Properties

Label 74.6.c.b
Level $74$
Weight $6$
Character orbit 74.c
Analytic conductor $11.868$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 1317 x^{14} - 2712 x^{13} + 1206654 x^{12} - 2683704 x^{11} + 572597881 x^{10} - 1472178324 x^{9} + 197936317446 x^{8} - 538366712748 x^{7} + 33466187997381 x^{6} - 131063526625884 x^{5} + 4041550479749833 x^{4} - 11057529820868736 x^{3} + 29709621477714456 x^{2} - 4628457422701056 x + 683643854869056\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 + 4 \beta_{3} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + 16 \beta_{3} q^{4} + ( -9 \beta_{3} + \beta_{4} - \beta_{9} ) q^{5} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{6} + ( \beta_{1} + 4 \beta_{3} + \beta_{11} ) q^{7} -64 q^{8} + ( -87 - 87 \beta_{3} - \beta_{9} + \beta_{12} ) q^{9} +O(q^{10})\) \( q + ( 4 + 4 \beta_{3} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + 16 \beta_{3} q^{4} + ( -9 \beta_{3} + \beta_{4} - \beta_{9} ) q^{5} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{6} + ( \beta_{1} + 4 \beta_{3} + \beta_{11} ) q^{7} -64 q^{8} + ( -87 - 87 \beta_{3} - \beta_{9} + \beta_{12} ) q^{9} + ( 36 + 4 \beta_{4} ) q^{10} + ( -15 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{11} + ( -16 - 16 \beta_{2} - 16 \beta_{3} ) q^{12} + ( 8 \beta_{1} - 202 \beta_{3} + \beta_{5} + 4 \beta_{11} - \beta_{12} + \beta_{14} ) q^{13} + ( -16 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{6} ) q^{14} + ( 93 + 16 \beta_{2} + 93 \beta_{3} + 2 \beta_{6} - \beta_{8} + 5 \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{15} + ( -256 - 256 \beta_{3} ) q^{16} + ( 27 + 5 \beta_{2} + 27 \beta_{3} + 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{17} + ( -348 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{9} + 4 \beta_{12} ) q^{18} + ( 7 \beta_{1} + 120 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{19} + ( 144 + 144 \beta_{3} + 16 \beta_{9} ) q^{20} + ( 217 + 15 \beta_{2} + 217 \beta_{3} + 7 \beta_{6} - 2 \beta_{8} + 7 \beta_{9} - 7 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{21} + ( -60 - 8 \beta_{2} - 60 \beta_{3} - 4 \beta_{6} + 4 \beta_{8} + 4 \beta_{9} + 4 \beta_{11} - 4 \beta_{14} ) q^{22} + ( 106 + 4 \beta_{1} + 4 \beta_{2} + 11 \beta_{4} - 8 \beta_{5} - \beta_{10} ) q^{23} + ( 64 \beta_{1} - 64 \beta_{3} ) q^{24} + ( -1470 - 79 \beta_{2} - 1470 \beta_{3} - 12 \beta_{6} - 3 \beta_{8} - \beta_{9} + 12 \beta_{11} - \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{25} + ( 808 + 32 \beta_{1} + 32 \beta_{2} + 4 \beta_{5} + 16 \beta_{6} + 4 \beta_{8} ) q^{26} + ( -5 + 29 \beta_{1} + 29 \beta_{2} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{27} + ( -64 + 16 \beta_{2} - 64 \beta_{3} + 16 \beta_{6} - 16 \beta_{11} ) q^{28} + ( 79 - 56 \beta_{1} - 56 \beta_{2} + 8 \beta_{4} + 19 \beta_{5} + 12 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{29} + ( -64 \beta_{1} + 372 \beta_{3} - 20 \beta_{4} + 4 \beta_{7} + 20 \beta_{9} - 8 \beta_{11} + 4 \beta_{13} + 4 \beta_{14} ) q^{30} + ( -1053 - 86 \beta_{1} - 86 \beta_{2} + 7 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{31} -1024 \beta_{3} q^{32} + ( 4 \beta_{1} - 603 \beta_{3} + 32 \beta_{4} + 2 \beta_{5} - 32 \beta_{9} - \beta_{10} + 23 \beta_{11} - 2 \beta_{12} - 4 \beta_{14} - \beta_{15} ) q^{33} + ( -20 \beta_{1} + 108 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} + 4 \beta_{7} - 16 \beta_{9} + 8 \beta_{12} + 4 \beta_{13} - 8 \beta_{14} ) q^{34} + ( -628 - 111 \beta_{2} - 628 \beta_{3} - 25 \beta_{6} + 13 \beta_{8} - 33 \beta_{9} + 25 \beta_{11} + 22 \beta_{12} - 4 \beta_{13} - 13 \beta_{14} ) q^{35} + ( 1392 + 16 \beta_{4} - 16 \beta_{5} ) q^{36} + ( 1145 - 53 \beta_{1} + 32 \beta_{2} + 1828 \beta_{3} + 58 \beta_{4} + 4 \beta_{5} + 33 \beta_{6} - 3 \beta_{7} + 8 \beta_{8} - 21 \beta_{9} - 7 \beta_{11} - 13 \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} ) q^{37} + ( -480 + 28 \beta_{1} + 28 \beta_{2} - 4 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{10} ) q^{38} + ( 2213 + 118 \beta_{2} + 2213 \beta_{3} + 8 \beta_{6} - 7 \beta_{8} + 8 \beta_{9} - 8 \beta_{11} - 12 \beta_{12} + 3 \beta_{13} + 7 \beta_{14} + 5 \beta_{15} ) q^{39} + ( 576 \beta_{3} - 64 \beta_{4} + 64 \beta_{9} ) q^{40} + ( -69 \beta_{1} + 1016 \beta_{3} + 7 \beta_{4} - 12 \beta_{5} - 3 \beta_{7} - 7 \beta_{9} - 5 \beta_{10} - 33 \beta_{11} + 12 \beta_{12} - 3 \beta_{13} + 13 \beta_{14} - 5 \beta_{15} ) q^{41} + ( -60 \beta_{1} + 868 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + 28 \beta_{9} + 4 \beta_{10} - 28 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} ) q^{42} + ( -1183 - 61 \beta_{1} - 61 \beta_{2} + 21 \beta_{4} - 22 \beta_{5} - 11 \beta_{6} - 5 \beta_{7} - 34 \beta_{8} - \beta_{10} ) q^{43} + ( 32 \beta_{1} - 240 \beta_{3} - 16 \beta_{4} + 16 \beta_{9} + 16 \beta_{11} - 16 \beta_{14} ) q^{44} + ( -3673 - 69 \beta_{1} - 69 \beta_{2} - 98 \beta_{4} + 