# Properties

 Label 285.10.a.d Level $285$ Weight $10$ Character orbit 285.a Self dual yes Analytic conductor $146.785$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,10,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("285.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 285.a (trivial)

## 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: yes Analytic conductor: $$146.785213307$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} - 4398 x^{10} + 11376 x^{9} + 7070146 x^{8} - 15274638 x^{7} - 5114407260 x^{6} + \cdots + 43\!\cdots\!00$$ x^12 - 3*x^11 - 4398*x^10 + 11376*x^9 + 7070146*x^8 - 15274638*x^7 - 5114407260*x^6 + 8284970600*x^5 + 1621868619141*x^4 - 1541179405015*x^3 - 176385539891230*x^2 + 130401229370000*x + 4344410152258400 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}\cdot 3^{3}\cdot 5^{2}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 3) q^{2} + 81 q^{3} + (\beta_{2} - 6 \beta_1 + 231) q^{4} + 625 q^{5} + (81 \beta_1 - 243) q^{6} + (\beta_{5} - \beta_{2} - 52 \beta_1 - 412) q^{7} + ( - \beta_{5} + \beta_{4} + \cdots - 3400) q^{8}+ \cdots + 6561 q^{9}+O(q^{10})$$ q + (b1 - 3) * q^2 + 81 * q^3 + (b2 - 6*b1 + 231) * q^4 + 625 * q^5 + (81*b1 - 243) * q^6 + (b5 - b2 - 52*b1 - 412) * q^7 + (-b5 + b4 - 8*b2 + 187*b1 - 3400) * q^8 + 6561 * q^9 $$q + (\beta_1 - 3) q^{2} + 81 q^{3} + (\beta_{2} - 6 \beta_1 + 231) q^{4} + 625 q^{5} + (81 \beta_1 - 243) q^{6} + (\beta_{5} - \beta_{2} - 52 \beta_1 - 412) q^{7} + ( - \beta_{5} + \beta_{4} + \cdots - 3400) q^{8}+ \cdots + (6561 \beta_{11} + 6561 \beta_{9} + \cdots - 33178977) q^{99}+O(q^{100})$$ q + (b1 - 3) * q^2 + 81 * q^3 + (b2 - 6*b1 + 231) * q^4 + 625 * q^5 + (81*b1 - 243) * q^6 + (b5 - b2 - 52*b1 - 412) * q^7 + (-b5 + b4 - 8*b2 + 187*b1 - 3400) * q^8 + 6561 * q^9 + (625*b1 - 1875) * q^10 + (b11 + b9 + b8 + b6 - 4*b5 + b4 + 3*b3 - 17*b2 + 69*b1 - 5057) * q^11 + (81*b2 - 486*b1 + 18711) * q^12 + (-3*b11 - 3*b10 + b9 - 4*b8 + 2*b7 - 5*b6 - 4*b5 - 2*b4 - 4*b3 - 73*b2 - 609*b1 - 13535) * q^13 + (b11 + 5*b10 + 7*b9 - 9*b8 + b7 - 13*b6 - b5 - 8*b4 + 3*b3 - 142*b2 - 655*b1 - 36932) * q^14 + 50625 * q^15 + (-10*b10 - 4*b9 + 10*b8 - 6*b7 + 30*b6 + 8*b5 + 2*b4 - 10*b3 + 179*b2 - 4832*b1 + 27863) * q^16 + (2*b11 + 16*b10 - 29*b9 + 26*b8 - b7 + 7*b6 + 19*b5 + 9*b3 + 29*b2 + 399*b1 - 46160) * q^17 + (6561*b1 - 19683) * q^18 - 130321 * q^19 + (625*b2 - 3750*b1 + 144375) * q^20 + (81*b5 - 81*b2 - 4212*b1 - 33372) * q^21 + (-13*b11 - 61*b10 + 3*b9 - 53*b8 + 15*b7 + 37*b6 + 61*b5 - 28*b4 - 95*b3 + 522*b2 - 13499*b1 + 62562) * q^22 + (23*b11 + 64*b10 + 18*b9 + 18*b8 - 52*b7 - 37*b6 - 67*b5 + 11*b4 - 27*b3 + 630*b2 - 3340*b1 - 228130) * q^23 + (-81*b5 + 81*b4 - 648*b2 + 15147*b1 - 275400) * q^24 + 390625 * q^25 + (-29*b11 - 55*b10 - 113*b9 + 57*b8 - 39*b7 + 43*b6 + 109*b5 - 78*b4 + 99*b3 + 246*b2 - 44309*b1 - 418768) * q^26 + 531441 * q^27 + (-23*b11 - 78*b10 + 99*b9 + 6*b8 + 108*b7 - 28*b6 + 204*b5 - 276*b4 - 70*b3 - 703*b2 - 72144*b1 - 180421) * q^28 + (-165*b11 + 134*b10 - 65*b9 - 20*b8 + 87*b7 - 99*b6 - 209*b5 - 161*b4 + 67*b3 - 848*b2 - 43586*b1 - 273985) * q^29 + (50625*b1 - 151875) * q^30 + (239*b11 - 99*b10 + 3*b9 - 78*b8 - 30*b7 + 284*b6 - 133*b5 - 232*b4 - 167*b3 + 1072*b2 - 43960*b1 - 807554) * q^31 + (272*b11 + 464*b10 - 136*b9 + 192*b8 - 16*b7 - 176*b6 - 813*b5 + 61*b4 + 368*b3 - 5352*b2 + 32963*b1 - 1866604) * q^32 + (81*b11 + 81*b9 + 81*b8 + 81*b6 - 324*b5 + 81*b4 + 243*b3 - 1377*b2 + 5589*b1 - 409617) * q^33 + (-292*b11 + 130*b10 + 150*b9 - 226*b8 - 220*b7 - 286*b6 - 266*b5 - 118*b4 - 126*b3 - 4450*b2 - 35258*b1 + 442240) * q^34 + (625*b5 - 625*b2 - 32500*b1 - 257500) * q^35 + (6561*b2 - 39366*b1 + 1515591) * q^36 + (-455*b11 - 761*b10 + 387*b9 - 80*b8 + 272*b7 - 131*b6 - 216*b5 - 114*b4 - 254*b3 + 887*b2 - 169509*b1 - 2003899) * q^37 + (-130321*b1 + 390963) * q^38 + (-243*b11 - 243*b10 + 81*b9 - 324*b8 + 162*b7 - 405*b6 - 324*b5 - 162*b4 - 324*b3 - 5913*b2 - 49329*b1 - 1096335) * q^39 + (-625*b5 + 625*b4 - 5000*b2 + 116875*b1 - 2125000) * q^40 + (394*b11 - 1123*b10 + 1510*b9 - 1013*b8 + 316*b7 - 654*b6 - 563*b5 - 745*b4 - 459*b3 - 807*b2 + 828*b1 - 2988522) * q^41 + (81*b11 + 405*b10 + 567*b9 - 729*b8 + 81*b7 - 1053*b6 - 81*b5 - 648*b4 + 243*b3 - 11502*b2 - 53055*b1 - 2991492) * q^42 + (-145*b11 - 805*b10 - 2203*b9 - 834*b8 - 424*b7 + 1303*b6 + 344*b5 + 644*b4 + 628*b3 - 171*b2 + 106137*b1 - 6179463) * q^43 + (831*b11 + 1254*b10 - 927*b9 + 950*b8 + 348*b7 - 1116*b6 - 1632*b5 + 1056*b4 + 1414*b3 - 14525*b2 + 327648*b1 - 7431171) * q^44 + 4100625 * q^45 + (526*b11 + 1380*b10 + 650*b9 + 2688*b8 + 140*b7 + 1320*b6 + 726*b5 + 1846*b4 - 492*b3 + 2204*b2 + 52514*b1 - 1677272) * q^46 + (-1837*b11 + 626*b10 + 1530*b9 - 677*b8 - 2593*b7 + 663*b6 - 1048*b5 + 943*b4 + 233*b3 - 14299*b2 + 245893*b1 - 11290496) * q^47 + (-810*b10 - 324*b9 + 810*b8 - 486*b7 + 2430*b6 + 648*b5 + 162*b4 - 810*b3 + 14499*b2 - 391392*b1 + 2256903) * q^48 + (1549*b11 + 79*b10 + 176*b9 + 1137*b8 + 1060*b7 + 618*b6 - 513*b5 + 802*b4 - 2213*b3 + 11384*b2 + 302147*b1 - 5434690) * q^49 + (390625*b1 - 1171875) * q^50 + (162*b11 + 1296*b10 - 2349*b9 + 2106*b8 - 81*b7 + 567*b6 + 1539*b5 + 729*b3 + 2349*b2 + 32319*b1 - 3738960) * q^51 + (-1249*b11 - 402*b10 + 681*b9 - 2746*b8 + 844*b7 - 2292*b6 + 1018*b5 - 1994*b4 + 782*b3 - 59677*b2 + 164206*b1 - 24248375) * q^52 + (2649*b11 - 222*b10 + 1158*b9 + 1738*b8 + 3258*b7 + 2613*b6 + 2265*b5 + 275*b4 - 407*b3 - 12068*b2 - 338674*b1 - 6153612) * q^53 + (531441*b1 - 1594323) * q^54 + (625*b11 + 625*b9 + 625*b8 + 625*b6 - 2500*b5 + 625*b4 + 1875*b3 - 10625*b2 + 43125*b1 - 3160625) * q^55 + (309*b11 - 56*b10 - 6213*b9 + 888*b8 - 1814*b7 - 838*b6 - 498*b5 + 680*b4 + 2680*b3 - 123877*b2 + 153594*b1 - 33673835) * q^56 - 10556001 * q^57 + (-3651*b11 + 21*b10 - 8123*b9 + 2593*b8 - 5987*b7 + 2643*b6 + 5767*b5 + 868*b4 + 2775*b3 - 71006*b2 - 482143*b1 - 31350920) * q^58 + (-638*b11 + 1450*b10 + 227*b9 - 4883*b8 + 206*b7 - 424*b6 + 4028*b5 - 10*b4 + 1116*b3 - 67804*b2 + 432787*b1 - 24639601) * q^59 + (50625*b2 - 303750*b1 + 11694375) * q^60 + (-5417*b11 - 4145*b10 + 1400*b9 - 1405*b8 - 1594*b7 - 438*b6 + 295*b5 - 1716*b4 - 4501*b3 + 11558*b2 - 505349*b1 - 15237987) * q^61 + (7110*b11 + 2768*b10 + 5452*b9 + 6424*b8 + 8722*b7 - 1028*b6 - 9620*b5 + 