# Properties

 Label 37.8.b.a Level $37$ Weight $8$ Character orbit 37.b Analytic conductor $11.558$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 37.b (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: $$11.5582459429$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96$$ x^20 + 1702*x^18 + 1194509*x^16 + 450999516*x^14 + 100204783492*x^12 + 13461378480848*x^10 + 1081011973644416*x^8 + 49304995250225664*x^6 + 1131877572418003968*x^4 + 9402469145336696832*x^2 + 130757963535876096 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{19}\cdot 3^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} + 4) q^{3} + (\beta_{2} - 42) q^{4} + \beta_{12} q^{5} + ( - \beta_{11} + 12 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{4} - 87) q^{7} + (\beta_{3} - 42 \beta_1) q^{8} + ( - \beta_{8} + \beta_{6} - 2 \beta_{4} + 2 \beta_{2} + 618) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b4 + 4) * q^3 + (b2 - 42) * q^4 + b12 * q^5 + (-b11 + 12*b1) * q^6 + (b8 + 2*b4 - 87) * q^7 + (b3 - 42*b1) * q^8 + (-b8 + b6 - 2*b4 + 2*b2 + 618) * q^9 $$q + \beta_1 q^{2} + (\beta_{4} + 4) q^{3} + (\beta_{2} - 42) q^{4} + \beta_{12} q^{5} + ( - \beta_{11} + 12 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{4} - 87) q^{7} + (\beta_{3} - 42 \beta_1) q^{8} + ( - \beta_{8} + \beta_{6} - 2 \beta_{4} + 2 \beta_{2} + 618) q^{9} + (\beta_{8} - \beta_{7} + 8 \beta_{4} - 43) q^{10} + (\beta_{10} + 6 \beta_{4} + 2 \beta_{2} + 176) q^{11} + (\beta_{8} - \beta_{5} - 17 \beta_{4} + 11 \beta_{2} - 1518) q^{12} + (\beta_{17} - 2 \beta_{12} - \beta_{11} - 14 \beta_1) q^{13} + (\beta_{15} - 13 \beta_{12} - \beta_{11} + \beta_{3} - 50 \beta_1) q^{14} + (\beta_{17} - \beta_{16} - 4 \beta_{12} - \beta_{11} - 2 \beta_{3} - 126 \beta_1) q^{15} + ( - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6 \beta_{4} + \cdots + 1792) q^{16}+ \cdots + (28 \beta_{14} + 219 \beta_{10} - 312 \beta_{9} - 6874 \beta_{8} + \cdots + 2156834) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b4 + 4) * q^3 + (b2 - 42) * q^4 + b12 * q^5 + (-b11 + 12*b1) * q^6 + (b8 + 2*b4 - 87) * q^7 + (b3 - 42*b1) * q^8 + (-b8 + b6 - 2*b4 + 2*b2 + 618) * q^9 + (b8 - b7 + 8*b4 - 43) * q^10 + (b10 + 6*b4 + 2*b2 + 176) * q^11 + (b8 - b5 - 17*b4 + 11*b2 - 1518) * q^12 + (b17 - 2*b12 - b11 - 14*b1) * q^13 + (b15 - 13*b12 - b11 + b3 - 50*b1) * q^14 + (b17 - b16 - 4*b12 - b11 - 2*b3 - 126*b1) * q^15 + (-b10 + b9 - 2*b8 + b7 - b6 + b5 - 6*b4 - 64*b2 + 1792) * q^16 + (b19 + 3*b12 - b11 - 3*b3 + 280*b1) * q^17 + (-b15 - b13 + 7*b12 - b11 + 3*b3 + 322*b1) * q^18 + (b18 - b16 + b15 - 30*b12 + 2*b11 - 301*b1) * q^19 + (b18 - 2*b17 - b13 - 16*b12 + 3*b11 + 6*b3 + 119*b1) * q^20 + (-b14 + 2*b8 - 2*b7 + 3*b6 - 282*b4 + 17*b2 + 5649) * q^21 + (-2*b19 + b18 - 2*b13 - 32*b12 + 13*b3 - 56*b1) * q^22 + (b19 - 2*b18 + b17 - b13 + 23*b12 - 13*b11 - b3 + 848*b1) * q^23 + (-4*b19 + 3*b18 - 4*b16 - b15 - 2*b13 - 67*b12 + 7*b11 + 19*b3 - 1564*b1) * q^24 + (b14 - b10 + 2*b9 - 7*b8 - 2*b6 - 2*b5 + 227*b4 - 101*b2 - 5403) * q^25 + (-b14 - 6*b10 + b9 + 15*b8 - 6*b7 - 3*b6 + b5 - 248*b4 - 57*b2 + 2430) * q^26 + (b14 + 6*b10 + 11*b8 + 2*b7 - 4*b6 - 6*b5 + 215*b4 + 65*b2 - 15168) * q^27 + (b14 - 8*b10 + 3*b9 - 16*b8 + 7*b7 + 3*b6 - b5 + 114*b4 - 144*b2 - 2081) * q^28 + (5*b19 + b18 + b17 + 6*b16 + 7*b15 + 5*b13 - 64*b12 - 15*b11 - 3*b3 - 105*b1) * q^29 + (-2*b14 + 9*b10 - 3*b9 + 10*b8 - 16*b7 + 7*b6 + 8*b5 - 237*b4 + 128*b2 + 21493) * q^30 + (b19 - 7*b18 + 12*b17 + 2*b16 + 7*b15 + 4*b13 + 49*b12 - 7*b11 + b3 - 1113*b1) * q^31 + (10*b19 - 5*b18 - 2*b17 - 8*b15 + 2*b13 + 250*b12 - 41*b11 - 69*b3 + 4536*b1) * q^32 + (2*b14 + 24*b10 - 6*b9 - 2*b8 - 14*b7 + 23*b6 + 8*b5 + 1015*b4 + 260*b2 + 