Properties

Label 6.19.b.a
Level 6
Weight 19
Character orbit 6.b
Analytic conductor 12.323
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 6.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(12.3231682626\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -1043 + 20 \beta_{1} - \beta_{2} ) q^{3} \) \( -131072 q^{4} \) \( + ( 553 \beta_{1} + 65 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{5} \) \( + ( -2625536 - 1047 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( 4705634 - 68 \beta_{1} - 1436 \beta_{2} - 21 \beta_{3} - 31 \beta_{4} - 26 \beta_{5} ) q^{7} \) \( -131072 \beta_{1} q^{8} \) \( + ( 115995273 - 401736 \beta_{1} - 2637 \beta_{2} - 144 \beta_{3} + 453 \beta_{4} + 210 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -1043 + 20 \beta_{1} - \beta_{2} ) q^{3} \) \( -131072 q^{4} \) \( + ( 553 \beta_{1} + 65 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{5} \) \( + ( -2625536 - 1047 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( 4705634 - 68 \beta_{1} - 1436 \beta_{2} - 21 \beta_{3} - 31 \beta_{4} - 26 \beta_{5} ) q^{7} \) \( -131072 \beta_{1} q^{8} \) \( + ( 115995273 - 401736 \beta_{1} - 2637 \beta_{2} - 144 \beta_{3} + 453 \beta_{4} + 210 \beta_{5} ) q^{9} \) \( + ( -72376320 - 632 \beta_{1} - 5456 \beta_{2} + 24 \beta_{3} - 1384 \beta_{4} - 680 \beta_{5} ) q^{10} \) \( + ( 3616267 \beta_{1} + 26324 \beta_{2} - 3478 \beta_{3} + 663 \beta_{5} ) q^{11} \) \( + ( 136708096 - 2621440 \beta_{1} + 131072 \beta_{2} ) q^{12} \) \( + ( 4927699370 + 13970 \beta_{1} + 354620 \beta_{2} + 5970 \beta_{3} - 1910 \beta_{4} + 2030 \beta_{5} ) q^{13} \) \( + ( 4699458 \beta_{1} - 76768 \beta_{2} - 15328 \beta_{3} + 13632 \beta_{5} ) q^{14} \) \( + ( 19937849280 - 16659288 \beta_{1} - 195078 \beta_{2} - 45729 \beta_{3} - 50367 \beta_{4} + 8682 \beta_{5} ) q^{15} \) \( + 17179869184 q^{16} \) \( + ( -126009100 \beta_{1} + 3697840 \beta_{2} - 323816 \beta_{3} - 7548 \beta_{5} ) q^{17} \) \( + ( 52656562176 + 116043009 \beta_{1} + 3801744 \beta_{2} - 97560 \beta_{3} + 4008 \beta_{4} - 104856 \beta_{5} ) q^{18} \) \( + ( -73135674502 - 159223 \beta_{1} - 1586614 \beta_{2} + 156 \beta_{3} - 319226 \beta_{4} - 159535 \beta_{5} ) q^{19} \) \( + ( -72482816 \beta_{1} - 8519680 \beta_{2} + 131072 \beta_{3} + 393216 \beta_{5} ) q^{20} \) \( + ( 479421234362 + 259561039 \beta_{1} - 5375801 \beta_{2} + 186381 \beta_{3} + 587412 \beta_{4} + 685827 \beta_{5} ) q^{21} \) \( + ( -474075488256 + 1264168 \beta_{1} + 31626352 \beta_{2} + 527352 \beta_{3} - 108424 \beta_{4} + 209464 \beta_{5} ) q^{22} \) \( + ( -1743910 \beta_{1} + 3352000 \beta_{2} + 2261716 \beta_{3} - 1568382 \beta_{5} ) q^{23} \) \( + ( 344134254592 + 137232384 \beta_{1} + 1048576 \beta_{2} + 1572864 \beta_{3} - 524288 \beta_{4} - 524288 \beta_{5} ) q^{24} \) \( + ( -4282674786215 - 5293230 \beta_{1} - 163996260 \beta_{2} - 3085110 \beta_{3} + 4839090 \beta_{4} + 876990 \beta_{5} ) q^{25} \) \( + ( 4928597290 \beta_{1} - 43199360 \beta_{2} + 5215360 \beta_{3} - 787200 \beta_{5} ) q^{26} \) \( + ( 1866210920469 - 3813969825 \beta_{1} - 165736233 \beta_{2} - 437292 \beta_{3} - 2834874 \beta_{4} - 8616573 \beta_{5} ) q^{27} \) \( + ( -616776859648 + 8912896 \beta_{1} + 188219392 \beta_{2} + 2752512 \beta_{3} + 4063232 \beta_{4} + 3407872 \beta_{5} ) q^{28} \) \( + ( -27023596949 \beta_{1} - 5915605 \beta_{2} - 3895387 \beta_{3} + 2709159 \beta_{5} ) q^{29} \) \( + ( 2178797936640 + 19958940840 \beta_{1} + 226383024 \beta_{2} + 13928760 \beta_{3} + 8584056 \beta_{4} + 22440888 \beta_{5} ) q^{30} \) \( + ( 3629302487858 - 23035830 \beta_{1} - 378197880 \beta_{2} - 4106655 \beta_{3} - 25538385 \beta_{4} - 14822520 \beta_{5} ) q^{31} \) \( + 17179869184 \beta_{1} q^{32} \) \( + ( -160426667328 + 76343916168 \beta_{1} + 570691785 \beta_{2} - 66371004 \beta_{3} + 12354339 \beta_{4} - 695490 \beta_{5} ) q^{33} \) \( + ( 16511268519936 + 96085088 \beta_{1} + 2628643904 \beta_{2} + 46327584 \beta_{3} - 39467744 \beta_{4} + 3429920 \beta_{5} ) q^{34} \) \( + ( -205898048554 \beta_{1} + 1437208540 \beta_{2} - 158710472 \beta_{3} + 17144814 \beta_{5} ) q^{35} \) \( + ( -15203732422656 + 52656340992 \beta_{1} + 345636864 \beta_{2} + 18874368 \beta_{3} - 59375616 \beta_{4} - 27525120 \beta_{5} ) q^{36} \) \( + ( 106407760802906 - 172297034 \beta_{1} - 4558794380 \beta_{2} - 78772890 \beta_{3} + 49270382 \beta_{4} - 14751254 \beta_{5} ) q^{37} \) \( + ( -73161185126 \beta_{1} - 1936612640 \beta_{2} + 25670368 \beta_{3} + 91902144 \beta_{5} ) q^{38} \) \( + ( -122756859184030 + 214659686830 \beta_{1} - 3579546170 \beta_{2} + 53846370 \beta_{3} - 97074450 \beta_{4} - 19523430 \beta_{5} ) q^{39} \) \( + ( 9486509015040 + 82837504 \beta_{1} + 715128832 \beta_{2} - 3145728 \beta_{3} + 181403648 \beta_{4} + 89128960 \beta_{5} ) q^{40} \) \( + ( -504081125930 \beta_{1} + 961235126 \beta_{2} + 393339434 \beta_{3} - 293776050 \beta_{5} ) q^{41} \) \( + ( -33992437080064 + 479537091282 \beta_{1} + 3823704848 \beta_{2} - 48600888 \beta_{3} + 183951944 \beta_{4} - 84366712 \beta_{5} ) q^{42} \) \( + ( -281385619813942 + 356601093 \beta_{1} + 8973104370 \beta_{2} + 150197040 \beta_{3} - 37783014 \beta_{4} + 56207013 \beta_{5} ) q^{43} \) \( + ( -473991348224 \beta_{1} - 3450339328 \beta_{2} + 455868416 \beta_{3} - 86900736 \beta_{5} ) q^{44} \) \( + ( 65098417345920 + 2603147199327 \beta_{1} - 3046828311 \beta_{2} + 639710001 \beta_{3} + 582853446 \beta_{4} + 451651671 \beta_{5} ) q^{45} \) \( + ( 329874653184 - 1176896848 \beta_{1} - 25752174304 \beta_{2} - 388422384 \beta_{3} - 411681776 \beta_{4} - 400052080 \beta_{5} ) q^{46} \) \( + ( -3263091145740 \beta_{1} - 18924713160 \beta_{2} + 945131520 \beta_{3} + 473792580 \beta_{5} ) q^{47} \) \( + ( -17918603558912 + 343597383680 \beta_{1} - 17179869184 \beta_{2} ) q^{48} \) \( + ( -149294649408045 + 1201779646 \beta_{1} + 25457721028 \beta_{2} + 373331238 \beta_{3} + 536903102 \beta_{4} + 455117170 \beta_{5} ) q^{49} \) \( + ( -4282830638375 \beta_{1} + 45682984320 \beta_{2} - 3003300480 \beta_{3} - 702593280 \beta_{5} ) q^{50} \) \( + ( 1606045587787008 + 6743000396448 \beta_{1} + 48952474236 \beta_{2} - 1332399888 \beta_{3} - 1828990764 \beta_{4} - 1680423480 \beta_{5} ) q^{51} \) \( + ( -645883411824640 - 1831075840 \beta_{1} - 46480752640 \beta_{2} - 782499840 \beta_{3} + 250347520 \beta_{4} - 266076160 \beta_{5} ) q^{52} \) \( + ( -3350514336515 \beta_{1} + 4914426005 \beta_{2} - 4061238421 \beta_{3} + 2208844257 \beta_{5} ) q^{53} \) \( + ( 498835300700160 + 1862727110793 \beta_{1} - 48710098872 \beta_{2} - 3429683748 \beta_{3} - 1957584276 \beta_{4} + 297066924 \beta_{5} ) q^{54} \) \( + ( -209993201297280 + 5155041942 \beta_{1} + 141969142836 \beta_{2} + 2511631206 \beta_{3} - 2248072146 \beta_{4} + 131779530 \beta_{5} ) q^{55} \) \( + ( -615967358976 \beta_{1} + 10062135296 \beta_{2} + 2009071616 \beta_{3} - 1786773504 \beta_{5} ) q^{56} \) \( + ( 670627886375666 + 3396347496130 \beta_{1} + 107628993763 \beta_{2} + 3181358430 \beta_{3} + 2078651079 \beta_{4} + 5221849032 \beta_{5} ) q^{57} \) \( + ( 3541862110126080 + 2029455256 \beta_{1} + 44389427728 \beta_{2} + 669302088 \beta_{3} + 712400072 \beta_{4} + 690851080 \beta_{5} ) q^{58} \) \( + ( -7373873304793 \beta_{1} - 67702787624 \beta_{2} + 5049907030 \beta_{3} + 675211683 \beta_{5} ) q^{59} \) \( + ( -2613293780828160 + 2183566196736 \beta_{1} + 25569263616 \beta_{2} + 5993791488 \beta_{3} + 6601703424 \beta_{4} - 1137967104 \beta_{5} ) q^{60} \) \( + ( -2713266212950006 - 13794308994 \beta_{1} - 305079221532 \beta_{2} - 4642670322 \beta_{3} - 4375266378 \beta_{4} - 4508968350 \beta_{5} ) q^{61} \) \( + ( 3626536645778 \beta_{1} - 133115921760 \beta_{2} - 1439372640 \beta_{3} + 8274945600 \beta_{5} ) q^{62} \) \( + ( -5420859869021838 + 1615515310092 \beta_{1} - 525582873864 \beta_{2} - 3454388091 \beta_{3} + 3989230143 \beta_{4} - 9316594254 \beta_{5} ) q^{63} \) \( -2251799813685248 q^{64} \) \( + ( -670617480310 \beta_{1} + 389087772850 \beta_{2} + 14289336670 \beta_{3} - 30348359790 \beta_{5} ) q^{65} \) \( + ( -10007958627115008 - 142785280056 \beta_{1} + 586653570672 \beta_{2} + 4155745176 \beta_{3} - 8968084776 \beta_{4} - 2711095272 \beta_{5} ) q^{66} \) \( + ( 1858954052435546 - 11897510061 \beta_{1} - 622096476906 \beta_{2} - 13975593786 \beta_{3} + 46082948808 \beta_{4} + 16053677511 \beta_{5} ) q^{67} \) \( + ( 16516264755200 \beta_{1} - 484683284480 \beta_{2} + 42443210752 \beta_{3} + 989331456 \beta_{5} ) q^{68} \) \( + ( 602006299124352 - 66049560175248 \beta_{1} - 519320671626 \beta_{2} + 3382548048 \beta_{3} - 17158860246 \beta_{4} + 13711602660 \beta_{5} ) q^{69} \) \( + ( 26984166456115200 + 53597165456 \beta_{1} + 1383415273568 \beta_{2} + 23540100528 \beta_{3} - 10506171728 \beta_{4} + 6516964400 \beta_{5} ) q^{70} \) \( + ( 87995155944686 \beta_{1} - 257146383560 \beta_{2} - 49648925612 \beta_{3} + 44626920294 \beta_{5} ) q^{71} \) \( + ( -6901800917532672 - 15209989275648 \beta_{1} - 498302189568 \beta_{2} + 12787384320 \beta_{3} - 525336576 \beta_{4} + 13743685632 \beta_{5} ) q^{72} \) \( + ( -13151707224630190 - 12811445856 \beta_{1} + 300407542272 \beta_{2} + 11903388912 \beta_{3} - 85139836272 \beta_{4} - 36618223680 \beta_{5} ) q^{73} \) \( + ( 106397838404826 \beta_{1} + 715956821888 \beta_{2} - 70741041280 \beta_{3} + 3455257344 \beta_{5} ) q^{74} \) \( + ( 58037638140848485 - 204259557581710 \beta_{1} + 3170124580895 \beta_{2} - 66204215190 \beta_{3} + 25047684270 \beta_{4} - 52938330030 \beta_{5} ) q^{75} \) \( + ( 9586039128326144 + 20869677056 \beta_{1} + 207960670208 \beta_{2} - 20447232 \beta_{3} + 41841590272 \beta_{4} + 20910571520 \beta_{5} ) q^{76} \) \( + ( 157954667775626 \beta_{1} + 952074839290 \beta_{2} - 84957610106 \beta_{3} - 974507118 \beta_{5} ) q^{77} \) \( + ( -28150651034705920 - 122779860594630 \beta_{1} - 915026030800 \beta_{2} - 22582755960 \beta_{3} + 26251104680 \beta_{4} + 41745562280 \beta_{5} ) q^{78} \) \( + ( -79419829404821038 - 96860422102 \beta_{1} - 1675099928056 \beta_{2} - 19624880751 \beta_{3} - 95596440449 \beta_{4} - 57610660600 \beta_{5} ) q^{79} \) \( + ( 9500467658752 \beta_{1} + 1116691496960 \beta_{2} - 17179869184 \beta_{3} - 51539607552 \beta_{5} ) q^{80} \) \( + ( 105113407684404177 - 97693528225158 \beta_{1} - 3289502779830 \beta_{2} + 24407167050 \beta_{3} - 8993584764 \beta_{4} + 91118986506 \beta_{5} ) q^{81} \) \( + ( 66089398878191616 - 211233315536 \beta_{1} - 4574442293216 \beta_{2} - 68391920496 \beta_{3} - 80507028592 \beta_{4} - 74449474544 \beta_{5} ) q^{82} \) \( + ( 163703405297793 \beta_{1} - 3482090799660 \beta_{2} + 231465592566 \beta_{3} + 51998293413 \beta_{5} ) q^{83} \) \( + ( -62838700030296064 - 34021184503808 \beta_{1} + 704616988672 \beta_{2} - 24429330432 \beta_{3} - 76993265664 \beta_{4} - 89892716544 \beta_{5} ) q^{84} \) \( + ( -161329778179822080 + 618591242856 \beta_{1} + 11237701841328 \beta_{2} + 140327483688 \beta_{3} + 535545067272 \beta_{4} + 337936275480 \beta_{5} ) q^{85} \) \( + ( -281362207776022 \beta_{1} - 1024559720736 \beta_{2} + 130389731040 \beta_{3} - 22762628928 \beta_{5} ) q^{86} \) \( + ( 69876085163661120 + 142151987457144 \beta_{1} + 1111546729974 \beta_{2} + 318573947637 \beta_{3} - 78424573389 \beta_{4} - 131748949986 \beta_{5} ) q^{87} \) \( + ( 62138022396690432 - 165697028096 \beta_{1} - 4145329209344 \beta_{2} - 69121081344 \beta_{3} + 14211350528 \beta_{4} - 27454865408 \beta_{5} ) q^{88} \) \( + ( 570688096880642 \beta_{1} - 975836991074 \beta_{2} + 441169722250 \beta_{3} - 215390552982 \beta_{5} ) q^{89} \) \( + ( -341141374440529920 + 65016403019208 \beta_{1} - 964247880528 \beta_{2} - 84820355496 \beta_{3} + 123743843928 \beta_{4} - 156764018664 \beta_{5} ) q^{90} \) \( + ( -172010027894260460 - 386848540140 \beta_{1} - 6290655536400 \beta_{2} - 67282503750 \beta_{3} - 437284561530 \beta_{4} - 252283532640 \beta_{5} ) q^{91} \) \( + ( 228577771520 \beta_{1} - 439353344000 \beta_{2} - 296447639552 \beta_{3} + 205570965504 \beta_{5} ) q^{92} \) \( + ( 127382356007834186 + 401255911499665 \beta_{1} - 1657718340833 \beta_{2} + 230488917795 \beta_{3} + 241865785350 \beta_{4} + 452786156145 \beta_{5} ) q^{93} \) \( + ( 427695961145180160 - 139496697120 \beta_{1} - 5612267455680 \beta_{2} - 117147235680 \beta_{3} + 306742784160 \beta_{4} + 94797774240 \beta_{5} ) q^{94} \) \( + ( -1905943821274906 \beta_{1} + 6868958240800 \beta_{2} - 657543207668 \beta_{3} + 20223169086 \beta_{5} ) q^{95} \) \( + ( -45106365017882624 - 17987323035648 \beta_{1} - 137438953472 \beta_{2} - 206158430208 \beta_{3} + 68719476736 \beta_{4} + 68719476736 \beta_{5} ) q^{96} \) \( + ( 139115902873885058 + 58138256222 \beta_{1} + 6751834235972 \beta_{2} + 171401435382 \beta_{3} - 740730664466 \beta_{4} - 284664614542 \beta_{5} ) q^{97} \) \( + ( -149185990861997 \beta_{1} + 1278618174080 \beta_{2} + 273632641664 \beta_{3} - 238254290688 \beta_{5} ) q^{98} \) \( + ( 126948025785635712 + 894516390677997 \beta_{1} + 6222958304076 \beta_{2} - 1643282130456 \beta_{3} - 4887357174 \beta_{4} - 75001151979 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 6258q^{3} \) \(\mathstrut -\mathstrut 786432q^{4} \) \(\mathstrut -\mathstrut 15753216q^{6} \) \(\mathstrut +\mathstrut 28233804q^{7} \) \(\mathstrut +\mathstrut 695971638q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 6258q^{3} \) \(\mathstrut -\mathstrut 786432q^{4} \) \(\mathstrut -\mathstrut 15753216q^{6} \) \(\mathstrut +\mathstrut 28233804q^{7} \) \(\mathstrut +\mathstrut 695971638q^{9} \) \(\mathstrut -\mathstrut 434257920q^{10} \) \(\mathstrut +\mathstrut 820248576q^{12} \) \(\mathstrut +\mathstrut 29566196220q^{13} \) \(\mathstrut +\mathstrut 119627095680q^{15} \) \(\mathstrut +\mathstrut 103079215104q^{16} \) \(\mathstrut +\mathstrut 315939373056q^{18} \) \(\mathstrut -\mathstrut 438814047012q^{19} \) \(\mathstrut +\mathstrut 2876527406172q^{21} \) \(\mathstrut -\mathstrut 2844452929536q^{22} \) \(\mathstrut +\mathstrut 2064805527552q^{24} \) \(\mathstrut -\mathstrut 25696048717290q^{25} \) \(\mathstrut +\mathstrut 11197265522814q^{27} \) \(\mathstrut -\mathstrut 3700661157888q^{28} \) \(\mathstrut +\mathstrut 13072787619840q^{30} \) \(\mathstrut +\mathstrut 21775814927148q^{31} \) \(\mathstrut -\mathstrut 962560003968q^{33} \) \(\mathstrut +\mathstrut 99067611119616q^{34} \) \(\mathstrut -\mathstrut 91222394535936q^{36} \) \(\mathstrut +\mathstrut 638446564817436q^{37} \) \(\mathstrut -\mathstrut 736541155104180q^{39} \) \(\mathstrut +\mathstrut 56919054090240q^{40} \) \(\mathstrut -\mathstrut 203954622480384q^{42} \) \(\mathstrut -\mathstrut 1688313718883652q^{43} \) \(\mathstrut +\mathstrut 390590504075520q^{45} \) \(\mathstrut +\mathstrut 1979247919104q^{46} \) \(\mathstrut -\mathstrut 107511621353472q^{48} \) \(\mathstrut -\mathstrut 895767896448270q^{49} \) \(\mathstrut +\mathstrut 9636273526722048q^{51} \) \(\mathstrut -\mathstrut 3875300470947840q^{52} \) \(\mathstrut +\mathstrut 2993011804200960q^{54} \) \(\mathstrut -\mathstrut 1259959207783680q^{55} \) \(\mathstrut +\mathstrut 4023767318253996q^{57} \) \(\mathstrut +\mathstrut 21251172660756480q^{58} \) \(\mathstrut -\mathstrut 15679762684968960q^{60} \) \(\mathstrut -\mathstrut 16279597277700036q^{61} \) \(\mathstrut -\mathstrut 32525159214131028q^{63} \) \(\mathstrut -\mathstrut 13510798882111488q^{64} \) \(\mathstrut -\mathstrut 60047751762690048q^{66} \) \(\mathstrut +\mathstrut 11153724314613276q^{67} \) \(\mathstrut +\mathstrut 3612037794746112q^{69} \) \(\mathstrut +\mathstrut 161904998736691200q^{70} \) \(\mathstrut -\mathstrut 41410805505196032q^{72} \) \(\mathstrut -\mathstrut 78910243347781140q^{73} \) \(\mathstrut +\mathstrut 348225828845090910q^{75} \) \(\mathstrut +\mathstrut 57516234769956864q^{76} \) \(\mathstrut -\mathstrut 168903906208235520q^{78} \) \(\mathstrut -\mathstrut 476518976428926228q^{79} \) \(\mathstrut +\mathstrut 630680446106425062q^{81} \) \(\mathstrut +\mathstrut 396536393269149696q^{82} \) \(\mathstrut -\mathstrut 377032200181776384q^{84} \) \(\mathstrut -\mathstrut 967978669078932480q^{85} \) \(\mathstrut +\mathstrut 419256510981966720q^{87} \) \(\mathstrut +\mathstrut 372828134380142592q^{88} \) \(\mathstrut -\mathstrut 2046848246643179520q^{90} \) \(\mathstrut -\mathstrut 1032060167365562760q^{91} \) \(\mathstrut +\mathstrut 764294136047005116q^{93} \) \(\mathstrut +\mathstrut 2566175766871080960q^{94} \) \(\mathstrut -\mathstrut 270638190107295744q^{96} \) \(\mathstrut +\mathstrut 834695417243310348q^{97} \) \(\mathstrut +\mathstrut 761688154713814272q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut -\mathstrut \) \(8797\) \(x^{4}\mathstrut +\mathstrut \) \(1809980\) \(x^{3}\mathstrut +\mathstrut \) \(107861490\) \(x^{2}\mathstrut -\mathstrut \) \(7180095600\) \(x\mathstrut +\mathstrut \) \(788142376800\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 7154 \nu^{5} - 39330 \nu^{4} - 61835324 \nu^{3} + 6710827644 \nu^{2} + 750396405600 \nu - 24732099541440 \)\()/\)\(169163330175\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(3792146903\) \(\nu^{5}\mathstrut +\mathstrut \) \(753972938703\) \(\nu^{4}\mathstrut +\mathstrut \) \(142676403256562\) \(\nu^{3}\mathstrut -\mathstrut \) \(15921411452300034\) \(\nu^{2}\mathstrut +\mathstrut \) \(1637154943766914320\) \(\nu\mathstrut +\mathstrut \) \(182744211516545607840\)\()/\)\(12564098858757600\)
\(\beta_{3}\)\(=\)\((\)\(161554484599\) \(\nu^{5}\mathstrut +\mathstrut \) \(49251373100433\) \(\nu^{4}\mathstrut -\mathstrut \) \(5692470420986290\) \(\nu^{3}\mathstrut -\mathstrut \) \(30747751600983102\) \(\nu^{2}\mathstrut +\mathstrut \) \(95099440503040251120\) \(\nu\mathstrut -\mathstrut \) \(1468147235715825720480\)\()/\)\(12564098858757600\)
