Properties

Label 84.12.i.b
Level $84$
Weight $12$
Character orbit 84.i
Analytic conductor $64.541$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.5408271670\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 581500324 x^{14} - 481772282104 x^{13} + 132272376701859942 x^{12} + \)\(18\!\cdots\!08\)\( x^{11} - \)\(14\!\cdots\!08\)\( x^{10} - \)\(25\!\cdots\!56\)\( x^{9} + \)\(80\!\cdots\!79\)\( x^{8} + \)\(11\!\cdots\!68\)\( x^{7} - \)\(19\!\cdots\!68\)\( x^{6} + \)\(59\!\cdots\!08\)\( x^{5} + \)\(21\!\cdots\!06\)\( x^{4} - \)\(37\!\cdots\!04\)\( x^{3} - \)\(31\!\cdots\!28\)\( x^{2} + \)\(25\!\cdots\!24\)\( x + \)\(79\!\cdots\!77\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{15}\cdot 7^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -243 + 243 \beta_{1} ) q^{3} + ( -269 \beta_{1} + \beta_{3} ) q^{5} + ( 1064 + 4186 \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} -59049 \beta_{1} q^{9} +O(q^{10})\) \( q + ( -243 + 243 \beta_{1} ) q^{3} + ( -269 \beta_{1} + \beta_{3} ) q^{5} + ( 1064 + 4186 \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} -59049 \beta_{1} q^{9} + ( -27848 + 27847 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{7} - \beta_{9} ) q^{11} + ( 168931 + 2 \beta_{1} + 36 \beta_{2} - 4 \beta_{4} - \beta_{8} + \beta_{9} ) q^{13} + ( 65367 + 243 \beta_{2} ) q^{15} + ( 639310 - 639291 \beta_{1} + 22 \beta_{2} + 19 \beta_{3} - 45 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} - \beta_{8} - 6 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{17} + ( -37 + 863778 \beta_{1} + 5 \beta_{2} - 175 \beta_{3} - 34 \beta_{4} + 61 \beta_{5} + \beta_{6} + 10 \beta_{7} - 5 \beta_{10} + 2 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{19} + ( -1275507 + 258552 \beta_{1} - 243 \beta_{5} ) q^{21} + ( -78 - 6423053 \beta_{1} + 8 \beta_{2} + 684 \beta_{3} + 142 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} - 8 \beta_{10} - \beta_{13} + 2 \beta_{14} - 5 \beta_{15} ) q^{23} + ( -23930950 + 23930931 \beta_{1} - 1812 \beta_{2} - 1809 \beta_{3} + 12 \beta_{4} - 399 \beta_{5} - 3 \beta_{6} + 19 \beta_{7} - 2 \beta_{8} - 19 \beta_{9} - 20 \beta_{10} - 12 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - 12 \beta_{14} + 5 \beta_{15} ) q^{25} + 14348907 q^{27} + ( 7427324 - 446 \beta_{1} + 2067 \beta_{2} + 82 \beta_{3} + 1045 \beta_{4} + 40 \beta_{5} - 31 \beta_{6} + 21 \beta_{8} + 89 \beta_{9} - 51 \beta_{10} + 11 \beta_{12} ) q^{29} + ( 20583436 - 20583280 \beta_{1} - 2049 \beta_{2} - 2041 \beta_{3} - 262 \beta_{4} - 73 \beta_{5} - 8 \beta_{6} - 136 \beta_{7} + 32 \beta_{8} + 136 \beta_{9} - 8 \beta_{10} + 42 \beta_{11} - 5 \beta_{12} - 32 \beta_{13} + 42 \beta_{14} + 5 \beta_{15} ) q^{31} + ( -243 - 6767550 \beta_{1} - 243 \beta_{3} + 486 \beta_{4} + 486 \beta_{5} - 243 \beta_{7} ) q^{33} + ( 1547228 + 3814468 \beta_{1} + 3443 \beta_{2} + 15150 \beta_{3} + 136 \beta_{4} - 42 \beta_{5} + 133 \beta_{6} + \beta_{7} - 89 \beta_{8} - 383 \beta_{9} - 99 \beta_{10} - 19 \beta_{11} + 15 \beta_{12} + 78 \beta_{13} + 27 \beta_{14} + 25 \beta_{15} ) q^{35} + ( -2204 + 9465150 \beta_{1} + 261 \beta_{2} + 11459 \beta_{3} - 491 \beta_{4} + 3862 \beta_{5} + 123 \beta_{6} + 116 \beta_{7} - 261 \beta_{10} - 20 \beta_{13} - 24 \beta_{14} + 10 \beta_{15} ) q^{37} + ( -41049747 + 41049261 \beta_{1} - 8748 \beta_{2} - 8748 \beta_{3} + 972 \beta_{4} - 972 \beta_{5} + 243 \beta_{7} + 243 \beta_{8} - 243 \beta_{9} - 243 \beta_{13} ) q^{39} + ( -113464461 - 1471 \beta_{1} - 15048 \beta_{2} + 111 \beta_{3} + 1010 \beta_{4} - 1807 \beta_{5} + 112 \beta_{6} - 398 \beta_{8} + 628 \beta_{9} - 223 \beta_{10} - 124 \beta_{11} + 45 \beta_{12} ) q^{41} + ( 649497 + 3022 \beta_{1} + 43662 \beta_{2} - 194 \beta_{3} - 4837 \beta_{4} + 1450 \beta_{5} + 82 \beta_{6} + 274 \beta_{8} + 452 \beta_{9} + 112 \beta_{10} + 33 \beta_{11} - 110 \beta_{12} ) q^{43} + ( -15884181 + 15884181 \beta_{1} - 59049 \beta_{2} - 59049 \beta_{3} ) q^{45} + ( 5536 - 129318552 \beta_{1} - 593 \beta_{2} - 29443 \beta_{3} + 7351 \beta_{4} - 10048 \beta_{5} - 30 \beta_{6} - 70 \beta_{7} + 593 \beta_{10} + 426 \beta_{13} - 162 \beta_{14} - 43 \beta_{15} ) q^{47} + ( 333639744 - 151879142 \beta_{1} + 76166 \beta_{2} + 73425 \beta_{3} - 5906 \beta_{4} + 857 \beta_{5} + 397 \beta_{6} + 803 \beta_{7} - 277 \beta_{8} - 1460 \beta_{9} + 607 \beta_{10} + 234 \beta_{11} - 51 \beta_{12} + 859 \beta_{13} + 30 \beta_{14} - 134 \beta_{15} ) q^{49} + ( 5832 + 155343582 \beta_{1} - 729 \beta_{2} - 5346 \beta_{3} + 9477 \beta_{4} - 10449 \beta_{5} - 243 \beta_{6} - 1458 \beta_{7} + 729 \beta_{10} - 243 \beta_{13} - 243 \beta_{14} ) q^{51} + ( -83145816 + 83143140 \beta_{1} - 10216 \beta_{2} - 10667 \beta_{3} + 5398 \beta_{4} + 11911 \beta_{5} + 451 \beta_{6} + 3088 \beta_{7} + 790 \beta_{8} - 3088 \beta_{9} + 850 \beta_{10} + 98 \beta_{11} + 325 \beta_{12} - 790 \beta_{13} + 98 \beta_{14} - 325 \beta_{15} ) q^{53} + ( -79101842 + 32483 \beta_{1} + 129383 \beta_{2} - 2354 \beta_{3} - 59708 \beta_{4} + 8261 \beta_{5} + 