Properties

Label 6.25.b.a
Level $6$
Weight $25$
Character orbit 6.b
Analytic conductor $21.898$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8980291355\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 9921984 x^{6} + 31297402621425 x^{4} + 35629505313218665424 x^{2} + 11190322069687119538557504\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{53}\cdot 3^{32}\cdot 17^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -16485 - \beta_{1} - \beta_{2} ) q^{3} -8388608 q^{4} + ( 27689 \beta_{1} - 155 \beta_{2} - \beta_{3} ) q^{5} + ( 4214784 - 16486 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{6} + ( -1270099330 - 760 \beta_{1} - 1537 \beta_{2} + 7 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{7} -8388608 \beta_{1} q^{8} + ( 36896131737 + 7676843 \beta_{1} + 13897 \beta_{2} - 896 \beta_{3} + 124 \beta_{4} + 26 \beta_{5} + 37 \beta_{6} - 107 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -16485 - \beta_{1} - \beta_{2} ) q^{3} -8388608 q^{4} + ( 27689 \beta_{1} - 155 \beta_{2} - \beta_{3} ) q^{5} + ( 4214784 - 16486 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{6} + ( -1270099330 - 760 \beta_{1} - 1537 \beta_{2} + 7 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{7} -8388608 \beta_{1} q^{8} + ( 36896131737 + 7676843 \beta_{1} + 13897 \beta_{2} - 896 \beta_{3} + 124 \beta_{4} + 26 \beta_{5} + 37 \beta_{6} - 107 \beta_{7} ) q^{9} + ( -232921178112 + 166446 \beta_{1} + 335078 \beta_{2} - 144 \beta_{3} + 622 \beta_{4} - 288 \beta_{5} - 40 \beta_{6} - 38 \beta_{7} ) q^{10} + ( 3213917 \beta_{1} - 3614787 \beta_{2} - 2479 \beta_{3} - 603 \beta_{4} - 2017 \beta_{5} - 759 \beta_{6} + 130 \beta_{7} ) q^{11} + ( 138286202880 + 8388608 \beta_{1} + 8388608 \beta_{2} ) q^{12} + ( 6321045459890 + 8669300 \beta_{1} + 17424910 \beta_{2} - 24320 \beta_{3} - 29160 \beta_{4} - 3080 \beta_{5} - 910 \beta_{6} + 2200 \beta_{7} ) q^{13} + ( -1287494990 \beta_{1} - 34995268 \beta_{2} + 76004 \beta_{3} - 2880 \beta_{4} - 24416 \beta_{5} - 14712 \beta_{6} + 9860 \beta_{7} ) q^{14} + ( -43504369595064 + 10810512930 \beta_{1} + 2342091 \beta_{2} - 212295 \beta_{3} + 39444 \beta_{4} - 108081 \beta_{5} - 2970 \beta_{6} - 9561 \beta_{7} ) q^{15} + 70368744177664 q^{16} + ( -56816550716 \beta_{1} + 247622722 \beta_{2} - 1217786 \beta_{3} - 103068 \beta_{4} + 121436 \beta_{5} + 219912 \beta_{6} - 269150 \beta_{7} ) q^{17} + ( -64342286622720 + 36896553179 \beta_{1} - 82406 \beta_{2} + 2184184 \beta_{3} + 773338 \beta_{4} - 12832 \beta_{5} - 234344 \beta_{6} - 15194 \beta_{7} ) q^{18} + ( -122260453917658 - 786946695 \beta_{1} - 1582524191 \beta_{2} + 1282263 \beta_{3} - 228277 \beta_{4} + 514635 \beta_{5} + 384415 \beta_{6} + 96254 \beta_{7} ) q^{19} + ( -232272166912 \beta_{1} + 1300234240 \beta_{2} + 8388608 \beta_{3} ) q^{20} + ( 455001538030242 + 1179586427263 \beta_{1} + 1226970745 \beta_{2} - 1988685 \beta_{3} - 3982176 \beta_{4} - 1279908 \beta_{5} + 4417524 \beta_{6} - 1588302 \beta_{7} ) q^{21} + ( -42056372428800 + 421618030 \beta_{1} + 855618854 \beta_{2} - 2432656 \beta_{3} + 1280430 \beta_{4} - 5248288 \beta_{5} + 3417304 \beta_{6} + 1356698 \beta_{7} ) q^{22} + ( -2084725845266 \beta_{1} - 1851141798 \beta_{2} + 58568074 \beta_{3} - 13954698 \beta_{4} - 16575134 \beta_{5} + 5012022 \beta_{6} - 15805600 \beta_{7} ) q^{23} + ( -35356170780672 + 138294591488 \beta_{1} + 8388608 \beta_{2} + 16777216 \beta_{3} - 8388608 \beta_{4} - 8388608 \beta_{7} ) q^{24} + ( 953827345876753 + 23755069376 \beta_{1} + 47836524138 \beta_{2} - 34413224 \beta_{3} + 64413172 \beta_{4} - 52978928 \beta_{5} + 7494010 \beta_{6} + 1426292 \beta_{7} ) q^{25} + ( 6304526735010 \beta_{1} - 33036798960 \beta_{2} - 239115920 \beta_{3} - 34087680 \beta_{4} - 26000000 \beta_{5} + 23109600 \beta_{6} - 