# Properties

 Label 12.9.d.a Level $12$ Weight $9$ Character orbit 12.d Analytic conductor $4.889$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,9,Mod(7,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("12.7");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 12.d (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: $$4.88854332073$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328$$ x^8 - 3*x^7 - 40*x^6 - 395*x^5 + 403*x^4 + 8998*x^3 + 74584*x^2 + 217224*x + 269328 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{22}\cdot 3^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} - 6) q^{4} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 17 \beta_1 - 37) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 142) q^{6} + ( - 2 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 10 \beta_{3} + \beta_{2} + \cdots - 7) q^{7}+ \cdots - 2187 q^{9}+O(q^{10})$$ q + (b1 + 1) * q^2 + (-b3 - b1) * q^3 + (-b3 + b2 - 6) * q^4 + (2*b5 - b4 - b3 + b2 + 17*b1 - 37) * q^5 + (-b6 - 2*b4 + b3 - 2*b2 + b1 + 142) * q^6 + (-2*b7 - 4*b6 + 3*b5 + 3*b4 + 10*b3 + b2 - 20*b1 - 7) * q^7 + (3*b7 - 5*b6 + 6*b5 + 4*b4 + 12*b3 + 3*b2 + 4*b1 - 1621) * q^8 - 2187 * q^9 $$q + (\beta_1 + 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} - 6) q^{4} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 17 \beta_1 - 37) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 142) q^{6} + ( - 2 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 10 \beta_{3} + \beta_{2} + \cdots - 7) q^{7}+ \cdots + (26244 \beta_{7} + 17496 \beta_{6} + 48114 \beta_{5} + \cdots - 170586) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^2 + (-b3 - b1) * q^3 + (-b3 + b2 - 6) * q^4 + (2*b5 - b4 - b3 + b2 + 17*b1 - 37) * q^5 + (-b6 - 2*b4 + b3 - 2*b2 + b1 + 142) * q^6 + (-2*b7 - 4*b6 + 3*b5 + 3*b4 + 10*b3 + b2 - 20*b1 - 7) * q^7 + (3*b7 - 5*b6 + 6*b5 + 4*b4 + 12*b3 + 3*b2 + 4*b1 - 1621) * q^8 - 2187 * q^9 + (4*b7 - 4*b6 + 24*b5 + 28*b2 - 66*b1 + 4570) * q^10 + (-12*b7 - 8*b6 - 22*b5 - 22*b4 - 48*b3 - 34*b2 + 268*b1 + 78) * q^11 + (9*b7 + 9*b6 + 18*b5 - 36*b4 + 5*b3 + 140*b1 - 1377) * q^12 + (-40*b7 + 40*b6 - 12*b5 + 54*b4 - 42*b3 + 66*b2 - 38*b1 - 364) * q^13 + (60*b7 + 16*b6 - 56*b5 + 24*b4 - 348*b3 - 20*b2 - 172*b1 + 6620) * q^14 + (-18*b7 + 4*b6 - 9*b5 - 73*b4 + 104*b3 - 91*b2 + 50*b1 - 19) * q^15 + (58*b7 + 26*b6 - 108*b5 + 24*b4 + 708*b3 + 14*b2 - 1352*b1 + 11906) * q^16 + (-48*b7 + 48*b6 - 164*b5 + 306*b4 - 142*b3 - 162*b2 + 846*b1 - 24032) * q^17 + (-2187*b1 - 2187) * q^18 + (-108*b7 - 24*b6 + 66*b5 - 318*b4 + 280*b3 - 426*b2 - 1532*b1 - 474) * q^19 + (120*b7 - 72*b6 + 112*b5 + 160*b4 + 1282*b3 + 86*b2 + 5024*b1 + 42884) * q^20 + (-72*b7 + 72*b6 + 18*b5 - 153*b4 + 135*b3 + 369*b2 - 2727*b1 + 14625) * q^21 + (72*b7 + 36*b6 + 240*b5 - 552*b4 - 2892*b3 + 112*b2 - 1196*b1 - 69040) * q^22 + (60*b7 - 200*b6 - 314*b5 + 710*b4 - 1412*b3 + 770*b2 + 5952*b1 + 1906) * q^23 + (-117*b7 - 13*b6 + 630*b5 - 188*b4 + 1840*b3 + 343*b2 - 860*b1 + 27163) * q^24 + (176*b7 - 176*b6 + 984*b5 - 780*b4 - 204*b3 + 252*b2 + 7084*b1 - 70257) * q^25 + (-216*b7 - 424*b6 - 16*b5 + 1280*b4 - 8576*b3 + 24*b2 - 3926*b1 + 3650) * q^26 + (2187*b3 + 2187*b1) * q^27 + (-532*b7 - 212*b6 - 936*b5 - 816*b4 + 12164*b3 - 776*b2 + 12688*b1 - 74140) * q^28 + (624*b7 - 624*b6 - 466*b5 - 439*b4 + 905*b3 - 1433*b2 - 3961*b1 + 256625) * q^29 + (-180*b7 + 342*b6 + 936*b5 - 828*b4 - 4466*b3 + 72*b2 - 2306*b1 + 6552) * q^30 + (782*b7 + 700*b6 + 2523*b5 + 987*b4 - 1934*b3 + 1769*b2 - 41564*b1 - 9887) * q^31 + (-552*b7 + 632*b6 - 656*b5 + 1504*b4 + 13224*b3 - 848*b2 + 19872*b1 - 449432) * q^32 + (288*b7 - 288*b6 - 1476*b5 - 414*b4 + 1890*b3 - 450*b2 - 23490*b1 - 121302) * q^33 + (-1224*b7 + 456*b6 - 5808*b5 + 1536*b4 - 15616*b3 - 1400*b2 - 30630*b1 + 191458) * q^34 + (1068*b7 + 1576*b6 - 1994*b5 - 202*b4 + 2500*b3 + 866*b2 + 29832*b1 + 6706) * q^35 + (2187*b3 - 2187*b2 + 13122) * q^36 + (40*b7 - 40*b6 + 2856*b5 + 828*b4 - 3684*b3 - 948*b2 + 54164*b1 + 947822) * q^37 + (-216*b7 + 1564*b6 + 3888*b5 - 4216*b4 - 23716*b3 - 1600*b2 - 12676*b1 + 460448) * q^38 + (324*b7 + 720*b6 - 4698*b5 - 666*b4 - 4526*b3 - 342*b2 + 67078*b1 + 17802) * q^39 + (554*b7 + 1594*b6 - 3756*b5 + 3384*b4 + 21128*b3 + 6090*b2 + 48696*b1 + 225818) * q^40 + (-816*b7 + 816*b6 - 4028*b5 - 1218*b4 + 5246*b3 + 3666*b2 - 87678*b1 - 1130396) * q^41 + (612*b7 - 1764*b6 + 5976*b5 + 2304*b4 - 8064*b3 - 324*b2 + 13860*b1 - 655776) * q^42 + (-1652*b7 - 2824*b6 + 7806*b5 + 1278*b4 - 4712*b3 - 374*b2 - 119060*b1 - 28294) * q^43 + (132*b7 - 3580*b6 + 10248*b5 - 8976*b4 + 20708*b3 - 880*b2 - 61264*b1 + 281916) * q^44 + (-4374*b5 + 2187*b4 + 2187*b3 - 2187*b2 - 37179*b1 + 80919) * q^45 + (3960*b7 - 3576*b6 - 9840*b5 + 4608*b4 + 14432*b3 + 3752*b2 + 12736*b1 - 1705976) * q^46 + (-5724*b7 - 5208*b6 - 1126*b5 - 7014*b4 + 21964*b3 - 12738*b2 + 35336*b1 + 3214) * q^47 + (1854*b7 + 702*b6 + 6300*b5 + 5832*b4 - 18700*b3 + 4338*b2 + 13160*b1 + 1371942) * q^48 + (-1616*b7 + 1616*b6 + 19248*b5 + 5064*b4 - 24312*b3 - 216*b2 + 331928*b1 - 2278151) * q^49 + (3120*b7 - 304*b6 + 13088*b5 - 5632*b4 + 40192*b3 + 12624*b2 - 66013*b1 + 1800819) * q^50 + (-1260*b7 - 4760*b6 - 6462*b5 + 7490*b4 + 12986*b3 + 6230*b2 + 147878*b1 + 34598) * q^51 + (5168*b7 - 6992*b6 - 23712*b5 - 10176*b4 - 43306*b3 - 18406*b2 - 12736*b1 + 2331444) * q^52 + (-6480*b7 + 6480*b6 - 354*b5 - 2159*b4 + 2513*b3 + 21599*b2 - 124097*b1 + 1061001) * q^53 + (2187*b6 + 4374*b4 - 2187*b3 + 4374*b2 - 2187*b1 - 310554) * q^54 + (-1264*b7 + 2752*b6 + 8664*b5 - 11304*b4 - 87016*b3 - 12568*b2 - 261784*b1 - 44696) * q^55 + (3372*b7 + 12876*b6 + 13016*b5 + 11280*b4 - 123680*b3 + 22716*b2 - 126832*b1 - 5696852) * q^56 + (-2160*b7 + 2160*b6 - 21492*b5 + 11610*b4 + 9882*b3 - 5130*b2 - 201690*b1 - 348246) * q^57 + (1756*b7 + 8228*b6 - 9432*b5 - 19968*b4 + 131328*b3 - 9212*b2 + 319590*b1 - 1054358) * q^58 + (3888*b7 + 9984*b6 - 17240*b5 - 11352*b4 + 57572*b3 - 7464*b2 + 278548*b1 + 53720) * q^59 + (-2682*b7 - 5690*b6 + 18828*b5 - 9112*b4 - 35242*b3 - 616*b2 - 19960*b1 + 3550874) * q^60 + (2312*b7 - 2312*b6 - 12240*b5 + 10104*b4 + 2136*b3 - 17040*b2 - 19880*b1 + 1670450) * q^61 + (-7908*b7 - 5336*b6 - 9208*b5 + 23560*b4 + 206380*b3 - 33924*b2 + 78428*b1 + 10052972) * q^62 + (4374*b7 + 8748*b6 - 6561*b5 - 6561*b4 - 21870*b3 - 2187*b2 + 43740*b1 + 15309) * q^63 + (-10056*b7 + 15096*b6 - 23952*b5 + 40608*b4 - 165952*b3 + 23640*b2 - 534112*b1 + 58552) * q^64 + (13104*b7 - 13104*b6 - 25684*b5 + 6090*b4 + 19594*b3 - 45402*b2 - 122634*b1 + 866522) * q^65 + (1656*b7 + 2952*b6 + 720*b5 - 9216*b4 + 87552*b3 - 25272*b2 - 51228*b1 - 6440148) * q^66 + (8624*b7 + 1408*b6 + 28968*b5 + 26664*b4 + 237644*b3 + 35288*b2 - 160900*b1 - 97832) * q^67 + (-6768*b7 - 17648*b6 - 23520*b5 - 29504*b4 - 255450*b3 - 68694*b2 + 157376*b1 - 14563212) * q^68 + (6192*b7 - 6192*b6 + 40356*b5 - 42354*b4 + 1998*b3 + 23778*b2 + 141426*b1 - 1026630) * q^69 + (-21960*b7 - 4176*b6 + 9744*b5 + 16080*b4 + 183032*b3 + 35768*b2 + 95320*b1 - 7562600) * q^70 + (16908*b7 - 6344*b6 + 12318*b5 + 75038*b4 - 323236*b3 + 91946*b2 - 311056*b1 + 8858) * q^71 + (-6561*b7 + 10935*b6 - 13122*b5 - 8748*b4 - 26244*b3 - 6561*b2 - 8748*b1 + 3545127) * q^72 + (5584*b7 - 5584*b6 + 64752*b5 - 86232*b4 + 21480*b3 + 69480*b2 - 71560*b1 + 11859434) * q^73 + (-3312*b7 + 3952*b6 - 21152*b5 - 1280*b4 - 65152*b3 + 49008*b2 + 847938*b1 + 