1045 \beta_{6} - 1034 \beta_{8} + 6182 \beta_{9} + 1309 \beta_{10} + 396 \beta_{11} - 330 \beta_{12} ) q^{55} + ( -209882016 - 1215 \beta_{1} - 43740 \beta_{2} - 1215 \beta_{3} - 7533 \beta_{4} - 7047 \beta_{5} + 972 \beta_{6} - 486 \beta_{8} - 2430 \beta_{9} + 243 \beta_{10} - 729 \beta_{11} + 243 \beta_{12} ) q^{57} + ( 130104384 - 130087498 \beta_{1} - 160281 \beta_{2} - 163448 \beta_{3} - 40421 \beta_{4} + 80218 \beta_{5} + 3167 \beta_{6} + 636 \beta_{7} + 122 \beta_{8} - 636 \beta_{9} + 182 \beta_{10} - 497 \beta_{11} - 181 \beta_{12} - 122 \beta_{13} - 497 \beta_{14} + 181 \beta_{15} ) q^{59} + ( 5376 - 1798876419 \beta_{1} + 437 \beta_{2} + 106323 \beta_{3} + 41573 \beta_{4} - 11350 \beta_{5} + 2967 \beta_{6} + 5353 \beta_{7} - 437 \beta_{10} + 1265 \beta_{13} + 276 \beta_{14} + 506 \beta_{15} ) q^{61} + ( 247120065 - 310007250 \beta_{1} + 59049 \beta_{4} ) q^{63} + ( 35850 - 2617237750 \beta_{1} - 4770 \beta_{2} + 266415 \beta_{3} + 144405 \beta_{4} - 60085 \beta_{5} + 3020 \beta_{6} - 18960 \beta_{7} + 4770 \beta_{10} - 3145 \beta_{13} + 2075 \beta_{14} + 285 \beta_{15} ) q^{65} + ( -4163529197 + 4163561323 \beta_{1} + 170307 \beta_{2} + 164062 \beta_{3} - 77137 \beta_{4} + 113118 \beta_{5} + 6245 \beta_{6} + 6766 \beta_{7} + 76 \beta_{8} - 6766 \beta_{9} - 572 \beta_{10} + 177 \beta_{11} - 1265 \beta_{12} - 76 \beta_{13} + 177 \beta_{14} + 1265 \beta_{15} ) q^{67} + ( 1560818889 + 17496 \beta_{1} + 164268 \beta_{2} - 1944 \beta_{3} - 35235 \beta_{4} + 1944 \beta_{5} + 729 \beta_{6} + 243 \beta_{8} + 1701 \beta_{9} + 1215 \beta_{10} + 486 \beta_{11} + 1215 \beta_{12} ) q^{69} + ( 4115817150 + 25304 \beta_{1} - 472412 \beta_{2} - 9245 \beta_{3} - 115436 \beta_{4} - 51878 \beta_{5} + 6433 \beta_{6} + 2936 \beta_{8} + 6326 \beta_{9} + 2812 \beta_{10} + 2728 \beta_{11} + 479 \beta_{12} ) q^{71} + ( 2213731729 - 2213708470 \beta_{1} - 226261 \beta_{2} - 228449 \beta_{3} - 47686 \beta_{4} + 202977 \beta_{5} + 2188 \beta_{6} - 20302 \beta_{7} + 1019 \beta_{8} + 20302 \beta_{9} + 3892 \beta_{10} - 684 \beta_{11} + 1695 \beta_{12} - 1019 \beta_{13} - 684 \beta_{14} - 1695 \beta_{15} ) q^{73} + ( -3159 - 5815310031 \beta_{1} + 729 \beta_{2} + 440316 \beta_{3} + 88452 \beta_{4} + 7776 \beta_{5} + 5589 \beta_{6} - 4617 \beta_{7} - 729 \beta_{10} - 486 \beta_{13} + 2916 \beta_{14} - 1215 \beta_{15} ) q^{75} + ( -3547475329 + 8168605099 \beta_{1} - 136136 \beta_{2} - 1593900 \beta_{3} + 6944 \beta_{4} - 35775 \beta_{5} + 9111 \beta_{6} + 3569 \beta_{7} + 2784 \beta_{8} - 13470 \beta_{9} - 537 \beta_{10} - 3978 \beta_{11} - 673 \beta_{12} - 407 \beta_{13} - 4052 \beta_{14} + 1263 \beta_{15} ) q^{77} + ( -69010 - 3328113107 \beta_{1} + 11510 \beta_{2} + 644839 \beta_{3} + 259309 \beta_{4} + 112690 \beta_{5} + 19878 \beta_{6} + 12572 \beta_{7} - 11510 \beta_{10} + 8548 \beta_{13} - 2310 \beta_{14} - 869 \beta_{15} ) q^{79} + ( -3486784401 + 3486784401 \beta_{1} ) q^{81} + ( -13144762778 - 150089 \beta_{1} + 1450452 \beta_{2} - 16864 \beta_{3} - 49910 \beta_{4} - 308789 \beta_{5} + 24710 \beta_{6} + 6391 \beta_{8} + 13937 \beta_{9} - 7846 \beta_{10} + 275 \beta_{11} - 3575 \beta_{12} ) q^{83} + ( -1615366049 - 174124 \beta_{1} - 2296594 \beta_{2} - 26783 \beta_{3} - 92861 \beta_{4} - 383938 \beta_{5} + 28840 \beta_{6} - 13133 \beta_{8} - 18601 \beta_{9} - 2057 \beta_{10} - 1548 \beta_{11} - 1860 \beta_{12} ) q^{85} + ( -1804997682 + 1805115780 \beta_{1} - 482355 \beta_{2} - 502281 \beta_{3} - 283581 \beta_{4} + 266328 \beta_{5} + 19926 \beta_{6} + 21627 \beta_{7} - 5103 \beta_{8} - 21627 \beta_{9} - 7533 \beta_{10} - 2673 \beta_{12} + 5103 \beta_{13} + 2673 \beta_{15} ) q^{87} + ( -37456 - 6993972508 \beta_{1} + 16320 \beta_{2} + 410250 \beta_{3} + 484670 \beta_{4} + 36232 \beta_{5} + 32012 \beta_{6} + 40478 \beta_{7} - 16320 \beta_{10} - 15202 \beta_{13} - 6040 \beta_{14} - 1590 \beta_{15} ) q^{89} + ( 8154876015 - 8053079604 \beta_{1} + 2321862 \beta_{2} - 1587888 \beta_{3} - 235318 \beta_{4} + 213447 \beta_{5} + 49928 \beta_{6} + 32818 \beta_{7} + 396 \beta_{8} + 5016 \beta_{9} + 776 \beta_{10} + 6780 \beta_{11} + 4706 \beta_{12} - 17722 \beta_{13} + 11373 \beta_{14} - 2806 \beta_{15} ) q^{91} + ( 23814 + 5001695487 \beta_{1} + 1944 \beta_{2} + 497907 \beta_{3} + 77517 \beta_{4} - 61722 \beta_{5} + 3888 \beta_{6} + 33048 \beta_{7} - 1944 \beta_{10} + 7776 \beta_{13} - 10206 \beta_{14} - 1215 \beta_{15} ) q^{93} + ( 13347829082 - 13347287704 \beta_{1} + 5050727 \beta_{2} + 4974876 \beta_{3} - 1243049 \beta_{4} + 1841478 \beta_{5} + 75851 \beta_{6} - 31608 \beta_{7} - 45186 \beta_{8} + 31608 \beta_{9} - 6665 \beta_{10} - 1926 \beta_{11} + 5555 \beta_{12} + 45186 \beta_{13} - 1926 \beta_{14} - 5555 \beta_{15} ) q^{95} + ( -9764625871 - 267288 \beta_{1} + 2715087 \beta_{2} - 52582 \beta_{3} - 38227 \beta_{4} - 469591 \beta_{5} + 40292 \beta_{6} + 25035 \beta_{8} - 148062 \beta_{9} + 12290 \beta_{10} - 10338 \beta_{11} + 6605 \beta_{12} ) q^{97} + ( 1644455601 + 177147 \beta_{1} - 59049 \beta_{2} - 236196 \beta_{4} + 118098 \beta_{5} + 59049 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 1944q^{3} - 2156q^{5} + 50512q^{7} - 472392q^{9} + O(q^{10}) \) \( 16q - 1944q^{3} - 2156q^{5} + 50512q^{7} - 472392q^{9} - 222796q^{11} + 2703176q^{13} + 1047816q^{15} + 5114600q^{17} + 6910556q^{19} - 18340668q^{21} - 51387712q^{23} - 191456372q^{25} + 229582512q^{27} + 118854616q^{29} + 164659160q^{31} - 54139428q^{33} + 55239344q^{35} + 75658364q^{37} - 328435884q^{39} - 1815568608q^{41} + 10754408q^{43} - 127309644q^{45} - 1034359464q^{47} + 4123496848q^{49} + 1242847800q^{51} - 665159988q^{53} - 1264543896q^{55} - 3358530216q^{57} + 1040514580q^{59} - 14391208024q^{61} + 1474099236q^{63} - 20938150200q^{65} - 33307097284q^{67} + 24974428032q^{69} + 65848902896q^{71} + 17709749204q^{73} - 46523898396q^{75} + 8594484604q^{77} - 26626784032q^{79} - 27894275208q^{81} - 210306955048q^{83} - 25867402032q^{85} - 14440835844q^{87} - 55951560072q^{89} + 66078280292q^{91} + 40012175880q^{93} + 106810047392q^{95} - 156216030712q^{97} + 26311762008q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 581500324 x^{14} - 481772282104 x^{13} + 132272376701859942 x^{12} + \)\(18\!\cdots\!08\)\( x^{11} - \)\(14\!\cdots\!08\)\( x^{10} - \)\(25\!\cdots\!56\)\( x^{9} + \)\(80\!\cdots\!79\)\( x^{8} + \)\(11\!\cdots\!68\)\( x^{7} - \)\(19\!\cdots\!68\)\( x^{6} + \)\(59\!\cdots\!08\)\( x^{5} + \)\(21\!\cdots\!06\)\( x^{4} - \)\(37\!\cdots\!04\)\( x^{3} - \)\(31\!\cdots\!28\)\( x^{2} + \)\(25\!\cdots\!24\)\( x + \)\(79\!\cdots\!77\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(68\!\cdots\!80\)\( \nu^{15} - \)\(16\!\cdots\!78\)\( \nu^{14} + \)\(39\!\cdots\!38\)\( \nu^{13} + \)\(12\!\cdots\!31\)\( \nu^{12} - \)\(87\!\cdots\!12\)\( \nu^{11} - \)\(34\!\cdots\!47\)\( \nu^{10} + \)\(92\!\cdots\!98\)\( \nu^{9} + \)\(39\!\cdots\!72\)\( \nu^{8} - \)\(45\!\cdots\!96\)\( \nu^{7} - \)\(18\!\cdots\!03\)\( \nu^{6} + \)\(92\!\cdots\!94\)\( \nu^{5} + \)\(21\!\cdots\!32\)\( \nu^{4} - \)\(95\!\cdots\!52\)\( \nu^{3} + \)\(25\!\cdots\!91\)\( \nu^{2} + \)\(83\!\cdots\!08\)\( \nu + \)\(22\!\cdots\!73\)\(\)\()/ \)\(11\!\cdots\!32\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(68\!\cdots\!80\)\( \nu^{15} - \)\(16\!\cdots\!78\)\( \nu^{14} + \)\(39\!\cdots\!38\)\( \nu^{13} + \)\(12\!\cdots\!31\)\( \nu^{12} - \)\(87\!\cdots\!12\)\( \nu^{11} - \)\(34\!\cdots\!47\)\( \nu^{10} + \)\(92\!\cdots\!98\)\( \nu^{9} + \)\(39\!\cdots\!72\)\( \nu^{8} - \)\(45\!\cdots\!96\)\( \nu^{7} - \)\(18\!\cdots\!03\)\( \nu^{6} + \)\(92\!\cdots\!94\)\( \nu^{5} + \)\(21\!\cdots\!32\)\( \nu^{4} - \)\(95\!\cdots\!52\)\( \nu^{3} + \)\(25\!\cdots\!91\)\( \nu^{2} + \)\(83\!\cdots\!40\)\( \nu + \)\(22\!\cdots\!73\)\(\)\()/ \)\(11\!\cdots\!32\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(54\!\cdots\!86\)\( \nu^{15} + \)\(13\!\cdots\!20\)\( \nu^{14} - \)\(31\!\cdots\!83\)\( \nu^{13} - \)\(10\!\cdots\!93\)\( \nu^{12} + \)\(70\!\cdots\!73\)\( \nu^{11} + \)\(27\!\cdots\!63\)\( \nu^{10} - \)\(74\!\cdots\!30\)\( \nu^{9} - \)\(31\!\cdots\!32\)\( \nu^{8} + \)\(36\!\cdots\!53\)\( \nu^{7} + \)\(14\!\cdots\!83\)\( \nu^{6} - \)\(73\!\cdots\!22\)\( \nu^{5} - \)\(17\!\cdots\!20\)\( \nu^{4} + \)\(76\!\cdots\!27\)\( \nu^{3} - \)\(20\!\cdots\!53\)\( \nu^{2} - \)\(66\!\cdots\!67\)\( \nu - \)\(18\!\cdots\!67\)\(\)\()/ \)\(38\!\cdots\!44\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(40\!\cdots\!01\)\( \nu^{15} - \)\(10\!\cdots\!79\)\( \nu^{14} + \)\(23\!\cdots\!71\)\( \nu^{13} + \)\(77\!\cdots\!66\)\( \nu^{12} - \)\(51\!\cdots\!20\)\( \nu^{11} - \)\(20\!\cdots\!55\)\( \nu^{10} + \)\(54\!\cdots\!39\)\( \nu^{9} + \)\(23\!\cdots\!79\)\( \nu^{8} - \)\(26\!\cdots\!36\)\( \nu^{7} - \)\(11\!\cdots\!35\)\( \nu^{6} + \)\(53\!\cdots\!75\)\( \nu^{5} + \)\(12\!\cdots\!91\)\( \nu^{4} - \)\(55\!\cdots\!19\)\( \nu^{3} + \)\(13\!\cdots\!16\)\( \nu^{2} + \)\(47\!\cdots\!40\)\( \nu + \)\(13\!\cdots\!49\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(17\!\cdots\!43\)\( \nu^{15} + \)\(41\!\cdots\!69\)\( \nu^{14} - \)\(99\!\cdots\!71\)\( \nu^{13} - \)\(32\!\cdots\!36\)\( \nu^{12} + \)\(22\!\cdots\!62\)\( \nu^{11} + \)\(86\!\cdots\!59\)\( \nu^{10} - \)\(23\!\cdots\!29\)\( \nu^{9} - \)\(99\!\cdots\!09\)\( \nu^{8} + \)\(11\!\cdots\!86\)\( \nu^{7} + \)\(47\!\cdots\!23\)\( \nu^{6} - \)\(23\!\cdots\!69\)\( \nu^{5} - \)\(54\!\cdots\!01\)\( \nu^{4} + \)\(24\!\cdots\!39\)\( \nu^{3} - \)\(65\!\cdots\!26\)\( \nu^{2} - \)\(21\!\cdots\!98\)\( \nu - \)\(57\!\cdots\!93\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(24\!\cdots\!46\)\( \nu^{15} + \)\(55\!\cdots\!65\)\( \nu^{14} - \)\(13\!\cdots\!90\)\( \nu^{13} - \)\(43\!\cdots\!85\)\( \nu^{12} + \)\(31\!\cdots\!96\)\( \nu^{11} + \)\(11\!\cdots\!32\)\( \nu^{10} - \)\(32\!\cdots\!00\)\( \nu^{9} - \)\(13\!\cdots\!75\)\( \nu^{8} + \)\(16\!\cdots\!60\)\( \nu^{7} + \)\(64\!\cdots\!04\)\( \nu^{6} - \)\(33\!\cdots\!52\)\( \nu^{5} - \)\(75\!\cdots\!35\)\( \nu^{4} + \)\(34\!\cdots\!10\)\( \nu^{3} - \)\(10\!\cdots\!65\)\( \nu^{2} - \)\(30\!