# Properties

 Label 15.6.e.a Level $15$ Weight $6$ Character orbit 15.e Analytic conductor $2.406$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 15.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.40575729719$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625$$ x^16 + 10768*x^12 + 16341006*x^8 + 4217167600*x^4 + 50906640625 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{10}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 5) q^{6} + (\beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} - 5 \beta_{3} - 5) q^{7} + ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{8} + (2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} - 6 \beta_{7} + \cdots + 7 \beta_{2}) q^{9}+O(q^{10})$$ q - b7 * q^2 - b4 * q^3 + (-b6 - 13*b3) * q^4 + (-b14 + b9 + b2) * q^5 + (b15 - b14 + b11 - b8 + b5 - b4 - b2 + b1 - 5) * q^6 + (b13 - b10 + b9 + b8 - 5*b3 - 5) * q^7 + (-2*b15 + 2*b14 + b12 - b11 - b5 - b4 - 7*b2) * q^8 + (2*b15 + b14 - 2*b13 - 2*b12 + b10 - 2*b9 - 6*b7 + 2*b6 + b5 - b4 + 37*b3 + 7*b2) * q^9 $$q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 5) q^{6} + (\beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} - 5 \beta_{3} - 5) q^{7} + ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{8} + (2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} - 6 \beta_{7} + \cdots + 7 \beta_{2}) q^{9}+ \cdots + ( - 865 \beta_{15} + 364 \beta_{14} - 1043 \beta_{13} + 865 \beta_{12} + \cdots + 2413 \beta_{2}) q^{99}+O(q^{100})$$ q - b7 * q^2 - b4 * q^3 + (-b6 - 13*b3) * q^4 + (-b14 + b9 + b2) * q^5 + (b15 - b14 + b11 - b8 + b5 - b4 - b2 + b1 - 5) * q^6 + (b13 - b10 + b9 + b8 - 5*b3 - 5) * q^7 + (-2*b15 + 2*b14 + b12 - b11 - b5 - b4 - 7*b2) * q^8 + (2*b15 + b14 - 2*b13 - 2*b12 + b10 - 2*b9 - 6*b7 + 2*b6 + b5 - b4 + 37*b3 + 7*b2) * q^9 + (-2*b15 + 2*b14 - 2*b13 + 2*b12 - 2*b11 + 3*b10 - 4*b9 + b8 - 5*b5 + 11*b4 + 10*b3 + 4*b2 - 5*b1 + 65) * q^10 + (-4*b15 + 3*b14 + b13 - 4*b12 - 3*b11 - 2*b9 + 7*b7 + b5 + 2*b4 + 7*b2) * q^11 + (3*b15 + 3*b14 + 5*b13 + 4*b12 + 4*b11 - 2*b10 - 9*b9 + 2*b8 + 15*b7 + 5*b6 + 25*b3 - 5*b1 + 25) * q^12 + (3*b15 - 3*b14 - 3*b12 + 3*b11 - 3*b10 - 3*b8 + 10*b6 + 2*b5 + 130*b3 - 6*b2 + 10*b1 - 130) * q^13 + (-4*b15 - 4*b14 - 6*b13 + 4*b12 - 4*b11 + 26*b9 + 21*b7 + 6*b5 + 26*b4 - 21*b2) * q^14 + (6*b15 - 4*b13 + b12 + b11 + b10 - 4*b9 + 2*b8 + 42*b7 - 10*b6 - 10*b5 + 4*b4 - 50*b3 - 18*b2 + 10*b1 - 185) * q^15 + (2*b15 - 2*b14 + 6*b13 - 2*b12 + 2*b11 + 22*b9 - 4*b8 + 10*b5 - 26*b4 - 4*b2 - 3*b1 - 29) * q^16 + (-8*b15 - 8*b14 + 14*b13 + b12 + b11 - 26*b9 - 38*b7) * q^17 + (8*b15 - 8*b14 - 7*b12 + 7*b11 - b10 - b8 - 20*b6 - 8*b5 - 4*b4 - 370*b3 + 103*b2 - 20*b1 + 370) * q^18 + (-2*b15 + 2*b14 - 19*b13 + 2*b12 - 2*b11 + 4*b10 - 53*b9 - 16*b6 + 15*b5 - 49*b4 - 296*b3 + 4*b2) * q^19 + (-5*b15 + 7*b14 - 15*b13 - 2*b12 - 10*b11 + 43*b9 - 147*b7 - 20*b5 - 52*b4 + 58*b2) * q^20 + (2*b15 + 4*b14 + 14*b13 + 9*b12 - 4*b11 - 3*b9 + 7*b8 - 147*b7 + 7*b5 + 10*b4 - 140*b2 + 14*b1 + 92) * q^21 + (22*b13 + 4*b10 + 74*b9 - 4*b8 - 15*b6 + 215*b3 + 15*b1 + 215) * q^22 + (b15 - b14 - 3*b12 + 3*b11 - 13*b5 - 45*b4 - 202*b2) * q^23 + (-7*b15 + b14 - 23*b13 + 7*b12 + 9*b11 - 8*b10 - 5*b9 - 123*b7 + 29*b6 + 31*b5 - 13*b4 + 685*b3 + 115*b2) * q^24 + (15*b15 - 15*b14 - 10*b13 - 15*b12 + 15*b11 - 25*b10 - 50*b9 - 5*b8 + 30*b6 - 10*b5 + 60*b4 - 465*b3 - 30*b2 - 10*b1 + 570) * q^25 + (19*b15 - 15*b14 + 27*b13 + 19*b12 + 15*b11 - 85*b9 + 342*b7 + 27*b5 + 85*b4 + 342*b2) * q^26 + (-9*b15 - 9*b14 + 21*b13 - 27*b12 - 27*b11 + 18*b10 + 42*b9 - 18*b8 + 198*b7 - 810*b3 - 810) * q^27 + (-28*b15 + 28*b14 + 28*b12 - 28*b11 + 28*b10 + 28*b8 - 15*b6 - 68*b5 + 148*b4 + 685*b3 + 56*b2 - 15*b1 - 685) * q^28 + (20*b15 + 23*b14 - 36*b13 - 20*b12 + 23*b11 + 65*b9 + 259*b7 + 36*b5 + 65*b4 - 259*b2) * q^29 + (-35*b15 - b14 - 30*b13 - 8*b12 + 5*b11 - 5*b10 + 16*b9 - 20*b8 + 477*b7 + 35*b6 - 10*b5 - 48*b4 + 1795*b3 - 319*b2 + 5*b1 - 835) * q^30 + (-17*b15 + 17*b14 + 14*b13 + 17*b12 - 17*b11 + 8*b9 + 34*b8 - 20*b5 + 26*b4 + 34*b2 - 40*b1 - 1358) * q^31 + (51*b15 + 51*b14 + 5*b13 - 22*b12 - 22*b11 - 117*b9 - 337*b7) * q^32 + (-45*b15 + 45*b14 + 48*b12 - 48*b11 + 6*b10 + 6*b8 + 60*b6 - 14*b5 + 50*b4 - 375*b3 + 438*b2 + 60*b1 + 375) * q^33 + (19*b15 - 19*b14 + 3*b13 - 19*b12 + 19*b11 - 38*b10 - 29*b9 + 54*b6 + 35*b5 - 67*b4 - 1090*b3 - 38*b2) * q^34 + (15*b15 - 12*b14 + 20*b13 + 21*b12 + 80*b11 - 63*b9 - 469*b7 - 15*b5 - 104*b4 + 517*b2) * q^35 + (-46*b15 - 2*b14 + 14*b13 - 54*b12 + 2*b11 + 26*b9 - 8*b8 - 714*b7 + 22*b5 - 34*b4 - 722*b2 - 79*b1 + 3851) * q^36 + (17*b10 + 34*b9 - 17*b8 + 10*b6 + 3480*b3 - 10*b1 + 3480) * q^37 + (70*b15 - 70*b14 - 6*b5 - 18*b4 - 648*b2) * q^38 + (-4*b15 - 47*b14 - 2*b13 + 4*b12 - 63*b11 + 16*b10 + 102*b9 - 573*b7 - 148*b6 - 14*b5 + 118*b4 + 394*b3 + 589*b2) * q^39 + (-26*b15 + 26*b14 - 26*b13 + 26*b12 - 26*b11 + 64*b10 - 2*b9 - 12*b8 - 125*b6 + 10*b5 - 82*b4 - 5245*b3 + 52*b2 + 110*b1 + 1370) * q^40 + (4*b15 - 8*b14 - 56*b13 + 4*b12 + 8*b11 + 172*b9 + 628*b7 - 56*b5 - 172*b4 + 628*b2) * q^41 + (-42*b15 - 42*b14 - 16*b13 + 59*b12 + 59*b11 - 55*b10 + 32*b9 + 55*b8 + 384*b7 - 95*b6 - 6415*b3 + 95*b1 - 6415) * q^42 + (92*b15 - 92*b14 - 92*b12 + 92*b11 - 92*b10 - 92*b8 - 80*b6 + 207*b5 - 437*b4 + 1480*b3 - 184*b2 - 80*b1 - 1480) * q^43 + (-11*b15 - 19*b14 + 97*b13 + 11*b12 - 19*b11 - 261*b9 + 335*b7 - 97*b5 - 261*b4 - 335*b2) * q^44 + (40*b15 + 11*b14 + 110*b13 + 20*b12 - 75*b11 - 10*b10 + 14*b9 + 75*b8 + 465*b7 - 20*b6 + 5*b5 + 190*b4 + 6110*b3 - 481*b2 - 210*b1 - 1380) * q^45 + (47*b15 - 47*b14 - 19*b13 - 47*b12 + 47*b11 + 37*b9 - 94*b8 + 75*b5 - 131*b4 - 94*b2 + 272*b1 - 9380) * q^46 + (-81*b15 - 81*b14 - 33*b13 + 107*b12 + 107*b11 + 261*b9 - 334*b7) * q^47 + (22*b15 - 22*b14 - 89*b12 + 89*b11 + 4*b10 + 4*b8 + 35*b6 + 17*b5 - 39*b4 - 4685*b3 + 839*b2 + 35*b1 + 4685) * q^48 + (-66*b15 + 66*b14 + 48*b13 + 66*b12 - 66*b11 + 132*b10 + 276*b9 + 104*b6 - 180*b5 + 408*b4 - 2269*b3 + 132*b2) * q^49 + (75*b15 - 15*b14 - 25*b13 - 85*b12 - 225*b11 + 15*b9 - 910*b7 + 175*b5 + 665*b4 - 85*b2) * q^50 + (120*b15 + 21*b14 - 175*b13 + 72*b12 - 21*b11 - 121*b9 - 48*b8 - 147*b7 - 127*b5 + 73*b4 - 195*b2 + 96*b1 + 8700) * q^51 + (-172*b13 - 112*b10 - 740*b9 + 112*b8 + 320*b6 + 12680*b3 - 320*b1 + 12680) * q^52 + (-252*b15 + 252*b14 - 59*b12 + 59*b11 + 148*b5 + 326*b4 - 104*b2) * q^53 + (-24*b15 + 168*b14 + 219*b13 + 24*b12 + 135*b11 + 33*b10 - 261*b9 + 522*b7 + 57*b6 - 252*b5 - 228*b4 + 8385*b3 - 489*b2) * q^54 + (-65*b15 + 65*b14 + 235*b13 + 65*b12 - 65*b11 + 5*b10 + 585*b9 + 125*b8 + 60*b6 + 5*b5 - 145*b4 - 11205*b3 + 130*b2 - 20*b1 + 5565) * q^55 + (-119*b15 + 181*b14 - 31*b13 - 119*b12 - 181*b11 + 31*b9 + 71*b7 - 31*b5 - 31*b4 + 71*b2) * q^56 + (189*b15 + 189*b14 - 180*b13 - 48*b12 - 48*b11 + 33*b10 - 444*b9 - 33*b8 + 216*b7 + 210*b6 - 10290*b3 - 210*b1 - 10290) * q^57 + (-84*b15 + 84*b14 + 84*b12 - 84*b11 + 84*b10 + 84*b8 + 55*b6 - 130*b5 + 222*b4 + 10205*b3 + 168*b2 + 55*b1 - 10205) * q^58 + (-65*b15 - 104*b14 + 88*b13 + 65*b12 - 104*b11 - 95*b9 + 263*b7 - 88*b5 - 95*b4 - 263*b2) * q^59 + (97*b15 - 43*b14 - 63*b13 - 9*b12 + 217*b11 + 102*b10 - 45*b9 - 106*b8 - 27*b7 + 90*b6 + 305*b5 - 203*b4 + 15570*b3 + 1277*b2 + 325*b1 - 13505) * q^60 + (-18*b15 + 18*b14 - 264*b13 + 18*b12 - 18*b11 - 828*b9 + 36*b8 - 300*b5 + 864*b4 + 36*b2 - 360*b1 - 11038) * q^61 + (-96*b15 - 96*b14 - 250*b13 - 98*b12 - 98*b11 + 942*b9 + 862*b7) * q^62 + (316*b15 - 316*b14 - 74*b12 + 74*b11 - 107*b10 - 107*b8 + 20*b6 + 431*b5 - 419*b4 - 8945*b3 - 1480*b2 + 20*b1 + 8945) * q^63 + (82*b15 - 82*b14 + 314*b13 - 82*b12 + 82*b11 - 164*b10 + 778*b9 - 387*b6 - 150*b5 + 614*b4 - 17843*b3 - 164*b2) * q^64 + (-335*b15 + 37*b14 + 320*b13 + 144*b12 + 180*b11 - 662*b9 + 2204*b7 + 10*b5 - 6*b4 - 532*b2) * q^65 + (-11*b15 - 157*b14 + 175*b13 + 99*b12 + 157*b11 + 423*b9 + 110*b8 + 1281*b7 + 65*b5 - 313*b4 + 1391*b2 - 50*b1 + 16450) * q^66 + (49*b13 + 110*b10 + 367*b9 - 110*b8 - 880*b6 + 8990*b3 + 880*b1 + 8990) * q^67 + (172*b15 - 172*b14 + 354*b12 - 354*b11 - 134*b5 + 306*b4 + 922*b2) * q^68 + (295*b15 - 178*b14 - 42*b13 - 295*b12 - 27*b11 - 151*b10 - 66*b9 - 234*b7 + 322*b6 + 193*b5 - 217*b4 + 12950*b3 + 83*b2) * q^69 + (235*b15 - 235*b14 - 215*b13 - 235*b12 + 235*b11 - 230*b10 - 635*b9 - 240*b8 + 55*b6 + 515*b5 - 1075*b4 - 18515*b3 - 470*b2 - 535*b1 + 23945) * q^70 + (-18*b15 - 394*b14 - 173*b13 - 18*b12 + 394*b11 + 931*b9 - 3296*b7 - 173*b5 - 931*b4 - 3296*b2) * q^71 + (-81*b15 - 81*b14 - 207*b13 + 102*b12 + 102*b11 + 156*b10 + 543*b9 - 156*b8 - 2673*b7 - 210*b6 - 20490*b3 + 210*b1 - 20490) * q^72 + (-174*b15 + 174*b14 + 174*b12 - 174*b11 + 174*b10 + 174*b8 + 300*b6 + 12*b5 - 384*b4 + 14945*b3 + 348*b2 + 300*b1 - 14945) * q^73 + (47*b15 + 109*b14 + 109*b13 - 47*b12 + 109*b11 - 483*b9 - 4012*b7 - 109*b5 - 483*b4 + 4012*b2) * q^74 + (-120*b15 + 45*b14 + 230*b13 + 10*b12 - 95*b11 - 140*b10 - 505*b9 - 70*b8 - 4035*b7 - 460*b6 - 180*b5 - 460*b4 + 16330*b3 - 435*b2 + 220*b1 - 12440) * q^75 + (-116*b15 + 116*b14 + 392*b13 + 116*b12 - 116*b11 + 944*b9 + 232*b8 + 160*b5 - 712*b4 + 232*b2 - 388*b1 - 17068) * q^76 + (303*b15 + 303*b14 + 104*b13 - 441*b12 - 441*b11 - 918*b9 - 146*b7) * q^77 + (-626*b15 + 626*b14 + 337*b12 - 337*b11 + 253*b10 + 253*b8 - 850*b6 - 888*b5 + 708*b4 - 26660*b3 - 2692*b2 - 850*b1 + 26660) * q^78 + (53*b15 - 53*b14 - 399*b13 - 53*b12 + 53*b11 - 106*b10 - 1303*b9 - 16*b6 + 505*b5 - 1409*b4 - 12146*b3 - 106*b2) * q^79 + (80*b15 + 52*b14 - 410*b13 + 9*b12 + 335*b11 + 1098*b9 + 3154*b7 - 5*b5 - 341*b4 - 2767*b2) * q^80 + (-231*b15 + 267*b14 - 186*b13 - 81*b12 - 267*b11 - 888*b9 + 150*b8 + 4788*b7 - 336*b5 + 1038*b4 + 4938*b2 + 300*b1 + 20451) * q^81 + (-152*b13 + 336*b10 + 216*b9 - 336*b8 + 340*b6 + 26860*b3 - 340*b1 + 26860) * q^82 + (419*b15 - 419*b14 - 487*b12 + 487*b11 + 269*b5 - 167*b4 + 5802*b2) * q^83 + (-337*b15 - 101*b14 - 77*b13 + 337*b12 - 117*b11 + 16*b10 + 929*b9 + 5061*b7 + 302*b6 + 61*b5 + 945*b4 + 22894*b3 - 5045*b2) * q^84 + (8*b15 - 8*b14 + 108*b13 - 8*b12 + 8*b11 + 93*b10 + 526*b9 - 109*b8 + 1090*b6 - 460*b5 + 1396*b4 - 12210*b3 - 16*b2 + 590*b1 + 300) * q^85 + (633*b15 + 263*b14 + 587*b13 + 633*b12 - 263*b11 - 2657*b9 - 2072*b7 + 587*b5 + 2657*b4 - 2072*b2) * q^86 + (-402*b15 - 402*b14 + 896*b13 - 266*b12 - 266*b11 - 281*b10 + 656*b9 + 281*b8 + 204*b7 + 500*b6 - 20855*b3 - 500*b1 - 20855) * q^87 + (356*b15 - 356*b14 - 356*b12 + 356*b11 - 356*b10 - 356*b8 + 405*b6 - 180*b5 + 1252*b4 + 25005*b3 - 712*b2 + 405*b1 - 25005) * q^88 + (-125*b15 + 509*b14 - 578*b13 + 125*b12 + 509*b11 + 1350*b9 + 2922*b7 + 578*b5 + 1350*b4 - 2922*b2) * q^89 + (-323*b15 + 143*b14 - 573*b13 - 182*b12 - 448*b11 - 278*b10 + 1509*b9 + 424*b8 - 2880*b7 + 425*b6 - 370*b5 + 1084*b4 + 20215*b3 + 5576*b2 - 70*b1 - 21790) * q^90 + (118*b15 - 118*b14 - 91*b13 - 118*b12 + 118*b11 - 37*b9 - 236*b8 + 145*b5 - 199*b4 - 236*b2 + 528*b1 - 30316) * q^91 + (8*b15 + 8*b14 + 476*b13 + 744*b12 + 744*b11 - 1444*b9 + 8484*b7) * q^92 + (-123*b15 + 123*b14 + 39*b12 - 39*b11 + 30*b10 + 30*b8 + 420*b6 + 180*b5 + 1546*b4 + 1290*b3 - 5646*b2 + 420*b1 - 1290) * q^93 + (-235*b15 + 235*b14 - 435*b13 + 235*b12 - 235*b11 + 470*b10 - 835*b9 + 324*b6 - 35*b5 - 