# Properties

 Label 1638.2.bj.i Level $1638$ Weight $2$ Character orbit 1638.bj Analytic conductor $13.079$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + 6655 x^{8} + 478 x^{7} + 1106 x^{6} + 11266 x^{5} + 55249 x^{4} + 8856 x^{3} + \cdots + 97344$$ x^16 - 2*x^15 + 2*x^14 + 18*x^13 + 143*x^12 - 148*x^11 + 172*x^10 + 1612*x^9 + 6655*x^8 + 478*x^7 + 1106*x^6 + 11266*x^5 + 55249*x^4 + 8856*x^3 + 288*x^2 + 7488*x + 97344 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_{5} q^{4} - \beta_{9} q^{5} - \beta_{4} q^{7} + (\beta_{4} - \beta_{3}) q^{8}+O(q^{10})$$ q - b3 * q^2 + b5 * q^4 - b9 * q^5 - b4 * q^7 + (b4 - b3) * q^8 $$q - \beta_{3} q^{2} + \beta_{5} q^{4} - \beta_{9} q^{5} - \beta_{4} q^{7} + (\beta_{4} - \beta_{3}) q^{8} - \beta_{13} q^{10} + (\beta_{15} - \beta_{14} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{15} - \beta_{14} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{13} + q^{14} + (\beta_{5} - 1) q^{16} + (\beta_{15} - 2 \beta_{14} + \beta_{10} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 1) q^{17} + (\beta_{9} - \beta_{7} - \beta_{2} - \beta_1) q^{19} + ( - \beta_{9} + \beta_{7}) q^{20} + ( - \beta_{12} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{22} + ( - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{23}+ \cdots - \beta_{4} q^{98}+O(q^{100})$$ q - b3 * q^2 + b5 * q^4 - b9 * q^5 - b4 * q^7 + (b4 - b3) * q^8 - b13 * q^10 + (b15 - b14 + b8 - b6 + b5 + b3 - b2 - b1 + 1) * q^11 + (b15 - b14 + b9 - b7 - b6 + b5 - b2 + 1) * q^13 + q^14 + (b5 - 1) * q^16 + (b15 - 2*b14 + b10 + b5 - b4 + 2*b3 + 1) * q^17 + (b9 - b7 - b2 - b1) * q^19 + (-b9 + b7) * q^20 + (-b12 + b8 - b6 - b5 + b4 - b3 - b2) * q^22 + (-b13 - b12 - b11 + b8 + b6 - b5 - b4 + b3 + b2 - b1 + 1) * q^23 + (-2*b15 + b14 + b13 + b11 - b10 + b6 - b5 - b4 - 2*b3 + b2 + b1 - 2) * q^25 + (b13 - b12 - b10 + b8 + b4 - b3) * q^26 - b3 * q^28 + (b12 + b11 - 2*b9 + 2*b8 + b7 + b6 + 3*b5 - b4 + 2*b2 - b1 - 3) * q^29 + (b14 - b13 - b10 + b8 + 3*b5 - 2*b4 + 2*b3 + b2 - b1 - 2) * q^31 + b4 * q^32 + (-2*b12 - b11 + b9 - 2*b5 + b4 + 1) * q^34 - b10 * q^35 + (b15 - b14 - b12 + b11 + b7 + b4 + 3*b3) * q^37 + (b13 - b10 + b8 - b2 - b1) * q^38 + (-b13 + b10) * q^40 + (2*b15 - 2*b14 - b13 + 2*b10 + 2*b8 - b7 - b6 - 2*b2 - b1) * q^41 + (b15 - 2*b14 + b10 - b5 + b4 - 2*b3 - b2 + b1 + 1) * q^43 + (-b14 + b8 - b6 + b5 - b4 + b3) * q^44 + (-b15 - b9 - b8 + b7 - b6 - b5 - b4 - b2 + 1) * q^46 + (-b14 - b13 + 2*b12 + b11 - b10 - 2*b9 - b8 + b6 + 3*b5 + b4 - 2*b3 - 1) * q^47 + (-b5 + 1) * q^49 + (b15 - b14 + b12 - b11 - b8 - b7 + 2*b5 - b4 + 2*b3 + b2 + 2) * q^50 + (-b14 - b7 - b6 + b5) * q^52 + (-b13 + b10 - b9 + 2*b7 + b6 - b4 - b3 + b2 + b1 + 2) * q^53 + (b15 + b14 + b12 + b11 - 4*b9 - b8 + 2*b7 - 2*b5 + 5*b4 - 3*b3 - b2 + 2) * q^55 + b5 * q^56 + (b15 - 2*b13 + b10 - 2*b8 - 2*b6 + 3*b4 - b2 + b1 + 1) * q^58 + (b15 - b9 + b8 + b7 + b6 + b5 - b4 + b2 - 1) * q^59 + (-b15 + 2*b14 + 2*b12 - 2*b10 - b9 - b8 - b7 + b6 - 2*b3 + b1 - 1) * q^61 + (b12 + b11 - 2*b9 - b8 + b7 - b6 - 2*b5 + 3*b4 - 2*b3 - b2 + b1 + 2) * q^62 - q^64 + (b15 - b14 + b12 + b11 + b10 - 3*b9 + 2*b7 - b6 + 4*b5 + 3*b4 - b3 - b2 - b1 + 2) * q^65 + (-b15 + b14 + 3*b7 + b6 - 2*b5 - 3*b3 + b1 - 2) * q^67 + (-b15 - b14 + b13 - 2*b4 + b3) * q^68 - b9 * q^70 + (-2*b15 + 4*b13 - b12 - 2*b11 - 2*b10 - b9 + b8 + b7 + b6 - b5 + b3 + b2) * q^71 + (b14 - b13 + 2*b12 + b11 - b10 - 2*b8 + 3*b6 + b5 - b4 + b2 - b1 - 1) * q^73 + (b15 - 2*b14 - b12 + b10 - 3*b5 + b3 + 1) * q^74 + (b8 - b7 - b6 - b2 - b1) * q^76 + (b11 + b8 - b3 - b2 - b1 - 1) * q^77 + (b13 - b11 - b10 + b9 + b8 - 2*b7 + b6 + 2*b4 + 3*b3 - 1) * q^79 + b7 * q^80 + (-2*b12 - b10 + b9 + 2*b8 + b7 - 2*b6 + 2*b3 - b2 - b1) * q^82 + (-2*b14 - 2*b12 - b11 + b9 + b8 - b6 + 2*b5 + b3) * q^83 + (-b15 + 4*b13 + b12 + 2*b11 - 2*b10 - 3*b9 + 3*b7 - b5 - b3 - 2*b2 - 2*b1 + 1) * q^85 + (-2*b12 - b11 + b9 + b8 + 2*b5 - b4 + 2*b3 + b2 - b1 - 1) * q^86 + (-b12 - b11 - b6 - b5 + b4 + b1 + 1) * q^88 + (b13 - b12 + b11 - 2*b10 + 3*b8 + b7 - 2*b5 + b4 - 6*b3 - 3*b2 - 2) * q^89 + (b13 + b11 + b8 - b3 - b1) * q^91 + (-b13 - b11 + b10 + b8 + b6 - b4 + 1) * q^92 + (b15 + b14 - 2*b13 - b12 - b11 - 2*b9 + b7 + b6 + 2*b5 + 3*b4 - b3 - b1 - 2) * q^94 + (2*b12 - b10 - b9 + b8 - b7 - b6 + 6*b5 - 2*b4 + 2*b3 - b1) * q^95 + (-b15 + 4*b13 + b12 + 2*b11 - 2*b10 + 3*b8 + 3*b6 + b5 - 6*b4 - b3 + 2*b2 - b1 - 3) * q^97 - b4 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4}+O(q^{10})$$ 16 * q + 8 * q^4 $$16 q + 8 q^{4} + 2 q^{10} + 12 q^{11} + 10 q^{13} + 16 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{22} + 12 q^{23} - 20 q^{25} + 2 q^{26} - 16 q^{29} - 2 q^{35} - 6 q^{37} + 4 q^{40} - 12 q^{41} - 6 q^{43} + 6 q^{46} + 8 q^{49} + 24 q^{50} - 4 q^{52} + 40 q^{53} + 20 q^{55} + 8 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} + 14 q^{62} - 16 q^{64} + 52 q^{65} - 30 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{74} - 8 q^{77} - 16 q^{79} + 2 q^{82} + 6 q^{85} + 4 q^{88} - 30 q^{89} + 4 q^{91} + 24 q^{92} - 8 q^{94} + 40 q^{95} - 24 q^{97}+O(q^{100})$$ 16 * q + 8 * q^4 + 2 * q^10 + 12 * q^11 + 10 * q^13 + 16 * q^14 - 8 * q^16 + 6 * q^17 - 4 * q^22 + 12 * q^23 - 20 * q^25 + 2 * q^26 - 16 * q^29 - 2 * q^35 - 6 * q^37 + 4 * q^40 - 12 * q^41 - 6 * q^43 + 6 * q^46 + 8 * q^49 + 24 * q^50 - 4 * q^52 + 40 * q^53 + 20 * q^55 + 8 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 + 14 * q^62 - 16 * q^64 + 52 * q^65 - 30 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^74 - 8 * q^77 - 16 * q^79 + 2 * q^82 + 6 * q^85 + 4 * q^88 - 30 * q^89 + 4 * q^91 + 24 * q^92 - 8 * q^94 + 40 * q^95 - 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + 6655 x^{8} + 478 x^{7} + 1106 x^{6} + 11266 x^{5} + 55249 x^{4} + 8856 x^{3} + \cdots + 97344$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 282646183413977 \nu^{15} + \cdots - 56\!\cdots\!92 ) / 11\!