Properties

Label 14.14.c.a
Level $14$
Weight $14$
Character orbit 14.c
Analytic conductor $15.012$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(15.0123300533\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + 48219450150076326 x^{2} + 340591169755393830 x + 2353882543131872025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + ( -64 - 64 \beta_{1} ) q^{2} + ( 45 \beta_{1} + \beta_{4} ) q^{3} + 4096 \beta_{1} q^{4} + ( -16102 - 16103 \beta_{1} - 5 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{7} ) q^{5} + ( 2880 - 64 \beta_{2} ) q^{6} + ( 11651 + 51746 \beta_{1} - 10 \beta_{2} - 9 \beta_{3} - 25 \beta_{4} - \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{7} + 262144 q^{8} + ( -996183 - 996177 \beta_{1} - 495 \beta_{2} - 5 \beta_{3} + 484 \beta_{4} + 22 \beta_{5} + 11 \beta_{6} - 6 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -64 - 64 \beta_{1} ) q^{2} + ( 45 \beta_{1} + \beta_{4} ) q^{3} + 4096 \beta_{1} q^{4} + ( -16102 - 16103 \beta_{1} - 5 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{7} ) q^{5} + ( 2880 - 64 \beta_{2} ) q^{6} + ( 11651 + 51746 \beta_{1} - 10 \beta_{2} - 9 \beta_{3} - 25 \beta_{4} - \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{7} + 262144 q^{8} + ( -996183 - 996177 \beta_{1} - 495 \beta_{2} - 5 \beta_{3} + 484 \beta_{4} + 22 \beta_{5} + 11 \beta_{6} - 6 \beta_{7} ) q^{9} + ( 1030592 \beta_{1} - 320 \beta_{4} - 64 \beta_{7} ) q^{10} + ( 251467 \beta_{1} + 10 \beta_{2} + 10 \beta_{3} + 1574 \beta_{4} + 10 \beta_{5} + 20 \beta_{6} - 113 \beta_{7} ) q^{11} + ( -184320 - 184320 \beta_{1} + 4096 \beta_{2} - 4096 \beta_{4} ) q^{12} + ( 6727617 + 3678 \beta_{2} + 260 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} ) q^{13} + ( 2566528 - 745216 \beta_{1} + 2304 \beta_{2} + 192 \beta_{3} - 512 \beta_{4} - 128 \beta_{5} + 64 \beta_{6} - 448 \beta_{7} ) q^{14} + ( -11122290 - 21479 \beta_{2} + 872 \beta_{3} + 251 \beta_{5} - 251 \beta_{6} ) q^{15} + ( -16777216 - 16777216 \beta_{1} ) q^{16} + ( 41332828 \beta_{1} - 945 \beta_{2} - 945 \beta_{3} - 940 \beta_{4} - 945 \beta_{5} - 1890 \beta_{6} + 1798 \beta_{7} ) q^{17} + ( 63755328 \beta_{1} - 704 \beta_{2} - 704 \beta_{3} - 30976 \beta_{4} - 704 \beta_{5} - 1408 \beta_{6} + 384 \beta_{7} ) q^{18} + ( -105896876 - 105894083 \beta_{1} - 87866 \beta_{2} + 2989 \beta_{3} + 88062 \beta_{4} - 392 \beta_{5} - 196 \beta_{6} - 2793 \beta_{7} ) q^{19} + ( 65953792 + 20480 \beta_{2} + 4096 \beta_{3} ) q^{20} + ( 71837991 + 85850670 \beta_{1} + 168041 \beta_{2} + 9671 \beta_{3} - 208249 \beta_{4} - 1300 \beta_{5} - 3179 \beta_{6} - 2667 \beta_{7} ) q^{21} + ( 16086656 - 102016 \beta_{2} + 5952 \beta_{3} + 640 \beta_{5} - 640 \beta_{6} ) q^{22} + ( 71699947 + 71717984 \beta_{1} + 300633 \beta_{2} + 17672 \beta_{3} - 300998 \beta_{4} + 730 \beta_{5} + 365 \beta_{6} - 18037 \beta_{7} ) q^{23} + ( 11796480 \beta_{1} + 262144 \beta_{4} ) q^{24} + ( 117915978 \beta_{1} - 1254 \beta_{2} - 1254 \beta_{3} + 212392 \beta_{4} - 1254 \beta_{5} - 2508 \beta_{6} - 35632 \beta_{7} ) q^{25} + ( -430567488 - 430585024 \beta_{1} - 235840 \beta_{2} - 17088 \beta_{3} + 236288 \beta_{4} - 896 \beta_{5} - 448 \beta_{6} + 17536 \beta_{7} ) q^{26} + ( -1301050206 - 662282 \beta_{2} - 47281 \beta_{3} + 11606 \beta_{5} - 11606 \beta_{6} ) q^{27} + ( -211980288 - 164257792 \beta_{1} - 106496 \beta_{2} + 24576 \beta_{3} + 135168 \beta_{4} + 12288 \beta_{5} + 8192 \beta_{6} ) q^{28} + ( 