(62\!\cdots\!72\)\(q^{48} + \)\(32\!\cdots\!07\)\(q^{49} + \)\(44\!\cdots\!86\)\(q^{50} + \)\(18\!\cdots\!54\)\(q^{51} + \)\(12\!\cdots\!40\)\(q^{52} - \)\(66\!\cdots\!14\)\(q^{53} - \)\(66\!\cdots\!30\)\(q^{54} + \)\(53\!\cdots\!32\)\(q^{55} + \)\(67\!\cdots\!20\)\(q^{56} - \)\(34\!\cdots\!36\)\(q^{57} - \)\(19\!\cdots\!40\)\(q^{58} + \)\(25\!\cdots\!80\)\(q^{59} - \)\(38\!\cdots\!08\)\(q^{60} + \)\(28\!\cdots\!86\)\(q^{61} + \)\(65\!\cdots\!80\)\(q^{62} + \)\(10\!\cdots\!23\)\(q^{63} + \)\(52\!\cdots\!04\)\(q^{64} + \)\(67\!\cdots\!64\)\(q^{65} + \)\(19\!\cdots\!00\)\(q^{66} - \)\(21\!\cdots\!12\)\(q^{67} - \)\(23\!\cdots\!68\)\(q^{68} - \)\(39\!\cdots\!48\)\(q^{69} - \)\(32\!\cdots\!72\)\(q^{70} + \)\(23\!\cdots\!20\)\(q^{71} + \)\(22\!\cdots\!80\)\(q^{72} + \)\(14\!\cdots\!06\)\(q^{73} - \)\(32\!\cdots\!80\)\(q^{74} + \)\(16\!\cdots\!19\)\(q^{75} - \)\(45\!\cdots\!88\)\(q^{76} + \)\(92\!\cdots\!16\)\(q^{77} - \)\(26\!\cdots\!00\)\(q^{78} + \)\(44\!\cdots\!48\)\(q^{79} + \)\(72\!\cdots\!92\)\(q^{80} + \)\(36\!\cdots\!47\)\(q^{81} + \)\(44\!\cdots\!40\)\(q^{82} + \)\(47\!\cdots\!76\)\(q^{83} - \)\(71\!\cdots\!56\)\(q^{84} - \)\(13\!\cdots\!08\)\(q^{85} + \)\(12\!\cdots\!00\)\(q^{86} - \)\(12\!\cdots\!14\)\(q^{87} + \)\(45\!\cdots\!40\)\(q^{88} + \)\(52\!\cdots\!86\)\(q^{89} - \)\(10\!\cdots\!08\)\(q^{90} + \)\(82\!\cdots\!30\)\(q^{91} + \)\(54\!\cdots\!36\)\(q^{92} - \)\(14\!\cdots\!88\)\(q^{93} - \)\(26\!\cdots\!00\)\(q^{94} - \)\(74\!\cdots\!28\)\(q^{95} - \)\(18\!\cdots\!20\)\(q^{96} + \)\(74\!\cdots\!42\)\(q^{97} + \)\(27\!\cdots\!70\)\(q^{98} + \)\(31\!\cdots\!24\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 678740466 x^{5} - 2954969748680 x^{4} + 92148363808982720 x^{3} + 418126965823508980224 x^{2} - 2978683903673490966301184 x - 14143029174281721602229028864\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 26128 \nu - 775699658 \)
\(\beta_{3}\)\(=\)\((\)\(-1805579 \nu^{6} + 8129637015 \nu^{5} + 1193361602560554 \nu^{4} - 215431344897505616 \nu^{3} - 165487976654590765959936 \nu^{2} + 57917753692735014466672128 \nu + 4788755976656875553490534045184\)\()/ 37522594970025984000 \)
\(\beta_{4}\)\(=\)\((\)\(19861369 \nu^{6} - 89426007165 \nu^{5} - 13126977628166094 \nu^{4} + 11750393536379057776 \nu^{3} + 1714629070574965202647296 \nu^{2} - 3766429430842992389914833408 \nu - 44052935678340281089314540737024\)\()/ 2345162185626624000 \)
\(\beta_{5}\)\(=\)\((\)\(-715708753 \nu^{6} + 11475545383605 \nu^{5} + 483454728032130078 \nu^{4} - 5150046832014358224112 \nu^{3} - 95706809698424030222852352 \nu^{2} + 432492726313806922361967223296 \nu + 3968868993332444765130177912447488\)\()/ 37522594970025984000 \)
\(\beta_{6}\)\(=\)\((\)\(1912820743 \nu^{6} - 11467240906755 \nu^{5} - 1206333174941152818 \nu^{4} + 1373032062976664797072 \nu^{3} + 156677507994496949964832512 \nu^{2} - 125793557861022481406559038976 \nu - 4243737937789549098874520350166528\)\()/ 37522594970025984000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 13064 \beta_{1} + 775699658\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{4} + 176 \beta_{3} + 11272 \beta_{2} + 814446688 \beta_{1} + 5066593285611\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2440 \beta_{6} + 844 \beta_{5} + 9869 \beta_{4} + 3987316 \beta_{3} + 583578350 \beta_{2} + 11697550465076 \beta_{1} + 315879862427642875\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3081016 \beta_{6} + 17120300 \beta_{5} + 601207983 \beta_{4} + 95762322036 \beta_{3} + 9828672269974 \beta_{2} + 433447351879040788 \beta_{1} + 4536730793945055353249\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(1598797265880 \beta_{6} + 634909365444 \beta_{5} + 9110350188545 \beta_{4} + 2962382943620572 \beta_{3} + 336959620208062418 \beta_{2} + 8452480029925229536892 \beta_{1} + 168109746203177430665189183\)\()/4\)

