# Properties

 Label 81.9.d.f Level $81$ Weight $9$ Character orbit 81.d Analytic conductor $32.998$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(26,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

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("81.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.d (of order $$6$$, degree $$2$$, not 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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2 x^{11} + 2 x^{10} - 84 x^{9} - 141 x^{8} + 1130 x^{7} + 1550 x^{6} + 8300 x^{5} + 44525 x^{4} + \cdots + 2500$$ x^12 - 2*x^11 + 2*x^10 - 84*x^9 - 141*x^8 + 1130*x^7 + 1550*x^6 + 8300*x^5 + 44525*x^4 + 29750*x^3 + 11250*x^2 + 7500*x + 2500 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 3^{42}$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + (\beta_{7} - 131 \beta_1) q^{4} + (\beta_{6} + 20 \beta_{2}) q^{5} + (\beta_{11} + 2 \beta_{7} + \beta_{5} + \cdots + 283) q^{7}+ \cdots + (\beta_{10} + 7 \beta_{9} + \cdots + 161 \beta_{2}) q^{8}+O(q^{10})$$ q - b4 * q^2 + (b7 - 131*b1) * q^4 + (b6 + 20*b2) * q^5 + (b11 + 2*b7 + b5 - 2*b3 + 283*b1 + 283) * q^7 + (b10 + 7*b9 - b8 - 161*b4 + 161*b2) * q^8 $$q - \beta_{4} q^{2} + (\beta_{7} - 131 \beta_1) q^{4} + (\beta_{6} + 20 \beta_{2}) q^{5} + (\beta_{11} + 2 \beta_{7} + \beta_{5} + \cdots + 283) q^{7}+ \cdots + (28240 \beta_{10} + \cdots + 1372456 \beta_{2}) q^{98}+O(q^{100})$$ q - b4 * q^2 + (b7 - 131*b1) * q^4 + (b6 + 20*b2) * q^5 + (b11 + 2*b7 + b5 - 2*b3 + 283*b1 + 283) * q^7 + (b10 + 7*b9 - b8 - 161*b4 + 161*b2) * q^8 + (-b5 - 53*b3 - 7713) * q^10 + (-5*b10 + 14*b9 - 14*b6 + 280*b4) * q^11 + (-2*b11 - 44*b7 + 6974*b1) * q^13 + (20*b8 - 4*b6 + 165*b2) * q^14 + (-16*b11 + 177*b7 - 16*b5 - 177*b3 - 28411*b1 - 28411) * q^16 + (42*b10 - 28*b9 - 42*b8 + 552*b4 - 552*b2) * q^17 + (14*b5 + 60*b3 - 6064) * q^19 + (-75*b10 - 133*b9 + 133*b6 + 17875*b4) * q^20 + (31*b11 - 23*b7 + 107883*b1) * q^22 + (46*b8 + 2*b6 + 13552*b2) * q^23 + (106*b11 - 1108*b7 + 106*b5 + 1108*b3 + 274448*b1 + 274448) * q^25 + (-272*b9 + 19310*b4 - 19310*b2) * q^26 + (-72*b5 + 1299*b3 - 139831) * q^28 + (230*b10 + 336*b9 - 336*b6 + 32280*b4) * q^29 + (-193*b11 + 2470*b7 + 343079*b1) * q^31 + (-273*b8 - 265*b6 + 39801*b2) * q^32 + (-350*b11 + 246*b7 - 350*b5 - 246*b3 + 220050*b1 + 220050) * q^34 + (-685*b10 + 2124*b9 + 685*b8 - 7480*b4 + 7480*b2) * q^35 + (110*b5 - 2100*b3 - 1565980) * q^37 + (368*b10 + 672*b9 - 672*b6 - 12832*b4) * q^38 + (552*b11 - 11771*b7 + 4926681*b1) * q^40 + (-1010*b8 + 2634*b6 + 60400*b2) * q^41 + (438*b11 + 10556*b7 + 438*b5 - 10556*b3 + 456214*b1 + 456214) * q^43 + (575*b10 - 4303*b9 - 575*b8 + 46625*b4 - 46625*b2) * q^44 + (412*b5 - 11732*b3 - 