Properties

Label 69.8.a.d
Level $69$
Weight $8$
Character orbit 69.a
Self dual yes
Analytic conductor $21.555$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 757 x^{6} - 1170 x^{5} + 170343 x^{4} + 424132 x^{3} - 9973075 x^{2} - 5161010 x + 130545120\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} ) q^{2} + 27 q^{3} + ( 70 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( 47 - 7 \beta_{1} - \beta_{5} ) q^{5} + ( 81 - 27 \beta_{1} ) q^{6} + ( 16 - 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + ( 522 - 58 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + ( 3 - \beta_{1} ) q^{2} + 27 q^{3} + ( 70 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( 47 - 7 \beta_{1} - \beta_{5} ) q^{5} + ( 81 - 27 \beta_{1} ) q^{6} + ( 16 - 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + ( 522 - 58 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + 729 q^{9} + ( 1461 - 13 \beta_{1} + 12 \beta_{2} + \beta_{3} - 10 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{10} + ( 867 - 12 \beta_{1} + \beta_{2} + 4 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( 1890 - 108 \beta_{1} + 27 \beta_{2} ) q^{12} + ( 1558 - 179 \beta_{1} - 24 \beta_{2} - \beta_{3} + 22 \beta_{4} + 12 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{13} + ( 3783 + 59 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 33 \beta_{4} + 15 \beta_{5} + 10 \beta_{6} - 9 \beta_{7} ) q^{14} + ( 1269 - 189 \beta_{1} - 27 \beta_{5} ) q^{15} + ( 3382 - 718 \beta_{1} - 3 \beta_{2} - 14 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{16} + ( 3072 - 67 \beta_{1} - 35 \beta_{2} - 2 \beta_{3} - 23 \beta_{4} + 23 \beta_{5} + 15 \beta_{6} + 4 \beta_{7} ) q^{17} + ( 2187 - 729 \beta_{1} ) q^{18} + ( -1791 - 457 \beta_{1} - 168 \beta_{2} + 4 \beta_{3} + 34 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} - 14 \beta_{7} ) q^{19} + ( -452 - 1368 \beta_{1} - 30 \beta_{2} + 18 \beta_{3} + 118 \beta_{4} + 44 \beta_{5} - 20 \beta_{6} + 4 \beta_{7} ) q^{20} + ( 432 - 513 \beta_{1} - 27 \beta_{2} + 27 \beta_{4} - 27 \beta_{5} + 27 \beta_{6} ) q^{21} + ( 4546 - 916 \beta_{1} - 94 \beta_{2} + 27 \beta_{3} - 111 \beta_{4} - 44 \beta_{5} - 40 \beta_{6} + 10 \beta_{7} ) q^{22} + 12167 q^{23} + ( 14094 - 1566 \beta_{1} + 135 \beta_{2} - 27 \beta_{3} + 27 \beta_{4} ) q^{24} + ( 18003 + 1223 \beta_{1} + 20 \beta_{2} + 17 \beta_{3} - 362 \beta_{4} - 284 \beta_{5} - 23 \beta_{6} + \beta_{7} ) q^{25} + ( 40612 - 170 \beta_{1} + 442 \beta_{2} - 12 \beta_{3} - 154 \beta_{4} + 44 \beta_{5} - 62 \beta_{6} + 62 \beta_{7} ) q^{26} + 19683 q^{27} + ( -2528 - 1488 \beta_{1} - 432 \beta_{2} - 3 \beta_{3} + 457 \beta_{4} + 152 \beta_{5} + 90 \beta_{6} - 98 \beta_{7} ) q^{28} + ( 31932 + 1560 \beta_{1} - 40 \beta_{2} - 50 \beta_{3} + 424 \beta_{4} + 10 \beta_{5} + 50 \beta_{6} - 122 \beta_{7} ) q^{29} + ( 39447 - 351 \beta_{1} + 324 \beta_{2} + 27 \beta_{3} - 270 \beta_{4} - 81 \beta_{5} - 54 \beta_{6} - 27 \beta_{7} ) q^{30} + ( 56270 + 3406 \beta_{1} + 228 \beta_{2} - 68 \beta_{3} - 256 \beta_{4} - 74 \beta_{5} + 68 \beta_{6} + 92 \beta_{7} ) q^{31} + ( 80754 + 2074 \beta_{1} + 973 \beta_{2} - 17 \beta_{3} + 125 \beta_{4} + 288 \beta_{5} - 76 \beta_{6} + 172 \beta_{7} ) q^{32} + ( 23409 - 324 \beta_{1} + 27 \beta_{2} + 108 \beta_{3} + 135 \beta_{4} + 54 \beta_{5} - 81 \beta_{6} + 54 \beta_{7} ) q^{33} + ( 24237 + 125 \beta_{1} - 482 \beta_{2} - 38 \beta_{3} + 435 \beta_{4} + 403 \beta_{5} + 190 \beta_{6} - 5 \beta_{7} ) q^{34} + ( 127708 + 12785 \beta_{1} + 286 \beta_{2} - 131 \beta_{3} - 640 \beta_{4} - 146 \beta_{5} + 87 \beta_{6} - 75 \beta_{7} ) q^{35} + ( 51030 - 2916 \beta_{1} + 729 \beta_{2} ) q^{36} + ( 25833 + 1906 \beta_{1} - 537 \beta_{2} + 200 \beta_{3} + 795 \beta_{4} - 260 \beta_{5} - 265 \beta_{6} + 90 \beta_{7} ) q^{37} + ( 92263 + 19089 \beta_{1} - 88 \beta_{2} + 183 \beta_{3} - 844 \beta_{4} - 135 \beta_{5} + 230 \beta_{6} - 277 \beta_{7} ) q^{38} + ( 42066 - 4833 \beta_{1} - 648 \beta_{2} - 27 \beta_{3} + 594 \beta_{4} + 324 \beta_{5} - 135 \beta_{6} + 81 \beta_{7} ) q^{39} + ( 73844 - 12 \beta_{1} - 146 \beta_{2} - 68 \beta_{3} - 184 \beta_{4} + 176 \beta_{5} + 108 \beta_{6} + 108 \beta_{7} ) q^{40} + ( 131926 + 23320 \beta_{1} - 1172 \beta_{2} + 84 \beta_{3} - 372 \beta_{4} - 64 \beta_{5} - 160 \beta_{6} + 32 \beta_{7} ) q^{41} + ( 102141 + 1593 \beta_{1} + 54 \beta_{2} - 54 \beta_{3} - 891 \beta_{4} + 405 \beta_{5} + 270 \beta_{6} - 243 \beta_{7} ) q^{42} + ( 198823 + 11003 \beta_{1} - 1212 \beta_{2} - 36 \beta_{3} - 640 \beta_{4} + 513 \beta_{5} - 300 \beta_{6} - 36 \beta_{7} ) q^{43} + ( 73336 + 14728 \beta_{1} - 104 \beta_{2} + 43 \beta_{3} + 123 \beta_{4} - 1276 \beta_{5} - 210 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 34263 - 5103 \beta_{1} - 729 \beta_{5} ) q^{45} + ( 36501 - 12167 \beta_{1} ) q^{46} + ( 56110 + 18273 \beta_{1} - 2336 \beta_{2} - 355 \beta_{3} + 2146 \beta_{4} + 638 \beta_{5} + 153 \beta_{6} + 253 \beta_{7} ) q^{47} + ( 91314 - 19386 \beta_{1} - 81 \beta_{2} - 378 \beta_{3} + 324 \beta_{4} + 324 \beta_{5} - 54 \beta_{6} + 162 \beta_{7} ) q^{48} + ( 243259 + 10297 \beta_{1} + 674 \beta_{2} - 345 \beta_{3} - 1056 \beta_{4} + 996 \beta_{5} + 645 \beta_{6} - 721 \beta_{7} ) q^{49} + ( -194385 + 6145 \beta_{1} - 2278 \beta_{2} + 828 \beta_{3} + 2074 \beta_{4} - 1772 \beta_{5} - 818 \beta_{6} + 130 \beta_{7} ) q^{50} + ( 82944 - 1809 \beta_{1} - 945 \beta_{2} - 54 \beta_{3} - 621 \beta_{4} + 621 \beta_{5} + 405 \beta_{6} + 108 \beta_{7} ) q^{51} + ( -78644 - 62672 \beta_{1} + 5382 \beta_{2} - 300 \beta_{3} + 908 \beta_{4} - 1240 \beta_{5} - 536 \beta_{6} + 952 \beta_{7} ) q^{52} + ( -47505 + 28645 \beta_{1} + 1448 \beta_{2} + 22 \beta_{3} - 1226 \beta_{4} - 1289 \beta_{5} + 460 \beta_{6} + 776 \beta_{7} ) q^{53} + ( 59049 - 19683 \beta_{1} ) q^{54} + ( 49486 - 22806 \beta_{1} + 4746 \beta_{2} + 938 \beta_{3} + 806 \beta_{4} - 820 \beta_{5} - 1324 \beta_{6} - 438 \beta_{7} ) q^{55} + ( -164660 + 17264 \beta_{1} - 264 \beta_{2} + 79 \beta_{3} - 1645 \beta_{4} - 1404 \beta_{5} + 882 \beta_{6} - 742 \beta_{7} ) q^{56} + ( -48357 - 12339 \beta_{1} - 4536 \beta_{2} + 108 \beta_{3} + 918 \beta_{4} - 135 \beta_{5} + 270 \beta_{6} - 378 \beta_{7} ) q^{57} + ( -176932 - 45420 \beta_{1} + 1368 \beta_{2} - 578 \beta_{3} - 5104 \beta_{4} - 586 \beta_{5} + 1496 \beta_{6} - 1634 \beta_{7} ) q^{58} + ( 79614 + 42727 \beta_{1} - 3168 \beta_{2} - 249 \beta_{3} - 986 \beta_{4} + 3338 \beta_{5} + 195 \beta_{6} - 501 \beta_{7} ) q^{59} + ( -12204 - 36936 \beta_{1} - 810 \beta_{2} + 486 \beta_{3} + 3186 \beta_{4} + 1188 \beta_{5} - 540 \beta_{6} + 108 \beta_{7} ) q^{60} + ( -251137 - 37436 \beta_{1} + 5965 \beta_{2} + 678 \beta_{3} - 591 \beta_{4} + 3020 \beta_{5} + 503 \beta_{6} + 696 \beta_{7} ) q^{61} + ( -487590 - 71826 \beta_{1} - 1636 \beta_{2} - 922 \beta_{3} + 3864 \beta_{4} + 3266 \beta_{5} - 204 \beta_{6} + 1102 \beta_{7} ) q^{62} + ( 11664 - 13851 \beta_{1} - 729 \beta_{2} + 729 \beta_{4} - 729 \beta_{5} + 729 \beta_{6} ) q^{63} + ( -635962 - 109302 \beta_{1} + 2973 \beta_{2} - 18 \beta_{3} + 1360 \beta_{4} + 1788 \beta_{5} - 1490 \beta_{6} + 1766 \beta_{7} ) q^{64} + ( 77966 + 44104 \beta_{1} + 10286 \beta_{2} - 50 \beta_{3} - 5312 \beta_{4} - 828 \beta_{5} - 34 \beta_{6} - 152 \beta_{7} ) q^{65} + ( 122742 - 24732 \beta_{1} - 2538 \beta_{2} + 729 \beta_{3} - 2997 \beta_{4} - 1188 \beta_{5} - 1080 \beta_{6} + 270 \beta_{7} ) q^{66} + ( -342299 - 44743 \beta_{1} + 7138 \beta_{2} - 148 \beta_{3} - 1832 \beta_{4} - 3551 \beta_{5} - 2216 \beta_{6} + 562 \beta_{7} ) q^{67} + ( -285964 + 424 \beta_{1} + 1426 \beta_{2} - 959 \beta_{3} - 315 \beta_{4} + 2248 \beta_{5} + 1130 \beta_{6} - 1658 \beta_{7} ) q^{68} + 328509 q^{69} + ( -2059416 - 130178 \beta_{1} - 11222 \beta_{2} - 584 \beta_{3} + 9916 \beta_{4} + 342 \beta_{5} + 1090 \beta_{6} + 52 \beta_{7} ) q^{70} + ( -784284 - 11414 \beta_{1} - 3428 \beta_{2} - 1046 \beta_{3} + 11704 \beta_{4} + 3368 \beta_{5} + 702 \beta_{6} - 1634 \beta_{7} ) q^{71} + ( 380538 - 42282 \beta_{1} + 3645 \beta_{2} - 729 \beta_{3} + 729 \beta_{4} ) q^{72} + ( -1292752 + 51812 \beta_{1} - 2482 \beta_{2} + 480 \beta_{3} + 4866 \beta_{4} + 3400 \beta_{5} + 1786 \beta_{6} + 996 \beta_{7} ) q^{73} + ( -251790 + 39652 \beta_{1} + 302 \beta_{2} + 2127 \beta_{3} - 16437 \beta_{4} - 4890 \beta_{5} - 4020 \beta_{6} + 300 \beta_{7} ) q^{74} + ( 486081 + 33021 \beta_{1} + 540 \beta_{2} + 459 \beta_{3} - 9774 \beta_{4} - 7668 \beta_{5} - 621 \beta_{6} + 27 \beta_{7} ) q^{75} + ( -3132824 + 30800 \beta_{1} - 20736 \beta_{2} + 2160 \beta_{3} + 620 \beta_{4} - 4252 \beta_{5} + 3792 \beta_{6} - 3160 \beta_{7} ) q^{76} + ( -267230 + 101644 \beta_{1} - 2570 \beta_{2} + 948 \beta_{3} + 7158 \beta_{4} + 7844 \beta_{5} - 1862 \beta_{6} + 1680 \beta_{7} ) q^{77} + ( 1096524 - 4590 \beta_{1} + 11934 \beta_{2} - 324 \beta_{3} - 4158 \beta_{4} + 1188 \beta_{5} - 1674 \beta_{6} + 1674 \beta_{7} ) q^{78} + ( -1102280 + 126311 \beta_{1} - 829 \beta_{2} - 2360 \beta_{3} - 4677 \beta_{4} - 7269 \beta_{5} - 33 \beta_{6} - 298 \beta_{7} ) q^{79} + ( 297004 + 110636 \beta_{1} + 3478 \beta_{2} - 3342 \beta_{3} - 10458 \beta_{4} - 544 \beta_{5} + 3208 \beta_{6} + 648 \beta_{7} ) q^{80} + 531441 q^{81} + ( -3969102 + 46762 \beta_{1} - 26532 \beta_{2} + 2664 \beta_{3} + 2880 \beta_{4} - 3568 \beta_{5} - 2136 \beta_{6} + 1160 \beta_{7} ) q^{82} + ( 45225 + 190966 \beta_{1} + 7115 \beta_{2} + 962 \beta_{3} - 6205 \beta_{4} - 5626 \beta_{5} + 205 \beta_{6} + 596 \beta_{7} ) q^{83} + ( -68256 - 40176 \beta_{1} - 11664 \beta_{2} - 81 \beta_{3} + 12339 \beta_{4} + 4104 \beta_{5} + 2430 \beta_{6} - 2646 \beta_{7} ) q^{84} + ( -2228050 + 83731 \beta_{1} - 4334 \beta_{2} - 4839 \beta_{3} + 2960 \beta_{4} - 1956 \beta_{5} + 3603 \beta_{6} - 1539 \beta_{7} ) q^{85} + ( -1444499 - 46529 \beta_{1} - 8792 \beta_{2} + 2059 \beta_{3} + 16070 \beta_{4} - 4525 \beta_{5} - 2358 \beta_{6} + 3145 \beta_{7} ) q^{86} + ( 862164 + 42120 \beta_{1} - 1080 \beta_{2} - 1350 \beta_{3} + 11448 \beta_{4} + 270 \beta_{5} + 1350 \beta_{6} - 3294 \beta_{7} ) q^{87} + ( -3153044 + 126472 \beta_{1} + 7596 \beta_{2} - 1231 \beta_{3} + 397 \beta_{4} - 2076 \beta_{5} + 118 \beta_{6} - 1490 \beta_{7} ) q^{88} + ( -418758 + 117113 \beta_{1} - 28373 \beta_{2} + 5428 \beta_{3} + 21493 \beta_{4} + 15587 \beta_{5} - 1655 \beta_{6} + 552 \beta_{7} ) q^{89} + ( 1065069 - 9477 \beta_{1} + 8748 \beta_{2} + 729 \beta_{3} - 7290 \beta_{4} - 2187 \beta_{5} - 1458 \beta_{6} - 729 \beta_{7} ) q^{90} + ( -2047308 + 318994 \beta_{1} + 26476 \beta_{2} + 3612 \beta_{3} - 26558 \beta_{4} - 6816 \beta_{5} - 1182 \beta_{6} + 3434 \beta_{7} ) q^{91} + ( 