# Properties

 Label 35.4.j.a Level $35$ Weight $4$ Character orbit 35.j Analytic conductor $2.065$ Analytic rank $0$ Dimension $20$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("35.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 35.j (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: $$2.06506685020$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 55 x^{18} + 2018 x^{16} - 42095 x^{14} + 639938 x^{12} - 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496$$ x^20 - 55*x^18 + 2018*x^16 - 42095*x^14 + 639938*x^12 - 5744691*x^10 + 35287093*x^8 - 51070316*x^6 + 53741776*x^4 - 27082496*x^2 + 9834496 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{4} - 3 \beta_{3} - \beta_{2}) q^{4} + \beta_{10} q^{5} + (\beta_{7} + \beta_{2} - 5) q^{6} + (\beta_{18} - \beta_{17} + \cdots + \beta_{15}) q^{7}+ \cdots + (\beta_{8} - \beta_{5} + 8 \beta_{3} + 8) q^{9}+O(q^{10})$$ q - b1 * q^2 - b14 * q^3 + (b4 - 3*b3 - b2) * q^4 + b10 * q^5 + (b7 + b2 - 5) * q^6 + (b18 - b17 + b16 + b15) * q^7 + (-b19 - b16 + 2*b15 - 2*b14 + b13 - b12 - b11 - b10 - b6) * q^8 + (b8 - b5 + 8*b3 + 8) * q^9 $$q - \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{4} - 3 \beta_{3} - \beta_{2}) q^{4} + \beta_{10} q^{5} + (\beta_{7} + \beta_{2} - 5) q^{6} + (\beta_{18} - \beta_{17} + \cdots + \beta_{15}) q^{7}+ \cdots + (62 \beta_{13} + 62 \beta_{12} + \cdots + 740) q^{99}+O(q^{100})$$ q - b1 * q^2 - b14 * q^3 + (b4 - 3*b3 - b2) * q^4 + b10 * q^5 + (b7 + b2 - 5) * q^6 + (b18 - b17 + b16 + b15) * q^7 + (-b19 - b16 + 2*b15 - 2*b14 + b13 - b12 - b11 - b10 - b6) * q^8 + (b8 - b5 + 8*b3 + 8) * q^9 + (b17 + b16 + 3*b14 + 2*b12 - b9 + b8 - b7 + b6 + b5 + 2*b3 - b1) * q^10 + (-b13 - b12 + 2*b9 - 2*b8 + b7 - b5 - 2*b4 - 2*b3 + 2*b2) * q^11 + (4*b19 - b18 - 5*b15 + b11 + b10 + 5*b1) * q^12 + (-b18 + b17 - 3*b15 + 3*b14 - b13 + b12 + b11 + b10 + 2*b6) * q^13 + (2*b11 - 2*b10 - b9 - b8 - 3*b7 + b5 - 4*b3 - 3*b2 + 1) * q^14 + (-4*b19 - b18 + b17 - 4*b16 + 2*b15 - 2*b14 + 2*b13 - 3*b12 - 3*b11 - 2*b10 - b9 - b7 - 4*b6 - 7) * q^15 + (2*b11 - 2*b10 - b8 + 2*b5 - 3*b4 - b3 - 1) * q^16 + (-4*b17 - b16 - 4*b14 - 2*b13 + 2*b12) * q^17 + (3*b17 + 