Properties

Label 7.17.b.b
Level 7
Weight 17
Character orbit 7.b
Analytic conductor 11.363
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(11.36271807\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{5}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -68 - \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( 2348 + 139 \beta_{2} + \beta_{5} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( -42 \beta_{1} + \beta_{4} ) q^{6} \) \( + ( -379309 - 119 \beta_{1} + 4512 \beta_{2} - 31 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( -4482976 + 10952 \beta_{2} - 208 \beta_{5} + 8 \beta_{7} ) q^{8} \) \( + ( -5241735 + 19868 \beta_{2} - 272 \beta_{5} + \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -68 - \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( 2348 + 139 \beta_{2} + \beta_{5} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( -42 \beta_{1} + \beta_{4} ) q^{6} \) \( + ( -379309 - 119 \beta_{1} + 4512 \beta_{2} - 31 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( -4482976 + 10952 \beta_{2} - 208 \beta_{5} + 8 \beta_{7} ) q^{8} \) \( + ( -5241735 + 19868 \beta_{2} - 272 \beta_{5} + \beta_{7} ) q^{9} \) \( + ( 500 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} + 15 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} + 5 \beta_{7} ) q^{10} \) \( + ( 53799838 + 10331 \beta_{2} + 2073 \beta_{5} - 246 \beta_{7} ) q^{11} \) \( + ( 13910 \beta_{1} - 39 \beta_{2} + 282 \beta_{3} - 104 \beta_{4} - 39 \beta_{5} + 78 \beta_{6} + 39 \beta_{7} ) q^{12} \) \( + ( 20572 \beta_{1} - 85 \beta_{2} - 561 \beta_{3} + 72 \beta_{4} - 85 \beta_{5} + 170 \beta_{6} + 85 \beta_{7} ) q^{13} \) \( + ( -260029280 + 33936 \beta_{1} + 1414473 \beta_{2} - 1078 \beta_{3} - 441 \beta_{4} - 11560 \beta_{5} + 174 \beta_{6} + 59 \beta_{7} ) q^{14} \) \( + ( 10424160 + 364690 \beta_{2} - 16510 \beta_{5} - 2365 \beta_{7} ) q^{15} \) \( + ( -544635104 + 3360200 \beta_{2} - 41256 \beta_{5} - 2176 \beta_{7} ) q^{16} \) \( + ( -151152 \beta_{1} - 543 \beta_{2} + 3912 \beta_{3} + 4536 \beta_{4} - 543 \beta_{5} + 1086 \beta_{6} + 543 \beta_{7} ) q^{17} \) \( + ( -904050108 + 15348929 \beta_{2} + 1510 \beta_{5} - 2240 \beta_{7} ) q^{18} \) \( + ( 518853 \beta_{1} - 429 \beta_{2} + 46764 \beta_{3} - 6720 \beta_{4} - 429 \beta_{5} + 858 \beta_{6} + 429 \beta_{7} ) q^{19} \) \( + ( 857850 \beta_{1} + 255 \beta_{2} - 50490 \beta_{3} + 2880 \beta_{4} + 255 \beta_{5} - 510 \beta_{6} - 255 \beta_{7} ) q^{20} \) \( + ( 5730904368 - 929824 \beta_{1} + 19958057 \beta_{2} - 59241 \beta_{3} - 6664 \beta_{4} + 534457 \beta_{5} - 1302 \beta_{6} - 7364 \beta_{7} ) q^{21} \) \( + ( -4284425496 - 140883638 \beta_{2} - 795428 \beta_{5} + 32328 \beta_{7} ) q^{22} \) \( + ( 11220635302 - 82768991 \beta_{2} + 313795 \beta_{5} + 23080 \beta_{7} ) q^{23} \) \( + ( -8026800 \beta_{1} + 9432 \beta_{2} + 56448 \beta_{3} - 15832 \beta_{4} + 9432 \beta_{5} - 18864 \beta_{6} - 9432 \beta_{7} ) q^{24} \) \( + ( 7745781385 + 182956400 \beta_{2} + 2231500 \beta_{5} + 85945 \beta_{7} ) q^{25} \) \( + ( -470880 \beta_{1} + 14787 \beta_{2} + 200198 \beta_{3} + 62433 \beta_{4} + 14787 \beta_{5} - 29574 \beta_{6} - 14787 \beta_{7} ) q^{26} \) \( + ( 33147738 \beta_{1} + 10773 \beta_{2} - 62316 \beta_{3} - 29088 \beta_{4} + 10773 \beta_{5} - 21546 \beta_{6} - 10773 \beta_{7} ) q^{27} \) \( + ( -47098215332 - 33458502 \beta_{1} + 350179462 \beta_{2} + 56742 \beta_{3} + 101136 \beta_{4} + 1373340 \beta_{5} - 9918 \beta_{6} + 134425 \beta_{7} ) q^{28} \) \( + ( -2804698958 - 1284223376 \beta_{2} - 5574824 \beta_{5} - 145850 \beta_{7} ) q^{29} \) \( + ( -24002287560 + 677356750 \beta_{2} - 5398150 \beta_{5} + 19280 \beta_{7} ) q^{30} \) \( + ( -69943436 \beta_{1} - 56281 \beta_{2} - 389664 \beta_{3} - 277536 \beta_{4} - 56281 \beta_{5} + 112562 \beta_{6} + 56281 \beta_{7} ) q^{31} \) \( + ( 117711173376 + 1349263936 \beta_{2} + 7438592 \beta_{5} - 715072 \beta_{7} ) q^{32} \) \( + ( 88141848 \beta_{1} - 121437 \beta_{2} - 2954862 \beta_{3} - 11576 \beta_{4} - 121437 \beta_{5} + 242874 \beta_{6} + 121437 \beta_{7} ) q^{33} \) \( + ( 329243268 \beta_{1} - 101982 \beta_{2} + 2473332 \beta_{3} - 24012 \beta_{4} - 101982 \beta_{5} + 203964 \beta_{6} + 101982 \beta_{7} ) q^{34} \) \( + ( 46490672760 - 181444305 \beta_{1} - 2555904260 \beta_{2} + 2373364 \beta_{3} - 211680 \beta_{4} + 4517240 \beta_{5} + 169680 \beta_{6} - 1055635 \beta_{7} ) q^{35} \) \( + ( -565957424916 - 1538322789 \beta_{2} - 3473727 \beta_{5} + 89904 \beta_{7} ) q^{36} \) \( + ( 717233316802 + 9923392964 \beta_{2} + 5140500 \beta_{5} - 1374370 \beta_{7} ) q^{37} \) \( + ( -545261670 \beta_{1} + 102906 \beta_{2} - 689532 \beta_{3} + 1913229 \beta_{4} + 102906 \beta_{5} - 205812 \beta_{6} - 102906 \beta_{7} ) q^{38} \) \( + ( -996975500976 + 674012378 \beta_{2} - 82291742 \beta_{5} + 2310061 \beta_{7} ) q^{39} \) \( + ( 144439840 \beta_{1} + 423440 \beta_{2} - 433776 \beta_{3} - 1225320 \beta_{4} + 423440 \beta_{5} - 846880 \beta_{6} - 423440 \beta_{7} ) q^{40} \) \( + ( 705417360 \beta_{1} + 232626 \beta_{2} - 1036346 \beta_{3} + 1198224 \beta_{4} + 232626 \beta_{5} - 465252 \beta_{6} - 232626 \beta_{7} ) q^{41} \) \( + ( -1645071067080 - 470563464 \beta_{1} - 29722904743 \beta_{2} - 3638250 \beta_{3} - 1824515 \beta_{4} - 73868123 \beta_{5} - 885654 \beta_{6} + 4257253 \beta_{7} ) q^{42} \) \( + ( -497119013858 - 14257052681 \beta_{2} + 213101365 \beta_{5} + 115980 \beta_{7} ) q^{43} \) \( + ( 5667201640056 + 47115976798 \beta_{2} + 144288122 \beta_{5} + 7689440 \beta_{7} ) q^{44} \) \( + ( -219322620 \beta_{1} + 253665 \beta_{2} + 4363281 \beta_{3} - 1652040 \beta_{4} + 253665 \beta_{5} - 507330 \beta_{6} - 253665 \beta_{7} ) q^{45} \) \( + ( 4477183719384 - 18777115866 \beta_{2} + 121355936 \beta_{5} + 1033240 \beta_{7} ) q^{46} \) \( + ( -70502232 \beta_{1} - 427929 \beta_{2} + 45337064 \beta_{3} + 699552 \beta_{4} - 427929 \beta_{5} + 855858 \beta_{6} + 427929 \beta_{7} ) q^{47} \) \( + ( -1230231152 \beta_{1} + 1249944 \beta_{2} - 