Properties

 Label 10.13.c.b Level 10 Weight 13 Character orbit 10.c Analytic conductor 9.140 Analytic rank 0 Dimension 6 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$13$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.13993817276$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{5}\cdot 5^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 32 + 32 \beta_{1} ) q^{2} + ( 49 - 49 \beta_{1} + \beta_{2} ) q^{3} + 2048 \beta_{1} q^{4} + ( 2412 + 2511 \beta_{1} + 5 \beta_{2} + 10 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{5} + ( 3136 + 32 \beta_{2} - 32 \beta_{3} ) q^{6} + ( -53685 - 53699 \beta_{1} - 31 \beta_{3} + 21 \beta_{4} - 7 \beta_{5} ) q^{7} + ( -65536 + 65536 \beta_{1} ) q^{8} + ( 36 - 190301 \beta_{1} - 113 \beta_{2} - 113 \beta_{3} + 36 \beta_{4} - 72 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 32 + 32 \beta_{1} ) q^{2} + ( 49 - 49 \beta_{1} + \beta_{2} ) q^{3} + 2048 \beta_{1} q^{4} + ( 2412 + 2511 \beta_{1} + 5 \beta_{2} + 10 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{5} + ( 3136 + 32 \beta_{2} - 32 \beta_{3} ) q^{6} + ( -53685 - 53699 \beta_{1} - 31 \beta_{3} + 21 \beta_{4} - 7 \beta_{5} ) q^{7} + ( -65536 + 65536 \beta_{1} ) q^{8} + ( 36 - 190301 \beta_{1} - 113 \beta_{2} - 113 \beta_{3} + 36 \beta_{4} - 72 \beta_{5} ) q^{9} + ( -3168 + 157536 \beta_{1} + 480 \beta_{2} + 160 \beta_{3} - 96 \beta_{4} - 32 \beta_{5} ) q^{10} + ( 274744 + 82 \beta_{1} + 1021 \beta_{2} - 1021 \beta_{3} - 164 \beta_{4} - 82 \beta_{5} ) q^{11} + ( 100352 + 100352 \beta_{1} - 2048 \beta_{3} ) q^{12} + ( -774227 + 774565 \beta_{1} - 388 \beta_{2} - 169 \beta_{4} - 507 \beta_{5} ) q^{13} + ( 448 - 3436288 \beta_{1} - 992 \beta_{2} - 992 \beta_{3} + 448 \beta_{4} - 896 \beta_{5} ) q^{14} + ( -6965297 - 3316961 \beta_{1} + 9195 \beta_{2} - 460 \beta_{3} + 1161 \beta_{4} - 783 \beta_{5} ) q^{15} -4194304 q^{16} + ( 8525361 + 8524769 \beta_{1} - 25214 \beta_{3} + 888 \beta_{4} - 296 \beta_{5} ) q^{17} + ( 6090784 - 6088480 \beta_{1} - 7232 \beta_{2} - 1152 \beta_{4} - 3456 \beta_{5} ) q^{18} + ( -2434 + 11094790 \beta_{1} + 40652 \beta_{2} + 40652 \beta_{3} - 2434 \beta_{4} + 4868 \beta_{5} ) q^{19} + ( -5142528 + 4939776 \beta_{1} + 20480 \beta_{2} - 10240 \beta_{3} - 4096 \beta_{4} + 2048 \beta_{5} ) q^{20} + ( 19595380 - 2538 \beta_{1} - 110983 \beta_{2} + 110983 \beta_{3} + 5076 \beta_{4} + 2538 \beta_{5} ) q^{21} + ( 8789184 + 8794432 \beta_{1} - 65344 \beta_{3} - 7872 \beta_{4} + 2624 \beta_{5} ) q^{22} + ( 17678893 - 17702851 \beta_{1} - 147189 \beta_{2} + 11979 \beta_{4} + 35937 \beta_{5} ) q^{23} + ( 6422528 \beta_{1} - 65536 \beta_{2} - 65536 \beta_{3} ) q^{24} + ( -26898905 - 69259290 \beta_{1} + 129425 \beta_{2} + 208475 \beta_{3} + 1490 \beta_{4} + 18680 \beta_{5} ) q^{25} + ( -49561344 + 10816 \beta_{1} - 12416 \beta_{2} + 12416 \beta_{3} - 