39 \beta_{5} - 68 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} + 2 \beta_{10} ) q^{45} + ( 424 + 16 \beta_{2} + 424 \beta_{3} + 44 \beta_{9} - 32 \beta_{12} + 4 \beta_{15} ) q^{46} + ( 1867 + 205 \beta_{1} + 205 \beta_{2} + 43 \beta_{4} + 20 \beta_{5} - 27 \beta_{6} - 9 \beta_{7} - 19 \beta_{8} - 4 \beta_{10} ) q^{47} + ( 256 + 256 \beta_{1} + 256 \beta_{2} ) q^{48} + ( 30 - 357 \beta_{2} + 30 \beta_{3} - 39 \beta_{6} - 16 \beta_{8} - 101 \beta_{9} + 39 \beta_{11} - 9 \beta_{12} - 7 \beta_{13} + 16 \beta_{14} + \beta_{15} ) q^{49} + ( 316 \beta_{1} - 5880 \beta_{3} + 4 \beta_{4} - 4 \beta_{7} - 4 \beta_{9} - 8 \beta_{10} + 48 \beta_{11} - 4 \beta_{13} + 12 \beta_{14} - 8 \beta_{15} ) q^{50} + ( -1135 + 379 \beta_{1} + 379 \beta_{2} - 196 \beta_{4} + 40 \beta_{5} + 22 \beta_{6} - 16 \beta_{7} - 8 \beta_{8} + 2 \beta_{10} ) q^{51} + ( 3232 + 128 \beta_{2} + 3232 \beta_{3} + 64 \beta_{6} + 16 \beta_{8} - 64 \beta_{11} + 16 \beta_{12} - 16 \beta_{14} ) q^{52} + ( -3688 + 51 \beta_{2} - 3688 \beta_{3} - 44 \beta_{6} - 22 \beta_{8} - 137 \beta_{9} + 44 \beta_{11} + 13 \beta_{12} - 19 \beta_{13} + 22 \beta_{14} + 4 \beta_{15} ) q^{53} + ( -20 + 116 \beta_{2} - 20 \beta_{3} + 16 \beta_{6} + 4 \beta_{8} - 16 \beta_{11} + 8 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{54} + ( -538 \beta_{1} - 1983 \beta_{3} + 5 \beta_{4} + 40 \beta_{5} - 5 \beta_{9} - 2 \beta_{10} - 91 \beta_{11} - 40 \beta_{12} + 39 \beta_{14} - 2 \beta_{15} ) q^{55} + ( -64 \beta_{1} - 256 \beta_{3} - 64 \beta_{11} ) q^{56} + ( 2008 + 190 \beta_{2} + 2008 \beta_{3} - 216 \beta_{6} - 15 \beta_{8} + 120 \beta_{9} + 216 \beta_{11} - 95 \beta_{12} - 2 \beta_{13} + 15 \beta_{14} - 4 \beta_{15} ) q^{57} + ( 316 - 224 \beta_{2} + 316 \beta_{3} + 48 \beta_{6} + 16 \beta_{8} + 32 \beta_{9} - 48 \beta_{11} + 76 \beta_{12} - 8 \beta_{13} - 16 \beta_{14} ) q^{58} + ( 7687 + 528 \beta_{2} + 7687 \beta_{3} + 10 \beta_{6} - 15 \beta_{8} + 40 \beta_{9} - 10 \beta_{11} - 28 \beta_{12} + 5 \beta_{13} + 15 \beta_{14} - \beta_{15} ) q^{59} + ( -1488 - 256 \beta_{1} - 256 \beta_{2} - 80 \beta_{4} - 32 \beta_{6} + 16 \beta_{7} + 16 \beta_{8} ) q^{60} + ( 334 \beta_{1} + 16430 \beta_{3} + 134 \beta_{4} + 46 \beta_{5} + 6 \beta_{7} - 134 \beta_{9} + \beta_{10} - 33 \beta_{11} - 46 \beta_{12} + 6 \beta_{13} - 67 \beta_{14} + \beta_{15} ) q^{61} + ( -4212 - 344 \beta_{2} - 4212 \beta_{3} - 36 \beta_{6} - 4 \beta_{8} + 28 \beta_{9} + 36 \beta_{11} - 24 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} ) q^{62} + ( -6561 - 273 \beta_{1} - 273 \beta_{2} - 282 \beta_{4} - 54 \beta_{5} - 109 \beta_{6} + 9 \beta_{7} + 5 \beta_{8} + 9 \beta_{10} ) q^{63} + 4096 q^{64} + ( -4786 - 28 \beta_{2} - 4786 \beta_{3} + 24 \beta_{6} + 21 \beta_{8} - 416 \beta_{9} - 24 \beta_{11} + 31 \beta_{12} - 28 \beta_{13} - 21 \beta_{14} ) q^{65} + ( 2412 + 16 \beta_{1} + 16 \beta_{2} + 128 \beta_{4} + 8 \beta_{5} + 92 \beta_{6} - 16 \beta_{8} - 4 \beta_{10} ) q^{66} + ( -242 \beta_{1} + 5792 \beta_{3} - 379 \beta_{4} + 24 \beta_{5} - 3 \beta_{7} + 379 \beta_{9} - \beta_{10} + 43 \beta_{11} - 24 \beta_{12} - 3 \beta_{13} + 14 \beta_{14} - \beta_{15} ) q^{67} + ( -432 - 80 \beta_{1} - 80 \beta_{2} + 64 \beta_{4} - 32 \beta_{5} + 16 \beta_{7} - 32 \beta_{8} ) q^{68} + ( -1655 \beta_{1} + 2999 \beta_{3} + 103 \beta_{4} - 84 \beta_{5} + 23 \beta_{7} - 103 \beta_{9} - 5 \beta_{10} - 263 \beta_{11} + 84 \beta_{12} + 23 \beta_{13} - 23 \beta_{14} - 5 \beta_{15} ) q^{69} + ( 444 \beta_{1} - 2512 \beta_{3} + 132 \beta_{4} - 88 \beta_{5} - 16 \beta_{7} - 132 \beta_{9} + 100 \beta_{11} + 88 \beta_{12} - 16 \beta_{13} - 52 \beta_{14} ) q^{70} + ( 702 \beta_{1} - 1716 \beta_{3} + 146 \beta_{4} + 26 \beta_{5} - 9 \beta_{7} - 146 \beta_{9} + 4 \beta_{10} - 113 \beta_{11} - 26 \beta_{12} - 9 \beta_{13} - 30 \beta_{14} + 4 \beta_{15} ) q^{71} + ( 5568 + 5568 \beta_{3} + 64 \beta_{9} - 64 \beta_{12} ) q^{72} + ( 17011 - 1675 \beta_{1} - 1675 \beta_{2} + 369 \beta_{4} - 14 \beta_{5} - 9 \beta_{6} + 5 \beta_{7} - 55 \beta_{8} + 5 \beta_{10} ) q^{73} + ( -2732 - 340 \beta_{1} - 212 \beta_{2} + 4580 \beta_{3} + 84 \beta_{4} + 52 \beta_{5} + 104 \beta_{6} - 20 \beta_{7} + 36 \beta_{8} + 148 \beta_{9} - 4 \beta_{10} - 132 \beta_{11} - 36 \beta_{12} - 8 \beta_{13} - 32 \beta_{14} - 4 \beta_{15} ) q^{74} + ( 26294 + 1747 \beta_{1} + 1747 \beta_{2} + 84 \beta_{4} + 58 \beta_{5} + 508 \beta_{6} - 11 \beta_{7} + 22 \beta_{8} - 18 \beta_{10} ) q^{75} + ( -1920 + 112 \beta_{2} - 1920 \beta_{3} + 16 \beta_{6} - 16 \beta_{9} - 16 \beta_{11} - 32 \beta_{12} - 16 \beta_{15} ) q^{76} + ( -1093 \beta_{1} + 23112 \beta_{3} - 434 \beta_{4} - 12 \beta_{5} + 15 \beta_{7} + 434 \beta_{9} + 6 \beta_{10} + 28 \beta_{11} + 12 \beta_{12} + 15 \beta_{13} - 82 \beta_{14} + 6 \beta_{15} ) q^{77} + ( -472 \beta_{1} + 8852 \beta_{3} - 32 \beta_{4} + 48 \beta_{5} + 12 \beta_{7} + 32 \beta_{9} + 20 \beta_{10} - 