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
25164.8
10372.4
7115.98
−5176.82
−6093.44
−13495.0
−17887.0
−41633.7 −4.78297e6 1.19649e9 9.35390e9 1.99133e11 6.78223e11 −2.74625e13 2.28768e13 −3.89437e14
1.2 −12048.8 −4.78297e6 −3.91697e8 −3.60173e9 5.76290e10 6.78223e11 1.11881e13 2.28768e13 4.33965e13
1.3 −5535.95 −4.78297e6 −5.06224e8 −9.04739e9 2.64783e10 6.78223e11 5.77452e12 2.28768e13 5.00859e13
1.4 19049.6 −4.78297e6 −1.73982e8 2.04816e10 −9.11138e10 6.78223e11 −1.35415e13 2.28768e13 3.90167e14
1.5 20882.9 −4.78297e6 −1.00776e8 −7.49509e9 −9.98822e10 6.78223e11 −1.33159e13 2.28768e13 −1.56519e14
1.6 35686.0 −4.78297e6 7.36619e8 −1.68211e10 −1.70685e11 6.78223e11 7.12823e12 2.28768e13 −6.00279e14
1.7 44469.9 −4.78297e6 1.44070e9 4.26822e9 −2.12698e11 6.78223e11 4.01934e13 2.28768e13 1.89808e14
\(n\): 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.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.a.c 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 60870 T_{2}^{6} - 1127037480 T_{2}^{5} + \)\(11\!\cdots\!80\)\( T_{2}^{4} - \)\(12\!\cdots\!60\)\( T_{2}^{3} - \)\(24\!\cdots\!64\)\( T_{2}^{2} + \)\(23\!\cdots\!20\)\( T_{2} + \)\(17\!\cdots\!60\)\( \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(17\!\cdots\!60\)\( + \)\(23\!\cdots\!20\)\( T - \)\(24\!\cdots\!64\)\( T^{2} - 1200857624504156160 T^{3} + 118672738634880 T^{4} - 1127037480 T^{5} - 60870 T^{6} + T^{7} \)
$3$ \( ( 4782969 + T )^{7} \)
$5$ \( -\)\(33\!\cdots\!00\)\( - \)\(64\!\cdots\!00\)\( T + \)\(23\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} - \)\(25\!\cdots\!00\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{5} + 2861618502 T^{6} + T^{7} \)
$7$ \( ( -678223072849 + T )^{7} \)
$11$ \( -\)\(10\!\cdots\!36\)\( + \)\(26\!\cdots\!76\)\( T + \)\(12\!\cdots\!96\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} - \)\(29\!\cdots\!20\)\( T^{4} - \)\(77\!\cdots\!56\)\( T^{5} - 135879674344284 T^{6} + T^{7} \)
$13$ \( -\)\(31\!\cdots\!20\)\( - \)\(16\!\cdots\!00\)\( T - \)\(56\!\cdots\!88\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!80\)\( T^{4} - \)\(83\!\cdots\!40\)\( T^{5} - 12138994948536170 T^{6} + T^{7} \)
$17$ \( \)\(17\!\cdots\!64\)\( - \)\(41\!\cdots\!84\)\( T - \)\(97\!\cdots\!44\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} - \)\(75\!\cdots\!60\)\( T^{4} - \)\(74\!\cdots\!16\)\( T^{5} + 385869676567281066 T^{6} + T^{7} \)
$19$ \( -\)\(33\!\cdots\!36\)\( - \)\(57\!\cdots\!44\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(56\!\cdots\!80\)\( T^{3} - \)\(96\!\cdots\!80\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{5} - 726075641163594644 T^{6} + T^{7} \)
$23$ \( \)\(35\!\cdots\!00\)\( - \)\(84\!\cdots\!00\)\( T - \)\(29\!\cdots\!00\)\( T^{2} + \)\(60\!\cdots\!00\)\( T^{3} + \)\(88\!\cdots\!40\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} - 81623233895572110192 T^{6} + T^{7} \)
$29$ \( \)\(52\!\cdots\!08\)\( - \)\(10\!\cdots\!92\)\( T + \)\(59\!\cdots\!72\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!80\)\( T^{4} - \)\(24\!\cdots\!76\)\( T^{5} - \)\(25\!\cdots\!06\)\( T^{6} + T^{7} \)
