# Properties

 Label 81.9.b.b Level $81$ Weight $9$ Character orbit 81.b Analytic conductor $32.998$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(80,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.80");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

 Self dual: no Analytic conductor: $$32.9976674150$$ 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: $$x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24$$ x^16 + 1024*x^14 + 419701*x^12 + 88203292*x^10 + 10121979748*x^8 + 629108384896*x^6 + 20221245756928*x^4 + 305638522445824*x^2 + 1623834669236224 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}\cdot 3^{84}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} + (\beta_1 - 128) q^{4} + ( - \beta_{9} - 2 \beta_{8}) q^{5} + (\beta_{3} + 231) q^{7} + ( - \beta_{11} - 101 \beta_{8}) q^{8}+O(q^{10})$$ q + b8 * q^2 + (b1 - 128) * q^4 + (-b9 - 2*b8) * q^5 + (b3 + 231) * q^7 + (-b11 - 101*b8) * q^8 $$q + \beta_{8} q^{2} + (\beta_1 - 128) q^{4} + ( - \beta_{9} - 2 \beta_{8}) q^{5} + (\beta_{3} + 231) q^{7} + ( - \beta_{11} - 101 \beta_{8}) q^{8} + ( - \beta_{4} - 3 \beta_1 + 672) q^{10} + ( - \beta_{12} + \beta_{11} + \cdots + 112 \beta_{8}) q^{11}+ \cdots + (236 \beta_{15} + \cdots + 1566103 \beta_{8}) q^{98}+O(q^{100})$$ q + b8 * q^2 + (b1 - 128) * q^4 + (-b9 - 2*b8) * q^5 + (b3 + 231) * q^7 + (-b11 - 101*b8) * q^8 + (-b4 - 3*b1 + 672) * q^10 + (-b12 + b11 - 3*b9 + 112*b8) * q^11 + (3*b3 + b2 - 11*b1 + 3992) * q^13 + (-b13 + 3*b11 + 317*b8) * q^14 + (b5 + 8*b3 + b2 - 84*b1 + 5947) * q^16 + (-b14 + b12 - 3*b11 + 25*b9 - 162*b8) * q^17 + (b7 + 2*b4 - 9*b3 + 63*b1 + 11567) * q^19 + (-2*b14 - b13 + 4*b11 + b10 + 56*b9 + 946*b8) * q^20 + (-2*b7 - b6 - 5*b5 - 5*b4 - 7*b3 - 5*b2 + 322*b1 - 43252) * q^22 + (b15 - 4*b14 + 2*b13 - 2*b12 + 10*b11 + 101*b9 + 714*b8) * q^23 + (-b7 - b6 + 6*b5 + 9*b4 - 5*b3 - 5*b2 - 339*b1 - 33857) * q^25 + (-b15 - 4*b14 - 10*b13 - 9*b12 + 55*b11 - 2*b10 + 48*b9 + 6815*b8) * q^26 + (3*b7 - 2*b6 - 5*b5 - 2*b4 - 154*b3 - 23*b2 + 876*b1 - 62361) * q^28 + (-b15 - 7*b14 - 14*b13 + 7*b12 - 39*b11 + 9*b10 + 202*b9 - 154*b8) * q^29 + (-b7 - 4*b6 - 6*b5 + 56*b4 - 62*b3 + 20*b2 + 301*b1 + 35678) * q^31 + (5*b15 + 4*b14 - 15*b13 - 49*b12 + 47*b11 - 5*b10 + 84*b9 + 52*b8) * q^32 + (b7 - 6*b6 + 12*b5 + 75*b4 + 120*b3 - 4*b2 - 700*b1 + 64260) * q^34 + (-5*b15 - 14*b14 + 4*b13 + 2*b12 - 50*b11 - 14*b10 - 315*b9 - 1900*b8) * q^35 + (2*b7 - 9*b6 + 10*b5 - 101*b4 - 180*b3 + 20*b2 - 1585*b1 + 272097) * q^37 + (-4*b15 - 2*b14 + 23*b13 - 122*b12 - 135*b11 + 34*b10 - 756*b9 - 3849*b8) * q^38 + (-3*b7 - 16*b6 + 4*b5 - 92*b4 - 138*b3 - 32*b2 + 1306*b1 - 185898) * q^40 + (20*b15 + 17*b14 + 98*b12 + 298*b11 - 17*b10 - 377*b9 + 6530*b8) * q^41 + (-2*b7 - 22*b6 - 26*b5 - 236*b4 + 189*b3 + 16*b2 + 1428*b1 + 