# Properties

 Label 912.2.ci.f Level $912$ Weight $2$ Character orbit 912.ci Analytic conductor $7.282$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.ci (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} - 3936 x^{10} - 1317 x^{9} + 4941 x^{8} + 21078 x^{7} - 14829 x^{6} + \cdots + 1367631$$ x^18 - 6*x^17 - 3*x^16 + 100*x^15 - 171*x^14 - 471*x^13 + 1537*x^12 + 321*x^11 - 3936*x^10 - 1317*x^9 + 4941*x^8 + 21078*x^7 - 14829*x^6 - 68589*x^5 - 45711*x^4 + 421452*x^3 - 61938*x^2 - 1219779*x + 1367631 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + (\beta_{16} + \beta_{12} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{16} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{2}) q^{7} + \beta_{6} q^{9}+O(q^{10})$$ q + b4 * q^3 + (b16 + b12 - b5 - b1) * q^5 + (-b16 + b10 + b7 - b6 - b2) * q^7 + b6 * q^9 $$q + \beta_{4} q^{3} + (\beta_{16} + \beta_{12} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{16} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{2}) q^{7} + \beta_{6} q^{9} + (\beta_{17} + \beta_{14} - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5}) q^{11} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{6} - \beta_{2}) q^{13} + (\beta_{17} - \beta_{7}) q^{15} + (\beta_{17} + \beta_{14} - \beta_{13} + 2 \beta_{11} + \beta_{9} - \beta_{3}) q^{17} + ( - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} - \beta_1 + 1) q^{19} + ( - \beta_{17} + \beta_{15} - \beta_{14} + \beta_{9} - \beta_{2} + 1) q^{21} + ( - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} - 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 1) q^{23}+ \cdots + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q + b4 * q^3 + (b16 + b12 - b5 - b1) * q^5 + (-b16 + b10 + b7 - b6 - b2) * q^7 + b6 * q^9 + (b17 + b14 - b10 - b9 - b6 + b5) * q^11 + (b15 - b13 + b12 + b6 - b2) * q^13 + (b17 - b7) * q^15 + (b17 + b14 - b13 + 2*b11 + b9 - b3) * q^17 + (-b14 - b12 + b11 + b9 + 2*b8 + b5 - b4 - b1 + 1) * q^19 + (-b17 + b15 - b14 + b9 - b2 + 1) * q^21 + (-b17 + b16 + b15 - b14 - 2*b11 + b10 + b9 + b8 - b7 + 2*b6 - b4 - b3 + b2 + b1 + 1) * q^23 + (-2*b17 + 2*b15 - b14 - b13 - b12 + 3*b10 + 3*b9 + 2*b8 + b7 - b4 - 2*b3 - b1 + 3) * q^25 + (-b8 - 1) * q^27 + (-2*b17 + b16 + b15 + b12 - b11 + 2*b10 - 2*b8 - b5 - 2*b4 - 2*b3 - b1) * q^29 + (-b17 + b16 - b14 + b13 - b10 - b8 - b6 - b5 - b3 - b2 - b1) * q^31 + (b15 + b8 + b5 - b3 + b2 - b1 + 1) * q^33 + (b16 + b15 + b13 + b12 - 2*b11 - 2*b9 + b7 + b6 - b5 + 2*b3 - 1) * q^35 + (-b17 + b16 + b15 + b12 + b11 + b7 + b6 - b5 + b4 - b3 + b2 - 2*b1) * q^37 + (-b16 - 2*b8 - b7 - b3 - 1) * q^39 + (-2*b16 + 3*b15 - 2*b13 - b12 + b11 + b9 + 2*b8 + 3*b7 - 2*b6 + 2*b5 + b3 - 2*b2 + b1 + 2) * q^41 + (2*b17 - b16 - b15 + b14 - b13 + b12 + 2*b11 - b9 - 2*b8 - b7 + b6 - b2 - 1) * q^43 + (b14 - b13) * q^45 + (-b17 + b16 - b15 + b14 + 2*b13 + 2*b12 - 2*b10 + b9 + b8 + b7 + 2*b6 + b4 + 4*b2 - 1) * q^47 + (2*b15 - 2*b13 - 2*b12 + 2*b11 - b10 + 2*b9 - b7 + b6 + 2*b5 - 2*b4 - 2*b2 + b1) * q^49 + (-b16 + b15 - 2*b13 - 2*b12 + b11 + b5) * q^51 + (-b17 + b16 + b15 - b14 - b13 + b11 + 4*b10 + 2*b9 + 4*b8 - b7 + 4*b6 + b5 - 2*b4 - 2*b3 + 2*b2 - 2*b1 + 2) * q^53 + (-2*b17 - 2*b16 - b14 - b13 - 2*b12 + b11 + b10 + b9 + b8 + 2*b7 + b6 + b4 + b3 + 2*b1 + 1) * q^55 + (-b13 - b12 + 2*b10 + b9 + b7 - b6 - b5 - b4 + b1) * q^57 + (-b17 + 2*b16 - b15 + b14 + 2*b10 + b9 + 3*b8 - 3*b6 - 2*b5 - 2*b4 + 2*b2 + 5) * q^59 + (b16 + b15 + b14 + 2*b12 + b10 - b9 - b8 - 2*b4 - b3 + b2 - b1 - 2) * q^61 + (-b15 - b8 - b5 + b4) * q^63 + (-2*b17 + b15 - b14 - b13 - b11 + 2*b9 - 2*b8 - 2*b5 + 4*b4 - 4*b2 + 2) * q^65 + (-b17 - 2*b15 + b14 + b12 + b11 + b9 - 2*b8 - b7 - b5 + 2*b4 - b1 - 2) * q^67 + (b17 - 2*b15 + b14 + 2*b12 + b11 + b10 - 2*b9 - b8 - b6 - b5 - b2 - 2) * q^69 + (b16 + b15 + 2*b14 - 3*b13 + b12 - 3*b10 - 2*b9 + 2*b8 + 