Properties

Label 10.18.b.a
Level 10
Weight 18
Character orbit 10.b
Analytic conductor 18.322
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3222087345\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{4}\cdot 5^{11} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} ) q^{3} -65536 q^{4} + ( -153195 - 272 \beta_{1} - 28 \beta_{2} - \beta_{3} ) q^{5} + ( -121856 - \beta_{3} - \beta_{5} - \beta_{6} ) q^{6} + ( 27250 \beta_{1} + 351 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + 65536 \beta_{1} q^{8} + ( -45397813 + 58 \beta_{1} + 53 \beta_{2} - 139 \beta_{3} + 32 \beta_{4} - 28 \beta_{5} + 5 \beta_{6} + 38 \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -2 \beta_{1} - \beta_{2} ) q^{3} -65536 q^{4} + ( -153195 - 272 \beta_{1} - 28 \beta_{2} - \beta_{3} ) q^{5} + ( -121856 - \beta_{3} - \beta_{5} - \beta_{6} ) q^{6} + ( 27250 \beta_{1} + 351 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + 65536 \beta_{1} q^{8} + ( -45397813 + 58 \beta_{1} + 53 \beta_{2} - 139 \beta_{3} + 32 \beta_{4} - 28 \beta_{5} + 5 \beta_{6} + 38 \beta_{7} ) q^{9} + ( -17597440 + 153665 \beta_{1} + 3184 \beta_{2} - 19 \beta_{3} + 64 \beta_{4} - 43 \beta_{5} - 59 \beta_{6} - 72 \beta_{7} ) q^{10} + ( 18279012 + 257 \beta_{1} + 178 \beta_{2} - 232 \beta_{3} + 66 \beta_{4} + 203 \beta_{5} + 65 \beta_{6} - 59 \beta_{7} ) q^{11} + ( 131072 \beta_{1} + 65536 \beta_{2} ) q^{12} + ( -5143437 \beta_{1} - 40033 \beta_{2} + 2610 \beta_{3} + 902 \beta_{4} + 854 \beta_{5} - 439 \beta_{6} + 415 \beta_{7} ) q^{13} + ( 1782561792 + 272 \beta_{1} + 64 \beta_{2} + 377 \beta_{3} - 96 \beta_{4} + 713 \beta_{5} - 55 \beta_{6} - 560 \beta_{7} ) q^{14} + ( -4976875340 + 8720275 \beta_{1} + 470649 \beta_{2} - 5089 \beta_{3} + 889 \beta_{4} - 1818 \beta_{5} + 1866 \beta_{6} + 953 \beta_{7} ) q^{15} + 4294967296 q^{16} + ( 26739554 \beta_{1} - 1899430 \beta_{2} - 2508 \beta_{3} + 7292 \beta_{4} - 4900 \beta_{5} - 598 \beta_{6} - 5498 \beta_{7} ) q^{17} + ( 45601585 \beta_{1} + 1407904 \beta_{2} - 21216 \beta_{3} - 1312 \beta_{4} - 9952 \beta_{5} + 2816 \beta_{6} - 7136 \beta_{7} ) q^{18} + ( -6783022260 - 15565 \beta_{1} - 6542 \beta_{2} + 8120 \beta_{3} + 1654 \beta_{4} - 13987 \beta_{5} + 15563 \beta_{6} + 20527 \beta_{7} ) q^{19} + ( 10039787520 + 17825792 \beta_{1} + 1835008 \beta_{2} + 65536 \beta_{3} ) q^{20} + ( 64430291632 - 34307 \beta_{1} - 25207 \beta_{2} + 103572 \beta_{3} - 10738 \beta_{4} + 44058 \beta_{5} + 55251 \beta_{6} + 2093 \beta_{7} ) q^{21} + ( -19302652 \beta_{1} - 7272000 \beta_{2} + 37568 \beta_{3} + 56640 \beta_{4} - 9536 \beta_{5} - 11776 \beta_{6} - 21312 \beta_{7} ) q^{22} + ( -258670082 \beta_{1} + 15207695 \beta_{2} + 12837 \beta_{3} + 182339 \beta_{4} - 84751 \beta_{5} - 24397 \beta_{6} - 109148 \beta_{7} ) q^{23} + ( 7985954816 + 65536 \beta_{3} + 65536 \beta_{5} + 65536 \beta_{6} ) q^{24} + ( 126653222325 + 1023060060 \beta_{1} + 12008905 \beta_{2} - 115635 \beta_{3} + 293740 \beta_{4} + 53620 \beta_{5} + 142685 \beta_{6} - 7020 \beta_{7} ) q^{25} + ( -336672946176 + 146192 \beta_{1} + 184384 \beta_{2} - 495082 \beta_{3} + 148384 \beta_{4} - 164506 \beta_{5} + 172646 \beta_{6} + 298960 \beta_{7} ) q^{26} + ( 3578087684 \beta_{1} + 70433858 \beta_{2} - 97314 \beta_{3} + 406546 \beta_{4} - 251930 \beta_{5} - 38654 \beta_{6} - 290584 \beta_{7} ) q^{27} + ( -1785856000 \beta_{1} - 23003136 \beta_{2} + 327680 \beta_{3} + 196608 \beta_{4} + 65536 \beta_{5} - 65536 \beta_{6} ) q^{28} + ( 207530385390 + 469146 \beta_{1} + 413628 \beta_{2} - 1423230 \beta_{3} + 238740 \beta_{4} - 540456 \beta_{5} - 