5982*b4 + 3020*b3 - 125334*b2 - 120482*b1 - 29745314) * q^62 + (6561*b5 - 6561*b2 - 341172*b1 - 2703132) * q^63 + (-3096*b11 - 2338*b10 + 6916*b9 - 910*b8 + 4882*b7 - 2506*b6 + 18464*b5 - 5118*b4 - 9858*b3 + 48427*b2 - 2075248*b1 + 14663911) * q^64 + (-1875*b11 - 1875*b10 + 625*b9 - 2500*b8 + 1250*b7 - 3125*b6 - 2500*b5 - 1250*b4 - 2500*b3 - 45625*b2 - 380625*b1 - 8459375) * q^65 + (-1053*b11 - 4941*b10 + 243*b9 - 4293*b8 + 1215*b7 + 2997*b6 + 4941*b5 - 2268*b4 - 7695*b3 + 42282*b2 - 1093419*b1 + 5067522) * q^66 + (3341*b11 - 2561*b10 - 2426*b9 - 4601*b8 - 9930*b7 - 3963*b6 + 4547*b5 - 348*b4 - 4628*b3 - 97646*b2 + 375860*b1 - 35751008) * q^67 + (-3440*b11 - 2752*b10 + 7076*b9 - 1388*b8 - 3040*b7 + 416*b6 + 3930*b5 + 854*b4 + 1032*b3 - 38644*b2 - 1600574*b1 - 4337964) * q^68 + (1863*b11 + 5184*b10 + 1458*b9 + 1458*b8 - 4212*b7 - 2997*b6 - 5427*b5 + 891*b4 - 2187*b3 + 51030*b2 - 270540*b1 - 18478530) * q^69 + (625*b11 + 3125*b10 + 4375*b9 - 5625*b8 + 625*b7 - 8125*b6 - 625*b5 - 5000*b4 + 1875*b3 - 88750*b2 - 409375*b1 - 23082500) * q^70 + (-4285*b11 + 6221*b10 - 92*b9 + 2186*b8 - 4507*b7 + 8185*b6 - 6390*b5 + 13766*b4 - 2084*b3 + 227*b2 + 127945*b1 - 40778082) * q^71 + (-6561*b5 + 6561*b4 - 52488*b2 + 1226907*b1 - 22307400) * q^72 + (12997*b11 + 17353*b10 + 9248*b9 + 8789*b8 + 17968*b7 - 1767*b6 + 5869*b5 - 1266*b4 + 10230*b3 + 77544*b2 - 795480*b1 - 15176720) * q^73 + (-4191*b11 - 5797*b10 - 21259*b9 - 12821*b8 - 10261*b7 - 6639*b6 - 11773*b5 - 4846*b4 + 8529*b3 - 197802*b2 - 973983*b1 - 118499848) * q^74 + 31640625 * q^75 + (-130321*b2 + 781926*b1 - 30104151) * q^76 + (-1164*b11 - 386*b10 + 13574*b9 - 11999*b8 + 115*b7 - 10717*b6 - 30229*b5 - 4882*b4 + 749*b3 - 85371*b2 + 1386080*b1 - 69319803) * q^77 + (-2349*b11 - 4455*b10 - 9153*b9 + 4617*b8 - 3159*b7 + 3483*b6 + 8829*b5 - 6318*b4 + 8019*b3 + 19926*b2 - 3589029*b1 - 33920208) * q^78 + (1978*b11 + 6000*b10 - 6785*b9 - 19338*b8 + 5521*b7 - 20966*b6 + 20529*b5 - 16102*b4 + 6326*b3 + 154789*b2 + 1159538*b1 - 51947925) * q^79 + (-6250*b10 - 2500*b9 + 6250*b8 - 3750*b7 + 18750*b6 + 5000*b5 + 1250*b4 - 6250*b3 + 111875*b2 - 3020000*b1 + 17414375) * q^80 + 43046721 * q^81 + (14937*b11 - 15441*b10 + 7695*b9 - 12935*b8 + 24763*b7 - 12179*b6 + 14355*b5 - 15590*b4 - 1815*b3 - 75406*b2 - 3242217*b1 + 9144246) * q^82 + (5565*b11 + 1652*b10 + 16885*b9 + 9784*b8 - 14715*b7 + 2853*b6 - 21432*b5 - 303*b4 - 9621*b3 + 182493*b2 - 867282*b1 - 2440755) * q^83 + (-1863*b11 - 6318*b10 + 8019*b9 + 486*b8 + 8748*b7 - 2268*b6 + 16524*b5 - 22356*b4 - 5670*b3 - 56943*b2 - 5843664*b1 - 14614101) * q^84 + (1250*b11 + 10000*b10 - 18125*b9 + 16250*b8 - 625*b7 + 4375*b6 + 11875*b5 + 5625*b3 + 18125*b2 + 249375*b1 - 28850000) * q^85 + (-16593*b11 - 39615*b10 + 31411*b9 - 5527*b8 + 42177*b7 + 6603*b6 - 28971*b5 - 918*b4 - 4125*b3 + 479162*b2 - 6698751*b1 + 97441210) * q^86 + (-13365*b11 + 10854*b10 - 5265*b9 - 1620*b8 + 7047*b7 - 8019*b6 - 16929*b5 - 13041*b4 + 5427*b3 - 68688*b2 - 3530466*b1 - 22192785) * q^87 + (-16949*b11 - 14672*b10 + 24237*b9 + 7128*b8 - 8130*b7 + 16734*b6 + 41702*b5 - 22164*b4 - 13808*b3 + 699665*b2 - 8974558*b1 + 228873919) * q^88 + (-7424*b11 - 3851*b10 - 16464*b9 + 32426*b8 - 529*b7 - 16678*b6 + 12386*b5 + 11829*b4 + 15097*b3 + 172836*b2 - 1151125*b1 + 39809044) * q^89 + (4100625*b1 - 12301875) * q^90 + (-1469*b11 + 5723*b10 - 51486*b9 + 37078*b8 - 30553*b7 + 50248*b6 + 10678*b5 + 53254*b4 + 20831*b3 + 318145*b2 + 6019834*b1 - 120611583) * q^91 + (17182*b11 + 33428*b10 - 33734*b9 + 7484*b8 - 39368*b7 + 39664*b6 - 21950*b5 + 32286*b4 + 14356*b3 - 194644*b2 + 611586*b1 + 159827152) * q^92 + (19359*b11 - 8019*b10 + 243*b9 - 6318*b8 - 2430*b7 + 23004*b6 - 10773*b5 - 18792*b4 - 13527*b3 + 86832*b2 - 3560760*b1 - 65411874) * q^93 + (-2590*b11 + 57576*b10 - 44490*b9 + 64756*b8 - 19816*b7 + 26756*b6 + 29786*b5 + 41958*b4 + 16640*b3 + 402940*b2 - 18046586*b1 + 211579232) * q^94 - 81450625 * q^95 + (22032*b11 + 37584*b10 - 11016*b9 + 15552*b8 - 1296*b7 - 14256*b6 - 65853*b5 + 4941*b4 + 29808*b3 - 433512*b2 + 2670003*b1 - 151194924) * q^96 + (-60718*b11 - 61840*b10 + 53559*b9 - 58611*b8 - 7712*b7 - 27872*b6 + 12203*b5 - 37128*b4 - 4298*b3 + 10973*b2 - 11048935*b1 + 29368521) * q^97 + (50106*b11 + 31110*b10 + 7106*b9 + 42016*b8 - 17068*b7 + 32594*b6 - 81372*b5 + 44452*b4 + 19466*b3 + 571742*b2 - 1591535*b1 + 239132181) * q^98 + (6561*b11 + 6561*b9 + 6561*b8 + 6561*b6 - 26244*b5 + 6561*b4 + 19683*b3 - 111537*b2 + 452709*b1 - 33178977) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 33 q^{2} + 972 q^{3} + 2751 q^{4} + 7500 q^{5} - 2673 q^{6} - 5100 q^{7} - 40215 q^{8} + 78732 q^{9}+O(q^{10})$$ 12 * q - 33 * q^2 + 972 * q^3 + 2751 * q^4 + 7500 * q^5 - 2673 * q^6 - 5100 * q^7 - 40215 * q^8 + 78732 * q^9 $$12 q - 33 q^{2} + 972 q^{3} + 2751 q^{4} + 7500 q^{5} - 2673 q^{6} - 5100 q^{7} - 40215 q^{8} + 78732 q^{9} - 20625 q^{10} - 60416 q^{11} + 222831 q^{12} - 164042 q^{13} - 444762 q^{14} + 607500 q^{15} + 319475 q^{16} - 552834 q^{17} - 216513 q^{18} - 1563852 q^{19} + 1719375 q^{20} - 413100 q^{21} + 708846 q^{22} - 2749174 q^{23} - 3257415 q^{24} + 4687500 q^{25} - 5159376 q^{26} + 6377292 q^{27} - 2379170 q^{28} - 3415632 q^{29} - 1670625 q^{30} - 9822574 q^{31} - 22282623 q^{32} - 4893696 q^{33} + 5214442 q^{34} - 3187500 q^{35} + 18049311 q^{36} - 24559054 q^{37} + 4300593 q^{38} - 13287402 q^{39} - 25134375 q^{40} - 35856950 q^{41} - 36025722 q^{42} - 73839462 q^{43} - 88153054 q^{44} + 49207500 q^{45} - 19966000 q^{46} - 134699358 q^{47} + 25877475 q^{48} - 64331236 q^{49} - 12890625 q^{50} - 44779554 q^{51} - 290325860 q^{52} - 74807846 q^{53} - 17537553 q^{54} - 37760000 q^{55} - 403278618 q^{56} - 126672012 q^{57} - 377474064 q^{58} - 294192102 q^{59} + 139269375 q^{60} - 184407884 q^{61} - 356900784 q^{62} - 33461100 q^{63} + 169573219 q^{64} - 102526250 q^{65} + 57416526 q^{66} - 427627864 q^{67} - 56755242 q^{68} - 222683094 q^{69} - 277976250 q^{70} - 488904216 q^{71} - 263850615 q^{72} - 184702472 q^{73} - 1424419204 q^{74} + 379687500 q^{75} - 358513071 q^{76} - 827371672 q^{77} - 417909456 q^{78} - 620531838 q^{79} + 199671875 q^{80} + 516560652 q^{81} + 100137664 q^{82} - 32273124 q^{83} - 192712770 q^{84} - 345521250 q^{85} + 1147824298 q^{86} - 276666192 q^{87} + 2717542210 