3*b7 + b6 + b5 + 3*b3 - b2 + b1 + 1) * q^71 + (2*b17 - 2*b15 + 2*b14 + 2*b13 + 2*b12 - b8 - 2*b3 - b2 - 2*b1) * q^73 + (-b16 - b15 - b14 - b13 + 2*b11 + 2*b10 + b9 + b7 - b6 - b5 + b4 + b3 - 3*b2) * q^75 + (b17 + 2*b16 + b15 + b12 - b11 - 2*b10 - b9 - 2*b7 + 2*b6 - 2*b4 - b3 + 4*b2 - b1) * q^77 + (2*b17 - b16 + 2*b14 - 3*b13 - b12 - 3*b8 - 2*b7 - 4*b6 - b5 - 2*b4 + b3 - 3*b2 + 2*b1 - 4) * q^79 - b10 * q^81 + (-2*b17 + b13 - b12 - b11 + b10 + b9 - 2*b7 + 2*b6 + 2*b5 - 3*b4 - 2*b3 - b2 + 2*b1) * q^83 + (-2*b17 + 3*b16 - 2*b15 - 3*b14 + 2*b13 + 3*b12 - 3*b10 + b9 - 3*b8 - b7 - 5*b5 + 3*b4 + b3) * q^85 + (b17 - 2*b15 + 2*b13 + b12 + 2*b11 - 2*b10 - b7 - 2*b6 + 2*b4 - 2*b2 - b1) * q^87 + (-2*b17 - b16 + b15 - b14 + b13 - b12 - b11 - 4*b10 + 2*b8 + 2*b7 + 2*b6 + b5 + 2*b4 + 2*b3 - 2*b2 + b1 + 4) * q^89 + (b16 + b14 - b13 + b12 - 2*b11 + 3*b10 - b9 - 3*b8 - b5 + b4 - b3 + 4*b2 - 3*b1 + 1) * q^91 + (b17 + b16 - b15 + b13 + b11 - b10 - b5 + b4 + b3 + b2 + 1) * q^93 + (-b17 + 2*b16 - b15 - b14 + 3*b13 + 3*b12 - 2*b11 - 2*b10 - 3*b9 + 2*b7 + b6 - 2*b5 - 3*b4 + b3 + b2 - b1 - 1) * q^95 + (3*b17 - 2*b16 - b15 + 2*b14 - 2*b13 - b12 - b11 + 2*b10 - 2*b9 - b8 + b7 - b6 - b5 + 2*b4 + 2*b3 - b2 + b1) * q^97 + (b11 + b10 + b9 + b8 - b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q+O(q^{10})$$ 18 * q $$18 q + 3 q^{13} + 12 q^{17} + 3 q^{19} + 15 q^{21} + 6 q^{23} + 24 q^{25} - 9 q^{27} + 12 q^{29} + 12 q^{31} + 6 q^{33} - 36 q^{35} + 12 q^{41} + 21 q^{43} - 6 q^{45} - 24 q^{47} - 3 q^{49} - 6 q^{51} + 6 q^{53} + 12 q^{55} + 54 q^{59} - 24 q^{61} + 12 q^{63} + 36 q^{65} - 21 q^{67} + 15 q^{73} + 18 q^{75} - 60 q^{79} + 54 q^{85} + 18 q^{87} + 36 q^{89} + 24 q^{91} + 24 q^{93} - 6 q^{95} - 6 q^{97} - 6 q^{99}+O(q^{100})$$ 18 * q + 3 * q^13 + 12 * q^17 + 3 * q^19 + 15 * q^21 + 6 * q^23 + 24 * q^25 - 9 * q^27 + 12 * q^29 + 12 * q^31 + 6 * q^33 - 36 * q^35 + 12 * q^41 + 21 * q^43 - 6 * q^45 - 24 * q^47 - 3 * q^49 - 6 * q^51 + 6 * q^53 + 12 * q^55 + 54 * q^59 - 24 * q^61 + 12 * q^63 + 36 * q^65 - 21 * q^67 + 15 * q^73 + 18 * q^75 - 60 * q^79 + 54 * q^85 + 18 * q^87 + 36 * q^89 + 24 * q^91 + 24 * q^93 - 6 * q^95 - 6 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} - 3936 x^{10} - 1317 x^{9} + 4941 x^{8} + 21078 x^{7} - 14829 x^{6} + \cdots + 1367631$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 97\!\cdots\!17 \nu^{17} + \cdots - 23\!\cdots\!81 ) / 64\!\cdots\!69$$ (97336440164367328190649021117*v^17 - 436435087292171261256035826579*v^16 - 1410309213189256678718187610745*v^15 + 10311937953612325826387212451919*v^14 - 2536615790401279009030640849658*v^13 - 84222874733864968603260401897267*v^12 + 124430500775759897783204308402491*v^11 + 252815364924229311203300039578773*v^10 - 549539512648041799115019568521278*v^9 - 546947406693265229202705520317045*v^8 + 179307112840199445605335465335630*v^7 + 4108851165535969466170887676890066*v^6 - 299570589966749065155736009480350*v^5 - 13911987476774468961626525516476188*v^4 - 11299710470531853578933626958189112*v^3 + 35163653702919517180130896562152743*v^2 + 66741026095948078126623380414683815*v - 236757217672847899677156101565120081) / 64889507508191236802088482696163369 $$\beta_{3}$$ $$=$$ $$( 12\!\cdots\!28 \nu^{17} + \cdots - 24\!\cdots\!68 ) / 64\!\cdots\!69$$ (128866953451703021987146669928*v^17 - 1161329720705882133342800814958*v^16 + 925401499089787652486550385425*v^15 + 17740705570566790730684527746371*v^14 - 46057671848396868357846123584374*v^13 - 72934621587381651007797198404505*v^12 + 347264573003311340905897370891288*v^11 + 12784465449707597048194532761781*v^10 - 923132938825022044454158656930774*v^9 - 367836570991481795163208512232638*v^8 + 1629565245239206613529711971626002*v^7 + 4831010944804023162642852223223871*v^6 - 4406041959028015653233311594391868*v^5 - 16968192812921247193795597112943588*v^4 - 5266868279877094144894488922282416*v^3 + 95076321140797092172191901817661309*v^2 - 32480491794641964425233134268838361*v - 248869087867947874200607815834477168) / 64889507508191236802088482696163369 $$\beta_{4}$$ $$=$$ $$( 18\!