340853) * q^43 + (-19*b15 + 2*b14 + 25*b13 + 257*b12 - 932*b11 - 47*b10 + 520*b9 - 88601*b8) * q^44 + (-14*b7 - 33*b6 - 11*b5 + 313*b4 + 1067*b3 - 9*b2 + 3936*b1 - 264372) * q^46 + (-15*b15 + 6*b14 + 4*b13 + 37*b12 + 463*b11 + 96*b10 + 2556*b9 + 89376*b8) * q^47 + (31*b7 - 30*b6 + 58*b5 - 86*b4 + 960*b3 - 72*b2 - 2477*b1 + 912907) * q^49 + (43*b15 + 112*b14 + 49*b13 - 49*b12 + 1182*b11 - 58*b10 - 2816*b9 + 42161*b8) * q^50 + (16*b7 - 50*b6 - 74*b5 + 154*b4 - 3292*b3 - 120*b2 + 15899*b1 - 1587414) * q^52 + (-38*b15 + 5*b14 + 50*b13 + 20*b12 + 1116*b11 - 111*b10 + 1355*b9 + 110956*b8) * q^53 + (-46*b7 - 58*b6 + 64*b5 + 182*b4 - 115*b3 + 36*b2 - 1236*b1 - 903313) * q^55 + (-23*b15 + 70*b14 + 117*b13 + 89*b12 - 2224*b11 + 161*b10 - 1288*b9 - 195973*b8) * q^56 + (13*b7 - 63*b6 + 91*b5 + 630*b4 - 3631*b3 - 177*b2 - 10200*b1 + 74916) * q^58 + (79*b15 + 94*b14 + 128*b13 - 169*b12 - 1499*b11 - 68*b10 + 9528*b9 - 223764*b8) * q^59 + (-17*b7 - 73*b6 - 176*b5 + 255*b4 + 1394*b3 - 104*b2 - 742*b1 + 1940161) * q^61 + (-60*b15 + 70*b14 + 14*b13 + 278*b12 - 604*b11 - 134*b10 - 17060*b9 - 43236*b8) * q^62 + (-29*b7 - 86*b6 - 122*b5 - 206*b4 - 4962*b3 - 248*b2 - 14606*b1 + 1514520) * q^64 + (-37*b15 - 61*b14 - 164*b13 - 32*b12 - 1090*b11 + 220*b10 - 13217*b9 - 95138*b8) * q^65 + (48*b7 - 48*b6 - 26*b5 - 182*b4 - 1339*b3 - 56*b2 + 40164*b1 - 3897655) * q^67 + (60*b15 + 120*b14 + 67*b13 - 166*b12 + 2188*b11 - 97*b10 - 17588*b9 + 185524*b8) * q^68 + (34*b7 - 53*b6 + 221*b5 + 73*b4 + 2171*b3 - 277*b2 - 11716*b1 + 692780) * q^70 + (-23*b15 - 200*b14 - 198*b13 - 266*b12 - 2010*b11 - 106*b10 + 13805*b9 - 107522*b8) * q^71 + (75*b7 + b6 + 166*b5 - 233*b4 + 7809*b3 - 73*b2 + 27659*b1 - 1327610) * q^73 + (14*b15 - 34*b14 + 71*b13 - 608*b12 + 3711*b11 + 58*b10 + 31524*b9 + 630152*b8) * q^74 + (-195*b7 - 62*b6 - 229*b5 + 50*b4 + 10954*b3 + 509*b2 - 19232*b1 + 4352283) * q^76 + (19*b15 - 350*b14 - 644*b13 - 23*b12 + 3643*b11 + 112*b10 - 3710*b9 + 195624*b8) * q^77 + (-105*b7 + 60*b6 + 322*b5 - 400*b4 - 3319*b3 + 872*b2 + 41033*b1 - 9441049) * q^79 + (36*b15 - 198*b14 + 170*b13 + 878*b12 - 331*b11 + 212*b10 + 40188*b9 - 247413*b8) * q^80 + (281*b7 + 77*b6 - 331*b5 - 835*b4 - 5893*b3 + 303*b2 + 76541*b1 - 2514108) * q^82 + (22*b15 - 564*b14 - 300*b13 - 1496*b12 + 1988*b11 - 166*b10 + 4586*b9 + 89624*b8) * q^83 + (-57*b7 + 145*b6 - 556*b5 - 2999*b4 - 8260*b3 - 36*b2 - 68188*b1 + 10060055) * q^85 + (-208*b15 - 292*b14 - 297*b13 + 1668*b12 - 3281*b11 + 100*b10 + 71176*b9 + 53913*b8) * q^86 + (117*b7 + 198*b6 + 1043*b5 - 1330*b4 + 11106*b3 + 297*b2 - 218012*b1 + 22939091) * q^88 + (212*b15 - 526*b14 - 500*b13 - 613*b12 + 1979*b11 + 343*b10 - 59942*b9 + 472660*b8) * q^89 + (162*b7 + 368*b6 + 542*b5 + 2118*b4 - 3241*b3 + 60*b2 - 227866*b1 + 23354467) * q^91 + (189*b15 + 