\cdots\!54\)\( \nu - \)\(84\!\cdots\!82\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(95\!\cdots\!24\)\( \nu^{15} + \)\(22\!\cdots\!47\)\( \nu^{14} - \)\(55\!\cdots\!53\)\( \nu^{13} - \)\(17\!\cdots\!33\)\( \nu^{12} + \)\(12\!\cdots\!51\)\( \nu^{11} + \)\(46\!\cdots\!82\)\( \nu^{10} - \)\(12\!\cdots\!12\)\( \nu^{9} - \)\(53\!\cdots\!67\)\( \nu^{8} + \)\(64\!\cdots\!63\)\( \nu^{7} + \)\(25\!\cdots\!14\)\( \nu^{6} - \)\(13\!\cdots\!32\)\( \nu^{5} - \)\(29\!\cdots\!83\)\( \nu^{4} + \)\(13\!\cdots\!47\)\( \nu^{3} - \)\(39\!\cdots\!13\)\( \nu^{2} - \)\(12\!\cdots\!59\)\( \nu - \)\(33\!\cdots\!74\)\(\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(35\!\cdots\!90\)\( \nu^{15} - \)\(85\!\cdots\!31\)\( \nu^{14} + \)\(20\!\cdots\!14\)\( \nu^{13} + \)\(66\!\cdots\!14\)\( \nu^{12} - \)\(45\!\cdots\!66\)\( \nu^{11} - \)\(17\!\cdots\!97\)\( \nu^{10} + \)\(48\!\cdots\!06\)\( \nu^{9} + \)\(20\!\cdots\!11\)\( \nu^{8} - \)\(23\!\cdots\!34\)\( \nu^{7} - \)\(97\!\cdots\!29\)\( \nu^{6} + \)\(48\!\cdots\!62\)\( \nu^{5} + \)\(11\!\cdots\!39\)\( \nu^{4} - \)\(49\!\cdots\!56\)\( \nu^{3} + \)\(13\!\cdots\!94\)\( \nu^{2} + \)\(43\!\cdots\!84\)\( \nu + \)\(11\!\cdots\!53\)\(\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(26\!\cdots\!18\)\( \nu^{15} - \)\(64\!\cdots\!72\)\( \nu^{14} + \)\(15\!\cdots\!18\)\( \nu^{13} + \)\(49\!\cdots\!23\)\( \nu^{12} - \)\(34\!\cdots\!70\)\( \nu^{11} - \)\(13\!\cdots\!65\)\( \nu^{10} + \)\(35\!\cdots\!72\)\( \nu^{9} + \)\(15\!\cdots\!62\)\( \nu^{8} - \)\(17\!\cdots\!98\)\( \nu^{7} - \)\(72\!\cdots\!65\)\( \nu^{6} + \)\(35\!\cdots\!20\)\( \nu^{5} + \)\(84\!\cdots\!58\)\( \nu^{4} - \)\(37\!\cdots\!12\)\( \nu^{3} + \)\(98\!\cdots\!43\)\( \nu^{2} + \)\(32\!\cdots\!90\)\( \nu + \)\(88\!\cdots\!67\)\(\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!87\)\( \nu^{15} - \)\(34\!\cdots\!61\)\( \nu^{14} + \)\(82\!\cdots\!24\)\( \nu^{13} + \)\(26\!\cdots\!24\)\( \nu^{12} - \)\(18\!\cdots\!93\)\( \nu^{11} - \)\(71\!\cdots\!31\)\( \nu^{10} + \)\(19\!\cdots\!91\)\( \nu^{9} + \)\(82\!\cdots\!11\)\( \nu^{8} - \)\(94\!\cdots\!49\)\( \nu^{7} - \)\(39\!\cdots\!87\)\( \nu^{6} + \)\(19\!\cdots\!31\)\( \nu^{5} + \)\(45\!\cdots\!79\)\( \nu^{4} - \)\(19\!\cdots\!66\)\( \nu^{3} + \)\(51\!\cdots\!74\)\( \nu^{2} + \)\(17\!\cdots\!87\)\( \nu + \)\(47\!\cdots\!87\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(16\!\cdots\!19\)\( \nu^{15} + \)\(38\!\cdots\!86\)\( \nu^{14} - \)\(92\!\cdots\!54\)\( \nu^{13} - \)\(30\!\cdots\!59\)\( \nu^{12} + \)\(20\!\cdots\!15\)\( \nu^{11} + \)\(79\!\cdots\!05\)\( \nu^{10} - \)\(21\!\cdots\!01\)\( \nu^{9} - \)\(92\!\cdots\!06\)\( \nu^{8} + \)\(10\!\cdots\!79\)\( \nu^{7} + \)\(43\!\cdots\!05\)\( \nu^{6} - \)\(21\!\cdots\!25\)\( \nu^{5} - \)\(50\!\cdots\!34\)\( \nu^{4} + \)\(22\!\cdots\!16\)\( \nu^{3} - \)\(59\!\cdots\!09\)\( \nu^{2} - \)\(19\!\cdots\!35\)\( \nu - \)\(53\!\cdots\!31\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(20\!\cdots\!17\)\( \nu^{15} - \)\(48\!\cdots\!19\)\( \nu^{14} + \)\(11\!\cdots\!16\)\( \nu^{13} + \)\(37\!\cdots\!36\)\( \nu^{12} - \)\(25\!\cdots\!51\)\( \nu^{11} - \)\(10\!\cdots\!97\)\( \nu^{10} + \)\(27\!\cdots\!69\)\( \nu^{9} + \)\(11\!\cdots\!49\)\( \nu^{8} - \)\(13\!\cdots\!11\)\( \nu^{7} - \)\(55\!\cdots\!09\)\( \nu^{6} + \)\(27\!\cdots\!37\)\( \nu^{5} + \)\(64\!\cdots\!01\)\( \nu^{4} - \)\(28\!\cdots\!94\)\( \nu^{3} + \)\(75\!\cdots\!46\)\( \nu^{2} + \)\(24\!\cdots\!09\)\( \nu + \)\(67\!\cdots\!21\)\(\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(88\!\cdots\!88\)\( \nu^{15} - \)\(21\!\cdots\!10\)\( \nu^{14} + \)\(50\!\cdots\!55\)\( \nu^{13} + \)\(16\!\cdots\!35\)\( \nu^{12} - \)\(11\!\cdots\!63\)\( \nu^{11} - \)\(44\!\cdots\!11\)\( \nu^{10} + \)\(11\!\cdots\!60\)\( \nu^{9} + \)\(50\!\cdots\!10\)\( \nu^{8} - \)\(58\!\cdots\!35\)\( \nu^{7} - \)\(24\!\cdots\!07\)\( \nu^{6} + \)\(11\!\cdots\!76\)\( \nu^{5} + \)\(28\!\cdots\!70\)\( \nu^{4} - \)\(12\!\cdots\!95\)\( \nu^{3} + \)\(32\!\cdots\!95\)\( \nu^{2} + \)\(10\!\cdots\!57\)\( \nu + \)\(29\!\cdots\!31\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(12\!\cdots\!84\)\( \nu^{15} - \)\(30\!\cdots\!97\)\( \nu^{14} + \)\(72\!\cdots\!18\)\( \nu^{13} + \)\(23\!\cdots\!83\)\( \nu^{12} - \)\(16\!\cdots\!66\)\( \nu^{11} - \)\(62\!\cdots\!22\)\( \nu^{10} + \)\(17\!\cdots\!22\)\( \nu^{9} + \)\(72\!\cdots\!47\)\( \nu^{8} - \)\(84\!\cdots\!78\)\( \nu^{7} - \)\(34\!\cdots\!34\)\( \nu^{6} + \)\(16\!\cdots\!02\)\( \nu^{5} + \)\(39\!\cdots\!63\)\( \nu^{4} - \)\(17\!\cdots\!42\)\( \nu^{3} + \)\(47\!\cdots\!23\)\( \nu^{2} + \)\(15\!\cdots\!64\)\( \nu + \)\(41\!\cdots\!84\)\(\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(29\!\cdots\!89\)\( \nu^{15} + \)\(70\!\cdots\!86\)\( \nu^{14} - \)\(16\!\cdots\!39\)\( \nu^{13} - \)\(54\!\cdots\!39\)\( \nu^{12} + \)\(37\!\cdots\!20\)\( \nu^{11} + \)\(14\!\cdots\!65\)\( \nu^{10} - \)\(39\!\cdots\!21\)\( \nu^{9} - \)\(16\!\cdots\!56\)\( \nu^{8} + \)\(19\!\cdots\!44\)\( \nu^{7} + \)\(80\!\cdots\!45\)\( \nu^{6} - \)\(39\!\cdots\!25\)\( \nu^{5} - \)\(93\!\cdots\!24\)\( \nu^{4} + \)\(41\!\cdots\!71\)\( \nu^{3} - \)\(11\!\cdots\!09\)\( \nu^{2} - \)\(35\!\cdots\!90\)\( \nu - \)\(98\!\cdots\!41\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} - \beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{12} + 12 \beta_{11} + 23 \beta_{10} + 19 \beta_{9} + 2 \beta_{8} - 20 \beta_{6} + 367 \beta_{5} - 376 \beta_{4} - \beta_{3} + 1271 \beta_{2} + 387 \beta_{1} + 72686726\)
