Properties

Label 14.10.c.b
Level $14$
Weight $10$
Character orbit 14.c
Analytic conductor $7.211$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.21050170629\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} + 4038 x^{4} - 137923 x^{3} + 16368349 x^{2} - 286546260 x + 5038160400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 16 + 16 \beta_{1} ) q^{2} + ( -24 \beta_{1} - \beta_{3} ) q^{3} + 256 \beta_{1} q^{4} + ( -364 - 364 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{5} + ( 384 - 16 \beta_{2} ) q^{6} + ( -1812 - 1353 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} - \beta_{4} - 11 \beta_{5} ) q^{7} -4096 q^{8} + ( -3033 - 3033 \beta_{1} - 34 \beta_{2} + 25 \beta_{3} + 18 \beta_{4} - 27 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 16 + 16 \beta_{1} ) q^{2} + ( -24 \beta_{1} - \beta_{3} ) q^{3} + 256 \beta_{1} q^{4} + ( -364 - 364 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{5} + ( 384 - 16 \beta_{2} ) q^{6} + ( -1812 - 1353 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} - \beta_{4} - 11 \beta_{5} ) q^{7} -4096 q^{8} + ( -3033 - 3033 \beta_{1} - 34 \beta_{2} + 25 \beta_{3} + 18 \beta_{4} - 27 \beta_{5} ) q^{9} + ( -5824 \beta_{1} - 16 \beta_{2} - 64 \beta_{3} + 48 \beta_{4} - 16 \beta_{5} ) q^{10} + ( -931 \beta_{1} - 37 \beta_{2} - 127 \beta_{3} + 111 \beta_{4} - 37 \beta_{5} ) q^{11} + ( 6144 + 6144 \beta_{1} - 256 \beta_{2} + 256 \beta_{3} ) q^{12} + ( 6211 - 565 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{13} + ( -7344 - 28992 \beta_{1} + 96 \beta_{2} + 192 \beta_{3} + 176 \beta_{4} - 192 \beta_{5} ) q^{14} + ( -118041 + 768 \beta_{2} - 141 \beta_{3} - 141 \beta_{4} - 282 \beta_{5} ) q^{15} + ( -65536 - 65536 \beta_{1} ) q^{16} + ( -7180 \beta_{1} + 167 \beta_{2} - 1282 \beta_{3} - 501 \beta_{4} + 167 \beta_{5} ) q^{17} + ( -48528 \beta_{1} - 144 \beta_{2} + 544 \beta_{3} + 432 \beta_{4} - 144 \beta_{5} ) q^{18} + ( 407143 + 407143 \beta_{1} + 39 \beta_{2} + 234 \beta_{3} - 546 \beta_{4} + 819 \beta_{5} ) q^{19} + ( 93184 - 1280 \beta_{2} + 256 \beta_{3} + 256 \beta_{4} + 512 \beta_{5} ) q^{20} + ( 446913 + 246411 \beta_{1} + 3685 \beta_{2} + 1875 \beta_{3} - 1251 \beta_{4} + 582 \beta_{5} ) q^{21} + ( 14896 - 2624 \beta_{2} + 592 \beta_{3} + 592 \beta_{4} + 1184 \beta_{5} ) q^{22} + ( -657482 - 657482 \beta_{1} + 13499 \beta_{2} - 12847 \beta_{3} - 1304 \beta_{4} + 1956 \beta_{5} ) q^{23} + ( 98304 \beta_{1} + 4096 \beta_{3} ) q^{24} + ( 292986 \beta_{1} + 1022 \beta_{2} + 12320 \beta_{3} - 3066 \beta_{4} + 1022 \beta_{5} ) q^{25} + ( 99376 + 99376 \beta_{1} - 9056 \beta_{2} + 9040 \beta_{3} + 32 \beta_{4} - 48 \beta_{5} ) q^{26} + ( 965655 - 9584 \beta_{2} - 639 \beta_{3} - 639 \beta_{4} - 1278 \beta_{5} ) q^{27} + ( 346368 - 117504 \beta_{1} + 4608 \beta_{2} - 1536 \beta_{3} + 3072 \beta_{4} - 256 \beta_{5} ) q^{28} + ( -865283 - 12191 \beta_{2} - 947 \beta_{3} - 947 \beta_{4} - 1894 \beta_{5} ) q^{29} + ( -1888656 - 1888656 \beta_{1} + 10032 \beta_{2} - 12288 \beta_{3} + 4512 \beta_{4} - 6768 \beta_{5} ) q^{30} + ( 3114553 \beta_{1} - 2455 \beta_{2} - 26965 \beta_{3} + 7365 \beta_{4} - 2455 \beta_{5} ) q^{31} -1048576 \beta_{1} q^{32} + ( -3601689 - 3601689 \beta_{1} + 11880 \beta_{2} - 16908 \beta_{3} + 10056 \beta_{4} - 15084 \beta_{5} ) q^{33} + ( 114880 - 17840 \beta_{2} - 2672 \beta_{3} - 2672 \beta_{4} - 5344 \beta_{5} ) q^{34} + ( 6950566 + 7100023 \beta_{1} - 37996 \beta_{2} + 43421 \beta_{3} - 749 \beta_{4} + 4893 \beta_{5} ) q^{35} + ( 776448 + 6400 \beta_{2} + 2304 \beta_{3} + 2304 \beta_{4} + 4608 \beta_{5} ) q^{36} + ( -8597411 - 8597411 \beta_{1} - 25866 \beta_{2} + 24804 \beta_{3} + 2124 \beta_{4} - 3186 \beta_{5} ) q^{37} + ( 6514288 \beta_{1} + 4368 \beta_{2} - 624 \beta_{3} - 13104 \beta_{4} + 4368 \beta_{5} ) q^{38} + ( -12659559 \beta_{1} - 4989 \beta_{2} + 27112 \beta_{3} + 14967 \beta_{4} - 4989 \beta_{5} ) q^{39} + ( 1490944 + 1490944 \beta_{1} - 16384 \beta_{2} + 20480 \beta_{3} - 8192 \beta_{4} + 12288 \beta_{5} ) q^{40} + ( 64929 + 32649 \beta_{2} + 9093 \beta_{3} + 9093 \beta_{4} + 18186 \beta_{5} ) q^{41} + ( 3208032 + 7150608 \beta_{1} + 88960 \beta_{2} - 58960 \beta_{3} - 9312 \beta_{4} - 10704 \beta_{5} ) q^{42} + ( 966848 + 40716 \beta_{2} - 1524 \beta_{3} - 1524 \beta_{4} - 3048 \beta_{5} ) q^{43} + ( 238336 + 238336 \beta_{1} - 32512 \beta_{2} + 41984 \beta_{3} - 18944 \beta_{4} + 28416 \beta_{5} ) q^{44} + ( 12475197 \beta_{1} + 765 \beta_{2} + 72738 \beta_{3} - 2295 \beta_{4} + 765 \beta_{5} ) q^{45} + ( -10519712 \beta_{1} + 10432 \beta_{2} - 215984 \beta_{3} - 31296 \beta_{4} + 10432 \beta_{5} ) q^{46} + ( 16121205 + 16121205 \beta_{1} + 101733 \beta_{2} - 77040 \beta_{3} - 49386 \beta_{4} + 74079 \beta_{5} ) q^{47} + ( -1572864 + 65536 \beta_{2} ) q^{48} + ( 28292860 + 20315538 \beta_{1} - 129925 \beta_{2} - 112843 \beta_{3} + 10695 \beta_{4} - 19464 \beta_{5} ) q^{49} + ( -4687776 + 213472 \beta_{2} - 16352 \beta_{3} - 16352 \beta_{4} - 32704 \beta_{5} ) q^{50} + ( -25091385 - 25091385 \beta_{1} - 149841 \beta_{2} + 155838 \beta_{3} - 11994 \beta_{4} + 17991 \beta_{5} ) q^{51} + ( 1590016 \beta_{1} - 256 \beta_{2} + 144896 \beta_{3} + 768 \beta_{4} - 256 \beta_{5} ) q^{52} + ( -34049691 \beta_{1} - 15204 \beta_{2} + 82950 \beta_{3} + 45612 \beta_{4} - 15204 \beta_{5} ) q^{53} + ( 15450480 + 15450480 \beta_{1} - 163568 \beta_{2} + 153344 \beta_{3} + 20448 \beta_{4} - 30672 \beta_{5} ) q^{54} + ( -76247234 + 417361 \beta_{2} - 22820 \beta_{3} - 22820 \beta_{4} - 45640 \beta_{5} ) q^{55} + ( 7421952 + 5541888 \beta_{1} + 49152 \beta_{2} - 73728 \beta_{3} + 4096 \beta_{4} + 45056 \beta_{5} ) q^{56} + ( 14571357 - 572854 \beta_{2} + 28314 \beta_{3} + 28314 \beta_{4} + 56628 \beta_{5} ) q^{57} + ( -13844528 - 13844528 \beta_{1} - 210208 \beta_{2} + 195056 \beta_{3} + 30304 \beta_{4} - 45456 \beta_{5} ) q^{58} + ( -48055016 \beta_{1} + 11500 \beta_{2} + 20587 \beta_{3} - 34500 \beta_{4} + 11500 \beta_{5} ) q^{59} + ( -30218496 \beta_{1} - 36096 \beta_{2} - 160512 \beta_{3} + 108288 \beta_{4} - 36096 \beta_{5} ) q^{60} + ( 93456245 + 93456245 \beta_{1} + 342266 \beta_{2} - 417556 \beta_{3} + 150580 \beta_{4} - 225870 \beta_{5} ) q^{61} + ( -49832848 - 470720 \beta_{2} + 39280 \beta_{3} + 39280 \beta_{4} + 78560 \beta_{5} ) q^{62} + ( 45965907 + 80539191 \beta_{1} - 219336 \beta_{2} - 15655 \beta_{3} + 24282 \beta_{4} - 26289 \beta_{5} ) q^{63} + 16777216 q^{64} + ( -62451039 - 62451039 \beta_{1} + 316650 \beta_{2} - 388893 \beta_{3} + 144486 \beta_{4} - 216729 \beta_{5} ) q^{65} + ( -57627024 \beta_{1} - 80448 \beta_{2} - 190080 \beta_{3} + 241344 \beta_{4} - 80448 \beta_{5} ) q^{66} + ( -56626530 \beta_{1} + 142306 \beta_{2} + 403891 \beta_{3} - 426918 \beta_{4} + 142306 \beta_{5} ) q^{67} + ( 1838080 + 1838080 \beta_{1} - 328192 \beta_{2} + 285440 \beta_{3} + 85504 \beta_{4} - 128256 \beta_{5} ) q^{68} + ( -301121688 - 395169 \beta_{2} - 53031 \beta_{3} - 53031 \beta_{4} - 106062 \beta_{5} ) q^{69} + ( -2391312 + 111209056 \beta_{1} + 86800 \beta_{2} + 607936 \beta_{3} - 78288 \beta_{4} + 66304 \beta_{5} ) q^{70} + ( 156335774 + 721994 \beta_{2} - 14686 \beta_{3} - 14686 \beta_{4} - 29372 \beta_{5} ) q^{71} + ( 12423168 + 12423168 \beta_{1} + 139264 \beta_{2} - 102400 \beta_{3} - 73728 \beta_{4} + 110592 \beta_{5} ) q^{72} + ( -204934671 \beta_{1} + 61354 \beta_{2} - 1149536 \beta_{3} - 184062 \beta_{4} + 61354 \beta_{5} ) q^{73} + ( -137558576 \beta_{1} - 16992 \beta_{2} + 413856 \beta_{3} + 50976 \beta_{4} - 16992 \beta_{5} ) q^{74} + ( 300997854 + 300997854 \beta_{1} - 84316 \beta_{2} + 302506 \beta_{3} - 436380 \beta_{4} + 654570 \beta_{5} ) q^{75} + ( -104228608 + 59904 \beta_{2} - 69888 \beta_{3} - 69888 \beta_{4} - 139776 \beta_{5} ) q^{76} + ( 1365567 + 246473164 \beta_{1} + 339934 \beta_{2} + 1294279 \beta_{3} - 10374 \beta_{4} + 129073 \beta_{5} ) q^{77} + ( 202552944 + 353968 \beta_{2} + 79824 \beta_{3} + 79824 \beta_{4} + 159648 \beta_{5} ) q^{78} + ( 65385458 + 65385458 \beta_{1} + 2047031 \beta_{2} - 1794499 \beta_{3} - 505064 \beta_{4} + 757596 \beta_{5} ) q^{79} + ( 23855104 \beta_{1} + 65536 \beta_{2} + 262144 \beta_{3} - 196608 \beta_{4} + 65536 \beta_{5} ) q^{80} + ( -295955424 \beta_{1} - 202059 \beta_{2} + 612806 \beta_{3} + 606177 \beta_{4} - 202059 \beta_{5} ) q^{81} + ( 1038864 + 1038864 \beta_{1} + 667872 \beta_{2} - 522384 \beta_{3} - 290976 \beta_{4} + 436464 \beta_{5} ) q^{82} + ( -357354022 - 1569286 \beta_{2} + 275042 \beta_{3} + 275042 \beta_{4} + 550084 \beta_{5} ) q^{83} + ( -63081216 + 51328512 \beta_{1} + 480000 \beta_{2} - 1423360 \beta_{3} + 171264 \beta_{4} - 320256 \beta_{5} ) q^{84} + ( 137215435 - 624674 \beta_{2} - 125444 \beta_{3} - 125444 \beta_{4} - 250888 \beta_{5} ) q^{85} + ( 15469568 + 15469568 \beta_{1} + 627072 \beta_{2} - 651456 \beta_{3} + 48768 \beta_{4} - 73152 \beta_{5} ) q^{86} + ( -250463013 \beta_{1} - 18807 \beta_{2} + 2096052 \beta_{3} + 56421 \beta_{4} - 18807 \beta_{5} ) q^{87} + ( 3813376 \beta_{1} + 151552 \beta_{2} + 520192 \beta_{3} - 454656 \beta_{4} + 151552 \beta_{5} ) q^{88} + ( 269250885 + 269250885 \beta_{1} - 3257028 \beta_{2} + 2845428 \beta_{3} + 823200 \beta_{4} - 1234800 \beta_{5} ) q^{89} + ( -199603152 + 1176048 \beta_{2} - 12240 \beta_{3} - 12240 \beta_{4} - 24480 \beta_{5} ) q^{90} + ( 134260423 + 264502932 \beta_{1} + 3211039 \beta_{2} - 2222915 \beta_{3} - 320491 \beta_{4} - 302426 \beta_{5} ) q^{91} + ( 168315392 - 3288832 \beta_{2} - 166912 \beta_{3} - 166912 \beta_{4} - 333824 \beta_{5} ) q^{92} + ( -573184803 - 573184803 \beta_{1} - 3463298 \beta_{2} + 2962838 \beta_{3} + 1000920 \beta_{4} - 1501380 \beta_{5} ) q^{93} + ( 257939280 \beta_{1} + 395088 \beta_{2} - 1627728 \beta_{3} - 1185264 \beta_{4} + 395088 \beta_{5} ) q^{94} + ( -601592992 \beta_{1} - 454450 \beta_{2} - 3993883 \beta_{3} + 1363350 \beta_{4} - 454450 \beta_{5} ) q^{95} + ( -25165824 - 25165824 \beta_{1} + 1048576 \beta_{2} - 1048576 \beta_{3} ) q^{96} + ( -754077081 - 1333445 \beta_{2} - 323297 \beta_{3} - 323297 \beta_{4} - 646594 \beta_{5} ) q^{97} + ( 127637152 + 452685760 \beta_{1} - 3884288 \beta_{2} + 2078800 \beta_{3} + 311424 \beta_{4} - 140304 \beta_{5} ) q^{98} + ( -435444723 + 2808357 \beta_{2} + 93411 \beta_{3} + 93411 \beta_{4} + 186822 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 48q^{2} + 71q^{3} - 768q^{4} - 1085q^{5} + 2272q^{6} - 6796q^{7} - 24576q^{8} - 9106q^{9} + O(q^{10}) \) \( 6q + 48q^{2} + 71q^{3} - 768q^{4} - 1085q^{5} + 2272q^{6} - 6796q^{7} - 24576q^{8} - 9106q^{9} + 17360q^{10} + 2555q^{11} + 18176q^{12} + 36140q^{13} + 43504q^{14} - 706146q^{15} - 196608q^{16} + 20759q^{17} + 145696q^{18} + 1220649q^{19} + 555520q^{20} + 1951577q^{21} + 81760q^{22} - 1960903q^{23} - 290816q^{24} - 863572q^{25} + 289120q^{26} + 5777318q^{27} + 2435840q^{28} - 5212292q^{29} - 5649168q^{30} - 9377989q^{31} + 3145728q^{32} - 10778103q^{33} + 664288q^{34} + 20361719q^{35} + 