348900 \beta_{6} + 247074 \beta_{7} ) q^{29} + ( 567061150720 + 4974643692 \beta_{1} - 14440512 \beta_{2} + 390681 \beta_{3} - 644320 \beta_{4} + 159465 \beta_{5} + 445545 \beta_{6} - 462640 \beta_{7} ) q^{30} + ( -756934624208 - 299336 \beta_{1} - 273772 \beta_{2} + 1269124 \beta_{3} - 165472 \beta_{4} + 696016 \beta_{5} + 524500 \beta_{6} - 197080 \beta_{7} ) q^{31} -4294967296 \beta_{1} q^{32} + ( -19441823632 \beta_{1} - 170236416 \beta_{2} + 617064 \beta_{3} - 537544 \beta_{4} + 577304 \beta_{5} - 9940 \beta_{6} + 567364 \beta_{7} ) q^{33} + ( 1769780805632 - 3681376 \beta_{1} - 2853248 \beta_{2} + 3962384 \beta_{3} - 1350080 \beta_{4} - 2572240 \beta_{5} - 2112976 \beta_{6} - 368864 \beta_{7} ) q^{34} + ( -1090654222620 - 2061228825 \beta_{1} - 397945543 \beta_{2} + 3979198 \beta_{3} + 23752 \beta_{4} + 3489901 \beta_{5} + 1483963 \beta_{6} + 1317029 \beta_{7} ) q^{35} + ( 2975191072768 - 3801088 \beta_{1} - 3473408 \beta_{2} + 9109504 \beta_{3} - 2097152 \beta_{4} + 1835008 \beta_{5} - 327680 \beta_{6} - 2490368 \beta_{7} ) q^{36} + ( 11536745455 \beta_{1} + 428714995 \beta_{2} + 54574 \beta_{3} - 7535702 \beta_{4} + 3795138 \beta_{5} + 935141 \beta_{6} + 4730279 \beta_{7} ) q^{37} + ( 6779515068 \beta_{1} - 39507776 \beta_{2} - 12085824 \beta_{3} - 8086976 \beta_{4} - 1999424 \beta_{5} + 2521600 \beta_{6} + 522176 \beta_{7} ) q^{38} + ( -7505904279096 + 12839130 \beta_{1} + 8396100 \beta_{2} - 34143048 \beta_{3} + 2635380 \beta_{4} - 12907818 \beta_{5} - 22283838 \beta_{6} - 4932990 \beta_{7} ) q^{39} + ( 1153265827840 - 10070589440 \beta_{1} - 208666624 \beta_{2} + 1245184 \beta_{3} - 4194304 \beta_{4} + 2818048 \beta_{5} + 3866624 \beta_{6} + 4718592 \beta_{7} ) q^{40} + ( -27490228746498 + 30999876 \beta_{1} + 22689441 \beta_{2} - 47054133 \beta_{3} + 9586004 \beta_{4} + 6635184 \beta_{5} - 3917115 \beta_{6} - 2241864 \beta_{7} ) q^{41} + ( -64921760596 \beta_{1} - 3491033952 \beta_{2} - 1668576 \beta_{3} - 5972512 \beta_{4} + 2151968 \beta_{5} + 955136 \beta_{6} + 3107104 \beta_{7} ) q^{42} + ( 56738827094 \beta_{1} + 1842524823 \beta_{2} + 31636898 \beta_{3} + 3474222 \beta_{4} + 14081338 \beta_{5} - 4388890 \beta_{6} + 9692448 \beta_{7} ) q^{43} + ( -1197933330432 - 16842752 \beta_{1} - 11665408 \beta_{2} + 15204352 \beta_{3} - 4325376 \beta_{4} - 13303808 \beta_{5} - 4259840 \beta_{6} + 3866624 \beta_{7} ) q^{44} + ( 62939735105135 - 62342445169 \beta_{1} + 6227939829 \beta_{2} + 6588263 \beta_{3} + 3428110 \beta_{4} + 34417930 \beta_{5} + 17756215 \beta_{6} - 16407405 \beta_{7} ) q^{45} + ( -17094148883456 - 76490704 \beta_{1} - 57445184 \beta_{2} + 131252585 \beta_{3} - 25599776 \beta_{4} - 2683303 \beta_{5} + 16053593 \beta_{6} - 308624 \beta_{7} ) q^{46} + ( -21709759994 \beta_{1} + 788625415 \beta_{2} + 30408315 \beta_{3} + 50528989 \beta_{4} - 10060337 \beta_{5} - 10117163 \beta_{6} - 20177500 \beta_{7} ) q^{47} + ( -8589934592 \beta_{1} - 4294967296 \beta_{2} ) q^{48} + ( 99651118005183 + 30940526 \beta_{1} + 27233401 \beta_{2} - 83109907 \beta_{3} + 15684184 \beta_{4} - 24935980 \beta_{5} - 12531079 \beta_{6} + 16112026 \beta_{7} ) q^{49} + ( 66926238566400 - 127805614605 \beta_{1} - 7840378240 \beta_{2} + 64063830 \beta_{3} - 18517920 \beta_{4} - 35269210 \beta_{5} + 24935270 \beta_{6} - 11129840 \beta_{7} ) q^{50} + ( -328671293941328 + 60301116 \beta_{1} + 59903748 \beta_{2} - 103047976 \beta_{3} + 39670920 \beta_{4} + 17156888 \beta_{5} + 75471164 \beta_{6} + 58711644 \beta_{7} ) q^{51} + ( 337080287232 \beta_{1} + 2623602688 \beta_{2} - 171048960 \beta_{3} - 