15183410) * q^74 + (-8424*b7 + 1152*b6 + 17172*b5 - 32364*b4 + 123045*b3 - 40788*b2 - 237891*b1 - 92628) * q^75 + (-20052*b7 - 21204*b6 + 77400*b5 - 66480*b4 - 51284*b3 - 15024*b2 + 385936*b1 + 18205140) * q^76 + (24576*b7 - 24576*b6 + 46552*b5 - 7276*b4 - 39276*b3 - 66452*b2 + 947756*b1 - 6883316) * q^77 + (-11016*b7 - 9314*b6 + 11664*b5 - 19924*b4 - 5302*b3 + 55892*b2 + 33578*b1 - 17547820) * q^78 + (3190*b7 + 2828*b6 - 48897*b5 + 4095*b4 - 395910*b3 + 7285*b2 + 389700*b1 + 196493) * q^79 + (-8052*b7 - 5684*b6 - 19240*b5 + 112080*b4 + 95288*b3 + 76324*b2 + 307856*b1 - 20436612) * q^80 + 4782969 * q^81 + (4872*b7 - 17928*b6 + 55344*b5 + 26112*b4 - 40192*b3 - 69320*b2 - 1034502*b1 - 23815886) * q^82 + (-7788*b7 - 12776*b6 + 17738*b5 + 4682*b4 + 507480*b3 - 3106*b2 + 268564*b1 - 58482) * q^83 + (17208*b7 + 6264*b6 - 58896*b5 - 7776*b4 + 116892*b3 + 3708*b2 - 682848*b1 + 24765408) * q^84 + (-10752*b7 + 10752*b6 - 198084*b5 + 77538*b4 + 120546*b3 - 45282*b2 - 2113794*b1 - 25742030) * q^85 + (40920*b7 + 8212*b6 - 28976*b5 - 7464*b4 - 203916*b3 - 120416*b2 - 143788*b1 + 30025664) * q^86 + (-558*b7 - 14996*b6 + 75537*b5 + 35537*b4 - 202300*b3 + 34979*b2 - 1280950*b1 - 266053) * q^87 + (52348*b7 + 32924*b6 + 133752*b5 - 51504*b4 + 188288*b3 + 5164*b2 + 288848*b1 + 19571516) * q^88 + (-62304*b7 + 62304*b6 - 8888*b5 + 98076*b4 - 89188*b3 + 88836*b2 + 269412*b1 + 23695798) * q^89 + (-8748*b7 + 8748*b6 - 52488*b5 - 61236*b2 + 144342*b1 - 9994590) * q^90 + (-36964*b7 + 9400*b6 - 193194*b5 - 152874*b4 + 1063572*b3 - 189838*b2 + 3737400*b1 + 656866) * q^91 + (37296*b7 + 9264*b6 - 113568*b5 + 16960*b4 + 898688*b3 - 3088*b2 - 1160128*b1 - 48979280) * q^92 + (-38376*b7 + 38376*b6 + 64674*b5 + 87903*b4 - 152577*b3 + 27225*b2 + 1575585*b1 - 6443055) * q^93 + (59400*b7 + 63744*b6 + 64368*b5 - 120048*b4 - 1256680*b3 + 21736*b2 - 597576*b1 - 4659448) * q^94 + (-39552*b7 + 29696*b6 + 35136*b5 - 212672*b4 - 1229960*b3 - 252224*b2 - 2415368*b1 - 313664) * q^95 + (-9360*b7 - 21616*b6 - 89568*b5 + 47488*b4 + 369880*b3 + 2488*b2 + 1463488*b1 + 31029088) * q^96 + (-15104*b7 + 15104*b6 - 138648*b5 + 123468*b4 + 15180*b3 - 78156*b2 - 686092*b1 - 15640954) * q^97 + (-20256*b7 - 5600*b6 - 69824*b5 + 51712*b4 - 754432*b3 + 328224*b2 - 3052399*b1 + 85881265) * q^98 + (26244*b7 + 17496*b6 + 48114*b5 + 48114*b4 + 104976*b3 + 74358*b2 - 586116*b1 - 170586) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{2} - 52 q^{4} - 336 q^{5} + 1134 q^{6} - 