32 \beta_{11} - 48 \beta_{12} + 12 \beta_{13} + 28 \beta_{14} + 20 \beta_{15} ) q^{78} + ( 1024 \beta_{1} - 15460 \beta_{3} + 169 \beta_{4} - 76 \beta_{5} + 15 \beta_{7} - 169 \beta_{9} - 15 \beta_{10} - 145 \beta_{11} + 76 \beta_{12} + 15 \beta_{13} + 134 \beta_{14} - 15 \beta_{15} ) q^{79} + ( -2304 - 256 \beta_{4} ) q^{80} + ( 106 \beta_{1} - 11922 \beta_{3} - 421 \beta_{4} - 68 \beta_{5} - 20 \beta_{7} + 421 \beta_{9} + \beta_{10} + 163 \beta_{11} + 68 \beta_{12} - 20 \beta_{13} + 19 \beta_{14} + \beta_{15} ) q^{81} + ( -4064 - 276 \beta_{1} - 276 \beta_{2} + 28 \beta_{4} - 48 \beta_{5} - 132 \beta_{6} - 12 \beta_{7} + 52 \beta_{8} - 20 \beta_{10} ) q^{82} + ( -2129 - 462 \beta_{2} - 2129 \beta_{3} - 230 \beta_{6} + 90 \beta_{8} + 196 \beta_{9} + 230 \beta_{11} - 88 \beta_{12} + 11 \beta_{13} - 90 \beta_{14} - 16 \beta_{15} ) q^{83} + ( -3472 - 240 \beta_{1} - 240 \beta_{2} - 112 \beta_{4} + 16 \beta_{5} - 112 \beta_{6} + 16 \beta_{7} + 32 \beta_{8} + 16 \beta_{10} ) q^{84} + ( -15581 + 2345 \beta_{1} + 2345 \beta_{2} + 149 \beta_{4} + 234 \beta_{5} - 8 \beta_{6} - 5 \beta_{7} + 122 \beta_{8} ) q^{85} + ( -4732 - 244 \beta_{2} - 4732 \beta_{3} - 44 \beta_{6} - 136 \beta_{8} + 84 \beta_{9} + 44 \beta_{11} - 88 \beta_{12} + 20 \beta_{13} + 136 \beta_{14} + 4 \beta_{15} ) q^{86} + ( 2741 \beta_{1} - 21084 \beta_{3} + 483 \beta_{4} - 94 \beta_{5} + 21 \beta_{7} - 483 \beta_{9} + 31 \beta_{10} + 62 \beta_{11} + 94 \beta_{12} + 21 \beta_{13} + 62 \beta_{14} + 31 \beta_{15} ) q^{87} + ( 960 + 128 \beta_{1} + 128 \beta_{2} - 64 \beta_{4} + 64 \beta_{6} - 64 \beta_{8} ) q^{88} + ( 6557 + 179 \beta_{2} + 6557 \beta_{3} + 548 \beta_{6} - 63 \beta_{8} - 219 \beta_{9} - 548 \beta_{11} - 38 \beta_{12} + 7 \beta_{13} + 63 \beta_{14} + 8 \beta_{15} ) q^{89} + ( -14692 - 276 \beta_{2} - 14692 \beta_{3} - 272 \beta_{6} + 40 \beta_{8} - 392 \beta_{9} + 272 \beta_{11} + 156 \beta_{12} + 12 \beta_{13} - 40 \beta_{14} - 8 \beta_{15} ) q^{90} + ( -54021 - 813 \beta_{2} - 54021 \beta_{3} - 468 \beta_{6} + 7 \beta_{8} + 176 \beta_{9} + 468 \beta_{11} + 26 \beta_{12} - 16 \beta_{13} - 7 \beta_{14} + 15 \beta_{15} ) q^{91} + ( -64 \beta_{1} + 1696 \beta_{3} - 176 \beta_{4} + 128 \beta_{5} + 176 \beta_{9} + 16 \beta_{10} - 128 \beta_{12} + 16 \beta_{15} ) q^{92} + ( -70 \beta_{1} - 27617 \beta_{3} + 624 \beta_{4} - 144 \beta_{5} + 24 \beta_{7} - 624 \beta_{9} - 15 \beta_{10} + 71 \beta_{11} + 144 \beta_{12} + 24 \beta_{13} - 16 \beta_{14} - 15 \beta_{15} ) q^{93} + ( 7468 + 820 \beta_{2} + 7468 \beta_{3} - 108 \beta_{6} - 76 \beta_{8} + 172 \beta_{9} + 108 \beta_{11} + 80 \beta_{12} + 36 \beta_{13} + 76 \beta_{14} + 16 \beta_{15} ) q^{94} + ( 6258 - 1443 \beta_{2} + 6258 \beta_{3} - 291 \beta_{6} + 44 \beta_{8} + 1439 \beta_{9} + 291 \beta_{11} + 42 \beta_{12} + 20 \beta_{13} - 44 \beta_{14} - 17 \beta_{15} ) q^{95} + ( 1024 + 1024 \beta_{2} + 1024 \beta_{3} ) q^{96} + ( -34388 - 1286 \beta_{1} - 1286 \beta_{2} - 1099 \beta_{4} + 33 \beta_{5} + 377 \beta_{6} - 40 \beta_{7} + 64 \beta_{8} + 15 \beta_{10} ) q^{97} + ( 1428 \beta_{1} + 120 \beta_{3} + 404 \beta_{4} + 36 \beta_{5} - 28 \beta_{7} - 404 \beta_{9} + 4 \beta_{10} + 156 \beta_{11} - 36 \beta_{12} - 28 \beta_{13} + 64 \beta_{14} + 4 \beta_{15} ) q^{98} + ( -1051 + 405 \beta_{2} - 1051 \beta_{3} + 275 \beta_{6} + 167 \beta_{8} + 549 \beta_{9} - 275 \beta_{11} + 100 \beta_{12} + 57 \beta_{13} - 167 \beta_{14} + 28 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 32q^{2} - 8q^{3} - 128q^{4} + 70q^{5} - 64q^{6} - 32q^{7} - 1024q^{8} - 698q^{9} + O(q^{10}) \) \( 16q + 32q^{2} - 8q^{3} - 128q^{4} + 70q^{5} - 64q^{6} - 32q^{7} - 1024q^{8} - 698q^{9} + 560q^{10} - 240q^{11} - 128q^{12} + 1614q^{13} - 256q^{14} + 728q^{15} - 2048q^{16} + 216q^{17} + 2792q^{18} - 950q^{19} + 1120q^{20} + 1718q^{21} - 480q^{22} + 1716q^{23} + 512q^{24} - 11760q^{25} + 12912q^{26} - 92q^{27} - 512q^{28} + 1112q^{29} - 2912q^{30} - 16816q^{31} + 8192q^{32} + 4744q^{33} - 864q^{34} - 5004q^{35} + 22336q^{36} + 3556q^{37} - 7600q^{38} + 17710q^{39} - 4480q^{40} - 8080q^{41} - 6872q^{42} - 19012q^{43} + 1920q^{44} - 58672q^{45} + 3432q^{46} + 29392q^{47} + 4096q^{48} + 474q^{49} + 47040q^{50} - 17856q^{51} + 25824q^{52} - 29250q^{53} - 184q^{54} + 15772q^{55} + 2048q^{56} + 16182q^{57} + 2224q^{58} + 61478q^{59} - 23296q^{60} - 132002q^{61} - 33632q^{62} - 103324q^{63} + 65536q^{64} - 37426q^{65} + 37952q^{66} - 45658q^{67} - 6912q^{68} - 23816q^{69} + 20016q^{70} + 13236q^{71} + 44672q^{72} + 270632q^{73} - 81304q^{74} + 419904q^{75} - 15200q^{76} - 184084q^{77} - 70840q^{78} + 123974q^{79} - 35840q^{80} + 96448q^{81} - 64640q^{82} - 16936q^{83} - 54976q^{84} - 251316q^{85} - 38024q^{86} + 168290q^{87} + 15360q^{88} + 52892q^{89} - 117344q^{90} - 432546q^{91} - 13728q^{92} + 220328q^{93} + 58784q^{94} + 47026q^{95} + 8192q^{96} - 546140q^{97} - 1896q^{98} - 9800q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 1317 x^{14} - 2712 x^{13} + 1206654 x^{12} - 2683704 x^{11} + 572597881 x^{10} - 1472178324 x^{9} + 197936317446 x^{8} - 538366712748 x^{7} + 33466187997381 x^{6} - 131063526625884 x^{5} + 4041550479749833 x^{4} - 11057529820868736 x^{3} + 29709621477714456 x^{2} - 4628457422701056 x + 683643854869056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(12\!