365*b4 - 10120*b3 + 470*b2) * q^94 + (1070*b15 - 8*b14 - 165*b13 - 410*b12 - 860*b11 - 567*b9 + 3870*b7 - 845*b5 - 2085*b4 + 1098*b2) * q^95 + (97*b15 - 7*b14 + 597*b13 - 459*b12 + 7*b11 + 159*b9 - 556*b8 + 225*b7 + 1153*b5 - 715*b4 - 331*b2 - 317*b1 + 19405) * q^96 + (948*b13 - 522*b10 + 1800*b9 + 522*b8 + 860*b6 + 27335*b3 - 860*b1 + 27335) * q^97 + (-476*b15 + 476*b14 - 632*b12 + 632*b11 - 916*b5 - 4012*b4 + 2547*b2) * q^98 + (-865*b15 + 364*b14 - 1043*b13 + 865*b12 - 216*b11 + 580*b10 - 800*b9 - 1833*b7 - 1000*b6 + 463*b5 - 220*b4 - 10400*b3 + 2413*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 84 q^{6} - 80 q^{7}+O(q^{10})$$ 16 * q - 84 * q^6 - 80 * q^7 $$16 q - 84 q^{6} - 80 q^{7} + 1060 q^{10} + 420 q^{12} - 2120 q^{13} - 3000 q^{15} - 452 q^{16} + 6000 q^{18} + 1416 q^{21} + 3380 q^{22} + 9160 q^{25} - 12960 q^{27} - 10900 q^{28} - 13380 q^{30} - 21568 q^{31} + 5760 q^{33} + 61932 q^{36} + 55720 q^{37} + 21480 q^{40} - 103020 q^{42} - 23360 q^{43} - 21240 q^{45} - 151168 q^{46} + 74820 q^{48} + 138816 q^{51} + 204160 q^{52} + 89120 q^{55} - 163800 q^{57} - 163500 q^{58} - 217380 q^{60} - 175168 q^{61} + 143040 q^{63} + 263400 q^{66} + 140320 q^{67} + 385260 q^{70} - 328680 q^{72} - 240320 q^{73} - 199920 q^{75} - 271536 q^{76} + 429960 q^{78} + 326016 q^{81} + 431120 q^{82} + 2440 q^{85} - 331680 q^{87} - 401700 q^{88} - 348360 q^{90} - 487168 q^{91} - 22320 q^{93} + 311748 q^{96} + 440800 q^{97}+O(q^{100})$$ 16 * q - 84 * q^6 - 80 * q^7 + 1060 * q^10 + 420 * q^12 - 2120 * q^13 - 3000 * q^15 - 452 * q^16 + 6000 * q^18 + 1416 * q^21 + 3380 * q^22 + 9160 * q^25 - 12960 * q^27 - 10900 * q^28 - 13380 * q^30 - 21568 * q^31 + 5760 * q^33 + 61932 * q^36 + 55720 * q^37 + 21480 * q^40 - 103020 * q^42 - 23360 * q^43 - 21240 * q^45 - 151168 * q^46 + 74820 * q^48 + 138816 * q^51 + 204160 * q^52 + 89120 * q^55 - 163800 * q^57 - 163500 * q^58 - 217380 * q^60 - 175168 * q^61 + 143040 * q^63 + 263400 * q^66 + 140320 * q^67 + 385260 * q^70 - 328680 * q^72 - 240320 * q^73 - 199920 * q^75 - 271536 * q^76 + 429960 * q^78 + 326016 * q^81 + 431120 * q^82 + 2440 * q^85 - 331680 * q^87 - 401700 * q^88 - 348360 * q^90 - 487168 * q^91 - 22320 * q^93 + 311748 * q^96 + 440800 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625$$ :

 $$\beta_{1}$$ $$=$$ $$( -983\nu^{12} - 10530009\nu^{8} - 14985743913\nu^{4} - 1365328353575 ) / 52711868160$$ (-983*v^12 - 10530009*v^8 - 14985743913*v^4 - 1365328353575) / 52711868160 $$\beta_{2}$$ $$=$$ $$( -193037\nu^{13} - 2063897891\nu^{9} - 3039554293747\nu^{5} - 800189026191325\nu ) / 175266961632000$$ (-193037*v^13 - 2063897891*v^9 - 3039554293747*v^5 - 800189026191325*v) / 175266961632000 $$\beta_{3}$$ $$=$$ $$( 65179\nu^{14} + 704329347\nu^{10} + 1093176906949\nu^{6} + 338344643638525\nu^{2} ) / 1156275094100000$$ (65179*v^14 + 704329347*v^10 + 1093176906949*v^6 + 338344643638525*v^2) / 1156275094100000 $$\beta_{4}$$ $$=$$ $$( 17652037 \nu^{14} + 44517950 \nu^{13} - 182440375 \nu^{12} + 186662751291 \nu^{10} + 495889999350 \nu^{9} - 1876904386125 \nu^{8} + \cdots - 16\!\cdots\!75 ) / 41\!\cdots\!00$$ (17652037*v^14 + 44517950*v^13 - 182440375*v^12 + 186662751291*v^10 + 495889999350*v^9 - 1876904386125*v^8 + 254505385192347*v^6 + 886646362656450*v^5 - 2152134880451625*v^4 + 43025506941324325*v^2 + 338120300528026250*v - 168881000149721875) / 41625903387600000 $$\beta_{5}$$ $$=$$ $$( - 17652037 \nu^{14} + 44517950 \nu^{13} + 182440375 \nu^{12} - 186662751291 \nu^{10} + 495889999350 \nu^{9} + \cdots + 16\!\cdots\!75 ) / 13\!\cdots\!00$$ (-17652037*v^14 + 44517950*v^13 + 182440375*v^12 - 186662751291*v^10 + 495889999350*v^9 + 1876904386125*v^8 - 254505385192347*v^6 + 886646362656450*v^5 + 2152134880451625*v^4 - 43025506941324325*v^2 + 338120300528026250*v + 168881000149721875) / 13875301129200000 $$\beta_{6}$$ $$=$$ $$( 40605037 \nu^{14} + 433934370291 \nu^{10} + 626362399315347 \nu^{6} + 11\!