q^{88} + 473499162 q^{89} - 135320625 q^{90} - 1430236924 q^{91} + 1920565904 q^{92} - 795628494 q^{93} + 2483643504 q^{94} - 977407500 q^{95} - 1804892463 q^{96} + 318965234 q^{97} + 2863543931 q^{98} - 396389376 q^{99}+O(q^{100})$$ 12 * q - 33 * q^2 + 972 * q^3 + 2751 * q^4 + 7500 * q^5 - 2673 * q^6 - 5100 * q^7 - 40215 * q^8 + 78732 * q^9 - 20625 * q^10 - 60416 * q^11 + 222831 * q^12 - 164042 * q^13 - 444762 * q^14 + 607500 * q^15 + 319475 * q^16 - 552834 * q^17 - 216513 * q^18 - 1563852 * q^19 + 1719375 * q^20 - 413100 * q^21 + 708846 * q^22 - 2749174 * q^23 - 3257415 * q^24 + 4687500 * q^25 - 5159376 * q^26 + 6377292 * q^27 - 2379170 * q^28 - 3415632 * q^29 - 1670625 * q^30 - 9822574 * q^31 - 22282623 * q^32 - 4893696 * q^33 + 5214442 * q^34 - 3187500 * q^35 + 18049311 * q^36 - 24559054 * q^37 + 4300593 * q^38 - 13287402 * q^39 - 25134375 * q^40 - 35856950 * q^41 - 36025722 * q^42 - 73839462 * q^43 - 88153054 * q^44 + 49207500 * q^45 - 19966000 * q^46 - 134699358 * q^47 + 25877475 * q^48 - 64331236 * q^49 - 12890625 * q^50 - 44779554 * q^51 - 290325860 * q^52 - 74807846 * q^53 - 17537553 * q^54 - 37760000 * q^55 - 403278618 * q^56 - 126672012 * q^57 - 377474064 * q^58 - 294192102 * q^59 + 139269375 * q^60 - 184407884 * q^61 - 356900784 * q^62 - 33461100 * q^63 + 169573219 * q^64 - 102526250 * q^65 + 57416526 * q^66 - 427627864 * q^67 - 56755242 * q^68 - 222683094 * q^69 - 277976250 * q^70 - 488904216 * q^71 - 263850615 * q^72 - 184702472 * q^73 - 1424419204 * q^74 + 379687500 * q^75 - 358513071 * q^76 - 827371672 * q^77 - 417909456 * q^78 - 620531838 * q^79 + 199671875 * q^80 + 516560652 * q^81 + 100137664 * q^82 - 32273124 * q^83 - 192712770 * q^84 - 345521250 * q^85 + 1147824298 * q^86 - 276666192 * q^87 + 2717542210 * q^88 + 473499162 * q^89 - 135320625 * q^90 - 1430236924 * q^91 + 1920565904 * q^92 - 795628494 * q^93 + 2483643504 * q^94 - 977407500 * q^95 - 1804892463 * q^96 + 318965234 * q^97 + 2863543931 * q^98 - 396389376 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 4398 x^{10} + 11376 x^{9} + 7070146 x^{8} - 15274638 x^{7} - 5114407260 x^{6} + \cdots + 43\!\cdots\!00$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 734$$ v^2 - 734 $$\beta_{3}$$ $$=$$ $$( - 60\!\cdots\!33 \nu^{11} + \cdots + 98\!\cdots\!80 ) / 66\!\cdots\!60$$ (-60591395331418431133*v^11 + 428988761957795306368078*v^10 + 3614596707357250280125160*v^9 - 1482984048958337654453537348*v^8 - 13874352939901387272471899354*v^7 + 1520844634177545310632867989816*v^6 + 17697267581825722255803789261472*v^5 - 351201042673457446477767273153276*v^4 - 7905651746656057494967678627440705*v^3 - 105197049120050736202499300409553190*v^2 + 612360742587190429521727267710561840*v + 9896612596442902471180859396863226080) / 669015141230611166661181580328960 $$\beta_{4}$$ $$=$$ $$( 25\!\cdots\!72 \nu^{11} + \cdots - 10\!\cdots\!20 ) / 33\!\cdots\!80$$ (2540371910509768924472*v^11 + 232436121896350987243873*v^10 - 7348231162284591896249425*v^9 - 864651308310085800928198973*v^8 + 4016517624077292143059618291*v^7 + 1054344316464661570773409954961*v^6 + 3039471107354818500515649175357*v^5 - 469171991504404806682052297619471*v^4 - 1728653100426519846910486685227815*v^3 + 53618598397833506674649137044585850*v^2 - 412687669170395115979623064947471600*v - 1025513909829049429461623413201010720) / 334507570615305583330590790164480 $$\beta_{5}$$ $$=$$ $$( 25\!