851690 - 48668 \beta_{1} + 12167 \beta_{2} ) q^{92} + ( 1519290 + 91962 \beta_{1} + 6156 \beta_{2} - 1836 \beta_{3} - 6912 \beta_{4} - 1998 \beta_{5} + 1836 \beta_{6} + 2484 \beta_{7} ) q^{93} + ( -3028540 + 104630 \beta_{1} + 6850 \beta_{2} - 4506 \beta_{3} - 20022 \beta_{4} + 15626 \beta_{5} + 378 \beta_{6} + 2436 \beta_{7} ) q^{94} + ( 3261018 + 188285 \beta_{1} + 26368 \beta_{2} + 1301 \beta_{3} - 31782 \beta_{4} - 12214 \beta_{5} - 791 \beta_{6} - 1803 \beta_{7} ) q^{95} + ( 2180358 + 55998 \beta_{1} + 26271 \beta_{2} - 459 \beta_{3} + 3375 \beta_{4} + 7776 \beta_{5} - 2052 \beta_{6} + 4644 \beta_{7} ) q^{96} + ( -3505132 - 65598 \beta_{1} + 2812 \beta_{2} - 348 \beta_{3} + 16924 \beta_{4} + 6618 \beta_{5} - 2104 \beta_{6} - 3820 \beta_{7} ) q^{97} + ( -1240067 - 356641 \beta_{1} - 24802 \beta_{2} - 3146 \beta_{3} + 28224 \beta_{4} + 2272 \beta_{5} + 14810 \beta_{6} - 7258 \beta_{7} ) q^{98} + ( 632043 - 8748 \beta_{1} + 729 \beta_{2} + 2916 \beta_{3} + 3645 \beta_{4} + 1458 \beta_{5} - 2187 \beta_{6} + 1458 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{2} + 216q^{3} + 562q^{4} + 378q^{5} + 648q^{6} + 126q^{7} + 4188q^{8} + 5832q^{9} + O(q^{10}) \) \( 8q + 24q^{2} + 216q^{3} + 562q^{4} + 378q^{5} + 648q^{6} + 126q^{7} + 4188q^{8} + 5832q^{9} + 11720q^{10} + 6932q^{11} + 15174q^{12} + 12404q^{13} + 30222q^{14} + 10206q^{15} + 27058q^{16} + 24434q^{17} + 17496q^{18} - 14682q^{19} - 3760q^{20} + 3402q^{21} + 36294q^{22} + 97336q^{23} + 113076q^{24} + 144644q^{25} + 325840q^{26} + 157464q^{27} - 21566q^{28} + 255356q^{29} + 316440q^{30} + 450764q^{31} + 647588q^{32} + 187164q^{33} + 191822q^{34} + 1022616q^{35} + 409698q^{36} + 206240q^{37} + 737372q^{38} + 334908q^{39} + 590028q^{40} + 1053344q^{41} + 815994q^{42} + 1587806q^{43} + 589366q^{44} + 275562q^{45} + 292008q^{46} + 443336q^{47} + 730566q^{48} + 1944828q^{49} - 1556112q^{50} + 659718q^{51} - 614236q^{52} - 375530q^{53} + 472392q^{54} + 407792q^{55} - 1316922q^{56} - 396414q^{57} - 1413384q^{58} + 624008q^{59} - 101520q^{60} - 2005568q^{61} - 3908272q^{62} + 91854q^{63} - 5082310q^{64} + 646124q^{65} + 979938q^{66} - 2712286q^{67} - 2289698q^{68} + 2628072q^{69} - 16499468q^{70} - 6287176q^{71} + 3053052q^{72} - 10358312q^{73} - 2000150q^{74} + 3905388q^{75} - 25107464q^{76} - 2156840q^{77} + 8797680q^{78} - 8800574q^{79} + 2384344q^{80} + 4251528q^{81} - 31799800q^{82} + 384948q^{83} - 582282q^{84} - 17826684q^{85} - 11563928q^{86} + 6894612q^{87} - 25202782q^{88} - 3445530q^{89} + 8543880q^{90} - 16316740q^{91} + 6837854q^{92} + 12170628q^{93} - 24237616q^{94} + 26164288q^{95} + 17484876q^{96} - 28043764q^{97} - 9998012q^{98} + 5053428q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 757 x^{6} - 1170 x^{5} + 170343 x^{4} + 424132 x^{3} - 9973075 x^{2} - 5161010 x + 130545120\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 189 \)
