# Properties

 Label 10.13.c.a 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: $$2^{5}\cdot 3^{2}\cdot 5^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -32 - 32 \beta_{1} ) q^{2} + ( -156 + 156 \beta_{1} + \beta_{3} ) q^{3} + 2048 \beta_{1} q^{4} + ( -2710 + 1905 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{5} + ( 9984 - 32 \beta_{2} - 32 \beta_{3} ) q^{6} + ( -7556 - 7556 \beta_{1} - 56 \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{7} + ( 65536 - 65536 \beta_{1} ) q^{8} + ( -45331 \beta_{1} + 581 \beta_{2} - 589 \beta_{3} - 24 \beta_{4} - 32 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -32 - 32 \beta_{1} ) q^{2} + ( -156 + 156 \beta_{1} + \beta_{3} ) q^{3} + 2048 \beta_{1} q^{4} + ( -2710 + 1905 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{5} + ( 9984 - 32 \beta_{2} - 32 \beta_{3} ) q^{6} + ( -7556 - 7556 \beta_{1} - 56 \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{7} + ( 65536 - 65536 \beta_{1} ) q^{8} + ( -45331 \beta_{1} + 581 \beta_{2} - 589 \beta_{3} - 24 \beta_{4} - 32 \beta_{5} ) q^{9} + ( 147680 + 25760 \beta_{1} - 512 \beta_{2} + 64 \beta_{3} + 32 \beta_{4} - 96 \beta_{5} ) q^{10} + ( -569668 + 1925 \beta_{2} + 1991 \beta_{3} + 264 \beta_{4} - 198 \beta_{5} ) q^{11} + ( -319488 - 319488 \beta_{1} + 2048 \beta_{2} ) q^{12} + ( 1351139 - 1351139 \beta_{1} - 133 \beta_{2} - 2959 \beta_{3} + 133 \beta_{4} - 931 \beta_{5} ) q^{13} + ( 483584 \beta_{1} + 1760 \beta_{2} - 1824 \beta_{3} - 192 \beta_{4} - 256 \beta_{5} ) q^{14} + ( 4448980 - 4743640 \beta_{1} + 7353 \beta_{2} - 10476 \beta_{3} + 467 \beta_{4} - 311 \beta_{5} ) q^{15} -4194304 q^{16} + ( 1795419 + 1795419 \beta_{1} - 24182 \beta_{2} + 308 \beta_{3} + 2156 \beta_{4} + 308 \beta_{5} ) q^{17} + ( -1450592 + 1450592 \beta_{1} + 256 \beta_{2} + 37440 \beta_{3} - 256 \beta_{4} + 1792 \beta_{5} ) q^{18} + ( 20457540 \beta_{1} + 16206 \beta_{2} - 15884 \beta_{3} + 966 \beta_{4} + 1288 \beta_{5} ) q^{19} + ( -3901440 - 5550080 \beta_{1} + 14336 \beta_{2} - 18432 \beta_{3} - 4096 \beta_{4} + 2048 \beta_{5} ) q^{20} + ( -28839428 + 30949 \beta_{2} + 30735 \beta_{3} - 856 \beta_{4} + 642 \beta_{5} ) q^{21} + ( 18229376 + 18229376 \beta_{1} - 125312 \beta_{2} - 2112 \beta_{3} - 14784 \beta_{4} - 2112 \beta_{5} ) q^{22} + ( 40191144 - 40191144 \beta_{1} + 1363 \beta_{2} + 74592 \beta_{3} - 1363 \beta_{4} + 9541 \beta_{5} ) q^{23} + ( 20447232 \beta_{1} - 65536 \beta_{2} + 65536 \beta_{3} ) q^{24} + ( -86823525 - 64412800 \beta_{1} - 95665 \beta_{2} - 264595 \beta_{3} + 11690 \beta_{4} + 5080 \beta_{5} ) q^{25} + ( -86472896 + 98944 \beta_{2} + 90432 \beta_{3} - 34048 \beta_{4} + 25536 \beta_{5} ) q^{26} + ( 239598540 + 239598540 \beta_{1} - 130812 \beta_{2} + 3744 \beta_{3} + 26208 \beta_{4} + 3744 \beta_{5} ) q^{27} + ( 15474688 - 15474688 \beta_{1} + 2048 \beta_{2} + 114688 \beta_{3} - 2048 \beta_{4} + 14336 \beta_{5} ) q^{28} + ( 3003800 \beta_{1} + 200828 \beta_{2} - 197246 \beta_{3} + 10746 \beta_{4} + 14328 \beta_{5} ) q^{29} + ( -294163840 + 9429120 \beta_{1} + 99936 \beta_{2} + 570528 \beta_{3} - 24896 \beta_{4} - 4992 \beta_{5} ) q^{30} + ( -284851368 + 104099 \beta_{2} + 132345 \beta_{3} + 112984 \beta_{4} - 84738 \beta_{5} ) q^{31} + ( 134217728 + 134217728 \beta_{1} ) q^{32} + ( 1069319108 - 1069319108 \beta_{1} - 23386 \beta_{2} - 1300948 \beta_{3} + 23386 \beta_{4} - 163702 \beta_{5} ) q^{33} + ( -114906816 \beta_{1} + 763968 \beta_{2} - 783680 \beta_{3} - 59136 \beta_{4} - 78848 \beta_{5} ) q^{34} + ( -637325360 + 51490980 \beta_{1} + 483499 \beta_{2} + 77072 \beta_{3} + 21586 \beta_{4} - 99608 \beta_{5} ) q^{35} + ( 92837888 - 1206272 \beta_{2} - 1189888 \beta_{3} + 65536 \beta_{4} - 49152 \beta_{5} ) q^{36} + ( -123769151 - 123769151 \beta_{1} + 4834563 \beta_{2} + 3053 \beta_{3} + 21371 \beta_{4} + 3053 \beta_{5} ) q^{37} + ( 654641280 - 654641280 \beta_{1} - 10304 \beta_{2} + 1026880 \beta_{3} + 10304 \beta_{4} - 72128 \beta_{5} ) q^{38} + ( 2200102168 \beta_{1} - 4536017 \beta_{2} + 4528567 \beta_{3} - 22350 \beta_{4} - 29800 \beta_{5} ) q^{39} + ( -52756480 + 302448640 \beta_{1} + 131072 \beta_{2} + 1048576 \beta_{3} + 196608 \beta_{4} + 65536 \beta_{5} ) q^{40} + ( -4124816588 - 1060681 \beta_{2} - 1190349 \beta_{3} - 518672 \beta_{4} + 389004 \beta_{5} ) q^{41} + ( 922861696 + 922861696 \beta_{1} - 1973888 \beta_{2} + 6848 \beta_{3} + 47936 \beta_{4} + 6848 \beta_{5} ) q^{42} + ( 2422160884 - 2422160884 \beta_{1} + 142378 \beta_{2} - 1171301 \beta_{3} - 142378 \beta_{4} + 996646 \beta_{5} ) q^{43} + ( -1166680064 \beta_{1} + 4077568 \beta_{2} - 3942400 \beta_{3} + 405504 \beta_{4} + 540672 \beta_{5} ) q^{44} + ( 2848582655 + 5643301710 \beta_{1} - 8630582 \beta_{2} + 2691434 \beta_{3} - 554798 \beta_{4} + 599799 \beta_{5} ) q^{45} + ( -2572233216 - 2430560 \beta_{2} - 2343328 \beta_{3} + 348928 \beta_{4} - 261696 \beta_{5} ) q^{46} + ( 1404547904 + 1404547904 \beta_{1} - 8011818 \beta_{2} - 152513 \beta_{3} - 1067591 \beta_{4} - 152513 \beta_{5} ) q^{47} + ( 654311424 - 654311424 \beta_{1} - 4194304 \beta_{3} ) q^{48} + ( -10213022829 \beta_{1} - 889149 \beta_{2} + 794113 \beta_{3} - 285108 \beta_{4} - 380144 \beta_{5} ) q^{49} + ( 717143200 + 4839562400 \beta_{1} + 11528320 \beta_{2} + 5405760 \beta_{3} - 211520 \beta_{4} - 536640 \beta_{5} ) q^{50} + ( -13830661328 + 16657381 \beta_{2} + 16535997 \beta_{3} - 485536 \beta_{4} + 364152 \beta_{5} ) q^{51} + ( 2767132672 + 2767132672 \beta_{1} - 6060032 \beta_{2} + 272384 \beta_{3} + 1906688 \beta_{4} + 272384 \beta_{5} ) q^{52} + ( 216169089 - 216169089 \beta_{1} - 342691 \beta_{2} + 14963325 \beta_{3} + 342691 \beta_{4} - 2398837 \beta_{5} ) q^{53} + ( -15334306560 \beta_{1} + 4066176 \beta_{2} - 4305792 \beta_{3} - 718848 \beta_{4} - 958464 \beta_{5} ) q^{54} + ( 12628862180 + 23736395760 \beta_{1} + 11574453 \beta_{2} - 41542831 \beta_{3} + 1502842 \beta_{4} - 1172666 \beta_{5} ) q^{55} + ( -990380032 - 3735552 \beta_{2} - 3604480 \beta_{3} + 524288 \beta_{4} - 393216 \beta_{5} ) q^{56} + ( 5020636960 + 5020636960 \beta_{1} + 11319358 \beta_{2} + 166034 \beta_{3} + 1162238 \beta_{4} + 166034 \beta_{5} ) q^{57} + ( 96121600 - 96121600 \beta_{1} - 114624 \beta_{2} + 12738368 \beta_{3} + 114624 \beta_{4} - 802368 \beta_{5} ) q^{58} + ( -51842651500 \beta_{1} + 10129686 \beta_{2} - 9883872 \beta_{3} + 737442 \beta_{4} + 983256 \beta_{5} ) q^{59} + ( 9714974720 + 9111511040 \beta_{1} - 21454848 \beta_{2} - 15058944 \beta_{3} + 636928 \beta_{4} + 956416 \beta_{5} ) q^{60} + ( -33546136988 - 12916257 \beta_{2} - 11507485 \beta_{3} + 5635088 \beta_{4} - 4226316 \beta_{5} ) q^{61} + ( 9115243776 + 9115243776 \beta_{1} - 7566208 \beta_{2} - 903872 \beta_{3} - 6327104 \beta_{4} - 903872 \beta_{5} ) q^{62} + ( 24975872264 - 24975872264 \beta_{1} + 309743 \beta_{2} - 28617204 \beta_{3} - 309743 \beta_{4} + 2168201 \beta_{5} ) q^{63} -8589934592 \beta_{1} q^{64} + ( 24359392905 + 77575225085 \beta_{1} + 28869783 \beta_{2} + 66180539 \beta_{3} - 1982388 \beta_{4} - 1210646 \beta_{5} ) q^{65} + ( -68436422912 + 42378688 \beta_{2} + 40881984 \beta_{3} - 5986816 \beta_{4} + 4490112 \beta_{5} ) q^{66} + ( 18050662924 + 18050662924 \beta_{1} - 47079021 \beta_{2} + 1251862 \beta_{3} + 8763034 \beta_{4} + 1251862 \beta_{5} ) q^{67} + ( -3677018112 + 3677018112 \beta_{1} + 630784 \beta_{2} + 49524736 \beta_{3} - 630784 \beta_{4} + 4415488 \beta_{5} ) q^{68} + ( -29064956172 \beta_{1} + 11261689 \beta_{2} - 11539483 \beta_{3} - 833382 \beta_{4} - 1111176 \beta_{5} ) q^{69} + ( 22042122880 + 18746700160 \beta_{1} - 17938272 \beta_{2} + 13005664 \beta_{3} - 3878208 \beta_{4} + 2496704 \beta_{5} ) q^{70} + ( -95066944568 - 81931333 \beta_{2} - 84341647 \beta_{3} - 9641256 \beta_{4} + 7230942 \beta_{5} ) q^{71} + ( -2970812416 - 2970812416 \beta_{1} + 76677120 \beta_{2} - 524288 \beta_{3} - 3670016 \beta_{4} - 524288 \beta_{5} ) q^{72} + ( 118373704689 - 118373704689 \beta_{1} + 58514 \beta_{2} - 55191070 \beta_{3} - 58514 \beta_{4} + 409598 \beta_{5} ) q^{73} + ( 7921225664 \beta_{1} - 154803712 \beta_{2} + 154608320 \beta_{3} - 586176 \beta_{4} - 781568 \beta_{5} ) q^{74} + ( -30001554800 + 136379526400 \beta_{1} - 104275230 \beta_{2} + 87741235 \beta_{3} + 5277530 \beta_{4} + 10056710 \beta_{5} ) q^{75} + ( -41897041920 - 32530432 \beta_{2} - 33189888 \beta_{3} - 2637824 \beta_{4} + 1978368 \beta_{5} ) q^{76} + ( 1533587308 + 1533587308 \beta_{1} + 116697658 \beta_{2} - 2834172 \beta_{3} - 19839204 \beta_{4} - 2834172 \beta_{5} ) q^{77} + ( 70403269376 - 70403269376 \beta_{1} + 238400 \beta_{2} - 290066688 \beta_{3} - 238400 \beta_{4} + 1668800 \beta_{5} ) q^{78} + ( 37092879520 \beta_{1} + 49895120 \beta_{2} - 45145536 \beta_{3} + 14248752 \beta_{4} + 18998336 \beta_{5} ) q^{79} + ( 11366563840 - 7990149120 \beta_{1} - 37748736 \beta_{2} - 29360128 \beta_{3} - 4194304 \beta_{4} - 8388608 \beta_{5} ) q^{80} + ( -125823444909 + 36599595 \beta_{2} + 40680723 \beta_{3} + 16324512 \beta_{4} - 12243384 \beta_{5} ) q^{81} + ( 131994130816 + 131994130816 \beta_{1} + 72032960 \beta_{2} + 4149376 \beta_{3} + 29045632 \beta_{4} + 4149376 \beta_{5} ) q^{82} + ( 121311248024 - 121311248024 \beta_{1} - 412816 \beta_{2} + 264327853 \beta_{3} + 412816 \beta_{4} - 2889712 \beta_{5} ) q^{83} + ( -59063148544 \beta_{1} + 62945280 \beta_{2} - 63383552 \beta_{3} - 1314816 \beta_{4} - 1753088 \beta_{5} ) q^{84} + ( -243222654885 - 17437980695 \beta_{1} + 270213684 \beta_{2} + 68641952 \beta_{3} + 1451051 \beta_{4} - 24794553 \beta_{5} ) q^{85} + ( -155018296576 + 32925536 \beta_{2} + 42037728 \beta_{3} + 36448768 \beta_{4} - 27336576 \beta_{5} ) q^{86} + ( 101735516600 + 101735516600 \beta_{1} - 116282278 \beta_{2} + 2011390 \beta_{3} + 14079730 \beta_{4} + 2011390 \beta_{5} ) q^{87} + ( -37333762048 + 37333762048 \beta_{1} - 4325376 \beta_{2} + 256638976 \beta_{3} + 4325376 \beta_{4} - 30277632 \beta_{5} ) q^{88} + ( -98027616480 \beta_{1} + 229057280 \beta_{2} - 251547820 \beta_{3} - 67471620 \beta_{4} - 89962160 \beta_{5} ) q^{89} + ( 89431009760 - 271740299680 \beta_{1} + 190052736 \beta_{2} - 362304512 \beta_{3} + 36947104 \beta_{4} - 1440032 \beta_{5} ) q^{90} + ( 306911090632 - 140900115 \beta_{2} - 148387807 \beta_{3} - 29950768 \beta_{4} + 22463076 \beta_{5} ) q^{91} + ( 82311462912 + 82311462912 \beta_{1} + 152764416 \beta_{2} - 2791424 \beta_{3} - 19539968 \beta_{4} - 2791424 \beta_{5} ) q^{92} + ( 83943985708 - 83943985708 \beta_{1} - 4250558 \beta_{2} + 19722160 \beta_{3} + 4250558 \beta_{4} - 29753906 \beta_{5} ) q^{93} + ( -89891065856 \beta_{1} + 261258592 \beta_{2} - 251497760 \beta_{3} + 29282496 \beta_{4} + 39043328 \beta_{5} ) q^{94} + ( -162016752100 + 52833887800 \beta_{1} - 141593410 \beta_{2} - 263555680 \beta_{3} - 45816490 \beta_{4} + 37645520 \beta_{5} ) q^{95} + ( -41875931136 + 134217728 \beta_{2} + 134217728 \beta_{3} ) q^{96} + ( -367973438391 - 367973438391 \beta_{1} - 890358044 \beta_{2} - 4060428 \beta_{3} - 28422996 \beta_{4} - 4060428 \beta_{5} ) q^{97} + ( -326816730528 + 326816730528 \beta_{1} + 3041152 \beta_{2} - 53864384 \beta_{3} - 3041152 \beta_{4} + 21288064 \beta_{5} ) q^{98} + ( 755875762708 \beta_{1} - 903009503 \beta_{2} + 943019869 \beta_{3} + 120031098 \beta_{4} + 160041464 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 192q^{2} - 936q^{3} - 16260q^{5} + 59904q^{6} - 45336q^{7} + 393216q^{8} + O(q^{10})$$ $$6q - 192q^{2} - 936q^{3} - 16260q^{5} + 59904q^{6} - 45336q^{7} + 393216q^{8} + 886080q^{10} - 3418008q^{11} - 1916928q^{12} + 8106834q^{13} + 26693880q^{15} - 25165824q^{16} + 10772514q^{17} - 8703552q^{18} - 23408640q^{20} - 173036568q^{21} + 109376256q^{22} + 241146864q^{23} - 520941150q^{25} - 518837376q^{26} + 1437591240q^{27} + 92848128q^{28} - 1764983040q^{30} - 1709108208q^{31} + 805306368q^{32} + 6415914648q^{33} - 3823952160q^{35} + 557027328q^{36} - 742614906q^{37} + 3927847680q^{38} - 316538880q^{40} - 24748899528q^{41} + 5537170176q^{42} + 14532965304q^{43} + 17091495930q^{45} - 15433399296q^{46} + 8427287424q^{47} + 3925868544q^{48} + 4302859200q^{50} - 82983967968q^{51} + 16602796032q^{52} + 1297014534q^{53} + 75773173080q^{55} - 5942280192q^{56} + 30123821760q^{57} + 576729600q^{58} + 58289848320q^{60} - 201276821928q^{61} + 54691462656q^{62} + 149855233584q^{63} + 146156357430q^{65} - 410618537472q^{66} + 108303977544q^{67} - 22062108672q^{68} + 132252737280q^{70} - 570401667408q^{71} - 17824874496q^{72} + 710242228134q^{73} - 180009328800q^{75} - 251382251520q^{76} + 9201523848q^{77} + 422419616256q^{78} + 68199383040q^{80} - 754940669454q^{81} + 791964784896q^{82} + 727867488144q^{83} - 1459335929310q^{85} - 930109779456q^{86} + 610413099600q^{87} - 224002572288q^{88} + 536586058560q^{90} + 1841466543792q^{91} + 493868777472q^{92} + 503663914248q^{93} - 972100512600q^{95} - 251255586816q^{96} - 2207840630346q^{97} - 1960900383168q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 2385 x^{4} + 1422264 x^{2} + 490000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 1685 \nu^{3} + 587164 \nu$$$$)/344400$$ $$\beta_{2}$$ $$=$$ $$($$$$17 \nu^{5} - 2450 \nu^{4} + 86045 \nu^{3} - 2865450 \nu^{2} + 77197188 \nu + 44319800$$$$)/91840$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{5} - 2450 \nu^{4} - 86045 \nu^{3} - 2865450 \nu^{2} - 77197188 \nu + 44319800$$$$)/91840$$ $$\beta_{4}$$ $$=$$ $$($$$$103 \nu^{5} + 22750 \nu^{4} + 575355 \nu^{3} + 27657350 \nu^{2} + 547516892 \nu + 422891000$$$$)/91840$$ $$\beta_{5}$$ $$=$$ $$($$$$79 \nu^{5} - 9870 \nu^{4} + 443075 \nu^{3} - 12015990 \nu^{2} + 422574076 \nu - 196948920$$$$)/55104$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{5} + 3 \beta_{4} + 29 \beta_{3} - 28 \beta_{2} + 500 \beta_{1}$$$$)/1500$$ $$\nu^{2}$$ $$=$$ $$($$$$-21 \beta_{5} + 28 \beta_{4} + 204 \beta_{3} + 197 \beta_{2} - 397500$$$$)/500$$ $$\nu^{3}$$ $$=$$ $$($$$$-4684 \beta_{5} - 3513 \beta_{4} - 35159 \beta_{3} + 33988 \beta_{2} - 738500 \beta_{1}$$$$)/1500$$ $$\nu^{4}$$ $$=$$ $$($$$$24561 \beta_{5} - 32748 \beta_{4} - 247964 \beta_{3} - 239777 \beta_{2} + 473949500$$$$)/500$$ $$\nu^{5}$$ $$=$$ $$($$$$5543884 \beta_{5} + 4157913 \beta_{4} + 42215159 \beta_{3} - 40829188 \beta_{2} + 1467390500 \beta_{1}$$$$)/1500$$