47664400 \beta_{7} ) q^{26} + ( -10824316075757205 - 4996047304032 \beta_{1} - 36143657418 \beta_{2} + 21791895 \beta_{3} - 43221387 \beta_{4} - 168195759 \beta_{5} + 42796797 \beta_{6} - 41089692 \beta_{7} ) q^{27} + ( 10654365400432640 + 6375342080 \beta_{1} + 12893290496 \beta_{2} - 58720256 \beta_{3} - 67108864 \beta_{4} - 58720256 \beta_{5} + 50331648 \beta_{6} + 25165824 \beta_{7} ) q^{28} + ( 27930439655843 \beta_{1} + 199416543283 \beta_{2} + 1460407241 \beta_{3} - 208918872 \beta_{4} - 183695016 \beta_{5} + 123377328 \beta_{6} - 276913500 \beta_{7} ) q^{29} + ( -90676273612529664 - 43385020645002 \beta_{1} + 239557834326 \beta_{2} + 1041405768 \beta_{3} + 82239894 \beta_{4} - 75815136 \beta_{5} + 112772520 \beta_{6} - 54512406 \beta_{7} ) q^{30} + ( 386514973352838638 - 203743711250 \beta_{1} - 409386848205 \beta_{2} + 160552535 \beta_{3} - 248737270 \beta_{4} - 105464785 \beta_{5} + 314592080 \beta_{6} + 124043875 \beta_{7} ) q^{31} + 70368744177664 \beta_{1} q^{32} + ( -1019835744083566200 + 28699253611773 \beta_{1} + 57701213817 \beta_{2} - 3570654354 \beta_{3} + 610989096 \beta_{4} + 412098570 \beta_{5} + 161210115 \beta_{6} - 381687207 \beta_{7} ) q^{33} + ( 477642254066417664 - 394424027480 \beta_{1} - 793683781304 \beta_{2} + 1354202432 \beta_{3} + 1298885032 \beta_{4} + 733420160 \beta_{5} - 409413600 \beta_{6} - 282304584 \beta_{7} ) q^{34} + ( -245924495126006 \beta_{1} - 651862497300 \beta_{2} - 3314258216 \beta_{3} + 408327030 \beta_{4} + 269039890 \beta_{5} - 308628870 \beta_{6} + 597463250 \beta_{7} ) q^{35} + ( -309507185858052096 - 64398026604544 \beta_{1} - 116576485376 \beta_{2} + 7516192768 \beta_{3} - 1040187392 \beta_{4} - 218103808 \beta_{5} - 310378496 \beta_{6} + 897581056 \beta_{7} ) q^{36} + ( 2823786652999097810 + 2913910009780 \beta_{1} + 5857760513570 \beta_{2} - 5135489296 \beta_{3} - 1555979712 \beta_{4} - 822110632 \beta_{5} - 1852459994 \beta_{6} - 387057664 \beta_{7} ) q^{37} + ( -122148687323022 \beta_{1} + 226969516196 \beta_{2} - 9675310148 \beta_{3} + 1869224256 \beta_{4} + 2344108128 \beta_{5} - 578449224 \beta_{6} + 2039727900 \beta_{7} ) q^{38} + ( -5023772072735183130 + 1102488084634480 \beta_{1} - 6058610109560 \beta_{2} - 14754006090 \beta_{3} - 5236092990 \beta_{4} + 2472190740 \beta_{5} - 6312467970 \beta_{6} + 5230946520 \beta_{7} ) q^{39} + ( 1953884458079748096 - 1396250247168 \beta_{1} - 2810837991424 \beta_{2} + 1207959552 \beta_{3} - 5217714176 \beta_{4} + 2415919104 \beta_{5} + 335544320 \beta_{6} + 318767104 \beta_{7} ) q^{40} + ( -1242357918025138 \beta_{1} - 7742983944474 \beta_{2} - 14568533518 \beta_{3} + 11508142032 \beta_{4} + 6832124144 \beta_{5} - 9261084432 \beta_{6} + 17307688720 \beta_{7} ) q^{41} + ( -9889985479445667840 + 455278497061324 \beta_{1} + 556104581458 \beta_{2} - 20660609200 \beta_{3} + 9962726762 \beta_{4} + 20058996384 \beta_{5} + 1716958728 \beta_{6} - 2139170578 \beta_{7} ) q^{42} + ( 28310933296475487110 - 2270728852515 \beta_{1} - 4593768637335 \beta_{2} + 14293608183 \beta_{3} + 6299381751 \beta_{4} + 20153028411 \beta_{5} - 14389587213 \beta_{6} - 6462366078 \beta_{7} ) q^{43} + ( -26960289857536 \beta_{1} + 30323031146496 \beta_{2} + 20795359232 \beta_{3} + 5058330624 \beta_{4} + 16919822336 \beta_{5} + 6366953472 \beta_{6} - 1090519040 \beta_{7} ) q^{44} + ( -43463749594647188016 - 9106694375370279 \beta_{1} + 46250425143789 \beta_{2} + 49969316661 \beta_{3} + 31867698336 \beta_{4} - 15697036284 \beta_{5} + 7977892830 \beta_{6} - 1146821214 \beta_{7} ) q^{45} + ( 17480266320140058624 - 48196113578380 \beta_{1} - 96920100293564 \beta_{2} + 82072557472 \beta_{3} - 4634797068 \beta_{4} + 31982313280 \beta_{5} + 21619652240 \beta_{6} + 4548545596 \beta_{7} ) q^{46} + ( 19315637872499580 \beta_{1} - 78830794234160 \beta_{2} - 154681903320 \beta_{3} - 41676000300 \beta_{4} - 59978211620 \beta_{5} + 7111341660 \beta_{6} - 40656118300 \beta_{7} ) q^{47} + ( -1160028747768791040 - 70368744177664 \beta_{1} - 70368744177664 \beta_{2} ) q^{48} + ( 13085837559202091475 + 123378526607440 \beta_{1} + 248245959918662 \beta_{2} - 289161945016 \beta_{3} - 87652411236 \beta_{4} - 182038862320 \beta_{5} + 38246505670 \beta_{6} + 32513638172 \beta_{7} ) q^{49} + ( 1076194478785633 \beta_{1} + 245460817238320 \beta_{2} + 939815542608 \beta_{3} + 5445953280 \beta_{4} + 51516557440 \beta_{5} + 31829976480 \beta_{6} - 21986686000 \beta_{7} ) q^{50} + ( 69761484367070817888 - 25772973764062896 \beta_{1} - 3568758758514 \beta_{2} + 10441482642 \beta_{3} - 239412489228 \beta_{4} - 286489533312 \beta_{5} + 3559143456 \beta_{6} - 80106767766 \beta_{7} ) q^{51} + ( -53024772513196933120 - 72723359334400 \beta_{1} - 146170739425280 \beta_{2} + 204010946560 \beta_{3} + 244611809280 \beta_{4} + 25836912640 \beta_{5} + 7633633280 \beta_{6} - 18454937600 \beta_{7} ) q^{52} + ( 30126208334301581 \beta_{1} - 479952467103527 \beta_{2} - 969459261349 \beta_{3} - 317223451872 \beta_{4} - 415103787936 \beta_{5} + 85201473888 \beta_{6} - 335354104800 \beta_{7} ) q^{53} + ( 41758346953349591040 - 10793167394592516 \beta_{1} + 62335147896825 \beta_{2} + 671956541790 \beta_{3} + 91113896067 \beta_{4} + 176532463776 \beta_{5} + 221361460872 \beta_{6} - 22009083321 \beta_{7} ) q^{54} + ( -276585958031308417296 + 314031420613118 \beta_{1} + 632019328291044 \beta_{2} - 515731626512 \beta_{3} + 445177808266 \beta_{4} - 506025433304 \beta_{5} + 18030024430 \beta_{6} + 10228286366 \beta_{7} ) q^{55} + ( 10800290773073920 \beta_{1} + 293561585106944 \beta_{2} - 637567762432 \beta_{3} + 24159191040 \beta_{4} + 204816252928 \beta_{5} + 123413200896 \beta_{6} - 82711674880 \beta_{7} ) q^{56} + ( 448609999455095090730 - 7700022492632165 \beta_{1} + 95686346626657 \beta_{2} - 1495570469382 \beta_{3} + 249015029436 \beta_{4} + 95981421366 \beta_{5} - 111367551753 \beta_{6} - 80686723293 \beta_{7} ) q^{57} + ( -233463294433714790400 - 901457247698790 \beta_{1} - 1813294287744766 \beta_{2} + 1573287780048 \beta_{3} - 351334107302 \beta_{4} + 903929396640 \beta_{5} + 214044833480 \beta_{6} + 23352618814 \beta_{7} ) q^{58} + ( 24953243885267569 \beta_{1} + 277247165093395 \beta_{2} + 5531364551035 \beta_{3} + 689620616469 \beta_{4} + 290838122879 \beta_{5} - 643897178427 \beta_{6} + 1111264829080 \beta_{7} ) q^{59} + ( 364941102820110630912 - 90685155248701440 \beta_{1} - 19646883299328 \beta_{2} + 1780859535360 \beta_{3} - 330880253952 \beta_{4} + 906649141248 \beta_{5} + 24914165760 \beta_{6} + 80203481088 \beta_{7} ) q^{60} + ( -813376347979761548206 + 348000342512660 \beta_{1} + 697561517942778 \beta_{2} - 897398673584 \beta_{3} - 2326933464944 \beta_{4} + 1009433801560 \beta_{5} - 712441243490 \beta_{6} - 117866562352 \beta_{7} ) q^{61} + ( 387213100213225778 \beta_{1} + 1404363269579180 \beta_{2} - 4419813644940 \beta_{3} + 693239855040 \beta_{4} + 1552724429600 \beta_{5} + 297993503400 \beta_{6} + 329371959700 \beta_{7} ) q^{62} + ( 1264122847201612422510 - 829638289122998876 \beta_{1} - 518521086550771 \beta_{2} - 4095582499855 \beta_{3} + 1126709276216 \beta_{4} - 929472917303 \beta_{5} - 1414868291026 \beta_{6} + 1606344627611 \beta_{7} ) q^{63} -590295810358705651712 q^{64} + ( 1577385524660049250 \beta_{1} - 4363952238326450 \beta_{2} - 22789998180950 \beta_{3} + 730642789800 \beta_{4} + 205860901400 \beta_{5} - 758907811200 \beta_{6} + 1241292167500 \beta_{7} ) q^{65} + ( -240514055292441305088 - 1019796729435412554 \beta_{1} + 74624043995286 \beta_{2} + 7823300912712 \beta_{3} + 3268319493078 \beta_{4} + 129933347616 \beta_{5} - 1410795626328 \beta_{6} + 