\cdots\!83\)\( \nu^{15} - \)\(35\!\cdots\!08\)\( \nu^{14} - \)\(16\!\cdots\!63\)\( \nu^{13} - \)\(13\!\cdots\!56\)\( \nu^{12} - \)\(14\!\cdots\!02\)\( \nu^{11} - \)\(10\!\cdots\!08\)\( \nu^{10} - \)\(68\!\cdots\!87\)\( \nu^{9} - \)\(22\!\cdots\!28\)\( \nu^{8} - \)\(23\!\cdots\!02\)\( \nu^{7} - \)\(40\!\cdots\!20\)\( \nu^{6} - \)\(39\!\cdots\!15\)\( \nu^{5} + \)\(43\!\cdots\!36\)\( \nu^{4} - \)\(46\!\cdots\!55\)\( \nu^{3} - \)\(48\!\cdots\!04\)\( \nu^{2} - \)\(36\!\cdots\!76\)\( \nu + \)\(57\!\cdots\!64\)\(\)\()/ \)\(36\!\cdots\!44\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(91\!\cdots\!04\)\( \nu^{15} - \)\(13\!\cdots\!03\)\( \nu^{14} - \)\(12\!\cdots\!96\)\( \nu^{13} + \)\(22\!\cdots\!65\)\( \nu^{12} - \)\(11\!\cdots\!12\)\( \nu^{11} + \)\(22\!\cdots\!34\)\( \nu^{10} - \)\(52\!\cdots\!52\)\( \nu^{9} + \)\(12\!\cdots\!29\)\( \nu^{8} - \)\(18\!\cdots\!32\)\( \nu^{7} + \)\(46\!\cdots\!10\)\( \nu^{6} - \)\(30\!\cdots\!44\)\( \nu^{5} + \)\(11\!\cdots\!21\)\( \nu^{4} - \)\(36\!\cdots\!56\)\( \nu^{3} + \)\(95\!\cdots\!89\)\( \nu^{2} - \)\(27\!\cdots\!88\)\( \nu + \)\(21\!\cdots\!08\)\(\)\()/ \)\(40\!\cdots\!04\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(74\!\cdots\!38\)\( \nu^{15} + \)\(18\!\cdots\!23\)\( \nu^{14} + \)\(98\!\cdots\!88\)\( \nu^{13} + \)\(45\!\cdots\!86\)\( \nu^{12} + \)\(89\!\cdots\!16\)\( \nu^{11} + \)\(30\!\cdots\!47\)\( \nu^{10} + \)\(42\!\cdots\!55\)\( \nu^{9} + \)\(14\!\cdots\!97\)\( \nu^{8} + \)\(14\!\cdots\!44\)\( \nu^{7} - \)\(84\!\cdots\!84\)\( \nu^{6} + \)\(23\!\cdots\!92\)\( \nu^{5} - \)\(29\!\cdots\!91\)\( \nu^{4} + \)\(27\!\cdots\!77\)\( \nu^{3} + \)\(27\!\cdots\!28\)\( \nu^{2} - \)\(42\!\cdots\!56\)\( \nu + \)\(62\!\cdots\!08\)\(\)\()/ \)\(86\!\cdots\!78\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(17\!\cdots\!77\)\( \nu^{15} + \)\(38\!\cdots\!78\)\( \nu^{14} + \)\(23\!\cdots\!73\)\( \nu^{13} + \)\(40\!\cdots\!40\)\( \nu^{12} + \)\(21\!\cdots\!38\)\( \nu^{11} + \)\(12\!\cdots\!14\)\( \nu^{10} + \)\(10\!\cdots\!91\)\( \nu^{9} - \)\(21\!\cdots\!46\)\( \nu^{8} + \)\(34\!\cdots\!30\)\( \nu^{7} - \)\(10\!\cdots\!00\)\( \nu^{6} + \)\(56\!\cdots\!93\)\( \nu^{5} - \)\(75\!\cdots\!26\)\( \nu^{4} + \)\(66\!\cdots\!27\)\( \nu^{3} + \)\(60\!\cdots\!20\)\( \nu^{2} - \)\(94\!\cdots\!60\)\( \nu + \)\(69\!\cdots\!16\)\(\)\()/ \)\(17\!\cdots\!56\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(12\!\cdots\!67\)\( \nu^{15} - \)\(35\!\cdots\!22\)\( \nu^{14} - \)\(17\!\cdots\!87\)\( \nu^{13} - \)\(11\!\cdots\!76\)\( \nu^{12} - \)\(15\!\cdots\!46\)\( \nu^{11} - \)\(76\!\cdots\!90\)\( \nu^{10} - \)\(73\!\cdots\!29\)\( \nu^{9} - \)\(11\!\cdots\!30\)\( \nu^{8} - \)\(25\!\cdots\!42\)\( \nu^{7} + \)\(29\!\cdots\!44\)\( \nu^{6} - \)\(41\!\cdots\!99\)\( \nu^{5} + \)\(48\!\cdots\!74\)\( \nu^{4} - \)\(47\!\cdots\!69\)\( \nu^{3} - \)\(49\!\cdots\!64\)\( \nu^{2} + \)\(77\!\cdots\!48\)\( \nu - \)\(14\!\cdots\!24\)\(\)\()/ \)\(86\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(51\!\cdots\!49\)\( \nu^{15} - \)\(14\!\cdots\!40\)\( \nu^{14} - \)\(67\!\cdots\!65\)\( \nu^{13} - \)\(46\!\cdots\!28\)\( \nu^{12} - \)\(61\!\cdots\!50\)\( \nu^{11} - \)\(34\!\cdots\!80\)\( \nu^{10} - \)\(28\!\cdots\!01\)\( \nu^{9} - \)\(62\!\cdots\!80\)\( \nu^{8} - \)\(99\!\cdots\!30\)\( \nu^{7} - \)\(97\!\cdots\!60\)\( \nu^{6} - \)\(16\!\cdots\!45\)\( \nu^{5} + \)\(18\!\cdots\!80\)\( \nu^{4} - \)\(18\!\cdots\!49\)\( \nu^{3} - \)\(19\!\cdots\!60\)\( \nu^{2} + \)\(30\!\cdots\!20\)\( \nu - \)\(13\!\cdots\!32\)\(\)\()/ \)\(34\!\cdots\!12\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(13\!\cdots\!91\)\( \nu^{15} + \)\(39\!\cdots\!56\)\( \nu^{14} + \)\(17\!\cdots\!51\)\( \nu^{13} + \)\(16\!\cdots\!18\)\( \nu^{12} + \)\(15\!\cdots\!58\)\( \nu^{11} + \)\(12\!\cdots\!20\)\( \nu^{10} + \)\(74\!\cdots\!57\)\( \nu^{9} + \)\(30\!\cdots\!40\)\( \nu^{8} + \)\(25\!\cdots\!66\)\( \nu^{7} + \)\(59\!\cdots\!58\)\( \nu^{6} + \)\(42\!\cdots\!27\)\( \nu^{5} - \)\(45\!\cdots\!52\)\( \nu^{4} + \)\(48\!\cdots\!17\)\( \nu^{3} + \)\(52\!\cdots\!72\)\( \nu^{2} - \)\(82\!\cdots\!04\)\( \nu + \)\(36\!\cdots\!72\)\(\)\()/ \)\(86\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(17\!\cdots\!74\)\( \nu^{15} - \)\(17\!\cdots\!01\)\( \nu^{14} - \)\(22\!\cdots\!93\)\( \nu^{13} + \)\(46\!\cdots\!38\)\( \nu^{12} - \)\(20\!\cdots\!96\)\( \nu^{11} + \)\(45\!\cdots\!00\)\( \nu^{10} - \)\(98\!\cdots\!65\)\( \nu^{9} + \)\(25\!\cdots\!29\)\( \nu^{8} - \)\(34\!\cdots\!45\)\( \nu^{7} + \)\(92\!\cdots\!64\)\( \nu^{6} - \)\(57\!