## 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
 − 34.7359i 34.3230i − 0.587129i 34.7359i − 34.3230i 0.587129i
−32.0000 + 32.0000i −864.774 864.774i 2048.00i −14634.8 5473.81i 55345.5 43339.1 43339.1i 65536.0 + 65536.0i 964226.i 643476. 293152.i
3.2 −32.0000 + 32.0000i 59.4453 + 59.4453i 2048.00i 3818.66 + 15151.2i −3804.50 −59325.1 + 59325.1i 65536.0 + 65536.0i 524374.i −607035. 362641.i
3.3 −32.0000 + 32.0000i 337.328 + 337.328i 2048.00i 2686.16 15392.4i −21589.0 −6681.98 + 6681.98i 65536.0 + 65536.0i 303860.i 406599. + 578513.i
7.1 −32.0000 32.0000i −864.774 + 864.774i 2048.00i −14634.8 + 5473.81i 55345.5 43339.1 + 43339.1i 65536.0 65536.0i 964226.i 643476. + 293152.i
7.2 −32.0000 32.0000i 59.4453 59.4453i 2048.00i 3818.66 15151.2i −3804.50 −59325.1 59325.1i 65536.0 65536.0i 524374.i −607035. + 362641.i
7.3 −32.0000 32.0000i 337.328 337.328i 2048.00i 2686.16 + 15392.4i −21589.0 −6681.98 6681.98i 65536.0 65536.0i 303860.i 406599. 578513.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.a 6
3.b odd 2 1 90.13.g.b 6
4.b odd 2 1 80.13.p.b 6
5.b even 2 1 50.13.c.d 6
5.c odd 4 1 inner 10.13.c.a 6
5.c odd 4 1 50.13.c.d 6
15.e even 4 1 90.13.g.b 6
20.e even 4 1 80.13.p.b 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 936 T_{3}^{5} + 438048 T_{3}^{4} - 674145288 T_{3}^{3} + 417489145956 T_{3}^{2} -$$$$44\!\cdots\!76$$$$T_{3} +$$$$24\!\cdots\!48$$ acting on $$S_{13}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 64 T + 2048 T^{2} )^{3}$$
$3$ $$1 + 936 T + 438048 T^{2} - 176716512 T^{3} - 108750640977 T^{4} + 273072500382744 T^{5} + 318848223943664352 T^{6} +$$$$14\!\cdots\!04$$$$T^{7} -$$$$30\!\cdots\!37$$$$T^{8} -$$$$26\!\cdots\!52$$$$T^{9} +$$$$34\!\cdots\!28$$$$T^{10} +$$$$39\!\cdots\!36$$$$T^{11} +$$$$22\!\cdots\!41$$$$T^{12}$$
$5$ $$1 + 16260 T + 392664375 T^{2} + 9140390625000 T^{3} + 95865325927734375 T^{4} +$$$$96\!\cdots\!00$$$$T^{5} +$$$$14\!\cdots\!25$$$$T^{6}$$
$7$ $$1 + 45336 T + 1027676448 T^{2} + 472787542763648 T^{3} +$$$$32\!\cdots\!23$$$$T^{4} +$$$$11\!\cdots\!64$$$$T^{5} +$$$$30\!\cdots\!52$$$$T^{6} +$$$$16\!\cdots\!64$$$$T^{7} +$$$$62\!\cdots\!23$$$$T^{8} +$$$$12\!\cdots\!48$$$$T^{9} +$$$$37\!\cdots\!48$$$$T^{10} +$$$$23\!\cdots\!36$$$$T^{11} +$$$$70\!\cdots\!01$$$$T^{12}$$
$11$ $$( 1 + 1709004 T - 531626769465 T^{2} - 9103105140673181000 T^{3} -$$$$16\!\cdots\!65$$$$T^{4} +$$$$16\!\cdots\!64$$$$T^{5} +$$$$30\!\cdots\!61$$$$T^{6} )^{2}$$