15436) * q^33 + (26*b10 - 8*b9 + 40*b8 + 17*b7 - 16*b6 - 23*b5 - 171*b4 + 656*b2 - 47895) * q^34 + (-4*b19 + 3*b18 + 12*b17 - 15*b16 - b15 + 9*b13 + 8*b12 + 17*b11 - 36*b3 - 6363*b1) * q^35 + (-b14 - 24*b10 - 3*b9 - 17*b8 + 25*b7 + 41*b6 + 2*b5 - 501*b4 + 125*b2 + 24195) * q^36 + (-b19 - 5*b18 - 13*b17 + 10*b16 - 7*b15 - b14 + 8*b13 + 217*b12 + 62*b11 + 8*b10 + 6*b9 + 32*b8 - 16*b7 + 18*b6 - 8*b5 - 241*b4 - 49*b3 - 93*b2 + 521*b1 + 5398) * q^37 + (b14 - b10 + 2*b9 - 162*b8 + 39*b7 + 24*b6 + 23*b5 + 197*b4 - 233*b2 + 52438) * q^38 + (-11*b19 - 3*b18 + 12*b17 + 19*b16 - 27*b15 - 5*b13 + 440*b12 - 60*b11 + 41*b3 + 3497*b1) * q^39 + (3*b14 - 29*b10 + 2*b9 + 51*b8 - 42*b7 - 68*b6 + 17*b5 + 1463*b4 - 627*b2 - 24773) * q^40 + (2*b14 - 35*b10 - 14*b9 + 189*b8 - 30*b7 - 27*b6 - 14*b5 + 649*b4 + 722*b2 - 78649) * q^41 + (-2*b19 + 9*b18 - 64*b17 + 20*b16 + 20*b15 - 3*b13 - 242*b12 + 198*b11 + 49*b3 + 1407*b1) * q^42 + (-16*b19 - 6*b18 - 26*b17 + 5*b16 + 28*b15 + 5*b13 - 411*b12 - 53*b11 + 56*b3 - 4372*b1) * q^43 + (b14 + 6*b10 + 17*b9 - 153*b8 + 69*b7 - 123*b6 + 62*b5 + 539*b4 - 1482*b2 + 34011) * q^44 + (4*b19 - 3*b18 - 63*b17 + 16*b16 - 49*b15 - 9*b13 + 696*b12 + 84*b11 + 128*b3 + 3759*b1) * q^45 + (-3*b14 + 4*b10 - 13*b9 + 49*b8 - 39*b7 - 125*b6 - 58*b5 - 1913*b4 + 901*b2 - 145283) * q^46 + (b14 - 27*b10 - 16*b9 - 219*b8 + 114*b7 + 42*b6 - 58*b5 - 1504*b4 + 1587*b2 - 75522) * q^47 + (-2*b14 - 129*b10 + 27*b9 - 23*b8 + 83*b7 - 39*b6 + 48*b5 - 1585*b4 - 2630*b2 + 75434) * q^48 + (-4*b14 - 3*b10 - 28*b9 - 284*b8 - 116*b7 + 163*b6 + 42*b5 - 322*b4 + 1292*b2 + 163645) * q^49 + (4*b19 - 2*b18 + 52*b17 - 36*b16 - 49*b15 - 10*b13 - 35*b12 + 270*b11 - 335*b3 + 9121*b1) * q^50 + (-6*b19 + 12*b18 - 52*b17 - 17*b16 + 6*b15 + 6*b13 + 157*b12 - 278*b11 - 298*b3 + 8682*b1) * q^51 + (18*b19 + 4*b18 + 52*b17 + 20*b16 + 22*b15 + 9*b13 - 908*b12 + 34*b11 - 183*b3 + 6951*b1) * q^52 + (-2*b14 + 159*b10 + 8*b9 + 288*b8 + 156*b7 - 165*b6 + 134*b5 + 2528*b4 - 2646*b2 + 149724) * q^53 + (-34*b19 + 15*b18 + 64*b17 - 44*b16 - 19*b15 - 20*b13 - 437*b12 + 622*b11 + 152*b3 - 22180*b1) * q^54 + (21*b19 + b18 + 65*b17 + 11*b16 - 41*b15 + 21*b13 + 425*b12 + 447*b11 + 99*b3 - 13295*b1) * q^55 + (26*b19 - 19*b18 + 62*b17 - 36*b16 + 70*b15 + 4*b13 - 416*b12 + 93*b11 - 474*b3 + 10136*b1) * q^56 + (29*b19 - 9*b18 - 128*b17 + 40*b16 + 21*b15 - b13 + 1320*b12 + 508*b11 + 365*b3 - 6377*b1) * q^57 + (13*b14 + 118*b10 + 21*b9 - 633*b8 + 95*b7 + 297*b6 - 154*b5 - 2251*b4 + 77*b2 + 19977) * q^58 + (4*b19 - 11*b18 + 76*b17 + 54*b16 + 55*b15 + 2*b13 + 35*b12 - 470*b11 - 456*b3 + 35361*b1) * q^59 + (-2*b19 + 15*b18 - 12*b17 - 44*b16 + 77*b15 - 9*b13 - 2511*b12 - 1075*b11 + 365*b3 - 11259*b1) * q^60 + (17*b19 - 24*b18 + 38*b17 - 34*b16 - 20*b15 + 20*b13 - 170*b12 - 807*b11 + 461*b3 - 21504*b1) * q^61 + (-10*b14 + 36*b10 + 28*b9 - 585*b8 - 420*b7 + 192*b6 - 113*b5 - 1013*b4 - 1746*b2 + 185964) * q^62 + (9*b14 - 309*b10 + 24*b9 + 1618*b8 - 390*b7 - 490*b6 - 42*b5 + 7852*b4 - 257*b2 - 598717) * q^63 + (-11*b14 + 330*b10 - 11*b9 + 1488*b8 - 63*b7 - 175*b6 - 231*b5 - 5754*b4 + 5381*b2 - 554735) * q^64 + (-12*b14 - 281*b10 + 18*b9 - 636*b8 + 150*b7 + 630*b6 - 42*b5 - 11613*b4 + 3314*b2 + 211844) * q^65 + (-36*b19 + 116*b17 + 16*b16 + 14*b15 - 49*b13 - 2416*b12 - 2031*b11 + 1190*b3 - 8921*b1) * q^66 + (26*b14 + 267*b10 + 8*b9 - 125*b8 - 388*b7 - 2*b6 + 192*b5 + 15914*b4 - 372*b2 + 179709) * q^67 + (-48*b19 + 78*b18 + 66*b17 - 60*b16 + 83*b15 - 33*b13 + 105*b12 + 2104*b11 + 1490*b3 - 99865*b1) * q^68 + (-85*b19 + 69*b18 - 64*b17 - 66*b16 - 69*b15 - 37*b13 + 1709*b12 - 2034*b11 - 501*b3 + 41473*b1) * q^69 + (-25*b14 + 305*b10 + 6*b9 + 370*b8 - 495*b7 + 948*b6 + 265*b5 + 2367*b4 - 1719*b2 + 1080118) * q^70 + (-b14 + 273*b10 + 16*b9 + 777*b8 + 654*b7 - 810*b6 - 254*b5 + 760*b4 + 345*b2 - 762648) * q^71 + (42*b19 - 48*b18 + 2*b17 + 40*b16 - 69*b15 - 78*b13 + 5083*b12 - 712*b11 + 503*b3 + 43668*b1) * q^72 + (15*b14 - 419*b10 + 8*b9 + 223*b8 + 670*b7 + 1057*b6 + 2*b5 + 3700*b4 - 2921*b2 + 553826) * q^73 + (-26*b19 + 55*b18 - 116*b17 - 36*b16 - 34*b15 + 11*b14 - 88*b13 - 3238*b12 + 1797*b11 + 282*b10 - 29*b9 + 1128*b8 - 342*b7 + 579*b6 - 134*b5 + 11383*b4 - 423*b3 + 7683*b2 + 16617*b1 - 99042) * q^74 + (-14*b14 + 33*b10 - 3873*b8 + 596*b7 + 602*b6 + 176*b5 - 8311*b4 + 5032*b2 + 743633) * q^75 + (104*b19 + 12*b18 + 130*b17 - 64*b16 + 13*b15 + 53*b13 + 4443*b12 - 2966*b11 - 940*b3 + 40229*b1) * q^76 + (-38*b14 - 235*b10 - 28*b9 + 1080*b8 - 744*b7 - 1065*b6 + 98*b5 + 1838*b4 - 7694*b2 - 121886) * q^77 + (-23*b14 - 543*b10 + 66*b9 + 3797*b8 - 337*b7 - 564*b6 + 26*b5 - 9240*b4 - 3657*b2 - 610096) * q^78 + (23*b19 - 110*b18 + 77*b17 - 65*b16 - 34*b15 + 41*b13 + 3082*b12 + 3737*b11 - 1279*b3 + 2438*b1) * q^79 + (140*b19 + 69*b18 - 168*b17 - 35*b15 - 24*b13 - 6149*b12 - 2163*b11 - 426*b3 + 88636*b1) * q^80 + (-10*b14 + 303*b10 - 78*b9 - 1710*b8 + 442*b7 - 1306*b6 + 102*b5 - 8949*b4 + 11388*b2 - 886847) * q^81 + (-38*b19 + 23*b18 + 116*b17 - 40*b16 + 217*b15 + 91*b13 - 6003*b12 + 318*b11 + 2176*b3 - 159631*b1) * q^82 + (39*b14 - 163*b10 + 72*b9 - 2253*b8 - 90*b7 - 1668*b6 - 18*b5 - 10434*b4 - 12995*b2 - 647210) * q^83 + (-15*b14 - 258*b10 + 81*b9 - 3826*b8 + 861*b7 + 353*b6 + 71*b5 - 1856*b4 + 738*b2 + 496519) * q^84 + (25*b14 - 51*b10 - 188*b9 + 1854*b8 + 246*b7 - 154*b6 - 102*b5 - 2880*b4 - 4613*b2 - 135977) * q^85 + (53*b14 - 347*b10 + 184*b9 - 4908*b8 - 105*b7 + 726*b6 + 109*b5 - 4859*b4 - 9301*b2 + 764276) * q^86 + (3*b19 + 18*b18 - 248*b17 + 200*b16 + 266*b15 + 110*b13 - 13260*b12 + 6145*b11 - 2105*b3 + 40408*b1) * q^87 + (50*b19 - 128*b18 + 130*b17 + 160*b16 - 133*b15 - 22*b13 + 10779*b12 - 6792*b11 - 2746*b3 + 216358*b1) * q^88 + (-171*b19 - 39*b18 - 120*b17 - 36*b16 - 197*b15 - 111*b13 - 9278*b12 + 2042*b11 - 1147*b3 + 14969*b1) * q^89 + (27*b14 + 246*b10 - 63*b9 + 5740*b8 + 651*b7 - 1095*b6 - 133*b5 + 16130*b4 - 15289*b2 - 661691) * q^90 + (226*b19 - 156*b18 - 142*b17 + 319*b16 + 74*b15 + 202*b13 + 3411*b12 + 8396*b11 - 686*b3 - 78206*b1) * q^91 + (-170*b19 - 18*b18 - 78*b17 - 120*b16 + 71*b15 - 108*b13 - 5133*b12 + 6330*b11 + 2637*b3 - 166590*b1) * q^92 + (-214*b19 + 117*b18 - 255*b17 + 200*b16 - 289*b15 - 23*b13 + 6374*b12 + 4516*b11 - 1874*b3 + 623*b1) * q^93 + (-240*b19 + 26*b18 + 352*b17 - 188*b16 - 97*b15 + 72*b13 + 15181*b12 + 5409*b11 + 2673*b3 - 307114*b1) * q^94 + (-68*b14 + 68*b10 - 184*b9 - 3978*b8 - 336*b7 + 1512*b6 + 296*b5 - 56014*b4 - 11900*b2 + 2440402) * q^95 + (42*b19 + 42*b18 - 62*b17 - 388*b16 - 175*b15 + 116*b13 + 9705*b12 - 2562*b11 - 5303*b3 + 198622*b1) * q^96 + (-46*b19 - 17*b18 - 120*b17 + 220*b16 - 259*b15 + 155*b13 - 9133*b12 - 9749*b11 + 1190*b3 + 13839*b1) * q^97 + (54*b19 + 15*b18 - 360*b17 + 360*b16 + 16*b15 - 69*b13 - 9890*b12 - 6868*b11 + 4611*b3 + 482*b1) * q^98 + (28*b14 + 219*b10 - 312*b9 - 6874*b8 - 496*b7 + 1080*b6 - 112*b5 + 45681*b4 - 12702*b2 + 2156834) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10})$$ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 $$20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100})$$ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 170$$ v^2 + 170 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 298\nu$$ v^3 + 298*v $$\beta_{4}$$ $$=$$ $$( - 24\!\cdots\!11 \nu^{18} + \cdots - 21\!\cdots\!80 ) / 18\!