\cdots\!56\)\( \nu^{5} + \)\(22\!\cdots\!38\)\( \nu^{4} - \)\(69\!\cdots\!47\)\( \nu^{3} + \)\(19\!\cdots\!88\)\( \nu^{2} - \)\(51\!\cdots\!80\)\( \nu + \)\(79\!\cdots\!76\)\(\)\()/ \)\(10\!\cdots\!22\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(95\!\cdots\!14\)\( \nu^{15} - \)\(26\!\cdots\!84\)\( \nu^{14} - \)\(12\!\cdots\!14\)\( \nu^{13} - \)\(89\!\cdots\!07\)\( \nu^{12} - \)\(11\!\cdots\!72\)\( \nu^{11} - \)\(66\!\cdots\!40\)\( \nu^{10} - \)\(53\!\cdots\!83\)\( \nu^{9} - \)\(12\!\cdots\!20\)\( \nu^{8} - \)\(18\!\cdots\!04\)\( \nu^{7} - \)\(23\!\cdots\!77\)\( \nu^{6} - \)\(30\!\cdots\!58\)\( \nu^{5} + \)\(34\!\cdots\!28\)\( \nu^{4} - \)\(34\!\cdots\!93\)\( \nu^{3} - \)\(36\!\cdots\!68\)\( \nu^{2} + \)\(57\!\cdots\!76\)\( \nu - \)\(47\!\cdots\!08\)\(\)\()/ \)\(43\!\cdots\!90\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(25\!\cdots\!96\)\( \nu^{15} - \)\(37\!\cdots\!39\)\( \nu^{14} - \)\(33\!\cdots\!20\)\( \nu^{13} + \)\(63\!\cdots\!61\)\( \nu^{12} - \)\(30\!\cdots\!04\)\( \nu^{11} + \)\(62\!\cdots\!54\)\( \nu^{10} - \)\(14\!\cdots\!44\)\( \nu^{9} + \)\(34\!\cdots\!65\)\( \nu^{8} - \)\(49\!\cdots\!56\)\( \nu^{7} + \)\(12\!\cdots\!74\)\( \nu^{6} - \)\(83\!\cdots\!72\)\( \nu^{5} + \)\(31\!\cdots\!41\)\( \nu^{4} - \)\(10\!\cdots\!28\)\( \nu^{3} + \)\(26\!\cdots\!45\)\( \nu^{2} - \)\(71\!\cdots\!48\)\( \nu + \)\(60\!\cdots\!36\)\(\)\()/ \)\(62\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(11\!\cdots\!98\)\( \nu^{15} - \)\(13\!\cdots\!83\)\( \nu^{14} - \)\(15\!\cdots\!54\)\( \nu^{13} + \)\(29\!\cdots\!13\)\( \nu^{12} - \)\(13\!\cdots\!96\)\( \nu^{11} + \)\(29\!\cdots\!22\)\( \nu^{10} - \)\(65\!\cdots\!14\)\( \nu^{9} + \)\(16\!\cdots\!65\)\( \nu^{8} - \)\(22\!\cdots\!64\)\( \nu^{7} + \)\(59\!\cdots\!98\)\( \nu^{6} - \)\(38\!\cdots\!58\)\( \nu^{5} + \)\(14\!\cdots\!93\)\( \nu^{4} - \)\(46\!\cdots\!38\)\( \nu^{3} + \)\(12\!\cdots\!89\)\( \nu^{2} - \)\(34\!\cdots\!80\)\( \nu + \)\(53\!\cdots\!12\)\(\)\()/ \)\(12\!\cdots\!64\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(50\!\cdots\!03\)\( \nu^{15} + \)\(14\!\cdots\!44\)\( \nu^{14} - \)\(66\!\cdots\!07\)\( \nu^{13} + \)\(15\!\cdots\!60\)\( \nu^{12} - \)\(60\!\cdots\!22\)\( \nu^{11} + \)\(15\!\cdots\!04\)\( \nu^{10} - \)\(28\!\cdots\!63\)\( \nu^{9} + \)\(81\!\cdots\!32\)\( \nu^{8} - \)\(99\!\cdots\!78\)\( \nu^{7} + \)\(29\!\cdots\!28\)\( \nu^{6} - \)\(16\!\cdots\!43\)\( \nu^{5} + \)\(70\!\cdots\!00\)\( \nu^{4} - \)\(20\!\cdots\!47\)\( \nu^{3} + \)\(60\!\cdots\!92\)\( \nu^{2} - \)\(14\!\cdots\!40\)\( \nu + \)\(23\!\cdots\!72\)\(\)\()/ \)\(54\!\cdots\!68\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(35\!\cdots\!84\)\( \nu^{15} + \)\(58\!\cdots\!81\)\( \nu^{14} + \)\(46\!\cdots\!90\)\( \nu^{13} - \)\(88\!\cdots\!09\)\( \nu^{12} + \)\(42\!\cdots\!76\)\( \nu^{11} - \)\(88\!\cdots\!46\)\( \nu^{10} + \)\(20\!\cdots\!36\)\( \nu^{9} - \)\(48\!\cdots\!05\)\( \nu^{8} + \)\(69\!\cdots\!94\)\( \nu^{7} - \)\(17\!\cdots\!16\)\( \nu^{6} + \)\(11\!\cdots\!88\)\( \nu^{5} - \)\(44\!\cdots\!19\)\( \nu^{4} + \)\(14\!\cdots\!72\)\( \nu^{3} - \)\(36\!\cdots\!25\)\( \nu^{2} + \)\(90\!\cdots\!52\)\( \nu - \)\(84\!\cdots\!44\)\(\)\()/ \)\(31\!\cdots\!60\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(26\!\cdots\!92\)\( \nu^{15} + \)\(24\!\cdots\!91\)\( \nu^{14} - \)\(34\!\cdots\!98\)\( \nu^{13} + \)\(75\!\cdots\!15\)\( \nu^{12} - \)\(32\!\cdots\!40\)\( \nu^{11} + \)\(74\!\cdots\!66\)\( \nu^{10} - \)\(15\!\cdots\!22\)\( \nu^{9} + \)\(40\!\cdots\!05\)\( \nu^{8} - \)\(52\!\cdots\!82\)\( \nu^{7} + \)\(14\!\cdots\!22\)\( \nu^{6} - \)\(88\!\cdots\!84\)\( \nu^{5} + \)\(35\!\cdots\!55\)\( \nu^{4} - \)\(10\!\cdots\!58\)\( \nu^{3} + \)\(30\!\cdots\!29\)\( \nu^{2} - \)\(78\!\cdots\!00\)\( \nu + \)\(12\!\cdots\!88\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} - \beta_{9} - 329 \beta_{3} + 2 \beta_{2} - 329\)
\(\nu^{3}\)\(=\)\(-\beta_{10} - \beta_{8} + 2 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 518 \beta_{2} - 518 \beta_{1} + 507\)
\(\nu^{4}\)\(=\)\(5 \beta_{15} + 23 \beta_{14} - 28 \beta_{13} - 667 \beta_{12} + 179 \beta_{11} + 5 \beta_{10} + 1156 \beta_{9} - 28 \beta_{7} + 667 \beta_{5} - 1156 \beta_{4} + 169652 \beta_{3} + 2170 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-836 \beta_{15} - 524 \beta_{14} + 1972 \beta_{13} + 4125 \beta_{12} - 3383 \beta_{11} - 9150 \beta_{9} + 524 \beta_{8} + 3383 \beta_{6} - 554370 \beta_{3} + 298692 \beta_{2} - 554370\)
\(\nu^{6}\)\(=\)\(-5836 \beta_{10} - 15922 \beta_{8} + 34928 \beta_{7} - 152512 \beta_{6} - 431725 \beta_{5} + 900559 \beta_{4} - 2034630 \beta_{2} - 2034630 \beta_{1} + 97046633\)
\(\nu^{7}\)\(=\)\(572565 \beta_{15} + 271263 \beta_{14} - 1587324 \beta_{13} - 3986667 \beta_{12} + 2441064 \beta_{11} + 572565 \beta_{10} + 10802697 \beta_{9} - 1587324 \beta_{7} + 3986667 \beta_{5} - 10802697 \beta_{4} + 551906403 \beta_{3} + 183300412 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-5282601 \beta_{15} - 8641761 \beta_{14} + 32099124 \beta_{13} + 281615599 \beta_{12} - 107371779 \beta_{11} - 647772562 \beta_{9} + 8641761 \beta_{8} + 107371779 \beta_{6} - 59048539556 \beta_{3} + 1775563910 \beta_{2} - 59048539556\)