$31$ \( \)\(21\!\cdots\!00\)\( - \)\(35\!\cdots\!00\)\( T - \)\(61\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!40\)\( T^{4} - \)\(82\!\cdots\!04\)\( T^{5} - \)\(30\!\cdots\!52\)\( T^{6} + T^{7} \)
$37$ \( \)\(10\!\cdots\!56\)\( + \)\(79\!\cdots\!36\)\( T - \)\(36\!\cdots\!36\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!80\)\( T^{4} - \)\(92\!\cdots\!16\)\( T^{5} - \)\(12\!\cdots\!66\)\( T^{6} + T^{7} \)
$41$ \( \)\(26\!\cdots\!92\)\( + \)\(90\!\cdots\!32\)\( T + \)\(70\!\cdots\!48\)\( T^{2} - \)\(70\!\cdots\!40\)\( T^{3} - \)\(11\!\cdots\!40\)\( T^{4} - \)\(23\!\cdots\!96\)\( T^{5} + \)\(26\!\cdots\!54\)\( T^{6} + T^{7} \)
$43$ \( -\)\(41\!\cdots\!60\)\( + \)\(60\!\cdots\!40\)\( T + \)\(93\!\cdots\!04\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} - \)\(28\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!80\)\( T^{6} + T^{7} \)
$47$ \( \)\(12\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T + \)\(69\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} - \)\(13\!\cdots\!80\)\( T^{4} - \)\(95\!\cdots\!04\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} + T^{7} \)
$53$ \( \)\(10\!\cdots\!16\)\( - \)\(40\!\cdots\!44\)\( T + \)\(47\!\cdots\!64\)\( T^{2} + \)\(73\!\cdots\!20\)\( T^{3} - \)\(18\!\cdots\!20\)\( T^{4} - \)\(46\!\cdots\!76\)\( T^{5} + \)\(66\!\cdots\!14\)\( T^{6} + T^{7} \)
$59$ \( -\)\(18\!\cdots\!20\)\( - \)\(14\!\cdots\!20\)\( T + \)\(14\!\cdots\!28\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} - \)\(41\!\cdots\!20\)\( T^{4} + \)\(21\!\cdots\!60\)\( T^{5} - \)\(25\!\cdots\!80\)\( T^{6} + T^{7} \)
$61$ \( -\)\(66\!\cdots\!32\)\( + \)\(25\!\cdots\!48\)\( T + \)\(18\!\cdots\!72\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!80\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} - \)\(28\!\cdots\!86\)\( T^{6} + T^{7} \)
$67$ \( -\)\(16\!\cdots\!64\)\( - \)\(19\!\cdots\!88\)\( T + \)\(21\!\cdots\!96\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} - \)\(21\!\cdots\!40\)\( T^{4} - \)\(32\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!12\)\( T^{6} + T^{7} \)
$71$ \( \)\(23\!\cdots\!00\)\( - \)\(27\!\cdots\!00\)\( T - \)\(22\!\cdots\!00\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!00\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{5} - \)\(23\!\cdots\!20\)\( T^{6} + T^{7} \)
$73$ \( -\)\(49\!\cdots\!36\)\( + \)\(39\!\cdots\!44\)\( T - \)\(68\!\cdots\!04\)\( T^{2} - \)\(88\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!20\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} - \)\(14\!\cdots\!06\)\( T^{6} + T^{7} \)
$79$ \( -\)\(23\!\cdots\!24\)\( - \)\(31\!\cdots\!48\)\( T + \)\(27\!\cdots\!96\)\( T^{2} + \)\(38\!\cdots\!60\)\( T^{3} - \)\(78\!\cdots\!40\)\( T^{4} - \)\(41\!\cdots\!04\)\( T^{5} - \)\(44\!\cdots\!48\)\( T^{6} + T^{7} \)
$83$ \( \)\(59\!\cdots\!48\)\( + \)\(21\!\cdots\!68\)\( T + \)\(25\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} - \)\(81\!\cdots\!20\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} - \)\(47\!\cdots\!76\)\( T^{6} + T^{7} \)
$89$ \( -\)\(14\!\cdots\!64\)\( + \)\(32\!\cdots\!76\)\( T - \)\(17\!\cdots\!16\)\( T^{2} - \)\(64\!\cdots\!60\)\( T^{3} + \)\(66\!\cdots\!00\)\( T^{4} - \)\(51\!\cdots\!76\)\( T^{5} - \)\(52\!\cdots\!86\)\( T^{6} + T^{7} \)
$97$ \( -\)\(62\!\cdots\!08\)\( + \)\(87\!\cdots\!36\)\( T - \)\(37\!\cdots\!68\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} - \)\(74\!\cdots\!42\)\( T^{6} + T^{7} \)
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