Properties

Label 8.13.d.b
Level 8
Weight 13
Character orbit 8.d
Analytic conductor 7.312
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(7.31195053821\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{3}\cdot 23 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -11 - \beta_{1} ) q^{2} \) \( + ( -66 + 3 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -244 + 12 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 48 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -13081 + 113 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{9} ) q^{6} \) \( + ( 323 \beta_{1} + 7 \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 2710 + 241 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} \) \( + ( 169275 - 763 \beta_{1} - 189 \beta_{2} + 11 \beta_{3} + 6 \beta_{4} - \beta_{6} - 4 \beta_{7} + 4 \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -11 - \beta_{1} ) q^{2} \) \( + ( -66 + 3 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -244 + 12 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 48 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} \) \( + ( -13081 + 113 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{9} ) q^{6} \) \( + ( 323 \beta_{1} + 7 \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 2710 + 241 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} \) \( + ( 169275 - 763 \beta_{1} - 189 \beta_{2} + 11 \beta_{3} + 6 \beta_{4} - \beta_{6} - 4 \beta_{7} + 4 \beta_{9} ) q^{9} \) \( + ( 187302 - 548 \beta_{1} - 92 \beta_{2} + 44 \beta_{3} + 9 \beta_{4} + 20 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} ) q^{10} \) \( + ( -459250 + 7249 \beta_{1} + 297 \beta_{2} + 242 \beta_{3} - 44 \beta_{4} - 22 \beta_{6} ) q^{11} \) \( + ( -542836 + 15674 \beta_{1} + 1218 \beta_{2} + 166 \beta_{3} - 38 \beta_{4} - 92 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} + 14 \beta_{8} + 36 \beta_{9} ) q^{12} \) \( + ( 64 + 3760 \beta_{1} + 79 \beta_{2} - 216 \beta_{3} + 32 \beta_{4} + 17 \beta_{5} + 8 \beta_{6} - 16 \beta_{7} + 8 \beta_{8} - 104 \beta_{9} ) q^{13} \) \( + ( 1272852 - 4480 \beta_{1} - 2456 \beta_{2} + 104 \beta_{3} + 6 \beta_{4} - 280 \beta_{5} + 84 \beta_{6} - 28 \beta_{7} - 20 \beta_{8} - 60 \beta_{9} ) q^{14} \) \( + ( 896 + 10231 \beta_{1} - 53 \beta_{2} - 2256 \beta_{3} + 61 \beta_{4} - 32 \beta_{5} - 144 \beta_{6} + 32 \beta_{7} - 19 \beta_{8} + 205 \beta_{9} ) q^{15} \) \( + ( -3829556 + 7178 \beta_{1} - 1472 \beta_{2} + 540 \beta_{3} - 220 \beta_{4} + 660 \beta_{5} + 84 \beta_{6} + 70 \beta_{7} - 10 \beta_{8} + 52 \beta_{9} ) q^{16} \) \( + ( 4287290 - 127881 \beta_{1} - 8607 \beta_{2} + 825 \beta_{3} + 706 \beta_{4} - 75 \beta_{6} + 20 \beta_{7} - 20 \beta_{9} ) q^{17} \) \( + ( 822855 - 167931 \beta_{1} + 18304 \beta_{2} + 568 \beta_{3} - 440 \beta_{4} + 1184 \beta_{5} + 416 \beta_{6} + 48 \beta_{7} - 80 \beta_{8} - 184 \beta_{9} ) q^{18} \) \( + ( 3604718 - 56143 \beta_{1} - 13399 \beta_{2} + 3586 \beta_{3} + 372 \beta_{4} - 326 \beta_{6} + 128 \beta_{7} - 128 \beta_{9} ) q^{19} \) \( + ( 1761384 - 173228 \beta_{1} + 49216 \beta_{2} + 1184 \beta_{3} - 20 \beta_{4} - 1432 \beta_{5} + 736 \beta_{6} + 28 \beta_{7} - 244 \beta_{8} - 216 \beta_{9} ) q^{20} \) \( + ( 7616 + 361568 \beta_{1} + 5508 \beta_{2} - 19560 \beta_{3} + 1632 \beta_{4} - 100 \beta_{5} - 1096 \beta_{6} + 144 \beta_{7} - 200 \beta_{8} + 808 \beta_{9} ) q^{21} \) \( + ( -22702361 + 511313 \beta_{1} - 60852 \beta_{2} + 5797 \beta_{3} - 2849 \beta_{4} - 704 \beta_{5} + 704 \beta_{6} + 352 \beta_{7} + 352 \beta_{8} + 671 \beta_{9} ) q^{22} \) \( + ( 6528 - 436207 \beta_{1} - 10659 \beta_{2} - 18192 \beta_{3} - 741 \beta_{4} + 864 \beta_{5} - 464 \beta_{6} - 352 \beta_{7} + 139 \beta_{8} - 2325 \beta_{9} ) q^{23} \) \( + ( 16908372 + 440942 \beta_{1} - 158152 \beta_{2} + 6084 \beta_{3} + 3760 \beta_{4} - 2020 \beta_{5} + 2068 \beta_{6} - 814 \beta_{7} + 370 \beta_{8} - 420 \beta_{9} ) q^{24} \) \( + ( -70709183 - 395986 \beta_{1} + 41026 \beta_{2} + 33330 \beta_{3} + 516 \beta_{4} - 3030 \beta_{6} + 104 \beta_{7} - 104 \beta_{9} ) q^{25} \) \( + ( 15122586 + 103012 \beta_{1} + 275804 \beta_{2} + 17748 \beta_{3} - 10217 \beta_{4} - 7700 \beta_{5} - 710 \beta_{6} - 122 \beta_{7} + 250 \beta_{8} + 2626 \beta_{9} ) q^{26} \) \( + ( 138237804 + 1408380 \beta_{1} - 160512 \beta_{2} + 8250 \beta_{3} - 4572 \beta_{4} - 750 \beta_{6} - 1920 \beta_{7} + 1920 \beta_{9} ) q^{27} \) \( + ( -13215568 - 1056360 \beta_{1} + 462848 \beta_{2} + 18496 \beta_{3} + 11880 \beta_{4} + 14768 \beta_{5} + 832 \beta_{6} - 1144 \beta_{7} + 1704 \beta_{8} - 1744 \beta_{9} ) q^{28} \) \( + ( -7680 + 4916368 \beta_{1} + 107051 \beta_{2} + 19776 \beta_{3} + 16768 \beta_{4} + 85 \beta_{5} + 1088 \beta_{6} - 128 \beta_{7} + 2240 \beta_{8} + 1344 \beta_{9} ) q^{29} \) \( + ( 45609796 - 500736 \beta_{1} - 777528 \beta_{2} - 11320 \beta_{3} - 20562 \beta_{4} + 20296 \beta_{5} - 2940 \beta_{6} - 1708 \beta_{7} - 2500 \beta_{8} - 972 \beta_{9} ) q^{30} \) \( + ( -21376 - 7559524 \beta_{1} - 156340 \beta_{2} + 59088 \beta_{3} - 30668 \beta_{4} - 10720 \beta_{5} + 1680 \beta_{6} + 992 \beta_{7} - 380 \beta_{8} + 6564 \beta_{9} ) q^{31} \) \( + ( -265117592 + 4521676 \beta_{1} - 311712 \beta_{2} - 14072 \beta_{3} + 26600 \beta_{4} - 21096 \beta_{5} - 8904 \beta_{6} + 3668 \beta_{7} - 3148 \beta_{8} - 3112 \beta_{9} ) q^{32} \) \( + ( -47726580 - 12099879 \beta_{1} + 215567 \beta_{2} - 172425 \beta_{3} + 58014 \beta_{4} + 15675 \beta_{6} - 660 \beta_{7} + 660 \beta_{9} ) q^{33} \) \( + ( 445516274 - 5646074 \beta_{1} + 108672 \beta_{2} - 131096 \beta_{3} - 24168 \beta_{4} - 8480 \beta_{5} + 480 \beta_{6} + 1040 \beta_{7} + 1680 \beta_{8} - 12648 \beta_{9} ) q^{34} \) \( + ( -283796064 + 21135412 \beta_{1} + 549468 \beta_{2} - 116820 \beta_{3} - 104712 \beta_{4} + 10620 \beta_{6} + 11392 \beta_{7} - 11392 \beta_{9} ) q^{35} \) \( + ( -407402940 + 499540 \beta_{1} + 745939 \beta_{2} - 182035 \beta_{3} + 97456 \beta_{4} - 17440 \beta_{5} - 22464 \beta_{6} + 7504 \beta_{7} - 5360 \beta_{8} + 23008 \beta_{9} ) q^{36} \) \( + ( -153664 + 19079120 \beta_{1} + 447305 \beta_{2} + 395736 \beta_{3} + 62944 \beta_{4} + 1623 \beta_{5} + 21752 \beta_{6} - 2544 \beta_{7} - 14600 \beta_{8} - 32408 \beta_{9} ) q^{37} \) \( + ( 164449943 - 4575199 \beta_{1} - 864756 \beta_{2} - 91723 \beta_{3} - 64241 \beta_{4} - 49344 \beta_{5} - 1856 \beta_{6} + 4192 \beta_{7} + 8288 \beta_{8} - 15153 \beta_{9} ) q^{38} \) \( + ( -134144 - 45421239 \beta_{1} - 931691 \beta_{2} + 360576 \beta_{3} - 157565 \beta_{4} + 80640 \beta_{5} + 13952 \beta_{6} + 2816 \beta_{7} - 765 \beta_{8} + 18947 \beta_{9} ) q^{39} \) \( + ( 818121872 - 3801480 \beta_{1} + 1117632 \beta_{2} - 227824 \beta_{3} + 225008 \beta_{4} + 98032 \beta_{5} - 18576 \beta_{6} - 4536 \beta_{7} + 11144 \beta_{8} + 45424 \beta_{9} ) q^{40} \) \( + ( -1233871822 - 37314966 \beta_{1} + 2102502 \beta_{2} - 121418 \beta_{3} + 149068 \beta_{4} + 11038 \beta_{6} - 1096 \beta_{7} + 1096 \beta_{9} ) q^{41} \) \( + ( 1469134648 - 5944336 \beta_{1} - 4292912 \beta_{2} + 283376 \beta_{3} - 236972 \beta_{4} + 136848 \beta_{5} - 37448 \beta_{6} - 11384 \beta_{7} - 16392 \beta_{8} + 16920 \beta_{9} ) q^{42} \) \( + ( 2508180734 + 48826271 \beta_{1} - 2469645 \beta_{2} - 71500 \beta_{3} - 193848 \beta_{4} + 6500 \beta_{6} - 29440 \beta_{7} + 29440 \beta_{9} ) q^{43} \) \( + ( -2795254132 + 30459770 \beta_{1} - 1913758 \beta_{2} + 444422 \beta_{3} + 200090 \beta_{4} - 141020 \beta_{5} + 16632 \beta_{6} - 15290 \beta_{7} + 1870 \beta_{8} - 103004 \beta_{9} ) q^{44} \) \( + ( 126912 + 100901936 \beta_{1} + 2113987 \beta_{2} - 311592 \beta_{3} + 349792 \beta_{4} - 6307 \beta_{5} - 23048 \beta_{6} + 7184 \beta_{7} + 59512 \beta_{8} + 109800 \beta_{9} ) q^{45} \) \( + ( -1681345892 + 7445760 \beta_{1} + 5560568 \beta_{2} - 36360 \beta_{3} - 434206 \beta_{4} - 148360 \beta_{5} - 42148 \beta_{6} - 8948 \beta_{7} - 6940 \beta_{8} + 81452 \beta_{9} ) q^{46} \) \( + ( 166272 - 63919238 \beta_{1} - 1393774 \beta_{2} - 495888 \beta_{3} - 271106 \beta_{4} - 408224 \beta_{5} - 976 \beta_{6} - 19808 \beta_{7} + 7790 \beta_{8} - 130866 \beta_{9} ) q^{47} \) \( + ( 1330597000 - 15229156 \beta_{1} + 11378240 \beta_{2} + 1010920 \beta_{3} + 159864 \beta_{4} + 