$13$ $$1 - 8106834 T + 32860378751778 T^{2} -$$$$17\!\cdots\!62$$$$T^{3} -$$$$52\!\cdots\!17$$$$T^{4} +$$$$83\!\cdots\!04$$$$T^{5} -$$$$34\!\cdots\!88$$$$T^{6} +$$$$19\!\cdots\!24$$$$T^{7} -$$$$28\!\cdots\!37$$$$T^{8} -$$$$22\!\cdots\!42$$$$T^{9} +$$$$96\!\cdots\!38$$$$T^{10} -$$$$55\!\cdots\!34$$$$T^{11} +$$$$15\!\cdots\!81$$$$T^{12}$$
$17$ $$1 - 10772514 T + 58023528940098 T^{2} +$$$$33\!\cdots\!58$$$$T^{3} -$$$$14\!\cdots\!77$$$$T^{4} -$$$$47\!\cdots\!16$$$$T^{5} +$$$$65\!\cdots\!52$$$$T^{6} -$$$$27\!\cdots\!76$$$$T^{7} -$$$$47\!\cdots\!17$$$$T^{8} +$$$$66\!\cdots\!98$$$$T^{9} +$$$$66\!\cdots\!18$$$$T^{10} -$$$$72\!\cdots\!14$$$$T^{11} +$$$$39\!\cdots\!61$$$$T^{12}$$
$19$ $$1 - 10998262041959766 T^{2} +$$$$53\!\cdots\!15$$$$T^{4} -$$$$15\!\cdots\!20$$$$T^{6} +$$$$26\!\cdots\!15$$$$T^{8} -$$$$26\!\cdots\!06$$$$T^{10} +$$$$11\!\cdots\!61$$$$T^{12}$$
$23$ $$1 - 241146864 T + 29075905008517248 T^{2} -$$$$57\!\cdots\!32$$$$T^{3} +$$$$14\!\cdots\!23$$$$T^{4} -$$$$20\!\cdots\!96$$$$T^{5} +$$$$23\!\cdots\!52$$$$T^{6} -$$$$44\!\cdots\!16$$$$T^{7} +$$$$69\!\cdots\!43$$$$T^{8} -$$$$60\!\cdots\!52$$$$T^{9} +$$$$67\!\cdots\!88$$$$T^{10} -$$$$12\!\cdots\!64$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12}$$
$29$ $$1 - 1972019965273100646 T^{2} +$$$$16\!\cdots\!15$$$$T^{4} -$$$$77\!\cdots\!20$$$$T^{6} +$$$$20\!\cdots\!15$$$$T^{8} -$$$$30\!\cdots\!06$$$$T^{10} +$$$$19\!\cdots\!41$$$$T^{12}$$
$31$ $$( 1 + 854554104 T + 1597560029054250855 T^{2} +$$$$12\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!55$$$$T^{4} +$$$$53\!\cdots\!84$$$$T^{5} +$$$$48\!\cdots\!81$$$$T^{6} )^{2}$$
$37$ $$1 + 742614906 T + 275738449306694418 T^{2} -$$$$35\!\cdots\!82$$$$T^{3} -$$$$16\!\cdots\!37$$$$T^{4} +$$$$10\!\cdots\!44$$$$T^{5} +$$$$62\!\cdots\!92$$$$T^{6} +$$$$70\!\cdots\!64$$$$T^{7} -$$$$70\!\cdots\!57$$$$T^{8} -$$$$10\!\cdots\!62$$$$T^{9} +$$$$51\!\cdots\!78$$$$T^{10} +$$$$91\!\cdots\!06$$$$T^{11} +$$$$81\!\cdots\!81$$$$T^{12}$$
$41$ $$( 1 + 12374449764 T + 96299783043291044775 T^{2} +$$$$53\!\cdots\!40$$$$T^{3} +$$$$21\!\cdots\!75$$$$T^{4} +$$$$62\!\cdots\!04$$$$T^{5} +$$$$11\!\cdots\!41$$$$T^{6} )^{2}$$
$43$ $$1 - 14532965304 T +$$$$10\!\cdots\!08$$$$T^{2} -$$$$46\!\cdots\!32$$$$T^{3} +$$$$24\!\cdots\!43$$$$T^{4} -$$$$25\!\cdots\!76$$$$T^{5} +$$$$21\!\cdots\!72$$$$T^{6} -$$$$10\!\cdots\!76$$$$T^{7} +$$$$39\!\cdots\!43$$$$T^{8} -$$$$29\!\cdots\!32$$$$T^{9} +$$$$26\!\cdots\!08$$$$T^{10} -$$$$14\!\cdots\!04$$$$T^{11} +$$$$40\!\cdots\!01$$$$T^{12}$$
$47$ $$1 - 8427287424 T + 35509586663354277888 T^{2} -$$$$10\!\cdots\!32$$$$T^{3} +$$$$63\!\cdots\!03$$$$T^{4} +$$$$80\!\cdots\!84$$$$T^{5} -$$$$34\!\cdots\!68$$$$T^{6} +$$$$93\!\cdots\!44$$$$T^{7} +$$$$85\!\cdots\!43$$$$T^{8} -$$$$16\!\cdots\!72$$$$T^{9} +$$$$64\!\cdots\!68$$$$T^{10} -$$$$17\!\cdots\!24$$$$T^{11} +$$$$24\!\cdots\!41$$$$T^{12}$$