\(\nu^{3}\)\(=\)\(-15 \beta_{15} + 36 \beta_{14} - 6 \beta_{13} - 6215 \beta_{12} + 69038 \beta_{11} + 20558 \beta_{10} + 202806 \beta_{9} - 109157 \beta_{8} - 57 \beta_{7} + 10334 \beta_{6} - 404016 \beta_{5} - 877977 \beta_{4} - 27013 \beta_{3} + 126629123 \beta_{2} - 217809794 \beta_{1} + 90378252009\)
\(\nu^{4}\)\(=\)\(24890 \beta_{15} + 276080 \beta_{14} + 436640 \beta_{13} + 643637515 \beta_{12} + 1754420396 \beta_{11} + 4025476974 \beta_{10} + 6031651252 \beta_{9} + 273003141 \beta_{8} - 811110 \beta_{7} - 3790045230 \beta_{6} + 76374085911 \beta_{5} - 77261681693 \beta_{4} + 271159228 \beta_{3} + 394588846773 \beta_{2} - 281603733669 \beta_{1} + 9199680358352342\)
\(\nu^{5}\)\(=\)\(-3218249875 \beta_{15} + 8771411960 \beta_{14} - 1366107335 \beta_{13} - 1460566381880 \beta_{12} + 14242035566564 \beta_{11} + 15636881447401 \beta_{10} + 54584348465293 \beta_{9} - 20940546595026 \beta_{8} - 30156228770 \beta_{7} + 1294989857905 \beta_{6} + 68709487126579 \beta_{5} - 488322419943257 \beta_{4} - 14940066144606 \beta_{3} + 17628689608115725 \beta_{2} - 45721151844183187 \beta_{1} + 28259692437719391999\)
\(\nu^{6}\)\(=\)\(8773053227580 \beta_{15} + 85425901234104 \beta_{14} + 125647381166961 \beta_{13} + 72815001702102170 \beta_{12} + 278551431443853452 \beta_{11} + 667577344298859007 \beta_{10} + 1286838026296585024 \beta_{9} + 27114591278759112 \beta_{8} - 327415628188773 \beta_{7} - 604888633842308449 \beta_{6} + 13103784385572315526 \beta_{5} - 14246103601469100073 \beta_{4} + 43171276864778397 \beta_{3} + 84555601172402729953 \beta_{2} - 155334999644262246407 \beta_{1} + 1279266236680765484604515\)
\(\nu^{7}\)\(=\)\(-509735751392635565 \beta_{15} + 1949561121552480758 \beta_{14} - 190241919129525228 \beta_{13} - 245105831210677023290 \beta_{12} + 2596818447840323503846 \beta_{11} + 4364532885897707384822 \beta_{10} + 12199084410116948439077 \beta_{9} - 3570963345181939003449 \beta_{8} - 9006720546017837146 \beta_{7} - 370821133820043928573 \beta_{6} + 39325998974079021254304 \beta_{5} - 128538326524604236915807 \beta_{4} - 3397519991502422756792 \beta_{3} + 2586169072836155249014800 \beta_{2} - 8870894206794174860552757 \beta_{1} + 6070626374179100007085274947\)
\(\nu^{8}\)\(=\)\(1962885715513732437390 \beta_{15} + 20766750534199001182872 \beta_{14} + 28568469488195375635398 \beta_{13} + 7727968103595048859809185 \beta_{12} + 45911113498479044290410996 \beta_{11} + 111009646866411186310465166 \beta_{10} + 246345925449675004416294502 \beta_{9} - 119308358206630480604899 \beta_{8} - 97556652982570327840164 \beta_{7} - 93816986524936276249369832 \beta_{6} + 2169149903321084794167821353 \beta_{5} - 2552988401677547782215050719 \beta_{4} + 3529225219299045764319632 \beta_{3} + 16349355204929160473913929357 \beta_{2} - 46106992006851514336762409121 \beta_{1} + 187424379926477240886005279083259\)
\(\nu^{9}\)\(=\)\(-69560555093318898251537175 \beta_{15} + 413106606201008022612403716 \beta_{14} + 945213686808199164773109 \beta_{13} - 39769348123402665226563070130 \beta_{12} + 462376505121435333100859564444 \beta_{11} + 954781589123528763684626286917 \beta_{10} + 2481501497998719648715113126603 \beta_{9} - 585829519692711070740774021446 \beta_{8} - 2216674486229624441497888992 \beta_{7} - 176724857511084299918187288361 \beta_{6} + 10716134432867102452372887421395 \beta_{5} - 27429243162339137943008122636785 \beta_{4} - 630006685699856035981065941044 \beta_{3} + 394487405039435137424277543508214 \beta_{2} - 1667606956017170900982378070494368 \beta_{1} + 1175266726244393105607754882141554399\)
\(\nu^{10}\)\(=\)\(\)\(39\!\cdots\!40\)\( \beta_{15} + \)\(46\!\cdots\!32\)\( \beta_{14} + \)\(58\!\cdots\!13\)\( \beta_{13} + \)\(76\!\cdots\!45\)\( \beta_{12} + \)\(76\!\cdots\!56\)\( \beta_{11} + \)\(18\!\cdots\!76\)\( \beta_{10} + \)\(44\!\cdots\!97\)\( \beta_{9} - \)\(85\!\cdots\!34\)\( \beta_{8} - \)\(24\!\cdots\!09\)\( \beta_{7} - \)\(14\!\cdots\!37\)\( \beta_{6} + \)\(35\!\cdots\!43\)\( \beta_{5} - \)\(45\!\cdots\!09\)\( \beta_{4} - \)\(17\!\cdots\!26\)\( \beta_{3} + \)\(30\!\cdots\!26\)\( \beta_{2} - \)\(11\!\cdots\!22\)\( \beta_{1} + \)\(28\!\cdots\!39\)\(\)
\(\nu^{11}\)\(=\)\(-\)\(83\!\cdots\!70\)\( \beta_{15} + \)\(84\!\cdots\!30\)\( \beta_{14} + \)\(93\!\cdots\!10\)\( \beta_{13} - \)\(65\!\cdots\!75\)\( \beta_{12} + \)\(81\!\cdots\!12\)\( \beta_{11} + \)\(18\!\cdots\!68\)\( \beta_{10} + \)\(47\!\cdots\!19\)\( \beta_{9} - \)\(94\!\cdots\!88\)\( \beta_{8} - \)\(49\!\cdots\!85\)\( \beta_{7} - \)\(47\!\cdots\!95\)\( \beta_{6} + \)\(23\!\cdots\!52\)\( \beta_{5} - \)\(53\!\cdots\!66\)\( \beta_{4} - \)\(10\!\cdots\!55\)\( \beta_{3} + \)\(62\!\cdots\!07\)\( \beta_{2} - \)\(31\!\cdots\!65\)\( \beta_{1} + \)\(21\!\cdots\!50\)\(\)
