LMFDB - Newform orbit 1100.6.b.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,6,Mod(749,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1100.749"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$1100 = 2^{2} \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	1100.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-754,0,1936] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$176.422201794$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 2231 x^{14} + 1950697 x^{12} + 841157794 x^{10} + 184347539486 x^{8} + 19011199757266 x^{6} + \cdots + 36\!\cdots\!25$ x^16 + 2231x^14 + 1950697x^12 + 841157794x^10 + 184347539486x^8 + 19011199757266x^6 + 761025943061793x^4 + 11029470554922471*x^2 + 36582879760245225
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$2^{14}\cdot 5^{10}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{9} + \beta_{8}) q^{3} + (\beta_{10} - 2 \beta_{9}) q^{7} + ( - \beta_{2} - \beta_1 - 47) q^{9} + 121 q^{11} + (\beta_{12} - \beta_{10} + \cdots - 5 \beta_{8}) q^{13} + ( - \beta_{12} + \beta_{11} + \cdots - 13 \beta_{9}) q^{17}+ \cdots + ( - 121 \beta_{2} - 121 \beta_1 - 5687) q^{99}+O(q^{100})$ q + (b9 + b8) * q^3 + (b10 - 2b9) q^7 + (-b2 - b1 - 47) * q^9 + 121 * q^11 + (b12 - b10 + 5b9 - 5b8) * q^13 + (-b12 + b11 - b10 - 13b9) q^17 + (-b5 + b3 - 2b1 + 44) q^19 + (-3b6 - 2b5 + b4 - b3 - 2b2 - 4b1 - 72) * q^21 + (-b15 + 2b13 - b12 - b11 + b10 + 86b9 + 28b8) q^23 + (b14 - 2b13 - b12 - 3b11 - 7b10 - 222b9 - 24b8) q^27 + (3b7 + 4b6 + b5 + b4 - b3 + b2 - 7b1 - 341) q^29 + (-4b7 - 3b6 + 2b5 - 2b4 + b3 - 9b2 - 10b1 + 473) * q^31 + (121b9 + 121b8) * q^33 + (2b15 - b14 + b13 - 4b12 - 5b11 - 30b10 - 278b9 - 116b8) * q^37 + (2b7 + 11b6 + 10b5 + 6b4 + 8b3 - 46b2 + 8b1 + 1247) q^39 + (6b7 - b6 + 8b5 + 2b4 + 2b3 - b2 - 13b1 + 1103) q^41 + (6b15 - b14 + 5b13 + 3b11 - 40b10 + 341b9 - 80b8) * q^43 + (-7b15 - 7b14 + b13 + 5b12 + 19b11 - 51b10 - 292b9 - 173b8) q^47 + (-7b7 - 3b6 - 4b5 - 10b3 - 91b2 - 17b1 - 3424) * q^49 + (-2b7 - 17b6 - 8b5 - 19b4 - 7b3 + 43b2 - 26b1 + 597) q^51 + (-6b15 + 7b14 + 10b13 - 