Properties

Label 8.14.b.b
Level 8
Weight 14
Character orbit 8.b
Analytic conductor 8.578
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(8.57847431615\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 11 + \beta_{1} ) q^{2} + ( 3 \beta_{1} + \beta_{3} ) q^{3} + ( -472 + 12 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} + ( 53 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{5} + ( -26769 + 26 \beta_{1} - \beta_{2} + 26 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + ( 58697 - 62 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{7} + ( -27077 - 442 \beta_{1} + \beta_{2} - 10 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{8} + ( 201402 - 2129 \beta_{1} - 4 \beta_{2} + 50 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} - 5 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( 11 + \beta_{1} ) q^{2} + ( 3 \beta_{1} + \beta_{3} ) q^{3} + ( -472 + 12 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} + ( 53 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{5} + ( -26769 + 26 \beta_{1} - \beta_{2} + 26 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6} + ( 58697 - 62 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{7} + ( -27077 - 442 \beta_{1} + \beta_{2} - 10 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{8} + ( 201402 - 2129 \beta_{1} - 4 \beta_{2} + 50 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} - 5 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9} + ( -454220 + 750 \beta_{1} - 20 \beta_{2} - 478 \beta_{3} + 47 \beta_{4} + 22 \beta_{5} - 3 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} ) q^{10} + ( 44 + 2651 \beta_{1} + 22 \beta_{2} - 1265 \beta_{3} - 99 \beta_{4} - 22 \beta_{5} - 11 \beta_{6} - 22 \beta_{7} ) q^{11} + ( 2798750 - 30960 \beta_{1} - 126 \beta_{2} - 204 \beta_{3} + 116 \beta_{4} + 28 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} - 16 \beta_{8} - 10 \beta_{9} ) q^{12} + ( -248 - 48385 \beta_{1} - 161 \beta_{2} + 6042 \beta_{3} + 494 \beta_{4} + 4 \beta_{5} + 134 \beta_{6} - 28 \beta_{7} - 16 \beta_{9} ) q^{13} + ( 141124 + 61292 \beta_{1} + 444 \beta_{2} - 5588 \beta_{3} - 494 \beta_{4} + 136 \beta_{5} - 50 \beta_{6} - 48 \beta_{7} - 48 \beta_{8} + 4 \beta_{9} ) q^{14} + ( -14592559 + 97418 \beta_{1} - 192 \beta_{2} - 928 \beta_{3} + 2822 \beta_{4} + 425 \beta_{5} + 240 \beta_{6} - 40 \beta_{7} + 31 \beta_{8} + 72 \beta_{9} ) q^{15} + ( 5662330 - 43092 \beta_{1} + 894 \beta_{2} + 24044 \beta_{3} + 488 \beta_{4} + 28 \beta_{5} - 136 \beta_{6} + 52 \beta_{7} + 96 \beta_{8} - 78 \beta_{9} ) q^{16} + ( 21730719 - 108095 \beta_{1} + 260 \beta_{2} + 3854 \beta_{3} + 4188 \beta_{4} - 1115 \beta_{5} - 91 \beta_{6} + 81 \beta_{7} - 66 \beta_{8} - 17 \beta_{9} ) q^{17} + ( -14759899 + 239599 \beta_{1} - 1912 \beta_{2} - 41528 \beta_{3} - 3860 \beta_{4} + 48 \beta_{5} - 236 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 72 \beta_{9} ) q^{18} + ( 6716 + 230823 \beta_{1} + 990 \beta_{2} + 27867 \beta_{3} - 16455 \beta_{4} - 478 \beta_{5} - 527 \beta_{6} + 290 \beta_{7} + 384 \beta_{9} ) q^{19} + ( 2167396 - 419952 \beta_{1} - 4292 \beta_{2} - 131112 \beta_{3} - 4312 \beta_{4} - 472 \beta_{5} - 1036 \beta_{6} + 440 \beta_{7} + 96 \beta_{8} - 396 \beta_{9} ) q^{20} + ( 9560 - 702076 \beta_{1} - 440 \beta_{2} + 144284 \beta_{3} - 