\cdots\!00$$ (-2464505834717504057037311*v^18 - 4062230331226013071114142142*v^16 - 2730941476640764596857601696187*v^14 - 972523355647324591990280529402392*v^12 - 199177511378201200972500208639297308*v^10 - 23775505782848904412782101038905822784*v^8 - 1589913372117868473220608533833621991040*v^6 - 52845024881973318142046045753052965016576*v^4 - 623414686985079517664020650512923038010368*v^2 - 211327012969517734318833939725568488570880) / 18367377317718507669652798578937823232000 $$\beta_{5}$$ $$=$$ $$( - 72\!\cdots\!59 \nu^{18} + \cdots + 28\!\cdots\!76 ) / 18\!\cdots\!00$$ (-72437390228236114481401559*v^18 - 112799828149121085988898999406*v^16 - 69752792845248289760031886510579*v^14 - 21835209857675357350108039622694104*v^12 - 3605155479800561631845347684396477308*v^10 - 282472056520527883521692248269326017600*v^8 - 4636145134099919875798984157928020959872*v^6 + 533930292250735347926698609694380110253056*v^4 + 18961059121622146897710082736570384660812800*v^2 + 28822489008580498058808841307397225184690176) / 18367377317718507669652798578937823232000 $$\beta_{6}$$ $$=$$ $$( - 10\!\cdots\!63 \nu^{18} + \cdots - 55\!\cdots\!48 ) / 18\!\cdots\!00$$ (-101067482367253264859276263*v^18 - 162305150863517882052894281742*v^16 - 105337021018987775713575067257923*v^14 - 35768085468339075101625706649539288*v^12 - 6872514902533616673995941872889047996*v^10 - 754953465509430144637985230470885469760*v^8 - 45806240196839285119260586577289831441024*v^6 - 1419291047654833611841014475136023845884928*v^4 - 19150657176574798473130880036550685696650240*v^2 - 55410934829190140382336869068041055632949248) / 18367377317718507669652798578937823232000 $$\beta_{7}$$ $$=$$ $$( 16\!\cdots\!05 \nu^{18} + \cdots - 31\!\cdots\!32 ) / 15\!\cdots\!00$$ (16619442297430347275982005*v^18 + 27283761149645033201740833066*v^16 + 18228281423432850953630707446217*v^14 + 6425249942322883123269485587423112*v^12 + 1292573216635505159764076557508921652*v^10 + 149248035661097057248573721848857581248*v^8 + 9356500178642552369820969763332920840064*v^6 + 273082982693398246556987855258438923484160*v^4 + 2418260399456169565892039579103286956082176*v^2 - 31600021201546169727353558919804946808832) / 1530614776476542305804399881578151936000 $$\beta_{8}$$ $$=$$ $$( - 14\!\cdots\!37 \nu^{18} + \cdots + 72\!\cdots\!16 ) / 91\!\cdots\!00$$ (-148716068399547684394225237*v^18 - 244445151347543262245234990442*v^16 - 163385861984356710921400503459305*v^14 - 57536667111050196571571778652234120*v^12 - 11542877301549046622542392341582745012*v^10 - 1327836725352669773138491119799966753472*v^8 - 83253080249132096883033614276778078463872*v^6 - 2481247060605210142141694366012318266555392*v^4 - 24300764537987168728439389547680442091193344*v^2 + 720553131452666542971219879063327302025216) / 9183688658859253834826399289468911616000 $$\beta_{9}$$ $$=$$ $$( - 11\!\cdots\!43 \nu^{18} + \cdots + 15\!\cdots\!04 ) / 61\!\cdots\!00$$ (-114067224462257589138622843*v^18 - 185581231536150839772344817078*v^16 - 122612795276310283962504197689895*v^14 - 42663850445321252821035703207982200*v^12 - 8471274478274206411133784188955697548*v^10 - 970045393477506642686243143876443668288*v^8 - 61352641342763749708877748348185159526528*v^6 - 1885162297743743462970134876919354197173248*v^4 - 16975494165485137648656619900241282500033536*v^2 + 159697596327172862838930910118954532532322304) / 6122459105906169223217599526312607744000 $$\beta_{10}$$ $$=$$ $$( 16\!\cdots\!83 \nu^{18} + \cdots + 31\!\cdots\!56 ) / 61\!\cdots\!00$$ (165171011642634770730971683*v^18 + 274106916426404600858491880838*v^16 + 185471438741091425592115567015535*v^14 + 66342044053553096868624394079108920*v^12 + 13576996287333963807307576976633100588*v^10 + 1602440530865773179690368432730607124288*v^8 + 103980658135798114308712700948305142594688*v^6 + 3266140417796693166430187885720753826505728*v^4 + 36306439943578942599770935450690609371749376*v^2 + 31159642359983761931280745581553362470240256) / 6122459105906169223217599526312607744000 $$\beta_{11}$$ $$=$$ $$( 24\!