Properties

Label 14.12.c.b
Level $14$
Weight $12$
Character orbit 14.c
Analytic conductor $10.757$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7568045278\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + 267736482219682 x^{2} - 4732823471679688 x + 3039499286293024324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32 \beta_{2} q^{2} + ( -67 + \beta_{1} + 67 \beta_{2} ) q^{3} + ( -1024 + 1024 \beta_{2} ) q^{4} + ( 952 \beta_{2} - \beta_{5} ) q^{5} + ( -2144 + 32 \beta_{1} - 32 \beta_{3} ) q^{6} + ( 12827 + 8 \beta_{1} + 1939 \beta_{2} + 32 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{7} -32768 q^{8} + ( -2 - 125998 \beta_{2} - 77 \beta_{3} - \beta_{4} + 25 \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{9} +O(q^{10})\) \( q + 32 \beta_{2} q^{2} + ( -67 + \beta_{1} + 67 \beta_{2} ) q^{3} + ( -1024 + 1024 \beta_{2} ) q^{4} + ( 952 \beta_{2} - \beta_{5} ) q^{5} + ( -2144 + 32 \beta_{1} - 32 \beta_{3} ) q^{6} + ( 12827 + 8 \beta_{1} + 1939 \beta_{2} + 32 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{7} -32768 q^{8} + ( -2 - 125998 \beta_{2} - 77 \beta_{3} - \beta_{4} + 25 \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{9} + ( -30464 + 30464 \beta_{2} + 32 \beta_{4} ) q^{10} + ( 229938 + 189 \beta_{1} - 229942 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{11} + ( -68608 \beta_{2} - 1024 \beta_{3} ) q^{12} + ( 124940 - 519 \beta_{1} - 14 \beta_{2} + 519 \beta_{3} - 123 \beta_{4} - 130 \beta_{5} - 35 \beta_{6} - 28 \beta_{7} ) q^{13} + ( -62048 + 1280 \beta_{1} + 472512 \beta_{2} - 256 \beta_{3} + 64 \beta_{4} + 32 \beta_{5} - 32 \beta_{6} - 32 \beta_{7} ) q^{14} + ( -65673 + 6230 \beta_{1} - 10 \beta_{2} - 6230 \beta_{3} + 379 \beta_{4} + 374 \beta_{5} - 25 \beta_{6} - 20 \beta_{7} ) q^{15} -1048576 \beta_{2} q^{16} + ( 336257 - 5757 \beta_{1} - 336299 \beta_{2} + 226 \beta_{4} - 21 \beta_{5} - 84 \beta_{6} + 21 \beta_{7} ) q^{17} + ( 4031936 - 2464 \beta_{1} - 4032000 \beta_{2} - 832 \beta_{4} - 32 \beta_{5} - 128 \beta_{6} + 32 \beta_{7} ) q^{18} + ( -56 - 5388298 \beta_{2} - 873 \beta_{3} - 28 \beta_{4} - 833 \beta_{5} + 28 \beta_{6} + 140 \beta_{7} ) q^{19} + ( -974848 + 1024 \beta_{4} + 1024 \beta_{5} ) q^{20} + ( -10550058 + 22454 \beta_{1} - 1521374 \beta_{2} + 741 \beta_{3} + 916 \beta_{4} + 2261 \beta_{5} + 143 \beta_{6} + 313 \beta_{7} ) q^{21} + ( 7358144 + 6048 \beta_{1} - 128 \beta_{2} - 6048 \beta_{3} - 32 \beta_{4} - 96 \beta_{5} - 320 \beta_{6} - 256 \beta_{7} ) q^{22} + ( 22 + 18118944 \beta_{2} - 17136 \beta_{3} + 11 \beta_{4} - 2998 \beta_{5} - 11 \beta_{6} - 55 \beta_{7} ) q^{23} + ( 2195456 - 32768 \beta_{1} - 2195456 \beta_{2} ) q^{24} + ( -19324176 + 72030 \beta_{1} + 19324596 \beta_{2} - 2632 \beta_{4} + 210 \beta_{5} + 840 \beta_{6} - 210 \beta_{7} ) q^{25} + ( 448 + 3997632 \beta_{2} + 16608 \beta_{3} + 224 \beta_{4} - 3936 \beta_{5} - 224 \beta_{6} - 1120 \beta_{7} ) q^{26} + ( 19661442 - 151843 \beta_{1} + 476 \beta_{2} + 151843 \beta_{3} - 8765 \beta_{4} - 8527 \beta_{5} + 1190 \beta_{6} + 952 \beta_{7} ) q^{27} + ( -15120384 + 32768 \beta_{1} + 13134848 \beta_{2} - 40960 \beta_{3} + 1024 \beta_{4} + 2048 \beta_{5} - 1024 \beta_{7} ) q^{28} + ( 26785510 - 176631 \beta_{1} + 1226 \beta_{2} + 176631 \beta_{3} + 4811 \beta_{4} + 5424 \beta_{5} + 3065 \beta_{6} + 2452 \beta_{7} ) q^{29} + ( 320 - 2101856 \beta_{2} - 199360 \beta_{3} + 160 \beta_{4} + 12128 \beta_{5} - 160 \beta_{6} - 800 \beta_{7} ) q^{30} + ( -48578741 - 19080 \beta_{1} + 48580071 \beta_{2} - 2076 \beta_{4} + 665 \beta_{5} + 2660 \beta_{6} - 665 \beta_{7} ) q^{31} + ( 33554432 - 33554432 \beta_{2} ) q^{32} + ( -1148 - 40948411 \beta_{2} + 373870 \beta_{3} - 574 \beta_{4} + 9616 \beta_{5} + 574 \beta_{6} + 2870 \beta_{7} ) q^{33} + ( 10761568 - 184224 \beta_{1} - 1344 \beta_{2} + 184224 \beta_{3} + 7904 \beta_{4} + 7232 \beta_{5} - 3360 \beta_{6} - 2688 \beta_{7} ) q^{34} + ( -153091869 + 485730 \beta_{1} + 60383792 \beta_{2} - 255465 \beta_{3} - 8253 \beta_{4} - 33824 \beta_{5} + 630 \beta_{6} + 420 \beta_{7} ) q^{35} + ( 129024000 - 78848 \beta_{1} - 2048 \beta_{2} + 78848 \beta_{3} - 25600 \beta_{4} - 26624 \beta_{5} - 5120 \beta_{6} - 4096 \beta_{7} ) q^{36} + ( -2238 + 42323890 \beta_{2} - 1116339 \beta_{3} - 1119 \beta_{4} - 14340 \beta_{5} + 1119 \beta_{6} + 5595 \beta_{7} ) q^{37} + ( 172425536 - 27936 \beta_{1} - 172427328 \beta_{2} + 25760 \beta_{4} - 896 \beta_{5} - 3584 \beta_{6} + 896 \beta_{7} ) q^{38} + ( -163920481 + 1271214 \beta_{1} + 163917629 \beta_{2} + 39721 \beta_{4} - 1426 \beta_{5} - 5704 \beta_{6} + 1426 \beta_{7} ) q^{39} + ( -31195136 \beta_{2} + 32768 \beta_{5} ) q^{40} + ( -413703884 - 615729 \beta_{1} + 574 \beta_{2} + 615729 \beta_{3} + 59507 \beta_{4} + 59794 \beta_{5} + 1435 \beta_{6} + 1148 \beta_{7} ) q^{41} + ( 48683968 + 742240 \beta_{1} - 386285824 \beta_{2} - 718528 \beta_{3} - 43040 \beta_{4} + 29312 \beta_{5} - 5440 \beta_{6} + 4576 \beta_{7} ) q^{42} + ( 106296798 - 26082 \beta_{1} - 4796 \beta_{2} + 26082 \beta_{3} + 16112 \beta_{4} + 13714 \beta_{5} - 11990 \beta_{6} - 9592 \beta_{7} ) q^{43} + ( 4096 + 235456512 \beta_{2} - 193536 \beta_{3} + 2048 \beta_{4} - 1024 \beta_{5} - 2048 \beta_{6} - 10240 \beta_{7} ) q^{44} + ( 1696878848 - 1776985 \beta_{1} - 1696889418 \beta_{2} - 140704 \beta_{4} - 5285 \beta_{5} - 21140 \beta_{6} + 5285 \beta_{7} ) q^{45} + ( -579806208 - 548352 \beta_{1} + 579806912 \beta_{2} + 96288 \beta_{4} + 352 \beta_{5} + 1408 \beta_{6} - 352 \beta_{7} ) q^{46} + ( 6202 + 555230677 \beta_{2} - 1463628 \beta_{3} + 3101 \beta_{4} + 43537 \beta_{5} - 3101 \beta_{6} - 15505 \beta_{7} ) q^{47} + ( 70254592 - 1048576 \beta_{1} + 1048576 \beta_{3} ) q^{48} + ( 4076533 + 849703 \beta_{1} - 1065806828 \beta_{2} + 1554613 \beta_{3} + 136993 \beta_{4} - 17582 \beta_{5} + 1023 \beta_{6} - 7676 \beta_{7} ) q^{49} + ( -618387072 + 2304960 \beta_{1} + 13440 \beta_{2} - 2304960 \beta_{3} - 90944 \beta_{4} - 84224 \beta_{5} + 33600 \beta_{6} + 26880 \beta_{7} ) q^{50} + ( 6004 + 1742365952 \beta_{2} + 565873 \beta_{3} + 3002 \beta_{4} + 5023 \beta_{5} - 3002 \beta_{6} - 15010 \beta_{7} ) q^{51} + ( -127924224 + 531456 \beta_{1} + 127938560 \beta_{2} + 133120 \beta_{4} + 7168 \beta_{5} + 28672 \beta_{6} - 7168 \beta_{7} ) q^{52} + ( -1796092730 - 538755 \beta_{1} + 1796099716 \beta_{2} - 285985 \beta_{4} + 3493 \beta_{5} + 13972 \beta_{6} - 3493 \beta_{7} ) q^{53} + ( -15232 + 629181376 \beta_{2} + 4858976 \beta_{3} - 7616 \beta_{4} - 280480 \beta_{5} + 7616 \beta_{6} + 38080 \beta_{7} ) q^{54} + ( 77057260 + 2112390 \beta_{1} - 3990 \beta_{2} - 2112390 \beta_{3} - 327670 \beta_{4} - 329665 \beta_{5} - 9975 \beta_{6} - 7980 \beta_{7} ) q^{55} + ( -420315136 - 262144 \beta_{1} - 63537152 \beta_{2} - 1048576 \beta_{3} - 32768 \beta_{4} + 32768 \beta_{5} + 32768 \beta_{6} ) q^{56} + ( 621417815 - 2968070 \beta_{1} + 564 \beta_{2} + 2968070 \beta_{3} + 367662 \beta_{4} + 367944 \beta_{5} + 1410 \beta_{6} + 1128 \beta_{7} ) q^{57} + ( -39232 + 857175552 \beta_{2} + 5652192 \beta_{3} - 19616 \beta_{4} + 153952 \beta_{5} + 19616 \beta_{6} + 98080 \beta_{7} ) q^{58} + ( 1748082751 + 2660199 \beta_{1} - 1748032771 \beta_{2} + 399086 \beta_{4} + 24990 \beta_{5} + 99960 \beta_{6} - 24990 \beta_{7} ) q^{59} + ( 67259392 - 6379520 \beta_{1} - 67249152 \beta_{2} - 382976 \beta_{4} + 5120 \beta_{5} + 20480 \beta_{6} - 5120 \beta_{7} ) q^{60} + ( 9310 - 1624036704 \beta_{2} - 9818493 \beta_{3} + 4655 \beta_{4} - 116604 \beta_{5} - 4655 \beta_{6} - 23275 \beta_{7} ) q^{61} + ( -1554562272 - 610560 \beta_{1} + 42560 \beta_{2} + 610560 \beta_{3} - 87712 \beta_{4} - 66432 \beta_{5} + 106400 \beta_{6} + 85120 \beta_{7} ) q^{62} + ( 251151194 - 21478638 \beta_{1} - 4799754125 \beta_{2} + 9943448 \beta_{3} - 315806 \beta_{4} + 421613 \beta_{5} - 40288 \beta_{6} - 8858 \beta_{7} ) q^{63} + 1073741824 q^{64} + ( 52350 + 7660600992 \beta_{2} + 8568735 \beta_{3} + 26175 \beta_{4} - 381921 \beta_{5} - 26175 \beta_{6} - 130875 \beta_{7} ) q^{65} + ( 1310349152 + 11963840 \beta_{1} - 1310385888 \beta_{2} - 326080 \beta_{4} - 18368 \beta_{5} - 73472 \beta_{6} + 18368 \beta_{7} ) q^{66} + ( -4668058747 + 5621511 \beta_{1} + 4668054791 \beta_{2} + 259148 \beta_{4} - 1978 \beta_{5} - 7912 \beta_{6} + 1978 \beta_{7} ) q^{67} + ( 43008 + 344327168 \beta_{2} + 5895168 \beta_{3} + 21504 \beta_{4} + 252928 \beta_{5} - 21504 \beta_{6} - 107520 \beta_{7} ) q^{68} + ( 3897681144 + 34544020 \beta_{1} - 68432 \beta_{2} - 34544020 \beta_{3} + 678857 \beta_{4} + 644641 \beta_{5} - 171080 \beta_{6} - 136864 \beta_{7} ) q^{69} + ( -1932281344 + 7368480 \beta_{1} - 2966658464 \beta_{2} - 15543360 \beta_{3} + 818272 \beta_{4} - 264096 \beta_{5} + 6720 \beta_{6} + 20160 \beta_{7} ) q^{70} + ( 2907905532 - 9710526 \beta_{1} - 23548 \beta_{2} + 9710526 \beta_{3} - 689262 \beta_{4} - 701036 \beta_{5} - 58870 \beta_{6} - 47096 \beta_{7} ) q^{71} + ( 65536 + 4128702464 \beta_{2} + 2523136 \beta_{3} + 32768 \beta_{4} - 819200 \beta_{5} - 32768 \beta_{6} - 163840 \beta_{7} ) q^{72} + ( 931178385 + 2964366 \beta_{1} - 931378221 \beta_{2} - 759764 \beta_{4} - 99918 \beta_{5} - 399672 \beta_{6} + 99918 \beta_{7} ) q^{73} + ( -1354364480 - 35722848 \beta_{1} + 1354292864 \beta_{2} + 423072 \beta_{4} - 35808 \beta_{5} - 143232 \beta_{6} + 35808 \beta_{7} ) q^{74} + ( -92680 - 22810946454 \beta_{2} - 19801520 \beta_{3} - 46340 \beta_{4} + 170282 \beta_{5} + 46340 \beta_{6} + 231700 \beta_{7} ) q^{75} + ( 5517674496 - 893952 \beta_{1} - 57344 \beta_{2} + 893952 \beta_{3} + 852992 \beta_{4} + 824320 \beta_{5} - 143360 \beta_{6} - 114688 \beta_{7} ) q^{76} + ( 3724465934 - 5607945 \beta_{1} - 14771926134 \beta_{2} + 21313509 \beta_{3} + 44758 \beta_{4} - 355628 \beta_{5} - 3521 \beta_{6} + 105224 \beta_{7} ) q^{77} + ( -5245364128 + 40678848 \beta_{1} - 91264 \beta_{2} - 40678848 \beta_{3} + 1316704 \beta_{4} + 1271072 \beta_{5} - 228160 \beta_{6} - 182528 \beta_{7} ) q^{78} + ( -259574 + 3061686078 \beta_{2} + 12144258 \beta_{3} - 129787 \beta_{4} + 2676870 \beta_{5} + 129787 \beta_{6} + 648935 \beta_{7} ) q^{79} + ( 998244352 - 998244352 \beta_{2} - 1048576 \beta_{4} ) q^{80} + ( -24351959011 + 37076963 \beta_{1} + 24351911193 \beta_{2} + 3292556 \beta_{4} - 23909 \beta_{5} - 95636 \beta_{6} + 23909 \beta_{7} ) q^{81} + ( -18368 - 13238505920 \beta_{2} + 19703328 \beta_{3} - 9184 \beta_{4} + 1904224 \beta_{5} + 9184 \beta_{6} + 45920 \beta_{7} ) q^{82} + ( 39607246566 - 29504256 \beta_{1} + 215768 \beta_{2} + 29504256 \beta_{3} - 300714 \beta_{4} - 192830 \beta_{5} + 539420 \beta_{6} + 431536 \beta_{7} ) q^{83} + ( 12361146368 + 758784 \beta_{1} - 10803259392 \beta_{2} - 23751680 \beta_{3} - 2315264 \beta_{4} - 1377280 \beta_{5} - 320512 \beta_{6} - 174080 \beta_{7} ) q^{84} + ( -18470571038 - 7501725 \beta_{1} + 143670 \beta_{2} + 7501725 \beta_{3} - 4754746 \beta_{4} - 4682911 \beta_{5} + 359175 \beta_{6} + 287340 \beta_{7} ) q^{85} + ( 153472 + 3401344064 \beta_{2} + 834624 \beta_{3} + 76736 \beta_{4} + 515584 \beta_{5} - 76736 \beta_{6} - 383680 \beta_{7} ) q^{86} + ( -54509950877 - 39916148 \beta_{1} + 54510585189 \beta_{2} + 4524817 \beta_{4} + 317156 \beta_{5} + 1268624 \beta_{6} - 317156 \beta_{7} ) q^{87} + ( -7534608384 - 6193152 \beta_{1} + 7534739456 \beta_{2} + 98304 \beta_{4} + 65536 \beta_{5} + 262144 \beta_{6} - 65536 \beta_{7} ) q^{88} + ( 157780 - 17808821033 \beta_{2} - 15044574 \beta_{3} + 78890 \beta_{4} - 517244 \beta_{5} - 78890 \beta_{6} - 394450 \beta_{7} ) q^{89} + ( 54300461376 - 56863520 \beta_{1} - 338240 \beta_{2} + 56863520 \beta_{3} - 4333408 \beta_{4} - 4502528 \beta_{5} - 845600 \beta_{6} - 676480 \beta_{7} ) q^{90} + ( 27632835369 + 51134128 \beta_{1} - 7813500294 \beta_{2} + 45724558 \beta_{3} - 1925559 \beta_{4} - 3107705 \beta_{5} + 742982 \beta_{6} - 206230 \beta_{7} ) q^{91} + ( -18553821184 - 17547264 \beta_{1} + 22528 \beta_{2} + 17547264 \beta_{3} + 3069952 \beta_{4} + 3081216 \beta_{5} + 56320 \beta_{6} + 45056 \beta_{7} ) q^{92} + ( 263450 + 2418918182 \beta_{2} - 80651823 \beta_{3} + 131725 \beta_{4} - 3268114 \beta_{5} - 131725 \beta_{6} - 658625 \beta_{7} ) q^{93} + ( -17767381664 - 46836096 \beta_{1} + 17767580128 \beta_{2} - 1293952 \beta_{4} + 99232 \beta_{5} + 396928 \beta_{6} - 99232 \beta_{7} ) q^{94} + ( -53806885066 + 68659500 \beta_{1} + 53807220136 \beta_{2} - 10900457 \beta_{4} + 167535 \beta_{5} + 670140 \beta_{6} - 167535 \beta_{7} ) q^{95} + ( 2248146944 \beta_{2} + 33554432 \beta_{3} ) q^{96} + ( 86074662586 - 86137617 \beta_{1} + 476518 \beta_{2} + 86137617 \beta_{3} + 4608085 \beta_{4} + 4846344 \beta_{5} + 1191295 \beta_{6} + 953036 \beta_{7} ) q^{97} + ( 34105818496 + 76938112 \beta_{1} - 33975369440 \beta_{2} - 27190496 \beta_{3} + 4946400 \beta_{4} + 4383776 \beta_{5} + 278368 \beta_{6} + 32736 \beta_{7} ) q^{98} + ( -68138460309 - 94818304 \beta_{1} + 353432 \beta_{2} + 94818304 \beta_{3} + 7037629 \beta_{4} + 7214345 \beta_{5} + 883580 \beta_{6} + 706864 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 