5252436) * q^46 + (126*b10 - 4796*b9 + 4796*b6 - 215920*b4) * q^47 + (-306*b11 + 21508*b7 + 4816776*b1) * q^49 + (3440*b8 - 9664*b6 - 604480*b2) * q^50 + (784*b11 - 17022*b7 + 784*b5 + 17022*b3 + 5680282*b1 + 5680282) * q^52 + (614*b10 - 3471*b9 - 614*b8 - 255644*b4 + 255644*b2) * q^53 + (-1979*b5 + 28678*b3 - 4445757) * q^55 + (-5405*b10 + 6773*b9 - 6773*b6 - 264995*b4) * q^56 + (-2406*b11 - 11762*b7 + 12540762*b1) * q^58 + (-630*b8 + 12604*b6 + 88400*b2) * q^59 + (-3624*b11 - 23440*b7 - 3624*b5 + 23440*b3 + 6696796*b1 + 6696796) * q^61 + (6716*b10 + 20764*b9 - 6716*b8 - 387273*b4 + 387273*b2) * q^62 + (1904*b5 + 3063*b3 - 8090243) * q^64 + (4370*b10 + 3372*b9 - 3372*b6 - 746040*b4) * q^65 + (5738*b11 - 10604*b7 + 18559214*b1) * q^67 + (2806*b8 + 15190*b6 + 35018*b2) * q^68 + (4041*b11 + 49487*b7 + 4041*b5 - 49487*b3 - 2954547*b1 - 2954547) * q^70 + (-12320*b10 - 1054*b9 + 12320*b8 - 201280*b4 + 201280*b2) * q^71 + (4926*b5 + 22428*b3 + 2136953) * q^73 + (320*b10 - 12720*b9 + 12720*b6 + 2152940*b4) * q^74 + (-400*b11 + 34736*b7 - 6437296*b1) * q^76 + (-15640*b8 - 68391*b6 + 883380*b2) * q^77 + (2468*b11 + 66848*b7 + 2468*b5 - 66848*b3 - 3636734*b1 - 3636734) * q^79 + (-4715*b10 - 58285*b9 + 4715*b8 + 3785635*b4 - 3785635*b2) * q^80 + (-11724*b5 - 188732*b3 - 23130972) * q^82 + (21301*b10 + 16014*b9 - 16014*b6 + 257704*b4) * q^83 + (-11822*b11 - 91284*b7 - 13974066*b1) * q^85 + (-920*b8 + 66008*b6 + 2508490*b2) * q^86 + (-8808*b11 - 159161*b7 - 8808*b5 + 159161*b3 - 9592029*b1 - 9592029) * q^88 + (-7400*b10 + 4606*b9 + 7400*b8 - 1376920*b4 + 1376920*b2) * q^89 + (-32*b5 - 90376*b3 + 26105494) * q^91 + (-14444*b10 - 74196*b9 + 74196*b6 + 5087388*b4) * q^92 + (3662*b11 + 62818*b7 - 83668986*b1) * q^94 + (2990*b8 + 19886*b6 + 1246640*b2) * q^95 + (4008*b11 + 36656*b7 + 4008*b5 - 36656*b3 - 8955851*b1 - 8955851) * q^97 + (28240*b10 + 156064*b9 - 28240*b8 - 1372456*b4 + 1372456*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 786 q^{4} + 1698 q^{7}+O(q^{10})$$ 12 * q + 786 * q^4 + 1698 * q^7 $$12 q + 786 q^{4} + 1698 q^{7} - 92556 q^{10} - 41844 q^{13} - 170466 q^{16} - 72768 q^{19} - 647298 q^{22} + 1646688 q^{25} - 1677972 q^{28} - 2058474 q^{31} + 1320300 q^{34} - 18791760 q^{37} - 29560086 q^{40} + 2737284 q^{43} - 63029232 q^{46} - 28900656 q^{49} + 34081692 q^{52} - 53349084 q^{55} - 75244572 q^{58} + 40180776 q^{61} - 97082916 q^{64} - 111355284 q^{67} - 17727282 q^{70} + 25643436 q^{73} + 38623776 q^{76} - 21820404 q^{79} - 277571664 q^{82} + 83844396 q^{85} - 57552174 q^{88} + 313265928 q^{91} + 502013916 q^{94} - 53735106 q^{97}+O(q^{100})$$ 12 * q + 786 * q^4 + 1698 * q^7 - 92556 * q^10 - 41844 * q^13 - 170466 * q^16 - 72768 * q^19 - 647298 * q^22 + 1646688 * q^25 - 1677972 * q^28 - 2058474 * q^31 + 1320300 * q^34 - 18791760 * q^37 - 29560086 * q^40 + 2737284 * q^43 - 63029232 * q^46 - 28900656 * q^49 + 34081692 * q^52 - 53349084 * q^55 - 75244572 * q^58 + 40180776 * q^61 - 97082916 * q^64 - 111355284 * q^67 - 17727282 * q^70 + 25643436 * q^73 + 38623776 * q^76 - 21820404 * q^79 - 277571664 * q^82 + 83844396 * q^85 - 57552174 * q^88 + 313265928 * q^91 + 502013916 * q^94 - 53735106 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 2 x^{10} - 84 x^{9} - 141 x^{8} + 1130 x^{7} + 1550 x^{6} + 8300 x^{5} + 44525 x^{4} + \cdots + 2500$$ :

 $$\beta_{1}$$ $$=$$ $$( - 36359663 \nu^{11} + 86582291 \nu^{10} - 109850966 \nu^{9} + 3107263582 \nu^{8} + \cdots - 254024381500 ) / 118206320250$$ (-36359663*v^11 + 86582291*v^10 - 109850966*v^9 + 3107263582*v^8 + 3922225423*v^7 - 42211411035*v^6 - 39967378450*v^5 - 290692531340*v^4 - 1511769690375*v^3 - 539522519125*v^2 - 381812194750*v - 254024381500) / 118206320250 $$\beta_{2}$$ $$=$$ $$( - 9565171136 \nu^{11} + 69328993967 \nu^{10} - 153334291592 \nu^{9} + 982947594424 \nu^{8} + \cdots - 4887640255000 ) / 25650771494250$$ (-9565171136*v^11 + 69328993967*v^10 - 153334291592*v^9 + 982947594424*v^8 - 2989202665964*v^7 - 14954954462655*v^6 + 45945445874180*v^5 - 39966901207400*v^4 - 42416069343900*v^3 + 1602789335899775*v^2 - 10131636970000*v - 4887640255000) / 25650771494250 $$\beta_{3}$$ $$=$$ $$( 545995374 \nu^{11} - 1664112534 \nu^{10} + 5414636868 \nu^{9} - 66349445169 \nu^{8} + \cdots - 112560389309850 ) / 855025716475$$ (545995374*v^11 - 1664112534*v^10 + 5414636868*v^9 - 66349445169*v^8 + 13828520346*v^7 + 391662830214*v^6 + 1031670326172*v^5 + 7823481754065*v^4 + 5378838107310*v^3 + 1984529113050*v^2 + 1347866456700*v - 112560389309850) / 855025716475 $$\beta_{4}$$ $$=$$ $$( 124544231765 \nu^{11} - 288279071309 \nu^{10} + 320627145188 \nu^{9} + \cdots + 467513871568750 ) / 25650771494250$$ (124544231765*v^11 - 288279071309*v^10 + 320627145188*v^9 - 10519064076418*v^8 - 14333800897729*v^7 + 147109014843189*v^6 + 148533678911380*v^5 + 967933865802650*v^4 + 5206597025689725*v^3 + 1859010396287275*v^2 + 165649040230000*v + 467513871568750) / 25650771494250 $$\beta_{5}$$ $$=$$ $$( 5109813018 \nu^{11} - 15779438388 \nu^{10} + 47454682626 \nu^{9} - 640213387758 \nu^{8} + \cdots - 37\!