\cdots\!72 \nu^{11} + \cdots - 12\!\cdots\!60 ) / 33\!\cdots\!80$$ (2540371910509768924472*v^11 + 232436121896350987243873*v^10 - 7348231162284591896249425*v^9 - 864651308310085800928198973*v^8 + 4016517624077292143059618291*v^7 + 1054344316464661570773409954961*v^6 + 3039471107354818500515649175357*v^5 - 469171991504404806682052297619471*v^4 - 2063160671041825430241077475392295*v^3 + 53953105968448812257979727834750330*v^2 - 16630705561873305316203569392727280*v - 1217186747791619528710051935965257760) / 334507570615305583330590790164480 $$\beta_{6}$$ $$=$$ $$( - 81\!\cdots\!91 \nu^{11} + \cdots + 17\!\cdots\!88 ) / 66\!\cdots\!96$$ (-819787061028667015391*v^11 + 5014340277527176689857*v^10 + 4487269974004478653844449*v^9 - 5697387889292840366554087*v^8 - 9047011482016283990454591337*v^7 - 23549645997395341221123784049*v^6 + 8083749399439107502331332080299*v^5 + 44659072949307320292930313050963*v^4 - 2980588324146304302141428186585412*v^3 - 22342109724752817397606138125800572*v^2 + 292291251719182125962484540932858112*v + 1767306721255496767049910232561508288) / 66901514123061116666118158032896 $$\beta_{7}$$ $$=$$ $$( - 16\!\cdots\!08 \nu^{11} + \cdots - 78\!\cdots\!60 ) / 11\!\cdots\!60$$ (-1689843662396140013108*v^11 + 84491020512174446830623*v^10 + 8894635655869215843164505*v^9 - 312054490742746266993498283*v^8 - 17676705913073953956244712419*v^7 + 380855414238664526597074731431*v^6 + 15879878083509995922485147715387*v^5 - 172894425934621138790523191512361*v^4 - 5849693552337301929466182585788205*v^3 + 22072872423952409325718299150994030*v^2 + 490866411269534242935169962828476080*v - 783581971557102435853822730433547360) / 111502523538435194443530263388160 $$\beta_{8}$$ $$=$$ $$( - 16\!\cdots\!11 \nu^{11} + \cdots - 24\!\cdots\!60 ) / 33\!\cdots\!80$$ (-16331709830699393093711*v^11 - 446404137777574521547594*v^10 + 52111763125092409506401020*v^9 + 1639514148591821801491466504*v^8 - 42509077013356648593911375818*v^7 - 1902750508524301715391905706908*v^6 - 4421139010418624598479181325396*v^5 + 715871693381331116912573572424808*v^4 + 11591422511998104814176257986883745*v^3 - 18008568079280218574126615416025770*v^2 - 1058711432271094228024867380983000880*v - 2405241999540041656496785498869844960) / 334507570615305583330590790164480 $$\beta_{9}$$ $$=$$ $$( 48\!\cdots\!39 \nu^{11} + \cdots + 97\!\cdots\!40 ) / 83\!\cdots\!20$$ (4867240843164657702839*v^11 + 64548199924697794578016*v^10 - 19252210839326320877349490*v^9 - 223700024552918964362203286*v^8 + 26224119222450717358826241052*v^7 + 233729145819246980964690681602*v^6 - 14631408657714439218433420758446*v^5 - 66746220767864371939121290932642*v^4 + 3079715122230322065415708625191485*v^3 - 6971882037029492225730916698084170*v^2 - 194497578827347086858814418234692080*v + 978665065421229729029178195213636640) / 83626892653826395832647697541120 $$\beta_{10}$$ $$=$$ $$( - 20\!