21632 \beta_{4} - 10816 \beta_{5} ) q^{26} + ( 104503552 + 104492896 \beta_{1} - 35716 \beta_{3} + 15984 \beta_{4} - 5328 \beta_{5} ) q^{27} + ( 109975552 - 109946880 \beta_{1} - 63488 \beta_{2} - 14336 \beta_{4} - 43008 \beta_{5} ) q^{28} + ( 2066 + 561953578 \beta_{1} + 258554 \beta_{2} + 258554 \beta_{3} + 2066 \beta_{4} - 4132 \beta_{5} ) q^{29} + ( -116746752 - 329032256 \beta_{1} + 279520 \beta_{2} - 308960 \beta_{3} + 12096 \beta_{4} - 62208 \beta_{5} ) q^{30} + ( -444881420 - 35978 \beta_{1} + 560827 \beta_{2} - 560827 \beta_{3} + 71956 \beta_{4} + 35978 \beta_{5} ) q^{31} + ( -134217728 - 134217728 \beta_{1} ) q^{32} + ( 730100026 - 729983710 \beta_{1} + 556234 \beta_{2} - 58158 \beta_{4} - 174474 \beta_{5} ) q^{33} + ( 18944 + 545604160 \beta_{1} - 806848 \beta_{2} - 806848 \beta_{3} + 18944 \beta_{4} - 37888 \beta_{5} ) q^{34} + ( -1292568389 - 715991597 \beta_{1} - 1951960 \beta_{2} - 388445 \beta_{3} + 17642 \beta_{4} - 28436 \beta_{5} ) q^{35} + ( 389736448 + 73728 \beta_{1} - 231424 \beta_{2} + 231424 \beta_{3} - 147456 \beta_{4} - 73728 \beta_{5} ) q^{36} + ( 291458451 + 291487149 \beta_{1} + 454626 \beta_{3} - 43047 \beta_{4} + 14349 \beta_{5} ) q^{37} + ( -355111168 + 354955392 \beta_{1} + 2601728 \beta_{2} + 77888 \beta_{4} + 233664 \beta_{5} ) q^{38} + ( 74250 + 417608880 \beta_{1} + 550383 \beta_{2} + 550383 \beta_{3} + 74250 \beta_{4} - 148500 \beta_{5} ) q^{39} + ( -322633728 - 6488064 \beta_{1} + 327680 \beta_{2} - 983040 \beta_{3} - 65536 \beta_{4} + 196608 \beta_{5} ) q^{40} + ( 3785486842 + 116744 \beta_{1} + 2235923 \beta_{2} - 2235923 \beta_{3} - 233488 \beta_{4} - 116744 \beta_{5} ) q^{41} + ( 627133376 + 626970944 \beta_{1} + 7102912 \beta_{3} + 243648 \beta_{4} - 81216 \beta_{5} ) q^{42} + ( -2647929031 + 2647589363 \beta_{1} - 1814987 \beta_{2} + 169834 \beta_{4} + 509502 \beta_{5} ) q^{43} + ( -167936 + 562675712 \beta_{1} - 2091008 \beta_{2} - 2091008 \beta_{3} - 167936 \beta_{4} + 335872 \beta_{5} ) q^{44} + ( -1347700189 - 5087153712 \beta_{1} - 5513885 \beta_{2} + 3316880 \beta_{3} - 257598 \beta_{4} - 48501 \beta_{5} ) q^{45} + ( 1132215808 - 766656 \beta_{1} - 4710048 \beta_{2} + 4710048 \beta_{3} + 1533312 \beta_{4} + 766656 \beta_{5} ) q^{46} + ( -3084891849 - 3084596667 \beta_{1} - 6319551 \beta_{3} - 442773 \beta_{4} + 147591 \beta_{5} ) q^{47} + ( -205520896 + 205520896 \beta_{1} - 4194304 \beta_{2} ) q^{48} + ( -622288 + 6916688017 \beta_{1} + 2974853 \beta_{2} + 2974853 \beta_{3} - 622288 \beta_{4} + 1244576 \beta_{5} ) q^{49} + ( 1355532320 - 3077062240 \beta_{1} + 10812800 \beta_{2} + 2529600 \beta_{3} + 645440 \beta_{4} + 550080 \beta_{5} ) q^{50} + ( 19023679898 + 753192 \beta_{1} + 2230165 \beta_{2} - 2230165 \beta_{3} - 1506384 \beta_{4} - 753192 \beta_{5} ) q^{51} + ( -1586309120 - 1585616896 \beta_{1} + 794624 \beta_{3} - 1038336 \beta_{4} + 