42942480 \beta_{3} - 5456960 \beta_{4} + 1249944 \beta_{5} - 2499888 \beta_{6} - 1249944 \beta_{7} ) q^{48} \) \( + ( -3431316848687 + 891050888 \beta_{1} - 81845788927 \beta_{2} - 34848114 \beta_{3} + 9968952 \beta_{4} - 77154567 \beta_{5} + 2022622 \beta_{6} - 8137871 \beta_{7} ) q^{49} \) \( + ( -12070595708540 - 115793488175 \beta_{2} - 112613450 \beta_{5} + 12351520 \beta_{7} ) q^{50} \) \( + ( 7333797573936 + 242849976132 \beta_{2} - 613005348 \beta_{5} - 12212076 \beta_{7} ) q^{51} \) \( + ( 4477402086 \beta_{1} - 438255 \beta_{2} + 23692218 \beta_{3} - 15580176 \beta_{4} - 438255 \beta_{5} + 876510 \beta_{6} + 438255 \beta_{7} ) q^{52} \) \( + ( -13584980188478 + 179101570888 \beta_{2} - 279030008 \beta_{5} - 32655332 \beta_{7} ) q^{53} \) \( + ( -3059478864 \beta_{1} - 428490 \beta_{2} - 31803300 \beta_{3} + 27078642 \beta_{4} - 428490 \beta_{5} + 856980 \beta_{6} + 428490 \beta_{7} ) q^{54} \) \( + ( -8624011630 \beta_{1} - 9383135 \beta_{2} + 87553464 \beta_{3} + 11008320 \beta_{4} - 9383135 \beta_{5} + 18766270 \beta_{6} + 9383135 \beta_{7} ) q^{55} \) \( + ( -1889639891776 + 7353052224 \beta_{1} - 129278948384 \beta_{2} + 88817008 \beta_{3} - 13170024 \beta_{4} + 671704728 \beta_{5} - 1221504 \beta_{6} + 3250728 \beta_{7} ) q^{56} \) \( + ( -24520381936920 - 389815394640 \beta_{2} - 1276813380 \beta_{5} - 89649795 \beta_{7} ) q^{57} \) \( + ( 81355960300584 + 331048518762 \beta_{2} + 1288217732 \beta_{5} - 35264192 \beta_{7} ) q^{58} \) \( + ( 12990798219 \beta_{1} - 7612347 \beta_{2} - 138016148 \beta_{3} + 23577120 \beta_{4} - 7612347 \beta_{5} + 15224694 \beta_{6} + 7612347 \beta_{7} ) q^{59} \) \( + ( -41972799093120 + 180597336960 \beta_{2} + 827435760 \beta_{5} + 110573520 \beta_{7} ) q^{60} \) \( + ( 816462608 \beta_{1} - 6156212 \beta_{2} - 276606093 \beta_{3} - 105434112 \beta_{4} - 6156212 \beta_{5} + 12312424 \beta_{6} + 6156212 \beta_{7} ) q^{61} \) \( + ( -22074078396 \beta_{1} + 22565118 \beta_{2} + 40738700 \beta_{3} - 22336200 \beta_{4} + 22565118 \beta_{5} - 45130236 \beta_{6} - 22565118 \beta_{7} ) q^{62} \) \( + ( 27967381222851 + 8276471973 \beta_{1} - 211520040037 \beta_{2} + 53946648 \beta_{3} - 3288096 \beta_{4} + 593545780 \beta_{5} + 805437 \beta_{6} - 5023580 \beta_{7} ) q^{63} \) \( + ( -57566742592768 - 744581945152 \beta_{2} - 1025111488 \beta_{5} + 247879680 \beta_{7} ) q^{64} \) \( + ( 71659931932680 + 381847993540 \beta_{2} - 2478464560 \beta_{5} + 139093175 \beta_{7} ) q^{65} \) \( + ( -13106923260 \beta_{1} + 36355812 \beta_{2} + 240824568 \beta_{3} + 122742058 \beta_{4} + 36355812 \beta_{5} - 72711624 \beta_{6} - 36355812 \beta_{7} ) q^{66} \) \( + ( -90265008205378 + 76265675719 \beta_{2} + 4346320085 \beta_{5} - 175726360 \beta_{7} ) q^{67} \) \( + ( -10453264200 \beta_{1} + 41900724 \beta_{2} - 28435032 \beta_{3} + 136237104 \beta_{4} + 41900724 \beta_{5} - 83801448 \beta_{6} - 41900724 \beta_{7} ) q^{68} \) \( + ( 17839173348 \beta_{1} - 8429805 \beta_{2} + 420565230 \beta_{3} + 100675816 \beta_{4} - 8429805 \beta_{5} + 16859610 \beta_{6} + 8429805 \beta_{7} ) q^{69} \) \( + ( 158580260859720 - 12546582790 \beta_{1} - 53147535490 \beta_{2} + 137461464 \beta_{3} - 78082725 \beta_{4} - 727212990 \beta_{5} - 22301720 \beta_{6} + 98656180 \beta_{7} ) q^{70} \) \( + ( -36136905833642 + 1118409067891 \beta_{2} + 5704875373 \beta_{5} - 148455611 \beta_{7} ) q^{71} \) \( + ( 195000839525664 - 184097280008 \beta_{2} + 1913700032 \beta_{5} + 113256968 \beta_{7} ) q^{72} \) \( + ( -13295071008 \beta_{1} - 52791543 \beta_{2} - 507801390 \beta_{3} - 312618600 \beta_{4} - 52791543 \beta_{5} + 105583086 \beta_{6} + 52791543 \beta_{7} ) q^{73} \) \( + ( -676458232955416 - 1631332812006 \beta_{2} - 13865193164 \beta_{5} + 129083680 \beta_{7} ) q^{74} \) \( + ( 35753375425 \beta_{1} - 72847575 \beta_{2} + 1865859660 \beta_{3} - 79070400 \beta_{4} - 72847575 \beta_{5} + 145695150 \beta_{6} + 72847575 \beta_{7} ) q^{75} \) \( + ( 143131916766 \beta_{1} - 60958971 \beta_{2} - 2751197454 \beta_{3} - 300094392 \beta_{4} - 60958971 \beta_{5} + 121917942 \beta_{6} + 60958971 \beta_{7} ) q^{76} \) \( + ( 560703532835362 - 97171164300 \beta_{1} + 496273903764 \beta_{2} - 1954472506 \beta_{3} + 385836192 \beta_{4} - 11343638500 \beta_{5} + 53849820 \beta_{6} - 183549655 \beta_{7} ) q^{77} \) \( + ( 24057446704200 + 4421709869330 \beta_{2} + 11033377030 \beta_{5} - 806177840 \beta_{7} ) q^{78} \) \( + ( 729097108320422 - 4671457972661 \beta_{2} + 5288549741 \beta_{5} - 1265320765 \beta_{7} ) q^{79} \) \( + ( -133181713680 \beta_{1} - 40433880 \beta_{2} + 2047833104 \beta_{3} - 247536000 \beta_{4} - 40433880 \beta_{5} + 80867760 \beta_{6} + 40433880 \beta_{7} ) q^{80} \) \( + ( -1827129366446823 + 249163471152 \beta_{2} - 8603694108 \beta_{5} + 145432899 \beta_{7} ) q^{81} \) \( + ( 76398197024 \beta_{1} - 82025930 \beta_{2} - 169860564 \beta_{3} + 465771138 \beta_{4} - 82025930 \beta_{5} + 164051860 \beta_{6} + 82025930 \beta_{7} ) q^{82} \) \( + ( 149053280853 \beta_{1} + 90090402 \beta_{2} + 3427436384 \beta_{3} + 396754272 \beta_{4} + 90090402 \beta_{5} - 180180804 \beta_{6} - 90090402 \beta_{7} ) q^{83} \) \( + ( 1615593445143552 - 37221432626 \beta_{1} + 5567591954925 \beta_{2} + 2391340434 \beta_{3} - 260465968 \beta_{4} + 12104133405 \beta_{5} + 60736998 \beta_{6} - 424981053 \beta_{7} ) q^{84} \) \( + ( -726922439292240 + 6531059395860 \beta_{2} - 19270419540 \beta_{5} + 1874778780 \beta_{7} ) q^{85} \) \( + ( 938556652219064 - 7519791888706 \beta_{2} - 144233704 \beta_{5} + 1697388200 \beta_{7} ) q^{86} \) \( + ( -52150373406 \beta_{1} + 193352886 \beta_{2} - 3670590168 \beta_{3} + 1045349536 \beta_{4} + 193352886 \beta_{5} - 386705772 \beta_{6} - 193352886 \beta_{7} ) q^{87} \) \( + ( -3083204399040192 - 5938692776400 \beta_{2} + 15126750592 \beta_{5} - 1456466992 \beta_{7} ) q^{88} \) \( + ( -155297423448 \beta_{1} + 454622625 \beta_{2} - 4422530730 \beta_{3} - 1003473576 \beta_{4} + 454622625 \beta_{5} - 909245250 \beta_{6} - 454622625 \beta_{7} ) q^{89} \) \( + ( -102259465200 \beta_{1} - 1524555 \beta_{2} - 986597334 \beta_{3} - 214807185 \beta_{4} - 1524555 \beta_{5} + 3049110 \beta_{6} + 1524555 \beta_{7} ) q^{90} \) \( + ( 2991224870409048 - 10959427367 \beta_{1} + 6317884027021 \beta_{2} + 1169273280 \beta_{3} - 882404544 \beta_{4} + 504443961 \beta_{5} - 335262970 \beta_{6} + 1565665262 \beta_{7} ) q^{91} \) \( + ( 149661027681336 - 2867348345714 \beta_{2} - 7464556438 \beta_{5} - 607850752 \beta_{7} ) q^{92} \) \( + ( 3376518945306480 - 20482804419700 \beta_{2} - 28233405020 \beta_{5} + 2203869880 \beta_{7} ) q^{93} \) \( + ( 55205405404 \beta_{1} - 179508202 \beta_{2} + 1391910972 \beta_{3} + 841767780 \beta_{4} - 179508202 \beta_{5} + 359016404 \beta_{6} + 179508202 \beta_{7} ) q^{94} \) \( + ( -6650587857138480 + 6245883608550 \beta_{2} + 134460072750 \beta_{5} + 4549979415 \beta_{7} ) q^{95} \) \( + ( 204677025408 \beta_{1} - 408091968 \beta_{2} - 8190772992 \beta_{3} - 1303434816 \beta_{4} - 408091968 \beta_{5} + 816183936 \beta_{6} + 408091968 \beta_{7} ) q^{96} \) \( + ( -670801115320 \beta_{1} + 120110527 \beta_{2} + 9754068660 \beta_{3} + 93966600 \beta_{4} + 120110527 \beta_{5} - 240221054 \beta_{6} - 120110527 \beta_{7} ) q^{97} \) \( + ( 5409802670912332 + 661029299532 \beta_{1} + 12523132196295 \beta_{2} + 4782542296 \beta_{3} + 401538438 \beta_{4} + 63180716816 \beta_{5} + 77160888 \beta_{6} + 14982044 \beta_{7} ) q^{98} \) \( + ( -1967325765598674 - 4988698499173 \beta_{2} - 7625446751 \beta_{5} - 1886147684 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 544q^{2} \) \(\mathstrut +\mathstrut 18784q^{4} \) \(\mathstrut -\mathstrut 3034472q^{7} \) \(\mathstrut -\mathstrut 35863808q^{8} \) \(\mathstrut -\mathstrut 41933880q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 544q^{2} \) \(\mathstrut +\mathstrut 18784q^{4} \) \(\mathstrut -\mathstrut 3034472q^{7} \) \(\mathstrut -\mathstrut 35863808q^{8} \) \(\mathstrut -\mathstrut 41933880q^{9} \) \(\mathstrut +\mathstrut 430398704q^{11} \) \(\mathstrut -\mathstrut 2080234240q^{14} \) \(\mathstrut +\mathstrut 83393280q^{15} \) \(\mathstrut -\mathstrut 4357080832q^{16} \) \(\mathstrut -\mathstrut 7232400864q^{18} \) \(\mathstrut +\mathstrut 45847234944q^{21} \) \(\mathstrut -\mathstrut 34275403968q^{22} \) \(\mathstrut +\mathstrut 89765082416q^{23} \) \(\mathstrut +\mathstrut 61966251080q^{25} \) \(\mathstrut -\mathstrut 376785722656q^{28} \) \(\mathstrut -\mathstrut 22437591664q^{29} \) \(\mathstrut -\mathstrut 192018300480q^{30} \) \(\mathstrut +\mathstrut 941689387008q^{32} \) \(\mathstrut +\mathstrut 371925382080q^{35} \) \(\mathstrut -\mathstrut 4527659399328q^{36} \) \(\mathstrut +\mathstrut 5737866534416q^{37} \) \(\mathstrut -\mathstrut 7975804007808q^{39} \) \(\mathstrut -\mathstrut 13160568536640q^{42} \) \(\mathstrut -\mathstrut 3976952110864q^{43} \) \(\mathstrut +\mathstrut 45337613120448q^{44} \) \(\mathstrut +\mathstrut 35817469755072q^{46} \) \(\mathstrut -\mathstrut 27450534789496q^{49} \) \(\mathstrut -\mathstrut 96564765668320q^{50} \) \(\mathstrut +\mathstrut 58670380591488q^{51} \) \(\mathstrut -\mathstrut 108679841507824q^{53} \) \(\mathstrut -\mathstrut 15117119134208q^{56} \) \(\mathstrut -\mathstrut 196163055495360q^{57} \) \(\mathstrut +\mathstrut 650847682404672q^{58} \) \(\mathstrut -\mathstrut 335782392744960q^{60} \) \(\mathstrut +\mathstrut 223739049782808q^{63} \) \(\mathstrut -\mathstrut 460533940742144q^{64} \) \(\mathstrut +\mathstrut 573279455461440q^{65} \) \(\mathstrut -\mathstrut 722120065643024q^{67} \) \(\mathstrut +\mathstrut 1268642086877760q^{70} \) \(\mathstrut -\mathstrut 289095246669136q^{71} \) \(\mathstrut +\mathstrut 1560006716205312q^{72} \) \(\mathstrut -\mathstrut 5411665863643328q^{74} \) \(\mathstrut +\mathstrut 4485628262682896q^{77} \) \(\mathstrut +\mathstrut 192459573633600q^{78} \) \(\mathstrut +\mathstrut 5832776866563376q^{79} \) \(\mathstrut -\mathstrut 14617034931574584q^{81} \) \(\mathstrut +\mathstrut 12924747561148416q^{84} \) \(\mathstrut -\mathstrut 5815379514337920q^{85} \) \(\mathstrut +\mathstrut 7508453217752512q^{86} \) \(\mathstrut -\mathstrut 24665635192321536q^{88} \) \(\mathstrut +\mathstrut 23929798963272384q^{91} \) \(\mathstrut +\mathstrut 1197288221450688q^{92} \) \(\mathstrut +\mathstrut 27012151562451840q^{93} \) \(\mathstrut -\mathstrut 53204702857107840q^{95} \) \(\mathstrut +\mathstrut 43278421367298656q^{98} \) \(\mathstrut -\mathstrut 15738606124789392q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(5365384\) \(x^{6}\mathstrut +\mathstrut \) \(10449491370210\) \(x^{4}\mathstrut +\mathstrut \) \(8743024230718881600\) \(x^{2}\mathstrut +\mathstrut \) \(2655236149032650377194000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2123049 \nu^{6} + 8379851087056 \nu^{4} + 10560594934645773100 \nu^{2} + 4267550274457843985049900 \)\()/\)\(36\!\cdots\!25\)
\(\beta_{3}\)\(=\)\((\)\( 20994056 \nu^{7} + 93059602745484 \nu^{5} + 131548429704596116180 \nu^{3} + 58436470612008109922333550 \nu \)\()/\)\(22\!\cdots\!75\)
\(\beta_{4}\)\(=\)\((\)\( -4246098 \nu^{7} - 16759702174112 \nu^{5} - 21121189869291546200 \nu^{3} - 8554298570598058199298300 \nu \)\()/\)\(12\!\cdots\!75\)
\(\beta_{5}\)\(=\)\((\)\( 543271063 \nu^{6} + 2190708740631772 \nu^{4} + 2681383660155330844500 \nu^{2} + 972429509087330792270572800 \)\()/\)\(11\!\cdots\!75\)
\(\beta_{6}\)\(=\)\((\)\(688547536\) \(\nu^{7}\mathstrut -\mathstrut \) \(320507083115\) \(\nu^{6}\mathstrut +\mathstrut \) \(2275489716337384\) \(\nu^{5}\mathstrut -\mathstrut \) \(1459856416958479910\) \(\nu^{4}\mathstrut +\mathstrut \) \(1904306695026691208700\) \(\nu^{3}\mathstrut -\mathstrut \) \(2124157260388749949974150\) \(\nu^{2}\mathstrut +\mathstrut \) \(180665932805429123003849100\) \(\nu\mathstrut -\mathstrut \) \(970854398946867402294493146750\)\()/\)\(34\!\cdots\!25\)
