Properties

Label 5.18.b.a
Level 5
Weight 18
Character orbit 5.b
Analytic conductor 9.161
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.16110436723\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{8}\cdot 5^{12}\cdot 11 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -72387 + \beta_{3} ) q^{4} + ( 47400 - 208 \beta_{1} + 24 \beta_{2} - \beta_{4} ) q^{5} + ( 44727 + 9 \beta_{3} + \beta_{6} ) q^{6} + ( 18407 \beta_{1} - 65 \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -65901 \beta_{1} + 1409 \beta_{2} + 3 \beta_{3} - 27 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{8} + ( -29364543 + 12 \beta_{1} + 15 \beta_{2} + 201 \beta_{3} + 60 \beta_{4} - 12 \beta_{6} - 3 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -72387 + \beta_{3} ) q^{4} + ( 47400 - 208 \beta_{1} + 24 \beta_{2} - \beta_{4} ) q^{5} + ( 44727 + 9 \beta_{3} + \beta_{6} ) q^{6} + ( 18407 \beta_{1} - 65 \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -65901 \beta_{1} + 1409 \beta_{2} + 3 \beta_{3} - 27 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{8} + ( -29364543 + 12 \beta_{1} + 15 \beta_{2} + 201 \beta_{3} + 60 \beta_{4} - 12 \beta_{6} - 3 \beta_{7} ) q^{9} + ( 41196275 + 59338 \beta_{1} + 1889 \beta_{2} - 1148 \beta_{3} - 11 \beta_{4} - 5 \beta_{5} - 45 \beta_{6} - 9 \beta_{7} ) q^{10} + ( 57912072 + 88 \beta_{1} + 110 \beta_{2} - 414 \beta_{3} + 440 \beta_{4} + 108 \beta_{6} - 22 \beta_{7} ) q^{11} + ( -1081527 \beta_{1} + 106083 \beta_{2} + 153 \beta_{3} - 1497 \beta_{4} + 33 \beta_{5} - 51 \beta_{7} ) q^{12} + ( 508024 \beta_{1} + 31157 \beta_{2} + 207 \beta_{3} - 2162 \beta_{4} - 92 \beta_{5} - 69 \beta_{7} ) q^{13} + ( -3742238499 + 416 \beta_{1} + 520 \beta_{2} + 49875 \beta_{3} + 2080 \beta_{4} + 135 \beta_{6} - 104 \beta_{7} ) q^{14} + ( 3830778300 - 1413099 \beta_{1} - 327627 \beta_{2} - 45366 \beta_{3} - 927 \beta_{4} + 165 \beta_{5} + 860 \beta_{6} - 78 \beta_{7} ) q^{15} + ( 3856000196 - 192 \beta_{1} - 240 \beta_{2} - 76876 \beta_{3} - 960 \beta_{4} - 2664 \beta_{6} + 48 \beta_{7} ) q^{16} + ( 21943312 \beta_{1} + 1943286 \beta_{2} - 1530 \beta_{3} + 14796 \beta_{4} - 504 \beta_{5} + 510 \beta_{7} ) q^{17} + ( -55012827 \beta_{1} - 2702550 \beta_{2} - 2610 \beta_{3} + 27354 \beta_{4} + 1254 \beta_{5} + 870 \beta_{7} ) q^{18} + ( -2576964160 - 7656 \beta_{1} - 9570 \beta_{2} + 99442 \beta_{3} - 38280 \beta_{4} + 1044 \beta_{6} + 1914 \beta_{7} ) q^{19} + ( -5944822425 + 157405901 \beta_{1} - 9095393 \beta_{2} + 48240 \beta_{3} + 17907 \beta_{4} - 2475 \beta_{5} - 5400 \beta_{6} + 2545 \beta_{7} ) q^{20} + ( -9379962378 - 11748 \beta_{1} - 14685 \beta_{2} + 489237 \beta_{3} - 58740 \beta_{4} + 24480 \beta_{6} + 2937 \beta_{7} ) q^{21} + ( 104066944 \beta_{1} + 21518012 \beta_{2} - 108 \beta_{3} + 5788 \beta_{4} + 4708 \beta_{5} + 36 \beta_{7} ) q^{22} + ( -662230271 \beta_{1} - 9349929 \beta_{2} + 2340 \beta_{3} - 33135 \beta_{4} - 9735 \beta_{5} - 780 \beta_{7} ) q^{23} + ( 221092642020 + 40128 \beta_{1} + 50160 \beta_{2} - 3748332 \beta_{3} + 200640 \beta_{4} - 30616 \beta_{6} - 10032 \beta_{7} ) q^{24} + ( -223656179375 + 787769500 \beta_{1} + 2795675 \beta_{2} + 2568825 \beta_{3} - 114950 \beta_{4} + 22000 \beta_{5} - 4500 \beta_{6} - 16275 \beta_{7} ) q^{25} + ( -104862649278 + 111136 \beta_{1} + 138920 \beta_{2} + 535374 \beta_{3} + 555680 \beta_{4} - 96822 \beta_{6} - 27784 \beta_{7} ) q^{26} + ( 2097357606 \beta_{1} - 15720174 \beta_{2} + 53352 \beta_{3} - 563418 \beta_{4} - 29898 \beta_{5} - 17784 \beta_{7} ) q^{27} + ( -7567092189 \beta_{1} + 81933057 \beta_{2} + 107667 \beta_{3} - 1032243 \beta_{4} + 44427 \beta_{5} - 35889 \beta_{7} ) q^{28} + ( -509877228090 - 29744 \beta_{1} - 37180 \beta_{2} + 8392668 \beta_{3} - 148720 \beta_{4} + 274968 \beta_{6} + 7436 \beta_{7} ) q^{29} + ( 302123000925 + 9492732204 \beta_{1} + 126456928 \beta_{2} - 5548965 \beta_{3} + 235128 \beta_{4} - 125400 \beta_{5} + 260775 \beta_{6} + 17280 \beta_{7} ) q^{30} + ( 1416166082312 - 366288 \beta_{1} - 457860 \beta_{2} - 7107356 \beta_{3} - 1831440 \beta_{4} - 28656 \beta_{6} + 91572 \beta_{7} ) q^{31} + ( 4819824508 \beta_{1} - 516963436 \beta_{2} - 145860 \beta_{3} + 1593636 \beta_{4} + 135036 \beta_{5} + 48620 \beta_{7} ) q^{32} + ( -16763626992 \beta_{1} - 85496568 \beta_{2} - 679068 \beta_{3} + 6696696 \beta_{4} - 93984 \beta_{5} + 226356 \beta_{7} ) q^{33} + ( -4550780454104 - 328896 \beta_{1} - 411120 \beta_{2} + 38744440 \beta_{3} - 1644480 \beta_{4} - 1375056 \beta_{6} + 82224 \beta_{7} ) q^{34} + ( 502457987400 + 8041746058 \beta_{1} + 284829869 \beta_{2} - 50895138 \beta_{3} + 740594 \beta_{4} + 449970 \beta_{5} - 1654020 \beta_{6} + 231446 \beta_{7} ) q^{35} + ( 7466219054091 + 132480 \beta_{1} + 165600 \beta_{2} - 11127897 \beta_{3} + 662400 \beta_{4} + 1951056 \beta_{6} - 33120 \beta_{7} ) q^{36} + ( 18472639092 \beta_{1} - 452104071 \beta_{2} - 870921 \beta_{3} + 8275194 \beta_{4} - 434016 \beta_{5} + 290307 \beta_{7} ) q^{37} + ( -14546171688 \beta_{1} + 560805932 \beta_{2} + 1515108 \beta_{3} - 15307764 \beta_{4} - 156684 \beta_{5} - 505036 \beta_{7} ) q^{38} + ( 5039807552784 - 184272 \beta_{1} - 230340 \beta_{2} - 107017116 \beta_{3} - 921360 \beta_{4} + 4198968 \beta_{6} + 46068 \beta_{7} ) q^{39} + ( -26218441521500 - 4435134775 \beta_{1} - 782185565 \beta_{2} + 217640365 \beta_{3} - 5084065 \beta_{4} - 799975 \beta_{5} + 4854600 \beta_{6} - 1097955 \beta_{7} ) q^{40} + ( 12152156605932 + 1249820 \beta_{1} + 1562275 \beta_{2} + 132045525 \beta_{3} + 6249100 \beta_{4} - 7966836 \beta_{6} - 312455 \beta_{7} ) q^{41} + ( -69801687078 \beta_{1} + 6470395962 \beta_{2} + 6217470 \beta_{3} - 61267926 \beta_{4} + 906774 \beta_{5} - 2072490 \beta_{7} ) q^{42} + ( 101997707882 \beta_{1} - 2526497897 \beta_{2} - 235872 \beta_{3} + 4072250 \beta_{4} + 1713530 \beta_{5} + 78624 \beta_{7} ) q^{43} + ( -14544731651364 + 9537792 \beta_{1} + 11922240 \beta_{2} - 287934324 \beta_{3} + 47688960 \beta_{4} - 7654176 \beta_{6} - 2384448 \beta_{7} ) q^{44} + ( -45855249750450 - 125773371096 \beta_{1} - 4455817917 \beta_{2} - 111602295 \beta_{3} + 11173533 \beta_{4} - 887700 \beta_{5} - 4411800 \beta_{6} + 1776765 \beta_{7} ) q^{45} + ( 135153032712517 + 4873440 \beta_{1} + 6091800 \beta_{2} - 306074165 \beta_{3} + 24367200 \beta_{4} + 9216279 \beta_{6} - 1218360 \beta_{7} ) q^{46} + ( -100019269969 \beta_{1} + 661266191 \beta_{2} - 13176924 \beta_{3} + 131211375 \beta_{4} - 557865 \beta_{5} + 4392308 \beta_{7} ) q^{47} + ( 544248926556 \beta_{1} + 773929780 \beta_{2} - 736740 \beta_{3} + 2681796 \beta_{4} - 4685604 \beta_{5} + 245580 \beta_{7} ) q^{48} + ( -107071794702707 - 33522180 \beta_{1} - 41902725 \beta_{2} + 636161085 \beta_{3} - 167610900 \beta_{4} + 7297524 \beta_{6} + 8380545 \beta_{7} ) q^{49} + ( -160408287738750 - 541776274375 \beta_{1} + 1886984300 \beta_{2} - 253052550 \beta_{3} - 18360700 \beta_{4} + 9289500 \beta_{5} - 13965750 \beta_{6} - 1177900 \beta_{7} ) q^{50} + ( 308538689303112 - 39554064 \beta_{1} - 49442580 \beta_{2} + 953034228 \beta_{3} - 197770320 \beta_{4} + 28859296 \beta_{6} + 9888516 \beta_{7} ) q^{51} + ( -110663296578 \beta_{1} - 19600836246 \beta_{2} + 784926 \beta_{3} - 12186750 \beta_{4} - 4337490 \beta_{5} - 261642 \beta_{7} ) q^{52} + ( 573385946560 \beta_{1} - 8956023429 \beta_{2} - 21326607 \beta_{3} + 214785258 \beta_{4} + 1519188 \beta_{5} + 7108869 \beta_{7} ) q^{53} + ( -426051221847210 + 31217472 \beta_{1} + 39021840 \beta_{2} + 2509203978 \beta_{3} + 156087360 \beta_{4} - 771246 \beta_{6} - 7804368 \beta_{7} ) q^{54} + ( -255339153344700 + 363364433324 \beta_{1} + 25329012778 \beta_{2} - 2082186450 \beta_{3} + 62499728 \beta_{4} - 22049500 \beta_{5} + 38479500 \beta_{6} + 7122150 \beta_{7} ) q^{55} + ( 1045377040834860 + 73943104 \beta_{1} + 92428880 \beta_{2} - 4457049156 \beta_{3} + 369715520 \beta_{4} - 104852232 \beta_{6} - 18485776 \beta_{7} ) q^{56} + ( -431096158824 \beta_{1} - 23799797864 \beta_{2} + 27158508 \beta_{3} - 251324400 \beta_{4} + 20260680 \beta_{5} - 9052836 \beta_{7} ) q^{57} + ( -1555204072682 \beta_{1} + 74272382696 \beta_{2} + 64040184 \beta_{3} - 615366104 \beta_{4} + 25035736 \beta_{5} - 21346728 \beta_{7} ) q^{58} + ( -136426189030080 + 62210888 \beta_{1} + 77763610 \beta_{2} - 5780641002 \beta_{3} + 311054440 \beta_{4} - 30027780 \beta_{6} - 15552722 \beta_{7} ) q^{59} + ( -1434922388513100 + 802390039203 \beta_{1} + 8189650689 \beta_{2} + 