61240 \beta_{5} + 83960 \beta_{6} - 18620 \beta_{7} - 4636 \beta_{8} - 204296 \beta_{9} ) q^{48} \) \( + ( -3953924639 - 48476552 \beta_{1} - 493496 \beta_{2} + 1549064 \beta_{3} + 199824 \beta_{4} - 140824 \beta_{6} + 10144 \beta_{7} - 10144 \beta_{9} ) q^{49} \) \( + ( 2489173909 + 62694079 \beta_{1} - 6641920 \beta_{2} - 716976 \beta_{3} - 430160 \beta_{4} - 128576 \beta_{5} + 86976 \beta_{6} + 47648 \beta_{7} + 50976 \beta_{8} + 52144 \beta_{9} ) q^{50} \) \( + ( 3675394188 + 34437752 \beta_{1} - 6036364 \beta_{2} + 769406 \beta_{3} - 94836 \beta_{4} - 69946 \beta_{6} + 512 \beta_{7} - 512 \beta_{9} ) q^{51} \) \( + ( -3616909672 - 11368404 \beta_{1} - 10933312 \beta_{2} - 609440 \beta_{3} + 520340 \beta_{4} + 263320 \beta_{5} + 162080 \beta_{6} - 33180 \beta_{7} + 39284 \beta_{8} + 241624 \beta_{9} ) q^{52} \) \( + ( 1286208 + 55817808 \beta_{1} + 811829 \beta_{2} - 3327192 \beta_{3} + 282016 \beta_{4} - 4181 \beta_{5} - 177144 \beta_{6} + 16368 \beta_{7} - 146040 \beta_{8} - 31464 \beta_{9} ) q^{53} \) \( + ( -7401626754 - 117756654 \beta_{1} + 6653208 \beta_{2} + 2273082 \beta_{3} - 278130 \beta_{4} + 559680 \beta_{5} + 208320 \beta_{6} + 27360 \beta_{7} - 34080 \beta_{8} - 112050 \beta_{9} ) q^{54} \) \( + ( 1255936 - 83577945 \beta_{1} - 2094565 \beta_{2} - 3258816 \beta_{3} + 18381 \beta_{4} + 1457280 \beta_{5} - 169664 \beta_{6} + 12672 \beta_{7} - 13299 \beta_{8} + 75405 \beta_{9} ) q^{55} \) \( + ( 15445917408 - 27044592 \beta_{1} + 5950080 \beta_{2} - 1802272 \beta_{3} + 160032 \beta_{4} - 829920 \beta_{5} + 110368 \beta_{6} + 68976 \beta_{7} - 94480 \beta_{8} + 416032 \beta_{9} ) q^{56} \) \( + ( 8041262796 - 144294263 \beta_{1} + 1871071 \beta_{2} - 916025 \beta_{3} + 675390 \beta_{4} + 83275 \beta_{6} + 4780 \beta_{7} - 4780 \beta_{9} ) q^{57} \) \( + ( 19325448418 - 53664268 \beta_{1} - 5206900 \beta_{2} + 4106084 \beta_{3} + 259459 \beta_{4} - 702628 \beta_{5} + 210178 \beta_{6} - 45826 \beta_{7} - 24830 \beta_{8} - 142422 \beta_{9} ) q^{58} \) \( + ( -10945426594 - 39068077 \beta_{1} + 9074319 \beta_{2} + 1429824 \beta_{3} + 23680 \beta_{4} - 129984 \beta_{6} + 194176 \beta_{7} - 194176 \beta_{9} ) q^{59} \) \( + ( -34737076240 + 58717240 \beta_{1} + 17283072 \beta_{2} + 2324288 \beta_{3} - 1543224 \beta_{4} + 791792 \beta_{5} - 164288 \beta_{6} + 209000 \beta_{7} - 107768 \beta_{8} - 362896 \beta_{9} ) q^{60} \) \( + ( -851392 - 179649552 \beta_{1} - 3498413 \beta_{2} + 1982568 \beta_{3} - 628320 \beta_{4} + 72845 \beta_{5} + 190536 \beta_{6} - 84112 \beta_{7} + 148680 \beta_{8} - 440104 \beta_{9} ) q^{61} \) \( + ( -29920635504 + 74842752 \beta_{1} - 7489248 \beta_{2} - 7896544 \beta_{3} + 1417464 \beta_{4} + 568608 \beta_{5} + 181392 \beta_{6} - 18864 \beta_{7} + 74992 \beta_{8} - 258736 \beta_{9} ) q^{62} \) \( + ( -224640 + 500967737 \beta_{1} + 10518565 \beta_{2} + 969360 \beta_{3} + 1030867 \beta_{4} - 3725920 \beta_{5} - 98480 \beta_{6} + 126560 \beta_{7} - 39197 \beta_{8} + 846723 \beta_{9} ) q^{63} \) \( + ( 27619504880 + 185724936 \beta_{1} - 28503616 \beta_{2} + 3863472 \beta_{3} - 1903440 \beta_{4} + 431120 \beta_{5} - 362416 \beta_{6} - 41032 \beta_{7} + 274680 \beta_{8} - 304240 \beta_{9} ) q^{64} \) \( + ( 9649843728 + 540701266 \beta_{1} - 45840066 \beta_{2} - 3025330 \beta_{3} - 1832196 \beta_{4} + 275030 \beta_{6} - 96104 \beta_{7} + 96104 \beta_{9} ) q^{65} \) \( + ( 48632714636 - 68918476 \beta_{1} + 49619328 \beta_{2} - 12110824 \beta_{3} + 2287208 \beta_{4} + 702240 \beta_{5} - 438240 \beta_{6} - 245520 \beta_{7} - 266640 \beta_{8} - 204952 \beta_{9} ) q^{66} \) \( + ( -4985168722 - 392498463 \beta_{1} + 15407865 \beta_{2} - 2110350 \beta_{3} + 1680084 \beta_{4} + 191850 \beta_{6} - 456960 \beta_{7} + 456960 \beta_{9} ) q^{67} \) \( + ( -29679598024 - 370192408 \beta_{1} + 17127874 \beta_{2} - 4511618 \beta_{3} - 1285488 \beta_{4} - 1439584 \beta_{5} - 414016 \beta_{6} - 226448 \beta_{7} + 28336 \beta_{8} + 545440 \beta_{9} ) q^{68} \) \( + ( -5304256 - 868818784 \beta_{1} - 17018580 \beta_{2} + 13852968 \beta_{3} - 3436128 \beta_{4} - 90700 \beta_{5} + 686600 \beta_{6} - 23568 \beta_{7} + 286600 \beta_{8} + 121624 \beta_{9} ) q^{69} \) \( + ( -79913865248 + 491135072 \beta_{1} - 65864960 \beta_{2} + 19069792 \beta_{3} + 2994720 \beta_{4} - 3123328 \beta_{5} - 1433472 \beta_{6} - 261056 \beta_{7} + 103488 \beta_{8} + 1022112 \beta_{9} ) q^{70} \) \( + ( -6964608 + 583061179 \beta_{1} + 14647167 \beta_{2} + 17548176 \beta_{3} + 2971177 \beta_{4} + 6729120 \beta_{5} + 1115216 \beta_{6} - 244640 \beta_{7} + 174905 \beta_{8} - 1537575 \beta_{9} ) q^{71} \) \( + ( 107759549538 + 61065283 \beta_{1} - 121024380 \beta_{2} - 6204854 \beta_{3} - 2335500 \beta_{4} + 2649318 \beta_{5} - 559718 \beta_{6} - 120179 \beta_{7} - 627 \beta_{8} + 84710 \beta_{9} ) q^{72} \) \( + ( -5884699894 + 1282756125 \beta_{1} + 134716907 \beta_{2} + 2613875 \beta_{3} - 8017578 \beta_{4} - 237625 \beta_{6} + 12060 \beta_{7} - 12060 \beta_{9} ) q^{73} \) \( + ( 74265226230 - 153033668 \beta_{1} + 95973380 \beta_{2} + 21403852 \beta_{3} + 4843585 \beta_{4} + 2586100 \beta_{5} - 784298 \beta_{6} + 746218 \beta_{7} + 691350 \beta_{8} + 712334 \beta_{9} ) q^{74} \) \( + ( -25173117378 - 1170803881 \beta_{1} + 43343771 \beta_{2} - 14109700 \beta_{3} + 5237016 \beta_{4} + 1282700 \beta_{6} + 209024 \beta_{7} - 209024 \beta_{9} ) q^{75} \) \( + ( -69972883380 + 27169946 \beta_{1} + 3174658 \beta_{2} - 2863130 \beta_{3} - 2387782 \beta_{4} - 4021660 \beta_{5} + 78456 \beta_{6} - 624858 \beta_{7} + 321710 \beta_{8} - 606492 \beta_{9} ) q^{76} \) \( + ( 2033856 - 1729492512 \beta_{1} - 37979260 \beta_{2} - 3822984 \beta_{3} - 6300448 \beta_{4} - 403172 \beta_{5} - 759528 \beta_{6} + 505296 \beta_{7} - 1329768 \beta_{8} + 2207304 \beta_{9} ) q^{77} \) \( + ( -179565812164 + 483456896 \beta_{1} + 20717368 \beta_{2} - 44813000 \beta_{3} + 5964562 \beta_{4} - 989000 \beta_{5} + 168700 \beta_{6} + 751020 \beta_{7} - 312892 \beta_{8} - 825012 \beta_{9} ) q^{78} \) \( + ( -4264448 + 1635638870 \beta_{1} + 36048414 \beta_{2} + 10254528 \beta_{3} + 4457458 \beta_{4} - 7926400 \beta_{5} + 846272 \beta_{6} - 313216 \beta_{7} - 35150 \beta_{8} - 2227662 \beta_{9} ) q^{79} \) \( + ( 198198530400 - 1429761968 \beta_{1} - 73352832 \beta_{2} - 11852064 \beta_{3} - 3659808 \beta_{4} - 993120 \beta_{5} + 1048864 \beta_{6} + 147760 \beta_{7} - 1322576 \beta_{8} - 736352 \beta_{9} ) q^{80} \) \( + ( 25957437165 + 1234872201 \beta_{1} - 304495137 \beta_{2} - 13382457 \beta_{3} - 1203138 \beta_{4} + 1216587 \beta_{6} + 625836 \beta_{7} - 625836 \beta_{9} ) q^{81} \) \( + ( 165428141818 + 814746830 \beta_{1} + 49554688 \beta_{2} - 42564880 \beta_{3} + 2902544 \beta_{4} + 686400 \beta_{5} - 248000 \beta_{6} - 167840 \beta_{7} - 202912 \beta_{8} + 1162768 \beta_{9} ) q^{82} \) \( + ( -14624160514 - 1492621005 \beta_{1} - 2810673 \beta_{2} + 1060400 \beta_{3} + 7060192 \beta_{4} - 96400 \beta_{6} + 913280 \beta_{7} - 913280 \beta_{9} ) q^{83} \) \( + ( -314427300064 - 630488560 \beta_{1} + 17017088 \beta_{2} + 8690816 \beta_{3} - 8889616 \beta_{4} + 6170400 \beta_{5} + 230784 \beta_{6} + 1819824 \beta_{7} - 493200 \beta_{8} - 1546720 \beta_{9} ) q^{84} \) \( + ( 7265600 + 90335168 \beta_{1} + 537746 \beta_{2} - 19913400 \beta_{3} + 864160 \beta_{4} + 1087054 \beta_{5} - 627800 \beta_{6} - 280400 \beta_{7} + 2102440 \beta_{8} + 139640 \beta_{9} ) q^{85} \) \( + ( -226214802313 - 1883859935 \beta_{1} + 96593356 \beta_{2} + 65040373 \beta_{3} - 740657 \beta_{4} + 9157760 \beta_{5} + 2618240 \beta_{6} + 131520 \beta_{7} - 810560 \beta_{8} - 1074097 \beta_{9} ) q^{86} \) \( + ( 24343168 + 389503101 \beta_{1} + 660505 \beta_{2} - 60258672 \beta_{3} + 2560063 \beta_{4} + 3440544 \beta_{5} - 4256944 \beta_{6} + 1214048 \beta_{7} - 901041 \beta_{8} + 7597295 \beta_{9} ) q^{87} \) \( + ( 217018007316 + 2448941198 \beta_{1} + 303182264 \beta_{2} + 43645316 \beta_{3} - 6224592 \beta_{4} - 3631012 \beta_{5} + 1030612 \beta_{6} - 294734 \beta_{7} + 1884498 \beta_{8} - 1460580 \beta_{9} ) q^{88} \) \( + ( 91323136010 - 3152696603 \beta_{1} + 416293731 \beta_{2} + 25243691 \beta_{3} + 8394310 \beta_{4} - 2294881 \beta_{6} - 298116 \beta_{7} + 298116 \beta_{9} ) q^{89} \) \( + ( 