59113472 \beta_{4} - 55967744 \beta_{5} + 28770304 \beta_{6} - 27197440 \beta_{7} ) q^{52} + ( 1075160608653 \beta_{1} + 1573012241 \beta_{2} - 74654682 \beta_{3} + 129962978 \beta_{4} - 102308830 \beta_{5} - 6913537 \beta_{6} - 109222367 \beta_{7} ) q^{53} + ( 233838102579200 - 196487648 \beta_{1} - 151252864 \beta_{2} + 380063150 \beta_{3} - 70678720 \beta_{4} + 32322638 \beta_{5} + 62008910 \beta_{6} - 15548512 \beta_{7} ) q^{54} + ( 171297515897060 - 78760269559 \beta_{1} - 5250316826 \beta_{2} - 149630362 \beta_{3} + 181933990 \beta_{4} - 53513755 \beta_{5} - 24130065 \beta_{6} + 76664105 \beta_{7} ) q^{55} + ( -116821969600512 - 17825792 \beta_{1} - 4194304 \beta_{2} - 24707072 \beta_{3} + 6291456 \beta_{4} - 46727168 \beta_{5} + 3604480 \beta_{6} + 36700160 \beta_{7} ) q^{56} + ( -86909936640 \beta_{1} + 49067818368 \beta_{2} + 365948928 \beta_{3} + 236727072 \beta_{4} + 64610928 \beta_{5} - 75334500 \beta_{6} - 10723572 \beta_{7} ) q^{57} + ( -202700123262 \beta_{1} + 34066006656 \beta_{2} - 136718208 \beta_{3} + 5949312 \beta_{4} - 71333760 \beta_{5} + 16346112 \beta_{6} - 54987648 \beta_{7} ) q^{58} + ( -19426319720220 + 442677001 \beta_{1} + 361477526 \beta_{2} - 469440392 \beta_{3} + 186852034 \beta_{4} + 334714135 \beta_{5} + 371393761 \beta_{6} + 117879101 \beta_{7} ) q^{59} + ( 326164502282240 - 571491942400 \beta_{1} - 30844452864 \beta_{2} + 333512704 \beta_{3} - 58261504 \beta_{4} + 119144448 \beta_{5} - 122290176 \beta_{6} - 62455808 \beta_{7} ) q^{60} + ( -793805508621118 + 45594239 \beta_{1} - 50833799 \beta_{2} + 50410370 \beta_{3} - 98174558 \beta_{4} + 45170810 \beta_{5} - 391375141 \beta_{6} - 340117913 \beta_{7} ) q^{61} + ( 750804865472 \beta_{1} - 43376263552 \beta_{2} + 110052480 \beta_{3} + 7035776 \beta_{4} + 51508352 \beta_{5} - 14636032 \beta_{6} + 36872320 \beta_{7} ) q^{62} + ( -5900156629526 \beta_{1} - 100869696839 \beta_{2} + 631762827 \beta_{3} - 292706611 \beta_{4} + 462234719 \beta_{5} - 42382027 \beta_{6} + 419852692 \beta_{7} ) q^{63} -281474976710656 q^{64} + ( 1313518010597760 + 2307876152060 \beta_{1} + 132544000889 \beta_{2} - 1321327459 \beta_{3} - 746705836 \beta_{4} - 904229868 \beta_{5} - 622272159 \beta_{6} - 60166872 \beta_{7} ) q^{65} + ( -1272553015939072 + 360409280 \beta_{1} + 289854208 \beta_{2} - 777006972 \beta_{3} + 146199424 \beta_{4} - 126743484 \beta_{5} - 119109564 \beta_{6} + 78188992 \beta_{7} ) q^{66} + ( 5692164418190 \beta_{1} - 121180969181 \beta_{2} - 1585764130 \beta_{3} - 1416102446 \beta_{4} - 84830842 \beta_{5} + 375233322 \beta_{6} + 290402480 \beta_{7} ) q^{67} + ( -1752403410944 \beta_{1} + 124481044480 \beta_{2} + 164364288 \beta_{3} - 477888512 \beta_{4} + 321126400 \beta_{5} + 39190528 \beta_{6} + 360316928 \beta_{7} ) q^{68} + ( 2602670028907424 - 1697282433 \beta_{1} - 1532590731 \beta_{2} + 4060555498 \beta_{3} - 911932686 \beta_{4} + 830682334 \beta_{5} - 43141589 \beta_{6} - 1038515625 \beta_{7} ) q^{69} + ( -131389709655040 + 1074801873956 \beta_{1} - 117241044416 \beta_{2} - 1774222417 \beta_{3} + 417255040 \beta_{4} - 472765105 \beta_{5} - 153578865 \beta_{6} + 307338080 \beta_{7} ) q^{70} + ( -4143628317966648 - 1366434226 \beta_{1} - 877436552 \beta_{2} + 570950228 \beta_{3} - 258959252 \beta_{4} - 1672920550 \beta_{5} - 594366406 \beta_{6} + 589556470 \beta_{7} ) q^{71} + ( -2988545474560 \beta_{1} - 92268396544 \beta_{2} + 1390411776 \beta_{3} + 85983232 \beta_{4} + 652214272 \beta_{5} - 184549376 \beta_{6} + 467664896 \beta_{7} ) q^{72} + ( -5247325937236 \beta_{1} - 378007509044 \beta_{2} - 2790213760 \beta_{3} - 1229920256 \beta_{4} - 780146752 \beta_{5} + 502516752 \beta_{6} - 277630000 \beta_{7} ) q^{73} + ( 752206043215872 + 3281736912 \beta_{1} + 2481751872 \beta_{2} - 4603639590 \beta_{3} + 1121177888 \beta_{4} + 1159849194 \beta_{5} + 441660906 \beta_{6} + 81796752 \beta_{7} ) q^{74} + ( 2178415054915400 - 21005919723280 \beta_{1} + 32734665235 \beta_{2} + 7392130380 \beta_{3} + 408242380 \beta_{4} + 2851502190 \beta_{5} + 182563470 \beta_{6} - 616367990 \beta_{7} ) q^{75} + ( 444532146831360 + 1020067840 \beta_{1} + 428736512 \beta_{2} - 532152320 \beta_{3} - 108396544 \beta_{4} + 916652032 \beta_{5} - 1019936768 \beta_{6} - 1345257472 \beta_{7} ) q^{76} + ( 16765349256664 \beta_{1} + 268926520960 \beta_{2} + 1270687668 \beta_{3} + 872415820 \beta_{4} + 199135924 \beta_{5} - 267887936 \beta_{6} - 68752012 \beta_{7} ) q^{77} + ( 7676685197064 \beta_{1} + 1216457532288 \beta_{2} + 3025745280 \beta_{3} + 3374951040 \beta_{4} - 174602880 \beta_{5} - 800087040 \beta_{6} - 974689920 \beta_{7} ) q^{78} + ( -2419333598963520 - 2298068412 \beta_{1} - 952470084 \beta_{2} - 272362452 \beta_{3} + 262085496 \beta_{4} - 3522900948 \beta_{5} + 907022280 \beta_{6} + 3084324900 \beta_{7} ) q^{79} + ( -657967514910720 - 1168231104512 \beta_{1} - 120259084288 \beta_{2} - 4294967296 \beta_{3} ) q^{80} + ( 6804046828104481 + 867737528 \beta_{1} + 1003385509 \beta_{2} + 1971846163 \beta_{3} + 759355660 \beta_{4} + 3842969200 \beta_{5} + 5388946633 \beta_{6} + 1410329452 \beta_{7} ) q^{81} + ( 27507951281582 \beta_{1} + 122834451488 \beta_{2} + 1710623648 \beta_{3} + 5493079136 \beta_{4} - 1891227744 \beta_{5} - 900462848 \beta_{6} - 2791690592 \beta_{7} ) q^{82} + ( 15962310614974 \beta_{1} - 1502958101637 \beta_{2} - 51062274 \beta_{3} + 4072431282 \beta_{4} - 2061746778 \beta_{5} - 502671126 \beta_{6} - 2564417904 \beta_{7} ) q^{83} + ( -4222503592394752 + 2248343552 \beta_{1} + 1651965952 \beta_{2} - 6787694592 \beta_{3} + 703725568 \beta_{4} - 2887385088 \beta_{5} - 3620929536 \beta_{6} - 137166848 \beta_{7} ) q^{84} + ( -8941279359611920 - 36348504535760 \beta_{1} + 454631788662 \beta_{2} - 6384592502 \beta_{3} + 1339761972 \beta_{4} - 4978536364 \beta_{5} + 6949382518 \beta_{6} + 2148806844 \beta_{7} ) q^{85} + ( 3702003555554304 + 5009388640 \beta_{1} + 4681644416 \beta_{2} - 8632813229 \beta_{3} + 2902600128 \beta_{4} + 1058219827 \beta_{5} + 4428887347 \beta_{6} + 3698411744 \beta_{7} ) q^{86} + ( 90256961796300 \beta_{1} + 1333424431674 \beta_{2} - 9010528128 \beta_{3} + 2142455616 \beta_{4} - 5576491872 \beta_{5} + 858509064 \beta_{6} - 4717982808 \beta_{7} ) q^{87} + ( 1265018601472 \beta_{1} + 476577792000 \beta_{2} - 2462056448 \beta_{3} - 3711959040 \beta_{4} + 624951296 \beta_{5} + 771751936 \beta_{6} + 1396703232 \beta_{7} ) q^{88} + ( -5446153787444790 - 4974284074 \beta_{1} - 4578852380 \beta_{2} + 4208284922 \beta_{3} - 2788947124 \beta_{4} - 5344851532 \beta_{5} - 8341977136 \beta_{6} - 3392557298 \beta_{7} ) q^{89} + ( -4144213335342080 - 63192296640465 \beta_{1} - 1775303594512 \beta_{2} + 17549473027 \beta_{3} + 5403623328 \beta_{4} + 6858542139 \beta_{5} + 4035011307 \beta_{6} - 1609138744 \beta_{7} ) q^{90} + ( 41677538486246472 - 20009426174 \beta_{1} - 16652115736 \beta_{2} + 38782904584 \beta_{3} - 8863203532 \beta_{4} + 2121362674 \beta_{5} - 1101511310 \beta_{6} - 6580184422 \beta_{7} ) q^{91} + ( 16952202493952 \beta_{1} - 996651499520 \beta_{2} - 841285632 \beta_{3} - 11949768704 \beta_{4} + 5554241536 \beta_{5} + 1598881792 \beta_{6} + 7153123328 \beta_{7} ) q^{92} + ( -113964863604128 \beta_{1} - 749786495152 \beta_{2} + 12178091448 \beta_{3} - 1386317304 \beta_{4} + 6782204376 \beta_{5} - 1348971768 \beta_{6} + 5433232608 \beta_{7} ) q^{93} + ( -1430037115286528 - 15665468336 \beta_{1} - 10978378432 \beta_{2} + 22095326677 \beta_{3} - 4194192352 \beta_{4} - 4548520091 \beta_{5} + 3221461093 \beta_{6} + 3082891280 \beta_{7} ) q^{94} + ( -16434754904399700 - 125004487248185 \beta_{1} + 2873013930270 \beta_{2} + 5609114710 \beta_{3} - 8828987190 \beta_{4} + 14983272155 \beta_{5} - 984348235 \beta_{6} - 5280760505 \beta_{7} ) q^{95} + ( -523367534821376 - 4294967296 \beta_{3} - 4294967296 \beta_{5} - 4294967296 \beta_{6} ) q^{96} + ( 249941178848846 \beta_{1} - 2876657579962 \beta_{2} + 22917590564 \beta_{3} - 4850062804 \beta_{4} + 13883826684 \beta_{5} - 2258440970 \beta_{6} + 11625385714 \beta_{7} ) q^{97} + ( -99430364985555 \beta_{1} + 1551318848032 \beta_{2} - 8910952032 \beta_{3} + 442539616 \beta_{4} - 4676745824 \beta_{5} + 1058551552 \beta_{6} - 3618194272 \beta_{7} ) q^{98} + ( -29585167052406356 + 49208069747 \beta_{1} + 36689494234 \beta_{2} - 88207904720 \beta_{3} + 16113945814 \beta_{4} - 2310340739 \beta_{5} - 15695148557 \beta_{6} - 866232305 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 524288q^{4} - 1225560q^{5} - 974848q^{6} - 363182504q^{9} + O(q^{10}) \) \( 8q - 524288q^{4} - 1225560q^{5} - 974848q^{6} - 363182504q^{9} - 140779520q^{10} + 146232096q^{11} + 14260494336q^{14} - 39815002720q^{15} + 34359738368q^{16} - 54264178080q^{19} + 80318300160q^{20} + 515442333056q^{21} + 63887638528q^{24} + 1013225778600q^{25} - 2693383569408q^{26} + 1660243083120q^{29} + 4536489205760q^{30} - 6055476993664q^{31} + 14158246445056q^{34} - 8725233780960q^{35} + 23801528582144q^{36} - 60047234232768q^{39} + 9226126622720q^{40} - 219921829971984q^{41} - 9583466643456q^{44} + 503517880841080q^{45} - 136753191067648q^{46} + 797208944041464q^{49} + 535409908531200q^{50} - 2629370351530624q^{51} + 1870704820633600q^{54} + 1370380127176480q^{55} - 934575756804096q^{56} - 155410557761760q^{59} + 2609316018257920q^{60} - 6350444068968944q^{61} - 2251799813685248q^{64} + 10508144084782080q^{65} - 10180424127512576q^{66} + 20821360231259392q^{69} - 1051117677240320q^{70} - 33149026543733184q^{71} + 6017648345726976q^{74} + 17427320439323200q^{75} + 3556257174650880q^{76} - 19354668791708160q^{79} - 5263740119285760q^{80} + 54432374624835848q^{81} - 33780028739158016q^{84} - 71530234876895360q^{85} + 29616028444434432q^{86} - 43569230299558320q^{89} - 33153706682736640q^{90} + 333420307889971776q^{91} - 11440296922292224q^{94} - 131478039235197600q^{95} - 4186940278571008q^{96} - 236681336419250848q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 556201 x^{6} + 76870744104 x^{4} + 1868329791349729 x^{2} + 78074963590050625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 520237696 \nu^{7} + 289353475332096 \nu^{5} + 40135518862867325184 \nu^{3} + 1012402155577289830662784 \nu \)\()/ \)\(25\!\cdots\!75\)\( \)
\(\beta_{2}\)\(=\)\((\)\(33035288673113 \nu^{7} + 18121192487852408472 \nu^{5} + 2419036458532416597476532 \nu^{3} + 46737489981645847313400688433 \nu\)\()/ \)\(12\!\cdots\!50\)\( \)
\(\beta_{3}\)\(=\)\((\)\(229165601212295 \nu^{7} - 14625588188161600 \nu^{6} + 122723677572398843136 \nu^{5} - 8664952963507573019000 \nu^{4} + 15399881171759226770508300 \nu^{3} - 1421902957035591255802011400 \nu^{2} + 250976422630515798926839126199 \nu - 41251997408073719474318796976500\)\()/ \)\(40\!\cdots\!50\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-2108820343600609 \nu^{7} + 32751068542893056 \nu^{6} - 1180889057065092454296 \nu^{5} + 17140122243018593008120 \nu^{4} - 165597129868211310875481876 \nu^{3} + 2067271302137333904705876104 \nu^{2} - 4175248388216129285483563165369 \nu + 25137560023818852374822000080500\)\()/ \)\(20\!\cdots\!75\)\( \)