2274622219 - 878300 \beta_{2} + 45966 \beta_{3} + 251 \beta_{5} - 251 \beta_{6} ) q^{29} + ( 711826560 + 711738624 \beta_{1} + 1358592 \beta_{2} - 71872 \beta_{3} - 1342528 \beta_{4} - 32128 \beta_{5} - 16064 \beta_{6} + 87936 \beta_{7} ) q^{30} + ( -376947098 \beta_{1} + 14567 \beta_{2} + 14567 \beta_{3} + 332713 \beta_{4} + 14567 \beta_{5} + 29134 \beta_{6} + 14360 \beta_{7} ) q^{31} + 1073741824 \beta_{1} q^{32} + ( -4222770669 - 4222930287 \beta_{1} - 1235388 \beta_{2} - 161564 \beta_{3} + 1233442 \beta_{4} + 3892 \beta_{5} + 1946 \beta_{6} + 159618 \beta_{7} ) q^{33} + ( 2645416064 + 181120 \beta_{2} + 5888 \beta_{3} - 60480 \beta_{5} + 60480 \beta_{6} ) q^{34} + ( 3205425859 + 7481616503 \beta_{1} + 3378299 \beta_{2} + 68428 \beta_{3} + 2188864 \beta_{4} - 12126 \beta_{5} + 5202 \beta_{6} - 171913 \beta_{7} ) q^{35} + ( 4080365568 + 2072576 \beta_{2} + 65536 \beta_{3} - 45056 \beta_{5} + 45056 \beta_{6} ) q^{36} + ( -4737768227 - 4737789404 \beta_{1} + 4217364 \beta_{2} - 98850 \beta_{3} - 4295037 \beta_{4} + 155346 \beta_{5} + 77673 \beta_{6} + 21177 \beta_{7} ) q^{37} + ( 6777221312 \beta_{1} + 12544 \beta_{2} + 12544 \beta_{3} - 5635968 \beta_{4} + 12544 \beta_{5} + 25088 \beta_{6} + 178752 \beta_{7} ) q^{38} + ( -8943975372 \beta_{1} + 89570 \beta_{2} + 89570 \beta_{3} + 8658807 \beta_{4} + 89570 \beta_{5} + 179140 \beta_{6} - 402441 \beta_{7} ) q^{39} + ( -4221042688 - 4221304832 \beta_{1} - 1310720 \beta_{2} - 262144 \beta_{3} + 1310720 \beta_{4} + 262144 \beta_{7} ) q^{40} + ( 14195216883 - 22591286 \beta_{2} - 47396 \beta_{3} + 55237 \beta_{5} - 55237 \beta_{6} ) q^{41} + ( 896640768 - 4598370624 \beta_{1} + 2656512 \beta_{2} - 365056 \beta_{3} + 10874880 \beta_{4} - 120256 \beta_{5} + 83200 \beta_{6} + 739200 \beta_{7} ) q^{42} + ( -11441235412 + 5839210 \beta_{2} - 1094822 \beta_{3} + 116314 \beta_{5} - 116314 \beta_{6} ) q^{43} + ( -1029545984 - 1030008832 \beta_{1} + 6488064 \beta_{2} - 421888 \beta_{3} - 6447104 \beta_{4} - 81920 \beta_{5} - 40960 \beta_{6} + 462848 \beta_{7} ) q^{44} + ( 29968330863 \beta_{1} - 131621 \beta_{2} - 131621 \beta_{3} - 33877402 \beta_{4} - 131621 \beta_{5} - 263242 \beta_{6} - 557940 \beta_{7} ) q^{45} + ( -4589950976 \beta_{1} - 23360 \beta_{2} - 23360 \beta_{3} + 19263872 \beta_{4} - 23360 \beta_{5} - 46720 \beta_{6} + 1154368 \beta_{7} ) q^{46} + ( -20349061206 - 20346826278 \beta_{1} - 20444900 \beta_{2} + 2076973 \beta_{3} + 20286945 \beta_{4} + 315910 \beta_{5} + 157955 \beta_{6} - 2234928 \beta_{7} ) q^{47} + ( 754974720 - 16777216 \beta_{2} ) q^{48} + ( -10758376370 - 6304773272 \beta_{1} + 4002194 \beta_{2} + 116340 \beta_{3} + 46538758 \beta_{4} + 389361 \beta_{5} - 280553 \beta_{6} - 1018318 \beta_{7} ) q^{49} + ( 7544342144 - 13432576 \beta_{2} + 2440960 \beta_{3} - 80256 \beta_{5} + 80256 \beta_{6} ) q^{50} + ( 3808820160 + 3812855487 \beta_{1} + 123562218 \beta_{2} + 4333969 \beta_{3} - 123263576 \beta_{4} - 597284 \beta_{5} - 298642 \beta_{6} - 4035327 \beta_{7} ) q^{51} + ( 27557441536 \beta_{1} + 28672 \beta_{2} + 28672 \beta_{3} - 15122432 \beta_{4} + 28672 \beta_{5} + 57344 \beta_{6} - 1122304 \beta_{7} ) q^{52} + ( -21903294390 \beta_{1} - 1064387 \beta_{2} - 1064387 \beta_{3} + 53619681 \beta_{4} - 1064387 \beta_{5} - 2128774 \beta_{6} + 3995379 \beta_{7} ) q^{53} + ( 83267213184 + 83268753600 \beta_{1} + 41643264 \beta_{2} + 2283200 \beta_{3} - 40900480 \beta_{4} - 1485568 \beta_{5} - 