\cdots\!57 \nu^{11} + \cdots + 40\!\cdots\!60 ) / 22\!\cdots\!20$$ (-20535809929802880323657*v^11 - 405738832999096008506568*v^10 + 83372317587702598109212550*v^9 + 1567008112695417230573031378*v^8 - 118267383507973968882877454156*v^7 - 2014386385366489490446684076486*v^6 + 70568275066423370265416507833018*v^5 + 984492986868423549637224225562966*v^4 - 16771681566940133610443720675484475*v^3 - 146446624373924168086378512620049850*v^2 + 1256645270893548331820402136830586000*v + 4033106371252393141391856358908585760) / 223005047076870388887060526776320 $$\beta_{11}$$ $$=$$ $$( 38\!\cdots\!33 \nu^{11} + \cdots - 82\!\cdots\!40 ) / 16\!\cdots\!40$$ (38962085296438874077133*v^11 + 932985777342519531762427*v^10 - 147648915068452854867990145*v^9 - 3537607007466007369843824017*v^8 + 186590464653260383021932064789*v^7 + 4421100297566025980548082678129*v^6 - 90048383893790172995713872174587*v^5 - 2060987749366642985949859592416539*v^4 + 13689285975566201611070955479887290*v^3 + 277215518068961189205100216090872640*v^2 - 668608216814837202751732901783189280*v - 8278822398677683121302155858416925440) / 167253785307652791665295395082240
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 734$$ b2 + 734 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{2} + 1184\beta _1 + 161$$ -b5 + b4 + b2 + 1184*b1 + 161 $$\nu^{4}$$ $$=$$ $$- 10 \beta_{10} - 4 \beta_{9} + 10 \beta_{8} - 6 \beta_{7} + 30 \beta_{6} - 4 \beta_{5} + 14 \beta_{4} + \cdots + 869182$$ -10*b10 - 4*b9 + 10*b8 - 6*b7 + 30*b6 - 4*b5 + 14*b4 - 10*b3 + 1673*b2 + 268*b1 + 869182 $$\nu^{5}$$ $$=$$ $$272 \beta_{11} + 314 \beta_{10} - 196 \beta_{9} + 342 \beta_{8} - 106 \beta_{7} + 274 \beta_{6} + \cdots + 459699$$ 272*b11 + 314*b10 - 196*b9 + 342*b8 - 106*b7 + 274*b6 - 2831*b5 + 2229*b4 + 218*b3 + 3539*b2 + 1623714*b1 + 459699 $$\nu^{6}$$ $$=$$ $$1800 \beta_{11} - 20936 \beta_{10} - 6312 \beta_{9} + 29496 \beta_{8} - 11576 \beta_{7} + \cdots + 1192208620$$ 1800*b11 - 20936*b10 - 6312*b9 + 29496*b8 - 11576*b7 + 75176*b6 - 12014*b5 + 38774*b4 - 30184*b3 + 2703135*b2 + 1230546*b1 + 1192208620 $$\nu^{7}$$ $$=$$ $$814672 \beta_{11} + 979580 \beta_{10} - 602248 \beta_{9} + 1029620 \beta_{8} - 476428 \beta_{7} + \cdots + 1318783089$$ 814672*b11 + 979580*b10 - 602248*b9 + 1029620*b8 - 476428*b7 + 843948*b6 - 5709353*b5 + 4250909*b4 + 604892*b3 + 8980001*b2 + 2411840184*b1 + 1318783089 $$\nu^{8}$$ $$=$$ $$6298704 \beta_{11} - 33653574 \beta_{10} - 4914508 \beta_{9} + 64963350 \beta_{8} - 18138650 \beta_{7} + \cdots + 1771384672930$$ 6298704*b11 - 33653574*b10 - 4914508*b9 + 64963350*b8 - 18138650*b7 + 147609522*b6 - 27316944*b5 + 82541126*b4 - 64829222*b3 + 4414841933*b2 + 3759501056*b1 + 1771384672930 $$\nu^{9}$$ $$=$$ $$1796504560 \beta_{11} + 2222145966 \beta_{10} - 1349736652 \beta_{9} + 2260779634 \beta_{8} + \cdots + 3425843328143$$ 1796504560*b11 + 2222145966*b10 - 1349736652*b9 + 2260779634*b8 - 1269094382*b7 + 1908966518*b6 - 10408936659*b5 + 7688043957*b4 + 1231536110*b3 + 19766486975*b2 + 3762694926302*b1 + 3425843328143 $$\nu^{10}$$ $$=$$ $$15881066552 \beta_{11} - 49671714164 \beta_{10} + 1717261456 \beta_{9} + 128958154260 \beta_{8} + \cdots + 27\!