\(\beta_{5}\)\(=\)\((\)\(20886453302745211 \nu^{7} + 5421414034731968 \nu^{6} + 11596221803461787998968 \nu^{5} + 20701945207894668735360 \nu^{4} + 1593947509783743169375643484 \nu^{3} + 6639158310913424108853328512 \nu^{2} + 37759920220963541776181327378707 \nu + 184737792047177364111343004624000\)\()/ \)\(12\!\cdots\!50\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-10786975053191048 \nu^{7} + 187778412492191488 \nu^{6} - 5982196418089492264188 \nu^{5} + 99624925564807873798260 \nu^{4} - 820073576649510424843584192 \nu^{3} + 11260748696925782778264463692 \nu^{2} - 19256424744427544586480922378652 \nu + 9216955677405457371678525712750\)\()/ \)\(61\!\cdots\!25\)\( \)
\(\beta_{7}\)\(=\)\((\)\(31348693698919781 \nu^{7} + 234961761787111168 \nu^{6} + 17404781619411251599296 \nu^{5} + 108133647140739608370360 \nu^{4} + 2392323353124384999788436324 \nu^{3} + 10030178373017353086787962312 \nu^{2} + 56668106581978595424536211449909 \nu + 89843560319956908560545386788500\)\()/ \)\(12\!\cdots\!50\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(28 \beta_{7} - 64 \beta_{6} + 92 \beta_{5} + 164 \beta_{4} + 348 \beta_{3} - 70420 \beta_{2} - 7161 \beta_{1}\)\()/2880000\)
\(\nu^{2}\)\(=\)\((\)\(16084 \beta_{7} - 170917 \beta_{6} - 93349 \beta_{5} + 135592 \beta_{4} - 781081 \beta_{3} + 297040 \beta_{2} + 390692 \beta_{1} - 800929440000\)\()/5760000\)
\(\nu^{3}\)\(=\)\((\)\(-3696256 \beta_{7} + 2758528 \beta_{6} - 6454784 \beta_{5} - 4579328 \beta_{4} - 17488896 \beta_{3} + 3215175040 \beta_{2} + 63326830647 \beta_{1}\)\()/480000\)
\(\nu^{4}\)\(=\)\((\)\(-3646813564 \beta_{7} + 12449717647 \beta_{6} + 11570450959 \beta_{5} - 8465982712 \beta_{4} + 50546639851 \beta_{3} - 17225054320 \beta_{2} - 21751134572 \beta_{1} + 44818002487584000\)\()/1152000\)
\(\nu^{5}\)\(=\)\((\)\(9421921710668 \beta_{7} - 4039916000384 \beta_{6} + 13461837711052 \beta_{5} + 2697826290484 \beta_{4} + 29621501712588 \beta_{3} - 6394364023434020 \beta_{2} - 175746759936799341 \beta_{1}\)\()/2880000\)
\(\nu^{6}\)\(=\)\((\)\(387644604611656 \beta_{7} - 879139240778953 \beta_{6} - 1012426488548041 \beta_{5} + 597572878864528 \beta_{4} - 3568217195669329 \beta_{3} + 1150716675139360 \beta_{2} + 1405074031981928 \beta_{1} - 2964259653292264560000\)\()/240000\)
\(\nu^{7}\)\(=\)\((\)\(-3583950474025161436 \beta_{7} + 1094629938641186368 \beta_{6} - 4678580412666347804 \beta_{5} + 300060658101602332 \beta_{4} - 9057100167231093276 \beta_{3} + 2205277530913233446740 \beta_{2} + 68591381255605553997057 \beta_{1}\)\()/2880000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
594.906i
6.46999i
414.151i
175.285i
175.285i
414.151i
6.46999i
594.906i
256.000i 20214.2i −65536.0 −765001. 421560.i −5.17482e6 1.76306e7i 1.67772e7i −2.79472e8 −1.07919e8 + 1.95840e8i
9.2 256.000i 2973.91i −65536.0 855091. 178210.i −761320. 7.58343e6i 1.67772e7i 1.20296e8 −4.56218e7 2.18903e8i
9.3 256.000i 5436.73i −65536.0 −679890. 548351.i 1.39180e6 7.62753e6i 1.67772e7i 9.95821e7 −1.40378e8 + 1.74052e8i
9.4 256.000i 15847.3i −65536.0 −22980.2 + 873162.i 4.05692e6 1.02660e7i 1.67772e7i −1.21998e8 2.23529e8 + 5.88293e6i
9.5 256.000i 15847.3i −65536.0 −22980.2 873162.i 4.05692e6 1.02660e7i 1.67772e7i −1.21998e8 2.23529e8 5.88293e6i