742784 \beta_{6} - 1540416 \beta_{7} ) q^{54} + ( 87656211869 - 92765798 \beta_{2} + 4163051 \beta_{3} + 404397 \beta_{5} - 404397 \beta_{6} ) q^{55} + ( 3054239744 + 13564903424 \beta_{1} - 2621440 \beta_{2} - 2359296 \beta_{3} - 6553600 \beta_{4} - 262144 \beta_{5} - 786432 \beta_{6} + 1835008 \beta_{7} ) q^{56} + ( -225978687855 - 129108872 \beta_{2} - 3465228 \beta_{3} + 288366 \beta_{5} - 288366 \beta_{6} ) q^{57} + ( -145575822016 - 145578795968 \beta_{1} + 56195136 \beta_{2} - 2957888 \beta_{3} - 56179072 \beta_{4} - 32128 \beta_{5} - 16064 \beta_{6} + 2973952 \beta_{7} ) q^{58} + ( 48491132435 \beta_{1} + 156282 \beta_{2} + 156282 \beta_{3} + 83719471 \beta_{4} + 156282 \beta_{5} + 312564 \beta_{6} + 4148210 \beta_{7} ) q^{59} + ( -45551271936 \beta_{1} + 1028096 \beta_{2} + 1028096 \beta_{3} + 85921792 \beta_{4} + 1028096 \beta_{5} + 2056192 \beta_{6} - 5627904 \beta_{7} ) q^{60} + ( 43862083413 + 43855469318 \beta_{1} - 190830916 \beta_{2} - 6841182 \beta_{3} + 190603829 \beta_{4} + 454174 \beta_{5} + 227087 \beta_{6} + 6614095 \beta_{7} ) q^{61} + ( -24123695232 - 23158208 \beta_{2} - 2783616 \beta_{3} + 932288 \beta_{5} - 932288 \beta_{6} ) q^{62} + ( 470328184089 + 129152509860 \beta_{1} + 287379399 \beta_{2} + 777347 \beta_{3} - 154160383 \beta_{4} - 460738 \beta_{5} + 2584084 \beta_{6} - 3446751 \beta_{7} ) q^{63} + 68719476736 q^{64} + ( -422458451709 - 422467214217 \beta_{1} + 6801387 \beta_{2} - 9682479 \beta_{3} - 7721358 \beta_{4} + 1839942 \beta_{5} + 919971 \beta_{6} + 8762508 \beta_{7} ) q^{65} + ( 270267538368 \beta_{1} - 124544 \beta_{2} - 124544 \beta_{3} - 78940288 \beta_{4} - 124544 \beta_{5} - 249088 \beta_{6} - 10215552 \beta_{7} ) q^{66} + ( 61052095491 \beta_{1} + 3955250 \beta_{2} + 3955250 \beta_{3} - 187363327 \beta_{4} + 3955250 \beta_{5} + 7910500 \beta_{6} - 9218276 \beta_{7} ) q^{67} + ( -169306628096 - 169299263488 \beta_{1} - 7720960 \beta_{2} + 3493888 \beta_{3} + 3850240 \beta_{4} + 7741440 \beta_{5} + 3870720 \beta_{6} - 7364608 \beta_{7} ) q^{68} + ( 755652156318 + 251346809 \beta_{2} - 13477631 \beta_{3} - 6598760 \beta_{5} + 6598760 \beta_{6} ) q^{69} + ( 273665198784 - 205150525376 \beta_{1} - 355522368 \beta_{2} + 7399104 \beta_{3} + 215102144 \beta_{4} + 1108992 \beta_{5} + 776064 \beta_{6} + 3270400 \beta_{7} ) q^{70} + ( -409198005718 + 455128476 \beta_{2} + 2470144 \beta_{3} - 3691390 \beta_{5} + 3691390 \beta_{6} ) q^{71} + ( -261143396352 - 261141823488 \beta_{1} - 129761280 \beta_{2} - 1310720 \beta_{3} + 126877696 \beta_{4} + 5767168 \beta_{5} + 2883584 \beta_{6} - 1572864 \beta_{7} ) q^{72} + ( 873078252549 \beta_{1} + 1283898 \beta_{2} + 1283898 \beta_{3} + 60762044 \beta_{4} + 1283898 \beta_{5} + 2567796 \beta_{6} + 28693156 \beta_{7} ) q^{73} + ( 303218521856 \beta_{1} - 4971072 \beta_{2} - 4971072 \beta_{3} + 274882368 \beta_{4} - 4971072 \beta_{5} - 9942144 \beta_{6} - 1355328 \beta_{7} ) q^{74} + ( -592433120394 - 592477579824 \beta_{1} + 203861266 \beta_{2} - 38961658 \beta_{3} - 198363494 \beta_{4} - 10995544 \beta_{5} - 5497772 \beta_{6} + 44459430 \beta_{7} ) q^{75} + ( 433753604096 + 359096320 \beta_{2} - 13045760 \beta_{3} + 802816 \beta_{5} - 802816 \beta_{6} ) q^{76} + ( 923759592333 + 252417728897 \beta_{1} - 178912220 \beta_{2} + 21855430 \beta_{3} - 266612699 \beta_{4} - 5027567 \beta_{5} - 5729077 \beta_{6} + 3098417 \beta_{7} ) q^{77} + ( -572440180032 - 565628608 \beta_{2} + 14291264 \beta_{3} + 5732480 \beta_{5} - 5732480 \beta_{6} ) q^{78} + ( -254927114361 - 254928612934 \beta_{1} - 1659026539 \beta_{2} + 5161560 \beta_{3} + 1665686672 \beta_{4} - 13320266 \beta_{5} - 6660133 \beta_{6} + 1498573 \beta_{7} ) q^{79} + ( 270163509248 \beta_{1} - 83886080 \beta_{4} - 16777216 \beta_{7} ) q^{80} + ( -3391264446 \beta_{1} - 2078609 \beta_{2} - 2078609 \beta_{3} - 1970484658 \beta_{4} - 2078609 \beta_{5} - 4157218 \beta_{6} + 6666504 \beta_{7} ) q^{81} + ( -908493880512 - 908497917504 \beta_{1} + 1442307136 \beta_{2} - 501824 \beta_{3} - 1438771968 \beta_{4} - 7070336 \beta_{5} - 3535168 \beta_{6} + 4036992 \beta_{7} ) q^{82} + ( 878277680162 - 280442670 \beta_{2} + 38132830 \beta_{3} + 22157996 \beta_{5} - 22157996 \beta_{6} ) q^{83} + ( -351633420288 - 57348624384 \beta_{1} - 858312704 \beta_{2} - 16248832 \beta_{3} + 156995584 \beta_{4} + 13021184 \beta_{5} + 7696384 \beta_{6} - 36384768 \beta_{7} ) q^{84} + ( 72296851091 + 2341333717 \beta_{2} + 26341667 \beta_{3} - 1108083 \beta_{5} + 1108083 \beta_{6} ) q^{85} + ( 732239066368 + 732294246784 \beta_{1} - 381153536 \beta_{2} + 62624512 \beta_{3} + 388597632 \beta_{4} - 14888192 \beta_{5} - 7444096 \beta_{6} - 55180416 \beta_{7} ) q^{86} + ( 2425822165698 \beta_{1} - 718228 \beta_{2} - 718228 \beta_{3} + 1891196803 \beta_{4} - 718228 \beta_{5} - 1436456 \beta_{6} - 57617673 \beta_{7} ) q^{87} + ( 65920565248 \beta_{1} + 2621440 \beta_{2} + 2621440 \beta_{3} + 412614656 \beta_{4} + 2621440 \beta_{5} + 5242880 \beta_{6} - 29622272 \beta_{7} ) q^{88} + ( -2009050981251 - 2008887235401 \beta_{1} - 884889932 \beta_{2} + 154858552 \beta_{3} + 876002634 \beta_{4} + 17774596 \beta_{5} + 8887298 \beta_{6} - 163745850 \beta_{7} ) q^{89} + ( 1917937467072 + 2185001216 \beta_{2} + 52555648 \beta_{3} - 8423744 \beta_{5} + 8423744 \beta_{6} ) q^{90} + ( -1939807285143 - 1020129394382 \beta_{1} + 545248893 \beta_{2} - 60299631 \beta_{3} - 1945017242 \beta_{4} + 2086280 \beta_{5} - 15578794 \beta_{6} + 20320202 \beta_{7} ) q^{91} + ( -293682982912 - 1229897728 \beta_{2} - 70889472 \beta_{3} - 1495040 \beta_{5} + 1495040 \beta_{6} ) q^{92} + ( -848041986087 - 848045730642 \beta_{1} - 1918372282 \beta_{2} - 20095384 \beta_{3} + 1902021453 \beta_{4} + 32701658 \beta_{5} + 16350829 \beta_{6} + 3744555 \beta_{7} ) q^{93} + ( 1302196881792 \beta_{1} - 10109120 \beta_{2} - 10109120 \beta_{3} - 1298364480 \beta_{4} - 10109120 \beta_{5} - 20218240 \beta_{6} + 143035392 \beta_{7} ) q^{94} + ( -355793550668 \beta_{1} - 21925839 \beta_{2} - 21925839 \beta_{3} - 3221346628 \beta_{4} - 21925839 \beta_{5} - 43851678 \beta_{6} + 90695245 \beta_{7} ) q^{95} + ( -48318382080 - 48318382080 \beta_{1} + 1073741824 \beta_{2} - 1073741824 \beta_{4} ) q^{96} + ( 6794839540729 + 1842491972 \beta_{2} - 53658078 \beta_{3} - 30222647 \beta_{5} + 30222647 \beta_{6} ) q^{97} + ( 284965425920 + 688485767424 \beta_{1} - 3259540032 \beta_{2} + 32807488 \beta_{3} + 299014912 \beta_{4} - 42874496 \beta_{5} - 24919104 \beta_{6} + 50320256 \beta_{7} ) q^{98} + ( -3231197379261 - 2958149165 \beta_{2} - 22034725 \beta_{3} + 30229148 \beta_{5} - 30229148 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 256q^{2} - 182q^{3} - 16384q^{4} - 64400q^{5} + 23296q^{6} - 113736q^{7} + 2097152q^{8} - 3983752q^{9} + O(q^{10}) \) \( 8q - 256q^{2} - 182q^{3} - 16384q^{4} - 64400q^{5} + 23296q^{6} - 113736q^{7} + 2097152q^{8} - 3983752q^{9} - 4121600q^{10} - 