294*b14 + 49*b13 + 2509*b12 + 1326*b11 - 931*b10 - 74608*b9 - 923059*b8) * q^92 + (-316*b7 + 192*b6 - 82*b5 + 3394*b4 + 5702*b3 + 980*b2 + 205818*b1 - 34044896) * q^94 + (-310*b15 - 806*b14 - 326*b13 - 3071*b12 - 4229*b11 + 616*b10 - 65189*b9 + 830518*b8) * q^95 + (-241*b7 + 426*b6 - 810*b5 + 5062*b4 + 4104*b3 - 16*b2 + 29995*b1 - 8366488) * q^97 + (236*b15 + 994*b14 + 190*b13 - 4734*b12 + 12040*b11 + 1022*b10 + 17892*b9 + 1566103*b8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2048 q^{4} + 3692 q^{7}+O(q^{10})$$ 16 * q - 2048 * q^4 + 3692 * q^7 $$16 q - 2048 q^{4} + 3692 q^{7} + 10752 q^{10} + 63860 q^{13} + 95116 q^{16} + 185108 q^{19} - 691980 q^{22} - 541712 q^{25} - 997132 q^{28} + 571136 q^{31} + 1027656 q^{34} + 4354268 q^{37} - 2973768 q^{40} + 5453084 q^{43} - 4234044 q^{46} + 14602560 q^{49} - 25384960 q^{52} - 14452572 q^{55} + 1213068 q^{58} + 31037996 q^{61} + 24253000 q^{64} - 62356828 q^{67} + 11075124 q^{70} - 21273664 q^{73} + 69593876 q^{76} - 151045036 q^{79} - 40201140 q^{82} + 160995564 q^{85} + 366976068 q^{88} + 373680796 q^{91} - 544741584 q^{94} - 133878688 q^{97}+O(q^{100})$$ 16 * q - 2048 * q^4 + 3692 * q^7 + 10752 * q^10 + 63860 * q^13 + 95116 * q^16 + 185108 * q^19 - 691980 * q^22 - 541712 * q^25 - 997132 * q^28 + 571136 * q^31 + 1027656 * q^34 + 4354268 * q^37 - 2973768 * q^40 + 5453084 * q^43 - 4234044 * q^46 + 14602560 * q^49 - 25384960 * q^52 - 14452572 * q^55 + 1213068 * q^58 + 31037996 * q^61 + 24253000 * q^64 - 62356828 * q^67 + 11075124 * q^70 - 21273664 * q^73 + 69593876 * q^76 - 151045036 * q^79 - 40201140 * q^82 + 160995564 * q^85 + 366976068 * q^88 + 373680796 * q^91 - 544741584 * q^94 - 133878688 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24$$ :

 $$\beta_{1}$$ $$=$$ $$3\nu^{2} + 384$$ 3*v^2 + 384 $$\beta_{2}$$ $$=$$ $$( - 16592055498945 \nu^{14} + \cdots + 31\!\cdots\!08 ) / 24\!\cdots\!60$$ (-16592055498945*v^14 - 15055659358671888*v^12 - 5081496386110238709*v^10 - 741541804832200946796*v^8 - 31046607770462708736996*v^6 + 2577092138472125753391168*v^4 + 217571758939484045885249280*v^2 + 3184032275120463151204343808) / 24748126752115897794560 $$\beta_{3}$$ $$=$$ $$( 2497774802274 \nu^{14} + \cdots + 11\!\cdots\!08 ) / 30\!\cdots\!20$$ (2497774802274*v^14 + 2512058700617379*v^12 + 995650378552289754*v^10 + 196784818377948395127*v^8 + 20138253382775852197740*v^6 + 1000020304704249240966828*v^4 + 20331500421407097329017536*v^2 + 119781211247943590650942208) / 3093515844014487224320 $$\beta_{4}$$ $$=$$ $$( - 25410686450643 \nu^{14} + \cdots - 98\!\cdots\!16 ) / 98\!\cdots\!24$$ (-25410686450643*v^14 - 24069115491533064*v^12 - 8913897159295895343*v^10 - 1636683235236669161916*v^8 - 155615626550615861930508*v^6 - 7302076158725742552677280*v^4 - 148744954403240646342017280*v^2 - 982095508798032548767266816) / 9899250700846359117824 $$\beta_{5}$$ $$=$$ $$( - 143265531846591 \nu^{14} + \cdots - 85\!