4662272q^{36} - 25814913q^{37} - 19530384q^{38} + 37990822q^{39} + 4444160q^{40} + 418500q^{41} - 2053952q^{42} + 5888616q^{43} + 654080q^{44} - 37350558q^{45} + 31374448q^{46} + 48391269q^{47} - 9306112q^{48} + 108466086q^{49} - 27634304q^{50} - 75441987q^{51} - 4625920q^{52} + 102186411q^{53} + 46218544q^{54} - 456557402q^{55} + 27836416q^{56} + 86169178q^{57} - 41698336q^{58} + 144220135q^{59} + 90386688q^{60} + 280936871q^{61} - 300095648q^{62} + 33751838q^{63} + 100663296q^{64} - 186819738q^{65} + 172449648q^{66} + 170710399q^{67} + 5314304q^{68} - 1807308342q^{69} - 347247824q^{70} + 939517376q^{71} + 37298176q^{72} + 613838539q^{73} + 413038608q^{74} + 902254676q^{75} - 624972288q^{76} - 729499715q^{77} + 1215706304q^{78} + 197445809q^{79} - 71106560q^{80} + 887872901q^{81} + 3348000q^{82} - 2148362872q^{83} - 532466944q^{84} + 822545038q^{85} + 47108928q^{86} + 753428670q^{87} - 10465280q^{88} + 805730427q^{89} - 1195217856q^{90} + 17178248q^{91} + 1003982336q^{92} - 1721516327q^{93} - 774260304q^{94} + 1799421743q^{95} - 74448896q^{96} - 4525836188q^{97} - 597954960q^{98} - 2607425268q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 4038 x^{4} - 137923 x^{3} + 16368349 x^{2} - 286546260 x + 5038160400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2716901 \nu^{5} + 2705071 \nu^{4} - 10923076698 \nu^{3} + 181829734103 \nu^{2} - 44277546197779 \nu - 3390681949200 \)\()/ 778518660033660 \)
\(\beta_{2}\)\(=\)\((\)\( -4027 \nu^{5} + 16261026 \nu^{4} + 146824014 \nu^{3} + 65915341423 \nu^{2} - 1153921789020 \nu + 166009397495697 \)\()/ 690992893521 \)
\(\beta_{3}\)\(=\)\((\)\(2081085611 \nu^{5} + 53013142539 \nu^{4} + 8366833294278 \nu^{3} - 139277523654233 \nu^{2} + 28003708179344409 \nu + 2597186800681200\)\()/ 2335555980100980 \)
\(\beta_{4}\)\(=\)\((\)\(694639239 \nu^{5} + 35795315171 \nu^{4} + 3747785917342 \nu^{3} + 26700567042283 \nu^{2} + 1908449358635081 \nu + 133935551353629480\)\()/ 778518660033660 \)
\(\beta_{5}\)\(=\)\((\)\(-352672159 \nu^{5} + 9199967499 \nu^{4} - 77151616502 \nu^{3} + 96406628067317 \nu^{2} - 4412132311683371 \nu + 105641374520591910\)\()/ 389259330016830 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} - 7 \beta_{1}\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(-15 \beta_{5} + 10 \beta_{4} - 172 \beta_{3} + 167 \beta_{2} - 37744 \beta_{1} - 37744\)\()/14\)
\(\nu^{3}\)\(=\)\((\)\(8054 \beta_{5} + 4027 \beta_{4} + 4027 \beta_{3} - 19841 \beta_{2} + 1883693\)\()/28\)
\(\nu^{4}\)\(=\)\((\)\(107323 \beta_{5} - 321969 \beta_{4} + 1616464 \beta_{3} + 107323 \beta_{2} + 302364503 \beta_{1}\)\()/28\)
\(\nu^{5}\)\(=\)\((\)\(-25289214 \beta_{5} + 16859476 \beta_{4} - 51395725 \beta_{3} + 42965987 \beta_{2} - 6330121189 \beta_{1} - 6330121189\)\()/14\)