\cdots\!00 ) / 855025716475$$ (5109813018*v^11 - 15779438388*v^10 + 47454682626*v^9 - 640213387758*v^8 + 188652984822*v^7 + 3815206572348*v^6 + 10261135116504*v^5 + 76889432068830*v^4 + 52820144616420*v^3 + 19501791155100*v^2 + 13238296889400*v - 3749279023579500) / 855025716475 $$\beta_{6}$$ $$=$$ $$( 20005351192 \nu^{11} - 147015937099 \nu^{10} + 325165769224 \nu^{9} - 2067073015928 \nu^{8} + \cdots + 10328107235000 ) / 1710051432950$$ (20005351192*v^11 - 147015937099*v^10 + 325165769224*v^9 - 2067073015928*v^8 + 6339892360108*v^7 + 31709346187035*v^6 - 96685288752460*v^5 + 84716654057800*v^4 + 89731407618300*v^3 - 3683321165355625*v^2 + 21430488590000*v + 10328107235000) / 1710051432950 $$\beta_{7}$$ $$=$$ $$( - 171819166089 \nu^{11} + 406679127573 \nu^{10} - 508959146298 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 4275128582375$$ (-171819166089*v^11 + 406679127573*v^10 - 508959146298*v^9 + 14604459660681*v^8 + 19080422071449*v^7 - 200289787014285*v^6 - 184896035286990*v^5 - 1386074148204180*v^4 - 7226137656345825*v^3 - 2578413591535875*v^2 - 1825093941263250*v - 1214456404870500) / 4275128582375 $$\beta_{8}$$ $$=$$ $$( - 1580212999724 \nu^{11} + 11535129466853 \nu^{10} - 25512624126728 \nu^{9} + \cdots - 811743383545000 ) / 25650771494250$$ (-1580212999724*v^11 + 11535129466853*v^10 - 25512624126728*v^9 + 162843738496216*v^8 - 497396455194476*v^7 - 2488102354818645*v^6 + 7614362587346120*v^5 - 6648360036536600*v^4 - 7048615590725100*v^3 + 283808872462627175*v^2 - 1683531563980000*v - 811743383545000) / 25650771494250 $$\beta_{9}$$ $$=$$ $$( 61359099583 \nu^{11} - 163761564148 \nu^{10} + 220848638212 \nu^{9} - 5276121240110 \nu^{8} + \cdots + 217040473279250 ) / 342010286590$$ (61359099583*v^11 - 163761564148*v^10 + 220848638212*v^9 - 5276121240110*v^8 - 5148674145791*v^7 + 73722698062374*v^6 + 47365085225648*v^5 + 463737349458310*v^4 + 2411668696916235*v^3 + 117884875453700*v^2 + 80865527292800*v + 217040473279250) / 342010286590 $$\beta_{10}$$ $$=$$ $$( 22087186843973 \nu^{11} - 51615953404919 \nu^{10} + 59261341856192 \nu^{9} + \cdots + 82\!\cdots\!50 ) / 25650771494250$$ (22087186843973*v^11 - 51615953404919*v^10 + 59261341856192*v^9 - 1870857917788978*v^8 - 2487647672656861*v^7 + 25980214804649133*v^6 + 25839931222402360*v^5 + 171957179902641050*v^4 + 922140119970545325*v^3 + 329211801767110675*v^2 + 29462503150744000*v + 82812042173998750) / 25650771494250 $$\beta_{11}$$ $$=$$ $$( 5754602301876 \nu^{11} - 13680306763182 \nu^{10} + 17291586766932 \nu^{9} + \cdots + 40\!\cdots\!00 ) / 4275128582375$$ (5754602301876*v^11 - 13680306763182*v^10 + 17291586766932*v^9 - 490912185136734*v^8 - 625843938452706*v^7 + 6688361672116830*v^6 + 6288640697723130*v^5 + 46122886080145800*v^4 + 240030876328333650*v^3 + 85658274539300250*v^2 + 60622721125942500*v + 40334896750545000) / 4275128582375