4b12 + 8b11 - 12b10 + 653b9 - 306b8) * q^53 + (5b15 + 7b14 - 4b12 - 13b11 - 27b10 - 182b9 - 389b8) q^57 + (-9b7 - 20b6 - 22b5 - 36b4 - 16b3 - 123b2 + 6b1 - 2581) q^59 + (20b7 - 14b6 - 3b5 + 14b4 + 3b3 + 61b2 - 11b1 + 1335) q^61 + (20b15 - 15b14 - 20b13 + 32b12 - 10b11 - 136b10 - 1758b9 - 405b8) * q^63 + (5b15 + 8b14 - 11b13 + 26b12 + 10b11 - 52b10 - 310b9 - 917b8) * q^67 + (-24b7 - 18b6 - 17b5 + 11b4 - 12b3 - 235b2 - 26b1 - 9077) q^69 + (-25b7 - 15b6 - 2b5 + 47b4 + 26b3 + 31b2 + 17b1 + 3995) q^71 + (-29b15 + 12b14 - 5b13 + 9b12 - 13b11 + 44b10 - 1326b9 - 1228b8) * q^73 + (121b10 - 242b9) * q^77 + (46b7 + 35b6 + 32b5 + 27b4 + 37b3 - 257b2 + 133b1 + 8904) q^79 + (7b7 + 25b6 + 8b5 + 5b4 - 3b3 + 361b2 - 53b1 - 847) q^81 + (-38b15 - 7b14 + 6b13 + 3b12 + 27b11 - 27b10 - 2176b9 - 1895b8) * q^83 + (-33b15 + 27b14 + 14b13 - b12 + 6b11 + 55b10 - 1411b9 - 2029b8) q^87 + (-9b7 - 15b6 + 5b5 - 17b4 + 22b3 - 680b2 + 141b1 + 1745) q^89 + (-8b7 - 9b6 - 33b5 - 102b4 - 61b3 + 130b2 + 225b1 + 16503) q^91 + (21b15 - 9b14 - 37b13 - 35b12 - 59b11 + 142b10 - 2890b9 - 1787b8) * q^93 + (72b15 - 71b14 + 34b13 + 9b12 - 67b11 + 24b10 - 4961b9 - 1028b8) * q^97 + (-121b2 - 121b1 - 5687) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 754 q^{9} + 1936 q^{11} + 708 q^{19} - 1176 q^{21} - 5486 q^{29} + 7536 q^{31} + 20054 q^{39} + 17512 q^{41} - 54682 q^{49} + 9414 q^{51} - 40800 q^{59} + 21022 q^{61} - 144904 q^{69} + 63724 q^{71}+ \cdots - 91234 q^{99}+O(q^{100})$ 16 * q - 754 * q^9 + 1936 * q^11 + 708 * q^19 - 1176 * q^21 - 5486 * q^29 + 7536 * q^31 + 20054 * q^39 + 17512 * q^41 - 54682 * q^49 + 9414 * q^51 - 40800 * q^59 + 21022 * q^61 - 144904 * q^69 + 63724 * q^71 + 143358 * q^79 - 14456 * q^81 + 29922 * q^89 + 265282 * q^91 - 91234 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 2231 x^{14} + 1950697 x^{12} + 841157794 x^{10} + 184347539486 x^{8} + 19011199757266 x^{6} + \cdots + 36\!\cdots\!25$ x^16 + 2231*x^14 + 1950697*x^12 + 841157794*x^10 + 184347539486*x^8 + 19011199757266*x^6 + 761025943061793*x^4 + 11029470554922471*x^2 + 36582879760245225 :