346112 \beta_{5} ) q^{52} + ( -14665660411 + 14667090577 \beta_{1} - 11795358 \beta_{2} - 715083 \beta_{4} - 2145249 \beta_{5} ) q^{53} + ( 340992 + 6687886336 \beta_{1} - 1142912 \beta_{2} - 1142912 \beta_{3} + 340992 \beta_{4} - 681984 \beta_{5} ) q^{54} + ( 6238401854 - 16927613688 \beta_{1} + 17301835 \beta_{2} - 2623455 \beta_{3} + 656758 \beta_{4} - 1330434 \beta_{5} ) q^{55} + ( 7037517824 + 917504 \beta_{1} - 2031616 \beta_{2} + 2031616 \beta_{3} - 1835008 \beta_{4} - 917504 \beta_{5} ) q^{56} + ( -29058173708 - 29059830104 \beta_{1} - 42440716 \beta_{3} + 2484594 \beta_{4} - 828198 \beta_{5} ) q^{57} + ( -17982448384 + 17982580608 \beta_{1} + 16547456 \beta_{2} - 66112 \beta_{4} - 198336 \beta_{5} ) q^{58} + ( 3534762 + 6417291586 \beta_{1} + 17956488 \beta_{2} + 17956488 \beta_{3} + 3534762 \beta_{4} - 7069524 \beta_{5} ) q^{59} + ( 6793136128 - 14264928256 \beta_{1} - 942080 \beta_{2} - 18831360 \beta_{3} - 1603584 \beta_{4} - 2377728 \beta_{5} ) q^{60} + ( 48647380818 - 2164256 \beta_{1} - 21266261 \beta_{2} + 21266261 \beta_{3} + 4328512 \beta_{4} + 2164256 \beta_{5} ) q^{61} + ( -14235054144 - 14237356736 \beta_{1} - 35892928 \beta_{3} + 3453888 \beta_{4} - 1151296 \beta_{5} ) q^{62} + ( -49597631795 + 49595756357 \beta_{1} + 20127955 \beta_{2} + 937719 \beta_{4} + 2813157 \beta_{5} ) q^{63} -8589934592 \beta_{1} q^{64} + ( 4494716183 - 32608566671 \beta_{1} + 4377645 \beta_{2} + 15635065 \beta_{3} - 2356404 \beta_{4} + 5556862 \beta_{5} ) q^{65} + ( 46722679552 + 3722112 \beta_{1} + 17799488 \beta_{2} - 17799488 \beta_{3} - 7444224 \beta_{4} - 3722112 \beta_{5} ) q^{66} + ( -608938723 - 604371991 \beta_{1} + 125943027 \beta_{3} - 6850098 \beta_{4} + 2283366 \beta_{5} ) q^{67} + ( -17458726912 + 17459939328 \beta_{1} - 51638272 \beta_{2} - 606208 \beta_{4} - 1818624 \beta_{5} ) q^{68} + ( -11551842 + 99287776372 \beta_{1} - 47911703 \beta_{2} - 47911703 \beta_{3} - 11551842 \beta_{4} + 23103684 \beta_{5} ) q^{69} + ( -18450457344 - 64273919552 \beta_{1} - 74892960 \beta_{2} + 50032480 \beta_{3} - 345408 \beta_{4} - 1474496 \beta_{5} ) q^{70} + ( 154852574612 - 5461898 \beta_{1} + 7622899 \beta_{2} - 7622899 \beta_{3} + 10923796 \beta_{4} + 5461898 \beta_{5} ) q^{71} + ( 12469207040 + 12473925632 \beta_{1} + 14811136 \beta_{3} - 7077888 \beta_{4} + 2359296 \beta_{5} ) q^{72} + ( 2116328009 - 2124630813 \beta_{1} - 18251788 \beta_{2} + 4151402 \beta_{4} + 12454206 \beta_{5} ) q^{73} + ( -918336 + 18654259200 \beta_{1} + 14548032 \beta_{2} + 14548032 \beta_{3} - 918336 \beta_{4} + 1836672 \beta_{5} ) q^{74} + ( -154707905395 - 97035694985 \beta_{1} - 46432675 \beta_{2} - 39053100 \beta_{3} + 15182910 \beta_{4} + 3450870 \beta_{5} ) q^{75} + ( -22722129920 - 4984832 \beta_{1} + 83255296 \beta_{2} - 83255296 \beta_{3} + 9969664 \beta_{4} + 4984832 \beta_{5} ) q^{76} + ( -72667730558 - 72674515470 \beta_{1} + 246715906 \beta_{3} + 10177368 \beta_{4} - 3392456 \beta_{5} ) q^{77} + ( -13361108160 + 13365860160 \beta_{1} + 35224512 \beta_{2} - 2376000 \beta_{4} - 7128000 \beta_{5} ) q^{78} + ( 17669312 + 115439724672 \beta_{1} - 114190768 \beta_{2} - 114190768 \beta_{3} + 17669312 \beta_{4} - 35338624 \beta_{5} ) q^{79} + ( -10116661248 - 10531897344 \beta_{1} - 20971520 \beta_{2} - 41943040 \beta_{3} + 4194304 \beta_{4} + 8388608 \beta_{5} ) q^{80} + ( 138984179357 + 17636436 \beta_{1} - 1124777 \beta_{2} + 1124777 \beta_{3} - 35272872 \beta_{4} - 17636436 \beta_{5} ) q^{81} + ( 121131843136 + 121139314752 \beta_{1} - 143099072 \beta_{3} - 11207424 \beta_{4} + 3735808 \beta_{5} ) q^{82} + ( 11306465713 - 11290607057 \beta_{1} + 434769245 \beta_{2} - 7929328 \beta_{4} - 23787984 \beta_{5} ) q^{83} + ( 5197824 + 40131338240 \beta_{1} + 227293184 \beta_{2} + 227293184 \beta_{3} + 5197824 \beta_{4} - 10395648 \beta_{5} ) q^{84} + ( 23675817701 - 149388692077 \beta_{1} + 50827765 \beta_{2} - 188093995 \beta_{3} - 46842803 \beta_{4} - 38578201 \beta_{5} ) q^{85} + ( -169456588608 - 10869376 \beta_{1} - 58079584 \beta_{2} + 58079584 \beta_{3} + 21738752 \beta_{4} + 10869376 \beta_{5} ) q^{86} + ( -157433716004 - 157453410344 \beta_{1} - 630298252 \beta_{3} + 29541510 \beta_{4} - 9847170 \beta_{5} ) q^{87} + ( -18010996736 + 18000248832 \beta_{1} - 133824512 \beta_{2} + 5373952 \beta_{4} + 16121856 \beta_{5} ) q^{88} + ( -7790100 + 284076200060 \beta_{1} - 50441340 \beta_{2} - 50441340 \beta_{3} - 7790100 \beta_{4} + 15580200 \beta_{5} ) q^{89} + ( 119662512736 - 205915324832 \beta_{1} - 70304160 \beta_{2} + 282584480 \beta_{3} - 9795168 \beta_{4} + 6691104 \beta_{5} ) q^{90} + ( -269840299122 + 13298116 \beta_{1} - 32918387 \beta_{2} + 32918387 \beta_{3} - 26596232 \beta_{4} - 13298116 \beta_{5} ) q^{91} + ( 36255438848 + 36206372864 \beta_{1} + 301443072 \beta_{3} + 73598976 \beta_{4} - 24532992 \beta_{5} ) q^{92} + ( 387054927226 - 387033328198 \beta_{1} - 895237094 \beta_{2} - 10799514 \beta_{4} - 32398542 \beta_{5} ) q^{93} + ( -9445824 - 197423632512 \beta_{1} - 202225632 \beta_{2} - 202225632 \beta_{3} - 9445824 \beta_{4} + 18891648 \beta_{5} ) q^{94} + ( -251992667370 + 467916738590 \beta_{1} + 706165200 \beta_{2} + 56783900 \beta_{3} + 78664710 \beta_{4} + 17221220 \beta_{5} ) q^{95} + ( -13153337344 - 134217728 \beta_{2} + 134217728 \beta_{3} ) q^{96} + ( 112465008313 + 112586171345 \beta_{1} - 230537920 \beta_{3} - 181744548 \beta_{4} + 60581516 \beta_{5} ) q^{97} + ( -221353929760 + 221314103328 \beta_{1} + 190390592 \beta_{2} + 19913216 \beta_{4} + 59739648 \beta_{5} ) q^{98} + ( 6804738 - 302452407656 \beta_{1} + 531596485 \beta_{2} + 531596485 \beta_{3} + 6804738 \beta_{4} - 13609476 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 192q^{2} + 