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} - 9 \beta_{9} + 18 \beta_{7} + 9 \beta_{6} + \beta_{5} + 68 \beta_{4} + \cdots + 972 ) / 2916$$ (b11 + b10 - 9*b9 + 18*b7 + 9*b6 + b5 + 68*b4 - 18*b3 + 972*b1 + 972) / 2916 $$\nu^{2}$$ $$=$$ $$( 10\beta_{8} + 27\beta_{6} - 805\beta_{2} ) / 2187$$ (10*b8 + 27*b6 - 805*b2) / 2187 $$\nu^{3}$$ $$=$$ $$( - 35 \beta_{10} + 171 \beta_{9} + 35 \beta_{8} + 23 \beta_{5} + 320 \beta_{4} - 306 \beta_{3} + \cdots + 60264 ) / 2916$$ (-35*b10 + 171*b9 + 35*b8 + 23*b5 + 320*b4 - 306*b3 - 320*b2 + 60264) / 2916 $$\nu^{4}$$ $$=$$ $$( 44\beta_{11} + 333\beta_{7} + 44\beta_{5} - 333\beta_{3} + 149202\beta _1 + 149202 ) / 729$$ (44*b11 + 333*b7 + 44*b5 - 333*b3 + 149202*b1 + 149202) / 729 $$\nu^{5}$$ $$=$$ $$( 1683\beta_{11} + 2615\beta_{8} + 18306\beta_{7} + 11151\beta_{6} - 78680\beta_{2} + 4968864\beta_1 ) / 8748$$ (1683*b11 + 2615*b8 + 18306*b7 + 11151*b6 - 78680*b2 + 4968864*b1) / 8748 $$\nu^{6}$$ $$=$$ $$( -4720\beta_{10} + 17793\beta_{9} + 4720\beta_{8} + 222415\beta_{4} - 222415\beta_{2} ) / 2187$$ (-4720*b10 + 17793*b9 + 4720*b8 + 222415*b4 - 222415*b2) / 2187 $$\nu^{7}$$ $$=$$ $$( 39861 \beta_{11} - 62305 \beta_{10} + 253017 \beta_{9} + 400302 \beta_{7} - 253017 \beta_{6} + \cdots + 122087088 ) / 8748$$ (39861*b11 - 62305*b10 + 253017*b9 + 400302*b7 - 253017*b6 + 39861*b5 + 2310760*b4 - 400302*b3 + 122087088*b1 + 122087088) / 8748 $$\nu^{8}$$ $$=$$ $$( 7672\beta_{11} + 71379\beta_{7} + 24247026\beta_1 ) / 243$$ (7672*b11 + 71379*b7 + 24247026*b1) / 243 $$\nu^{9}$$ $$=$$ $$( - 487645 \beta_{10} + 1945413 \beta_{9} + 487645 \beta_{8} - 311329 \beta_{5} + 19287640 \beta_{4} + \cdots - 965736432 ) / 2916$$ (-487645*b10 + 1945413*b9 + 487645*b8 - 311329*b5 + 19287640*b4 + 3034278*b3 - 19287640*b2 - 965736432) / 2916 $$\nu^{10}$$ $$=$$ $$( -2514580\beta_{10} + 9869877\beta_{9} - 9869877\beta_{6} + 105032935\beta_{4} ) / 2187$$ (-2514580*b10 + 9869877*b9 - 9869877*b6 + 105032935*b4) / 2187 $$\nu^{11}$$ $$=$$ $$( 7266743 \beta_{11} - 11390715 \beta_{8} + 70048026 \beta_{7} - 45149571 \beta_{6} + \cdots + 22643762544 \beta_1 ) / 2916$$ (7266743*b11 - 11390715*b8 + 70048026*b7 - 45149571*b6 + 460613880*b2 + 22643762544*b1) / 2916

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.752622 + 2.80882i −0.486700 + 0.130411i −1.24906 − 4.66155i 4.66155 − 1.24906i 0.130411 + 0.486700i −2.80882 + 0.752622i 0.752622 − 2.80882i −0.486700 − 0.130411i −1.24906 + 4.66155i 4.66155 + 1.24906i 0.130411 − 0.486700i −2.80882 − 0.752622i
−25.1513 14.5211i 0 293.725 + 508.746i 988.609 570.773i 0 309.155 535.472i 9626.03i 0 −33153.0
26.2 −13.8243 7.98144i 0 −0.593312 1.02765i −199.822 + 115.367i 0 −1921.46 + 3328.07i 4105.44i 0 3683.19
26.3 −6.85950 3.96034i 0 −96.6315 167.371i −692.198 + 399.641i 0 2036.81 3527.85i 3558.46i 0 6330.85
26.4 6.85950 + 3.96034i 0 −96.6315 167.371i 692.198 399.641i 0 2036.81 3527.85i 3558.46i 0 6330.85