\cdots\!57 \nu^{17} + \cdots + 21\!\cdots\!99 ) / 64\!\cdots\!69$$ (184410549204258523036849788157*v^17 - 547871806121317289525303665138*v^16 - 1233213633639964795980823063129*v^15 + 4498523070557069126343017502595*v^14 + 4138221659680904949527666313512*v^13 + 22914271655599888638902558924501*v^12 - 117412393956707247865792776201881*v^11 - 169980623952548286277723385135053*v^10 + 883284976909686931719050673577430*v^9 + 181892018495183206781152888286454*v^8 - 1180809603774687456926040824802060*v^7 - 3124481240357495496193671212345145*v^6 + 2744327059147886376892560025727982*v^5 + 14155765136826715776156463662881325*v^4 - 1491160064495183797687618644429114*v^3 - 13810755941525880750106251710166294*v^2 - 56221285496123647964524685081718861*v + 210529275608145388187854429199406099) / 64889507508191236802088482696163369 $$\beta_{5}$$ $$=$$ $$( - 22\!\cdots\!40 \nu^{17} + \cdots + 18\!\cdots\!98 ) / 64\!\cdots\!69$$ (-223763635972848348007671737940*v^17 - 731429549818403086323451942769*v^16 + 9226494133524370332035644914802*v^15 - 6896662535721599745599425313619*v^14 - 93452862130407651621039377604491*v^13 + 181790939388419476474014885006181*v^12 + 243740526458689661695590538663521*v^11 - 815314922087339327240629491833714*v^10 - 200319908678130712656590326282388*v^9 + 527916964398368262392719531803195*v^8 + 3650118308066397665232792554291250*v^7 - 3097782300294133787239303930750908*v^6 - 13155222261347610811293514189911426*v^5 - 8824782416873350921011653144840745*v^4 + 50499558338470167494148295235658708*v^3 + 46513509239272694770939752719015778*v^2 - 271980157558661762996300695044517386*v + 180316241185612557838639873612210398) / 64889507508191236802088482696163369 $$\beta_{6}$$ $$=$$ $$( 36\!\cdots\!58 \nu^{17} + \cdots + 16\!\cdots\!11 ) / 64\!\cdots\!69$$ (366441140915945537697647846858*v^17 - 1478195617849484073036945058243*v^16 - 2073302615581201457937150419861*v^15 + 19328363113845863753563976610845*v^14 - 15222014211634379264196712822024*v^13 - 49313263673018864943813087266759*v^12 + 52359756454883978739874047221414*v^11 - 7355672413067778750713455126588*v^10 + 441784880346870724943973457640209*v^9 - 526344316840625835572900866499916*v^8 - 435100165156684552175470201019997*v^7 - 180707279443599617410998547376064*v^6 + 5367697586903115182368620600391977*v^5 + 3692715138381336651621456332733744*v^4 - 19779269178910737103880153633152593*v^3 + 31215700580457772751155686727338384*v^2 - 44730350655702340271858659328469255*v + 160853955261596127836740250422749711) / 64889507508191236802088482696163369 $$\beta_{7}$$ $$=$$ $$( - 41\!\cdots\!50 \nu^{17} + \cdots + 38\!\cdots\!37 ) / 64\!\cdots\!69$$ (-411820212356401560977660675150*v^17 + 4723737729517095906602644825131*v^16 - 6921345360081046849621560819372*v^15 - 59250367193852525275485741273587*v^14 + 184550140727347413051232378643742*v^13 + 163320693135767627660794241357928*v^12 - 1006361056010876447935373989036376*v^11 + 4629917905117158115599629174697*v^10 + 1835323518582222296847361378599066*v^9 + 3295400904837117383061864212595360*v^8 - 4806742738119393894051191211125118*v^7 - 13998933478140517444715768774204172*v^6 + 7831710764724190236478907530289880*v^5 + 53493195414827831475114803210261274*v^4 + 34609678134499276668663587302842486*v^3 - 281871253299705865831638262663030332*v^2 + 189885082142350170758836933813513323*v + 389520534298600793547264148921443537) / 64889507508191236802088482696163369 $$\beta_{8}$$ $$=$$ $$( - 21\!\cdots\!87 \nu^{17} + \cdots - 68\!\cdots\!41 ) / 32\!