Properties

Label 69.8.a.c
Level $69$
Weight $8$
Character orbit 69.a
Self dual yes
Analytic conductor $21.555$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 775 x^{5} - 474 x^{4} + 167184 x^{3} - 33920 x^{2} - 9348928 x + 28965760\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -27 q^{3} + ( 93 + \beta_{1} + \beta_{2} ) q^{4} + ( -74 - 10 \beta_{1} - \beta_{3} ) q^{5} -27 \beta_{1} q^{6} + ( 143 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{7} + ( 199 + 91 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -27 q^{3} + ( 93 + \beta_{1} + \beta_{2} ) q^{4} + ( -74 - 10 \beta_{1} - \beta_{3} ) q^{5} -27 \beta_{1} q^{6} + ( 143 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{7} + ( 199 + 91 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{8} + 729 q^{9} + ( -2181 - 107 \beta_{1} - \beta_{2} + 13 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} ) q^{10} + ( 1291 + 157 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - 11 \beta_{6} ) q^{11} + ( -2511 - 27 \beta_{1} - 27 \beta_{2} ) q^{12} + ( 533 + 255 \beta_{1} + 17 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} - 25 \beta_{5} - 7 \beta_{6} ) q^{13} + ( 641 + 729 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} - 5 \beta_{4} + 38 \beta_{5} + 26 \beta_{6} ) q^{14} + ( 1998 + 270 \beta_{1} + 27 \beta_{3} ) q^{15} + ( 7399 + 979 \beta_{1} + 79 \beta_{2} + 68 \beta_{3} - 32 \beta_{4} + 26 \beta_{5} + 14 \beta_{6} ) q^{16} + ( -5833 + 625 \beta_{1} + 4 \beta_{2} - 27 \beta_{3} - 12 \beta_{4} - 36 \beta_{5} - 5 \beta_{6} ) q^{17} + 729 \beta_{1} q^{18} + ( 11668 + 510 \beta_{1} - 24 \beta_{2} - 47 \beta_{3} + 16 \beta_{4} + 20 \beta_{5} + 72 \beta_{6} ) q^{19} + ( -12592 - 1428 \beta_{1} - 309 \beta_{2} - 94 \beta_{3} + 6 \beta_{4} - 93 \beta_{5} - 48 \beta_{6} ) q^{20} + ( -3861 - 81 \beta_{1} - 162 \beta_{2} + 27 \beta_{3} + 54 \beta_{5} - 27 \beta_{6} ) q^{21} + ( 34942 + 1134 \beta_{1} + 169 \beta_{2} - 10 \beta_{3} + 46 \beta_{4} - 39 \beta_{5} - 90 \beta_{6} ) q^{22} -12167 q^{23} + ( -5373 - 2457 \beta_{1} - 135 \beta_{2} + 108 \beta_{4} - 108 \beta_{5} - 54 \beta_{6} ) q^{24} + ( 45870 - 361 \beta_{1} - 23 \beta_{2} + 90 \beta_{3} - 146 \beta_{4} + 83 \beta_{5} + 37 \beta_{6} ) q^{25} + ( 59000 + 1448 \beta_{1} + 42 \beta_{2} - 88 \beta_{3} + 216 \beta_{4} + 156 \beta_{5} + 62 \beta_{6} ) q^{26} -19683 q^{27} + ( 137638 + 4062 \beta_{1} + 570 \beta_{2} + 288 \beta_{3} - 380 \beta_{4} + 96 \beta_{5} - 112 \beta_{6} ) q^{28} + ( 22138 - 880 \beta_{1} - 88 \beta_{2} - 326 \beta_{3} + 376 \beta_{4} + 12 \beta_{5} + 112 \beta_{6} ) q^{29} + ( 58887 + 2889 \beta_{1} + 27 \beta_{2} - 351 \beta_{4} + 135 \beta_{5} + 108 \beta_{6} ) q^{30} + ( 