145185527466 \beta_{7} ) q^{66} + ( -880265014768757518090 + 1349447454769875 \beta_{1} + 2711242539483213 \beta_{2} - 193140588345 \beta_{3} + 3653845709769 \beta_{4} + 1051460736483 \beta_{5} - 2691113439639 \beta_{6} - 1189981554216 \beta_{7} ) q^{67} + ( 476611771868643328 \beta_{1} - 2077209946750976 \beta_{2} + 10215529381888 \beta_{3} + 864597049344 \beta_{4} - 1018679001088 \beta_{5} - 1844755562496 \beta_{6} + 2257793843200 \beta_{7} ) q^{68} + ( -540416995921091418192 - 3226208932469754546 \beta_{1} + 70254008784426 \beta_{2} - 19401535121808 \beta_{3} - 8483395643688 \beta_{4} + 4561963017588 \beta_{5} + 2677929473106 \beta_{6} - 2019338928606 \beta_{7} ) q^{69} + ( 2060238779554444197888 + 1953987019731596 \beta_{1} + 3930943532718268 \beta_{2} - 3504913756064 \beta_{3} + 859611536652 \beta_{4} - 2258038599488 \beta_{5} - 251098026640 \beta_{6} + 30310381252 \beta_{7} ) q^{70} + ( 4414863253253793802 \beta_{1} + 9766794098829802 \beta_{2} + 40595412831154 \beta_{3} - 1095890867478 \beta_{4} + 741611181086 \beta_{5} + 1926071970162 \beta_{6} - 2518302364700 \beta_{7} ) q^{71} + ( 539742220301641973760 - 309510721169784832 \beta_{1} + 691271630848 \beta_{2} - 18322263375872 \beta_{3} - 6487229333504 \beta_{4} + 107642617856 \beta_{5} + 1965819953152 \beta_{6} + 127456509952 \beta_{7} ) q^{72} + ( 4532477540594822085890 - 8142346209064760 \beta_{1} - 16355192391004104 \beta_{2} + 8164058430992 \beta_{3} - 1734616815928 \beta_{4} - 6942185448880 \beta_{5} + 13289779427840 \beta_{6} + 4756609228936 \beta_{7} ) q^{73} + ( 2819558995731636322 \beta_{1} - 8489511987104400 \beta_{2} + 9077792422160 \beta_{3} - 8956863030528 \beta_{4} - 11997739024768 \beta_{5} + 2197774519584 \beta_{6} - 9295531291760 \beta_{7} ) q^{74} + ( -13510247306076613581909 - 8244513137133343027 \beta_{1} + 18998273526101 \beta_{2} + 141329085116598 \beta_{3} + 19231650534294 \beta_{4} - 7223577744636 \beta_{5} + 5333044101030 \beta_{6} - 3699700417476 \beta_{7} ) q^{75} + ( 1025595021817297240064 + 6601387341250560 \beta_{1} + 13275175088816128 \beta_{2} - 10756401659904 \beta_{3} + 1914926268416 \beta_{4} - 4317071278080 \beta_{5} - 3224706744320 \beta_{6} - 807437074432 \beta_{7} ) q^{76} + ( 11520502746843831610 \beta_{1} + 26943039985223362 \beta_{2} - 3371682434906 \beta_{3} - 12864174078528 \beta_{4} + 2423898443456 \beta_{5} + 17898141430752 \beta_{6} - 25635262924400 \beta_{7} ) q^{77} + ( -9273612253416197283840 - 5023522932337893740 \beta_{1} + 588206862894670 \beta_{2} - 110558579398420 \beta_{3} - 14026762608190 \beta_{4} - 38588590961280 \beta_{5} + 4733935043040 \beta_{6} - 5531945934670 \beta_{7} ) q^{78} + ( 39600881597166251914478 - 17060029207952130 \beta_{1} - 34311311176475549 \beta_{2} + 38162742699447 \beta_{3} + 17970459269402 \beta_{4} + 16164712289295 \beta_{5} + 644651143120 \beta_{6} - 2427428229709 \beta_{7} ) q^{79} + ( 1948440157535338496 \beta_{1} - 10907155347537920 \beta_{2} - 70368744177664 \beta_{3} ) q^{80} + ( -33564832563049597101663 - 23014104586876125090 \beta_{1} + 13129629924096648 \beta_{2} - 22792111738164 \beta_{3} + 17828669137788 \beta_{4} + 12403406291148 \beta_{5} - 4180278111444 \beta_{6} - 10624161041046 \beta_{7} ) q^{81} + ( 10389367691206830735360 + 42911595962649700 \beta_{1} + 86312197714618868 \beta_{2} - 90461478435040 \beta_{3} - 25754518780956 \beta_{4} - 43926613193152 \beta_{5} - 2375209617584 \beta_{6} + 4629253346444 \beta_{7} ) q^{82} + ( 35134288308427466991 \beta_{1} - 37822345736135333 \beta_{2} + 63147049529679 \beta_{3} + 18606480696207 \beta_{4} - 7224046764899 \beta_{5} - 28676135943933 \beta_{6} + 39402180533450 \beta_{7} ) q^{83} + ( -3816829541932792283136 - 9895088140429819904 \beta_{1} - 10292576607272960 \beta_{2} + 16682298900480 \beta_{3} + 