\cdots\!61$$ (-21358157874954387*v^17 + 77903753943763154*v^16 + 157794005839271082*v^15 - 1043988330015655896*v^14 + 410790551332969133*v^13 + 2840933642425807032*v^12 - 1146902260107760230*v^11 + 1180915341374941364*v^10 - 28743684830680112700*v^9 + 21335524254141352509*v^8 + 41845300031249174727*v^7 - 2130654551100964551*v^6 - 194654920955344120728*v^5 - 165701214956773379037*v^4 + 799371241749399120489*v^3 - 1604342215780527340215*v^2 + 1930095780959745732087*v - 6818051868404325484041) / 3228423079708188707661 $$\beta_{9}$$ $$=$$ $$( - 15\!\cdots\!92 \nu^{17} + \cdots + 68\!\cdots\!91 ) / 17\!\cdots\!37$$ (-15097067273087401316102569292*v^17 + 18377891514248357482980370234*v^16 + 376825185131588734522755711165*v^15 - 964119610097543578076458921307*v^14 - 2966801090021774405115102949904*v^13 + 10833821840098718858741375691870*v^12 + 6193957033705818166825734246850*v^11 - 43489862123720229145732201074686*v^10 - 11480019237761937341099569169979*v^9 + 56540057497106184304084206164481*v^8 + 189499643148239368777415974246443*v^7 - 148079759278319892297459609035355*v^6 - 724440548545881178722458210345454*v^5 - 187525150004883176995941135730149*v^4 + 2473809479047541724352234287257034*v^3 + 1210790943230007663909953591968935*v^2 - 11769442997458557977276216944534146*v + 6816367130237007787011078719647191) / 1753770473194357751407796829626037 $$\beta_{10}$$ $$=$$ $$( - 57\!\cdots\!91 \nu^{17} + \cdots + 19\!\cdots\!65 ) / 64\!\cdots\!69$$ (-570158591707351016080718854091*v^17 + 2242326171173063501959631788314*v^16 + 5694099207337725193926188110303*v^15 - 38851262851444446225265002134936*v^14 + 7121949359685756561873262267119*v^13 + 221423600876662549694368585403212*v^12 - 198353977019880299615346371708402*v^11 - 578537561578599457439759294102631*v^10 + 243381303267227291568940867999759*v^9 + 1701069963570151024245274030298124*v^8 + 1369043861331023960447204090699424*v^7 - 5438847867206607418968586290998637*v^6 - 10752707870377617424195254743915991*v^5 + 11086819027799081817694979836943973*v^4 + 59053194989085503455857498471889332*v^3 - 69525186953734036150655182264342575*v^2 - 103170819094004321436814063689620862*v + 197880417471399388698253204917172665) / 64889507508191236802088482696163369 $$\beta_{11}$$ $$=$$ $$( - 66\!\cdots\!55 \nu^{17} + \cdots + 58\!\cdots\!23 ) / 64\!\cdots\!69$$ (-661431025149764313532736167455*v^17 + 4925129389049861684160111055867*v^16 - 1103874554075180168343712319881*v^15 - 72621210796017693045743305355973*v^14 + 144254037194769934930016443186360*v^13 + 322723152647197340002537318375640*v^12 - 978483080354210433739293296901528*v^11 - 645643168882423834669325973101225*v^10 + 2022684829001894873558395963568156*v^9 + 4602042841143544636416384817436016*v^8 - 3003130160556929739676910914434285*v^7 - 19260416972681105294133038255635674*v^6 - 1928726614664627410538018348422707*v^5 + 60244199277458311802125392707243820*v^4 + 87463724716984972222171421914347213*v^3 - 301229291218930418661963097475898954*v^2 + 6046903579661463301219139890243212*v + 586497773618389248565299716150863623) / 64889507508191236802088482696163369 $$\beta_{12}$$ $$=$$ $$( 68\!\cdots\!21 \nu^{17} + \cdots - 50\!\cdots\!22 ) / 64\!\cdots\!69$$ (685316935833707184964267389621*v^17 - 5373948727874619782193124717925*v^16 + 3092701110573235308332291334038*v^15 + 73204107453939000592186302951102*v^14 - 174979748745348801171444604544480*v^13 - 253796155079189944359059638668490*v^12 + 1078474847864839718314775828778060*v^11 + 156541931148080412289505967669673*v^10 - 2082091535797533128826029875812685*v^9 - 3354868109494573278886748308331661*v^8 + 4316802600587325354527675474318817*v^7 + 15251567337955278845106974952831999*v^6 - 3820180577053438339811542092810852*v^5 - 63090983376036100597161984244758663*v^4 - 60876917173600354207703888147578224*v^3 + 328671344146455742581103689584423325*v^2 - 129000476320222983227440044950694843*v - 509112634301787493572421751912546622) / 64889507508191236802088482696163369 $$\beta_{13}$$ $$=$$ $$( 19\!