34686 - 3982 \beta_{1} + 98 \beta_{2} - 698 \beta_{3} + 324 \beta_{4} - 366 \beta_{5} + 22 \beta_{6} ) q^{31} + ( 182131 + 9463 \beta_{1} + 725 \beta_{2} + 544 \beta_{3} - 756 \beta_{4} + 244 \beta_{5} + 230 \beta_{6} ) q^{32} + ( -34857 - 4239 \beta_{1} + 216 \beta_{2} + 108 \beta_{4} - 108 \beta_{5} + 297 \beta_{6} ) q^{33} + ( 140735 - 6879 \beta_{1} + 504 \beta_{2} + 234 \beta_{3} + 513 \beta_{4} + 320 \beta_{5} + 134 \beta_{6} ) q^{34} + ( -18343 - 8671 \beta_{1} - 1063 \beta_{2} + 68 \beta_{3} - 418 \beta_{4} - 849 \beta_{5} - 1025 \beta_{6} ) q^{35} + ( 67797 + 729 \beta_{1} + 729 \beta_{2} ) q^{36} + ( 82589 - 1511 \beta_{1} + 1312 \beta_{2} + 590 \beta_{3} - 260 \beta_{4} + 300 \beta_{5} - 155 \beta_{6} ) q^{37} + ( 110153 + 6197 \beta_{1} + 1241 \beta_{2} + 136 \beta_{3} + 451 \beta_{4} - 599 \beta_{5} + 12 \beta_{6} ) q^{38} + ( -14391 - 6885 \beta_{1} - 459 \beta_{2} + 378 \beta_{3} - 162 \beta_{4} + 675 \beta_{5} + 189 \beta_{6} ) q^{39} + ( -17104 - 44300 \beta_{1} - 3145 \beta_{2} - 198 \beta_{3} + 1250 \beta_{4} - 337 \beta_{5} - 236 \beta_{6} ) q^{40} + ( 17544 - 13362 \beta_{1} - 2694 \beta_{2} + 716 \beta_{3} + 340 \beta_{4} + 1050 \beta_{5} + 262 \beta_{6} ) q^{41} + ( -17307 - 19683 \beta_{1} - 378 \beta_{2} - 270 \beta_{3} + 135 \beta_{4} - 1026 \beta_{5} - 702 \beta_{6} ) q^{42} + ( -94122 - 18040 \beta_{1} - 1010 \beta_{2} - 541 \beta_{3} - 548 \beta_{4} + 166 \beta_{5} + 786 \beta_{6} ) q^{43} + ( 93428 + 37584 \beta_{1} + 1643 \beta_{2} - 1198 \beta_{3} + 210 \beta_{4} - 85 \beta_{5} + 1216 \beta_{6} ) q^{44} + ( -53946 - 7290 \beta_{1} - 729 \beta_{3} ) q^{45} -12167 \beta_{1} q^{46} + ( -84727 + 21469 \beta_{1} - 253 \beta_{2} - 300 \beta_{3} - 438 \beta_{4} - 47 \beta_{5} - 949 \beta_{6} ) q^{47} + ( -199773 - 26433 \beta_{1} - 2133 \beta_{2} - 1836 \beta_{3} + 864 \beta_{4} - 702 \beta_{5} - 378 \beta_{6} ) q^{48} + ( 465658 + 461 \beta_{1} - 1871 \beta_{2} + 1638 \beta_{3} - 3410 \beta_{4} + 943 \beta_{5} - 1345 \beta_{6} ) q^{49} + ( -100828 + 49741 \beta_{1} + 1278 \beta_{2} + 2392 \beta_{3} - 1192 \beta_{4} + 1108 \beta_{5} + 622 \beta_{6} ) q^{50} + ( 157491 - 16875 \beta_{1} - 108 \beta_{2} + 729 \beta_{3} + 324 \beta_{4} + 972 \beta_{5} + 135 \beta_{6} ) q^{51} + ( 255034 + 33922 \beta_{1} + 564 \beta_{2} - 1604 \beta_{3} - 972 \beta_{4} - 574 \beta_{5} - 800 \beta_{6} ) q^{52} + ( 30550 + 25762 \beta_{1} - 1620 \beta_{2} - 1125 \beta_{3} + 1344 \beta_{4} + 1660 \beta_{5} + 1740 \beta_{6} ) q^{53} -19683 \beta_{1} q^{54} + ( 26982 - 27390 \beta_{1} + 2340 \beta_{2} - 3344 \beta_{3} + 4344 \beta_{4} - 1408 \beta_{5} + 3478 \beta_{6} ) q^{55} + ( 766150 + 140926 \beta_{1} + 6178 \beta_{2} + 3936 \beta_{3} - 4784 \beta_{4} + 1680 \beta_{5} - 764 \beta_{6} ) q^{56} + ( -315036 - 13770 \beta_{1} + 648 \beta_{2} + 1269 \beta_{3} - 432 \beta_{4} - 540 \beta_{5} - 1944 \beta_{6} ) q^{57} + ( -162338 - 6124 \beta_{1} - 1078 \beta_{2} - 5368 \beta_{3} + 3566 \beta_{4} - 5738 \beta_{5} - 2384 \beta_{6} ) q^{58} + ( 64293 - 6147 \beta_{1} + 2703 \beta_{2} + 1472 \beta_{3} + 2250 \beta_{4} - 287 \beta_{5} + 1031 \beta_{6} ) q^{59} + ( 339984 + 38556 \beta_{1} + 8343 \beta_{2} + 2538 \beta_{3} - 162 \beta_{4} + 2511 \beta_{5} + 1296 \beta_{6} ) q^{60} + ( 321777 - 26607 \beta_{1} + 426 \beta_{2} + 3974 \beta_{3} - 1344 \beta_{4} - 2062 \beta_{5} - 1231 \beta_{6} ) q^{61} + ( -813314 + 2130 \beta_{1} - 4894 \beta_{2} - 4320 \beta_{3} + 9454 \beta_{4} - 3022 \beta_{5} - 1564 \beta_{6} ) q^{62} + ( 104247 + 2187 \beta_{1} + 4374 \beta_{2} - 729 \beta_{3} - 1458 \beta_{5} + 729 \beta_{6} ) q^{63} + ( 1025175 + 184195 \beta_{1} + 8251 \beta_{2} + 4284 \beta_{3} - 5800 \beta_{4} + 7886 \beta_{5} + 4774 \beta_{6} ) q^{64} + ( -119886 - 15602 \beta_{1} + 5724 \beta_{2} - 572 \beta_{3} + 1256 \beta_{4} - 8324 \beta_{5} - 5882 \beta_{6} ) q^{65} + ( -943434 - 30618 \beta_{1} - 4563 \beta_{2} + 270 \beta_{3} - 1242 \beta_{4} + 1053 \beta_{5} + 2430 \beta_{6} ) q^{66} + ( 639922 - 29556 \beta_{1} - 7424 \beta_{2} + 6383 \beta_{3} - 7192 \beta_{4} + 1436 \beta_{5} - 142 \beta_{6} ) q^{67} + ( -783340 + 126924 \beta_{1} - 7402 \beta_{2} - 4588 \beta_{3} - 6096 \beta_{4} - 82 \beta_{5} - 584 \beta_{6} ) q^{68} + 328509 q^{69} + ( -1806322 - 158520 \beta_{1} - 24910 \beta_{2} + 2236 \beta_{3} + 9650 \beta_{4} + 6884 \beta_{5} - 1238 \beta_{6} ) q^{70} + ( -734368 - 114856 \beta_{1} - 6512 \beta_{2} + 1128 \beta_{3} - 504 \beta_{4} + 2412 \beta_{5} - 3836 \beta_{6} ) q^{71} + ( 145071 + 66339 \beta_{1} + 3645 \beta_{2} - 2916 \beta_{4} + 2916 \beta_{5} + 1458 \beta_{6} ) q^{72} + ( -1891514 + 2608 \beta_{1} + 7414 \beta_{2} - 320 \beta_{3} + 5236 \beta_{4} + 462 \beta_{5} + 6288 \beta_{6} ) q^{73} + ( -411260 + 284078 \beta_{1} + 4047 \beta_{2} + 2630 \beta_{3} - 13288 \beta_{4} + 7943 \beta_{5} + 3294 \beta_{6} ) q^{74} + ( -1238490 + 9747 \beta_{1} + 621 \beta_{2} - 2430 \beta_{3} + 3942 \beta_{4} - 2241 \beta_{5} - 999 \beta_{6} ) q^{75} + ( -80166 + 170806 \beta_{1} + 1211 \beta_{2} + 70 \beta_{3} - 7310 \beta_{4} + 3977 \beta_{5} - 4328 \beta_{6} ) q^{76} + ( -2642758 + 67334 \beta_{1} + 28396 \beta_{2} - 8664 \beta_{3} + 3648 \beta_{4} - 76 \beta_{5} + 8102 \beta_{6} ) q^{77} + ( -1593000 - 39096 \beta_{1} - 1134 \beta_{2} + 2376 \beta_{3} - 5832 \beta_{4} - 4212 \beta_{5} - 1674 \beta_{6} ) q^{78} + ( 1372849 - 38903 \beta_{1} - 