128q^{2} - 266q^{3} - 4096q^{4} + 3808q^{5} - 17024q^{6} + 110328q^{7} - 262144q^{8} - 503848q^{9} + O(q^{10}) \) \( 8q + 128q^{2} - 266q^{3} - 4096q^{4} + 3808q^{5} - 17024q^{6} + 110328q^{7} - 262144q^{8} - 503848q^{9} - 121856q^{10} + 920150q^{11} - 272384q^{12} + 997472q^{13} + 1396800q^{14} - 500444q^{15} - 4194304q^{16} + 1333724q^{17} + 16123136q^{18} - 21551726q^{19} - 7798784q^{20} - 90442480q^{21} + 58889600q^{22} + 72510158q^{23} + 8716288q^{24} - 77154744q^{25} + 15959552q^{26} + 156683212q^{27} - 68278272q^{28} + 213575104q^{29} - 8007104q^{30} - 194359774q^{31} + 134217728q^{32} - 164547124q^{33} + 85358336q^{34} - 981719074q^{35} + 1031880704q^{36} + 171517048q^{37} + 689655232q^{38} - 653125236q^{39} - 124780544q^{40} - 3312095136q^{41} - 1152719104q^{42} + 850279648q^{43} + 942233600q^{44} + 6784014272q^{45} - 2320325056q^{46} + 2223880974q^{47} + 557842432q^{48} - 4232044312q^{49} - 4937903616q^{50} + 6968362082q^{51} - 510705664q^{52} - 7185483360q^{53} + 2506931392q^{54} + 624915620q^{55} - 3615227904q^{56} + 4959469112q^{57} + 3417201664q^{58} + 6997401502q^{59} + 256227328q^{60} - 6476463280q^{61} - 12439025536q^{62} - 17252507684q^{63} + 8589934592q^{64} + 30625528248q^{65} + 5265507968q^{66} - 18660972186q^{67} + 1365733376q^{68} + 31319762096q^{69} - 27279047488q^{70} + 23224449248q^{71} + 16510091264q^{72} + 3731641452q^{73} - 5488545536q^{74} - 91204646176q^{75} + 44137934848q^{76} - 29345595440q^{77} - 41800015104q^{78} + 12221157926q^{79} + 3992977408q^{80} - 97333443028q^{81} - 52993522176q^{82} + 316739523968q^{83} + 55726088192q^{84} - 147794862544q^{85} + 13604474368q^{86} - 218122807364q^{87} - 30151475200q^{88} - 71204406084q^{89} + 434176913408q^{90} + 189816116528q^{91} - 148500803584q^{92} + 9838293624q^{93} - 71164191168q^{94} - 215091896614q^{95} + 8925478912q^{96} + 688251797184q^{97} + 137152279424q^{98} - 545487662552q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + 267736482219682 x^{2} - 4732823471679688 x + 3039499286293024324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-35869944812371155025 \nu^{7} - 21209640790858534949472 \nu^{6} - 4918634136858061026739296 \nu^{5} - 3125365729265160241831479801 \nu^{4} - 798481013014520440296578811936 \nu^{3} - 450216271942383165233821223326560 \nu^{2} - 8968490297199261493238481427512776 \nu - 261959263537250221493613159712766844\)\()/ \)\(52\!\cdots\!04\)\( \)
\(\beta_{3}\)\(=\)\((\)\(170605955991169 \nu^{7} - 3519857957118208 \nu^{6} + 23490439374665485202 \nu^{5} + 490608600836399435497 \nu^{4} + 3497549540077942970448320 \nu^{3} - 5100811115489544633439898 \nu^{2} + 45777575111378069135800018696 \nu - 875507290830842081809384533700\)\()/ \)\(21\!\cdots\!54\)\( \)
\(\beta_{4}\)\(=\)\((\)\(763941504366608166441475775 \nu^{7} + 278280356085554103425660348152 \nu^{6} + 64534768530797825143586472412536 \nu^{5} + 51533380896120194140529312736306615 \nu^{4} + 10476442426361352652437663684948916776 \nu^{3} + 5907046974866076547412054979345174267960 \nu^{2} - 501030786210071012943051817184728141130744 \nu + 72568010064580969948258637038469208235937168\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(2039661842683272150115872837 \nu^{7} - 400401774324977752442646550064 \nu^{6} + 