2*b16 + 15*b14 - 3*b13 + 3*b12) * q^18 + (-3*b11 + 3*b10 + b8 + 3*b5 + 2*b4 - 22*b3 - 22) * q^19 + (-2*b19 - 3*b18 + 3*b17 - 2*b16 + b15 - b14 - b13 + b12 + b11 + b10 + 2*b9 - 3*b7 + 8*b6 - 5*b2 + 29) * q^20 + (-b13 - b12 + 3*b11 - 3*b10 - b9 + b8 + 2*b7 - 3*b5 + 2*b4 + 5*b3 + 6*b2 - 17) * q^21 + (5*b19 + 5*b18 - 5*b17 + 5*b16 - b15 + b14 + 6*b13 - 6*b12 - 6*b11 - 6*b10 + 6*b6) * q^22 + (-b18 - 6*b15 + 5*b11 + 5*b10 + 6*b1) * q^23 + (4*b13 + 4*b12 + b9 - b8 + 3*b7 - 3*b5 - 9*b4 + 43*b3 + 9*b2) * q^24 + (5*b17 - 5*b14 - b13 + 5*b7 - 10*b6 - 5*b5 + 20*b3 + 10*b1) * q^25 + (-4*b11 + 4*b10 + 3*b8 - b5 - 4*b4 + 40*b3 + 40) * q^26 + (7*b19 + 6*b18 - 6*b17 + 7*b16 + b15 - b14 - 3*b13 + 3*b12 + 3*b11 + 3*b10 - 26*b6) * q^27 + (-5*b19 + b18 + b17 - 6*b16 + 27*b15 - 19*b14 + 7*b13 - 7*b12 - 7*b1) * q^28 + (-7*b13 - 7*b12 - 7*b11 + 7*b10 - b9 + b7 + 6*b2 + 9) * q^29 + (-5*b19 + 10*b15 + 5*b11 - b10 + 5*b5 + 15*b4 - 65*b3 + 10*b1 - 65) * q^30 + (9*b13 + 9*b12 - b9 + b8 - 4*b7 + 4*b5 + 24*b4 - 90*b3 - 24*b2) * q^31 + (-8*b17 + 2*b16 - 16*b14 - 4*b13 + 4*b12 + 35*b6 - 35*b1) * q^32 + (-8*b19 - 2*b18 - 18*b15 - 9*b11 - 9*b10 - 16*b1) * q^33 + (4*b13 + 4*b12 + 4*b11 - 4*b10 + b9 + 10*b7 + 2*b2 - 8) * q^34 + (-4*b19 - b18 + 4*b17 - b16 - 18*b15 + 17*b14 - 2*b13 - 2*b12 + 7*b11 + b9 - 2*b8 + 6*b7 + 14*b6 - 2*b5 - 10*b4 + b3 - 28*b1 + 33) * q^35 + (-12*b13 - 12*b12 - 12*b11 + 12*b10 - b9 - 5*b7 - 16*b2 + 20) * q^36 + (11*b19 - b18 - 7*b15 + 9*b11 + 9*b10 - 22*b1) * q^37 + (5*b17 - 6*b16 - 15*b14 - 7*b13 + 7*b12 - 12*b6 + 12*b1) * q^38 + (b9 - b8 - 8*b7 + 8*b5 + 8*b4 - 88*b3 - 8*b2) * q^39 + (-4*b19 - b18 + 27*b15 - 3*b11 - 3*b10 - b8 - b5 + 68*b3 - 29*b1 + 68) * q^40 + (5*b13 + 5*b12 + 5*b11 - 5*b10 - 3*b9 - 10*b7 + 14*b2 - 55) * q^41 + (9*b19 - 2*b18 - 3*b17 - 2*b16 - 32*b15 + 9*b14 + 7*b13 - 7*b12 + 7*b11 + 7*b10 - 35*b6 + 77*b1) * q^42 + (8*b19 - 4*b18 + 4*b17 + 8*b16 + 23*b15 - 23*b14 + 7*b13 - 7*b12 - 7*b11 - 7*b10) * q^43 + (14*b11 - 14*b10 - 6*b8 - 15*b5 - 6*b4 + 68*b3 + 68) * q^44 + (3*b17 - 7*b16 + 9*b14 - 2*b13 + 6*b12 + 7*b9 - 7*b8 + 2*b7 - 22*b6 - 2*b5 - 30*b4 - 4*b3 + 30*b2 + 22*b1) * q^45 + (12*b13 + 12*b12 - 11*b9 + 11*b8 - 4*b7 + 4*b5 - 11*b4 + 115*b3 + 11*b2) * q^46 + (-10*b19 + 5*b18 + 39*b15 - 16*b11 - 16*b10 + 56*b1) * q^47 + (-5*b19 - b18 + b17 - 5*b16 - 31*b15 + 31*b14 - 6*b13 + 6*b12 + 6*b11 + 6*b10 + 49*b6) * q^48 + (-9*b13 - 9*b12 + 8*b9 + 5*b8 - b7 + 5*b5 + 4*b4 + 85*b3 - 14*b2 + 19) * q^49 + (14*b19 - 4*b18 + 4*b17 + 14*b16 - 52*b15 + 52*b14 - 10*b13 - 12*b12 - 12*b11 + 10*b10 + 6*b9 + 6*b7 - 11*b6 + 20*b2 - 138) * q^50 + (3*b11 - 3*b10 + 7*b8 - 2*b5 - 32*b4 + 98*b3 + 98) * q^51 + (7*b17 + 12*b16 + 31*b14 - 13*b13 + 13*b12 + 44*b6 - 44*b1) * q^52 + (-13*b17 - 8*b16 + 53*b14 + 10*b13 - 10*b12 - 62*b6 + 62*b1) * q^53 + (6*b11 - 6*b10 - 7*b8 - 7*b5 + 7*b4 - 273*b3 - 273) * q^54 + (-3*b19 + 8*b18 - 8*b17 - 3*b16 + 44*b15 - 44*b14 + 9*b13 + 4*b12 + 4*b11 - 9*b10 - 12*b9 + 3*b7 + 92*b6 - 20*b2 - 29) * q^55 + (-6*b13 - 6*b12 - 4*b11 + 4*b10 + 5*b9 - 4*b8 - 4*b7 + 13*b5 + 19*b4 - 171*b3 + 13*b2 - 92) * q^56 + (-17*b19 - 10*b18 + 10*b17 - 17*b16 - 26*b15 + 26*b14 + 2*b13 - 2*b12 - 2*b11 - 2*b10 + 48*b6) * q^57 + (-6*b19 + 11*b18 + 15*b15 - 5*b11 - 5*b10 + 5*b1) * q^58 + (9*b13 + 9*b12 - 9*b9 + 9*b8 - 2*b7 + 2*b5 + 242*b3) * q^59 + (-19*b17 - 4*b16 - 87*b14 + 9*b13 + 7*b12 - b9 + b8 - 16*b7 - 99*b6 + 16*b5 + 10*b4 + 12*b3 - 10*b2 + 99*b1) * q^60 + (4*b11 - 4*b10 - 18*b8 - b5 - 4*b4 + 61*b3 + 61) * q^61 + (-19*b19 - 16*b18 + 16*b17 - 19*b16 + 34*b15 - 34*b14 + 3*b13 - 3*b12 - 3*b11 - 3*b10 - 194*b6) * q^62 + (24*b19 - 11*b18 + 31*b16 + 21*b15 + 10*b14 - 21*b13 + 21*b12 + 21*b11 + 21*b10 + 42*b6 - 70*b1) * q^63 + (-8*b13 - 8*b12 - 8*b11 + 8*b10 + 6*b9 + 4*b7 - 7*b2 + 337) * q^64 + (13*b19 + 2*b18 - 14*b15 + b11 + 5*b10 + 2*b8 - 3*b5 + 10*b4 - 96*b3 + 48*b1 - 96) * q^65 + (-14*b13 - 14*b12 + 8*b9 - 8*b8 + 20*b7 - 20*b5 + 16*b4 - 124*b3 - 16*b2) * q^66 + (24*b17 - 13*b16 - 43*b14 + 19*b13 - 19*b12 + 162*b6 - 162*b1) * q^67 + (33*b19 + 16*b18 - 102*b15 + 23*b11 + 23*b10 - 30*b1) * q^68 + (-4*b13 - 4*b12 - 4*b11 + 4*b10 - 4*b9 - 23*b7 - 12*b2 + 157) * q^69 + (15*b19 - 13*b17 + 7*b16 - 50*b15 + b14 + 8*b13 - 6*b12 - 15*b11 + b10 - 7*b9 + 12*b8 - 12*b7 + 42*b6 - 3*b5 + 5*b4 - 51*b3 - 30*b2 - 112*b1 + 235) * q^70 + (8*b13 + 8*b12 + 8*b11 - 8*b10 + 9*b9 + 30*b7 + 8*b2 - 152) * q^71 + (-40*b19 + 3*b18 + 35*b15 - 13*b11 - 13*b10 - 112*b1) * q^72 + (-22*b17 + 33*b16 - 22*b14 - 10*b13 + 10*b12 + 28*b6 - 28*b1) * q^73 + (20*b13 + 20*b12 - 8*b9 + 8*b8 + 11*b7 - 11*b5 - 6*b4 - 172*b3 + 6*b2) * q^74 + (14*b19 + 11*b18 + 58*b15 - 12*b11 - 3*b10 + 6*b8 - 9*b5 + 20*b4 + 297*b3 - 106*b1 + 297) * q^75 + (17*b9 + 17*b7 + 28*b2 - 8) * q^76 + (-30*b19 - 6*b18 + 25*b17 + b16 + 10*b15 + 19*b14 - 21*b13 + 21*b12 - 70*b6 + 182*b1) * q^77 + (-31*b19 + 10*b18 - 10*b17 - 31*b16 + 108*b15 - 108*b14 + 11*b13 - 11*b12 - 11*b11 - 11*b10 - 86*b6) * q^78 + (-29*b11 + 29*b10 + 7*b8 + 52*b5 + 56*b4 - 26*b3 - 26) * q^79 + (-25*b17 + 20*b16 + 15*b14 + 7*b13 - 15*b12 - 15*b9 + 15*b8 - 5*b7 + 5*b5 + 25*b4 - 235*b3 - 25*b2) * q^80 + (-30*b13 - 30*b12 + 13*b9 - 13*b8 + 14*b7 - 14*b5 - 6*b4 + 39*b3 + 6*b2) * q^81 + (-9*b19 - 6*b18 + 40*b15 + 33*b11 + 33*b10 + 203*b1) * q^82 + (17*b19 + 22*b18 - 22*b17 + 17*b16 - b15 + b14 + 30*b13 - 30*b12 - 30*b11 - 30*b10 + 58*b6) * q^83 + (14*b13 + 14*b12 - 12*b11 + 12*b10 - 22*b9 - b8 + 25*b7 - 20*b5 - 42*b4 + 472*b3 + 53*b2 - 405) * q^84 + (-7*b19 + 27*b18 - 27*b17 - 7*b16 + b15 - b14 + 10*b13 + 41*b12 + 41*b11 - 10*b10 - 8*b9 - 23*b7 - 102*b6 - 40*b2 - 231) * q^85 + (6*b11 - 6*b10 - 18*b8 - 7*b5 + 11*b4 - 147*b3 - 147) * q^86 + (26*b17 - 39*b16 - 5*b14 + 32*b13 - 32*b12) * q^87 + (15*b17 - 11*b16 + 61*b14 - 8*b13 + 8*b12 - 50*b6 + 50*b1) * q^88 + (-14*b11 + 14*b10 + 12*b8 - 9*b5 - 42*b4 - 599*b3 - 599) * q^89 + (47*b19 + 8*b18 - 8*b17 + 47*b16 - 66*b15 + 66*b14 - 27*b13 - b12 - b11 + 27*b10 + 18*b9 + 13*b7 + 182*b6 + 30*b2 - 184) * q^90 + (14*b13 + 14*b12 + 6*b11 - 6*b10 - 3*b9 - 3*b8 + 5*b7 - 18*b5 - 28*b4 + 44*b3 - 2*b2 + 248) * q^91 + (12*b19 - 34*b18 + 34*b17 + 12*b16 - 46*b15 + 46*b14 - 28*b13 + 28*b12 + 28*b11 + 28*b10 + 159*b6) * q^92 + (21*b19 - 46*b18 - 122*b15 - 8*b11 - 8*b10 + 132*b1) * q^93 + (-42*b13 - 42*b12 + 27*b9 - 27*b8 - 27*b7 + 27*b5 - 2*b4 + 362*b3 + 2*b2) * q^94 + (12*b17 + 17*b16 + 36*b14 - 23*b13 - 16*b12 + 3*b9 - 3*b8 + 3*b7 - 108*b6 - 3*b5 + 20*b4 + 229*b3 - 20*b2 + 108*b1) * q^95 + (18*b11 - 18*b10 + 26*b8 + 15*b5 + 3*b4 + 373*b3 + 373) * q^96 + (-26*b19 - 16*b18 + 16*b17 - 26*b16 - 48*b15 + 48*b14 + 6*b13 - 6*b12 - 6*b11 - 6*b10 - 292*b6) * q^97 + (-27*b19 + 35*b18 - 13*b17 - 42*b16 + 125*b15 - 97*b14 + 7*b13 - 7*b12 - 56*b11 - 56*b10 + 63*b6 - 77*b1) * q^98 + (62*b13 + 62*b12 + 62*b11 - 62*b10 - 10*b9 - 29*b7 + 14*b2 + 740) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 30 q^{4} + 3 q^{5} - 96 q^{6} + 82 q^{9}+O(q^{10})$$ 20 * q + 30 * q^4 + 3 * q^5 - 96 * q^6 + 82 * q^9 $$20 q + 30 q^{4} + 3 q^{5} - 96 q^{6} + 82 q^{9} - 32 q^{10} + 36 q^{11} + 26 q^{14} - 146 q^{15} - 22 q^{16} - 192 q^{19} + 584 q^{20} - 404 q^{21} - 444 q^{24} - 187 q^{25} + 434 q^{26} + 260 q^{29} - 658 q^{30} + 834 q^{31} - 160 q^{34} + 661 q^{35} + 516 q^{36} + 868 q^{39} + 674 q^{40} - 1224 q^{41} + 542 q^{44} + 60 q^{45} - 1274 q^{46} - 326 q^{49} - 2556 q^{50} + 986 q^{51} - 2808 q^{54} - 742 q^{55} - 36 q^{56} - 2514 q^{59} - 204 q^{60} + 512 q^{61} + 6900 q^{64} - 946 q^{65} + 1396 q^{66} + 3064 q^{69} + 5190 q^{70} - 2944 q^{71} + 1590 q^{74} + 3003 q^{75} + 44 q^{76} + 46 q^{79} + 2304 q^{80} - 130 q^{81} - 12952 q^{84} - 5082 q^{85} - 1592 q^{86} - 5876 q^{89} - 3316 q^{90} + 4348 q^{91} - 3314 q^{94} - 2155 q^{95} + 3756 q^{96} + 13860 q^{99}+O(q^{100})$$ 20 * q + 30 * q^4 + 3 * q^5 - 96 * q^6 + 82 * q^9 - 32 * q^10 + 36 * q^11 + 26 * q^14 - 146 * q^15 - 22 * q^16 - 192 * q^19 + 584 * q^20 - 404 * q^21 - 444 * q^24 - 187 * q^25 + 434 * q^26 + 260 * q^29 - 658 * q^30 + 834 * q^31 - 160 * q^34 + 661 * q^35 + 516 * q^36 + 868 * q^39 + 674 * q^40 - 1224 * q^41 + 542 * q^44 + 60 * q^45 - 1274 * q^46 - 326 * q^49 - 2556 * q^50 + 986 * q^51 - 2808 * q^54 - 742 * q^55 - 36 * q^56 - 2514 * q^59 - 204 * q^60 + 512 * q^61 + 6900 * q^64 - 946 * q^65 + 1396 * q^66 + 3064 * q^69 + 5190 * q^70 - 2944 * q^71 + 1590 * q^74 + 3003 * q^75 + 44 * q^76 + 46 * q^79 + 2304 * q^80 - 130 * q^81 - 12952 * q^84 - 5082 * q^85 - 1592 * q^86 - 5876 * q^89 - 3316 * q^90 + 4348 * q^91 - 3314 * q^94 - 2155 * q^95 + 3756 * q^96 + 13860 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 55 x^{18} + 2018 x^{16} - 42095 x^{14} + 639938 x^{12} - 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 77\!