\(\beta_{3}\)\(=\)\((\)\( 105 \nu^{7} - 413 \nu^{6} - 99508 \nu^{5} - 294286 \nu^{4} + 29391029 \nu^{3} + 156833675 \nu^{2} - 2118757034 \nu - 2892408736 \)\()/5195392\)
\(\beta_{4}\)\(=\)\((\)\( 105 \nu^{7} - 413 \nu^{6} - 99508 \nu^{5} - 294286 \nu^{4} + 24195637 \nu^{3} + 177615243 \nu^{2} - 575725610 \nu - 4544543392 \)\()/5195392\)
\(\beta_{5}\)\(=\)\((\)\( 331 \nu^{7} + 1404 \nu^{6} - 231736 \nu^{5} - 1304986 \nu^{4} + 44182269 \nu^{3} + 300757102 \nu^{2} - 1271589824 \nu - 6314973632 \)\()/2597696\)
\(\beta_{6}\)\(=\)\((\)\( -528 \nu^{7} - 6041 \nu^{6} + 393692 \nu^{5} + 4513576 \nu^{4} - 78508306 \nu^{3} - 895044703 \nu^{2} + 1971608022 \nu + 18986484320 \)\()/2597696\)
\(\beta_{7}\)\(=\)\((\)\( -1641 \nu^{7} - 9781 \nu^{6} + 1156236 \nu^{5} + 8996798 \nu^{4} - 219270937 \nu^{3} - 2072158985 \nu^{2} + 5126482774 \nu + 48102725728 \)\()/5195392\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 189\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + 4 \beta_{2} + 305 \beta_{1} + 438\)
\(\nu^{4}\)\(=\)\(6 \beta_{7} - 2 \beta_{6} + 12 \beta_{5} - 2 \beta_{3} + 375 \beta_{2} + 1406 \beta_{1} + 57999\)
\(\nu^{5}\)\(=\)\(-82 \beta_{7} + 46 \beta_{6} - 108 \beta_{5} - 547 \beta_{4} + 409 \beta_{3} + 2002 \beta_{2} + 103317 \beta_{1} + 288060\)
\(\nu^{6}\)\(=\)\(3320 \beta_{7} - 1672 \beta_{6} + 5904 \beta_{5} - 1346 \beta_{4} - 806 \beta_{3} + 134865 \beta_{2} + 674978 \beta_{1} + 19697569\)
\(\nu^{7}\)\(=\)\(-47836 \beta_{7} + 31412 \beta_{6} - 45496 \beta_{5} - 243769 \beta_{4} + 148397 \beta_{3} + 865464 \beta_{2} + 36325961 \beta_{1} + 135668930\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
19.5036
19.1553
5.38924
4.96449
−4.11469
−11.6014
−14.7586
−18.5379
−16.5036 27.0000 144.368 −425.514 −445.597 −1501.62 −270.137 729.000 7022.50
1.2 −16.1553 27.0000 132.994 118.315 −436.193 738.380 −80.6798 729.000 −1911.41
1.3 −2.38924 27.0000 −122.292 −147.281 −64.5095 −1223.93 598.006 729.000 351.889
1.4 −1.96449 27.0000 −124.141 495.444 −53.0413 824.524 495.329 729.000 −973.296
1.5 7.11469 27.0000 −77.3812 −244.007 192.097 549.357 −1461.22 729.000 −1736.03
1.6 14.6014 27.0000 85.2018 493.797 394.239 −368.336 −624.915 729.000 7210.15
1.7 17.7586 27.0000 187.368 32.5148 479.482 1672.82 1054.29 729.000 577.418
1.8 21.5379 27.0000 335.881 54.7305 581.523 −565.198 4477.33 729.000 1178.78
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.8.a.d 8