$\beta_{1}$	$=$	$( 11678598862180 \nu^{14} + \cdots - 10\!\cdots\!11 ) / 36\!\cdots\!24$ (11678598862180v^14 + 23185297939504341v^12 + 18003410340768016076v^10 + 6490191635647275020068v^8 + 893484911202881828572026v^6 - 18846392353777716916215892v^4 - 42283378596194345044367536320*v^2 - 10013627964516016502178508354011) / 36067963071986923752007955424
$\beta_{2}$	$=$	$( 58392994310900 \nu^{14} + \cdots + 17\!\cdots\!77 ) / 36\!\cdots\!24$ (58392994310900v^14 + 115926489697521705v^12 + 90017051703840080380v^10 + 32450958178236375100340v^8 + 4467424556014409142860130v^6 - 94231961768888584581079460v^4 - 31077077621037106461797904480*v^2 + 174532736697702275654540135577) / 36067963071986923752007955424
$\beta_{3}$	$=$	$( 26\!\cdots\!95 \nu^{14} + \cdots - 41\!\cdots\!09 ) / 14\!\cdots\!68$ (26071504441571960777760995v^14 + 52078720401536568278310355933v^12 + 38813075628541417699720809706022v^10 + 12790860097546787943225038269366532v^8 + 1488454787503013562602188435049290012v^6 - 87024048663089552047281381526095667738v^4 - 23102364599667285803552545796587177203327*v^2 - 414924080215088292601665340989145907832609) / 142674860068246611493058959215261169968
$\beta_{4}$	$=$	$( 47\!\cdots\!93 \nu^{14} + \cdots + 12\!\cdots\!63 ) / 14\!\cdots\!68$ (47512443124349155950149593v^14 + 107319573221886120188684645405v^12 + 94341139054338407049661164477166v^10 + 40329475985805523301984390323844320v^8 + 8485413303562096945291574453528770340v^6 + 769306236955546621757325992195412235438v^4 + 19724925651042727820082486305428559042679*v^2 + 120474060830772211408074337915024899407463) / 142674860068246611493058959215261169968
$\beta_{5}$	$=$	$( 19\!\cdots\!02 \nu^{14} + \cdots + 80\!\cdots\!85 ) / 28\!\cdots\!36$ (198159958929481659169640402v^14 + 431675917701365584832796580801v^12 + 366203365713566883966390100441520v^10 + 151637773568204999545657698612292916v^8 + 31320354178353629633813058046554399478v^6 + 2923692817866372688153015981313918966768v^4 + 94444292205062782832599848129348314571330*v^2 + 800122425685228380049753508922594761708085) / 285349720136493222986117918430522339936
$\beta_{6}$	$=$	$( - 67\!\cdots\!65 \nu^{14} + \cdots - 80\!\cdots\!12 ) / 95\!\cdots\!12$ (-67675308395995488835347265v^14 - 149877613374580763979038027688v^12 - 128907264169837621446878065008788v^10 - 53768836797132162523426763840208154v^8 - 11008576416660071677699842272910714588v^6 - 970133679589493716818341557268570848724v^4 - 23605633073383295244691546752084967441785*v^2 - 80773204057332880973288476004393416981812) / 95116573378831074328705972810174113312
$\beta_{7}$	$=$	$( 41\!\cdots\!13 \nu^{14} + \cdots + 69\!\cdots\!86 ) / 28\!\cdots\!36$ (412582842935335794809450713v^14 + 910917657591225453299416475510v^12 + 783883262219442777228369323088628v^10 + 329081923457797306868506483090793050v^8 + 68497832221951815106767676584506952656v^6 + 6253110688373938530722975201860637331988v^4 + 165708572639196196367818063809126084820161*v^2 + 697607309270727178756263866446272660388986) / 285349720136493222986117918430522339936
$\beta_{8}$	$=$	$( - 78\!\cdots\!43 \nu^{15} + \cdots - 17\!\cdots\!73 \nu ) / 10\!\cdots\!60$ (-7816117698280095143v^15 - 17370069951394645782133v^13 - 15112496402162926872424016v^11 - 6470241364548016337980221962v^9 - 1403265207349325271773091091558v^7 - 143415176549794132459559808604208v^5 - 6057501184410419972510242848488259v^3 - 17707763238147346882310396920979073v) / 104524145453448985327534634639754960