\cdots\!65 \nu^{17} + \cdots - 13\!\cdots\!54 ) / 17\!\cdots\!37$$ (19471654801248355490511946565*v^17 - 26323761968469320130924510251*v^16 - 467993269668883513951373191215*v^15 + 1282146510405197504921650513262*v^14 + 3331905775632202251669704016307*v^13 - 13807034517106062505443532254036*v^12 - 3377926449921251252747524701838*v^11 + 50921546243027901657349064401981*v^10 - 1182198223089880065543206815890*v^9 - 60694211957361390647014816549875*v^8 - 213636585072159450297351888201324*v^7 + 291936574744477609753703797877007*v^6 + 779095852774733111804470932456138*v^5 - 81861572608160827842783051498015*v^4 - 3330314895698629891961983611531336*v^3 - 595505926206770420484912080804823*v^2 + 16427869288835331347676741713928489*v - 13544763891676094909918157363412254) / 1753770473194357751407796829626037 $$\beta_{14}$$ $$=$$ $$( 78\!\cdots\!46 \nu^{17} + \cdots - 83\!\cdots\!67 ) / 64\!\cdots\!69$$ (788610273208330535505965253146*v^17 - 5485914181460586609583929239746*v^16 - 2111758542413347813605279431251*v^15 + 89651758967586415868841147241748*v^14 - 128576174391122959632340178037553*v^13 - 481171687269250646951451991921555*v^12 + 1051281140824842409042140321714398*v^11 + 1163040243628978093938920018101313*v^10 - 2445481838298479930124832161054997*v^9 - 5092694858788866502835629232681325*v^8 + 1903040669469964538641822896425385*v^7 + 22841335177060677887192386355311896*v^6 + 7087869901630734234658614874917483*v^5 - 71617199110082885083642733689939497*v^4 - 126882804195229544472408800046143043*v^3 + 331283999194431583061191763162722788*v^2 + 180986478082023966137798327404624323*v - 838884943971190541411071136594114967) / 64889507508191236802088482696163369 $$\beta_{15}$$ $$=$$ $$( 23\!\cdots\!44 \nu^{17} + \cdots - 17\!\cdots\!25 ) / 17\!\cdots\!37$$ (23460399495681671919913522244*v^17 - 56548083392689717270012092613*v^16 - 452363703799272892131851290128*v^15 + 1663441523316157886315443533839*v^14 + 2294678440052527737467955017627*v^13 - 14487456348372891847222538660910*v^12 + 2610461864442732500066034860330*v^11 + 46423619644079469350413253012663*v^10 - 12499909439219432983803227343078*v^9 - 68846434606332729106059717847266*v^8 - 158036755889688236173881952801704*v^7 + 322850939102062854201903752597646*v^6 + 583534274873633698605603370102563*v^5 - 267006557856227277572806552955040*v^4 - 3488665805894127784407240765197529*v^3 + 1371246790735085301139298716077030*v^2 + 13237915990846247983782297680521815*v - 17142610729471604461829117130959925) / 1753770473194357751407796829626037 $$\beta_{16}$$ $$=$$ $$( 13\!\cdots\!64 \nu^{17} + \cdots - 78\!\cdots\!81 ) / 87\!\cdots\!53$$ (1357978197458464*v^17 - 2532960330119133*v^16 - 29508850202156292*v^15 + 87606876899573812*v^14 + 195101586937235385*v^13 - 856232064694517097*v^12 - 217213081600954043*v^11 + 3048902546662177836*v^10 + 183599180193880410*v^9 - 3983134002470237862*v^8 - 12109637760464529855*v^7 + 13820947137379533123*v^6 + 44071240687622157540*v^5 + 4781932780341644964*v^4 - 199921517214452674857*v^3 - 16411194554076240813*v^2 + 888386062782657100122*v - 789461589531939006681) / 87254677829951046153 $$\beta_{17}$$ $$=$$ $$( 31\!\cdots\!36 \nu^{17} + \cdots - 21\!\cdots\!33 ) / 17\!\cdots\!37$$ (31854739974893043095261657736*v^17 - 107665498167991139072541393190*v^16 - 490935035656504199535321169572*v^15 + 2442572157358682896971071940066*v^14 + 1273543130202696726477026888477*v^13 - 18323777795522113840559959649445*v^12 + 10689639287582156250752663295660*v^11 + 54074673343051521830399149775741*v^10 - 25680299953826209082891008093521*v^9 - 113140471971812035970325295912515*v^8 - 177809592670295602648129884203553*v^7 + 519124043967727720044763098141390*v^6 + 757291584292335595250417449900098*v^5 - 891639881177048139453831322500963*v^4 - 4615386265905742277799890276486961*v^3 + 3742845998572276450562746704711060*v^2 + 13448298901697338549882971843434952*v - 21074772025278815607391611003765633) / 1753770473194357751407796829626037