1008790q^{11} - 745472q^{12} + 53807264q^{13} + 23506560q^{14} - 88888916q^{15} - 67108864q^{16} - 165333028q^{17} - 254960128q^{18} - 423405794q^{19} + 527564800q^{20} + 231089600q^{21} + 129125120q^{22} + 286233866q^{23} - 47710208q^{24} - 472017432q^{25} - 1721832448q^{26} - 10405941644q^{27} - 1038557184q^{28} + 18200674816q^{29} + 2844445312q^{30} + 1507094246q^{31} - 4294967296q^{32} - 16888935028q^{33} + 21162627584q^{34} - 4300332526q^{35} + 32634896384q^{36} - 18959705336q^{37} - 27097970816q^{38} + 35759388756q^{39} - 16882073600q^{40} + 113651910624q^{41} + 25531294208q^{42} - 91557619424q^{43} - 4132003840q^{44} - 119804452768q^{45} + 18318967424q^{46} - 81351201078q^{47} + 6106906624q^{48} - 60954502168q^{49} + 60418231296q^{50} + 14996824142q^{51} - 110197276672q^{52} + 87497947440q^{53} + 332990132608q^{54} + 701637410348q^{55} - 29815209984q^{56} - 1807326928264q^{57} - 582421594112q^{58} - 194140265102q^{59} + 182044499968q^{60} + 175816313120q^{61} - 192908063488q^{62} + 3245184239332q^{63} + 549755813888q^{64} - 1689866774568q^{65} - 1080891841792q^{66} - 243815218758q^{67} - 677204082688q^{68} + 6044157952784q^{69} + 3010938632576q^{70} - 3275394679072q^{71} - 1044316684288q^{72} - 3492491920596q^{73} - 1213421141504q^{74} - 2370218127424q^{75} + 3468540264448q^{76} + 6381735922240q^{77} - 4577201760768q^{78} - 1016380081246q^{79} - 1080452710400q^{80} + 17492694092q^{81} - 3636861139968q^{82} + 7027495743296q^{83} - 2580545830912q^{84} + 569114840528q^{85} + 2929843821568q^{86} - 9706955821052q^{87} - 264448245760q^{88} - 8034124428036q^{89} + 15334969954304q^{90} - 11436513503632q^{91} - 2344827830272q^{92} - 3388371390552q^{93} - 5206476868992q^{94} + 1429435505438q^{95} - 195421011968q^{96} + 54351131725632q^{97} - 461748942592q^{98} - 25837834576328q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + 48219450150076326 x^{2} + 340591169755393830 x + 2353882543131872025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-2725633210820746361507626 \nu^{7} + 24366373153557504504493202 \nu^{6} - 1892474572652302255981857331644 \nu^{5} - 1200775420548354246138219704800663 \nu^{4} - 1304293656225820282563152576931543652 \nu^{3} - 411017803484651058911794859641331539170 \nu^{2} - 131416590523498454214760714210465635758166 \nu - 928243654015606607913572166220307666610405\)\()/ \)\(90\!\cdots\!05\)\( \)
\(\beta_{2}\)\(=\)\((\)\(1497539527755016787080970 \nu^{7} - 649774079230724020048958011 \nu^{6} + 1040718590427502721052507134499 \nu^{5} + 334508904560983794369850438465742 \nu^{4} + 377114080808092465832636282858779397 \nu^{3} + 945642625562751927830056719705805506 \nu^{2} + 6567252280857293171367901837984522605 \nu - 4261750953452276958443698613494890145218405\)\()/ \)\(25\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-1693049396148364941651230 \nu^{7} + 2229815004787404124450149049 \nu^{6} - 1176588629834087076886283291241 \nu^{5} - 378180400835384494765957070275178 \nu^{4} + 874893751833243004297971676829422417 \nu^{3} - 1069100111555174325078601027390890054 \nu^{2} - 7424633742474624814321512652648474695 \nu - 84044607558473812127025961807134661031787045\)\()/ \)\(25\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-990045188086599135987844519439 \nu^{7} - 40632807802321073635846656863312 \nu^{6} - 665294430705948956131641556847685609 \nu^{5} - 470552267283055805337967139216545737065 \nu^{4} - 484818265485294129110796510783037975473167 \nu^{3} - 161893630306252968191621546598236684135886399 \nu^{2} - 47766343113653987724669892697036902887379266576 \nu - 337387521686781712761182293596151332524743065705\)\()/ \)\(19\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(17658325854018876196147835770337 \nu^{7} + 521617787596639463326393069755815 \nu^{6} + 12492210885102845865116586009471074370 \nu^{5} + 7881819489725579971356446242180806276041 \nu^{4} + 9012893144256742395433683210521746353517550 \nu^{3} + 2880510087466831911171137325998567513117658405 \nu^{2} + 957584381924393648547897697691328051234838165899 \nu - 14447449981092188381964466378482695623342561016670\)\()/ \)\(23\!\cdots\!70\)\( \)
\(\beta_{6}\)\(=\)\((\)\(20981655824344192223728750801867 \nu^{7} - 1413142947425737351076058093368384 \nu^{6} + 14801766805424596312143464800923093621 \nu^{5} + 8624159472921047970180012783496085727799 \nu^{4} + 9516231102347956268577108308277720344442163 \nu^{3} + 2882608651424913909285085721334093157047555399 \nu^{2} + 957598955927899505067978529527608640793848749544 \nu + 21351991572688577658918669087743486312333040703815\)\()/ \)\(23\!\cdots\!70\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-10502584879251467487511674385081 \nu^{7} + 187006872869982294333126057385952 \nu^{6} - 7363850479008466564154768123373583071 \nu^{5} - 4562193205591013836350515963421039436375 \nu^{4} - 5004988337391987576293265536248288786722313 \nu^{3} - 1588727362914455567866229252030520865238507601 \nu^{2} - 506324043896508701557418975979737880239833973664 \nu - 3576359691991709701117167982015984890295996263495\)\()/ \)\(10\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} + \beta_{6} + 2 \beta_{5} - 9 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} + 171 \beta_{1} + 174\)\()/336\)
\(\nu^{2}\)\(=\)\((\)\(-87 \beta_{7} + 386 \beta_{6} + 193 \beta_{5} - 5751 \beta_{4} + 193 \beta_{3} + 193 \beta_{2} + 29070783 \beta_{1}\)\()/84\)
\(\nu^{3}\)\(=\)\((\)\(813235 \beta_{6} - 813235 \beta_{5} + 2971559 \beta_{3} - 16649389 \beta_{2} - 56301080754\)\()/336\)
\(\nu^{4}\)\(=\)\((\)\(-26739030 \beta_{7} - 47369614 \beta_{6} - 94739228 \beta_{5} + 1122208434 \beta_{4} + 74108644 \beta_{3} - 1074838820 \beta_{2} - 5071528550799 \beta_{1} - 5071555289829\)\()/21\)
\(\nu^{5}\)\(=\)\((\)\(-859127137203 \beta_{7} - 1474522292642 \beta_{6} - 737261146321 \beta_{5} + 17805005133849 \beta_{4} - 737261146321 \beta_{3} - 737261146321 \beta_{2} - 64941178278040731 \beta_{1}\)\()/336\)
\(\nu^{6}\)\(=\)\((\)\(-176934179918233 \beta_{6} + 176934179918233 \beta_{5} - 502838085390899 \beta_{3} + 3784816663908685 \beta_{2} + 17241586909054883976\)\()/84\)
\(\nu^{7}\)\(=\)\((\)\(692823156934845009 \beta_{7} + 681166461204542755 \beta_{6} + 1362332922409085510 \beta_{5} - 16369789341799680579 \beta_{4} - 1373989618139387764 \beta_{3} + 15688622880595137824 \beta_{2} + 63183021025465473833025 \beta_{1} + 63183713848622408678034\)\()/336\)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−3.53342 + 6.12007i
−258.759 + 448.183i
481.229 833.513i
−217.937 + 377.477i
−3.53342 6.12007i
−258.759 448.183i
481.229 + 833.513i
−217.937 377.477i
−32.0000 + 55.4256i −855.669 1482.06i −2048.00 3547.24i 12545.8 21729.9i 109526. 48813.9 307419.i 262144. −667176. + 1.15558e6i 802928. + 1.39071e6i
9.2 −32.0000 + 55.4256i −679.656 1177.20i −2048.00 3547.24i −29011.3 + 50249.0i 86996.0 −173759. + 258257.i 262144. −126703. + 219456.i −1.85672e6 3.21593e6i