\(\beta_{4}\)\(=\)\((\)\(612110098999\) \(\nu^{5}\mathstrut +\mathstrut \) \(68045562827217\) \(\nu^{4}\mathstrut -\mathstrut \) \(331993319994418\) \(\nu^{3}\mathstrut +\mathstrut \) \(751576414500006210\) \(\nu^{2}\mathstrut +\mathstrut \) \(93646522811075082480\) \(\nu\mathstrut +\mathstrut \) \(5261332185386445967200\)\()/\)\(12564098858757600\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(25222077880\) \(\nu^{5}\mathstrut -\mathstrut \) \(119659606848\) \(\nu^{4}\mathstrut +\mathstrut \) \(235802762330896\) \(\nu^{3}\mathstrut -\mathstrut \) \(55572584104020744\) \(\nu^{2}\mathstrut -\mathstrut \) \(4151770157654853120\) \(\nu\mathstrut +\mathstrut \) \(184718236679707465440\)\()/\)\(392628089336175\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(190\) \(\beta_{2}\mathstrut -\mathstrut \) \(49\) \(\beta_{1}\mathstrut +\mathstrut \) \(13824\)\()/41472\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(197\) \(\beta_{5}\mathstrut +\mathstrut \) \(49\) \(\beta_{4}\mathstrut -\mathstrut \) \(54\) \(\beta_{3}\mathstrut -\mathstrut \) \(3301\) \(\beta_{2}\mathstrut -\mathstrut \) \(362831\) \(\beta_{1}\mathstrut +\mathstrut \) \(60818688\)\()/20736\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(3314\) \(\beta_{5}\mathstrut +\mathstrut \) \(10500\) \(\beta_{4}\mathstrut -\mathstrut \) \(27671\) \(\beta_{3}\mathstrut +\mathstrut \) \(832333\) \(\beta_{2}\mathstrut -\mathstrut \) \(2776374\) \(\beta_{1}\mathstrut -\mathstrut \) \(18583430400\)\()/20736\)
\(\nu^{4}\)\(=\)\((\)\(1337893\) \(\beta_{5}\mathstrut +\mathstrut \) \(3415903\) \(\beta_{4}\mathstrut +\mathstrut \) \(1849470\) \(\beta_{3}\mathstrut -\mathstrut \) \(88763971\) \(\beta_{2}\mathstrut -\mathstrut \) \(3098725745\) \(\beta_{1}\mathstrut -\mathstrut \) \(1005732681984\)\()/20736\)
\(\nu^{5}\)\(=\)\((\)\(320844943\) \(\beta_{5}\mathstrut +\mathstrut \) \(220908921\) \(\beta_{4}\mathstrut -\mathstrut \) \(230796632\) \(\beta_{3}\mathstrut -\mathstrut \) \(161968511\) \(\beta_{2}\mathstrut +\mathstrut \) \(792214280397\) \(\beta_{1}\mathstrut -\mathstrut \) \(152243998149888\)\()/20736\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
27.6459 55.6125i
85.4804 + 88.2031i
−112.126 31.1763i
27.6459 + 55.6125i
85.4804 88.2031i
−112.126 + 31.1763i
362.039i −19019.3 5068.08i −131072. 2.98937e6i −1.83484e6 + 6.88573e6i −4.92752e7 4.74531e7i 3.36050e8 + 1.92783e8i −1.08227e9
5.2 362.039i −3612.02 19348.7i −131072. 3.69488e6i −7.00499e6 + 1.30769e6i 3.18077e7 4.74531e7i −3.61327e8 + 1.39776e8i 1.33769e9
5.3 362.039i 19502.4 + 2660.56i −131072. 1.30525e6i 963226. 7.06061e6i 3.15845e7 4.74531e7i 3.73263e8 + 1.03774e8i −4.72550e8
5.4 362.039i −19019.3 + 5068.08i −131072. 2.98937e6i −1.83484e6 6.88573e6i −4.92752e7 4.74531e7i 3.36050e8 1.92783e8i −1.08227e9
5.5 362.039i −3612.02 + 19348.7i −131072. 3.69488e6i −7.00499e6 1.30769e6i 3.18077e7 4.74531e7i −3.61327e8 1.39776e8i 1.33769e9
5.6 362.039i 19502.4 2660.56i −131072. 1.30525e6i 963226. + 7.06061e6i 3.15845e7 4.74531e7i 3.73263e8 1.03774e8i −4.72550e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{19}^{\mathrm{new}}(6, [\chi])\).