296q^{3} + 14460q^{5} + 18944q^{6} - 322104q^{7} - 393216q^{8} + O(q^{10})$$ $$6q + 192q^{2} + 296q^{3} + 14460q^{5} + 18944q^{6} - 322104q^{7} - 393216q^{8} - 18240q^{10} + 1652712q^{11} + 606208q^{12} - 4646814q^{13} - 41776360q^{15} - 25165824q^{16} + 51200226q^{17} + 36525632q^{18} - 30781440q^{20} + 117123272q^{21} + 52886784q^{22} + 105826896q^{23} - 161517150q^{25} - 297396096q^{26} + 627050120q^{27} + 659668992q^{28} - 699452160q^{30} - 2667117168q^{31} - 805306368q^{32} + 4381479992q^{33} - 7758629520q^{35} + 2337640448q^{36} + 1747956246q^{37} - 2125152000q^{38} - 1932656640q^{40} + 22722098232q^{41} + 3747944704q^{42} - 15890524824q^{43} - 8103444470q^{45} + 6772921344q^{46} - 18495531264q^{47} - 1241513984q^{48} + 8149569600q^{50} + 114152506432q^{51} - 9516675072q^{52} - 88020413514q^{53} + 37466287320q^{55} + 42218815488q^{56} - 174270786400q^{57} - 107861859840q^{58} + 40793047040q^{60} + 291794891352q^{61} - 85347749376q^{62} - 297541783984q^{63} + 26961608790q^{65} + 280414719488q^{66} - 3887251464q^{67} - 104858062848q^{68} - 110954853120q^{70} + 929135015472q^{71} + 74804494336q^{72} + 12678070086q^{73} - 928285655600q^{75} - 136009728000q^{76} - 436526954808q^{77} - 80105703936q^{78} - 60649635840q^{80} + 833935849906q^{81} + 727107143424q^{82} + 68676615456q^{83} + 142549278930q^{85} - 1016993588736q^{86} - 943420476880q^{87} - 108312133632q^{88} + 717302271680q^{90} - 1619146872048q^{91} + 216733483008q^{92} + 2320495891112q^{93} - 1510780128600q^{95} - 79456894976q^{96} + 675735777846q^{97} - 1327663144512q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 43009 x^{4} + 461169144 x^{2} + 392422062096$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 669445 \nu^{3} - 13932675324 \nu$$$$)/ 405892941840$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 1566090 \nu^{4} + 669445 \nu^{3} - 33678765450 \nu^{2} + 1028665029924 \nu + 1386948097080$$$$)/ 202946470920$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 1566090 \nu^{4} + 669445 \nu^{3} + 33678765450 \nu^{2} + 1028665029924 \nu - 1386948097080$$$$)/ 202946470920$$ $$\beta_{4}$$ $$=$$ $$($$$$-266983 \nu^{5} + 22447290 \nu^{4} - 9608375335 \nu^{3} + 820973089650 \nu^{2} - 75714448599792 \nu + 4830067523831040$$$$)/ 608839412760$$ $$\beta_{5}$$ $$=$$ $$($$$$1067929 \nu^{5} + 22447290 \nu^{4} + 38431493005 \nu^{3} + 820973089650 \nu^{2} + 302815996373196 \nu + 4830676363243800$$$$)/ 1217678825520$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 4 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$36 \beta_{5} + 72 \beta_{4} - 215 \beta_{3} + 215 \beta_{2} - 36 \beta_{1} - 716948$$$$)/50$$ $$\nu^{3}$$ $$=$$ $$($$$$-72 \beta_{5} + 36 \beta_{4} - 107735 \beta_{3} - 107735 \beta_{2} - 32468828 \beta_{1} + 36$$$$)/50$$ $$\nu^{4}$$ $$=$$ $$($$$$-154836 \beta_{5} - 309672 \beta_{4} + 1572655 \beta_{3} - 1572655 \beta_{2} + 154836 \beta_{1} + 3092449468$$$$)/10$$ $$\nu^{5}$$ $$=$$ $$($$$$9640008 \beta_{5} - 4820004 \beta_{4} + 491856091 \beta_{3} + 491856091 \beta_{2} + 232558792396 \beta_{1} - 4820004$$$$)/10$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$\beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 159.940i 30.4929i 128.447i 159.940i − 30.4929i − 128.447i