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
9.63015 + 16.6799i
26.1296 + 45.2579i
−35.2598 61.0717i
9.63015 16.6799i
26.1296 45.2579i
−35.2598 + 61.0717i
8.00000 13.8564i −89.6637 155.302i −128.000 221.703i 84.1442 145.742i −2869.24 −3162.16 + 5509.48i −4096.00 −6237.66 + 10803.9i −1346.31 2331.87i
9.2 8.00000 13.8564i 38.4345 + 66.5705i −128.000 221.703i 546.130 945.925i 1229.90 6111.84 1731.76i −4096.00 6887.08 11928.8i −8738.08 15134.8i
9.3 8.00000 13.8564i 86.7292 + 150.219i −128.000 221.703i −1172.77 + 2031.30i 2775.34 −6347.68 246.066i −4096.00 −5202.42 + 9010.85i 18764.4 + 32500.9i
11.1 8.00000 + 13.8564i −89.6637 + 155.302i −128.000 + 221.703i 84.1442 + 145.742i −2869.24 −3162.16 5509.48i −4096.00 −6237.66 10803.9i −1346.31 + 2331.87i
11.2 8.00000 + 13.8564i 38.4345 66.5705i −128.000 + 221.703i 546.130 + 945.925i 1229.90 6111.84 + 1731.76i −4096.00 6887.08 + 11928.8i −8738.08 + 15134.8i
11.3 8.00000 + 13.8564i 86.7292 150.219i −128.000 + 221.703i −1172.77 2031.30i 2775.34 −6347.68 + 246.066i −4096.00 −5202.42 9010.85i 18764.4 32500.9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
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 14.10.c.b 6
3.b odd 2 1 126.10.g.e 6
4.b odd 2 1 112.10.i.a 6
7.b odd 2 1 98.10.c.l 6
7.c even 3 1 inner 14.10.c.b 6
7.c even 3 1 98.10.a.g 3
7.d odd 6 1 98.10.a.h 3
7.d odd 6 1 98.10.c.l 6
21.h odd 6 1 126.10.g.e 6
28.g odd 6 1 112.10.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.c.b 6 1.a even 1 1 trivial
14.10.c.b 6 7.c even 3 1 inner
98.10.a.g 3 7.c even 3 1
98.10.a.h 3 7.d odd 6 1
98.10.c.l 6 7.b odd 2 1
98.10.c.l 6 7.d odd 6 1
112.10.i.a 6 4.b odd 2 1
112.10.i.a 6 28.g odd 6 1
126.10.g.e 6 3.b odd 2 1
126.10.g.e 6 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 71 T_{3}^{5} + 36598 T_{3}^{4} - 2541603 T_{3}^{3} + 1165610574 T_{3}^{2} - 75455153775 T_{3} + \)\(57\!\cdots\!25\)\( \) acting on \(S_{10}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 - 16 T + T^{2} )^{3} \)
$3$ \( 5717239655625 - 75455153775 T + 1165610574 T^{2} - 2541603 T^{3} + 36598 T^{4} - 71 T^{5} + T^{6} \)
$5$ \( 185886748283681025 - 1195507526641155 T + 7220964872646 T^{2} - 3870845895 T^{3} + 3950086 T^{4} + 1085 T^{5} + T^{6} \)
$7$ \( \)\(65\!\cdots\!43\)\( + 11066698811399411404 T - 1256620805077645 T^{2} - 432947916632 T^{3} - 31140235 T^{4} + 6796 T^{5} + T^{6} \)
$11$ \( \)\(31\!\cdots\!41\)\( + \)\(22\!\cdots\!93\)\( T + 16596783693336655494 T^{2} + 121888868945193 T^{3} + 4097877742 T^{4} - 2555 T^{5} + T^{6} \)
$13$ \( ( 368974841338200 - 10506518276 T - 18070 T^{2} + T^{3} )^{2} \)
$17$ \( \)\(11\!\cdots\!25\)\( + \)\(37\!\cdots\!05\)\( T + \)\(11\!\cdots\!94\)\( T^{2} + 9098508778892973 T^{3} + 108332373958 T^{4} - 20759 T^{5} + T^{6} \)
