# Properties

 Label 18.7.d.a Level $18$ Weight $7$ Character orbit 18.d Analytic conductor $4.141$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,7,Mod(5,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("18.5");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 18.d (of order $$6$$, degree $$2$$, 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: $$4.14097350516$$ 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: $$x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$$ x^12 + 370*x^10 + 51793*x^8 + 3491832*x^6 + 117603792*x^4 + 1832032512*x^2 + 10453017600 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 3^{10}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} + \cdots - 6) q^{3}+ \cdots + (2 \beta_{10} - \beta_{9} - 5 \beta_{8} + \cdots + 181) q^{9}+O(q^{10})$$ q - b3 * q^2 + (b7 - b3 + b2 + 4*b1 - 6) * q^3 + (-32*b1 + 32) * q^4 + (b10 + b8 + b5 + 24*b1 + 24) * q^5 + (b11 + b10 - b9 + b8 + b4 + 6*b3 - 4*b2 - 13*b1 - 5) * q^6 + (4*b10 - 2*b9 + b8 - b7 - b6 - 7*b5 - 3*b4 - 3*b3 + 14*b2 + 35*b1 + 10) * q^7 + (-32*b3 + 32*b2) * q^8 + (2*b10 - b9 - 5*b8 - 4*b7 - 6*b6 - 5*b5 + 5*b4 - 33*b3 + 22*b2 + 9*b1 + 181) * q^9 $$q - \beta_{3} q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} + \cdots - 6) q^{3}+ \cdots + (2637 \beta_{11} + 750 \beta_{10} + \cdots + 190140) q^{99}+O(q^{100})$$ q - b3 * q^2 + (b7 - b3 + b2 + 4*b1 - 6) * q^3 + (-32*b1 + 32) * q^4 + (b10 + b8 + b5 + 24*b1 + 24) * q^5 + (b11 + b10 - b9 + b8 + b4 + 6*b3 - 4*b2 - 13*b1 - 5) * q^6 + (4*b10 - 2*b9 + b8 - b7 - b6 - 7*b5 - 3*b4 - 3*b3 + 14*b2 + 35*b1 + 10) * q^7 + (-32*b3 + 32*b2) * q^8 + (2*b10 - b9 - 5*b8 - 4*b7 - 6*b6 - 5*b5 + 5*b4 - 33*b3 + 22*b2 + 9*b1 + 181) * q^9 + (-5*b11 + b10 + b9 - 5*b8 + 12*b7 + 8*b6 + 16*b5 - 7*b4 - 38*b3 - 19*b2 + 5*b1 - 15) * q^10 + (15*b11 + 2*b9 + 10*b8 - 23*b7 - 4*b6 - 33*b5 + 14*b4 + 53*b3 - 14*b2 - 30*b1 + 75) * q^11 + (32*b7 + 32*b5 - 32*b3 + 32*b2 + 192*b1 - 96) * q^12 + (-20*b11 - b10 + 2*b9 + 8*b8 - 16*b7 - 5*b6 - 6*b5 - 22*b4 + 115*b3 - 53*b2 - 301*b1 + 302) * q^13 + (-6*b11 - 11*b10 + 4*b8 + 12*b7 + 24*b6 + 40*b5 - 3*b4 - 21*b3 - 28*b2 - 246*b1 - 306) * q^14 + (27*b11 - 12*b10 - 12*b9 + 3*b8 + 24*b7 - 9*b6 - 24*b5 + 6*b4 + 351*b3 - 258*b2 + 27*b1 - 915) * q^15 - 1024*b1 * q^16 + (-66*b11 - 53*b10 + 53*b9 - 58*b8 - 38*b7 + 20*b6 + 22*b5 + 29*b4 - 145*b3 + 136*b2 + 2586*b1 - 1359) * q^17 + (24*b11 - 31*b10 - 10*b9 - 14*b8 - 4*b7 + 12*b6 + 28*b5 + 17*b4 - 198*b3 - 5*b2 - 1104*b1 + 280) * q^18 + (35*b11 + 20*b10 + 20*b9 + 35*b8 - 57*b7 - 83*b6 - 13*b5 - 50*b4 - 166*b3 - 173*b2 - 35*b1 - 130) * q^19 + (32*b9 - 32*b8 - 32*b7 + 32*b6 + 32*b4 + 32*b3 - 32*b2 - 768*b1 + 1536) * q^20 + (54*b11 - 12*b10 - 39*b9 - 78*b8 - 57*b7 + 72*b6 - 109*b5 - 21*b4 + 432*b3 + 336*b2 + 3583*b1 + 396) * q^21 + (-64*b11 + 22*b10 - 44*b9 + 40*b8 + 64*b7 - 52*b6 + 132*b5 - 74*b4 - 172*b3 + 95*b2 + 612*b1 - 688) * q^22 + (-123*b11 - 10*b10 - 13*b8 - 105*b7 + 6*b6 - 247*b5 + 114*b4 + 231*b3 + 267*b2 - 4353*b1 - 4125) * q^23 + (64*b11 + 32*b10 + 32*b4 + 64*b3 - 192*b2 + 192*b1 - 576) * q^24 + (120*b11 + 174*b10 - 87*b9 + 291*b8 + 117*b7 - 147*b6 + 414*b5 - 27*b4 + 273*b3 - 933*b2 + 1499*b1 - 180) * q^25 + (-51*b11 - 11*b10 + 11*b9 - 7*b8 - 116*b7 + 