26.5 13.8243 + 7.98144i 0 −0.593312 1.02765i 199.822 115.367i 0 −1921.46 + 3328.07i 4105.44i 0 3683.19
26.6 25.1513 + 14.5211i 0 293.725 + 508.746i −988.609 + 570.773i 0 309.155 535.472i 9626.03i 0 −33153.0
53.1 −25.1513 + 14.5211i 0 293.725 508.746i 988.609 + 570.773i 0 309.155 + 535.472i 9626.03i 0 −33153.0
53.2 −13.8243 + 7.98144i 0 −0.593312 + 1.02765i −199.822 115.367i 0 −1921.46 3328.07i 4105.44i 0 3683.19
53.3 −6.85950 + 3.96034i 0 −96.6315 + 167.371i −692.198 399.641i 0 2036.81 + 3527.85i 3558.46i 0 6330.85
53.4 6.85950 3.96034i 0 −96.6315 + 167.371i 692.198 + 399.641i 0 2036.81 + 3527.85i 3558.46i 0 6330.85
53.5 13.8243 7.98144i 0 −0.593312 + 1.02765i 199.822 + 115.367i 0 −1921.46 3328.07i 4105.44i 0 3683.19
53.6 25.1513 14.5211i 0 293.725 508.746i −988.609 570.773i 0 309.155 + 535.472i 9626.03i 0 −33153.0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.6 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
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.d.f 12
3.b odd 2 1 inner 81.9.d.f 12
9.c even 3 1 27.9.b.d 6
9.c even 3 1 inner 81.9.d.f 12
9.d odd 6 1 27.9.b.d 6
9.d odd 6 1 inner 81.9.d.f 12
36.f odd 6 1 432.9.e.k 6
36.h even 6 1 432.9.e.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.d 6 9.c even 3 1
27.9.b.d 6 9.d odd 6 1
81.9.d.f 12 1.a even 1 1 trivial
81.9.d.f 12 3.b odd 2 1 inner
81.9.d.f 12 9.c even 3 1 inner
81.9.d.f 12 9.d odd 6 1 inner
432.9.e.k 6 36.f odd 6 1
432.9.e.k 6 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 1161 T_{2}^{10} + 1064097 T_{2}^{8} - 302552496 T_{2}^{6} + 64901621952 T_{2}^{4} + \cdots + 181807037485056$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + \cdots + 181807037485056$$
$3$ $$T^{12}$$
$5$ $$T^{12} + \cdots + 19\!\cdots\!25$$
$7$ $$(T^{6} + \cdots + 93\!\cdots\!25)^{2}$$
$11$ $$T^{12} + \cdots + 48\!\cdots\!25$$
$13$ $$(T^{6} + \cdots + 10\!\cdots\!00)^{2}$$
$17$ $$(T^{6} + \cdots + 16\!\cdots\!76)^{2}$$
$19$ $$(T^{3} + \cdots + 54072465923584)^{4}$$
$23$ $$T^{12} + \cdots + 29\!\cdots\!00$$
$29$ $$T^{12} + \cdots + 10\!\cdots\!00$$
$31$ $$(T^{6} + \cdots + 69\!\cdots\!01)^{2}$$
$37$ $$(T^{3} + \cdots + 22\!\cdots\!00)^{4}$$
$41$ $$T^{12} + \cdots + 23\!\cdots\!00$$
$43$ $$(T^{6} + \cdots + 20\!\cdots\!00)^{2}$$
$47$ $$T^{12} + \cdots + 37\!\cdots\!16$$
$53$ $$(T^{6} + \cdots + 28\!\cdots\!89)^{2}$$
$59$ $$T^{12} + \cdots + 34\!\cdots\!00$$
$61$ $$(T^{6} + \cdots + 95\!\cdots\!16)^{2}$$
$67$ $$(T^{6} + \cdots + 66\!\cdots\!00)^{2}$$
$71$ $$(T^{6} + \cdots + 33\!\cdots\!00)^{2}$$
$73$ $$(T^{3} + \cdots + 43\!\cdots\!75)^{4}$$
$79$ $$(T^{6} + \cdots + 20\!\cdots\!00)^{2}$$
$83$ $$T^{12} + \cdots + 21\!\cdots\!01$$
$89$ $$(T^{6} + \cdots + 10\!\cdots\!00)^{2}$$
$97$ $$(T^{6} + \cdots + 48\!\cdots\!25)^{2}$$