$\beta_{9}$	$=$	$( 78\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!33 \nu ) / 41\!\cdots\!84$ (7816117698280095143v^15 + 17370069951394645782133v^13 + 15112496402162926872424016v^11 + 6470241364548016337980221962v^9 + 1403265207349325271773091091558v^7 + 143415176549794132459559808604208v^5 + 6057501184410419972510242848488259v^3 + 122231908691596332209845031560734033v) / 41809658181379594131013853855901984
$\beta_{10}$	$=$	$( - 29\!\cdots\!29 \nu^{15} + \cdots - 64\!\cdots\!07 \nu ) / 11\!\cdots\!92$ (-2999588625420019785230301920329v^15 - 6641208364935264603065755153420295v^13 - 5726885529203628647802824384101329640v^11 - 2405875658499984557307778346484686483998v^9 - 499746084764690066090210281796184341224922v^7 - 45260163791715978333989165407375232439972712v^5 - 1180093434902656846409577930464937425098377141v^3 - 6481449618751364356608744699529448346158264307v) / 110258275811580571882167005327798540584250592
$\beta_{11}$	$=$	$( - 87\!\cdots\!76 \nu^{15} + \cdots - 41\!\cdots\!41 \nu ) / 91\!\cdots\!60$ (-8721566958001256909003260706576v^15 - 19250029711209628903103633635455811v^13 - 16562684437953184871586057742106185202v^11 - 6959388212121426077210388652099311292814v^9 - 1456197269877215230480190536177140437536996v^7 - 136130073999113839689928322361491320703205266v^5 - 4148110711526653491514262157424890302444884158v^3 - 41679562781918312556261296366711629729509741441v) / 91881896509650476568472504439832117153542160
$\beta_{12}$	$=$	$( - 30\!\cdots\!31 \nu^{15} + \cdots - 25\!\cdots\!59 \nu ) / 27\!\cdots\!48$ (-3049569047035759046539842626831v^15 - 6683004752304122114247568328948146v^13 - 5690609445474277531277487989535403757v^11 - 2350788465835085643979078689446917779089v^9 - 475923382687727588627227379463102656413705v^7 - 40967538130336594611306818918721588192540629v^5 - 891547044751089829219056710410325180854926090v^3 - 2587040205156777672864170136806487853087052559v) / 27564568952895142970541751331949635146062648
$\beta_{13}$	$=$	$( 12\!\cdots\!71 \nu^{15} + \cdots - 65\!\cdots\!66 \nu ) / 11\!\cdots\!92$ (12625137100570072594357832624371v^15 + 27588538774152650451461930990065898v^13 + 23372150087540570102737855202095032016v^11 + 9553533028880452837714968774214600062298v^9 + 1884369819885989662533071865602493281907188v^7 + 148582632521245977884940344490947992341608944v^5 + 1374479851290636393967467578959907703209317959v^3 - 65868773559030981645065284542727783346610450966v) / 110258275811580571882167005327798540584250592
$\beta_{14}$	$=$	$( - 14\!\cdots\!56 \nu^{15} + \cdots - 62\!\cdots\!21 \nu ) / 55\!\cdots\!60$ (-146933990752964379483675922584056v^15 - 325708996353595625975514285321721801v^13 - 281780842804353029962608889888788818732v^11 - 119309859297699206554196261578348757570204v^9 - 25274701492225341881610076864935026987519506v^7 - 2422508601859954192739554805302238224160553356v^5 - 78824598793344547849986751684621514344588064388v^3 - 624421354207299070811922582912074831635117612521v) / 551291379057902859410835026638992702921252960
$\beta_{15}$	$=$	$( - 21\!\cdots\!39 \nu^{15} + \cdots - 12\!\cdots\!54 \nu ) / 55\!\cdots\!60$ (-210970999371661450583542448050939v^15 - 468249349021097647907990359579944254v^13 - 406299101535534033197735751865301913688v^11 - 173052070610917613492381053347503849213986v^9 - 37080433492191188995198401456633339830648564v^7 - 3638395882525187858198699631243908468992081144v^5 - 125963787513079970892466591175282687593765831047v^3 - 1270314883803361565874543269961805268267650060854v) / 551291379057902859410835026638992702921252960