80*b6 - 320*b5 - 91*b4 - 2*b3 + 173*b2 + 3171*b1 - 1353) * q^26 + (84*b11 + 66*b10 - 66*b9 + 21*b8 + 198*b7 + 135*b6 - 60*b5 - 123*b4 - 612*b3 + 771*b2 - 7899*b1 + 14601) * q^27 + (-32*b11 + 64*b10 + 64*b9 - 32*b8 + 192*b7 - 64*b6 - 32*b5 + 32*b4 + 160*b3 + 320*b2 + 32*b1 + 1184) * q^28 + (-90*b11 + 83*b9 - 236*b8 + 448*b7 + 119*b6 + 360*b5 + 173*b4 - 1264*b3 + 151*b2 - 5097*b1 + 10104) * q^29 + (171*b11 + 51*b10 - 75*b9 + 255*b8 - 48*b7 - 252*b6 - 60*b5 + 123*b4 + 936*b3 + 39*b2 + 11565*b1 - 2823) * q^30 + (45*b11 - 24*b10 + 48*b9 + 147*b8 + 87*b7 - 84*b6 - 153*b5 + 6*b4 - 4023*b3 + 2097*b2 - 3577*b1 + 3619) * q^31 + 1024*b2 * q^32 + (-210*b11 - 33*b10 - 99*b9 - 288*b8 + 42*b7 + 288*b6 + 156*b5 - 285*b4 - 1467*b3 - 1722*b2 - 8529*b1 - 16335) * q^33 + (-189*b11 - 46*b10 + 23*b9 - 403*b8 - 488*b7 + 52*b6 - 932*b5 - 6*b4 + 2046*b3 - 3023*b2 - 5083*b1 + 857) * q^34 + (93*b11 + 40*b10 - 40*b9 + 167*b8 + 376*b7 - 196*b6 + 667*b5 + 119*b4 + 173*b3 - 488*b2 + 26193*b1 - 13611) * q^35 + (192*b11 + 32*b10 + 32*b9 + 160*b8 + 32*b7 - 384*b6 - 128*b5 + 224*b4 - 384*b3 + 1216*b2 - 5568*b1 + 6016) * q^36 + (106*b11 - 122*b10 - 122*b9 + 106*b8 - 753*b7 + 329*b6 + 301*b5 - 94*b4 + 4204*b3 + 3545*b2 - 106*b1 - 1810) * q^37 + (117*b11 - 107*b9 + 287*b8 - 1504*b7 + 100*b6 - 684*b5 - 440*b4 + 1550*b3 - 667*b2 - 6261*b1 + 12639) * q^38 + (-477*b11 - 186*b10 + 336*b9 - 300*b8 + 256*b7 + 198*b6 + 97*b5 + 12*b4 + 6290*b3 - 1487*b2 + 14064*b1 - 4029) * q^39 + (-256*b11 - 32*b10 + 64*b9 - 320*b8 - 128*b7 + 128*b6 + 384*b5 - 32*b4 - 1696*b3 + 704*b2 - 256) * q^40 + (456*b11 - 128*b10 + 55*b8 - 480*b7 - 285*b6 + 397*b5 + 12*b4 - 159*b3 - 2829*b2 - 22638*b1 - 22695) * q^41 + (140*b11 - 219*b10 + 218*b9 + 286*b8 - 156*b7 - 468*b6 + 588*b5 + 717*b4 - 641*b3 - 3382*b2 + 11084*b1 - 24668) * q^42 + (81*b11 - 660*b10 + 330*b9 - 435*b8 + 786*b7 + 570*b6 + 291*b5 + 90*b4 + 5004*b3 - 10950*b2 + 12541*b1 - 1191) * q^43 + (384*b11 - 64*b10 + 64*b9 + 64*b8 + 320*b7 + 256*b6 - 736*b5 - 32*b4 + 928*b3 - 640*b2 - 1344*b1 + 1056) * q^44 + (-1098*b11 - 243*b10 + 252*b9 - 522*b8 - 954*b7 + 189*b6 + 180*b5 - 216*b4 - 12123*b3 + 11097*b2 - 18513*b1 + 10224) * q^45 + (-667*b11 - 241*b10 - 241*b9 - 667*b8 + 2256*b7 + 412*b6 + 308*b5 - 137*b4 + 2732*b3 + 5125*b2 + 667*b1 - 13257) * q^46 + (102*b11 - 408*b9 + 927*b8 + 2031*b7 - 1437*b6 - 111*b5 - 213*b4 + 2187*b3 + 1428*b2 - 18711*b1 + 37524) * q^47 + (1024*b5 + 2048*b1 + 3072) * q^48 + (1280*b11 + b10 - 2*b9 - 386*b8 - 1208*b7 + 815*b6 - 2100*b5 + 850*b4 - 16621*b3 + 8117*b2 + 22266*b1 - 20593) * q^49 + (426*b11 + 441*b10 + 240*b8 + 1956*b7 + 564*b6 + 2220*b5 - 1191*b4 - 2181*b3 - 365*b2 + 18270*b1 + 15726) * q^50 + (-507*b11 + 636*b10 + 444*b9 + 834*b8 - 510*b7 - 396*b6 - 2505*b5 - 435*b4 + 9375*b3 - 20793*b2 - 17784*b1 + 38715) * q^51 + (-768*b11 - 64*b10 + 32*b9 + 128*b8 - 320*b7 + 160*b6 - 512*b5 - 96*b4 + 2304*b3 - 3488*b2 - 9504*b1 + 320) * q^52 + (648*b11 + 1220*b10 - 1220*b9 + 10*b8 - 805*b7 - 977*b6 + 1787*b5 + 76*b4 + 124*b3 - 1177*b2 + 22758*b1 - 10974) * q^53 + (-42*b11 + 216*b10 + 246*b9 + 582*b8 + 420*b7 - 72*b6 + 1752*b5 + 576*b4 - 15525*b3 + 8406*b2 - 18330*b1 - 6582) * q^54 + (945*b11 + 90*b10 + 90*b9 + 945*b8 - 2979*b7 - 801*b6 - 3000*b5 + 