398319828498 - 1240312556 \beta_{1} - 386274772 \beta_{2} + 74360772 \beta_{3} - 1271797 \beta_{4} - 13615748 \beta_{5} + 4988978 \beta_{6} - 2015474 \beta_{7} - 1693582 \beta_{8} - 4340070 \beta_{9} ) q^{90} \) \( + ( 36126681696 + 1813309132 \beta_{1} - 112186204 \beta_{2} + 64127316 \beta_{3} - 8461560 \beta_{4} - 5829756 \beta_{6} - 2293376 \beta_{7} + 2293376 \beta_{9} ) q^{91} \) \( + ( -264938071920 + 2310023688 \beta_{1} - 307742720 \beta_{2} - 1515840 \beta_{3} + 6585848 \beta_{4} + 13796240 \beta_{5} + 2944960 \beta_{6} - 529320 \beta_{7} + 273592 \beta_{8} + 7108112 \beta_{9} ) q^{92} \) \( + ( 6214144 + 1884616448 \beta_{1} + 39951248 \beta_{2} - 21648576 \beta_{3} + 8861696 \beta_{4} + 1085552 \beta_{5} + 1002048 \beta_{6} - 1778816 \beta_{7} - 1134272 \beta_{8} - 13585984 \beta_{9} ) q^{93} \) \( + ( -251725774760 + 856940672 \beta_{1} + 369088304 \beta_{2} - 39059920 \beta_{3} - 12279500 \beta_{4} - 7120 \beta_{5} + 1170904 \beta_{6} - 2841544 \beta_{7} + 2753320 \beta_{8} + 2867448 \beta_{9} ) q^{94} \) \( + ( 29283840 - 2857925673 \beta_{1} - 68089461 \beta_{2} - 77313600 \beta_{3} - 6702115 \beta_{4} + 10187136 \beta_{5} - 3512640 \beta_{6} - 147840 \beta_{7} + 1204125 \beta_{8} + 169245 \beta_{9} ) q^{95} \) \( + ( 283668065392 - 1949460792 \beta_{1} + 466679104 \beta_{2} - 18072016 \beta_{3} + 14632176 \beta_{4} - 13347440 \beta_{5} - 1716144 \beta_{6} + 1455608 \beta_{7} + 1834040 \beta_{8} + 13719312 \beta_{9} ) q^{96} \) \( + ( -147448328326 + 322170951 \beta_{1} - 951621615 \beta_{2} + 58859625 \beta_{3} + 10569762 \beta_{4} - 5350875 \beta_{6} - 2957100 \beta_{7} + 2957100 \beta_{9} ) q^{97} \) \( + ( 237815190229 + 3283451935 \beta_{1} - 243477504 \beta_{2} - 62522048 \beta_{3} - 23368000 \beta_{4} - 7590144 \beta_{5} + 3532544 \beta_{6} + 2172032 \beta_{7} + 2496640 \beta_{8} - 1079616 \beta_{9} ) q^{98} \) \( + ( -71728800198 + 6632125049 \beta_{1} + 74905677 \beta_{2} + 29189072 \beta_{3} - 33005280 \beta_{4} - 2653552 \beta_{6} + 3014528 \beta_{7} - 3014528 \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 110q^{2} \) \(\mathstrut -\mathstrut 660q^{3} \) \(\mathstrut -\mathstrut 2444q^{4} \) \(\mathstrut -\mathstrut 130788q^{6} \) \(\mathstrut +\mathstrut 27160q^{8} \) \(\mathstrut +\mathstrut 1692798q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 110q^{2} \) \(\mathstrut -\mathstrut 660q^{3} \) \(\mathstrut -\mathstrut 2444q^{4} \) \(\mathstrut -\mathstrut 130788q^{6} \) \(\mathstrut +\mathstrut 27160q^{8} \) \(\mathstrut +\mathstrut 1692798q^{9} \) \(\mathstrut +\mathstrut 1873200q^{10} \) \(\mathstrut -\mathstrut 4591444q^{11} \) \(\mathstrut -\mathstrut 5427720q^{12} \) \(\mathstrut +\mathstrut 12728736q^{14} \) \(\mathstrut -\mathstrut 38294000q^{16} \) \(\mathstrut +\mathstrut 42876500q^{17} \) \(\mathstrut +\mathstrut 8229270q^{18} \) \(\mathstrut +\mathstrut 36062828q^{19} \) \(\mathstrut +\mathstrut 17615520q^{20} \) \(\mathstrut -\mathstrut 227004580q^{22} \) \(\mathstrut +\mathstrut 169102992q^{24} \) \(\mathstrut -\mathstrut 706946390q^{25} \) \(\mathstrut +\mathstrut 151295184q^{26} \) \(\mathstrut +\mathstrut 1382414040q^{27} \) \(\mathstrut -\mathstrut 132075840q^{28} \) \(\mathstrut +\mathstrut 456064800q^{30} \) \(\mathstrut -\mathstrut 2651204000q^{32} \) \(\mathstrut -\mathstrut 478018200q^{33} \) \(\mathstrut +\mathstrut 4454663012q^{34} \) \(\mathstrut -\mathstrut 2838470400q^{35} \) \(\mathstrut -\mathstrut 4074739428q^{36} \) \(\mathstrut +\mathstrut 1644178460q^{38} \) \(\mathstrut +\mathstrut 8180322240q^{40} \) \(\mathstrut -\mathstrut 12339248044q^{41} \) \(\mathstrut +\mathstrut 14692585920q^{42} \) \(\mathstrut +\mathstrut 25081495340q^{43} \) \(\mathstrut -\mathstrut 27950589832q^{44} \) \(\mathstrut -\mathstrut 16813594656q^{46} \) \(\mathstrut +\mathstrut 13310114400q^{48} \) \(\mathstrut -\mathstrut 39532486838q^{49} \) \(\mathstrut +\mathstrut 24888425650q^{50} \) \(\mathstrut +\mathstrut 36757299288q^{51} \) \(\mathstrut -\mathstrut 36172521120q^{52} \) \(\mathstrut -\mathstrut 74007907272q^{54} \) \(\mathstrut +\mathstrut 154450364544q^{56} \) \(\mathstrut +\mathstrut 80408630760q^{57} \) \(\mathstrut +\mathstrut 193270394640q^{58} \) \(\mathstrut -\mathstrut 109448026708q^{59} \) \(\mathstrut -\mathstrut 347360715840q^{60} \) \(\mathstrut -\mathstrut 299237961600q^{62} \) \(\mathstrut +\mathstrut 276213192256q^{64} \) \(\mathstrut +\mathstrut 96485235840q^{65} \) \(\mathstrut +\mathstrut 486280823688q^{66} \) \(\mathstrut -\mathstrut 49860896020q^{67} \) \(\mathstrut -\mathstrut 296812951960q^{68} \) \(\mathstrut -\mathstrut 799057954560q^{70} \) \(\mathstrut +\mathstrut 1077572984520q^{72} \) \(\mathstrut -\mathstrut 58835592940q^{73} \) \(\mathstrut +\mathstrut 742739480496q^{74} \) \(\mathstrut -\mathstrut 251792743380q^{75} \) \(\mathstrut -\mathstrut 699737494024q^{76} \) \(\mathstrut -\mathstrut 1795838526240q^{78} \) \(\mathstrut +\mathstrut 1981932232320q^{80} \) \(\mathstrut +\mathstrut 259515975474q^{81} \) \(\mathstrut +\mathstrut 1654109754980q^{82} \) \(\mathstrut -\mathstrut 146236977940q^{83} \) \(\mathstrut -\mathstrut 3144240693120q^{84} \) \(\mathstrut -\mathstrut 2261898070564q^{86} \) \(\mathstrut +\mathstrut 2170357811600q^{88} \) \(\mathstrut +\mathstrut 913341514388q^{89} \) \(\mathstrut +\mathstrut 3983485096080q^{90} \) \(\mathstrut +\mathstrut 361546645248q^{91} \) \(\mathstrut -\mathstrut 2649411172800q^{92} \) \(\mathstrut -\mathstrut 2517413216064q^{94} \) \(\mathstrut +\mathstrut 2836588548672q^{96} \) \(\mathstrut -\mathstrut 1474226441260q^{97} \) \(\mathstrut +\mathstrut 2377890492370q^{98} \) \(\mathstrut -\mathstrut 717160631484q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(5\) \(x^{9}\mathstrut +\mathstrut \) \(468\) \(x^{8}\mathstrut +\mathstrut \) \(1496\) \(x^{7}\mathstrut +\mathstrut \) \(710096\) \(x^{6}\mathstrut +\mathstrut \) \(29155008\) \(x^{5}\mathstrut +\mathstrut \) \(143571008\) \(x^{4}\mathstrut +\mathstrut \) \(28213427840\) \(x^{3}\mathstrut +\mathstrut \) \(1335549648384\) \(x^{2}\mathstrut +\mathstrut \) \(41051831642368\) \(x\mathstrut +\mathstrut \) \(824967906703360\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{9} + \nu^{8} + 474 \nu^{7} + 4340 \nu^{6} + 736136 \nu^{5} + 33571824 \nu^{4} + 345001952 \nu^{3} + 30283439552 \nu^{2} + 1517250285696 \nu + 44108019403776 \)\()/\)\(549755813888\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(4493\) \(\nu^{9}\mathstrut -\mathstrut \) \(30093\) \(\nu^{8}\mathstrut +\mathstrut \) \(3196142\) \(\nu^{7}\mathstrut -\mathstrut \) \(128916068\) \(\nu^{6}\mathstrut -\mathstrut \) \(3415507432\) \(\nu^{5}\mathstrut -\mathstrut \) \(57254189872\) \(\nu^{4}\mathstrut -\mathstrut \) \(872932384352\) \(\nu^{3}\mathstrut -\mathstrut \) \(76346388882112\) \(\nu^{2}\mathstrut -\mathstrut \) \(7765245353563776\) \(\nu\mathstrut -\mathstrut \) \(145725105007139840\)\()/\)\(23639499997184\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(8707\) \(\nu^{9}\mathstrut +\mathstrut \) \(53757\) \(\nu^{8}\mathstrut +\mathstrut \) \(1815154\) \(\nu^{7}\mathstrut -\mathstrut \) \(101763804\) \(\nu^{6}\mathstrut -\mathstrut \) \(5862740632\) \(\nu^{5}\mathstrut -\mathstrut \) \(129969712080\) \(\nu^{4}\mathstrut +\mathstrut \) \(1042235363424\) \(\nu^{3}\mathstrut -\mathstrut \) \(153364515412288\) \(\nu^{2}\mathstrut -\mathstrut \) \(11188479510327680\) \(\nu\mathstrut -\mathstrut \) \(214530251328070656\)\()/\)\(23639499997184\)
\(\beta_{4}\)\(=\)\((\)\(1615\) \(\nu^{9}\mathstrut +\mathstrut \) \(11215\) \(\nu^{8}\mathstrut -\mathstrut \) \(1231674\) \(\nu^{7}\mathstrut +\mathstrut \) \(48040268\) \(\nu^{6}\mathstrut +\mathstrut \) \(1229377784\) \(\nu^{5}\mathstrut +\mathstrut \) \(19124490000\) \(\nu^{4}\mathstrut +\mathstrut \) \(303242632736\) \(\nu^{3}\mathstrut +\mathstrut \) \(26513840492096\) \(\nu^{2}\mathstrut +\mathstrut \) \(3562413143783296\) \(\nu\mathstrut +\mathstrut \) \(51186634521883648\)\()/\)\(1477468749824\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(27321\) \(\nu^{9}\mathstrut +\mathstrut \) \(1186119\) \(\nu^{8}\mathstrut -\mathstrut \) \(6815370\) \(\nu^{7}\mathstrut -\mathstrut \) \(965949524\) \(\nu^{6}\mathstrut +\mathstrut \) \(20225317560\) \(\nu^{5}\mathstrut -\mathstrut \) \(647936292976\) \(\nu^{4}\mathstrut +\mathstrut \) \(15420159870752\) \(\nu^{3}\mathstrut -\mathstrut \) \(704336582180800\) \(\nu^{2}\mathstrut -\mathstrut \) \(21242122029100160\) \(\nu\mathstrut +\mathstrut \) \(644305990841488384\)\()/\)\(23639499997184\)