\cdots\!25 \nu^{2} ) / 16\!\cdots\!00$$ (40605037*v^14 + 433934370291*v^10 + 626362399315347*v^6 + 112231779807789325*v^2) / 16650361355040000 $$\beta_{7}$$ $$=$$ $$( - 2501702961 \nu^{15} - 26901148843423 \nu^{11} + \cdots - 10\!\cdots\!25 \nu^{3} ) / 79\!\cdots\!00$$ (-2501702961*v^15 - 26901148843423*v^11 - 40512989266909391*v^7 - 10161642791532304225*v^3) / 7908921643644000000 $$\beta_{8}$$ $$=$$ $$( - 2077147966 \nu^{15} - 5030830545 \nu^{14} - 16232545500 \nu^{13} - 120958465000 \nu^{12} - 22414270741638 \nu^{11} + \cdots - 57\!\cdots\!00 ) / 39\!\cdots\!00$$ (-2077147966*v^15 - 5030830545*v^14 - 16232545500*v^13 - 120958465000*v^12 - 22414270741638*v^11 - 53198884117935*v^10 - 174687878881500*v^9 - 1333178634307500*v^8 - 34409367948012546*v^7 - 72534034779818895*v^6 - 259797537872710500*v^5 - 2286353916174915000*v^4 - 9152134028790657850*v^3 - 12262269478277432625*v^2 - 13090727261115862500*v - 577443498252887312500) / 3954460821822000000 $$\beta_{9}$$ $$=$$ $$( - 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} - 22414270741638 \nu^{11} + 17732961372645 \nu^{10} + \cdots + 16\!\cdots\!25 ) / 39\!\cdots\!00$$ (-2077147966*v^15 + 1676943515*v^14 + 17331835625*v^12 - 22414270741638*v^11 + 17732961372645*v^10 + 178305916681875*v^8 - 34409367948012546*v^7 + 24178011593272965*v^6 + 204452813642904375*v^4 - 9152134028790657850*v^3 + 4087423159425810875*v^2 + 16043695014223578125) / 3954460821822000000 $$\beta_{10}$$ $$=$$ $$( 2077147966 \nu^{15} + 39891462730 \nu^{14} - 16232545500 \nu^{13} + 51995506875 \nu^{12} + 22414270741638 \nu^{11} + \cdots + 48\!\cdots\!75 ) / 39\!\cdots\!00$$ (2077147966*v^15 + 39891462730*v^14 - 16232545500*v^13 + 51995506875*v^12 + 22414270741638*v^11 + 429932673945390*v^10 - 174687878881500*v^9 + 534917750045625*v^8 + 34409367948012546*v^7 + 654934582398012630*v^6 - 259797537872710500*v^5 + 613358440928713125*v^4 + 9152134028790657850*v^3 + 162255391807988379250*v^2 - 13090727261115862500*v + 48131085042670734375) / 3954460821822000000 $$\beta_{11}$$ $$=$$ $$( 4963927071 \nu^{15} + 3353887030 \nu^{14} - 82856539500 \nu^{13} - 34663671250 \nu^{12} + 52964213316153 \nu^{11} + \cdots - 32\!\cdots\!50 ) / 79\!\cdots\!00$$ (4963927071*v^15 + 3353887030*v^14 - 82856539500*v^13 - 34663671250*v^12 + 52964213316153*v^11 + 35465922745290*v^10 - 882846885098500*v^9 - 356611833363750*v^8 + 76188822451032801*v^7 + 48356023186545930*v^6 - 1250904083702724500*v^5 - 408905627285808750*v^4 + 16416247801819083975*v^3 + 8174846318851621750*v^2 - 186945138726007637500*v - 32087390028447156250) / 7908921643644000000 $$\beta_{12}$$ $$=$$ $$( 4963927071 \nu^{15} - 3353887030 \nu^{14} + 82856539500 \nu^{13} + 34663671250 \nu^{12} + 52964213316153 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00$$ (4963927071*v^15 - 3353887030*v^14 + 82856539500*v^13 + 34663671250*v^12 + 52964213316153*v^11 - 35465922745290*v^10 + 882846885098500*v^9 + 356611833363750*v^8 + 76188822451032801*v^7 - 48356023186545930*v^6 + 1250904083702724500*v^5 + 408905627285808750*v^4 + 16416247801819083975*v^3 - 8174846318851621750*v^2 + 186945138726007637500*v + 32087390028447156250) / 7908921643644000000 $$\beta_{13}$$ $$=$$ $$( 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} + 22414270741638 \nu^{11} + 17732961372645 \nu^{10} + \cdots + 16\!\cdots\!25 ) / 13\!