32.0000 32.0000i −748.698 748.698i 2048.00i 11268.2 10824.4i −47916.6 −121018. + 121018.i −65536.0 65536.0i 589655.i 14201.0 706964.i
3.2 32.0000 32.0000i 203.464 + 203.464i 2048.00i −11784.0 10260.5i 13021.7 68548.2 68548.2i −65536.0 65536.0i 448646.i −705424. + 48750.4i
3.3 32.0000 32.0000i 693.233 + 693.233i 2048.00i 7745.77 + 13570.0i 44366.9 −108582. + 108582.i −65536.0 65536.0i 429704.i 682103. + 186374.i
7.1 32.0000 + 32.0000i −748.698 + 748.698i 2048.00i 11268.2 + 10824.4i −47916.6 −121018. 121018.i −65536.0 + 65536.0i 589655.i 14201.0 + 706964.i
7.2 32.0000 + 32.0000i 203.464 203.464i 2048.00i −11784.0 + 10260.5i 13021.7 68548.2 + 68548.2i −65536.0 + 65536.0i 448646.i −705424. 48750.4i
7.3 32.0000 + 32.0000i 693.233 693.233i 2048.00i 7745.77 13570.0i 44366.9 −108582. 108582.i −65536.0 + 65536.0i 429704.i 682103. 186374.i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.13.c.b 6
3.b odd 2 1 90.13.g.a 6
4.b odd 2 1 80.13.p.a 6
5.b even 2 1 50.13.c.c 6
5.c odd 4 1 inner 10.13.c.b 6
5.c odd 4 1 50.13.c.c 6
15.e even 4 1 90.13.g.a 6
20.e even 4 1 80.13.p.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.13.c.b 6 1.a even 1 1 trivial
10.13.c.b 6 5.c odd 4 1 inner
50.13.c.c 6 5.b even 2 1
50.13.c.c 6 5.c odd 4 1
80.13.p.a 6 4.b odd 2 1
80.13.p.a 6 20.e even 4 1
90.13.g.a 6 3.b odd 2 1
90.13.g.a 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 296 T_{3}^{5} + 43808 T_{3}^{4} - 108468072 T_{3}^{3} +$$$$11\!\cdots\!96$$$$T_{3}^{2} -$$$$44\!\cdots\!24$$$$T_{3} +$$$$89\!\cdots\!28$$ acting on $$S_{13}^{\mathrm{new}}(10, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 64 T + 2048 T^{2} )^{3}$$
$3$ $$1 - 296 T + 43808 T^{2} - 265774608 T^{3} - 282424392657 T^{4} + 225087149263896 T^{5} - 18935277259818528 T^{6} +$$$$11\!\cdots\!36$$$$T^{7} -$$$$79\!\cdots\!17$$$$T^{8} -$$$$39\!\cdots\!68$$$$T^{9} +$$$$34\!\cdots\!88$$$$T^{10} -$$$$12\!\cdots\!96$$$$T^{11} +$$$$22\!\cdots\!41$$$$T^{12}$$
$5$ $$1 - 14460 T + 185304375 T^{2} + 1167590625000 T^{3} + 45240325927734375 T^{4} -$$$$86\!\cdots\!00$$$$T^{5} +$$$$14\!\cdots\!25$$$$T^{6}$$
$7$ $$1 + 322104 T + 51875493408 T^{2} + 6387501326203232 T^{3} +$$$$31\!\cdots\!43$$$$T^{4} -$$$$30\!\cdots\!24$$$$T^{5} -$$$$57\!\cdots\!28$$$$T^{6} -$$$$42\!\cdots\!24$$$$T^{7} +$$$$60\!\cdots\!43$$$$T^{8} +$$$$16\!\cdots\!32$$$$T^{9} +$$$$19\!\cdots\!08$$$$T^{10} +$$$$16\!\cdots\!04$$$$T^{11} +$$$$70\!\cdots\!01$$$$T^{12}$$