321488533363149087523466821182160 \nu^{5} - 43471107176511096153811631885610259 \nu^{4} + 40784175391277837937658617892269179920 \nu^{3} - 5990869670462917018654298980099656828064 \nu^{2} + 558002197074193522845579893423897107957768 \nu - 17748388865676875203818500100565898161683860\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-4394527928097847786955155445 \nu^{7} + 908160239736686654710771347952 \nu^{6} - 415854448090089161117961525328564 \nu^{5} + 116188754641682762680813509581326435 \nu^{4} - 50766387194063743827193878880910428144 \nu^{3} + 17704791230277567491200670294540088987220 \nu^{2} + 3571728299655536862334961072832998258100256 \nu + 270471975097904231977109463827092632346198848\)\()/ \)\(11\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-102821913213394545663127073281 \nu^{7} - 5362809594472693060136045373848 \nu^{6} - 15884190517755518156474531764291080 \nu^{5} - 1336549739362583494409031335145491913 \nu^{4} - 2189654265267293270918372381476692999720 \nu^{3} - 140312460486311194111287830851831301826168 \nu^{2} - 35027331419216752639380198650985741298989584 \nu + 150490056442413630132422354693154261872888540\)\()/ \)\(77\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{7} + \beta_{6} + 25 \beta_{5} - \beta_{4} + 57 \beta_{3} - 298656 \beta_{2} - 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(1756 \beta_{7} + 2195 \beta_{6} - 3301 \beta_{5} - 3740 \beta_{4} + 504127 \beta_{3} + 878 \beta_{2} - 504127 \beta_{1} - 16931881\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-299125 \beta_{7} + 1196500 \beta_{6} + 299125 \beta_{5} + 7762535 \beta_{4} + 75274567413 \beta_{2} - 28387965 \beta_{1} - 75273969163\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-178225175 \beta_{7} - 35645035 \beta_{6} + 207672605 \beta_{5} + 35645035 \beta_{4} - 34326018287 \beta_{3} + 2112657039800 \beta_{2} + 71290070\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-169793717028 \beta_{7} - 212242146285 \beta_{6} - 1059317406417 \beta_{5} - 1016868977160 \beta_{4} - 5477991972489 \beta_{3} - 84896858514 \beta_{2} + 5477991972489 \beta_{1} + 10250549623555691\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(10735791588669 \beta_{7} - 42943166354676 \beta_{6} - 10735791588669 \beta_{5} + 23495063646081 \beta_{4} - 816120275381376861 \beta_{2} + 9456340375663717 \beta_{1} + 816098803798199523\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−178.705 309.527i
−62.4133 108.103i
51.3434 + 88.9294i
190.275 + 329.566i
−178.705 + 309.527i
−62.4133 + 108.103i
51.3434 88.9294i
190.275 329.566i
16.0000 27.7128i −390.910 677.077i −512.000 886.810i 4992.28 8646.88i −25018.3 24330.9 37220.1i −32768.0 −217048. + 375939.i −159753. 276700.i
9.2 16.0000 27.7128i −158.327 274.230i −512.000 886.810i −6009.47 + 10408.7i −10132.9 34168.3 + 28457.9i −32768.0 38438.9 66578.1i 192303. + 333079.i
9.3 16.0000 27.7128i 69.1868 + 119.835i −512.000 886.810i 274.515 475.474i 4427.96 −25454.3 36461.0i −32768.0 78999.9 136832.i −8784.49 15215.2i
9.4 16.0000 27.7128i 347.050 + 601.109i −512.000 886.810i 2646.67 4584.17i 22211.2 22119.1 + 38575.5i −32768.0 −152314. + 263816.i −84693.5 146693.i
11.1 16.0000 + 27.7128i −390.910 + 677.077i −512.000 + 886.810i 4992.28 + 8646.88i −25018.3 24330.9 + 37220.1i −32768.0 −217048. 375939.i −159753. + 276700.i