12960 q^{8} - 17496 q^{9}+O(q^{10})$$ 8 * q + 6 * q^2 - 52 * q^4 - 336 * q^5 + 1134 * q^6 - 12960 * q^8 - 17496 * q^9 $$8 q + 6 q^{2} - 52 q^{4} - 336 q^{5} + 1134 q^{6} - 12960 q^{8} - 17496 q^{9} + 36628 q^{10} - 11340 q^{12} - 2864 q^{13} + 52728 q^{14} + 99440 q^{16} - 193200 q^{17} - 13122 q^{18} + 335592 q^{20} + 121824 q^{21} - 556968 q^{22} + 221616 q^{24} - 579048 q^{25} + 21564 q^{26} - 594672 q^{28} + 2063472 q^{29} + 46980 q^{30} - 3602784 q^{32} - 920160 q^{33} + 1568476 q^{34} + 113724 q^{36} + 7470352 q^{37} + 3659400 q^{38} + 1749184 q^{40} - 8865456 q^{41} - 5288328 q^{42} + 2395920 q^{44} + 734832 q^{45} - 13649856 q^{46} + 10916208 q^{48} - 18923896 q^{49} + 14581842 q^{50} + 18592888 q^{52} + 8706672 q^{53} - 2480058 q^{54} - 45565632 q^{56} - 2325024 q^{57} - 8816444 q^{58} + 28348056 q^{60} + 13457296 q^{61} + 80783976 q^{62} + 1268864 q^{64} + 7293408 q^{65} - 51205608 q^{66} - 117288264 q^{68} - 8636544 q^{69} - 60373104 q^{70} + 28343520 q^{72} + 94738960 q^{73} + 119548428 q^{74} + 144621360 q^{76} - 56971392 q^{77} - 140630580 q^{78} - 163857888 q^{80} + 38263752 q^{81} - 188383460 q^{82} + 199712304 q^{84} - 201200416 q^{85} + 240327384 q^{86} + 156323520 q^{88} + 188992272 q^{89} - 80105436 q^{90} - 387657984 q^{92} - 54802656 q^{93} - 38749872 q^{94} + 246092256 q^{96} - 123291632 q^{97} + 691081830 q^{98}+O(q^{100})$$ 8 * q + 6 * q^2 - 52 * q^4 - 336 * q^5 + 1134 * q^6 - 12960 * q^8 - 17496 * q^9 + 36628 * q^10 - 11340 * q^12 - 2864 * q^13 + 52728 * q^14 + 99440 * q^16 - 193200 * q^17 - 13122 * q^18 + 335592 * q^20 + 121824 * q^21 - 556968 * q^22 + 221616 * q^24 - 579048 * q^25 + 21564 * q^26 - 594672 * q^28 + 2063472 * q^29 + 46980 * q^30 - 3602784 * q^32 - 920160 * q^33 + 1568476 * q^34 + 113724 * q^36 + 7470352 * q^37 + 3659400 * q^38 + 1749184 * q^40 - 8865456 * q^41 - 5288328 * q^42 + 2395920 * q^44 + 734832 * q^45 - 13649856 * q^46 + 10916208 * q^48 - 18923896 * q^49 + 14581842 * q^50 + 18592888 * q^52 + 8706672 * q^53 - 2480058 * q^54 - 45565632 * q^56 - 2325024 * q^57 - 8816444 * q^58 + 28348056 * q^60 + 13457296 * q^61 + 80783976 * q^62 + 1268864 * q^64 + 7293408 * q^65 - 51205608 * q^66 - 117288264 * q^68 - 8636544 * q^69 - 60373104 * q^70 + 28343520 * q^72 + 94738960 * q^73 + 119548428 * q^74 + 144621360 * q^76 - 56971392 * q^77 - 140630580 * q^78 - 163857888 * q^80 + 38263752 * q^81 - 188383460 * q^82 + 199712304 * q^84 - 201200416 * q^85 + 240327384 * q^86 + 156323520 * q^88 + 188992272 * q^89 - 80105436 * q^90 - 387657984 * q^92 - 54802656 * q^93 - 38749872 * q^94 + 246092256 * q^96 - 123291632 * q^97 + 691081830 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328$$ :