\(\nu^{9}\)\(=\)\(-378422176 \beta_{10} - 143807242 \beta_{8} + 1196237150 \beta_{7} - 1724105347 \beta_{6} - 3423291789 \beta_{5} + 9944321700 \beta_{4} - 117156839426 \beta_{2} - 117156839426 \beta_{1} + 503292419706\)
\(\nu^{10}\)\(=\)\(4390552784 \beta_{15} + 4481982956 \beta_{14} - 26315805496 \beta_{13} - 186754914721 \beta_{12} + 73021948952 \beta_{11} + 4390552784 \beta_{10} + 456663160957 \beta_{9} - 26315805496 \beta_{7} + 186754914721 \beta_{5} - 456663160957 \beta_{4} + 37443627318449 \beta_{3} + 1471564841938 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-250995758105 \beta_{15} - 80324939597 \beta_{14} + 877432198318 \beta_{13} + 2773143081867 \beta_{12} - 1233043584356 \beta_{11} - 8238282022815 \beta_{9} + 80324939597 \beta_{8} + 1233043584356 \beta_{6} - 428983578680523 \beta_{3} + 77100304225314 \beta_{2} - 428983578680523\)
\(\nu^{12}\)\(=\)\(-3504195150013 \beta_{10} - 2376500679691 \beta_{8} + 20434118734244 \beta_{7} - 49731246033331 \beta_{6} - 125973043133395 \beta_{5} + 320958151654384 \beta_{4} - 1174618753023786 \beta_{2} - 1174618753023786 \beta_{1} + 24471154459389476\)
\(\nu^{13}\)\(=\)\(168695898866700 \beta_{15} + 47904700551432 \beta_{14} - 635992035431184 \beta_{13} - 2170188177901677 \beta_{12} + 895408747272591 \beta_{11} + 168695898866700 \beta_{10} + 6470068879032954 \beta_{9} - 635992035431184 \beta_{7} + 2170188177901677 \beta_{5} - 6470068879032954 \beta_{4} + 348594831644807490 \beta_{3} + 51872092159731784 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-2728205127440868 \beta_{15} - 1329857678950950 \beta_{14} + 15415544294871864 \beta_{13} + 86250923196750757 \beta_{12} - 34191130563956208 \beta_{11} - 226128810539166427 \beta_{9} + 1329857678950950 \beta_{8} + 34191130563956208 \beta_{6} - 16367314578724777577 \beta_{3} + 912528616328787542 \beta_{2} - 16367314578724777577\)
\(\nu^{15}\)\(=\)\(-114985643505825229 \beta_{10} - 30372185328537499 \beta_{8} + 458672410925314952 \beta_{7} - 656592124316741056 \beta_{6} - 1660400717417678811 \beta_{5} + 4931315898769633989 \beta_{4} - 35497073202576216176 \beta_{2} - 35497073202576216176 \beta_{1} + 274002749435241272739\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
11.6855 + 20.2399i
11.5099 + 19.9358i
6.66586 + 11.5456i
1.38002 + 2.39026i
0.0781534 + 0.135366i
−7.78646 13.4866i
−10.0937 17.4828i
−13.4393 23.2776i
11.6855 20.2399i
11.5099 19.9358i
6.66586 11.5456i
1.38002 2.39026i
0.0781534 0.135366i
−7.78646 + 13.4866i
−10.0937 + 17.4828i
−13.4393 + 23.2776i
2.00000 3.46410i −12.1855 21.1059i −8.00000 13.8564i −5.21821 9.03821i −97.4840 −70.8626 122.738i −64.0000 −175.473 + 303.928i −41.7457
47.2 2.00000 3.46410i −12.0099 20.8018i −8.00000 13.8564i 51.5512 + 89.2893i −96.0793 102.698 + 177.878i −64.0000 −166.976 + 289.211i 412.410
47.3 2.00000 3.46410i −7.16586 12.4116i −8.00000 13.8564i −51.9940 90.0563i −57.3269 36.2078 + 62.7137i −64.0000 18.8008 32.5639i −415.952
47.4 2.00000 3.46410i −1.88002 3.25629i −8.00000 13.8564i 7.78936 + 13.4916i −15.0401 −28.3238 49.0583i −64.0000 114.431 198.200i 62.3149
47.5 2.00000 3.46410i −0.578153 1.00139i −8.00000 13.8564i 40.4037 + 69.9813i −4.62523 −85.9598 148.887i −64.0000 120.831 209.286i 323.230
47.6 2.00000 3.46410i 7.28646 + 12.6205i −8.00000 13.8564i −3.95824 6.85587i 58.2917 79.3358 + 137.414i −64.0000 15.3149 26.5262i −31.6659
47.7 2.00000 3.46410i 9.59367 + 16.6167i −8.00000 13.8564i −34.0877 59.0415i 76.7494 −46.0584 79.7755i −64.0000 −62.5771 + 108.387i −272.701
47.8 2.00000 3.46410i 12.9393 + 22.4116i −8.00000 13.8564i 30.5138 + 52.8515i 103.515 −3.03701 5.26025i −64.0000 −213.352 + 369.536i 244.111
63.1 2.00000 + 3.46410i −12.1855 + 21.1059i −8.00000 + 13.8564i −5.21821 + 9.03821i −97.4840 −70.8626 + 122.738i −64.0000 −175.473 303.928i −41.7457
63.2 2.00000 + 3.46410i −12.0099 + 20.8018i −8.00000 + 13.8564i 51.5512 89.2893i −96.0793 102.698 177.878i −64.0000 −166.976 289.211i 412.410
63.3 2.00000 + 3.46410i −7.16586 + 12.4116i −8.00000 + 13.8564i −51.9940 + 90.0563i −57.3269 36.2078 62.7137i −64.0000 18.8008 + 32.5639i −415.952
63.4 2.00000 + 3.46410i −1.88002 + 3.25629i −8.00000 + 13.8564i 7.78936 13.4916i −15.0401 −28.3238 + 49.0583i −64.0000 114.431 + 198.200i 62.3149
63.5 2.00000 + 3.46410i −0.578153 + 1.00139i −8.00000 + 13.8564i 40.4037 69.9813i −4.62523 −85.9598 + 148.887i −64.0000 120.831 + 209.286i 323.230
63.6 2.00000 + 3.46410i 7.28646 12.6205i −8.00000 + 13.8564i −3.95824 + 6.85587i 58.2917 79.3358 137.414i −64.0000 15.3149 + 26.5262i −31.6659
63.7 2.00000 + 3.46410i 9.59367 16.6167i −8.00000 + 13.8564i −34.0877 + 59.0415i 76.7494 −46.0584 + 79.7755i −64.0000 −62.5771 108.387i −272.701