1628 \beta_{2} - 3245 \beta_{3} + 9260 \beta_{4} + 4672 \beta_{5} - 6005 \beta_{6} ) q^{79} + ( -7972328 - 302692 \beta_{1} - 32121 \beta_{2} - 8710 \beta_{3} + 13706 \beta_{4} - 11369 \beta_{5} - 5332 \beta_{6} ) q^{80} + 531441 q^{81} + ( -2977956 - 275166 \beta_{1} - 18944 \beta_{2} - 5968 \beta_{3} - 3936 \beta_{4} - 19784 \beta_{5} - 8612 \beta_{6} ) q^{82} + ( -84733 - 181271 \beta_{1} + 1006 \beta_{2} - 8468 \beta_{3} - 2712 \beta_{4} + 5814 \beta_{5} + 14493 \beta_{6} ) q^{83} + ( -3716226 - 109674 \beta_{1} - 15390 \beta_{2} - 7776 \beta_{3} + 10260 \beta_{4} - 2592 \beta_{5} + 3024 \beta_{6} ) q^{84} + ( 1064971 - 137997 \beta_{1} + 16579 \beta_{2} + 5538 \beta_{3} + 4170 \beta_{4} - 14123 \beta_{5} - 10447 \beta_{6} ) q^{85} + ( -4048025 - 264253 \beta_{1} - 6977 \beta_{2} + 13152 \beta_{3} + 9933 \beta_{4} - 1973 \beta_{5} + 1728 \beta_{6} ) q^{86} + ( -597726 + 23760 \beta_{1} + 2376 \beta_{2} + 8802 \beta_{3} - 10152 \beta_{4} - 324 \beta_{5} - 3024 \beta_{6} ) q^{87} + ( 3782040 + 110028 \beta_{1} + 35175 \beta_{2} + 5386 \beta_{3} + 602 \beta_{4} + 5455 \beta_{5} + 16980 \beta_{6} ) q^{88} + ( -555995 - 24657 \beta_{1} + 11712 \beta_{2} - 3471 \beta_{3} - 13364 \beta_{4} - 1440 \beta_{5} - 695 \beta_{6} ) q^{89} + ( -1589949 - 78003 \beta_{1} - 729 \beta_{2} + 9477 \beta_{4} - 3645 \beta_{5} - 2916 \beta_{6} ) q^{90} + ( 5153050 - 136994 \beta_{1} - 11930 \beta_{2} - 5448 \beta_{3} - 12916 \beta_{4} + 11714 \beta_{5} + 4850 \beta_{6} ) q^{91} + ( -1131531 - 12167 \beta_{1} - 12167 \beta_{2} ) q^{92} + ( -936522 + 107514 \beta_{1} - 2646 \beta_{2} + 18846 \beta_{3} - 8748 \beta_{4} + 9882 \beta_{5} - 594 \beta_{6} ) q^{93} + ( 4785346 - 62080 \beta_{1} + 22692 \beta_{2} + 1408 \beta_{3} + 7874 \beta_{4} + 1342 \beta_{5} - 5366 \beta_{6} ) q^{94} + ( 1500081 - 322891 \beta_{1} - 15053 \beta_{2} + 1048 \beta_{3} - 3350 \beta_{4} + 19193 \beta_{5} - 14765 \beta_{6} ) q^{95} + ( -4917537 - 255501 \beta_{1} - 19575 \beta_{2} - 14688 \beta_{3} + 20412 \beta_{4} - 6588 \beta_{5} - 6210 \beta_{6} ) q^{96} + ( 7068666 + 177124 \beta_{1} - 10036 \beta_{2} - 17662 \beta_{3} - 6920 \beta_{4} + 2740 \beta_{5} + 11400 \beta_{6} ) q^{97} + ( -146864 + 418747 \beta_{1} + 15804 \beta_{2} + 44604 \beta_{3} - 8072 \beta_{4} + 24974 \beta_{5} + 2722 \beta_{6} ) q^{98} + ( 941139 + 114453 \beta_{1} - 5832 \beta_{2} - 2916 \beta_{4} + 2916 \beta_{5} - 8019 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 189q^{3} + 654q^{4} - 516q^{5} + 1018q^{7} + 1422q^{8} + 5103q^{9} + O(q^{10}) \) \( 7q - 189q^{3} + 654q^{4} - 516q^{5} + 1018q^{7} + 1422q^{8} + 5103q^{9} - 15310q^{10} + 9040q^{11} - 17658q^{12} + 3774q^{13} + 4536q^{14} + 