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( 2\beta_{9} + 5\beta_{8} ) / 5$ (2b9 + 5b8) / 5
$\nu^{2}$	$=$	$( \beta_{2} - 5\beta _1 - 1393 ) / 5$ (b2 - 5*b1 - 1393) / 5
$\nu^{3}$	$=$	$( 5\beta_{14} - \beta_{13} + 4\beta_{12} - 15\beta_{11} - 17\beta_{10} - 948\beta_{9} - 2415\beta_{8} ) / 5$ (5b14 - b13 + 4b12 - 15b11 - 17b10 - 948b9 - 2415b8) / 5
$\nu^{4}$	$=$	$( -49\beta_{7} + 77\beta_{6} + 88\beta_{5} + 181\beta_{4} + 21\beta_{3} - 340\beta_{2} + 2966\beta _1 + 671767 ) / 5$ (-49b7 + 77b6 + 88b5 + 181b4 + 21b3 - 340b2 + 2966*b1 + 671767) / 5
$\nu^{5}$	$=$	$( - 135 \beta_{15} - 3205 \beta_{14} - 460 \beta_{13} - 4895 \beta_{12} + 12390 \beta_{11} + \cdots + 1286665 \beta_{8} ) / 5$ (-135b15 - 3205b14 - 460b13 - 4895b12 + 12390b11 + 8395b10 + 566572b9 + 1286665b8) / 5
$\nu^{6}$	$=$	$( 37835 \beta_{7} - 86599 \beta_{6} - 75755 \beta_{5} - 176645 \beta_{4} - 26670 \beta_{3} + \cdots - 357061603 ) / 5$ (37835b7 - 86599b6 - 75755b5 - 176645b4 - 26670b3 + 332431b2 - 1705244*b1 - 357061603) / 5
$\nu^{7}$	$=$	$( 205665 \beta_{15} + 1800585 \beta_{14} + 824598 \beta_{13} + 3978813 \beta_{12} + \cdots - 713544890 \beta_{8} ) / 5$ (205665b15 + 1800585b14 + 824598b13 + 3978813b12 - 8485170b11 - 3555354b10 - 273487109b9 - 713544890b8) / 5
$\nu^{8}$	$=$	$( - 24403371 \beta_{7} + 69791979 \beta_{6} + 53575827 \beta_{5} + 131839674 \beta_{4} + \cdots + 197482475914 ) / 5$ (-24403371b7 + 69791979b6 + 53575827b5 + 131839674b4 + 22306329b3 - 348063967b2 + 985302500*b1 + 197482475914) / 5
$\nu^{9}$	$=$	$( - 186236361 \beta_{15} - 1001824847 \beta_{14} - 790011254 \beta_{13} - 2875838770 \beta_{12} + \cdots + 404625580416 \beta_{8} ) / 5$ (-186236361b15 - 1001824847b14 - 790011254b13 - 2875838770b12 + 5458934553b11 + 1401098609b10 + 109962737316b9 + 404625580416b8) / 5
$\nu^{10}$	$=$	$( 15473594935 \beta_{7} - 49189316903 \beta_{6} - 35702615110 \beta_{5} - 89838206710 \beta_{4} + \cdots - 111689955247228 ) / 5$ (15473594935b7 - 49189316903b6 - 35702615110b5 - 89838206710b4 - 16304835090b3 + 311683996951b2 - 573179302463*b1 - 111689955247228) / 5
$\nu^{11}$	$=$	$( 137132352009 \beta_{15} + 566878487458 \beta_{14} + 638666427451 \beta_{13} + \cdots - 232765266333919 \beta_{8} ) / 5$ (137132352009b15 + 566878487458b14 + 638666427451b13 + 1973258333780b12 - 3418674733482b11 - 469861233841b10 - 31822132955473b9 - 232765266333919b8) / 5
$\nu^{12}$	$=$	$( - 9903842142170 \beta_{7} + 32383511642434 \beta_{6} + 23100578496890 \beta_{5} + 58783704339230 \beta_{4} + \cdots + 64\!\cdots\!53 ) / 5$ (-9903842142170b7 + 32383511642434b6 + 23100578496890b5 + 58783704339230b4 + 11156042923500b3 - 249757419180346b2 + 335461066502864*b1 + 64088776180787953) / 5
$\nu^{13}$	$=$	$( - 90993771245166 \beta_{15} - 327663206213142 \beta_{14} - 476369588987724 \beta_{13} + \cdots + 13\!\cdots\!31 \beta_{8} ) / 5$ (-90993771245166b15 - 327663206213142b14 - 476369588987724b13 - 1313910603767790b12 + 2113997619449148b11 + 74731649405244b10 - 833249708902204b9 + 135284688731466731b8) / 5
$\nu^{14}$	$=$	$( 63\!\cdots\!54 \beta_{7} + \cdots - 37\!\cdots\!87 ) / 5$ (6396326488908354b7 - 20497592074186626b6 - 14659114442988018b5 - 37671257904442356b4 - 7334858952764766b3 + 186424816727157085b2 - 197365587957936455*b1 - 37158520556810423887) / 5
$\nu^{15}$	$=$	$( 56\!\cdots\!82 \beta_{15} + \cdots - 79\!\cdots\!37 \beta_{8} ) / 5$ (56794015768350582b15 + 193099974328226009b14 + 338939533883690753b13 + 857674908281107600b12 - 1299277869198236331b11 + 80007461194012447b10 + 11786803174053422580b9 - 79257031735358267637b8) / 5

