# Properties

 Label 35.4.a.c Level $35$ Weight $4$ Character orbit 35.a Self dual yes Analytic conductor $2.065$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(1,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("35.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 35.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$2.06506685020$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.14360.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 17x - 14$$ x^3 - 17*x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + (3 \beta_{2} - 4 \beta_1 + 7) q^{6} + 7 q^{7} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (-b2 + b1 + 1) * q^3 + (b2 - b1 + 4) * q^4 + 5 * q^5 + (3*b2 - 4*b1 + 7) * q^6 + 7 * q^7 + (-3*b2 + b1 - 4) * q^8 + (-3*b2 - 9*b1 + 28) * q^9 $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + (3 \beta_{2} - 4 \beta_1 + 7) q^{6} + 7 q^{7} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9} + (5 \beta_1 - 5) q^{10} + (\beta_{2} + 3 \beta_1 - 25) q^{11} + ( - 2 \beta_{2} + 14 \beta_1 - 50) q^{12} + (5 \beta_{2} - 13 \beta_1 + 13) q^{13} + (7 \beta_1 - 7) q^{14} + ( - 5 \beta_{2} + 5 \beta_1 + 5) q^{15} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + (11 \beta_{2} + 13 \beta_1 - 21) q^{17} + ( - 3 \beta_{2} + 13 \beta_1 - 136) q^{18} + (6 \beta_{2} - 10 \beta_1 + 54) q^{19} + (5 \beta_{2} - 5 \beta_1 + 20) q^{20} + ( - 7 \beta_{2} + 7 \beta_1 + 7) q^{21} + (\beta_{2} - 20 \beta_1 + 61) q^{22} + (2 \beta_{2} - 14 \beta_1 - 42) q^{23} + ( - 6 \beta_{2} - 28 \beta_1 + 142) q^{24} + 25 q^{25} + ( - 23 \beta_{2} + 38 \beta_1 - 141) q^{26} + ( - 31 \beta_{2} + 7 \beta_1 + 67) q^{27} + (7 \beta_{2} - 7 \beta_1 + 28) q^{28} + (17 \beta_{2} + 19 \beta_1 + 105) q^{29} + (15 \beta_{2} - 20 \beta_1 + 35) q^{30} + ( - 4 \beta_{2} + 24 \beta_1 + 108) q^{31} + (15 \beta_{2} - 39 \beta_1 - 66) q^{32} + (35 \beta_{2} - 27 \beta_1 - 47) q^{33} + ( - 9 \beta_{2} + 34 \beta_1 + 197) q^{34} + 35 q^{35} + (43 \beta_{2} - 79 \beta_1 + 46) q^{36} + ( - 12 \beta_{2} + 16 \beta_1 - 14) q^{37} + ( - 22 \beta_{2} + 84 \beta_1 - 146) q^{38} + ( - 19 \beta_{2} + 87 \beta_1 - 321) q^{39} + ( - 15 \beta_{2} + 5 \beta_1 - 20) q^{40} + (2 \beta_{2} + 10 \beta_1 + 120) q^{41} + (21 \beta_{2} - 28 \beta_1 + 49) q^{42} + ( - 34 \beta_{2} + 30 \beta_1 + 6) q^{43} + ( - 30 \beta_{2} + 42 \beta_1 - 78) q^{44} + ( - 15 \beta_{2} - 45 \beta_1 + 140) q^{45} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + ( - 13 \beta_{2} - 51 \beta_1 - 239) q^{47} - 68 q^{48} + 49 q^{49} + (25 \beta_1 - 25) q^{50} + (91 \beta_{2} + 17 \beta_1 - 