3.b odd 2 1 207.8.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.d 8 1.a even 1 1 trivial
207.8.a.e 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49724160 + 36710912 T + 1827392 T^{2} - 1967776 T^{3} + 56268 T^{4} + 13284 T^{5} - 505 T^{6} - 24 T^{7} + T^{8} \)
$3$ \( ( -27 + T )^{8} \)
$5$ \( -787687446753720000 + 38032671345732000 T - 355872020101600 T^{2} - 3073894582440 T^{3} + 27011247444 T^{4} + 86758768 T^{5} - 313380 T^{6} - 378 T^{7} + T^{8} \)
$7$ \( \)\(21\!\cdots\!72\)\( + \)\(21\!\cdots\!24\)\( T - 1886354075565662800 T^{2} - 786960979979224 T^{3} + 5050613893140 T^{4} + 417223216 T^{5} - 4258648 T^{6} - 126 T^{7} + T^{8} \)
$11$ \( -\)\(27\!\cdots\!60\)\( - \)\(97\!\cdots\!20\)\( T - \)\(10\!\cdots\!20\)\( T^{2} - 2185164143859716608 T^{3} + 1168194316296416 T^{4} + 288689144008 T^{5} - 54793352 T^{6} - 6932 T^{7} + T^{8} \)
$13$ \( -\)\(32\!\cdots\!68\)\( - \)\(40\!\cdots\!60\)\( T - \)\(12\!\cdots\!64\)\( T^{2} - 64196892435624514560 T^{3} + 9716850182040816 T^{4} + 1989972799888 T^{5} - 184583456 T^{6} - 12404 T^{7} + T^{8} \)
$17$ \( \)\(47\!\cdots\!40\)\( + \)\(14\!\cdots\!48\)\( T - \)\(21\!\cdots\!56\)\( T^{2} - \)\(31\!\cdots\!48\)\( T^{3} + 259043976820345636 T^{4} + 18380313259944 T^{5} - 1039232716 T^{6} - 24434 T^{7} + T^{8} \)
$19$ \( -\)\(91\!\cdots\!40\)\( + \)\(17\!\cdots\!04\)\( T - \)\(42\!\cdots\!64\)\( T^{2} - \)\(87\!\cdots\!48\)\( T^{3} + 4912761877471106324 T^{4} + 13012704914776 T^{5} - 4070992632 T^{6} + 14682 T^{7} + T^{8} \)
$23$ \( ( -12167 + T )^{8} \)
$29$ \( -\)\(21\!\cdots\!00\)\( - \)\(55\!\cdots\!00\)\( T - \)\(26\!\cdots\!60\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!80\)\( T^{4} + 20581382990821264 T^{5} - 85498221560 T^{6} - 255356 T^{7} + T^{8} \)
$31$ \( -\)\(28\!\cdots\!12\)\( + \)\(25\!\cdots\!28\)\( T + \)\(59\!\cdots\!68\)\( T^{2} - \)\(59\!\cdots\!92\)\( T^{3} - \)\(23\!\cdots\!96\)\( T^{4} + 33934540026989888 T^{5} - 26520389696 T^{6} - 450764 T^{7} + T^{8} \)
$37$ \( \)\(93\!\cdots\!08\)\( + \)\(63\!\cdots\!76\)\( T - \)\(15\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!04\)\( T^{3} + \)\(55\!\cdots\!24\)\( T^{4} + 45650518456397768 T^{5} - 439892867128 T^{6} - 206240 T^{7} + T^{8} \)
$41$ \( \)\(42\!\cdots\!40\)\( - \)\(66\!\cdots\!40\)\( T + \)\(35\!\cdots\!20\)\( T^{2} - \)\(53\!\cdots\!56\)\( T^{3} - \)\(15\!\cdots\!20\)\( T^{4} + 567796466811702848 T^{5} - 187972280448 T^{6} - 1053344 T^{7} + T^{8} \)
$43$ \( \)\(94\!\cdots\!60\)\( - \)\(13\!\cdots\!32\)\( T + \)\(13\!\cdots\!68\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} - \)\(15\!\cdots\!20\)\( T^{4} + 471057097034865512 T^{5} + 300896770584 T^{6} - 1587806 T^{7} + T^{8} \)