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

749.1

	−	23.6316i
	−	22.3000i
	−	24.8760i
	−	17.8741i
	−	13.1518i
	−	6.06056i
	−	4.75672i
	−	2.15294i
		2.15294i
		4.75672i
		6.06056i
		13.1518i
		17.8741i
		24.8760i
		22.3000i
		23.6316i

−

26.6316i

−

210.828i

−466.240

749.2

−

25.3000i

150.442i

−397.092

749.3

−

21.8760i

48.9354i

−235.558

749.4

−

14.8741i

−

166.492i

21.7626

749.5

−

10.1518i

122.703i

139.941

749.6

−

9.06056i

199.386i

160.906

749.7

−

7.75672i

−

86.6415i

182.833

749.8

−

5.15294i

47.7885i

216.447

749.9

5.15294i

−

47.7885i

216.447

749.10

7.75672i

86.6415i

182.833

749.11

9.06056i

−

199.386i

160.906

749.12

10.1518i

−

122.703i

139.941

749.13

14.8741i

166.492i

21.7626

749.14

21.8760i

−

48.9354i

−235.558

749.15

25.3000i

−

150.442i

−397.092

749.16

26.6316i

210.828i

−466.240

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1100.6.b.i		16
5.b	even	2	1	inner	1100.6.b.i		16
5.c	odd	4	1		1100.6.a.h	✓	8
5.c	odd	4	1		1100.6.a.k	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1100.6.a.h	✓	8	5.c	odd	4	1
1100.6.a.k	yes	8	5.c	odd	4	1
1100.6.b.i		16	1.a	even	1	1	trivial
1100.6.b.i		16	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{16} + 2321 T_{3}^{14} + 2087326 T_{3}^{12} + 924020173 T_{3}^{10} + 214678632158 T_{3}^{8} + \cdots + 64\!\cdots\!76$ T3^16 + 2321*T3^14 + 2087326*T3^12 + 924020173*T3^10 + 214678632158*T3^8 + 26670782667673*T3^6 + 1749655004415201*T3^4 + 55707429442387896*T3^2 + 649668040507055376 acting on $S_{6}^{\mathrm{new}}(1100, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16} + \cdots + 64\!\cdots\!76$ T^16 + 2321T^14 + 2087326T^12 + 924020173T^10 + 214678632158T^8 + 26670782667673T^6 + 1749655004415201T^4 + 55707429442387896*T^2 + 649668040507055376
$5$	$T^{16}$ T^16
$7$	$T^{16} + \cdots + 68\!\cdots\!00$ T^16 + 161797T^14 + 10523768910T^12 + 353326091441185T^10 + 6546208697846961790T^8 + 66379643110022999910117T^6 + 344956280922057608934796809T^4 + 810621780659719306628885370000*T^2 + 685217015406263953838030400000000
$11$	$(T - 121)^{16}$ (T - 121)^16
$13$	$T^{16} + \cdots + 79\!\cdots\!25$ T^16 + 2733152T^14 + 2802642976996T^12 + 1386765416888153008T^10 + 365077898697567774596846T^8 + 51610390875472629368383317952T^6 + 3611848386025712525363575132946676T^4 + 98470937479864334742883832187292916400*T^2 + 794869828107341312117736850252431884765625
$17$	$T^{16} + \cdots + 56\!\cdots\!64$ T^16 + 6841589T^14 + 17340184796710T^12 + 20949287222325066817T^10 + 13712472259996215235969670T^8 + 5134407505838322883043838496357T^6 + 1096497579602822284955806629426998913T^4 + 123744522221883873828930562736210932435920*T^2 + 5692878526929904206998284537972670274486641664
$19$	$(T^{8} + \cdots - 11\!\cdots\!89)^{2}$ (T^8 - 354T^7 - 11308006T^6 + 6119322094T^5 + 27781233806340T^4 - 2929891334233846T^3 - 22671404637401809946T^2 - 10065684003726792063894*T - 1100875006767821221712389)^2