\cdots\!20 ) / 24\!\cdots\!60$$ (-143265531846591*v^14 - 145716097480840368*v^12 - 58640127841236305547*v^10 - 11852686571356496341332*v^8 - 1257801608727191831918364*v^6 - 66355658498775034095117120*v^4 - 1455531573931130040179476224*v^2 - 8577038093440768319923281920) / 24748126752115897794560 $$\beta_{6}$$ $$=$$ $$( - 23\!\cdots\!33 \nu^{14} + \cdots - 14\!\cdots\!52 ) / 49\!\cdots\!20$$ (-2301885889943733*v^14 - 2310536291091592392*v^12 - 915834193856881948665*v^10 - 181667754852287486009652*v^8 - 18788516775737181380517588*v^6 - 958561680867857786075370912*v^4 - 20939778282050371193330572032*v^2 - 143547839369223491456803428352) / 49496253504231795589120 $$\beta_{7}$$ $$=$$ $$( 594985394614059 \nu^{14} + \cdots + 28\!\cdots\!08 ) / 12\!\cdots\!80$$ (594985394614059*v^14 + 578295521189373744*v^12 + 221767183556168694759*v^10 + 42637332823587679522212*v^8 + 4295229626156564921299020*v^6 + 214708423273221658217594688*v^4 + 4539197259920006053757043456*v^2 + 28685032908362331811732267008) / 12374063376057948897280 $$\beta_{8}$$ $$=$$ $$( - 55090601 \nu^{15} - 55395890770 \nu^{13} - 21996486413661 \nu^{11} + \cdots - 34\!\cdots\!12 \nu ) / 83\!\cdots\!04$$ (-55090601*v^15 - 55395890770*v^13 - 21996486413661*v^11 - 4369143077181302*v^9 - 451893434941315228*v^7 - 22995865393769474952*v^5 - 499698741959542661504*v^3 - 3419545991509080474112*v) / 83501732862595989504 $$\beta_{9}$$ $$=$$ $$( - 69\!\cdots\!75 \nu^{15} + \cdots - 63\!\cdots\!48 \nu ) / 60\!\cdots\!64$$ (-69883909169728670075*v^15 - 71027057285269697516104*v^13 - 28644818869283692802504439*v^11 - 5821106649038210835024216668*v^9 - 622749860960574671227477195948*v^7 - 33291414422639606284018277811360*v^5 - 774758697652078586178055309760768*v^3 - 6392155350227584324044844423886848*v) / 6058421969221673866086070616064 $$\beta_{10}$$ $$=$$ $$( - 11\!\cdots\!79 \nu^{15} + \cdots + 35\!\cdots\!56 \nu ) / 50\!\cdots\!20$$ (-1179078217397867680579*v^15 - 1003430685985354837225928*v^13 - 307285211064588194377559391*v^11 - 36688051314632334757084420540*v^9 + 2419620181601713897743141172*v^7 + 340807727088118770385456331424096*v^5 + 22269349237259665259625035217346304*v^3 + 350004516143389815845270244219897856*v) / 5048684974351394888405058846720 $$\beta_{11}$$ $$=$$ $$( 30719884451 \nu^{15} + 30582399292090 \nu^{13} + \cdots + 18\!\cdots\!84 \nu ) / 83\!\cdots\!04$$ (30719884451*v^15 + 30582399292090*v^13 + 12013758303742623*v^11 + 2361087168254638166*v^9 + 242024394746154772132*v^7 + 12253559967105725828904*v^5 + 266060537124849590623616*v^3 + 1827807609236486957238784*v) / 83501732862595989504 $$\beta_{12}$$ $$=$$ $$( 11\!\cdots\!63 \nu^{15} + \cdots + 94\!