423) q^{51} + (44 \beta_{2} - 152 \beta_1 + 386) q^{52} + ( - 22 \beta_{2} - 130 \beta_1 + 44) q^{53} + (69 \beta_{2} - 88 \beta_1 - 83) q^{54} + (5 \beta_{2} + 15 \beta_1 - 125) q^{55} + ( - 21 \beta_{2} + 7 \beta_1 - 28) q^{56} + ( - 50 \beta_{2} + 126 \beta_1 - 302) q^{57} + ( - 15 \beta_{2} + 190 \beta_1 + 155) q^{58} + (48 \beta_{2} - 176 \beta_1 - 76) q^{59} + ( - 10 \beta_{2} + 70 \beta_1 - 250) q^{60} + ( - 26 \beta_{2} - 34 \beta_1 + 416) q^{61} + (32 \beta_{2} + 88 \beta_1 + 144) q^{62} + ( - 21 \beta_{2} - 63 \beta_1 + 196) q^{63} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + (25 \beta_{2} - 65 \beta_1 + 65) q^{65} + ( - 97 \beta_{2} + 128 \beta_1 - 145) q^{66} + (108 \beta_{2} - 12 \beta_1 + 32) q^{67} + ( - 36 \beta_{2} + 48 \beta_1 + 318) q^{68} + (22 \beta_{2} + 14 \beta_1 - 246) q^{69} + (35 \beta_1 - 35) q^{70} + ( - 40 \beta_{2} + 72 \beta_1 - 32) q^{71} + ( - 141 \beta_{2} + 157 \beta_1 + 302) q^{72} + (76 \beta_{2} + 124 \beta_1 + 78) q^{73} + (40 \beta_{2} - 74 \beta_1 + 154) q^{74} + ( - 25 \beta_{2} + 25 \beta_1 + 25) q^{75} + (80 \beta_{2} - 176 \beta_1 + 572) q^{76} + (7 \beta_{2} + 21 \beta_1 - 175) q^{77} + (125 \beta_{2} - 416 \beta_1 + 1221) q^{78} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + ( - 5 \beta_{2} - 55 \beta_1 - 130) q^{80} + ( - 96 \beta_{2} + 72 \beta_1 + 793) q^{81} + (6 \beta_{2} + 130 \beta_1 - 4) q^{82} + (8 \beta_{2} - 160 \beta_1 - 556) q^{83} + ( - 14 \beta_{2} + 98 \beta_1 - 350) q^{84} + (55 \beta_{2} + 65 \beta_1 - 105) q^{85} + (98 \beta_{2} - 164 \beta_1 + 222) q^{86} + (\beta_{2} + 167 \beta_1 - 525) q^{87} + (94 \beta_{2} - 68 \beta_1 - 38) q^{88} + ( - 82 \beta_{2} - 98 \beta_1 + 108) q^{89} + ( - 15 \beta_{2} + 65 \beta_1 - 680) q^{90} + (35 \beta_{2} - 91 \beta_1 + 91) q^{91} + ( - 12 \beta_{2} - 84 \beta_1 + 36) q^{92} + ( - 76 \beta_{2} + 8 \beta_1 + 484) q^{93} + ( - 25 \beta_{2} - 304 \beta_1 - 361) q^{94} + (30 \beta_{2} - 50 \beta_1 + 270) q^{95} + (48 \beta_{2} + 156 \beta_1 - 1068) q^{96} + ( - 65 \beta_{2} - 87 \beta_1 + 55) q^{97} + (49 \beta_1 - 49) q^{98} + (106 \beta_{2} + 198 \beta_1 - 1198) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (-b2 + b1 + 1) * q^3 + (b2 - b1 + 4) * q^4 + 5 * q^5 + (3*b2 - 4*b1 + 7) * q^6 + 7 * q^7 + (-3*b2 + b1 - 4) * q^8 + (-3*b2 - 9*b1 + 28) * q^9 + (5*b1 - 5) * q^10 + (b2 + 3*b1 - 25) * q^11 + (-2*b2 + 14*b1 - 50) * q^12 + (5*b2 - 13*b1 + 13) * q^13 + (7*b1 - 7) * q^14 + (-5*b2 + 5*b1 + 5) * q^15 + (-b2 - 11*b1 - 26) * q^16 + (11*b2 + 13*b1 - 21) * q^17 + (-3*b2 + 13*b1 - 136) * q^18 + (6*b2 - 10*b1 + 54) * q^19 + (5*b2 - 5*b1 + 20) * q^20 + (-7*b2 + 7*b1 + 7) * q^21 + (b2 - 20*b1 + 61) * q^22 + (2*b2 - 14*b1 - 42) * q^23 + (-6*b2 - 28*b1 + 142) * q^24 + 25 * q^25 + (-23*b2 + 38*b1 - 141) * q^26 + (-31*b2 + 7*b1 + 67) * q^27 + (7*b2 - 7*b1 + 28) * q^28 + (17*b2 + 19*b1 + 105) * q^29 + (15*b2 - 20*b1 + 35) * q^30 + (-4*b2 + 24*b1 + 108) * q^31 + (15*b2 - 39*b1 - 66) * q^32 + (35*b2 - 27*b1 - 47) * q^33 + (-9*b2 + 34*b1 + 197) * q^34 + 35 * q^35 + (43*b2 - 79*b1 + 46) * q^36 + (-12*b2 + 16*b1 - 14) * q^37 + (-22*b2 + 84*b1 - 146) * q^38 + (-19*b2 + 87*b1 - 321) * q^39 + (-15*b2 + 5*b1 - 20) * q^40 + (2*b2 + 10*b1 + 120) * q^41 + (21*b2 - 28*b1 + 49) * q^42 + (-34*b2 + 30*b1 + 6) * q^43 + (-30*b2 + 42*b1 - 78) * q^44 + (-15*b2 - 45*b1 + 140) * q^45 + (-18*b2 - 32*b1 - 106) * q^46 + (-13*b2 - 51*b1 - 239) * q^47 - 68 * q^48 + 49 * q^49 + (25*b1 - 25) * q^50 + (91*b2 + 17*b1 - 423) * q^51 + (44*b2 - 152*b1 + 386) * q^52 + (-22*b2 - 130*b1 + 44) * q^53 + (69*b2 - 88*b1 - 83) * q^54 + (5*b2 + 15*b1 - 125) * q^55 + (-21*b2 + 7*b1 - 28) * q^56 + (-50*b2 + 126*b1 - 302) * q^57 + (-15*b2 + 190*b1 + 155) * q^58 + (48*b2 - 176*b1 - 76) * q^59 + (-10*b2 + 70*b1 - 250) * q^60 + (-26*b2 - 34*b1 + 416) * q^61 + (32*b2 + 88*b1 + 144) * q^62 + (-21*b2 - 63*b1 + 196) * q^63 + (-61*b2 + 97*b1 - 110) * q^64 + (25*b2 - 65*b1 + 65) * q^65 + (-97*b2 + 128*b1 - 145) * q^66 + (108*b2 - 12*b1 + 32) * q^67 + (-36*b2 + 48*b1 + 318) * q^68 + (22*b2 + 14*b1 - 246) * q^69 + (35*b1 - 35) * q^70 + (-40*b2 + 72*b1 - 32) * q^71 + (-141*b2 + 157*b1 + 302) * q^72 + (76*b2 + 124*b1 + 78) * q^73 + (40*b2 - 74*b1 + 154) * q^74 + (-25*b2 + 25*b1 + 25) * q^75 + (80*b2 - 176*b1 + 572) * q^76 + (7*b2 + 21*b1 - 175) * q^77 + (125*b2 - 416*b1 + 1221) * q^78 + (-89*b2 - 83*b1 - 315) * q^79 + (-5*b2 - 55*b1 - 130) * q^80 + (-96*b2 + 72*b1 + 793) * q^81 + (6*b2 + 130*b1 - 4) * q^82 + (8*b2 - 160*b1 - 556) * q^83 + (-14*b2 + 98*b1 - 350) * q^84 + (55*b2 + 65*b1 - 105) * q^85 + (98*b2 - 164*b1 + 222) * q^86 + (b2 + 167*b1 - 525) * q^87 + (94*b2 - 68*b1 - 38) * q^88 + (-82*b2 - 98*b1 + 108) * q^89 + (-15*b2 + 65*b1 - 680) * q^90 + (35*b2 - 91*b1 + 91) * q^91 + (-12*b2 - 84*b1 + 36) * q^92 + (-76*b2 + 8*b1 + 484) * q^93 + (-25*b2 - 304*b1 - 361) * q^94 + (30*b2 - 50*b1 + 270) * q^95 + (48*b2 + 156*b1 - 1068) * q^96 + (-65*b2 - 87*b1 + 55) * q^97 + (49*b1 - 49) * q^98 + (106*b2 + 198*b1 - 1198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 2 q^{3} + 13 q^{4} + 15 q^{5} + 24 q^{6} + 21 q^{7} - 15 q^{8} + 81 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 2 * q^3 + 13 * q^4 + 15 * q^5 + 24 * q^6 + 21 * q^7 - 15 * q^8 + 81 * q^9 $$3 q - 3 q^{2} + 2 q^{3} + 13 q^{4} + 15 q^{5} + 24 q^{6} + 21 q^{7} - 15 q^{8} + 81 q^{9} - 15 q^{10} - 74 q^{11} - 152 q^{12} + 44 q^{13} - 21 q^{14} + 10 q^{15} - 79 q^{16} - 52 q^{17} - 411 q^{18} + 168 q^{19} + 65 q^{20} + 14 q^{21} + 184 q^{22} - 124 q^{23} + 420 q^{24} + 75 q^{25} - 446 q^{26} + 170 q^{27} + 91 q^{28} + 332 q^{29} + 120 q^{30} + 320 q^{31} - 183 q^{32} - 106 q^{33} + 582 q^{34} + 105 q^{35} + 181 q^{36} - 54 q^{37} - 460 q^{38} - 982 q^{39} - 75 q^{40} + 362 q^{41} + 168 q^{42} - 16 q^{43} - 264 q^{44} + 405 q^{45} - 336 q^{46} - 730 q^{47} - 204 q^{48} + 147 q^{49} - 75 q^{50} - 1178 q^{51} + 1202 q^{52} + 110 q^{53} - 180 q^{54} - 370 q^{55} - 105 q^{56} - 956 q^{57} + 450 q^{58} - 180 q^{59} - 760 q^{60} + 1222 q^{61} + 464 q^{62} + 567 q^{63} - 391 q^{64} + 220 q^{65} - 532 q^{66} + 204 q^{67} + 918 q^{68} - 716 q^{69} - 105 q^{70} - 136 q^{71} + 765 q^{72} + 310 q^{73} + 502 q^{74} + 50 q^{75} + 1796 q^{76} - 518 q^{77} + 3788 q^{78} - 1034 q^{79} - 395 q^{80} + 2283 q^{81} - 6 q^{82} - 1660 q^{83} - 1064 q^{84} - 260 q^{85} + 764 q^{86} - 1574 q^{87} - 20 q^{88} + 242 q^{89} - 2055 q^{90} + 308 q^{91} + 96 q^{92} + 1376 q^{93} - 1108 q^{94} + 840 q^{95} - 3156 q^{96} + 100 q^{97} - 147 q^{98} - 3488 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 2 * q^3 + 13 * q^4 + 15 * q^5 + 24 * q^6 + 21 * q^7 - 15 * q^8 + 81 * q^9 - 15 * q^10 - 74 * q^11 - 152 * q^12 + 44 * q^13 - 21 * q^14 + 10 * q^15 - 79 * q^16 - 52 * q^17 - 411 * q^18 + 168 * q^19 + 65 * q^20 + 14 * q^21 + 184 * q^22 - 124 * q^23 + 420 * q^24 + 75 * q^25 - 446 * q^26 + 170 * q^27 + 91 * q^28 + 332 * q^29 + 120 * q^30 + 320 * q^31 - 183 * q^32 - 106 * q^33 + 582 * q^34 + 105 * q^35 + 181 * q^36 - 54 * q^37 - 460 * q^38 - 982 * q^39 - 75 * q^40 + 362 * q^41 + 168 * q^42 - 16 * q^43 - 264 * q^44 + 405 * q^45 - 336 * q^46 - 730 * q^47 - 204 * q^48 + 147 * q^49 - 75 * q^50 - 1178 * q^51 + 1202 * q^52 + 110 * q^53 - 180 * q^54 - 370 * q^55 - 105 * q^56 - 956 * q^57 + 450 * q^58 - 180 * q^59 - 760 * q^60 + 1222 * q^61 + 464 * q^62 + 567 * q^63 - 391 * q^64 + 220 * q^65 - 532 * q^66 + 204 * q^67 + 918 * q^68 - 716 * q^69 - 105 * q^70 - 136 * q^71 + 765 * q^72 + 310 * q^73 + 502 * q^74 + 50 * q^75 + 1796 * q^76 - 518 * q^77 + 3788 * q^78 - 1034 * q^79 - 395 * q^80 + 2283 * q^81 - 6 * q^82 - 1660 * q^83 - 1064 * q^84 - 260 * q^85 + 764 * q^86 - 1574 * q^87 - 20 * q^88 + 242 * q^89 - 2055 * q^90 + 308 * q^91 + 96 * q^92 + 1376 * q^93 - 1108 * q^94 + 840 * q^95 - 3156 * q^96 + 100 * q^97 - 147 * q^98 - 3488 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 17x - 14$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 11$$ v^2 - v - 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11