9.6 256.000i 5436.73i −65536.0 −679890. + 548351.i 1.39180e6 7.62753e6i 1.67772e7i 9.95821e7 −1.40378e8 1.74052e8i
9.7 256.000i 2973.91i −65536.0 855091. + 178210.i −761320. 7.58343e6i 1.67772e7i 1.20296e8 −4.56218e7 + 2.18903e8i
9.8 256.000i 20214.2i −65536.0 −765001. + 421560.i −5.17482e6 1.76306e7i 1.67772e7i −2.79472e8 −1.07919e8 1.95840e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.18.b.a 8
3.b odd 2 1 90.18.c.b 8
4.b odd 2 1 80.18.c.a 8
5.b even 2 1 inner 10.18.b.a 8
5.c odd 4 1 50.18.a.j 4
5.c odd 4 1 50.18.a.k 4
15.d odd 2 1 90.18.c.b 8
20.d odd 2 1 80.18.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.b.a 8 1.a even 1 1 trivial
10.18.b.a 8 5.b even 2 1 inner
50.18.a.j 4 5.c odd 4 1
50.18.a.k 4 5.c odd 4 1
80.18.c.a 8 4.b odd 2 1
80.18.c.a 8 20.d odd 2 1
90.18.c.b 8 3.b odd 2 1
90.18.c.b 8 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 65536 T^{2} )^{4} \)
$3$ \( 1 - 334969400 T^{2} + 54219517753298076 T^{4} - \)\(80\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(77\!\cdots\!21\)\( T^{16} \)
$5$ \( 1 + 1225560 T + 244385767500 T^{2} - 884410161796875000 T^{3} - \)\(95\!\cdots\!50\)\( T^{4} - \)\(67\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!00\)\( T^{7} + \)\(33\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 1329126527969560 T^{2} + \)\(85\!\cdots\!96\)\( T^{4} - \)\(34\!\cdots\!20\)\( T^{6} + \)\(96\!\cdots\!06\)\( T^{8} - \)\(18\!\cdots\!80\)\( T^{10} + \)\(25\!\cdots\!96\)\( T^{12} - \)\(21\!\cdots\!40\)\( T^{14} + \)\(85\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 - 73116048 T + 1370043014096852748 T^{2} - \)\(31\!\cdots\!36\)\( T^{3} + \)\(92\!\cdots\!70\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{5} + \)\(35\!\cdots\!68\)\( T^{6} - \)\(94\!\cdots\!28\)\( T^{7} + \)\(65\!\cdots\!81\)\( T^{8} )^{2} \)
$13$ \( 1 + 5551601366327032600 T^{2} + \)\(60\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{6} - \)\(41\!\cdots\!74\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{10} + \)\(34\!\cdots\!76\)\( T^{12} + \)\(23\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!41\)\( T^{16} \)
$17$ \( 1 - \)\(22\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!16\)\( T^{4} - \)\(42\!\cdots\!60\)\( T^{6} + \)\(40\!\cdots\!46\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!56\)\( T^{12} - \)\(72\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!81\)\( T^{16} \)
$19$ \( ( 1 + 27132089040 T + \)\(10\!\cdots\!56\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} - \)\(70\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!76\)\( T^{6} + \)\(44\!\cdots\!60\)\( T^{7} + \)\(90\!\cdots\!41\)\( T^{8} )^{2} \)
$23$ \( 1 - \)\(25\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!86\)\( T^{8} - \)\(39\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(20\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 830121541560 T + \)\(21\!\cdots\!36\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!86\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!16\)\( T^{6} - \)\(31\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} )^{2} \)
$31$ \( ( 1 + 3027738496832 T + \)\(82\!\cdots\!28\)\( T^{2} + \)\(17\!\cdots\!04\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} + \)\(38\!\cdots\!44\)\( T^{5} + \)\(41\!\cdots\!88\)\( T^{6} + \)\(34\!\cdots\!92\)\( T^{7} + \)\(25\!\cdots\!41\)\( T^{8} )^{2} \)
$37$ \( 1 - \)\(22\!\cdots\!60\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(73\!\cdots\!26\)\( T^{8} - \)\(30\!\cdots\!80\)\( T^{10} + \)\(98\!\cdots\!76\)\( T^{12} - \)\(20\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!41\)\( T^{16} \)