 $$\beta_{1}$$ $$=$$ $$( - 13071742 \nu^{7} - 40510221 \nu^{6} + 365720344 \nu^{5} + 4477637544 \nu^{4} + 21313950753 \nu^{3} + 10344779632 \nu^{2} + \cdots - 864197730939 ) / 48999153875$$ (-13071742*v^7 - 40510221*v^6 + 365720344*v^5 + 4477637544*v^4 + 21313950753*v^3 + 10344779632*v^2 - 18901658816*v - 864197730939) / 48999153875 $$\beta_{2}$$ $$=$$ $$( - 442676109 \nu^{7} + 14876179533 \nu^{6} - 20899272462 \nu^{5} - 244299281437 \nu^{4} - 3890047326769 \nu^{3} + \cdots + 257458199902472 ) / 783986462000$$ (-442676109*v^7 + 14876179533*v^6 - 20899272462*v^5 - 244299281437*v^4 - 3890047326769*v^3 - 319558226336*v^2 + 19966153994968*v + 257458199902472) / 783986462000 $$\beta_{3}$$ $$=$$ $$( 16258001 \nu^{7} - 126478257 \nu^{6} - 1086101402 \nu^{5} - 4794860127 \nu^{4} + 44454963621 \nu^{3} + 311217070624 \nu^{2} + \cdots + 2529327015672 ) / 22399613200$$ (16258001*v^7 - 126478257*v^6 - 1086101402*v^5 - 4794860127*v^4 + 44454963621*v^3 + 311217070624*v^2 + 1515996537928*v + 2529327015672) / 22399613200 $$\beta_{4}$$ $$=$$ $$( 909414791 \nu^{7} + 1589452833 \nu^{6} + 7394147338 \nu^{5} - 139320190137 \nu^{4} - 1333799641269 \nu^{3} + \cdots - 157888370611128 ) / 391993231000$$ (909414791*v^7 + 1589452833*v^6 + 7394147338*v^5 - 139320190137*v^4 - 1333799641269*v^3 - 15999239006336*v^2 - 53245255804232*v - 157888370611128) / 391993231000 $$\beta_{5}$$ $$=$$ $$( 1924762227 \nu^{7} + 1444346301 \nu^{6} + 32579772786 \nu^{5} - 555685145789 \nu^{4} - 1808597634593 \nu^{3} + \cdots - 343266156881416 ) / 783986462000$$ (1924762227*v^7 + 1444346301*v^6 + 32579772786*v^5 - 555685145789*v^4 - 1808597634593*v^3 - 28806873474592*v^2 - 97146730558504*v - 343266156881416) / 783986462000 $$\beta_{6}$$ $$=$$ $$( - 2088295459 \nu^{7} + 1743279843 \nu^{6} + 40266101998 \nu^{5} + 817766792173 \nu^{4} + 814888424641 \nu^{3} + \cdots - 153540313829368 ) / 156797292400$$ (-2088295459*v^7 + 1743279843*v^6 + 40266101998*v^5 + 817766792173*v^4 + 814888424641*v^3 - 2033961609696*v^2 - 83293581219672*v - 153540313829368) / 156797292400 $$\beta_{7}$$ $$=$$ $$( - 844419621 \nu^{7} + 4157547477 \nu^{6} + 19453004322 \nu^{5} + 248245173547 \nu^{4} - 610375035161 \nu^{3} + \cdots - 74886881534632 ) / 55999033000$$ (-844419621*v^7 + 4157547477*v^6 + 19453004322*v^5 + 248245173547*v^4 - 610375035161*v^3 - 3612345119584*v^2 - 37207566428008*v - 74886881534632) / 55999033000