2289*b4 + 17127*b3 + 11859*b2 - 945*b1 + 51201) * q^55 + (-96*b11 - 352*b9 - 416*b8 - 896*b7 + 896*b6 + 384*b5 - 256*b4 - 672*b3 - 608*b2 + 8160*b1 - 16416) * q^56 + (-954*b11 + 1290*b10 - 501*b9 + 537*b8 - 871*b7 + 387*b6 - 822*b5 - 2049*b4 + 19549*b3 + 13289*b2 - 18013*b1 - 32478) * q^57 + (272*b11 - 65*b10 + 130*b9 - 1514*b8 + 532*b7 + 692*b6 + 1644*b5 + 1159*b4 - 11941*b3 + 5246*b2 - 25320*b1 + 24368) * q^58 + (-255*b11 + 1010*b10 - 877*b8 + 1185*b7 - 1167*b6 - 3721*b5 - 465*b4 + 957*b3 - 3123*b2 + 19119*b1 + 23130) * q^59 + (576*b11 + 384*b10 - 768*b9 + 1056*b8 + 1536*b7 - 576*b6 + 768*b5 - 1056*b4 + 1440*b3 - 10464*b2 + 29664*b1 - 29184) * q^60 + (462*b11 - 426*b10 + 213*b9 - 156*b8 - 768*b7 - 537*b6 + 1752*b5 + 963*b4 + 9246*b3 - 20169*b2 + 23569*b1 - 1428) * q^61 + (-255*b11 + 633*b10 - 633*b9 + 693*b8 - 648*b7 - 180*b6 - 1020*b5 - 927*b4 - 2446*b3 + 3193*b2 - 130617*b1 + 66795) * q^62 + (336*b11 + 4*b10 + 244*b9 - 76*b8 - 1136*b7 + 699*b6 - 3793*b5 - 953*b4 - 41448*b3 + 27638*b2 + 115302*b1 - 63190) * q^63 - 32768 * q^64 + (-768*b11 + 163*b9 - 1150*b8 + 1868*b7 + 697*b6 + 2208*b5 + 67*b4 - 2786*b3 + 743*b2 + 43005*b1 - 86778) * q^65 + (369*b11 - 876*b10 + 1329*b9 - 501*b8 - 288*b7 + 180*b6 - 948*b5 + 2220*b4 + 17034*b3 + 8142*b2 - 43497*b1 + 96771) * q^66 + (273*b11 + 420*b10 - 840*b9 + 3768*b8 + 5073*b7 - 3084*b6 + 1872*b5 - 831*b4 - 34476*b3 + 18912*b2 + 50312*b1 - 52028) * q^67 + (-768*b11 - 1696*b10 - 640*b8 - 1920*b7 + 480*b6 - 1216*b5 + 1344*b4 + 1632*b3 + 3936*b2 + 41088*b1 + 41184) * q^68 + (-690*b11 - 2013*b10 + 726*b9 - 3876*b8 - 2481*b7 + 3168*b6 + 4533*b5 + 1128*b4 + 10995*b3 - 55386*b2 - 87387*b1 + 212358) * q^69 + (1473*b11 + 614*b10 - 307*b9 + 203*b8 + 1252*b7 - 212*b6 + 556*b5 - 402*b4 + 11799*b3 - 25415*b2 + 26747*b1 - 37) * q^70 + (-1578*b11 - 3056*b10 + 3056*b9 - 664*b8 + 2089*b7 + 1781*b6 - 1469*b5 + 818*b4 + 13730*b3 - 12767*b2 - 107700*b1 + 50814) * q^71 + (864*b11 + 320*b10 - 1312*b9 + 928*b8 - 1024*b7 - 384*b6 - 128*b5 - 1216*b4 - 5472*b3 + 5440*b2 - 10848*b1 - 24032) * q^72 + (-1098*b11 + 27*b10 + 27*b9 - 1098*b8 + 2466*b7 + 1926*b6 + 6582*b5 - 4629*b4 + 12771*b3 + 19866*b2 + 1098*b1 - 84769) * q^73 + (-804*b11 + 1700*b9 + 76*b8 + 2968*b7 - 1888*b6 + 840*b5 + 128*b4 - 930*b3 + 1924*b2 + 123108*b1 - 247020) * q^74 + (2862*b11 - 2862*b10 - 810*b9 + 891*b8 + 270*b7 - 1620*b6 - 89*b5 + 2565*b4 + 46089*b3 + 10557*b2 - 221281*b1 - 67011) * q^75 + (-2240*b11 - 640*b10 + 1280*b9 + 1376*b8 - 1408*b7 - 896*b6 - 1824*b5 - 2080*b4 - 9440*b3 + 5728*b2 + 5216*b1 - 5152) * q^76 + (-624*b11 - 2101*b10 + 2159*b8 - 5772*b7 + 1686*b6 + 4283*b5 + 3198*b4 + 2136*b3 + 17208*b2 - 7782*b1 - 11592) * q^77 + (-1627*b11 + 121*b10 + 3*b9 - 2559*b8 - 2976*b7 + 1764*b6 - 4548*b5 - 167*b4 + 8282*b3 - 17019*b2 + 204603*b1 - 151833) * q^78 + (-474*b11 + 2156*b10 - 1078*b9 + 692*b8 - 1472*b7 - 845*b6 - 3635*b5 - 1311*b4 + 45078*b3 - 85202*b2 - 130482*b1 + 5558) * q^79 + (-1024*b10 + 1024*b9 - 2048*b8 - 1024*b7 + 1024*b6 - 1024*b5 + 1024*b4 + 1024*b3 - 1024*b2 - 49152*b1 + 24576) * q^80 + (3276*b11 + 363*b10 - 1320*b9 - 579*b8 + 12072*b7 - 1926*b6 + 7161*b5 + 2460*b4 - 40518*b3 + 40128*b2 + 348543*b1 - 302322) * q^81 + (2176*b11 + 112*b10 + 112*b9 + 2176*b8 - 5856*b7 - 2848*b6 - 2408*b5 - 328*b4 + 26551*b3 + 21023*b2 - 2176*b1 + 95952) * q^82 + (1344*b11 + 678*b9 - 2211*b8 - 14985*b7 + 6507*b6 - 4107*b5 + 603*b4 - 16347*b3 - 9270*b2 - 27765*b1 + 56874) * q^83 + (576*b11 + 1248*b10 - 1632*b9 - 672*b8 + 1664*b7 - 1440*b6 - 1824*b5 - 2784*b4 + 22912*b3 - 10336*b2 - 9568*b1 + 127296) * q^84 + (-8352*b11 - 708*b10 + 1416*b9 - 6747*b8 - 10974*b7 + 4029*b6 + 2709*b5 - 4926*b4 - 82269*b3 + 38115*b2 - 276873*b1 + 275466) * q^85 + (504*b11 + 2112*b10 + 1284*b8 - 6408*b7 - 4284*b6 - 6276*b5 + 2952*b4 + 6732*b3 - 15397*b2 + 177588*b1 + 189432) * q^86 + (5874*b11 + 2298*b10 - 3618*b9 + 6066*b8 + 12813*b7 - 6354*b6 + 5232*b5 + 993*b4 - 20772*b3 - 16311*b2 + 2100*b1 + 249570) * q^87 + (-960*b11 + 1408*b10 - 704*b9 + 2368*b8 - 2176*b7 - 1792*b6 + 2048*b5 + 384*b4 - 288*b3 + 1280*b2 + 18496*b1 + 448) * q^88 + (-624*b11 - 716*b10 + 716*b9 + 1334*b8 + 1333*b7 + 809*b6 - 2783*b5 - 1072*b4 - 60124*b3 + 62005*b2 + 279858*b1 - 139338) * q^89 + (-2475*b11 - 2439*b10 + 1755*b9 - 5319*b8 - 1476*b7 + 3456*b6 - 9504*b5 - 279*b4 - 3942*b3 + 14751*b2 - 404181*b1 + 39663) * q^90 + (-3651*b11 + 2442*b10 + 2442*b9 - 3651*b8 + 17181*b7 - 2577*b6 + 1614*b5 - 1749*b4 + 13827*b3 + 32757*b2 + 3651*b1 + 18679) * q^91 + (3648*b11 - 320*b9 + 128*b8 + 4544*b7 - 224*b6 - 3360*b5 + 7264*b4 + 11392*b3 + 512*b2 + 139584*b1 - 275520) * q^92 + (2592*b11 + 4545*b10 - 1530*b9 - 1764*b8 + 4255*b7 + 108*b6 + 3859*b5 + 3924*b4 + 30395*b3 - 29918*b2 - 45453*b1 - 190272) * q^93 + (7344*b11 + 2661*b10 - 5322*b9 + 8214*b8 - 924*b7 - 2040*b6 - 11160*b5 - 1491*b4 - 30519*b3 + 18036*b2 + 119844*b1 - 109536) * q^94 + (-3738*b11 + 782*b10 - 700*b8 + 15522*b7 + 8148*b6 + 8120*b5 - 5892*b4 - 10302*b3 + 13650*b2 - 111660*b1 - 128628) * q^95 + (1024*b11 + 1024*b9 - 1024*b8 - 4096*b3 - 2048*b2 + 19456*b1 - 13312) * q^96 + (-1290*b11 + 5768*b10 - 2884*b9 + 6473*b8 + 8635*b7 + 1726*b6 - 4490*b5 - 7494*b4 + 31023*b3 - 57992*b2 - 9948*b1 + 5648) * q^97 + (-3693*b11 - 277*b10 + 277*b9 + 1447*b8 + 3428*b7 - 5120*b6 + 18464*b5 + 2827*b4 + 8377*b3 - 16324*b2 - 597603*b1 + 284457) * q^98 + (2637*b11 + 750*b10 - 1626*b9 + 7314*b8 - 19158*b7 - 1494*b6 + 6162*b5 + 5223*b4 - 2673*b3 + 24279*b2 + 364014*b1 + 190140) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10})$$ 12 * q - 42 * q^3 + 192 * q^4 + 432 * q^5 - 144 * q^6 + 240 * q^7 + 2190 * q^9 $$12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9} + 378 q^{11} + 384 q^{12} + 1680 q^{13} - 4752 q^{14} - 10872 q^{15} - 6144 q^{16} - 2976 q^{18} - 2820 q^{19} + 13824 q^{20} + 24876 q^{21} - 3600 q^{22} - 76248 q^{23} - 6144 q^{24} + 8094 q^{25} + 127008 q^{27} + 15360 q^{28} + 97092 q^{29} + 34272 q^{30} + 21480 q^{31} - 246258 q^{33} - 27360 q^{34} + 38208 q^{36} - 25536 q^{37} + 97632 q^{38} + 42204 q^{39} - 410562 q^{41} - 222144 q^{42} + 71430 q^{43} + 13716 q^{45} - 135072 q^{46} + 347652 q^{47} + 55296 q^{48} - 135954 q^{49} + 311040 q^{50} + 336402 q^{51} - 53760 q^{52} - 173520 q^{54} + 580392 q^{55} - 152064 q^{56} - 522282 q^{57} + 159264 q^{58} + 369738 q^{59} - 170496 q^{60} + 135744 q^{61} - 103800 q^{63} - 393216 q^{64} - 753840 q^{65} + 909216 q^{66} - 289938 q^{67} + 744768 q^{68} + 2059272 q^{69} + 155952 q^{70} - 