\cdots\!84 \nu ) / 30\!\cdots\!20$$ (11624807649009095292863*v^15 + 11199434524704160281692200*v^13 + 4257506304525844206024272139*v^11 + 814064968574475867139964500268*v^9 + 82518271608547707470651259647164*v^7 + 4301081934104193425974584921544224*v^5 + 104982495326877277113542995337798912*v^3 + 943472296728672240404031503920912384*v) / 30292109846108369330430353080320 $$\beta_{13}$$ $$=$$ $$( 636078186037275 \nu^{15} + \cdots + 30\!\cdots\!84 \nu ) / 33\!\cdots\!60$$ (636078186037275*v^15 + 631955223204357116*v^13 + 247107186256660281303*v^11 + 48128702841581003015712*v^9 + 4851562030197088467842972*v^7 + 237887863459264761210392304*v^5 + 4845882884983838126877304320*v^3 + 30007679361618968990444432384*v) / 334099711153564620226560 $$\beta_{14}$$ $$=$$ $$( - 74\!\cdots\!17 \nu^{15} + \cdots - 39\!\cdots\!36 \nu ) / 30\!\cdots\!20$$ (-74051097268330470330617*v^15 - 73155658029485982863182360*v^13 - 28462913218841551578830866701*v^11 - 5525112062594395272381171335732*v^9 - 556846153951411788627547357020196*v^7 - 27463866758114836490707560933433056*v^5 - 569442473136236948002484389654446848*v^3 - 3925956532612184238311929637898151936*v) / 30292109846108369330430353080320 $$\beta_{15}$$ $$=$$ $$( - 26\!\cdots\!27 \nu^{15} + \cdots - 12\!\cdots\!72 \nu ) / 30\!\cdots\!20$$ (-264055155989373352710527*v^15 - 262105100295719124127972264*v^13 - 102694175558180478943196980683*v^11 - 20141385925404283330558610453420*v^9 - 2060923803775707650712522513710524*v^7 - 103805462650438486850004309386368032*v^5 - 2186089269477592162187579526765970688*v^3 - 12952354789986531698252712285544732672*v) / 30292109846108369330430353080320
 $$\nu$$ $$=$$ $$( \beta_{15} - 26 \beta_{14} - 27 \beta_{13} - 33 \beta_{12} + 62 \beta_{11} + 4 \beta_{10} + \cdots + 10850 \beta_{8} ) / 177147$$ (b15 - 26*b14 - 27*b13 - 33*b12 + 62*b11 + 4*b10 + 476*b9 + 10850*b8) / 177147 $$\nu^{2}$$ $$=$$ $$( \beta _1 - 384 ) / 3$$ (b1 - 384) / 3 $$\nu^{3}$$ $$=$$ $$( - 101 \beta_{15} + 3922 \beta_{14} + 6453 \beta_{13} + 7959 \beta_{12} - 31039 \beta_{11} + \cdots - 3536290 \beta_{8} ) / 177147$$ (-101*b15 + 3922*b14 + 6453*b13 + 7959*b12 - 31039*b11 - 575*b10 - 201040*b9 - 3536290*b8) / 177147 $$\nu^{4}$$ $$=$$ $$( \beta_{5} + 8\beta_{3} + \beta_{2} - 852\beta _1 + 235323 ) / 9$$ (b5 + 8*b3 + b2 - 852*b1 + 235323) / 9 $$\nu^{5}$$ $$=$$ $$( 69514 \beta_{15} - 726014 \beta_{14} - 1602774 \beta_{13} - 1954284 \beta_{12} + \cdots + 1324336535 \beta_{8} ) / 177147$$ (69514*b15 - 726014*b14 - 1602774*b13 - 1954284*b12 + 10948835*b11 + 30232*b10 + 44615756*b9 + 1324336535*b8) / 177147 $$\nu^{6}$$ $$=$$ $$( - 29 \beta_{7} - 86 \beta_{6} - 1402 \beta_{5} - 206 \beta_{4} - 15202 \beta_{3} - 1528 \beta_{2} + \cdots - 165481192 ) / 27$$ (-29*b7 - 86*b6 - 1402*b5 - 206*b4 - 15202*b3 - 1528*b2 + 