$19$ \( \)\(37\!\cdots\!25\)\( + \)\(61\!\cdots\!75\)\( T + \)\(12\!\cdots\!74\)\( T^{2} - 428263259498975557 T^{3} + 1170655553358 T^{4} - 1220649 T^{5} + T^{6} \)
$23$ \( \)\(97\!\cdots\!61\)\( + \)\(51\!\cdots\!55\)\( T + \)\(46\!\cdots\!82\)\( T^{2} + 9499973607852847203 T^{3} + 9093716677654 T^{4} + 1960903 T^{5} + T^{6} \)
$29$ \( ( -1136027665699119432 - 4849005669540 T + 2606146 T^{2} + T^{3} )^{2} \)
$31$ \( \)\(38\!\cdots\!09\)\( + \)\(26\!\cdots\!29\)\( T + \)\(20\!\cdots\!66\)\( T^{2} + \)\(26\!\cdots\!29\)\( T^{3} + 101258784791214 T^{4} + 9377989 T^{5} + T^{6} \)
$37$ \( \)\(22\!\cdots\!69\)\( + \)\(95\!\cdots\!33\)\( T + \)\(27\!\cdots\!50\)\( T^{2} + \)\(41\!\cdots\!93\)\( T^{3} + 467514029705910 T^{4} + 25814913 T^{5} + T^{6} \)
$41$ \( ( \)\(12\!\cdots\!24\)\( - 231960363901092 T - 209250 T^{2} + T^{3} )^{2} \)
$43$ \( ( 85536171172243734592 - 57381514505424 T - 2944308 T^{2} + T^{3} )^{2} \)
$47$ \( \)\(21\!\cdots\!25\)\( - \)\(40\!\cdots\!45\)\( T + \)\(30\!\cdots\!14\)\( T^{2} - \)\(51\!\cdots\!37\)\( T^{3} + 3206409899672718 T^{4} - 48391269 T^{5} + T^{6} \)
$53$ \( \)\(10\!\cdots\!21\)\( - \)\(28\!\cdots\!11\)\( T + \)\(66\!\cdots\!22\)\( T^{2} - \)\(26\!\cdots\!11\)\( T^{3} + 7664409827848422 T^{4} - 102186411 T^{5} + T^{6} \)
$59$ \( \)\(88\!\cdots\!25\)\( - \)\(62\!\cdots\!75\)\( T + \)\(29\!\cdots\!26\)\( T^{2} - \)\(76\!\cdots\!15\)\( T^{3} + 14210689521501526 T^{4} - 144220135 T^{5} + T^{6} \)
$61$ \( \)\(31\!\cdots\!25\)\( + \)\(12\!\cdots\!65\)\( T + \)\(54\!\cdots\!94\)\( T^{2} - \)\(55\!\cdots\!03\)\( T^{3} + 71629017810652758 T^{4} - 280936871 T^{5} + T^{6} \)
$67$ \( \)\(29\!\cdots\!25\)\( - \)\(25\!\cdots\!95\)\( T + \)\(31\!\cdots\!86\)\( T^{2} - \)\(26\!\cdots\!31\)\( T^{3} + 76733529465554342 T^{4} - 170710399 T^{5} + T^{6} \)
$71$ \( ( -\)\(12\!\cdots\!40\)\( + 55775894095701696 T - 469758688 T^{2} + T^{3} )^{2} \)
$73$ \( \)\(24\!\cdots\!41\)\( + \)\(38\!\cdots\!81\)\( T + \)\(89\!\cdots\!02\)\( T^{2} - \)\(57\!\cdots\!79\)\( T^{3} + 299769435047295782 T^{4} - 613838539 T^{5} + T^{6} \)
$79$ \( \)\(77\!\cdots\!25\)\( - \)\(68\!\cdots\!45\)\( T + \)\(66\!\cdots\!74\)\( T^{2} - \)\(71\!\cdots\!97\)\( T^{3} + 285731562527913078 T^{4} - 197445809 T^{5} + T^{6} \)
$83$ \( ( -\)\(87\!\cdots\!00\)\( + 126447385266753648 T + 1074181436 T^{2} + T^{3} )^{2} \)
$89$ \( \)\(11\!\cdots\!25\)\( - \)\(49\!\cdots\!75\)\( T + \)\(29\!\cdots\!74\)\( T^{2} + \)\(15\!\cdots\!61\)\( T^{3} + 1107424475695806822 T^{4} - 805730427 T^{5} + T^{6} \)
$97$ \( ( \)\(14\!\cdots\!40\)\( + 1399099087623178268 T + 2262918094 T^{2} + T^{3} )^{2} \)
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