33404913451008 \beta_{4} + 10736646488064 \beta_{5} - 37056877166592 \beta_{6} + 13323642863616 \beta_{7} ) q^{84} + ( -27814107818190525084864 - 14870939110075088 \beta_{1} - 29934674977533384 \beta_{2} + 19007097616832 \beta_{3} - 38461182630016 \beta_{4} + 25793749055264 \beta_{5} + 655649994920 \beta_{6} + 1176156427264 \beta_{7} ) q^{85} + ( 28257047970683808434 \beta_{1} - 108298102784778300 \beta_{2} + 36303755192220 \beta_{3} - 13593726680256 \beta_{4} - 64851260312736 \beta_{5} - 31646286884232 \beta_{6} + 15043800169980 \beta_{7} ) q^{86} + ( 56182697150225917766040 - 61459644311827326426 \beta_{1} - 2515385768662083 \beta_{2} - 229985611569681 \beta_{3} - 115624136354460 \beta_{4} + 93888698611437 \beta_{5} + 38833533423954 \beta_{6} + 60612782291913 \beta_{7} ) q^{87} + ( 352794422207211110400 - 3536788379402240 \beta_{1} - 7177451163615232 \beta_{2} + 20406597582848 \beta_{3} - 10741025341440 \beta_{4} + 44025830703104 \beta_{5} - 28666423672832 \beta_{6} - 11380807696384 \beta_{7} ) q^{88} + ( 59465272753218855274 \beta_{1} + 241998637136851600 \beta_{2} - 86827062835280 \beta_{3} + 118373156741844 \beta_{4} + 217351306485164 \beta_{5} + 15047033936568 \beta_{6} + 86105102337730 \beta_{7} ) q^{89} + ( 76585496942922753171456 - 43420268583814360434 \beta_{1} + 87096809080417686 \beta_{2} + 457522348279776 \beta_{3} - 32483553557106 \beta_{4} + 68423434188384 \beta_{5} + 1120654303800 \beta_{6} - 13075706562006 \beta_{7} ) q^{90} + ( -149604950291961346795940 - 59945714394225900 \beta_{1} - 120766792522081890 \beta_{2} + 207840035254110 \beta_{3} + 102655861497900 \beta_{4} + 194093288252430 \beta_{5} - 109764274349040 \beta_{6} - 56600480549730 \beta_{7} ) q^{91} + ( 17487947903405129728 \beta_{1} + 15528502895837184 \beta_{2} - 491304614100992 \beta_{3} + 117060491280384 \beta_{4} + 139042301673472 \beta_{5} - 42043887845376 \beta_{6} + 132586982604800 \beta_{7} ) q^{92} + ( 109154692181353597968210 - 47707436090237210273 \beta_{1} - 392679107388719903 \beta_{2} - 267738408754155 \beta_{3} + 222075918511920 \beta_{4} + 62158040032680 \beta_{5} - 164969464721490 \beta_{6} + 18692723487240 \beta_{7} ) q^{93} + ( -162360889702721675427840 - 75625692973348600 \beta_{1} - 151912154319973080 \beta_{2} + 155304605577280 \beta_{3} + 177995991437320 \beta_{4} - 41187174350720 \beta_{5} + 72996130118560 \beta_{6} + 11936592568280 \beta_{7} ) q^{94} + ( 50107767560174853538 \beta_{1} - 241686235930795610 \beta_{2} - 427850308433642 \beta_{3} - 66603271954950 \beta_{4} - 137212646357650 \beta_{5} - 19655394824550 \beta_{6} - 39123230942000 \beta_{7} ) q^{95} + ( 296589057060111384576 - 1160099116512968704 \beta_{1} - 70368744177664 \beta_{2} - 140737488355328 \beta_{3} + 70368744177664 \beta_{4} + 70368744177664 \beta_{7} ) q^{96} + ( 102064697031025158214610 + 243976801849089800 \beta_{1} + 490949337330467374 \beta_{2} - 653200070509640 \beta_{3} - 339872945053548 \beta_{4} - 410356685216576 \beta_{5} + 142243651667558 \beta_{6} + 101477248995412 \beta_{7} ) q^{97} + ( 13342234597418808323 \beta_{1} + 515469516318984528 \beta_{2} + 237540614117936 \beta_{3} - 218479587892992 \beta_{4} + 46913179128704 \beta_{5} + 308284369212768 \beta_{6} - 438969964254800 \beta_{7} ) q^{98} + ( 86982360420901115289168 + 4998634484003155023 \beta_{1} + 1016052289464422631 \beta_{2} + 164090546821587 \beta_{3} - 190207038327585 \beta_{4} - 721634162777877 \beta_{5} + 172071962096331 \beta_{6} - 168291602235474 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 131880q^{3} - 67108864q^{4} + 33718272q^{6} - 10160794640q^{7} + 295169053896q^{9} + O(q^{10}) \) \( 8q - 131880q^{3} - 67108864q^{4} + 33718272q^{6} - 10160794640q^{7} + 295169053896q^{9} - 1863369424896q^{10} + 1106289623040q^{12} + 50568363679120q^{13} - 