9.3 −32.0000 + 55.4256i 301.970 + 523.027i −2048.00 3547.24i 2521.96 4368.16i −38652.2 302038. + 75247.5i 262144. 614790. 1.06485e6i 161406. + 279563.i
9.4 −32.0000 + 55.4256i 1142.35 + 1978.62i −2048.00 3547.24i −18256.5 + 31621.1i −146221. −233961. 205308.i 262144. −1.81279e6 + 3.13984e6i −1.16841e6 2.02375e6i
11.1 −32.0000 55.4256i −855.669 + 1482.06i −2048.00 + 3547.24i 12545.8 + 21729.9i 109526. 48813.9 + 307419.i 262144. −667176. 1.15558e6i 802928. 1.39071e6i
11.2 −32.0000 55.4256i −679.656 + 1177.20i −2048.00 + 3547.24i −29011.3 50249.0i 86996.0 −173759. 258257.i 262144. −126703. 219456.i −1.85672e6 + 3.21593e6i
11.3 −32.0000 55.4256i 301.970 523.027i −2048.00 + 3547.24i 2521.96 + 4368.16i −38652.2 302038. 75247.5i 262144. 614790. + 1.06485e6i 161406. 279563.i
11.4 −32.0000 55.4256i 1142.35 1978.62i −2048.00 + 3547.24i −18256.5 31621.1i −146221. −233961. + 205308.i 262144. −1.81279e6 3.13984e6i −1.16841e6 + 2.02375e6i
\(n\): 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 14.14.c.a 8
3.b odd 2 1 126.14.g.d 8
7.b odd 2 1 98.14.c.n 8
7.c even 3 1 inner 14.14.c.a 8
7.c even 3 1 98.14.a.l 4
7.d odd 6 1 98.14.a.k 4
7.d odd 6 1 98.14.c.n 8
21.h odd 6 1 126.14.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.a 8 1.a even 1 1 trivial
14.14.c.a 8 7.c even 3 1 inner
98.14.a.k 4 7.d odd 6 1
98.14.a.l 4 7.c even 3 1
98.14.c.n 8 7.b odd 2 1
98.14.c.n 8 7.d odd 6 1
126.14.g.d 8 3.b odd 2 1
126.14.g.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(23\!\cdots\!37\)\( T_{3}^{4} + \)\(11\!\cdots\!80\)\( T_{3}^{3} + \)\(22\!\cdots\!76\)\( T_{3}^{2} - \)\(79\!\cdots\!70\)\( T_{3} + \)\(10\!\cdots\!25\)\( \)">\(T_{3}^{8} + \cdots\) acting on \(S_{14}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4096 + 64 T + T^{2} )^{4} \)
$3$ \( \)\(10\!\cdots\!25\)\( - \)\(79\!\cdots\!70\)\( T + 22739280266229742476 T^{2} + 11652343744019580 T^{3} + 23908528667637 T^{4} + 4025617932 T^{5} + 5197084 T^{6} + 182 T^{7} + T^{8} \)
$5$ \( \)\(71\!\cdots\!25\)\( - \)\(13\!\cdots\!00\)\( T + \)\(28\!\cdots\!50\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + 3437420980162575031 T^{4} + 64878800757680 T^{5} + 4751094966 T^{6} + 64400 T^{7} + T^{8} \)
$7$ \( \)\(88\!\cdots\!01\)\( + \)\(10\!\cdots\!48\)\( T + \)\(34\!\cdots\!68\)\( T^{2} - \)\(31\!\cdots\!60\)\( T^{3} - \)\(20\!\cdots\!30\)\( T^{4} - 32955293602784280 T^{5} + 36945189932 T^{6} + 113736 T^{7} + T^{8} \)
$11$ \( \)\(29\!\cdots\!25\)\( - \)\(47\!\cdots\!70\)\( T + \)\(57\!\cdots\!76\)\( T^{2} - \)\(32\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!09\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + 39034557572652 T^{6} + 1008790 T^{7} + T^{8} \)
$13$ \( ( -\)\(94\!\cdots\!80\)\( + \)\(24\!\cdots\!92\)\( T + 59210836770904 T^{2} - 26903632 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(17\!\cdots\!25\)\( - \)\(47\!\cdots\!20\)\( T + \)\(16\!\cdots\!66\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} + \)\(27\!\cdots\!48\)\( T^{5} + 53919413078235954 T^{6} + 165333028 T^{7} + T^{8} \)
$19$ \( \)\(13\!\cdots\!25\)\( + \)\(40\!\cdots\!90\)\( T + \)\(12\!\cdots\!16\)\( T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(50\!\cdots\!65\)\( T^{4} + \)\(24\!\cdots\!56\)\( T^{5} + 173691993370688620 T^{6} + 423405794 T^{7} + T^{8} \)
$23$ \( \)\(17\!\cdots\!81\)\( + \)\(17\!\cdots\!86\)\( T + \)\(22\!\cdots\!88\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!09\)\( T^{4} - \)\(53\!\cdots\!20\)\( T^{5} + 1134956096106046248 T^{6} - 286233866 T^{7} + T^{8} \)