$23$	$T^{16} + \cdots + 12\!\cdots\!29$ T^16 + 46895791T^14 + 864899302939353T^12 + 8017697693376767526786T^10 + 39199179588816454148164608558T^8 + 95340514588194567319186923732710610T^6 + 94174010747365348836322857397878510421953T^4 + 26274922818199752859892306208902328544285859007*T^2 + 1236706299923400308278524979251303260158198736806329
$29$	$(T^{8} + \cdots - 14\!\cdots\!19)^{2}$ (T^8 + 2743T^7 - 57702581T^6 - 234480376864T^5 + 627468091772648T^4 + 4285929644426353184T^3 + 5650159866888782520999T^2 - 6020400749308731419943*T - 1498738735866810302369226819)^2
$31$	$(T^{8} + \cdots + 62\!\cdots\!25)^{2}$ (T^8 - 3768T^7 - 104259904T^6 + 245310001048T^5 + 3172583179546026T^4 - 1910619633030715528T^3 - 26578345674921150247704T^2 + 5356344429180936709930920*T + 62398456862040213736476862125)^2
$37$	$T^{16} + \cdots + 91\!\cdots\!04$ T^16 + 416373926T^14 + 58602163718481823T^12 + 3310868094385445754401716T^10 + 84399334010019230377569072741743T^8 + 1006296392413473804343431957616587308710T^6 + 5041599521039185634618161144863880674667770513T^4 + 6230983745731163630472824801081103334081642955555232*T^2 + 912166572744010645208145685408718142022139312933528987904
$41$	$(T^{8} + \cdots - 31\!\cdots\!00)^{2}$ (T^8 - 8756T^7 - 355718271T^6 + 1881514716244T^5 + 45427691845325851T^4 - 35414197539837661836T^3 - 2197578108950692765072701T^2 - 7137093357563808706937791380*T - 3176394822850950034213626511200)^2
$43$	$T^{16} + \cdots + 12\!\cdots\!24$ T^16 + 1226413834T^14 + 537891756601175985T^12 + 103025254160988518937606872T^10 + 8718537268184361389384500913271920T^8 + 331233890672859830700239266417056087512832T^6 + 4911212962582298463466037929392979762504913628928T^4 + 14315756242616784487228246119551504959450711367853905920*T^2 + 1299580743427995264234383716154159506555401224328773875175424
$47$	$T^{16} + \cdots + 93\!\cdots\!00$ T^16 + 3034231942T^14 + 3817901123997344511T^12 + 2599019472326378197515104788T^10 + 1047306756852269229692342849398277551T^8 + 255906316235515100647556836212544507577608902T^6 + 37014679109037096315896615522458674082486284300769521T^4 + 2893098907607022826289999174562183142186044487563778041000000*T^2 + 93237045176535648868323178845435993727746574173261088537308733440000
$53$	$T^{16} + \cdots + 30\!\cdots\!00$ T^16 + 2503727573T^14 + 2348009888441408326T^12 + 1128956057258356865656747225T^10 + 309379300338687496220447843430143750T^8 + 49524298717264167546048350293735427512328125T^6 + 4463836619638421935772798139526165125779648506640625T^4 + 199739415052136741110284179906079542698619256288954609375000*T^2 + 3057512036790379583460565526640799943345254360107270966597656250000
$59$	$(T^{8} + \cdots + 75\!\cdots\!36)^{2}$ (T^8 + 20400T^7 - 3045878467T^6 - 42042272145388T^5 + 2794917475159660131T^4 + 24698950229484726777832T^3 - 877890243251729424281956737T^2 - 4714882853825962765748824452780*T + 75742589574397792813009044256775136)^2