13932q^{15} + 52002q^{16} - 40760q^{17} + 81598q^{19} - 88946q^{20} - 27486q^{21} + 245034q^{22} - 85169q^{23} - 38394q^{24} + 321325q^{25} + 412748q^{26} - 137781q^{27} + 965948q^{28} + 154126q^{29} + 413370q^{30} + 243132q^{31} + 1278286q^{32} - 244080q^{33} + 984836q^{34} - 130296q^{35} + 476766q^{36} + 582114q^{37} + 772558q^{38} - 101898q^{39} - 132618q^{40} + 113062q^{41} - 122472q^{42} - 659778q^{43} + 659390q^{44} - 376164q^{45} - 591032q^{47} - 1404054q^{48} + 3263235q^{49} - 702684q^{50} + 1100520q^{51} + 1793280q^{52} + 207128q^{53} + 184664q^{55} + 5390508q^{56} - 2203146q^{57} - 1142916q^{58} + 447148q^{59} + 2401542q^{60} + 2248970q^{61} - 5729060q^{62} + 742122q^{63} + 7212922q^{64} - 827096q^{65} - 6615918q^{66} + 4467570q^{67} - 5477620q^{68} + 2299563q^{69} - 12744284q^{70} - 5154608q^{71} + 1036638q^{72} - 13239250q^{73} - 2827426q^{74} - 8675775q^{75} - 527434q^{76} - 18415912q^{77} - 11144196q^{78} + 9594446q^{79} - 55932394q^{80} + 3720087q^{81} - 20889952q^{82} - 573720q^{83} - 26080596q^{84} + 7477272q^{85} - 28416910q^{86} - 4161402q^{87} + 26555702q^{88} - 3810540q^{89} - 11160990q^{90} + 36092068q^{91} - 7957218q^{92} - 6564564q^{93} + 33545768q^{94} + 10497320q^{95} - 34513722q^{96} + 49497978q^{97} - 1023376q^{98} + 6590160q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 775 x^{5} - 474 x^{4} + 167184 x^{3} - 33920 x^{2} - 9348928 x + 28965760\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 221 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 52 \nu^{5} + 4145 \nu^{4} - 3278 \nu^{3} - 1938192 \nu^{2} + 3642512 \nu + 99283584 \)\()/301376\)
\(\beta_{4}\)\(=\)\((\)\( 25 \nu^{6} - 192 \nu^{5} - 16039 \nu^{4} + 60982 \nu^{3} + 2595192 \nu^{2} + 641136 \nu - 63110496 \)\()/150688\)
\(\beta_{5}\)\(=\)\((\)\( -33 \nu^{6} - 500 \nu^{5} + 27199 \nu^{4} + 349718 \nu^{3} - 5892416 \nu^{2} - 51923504 \nu + 299476832 \)\()/150688\)
\(\beta_{6}\)\(=\)\((\)\( 29 \nu^{6} + 154 \nu^{5} - 21619 \nu^{4} - 125532 \nu^{3} + 4149624 \nu^{2} + 19840408 \nu - 164228248 \)\()/37672\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 221\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 5 \beta_{2} + 347 \beta_{1} + 199\)
\(\nu^{4}\)\(=\)\(14 \beta_{6} + 26 \beta_{5} - 32 \beta_{4} + 68 \beta_{3} + 463 \beta_{2} + 1363 \beta_{1} + 75879\)
\(\nu^{5}\)\(=\)\(1254 \beta_{6} + 2292 \beta_{5} - 2804 \beta_{4} + 544 \beta_{3} + 3285 \beta_{2} + 137975 \beta_{1} + 284019\)
\(\nu^{6}\)\(=\)\(13734 \beta_{6} + 24526 \beta_{5} - 26280 \beta_{4} + 47804 \beta_{3} + 206267 \beta_{2} + 958211 \beta_{1} + 29959703\)

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
−19.6379
−16.2020
−11.3612
4.39427
5.40652
15.1764