6817672407 \beta_{3} - 132236211 \beta_{4} + 5910795 \beta_{5} + 9785280 \beta_{6} - 25579569 \beta_{7} ) q^{60} + ( 1008091376346872 + 18409116 \beta_{1} + 23011395 \beta_{2} + 9653950837 \beta_{3} + 92045580 \beta_{4} + 35598600 \beta_{6} - 4602279 \beta_{7} ) q^{61} + ( 2322122703000 \beta_{1} - 7531616344 \beta_{2} + 26988600 \beta_{3} - 313448856 \beta_{4} - 43562856 \beta_{5} - 8996200 \beta_{7} ) q^{62} + ( -2638880795169 \beta_{1} - 95888994399 \beta_{2} + 1966356 \beta_{3} - 84710529 \beta_{4} - 65046969 \beta_{5} - 655452 \beta_{7} ) q^{63} + ( -452020040533232 - 132683520 \beta_{1} - 165854400 \beta_{2} - 2988931888 \beta_{3} - 663417600 \beta_{4} + 209143008 \beta_{6} + 33170880 \beta_{7} ) q^{64} + ( -1498688694737700 + 2005953285596 \beta_{1} + 15069190963 \beta_{2} - 1800954291 \beta_{3} - 167654462 \beta_{4} + 89279040 \beta_{5} - 99301140 \beta_{6} - 6157103 \beta_{7} ) q^{65} + ( 3414855863369844 - 199934592 \beta_{1} - 249918240 \beta_{2} - 4780536756 \beta_{3} - 999672960 \beta_{4} + 328617468 \beta_{6} + 49983648 \beta_{7} ) q^{66} + ( 1391989209526 \beta_{1} + 183517901891 \beta_{2} - 136358352 \beta_{3} + 1386611206 \beta_{4} + 23027686 \beta_{5} + 45452784 \beta_{7} ) q^{67} + ( -6476532887336 \beta_{1} + 8642059976 \beta_{2} - 210942888 \beta_{3} + 2131535592 \beta_{4} + 22106712 \beta_{5} + 70314296 \beta_{7} ) q^{68} + ( -1509873699670626 + 5411628 \beta_{1} + 6764535 \beta_{2} - 2571511455 \beta_{3} + 27058140 \beta_{4} - 677345384 \beta_{6} - 1352907 \beta_{7} ) q^{69} + ( -1648866401712225 + 6844000882432 \beta_{1} - 429989849716 \beta_{2} - 5095747755 \beta_{3} + 1198409884 \beta_{4} - 193790300 \beta_{5} - 214174575 \beta_{6} + 167693460 \beta_{7} ) q^{70} + ( 3155186507267592 + 165941088 \beta_{1} + 207426360 \beta_{2} - 3221312184 \beta_{3} + 829705440 \beta_{4} - 312136200 \beta_{6} - 41485272 \beta_{7} ) q^{71} + ( 1624886221989 \beta_{1} + 64829040231 \beta_{2} - 148427595 \beta_{3} + 1634944803 \beta_{4} + 150668853 \beta_{5} + 49475865 \beta_{7} ) q^{72} + ( -3716713373280 \beta_{1} - 349424653620 \beta_{2} + 96501168 \beta_{3} - 780344064 \beta_{4} + 184667616 \beta_{5} - 32167056 \beta_{7} ) q^{73} + ( -3737815438791114 - 126013536 \beta_{1} - 157516920 \beta_{2} + 46604493978 \beta_{3} - 630067680 \beta_{4} + 785856006 \beta_{6} + 31503384 \beta_{7} ) q^{74} + ( 479904464610000 - 1443682275000 \beta_{1} + 420910579475 \beta_{2} - 17896356600 \beta_{3} - 1343132400 \beta_{4} + 20889000 \beta_{5} + 1030221000 \beta_{6} - 221467800 \beta_{7} ) q^{75} + ( 2595954341834820 - 404988672 \beta_{1} - 506235840 \beta_{2} - 20263898092 \beta_{3} - 2024943360 \beta_{4} - 940761504 \beta_{6} + 101247168 \beta_{7} ) q^{76} + ( -5395542014100 \beta_{1} + 265783789992 \beta_{2} + 839844180 \beta_{3} - 8860100844 \beta_{4} - 