\(\beta_{6}\)\(=\)\((\)\(136211\) \(\nu^{9}\mathstrut +\mathstrut \) \(361491\) \(\nu^{8}\mathstrut -\mathstrut \) \(76989650\) \(\nu^{7}\mathstrut +\mathstrut \) \(3208778268\) \(\nu^{6}\mathstrut +\mathstrut \) \(99804859160\) \(\nu^{5}\mathstrut +\mathstrut \) \(1795886598864\) \(\nu^{4}\mathstrut +\mathstrut \) \(12541834980768\) \(\nu^{3}\mathstrut +\mathstrut \) \(3832239415276864\) \(\nu^{2}\mathstrut +\mathstrut \) \(211929337247906176\) \(\nu\mathstrut +\mathstrut \) \(4199044269233464320\)\()/\)\(23639499997184\)
\(\beta_{7}\)\(=\)\((\)\(38493\) \(\nu^{9}\mathstrut -\mathstrut \) \(13015459\) \(\nu^{8}\mathstrut +\mathstrut \) \(358357042\) \(\nu^{7}\mathstrut -\mathstrut \) \(2234490716\) \(\nu^{6}\mathstrut -\mathstrut \) \(180252307352\) \(\nu^{5}\mathstrut -\mathstrut \) \(795380527568\) \(\nu^{4}\mathstrut -\mathstrut \) \(192214906747808\) \(\nu^{3}\mathstrut +\mathstrut \) \(4323727691730624\) \(\nu^{2}\mathstrut -\mathstrut \) \(235260934463488384\) \(\nu\mathstrut -\mathstrut \) \(10401552655163827200\)\()/\)\(23639499997184\)
\(\beta_{8}\)\(=\)\((\)\(26341\) \(\nu^{9}\mathstrut -\mathstrut \) \(1612827\) \(\nu^{8}\mathstrut +\mathstrut \) \(35745538\) \(\nu^{7}\mathstrut -\mathstrut \) \(95398332\) \(\nu^{6}\mathstrut +\mathstrut \) \(3422899368\) \(\nu^{5}\mathstrut +\mathstrut \) \(20630795696\) \(\nu^{4}\mathstrut -\mathstrut \) \(6505053142176\) \(\nu^{3}\mathstrut +\mathstrut \) \(681614306008256\) \(\nu^{2}\mathstrut +\mathstrut \) \(5123512746181248\) \(\nu\mathstrut -\mathstrut \) \(553898363154809856\)\()/\)\(2954937499648\)
\(\beta_{9}\)\(=\)\((\)\(123429\) \(\nu^{9}\mathstrut -\mathstrut \) \(4683739\) \(\nu^{8}\mathstrut +\mathstrut \) \(61948546\) \(\nu^{7}\mathstrut +\mathstrut \) \(2245161796\) \(\nu^{6}\mathstrut -\mathstrut \) \(24203267416\) \(\nu^{5}\mathstrut +\mathstrut \) \(1104372863408\) \(\nu^{4}\mathstrut -\mathstrut \) \(29994699314336\) \(\nu^{3}\mathstrut +\mathstrut \) \(3078909051456704\) \(\nu^{2}\mathstrut +\mathstrut \) \(86730240289333888\) \(\nu\mathstrut -\mathstrut \) \(1405108855990273024\)\()/\)\(11819749998592\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(26\) \(\beta_{1}\mathstrut +\mathstrut \) \(256\)\()/512\)
\(\nu^{2}\)\(=\)\((\)\(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(40\) \(\beta_{3}\mathstrut +\mathstrut \) \(186\) \(\beta_{2}\mathstrut +\mathstrut \) \(390\) \(\beta_{1}\mathstrut -\mathstrut \) \(46656\)\()/512\)
\(\nu^{3}\)\(=\)\((\)\(32\) \(\beta_{9}\mathstrut +\mathstrut \) \(64\) \(\beta_{8}\mathstrut -\mathstrut \) \(96\) \(\beta_{7}\mathstrut +\mathstrut \) \(72\) \(\beta_{6}\mathstrut -\mathstrut \) \(128\) \(\beta_{5}\mathstrut -\mathstrut \) \(193\) \(\beta_{4}\mathstrut +\mathstrut \) \(2024\) \(\beta_{3}\mathstrut +\mathstrut \) \(2498\) \(\beta_{2}\mathstrut +\mathstrut \) \(66014\) \(\beta_{1}\mathstrut -\mathstrut \) \(583616\)\()/512\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(4320\) \(\beta_{9}\mathstrut +\mathstrut \) \(1856\) \(\beta_{8}\mathstrut -\mathstrut \) \(1120\) \(\beta_{7}\mathstrut +\mathstrut \) \(552\) \(\beta_{6}\mathstrut -\mathstrut \) \(24192\) \(\beta_{5}\mathstrut -\mathstrut \) \(8449\) \(\beta_{4}\mathstrut -\mathstrut \) \(12984\) \(\beta_{3}\mathstrut +\mathstrut \) \(18594\) \(\beta_{2}\mathstrut +\mathstrut \) \(3979134\) \(\beta_{1}\mathstrut -\mathstrut \) \(126891712\)\()/512\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(193888\) \(\beta_{9}\mathstrut +\mathstrut \) \(83008\) \(\beta_{8}\mathstrut +\mathstrut \) \(44832\) \(\beta_{7}\mathstrut -\mathstrut \) \(11416\) \(\beta_{6}\mathstrut -\mathstrut \) \(342144\) \(\beta_{5}\mathstrut -\mathstrut \) \(317977\) \(\beta_{4}\mathstrut +\mathstrut \) \(973192\) \(\beta_{3}\mathstrut -\mathstrut \) \(8415534\) \(\beta_{2}\mathstrut -\mathstrut \) \(6247474\) \(\beta_{1}\mathstrut -\mathstrut \) \(7937852096\)\()/512\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(2658400\) \(\beta_{9}\mathstrut +\mathstrut \) \(65088\) \(\beta_{8}\mathstrut +\mathstrut \) \(2715680\) \(\beta_{7}\mathstrut -\mathstrut \) \(4321240\) \(\beta_{6}\mathstrut -\mathstrut \) \(3255424\) \(\beta_{5}\mathstrut -\mathstrut \) \(15187809\) \(\beta_{4}\mathstrut +\mathstrut \) \(92441928\) \(\beta_{3}\mathstrut -\mathstrut \) \(477634078\) \(\beta_{2}\mathstrut +\mathstrut \) \(2069551038\) \(\beta_{1}\mathstrut +\mathstrut \) \(2085743424\)\()/512\)