\cdots\!11 \nu^{19} + \cdots + 35\!\cdots\!80 \nu ) / 18\!\cdots\!00$$ (2464505834717504057037311*v^19 + 4062230331226013071114142142*v^17 + 2730941476640764596857601696187*v^15 + 972523355647324591990280529402392*v^13 + 199177511378201200972500208639297308*v^11 + 23775505782848904412782101038905822784*v^9 + 1589913372117868473220608533833621991040*v^7 + 52845024881973318142046045753052965016576*v^5 + 623414686985079517664020650512923038010368*v^3 + 358266031511265795676056328357071074426880*v) / 18367377317718507669652798578937823232000 $$\beta_{12}$$ $$=$$ $$( 14\!\cdots\!31 \nu^{19} + \cdots - 10\!\cdots\!64 \nu ) / 54\!\cdots\!00$$ (149205623889522174853968331*v^19 + 238680118902384976575227678934*v^17 + 153140903669826030528221896145591*v^15 + 50523064565145096384152947691643256*v^13 + 9041291190994924209985118746589434572*v^11 + 820674021743754356805991733214979476800*v^9 + 24248272179242725519693513439418885314688*v^7 - 1258964413779277508063336113195278414019584*v^5 - 87135115079495926056346188459433204109460480*v^3 - 1038623212087779826978324643826625958792331264*v) / 542858040723680337791960491333051219968000 $$\beta_{13}$$ $$=$$ $$( 72\!\cdots\!43 \nu^{19} + \cdots + 30\!\cdots\!36 \nu ) / 20\!\cdots\!00$$ (729822063149394118837889443*v^19 + 1171805715531447039048952527558*v^17 + 762382413637802330239947558182255*v^15 + 261195117618999253722889564657776760*v^13 + 51412366821125545376098083953189630508*v^11 + 5993860648952919419107518331622864395328*v^9 + 417884136696393193162770940159624519045248*v^7 + 17391747657627371705565062673254299998901248*v^5 + 394895243938276680982437690387624803456572416*v^3 + 3063816581328457356584465379762302296771854336*v) / 203571765271380126671985184249894207488000 $$\beta_{14}$$ $$=$$ $$( - 43\!\cdots\!81 \nu^{18} + \cdots + 17\!\cdots\!36 ) / 36\!\cdots\!00$$ (-4350166997645308103506370681*v^18 - 7141291471039086937201056696498*v^16 - 4765331788029016020751164009839005*v^14 - 1674560577794386654585190655386659880*v^12 - 335026570812640896424810073773872726468*v^10 - 38397705895864666193484618182376914983360*v^8 - 2393675572935401817000950052688222395878784*v^6 - 70501807823924255345181808791712819805039616*v^4 - 668945029458379111919865599051965164524835840*v^2 + 1725306828150691736179408499614108811329536) / 3673475463543701533930559715787564646400 $$\beta_{15}$$ $$=$$ $$( - 62\!\cdots\!83 \nu^{19} + \cdots - 12\!\cdots\!56 \nu ) / 48\!\cdots\!00$$ (-62315448945520129718985456083*v^19 - 103199799873420092732079738289638*v^17 - 69730223279094568021147228042501855*v^15 - 24956999561558916640599705983887327160*v^13 - 5135960873104288190082978887869109585388*v^11 - 616714561881838868137176281823778662922048*v^9 - 41876507804550229669846423592833671610578048*v^7 - 1481379049272752166688575976132712980942169088*v^5 - 23293529227614741386861016426578115062412205056*v^3 - 122789425747637104107053843906013718012351021056*v) / 4885722366513123040127644421997460979712000 $$\beta_{16}$$ $$=$$ $$( 92\!\cdots\!41 \nu^{19} + \cdots - 43\!\cdots\!64 \nu ) / 48\!\cdots\!00$$ (92697198970605567323807656241*v^19 + 150464369228458455738870660527394*v^17 + 98682168353084453998552138092900661*v^15 + 33724169481757103374160919133967037416*v^13 + 6428309090677878028492878078689385111972*v^11 + 670202260930524373751896232362062674960320*v^9 + 33162699332908467965920029924017091569578368*v^7 + 313775659666187653224044121183354600031620096*v^5 - 24307664953523674676975011262836680502568094720*v^3 - 437293503148256512793832612440447330452298072064*v) / 4885722366513123040127644421997460979712000 $$\beta_{17}$$ $$=$$ $$( - 93\!