\cdots\!16$$ (282646183413977*v^15 - 55214164674845155*v^14 + 299583605589306334*v^13 - 636632814214148793*v^12 - 377759795822749871*v^11 - 3533639551957762244*v^10 + 27498294293097538484*v^9 - 51590047455907257838*v^8 - 39241111567554977845*v^7 + 16212479444246061545*v^6 + 533807150694494999134*v^5 - 388304314020910565305*v^4 - 467910520445787298453*v^3 + 394453724165083618062*v^2 + 1284563243127956350320*v - 56299713193433943792) / 1107554912919410126016 $$\beta_{3}$$ $$=$$ $$( - 30\!\cdots\!11 \nu^{15} + \cdots - 67\!\cdots\!08 ) / 57\!\cdots\!32$$ (-30074633116150611*v^15 + 45451664694774418*v^14 + 2810987296859646838*v^13 - 16119690886734640366*v^12 + 28804233803526199863*v^11 + 24094555083973283720*v^10 + 178576419805825731596*v^9 - 1478391611824306786100*v^8 + 2482535784319195091371*v^7 + 2026162126883338855882*v^6 - 876311475327257776106*v^5 - 28096792652800292738494*v^4 + 18530230924053144288721*v^3 + 24065006112304309708540*v^2 - 20520255150921799515192*v - 67022487495427465991808) / 57592855471809326552832 $$\beta_{4}$$ $$=$$ $$( 39\!\cdots\!57 \nu^{15} + \cdots - 66\!\cdots\!28 ) / 57\!\cdots\!32$$ (39517753222862357*v^15 - 648729221206109474*v^14 + 4075635642617830878*v^13 - 16029719207645527418*v^12 + 29062895761648473127*v^11 - 67048936194646430848*v^10 + 277351399516524139212*v^9 - 1465656548682883548524*v^8 + 2069394095569353712483*v^7 - 1857947376226502418458*v^6 - 761377044207872614482*v^5 - 27925559223479363555322*v^4 + 15820109597234308213249*v^3 - 21384703509084770905060*v^2 - 23591819177356590025656*v - 66502968233470533318528) / 57592855471809326552832 $$\beta_{5}$$ $$=$$ $$( 209664617 \nu^{15} - 2025754990 \nu^{14} + 4591800238 \nu^{13} - 476218986 \nu^{12} - 6913891517 \nu^{11} - 201926523452 \nu^{10} + \cdots + 19272755774688 ) / 35896513665792$$ (209664617*v^15 - 2025754990*v^14 + 4591800238*v^13 - 476218986*v^12 - 6913891517*v^11 - 201926523452*v^10 + 343932956684*v^9 - 26293832188*v^8 - 1807202553409*v^7 - 5305525364014*v^6 + 382854118510*v^5 + 79019616230*v^4 - 13644134133259*v^3 - 14135317026696*v^2 - 15902015471304*v + 19272755774688) / 35896513665792 $$\beta_{6}$$ $$=$$ $$( - 10\!\cdots\!30 \nu^{15} + \cdots - 73\!\cdots\!04 ) / 11\!\cdots\!16$$ (-10955648360776630*v^15 + 76857694926386657*v^14 - 321943053185712497*v^13 + 450228020207291463*v^12 - 1176929013801207731*v^11 + 5202968191580611804*v^10 - 29410753209194955154*v^9 + 34738527843677013974*v^8 - 36093016582058281252*v^7 - 15482378447545353487*v^6 - 545591658294002516717*v^5 + 262246024027392035603*v^4 - 417974475608200768051*v^3 - 453907697890091815086*v^2 - 1284593791723140897072*v - 73977234033198332304) / 1107554912919410126016 $$\beta_{7}$$ $$=$$ $$( 40\!\cdots\!40 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 14\!