374784 q^{72} - 977700 q^{73} - 2197152 q^{74} - 2115342 q^{75} - 45120 q^{76} - 159192 q^{77} - 631488 q^{78} - 764796 q^{79} - 1428282 q^{81} + 1073088 q^{82} + 396900 q^{83} + 1441536 q^{84} + 1619568 q^{85} + 3264624 q^{86} + 3072636 q^{87} + 115200 q^{88} - 1987200 q^{90} + 355584 q^{91} - 2439936 q^{92} - 2526576 q^{93} - 736848 q^{94} - 2089260 q^{95} - 49152 q^{96} - 38874 q^{97} + 4398804 q^{99}+O(q^{100})$$ 12 * q - 42 * q^3 + 192 * q^4 + 432 * q^5 - 144 * q^6 + 240 * q^7 + 2190 * q^9 + 378 * q^11 + 384 * q^12 + 1680 * q^13 - 4752 * q^14 - 10872 * q^15 - 6144 * q^16 - 2976 * q^18 - 2820 * q^19 + 13824 * q^20 + 24876 * q^21 - 3600 * q^22 - 76248 * q^23 - 6144 * q^24 + 8094 * q^25 + 127008 * q^27 + 15360 * q^28 + 97092 * q^29 + 34272 * q^30 + 21480 * q^31 - 246258 * q^33 - 27360 * q^34 + 38208 * q^36 - 25536 * q^37 + 97632 * q^38 + 42204 * q^39 - 410562 * q^41 - 222144 * q^42 + 71430 * q^43 + 13716 * q^45 - 135072 * q^46 + 347652 * q^47 + 55296 * q^48 - 135954 * q^49 + 311040 * q^50 + 336402 * q^51 - 53760 * q^52 - 173520 * q^54 + 580392 * q^55 - 152064 * q^56 - 522282 * q^57 + 159264 * q^58 + 369738 * q^59 - 170496 * q^60 + 135744 * q^61 - 103800 * q^63 - 393216 * q^64 - 753840 * q^65 + 909216 * q^66 - 289938 * q^67 + 744768 * q^68 + 2059272 * q^69 + 155952 * q^70 - 374784 * q^72 - 977700 * q^73 - 2197152 * q^74 - 2115342 * q^75 - 45120 * q^76 - 159192 * q^77 - 631488 * q^78 - 764796 * q^79 - 1428282 * q^81 + 1073088 * q^82 + 396900 * q^83 + 1441536 * q^84 + 1619568 * q^85 + 3264624 * q^86 + 3072636 * q^87 + 115200 * q^88 - 1987200 * q^90 + 355584 * q^91 - 2439936 * q^92 - 2526576 * q^93 - 736848 * q^94 - 2089260 * q^95 - 49152 * q^96 - 38874 * q^97 + 4398804 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$$ :

 $$\beta_{1}$$ $$=$$ $$( - 119 \nu^{11} - 59366 \nu^{9} - 10447223 \nu^{7} - 794976432 \nu^{5} - 25420007664 \nu^{3} + \cdots + 25705589760 ) / 51411179520$$ (-119*v^11 - 59366*v^9 - 10447223*v^7 - 794976432*v^5 - 25420007664*v^3 - 246402608256*v + 25705589760) / 51411179520 $$\beta_{2}$$ $$=$$ $$( - 79977 \nu^{11} - 234158 \nu^{10} - 22976562 \nu^{9} - 160687484 \nu^{8} + \cdots - 10\!\cdots\!40 ) / 10520012609280$$ (-79977*v^11 - 234158*v^10 - 22976562*v^9 - 160687484*v^8 - 2207867961*v^7 - 34293527246*v^6 - 86986802664*v^5 - 2987464567824*v^4 - 1232368368624*v^3 - 104865602354208*v^2 + 2418745339008*v - 1060766629021440) / 10520012609280 $$\beta_{3}$$ $$=$$ $$( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + \cdots - 10\!\cdots\!40 ) / 10520012609280$$ (79977*v^11 - 234158*v^10 + 22976562*v^9 - 160687484*v^8 + 2207867961*v^7 - 34293527246*v^6 + 86986802664*v^5 - 2987464567824*v^4 + 1232368368624*v^3 - 104865602354208*v^2 - 2418745339008*v - 1060766629021440) / 10520012609280 $$\beta_{4}$$ $$=$$ $$( 2018785 \nu^{11} + 50427040 \nu^{10} + 441850954 \nu^{9} + 13150349632 \nu^{8} + \cdots + 62\!\cdots\!40 ) / 84160100874240$$ (2018785*v^11 + 50427040*v^10 + 441850954*v^9 + 13150349632*v^8 + 16441271137*v^7 + 1021898930848*v^6 - 1294732628592*v^5 + 23706651025152*v^4 - 67927494811248*v^3 + 113520656452608*v^2 - 203524169740416*v + 6298447508720640) / 84160100874240 $$\beta_{5}$$ $$=$$ $$( 2658601 \nu^{11} + 49792016 \nu^{10} + 625663450 \nu^{9} + 13943243552 \nu^{8} + \cdots - 76\!