11.2 16.0000 + 27.7128i −158.327 + 274.230i −512.000 + 886.810i −6009.47 10408.7i −10132.9 34168.3 28457.9i −32768.0 38438.9 + 66578.1i 192303. 333079.i
11.3 16.0000 + 27.7128i 69.1868 119.835i −512.000 + 886.810i 274.515 + 475.474i 4427.96 −25454.3 + 36461.0i −32768.0 78999.9 + 136832.i −8784.49 + 15215.2i
11.4 16.0000 + 27.7128i 347.050 601.109i −512.000 + 886.810i 2646.67 + 4584.17i 22211.2 22119.1 38575.5i −32768.0 −152314. 263816.i −84693.5 + 146693.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
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 14.12.c.b 8
3.b odd 2 1 126.12.g.c 8
4.b odd 2 1 112.12.i.b 8
7.b odd 2 1 98.12.c.n 8
7.c even 3 1 inner 14.12.c.b 8
7.c even 3 1 98.12.a.i 4
7.d odd 6 1 98.12.a.h 4
7.d odd 6 1 98.12.c.n 8
21.h odd 6 1 126.12.g.c 8
28.g odd 6 1 112.12.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.c.b 8 1.a even 1 1 trivial
14.12.c.b 8 7.c even 3 1 inner
98.12.a.h 4 7.d odd 6 1
98.12.a.i 4 7.c even 3 1
98.12.c.n 8 7.b odd 2 1
98.12.c.n 8 7.d odd 6 1
112.12.i.b 8 4.b odd 2 1
112.12.i.b 8 28.g odd 6 1
126.12.g.c 8 3.b odd 2 1
126.12.g.c 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(44\!\cdots\!80\)\( T_{3}^{3} + \)\(23\!\cdots\!04\)\( T_{3}^{2} - \)\(23\!\cdots\!50\)\( T_{3} + \)\(56\!\cdots\!25\)\( \)">\(T_{3}^{8} + \cdots\) acting on \(S_{12}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1024 - 32 T + T^{2} )^{4} \)
$3$ \( \)\(56\!\cdots\!25\)\( - 2391762946776893550 T + 23691353797437804 T^{2} + 44770693165380 T^{3} + 328837501557 T^{4} + 49334964 T^{5} + 641596 T^{6} + 266 T^{7} + T^{8} \)
$5$ \( \)\(12\!\cdots\!25\)\( - \)\(24\!\cdots\!00\)\( T + \)\(45\!\cdots\!50\)\( T^{2} - 93849758229006822000 T^{3} + 19677749560043575 T^{4} - 922871052880 T^{5} + 143484054 T^{6} - 3808 T^{7} + T^{8} \)
$7$ \( \)\(15\!\cdots\!01\)\( - \)\(85\!\cdots\!96\)\( T + \)\(32\!\cdots\!52\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + 17127928136004183750 T^{4} - 368789009960280 T^{5} + 8202155948 T^{6} - 110328 T^{7} + T^{8} \)
$11$ \( \)\(87\!\cdots\!25\)\( - \)\(56\!\cdots\!50\)\( T + \)\(47\!\cdots\!96\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(77\!\cdots\!09\)\( T^{4} - 221336924541889772 T^{5} + 738532685772 T^{6} - 920150 T^{7} + T^{8} \)
$13$ \( ( \)\(24\!\cdots\!40\)\( + 1751795111703572544 T - 4054630895144 T^{2} - 498736 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(87\!\cdots\!25\)\( + \)\(68\!\cdots\!60\)\( T + \)\(15\!\cdots\!74\)\( T^{2} - \)\(36\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!87\)\( T^{4} + 21030874291181409424 T^{5} + 52088181193266 T^{6} - 1333724 T^{7} + T^{8} \)
$19$ \( \)\(13\!\cdots\!25\)\( - \)\(21\!\cdots\!70\)\( T + \)\(32\!\cdots\!96\)\( T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(51\!\cdots\!25\)\( T^{4} + \)\(11\!\cdots\!64\)\( T^{5} + 430735076074060 T^{6} + 21551726 T^{7} + T^{8} \)
$23$ \( \)\(40\!\cdots\!81\)\( - \)\(27\!\cdots\!78\)\( T + \)\(22\!\cdots\!92\)\( T^{2} - \)\(67\!\cdots\!80\)\( T^{3} + \)\(40\!\cdots\!09\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + 4661790384010872 T^{6} - 72510158 T^{7} + T^{8} \)