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$- \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta _1 + 2$$ -b15 + b14 - b13 - b10 - b9 - b8 + b7 - 3*b6 - b5 + 2*b4 + 2*b3 - b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$2 \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 5 \beta _1 - 4$$ 2*b17 - b16 - b15 - 3*b13 - 2*b12 - b11 + 2*b10 - 4*b9 - 2*b8 + b7 + 2*b6 + b5 - 2*b4 + 4*b3 + 2*b2 + 5*b1 - 4 $$\nu^{4}$$ $$=$$ $$2 \beta_{17} - 4 \beta_{16} + \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} - 4 \beta_{11} - 11 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 14 \beta_{3} - 9 \beta_{2} + 9 \beta _1 + 5$$ 2*b17 - 4*b16 + b15 + 3*b14 - 6*b13 - 10*b12 - 4*b11 - 11*b10 - 5*b9 - 5*b8 + 3*b7 - 11*b6 + 2*b5 + 3*b4 + 14*b3 - 9*b2 + 9*b1 + 5 $$\nu^{5}$$ $$=$$ $$15 \beta_{17} - 8 \beta_{16} - 7 \beta_{15} + 8 \beta_{14} - 14 \beta_{13} - 14 \beta_{12} - 6 \beta_{11} + 10 \beta_{10} - 17 \beta_{9} - 4 \beta_{8} + 15 \beta_{7} + 26 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} + 23 \beta_{3} - 4 \beta_{2} + 21 \beta _1 - 44$$ 15*b17 - 8*b16 - 7*b15 + 8*b14 - 14*b13 - 14*b12 - 6*b11 + 10*b10 - 17*b9 - 4*b8 + 15*b7 + 26*b6 + 10*b5 - 18*b4 + 23*b3 - 4*b2 + 21*b1 - 44 $$\nu^{6}$$ $$=$$ $$11 \beta_{17} - 20 \beta_{16} - 7 \beta_{15} + 27 \beta_{14} - 3 \beta_{13} - 50 \beta_{12} - 13 \beta_{11} - 19 \beta_{10} - 13 \beta_{9} - 84 \beta_{8} + 15 \beta_{7} - 46 \beta_{6} + 19 \beta_{5} - 44 \beta_{4} + 50 \beta_{3} - 55 \beta_{2} + 7 \beta _1 - 30$$ 11*b17 - 20*b16 - 7*b15 + 27*b14 - 3*b13 - 50*b12 - 13*b11 - 19*b10 - 13*b9 - 84*b8 + 15*b7 - 46*b6 + 19*b5 - 44*b4 + 50*b3 - 55*b2 + 7*b1 - 30 $$\nu^{7}$$ $$=$$ $$13 \beta_{17} + 59 \beta_{16} - 20 \beta_{15} + 16 \beta_{14} + \beta_{13} - 17 \beta_{12} - 32 \beta_{11} + 192 \beta_{10} + 24 \beta_{9} - 38 \beta_{8} + 36 \beta_{7} + 242 \beta_{6} + 49 \beta_{5} - 190 \beta_{4} - \beta_{3} - 16 \beta_{2} - 27 \beta _1 - 292$$ 13*b17 + 59*b16 - 20*b15 + 16*b14 + b13 - 17*b12 - 32*b11 + 192*b10 + 24*b9 - 38*b8 + 36*b7 + 242*b6 + 49*b5 - 190*b4 - b3 - 16*b2 - 27*b1 - 292 $$\nu^{8}$$ $$=$$ $$- 121 \beta_{17} + 135 \beta_{16} - 144 \beta_{15} + 78 \beta_{14} + 249 \beta_{13} + 44 \beta_{12} + 119 \beta_{11} - 27 \beta_{10} + 186 \beta_{9} - 503 \beta_{8} - 103 \beta_{7} - 332 \beta_{6} - 61 \beta_{5} - 379 \beta_{4} + \cdots - 256$$ -121*b17 + 135*b16 - 144*b15 + 78*b14 + 249*b13 + 44*b12 + 119*b11 - 27*b10 + 186*b9 - 503*b8 - 103*b7 - 332*b6 - 61*b5 - 379*b4 - 33*b3 - 501*b2 - 340*b1 - 256 $$\nu^{9}$$ $$=$$ $$- 208 \beta_{17} + 952 \beta_{16} - 496 \beta_{15} - 327 \beta_{14} + 693 \beta_{13} + 688 \beta_{12} + 137 \beta_{11} + 1478 \beta_{10} + 836 \beta_{9} + 306 \beta_{8} - 420 \beta_{7} + 1388 \beta_{6} - 296 \beta_{5} + \cdots - 1434$$ -208*b17 + 952*b16 - 496*b15 - 327*b14 + 693*b13 + 688*b12 + 137*b11 + 1478*b10 + 836*b9 + 306*b8 - 420*b7 + 1388*b6 - 296*b5 - 632*b4 - 859*b3 + 128*b2 - 965*b1 - 1434 $$\nu^{10}$$ $$=$$ $$- 1427 \beta_{17} + 1967 \beta_{16} - 1439 \beta_{15} - 416 \beta_{14} + 2290 \beta_{13} + 1723 \beta_{12} + 1723 \beta_{11} - 240 \beta_{10} + 2304 \beta_{9} - 1922 \beta_{8} - 2532 \beta_{7} - 2302 \beta_{6} + \cdots - 298$$ -1427*b17 + 1967*b16 - 1439*b15 - 416*b14 + 2290*b13 + 1723*b12 + 1723*b11 - 240*b10 + 2304*b9 - 1922*b8 - 2532*b7 - 2302*b6 - 2339*b5 - 1579*b4 - 2299*b3 - 2347*b2 - 3105*b1 - 298 $$\nu^{11}$$ $$=$$ $$- 1510 \beta_{17} + 7056 \beta_{16} - 4053 \beta_{15} - 3543 \beta_{14} + 4563 \beta_{13} + 6119 \beta_{12} + 2858 \beta_{11} + 