\cdots\!00$$ (2077147966*v^15 + 1676943515*v^14 + 17331835625*v^12 + 22414270741638*v^11 + 17732961372645*v^10 + 178305916681875*v^8 + 34409367948012546*v^7 + 24178011593272965*v^6 + 204452813642904375*v^4 + 9152134028790657850*v^3 + 4087423159425810875*v^2 + 16043695014223578125) / 1318153607274000000 $$\beta_{14}$$ $$=$$ $$( 14854266548 \nu^{15} + 3353887030 \nu^{14} - 33222243375 \nu^{13} + 34663671250 \nu^{12} + 159176103876364 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00$$ (14854266548*v^15 + 3353887030*v^14 - 33222243375*v^13 + 34663671250*v^12 + 159176103876364*v^11 + 35465922745290*v^10 - 346118635127625*v^9 + 356611833363750*v^8 + 234466659299379788*v^7 + 48356023186545930*v^6 - 425686311547244625*v^5 + 408905627285808750*v^4 + 51012121813586157300*v^3 + 8174846318851621750*v^2 - 60412297296567384375*v + 32087390028447156250) / 7908921643644000000 $$\beta_{15}$$ $$=$$ $$( 14854266548 \nu^{15} + 3353887030 \nu^{14} + 33222243375 \nu^{13} + 34663671250 \nu^{12} + 159176103876364 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00$$ (14854266548*v^15 + 3353887030*v^14 + 33222243375*v^13 + 34663671250*v^12 + 159176103876364*v^11 + 35465922745290*v^10 + 346118635127625*v^9 + 356611833363750*v^8 + 234466659299379788*v^7 + 48356023186545930*v^6 + 425686311547244625*v^5 + 408905627285808750*v^4 + 51012121813586157300*v^3 + 8174846318851621750*v^2 + 60412297296567384375*v + 32087390028447156250) / 7908921643644000000
 $$\nu$$ $$=$$ $$( 6\beta_{15} - 6\beta_{14} - 3\beta_{12} + 3\beta_{11} + 5\beta_{5} + 9\beta_{4} + 12\beta_{2} ) / 90$$ (6*b15 - 6*b14 - 3*b12 + 3*b11 + 5*b5 + 9*b4 + 12*b2) / 90 $$\nu^{2}$$ $$=$$ $$( 3 \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 12 \beta_{6} + 5 \beta_{5} - 9 \beta_{4} + 690 \beta_{3} - 6 \beta_{2} ) / 18$$ (3*b15 - 3*b14 + b13 - 3*b12 + 3*b11 - 6*b10 - 3*b9 + 12*b6 + 5*b5 - 9*b4 + 690*b3 - 6*b2) / 18 $$\nu^{3}$$ $$=$$ $$( -27\beta_{15} - 27\beta_{14} + 95\beta_{13} + 134\beta_{12} + 134\beta_{11} - 231\beta_{9} + 1068\beta_{7} ) / 30$$ (-27*b15 - 27*b14 + 95*b13 + 134*b12 + 134*b11 - 231*b9 + 1068*b7) / 30 $$\nu^{4}$$ $$=$$ $$( 129 \beta_{15} - 129 \beta_{14} + 2 \beta_{13} - 129 \beta_{12} + 129 \beta_{11} + 264 \beta_{9} - 258 \beta_{8} + 260 \beta_{5} - 522 \beta_{4} - 258 \beta_{2} + 732 \beta _1 - 24045 ) / 9$$ (129*b15 - 129*b14 + 2*b13 - 129*b12 + 129*b11 + 264*b9 - 258*b8 + 260*b5 - 522*b4 - 258*b2 + 732*b1 - 24045) / 9 $$\nu^{5}$$ $$=$$ $$( - 31344 \beta_{15} + 31344 \beta_{14} + 1707 \beta_{12} - 1707 \beta_{11} - 25195 \beta_{5} - 72171 \beta_{4} - 350088 \beta_{2} ) / 90$$ (-31344*b15 + 31344*b14 + 1707*b12 - 1707*b11 - 25195*b5 - 72171*b4 - 350088*b2) / 90 $$\nu^{6}$$ $$=$$ $$( - 2535 \beta_{15} + 2535 \beta_{14} + 685 \beta_{13} + 2535 \beta_{12} - 2535 \beta_{11} + 5070 \beta_{10} + 7125 \beta_{9} - 16332 \beta_{6} - 5755 \beta_{5} + 12195 \beta_{4} - 453290 \beta_{3} + 5070 \beta_{2} ) / 2$$ (-2535*b15 + 2535*b14 + 685*b13 + 2535*b12 - 2535*b11 + 5070*b10 + 7125*b9 - 16332*b6 - 5755*b5 + 12195*b4 - 453290*b3 + 5070*b2) / 2 $$\nu^{7}$$ $$=$$ $$( - 15171 \beta_{15} - 15171 \beta_{14} - 2388515 \beta_{13} - 2702868 \beta_{12} - 2702868 \beta_{11} + 7195887 \beta_{9} - 34754736 \beta_{7} ) / 90$$ (-15171*b15 - 15171*b14 - 2388515*b13 - 2702868*b12 - 2702868*b11 + 7195887*b9 - 34754736*b7) / 90 $$\nu^{8}$$ $$=$$ $$( - 1048344 \beta_{15} + 1048344 \beta_{14} - 419512 \beta_{13} + 1048344 \beta_{12} - 1048344 \beta_{11} - 3355224 \beta_{9} + 2096688 \beta_{8} - 2516200 \beta_{5} + \cdots + 185581065 ) / 9$$ (-1048344*b15 + 1048344*b14 - 419512*b13 + 1048344*b12 - 1048344*b11 - 3355224*b9 + 2096688*b8 - 2516200*b5 + 5451912*b4 + 2096688*b2 - 7086048*b1 + 185581065) / 9 $$\nu^{9}$$ $$=$$ $$( 81977282 \beta_{15} - 81977282 \beta_{14} + 2406879 \beta_{12} - 2406879 \beta_{11} + 75890135 \beta_{5} + 232484163 \beta_{4} + 1117919364 \beta_{2} ) / 30$$ (81977282*b15 - 81977282*b14 + 2406879*b12 - 2406879*b11 + 75890135*b5 + 232484163*b4 + 1117919364*b2) / 30 $$\nu^{10}$$ $$=$$ $$( 196307187 \beta_{15} - 196307187 \beta_{14} - 89861071 \beta_{13} - 196307187 \beta_{12} + 196307187 \beta_{11} - 392614374 \beta_{10} - 662197587 \beta_{9} + \cdots - 392614374 \beta_{2} ) / 18$$ (196307187*b15 - 196307187*b14 - 89861071*b13 - 196307187*b12 + 196307187*b11 - 392614374*b10 - 662197587*b9 + 1351888332*b6 + 482475445*b5 - 1054811961*b4 + 34655015730*b3 - 392614374*b2) / 18 $$\nu^{11}$$ $$=$$ $$( 887812239 \beta_{15} + 887812239 \beta_{14} + 21662418685 \beta_{13} + 22932690162 \beta_{12} + 22932690162 \beta_{11} - 66762880533 \beta_{9} + \cdots + 320332617924 \beta_{7} ) / 90$$ (887812239*b15 + 887812239*b14 + 21662418685*b13 + 22932690162*b12 + 22932690162*b11 - 66762880533*b9 + 320332617924*b7) / 90 $$\nu^{12}$$ $$=$$ $$1029265377 \beta_{15} - 1029265377 \beta_{14} + 495930106 \beta_{13} - 1029265377 \beta_{12} + 1029265377 \beta_{11} + 3546321072 \beta_{9} - 2058530754 \beta_{8} + \cdots - 181544707525$$ 1029265377*b15 - 1029265377*b14 + 495930106*b13 - 1029265377*b12 + 1029265377*b11 + 3546321072*b9 - 2058530754*b8 + 2554460860*b5 - 5604851826*b4 - 2058530754*b2 + 7140519708*b1 - 181544707525 $$\nu^{13}$$ $$=$$ $$( - 2160764841264 \beta_{15} + 2160764841264 \beta_{14} - 91643620533 \beta_{12} + 91643620533 \beta_{11} - 2058195285995 \beta_{5} + \cdots - 30476432312328 \beta_{2} ) / 90$$ (-2160764841264*b15 + 2160764841264*b14 - 91643620533*b12 + 91643620533*b11 - 2058195285995*b5 - 6357873099051*b4 - 30476432312328*b2) / 90 $$\nu^{14}$$ $$=$$ $$( - 1754232431151 \beta_{15} + 1754232431151 \beta_{14} + 862455839813 \beta_{13} + 1754232431151 \beta_{12} - 1754232431151 \beta_{11} + \cdots + 3508464862302 \beta_{2} ) / 18$$ (-1754232431151*b15 + 1754232431151*b14 + 862455839813*b13 + 1754232431151*b12 - 1754232431151*b11 + 3508464862302*b10 + 6095832381741*b9 - 12205631792748*b6 - 4370920702115*b5 + 9604297244043*b4 - 309324383780730*b3 + 3508464862302*b2) / 18 $$\nu^{15}$$ $$=$$ $$( - 2990689930017 \beta_{15} - 2990689930017 \beta_{14} - 65138846432705 \beta_{13} - 68153498840636 \beta_{12} - 68153498840636 \beta_{11} + \cdots - 965018230513872 \beta_{7} ) / 30$$ (-2990689930017*b15 - 2990689930017*b14 - 65138846432705*b13 - 68153498840636*b12 - 68153498840636*b11 + 201397919158149*b9 - 965018230513872*b7) / 30

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −6.88827 − 6.88827i 1.33460 + 1.33460i 4.35647 + 4.35647i 2.96509 + 2.96509i −2.96509 − 2.96509i −4.35647 − 4.35647i −1.33460 − 1.33460i 6.88827 + 6.88827i −6.88827 + 6.88827i 1.33460 − 1.33460i 4.35647 − 4.35647i 2.96509 − 2.96509i −2.96509 + 2.96509i −4.35647 + 4.35647i −1.33460 + 1.33460i 6.88827 − 6.88827i
−6.77944 + 6.77944i 3.70445 + 15.1419i 59.9217i −52.1700 20.0821i −127.768 77.5396i 47.9711 + 47.9711i 189.294 + 189.294i −215.554 + 112.185i 489.829 217.538i
2.2 −5.80199 + 5.80199i −7.07428 13.8908i 35.3261i 42.8643 35.8839i 121.639 + 39.5494i −140.332 140.332i 19.2981 + 19.2981i −142.909 + 196.535i −40.5000 + 456.896i
2.3 −3.17835 + 3.17835i 14.5679 5.54767i 11.7962i 11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 + 83.6967i −139.200 139.200i 181.447 161.636i −211.671 135.397i
2.4 −0.879873 + 0.879873i −15.5882 0.0935382i 30.4516i −51.8842 + 20.8094i 13.7979 13.6333i −11.3356 11.3356i −54.9495 54.9495i 242.983 + 2.91618i 27.3419 63.9612i
2.5 0.879873 0.879873i 0.0935382 + 15.5882i 30.4516i 51.8842 20.8094i 13.7979 + 13.6333i −11.3356 11.3356i 54.9495 + 54.9495i −242.983 + 2.91618i 27.3419 63.9612i
2.6 3.17835 3.17835i 5.54767 14.5679i 11.7962i −11.9990 54.5987i −28.6694 63.9343i 83.6967 + 83.6967i 139.200 + 139.200i −181.447 161.636i −211.671 135.397i