25926 \beta_{4} - 340 \beta_{5} + 1730 \beta_{6} + 844 \beta_{7} + 592 \beta_{9} ) q^{21} + ( -17794645 + 84370 \beta_{1} + 7227 \beta_{2} - 178574 \beta_{3} - 9823 \beta_{4} + 693 \beta_{5} - 803 \beta_{6} + 704 \beta_{7} + 704 \beta_{8} - 528 \beta_{9} ) q^{22} + ( -7892749 - 708114 \beta_{1} + 832 \beta_{2} + 36960 \beta_{3} + 58658 \beta_{4} - 6469 \beta_{5} - 16 \beta_{6} + 408 \beta_{7} - 435 \beta_{8} + 392 \beta_{9} ) q^{23} + ( 32033310 + 2752716 \beta_{1} + 9562 \beta_{2} + 173356 \beta_{3} - 26948 \beta_{4} + 1668 \beta_{5} - 6196 \beta_{6} + 44 \beta_{7} - 1600 \beta_{8} - 890 \beta_{9} ) q^{24} + ( -307689609 + 4426842 \beta_{1} - 4184 \beta_{2} - 45428 \beta_{3} + 115912 \beta_{4} + 4866 \beta_{5} + 11314 \beta_{6} - 678 \beta_{7} + 1004 \beta_{8} + 2150 \beta_{9} ) q^{25} + ( 373492924 - 705206 \beta_{1} - 1692 \beta_{2} + 637286 \beta_{3} - 13747 \beta_{4} + 6242 \beta_{5} + 1271 \beta_{6} + 488 \beta_{7} - 744 \beta_{8} - 2106 \beta_{9} ) q^{26} + ( 956 - 748054 \beta_{1} - 18530 \beta_{2} + 995676 \beta_{3} - 8871 \beta_{4} - 1438 \beta_{5} + 6545 \beta_{6} - 1694 \beta_{7} - 128 \beta_{9} ) q^{27} + ( -165361968 + 871616 \beta_{1} + 23600 \beta_{2} - 1579184 \beta_{3} - 17400 \beta_{4} + 14176 \beta_{5} - 21504 \beta_{6} - 4576 \beta_{7} + 1280 \beta_{8} - 176 \beta_{9} ) q^{28} + ( 53440 - 9290205 \beta_{1} + 11415 \beta_{2} - 443578 \beta_{3} - 149296 \beta_{4} - 12320 \beta_{5} + 14736 \beta_{6} - 8480 \beta_{7} + 1920 \beta_{9} ) q^{29} + ( 633816900 - 14826388 \beta_{1} - 52676 \beta_{2} - 2643540 \beta_{3} - 10670 \beta_{4} - 3704 \beta_{5} + 24974 \beta_{6} - 5168 \beta_{7} - 3632 \beta_{8} - 4732 \beta_{9} ) q^{30} + ( 64793900 - 20926432 \beta_{1} - 6976 \beta_{2} + 468896 \beta_{3} + 126736 \beta_{4} + 15212 \beta_{5} - 43504 \beta_{6} - 1560 \beta_{7} + 3596 \beta_{8} + 2296 \beta_{9} ) q^{31} + ( -1129872988 + 6405080 \beta_{1} - 94644 \beta_{2} + 2620760 \beta_{3} + 142208 \beta_{4} + 21784 \beta_{5} - 40224 \beta_{6} - 5624 \beta_{7} + 10560 \beta_{8} + 1620 \beta_{9} ) q^{32} + ( 1548463631 + 26283367 \beta_{1} + 4444 \beta_{2} - 606078 \beta_{3} - 193644 \beta_{4} + 2211 \beta_{5} + 54131 \beta_{6} + 759 \beta_{7} - 9262 \beta_{8} - 2167 \beta_{9} ) q^{33} + ( -609724706 + 21655034 \beta_{1} + 116344 \beta_{2} + 3892024 \beta_{3} - 8556 \beta_{4} - 41520 \beta_{5} + 97644 \beta_{6} - 6176 \beta_{7} + 7200 \beta_{8} + 1352 \beta_{9} ) q^{34} + ( -2648 + 75559352 \beta_{1} + 88468 \beta_{2} - 1354980 \beta_{3} + 157190 \beta_{4} + 4204 \beta_{5} - 150762 \beta_{6} + 4972 \beta_{7} + 384 \beta_{9} ) q^{35} + ( 400375928 - 15910060 \beta_{1} + 74656 \beta_{2} + 792978 \beta_{3} + 286167 \beta_{4} - 45760 \beta_{5} - 81664 \beta_{6} + 20416 \beta_{7} - 16896 \beta_{8} + 352 \beta_{9} ) q^{36} + ( -317480 - 93260595 \beta_{1} - 56499 \beta_{2} - 933618 \beta_{3} + 618906 \beta_{4} + 63340 \beta_{5} + 181154 \beta_{6} + 37900 \beta_{7} - 12720 \beta_{9} ) q^{37} + ( -1876499561 + 4914394 \beta_{1} - 24985 \beta_{2} - 7228870 \beta_{3} - 187227 \beta_{4} + 46793 \beta_{5} + 221521 \beta_{6} + 26560 \beta_{7} + 4032 \beta_{8} + 34864 \beta_{9} ) q^{38} + ( -6349205925 - 93530234 \beta_{1} + 35840 \beta_{2} + 1436672 \beta_{3} - 1164358 \beta_{4} - 12925 \beta_{5} - 217856 \beta_{6} + 4928 \beta_{7} - 18883 \beta_{8} - 21056 \beta_{9} ) q^{39} + ( 746682892 - 2861192 \beta_{1} + 107940 \beta_{2} + 12579384 \beta_{3} + 175864 \beta_{4} - 217368 \beta_{5} - 308072 \beta_{6} + 40184 \beta_{7} - 30592 \beta_{8} + 6300 \beta_{9} ) q^{40} + ( 5933317952 + 146320078 \beta_{1} + 69624 \beta_{2} - 3847900 \beta_{3} - 2484520 \beta_{4} - 137274 \beta_{5} + 256726 \beta_{6} + 10446 \beta_{7} + 57156 \beta_{8} - 38286 \beta_{9} ) q^{41} + ( 5389922032 - 6465896 \beta_{1} - 255152 \beta_{2} - 1713880 \beta_{3} - 797140 \beta_{4} + 142744 \beta_{5} + 478340 \beta_{6} + 40800 \beta_{7} - 19040 \beta_{8} + 60136 \beta_{9} ) q^{42} + ( -379016 + 334310023 \beta_{1} - 210628 \beta_{2} - 9561015 \beta_{3} + 1655986 \beta_{4} + 41668 \beta_{5} - 698590 \beta_{6} + 2244 \beta_{7} - 19712 \beta_{9} ) q^{43} + ( 1332537910 - 21952304 \beta_{1} - 437910 \beta_{2} + 5775044 \beta_{3} + 754820 \beta_{4} - 149556 \beta_{5} - 714494 \beta_{6} - 30844 \beta_{7} + 73392 \beta_{8} + 36718 \beta_{9} ) q^{44} + ( -596104 - 383910733 \beta_{1} + 45451 \beta_{2} - 19558710 \beta_{3} + 848370 \beta_{4} + 21692 \beta_{5} + 660314 \beta_{6} - 52004 \beta_{7} - 36848 \beta_{9} ) q^{45} + ( -5504873204 - 4840700 \beta_{1} + 662132 \beta_{2} + 14726596 \beta_{3} - 723178 \beta_{4} - 378600 \beta_{5} + 1036938 \beta_{6} - 97168 \beta_{7} + 31344 \beta_{8} - 23348 \beta_{9} ) q^{46} + ( -1016950802 - 563994860 \beta_{1} - 85952 \beta_{2} + 11213792 \beta_{3} - 330884 \beta_{4} + 349022 \beta_{5} - 1236048 \beta_{6} - 27720 \beta_{7} + 61194 \beta_{8} + 2792 \beta_{9} ) q^{47} + ( -30184952396 + 71893528 \beta_{1} + 526268 \beta_{2} + 839320 \beta_{3} + 3032304 \beta_{4} + 598840 \beta_{5} - 764528 \beta_{6} - 102040 \beta_{7} + 5568 \beta_{8} + 44324 \beta_{9} ) q^{48} + ( 18270820881 + 652777880 \beta_{1} - 142240 \beta_{2} - 13661872 \beta_{3} - 941792 \beta_{4} + 668408 \beta_{5} + 1399160 \beta_{6} - 33768 \beta_{7} - 243760 \beta_{8} + 40936 \beta_{9} ) q^{49} + ( 32643907847 - 339422027 \beta_{1} - 1045840 \beta_{2} - 57050064 \beta_{3} + 2099656 \beta_{4} - 484832 \beta_{5} + 1518136 \beta_{6} - 183616 \beta_{7} - 38592 \beta_{8} - 141616 \beta_{9} ) q^{50} + ( -933116 + 680235082 \beta_{1} + 1005698 \beta_{2} + 65691264 \beta_{3} + 3209271 \beta_{4} + 32638 \beta_{5} - 847553 \beta_{6} - 83074 \beta_{7} - 57856 \beta_{9} ) q^{51} + ( -5330279700 + 366987952 \beta_{1} - 878092 \beta_{2} + 56565256 \beta_{3} + 3218936 \beta_{4} + 973304 \beta_{5} - 1128932 \beta_{6} - 71768 \beta_{7} - 84192 \beta_{8} - 13796 \beta_{9} ) q^{52} + ( 2709608 - 855035743 \beta_{1} + 544009 \beta_{2} - 12830354 \beta_{3} - 8358570 \beta_{4} - 455884 \beta_{5} + 1524622 \beta_{6} - 216172 \beta_{7} + 119856 \beta_{9} ) q^{53} + ( 3543962566 - 18872460 \beta_{1} - 437466 \beta_{2} + 41775316 \beta_{3} + 1581034 \beta_{4} + 965642 \beta_{5} + 1293714 \beta_{6} + 200640 \beta_{7} - 108608 \beta_{8} - 89552 \beta_{9} ) q^{54} + ( -12305069843 - 928200086 \beta_{1} + 244992 \beta_{2} + 22940544 \beta_{3} + 10952854 \beta_{4} - 1393403 \beta_{5} - 1691712 \beta_{6} + 93984 \beta_{7} - 90277 \beta_{8} + 36960 \beta_{9} ) q^{55} + ( -46211452808 - 58941968 \beta_{1} + 389864 \beta_{2} - 136026256 \beta_{3} - 6205872 \beta_{4} - 667248 \beta_{5} - 3623664 \beta_{6} - 68816 \beta_{7} + 201216 \beta_{8} - 100840 \beta_{9} ) q^{56} + ( -51144839445 + 1218278435 \beta_{1} - 399284 \beta_{2} - 18101558 \beta_{3} + 16122916 \beta_{4} - 521009 \beta_{5} + 2786975 \beta_{6} - 46733 \beta_{7} + 703834 \beta_{8} + 259085 \beta_{9} ) q^{57} + ( 76654198700 - 70779742 \beta_{1} + 2960308 \beta_{2} - 85819954 \beta_{3} - 15738407 \beta_{4} + 716602 \beta_{5} + 1001627 \beta_{6} + 478280 \beta_{7} + 310200 \beta_{8} - 70930 \beta_{9} ) q^{58} + ( 5925120 + 549676647 \beta_{1} - 6205632 \beta_{2} + 5423117 \beta_{3} - 13419904 \beta_{4} - 425280 \beta_{5} - 1562048 \beta_{6} + 251328 \beta_{7} + 338304 \beta_{9} ) q^{59} + ( 181315037392 + 384106176 \beta_{1} + 6153776 \beta_{2} + 56893008 \beta_{3} - 14161272 \beta_{4} - 2159776 \beta_{5} - 4238336 \beta_{6} + 338464 \beta_{7} - 416512 \beta_{8} - 334512 \beta_{9} ) q^{60} + ( 3922824 - 1276639589 \beta_{1} - 1251637 \beta_{2} + 75366930 \beta_{3} - 12567986 \beta_{4} + 403908 \beta_{5} + 2603238 \beta_{6} + 1034660 \beta_{7} + 315376 \beta_{9} ) q^{61} + ( -166520982448 + 314321008 \beta_{1} - 3146960 \beta_{2} - 31498000 \beta_{3} - 25385816 \beta_{4} - 272480 \beta_{5} + 503512 \beta_{6} - 21440 \beta_{7} - 27072 \beta_{8} - 161328 \beta_{9} ) q^{62} + ( -89907478473 + 397056550 \beta_{1} - 227392 \beta_{2} - 623584 \beta_{3} + 19508234 \beta_{4} - 496993 \beta_{5} + 1281616 \beta_{6} + 9416 \beta_{7} - 155527 \beta_{8} + 255640 \beta_{9} ) q^{63} + ( -218060179128 - 865322704 \beta_{1} - 8099432 \beta_{2} + 37970992 \beta_{3} + 10143168 \beta_{4} + 666928 \beta_{5} + 2628992 \beta_{6} + 980240 \beta_{7} - 432000 \beta_{8} - 539992 \beta_{9} ) q^{64} + ( 157722994006 - 1043371650 \beta_{1} + 1301432 \beta_{2} + 21040452 \beta_{3} + 369112 \beta_{4} - 3335882 \beta_{5} - 1835258 \beta_{6} + 339966 \beta_{7} - 1200540 \beta_{8} - 281534 \beta_{9} ) q^{65} + ( 226942670020 + 1265834284 \beta_{1} + 4799432 \beta_{2} - 51943672 \beta_{3} + 27399372 \beta_{4} + 1557424 \beta_{5} - 1830092 \beta_{6} - 215776 \beta_{7} - 347424 \beta_{8} + 175736 \beta_{9} ) q^{66} + ( 8425980 + 781961615 \beta_{1} + 21797758 \beta_{2} + 29706099 \beta_{3} - 20584951 \beta_{4} - 747390 \beta_{5} - 711039 \beta_{6} + 176770 \beta_{7} + 462080 \beta_{9} ) q^{67} + ( 233813222896 - 976593000 \beta_{1} - 4994976 \beta_{2} + 79067292 \beta_{3} + 32163090 \beta_{4} + 4524736 \beta_{5} + 7433984 \beta_{6} - 313280 \beta_{7} + 1638912 \beta_{8} + 12960 \beta_{9} ) q^{68} + ( -10085720 + 2314320460 \beta_{1} - 2106808 \beta_{2} + 207661572 \beta_{3} + 28413510 \beta_{4} + 188500 \beta_{5} - 2875586 \beta_{6} - 1105996 \beta_{7} - 647248 \beta_{9} ) q^{69} + ( -607035216816 + 886920896 \beta_{1} + 693200 \beta_{2} - 276100352 \beta_{3} + 63856608 \beta_{4} - 2172592 \beta_{5} - 3983488 \beta_{6} - 860544 \beta_{7} + 591488 \beta_{8} + 331552 \beta_{9} ) q^{70} + ( 72648964185 + 2923311210 \beta_{1} - 2964288 \beta_{2} - 71313504 \beta_{3} - 30956154 \beta_{4} + 11007537 \beta_{5} + 4967952 \beta_{6} - 858648 \beta_{7} + 1095495 \beta_{8} + 388344 \beta_{9} ) q^{71} + ( -360070773683 + 840966474 \beta_{1} + 8813847 \beta_{2} + 196613146 \beta_{3} - 15564698 \beta_{4} - 1389706 \beta_{5} + 4688814 \beta_{6} - 1829934 \beta_{7} - 41984 \beta_{8} + 782569 \beta_{9} ) q^{72} + ( -63300715771 - 5238266937 \beta_{1} + 237148 \beta_{2} + 88267906 \beta_{3} - 44908748 \beta_{4} + 2828163 \beta_{5} - 12222989 \beta_{6} - 125289 \beta_{7} + 277202 \beta_{8} - 613015 \beta_{9} ) q^{73} + ( 752874413684 - 1228039234 \beta_{1} - 13673492 \beta_{2} + 715892050 \beta_{3} - 51022761 \beta_{4} - 4103546 \beta_{5} - 7256011 \beta_{6} - 2590408 \beta_{7} - 1463352 \beta_{8} + 32882 \beta_{9} ) q^{74} + ( -37591448 - 10424934857 \beta_{1} - 44716300 \beta_{2} - 608964015 \beta_{3} + 78747990 \beta_{4} + 2075404 \beta_{5} + 16345318 \beta_{6} - 2383348 \beta_{7} - 2229376 \beta_{9} ) q^{75} + ( 1033849336974 - 3022119920 \beta_{1} - 6310510 \beta_{2} - 867908908 \beta_{3} - 35134572 \beta_{4} - 8398340 \beta_{5} + 13575050 \beta_{6} - 371052 \beta_{7} - 1116816 \beta_{8} + 1170598 \beta_{9} ) q^{76} + ( -19687272 + 9840301460 \beta_{1} + 5847160 \beta_{2} - 959839716 \beta_{3} + 73231466 \beta_{4} - 305844 \beta_{5} - 23336654 \beta_{6} - 3012372 \beta_{7} - 1353264 \beta_{9} ) q^{77} + ( -825161592628 - 5455859900 \beta_{1} + 12398132 \beta_{2} + 532579204 \beta_{3} - 80693834 \beta_{4} + 7184280 \beta_{5} - 17867798 \beta_{6} + 1473392 \beta_{7} - 275600 \beta_{8} + 1423116 \beta_{9} ) q^{78} + ( 544561634522 + 8223218036 \beta_{1} + 7924480 \beta_{2} - 222874496 \beta_{3} - 154133556 \beta_{4} - 16020598 \beta_{5} + 15582784 \beta_{6} + 1616736 \beta_{7} - 2081898 \beta_{8} - 3074272 \beta_{9} ) q^{79} + ( -1540702585336 + 2590746096 \beta_{1} + 19221080 \beta_{2} + 816388848 \beta_{3} + 11831008 \beta_{4} + 4414512 \beta_{5} + 30548768 \beta_{6} - 1039856 \beta_{7} + 685440 \beta_{8} + 2319464 \beta_{9} ) q^{80} + ( -967434570912 - 8273452465 \beta_{1} - 5636612 \beta_{2} + 155994034 \beta_{3} - 27024716 \beta_{4} + 18754603 \beta_{5} - 19836997 \beta_{6} - 1639457 \beta_{7} + 4400770 \beta_{8} + 718241 \beta_{9} ) q^{81} + ( 1227249279038 + 3968133690 \beta_{1} - 16462320 \beta_{2} + 885031568 \beta_{3} + 233468504 \beta_{4} - 425376 \beta_{5} - 25704024 \beta_{6} + 6975552 \beta_{7} + 4080576 \beta_{8} + 2221680 \beta_{9} ) q^{82} + ( -34502432 - 12072829957 \beta_{1} + 58375600 \beta_{2} + 320134105 \beta_{3} + 58109256 \beta_{4} + 5728336 \beta_{5} + 28915016 \beta_{6} + 2655568 \beta_{7} - 1536384 \beta_{9} ) q^{83} + ( 2036275648528 + 3115986752 \beta_{1} - 11114384 \beta_{2} - 1248455072 \beta_{3} - 59372896 \beta_{4} + 986016 \beta_{5} + 32338640 \beta_{6} + 282592 \beta_{7} - 5078656 \beta_{8} + 1787472 \beta_{9} ) q^{84} + ( 22052104 + 24794090250 \beta_{1} + 9744798 \beta_{2} + 1530600720 \beta_{3} - 15668370 \beta_{4} + 5743108 \beta_{5} - 40579834 \beta_{6} + 10214884 \beta_{7} + 2235888 \beta_{9} ) q^{85} + ( -2679572959125 + 3695196066 \beta_{1} + 13539515 \beta_{2} - 564299166 \beta_{3} + 311314345 \beta_{4} - 21111387 \beta_{5} - 25348939 \beta_{6} - 770176 \beta_{7} - 1896576 \beta_{8} - 1663904 \beta_{9} ) q^{86} + ( 763225810151 + 12283473382 \beta_{1} + 5765824 \beta_{2} - 262690656 \beta_{3} - 45730806 \beta_{4} - 18241713 \beta_{5} + 26075792 \beta_{6} + 1430952 \beta_{7} - 276631 \beta_{8} - 1472968 \beta_{9} ) q^{87} + ( -2767741945738 + 5194601148 \beta_{1} - 14405534 \beta_{2} - 115046756 \beta_{3} - 67283348 \beta_{4} - 17079084 \beta_{5} + 35548348 \beta_{6} + 8820636 \beta_{7} + 828608 \beta_{8} - 1414402 \beta_{9} ) q^{88} + ( 550637710917 - 17626489609 \beta_{1} + 