\(\nu^{12}\)\(=\)\(\)\(78\!\cdots\!60\)\( \beta_{15} + \)\(98\!\cdots\!08\)\( \beta_{14} + \)\(11\!\cdots\!22\)\( \beta_{13} + \)\(64\!\cdots\!40\)\( \beta_{12} + \)\(12\!\cdots\!40\)\( \beta_{11} + \)\(31\!\cdots\!18\)\( \beta_{10} + \)\(80\!\cdots\!80\)\( \beta_{9} - \)\(27\!\cdots\!40\)\( \beta_{8} - \)\(57\!\cdots\!46\)\( \beta_{7} - \)\(22\!\cdots\!78\)\( \beta_{6} + \)\(59\!\cdots\!48\)\( \beta_{5} - \)\(80\!\cdots\!54\)\( \beta_{4} - \)\(15\!\cdots\!90\)\( \beta_{3} + \)\(54\!\cdots\!10\)\( \beta_{2} - \)\(25\!\cdots\!26\)\( \beta_{1} + \)\(44\!\cdots\!81\)\(\)
\(\nu^{13}\)\(=\)\(-\)\(83\!\cdots\!30\)\( \beta_{15} + \)\(16\!\cdots\!32\)\( \beta_{14} + \)\(36\!\cdots\!28\)\( \beta_{13} - \)\(11\!\cdots\!60\)\( \beta_{12} + \)\(14\!\cdots\!48\)\( \beta_{11} + \)\(35\!\cdots\!04\)\( \beta_{10} + \)\(89\!\cdots\!26\)\( \beta_{9} - \)\(15\!\cdots\!82\)\( \beta_{8} - \)\(10\!\cdots\!84\)\( \beta_{7} - \)\(10\!\cdots\!42\)\( \beta_{6} + \)\(47\!\cdots\!60\)\( \beta_{5} - \)\(99\!\cdots\!10\)\( \beta_{4} - \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(10\!\cdots\!31\)\( \beta_{2} - \)\(57\!\cdots\!89\)\( \beta_{1} + \)\(39\!\cdots\!78\)\(\)
\(\nu^{14}\)\(=\)\(\)\(15\!\cdots\!40\)\( \beta_{15} + \)\(20\!\cdots\!00\)\( \beta_{14} + \)\(21\!\cdots\!20\)\( \beta_{13} + \)\(33\!\cdots\!25\)\( \beta_{12} + \)\(21\!\cdots\!88\)\( \beta_{11} + \)\(54\!\cdots\!17\)\( \beta_{10} + \)\(14\!\cdots\!81\)\( \beta_{9} - \)\(67\!\cdots\!12\)\( \beta_{8} - \)\(12\!\cdots\!00\)\( \beta_{7} - \)\(36\!\cdots\!80\)\( \beta_{6} + \)\(99\!\cdots\!63\)\( \beta_{5} - \)\(14\!\cdots\!34\)\( \beta_{4} - \)\(41\!\cdots\!91\)\( \beta_{3} + \)\(98\!\cdots\!31\)\( \beta_{2} - \)\(54\!\cdots\!41\)\( \beta_{1} + \)\(72\!\cdots\!32\)\(\)
\(\nu^{15}\)\(=\)\(-\)\(50\!\cdots\!55\)\( \beta_{15} + \)\(32\!\cdots\!20\)\( \beta_{14} + \)\(10\!\cdots\!40\)\( \beta_{13} - \)\(18\!\cdots\!05\)\( \beta_{12} + \)\(25\!\cdots\!06\)\( \beta_{11} + \)\(64\!\cdots\!04\)\( \beta_{10} + \)\(16\!\cdots\!72\)\( \beta_{9} - \)\(24\!\cdots\!59\)\( \beta_{8} - \)\(21\!\cdots\!15\)\( \beta_{7} - \)\(22\!\cdots\!60\)\( \beta_{6} + \)\(90\!\cdots\!86\)\( \beta_{5} - \)\(17\!\cdots\!73\)\( \beta_{4} - \)\(27\!\cdots\!23\)\( \beta_{3} + \)\(16\!\cdots\!93\)\( \beta_{2} - \)\(10\!\cdots\!66\)\( \beta_{1} + \)\(70\!\cdots\!57\)\(\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
13166.9 0.866025i
10810.2 0.866025i
2410.13 0.866025i
2396.04 0.866025i
−438.744 0.866025i
−6748.51 0.866025i
−10168.9 0.866025i
−11427.1 0.866025i
13166.9 + 0.866025i
10810.2 + 0.866025i
2410.13 + 0.866025i
2396.04 + 0.866025i
−438.744 + 0.866025i
−6748.51 + 0.866025i
−10168.9 + 0.866025i
−11427.1 + 0.866025i
0 −121.500 + 210.444i 0 −6718.22 11636.3i 0 41445.4 16112.4i 0 −29524.5 51137.9i 0
25.2 0 −121.500 + 210.444i 0 −5539.86 9595.32i 0 −32020.8 + 30854.4i 0 −29524.5 51137.9i 0
25.3 0 −121.500 + 210.444i 0 −1339.81 2320.63i 0 −28039.7 34512.4i 0 −29524.5 51137.9i 0
25.4 0 −121.500 + 210.444i 0 −1332.77 2308.42i 0 43648.5 + 8493.30i 0 −29524.5 51137.9i 0
25.5 0 −121.500 + 210.444i 0 84.6221 + 146.570i 0 −9851.70 + 43362.1i 0 −29524.5 51137.9i 0
25.6 0 −121.500 + 210.444i 0 3239.51 + 5610.99i 0 −44011.6 6348.37i 0 −29524.5 51137.9i 0
25.7 0 −121.500 + 210.444i 0 4949.72 + 8573.16i 0 29163.1 33568.4i 0 −29524.5 51137.9i 0
25.8 0 −121.500 + 210.444i 0 5578.82 + 9662.79i 0 24922.9 + 36826.3i 0 −29524.5 51137.9i 0
37.1 0 −121.500 210.444i 0 −6718.22 + 11636.3i 0 41445.4 + 16112.4i 0 −29524.5 + 51137.9i 0
37.2 0 −121.500 210.444i 0 −5539.86 + 9595.32i 0 −32020.8 30854.4i 0 −29524.5 + 51137.9i 0
37.3 0 −121.500 210.444i 0 −1339.81 + 2320.63i 0 −28039.7 + 34512.4i 0 −29524.5 + 51137.9i 0
37.4 0 −121.500 210.444i 0 −1332.77 + 2308.42i 0 43648.5 8493.30i 0 −29524.5 + 51137.9i 0
37.5 0 −121.500 210.444i 0 84.6221 146.570i 0 −9851.70 43362.1i 0 −29524.5 + 51137.9i 0
37.6 0 −121.500 210.444i 0 3239.51 5610.99i 0 −44011.6 + 6348.37i 0 −29524.5 + 51137.9i 0
37.7 0 −121.500 210.444i 0 4949.72 8573.16i 0 29163.1 + 33568.4i 0 −29524.5 + 51137.9i 0
37.8 0 −121.500 210.444i 0 5578.82 9662.79i 0 24922.9 36826.3i 0 −29524.5 + 51137.9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.i.b 16
3.b odd 2 1 252.12.k.d 16
7.c even 3 1 inner 84.12.i.b 16
21.h odd 6 1 252.12.k.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.i.b 16 1.a even 1 1 trivial
84.12.i.b 16 7.c even 3 1 inner
252.12.k.d 16 3.b odd 2 1
252.12.k.d 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(59\!\cdots\!19\)\( T_{5}^{12} - \)\(36\!\cdots\!40\)\( T_{5}^{11} + \)\(60\!\cdots\!50\)\( T_{5}^{10} - \)\(65\!\cdots\!00\)\( T_{5}^{9} + \)\(44\!\cdots\!25\)\( T_{5}^{8} + \)\(23\!\cdots\!00\)\( T_{5}^{7} + \)\(10\!\cdots\!00\)\( T_{5}^{6} + \)\(32\!\cdots\!00\)\( T_{5}^{5} + \)\(18\!\cdots\!00\)\( T_{5}^{4} + \)\(29\!\cdots\!00\)\( T_{5}^{3} + \)\(52\!\cdots\!00\)\( T_{5}^{2} - \)\(88\!\cdots\!00\)\( T_{5} + \)\(16\!\cdots\!00\)\( \)">\(T_{5}^{16} + \cdots\) acting on \(S_{12}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 59049 + 243 T + T^{2} )^{8} \)