$29$ \( ( -\)\(20\!\cdots\!32\)\( + \)\(54\!\cdots\!52\)\( T - 33972485772246344 T^{2} - 106787552 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(45\!\cdots\!25\)\( + \)\(18\!\cdots\!90\)\( T + \)\(70\!\cdots\!24\)\( T^{2} + \)\(50\!\cdots\!64\)\( T^{3} + \)\(67\!\cdots\!81\)\( T^{4} + \)\(38\!\cdots\!44\)\( T^{5} + 46461345530648904 T^{6} + 194359774 T^{7} + T^{8} \)
$37$ \( \)\(25\!\cdots\!41\)\( + \)\(87\!\cdots\!52\)\( T + \)\(13\!\cdots\!62\)\( T^{2} + \)\(86\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!19\)\( T^{4} + \)\(31\!\cdots\!60\)\( T^{5} + 856340285762585662 T^{6} - 171517048 T^{7} + T^{8} \)
$41$ \( ( -\)\(13\!\cdots\!48\)\( - \)\(38\!\cdots\!12\)\( T + 329083932629864088 T^{2} + 1656047568 T^{3} + T^{4} )^{2} \)
$43$ \( ( -\)\(42\!\cdots\!40\)\( + \)\(10\!\cdots\!24\)\( T - 254940072096315936 T^{2} - 425139824 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(13\!\cdots\!25\)\( - \)\(19\!\cdots\!90\)\( T + \)\(26\!\cdots\!64\)\( T^{2} - \)\(83\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!97\)\( T^{4} - \)\(28\!\cdots\!96\)\( T^{5} + 5135110722683396856 T^{6} - 2223880974 T^{7} + T^{8} \)
$53$ \( \)\(10\!\cdots\!41\)\( + \)\(88\!\cdots\!60\)\( T + \)\(14\!\cdots\!38\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + 44703277939648770678 T^{6} + 7185483360 T^{7} + T^{8} \)
$59$ \( \)\(18\!\cdots\!25\)\( + \)\(14\!\cdots\!50\)\( T + \)\(14\!\cdots\!24\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!45\)\( T^{4} + \)\(42\!\cdots\!68\)\( T^{5} + 79213588806023076420 T^{6} - 6997401502 T^{7} + T^{8} \)
$61$ \( \)\(51\!\cdots\!25\)\( + \)\(69\!\cdots\!40\)\( T + \)\(84\!\cdots\!74\)\( T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(41\!\cdots\!59\)\( T^{4} + \)\(32\!\cdots\!04\)\( T^{5} + 86429631877407322998 T^{6} + 6476463280 T^{7} + T^{8} \)
$67$ \( \)\(18\!\cdots\!25\)\( + \)\(28\!\cdots\!10\)\( T + \)\(29\!\cdots\!36\)\( T^{2} + \)\(16\!\cdots\!96\)\( T^{3} + \)\(64\!\cdots\!37\)\( T^{4} + \)\(14\!\cdots\!96\)\( T^{5} + \)\(24\!\cdots\!12\)\( T^{6} + 18660972186 T^{7} + T^{8} \)
$71$ \( ( \)\(23\!\cdots\!40\)\( - \)\(92\!\cdots\!36\)\( T - 72768654772381942976 T^{2} - 11612224624 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(40\!\cdots\!25\)\( - \)\(28\!\cdots\!80\)\( T + \)\(16\!\cdots\!54\)\( T^{2} - \)\(22\!\cdots\!92\)\( T^{3} + \)\(27\!\cdots\!27\)\( T^{4} - \)\(69\!\cdots\!68\)\( T^{5} + \)\(51\!\cdots\!58\)\( T^{6} - 3731641452 T^{7} + T^{8} \)
$79$ \( \)\(28\!\cdots\!25\)\( + \)\(76\!\cdots\!50\)\( T + \)\(10\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!89\)\( T^{4} - \)\(85\!\cdots\!28\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} - 12221157926 T^{7} + T^{8} \)
$83$ \( ( \)\(54\!\cdots\!00\)\( - \)\(15\!\cdots\!44\)\( T + \)\(83\!\cdots\!76\)\( T^{2} - 158369761984 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(87\!\cdots\!25\)\( - \)\(36\!\cdots\!00\)\( T + \)\(15\!\cdots\!86\)\( T^{2} + \)\(19\!\cdots\!64\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} + \)\(95\!\cdots\!16\)\( T^{5} + \)\(36\!\cdots\!50\)\( T^{6} + 71204406084 T^{7} + T^{8} \)
$97$ \( ( -\)\(56\!\cdots\!00\)\( - \)\(48\!\cdots\!80\)\( T + \)\(34\!\cdots\!92\)\( T^{2} - 344125898592 T^{3} + T^{4} )^{2} \)
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