\cdots\!20 ) / 84160100874240$$ (2658601*v^11 + 49792016*v^10 + 625663450*v^9 + 13943243552*v^8 + 34104214825*v^7 + 1231527538448*v^6 - 598838207280*v^5 + 35365881013632*v^4 - 58068547862256*v^3 - 78947558385408*v^2 - 222874132452480*v - 7692497976683520) / 84160100874240 $$\beta_{6}$$ $$=$$ $$( 3789367 \nu^{11} + 91688548 \nu^{10} + 1383048298 \nu^{9} + 27588769336 \nu^{8} + \cdots + 13\!\cdots\!40 ) / 21040025218560$$ (3789367*v^11 + 91688548*v^10 + 1383048298*v^9 + 27588769336*v^8 + 185839870831*v^7 + 2837672924164*v^6 + 11296153681044*v^5 + 122317059365136*v^4 + 296761361759424*v^3 + 2132930705414976*v^2 + 2413162191350016*v + 13236687923811840) / 21040025218560 $$\beta_{7}$$ $$=$$ $$( 18455885 \nu^{11} + 113412560 \nu^{10} + 6341669138 \nu^{9} + 38860753760 \nu^{8} + \cdots + 60\!\cdots\!80 ) / 84160100874240$$ (18455885*v^11 + 113412560*v^10 + 6341669138*v^9 + 38860753760*v^8 + 795126641837*v^7 + 4853986176080*v^6 + 45281670938208*v^5 + 276298629295680*v^4 + 1138835846124432*v^3 + 7056142761081600*v^2 + 9410424670235520*v + 60698510331540480) / 84160100874240 $$\beta_{8}$$ $$=$$ $$( - 3697973 \nu^{11} - 10394400 \nu^{10} - 1223942794 \nu^{9} - 4111867872 \nu^{8} + \cdots - 95\!\cdots\!60 ) / 14026683479040$$ (-3697973*v^11 - 10394400*v^10 - 1223942794*v^9 - 4111867872*v^8 - 146365636165*v^7 - 598349805408*v^6 - 7967883155720*v^5 - 39135417500832*v^4 - 198554287555632*v^3 - 1103232300438528*v^2 - 1716906782814336*v - 9525411508239360) / 14026683479040 $$\beta_{9}$$ $$=$$ $$( 2357021 \nu^{11} - 26258072 \nu^{10} + 793738762 \nu^{9} - 9474316112 \nu^{8} + \cdots - 16\!\cdots\!60 ) / 7013341739520$$ (2357021*v^11 - 26258072*v^10 + 793738762*v^9 - 9474316112*v^8 + 95300421325*v^7 - 1246411646168*v^6 + 4977096692600*v^5 - 73474166288352*v^4 + 106846156334064*v^3 - 1892072031975936*v^2 + 682461855116928*v - 16379630739624960) / 7013341739520 $$\beta_{10}$$ $$=$$ $$( - 30942853 \nu^{11} + 63612592 \nu^{10} - 10150528594 \nu^{9} + 6352050784 \nu^{8} + \cdots - 76\!\cdots\!60 ) / 84160100874240$$ (-30942853*v^11 + 63612592*v^10 - 10150528594*v^9 + 6352050784*v^8 - 1177709270725*v^7 - 1489665176144*v^6 - 59126322103920*v^5 - 210754607273856*v^4 - 1224085328146512*v^3 - 7798028503032576*v^2 - 7966668128950656*v - 76011443927516160) / 84160100874240 $$\beta_{11}$$ $$=$$ $$( 93257311 \nu^{11} - 164458720 \nu^{10} + 31877901190 \nu^{9} - 53050300288 \nu^{8} + \cdots - 64\!\cdots\!80 ) / 84160100874240$$ (93257311*v^11 - 164458720*v^10 + 31877901190*v^9 - 53050300288*v^8 + 3970946505535*v^7 - 6117873519712*v^6 + 224849028582720*v^5 - 317784753586368*v^4 + 5665620769489584*v^3 - 7492891719532032*v^2 + 47617178158961280*v - 64202471563207680) / 84160100874240