$11$ $$( 1 - 826356 T + 6038845083975 T^{2} - 6844626283938959560 T^{3} +$$$$18\!\cdots\!75$$$$T^{4} -$$$$81\!\cdots\!96$$$$T^{5} +$$$$30\!\cdots\!61$$$$T^{6} )^{2}$$
$13$ $$1 + 4646814 T + 10796440175298 T^{2} + 59958496622356841022 T^{3} +$$$$87\!\cdots\!23$$$$T^{4} +$$$$44\!\cdots\!76$$$$T^{5} +$$$$13\!\cdots\!52$$$$T^{6} +$$$$10\!\cdots\!56$$$$T^{7} +$$$$47\!\cdots\!03$$$$T^{8} +$$$$75\!\cdots\!02$$$$T^{9} +$$$$31\!\cdots\!58$$$$T^{10} +$$$$31\!\cdots\!14$$$$T^{11} +$$$$15\!\cdots\!81$$$$T^{12}$$
$17$ $$1 - 51200226 T + 1310731571225538 T^{2} -$$$$34\!\cdots\!38$$$$T^{3} +$$$$42\!\cdots\!03$$$$T^{4} +$$$$32\!\cdots\!76$$$$T^{5} -$$$$10\!\cdots\!68$$$$T^{6} +$$$$18\!\cdots\!36$$$$T^{7} +$$$$14\!\cdots\!63$$$$T^{8} -$$$$69\!\cdots\!78$$$$T^{9} +$$$$15\!\cdots\!58$$$$T^{10} -$$$$34\!\cdots\!26$$$$T^{11} +$$$$39\!\cdots\!61$$$$T^{12}$$
$19$ $$1 - 3019774946894166 T^{2} -$$$$27\!\cdots\!85$$$$T^{4} +$$$$12\!\cdots\!80$$$$T^{6} -$$$$13\!\cdots\!85$$$$T^{8} -$$$$72\!\cdots\!06$$$$T^{10} +$$$$11\!\cdots\!61$$$$T^{12}$$
$23$ $$1 - 105826896 T + 5599665958497408 T^{2} -$$$$36\!\cdots\!88$$$$T^{3} +$$$$94\!\cdots\!43$$$$T^{4} -$$$$70\!\cdots\!64$$$$T^{5} +$$$$90\!\cdots\!72$$$$T^{6} -$$$$15\!\cdots\!44$$$$T^{7} +$$$$45\!\cdots\!63$$$$T^{8} -$$$$38\!\cdots\!68$$$$T^{9} +$$$$12\!\cdots\!48$$$$T^{10} -$$$$53\!\cdots\!96$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12}$$
$29$ $$1 - 887923775060223846 T^{2} +$$$$42\!\cdots\!15$$$$T^{4} -$$$$15\!\cdots\!20$$$$T^{6} +$$$$53\!\cdots\!15$$$$T^{8} -$$$$13\!\cdots\!06$$$$T^{10} +$$$$19\!\cdots\!41$$$$T^{12}$$
$31$ $$( 1 + 1333558584 T + 1976764301333685735 T^{2} +$$$$20\!\cdots\!00$$$$T^{3} +$$$$15\!\cdots\!35$$$$T^{4} +$$$$82\!\cdots\!64$$$$T^{5} +$$$$48\!\cdots\!81$$$$T^{6} )^{2}$$
$37$ $$1 - 1747956246 T + 1527675518965206258 T^{2} -$$$$11\!\cdots\!98$$$$T^{3} +$$$$13\!\cdots\!43$$$$T^{4} -$$$$15\!\cdots\!84$$$$T^{5} +$$$$13\!\cdots\!72$$$$T^{6} -$$$$10\!\cdots\!04$$$$T^{7} +$$$$58\!\cdots\!23$$$$T^{8} -$$$$33\!\cdots\!18$$$$T^{9} +$$$$28\!\cdots\!18$$$$T^{10} -$$$$21\!\cdots\!46$$$$T^{11} +$$$$81\!\cdots\!81$$$$T^{12}$$
$41$ $$( 1 - 11361049116 T + 97304369679810531495 T^{2} -$$$$53\!\cdots\!40$$$$T^{3} +$$$$21\!\cdots\!95$$$$T^{4} -$$$$57\!\cdots\!76$$$$T^{5} +$$$$11\!\cdots\!41$$$$T^{6} )^{2}$$
$43$ $$1 + 15890524824 T +$$$$12\!\cdots\!88$$$$T^{2} +$$$$10\!\cdots\!72$$$$T^{3} +$$$$97\!\cdots\!03$$$$T^{4} +$$$$68\!\cdots\!96$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6} +$$$$27\!\cdots\!96$$$$T^{7} +$$$$15\!\cdots\!03$$$$T^{8} +$$$$67\!\cdots\!72$$$$T^{9} +$$$$32\!\cdots\!88$$$$T^{10} +$$$$16\!\cdots\!24$$$$T^{11} +$$$$40\!\cdots\!01$$$$T^{12}$$
$47$ $$1 + 18495531264 T +$$$$17\!\cdots\!48$$$$T^{2} +$$$$25\!\cdots\!12$$$$T^{3} +$$$$43\!\cdots\!23$$$$T^{4} +$$$$39\!\cdots\!56$$$$T^{5} +$$$$32\!\cdots\!52$$$$T^{6} +$$$$45\!\cdots\!96$$$$T^{7} +$$$$58\!\cdots\!63$$$$T^{8} +$$$$40\!\cdots\!52$$$$T^{9} +$$$$31\!\cdots\!28$$$$T^{10} +$$$$39\!\cdots\!64$$$$T^{11} +$$$$24\!\cdots\!41$$$$T^{12}$$