682738*b1 - 165481192) / 27 $$\nu^{7}$$ $$=$$ $$( - 27108917 \beta_{15} + 145439524 \beta_{14} + 412743735 \beta_{13} + \cdots - 477292582570 \beta_{8} ) / 177147$$ (-27108917*b15 + 145439524*b14 + 412743735*b13 + 516431721*b12 - 3487518469*b11 + 15875221*b10 - 9284168548*b9 - 477292582570*b8) / 177147 $$\nu^{8}$$ $$=$$ $$( 44142 \beta_{7} + 151412 \beta_{6} + 1491915 \beta_{5} + 256900 \beta_{4} + 19087316 \beta_{3} + \cdots + 124022951365 ) / 81$$ (44142*b7 + 151412*b6 + 1491915*b5 + 256900*b4 + 19087316*b3 + 1784447*b2 - 555547234*b1 + 124022951365) / 81 $$\nu^{9}$$ $$=$$ $$( 8996587270 \beta_{15} - 30640114838 \beta_{14} - 109372233150 \beta_{13} + \cdots + 162516241907297 \beta_{8} ) / 177147$$ (8996587270*b15 - 30640114838*b14 - 109372233150*b13 - 142710481752*b12 + 1071654353345*b11 - 8620187120*b10 + 1953406211708*b9 + 162516241907297*b8) / 177147 $$\nu^{10}$$ $$=$$ $$( - 15281869 \beta_{7} - 62299606 \beta_{6} - 481409782 \beta_{5} - 66055758 \beta_{4} + \cdots - 32351942692828 ) / 81$$ (-15281869*b7 - 62299606*b6 - 481409782*b5 - 66055758*b4 - 6855618242*b3 - 621157268*b2 + 153821726310*b1 - 32351942692828) / 81 $$\nu^{11}$$ $$=$$ $$( - 2819717649515 \beta_{15} + 6750809798956 \beta_{14} + 29650652884593 \beta_{13} + \cdots - 53\!\cdots\!50 \beta_{8} ) / 177147$$ (-2819717649515*b15 + 6750809798956*b14 + 29650652884593*b13 + 40544038861815*b12 - 324139187253391*b11 + 3141966474307*b10 - 425653728895132*b9 - 53030181988772650*b8) / 177147 $$\nu^{12}$$ $$=$$ $$( 4527415742 \beta_{7} + 22266002740 \beta_{6} + 148886537917 \beta_{5} + 11886560580 \beta_{4} + \cdots + 87\!\cdots\!59 ) / 81$$ (4527415742*b7 + 22266002740*b6 + 148886537917*b5 + 11886560580*b4 + 2287313440548*b3 + 203755437777*b2 - 43335357263438*b1 + 8720530036690859) / 81 $$\nu^{13}$$ $$=$$ $$( 862090036459174 \beta_{15} + \cdots + 16\!\cdots\!71 \beta_{8} ) / 177147$$ (862090036459174*b15 - 1554468943605290*b14 - 8189925697660950*b13 - 11725080376696980*b12 + 97245357108901163*b11 - 1012612976125052*b10 + 97003184357908628*b9 + 16804475589383865671*b8) / 177147 $$\nu^{14}$$ $$=$$ $$( - 3713925118737 \beta_{7} - 22225534276430 \beta_{6} - 135217317246954 \beta_{5} + \cdots - 72\!\cdots\!20 ) / 243$$ (-3713925118737*b7 - 22225534276430*b6 - 135217317246954*b5 - 2642203831846*b4 - 2197339542709514*b3 - 193646752057184*b2 + 37117015609737094*b1 - 7230966633225361120) / 243 $$\nu^{15}$$ $$=$$ $$( - 26\!\cdots\!65 \beta_{15} + \cdots - 52\!\cdots\!02 \beta_{8} ) / 177147$$ (-260314539934423565*b15 + 373736806412308540*b14 + 2297120412499709607*b13 + 3429286884098335929*b12 - 29038968895786888525*b11 + 309485374369484653*b10 - 23181702805706510644*b9 - 5219250093169944560002*b8) / 177147