$29$ \( ( \)\(47\!\cdots\!28\)\( - \)\(18\!\cdots\!72\)\( T + 22515128502490853176 T^{2} - 9100337408 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(34\!\cdots\!25\)\( - \)\(96\!\cdots\!10\)\( T + \)\(20\!\cdots\!04\)\( T^{2} - \)\(19\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} - \)\(17\!\cdots\!16\)\( T^{5} + 12765136729777969464 T^{6} - 1507094246 T^{7} + T^{8} \)
$37$ \( \)\(55\!\cdots\!61\)\( + \)\(13\!\cdots\!24\)\( T + \)\(28\!\cdots\!18\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!39\)\( T^{4} + \)\(72\!\cdots\!80\)\( T^{5} + \)\(59\!\cdots\!38\)\( T^{6} + 18959705336 T^{7} + T^{8} \)
$41$ \( ( -\)\(10\!\cdots\!08\)\( + \)\(11\!\cdots\!48\)\( T - \)\(15\!\cdots\!52\)\( T^{2} - 56825955312 T^{3} + T^{4} )^{2} \)
$43$ \( ( -\)\(18\!\cdots\!00\)\( - \)\(63\!\cdots\!08\)\( T - \)\(20\!\cdots\!84\)\( T^{2} + 45778809712 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(82\!\cdots\!25\)\( + \)\(32\!\cdots\!30\)\( T + \)\(98\!\cdots\!36\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!77\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(15\!\cdots\!24\)\( T^{6} + 81351201078 T^{7} + T^{8} \)
$53$ \( \)\(64\!\cdots\!41\)\( - \)\(26\!\cdots\!60\)\( T + \)\(89\!\cdots\!62\)\( T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!63\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(78\!\cdots\!22\)\( T^{6} - 87497947440 T^{7} + T^{8} \)
$59$ \( \)\(14\!\cdots\!25\)\( - \)\(24\!\cdots\!50\)\( T + \)\(62\!\cdots\!84\)\( T^{2} + \)\(22\!\cdots\!92\)\( T^{3} + \)\(44\!\cdots\!65\)\( T^{4} + \)\(79\!\cdots\!72\)\( T^{5} + \)\(98\!\cdots\!40\)\( T^{6} + 194140265102 T^{7} + T^{8} \)
$61$ \( \)\(12\!\cdots\!25\)\( + \)\(33\!\cdots\!00\)\( T + \)\(11\!\cdots\!94\)\( T^{2} - \)\(76\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!99\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!58\)\( T^{6} - 175816313120 T^{7} + T^{8} \)
$67$ \( \)\(98\!\cdots\!25\)\( + \)\(55\!\cdots\!70\)\( T + \)\(28\!\cdots\!84\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(71\!\cdots\!37\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + \)\(87\!\cdots\!28\)\( T^{6} + 243815218758 T^{7} + T^{8} \)
$71$ \( ( -\)\(79\!\cdots\!40\)\( - \)\(20\!\cdots\!56\)\( T - \)\(60\!\cdots\!56\)\( T^{2} + 1637697339536 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(21\!\cdots\!25\)\( + \)\(47\!\cdots\!20\)\( T + \)\(22\!\cdots\!46\)\( T^{2} + \)\(48\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(95\!\cdots\!22\)\( T^{6} + 3492491920596 T^{7} + T^{8} \)
$79$ \( \)\(33\!\cdots\!25\)\( - \)\(58\!\cdots\!50\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!49\)\( T^{4} + \)\(38\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!48\)\( T^{6} + 1016380081246 T^{7} + T^{8} \)
$83$ \( ( \)\(13\!\cdots\!00\)\( + \)\(45\!\cdots\!68\)\( T - \)\(21\!\cdots\!16\)\( T^{2} - 3513747871648 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(62\!\cdots\!25\)\( + \)\(30\!\cdots\!20\)\( T + \)\(12\!\cdots\!46\)\( T^{2} + \)\(24\!\cdots\!16\)\( T^{3} + \)\(48\!\cdots\!95\)\( T^{4} + \)\(52\!\cdots\!04\)\( T^{5} + \)\(95\!\cdots\!10\)\( T^{6} + 8034124428036 T^{7} + T^{8} \)
$97$ \( ( -\)\(27\!\cdots\!00\)\( - \)\(35\!\cdots\!60\)\( T + \)\(21\!\cdots\!28\)\( T^{2} - 27175565862816 T^{3} + T^{4} )^{2} \)
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