2.7 5.80199 5.80199i 13.8908 + 7.07428i 35.3261i −42.8643 + 35.8839i 121.639 39.5494i −140.332 140.332i −19.2981 19.2981i 142.909 + 196.535i −40.5000 + 456.896i
2.8 6.77944 6.77944i −15.1419 3.70445i 59.9217i 52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 + 47.9711i −189.294 189.294i 215.554 + 112.185i 489.829 217.538i
8.1 −6.77944 6.77944i 3.70445 15.1419i 59.9217i −52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 47.9711i 189.294 189.294i −215.554 112.185i 489.829 + 217.538i
8.2 −5.80199 5.80199i −7.07428 + 13.8908i 35.3261i 42.8643 + 35.8839i 121.639 39.5494i −140.332 + 140.332i 19.2981 19.2981i −142.909 196.535i −40.5000 456.896i
8.3 −3.17835 3.17835i 14.5679 + 5.54767i 11.7962i 11.9990 54.5987i −28.6694 63.9343i 83.6967 83.6967i −139.200 + 139.200i 181.447 + 161.636i −211.671 + 135.397i
8.4 −0.879873 0.879873i −15.5882 + 0.0935382i 30.4516i −51.8842 20.8094i 13.7979 + 13.6333i −11.3356 + 11.3356i −54.9495 + 54.9495i 242.983 2.91618i 27.3419 + 63.9612i
8.5 0.879873 + 0.879873i 0.0935382 15.5882i 30.4516i 51.8842 + 20.8094i 13.7979 13.6333i −11.3356 + 11.3356i 54.9495 54.9495i −242.983 2.91618i 27.3419 + 63.9612i
8.6 3.17835 + 3.17835i 5.54767 + 14.5679i 11.7962i −11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 83.6967i 139.200 139.200i −181.447 + 161.636i −211.671 + 135.397i
8.7 5.80199 + 5.80199i 13.8908 7.07428i 35.3261i −42.8643 35.8839i 121.639 + 39.5494i −140.332 + 140.332i −19.2981 + 19.2981i 142.909 196.535i −40.5000 456.896i
8.8 6.77944 + 6.77944i −15.1419 + 3.70445i 59.9217i 52.1700 20.0821i −127.768 77.5396i 47.9711 47.9711i −189.294 + 189.294i 215.554 112.185i 489.829 + 217.538i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.6.e.a 16
3.b odd 2 1 inner 15.6.e.a 16
5.b even 2 1 75.6.e.e 16
5.c odd 4 1 inner 15.6.e.a 16
5.c odd 4 1 75.6.e.e 16
15.d odd 2 1 75.6.e.e 16
15.e even 4 1 inner 15.6.e.a 16
15.e even 4 1 75.6.e.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.e.a 16 1.a even 1 1 trivial
15.6.e.a 16 3.b odd 2 1 inner
15.6.e.a 16 5.c odd 4 1 inner
15.6.e.a 16 15.e even 4 1 inner
75.6.e.e 16 5.b even 2 1
75.6.e.e 16 5.c odd 4 1
75.6.e.e 16 15.d odd 2 1
75.6.e.e 16 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 13393 T^{12} + \cdots + 37480960000$$
$3$ $$T^{16} + 4320 T^{13} + \cdots + 12\!\cdots\!01$$
$5$ $$T^{16} - 4580 T^{14} + \cdots + 90\!\cdots\!25$$
$7$ $$(T^{8} + 40 T^{7} + \cdots + 652674756250000)^{2}$$
$11$ $$(T^{8} + 532700 T^{6} + \cdots + 61\!\cdots\!00)^{2}$$
$13$ $$(T^{8} + 1060 T^{7} + \cdots + 51\!\cdots\!00)^{2}$$
$17$ $$T^{16} + 9871879225768 T^{12} + \cdots + 26\!\cdots\!00$$
$19$ $$(T^{8} + 9996984 T^{6} + \cdots + 11\!\cdots\!56)^{2}$$
$23$ $$T^{16} + 92370076420168 T^{12} + \cdots + 13\!\cdots\!00$$
$29$ $$(T^{8} - 69254700 T^{6} + \cdots + 14\!\cdots\!00)^{2}$$
$31$ $$(T^{4} + 5392 T^{3} + \cdots + 12072937186816)^{4}$$
$37$ $$(T^{8} - 27860 T^{7} + \cdots + 16\!\cdots\!00)^{2}$$
$41$ $$(T^{8} + 257067200 T^{6} + \cdots + 46\!\cdots\!00)^{2}$$
$43$ $$(T^{8} + 11680 T^{7} + \cdots + 32\!\cdots\!00)^{2}$$
$47$ $$T^{16} + \cdots + 66\!\cdots\!00$$
$53$ $$T^{16} + \cdots + 35\!\cdots\!00$$
$59$ $$(T^{8} - 525588900 T^{6} + \cdots + 23\!\cdots\!00)^{2}$$
$61$ $$(T^{4} + 43792 T^{3} + \cdots - 18\!\cdots\!84)^{4}$$
$67$ $$(T^{8} - 70160 T^{7} + \cdots + 26\!\cdots\!00)^{2}$$
$71$ $$(T^{8} + 8891090600 T^{6} + \cdots + 10\!\cdots\!00)^{2}$$
$73$ $$(T^{8} + 120160 T^{7} + \cdots + 93\!\cdots\!00)^{2}$$
$79$ $$(T^{8} + 6750333984 T^{6} + \cdots + 33\!\cdots\!56)^{2}$$
$83$ $$T^{16} + \cdots + 79\!\cdots\!00$$
$89$ $$(T^{8} - 17794791000 T^{6} + \cdots + 48\!\cdots\!00)^{2}$$
$97$ $$(T^{8} - 220400 T^{7} + \cdots + 16\!\cdots\!00)^{2}$$