 $$\nu$$ $$=$$ $$( 3\beta_{7} - 5\beta_{6} - 12\beta_{5} + 8\beta_{4} - 7\beta_{2} + 54\beta _1 + 335 ) / 864$$ (3*b7 - 5*b6 - 12*b5 + 8*b4 - 7*b2 + 54*b1 + 335) / 864 $$\nu^{2}$$ $$=$$ $$( 12\beta_{7} - 11\beta_{6} - 12\beta_{5} - 22\beta_{4} - 16\beta_{3} - 55\beta_{2} - 358\beta _1 + 4715 ) / 432$$ (12*b7 - 11*b6 - 12*b5 - 22*b4 - 16*b3 - 55*b2 - 358*b1 + 4715) / 432 $$\nu^{3}$$ $$=$$ $$( 207\beta_{7} - 175\beta_{6} - 18\beta_{5} + 64\beta_{4} + 914\beta_{3} - 413\beta_{2} + 788\beta _1 + 169669 ) / 864$$ (207*b7 - 175*b6 - 18*b5 + 64*b4 + 914*b3 - 413*b2 + 788*b1 + 169669) / 864 $$\nu^{4}$$ $$=$$ $$( 1395 \beta_{7} - 1505 \beta_{6} - 3942 \beta_{5} + 1940 \beta_{4} + 374 \beta_{3} - 4999 \beta_{2} - 11524 \beta _1 + 843359 ) / 864$$ (1395*b7 - 1505*b6 - 3942*b5 + 1940*b4 + 374*b3 - 4999*b2 - 11524*b1 + 843359) / 864 $$\nu^{5}$$ $$=$$ $$( 6099 \beta_{7} - 6978 \beta_{6} - 7368 \beta_{5} - 2886 \beta_{4} - 5956 \beta_{3} - 22716 \beta_{2} - 57400 \beta _1 + 4054608 ) / 432$$ (6099*b7 - 6978*b6 - 7368*b5 - 2886*b4 - 5956*b3 - 22716*b2 - 57400*b1 + 4054608) / 432 $$\nu^{6}$$ $$=$$ $$( 114921 \beta_{7} - 109501 \beta_{6} - 117126 \beta_{5} + 41656 \beta_{4} + 242822 \beta_{3} - 286079 \beta_{2} - 478564 \beta _1 + 70038967 ) / 864$$ (114921*b7 - 109501*b6 - 117126*b5 + 41656*b4 + 242822*b3 - 286079*b2 - 478564*b1 + 70038967) / 864 $$\nu^{7}$$ $$=$$ $$( 815151 \beta_{7} - 862161 \beta_{6} - 1430598 \beta_{5} + 431916 \beta_{4} + 507302 \beta_{3} - 2847231 \beta_{2} - 8274772 \beta _1 + 525219687 ) / 864$$ (815151*b7 - 862161*b6 - 1430598*b5 + 431916*b4 + 507302*b3 - 2847231*b2 - 8274772*b1 + 525219687) / 864

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.97054 − 1.25304i −1.97054 + 1.25304i 8.23534 + 0.0522875i 8.23534 − 0.0522875i −1.11858 − 4.64627i −1.11858 + 4.64627i −3.64622 − 4.31154i −3.64622 + 4.31154i
−15.8645 2.07809i 46.7654i 247.363 + 65.9356i −374.901 97.1827 741.908i 4472.52i −3787.26 1560.08i −2187.00 5947.61 + 779.079i
7.2 −15.8645 + 2.07809i 46.7654i 247.363 65.9356i −374.901 97.1827 + 741.908i 4472.52i −3787.26 + 1560.08i −2187.00 5947.61 779.079i
7.3 −2.26258 15.8392i 46.7654i −245.761 + 71.6749i −538.046 740.727 105.810i 3350.97i 1691.33 + 3730.50i −2187.00 1217.37 + 8522.23i