 $$\nu$$ $$=$$ $$( 2\beta_{8} + 3\beta_{7} - \beta_{6} + 3\beta_{5} - 5\beta_{3} + 4\beta_{2} - 10\beta _1 + 2 ) / 18$$ (2*b8 + 3*b7 - b6 + 3*b5 - 5*b3 + 4*b2 - 10*b1 + 2) / 18 $$\nu^{2}$$ $$=$$ $$( 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} + 3 \beta_{6} - 7 \beta_{5} + \cdots - 1104 ) / 18$$ (2*b11 - 2*b10 - 2*b9 + 2*b8 - 11*b7 + 3*b6 - 7*b5 + 8*b4 - 2*b3 - 21*b2 - 2*b1 - 1104) / 18 $$\nu^{3}$$ $$=$$ $$( 36 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} - 154 \beta_{8} - 249 \beta_{7} + 83 \beta_{6} + \cdots - 1498 ) / 18$$ (36*b11 + 24*b10 - 24*b9 - 154*b8 - 249*b7 + 83*b6 - 285*b5 - 12*b4 - 368*b3 + 463*b2 + 3602*b1 - 1498) / 18 $$\nu^{4}$$ $$=$$ $$( - 394 \beta_{11} + 346 \beta_{10} + 346 \beta_{9} - 394 \beta_{8} + 2191 \beta_{7} - 615 \beta_{6} + \cdots + 98760 ) / 18$$ (-394*b11 + 346*b10 + 346*b9 - 394*b8 + 2191*b7 - 615*b6 + 671*b5 - 940*b4 + 1450*b3 + 4581*b2 + 394*b1 + 98760) / 18 $$\nu^{5}$$ $$=$$ $$( - 6804 \beta_{11} - 4536 \beta_{10} + 4536 \beta_{9} + 13802 \beta_{8} + 25677 \beta_{7} + \cdots + 278906 ) / 18$$ (-6804*b11 - 4536*b10 + 4536*b9 + 13802*b8 + 25677*b7 - 9607*b6 + 37197*b5 + 3840*b4 + 81832*b3 - 95279*b2 - 635866*b1 + 278906) / 18 $$\nu^{6}$$ $$=$$ $$( 61598 \beta_{11} - 49382 \beta_{10} - 49382 \beta_{9} + 61598 \beta_{8} - 344663 \beta_{7} + \cdots - 10730004 ) / 18$$ (61598*b11 - 49382*b10 - 49382*b9 + 61598*b8 - 344663*b7 + 98271*b6 - 58987*b5 + 107876*b4 - 333770*b3 - 786309*b2 - 61598*b1 - 10730004) / 18 $$\nu^{7}$$ $$=$$ $$( 1082880 \beta_{11} + 738660 \beta_{10} - 738660 \beta_{9} - 1382062 \beta_{8} - 2994321 \beta_{7} + \cdots - 43323814 ) / 18$$ (1082880*b11 + 738660*b10 - 738660*b9 - 1382062*b8 - 2994321*b7 + 1217819*b6 - 5116125*b5 - 724008*b4 - 12377120*b3 + 14318947*b2 + 97303622*b1 - 43323814) / 18 $$\nu^{8}$$ $$=$$ $$( - 8984866 \beta_{11} + 6747970 \beta_{10} + 6747970 \beta_{9} - 8984866 \beta_{8} + 50477575 \beta_{7} + \cdots + 1299912672 ) / 18$$ (-8984866*b11 + 6747970*b10 + 6747970*b9 - 8984866*b8 + 50477575*b7 - 14538111*b6 + 5493695*b5 - 13283836*b4 + 58213498*b3 + 121974909*b2 + 8984866*b1 + 1299912672) / 18 $$\nu^{9}$$ $$=$$ $$( - 161899740 \beta_{11} - 112694880 \beta_{10} + 112694880 \beta_{9} + 152703602 \beta_{8} + \cdots + 6352973042 ) / 18$$ (-161899740*b11 - 112694880*b10 + 112694880*b9 + 152703602*b8 + 376483773*b7 - 160290151*b6 + 709248693*b5 + 115683360*b4 + 1746399160*b3 - 2022372671*b2 - 14181956530*b1 + 6352973042) / 18 $$\nu^{10}$$ $$=$$ $$( 1275517622 \beta_{11} - 917170862 \beta_{10} - 917170862 \beta_{9} + 1275517622 \beta_{8} + \cdots - 167725156524 ) / 18$$ (1275517622*b11 - 917170862*b10 - 917170862*b9 + 1275517622*b8 - 7189238831*b7 + 2087168343*b6 - 556996747*b5 + 1726994228*b4 - 9074225402*b3 - 17990458461*b2 - 1275517622*b1 - 167725156524) / 18 $$\nu^{11}$$ $$=$$ $$( 23463449496 \beta_{11} + 16548163164 \beta_{10} - 16548163164 \beta_{9} - 18294797254 \beta_{8} + \cdots - 909341860558 ) / 18$$ (23463449496*b11 + 16548163164*b10 - 16548163164*b9 - 18294797254*b8 - 49497749649*b7 + 21570075563*b6 - 98497505205*b5 - 17239115352*b4 - 242337271568*b3 + 281146462483*b2 + 2023929703982*b1 - 909341860558) / 18

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$1 - \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 4.28281i 7.20150i − 11.8022i − 8.15670i 8.88570i − 3.87527i − 4.28281i − 7.20150i 11.8022i 8.15670i − 8.88570i 3.87527i
−4.89898 + 2.82843i −22.5447 14.8572i 16.0000 27.7128i 156.951 + 90.6160i 152.469 + 9.01919i 104.306 + 180.663i 181.019i 287.526 + 669.903i −1025.20
5.2 −4.89898 + 2.82843i −11.1983 + 24.5683i 16.0000 27.7128i −39.5602 22.8401i −14.6295 152.033i −245.097 424.521i 181.019i −478.198 550.243i 258.406