## Embeddings

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

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.62456 −0.861086 4.48565
−4.62456 −8.38660 13.3866 5.00000 38.7844 7.00000 −24.9107 43.3350 −23.1228
1.2 −1.86109 9.53636 −4.53636 5.00000 −17.7480 7.00000 23.3312 63.9421 −9.30543
1.3 3.48565 0.850238 4.14976 5.00000 2.96363 7.00000 −13.4206 −26.2771 17.4283
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.a.c 3
3.b odd 2 1 315.4.a.p 3
4.b odd 2 1 560.4.a.u 3
5.b even 2 1 175.4.a.f 3
5.c odd 4 2 175.4.b.e 6
7.b odd 2 1 245.4.a.l 3
7.c even 3 2 245.4.e.m 6
7.d odd 6 2 245.4.e.n 6
8.b even 2 1 2240.4.a.bt 3
8.d odd 2 1 2240.4.a.bv 3
15.d odd 2 1 1575.4.a.ba 3
21.c even 2 1 2205.4.a.bm 3
35.c odd 2 1 1225.4.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 1.a even 1 1 trivial
175.4.a.f 3 5.b even 2 1
175.4.b.e 6 5.c odd 4 2
245.4.a.l 3 7.b odd 2 1
245.4.e.m 6 7.c even 3 2
245.4.e.n 6 7.d odd 6 2
315.4.a.p 3 3.b odd 2 1
560.4.a.u 3 4.b odd 2 1
1225.4.a.y 3 35.c odd 2 1
1575.4.a.ba 3 15.d odd 2 1
2205.4.a.bm 3 21.c even 2 1
2240.4.a.bt 3 8.b even 2 1
2240.4.a.bv 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 3T_{2}^{2} - 14T_{2} - 30$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(35))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3 T^{2} + \cdots - 30$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 68$$
$5$ $$(T - 5)^{3}$$
$7$ $$(T - 7)^{3}$$
$11$ $$T^{3} + 74 T^{2} + \cdots + 7692$$
$13$ $$T^{3} - 44 T^{2} + \cdots - 44870$$
$17$ $$T^{3} + 52 T^{2} + \cdots - 56706$$
$19$ $$T^{3} - 168 T^{2} + \cdots - 28720$$
$23$ $$T^{3} + 124 T^{2} + \cdots - 94368$$
$29$ $$T^{3} - 332 T^{2} + \cdots + 2565450$$
$31$ $$T^{3} - 320 T^{2} + \cdots + 50176$$
$37$ $$T^{3} + 54 T^{2} + \cdots + 25736$$
$41$ $$T^{3} - 362 T^{2} + \cdots - 1536192$$
$43$ $$T^{3} + 16 T^{2} + \cdots - 1524560$$
$47$ $$T^{3} + 730 T^{2} + \cdots + 4968912$$
$53$ $$T^{3} - 110 T^{2} + \cdots + 90318336$$
$59$ $$T^{3} + 180 T^{2} + \cdots - 202459200$$
$61$ $$T^{3} - 1222 T^{2} + \cdots - 38393792$$
$67$ $$T^{3} - 204 T^{2} + \cdots + 324944128$$
$71$ $$T^{3} + 136 T^{2} + \cdots + 15575040$$
$73$ $$T^{3} - 310 T^{2} + \cdots + 48718616$$
$79$ $$T^{3} + 1034 T^{2} + \cdots - 343615600$$
$83$ $$T^{3} + 1660 T^{2} + \cdots - 42727104$$
$89$ $$T^{3} - 242 T^{2} + \cdots - 6359520$$
$97$ $$T^{3} - 100 T^{2} + \cdots - 1978018$$