\cdots\!08$$ (408313565548094940*v^15 + 114088097456166367*v^14 - 4647944585615334641*v^13 + 24964519130544138227*v^12 + 43206223832315548641*v^11 + 36724246385156610848*v^10 - 403760627234778514774*v^9 + 2266094682963606617122*v^8 + 1576905497216767636042*v^7 + 2824503058282255431427*v^6 - 7207881507584697470057*v^5 + 33833924199208404765035*v^4 + 9598542624756210644617*v^3 + 15311865936965786362894*v^2 - 49205521739698931456448*v + 110391889191684991414992) / 14398213867952331638208 $$\beta_{8}$$ $$=$$ $$( 10297601 \nu^{15} - 26746609 \nu^{14} + 27244757 \nu^{13} + 236512383 \nu^{12} + 1095488206 \nu^{11} - 1973529760 \nu^{10} + \cdots + 130830721008 ) / 230105856832$$ (10297601*v^15 - 26746609*v^14 + 27244757*v^13 + 236512383*v^12 + 1095488206*v^11 - 1973529760*v^10 + 2335084582*v^9 + 20528978074*v^8 + 34652211865*v^7 - 967724693*v^6 + 14635012557*v^5 + 161717272807*v^4 + 102513505608*v^3 + 102323069750*v^2 + 116626452200*v + 130830721008) / 230105856832 $$\beta_{9}$$ $$=$$ $$( - 66\!\cdots\!76 \nu^{15} + \cdots - 77\!\cdots\!28 ) / 14\!\cdots\!08$$ (-660907316906610476*v^15 + 2446201482420671640*v^14 - 4109013180543494927*v^13 - 8487090366137862614*v^12 - 76365349316650189861*v^11 + 252296575315434561228*v^10 - 369458931097981206862*v^9 - 782393670615510605184*v^8 - 2713535221472810302086*v^7 + 6198648419396069802324*v^6 - 4726103380790664836939*v^5 - 6147618342605697618682*v^4 - 22068404703749564010057*v^3 + 27201171405731373852344*v^2 - 32833088941000481103960*v - 7713496531067974684128) / 14398213867952331638208 $$\beta_{10}$$ $$=$$ $$( 88\!\cdots\!08 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 14\!\cdots\!08$$ (883834513713952708*v^15 - 4787039289291129698*v^14 + 10377071398763163779*v^13 + 5219493265937510160*v^12 + 65052764096699429531*v^11 - 466710621731819594158*v^10 + 880562450252544180622*v^9 + 459279386334980334604*v^8 + 452700544956609079834*v^7 - 11096079191206940833754*v^6 + 9490317189397232697455*v^5 - 707685118469686992092*v^4 + 1467370293529741991431*v^3 - 49691169994610999008134*v^2 + 45159568905471194446944*v - 18074855881252503387600) / 14398213867952331638208 $$\beta_{11}$$ $$=$$ $$( - 40\!\cdots\!63 \nu^{15} + \cdots - 57\!\cdots\!04 ) / 57\!\cdots\!32$$ (-4042961793344474963*v^15 + 6370653498730640186*v^14 + 12678855019045914918*v^13 - 148853707820713736326*v^12 - 460602900834211537753*v^11 + 533423188766255558152*v^10 + 1224255212169639475836*v^9 - 13457629546224772248340*v^8 - 16686332908623221270725*v^7 + 3594614060710313449058*v^6 + 35816259489191874438966*v^5 - 172483976292755482519302*v^4 - 84610589276717757594223*v^3 + 14365008843455923139068*v^2 + 158019713813171160226248*v - 572250267913665698757504) / 57592855471809326552832 $$\beta_{12}$$ $$=$$ $$( - 46\!\cdots\!27 \nu^{15} + \cdots + 17\!\cdots\!64 ) / 57\!\cdots\!32$$ (-4655712139922920527*v^15 + 20676019676668269334*v^14 - 52652594422613273210*v^13 - 11769248677183450654*v^12 - 482012269843503086181*v^11 + 1770951776951990089580*v^10 - 4819082238532432311988*v^9 - 1267105736508227809700*v^8 - 14148137946423422304185*v^7 + 24836557510455048011590*v^6 - 82587986989531205017130*v^5 + 14087363660144992224962*v^4 - 75144435936148901175923*v^3 + 86043706206301487985496*v^2 - 320900374303127780924040*v + 174897622298969788349664) / 57592855471809326552832 $$\beta_{13}$$ $$=$$ $$( 13\!