8312860 \beta_{2} + 357989794 \beta_{3} + 15525684 \beta_{4} - 17101453 \beta_{5} - 34139741 \beta_{6} + 2168391 \beta_{7} - 11809038 \beta_{8} - 1807687 \beta_{9} ) q^{89} + ( 3145563778252 - 4383604542 \beta_{1} + 18452820 \beta_{2} + 807197998 \beta_{3} - 357936407 \beta_{4} - 11609542 \beta_{5} - 32926965 \beta_{6} - 2705208 \beta_{7} + 1316920 \beta_{8} - 3693170 \beta_{9} ) q^{90} + ( 105789112 - 15362091032 \beta_{1} - 54485860 \beta_{2} - 953856396 \beta_{3} - 284560862 \beta_{4} - 3653276 \beta_{5} + 16805330 \beta_{6} + 9477732 \beta_{7} + 6565504 \beta_{9} ) q^{91} + ( 3397128056688 - 10620099520 \beta_{1} + 4142480 \beta_{2} + 1145177584 \beta_{3} + 85968152 \beta_{4} + 27777056 \beta_{5} + 30190592 \beta_{6} + 434272 \beta_{7} + 11321088 \beta_{8} - 1891088 \beta_{9} ) q^{92} + ( 47094720 - 1894796592 \beta_{1} - 12318288 \beta_{2} - 1945353280 \beta_{3} - 109297008 \beta_{4} - 10140960 \beta_{5} - 9333552 \beta_{6} - 6565920 \beta_{7} + 1787520 \beta_{9} ) q^{93} + ( -4535337247272 + 5362366024 \beta_{1} - 57150808 \beta_{2} + 205803336 \beta_{3} - 621379124 \beta_{4} + 10120368 \beta_{5} - 34971276 \beta_{6} + 4354016 \beta_{7} - 610848 \beta_{8} - 423464 \beta_{9} ) q^{94} + ( -1421631768543 + 17309214802 \beta_{1} - 31674112 \beta_{2} - 201341056 \beta_{3} + 401030574 \beta_{4} + 71677849 \beta_{5} + 40231616 \beta_{6} - 6826272 \beta_{7} + 8325639 \beta_{8} + 11195296 \beta_{9} ) q^{95} + ( -3539806193272 - 24598612432 \beta_{1} - 42293416 \beta_{2} - 3662008016 \beta_{3} - 114637952 \beta_{4} + 13970864 \beta_{5} - 26872256 \beta_{6} - 10691952 \beta_{7} + 920192 \beta_{8} - 5181976 \beta_{9} ) q^{96} + ( 135946286999 - 8942656439 \beta_{1} + 9838116 \beta_{2} + 341106174 \beta_{3} + 392173180 \beta_{4} - 71273651 \beta_{5} - 11423859 \beta_{6} + 3692505 \beta_{7} + 13407278 \beta_{8} + 1239399 \beta_{9} ) q^{97} + ( 5432359791379 + 12421831961 \beta_{1} + 74109760 \beta_{2} - 4805274304 \beta_{3} + 505846240 \beta_{4} + 41204608 \beta_{5} - 15276512 \beta_{6} - 14779136 \beta_{7} - 16022784 \beta_{8} - 1644864 \beta_{9} ) q^{98} + ( 77120736 - 2787378231 \beta_{1} + 27909552 \beta_{2} + 1836562035 \beta_{3} - 210226104 \beta_{4} - 25223088 \beta_{5} + 16535112 \beta_{6} - 21666480 \beta_{7} + 1778304 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 110q^{2} - 4716q^{4} - 267668q^{6} + 586960q^{7} - 270712q^{8} + 2014054q^{9} + O(q^{10}) \) \( 10q + 110q^{2} - 4716q^{4} - 267668q^{6} + 586960q^{7} - 270712q^{8} + 2014054q^{9} - 4542088q^{10} + 27987880q^{12} + 1408688q^{14} - 145914416q^{15} + 56624912q^{16} + 217326004q^{17} - 147615262q^{18} + 21655184q^{20} - 177987876q^{22} - 78679952q^{23} + 320199056q^{24} - 3076402574q^{25} + 3734872040q^{26} - 1653812448q^{28} + 6338232752q^{30} + 648233792q^{31} - 11298380000q^{32} + 15484079688q^{33} - 6096822724q^{34} + 4004708940q^{36} - 18764968628q^{38} - 63497510288q^{39} + 7466802592q^{40} + 59324640356q^{41} + 53897620960q^{42} + 13325704392q^{44} - 55046867440q^{46} - 10176534816q^{47} - 301841943264q^{48} + 182708552058q^{49} + 326454435302q^{50} - 53296499536q^{52} + 35449773752q^{54} - 123010753008q^{55} - 462152447680q^{56} - 511372324504q^{57} + 766482705096q^{58} + 1813082440992q^{60} - 1665308528960q^{62} - 898991123792q^{63} - 2180548996032q^{64} + 