\cdots\!19 \nu^{19} + \cdots - 40\!\cdots\!12 \nu ) / 48\!\cdots\!00$$ (-93494584868252911760372481719*v^19 - 155815042287133465143144571744302*v^17 - 106256640275374041915989412954038611*v^15 - 38560071452454961890404077058819736536*v^13 - 8107525082435521443330911289283372820220*v^11 - 1007831820003429717817375217706094073846848*v^9 - 72557751916385203650968031617891967282877056*v^7 - 2849677431711812800735695257106267735991809024*v^5 - 54653896471703559060129207815783472885427055616*v^3 - 405774174175514817465375356122020925659481178112*v) / 4885722366513123040127644421997460979712000 $$\beta_{18}$$ $$=$$ $$( - 28\!\cdots\!65 \nu^{19} + \cdots - 52\!\cdots\!32 \nu ) / 12\!\cdots\!00$$ (-28870075983610019882861554465*v^19 - 50799496339937136862030236224514*v^17 - 37210544776818150001556033919683813*v^15 - 14834519115805829009828132356139423848*v^13 - 3528284451962962502174943843969842371428*v^11 - 515148362472635435878632433180400037482432*v^9 - 45617299399335146062848171526066967437719936*v^7 - 2314974914925616314625206494833109108446694400*v^5 - 58814483967376471592621851792871466936295732224*v^3 - 523462747630686718483457689547495033161979461632*v) / 1221430591628280760031911105499365244928000 $$\beta_{19}$$ $$=$$ $$( - 84\!\cdots\!35 \nu^{19} + \cdots - 51\!\cdots\!64 \nu ) / 25\!\cdots\!00$$ (-8456739437484389320049780135*v^19 - 14222066674967869294729331257998*v^17 - 9820756335196560683573500387702531*v^15 - 3626707594284718663920869900355972056*v^13 - 781615237036981798965987728921450649276*v^11 - 100669860939577454640577105470166908480064*v^9 - 7630268473601477358104475453663987598119552*v^7 - 322477926047112602871112963969255043996881920*v^5 - 6764258440109592244326335249383305383336537088*v^3 - 51320688494251936745231981397930113856559448064*v) / 257143282448059107375139180105129525248000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 170$$ b2 - 170 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 298\beta_1$$ b3 - 298*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6\beta_{4} - 448\beta_{2} + 50688$$ -b10 + b9 - 2*b8 + b7 - b6 + b5 - 6*b4 - 448*b2 + 50688 $$\nu^{5}$$ $$=$$ $$10 \beta_{19} - 5 \beta_{18} - 2 \beta_{17} - 8 \beta_{15} + 2 \beta_{13} + 250 \beta_{12} - 41 \beta_{11} - 581 \beta_{3} + 107960 \beta_1$$ 10*b19 - 5*b18 - 2*b17 - 8*b15 + 2*b13 + 250*b12 - 41*b11 - 581*b3 + 107960*b1 $$\nu^{6}$$ $$=$$ $$- 11 \beta_{14} + 970 \beta_{10} - 651 \beta_{9} + 2768 \beta_{8} - 703 \beta_{7} + 465 \beta_{6} - 871 \beta_{5} - 1914 \beta_{4} + 193797 \beta_{2} - 18380527$$ -11*b14 + 970*b10 - 651*b9 + 2768*b8 - 703*b7 + 465*b6 - 871*b5 - 1914*b4 + 193797*b2 - 18380527 $$\nu^{7}$$ $$=$$ $$- 8050 \beta_{19} + 4363 \beta_{18} + 538 \beta_{17} - 660 \beta_{16} + 6402 \beta_{15} - 2224 \beta_{13} - 208888 \beta_{12} + 61423 \beta_{11} + 286937 \beta_{3} - 43171486 \beta_1$$ -8050*b19 + 4363*b18 + 538*b17 - 660*b16 + 6402*b15 - 2224*b13 - 208888*b12 + 61423*b11 + 286937*b3 - 43171486*b1 $$\nu^{8}$$ $$=$$ $$9567 \beta_{14} - 604362 \beta_{10} + 339039 \beta_{9} - 1930120 \beta_{8} + 417323 \beta_{7} - 230277 \beta_{6} + 556419 \beta_{5} + 6774826 \beta_{4} - 85047949 \beta_{2} + \cdots + 7356193323$$ 9567*b14 - 604362*b10 + 339039*b9 - 1930120*b8 + 417323*b7 - 230277*b6 + 556419*b5 + 6774826*b4 - 85047949*b2 + 7356193323 $$\nu^{9}$$ $$=$$ $$4809690 \beta_{19} - 2757911 \beta_{18} + 52510 \beta_{17} + 678180 \beta_{16} - 3642650 \beta_{15} + 1593872 \beta_{13} + 130255496 \beta_{12} - 51030763 \beta_{11} + \cdots + 18274909902 \beta_1$$ 4809690*b19 - 2757911*b18 + 52510*b17 + 678180*b16 - 3642650*b15 + 1593872*b13 + 130255496*b12 - 51030763*b11 - 136140225*b3 + 18274909902*b1 $$\nu^{10}$$ $$=$$ $$- 5774891 \beta_{14} + 327064038 \beta_{10} - 165205439 \beta_{9} + 1091155264 \beta_{8} - 230683451 \beta_{7} + 119017373 \beta_{6} - 315170019 \beta_{5} + \cdots - 3116047703047$$ -5774891*b14 + 327064038*b10 - 165205439*b9 + 1091155264*b8 - 230683451*b7 + 119017373*b6 - 315170019*b5 - 6306548266*b4 + 37965245697*b2 - 3116047703047 $$\nu^{11}$$ $$=$$ $$- 2587179690 \beta_{19} + 1543681783 \beta_{18} - 147038606 \beta_{17} - 484360500 \beta_{16} + 1832478546 \beta_{15} - 961797400 \beta_{13} + \cdots - 8013645796550 \beta_1$$ -2587179690*b19 + 1543681783*b18 - 147038606*b17 - 484360500*b16 + 1832478546*b15 - 961797400*b13 - 72399497408*b12 + 33518622891*b11 + 63844372521*b3 - 8013645796550*b1 $$\nu^{12}$$ $$=$$ $$3038838435 \beta_{14} - 166734036758 \beta_{10} + 78569642423 \beta_{9} - 564579581136 \beta_{8} + 121539360243 \beta_{7} - 61461655013 \beta_{6} + \cdots + 13\!\cdots\!47$$ 3038838435*b14 - 166734036758*b10 + 78569642423*b9 - 564579581136*b8 + 121539360243*b7 - 61461655013*b6 + 167858164043*b5 + 4306279184522*b4 - 17187322078969*b2 + 1367118555542847 $$\nu^{13}$$ $$=$$ $$1325256976250 \beta_{19} - 813119758175 \beta_{18} + 111130456894 \beta_{17} + 296377317780 \beta_{16} - 875227443314 \beta_{15} + 531829170296 \beta_{13} + \cdots + 35\!\cdots\!58 \beta_1$$ 1325256976250*b19 - 813119758175*b18 + 111130456894*b17 + 296377317780*b16 - 875227443314*b15 + 531829170296*b13 + 37913874693840*b12 - 19542265379763*b11 - 29862529158913*b3 + 3595118956976758*b1 $$\nu^{14}$$ $$=$$ $$- 1503100340603 \beta_{14} + 82533398918870 \beta_{10} - 37034306669503 \beta_{9} + 280047778113504 \beta_{8} - 61935527232219 \beta_{7} + \cdots - 61\!\cdots\!31$$ -1503100340603*b14 + 82533398918870*b10 - 37034306669503*b9 + 280047778113504*b8 - 61935527232219*b7 + 31266223449565*b6 - 86193312276323*b5 - 2556428038705322*b4 + 7865537953198209*b2 - 613565538221477431 $$\nu^{15}$$ $$=$$ $$- 660983474302250 \beta_{19} + 413570565364279 \beta_{18} - 65919848222606 \beta_{17} - 166574015615220 \beta_{16} + 409096393166226 \beta_{15} + \cdots - 16\!\cdots\!62 \beta_1$$ -660983474302250*b19 + 413570565364279*b18 - 65919848222606*b17 - 166574015615220*b16 + 409096393166226*b15 - 279511745712952*b13 - 19178116036789824*b12 + 10637856164447339*b11 + 13975696300149529*b3 - 1637644586730934662*b1 $$\nu^{16}$$ $$=$$ $$722012791137891 \beta_{14} + \cdots + 27\!\cdots\!43$$ 722012791137891*b14 - 40202332506443974*b10 + 17408832726416935*b9 - 135974888525503376*b8 + 30851229410608867*b7 - 15648365918817109*b6 + 43227374131287835*b5 + 1405488490558677450*b4 - 3629185983177290041*b2 + 279572994164221596543 $$\nu^{17}$$ $$=$$ $$32\!\cdots\!10 \beta_{19} + \cdots + 75\!\cdots\!46 \beta_1$$ 324393518025982810*b19 - 205789773848881583*b18 + 35387895383823454*b17 + 88833909796725780*b16 - 189788661212829074*b15 + 142279652116946072*b13 + 9495027656523550288*b12 - 5548134816166330307*b11 - 6550975475159496113*b3 + 753838420219553115446*b1 $$\nu^{18}$$ $$=$$ $$- 34\!\cdots\!31 \beta_{14} + \cdots - 12\!\cdots\!47$$ -342132420648808331*b14 + 19397737494803189494*b10 - 8181925611967772175*b9 + 65284832893688088896*b8 - 15131882120342702827*b7 + 7724403906458336589*b6 - 21334515277928070355*b5 - 737482622129337657450*b4 + 1684923278142934179057*b2 - 128720745678166304922247 $$\nu^{19}$$ $$=$$ $$- 15\!\cdots\!70 \beta_{19} + \cdots - 34\!\cdots\!98 \beta_1$$ -157545503390487365770*b19 + 100928092393471761703*b18 - 18064007031742816814*b17 - 45767710250865026100*b16 + 87963899138175361554*b15 - 70908824283094853752*b13 - 4636402684353851147488*b12 + 2813303336205830229787*b11 + 3076107017597251199369*b3 - 349574716089700914349798*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 − 21.7545i − 18.8806i − 16.0010i − 14.8861i − 12.3896i − 10.3219i − 7.77325i − 7.52533i − 4.18617i − 0.118026i 0.118026i 4.18617i 7.52533i 7.77325i 10.3219i 12.3896i 14.8861i 16.0010i 18.8806i 21.7545i