\cdots\!15 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 14\!\cdots\!08$$ (1389004728106834915*v^15 - 5707072215973687451*v^14 + 10464821837010805949*v^13 + 17381105895348833355*v^12 + 135324009286701961232*v^11 - 539190536576300063380*v^10 + 859965003678575982142*v^9 + 1661977215558551182330*v^8 + 3621677534400842843695*v^7 - 11226110831858809899695*v^6 + 6418510862957604190973*v^5 + 18010158180865164466747*v^4 + 26727614738255910023950*v^3 - 47728380563540951629842*v^2 + 9272082696190925446104*v + 42522752294979490201200) / 14398213867952331638208 $$\beta_{14}$$ $$=$$ $$( 59\!\cdots\!81 \nu^{15} + \cdots + 94\!\cdots\!84 ) / 57\!\cdots\!32$$ (5920048932267204181*v^15 - 26071365133088918354*v^14 + 52523838828502146206*v^13 + 59725596749121609162*v^12 + 550206583049233591871*v^11 - 2466578531453491503892*v^10 + 4537877522090572521292*v^9 + 5606548987304473337116*v^8 + 12521445239415270735715*v^7 - 52126559808514785178418*v^6 + 51514289581899984203486*v^5 + 45857573994379389510490*v^4 + 54158784566065240927081*v^3 - 255017389587749948710536*v^2 + 128855821261818828222744*v + 94190897396669655045984) / 57592855471809326552832 $$\beta_{15}$$ $$=$$ $$( 80\!\cdots\!01 \nu^{15} + \cdots + 35\!\cdots\!68 ) / 57\!\cdots\!32$$ (8036474130703798501*v^15 - 19805312696885386022*v^14 + 11075160454875415250*v^13 + 197537106076707726234*v^12 + 929445970952101320107*v^11 - 1745817760735834084360*v^10 + 751102224385844721892*v^9 + 17606914726801358758540*v^8 + 33738603578120391202219*v^7 - 23802966435448555129406*v^6 - 22123128289384736683582*v^5 + 170958557022024542732602*v^4 + 182575182215410744277797*v^3 - 102817180123678203728628*v^2 - 167706224321404297736856*v + 359176511227861954992768) / 57592855471809326552832
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{9} - \beta_{8} - \beta_{5} - 5\beta_{4} + 5\beta_{3} - \beta_{2} + \beta_1$$ b14 + b9 - b8 - b5 - 5*b4 + 5*b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + \beta _1 - 4$$ 2*b14 - 2*b13 - 2*b11 + 2*b9 - 2*b8 - 2*b7 - 8*b6 + 2*b5 - 4*b4 + 2*b3 - 8*b2 + b1 - 4 $$\nu^{4}$$ $$=$$ $$2 \beta_{15} - \beta_{14} - 14 \beta_{13} - 12 \beta_{11} + 14 \beta_{10} + \beta_{9} - 13 \beta_{8} - 2 \beta_{7} - 16 \beta_{6} + \beta_{5} - \beta_{4} + 11 \beta_{3} - 3 \beta_{2} - 3 \beta _1 - 38$$ 2*b15 - b14 - 14*b13 - 12*b11 + 14*b10 + b9 - 13*b8 - 2*b7 - 16*b6 + b5 - b4 + 11*b3 - 3*b2 - 3*b1 - 38 $$\nu^{5}$$ $$=$$ $$4 \beta_{15} - 30 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 26 \beta_{11} + 30 \beta_{10} - 2 \beta_{9} - 28 \beta_{7} - 18 \beta_{6} - 22 \beta_{5} + 8 \beta_{4} - 34 \beta_{3} + 18 \beta_{2} - 81 \beta _1 - 8$$ 4*b15 - 30*b14 - 2*b13 + 4*b12 - 26*b11 + 30*b10 - 2*b9 - 28*b7 - 18*b6 - 22*b5 + 8*b4 - 34*b3 + 18*b2 - 81*b1 - 8 $$\nu^{6}$$ $$=$$ $$- 135 \beta_{14} + 24 \beta_{13} + 40 \beta_{12} + 20 \beta_{11} + 24 \beta_{10} - 163 \beta_{9} + 155 \beta_{8} + 56 \beta_{6} + 63 \beta_{5} + 355 \beta_{4} - 375 \beta_{3} + 211 \beta_{2} - 211 \beta _1 + 36$$ -135*b14 + 24*b13 + 40*b12 + 20*b11 + 24*b10 - 163*b9 + 155*b8 + 56*b6 + 63*b5 + 355*b4 - 375*b3 + 211*b2 - 211*b1 + 36 $$\nu^{7}$$ $$=$$ $$- 88 \beta_{15} - 290 \beta_{14} + 390 \beta_{13} + 88 \beta_{12} + 378 \beta_{11} - 56 \beta_{10} - 390 \beta_{9} + 534 \beta_{8} + 334 \beta_{7} + 896 \beta_{6} - 322 \beta_{5} + 760 \beta_{4} - 526 \beta_{3} + 896 \beta_{2} + \cdots + 760$$ -88*b15 - 290*b14 + 390*b13 + 88*b12 + 378*b11 - 56*b10 - 390*b9 + 534*b8 + 334*b7 + 896*b6 - 322*b5 + 760*b4 - 526*b3 + 896*b2 - 267*b1 + 760 $$\nu^{8}$$ $$=$$ $$- 622 \beta_{15} + 311 \beta_{14} + 1842 \beta_{13} + 1544 \beta_{11} - 1842 \beta_{10} - 411 \beta_{9} + 1863 \beta_{8} + 822 \beta_{7} + 2656 \beta_{6} - 311 \beta_{5} + 711 \beta_{4} - 833 \beta_{3} + 793 \beta_{2} + \cdots + 3614$$ -622*b15 + 311*b14 + 1842*b13 + 1544*b11 - 1842*b10 - 411*b9 + 1863*b8 + 822*b7 + 2656*b6 - 311*b5 + 711*b4 - 833*b3 + 793*b2 + 793*b1 + 3614 $$\nu^{9}$$ $$=$$ $$- 1452 \beta_{15} + 4598 \beta_{14} + 1006 \beta_{13} - 1452 \beta_{12} + 3146 \beta_{11} - 4922 \beta_{10} + 1006 \beta_{9} + 3916 \beta_{7} + 3678 \beta_{6} + 2662 \beta_{5} - 2004 \beta_{4} + 6118 \beta_{3} + \cdots + 2004$$ -1452*b15 + 4598*b14 + 1006*b13 - 1452*b12 + 3146*b11 - 4922*b10 + 1006*b9 + 3916*b7 + 3678*b6 + 2662*b5 - 2004*b4 + 6118*b3 - 3678*b2 + 10299*b1 + 2004 $$\nu^{10}$$ $$=$$ $$17921 \beta_{14} - 6136 \beta_{13} - 8896 \beta_{12} - 4448 \beta_{11} - 6136 \beta_{10} + 20953 \beta_{9} - 22689 \beta_{8} - 10280 \beta_{6} + 4735 \beta_{5} - 38365 \beta_{4} + 42813 \beta_{3} + \cdots - 11328$$ 17921*b14 - 6136*b13 - 8896*b12 - 4448*b11 - 6136*b10 + 20953*b9 - 22689*b8 - 10280*b6 + 4735*b5 - 38365*b4 + 42813*b3 - 32969*b2 + 32969*b1 - 11328 $$\nu^{11}$$ $$=$$ $$21488 \beta_{15} + 34122 \beta_{14} - 61474 \beta_{13} - 21488 \beta_{12} - 55610 \beta_{11} + 15072 \beta_{10} + 61474 \beta_{9} - 97490 \beta_{8} - 46402 \beta_{7} - 120840 \beta_{6} + 53578 \beta_{5} + \cdots - 111820$$ 21488*b15 + 34122*b14 - 61474*b13 - 21488*b12 - 55610*b11 + 15072*b10 + 61474*b9 - 97490*b8 - 46402*b7 - 120840*b6 + 53578*b5 - 111820*b4 + 79730*b3 - 120840*b2 + 48745*b1 - 111820 $$\nu^{12}$$ $$=$$ $$122050 \beta_{15} - 61025 \beta_{14} - 241902 \beta_{13} - 210020 \beta_{11} + 241902 \beta_{10} + 85305 \beta_{9} - 278773 \beta_{8} - 170610 \beta_{7} - 407248 \beta_{6} + 61025 \beta_{5} + \cdots - 418238$$ 122050*b15 - 61025*b14 - 241902*b13 - 210020*b11 + 241902*b10 + 85305*b9 - 278773*b8 - 170610*b7 - 407248*b6 + 61025*b5 - 163201*b4 + 46819*b3 - 128475*b2 - 128475*b1 - 418238 $$\nu^{13}$$ $$=$$ $$300388 \beta_{15} - 673430 \beta_{14} - 206178 \beta_{13} + 300388 \beta_{12} - 373042 \beta_{11} + 763598 \beta_{10} - 206178 \beta_{9} - 557420 \beta_{7} - 631474 \beta_{6} - 400830 \beta_{5} + \cdots - 273360$$ 300388*b15 - 673430*b14 - 206178*b13 + 300388*b12 - 373042*b11 + 763598*b10 - 206178*b9 - 557420*b7 - 631474*b6 - 400830*b5 + 273360*b4 - 974578*b3 + 631474*b2 - 1436329*b1 - 273360 $$\nu^{14}$$ $$=$$ $$- 2478759 \beta_{14} + 1138040 \beta_{13} + 1631704 \beta_{12} + 815852 \beta_{11} + 1138040 \beta_{10} - 2834043 \beta_{9} + 3442451 \beta_{8} + 1578264 \beta_{6} - 1975057 \beta_{5} + \cdots + 2226908$$ -2478759*b14 + 1138040*b13 + 1631704*b12 + 815852*b11 + 1138040*b10 - 2834043*b9 + 3442451*b8 + 1578264*b6 - 1975057*b5 + 4652819*b4 - 5468671*b3 + 5020715*b2 - 5020715*b1 + 2226908 $$\nu^{15}$$ $$=$$ $$- 4055624 \beta_{15} - 4121322 \beta_{14} + 9453366 \beta_{13} + 4055624 \beta_{12} + 8176946 \beta_{11} - 2683208 \beta_{10} - 9453366 \beta_{9} + 16126950 \beta_{8} + \cdots + 16260016$$ -4055624*b15 - 4121322*b14 + 9453366*b13 + 4055624*b12 + 8176946*b11 - 2683208*b10 - 9453366*b9 + 16126950*b8 + 6770158*b7 + 17226448*b6 - 9151946*b5 + 16260016*b4 - 11163694*b3 + 17226448*b2 - 8063475*b1 + 16260016

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.15585 − 1.15585i −1.29491 + 1.29491i 2.48294 − 2.48294i −0.977855 + 0.977855i −2.02798 − 2.02798i 2.24849 + 2.24849i 0.830471 + 0.830471i −1.41701 − 1.41701i −0.977855 − 0.977855i 2.48294 + 2.48294i −1.29491 − 1.29491i 1.15585 + 1.15585i −1.41701 + 1.41701i 0.830471 − 0.830471i 2.24849 − 2.24849i −2.02798 + 2.02798i
−0.866025 + 0.500000i 0 0.500000 0.866025i 3.62374i 0 −0.866025 0.500000i 1.00000i 0 1.81187 + 3.13825i
127.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.34861i 0 −0.866025 0.500000i 1.00000i 0 0.674306 + 1.16793i
127.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.145508i 0 −0.866025 0.500000i 1.00000i 0 0.0727538 + 0.126013i
127.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 4.11786i 0 −0.866025 0.500000i 1.00000i 0 −2.05893 3.56617i
127.5 0.866025 0.500000i 0 0.500000 0.866025i 2.19147i 0 0.866025 + 0.500000i 1.00000i 0 −1.09573 1.89787i
127.6 0.866025 0.500000i 0 0.500000 0.866025i 1.24768i 0 0.866025 + 0.500000i 1.00000i 0 −0.623842 1.08053i
127.7 0.866025 0.500000i 0 0.500000 0.866025i 1.25534i 0 0.866025 + 0.500000i 1.00000i 0 0.627670 + 1.08716i
127.8 0.866025 0.500000i 0 0.500000 0.866025i 3.18381i 0 0.866025 + 0.500000i 1.00000i 0 1.59191 + 2.75726i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 4.11786i 0 −0.866025 + 0.500000i 1.00000i 0 −2.05893 + 3.56617i
1135.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.145508i 0 −0.866025 + 0.500000i 1.00000i 0 0.0727538 0.126013i
1135.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.34861i 0 −0.866025 + 0.500000i 1.00000i 0 0.674306 1.16793i
1135.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.62374i 0 −0.866025 + 0.500000i 1.00000i 0 1.81187 3.13825i
