# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{17}^{\mathrm{new}}(7, [\chi])$$.