7194 \beta_{10} + 6876 \beta_{9} + 9454 \beta_{8} - 5500 \beta_{7} + 10396 \beta_{6} + \cdots - 1030$$ -1510*b17 + 7056*b16 - 4053*b15 - 3543*b14 + 4563*b13 + 6119*b12 + 2858*b11 + 7194*b10 + 6876*b9 + 9454*b8 - 5500*b7 + 10396*b6 - 5311*b5 + 2330*b4 - 7410*b3 + 6564*b2 - 6142*b1 - 1030 $$\nu^{12}$$ $$=$$ $$- 6211 \beta_{17} + 7531 \beta_{16} - 6772 \beta_{15} - 2826 \beta_{14} + 9171 \beta_{13} + 10096 \beta_{12} + 14735 \beta_{11} - 7096 \beta_{10} + 11228 \beta_{9} - 738 \beta_{8} - 19284 \beta_{7} + \cdots + 9582$$ -6211*b17 + 7531*b16 - 6772*b15 - 2826*b14 + 9171*b13 + 10096*b12 + 14735*b11 - 7096*b10 + 11228*b9 - 738*b8 - 19284*b7 - 12523*b6 - 19949*b5 + 2155*b4 - 15157*b3 - 3181*b2 - 11015*b1 + 9582 $$\nu^{13}$$ $$=$$ $$- 755 \beta_{17} + 22457 \beta_{16} - 12245 \beta_{15} - 14276 \beta_{14} + 5827 \beta_{13} + 22486 \beta_{12} + 14605 \beta_{11} + 27444 \beta_{10} + 22155 \beta_{9} + 96934 \beta_{8} - 21933 \beta_{7} + \cdots + 37664$$ -755*b17 + 22457*b16 - 12245*b15 - 14276*b14 + 5827*b13 + 22486*b12 + 14605*b11 + 27444*b10 + 22155*b9 + 96934*b8 - 21933*b7 + 78182*b6 - 26513*b5 + 61316*b4 - 28687*b3 + 87950*b2 - 12234*b1 + 37664 $$\nu^{14}$$ $$=$$ $$- 11569 \beta_{17} - 47250 \beta_{16} + 33870 \beta_{15} + 4347 \beta_{14} - 13107 \beta_{13} + 8345 \beta_{12} + 60185 \beta_{11} - 71460 \beta_{10} - 22863 \beta_{9} + 21253 \beta_{8} - 60457 \beta_{7} + \cdots + 97034$$ -11569*b17 - 47250*b16 + 33870*b15 + 4347*b14 - 13107*b13 + 8345*b12 + 60185*b11 - 71460*b10 - 22863*b9 + 21253*b8 - 60457*b7 - 59342*b6 - 64726*b5 + 58148*b4 - 29238*b3 + 31596*b2 + 23969*b1 + 97034 $$\nu^{15}$$ $$=$$ $$25748 \beta_{17} - 116042 \beta_{16} + 136559 \beta_{15} - 31065 \beta_{14} - 93582 \beta_{13} - 23678 \beta_{12} - 91438 \beta_{11} + 50984 \beta_{10} - 95947 \beta_{9} + 676734 \beta_{8} + \cdots + 373716$$ 25748*b17 - 116042*b16 + 136559*b15 - 31065*b14 - 93582*b13 - 23678*b12 - 91438*b11 + 50984*b10 - 95947*b9 + 676734*b8 + 121005*b7 + 490310*b6 + 97468*b5 + 555544*b4 + 48227*b3 + 670544*b2 + 118756*b1 + 373716 $$\nu^{16}$$ $$=$$ $$- 47696 \beta_{17} - 956557 \beta_{16} + 854071 \beta_{15} + 63757 \beta_{14} - 322373 \beta_{13} - 223460 \beta_{12} - 188606 \beta_{11} - 531198 \beta_{10} - 759699 \beta_{9} - 250835 \beta_{8} + \cdots + 372131$$ -47696*b17 - 956557*b16 + 854071*b15 + 63757*b14 - 322373*b13 - 223460*b12 - 188606*b11 - 531198*b10 - 759699*b9 - 250835*b8 + 314544*b7 - 560962*b6 + 292309*b5 + 158903*b4 + 285251*b3 - 94753*b2 + 576507*b1 + 372131 $$\nu^{17}$$ $$=$$ $$- 150682 \beta_{17} - 1835490 \beta_{16} + 2049249 \beta_{15} - 317313 \beta_{14} - 557694 \beta_{13} - 635779 \beta_{12} - 2201278 \beta_{11} - 377322 \beta_{10} - 1456380 \beta_{9} + \cdots + 2261204$$ -150682*b17 - 1835490*b16 + 2049249*b15 - 317313*b14 - 557694*b13 - 635779*b12 - 2201278*b11 - 377322*b10 - 1456380*b9 + 3142216*b8 + 2498588*b7 + 2057080*b6 + 2379950*b5 + 3114986*b4 + 1326225*b3 + 3407826*b2 + 1112882*b1 + 2261204

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 1.12178 + 1.48644i −0.489413 − 1.79677i −1.33811 + 1.29514i 1.12178 − 1.48644i −0.489413 + 1.79677i −1.33811 − 1.29514i 2.46004 + 0.909722i −2.61262 − 0.231530i 2.26592 − 1.32098i 2.46004 − 0.909722i −2.61262 + 0.231530i 2.26592 + 1.32098i 1.98709 − 0.839158i 1.76137 + 1.24510i −2.15607 − 0.0639189i 1.98709 + 0.839158i 1.76137 − 1.24510i −2.15607 + 0.0639189i
0 −0.939693 + 0.342020i 0 −0.189533 + 1.07490i 0 1.46125 0.843655i 0 0.766044 0.642788i 0
79.2 0 −0.939693 + 0.342020i 0 −0.0537040 + 0.304570i 0 −4.22544 + 2.43956i 0 0.766044 0.642788i 0
79.3 0 −0.939693 + 0.342020i 0 0.590533 3.34908i 0 1.12990 0.652348i 0 0.766044 0.642788i 0
127.1 0 −0.939693 0.342020i 0 −0.189533 1.07490i 0 1.46125 + 0.843655i 0 0.766044 + 0.642788i 0
127.2 0 −0.939693 0.342020i 0 −0.0537040 0.304570i 0 −4.22544 2.43956i 0 0.766044 + 0.642788i 0