\(\beta_{7}\)\(=\)\((\)\( 4245503308 \nu^{6} + 19280026651791232 \nu^{4} + 27951093868201530406620 \nu^{2} + 12724199966780429926709572200 \)\()/\)\(22\!\cdots\!75\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(272\) \(\beta_{5}\mathstrut +\mathstrut \) \(19868\) \(\beta_{2}\mathstrut -\mathstrut \) \(48288456\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1197\) \(\beta_{7}\mathstrut -\mathstrut \) \(2394\) \(\beta_{6}\mathstrut +\mathstrut \) \(1197\) \(\beta_{5}\mathstrut -\mathstrut \) \(3232\) \(\beta_{4}\mathstrut -\mathstrut \) \(6924\) \(\beta_{3}\mathstrut +\mathstrut \) \(1197\) \(\beta_{2}\mathstrut -\mathstrut \) \(5882856\) \(\beta_{1}\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(452576\) \(\beta_{7}\mathstrut +\mathstrut \) \(736734173\) \(\beta_{5}\mathstrut -\mathstrut \) \(64349813537\) \(\beta_{2}\mathstrut +\mathstrut \) \(70995264543324\)\()/36\)
\(\nu^{5}\)\(=\)\((\)\(3184217523\) \(\beta_{7}\mathstrut +\mathstrut \) \(6368435046\) \(\beta_{6}\mathstrut -\mathstrut \) \(3184217523\) \(\beta_{5}\mathstrut +\mathstrut \) \(10030142488\) \(\beta_{4}\mathstrut +\mathstrut \) \(23807855586\) \(\beta_{3}\mathstrut -\mathstrut \) \(3184217523\) \(\beta_{2}\mathstrut +\mathstrut \) \(9315915656814\) \(\beta_{1}\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(1690153456211\) \(\beta_{7}\mathstrut -\mathstrut \) \(388738182475028\) \(\beta_{5}\mathstrut +\mathstrut \) \(40356451128322157\) \(\beta_{2}\mathstrut -\mathstrut \) \(28097176608290600964\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(1653545977361253\) \(\beta_{7}\mathstrut -\mathstrut \) \(3307091954722506\) \(\beta_{6}\mathstrut +\mathstrut \) \(1653545977361253\) \(\beta_{5}\mathstrut -\mathstrut \) \(6052148172423493\) \(\beta_{4}\mathstrut -\mathstrut \) \(14882455042880196\) \(\beta_{3}\mathstrut +\mathstrut \) \(1653545977361253\) \(\beta_{2}\mathstrut -\mathstrut \) \(3891587501044539054\) \(\beta_{1}\)\()/6\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
1246.39i
1246.39i
923.458i
923.458i
1024.53i
1024.53i
1381.83i
1381.83i
−408.297 7478.36i 101171. 25884.6i 3.05340e6i 879246. + 5.69736e6i −1.45496e7 −1.28792e7 1.05686e7i
6.2 −408.297 7478.36i 101171. 25884.6i 3.05340e6i 879246. 5.69736e6i −1.45496e7 −1.28792e7 1.05686e7i
6.3 −174.041 5540.75i −35245.9 367486.i 964315.i −2.61707e6 5.13652e6i 1.75401e7 1.23468e7 6.39575e7i
6.4 −174.041 5540.75i −35245.9 367486.i 964315.i −2.61707e6 + 5.13652e6i 1.75401e7 1.23468e7 6.39575e7i
6.5 40.3121 6147.19i −63910.9 623292.i 247806.i 5.20719e6 2.47347e6i −5.21828e6 5.25876e6 2.51262e7i
6.6 40.3121 6147.19i −63910.9 623292.i 247806.i 5.20719e6 + 2.47347e6i −5.21828e6 5.25876e6 2.51262e7i
6.7 270.026 8290.96i 7378.00 234861.i 2.23878e6i −4.98661e6 + 2.89252e6i −1.57042e7 −2.56934e7 6.34186e7i
6.8 270.026 8290.96i 7378.00 234861.i 2.23878e6i −4.98661e6 2.89252e6i −1.57042e7 −2.56934e7 6.34186e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 272 T_{2}^{3} \) \(\mathstrut -\mathstrut 98776 T_{2}^{2} \) \(\mathstrut -\mathstrut 15713792 T_{2} \) \(\mathstrut +\mathstrut 773514240 \) acting on \(S_{17}^{\mathrm{new}}(7, [\chi])\).