$53$ $$1 - 1297014534 T + 841123350703618578 T^{2} +$$$$45\!\cdots\!98$$$$T^{3} +$$$$45\!\cdots\!83$$$$T^{4} -$$$$50\!\cdots\!76$$$$T^{5} +$$$$16\!\cdots\!12$$$$T^{6} -$$$$24\!\cdots\!16$$$$T^{7} +$$$$10\!\cdots\!23$$$$T^{8} +$$$$54\!\cdots\!58$$$$T^{9} +$$$$48\!\cdots\!58$$$$T^{10} -$$$$37\!\cdots\!34$$$$T^{11} +$$$$14\!\cdots\!41$$$$T^{12}$$
$59$ $$1 -$$$$21\!\cdots\!86$$$$T^{2} +$$$$89\!\cdots\!15$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} +$$$$28\!\cdots\!15$$$$T^{8} -$$$$21\!\cdots\!06$$$$T^{10} +$$$$31\!\cdots\!81$$$$T^{12}$$
$61$ $$( 1 + 100638410964 T +$$$$85\!\cdots\!95$$$$T^{2} +$$$$53\!\cdots\!60$$$$T^{3} +$$$$22\!\cdots\!95$$$$T^{4} +$$$$70\!\cdots\!24$$$$T^{5} +$$$$18\!\cdots\!61$$$$T^{6} )^{2}$$
$67$ $$1 - 108303977544 T +$$$$58\!\cdots\!68$$$$T^{2} -$$$$78\!\cdots\!52$$$$T^{3} +$$$$85\!\cdots\!63$$$$T^{4} -$$$$40\!\cdots\!96$$$$T^{5} +$$$$24\!\cdots\!92$$$$T^{6} -$$$$33\!\cdots\!56$$$$T^{7} +$$$$57\!\cdots\!23$$$$T^{8} -$$$$42\!\cdots\!12$$$$T^{9} +$$$$26\!\cdots\!88$$$$T^{10} -$$$$39\!\cdots\!44$$$$T^{11} +$$$$30\!\cdots\!61$$$$T^{12}$$
$71$ $$( 1 + 285200833704 T +$$$$59\!\cdots\!95$$$$T^{2} +$$$$93\!\cdots\!60$$$$T^{3} +$$$$96\!\cdots\!95$$$$T^{4} +$$$$76\!\cdots\!24$$$$T^{5} +$$$$44\!\cdots\!21$$$$T^{6} )^{2}$$
$73$ $$1 - 710242228134 T +$$$$25\!\cdots\!78$$$$T^{2} -$$$$68\!\cdots\!22$$$$T^{3} +$$$$15\!\cdots\!83$$$$T^{4} -$$$$29\!\cdots\!16$$$$T^{5} +$$$$48\!\cdots\!12$$$$T^{6} -$$$$68\!\cdots\!36$$$$T^{7} +$$$$82\!\cdots\!03$$$$T^{8} -$$$$81\!\cdots\!42$$$$T^{9} +$$$$69\!\cdots\!18$$$$T^{10} -$$$$44\!\cdots\!34$$$$T^{11} +$$$$14\!\cdots\!21$$$$T^{12}$$
$79$ $$1 -$$$$28\!\cdots\!46$$$$T^{2} +$$$$37\!\cdots\!15$$$$T^{4} -$$$$28\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!15$$$$T^{8} -$$$$35\!\cdots\!06$$$$T^{10} +$$$$42\!\cdots\!41$$$$T^{12}$$
$83$ $$1 - 727867488144 T +$$$$26\!\cdots\!68$$$$T^{2} -$$$$11\!\cdots\!52$$$$T^{3} +$$$$49\!\cdots\!63$$$$T^{4} -$$$$15\!\cdots\!96$$$$T^{5} +$$$$44\!\cdots\!92$$$$T^{6} -$$$$16\!\cdots\!56$$$$T^{7} +$$$$56\!\cdots\!23$$$$T^{8} -$$$$13\!\cdots\!12$$$$T^{9} +$$$$34\!\cdots\!88$$$$T^{10} -$$$$10\!\cdots\!44$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$89$ $$1 +$$$$54\!\cdots\!74$$$$T^{2} +$$$$10\!\cdots\!15$$$$T^{4} +$$$$79\!\cdots\!80$$$$T^{6} +$$$$64\!\cdots\!15$$$$T^{8} +$$$$20\!\cdots\!94$$$$T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$
$97$ $$1 + 2207840630346 T +$$$$24\!\cdots\!58$$$$T^{2} +$$$$22\!\cdots\!58$$$$T^{3} +$$$$19\!\cdots\!43$$$$T^{4} +$$$$16\!\cdots\!04$$$$T^{5} +$$$$14\!\cdots\!72$$$$T^{6} +$$$$11\!\cdots\!64$$$$T^{7} +$$$$92\!\cdots\!83$$$$T^{8} +$$$$73\!\cdots\!18$$$$T^{9} +$$$$56\!\cdots\!38$$$$T^{10} +$$$$35\!\cdots\!46$$$$T^{11} +$$$$11\!\cdots\!41$$$$T^{12}$$