348034956760512q^{15} + 562949953421312q^{16} - 514738292981760q^{18} - 978083631341264q^{19} + 3640012304241936q^{21} - 336450979430400q^{22} - 282849366245376q^{24} + 7630618767014024q^{25} - 86594528606057640q^{27} + 85234923203461120q^{28} - 725410188900237312q^{30} + 3092119786822709104q^{31} - 8158685952668529600q^{33} + 3821138032531341312q^{34} - 2476057486864416768q^{36} + 22590293223992782480q^{37} - 40190176581881465040q^{39} + 15631075664637984768q^{40} - 79119883835565342720q^{42} + 226487466371803896880q^{43} - 347709996757177504128q^{45} + 139842130561120468992q^{46} - 9280229982150328320q^{48} + 104686700473616731800q^{49} + 558091874936566543104q^{51} - 424198180105575464960q^{52} + 334066775626796728320q^{54} - 2212687664250467338368q^{55} + 3588879995640760725840q^{57} - 1867706355469718323200q^{58} + 2919528822560885047296q^{60} - 6507010783838092385648q^{61} + 10112982777612899380080q^{63} - 4722366482869645213696q^{64} - 1924112442339530440704q^{66} - 7042120118150060144720q^{67} - 4323335967368731345536q^{69} + 16481910236435553583104q^{70} + 4317937762413135790080q^{72} + 36259820324758576687120q^{73} - 108081978448612908655272q^{75} + 8204760174538377920512q^{76} - 74188898027329578270720q^{78} + 316807052777330015315824q^{79} - 268518660504396776813304q^{81} + 83114941529654645882880q^{82} - 30534636335462338265088q^{84} - 222512862545524200678912q^{85} + 449461577201807342128320q^{87} + 2822355377657688883200q^{88} + 612683975543382025371648q^{90} - 1196839602335690774367520q^{91} + 873237537450828783745680q^{93} - 1298887117621773403422720q^{94} + 2372712456480891076608q^{96} + 816517576248201265716880q^{97} + 695858883367208922313344q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9921984 x^{6} + 31297402621425 x^{4} + 35629505313218665424 x^{2} + 11190322069687119538557504\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3943712992 \nu^{7} + 39117563184545024 \nu^{5} + 110176482620237363993312 \nu^{3} + 75024984334667741002185220864 \nu \)\()/ \)\(71\!\cdots\!25\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-11420728189106343935 \nu^{7} + 23411849388295723301784 \nu^{6} - 100151413680080815081352200 \nu^{5} + 187889840842995389939480948928 \nu^{4} - 261018807890632312352793928059535 \nu^{3} + 347542742210158476690821458447257624 \nu^{2} - 248123996605684458316461010979895164600 \nu + 68123794807884840843359457272710645982208\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-3564304444303877240677 \nu^{7} - 3628836655185837111776520 \nu^{6} - 37524128983984273252312560344 \nu^{5} - 29122925330664285440619547083840 \nu^{4} - 133359008260529281515219521762907797 \nu^{3} - 53869125042574563887077326059324931720 \nu^{2} - 151155782626340829456294511317407681972584 \nu - 10559188195222150330720715877270150127242240\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(122941344043294481431 \nu^{7} - 2287938647500248336992904 \nu^{6} + 1053663402283439808550565192 \nu^{5} - 19156066115086210768367656392768 \nu^{4} + 2610292899712306229978128337322791 \nu^{3} - 43927655001185298392314438519217191944 \nu^{2} + 1367648908449032658660533398711521217912 \nu - 24252577300183255459211788675782807305576448\)\()/ \)\(58\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-3225874100802829593635 \nu^{7} - 3285418527515449456715184 \nu^{6} - 25562862033946557711091473640 \nu^{5} - 378495552286107786411397385088 \nu^{4} - 51274656839946437051179563964370035 \nu^{3} + 118448621373550051591770720764945111376 \nu^{2} - 57440503815354837769708907910475546778840 \nu + 172301233053063110073428820863156343146802432\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-14813324883410082408601 \nu^{7} - 1312324715884710261674744 \nu^{6} - 120257498048867839364259090232 \nu^{5} - 52710994722596742903989066299328 \nu^{4} - 259156034844656115520191079045351561 \nu^{3} - 231033739553879127402243388718549444984 \nu^{2} - 159432809172802121481673225038368660767752 \nu - 150534237664291160730372531172844545629393408\)\()/ \)\(72\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2318338105568338239985 \nu^{7} + 1783167064982648189299308 \nu^{6} + 18601832111642368152693837560 \nu^{5} + 12402830145651414989327607033696 \nu^{4} + 38747531457589727531226698188751585 \nu^{3} + 20179365049222805171172476424469500588 \nu^{2} + 16592477294695454774108454940867378933960 \nu + 6682530588959596156082807810408683708121856\)\()/ \)\(60\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-965 \beta_{7} - 618 \beta_{6} - 3784 \beta_{5} - 1776 \beta_{4} + 3651 \beta_{3} - 4375819 \beta_{2} - 1842900 \beta_{1}\)\()/ 2741133312 \)
\(\nu^{2}\)\(=\)\((\)\(18797 \beta_{7} + 69228 \beta_{6} + 135280 \beta_{5} - 102729 \beta_{4} + 261816 \beta_{3} - 332801933 \beta_{2} - 165468777 \beta_{1} - 399962953875456\)\()/ 161243136 \)
\(\nu^{3}\)\(=\)\((\)\(3528444415 \beta_{7} + 2240481438 \beta_{6} + 13778333144 \beta_{5} + 6474614736 \beta_{4} - 38026095513 \beta_{3} + 12086216753009 \beta_{2} - 1107412181540508 \beta_{1}\)\()/ 2741133312 \)
\(\nu^{4}\)\(=\)\((\)\(-10151333995 \beta_{7} - 33506446020 \beta_{6} - 34594106480 \beta_{5} + 32535664335 \beta_{4} - 87409218600 \beta_{3} + 116586270157915 \beta_{2} + 57976542868815 \beta_{1} + 90323772204901158144\)\()/10077696\)
\(\nu^{5}\)\(=\)\((\)\(-17865749256948935 \beta_{7} - 7911057782925102 \beta_{6} - 59464671862673176 \beta_{5} - 29349956891263824 \beta_{4} + 200798920683315729 \beta_{3} - 44034153794096165161 \beta_{2} + 9178644417483035893500 \beta_{1}\)\()/ 2741133312 \)
\(\nu^{6}\)\(=\)\((\)\(1024462405148585123 \beta_{7} + 3161105560117832532 \beta_{6} + 2092896012211995280 \beta_{5} - 2766485038672271751 \beta_{4} + 6541619011494644424 \beta_{3} - 9187659363129781518467 \beta_{2} - 4569345555509101345383 \beta_{1} - 6130014639216231125637955584\)\()/ 161243136 \)
\(\nu^{7}\)\(=\)\((\)\(96992882415914802746785 \beta_{7} + 27633445690718407365282 \beta_{6} + 276886101903984827589416 \beta_{5} + 144024904589816170661424 \beta_{4} - 998830097831923851922887 \beta_{3} + 182363092007825665672407791 \beta_{2} - 55125628618782747495241378212 \beta_{1}\)\()/ 2741133312 \)

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
2234.51i
1694.95i
708.500i
1246.64i
2234.51i
1694.95i
708.500i
1246.64i
2896.31i −490828. 203759.i −8.38861e6 1.24013e8i −5.90150e8 + 1.42159e9i 1.50611e10 2.42960e10i 1.99394e11 + 2.00021e11i 3.59180e11
5.2 2896.31i −317945. + 425841.i −8.38861e6 6.76938e7i 1.23337e9 + 9.20867e8i −2.41596e10 2.42960e10i −8.02515e10 2.70788e11i −1.96062e11
5.3 2896.31i 269557. 458005.i −8.38861e6 4.56581e8i −1.32652e9 7.80721e8i 1.81935e9 2.42960e10i −1.37107e11 2.46917e11i −1.32240e12
5.4 2896.31i 473275. + 241744.i −8.38861e6 7.85820e7i 7.00165e8 1.37075e9i 2.19870e9 2.42960e10i 1.65549e11 + 2.28823e11i 2.27598e11
5.5 2896.31i −490828. + 203759.i −8.38861e6 1.24013e8i −5.90150e8 1.42159e9i 1.50611e10 2.42960e10i 1.99394e11 2.00021e11i 3.59180e11
5.6 2896.31i −317945. 425841.i −8.38861e6 6.76938e7i 1.23337e9 9.20867e8i −2.41596e10 2.42960e10i −8.02515e10 + 2.70788e11i −1.96062e11
5.7 2896.31i 269557. + 458005.i −8.38861e6 4.56581e8i −1.32652e9 + 7.80721e8i 1.81935e9 2.42960e10i −1.37107e11 + 2.46917e11i −1.32240e12
5.8 2896.31i 473275. 241744.i −8.38861e6 7.85820e7i 7.00165e8 + 1.37075e9i 2.19870e9 2.42960e10i 1.65549e11 2.28823e11i 2.27598e11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.25.b.a 8
3.b odd 2 1 inner 6.25.b.a 8
4.b odd 2 1 48.25.e.d 8
12.b even 2 1 48.25.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.25.b.a 8 1.a even 1 1 trivial