21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
36.2 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.3 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.4 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.5 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.6 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.7 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.8 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.9 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.10 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.11 0.118026i −7.03331 127.986 225.549i 0.830113i 6.20125 30.2130i −2137.53 26.6207
36.12 4.18617i 63.9664 110.476 270.412i 267.774i 565.484 998.302i 1904.70 −1131.99
36.13 7.52533i −79.9440 71.3694 468.997i 601.605i −1534.47 1500.32i 4204.05 −3529.36
36.14 7.77325i −62.8536 67.5766 184.143i 488.577i 1071.60 1520.27i 1763.58 1431.39
36.15 10.3219i 6.68770 21.4577 296.142i 69.0300i −1262.47 1542.69i −2142.27 3056.76
36.16 12.3896i 76.2286 −25.5018 93.6237i 944.440i −1237.82 1269.91i 3623.79 −1159.96
36.17 14.8861i −0.807131 −93.5969 387.405i 12.0151i 696.938 512.130i −2186.35 −5766.96
36.18 16.0010i 52.6587 −128.032 434.946i 842.592i 1479.04 0.510187i 585.941 6959.56
36.19 18.8806i −54.3989 −228.476 103.526i 1027.08i −202.536 1897.05i 772.237 1954.63
36.20 21.7545i 44.4956 −345.259 104.884i 967.981i −454.964 4726.36i −207.137 −2281.70
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 36.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.b.a 20
37.b even 2 1 inner 37.8.b.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.b.a 20 1.a even 1 1 trivial
37.8.b.a 20 37.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 1702 T^{18} + \cdots + 13\!\cdots\!96$$
$3$ $$(T^{10} - 39 T^{9} + \cdots - 118561742732844)^{2}$$
$5$ $$T^{20} + 835306 T^{18} + \cdots + 71\!\cdots\!00$$
$7$ $$(T^{10} + 873 T^{9} + \cdots - 85\!\cdots\!36)^{2}$$
$11$ $$(T^{10} - 1749 T^{9} + \cdots - 11\!\cdots\!04)^{2}$$
$13$ $$T^{20} + 632138730 T^{18} + \cdots + 38\!\cdots\!56$$
$17$ $$T^{20} + 3321595492 T^{18} + \cdots + 26\!\cdots\!56$$
$19$ $$T^{20} + 6260069400 T^{18} + \cdots + 50\!\cdots\!76$$
$23$ $$T^{20} + 28047457810 T^{18} + \cdots + 33\!\cdots\!96$$
$29$ $$T^{20} + 224854622446 T^{18} + \cdots + 12\!\cdots\!96$$
$31$ $$T^{20} + 263739369210 T^{18} + \cdots + 14\!\cdots\!00$$
$37$ $$T^{20} - 108732 T^{19} + \cdots + 59\!\cdots\!49$$
$41$ $$(T^{10} + 788871 T^{9} + \cdots + 23\!\cdots\!26)^{2}$$
$43$ $$T^{20} + 2779292121648 T^{18} + \cdots + 79\!\cdots\!96$$
$47$ $$(T^{10} + 756393 T^{9} + \cdots + 74\!\cdots\!84)^{2}$$
$53$ $$(T^{10} - 1499679 T^{9} + \cdots + 14\!\cdots\!64)^{2}$$
$59$ $$T^{20} + 23780290980220 T^{18} + \cdots + 12\!\cdots\!56$$
$61$ $$T^{20} + 19651838390478 T^{18} + \cdots + 27\!\cdots\!00$$
$67$ $$(T^{10} - 1781112 T^{9} + \cdots + 25\!\cdots\!76)^{2}$$
$71$ $$(T^{10} + 7629543 T^{9} + \cdots + 12\!\cdots\!76)^{2}$$
$73$ $$(T^{10} - 5544009 T^{9} + \cdots + 35\!\cdots\!06)^{2}$$
$79$ $$T^{20} + 170634215807934 T^{18} + \cdots + 16\!\cdots\!36$$
$83$ $$(T^{10} + 6436911 T^{9} + \cdots + 50\!\cdots\!16)^{2}$$
$89$ $$T^{20} + 261084499656316 T^{18} + \cdots + 13\!\cdots\!56$$
$97$ $$T^{20} + 788101919779560 T^{18} + \cdots + 27\!\cdots\!96$$