461659044 \beta_{5} - 279948060 \beta_{7} ) q^{77} + ( 18361962163968 \beta_{1} + 796257068664 \beta_{2} + 234139176 \beta_{3} - 2648351688 \beta_{4} - 306959928 \beta_{5} - 78046392 \beta_{7} ) q^{78} + ( 403731542216360 + 648874512 \beta_{1} + 811093140 \beta_{2} + 8718773260 \beta_{3} + 3244372560 \beta_{4} + 1330370784 \beta_{6} - 162218628 \beta_{7} ) q^{79} + ( 157925908173900 - 32656270129828 \beta_{1} + 147331171364 \beta_{2} + 15320083320 \beta_{3} - 1751567036 \beta_{4} + 491036700 \beta_{5} - 415222200 \beta_{6} - 199827940 \beta_{7} ) q^{80} + ( -6169192625650329 + 1709696268 \beta_{1} + 2137120335 \beta_{2} + 22767123465 \beta_{3} + 8548481340 \beta_{4} + 1256362380 \beta_{6} - 427424067 \beta_{7} ) q^{81} + ( -4328584209568 \beta_{1} - 1632715080974 \beta_{2} - 784846746 \beta_{3} + 8272104674 \beta_{4} + 423637214 \beta_{5} + 261615582 \beta_{7} ) q^{82} + ( 42284768665366 \beta_{1} - 92403518373 \beta_{2} + 1088056080 \beta_{3} - 10918393050 \beta_{4} - 37832250 \beta_{5} - 362685360 \beta_{7} ) q^{83} + ( 12679993416547836 + 271497600 \beta_{1} + 339372000 \beta_{2} - 168009495444 \beta_{3} + 1357488000 \beta_{4} - 5490979632 \beta_{6} - 67874400 \beta_{7} ) q^{84} + ( 3211712796223400 - 1893274171052 \beta_{1} + 970569612554 \beta_{2} + 50667180682 \beta_{3} + 4159347904 \beta_{4} - 517984580 \beta_{5} - 3335553720 \beta_{6} + 459943506 \beta_{7} ) q^{85} + ( -20639104846194093 - 795855424 \beta_{1} - 994819280 \beta_{2} + 74704079373 \beta_{3} - 3979277120 \beta_{4} + 2488712877 \beta_{6} + 198963856 \beta_{7} ) q^{86} + ( -57586162586160 \beta_{1} - 125010710070 \beta_{2} + 250354728 \beta_{3} - 1901361696 \beta_{4} + 602185584 \beta_{5} - 83451576 \beta_{7} ) q^{87} + ( 34476681800420 \beta_{1} + 464403198988 \beta_{2} - 3194366940 \beta_{3} + 32225680316 \beta_{4} + 282010916 \beta_{5} + 1064788980 \beta_{7} ) q^{88} + ( 11620498441953330 - 1264942416 \beta_{1} - 1581178020 \beta_{2} + 44887052676 \beta_{3} - 6324712080 \beta_{4} + 5346656424 \beta_{6} + 316235604 \beta_{7} ) q^{89} + ( 25789476802727925 - 32102438957544 \beta_{1} - 1064049763587 \beta_{2} - 23752396146 \beta_{3} + 413179113 \beta_{4} - 551495385 \beta_{5} + 5508600285 \beta_{6} + 297137307 \beta_{7} ) q^{90} + ( -28198958779601208 - 5655431136 \beta_{1} - 7069288920 \beta_{2} - 116880532776 \beta_{3} - 28277155680 \beta_{4} - 3971913912 \beta_{6} + 1413857784 \beta_{7} ) q^{91} + ( 86169264784315 \beta_{1} + 293426745353 \beta_{2} - 141072981 \beta_{3} - 434556171 \beta_{4} - 1845285981 \beta_{5} + 47024327 \beta_{7} ) q^{92} + ( 5917370374488 \beta_{1} - 2845164562160 \beta_{2} - 202595616 \beta_{3} + 2528524248 \beta_{4} + 502568088 \beta_{5} + 67531872 \beta_{7} ) q^{93} + ( 20326570585233481 - 4406205408 \beta_{1} - 5507756760 \beta_{2} + 72530251943 \beta_{3} - 22031027040 \beta_{4} + 3967746219 \beta_{6} + 1101551352 \beta_{7} ) q^{94} + ( 25539964438156500 + 62729794344900 \beta_{1} - 610860219710 \beta_{2} + 121745969910 \beta_{3} - 3313574960 \beta_{4} + 684707100 \beta_{5} - 460331100 \beta_{6} - 516937970 \beta_{7} ) q^{95} + ( -81787902874192368 + 6949536000 \beta_{1} + 8686920000 \beta_{2} + 199704655824 \beta_{3} + 34747680000 \beta_{4} - 4202206752 \beta_{6} - 1737384000 \beta_{7} ) q^{96} + ( -63862332831592 \beta_{1} + 2533572101914 \beta_{2} - 4764817998 \beta_{3} + 48740790092 \beta_{4} + 1092610112 \beta_{5} + 1588272666 \beta_{7} ) q^{97} + ( -184503699591167 \beta_{1} + 3350360090706 \beta_{2} + 7579740294 \beta_{3} - 75864845886 \beta_{4} - 67442946 \beta_{5} - 2526580098 \beta_{7} ) q^{98} + ( -7040916404994096 + 6809784696 \beta_{1} + 8512230870 \beta_{2} + 281271892986 \beta_{3} + 34048923480 \beta_{4} - 12011773068 \beta_{6} - 1702446174 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 579096q^{4} + 379200q^{5} + 357816q^{6} - 234916344q^{9} + O(q^{10}) \) \( 8q - 579096q^{4} + 379200q^{5} + 357816q^{6} - 234916344q^{9} + 329570200q^{10} + 463296576q^{11} - 29937907992q^{14} + 30646226400q^{15} + 30848001568q^{16} - 20615713280q^{19} - 47558579400q^{20} - 75039699024q^{21} + 1768741136160q^{24} - 1789249435000q^{25} - 838901194224q^{26} - 4079017824720q^{29} + 2416984007400q^{30} + 11329328658496q^{31} - 36406243632832q^{34} + 4019663899200q^{35} + 59729752432728q^{36} + 40318460422272q^{39} - 209747532172000q^{40} + 97217252847456q^{41} - 116357853210912q^{44} - 366841998003600q^{45} + 1081224261700136q^{46} - 856574357621656q^{49} - 1283266301910000q^{50} + 2468309514424896q^{51} - 3408409774777680q^{54} - 2042713226757600q^{55} + 8363016326678880q^{56} - 1091409512240640q^{59} - 11479379108104800q^{60} + 8064731010774976q^{61} - 3616160324265856q^{64} - 11989509557901600q^{65} + 27318846906958752q^{66} - 12078989597365008q^{69} - 13190931213697800q^{70} + 25241492058140736q^{71} - 29902523510328912q^{74} + 3839235716880000q^{75} + 20767634734678560q^{76} + 3229852337730880q^{79} + 1263407265391200q^{80} - 49353541005202632q^{81} + 101439947332382688q^{84} + 25693702369787200q^{85} - 165112838769552744q^{86} + 92963987535626640q^{89} + 206315814421823400q^{90} - 225591670236809664q^{91} + 162612564681867848q^{94} + 204319715505252000q^{95} - 654303222993538944q^{96} - 56327331239952768q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 203459 x^{6} + 12362849196 x^{4} + 237701205446144 x^{2} + 1320400799499206656\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -19 \nu^{7} - 3990585 \nu^{5} - 233779723524 \nu^{3} - 2952850120132352 \nu \)\()/ 7999594960896 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 203459 \)