\(\nu^{7}\)\(=\)\((\)\(35934880\) \(\beta_{9}\mathstrut -\mathstrut \) \(57652160\) \(\beta_{8}\mathstrut +\mathstrut \) \(98783008\) \(\beta_{7}\mathstrut -\mathstrut \) \(216557976\) \(\beta_{6}\mathstrut +\mathstrut \) \(485832576\) \(\beta_{5}\mathstrut -\mathstrut \) \(53454153\) \(\beta_{4}\mathstrut +\mathstrut \) \(1358153864\) \(\beta_{3}\mathstrut -\mathstrut \) \(10219468110\) \(\beta_{2}\mathstrut +\mathstrut \) \(221797639662\) \(\beta_{1}\mathstrut -\mathstrut \) \(4460137581248\)\()/512\)
\(\nu^{8}\)\(=\)\((\)\(4710587296\) \(\beta_{9}\mathstrut -\mathstrut \) \(4144748992\) \(\beta_{8}\mathstrut +\mathstrut \) \(2120907808\) \(\beta_{7}\mathstrut -\mathstrut \) \(3951195992\) \(\beta_{6}\mathstrut +\mathstrut \) \(24607916928\) \(\beta_{5}\mathstrut -\mathstrut \) \(10898687281\) \(\beta_{4}\mathstrut -\mathstrut \) \(17279952952\) \(\beta_{3}\mathstrut -\mathstrut \) \(176195721854\) \(\beta_{2}\mathstrut +\mathstrut \) \(4161762999326\) \(\beta_{1}\mathstrut -\mathstrut \) \(451877316724928\)\()/512\)
\(\nu^{9}\)\(=\)\((\)\(266511889568\) \(\beta_{9}\mathstrut -\mathstrut \) \(114305216448\) \(\beta_{8}\mathstrut -\mathstrut \) \(23011923680\) \(\beta_{7}\mathstrut -\mathstrut \) \(151881717016\) \(\beta_{6}\mathstrut +\mathstrut \) \(867430313856\) \(\beta_{5}\mathstrut -\mathstrut \) \(921641522905\) \(\beta_{4}\mathstrut -\mathstrut \) \(3217809582328\) \(\beta_{3}\mathstrut -\mathstrut \) \(2934137801262\) \(\beta_{2}\mathstrut -\mathstrut \) \(39822522506034\) \(\beta_{1}\mathstrut -\mathstrut \) \(8697213876274880\)\()/512\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−25.4137 + 6.09761i
−25.4137 6.09761i
−11.5862 + 26.7344i
−11.5862 26.7344i
−10.5406 + 27.3936i
−10.5406 27.3936i
17.4695 + 29.8739i
17.4695 29.8739i
32.5710 + 17.8321i
32.5710 17.8321i
−62.8274 12.1952i 24.0930 3798.55 + 1532.39i 25133.2i −1513.70 293.819i 190438.i −219965. 142600.i −530861. −306505. + 1.57906e6i
3.2 −62.8274 + 12.1952i 24.0930 3798.55 1532.39i 25133.2i −1513.70 + 293.819i 190438.i −219965. + 142600.i −530861. −306505. 1.57906e6i
3.3 −35.1724 53.4687i 1272.20 −1621.80 + 3761.25i 19079.0i −44746.4 68023.0i 136156.i 258152. 45576.2i 1.08706e6 1.02013e6 671054.i
3.4 −35.1724 + 53.4687i 1272.20 −1621.80 3761.25i 19079.0i −44746.4 + 68023.0i 136156.i 258152. + 45576.2i 1.08706e6 1.02013e6 + 671054.i
3.5 −33.0812 54.7872i −875.418 −1907.27 + 3624.85i 2272.99i 28959.9 + 47961.7i 92079.6i 261690. 15420.3i 234916. 124531. 75193.1i
3.6 −33.0812 + 54.7872i −875.418 −1907.27 3624.85i 2272.99i 28959.9 47961.7i 92079.6i 261690. + 15420.3i 234916. 124531. + 75193.1i
3.7 22.9390 59.7478i 271.191 −3043.60 2741.11i 11036.2i 6220.87 16203.1i 58770.2i −233593. + 118970.i −457896. −659388. 253159.i
3.8 22.9390 + 59.7478i 271.191 −3043.60 + 2741.11i 11036.2i 6220.87 + 16203.1i 58770.2i −233593. 118970.i −457896. −659388. + 253159.i
3.9 53.1419 35.6642i −1022.07 1552.12 3790.53i 21249.1i −54314.6 + 36451.3i 149114.i −52703.6 256791.i 513182. 757833. + 1.12922e6i
3.10 53.1419 + 35.6642i −1022.07 1552.12 + 3790.53i 21249.1i −54314.6 36451.3i 149114.i −52703.6 + 256791.i 513182. 757833. 1.12922e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{5} \) \(\mathstrut +\mathstrut 330 T_{3}^{4} \) \(\mathstrut -\mathstrut 1697352 T_{3}^{3} \) \(\mathstrut -\mathstrut 685590480 T_{3}^{2} \) \(\mathstrut +\mathstrut 326191796496 T_{3} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!00\)\( \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\).