1577231990240q^{65} + 2269525079448q^{66} + 2338280915304q^{68} - 6070110714688q^{70} + 726361179984q^{71} - 3600753685960q^{72} - 633240365532q^{73} + 7528513982264q^{74} + 10338420845032q^{76} - 8252024440816q^{78} + 5445103565344q^{79} - 15406871881920q^{80} - 9674575380574q^{81} + 12273334206796q^{82} + 20362643366464q^{84} - 26794541719396q^{86} + 7632221772720q^{87} - 27677491769136q^{88} + 5506344808004q^{89} + 31454099524040q^{90} + 33971694298464q^{92} - 45356008560096q^{94} - 14214732035504q^{95} - 35398666935232q^{96} + 1361133320788q^{97} + 54325451514942q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 5 x^{9} + 752 x^{8} + 708 x^{7} - 743866 x^{6} + 96647426 x^{5} + 2540283092 x^{4} - 180067834748 x^{3} + 15101451375489 x^{2} - 802096030557765 x + 31616997813655668\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\(4673 \nu^{9} + 7354742 \nu^{8} + 521346978 \nu^{7} - 35423339110 \nu^{6} - 3119603134524 \nu^{5} + 27286958100142 \nu^{4} + 135650155419758 \nu^{3} + 72918037609395038 \nu^{2} + 438921715422012651 \nu - 240168537249774828060\)\()/ 17492130486288384 \)
\(\beta_{3}\)\(=\)\((\)\(-53727 \nu^{9} - 600394 \nu^{8} + 135456610 \nu^{7} + 4741322330 \nu^{6} + 316521478980 \nu^{5} + 11136753949934 \nu^{4} - 26417243554130 \nu^{3} + 1817150962183070 \nu^{2} - 42069080694288501 \nu + 17999672649818073444\)\()/ 69968521945153536 \)
\(\beta_{4}\)\(=\)\((\)\(-53727 \nu^{9} - 600394 \nu^{8} + 135456610 \nu^{7} + 4741322330 \nu^{6} + 316521478980 \nu^{5} + 11136753949934 \nu^{4} - 26417243554130 \nu^{3} + 141754194852490142 \nu^{2} + 517679094866939787 \nu + 38430481057802905956\)\()/ 34984260972576768 \)
\(\beta_{5}\)\(=\)\((\)\(1098741 \nu^{9} + 90806382 \nu^{8} + 2440766858 \nu^{7} + 54449976802 \nu^{6} + 1297531495828 \nu^{5} + 78627040368902 \nu^{4} - 1466142409638458 \nu^{3} + 36672486504925558 \nu^{2} + 1802119555982310951 \nu + 45775038388159863348\)\()/ 17492130486288384 \)
\(\beta_{6}\)\(=\)\((\)\(-393119 \nu^{9} + 2793526 \nu^{8} - 136735774 \nu^{7} + 5861994714 \nu^{6} + 563380286532 \nu^{5} - 22898903292818 \nu^{4} - 718390716500434 \nu^{3} + 89713441546092318 \nu^{2} - 5000468461985800117 \nu + 287833874783644461668\)\()/ 5830710162096128 \)
\(\beta_{7}\)\(=\)\((\)\(-712491 \nu^{9} - 83201106 \nu^{8} - 2967283894 \nu^{7} + 68951707810 \nu^{6} - 515357778668 \nu^{5} + 149753232903494 \nu^{4} - 14258798073026042 \nu^{3} - 318818021959084106 \nu^{2} + 1662894743218891719 \nu + 80019370682910065076\)\()/ 8746065243144192 \)
\(\beta_{8}\)\(=\)\((\)\(-1412391 \nu^{9} - 53876346 \nu^{8} - 6991130158 \nu^{7} - 262411250934 \nu^{6} + 3603178854948 \nu^{5} + 358142733536254 \nu^{4} + 3965347224942526 \nu^{3} + 110417944407472430 \nu^{2} - 33149970901662232461 \nu + 774492335446627231044\)\()/ 8746065243144192 \)
\(\beta_{9}\)\(=\)\((\)\(-12461681 \nu^{9} - 378837910 \nu^{8} - 5471783426 \nu^{7} - 1181809514106 \nu^{6} - 9343551156356 \nu^{5} - 2306079501218638 \nu^{4} - 156691468785680654 \nu^{3} + 543216588390237954 \nu^{2} - 140585877661687524603 \nu + 1111730392134764711772\)\()/ 34984260972576768 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - 2 \beta_{3} - 8 \beta_{1} - 592\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - 20 \beta_{4} + 50 \beta_{3} + \beta_{2} - 502 \beta_{1} - 