127.3 0 −0.939693 0.342020i 0 0.590533 + 3.34908i 0 1.12990 + 0.652348i 0 0.766044 + 0.642788i 0
223.1 0 0.173648 0.984808i 0 −0.718121 0.602575i 0 0.983288 0.567702i 0 −0.939693 0.342020i 0
223.2 0 0.173648 0.984808i 0 −0.345737 0.290108i 0 −0.993418 + 0.573550i 0 −0.939693 0.342020i 0
223.3 0 0.173648 0.984808i 0 2.59595 + 2.17826i 0 −2.88040 + 1.66300i 0 −0.939693 0.342020i 0
319.1 0 0.173648 + 0.984808i 0 −0.718121 + 0.602575i 0 0.983288 + 0.567702i 0 −0.939693 + 0.342020i 0
319.2 0 0.173648 + 0.984808i 0 −0.345737 + 0.290108i 0 −0.993418 0.573550i 0 −0.939693 + 0.342020i 0
319.3 0 0.173648 + 0.984808i 0 2.59595 2.17826i 0 −2.88040 1.66300i 0 −0.939693 + 0.342020i 0
751.1 0 0.766044 0.642788i 0 −3.87454 1.41022i 0 3.15920 + 1.82397i 0 0.173648 0.984808i 0
751.2 0 0.766044 0.642788i 0 −1.03170 0.375509i 0 −0.450834 0.260289i 0 0.173648 0.984808i 0
751.3 0 0.766044 0.642788i 0 3.02685 + 1.10168i 0 1.81645 + 1.04873i 0 0.173648 0.984808i 0
895.1 0 0.766044 + 0.642788i 0 −3.87454 + 1.41022i 0 3.15920 1.82397i 0 0.173648 + 0.984808i 0
895.2 0 0.766044 + 0.642788i 0 −1.03170 + 0.375509i 0 −0.450834 + 0.260289i 0 0.173648 + 0.984808i 0
895.3 0 0.766044 + 0.642788i 0 3.02685 1.10168i 0 1.81645 1.04873i 0 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.f yes 18
4.b odd 2 1 912.2.ci.e 18
19.f odd 18 1 912.2.ci.e 18
76.k even 18 1 inner 912.2.ci.f yes 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.e 18 4.b odd 2 1
912.2.ci.e 18 19.f odd 18 1
912.2.ci.f yes 18 1.a even 1 1 trivial
912.2.ci.f yes 18 76.k even 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{18} - 12 T_{5}^{16} + 38 T_{5}^{15} + 48 T_{5}^{14} - 486 T_{5}^{13} + 2238 T_{5}^{12} - 306 T_{5}^{11} - 10332 T_{5}^{10} + 332 T_{5}^{9} + 61155 T_{5}^{8} + 159078 T_{5}^{7} + 239065 T_{5}^{6} + 237444 T_{5}^{5} + \cdots + 576$$ T5^18 - 12*T5^16 + 38*T5^15 + 48*T5^14 - 486*T5^13 + 2238*T5^12 - 306*T5^11 - 10332*T5^10 + 332*T5^9 + 61155*T5^8 + 159078*T5^7 + 239065*T5^6 + 237444*T5^5 + 158904*T5^4 + 70032*T5^3 + 21312*T5^2 + 4320*T5 + 576 $$T_{7}^{18} - 30 T_{7}^{16} + 675 T_{7}^{14} - 855 T_{7}^{13} - 5477 T_{7}^{12} + 10431 T_{7}^{11} + 33117 T_{7}^{10} - 120555 T_{7}^{9} + 85032 T_{7}^{8} + 130302 T_{7}^{7} - 169952 T_{7}^{6} - 139707 T_{7}^{5} + \cdots + 34347$$ T7^18 - 30*T7^16 + 675*T7^14 - 855*T7^13 - 5477*T7^12 + 10431*T7^11 + 33117*T7^10 - 120555*T7^9 + 85032*T7^8 + 130302*T7^7 - 169952*T7^6 - 139707*T7^5 + 302658*T7^4 - 59508*T7^3 - 108459*T7^2 + 18297*T7 + 34347

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$(T^{6} + T^{3} + 1)^{3}$$
$5$ $$T^{18} - 12 T^{16} + 38 T^{15} + \cdots + 576$$
$7$ $$T^{18} - 30 T^{16} + 675 T^{14} + \cdots + 34347$$
$11$ $$T^{18} - 33 T^{16} + 813 T^{14} + \cdots + 1728$$
$13$ $$T^{18} - 3 T^{17} - 18 T^{16} + \cdots + 162867$$
$17$ $$T^{18} - 12 T^{17} + \cdots + 268697664$$
$19$ $$T^{18} - 3 T^{17} + \cdots + 322687697779$$
$23$ $$T^{18} - 6 T^{17} + \cdots + 9032809152$$
$29$ $$T^{18} - 12 T^{17} + \cdots + 122712760512$$
$31$ $$T^{18} - 12 T^{17} + \cdots + 544949574849$$
$37$ $$T^{18} + 282 T^{16} + \cdots + 13733220843$$
$41$ $$T^{18} - 12 T^{17} + \cdots + 108865728$$
$43$ $$T^{18} - 21 T^{17} + \cdots + 11432149083$$
$47$ $$T^{18} + 24 T^{17} + \cdots + 7316338368$$
$53$ $$T^{18} - 6 T^{17} + \cdots + 3614768832$$
$59$ $$T^{18} - 54 T^{17} + \cdots + 7831445534784$$
$61$ $$T^{18} + 24 T^{17} + \cdots + 573075721$$
$67$ $$T^{18} + 21 T^{17} + \cdots + 7453904896$$
$71$ $$T^{18} + 378 T^{16} + \cdots + 4653477324864$$
$73$ $$T^{18} + \cdots + 855648672669729$$
$79$ $$T^{18} + 60 T^{17} + \cdots + 11\!\cdots\!01$$
$83$ $$T^{18} - 381 T^{16} + \cdots + 26\!\cdots\!88$$
$89$ $$T^{18} - 36 T^{17} + \cdots + 123685470912$$
$97$ $$T^{18} + 6 T^{17} + \cdots + 56816030267643$$