$41$ \( ( 1 + 109960914985992 T + \)\(10\!\cdots\!48\)\( T^{2} + \)\(67\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(17\!\cdots\!44\)\( T^{5} + \)\(74\!\cdots\!28\)\( T^{6} + \)\(19\!\cdots\!72\)\( T^{7} + \)\(46\!\cdots\!21\)\( T^{8} )^{2} \)
$43$ \( 1 - \)\(31\!\cdots\!20\)\( T^{2} + \)\(49\!\cdots\!96\)\( T^{4} - \)\(50\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!06\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{10} + \)\(59\!\cdots\!96\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(17\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!76\)\( T^{4} - \)\(68\!\cdots\!40\)\( T^{6} + \)\(22\!\cdots\!66\)\( T^{8} - \)\(48\!\cdots\!60\)\( T^{10} + \)\(70\!\cdots\!36\)\( T^{12} - \)\(62\!\cdots\!80\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 - \)\(69\!\cdots\!40\)\( T^{2} + \)\(33\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(25\!\cdots\!66\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{10} + \)\(60\!\cdots\!36\)\( T^{12} - \)\(52\!\cdots\!60\)\( T^{14} + \)\(31\!\cdots\!21\)\( T^{16} \)
$59$ \( ( 1 + 77705278880880 T + \)\(21\!\cdots\!76\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!66\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!36\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \)
$61$ \( ( 1 + 3175222034484472 T + \)\(93\!\cdots\!28\)\( T^{2} + \)\(16\!\cdots\!64\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(37\!\cdots\!44\)\( T^{5} + \)\(47\!\cdots\!48\)\( T^{6} + \)\(35\!\cdots\!92\)\( T^{7} + \)\(25\!\cdots\!81\)\( T^{8} )^{2} \)
$67$ \( 1 - \)\(34\!\cdots\!20\)\( T^{2} + \)\(52\!\cdots\!16\)\( T^{4} - \)\(43\!\cdots\!40\)\( T^{6} + \)\(31\!\cdots\!46\)\( T^{8} - \)\(53\!\cdots\!60\)\( T^{10} + \)\(78\!\cdots\!56\)\( T^{12} - \)\(62\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!81\)\( T^{16} \)
$71$ \( ( 1 + 16574513271866592 T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(85\!\cdots\!70\)\( T^{4} + \)\(40\!\cdots\!44\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + \)\(43\!\cdots\!32\)\( T^{7} + \)\(76\!\cdots\!61\)\( T^{8} )^{2} \)
$73$ \( 1 - \)\(19\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!36\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(85\!\cdots\!86\)\( T^{8} - \)\(35\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!16\)\( T^{12} - \)\(22\!\cdots\!40\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} \)
$79$ \( ( 1 + 9677334395854080 T + \)\(45\!\cdots\!36\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(87\!\cdots\!86\)\( T^{4} + \)\(26\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(58\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} )^{2} \)
$83$ \( 1 - \)\(13\!\cdots\!40\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{4} - \)\(76\!\cdots\!80\)\( T^{6} + \)\(37\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(39\!\cdots\!56\)\( T^{12} - \)\(75\!\cdots\!60\)\( T^{14} + \)\(98\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 + 21784615149779160 T + \)\(45\!\cdots\!16\)\( T^{2} + \)\(61\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!46\)\( T^{4} + \)\(85\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!56\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(36\!\cdots\!81\)\( T^{8} )^{2} \)
$97$ \( 1 - \)\(15\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(76\!\cdots\!66\)\( T^{8} - \)\(41\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!36\)\( T^{12} - \)\(69\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!21\)\( T^{16} \)
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