$5$ \( \)\(16\!\cdots\!00\)\( - \)\(88\!\cdots\!00\)\( T + \)\(52\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!00\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!25\)\( T^{8} - \)\(65\!\cdots\!00\)\( T^{9} + \)\(60\!\cdots\!50\)\( T^{10} - 36135501613992262840 T^{11} + 59972577625827719 T^{12} - 166151332768 T^{13} + 293364854 T^{14} + 2156 T^{15} + T^{16} \)
$7$ \( \)\(23\!\cdots\!01\)\( - \)\(59\!\cdots\!84\)\( T - \)\(46\!\cdots\!48\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{3} + \)\(69\!\cdots\!42\)\( T^{4} - \)\(94\!\cdots\!24\)\( T^{5} - \)\(43\!\cdots\!64\)\( T^{6} + \)\(64\!\cdots\!24\)\( T^{7} + \)\(88\!\cdots\!39\)\( T^{8} + \)\(32\!\cdots\!68\)\( T^{9} - \)\(11\!\cdots\!36\)\( T^{10} - \)\(12\!\cdots\!32\)\( T^{11} + 4524021951195991342 T^{12} + 39745685096936 T^{13} - 786017352 T^{14} - 50512 T^{15} + T^{16} \)
$11$ \( \)\(11\!\cdots\!84\)\( - \)\(19\!\cdots\!00\)\( T + \)\(41\!\cdots\!04\)\( T^{2} + \)\(87\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!08\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!84\)\( T^{6} + \)\(20\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!77\)\( T^{8} + \)\(93\!\cdots\!16\)\( T^{9} + \)\(40\!\cdots\!34\)\( T^{10} + \)\(28\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!67\)\( T^{12} + 335100284799754240 T^{13} + 1215430595858 T^{14} + 222796 T^{15} + T^{16} \)
$13$ \( ( \)\(10\!\cdots\!00\)\( + \)\(25\!\cdots\!00\)\( T - \)\(36\!\cdots\!44\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(30\!\cdots\!37\)\( T^{4} + 9797327173900188700 T^{5} - 9885557561546 T^{6} - 1351588 T^{7} + T^{8} )^{2} \)
$17$ \( \)\(39\!\cdots\!36\)\( + \)\(75\!\cdots\!40\)\( T + \)\(17\!\cdots\!80\)\( T^{2} - \)\(48\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!36\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!36\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!32\)\( T^{8} - \)\(67\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!68\)\( T^{10} - \)\(23\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} - \)\(32\!\cdots\!40\)\( T^{13} + 145405856269568 T^{14} - 5114600 T^{15} + T^{16} \)
$19$ \( \)\(11\!\cdots\!36\)\( + \)\(90\!\cdots\!76\)\( T + \)\(51\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(42\!\cdots\!08\)\( T^{4} + \)\(63\!\cdots\!20\)\( T^{5} + \)\(83\!\cdots\!28\)\( T^{6} + \)\(51\!\cdots\!36\)\( T^{7} + \)\(82\!\cdots\!45\)\( T^{8} + \)\(26\!\cdots\!80\)\( T^{9} + \)\(66\!\cdots\!38\)\( T^{10} - \)\(79\!\cdots\!88\)\( T^{11} + \)\(31\!\cdots\!39\)\( T^{12} - \)\(25\!\cdots\!64\)\( T^{13} + 678373754866794 T^{14} - 6910556 T^{15} + T^{16} \)
$23$ \( \)\(55\!\cdots\!44\)\( + \)\(82\!\cdots\!80\)\( T + \)\(11\!\cdots\!80\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!36\)\( T^{4} + \)\(94\!\cdots\!12\)\( T^{5} + \)\(62\!\cdots\!52\)\( T^{6} + \)\(13\!\cdots\!44\)\( T^{7} + \)\(66\!\cdots\!52\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(49\!\cdots\!20\)\( T^{10} + \)\(70\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!00\)\( T^{12} + \)\(22\!\cdots\!44\)\( T^{13} + 6666026666325056 T^{14} + 51387712 T^{15} + T^{16} \)
$29$ \( ( -\)\(52\!\cdots\!48\)\( + \)\(92\!\cdots\!20\)\( T - \)\(28\!\cdots\!08\)\( T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(76\!\cdots\!33\)\( T^{4} + \)\(22\!\cdots\!72\)\( T^{5} - 51378044248844398 T^{6} - 59427308 T^{7} + T^{8} )^{2} \)
$31$ \( \)\(15\!\cdots\!69\)\( + \)\(10\!\cdots\!04\)\( T + \)\(51\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(20\!\cdots\!90\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!28\)\( T^{6} + \)\(89\!\cdots\!48\)\( T^{7} + \)\(85\!\cdots\!31\)\( T^{8} + \)\(26\!\cdots\!44\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} + \)\(24\!\cdots\!68\)\( T^{11} + \)\(68\!\cdots\!34\)\( T^{12} - \)\(21\!\cdots\!84\)\( T^{13} + 120948747053994744 T^{14} - 164659160 T^{15} + T^{16} \)
$37$ \( \)\(20\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( T + \)\(17\!\cdots\!00\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!76\)\( T^{4} + \)\(18\!\cdots\!44\)\( T^{5} + \)\(29\!\cdots\!96\)\( T^{6} - \)\(34\!\cdots\!20\)\( T^{7} + \)\(62\!\cdots\!45\)\( T^{8} - \)\(49\!\cdots\!96\)\( T^{9} + \)\(76\!\cdots\!82\)\( T^{10} - \)\(96\!\cdots\!64\)\( T^{11} + \)\(66\!\cdots\!55\)\( T^{12} - \)\(66\!\cdots\!40\)\( T^{13} + 317539801427310666 T^{14} - 75658364 T^{15} + T^{16} \)
$41$ \( ( -\)\(46\!\cdots\!68\)\( + \)\(43\!\cdots\!16\)\( T + \)\(52\!\cdots\!96\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(27\!\cdots\!92\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{5} - 2126262369348540072 T^{6} + 907784304 T^{7} + T^{8} )^{2} \)