1135.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.18381i 0 0.866025 0.500000i 1.00000i 0 1.59191 2.75726i
1135.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.25534i 0 0.866025 0.500000i 1.00000i 0 0.627670 1.08716i
1135.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.24768i 0 0.866025 0.500000i 1.00000i 0 −0.623842 + 1.08053i
1135.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.19147i 0 0.866025 0.500000i 1.00000i 0 −1.09573 + 1.89787i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1135.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.i yes 16
3.b odd 2 1 1638.2.bj.h 16
13.e even 6 1 inner 1638.2.bj.i yes 16
39.h odd 6 1 1638.2.bj.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.bj.h 16 3.b odd 2 1
1638.2.bj.h 16 39.h odd 6 1
1638.2.bj.i yes 16 1.a even 1 1 trivial
1638.2.bj.i yes 16 13.e even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{16} + 50 T_{5}^{14} + 953 T_{5}^{12} + 8752 T_{5}^{10} + 40824 T_{5}^{8} + 96800 T_{5}^{6} + 111760 T_{5}^{4} + 50688 T_{5}^{2} + 1024$$ T5^16 + 50*T5^14 + 953*T5^12 + 8752*T5^10 + 40824*T5^8 + 96800*T5^6 + 111760*T5^4 + 50688*T5^2 + 1024 $$T_{11}^{16} - 12 T_{11}^{15} + 18 T_{11}^{14} + 360 T_{11}^{13} - 881 T_{11}^{12} - 9276 T_{11}^{11} + 41282 T_{11}^{10} + 41736 T_{11}^{9} - 365343 T_{11}^{8} - 211620 T_{11}^{7} + 2108476 T_{11}^{6} + 1784016 T_{11}^{5} + \cdots + 11075584$$ T11^16 - 12*T11^15 + 18*T11^14 + 360*T11^13 - 881*T11^12 - 9276*T11^11 + 41282*T11^10 + 41736*T11^9 - 365343*T11^8 - 211620*T11^7 + 2108476*T11^6 + 1784016*T11^5 - 5632240*T11^4 - 6716160*T11^3 + 9853952*T11^2 + 21086208*T11 + 11075584

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 50 T^{14} + 953 T^{12} + \cdots + 1024$$
$7$ $$(T^{4} - T^{2} + 1)^{4}$$
$11$ $$T^{16} - 12 T^{15} + 18 T^{14} + \cdots + 11075584$$
$13$ $$T^{16} - 10 T^{15} + \cdots + 815730721$$
$17$ $$T^{16} - 6 T^{15} + 108 T^{14} + \cdots + 184226329$$
$19$ $$T^{16} - 62 T^{14} + 3055 T^{12} + \cdots + 11075584$$
$23$ $$T^{16} - 12 T^{15} + \cdots + 112123183104$$
$29$ $$T^{16} + 16 T^{15} + \cdots + 37903417344$$
$31$ $$T^{16} + 330 T^{14} + \cdots + 645566464$$
$37$ $$T^{16} + 6 T^{15} - 137 T^{14} + \cdots + 897122304$$
$41$ $$T^{16} + 12 T^{15} + \cdots + 887876521984$$
$43$ $$T^{16} + 6 T^{15} + \cdots + 4269838336$$
$47$ $$T^{16} + 580 T^{14} + \cdots + 7653323063296$$
$53$ $$(T^{8} - 20 T^{7} + 64 T^{6} + 876 T^{5} + \cdots - 4239)^{2}$$
$59$ $$T^{16} + 6 T^{15} - 81 T^{14} + \cdots + 37748736$$
$61$ $$T^{16} + 2 T^{15} + \cdots + 3444868249$$
$67$ $$T^{16} + 30 T^{15} + \cdots + 102072582144$$
$71$ $$T^{16} + 12 T^{15} + \cdots + 72\!\cdots\!04$$
$73$ $$T^{16} + 678 T^{14} + \cdots + 5464074551296$$
$79$ $$(T^{8} + 8 T^{7} - 214 T^{6} + \cdots + 975168)^{2}$$
$83$ $$T^{16} + 374 T^{14} + 51953 T^{12} + \cdots + 331776$$
$89$ $$T^{16} + 30 T^{15} + \cdots + 13\!\cdots\!36$$
$97$ $$T^{16} + 24 T^{15} + \cdots + 41\!\cdots\!76$$
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