$53$ $$1 + 88020413514 T +$$$$38\!\cdots\!98$$$$T^{2} +$$$$12\!\cdots\!62$$$$T^{3} +$$$$33\!\cdots\!23$$$$T^{4} +$$$$85\!\cdots\!56$$$$T^{5} +$$$$20\!\cdots\!52$$$$T^{6} +$$$$41\!\cdots\!96$$$$T^{7} +$$$$80\!\cdots\!63$$$$T^{8} +$$$$14\!\cdots\!02$$$$T^{9} +$$$$22\!\cdots\!78$$$$T^{10} +$$$$25\!\cdots\!14$$$$T^{11} +$$$$14\!\cdots\!41$$$$T^{12}$$
$59$ $$1 -$$$$38\!\cdots\!86$$$$T^{2} +$$$$11\!\cdots\!15$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$35\!\cdots\!15$$$$T^{8} -$$$$39\!\cdots\!06$$$$T^{10} +$$$$31\!\cdots\!81$$$$T^{12}$$
$61$ $$( 1 - 145897445676 T +$$$$13\!\cdots\!55$$$$T^{2} -$$$$76\!\cdots\!80$$$$T^{3} +$$$$34\!\cdots\!55$$$$T^{4} -$$$$10\!\cdots\!16$$$$T^{5} +$$$$18\!\cdots\!61$$$$T^{6} )^{2}$$
$67$ $$1 + 3887251464 T + 7555361972185071648 T^{2} +$$$$14\!\cdots\!92$$$$T^{3} -$$$$56\!\cdots\!77$$$$T^{4} -$$$$75\!\cdots\!84$$$$T^{5} +$$$$10\!\cdots\!52$$$$T^{6} -$$$$62\!\cdots\!24$$$$T^{7} -$$$$37\!\cdots\!17$$$$T^{8} +$$$$80\!\cdots\!52$$$$T^{9} +$$$$33\!\cdots\!68$$$$T^{10} +$$$$14\!\cdots\!64$$$$T^{11} +$$$$30\!\cdots\!61$$$$T^{12}$$
$71$ $$( 1 - 464567507736 T +$$$$11\!\cdots\!55$$$$T^{2} -$$$$18\!\cdots\!80$$$$T^{3} +$$$$18\!\cdots\!55$$$$T^{4} -$$$$12\!\cdots\!16$$$$T^{5} +$$$$44\!\cdots\!21$$$$T^{6} )^{2}$$
$73$ $$1 - 12678070086 T + 80366730552764023698 T^{2} -$$$$64\!\cdots\!18$$$$T^{3} +$$$$92\!\cdots\!23$$$$T^{4} -$$$$18\!\cdots\!04$$$$T^{5} +$$$$16\!\cdots\!52$$$$T^{6} -$$$$43\!\cdots\!84$$$$T^{7} +$$$$48\!\cdots\!43$$$$T^{8} -$$$$77\!\cdots\!98$$$$T^{9} +$$$$22\!\cdots\!38$$$$T^{10} -$$$$79\!\cdots\!86$$$$T^{11} +$$$$14\!\cdots\!21$$$$T^{12}$$
$79$ $$1 -$$$$11\!\cdots\!46$$$$T^{2} +$$$$39\!\cdots\!15$$$$T^{4} -$$$$19\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!15$$$$T^{8} -$$$$14\!\cdots\!06$$$$T^{10} +$$$$42\!\cdots\!41$$$$T^{12}$$
$83$ $$1 - 68676615456 T +$$$$23\!\cdots\!68$$$$T^{2} +$$$$31\!\cdots\!52$$$$T^{3} -$$$$11\!\cdots\!37$$$$T^{4} -$$$$21\!\cdots\!04$$$$T^{5} +$$$$67\!\cdots\!92$$$$T^{6} -$$$$22\!\cdots\!44$$$$T^{7} -$$$$12\!\cdots\!77$$$$T^{8} +$$$$38\!\cdots\!12$$$$T^{9} +$$$$30\!\cdots\!88$$$$T^{10} -$$$$95\!\cdots\!56$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$89$ $$1 -$$$$12\!\cdots\!26$$$$T^{2} +$$$$65\!\cdots\!15$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} +$$$$40\!\cdots\!15$$$$T^{8} -$$$$44\!\cdots\!06$$$$T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$
$97$ $$1 - 675735777846 T +$$$$22\!\cdots\!58$$$$T^{2} -$$$$35\!\cdots\!58$$$$T^{3} -$$$$45\!\cdots\!57$$$$T^{4} +$$$$57\!\cdots\!96$$$$T^{5} -$$$$21\!\cdots\!28$$$$T^{6} +$$$$39\!\cdots\!36$$$$T^{7} -$$$$21\!\cdots\!17$$$$T^{8} -$$$$11\!\cdots\!18$$$$T^{9} +$$$$52\!\cdots\!38$$$$T^{10} -$$$$10\!\cdots\!46$$$$T^{11} +$$$$11\!\cdots\!41$$$$T^{12}$$