$47$ \( -\)\(32\!\cdots\!00\)\( + \)\(18\!\cdots\!88\)\( T - \)\(72\!\cdots\!60\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!56\)\( T^{4} + 1549675860286664864 T^{5} - 2338367716296 T^{6} - 443336 T^{7} + T^{8} \)
$53$ \( -\)\(11\!\cdots\!40\)\( + \)\(15\!\cdots\!40\)\( T - \)\(36\!\cdots\!40\)\( T^{2} - \)\(21\!\cdots\!72\)\( T^{3} + \)\(79\!\cdots\!68\)\( T^{4} + 193466558821137696 T^{5} - 5207142091452 T^{6} + 375530 T^{7} + T^{8} \)
$59$ \( \)\(16\!\cdots\!60\)\( - \)\(12\!\cdots\!68\)\( T - \)\(92\!\cdots\!24\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!40\)\( T^{4} + 1341027425051652192 T^{5} - 6953041735192 T^{6} - 624008 T^{7} + T^{8} \)
$61$ \( \)\(13\!\cdots\!56\)\( + \)\(27\!\cdots\!72\)\( T - \)\(57\!\cdots\!68\)\( T^{2} - \)\(54\!\cdots\!44\)\( T^{3} + \)\(56\!\cdots\!36\)\( T^{4} - 13006977760131140136 T^{5} - 13046882887880 T^{6} + 2005568 T^{7} + T^{8} \)
$67$ \( \)\(30\!\cdots\!40\)\( + \)\(15\!\cdots\!80\)\( T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!52\)\( T^{3} + \)\(88\!\cdots\!08\)\( T^{4} - 55933415405314697608 T^{5} - 21285931031464 T^{6} + 2712286 T^{7} + T^{8} \)
$71$ \( -\)\(28\!\cdots\!80\)\( - \)\(47\!\cdots\!40\)\( T + \)\(52\!\cdots\!60\)\( T^{2} + \)\(17\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!96\)\( T^{4} - \)\(18\!\cdots\!64\)\( T^{5} - 23960029483600 T^{6} + 6287176 T^{7} + T^{8} \)
$73$ \( -\)\(19\!\cdots\!64\)\( + \)\(16\!\cdots\!20\)\( T + \)\(16\!\cdots\!36\)\( T^{2} + \)\(84\!\cdots\!72\)\( T^{3} - \)\(42\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{5} + 18426710185504 T^{6} + 10358312 T^{7} + T^{8} \)
$79$ \( -\)\(56\!\cdots\!00\)\( - \)\(26\!\cdots\!40\)\( T + \)\(13\!\cdots\!36\)\( T^{2} + \)\(66\!\cdots\!72\)\( T^{3} - \)\(68\!\cdots\!24\)\( T^{4} - \)\(36\!\cdots\!68\)\( T^{5} - 23968759546760 T^{6} + 8800574 T^{7} + T^{8} \)
$83$ \( \)\(68\!\cdots\!40\)\( + \)\(50\!\cdots\!40\)\( T - \)\(30\!\cdots\!60\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(68\!\cdots\!48\)\( T^{4} + 3251237833480599736 T^{5} - 47266486881160 T^{6} - 384948 T^{7} + T^{8} \)
$89$ \( \)\(42\!\cdots\!20\)\( + \)\(87\!\cdots\!32\)\( T - \)\(46\!\cdots\!28\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!20\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{5} - 271250340256164 T^{6} + 3445530 T^{7} + T^{8} \)
$97$ \( \)\(70\!\cdots\!48\)\( + \)\(31\!\cdots\!24\)\( T + \)\(22\!\cdots\!80\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} - \)\(98\!\cdots\!84\)\( T^{4} - \)\(82\!\cdots\!48\)\( T^{5} + 183433892802200 T^{6} + 28043764 T^{7} + T^{8} \)
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