5.3 −4.89898 + 2.82843i 25.6924 + 8.30052i 16.0000 27.7128i −9.39126 5.42205i −149.344 + 32.0051i 322.041 + 557.792i 181.019i 591.203 + 426.521i 61.3435
5.4 4.89898 2.82843i −23.6571 13.0130i 16.0000 27.7128i −95.8504 55.3393i −152.702 + 3.16189i −163.169 282.617i 181.019i 390.321 + 615.703i −626.092
5.5 4.89898 2.82843i −14.9408 + 22.4894i 16.0000 27.7128i 202.253 + 116.771i −9.58533 + 152.434i 95.5752 + 165.541i 181.019i −282.543 672.020i 1321.11
5.6 4.89898 2.82843i 25.6485 8.43532i 16.0000 27.7128i 1.59771 + 0.922438i 101.793 113.869i 6.34411 + 10.9883i 181.019i 586.691 432.707i 10.4362
11.1 −4.89898 2.82843i −22.5447 + 14.8572i 16.0000 + 27.7128i 156.951 90.6160i 152.469 9.01919i 104.306 180.663i 181.019i 287.526 669.903i −1025.20
11.2 −4.89898 2.82843i −11.1983 24.5683i 16.0000 + 27.7128i −39.5602 + 22.8401i −14.6295 + 152.033i −245.097 + 424.521i 181.019i −478.198 + 550.243i 258.406
11.3 −4.89898 2.82843i 25.6924 8.30052i 16.0000 + 27.7128i −9.39126 + 5.42205i −149.344 32.0051i 322.041 557.792i 181.019i 591.203 426.521i 61.3435
11.4 4.89898 + 2.82843i −23.6571 + 13.0130i 16.0000 + 27.7128i −95.8504 + 55.3393i −152.702 3.16189i −163.169 + 282.617i 181.019i 390.321 615.703i −626.092
11.5 4.89898 + 2.82843i −14.9408 22.4894i 16.0000 + 27.7128i 202.253 116.771i −9.58533 152.434i 95.5752 165.541i 181.019i −282.543 + 672.020i 1321.11
11.6 4.89898 + 2.82843i 25.6485 + 8.43532i 16.0000 + 27.7128i 1.59771 0.922438i 101.793 + 113.869i 6.34411 10.9883i 181.019i 586.691 + 432.707i 10.4362
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 18.7.d.a 12
3.b odd 2 1 54.7.d.a 12
4.b odd 2 1 144.7.q.c 12
9.c even 3 1 54.7.d.a 12
9.c even 3 1 162.7.b.c 12
9.d odd 6 1 inner 18.7.d.a 12
9.d odd 6 1 162.7.b.c 12
12.b even 2 1 432.7.q.b 12
36.f odd 6 1 432.7.q.b 12
36.h even 6 1 144.7.q.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a 12 1.a even 1 1 trivial
18.7.d.a 12 9.d odd 6 1 inner
54.7.d.a 12 3.b odd 2 1
54.7.d.a 12 9.c even 3 1
144.7.q.c 12 4.b odd 2 1
144.7.q.c 12 36.h even 6 1
162.7.b.c 12 9.c even 3 1
162.7.b.c 12 9.d odd 6 1
432.7.q.b 12 12.b even 2 1
432.7.q.b 12 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 32 T^{2} + 1024)^{3}$$
$3$ $$T^{12} + \cdots + 15\!\cdots\!21$$
$5$ $$T^{12} + \cdots + 18\!\cdots\!00$$
$7$ $$T^{12} + \cdots + 27\!\cdots\!00$$
$11$ $$T^{12} + \cdots + 83\!\cdots\!09$$
$13$ $$T^{12} + \cdots + 98\!\cdots\!16$$
$17$ $$T^{12} + \cdots + 64\!\cdots\!00$$
$19$ $$(T^{6} + \cdots - 71\!\cdots\!00)^{2}$$
$23$ $$T^{12} + \cdots + 18\!\cdots\!84$$
$29$ $$T^{12} + \cdots + 50\!\cdots\!00$$
$31$ $$T^{12} + \cdots + 22\!\cdots\!00$$
$37$ $$(T^{6} + \cdots + 54\!\cdots\!48)^{2}$$
$41$ $$T^{12} + \cdots + 10\!\cdots\!25$$
$43$ $$T^{12} + \cdots + 19\!\cdots\!25$$
$47$ $$T^{12} + \cdots + 11\!\cdots\!84$$
$53$ $$T^{12} + \cdots + 27\!\cdots\!00$$
$59$ $$T^{12} + \cdots + 10\!\cdots\!25$$
$61$ $$T^{12} + \cdots + 10\!\cdots\!04$$
$67$ $$T^{12} + \cdots + 50\!\cdots\!25$$
$71$ $$T^{12} + \cdots + 58\!\cdots\!76$$
$73$ $$(T^{6} + \cdots - 49\!\cdots\!60)^{2}$$
$79$ $$T^{12} + \cdots + 34\!\cdots\!00$$
$83$ $$T^{12} + \cdots + 42\!\cdots\!36$$
$89$ $$T^{12} + \cdots + 74\!\cdots\!00$$
$97$ $$T^{12} + \cdots + 16\!\cdots\!25$$