\(\beta_{4}\)\(=\)\((\)\(-19613 \nu^{7} - 2309984 \nu^{6} - 3663672279 \nu^{5} - 423805335072 \nu^{4} - 186725279011644 \nu^{3} - 20625689117860992 \nu^{2} - 2138120976475779328 \nu - 200800987462872315904\)\()/ 66663291340800 \)
\(\beta_{5}\)\(=\)\((\)\(-699511 \nu^{7} + 6929952 \nu^{6} - 125502752613 \nu^{5} + 1271416005216 \nu^{4} - 5507559017590068 \nu^{3} + 61877067353582976 \nu^{2} - 22205687366696911616 \nu + 602402962388616947712\)\()/ 199989874022400 \)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 8925 \nu^{4} - 13674601908 \nu^{2} - 261009316476832 \)\()/32033232\)
\(\beta_{7}\)\(=\)\((\)\(-235831 \nu^{7} + 41579712 \nu^{6} - 44063831973 \nu^{5} + 7628496031296 \nu^{4} - 2246547841227828 \nu^{3} + 371742379819151616 \nu^{2} - 25730952986914224896 \nu + 3638831618198935775232\)\()/ 39997974804480 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 203459\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{5} - 27 \beta_{4} + 3 \beta_{3} + 1409 \beta_{2} - 328045 \beta_{1}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{7} - 666 \beta_{6} - 240 \beta_{4} - 117523 \beta_{3} - 60 \beta_{2} - 48 \beta_{1} + 16669866289\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(143227 \beta_{7} - 359457 \beta_{5} + 3937353 \beta_{4} - 429681 \beta_{3} - 313921307 \beta_{2} + 31317568479 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(107100 \beta_{7} + 122188878 \beta_{6} - 2142000 \beta_{4} + 12625709133 \beta_{3} - 535500 \beta_{2} - 428400 \beta_{1} - 1589405007063119\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-17777883909 \beta_{7} + 38584449567 \beta_{5} - 494752067703 \beta_{4} + 53333651727 \beta_{3} + 45228277303269 \beta_{2} - 3162976320308897 \beta_{1}\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
330.281i
256.320i
137.667i
98.5951i
98.5951i
137.667i
256.320i
330.281i
660.562i 11309.3i −305271. −21187.7 + 873207.i −7.47047e6 2.37065e7i 1.15069e8i 1.24087e6 5.76808e8 + 1.39958e7i
4.2 512.640i 18245.5i −131728. 346646. 801733.i 9.35337e6 1.34806e7i 336370.i −2.03757e8 −4.11001e8 1.77705e8i
4.3 275.335i 2990.60i 55262.8 −756669. + 436339.i 823415. 2.43909e7i 5.13044e7i 1.20196e8 1.20139e8 + 2.08337e8i
4.4 197.190i 12817.1i 92188.0 620811. 614437.i −2.52741e6 4.49140e6i 4.40247e7i −3.51381e7 −1.21161e8 1.22418e8i
4.5 197.190i 12817.1i 92188.0 620811. + 614437.i −2.52741e6 4.49140e6i 4.40247e7i −3.51381e7 −1.21161e8 + 1.22418e8i
4.6 275.335i 2990.60i 55262.8 −756669. 436339.i 823415. 2.43909e7i 5.13044e7i 1.20196e8 1.20139e8 2.08337e8i
4.7 512.640i 18245.5i −131728. 346646. + 801733.i 9.35337e6 1.34806e7i 336370.i −2.03757e8 −4.11001e8 + 1.77705e8i
4.8 660.562i 11309.3i −305271. −21187.7 873207.i −7.47047e6 2.37065e7i 1.15069e8i 1.24087e6 5.76808e8 1.39958e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{18}^{\mathrm{new}}(5, [\chi])\).