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 − 17.2054i 14.8268i 13.9244i − 12.9444i − 8.62663i 5.78233i − 5.26228i 3.33872i − 3.33872i 5.26228i − 5.78233i 8.62663i 12.9444i − 13.9244i − 14.8268i 17.2054i
29.8006i 0 −632.074 177.778i 0 3390.81 11207.2i 0 5297.89
80.2 25.6809i 0 −403.506 204.369i 0 −3702.42 3788.08i 0 −5248.38
80.3 24.1178i 0 −325.666 436.510i 0 550.753 1680.18i 0 −10527.6
80.4 22.4204i 0 −246.676 1080.12i 0 −1006.64 209.045i 0 24216.8
80.5 14.9418i 0 32.7440 1089.48i 0 1308.49 4314.34i 0 −16278.7
80.6 10.0153i 0 155.694 594.387i 0 4409.51 4123.23i 0 5952.96
80.7 9.11454i 0 172.925 181.837i 0 −873.072 3909.46i 0 −1657.36
80.8 5.78284i 0 222.559 626.068i 0 −2231.44 2767.43i 0 3620.45
80.9 5.78284i 0 222.559 626.068i 0 −2231.44 2767.43i 0 3620.45
80.10 9.11454i 0 172.925 181.837i 0 −873.072 3909.46i 0 −1657.36
80.11 10.0153i 0 155.694 594.387i 0 4409.51 4123.23i 0 5952.96
80.12 14.9418i 0 32.7440 1089.48i 0 1308.49 4314.34i 0 −16278.7
80.13 22.4204i 0 −246.676 1080.12i 0 −1006.64 209.045i 0 24216.8
80.14 24.1178i 0 −325.666 436.510i 0 550.753 1680.18i 0 −10527.6
80.15 25.6809i 0 −403.506 204.369i 0 −3702.42 3788.08i 0 −5248.38
80.16 29.8006i 0 −632.074 177.778i 0 3390.81 11207.2i 0 5297.89
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 80.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 81.9.b.b 16
3.b odd 2 1 inner 81.9.b.b 16
9.c even 3 2 81.9.d.g 32
9.d odd 6 2 81.9.d.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.9.b.b 16 1.a even 1 1 trivial
81.9.b.b 16 3.b odd 2 1 inner
81.9.d.g 32 9.c even 3 2
81.9.d.g 32 9.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 3072 T_{2}^{14} + 3777309 T_{2}^{12} + 2381488884 T_{2}^{10} + 819880359588 T_{2}^{8} + \cdots + 10\!\cdots\!64$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + \cdots + 10\!\cdots\!64$$
$3$ $$T^{16}$$
$5$ $$T^{16} + \cdots + 15\!\cdots\!25$$
$7$ $$(T^{8} + \cdots + 78\!\cdots\!56)^{2}$$
$11$ $$T^{16} + \cdots + 11\!\cdots\!64$$
$13$ $$(T^{8} + \cdots + 40\!\cdots\!93)^{2}$$
$17$ $$T^{16} + \cdots + 10\!\cdots\!69$$
$19$ $$(T^{8} + \cdots + 24\!\cdots\!92)^{2}$$
$23$ $$T^{16} + \cdots + 30\!\cdots\!36$$
$29$ $$T^{16} + \cdots + 11\!\cdots\!09$$
$31$ $$(T^{8} + \cdots - 14\!\cdots\!44)^{2}$$
$37$ $$(T^{8} + \cdots - 11\!\cdots\!63)^{2}$$
$41$ $$T^{16} + \cdots + 25\!\cdots\!36$$
$43$ $$(T^{8} + \cdots + 24\!\cdots\!52)^{2}$$
$47$ $$T^{16} + \cdots + 35\!\cdots\!16$$
$53$ $$T^{16} + \cdots + 12\!\cdots\!44$$
$59$ $$T^{16} + \cdots + 46\!\cdots\!44$$
$61$ $$(T^{8} + \cdots + 11\!\cdots\!77)^{2}$$
$67$ $$(T^{8} + \cdots + 33\!\cdots\!68)^{2}$$
$71$ $$T^{16} + \cdots + 19\!\cdots\!16$$
$73$ $$(T^{8} + \cdots - 97\!\cdots\!99)^{2}$$
$79$ $$(T^{8} + \cdots - 12\!\cdots\!84)^{2}$$
$83$ $$T^{16} + \cdots + 28\!\cdots\!76$$
$89$ $$T^{16} + \cdots + 62\!\cdots\!81$$
$97$ $$(T^{8} + \cdots - 17\!\cdots\!84)^{2}$$