$43$ \( ( \)\(45\!\cdots\!92\)\( + \)\(31\!\cdots\!72\)\( T - \)\(10\!\cdots\!48\)\( T^{2} - \)\(62\!\cdots\!92\)\( T^{3} + \)\(35\!\cdots\!41\)\( T^{4} + \)\(55\!\cdots\!20\)\( T^{5} - 3524898654205841186 T^{6} - 5377204 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(94\!\cdots\!76\)\( + \)\(20\!\cdots\!12\)\( T + \)\(32\!\cdots\!32\)\( T^{2} + \)\(25\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} + \)\(47\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} - \)\(62\!\cdots\!56\)\( T^{7} + \)\(61\!\cdots\!60\)\( T^{8} - \)\(30\!\cdots\!12\)\( T^{9} + \)\(10\!\cdots\!68\)\( T^{10} - \)\(49\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!92\)\( T^{12} - \)\(34\!\cdots\!72\)\( T^{13} + 4910740742278397664 T^{14} + 1034359464 T^{15} + T^{16} \)
$53$ \( \)\(42\!\cdots\!44\)\( - \)\(70\!\cdots\!12\)\( T + \)\(21\!\cdots\!48\)\( T^{2} + \)\(13\!\cdots\!28\)\( T^{3} + \)\(24\!\cdots\!56\)\( T^{4} + \)\(57\!\cdots\!08\)\( T^{5} + \)\(71\!\cdots\!40\)\( T^{6} + \)\(84\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!85\)\( T^{8} + \)\(71\!\cdots\!32\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} + \)\(59\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!99\)\( T^{12} + \)\(21\!\cdots\!96\)\( T^{13} + 40070196965083418118 T^{14} + 665159988 T^{15} + T^{16} \)
$59$ \( \)\(16\!\cdots\!96\)\( - \)\(14\!\cdots\!80\)\( T + \)\(10\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!88\)\( T^{4} - \)\(48\!\cdots\!40\)\( T^{5} + \)\(80\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!17\)\( T^{8} - \)\(55\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!66\)\( T^{10} - \)\(14\!\cdots\!00\)\( T^{11} + \)\(71\!\cdots\!47\)\( T^{12} - \)\(24\!\cdots\!40\)\( T^{13} + 93740801099549624642 T^{14} - 1040514580 T^{15} + T^{16} \)
$61$ \( \)\(20\!\cdots\!76\)\( + \)\(12\!\cdots\!32\)\( T + \)\(59\!\cdots\!88\)\( T^{2} + \)\(70\!\cdots\!04\)\( T^{3} + \)\(64\!\cdots\!20\)\( T^{4} + \)\(28\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(36\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!32\)\( T^{8} + \)\(30\!\cdots\!48\)\( T^{9} + \)\(73\!\cdots\!96\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(16\!\cdots\!56\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(23\!\cdots\!84\)\( T^{14} + 14391208024 T^{15} + T^{16} \)
$67$ \( \)\(39\!\cdots\!64\)\( - \)\(19\!\cdots\!68\)\( T + \)\(16\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!56\)\( T^{4} + \)\(41\!\cdots\!52\)\( T^{5} + \)\(21\!\cdots\!36\)\( T^{6} + \)\(48\!\cdots\!72\)\( T^{7} + \)\(90\!\cdots\!13\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(98\!\cdots\!70\)\( T^{10} + \)\(63\!\cdots\!56\)\( T^{11} + \)\(41\!\cdots\!67\)\( T^{12} + \)\(19\!\cdots\!68\)\( T^{13} + \)\(10\!\cdots\!42\)\( T^{14} + 33307097284 T^{15} + T^{16} \)
$71$ \( ( -\)\(14\!\cdots\!24\)\( + \)\(48\!\cdots\!76\)\( T + \)\(47\!\cdots\!32\)\( T^{2} - \)\(42\!\cdots\!08\)\( T^{3} - \)\(16\!\cdots\!40\)\( T^{4} + \)\(28\!\cdots\!56\)\( T^{5} - \)\(51\!\cdots\!12\)\( T^{6} - 32924451448 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(51\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T + \)\(20\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!04\)\( T^{4} - \)\(10\!\cdots\!08\)\( T^{5} + \)\(41\!\cdots\!44\)\( T^{6} - \)\(13\!\cdots\!64\)\( T^{7} + \)\(21\!\cdots\!77\)\( T^{8} - \)\(68\!\cdots\!72\)\( T^{9} + \)\(80\!\cdots\!26\)\( T^{10} - \)\(23\!\cdots\!76\)\( T^{11} + \)\(15\!\cdots\!43\)\( T^{12} - \)\(25\!\cdots\!08\)\( T^{13} + \)\(15\!\cdots\!74\)\( T^{14} - 17709749204 T^{15} + T^{16} \)
$79$ \( \)\(34\!\cdots\!09\)\( - \)\(53\!\cdots\!76\)\( T + \)\(17\!\cdots\!24\)\( T^{2} + \)\(45\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!88\)\( T^{5} + \)\(12\!\cdots\!72\)\( T^{6} + \)\(39\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!23\)\( T^{8} + \)\(13\!\cdots\!24\)\( T^{9} + \)\(51\!\cdots\!08\)\( T^{10} + \)\(49\!\cdots\!36\)\( T^{11} + \)\(53\!\cdots\!06\)\( T^{12} + \)\(35\!\cdots\!08\)\( T^{13} + \)\(31\!\cdots\!76\)\( T^{14} + 26626784032 T^{15} + T^{16} \)
$83$ \( ( \)\(15\!\cdots\!36\)\( - \)\(70\!\cdots\!32\)\( T + \)\(29\!\cdots\!84\)\( T^{2} + \)\(23\!\cdots\!52\)\( T^{3} - \)\(53\!\cdots\!63\)\( T^{4} - \)\(32\!\cdots\!36\)\( T^{5} - \)\(52\!\cdots\!14\)\( T^{6} + 105153477524 T^{7} + T^{8} )^{2} \)
$89$ \( \)\(10\!\cdots\!76\)\( + \)\(51\!\cdots\!96\)\( T + \)\(11\!\cdots\!08\)\( T^{2} + \)\(86\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!76\)\( T^{4} + \)\(60\!\cdots\!08\)\( T^{5} + \)\(34\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(39\!\cdots\!00\)\( T^{8} + \)\(89\!\cdots\!28\)\( T^{9} + \)\(27\!\cdots\!84\)\( T^{10} + \)\(45\!\cdots\!48\)\( T^{11} + \)\(96\!\cdots\!96\)\( T^{12} + \)\(55\!\cdots\!80\)\( T^{13} + \)\(11\!\cdots\!68\)\( T^{14} + 55951560072 T^{15} + T^{16} \)
$97$ \( ( \)\(60\!\cdots\!52\)\( - \)\(79\!\cdots\!92\)\( T - \)\(29\!\cdots\!04\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!01\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{5} - \)\(42\!\cdots\!54\)\( T^{6} + 78108015356 T^{7} + T^{8} )^{2} \)
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