6.25.b.a 8 3.b odd 2 1 inner
48.25.e.d 8 4.b odd 2 1
48.25.e.d 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{25}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8388608 + T^{2} )^{4} \)
$3$ \( \)\(63\!\cdots\!21\)\( + \)\(29\!\cdots\!80\)\( T - \)\(11\!\cdots\!28\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!78\)\( T^{4} + 9783678964895640 T^{5} - 138888359748 T^{6} + 131880 T^{7} + T^{8} \)
$5$ \( \)\(90\!\cdots\!00\)\( + \)\(40\!\cdots\!00\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{4} + 234603269718055488 T^{6} + T^{8} \)
$7$ \( ( -\)\(14\!\cdots\!04\)\( + \)\(14\!\cdots\!60\)\( T - \)\(39\!\cdots\!52\)\( T^{2} + 5080397320 T^{3} + T^{4} )^{2} \)
$11$ \( \)\(22\!\cdots\!36\)\( + \)\(16\!\cdots\!12\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{4} + \)\(34\!\cdots\!48\)\( T^{6} + T^{8} \)
$13$ \( ( -\)\(14\!\cdots\!00\)\( + \)\(37\!\cdots\!00\)\( T - \)\(12\!\cdots\!00\)\( T^{2} - 25284181839560 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(73\!\cdots\!56\)\( + \)\(61\!\cdots\!72\)\( T^{2} + \)\(46\!\cdots\!84\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{6} + T^{8} \)
$19$ \( ( \)\(54\!\cdots\!36\)\( + \)\(87\!\cdots\!08\)\( T - \)\(28\!\cdots\!56\)\( T^{2} + 489041815670632 T^{3} + T^{4} )^{2} \)
$23$ \( \)\(36\!\cdots\!76\)\( + \)\(62\!\cdots\!52\)\( T^{2} + \)\(22\!\cdots\!24\)\( T^{4} + \)\(26\!\cdots\!48\)\( T^{6} + T^{8} \)
$29$ \( \)\(10\!\cdots\!00\)\( + \)\(34\!\cdots\!00\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{4} + \)\(96\!\cdots\!60\)\( T^{6} + T^{8} \)
$31$ \( ( -\)\(23\!\cdots\!64\)\( + \)\(72\!\cdots\!12\)\( T + \)\(35\!\cdots\!64\)\( T^{2} - 1546059893411354552 T^{3} + T^{4} )^{2} \)
$37$ \( ( -\)\(34\!\cdots\!64\)\( + \)\(19\!\cdots\!60\)\( T - \)\(10\!\cdots\!92\)\( T^{2} - 11295146611996391240 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(27\!\cdots\!96\)\( + \)\(36\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{4} + \)\(20\!\cdots\!48\)\( T^{6} + T^{8} \)
$43$ \( ( -\)\(18\!\cdots\!24\)\( + \)\(68\!\cdots\!20\)\( T + \)\(25\!\cdots\!32\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + T^{4} )^{2} \)
$47$ \( \)\(18\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{4} + \)\(39\!\cdots\!00\)\( T^{6} + T^{8} \)
$53$ \( \)\(51\!\cdots\!36\)\( + \)\(70\!\cdots\!92\)\( T^{2} + \)\(54\!\cdots\!44\)\( T^{4} + \)\(13\!\cdots\!68\)\( T^{6} + T^{8} \)
$59$ \( \)\(14\!\cdots\!56\)\( + \)\(57\!\cdots\!72\)\( T^{2} + \)\(49\!\cdots\!84\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{6} + T^{8} \)
$61$ \( ( \)\(20\!\cdots\!36\)\( - \)\(26\!\cdots\!56\)\( T - \)\(11\!\cdots\!44\)\( T^{2} + \)\(32\!\cdots\!24\)\( T^{3} + T^{4} )^{2} \)
$67$ \( ( -\)\(23\!\cdots\!24\)\( - \)\(27\!\cdots\!80\)\( T - \)\(66\!\cdots\!72\)\( T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + T^{4} )^{2} \)
$71$ \( \)\(15\!\cdots\!00\)\( + \)\(58\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{6} + T^{8} \)
$73$ \( ( \)\(15\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T - \)\(13\!\cdots\!40\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + T^{4} )^{2} \)
$79$ \( ( \)\(50\!\cdots\!96\)\( - \)\(13\!\cdots\!68\)\( T + \)\(78\!\cdots\!64\)\( T^{2} - \)\(15\!\cdots\!12\)\( T^{3} + T^{4} )^{2} \)
$83$ \( \)\(78\!\cdots\!76\)\( + \)\(59\!\cdots\!08\)\( T^{2} + \)\(10\!\cdots\!64\)\( T^{4} + \)\(56\!\cdots\!92\)\( T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!96\)\( + \)\(72\!\cdots\!32\)\( T^{2} + \)\(34\!\cdots\!64\)\( T^{4} + \)\(35\!\cdots\!88\)\( T^{6} + T^{8} \)
$97$ \( ( -\)\(78\!\cdots\!44\)\( - \)\(14\!\cdots\!80\)\( T - \)\(63\!\cdots\!48\)\( T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + T^{4} )^{2} \)
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