10617\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-19 \beta_{9} + 48 \beta_{8} + 66 \beta_{7} - 108 \beta_{6} + 134 \beta_{5} + 344 \beta_{4} + 11622 \beta_{3} + 427 \beta_{2} - 11106 \beta_{1} + 3214105\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(565 \beta_{9} + 720 \beta_{8} - 1278 \beta_{7} - 3928 \beta_{6} + 1798 \beta_{5} + 14726 \beta_{4} + 178570 \beta_{3} - 17293 \beta_{2} + 1005960 \beta_{1} - 179362061\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-45887 \beta_{9} - 57600 \beta_{8} + 89730 \beta_{7} + 299902 \beta_{6} - 29882 \beta_{5} + 143293 \beta_{4} - 5206788 \beta_{3} - 68737 \beta_{2} - 81514094 \beta_{1} - 8831822803\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(2487037 \beta_{9} - 3107424 \beta_{8} - 5870294 \beta_{7} - 24474454 \beta_{6} - 5731714 \beta_{5} + 9846400 \beta_{4} + 1790387310 \beta_{3} + 21306707 \beta_{2} - 10335222690 \beta_{1} + 1283839484109\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-51318481 \beta_{9} + 98556528 \beta_{8} - 12162394 \beta_{7} + 1717394936 \beta_{6} + 1245638962 \beta_{5} - 7853870680 \beta_{4} - 28600747182 \beta_{3} + 633178633 \beta_{2} + 798648764194 \beta_{1} - 69547145106093\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(488837006 \beta_{9} - 167230080 \beta_{8} + 194693340 \beta_{7} - 24404269452 \beta_{6} + 4566636116 \beta_{5} + 115043898562 \beta_{4} - 537655247824 \beta_{3} - 20306671118 \beta_{2} - 10811163478243 \beta_{1} + 912576881846255\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−46.7129 17.5509i
−46.7129 + 17.5509i
−17.5296 43.4857i
−17.5296 + 43.4857i
1.33949 44.8086i
1.33949 + 44.8086i
27.9424 31.0290i
27.9424 + 31.0290i
37.4606 15.6556i
37.4606 + 15.6556i
−83.4258 35.1018i 622.159i 5727.73 + 5856.79i 26675.4i −21838.9 + 51904.1i 50480.5 −272257. 689661.i 1.20724e6 936352. 2.22541e6i
5.2 −83.4258 + 35.1018i 622.159i 5727.73 5856.79i 26675.4i −21838.9 51904.1i 50480.5 −272257. + 689661.i 1.20724e6 936352. + 2.22541e6i
5.3 −25.0592 86.9715i 1746.24i −6936.08 + 4358.86i 64905.5i −151873. + 43759.4i 201238. 552909. + 494011.i −1.45504e6 −5.64492e6 + 1.62648e6i
5.4 −25.0592 + 86.9715i 1746.24i −6936.08 4358.86i 64905.5i −151873. 43759.4i 201238. 552909. 494011.i −1.45504e6 −5.64492e6 1.62648e6i
5.5 12.6790 89.6172i 86.8898i −7870.49 2272.51i 45531.7i 7786.82 + 1101.67i −249036. −303446. + 676518.i 1.58677e6 4.08042e6 + 577295.i
5.6 12.6790 + 89.6172i 86.8898i −7870.49 + 2272.51i 45531.7i 7786.82 1101.67i −249036. −303446. 676518.i 1.58677e6 4.08042e6 577295.i
5.7 65.8848 62.0580i 1231.41i 489.620 8177.36i 25270.7i 76418.8 + 81131.3i 608245. −475211. 569148.i 77950.5 −1.56825e6 1.66495e6i
5.8 65.8848 + 62.0580i 1231.41i 489.620 + 8177.36i 25270.7i 76418.8 81131.3i 608245. −475211. + 569148.i 77950.5 −1.56825e6 + 1.66495e6i
5.9 84.9212 31.3112i 1415.71i 6231.21 5317.98i 2384.10i −44327.5 120223.i −317448. 362649. 646716.i −409898. −74649.0 202460.i
5.10 84.9212 + 31.3112i 1415.71i 6231.21 + 5317.98i 2384.10i −44327.5 + 120223.i −317448. 362649. + 646716.i −409898. −74649.0 + 202460.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{10} + 6964588 T_{3}^{8} + \)\(16\!\cdots\!52\)\( T_{3}^{6} + \)\(14\!\cdots\!28\)\( T_{3}^{4} + \)\(36\!\cdots\!44\)\( T_{3}^{2} + \)\(27\!\cdots\!16\)\( \) acting on \(S_{14}^{\mathrm{new}}(8, [\chi])\).