\cdots\!56$$ 15881066552*b11 - 49671714164*b10 + 1717261456*b9 + 128958154260*b8 - 27310730620*b7 + 268773497804*b6 - 56284595026*b5 + 160435144790*b4 - 123220388788*b3 + 7295582519771*b2 + 9416120235246*b1 + 2764368350814056 $$\nu^{11}$$ $$=$$ $$3526759027120 \beta_{11} + 4435915245424 \beta_{10} - 2666403632272 \beta_{9} + 4419288994480 \beta_{8} + \cdots + 79\!\cdots\!01$$ 3526759027120*b11 + 4435915245424*b10 - 2666403632272*b9 + 4419288994480*b8 - 2769430233520*b7 + 3848589391376*b6 - 18275402789277*b5 + 13570250782573*b4 + 2234418724272*b3 + 40531629516925*b2 + 6052839956752260*b1 + 7997250520583901

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −40.7797 −34.2245 −28.9288 −22.0115 −12.3423 −5.36255 6.42527 11.5013 26.4167 26.9112 33.7980 41.5969
−43.7797 81.0000 1404.66 625.000 −3546.16 6872.55 −39080.6 6561.00 −27362.3
1.2 −37.2245 81.0000 873.663 625.000 −3015.18 −2357.55 −13462.7 6561.00 −23265.3
1.3 −31.9288 81.0000 507.448 625.000 −2586.23 −632.496 145.335 6561.00 −19955.5
1.4 −25.0115 81.0000 113.577 625.000 −2025.93 −7116.10 9965.17 6561.00 −15632.2
1.5 −15.3423 81.0000 −276.614 625.000 −1242.73 8224.35 12099.1 6561.00 −9588.93
1.6 −8.36255 81.0000 −442.068 625.000 −677.366 1660.91 7978.44 6561.00 −5226.59
1.7 3.42527 81.0000 −500.268 625.000 277.446 −1060.52 −3467.29 6561.00 2140.79
1.8 8.50131 81.0000 −439.728 625.000 688.606 −8602.42 −8090.93 6561.00 5313.32
1.9 23.4167 81.0000 36.3418 625.000 1896.75 2524.82 −11138.3 6561.00 14635.4
1.10 23.9112 81.0000 59.7445 625.000 1936.81 7881.01 −10814.0 6561.00 14944.5
1.11 30.7980 81.0000 436.517 625.000 2494.64 −2782.02 −2324.71 6561.00 19248.8
1.12 38.5969 81.0000 977.722 625.000 3126.35 −9712.55 17975.4 6561.00 24123.1
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.10.a.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.10.a.d 12 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 33 T_{2}^{11} - 3903 T_{2}^{10} - 116109 T_{2}^{9} + 5622838 T_{2}^{8} + \cdots + 32\!\cdots\!12$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(285))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + \cdots + 32\!\cdots\!12$$
$3$ $$(T - 81)^{12}$$
$5$ $$(T - 625)^{12}$$
$7$ $$T^{12} + \cdots - 48\!\cdots\!00$$
$11$ $$T^{12} + \cdots - 32\!\cdots\!00$$
$13$ $$T^{12} + \cdots - 12\!\cdots\!64$$
$17$ $$T^{12} + \cdots + 45\!\cdots\!88$$
$19$ $$(T + 130321)^{12}$$
$23$ $$T^{12} + \cdots + 55\!\cdots\!44$$
$29$ $$T^{12} + \cdots + 40\!\cdots\!00$$
$31$ $$T^{12} + \cdots - 27\!\cdots\!00$$
$37$ $$T^{12} + \cdots + 31\!\cdots\!00$$
$41$ $$T^{12} + \cdots + 66\!\cdots\!00$$
$43$ $$T^{12} + \cdots - 52\!\cdots\!20$$
$47$ $$T^{12} + \cdots + 19\!\cdots\!08$$
$53$ $$T^{12} + \cdots + 85\!\cdots\!00$$
$59$ $$T^{12} + \cdots + 28\!\cdots\!00$$
$61$ $$T^{12} + \cdots + 36\!\cdots\!24$$
$67$ $$T^{12} + \cdots - 80\!\cdots\!88$$
$71$ $$T^{12} + \cdots + 17\!\cdots\!00$$
$73$ $$T^{12} + \cdots + 99\!\cdots\!00$$
$79$ $$T^{12} + \cdots - 88\!\cdots\!00$$
$83$ $$T^{12} + \cdots - 28\!\cdots\!60$$
$89$ $$T^{12} + \cdots - 33\!\cdots\!00$$
$97$ $$T^{12} + \cdots + 17\!\cdots\!00$$