63.8 2.00000 + 3.46410i 12.9393 22.4116i −8.00000 + 13.8564i 30.5138 52.8515i 103.515 −3.03701 + 5.26025i −64.0000 −213.352 369.536i 244.111
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.6.c.b 16
37.c even 3 1 inner 74.6.c.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.c.b 16 1.a even 1 1 trivial
74.6.c.b 16 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(35\!\cdots\!84\)\( T_{3}^{6} + \)\(13\!\cdots\!28\)\( T_{3}^{5} + \)\(41\!\cdots\!52\)\( T_{3}^{4} + \)\(18\!\cdots\!40\)\( T_{3}^{3} + \)\(73\!\cdots\!56\)\( T_{3}^{2} + \)\(78\!\cdots\!36\)\( T_{3} + \)\(69\!\cdots\!84\)\( \)">\(T_{3}^{16} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 4 T + T^{2} )^{8} \)
$3$ \( 69665869241213184 + 78293103562734336 T + 73109589781395456 T^{2} + 18289188965946240 T^{3} + 4190985837670752 T^{4} + 132480919789728 T^{5} + 35031371484784 T^{6} + 831505714520 T^{7} + 201549550188 T^{8} + 2293763524 T^{9} + 594442958 T^{10} + 6808806 T^{11} + 1233767 T^{12} + 8092 T^{13} + 1353 T^{14} + 8 T^{15} + T^{16} \)
$5$ \( \)\(21\!\cdots\!81\)\( + \)\(30\!\cdots\!26\)\( T + \)\(54\!\cdots\!46\)\( T^{2} + \)\(19\!\cdots\!64\)\( T^{3} + \)\(26\!\cdots\!64\)\( T^{4} + 8554663569070451490 T^{5} + 13853682253767833148 T^{6} - 174190420812579726 T^{7} + 6865089107277385 T^{8} - 52003792368166 T^{9} + 1807569881596 T^{10} - 13025134662 T^{11} + 272758632 T^{12} - 1069272 T^{13} + 20830 T^{14} - 70 T^{15} + T^{16} \)
$7$ \( \)\(33\!\cdots\!36\)\( + \)\(60\!\cdots\!00\)\( T + \)\(10\!\cdots\!08\)\( T^{2} + \)\(17\!\cdots\!84\)\( T^{3} + \)\(49\!\cdots\!60\)\( T^{4} + \)\(44\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + 90368136537017166816 T^{7} + 1450186440125167192 T^{8} + 6183297696583896 T^{9} + 81620546294608 T^{10} + 238535800054 T^{11} + 3316853213 T^{12} + 4914548 T^{13} + 67503 T^{14} + 32 T^{15} + T^{16} \)
$11$ \( ( 384628853882268672 + 5590149264449024 T - 904414241063008 T^{2} + 1097282666720 T^{3} + 62942708960 T^{4} - 46256112 T^{5} - 573286 T^{6} + 120 T^{7} + T^{8} )^{2} \)
$13$ \( \)\(38\!\cdots\!04\)\( + \)\(31\!\cdots\!24\)\( T + \)\(27\!\cdots\!48\)\( T^{2} + \)\(71\!\cdots\!12\)\( T^{3} + \)\(38\!\cdots\!08\)\( T^{4} + \)\(54\!\cdots\!16\)\( T^{5} + \)\(36\!\cdots\!88\)\( T^{6} + \)\(23\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!00\)\( T^{8} - 72639978445637068896 T^{9} + 671462671676430472 T^{10} - 490969748032380 T^{11} + 1727919155121 T^{12} - 1770199510 T^{13} + 2792595 T^{14} - 1614 T^{15} + T^{16} \)
$17$ \( \)\(59\!\cdots\!41\)\( - \)\(24\!\cdots\!40\)\( T + \)\(15\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!50\)\( T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(53\!\cdots\!48\)\( T^{6} - \)\(53\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!55\)\( T^{8} - \)\(43\!\cdots\!04\)\( T^{9} + 84283837660222299312 T^{10} - 22458174406253832 T^{11} + 37544878436002 T^{12} - 4741339632 T^{13} + 6996716 T^{14} - 216 T^{15} + T^{16} \)
$19$ \( \)\(71\!\cdots\!64\)\( + \)\(64\!\cdots\!92\)\( T + \)\(37\!\cdots\!68\)\( T^{2} + \)\(13\!\cdots\!16\)\( T^{3} + \)\(34\!\cdots\!80\)\( T^{4} + \)\(62\!\cdots\!08\)\( T^{5} + \)\(84\!\cdots\!56\)\( T^{6} + \)\(75\!\cdots\!16\)\( T^{7} + \)\(52\!\cdots\!96\)\( T^{8} + \)\(22\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!86\)\( T^{10} + 336525195920696968 T^{11} + 182238870243527 T^{12} + 29950315106 T^{13} + 15177577 T^{14} + 950 T^{15} + T^{16} \)
$23$ \( ( \)\(41\!\cdots\!16\)\( + \)\(10\!\cdots\!28\)\( T - \)\(27\!\cdots\!52\)\( T^{2} - 112317952838168592 T^{3} + 198205183306896 T^{4} + 20906590516 T^{5} - 27455214 T^{6} - 858 T^{7} + T^{8} )^{2} \)
$29$ \( ( -\)\(32\!\cdots\!16\)\( + \)\(79\!\cdots\!40\)\( T - \)\(34\!\cdots\!01\)\( T^{2} - 7177553414587432140 T^{3} + 3713827907601399 T^{4} + 140283111308 T^{5} - 110701823 T^{6} - 556 T^{7} + T^{8} )^{2} \)
$31$ \( ( -\)\(52\!\cdots\!36\)\( - \)\(56\!\cdots\!16\)\( T - \)\(16\!\cdots\!20\)\( T^{2} - 1942055572524300256 T^{3} - 1082949080157920 T^{4} - 263413976944 T^{5} - 5213350 T^{6} + 8408 T^{7} + T^{8} )^{2} \)
$37$ \( \)\(53\!\cdots\!01\)\( - \)\(27\!\cdots\!08\)\( T + \)\(47\!\cdots\!37\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} - \)\(39\!\cdots\!31\)\( T^{4} - \)\(98\!\cdots\!08\)\( T^{5} - \)\(23\!\cdots\!06\)\( T^{6} + \)\(22\!\cdots\!88\)\( T^{7} + \)\(13\!\cdots\!62\)\( T^{8} + \)\(33\!\cdots\!84\)\( T^{9} - \)\(48\!\cdots\!94\)\( T^{10} - 29659406451915847356 T^{11} - 1705569418281431 T^{12} - 897318691240 T^{13} + 42275613 T^{14} - 3556 T^{15} + T^{16} \)
$41$ \( \)\(21\!\cdots\!21\)\( + \)\(60\!\cdots\!08\)\( T + \)\(42\!\cdots\!76\)\( T^{2} - \)\(86\!\cdots\!48\)\( T^{3} + \)\(27\!\cdots\!14\)\( T^{4} - \)\(25\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(21\!\cdots\!92\)\( T^{7} + \)\(43\!\cdots\!87\)\( T^{8} - \)\(29\!\cdots\!04\)\( T^{9} + \)\(34\!\cdots\!52\)\( T^{10} - \)\(91\!\cdots\!72\)\( T^{11} + 167359749151963198 T^{12} - 1907576189008 T^{13} + 547526144 T^{14} + 8080 T^{15} + T^{16} \)