22.2240
−19.6379 −27.0000 257.648 −311.660 530.224 487.750 −2546.02 729.000 6120.36
1.2 −16.2020 −27.0000 134.506 395.951 437.455 1527.96 −105.404 729.000 −6415.21
1.3 −11.3612 −27.0000 1.07734 392.713 306.753 −1170.86 1442.00 729.000 −4461.70
1.4 4.39427 −27.0000 −108.690 −380.251 −118.645 −1696.01 −1040.08 729.000 −1670.93
1.5 5.40652 −27.0000 −98.7695 −344.177 −145.976 525.740 −1226.03 729.000 −1860.80
1.6 15.1764 −27.0000 102.322 149.395 −409.762 −45.6786 −389.696 729.000 2267.27
1.7 22.2240 −27.0000 365.907 −417.971 −600.048 1389.11 5287.24 729.000 −9288.99
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.c 7
3.b odd 2 1 207.8.a.d 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.c 7 1.a even 1 1 trivial
207.8.a.d 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 775 T_{2}^{5} - 474 T_{2}^{4} + 167184 T_{2}^{3} - 33920 T_{2}^{2} - 9348928 T_{2} + 28965760 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 28965760 - 9348928 T - 33920 T^{2} + 167184 T^{3} - 474 T^{4} - 775 T^{5} + T^{7} \)
$3$ \( ( 27 + T )^{7} \)
$5$ \( -396033586907868000 + 249170572282400 T + 17554034382960 T^{2} + 21012988548 T^{3} - 176741992 T^{4} - 300972 T^{5} + 516 T^{6} + T^{7} \)
$7$ \( 49369886575152432928 + 889108027085875952 T - 4067598878965608 T^{2} + 3037397232308 T^{3} + 4411014448 T^{4} - 3995856 T^{5} - 1018 T^{6} + T^{7} \)
$11$ \( \)\(78\!\cdots\!60\)\( + \)\(25\!\cdots\!20\)\( T - 5844393456000573440 T^{2} + 327027275976704 T^{3} + 532355334104 T^{4} - 53173472 T^{5} - 9040 T^{6} + T^{7} \)
$13$ \( \)\(21\!\cdots\!84\)\( - \)\(37\!\cdots\!88\)\( T - 40168572623250815232 T^{2} + 7830902457345024 T^{3} + 1271146805784 T^{4} - 237324884 T^{5} - 3774 T^{6} + T^{7} \)
$17$ \( \)\(11\!\cdots\!60\)\( - \)\(59\!\cdots\!72\)\( T + \)\(47\!\cdots\!24\)\( T^{2} + 131112182279438116 T^{3} - 15350732220480 T^{4} - 356602516 T^{5} + 40760 T^{6} + T^{7} \)
$19$ \( -\)\(87\!\cdots\!96\)\( + \)\(34\!\cdots\!16\)\( T - \)\(26\!\cdots\!68\)\( T^{2} - 1013975580949412236 T^{3} + 95149313141016 T^{4} + 221185648 T^{5} - 81598 T^{6} + T^{7} \)
$23$ \( ( 12167 + T )^{7} \)
$29$ \( \)\(22\!\cdots\!00\)\( - \)\(44\!\cdots\!20\)\( T - \)\(58\!\cdots\!40\)\( T^{2} + \)\(97\!\cdots\!40\)\( T^{3} + 5779838675472584 T^{4} - 55222920380 T^{5} - 154126 T^{6} + T^{7} \)
$31$ \( \)\(13\!\cdots\!36\)\( - \)\(79\!\cdots\!96\)\( T - \)\(57\!\cdots\!72\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + 24364400532707008 T^{4} - 117464166560 T^{5} - 243132 T^{6} + T^{7} \)
$37$ \( \)\(96\!\cdots\!32\)\( - \)\(22\!\cdots\!92\)\( T - \)\(77\!\cdots\!92\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + 92935836009951360 T^{4} - 196707772012 T^{5} - 582114 T^{6} + T^{7} \)