7.4 −2.26258 + 15.8392i 46.7654i −245.761 71.6749i −538.046 740.727 + 105.810i 3350.97i 1691.33 3730.50i −2187.00 1217.37 8522.23i
7.5 7.47945 14.1442i 46.7654i −144.116 211.581i −159.249 −661.458 349.779i 707.133i −4070.55 + 455.883i −2187.00 −1191.10 + 2252.45i
7.6 7.47945 + 14.1442i 46.7654i −144.116 + 211.581i −159.249 −661.458 + 349.779i 707.133i −4070.55 455.883i −2187.00 −1191.10 2252.45i
7.7 13.6476 8.35123i 46.7654i 116.514 227.948i 904.196 390.548 + 638.235i 888.085i −313.513 4083.98i −2187.00 12340.1 7551.15i
7.8 13.6476 + 8.35123i 46.7654i 116.514 + 227.948i 904.196 390.548 638.235i 888.085i −313.513 + 4083.98i −2187.00 12340.1 + 7551.15i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.9.d.a 8
3.b odd 2 1 36.9.d.c 8
4.b odd 2 1 inner 12.9.d.a 8
8.b even 2 1 192.9.g.e 8
8.d odd 2 1 192.9.g.e 8
12.b even 2 1 36.9.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.d.a 8 1.a even 1 1 trivial
12.9.d.a 8 4.b odd 2 1 inner
36.9.d.c 8 3.b odd 2 1
36.9.d.c 8 12.b even 2 1
192.9.g.e 8 8.b even 2 1
192.9.g.e 8 8.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 6 T^{7} + \cdots + 4294967296$$
$3$ $$(T^{2} + 2187)^{4}$$
$5$ $$(T^{4} + 168 T^{3} + \cdots - 29045327600)^{2}$$
$7$ $$T^{8} + 32521152 T^{6} + \cdots + 88\!\cdots\!36$$
$11$ $$T^{8} + 731054784 T^{6} + \cdots + 71\!\cdots\!04$$
$13$ $$(T^{4} + 1432 T^{3} + \cdots + 46\!\cdots\!76)^{2}$$
$17$ $$(T^{4} + 96600 T^{3} + \cdots + 31\!\cdots\!00)^{2}$$
$19$ $$T^{8} + 68043402432 T^{6} + \cdots + 20\!\cdots\!56$$
$23$ $$T^{8} + 372146313216 T^{6} + \cdots + 83\!\cdots\!76$$
$29$ $$(T^{4} - 1031736 T^{3} + \cdots - 15\!\cdots\!24)^{2}$$
$31$ $$T^{8} + 5136946069440 T^{6} + \cdots + 17\!\cdots\!64$$
$37$ $$(T^{4} - 3735176 T^{3} + \cdots - 48\!\cdots\!84)^{2}$$
$41$ $$(T^{4} + 4432728 T^{3} + \cdots - 44\!\cdots\!92)^{2}$$
$43$ $$T^{8} + 42553473302208 T^{6} + \cdots + 21\!\cdots\!16$$
$47$ $$T^{8} + 108463214970624 T^{6} + \cdots + 26\!\cdots\!00$$
$53$ $$(T^{4} - 4353336 T^{3} + \cdots - 39\!\cdots\!44)^{2}$$
$59$ $$T^{8} + 345827707315392 T^{6} + \cdots + 15\!\cdots\!96$$
$61$ $$(T^{4} - 6728648 T^{3} + \cdots + 34\!\cdots\!04)^{2}$$
$67$ $$T^{8} + \cdots + 10\!\cdots\!76$$
$71$ $$T^{8} + \cdots + 30\!\cdots\!00$$
$73$ $$(T^{4} - 47369480 T^{3} + \cdots + 42\!\cdots\!28)^{2}$$
$79$ $$T^{8} + \cdots + 29\!\cdots\!24$$
$83$ $$T^{8} + \cdots + 25\!\cdots\!84$$
$89$ $$(T^{4} - 94496136 T^{3} + \cdots - 16\!\cdots\!88)^{2}$$
$97$ $$(T^{4} + 61645816 T^{3} + \cdots + 30\!\cdots\!48)^{2}$$