$61$	$(T^{8} + \cdots + 30\!\cdots\!04)^{2}$ (T^8 - 10511T^7 - 2325908022T^6 + 20662392339609T^5 + 857491594857092358T^4 - 553477079673106513395T^3 - 73362305997996069462079927T^2 - 233550183469848442136119827892*T + 308388166900774117261899860564604)^2
$67$	$T^{16} + \cdots + 15\!\cdots\!76$ T^16 + 7618073992T^14 + 20660843064150250128T^12 + 26938278894659869539510777344T^10 + 18307167597598788225009620186740920320T^8 + 6441591984882029537177623344282865928949202944T^6 + 1109944771289240485794716962985552323756192143694102528T^4 + 81612596606047605791205901990925193178080655864132285751099392*T^2 + 1529672748785238523902154887638071425824408592845120529347347507838976
$71$	$(T^{8} + \cdots + 36\!\cdots\!12)^{2}$ (T^8 - 31862T^7 - 9670688481T^6 + 194430744120902T^5 + 31307686538856805472T^4 - 312266714764719636827016T^3 - 33282125696041971719016552192T^2 + 146420027406260792467432116955488*T + 3695111656175761302693393261408117312)^2
$73$	$T^{16} + \cdots + 17\!\cdots\!44$ T^16 + 15182748821T^14 + 77726500693309839838T^12 + 158319450879239458553421640241T^10 + 145014433570552141892444275692589625518T^8 + 62468919125326441439966763055312001381458045205T^6 + 11954288771096919774983494586266869528681364145014935673T^4 + 820840620646786592542898690308491445733059033976764331869441232*T^2 + 17573500622424702558271695632059465445271707947204220805691744543744
$79$	$(T^{8} + \cdots - 89\!\cdots\!08)^{2}$ (T^8 - 71679T^7 - 13023850944T^6 + 1200494095111701T^5 + 25673362585007064552T^4 - 4410400647742499452536147T^3 + 41677761291387432295080271367T^2 + 4435923119017605774606329733741444*T - 89445275450693214466451475315715831008)^2
$83$	$T^{16} + \cdots + 64\!\cdots\!09$ T^16 + 24102902147T^14 + 230013021434293551577T^12 + 1122475671106753464858373866450T^10 + 3030468140920671973724171769923012454582T^8 + 4569200291928441416122664922703254994813717959426T^6 + 3625855201691298061300339591990443106260156551611823005137T^4 + 1235287636632724826877605442263459383112993528647557791046431397171*T^2 + 64336517742469473820546865558839687608820077120612309201760035536904756009
$89$	$(T^{8} + \cdots - 34\!\cdots\!19)^{2}$ (T^8 - 14961T^7 - 20046089679T^6 - 455787474947838T^5 + 95473976846995236618T^4 + 5801515231120842185690202T^3 + 110070591243811703178840302661T^2 + 464126498744364926221067808401079*T - 3414853333833232004113270952982089619)^2
$97$	$T^{16} + \cdots + 57\!\cdots\!81$ T^16 + 146571819299T^14 + 9046513261505539200361T^12 + 304263257765401300581394406676322T^10 + 6013579377724596818654073987247321552229078T^8 + 69836989258290229996866865320137332122802732278380242T^6 + 444003134278733231033972190301758731392956198927490014526068321T^4 + 1248368092484026072337804476250264735746540161548773614785159119389553779*T^2 + 570506435374902572516699077841993095448787585091403237221065832843030918847515881
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 749.16
Significant digits:
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