$41$ \( \)\(72\!\cdots\!60\)\( - \)\(28\!\cdots\!20\)\( T - \)\(92\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!48\)\( T^{3} + 197288333373210664 T^{4} - 1181463300732 T^{5} - 113062 T^{6} + T^{7} \)
$43$ \( -\)\(80\!\cdots\!80\)\( - \)\(10\!\cdots\!16\)\( T + \)\(11\!\cdots\!48\)\( T^{2} + \)\(16\!\cdots\!96\)\( T^{3} - 502424433883034392 T^{4} - 748807338960 T^{5} + 659778 T^{6} + T^{7} \)
$47$ \( -\)\(95\!\cdots\!80\)\( - \)\(97\!\cdots\!52\)\( T + \)\(31\!\cdots\!88\)\( T^{2} + \)\(16\!\cdots\!08\)\( T^{3} - 292550186224313888 T^{4} - 696656534184 T^{5} + 591032 T^{6} + T^{7} \)
$53$ \( \)\(76\!\cdots\!40\)\( - \)\(62\!\cdots\!20\)\( T - \)\(55\!\cdots\!40\)\( T^{2} + \)\(26\!\cdots\!64\)\( T^{3} + 874020420490291848 T^{4} - 3304817863308 T^{5} - 207128 T^{6} + T^{7} \)
$59$ \( -\)\(59\!\cdots\!08\)\( - \)\(23\!\cdots\!84\)\( T + \)\(12\!\cdots\!04\)\( T^{2} + \)\(30\!\cdots\!60\)\( T^{3} - 1031061816096984768 T^{4} - 4332298484392 T^{5} - 447148 T^{6} + T^{7} \)
$61$ \( -\)\(41\!\cdots\!28\)\( + \)\(24\!\cdots\!00\)\( T - \)\(32\!\cdots\!76\)\( T^{2} + \)\(23\!\cdots\!44\)\( T^{3} + 17726645067959382864 T^{4} - 6263277418140 T^{5} - 2248970 T^{6} + T^{7} \)
$67$ \( -\)\(18\!\cdots\!80\)\( + \)\(77\!\cdots\!40\)\( T - \)\(90\!\cdots\!40\)\( T^{2} - \)\(13\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!64\)\( T^{4} - 15915357789424 T^{5} - 4467570 T^{6} + T^{7} \)
$71$ \( -\)\(28\!\cdots\!20\)\( - \)\(87\!\cdots\!60\)\( T + \)\(90\!\cdots\!20\)\( T^{2} + \)\(65\!\cdots\!32\)\( T^{3} - 50362296862972444032 T^{4} - 10444682714224 T^{5} + 5154608 T^{6} + T^{7} \)
$73$ \( \)\(11\!\cdots\!12\)\( + \)\(12\!\cdots\!76\)\( T - \)\(15\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + 23064712326395258536 T^{4} + 50669339671764 T^{5} + 13239250 T^{6} + T^{7} \)
$79$ \( \)\(37\!\cdots\!80\)\( - \)\(39\!\cdots\!24\)\( T - \)\(91\!\cdots\!68\)\( T^{2} + \)\(77\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!12\)\( T^{4} - 50722254327120 T^{5} - 9594446 T^{6} + T^{7} \)
$83$ \( -\)\(58\!\cdots\!40\)\( - \)\(36\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!56\)\( T^{3} - \)\(12\!\cdots\!96\)\( T^{4} - 132446610346240 T^{5} + 573720 T^{6} + T^{7} \)
$89$ \( \)\(11\!\cdots\!32\)\( + \)\(23\!\cdots\!72\)\( T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} - \)\(62\!\cdots\!96\)\( T^{4} - 97136743222404 T^{5} + 3810540 T^{6} + T^{7} \)
$97$ \( \)\(24\!\cdots\!52\)\( - \)\(30\!\cdots\!24\)\( T + \)\(83\!\cdots\!76\)\( T^{2} - \)\(73\!\cdots\!44\)\( T^{3} - \)\(24\!\cdots\!00\)\( T^{4} + 814042966825412 T^{5} - 49497978 T^{6} + T^{7} \)
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