$43$ \( ( -\)\(37\!\cdots\!56\)\( - \)\(73\!\cdots\!44\)\( T - \)\(10\!\cdots\!04\)\( T^{2} + \)\(11\!\cdots\!56\)\( T^{3} + 145671773493025704 T^{4} - 5887218886816 T^{5} - 671523324 T^{6} + 9506 T^{7} + T^{8} )^{2} \)
$47$ \( ( \)\(17\!\cdots\!88\)\( + \)\(24\!\cdots\!36\)\( T - \)\(34\!\cdots\!96\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + 275681353662731808 T^{4} + 12279616491040 T^{5} - 896915078 T^{6} - 14696 T^{7} + T^{8} )^{2} \)
$53$ \( \)\(28\!\cdots\!16\)\( + \)\(43\!\cdots\!32\)\( T + \)\(69\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!68\)\( T^{3} + \)\(48\!\cdots\!88\)\( T^{4} + \)\(27\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!64\)\( T^{6} + \)\(90\!\cdots\!80\)\( T^{7} + \)\(50\!\cdots\!24\)\( T^{8} + \)\(14\!\cdots\!76\)\( T^{9} + \)\(69\!\cdots\!36\)\( T^{10} + \)\(15\!\cdots\!84\)\( T^{11} + 6140356029217106089 T^{12} + 82266864154922 T^{13} + 3128497859 T^{14} + 29250 T^{15} + T^{16} \)
$59$ \( \)\(45\!\cdots\!56\)\( + \)\(94\!\cdots\!52\)\( T + \)\(14\!\cdots\!28\)\( T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(54\!\cdots\!96\)\( T^{4} + \)\(29\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(25\!\cdots\!80\)\( T^{7} + \)\(66\!\cdots\!32\)\( T^{8} - \)\(29\!\cdots\!20\)\( T^{9} + \)\(98\!\cdots\!16\)\( T^{10} - \)\(58\!\cdots\!08\)\( T^{11} + 1112378861521150969 T^{12} - 61747284851550 T^{13} + 2979142267 T^{14} - 61478 T^{15} + T^{16} \)
$61$ \( \)\(33\!\cdots\!89\)\( - \)\(63\!\cdots\!22\)\( T + \)\(13\!\cdots\!22\)\( T^{2} + \)\(26\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!84\)\( T^{4} + \)\(26\!\cdots\!74\)\( T^{5} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(72\!\cdots\!82\)\( T^{7} + \)\(26\!\cdots\!17\)\( T^{8} + \)\(74\!\cdots\!34\)\( T^{9} + \)\(17\!\cdots\!52\)\( T^{10} + \)\(35\!\cdots\!82\)\( T^{11} + 63372741788610256692 T^{12} + 960406131787088 T^{13} + 13349113610 T^{14} + 132002 T^{15} + T^{16} \)
$67$ \( \)\(12\!\cdots\!04\)\( + \)\(11\!\cdots\!92\)\( T + \)\(16\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} + \)\(56\!\cdots\!72\)\( T^{5} + \)\(39\!\cdots\!08\)\( T^{6} + \)\(15\!\cdots\!24\)\( T^{7} + \)\(80\!\cdots\!28\)\( T^{8} + \)\(24\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} + \)\(24\!\cdots\!84\)\( T^{11} + 8533107028757861591 T^{12} + 129006533987766 T^{13} + 4070918873 T^{14} + 45658 T^{15} + T^{16} \)
$71$ \( \)\(78\!\cdots\!16\)\( - \)\(16\!\cdots\!36\)\( T + \)\(32\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!12\)\( T^{4} - \)\(65\!\cdots\!96\)\( T^{5} + \)\(60\!\cdots\!64\)\( T^{6} - \)\(17\!\cdots\!88\)\( T^{7} + \)\(11\!\cdots\!48\)\( T^{8} - \)\(20\!\cdots\!28\)\( T^{9} + \)\(11\!\cdots\!24\)\( T^{10} - \)\(15\!\cdots\!86\)\( T^{11} + 8328205614665344597 T^{12} - 64276029222744 T^{13} + 3483278135 T^{14} - 13236 T^{15} + T^{16} \)
$73$ \( ( \)\(87\!\cdots\!08\)\( - \)\(19\!\cdots\!56\)\( T + \)\(11\!\cdots\!44\)\( T^{2} + \)\(52\!\cdots\!48\)\( T^{3} - 23596937207750781168 T^{4} + 578550085039040 T^{5} + 490263492 T^{6} - 135316 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(17\!\cdots\!76\)\( - \)\(31\!\cdots\!16\)\( T + \)\(55\!\cdots\!64\)\( T^{2} - \)\(29\!\cdots\!48\)\( T^{3} + \)\(50\!\cdots\!52\)\( T^{4} - \)\(59\!\cdots\!40\)\( T^{5} + \)\(30\!\cdots\!96\)\( T^{6} - \)\(32\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!28\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!50\)\( T^{10} - \)\(20\!\cdots\!52\)\( T^{11} + \)\(31\!\cdots\!35\)\( T^{12} - 2030655706103306 T^{13} + 26417529705 T^{14} - 123974 T^{15} + T^{16} \)
$83$ \( \)\(27\!\cdots\!76\)\( - \)\(52\!\cdots\!60\)\( T + \)\(73\!\cdots\!36\)\( T^{2} - \)\(53\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!32\)\( T^{4} - \)\(98\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!52\)\( T^{6} - \)\(42\!\cdots\!08\)\( T^{7} + \)\(17\!\cdots\!56\)\( T^{8} - \)\(13\!\cdots\!84\)\( T^{9} + \)\(68\!\cdots\!50\)\( T^{10} + \)\(21\!\cdots\!46\)\( T^{11} + \)\(16\!\cdots\!67\)\( T^{12} + 287828390304756 T^{13} + 14797117105 T^{14} + 16936 T^{15} + T^{16} \)
$89$ \( \)\(11\!\cdots\!81\)\( + \)\(88\!\cdots\!60\)\( T + \)\(85\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!02\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(86\!\cdots\!32\)\( T^{6} - \)\(17\!\cdots\!84\)\( T^{7} + \)\(34\!\cdots\!79\)\( T^{8} - \)\(33\!\cdots\!72\)\( T^{9} + \)\(51\!\cdots\!08\)\( T^{10} - \)\(35\!\cdots\!68\)\( T^{11} + \)\(56\!\cdots\!90\)\( T^{12} - 1794383613225256 T^{13} + 27348500600 T^{14} - 52892 T^{15} + T^{16} \)
$97$ \( ( -\)\(49\!\cdots\!16\)\( + \)\(32\!\cdots\!32\)\( T + \)\(54\!\cdots\!21\)\( T^{2} + \)\(57\!\cdots\!54\)\( T^{3} - \)\(45\!\cdots\!53\)\( T^{4} - 8466211436628352 T^{5} - 10543123481 T^{6} + 273070 T^{7} + T^{8} )^{2} \)
show more
show less