\cdots\!35 \nu^{18} + \cdots + 31\!\cdots\!44 ) / 29\!\cdots\!40$$ (7739632444020935*v^18 - 414686284259737693*v^16 + 15035111378414269442*v^14 - 304645306677022433553*v^12 + 4526633566471324344706*v^10 - 38092787485268312110189*v^8 + 219512275476062741945031*v^6 - 61171471704175030517776*v^4 + 31064575702075290434304*v^2 + 3120990522831004887032144) / 298808271176611818624240 $$\beta_{3}$$ $$=$$ $$( - 26\!\cdots\!41 \nu^{18} + \cdots + 20\!\cdots\!32 ) / 47\!\cdots\!40$$ (-269907818538703141*v^18 + 14721095900524337795*v^16 - 538038997262947135450*v^14 + 11121207839332080409323*v^12 - 167849944673188251708410*v^10 + 1478110878925379896259135*v^8 - 8914777694438048842096089*v^6 + 10272081176025223769942060*v^4 - 13526581977288831045834000*v^2 + 2031811865884159916383232) / 4780932338825789097987840 $$\beta_{4}$$ $$=$$ $$( - 94\!\cdots\!97 \nu^{18} + \cdots + 24\!\cdots\!52 ) / 15\!\cdots\!80$$ (-948383961607133197*v^18 + 51765691452537304219*v^16 - 1892622395945930059626*v^14 + 39152987108606841855235*v^12 - 591307751447176526425738*v^10 + 5216578356138295288362487*v^8 - 31516786077067177797312161*v^6 + 37338049796336886993692748*v^4 - 47838145400039382586412432*v^2 + 24095259630007279090909952) / 1593644112941929699329280 $$\beta_{5}$$ $$=$$ $$( - 99\!\cdots\!73 \nu^{18} + \cdots + 33\!\cdots\!52 ) / 79\!\cdots\!40$$ (-992807843266452473*v^18 + 56327324813977119103*v^16 - 2095853077211783224162*v^14 + 45141153330489994730647*v^12 - 703236624197915363614946*v^10 + 6711632565966839265241659*v^8 - 43518055875544794619236893*v^6 + 99098009356310123781773196*v^4 - 66980535241783485111746384*v^2 + 33807010059944943861905152) / 796822056470964849664640 $$\beta_{6}$$ $$=$$ $$( 26\!\cdots\!41 \nu^{19} + \cdots - 20\!\cdots\!32 \nu ) / 47\!\cdots\!40$$ (269907818538703141*v^19 - 14721095900524337795*v^17 + 538038997262947135450*v^15 - 11121207839332080409323*v^13 + 167849944673188251708410*v^11 - 1478110878925379896259135*v^9 + 8914777694438048842096089*v^7 - 10272081176025223769942060*v^5 + 13526581977288831045834000*v^3 - 2031811865884159916383232*v) / 4780932338825789097987840 $$\beta_{7}$$ $$=$$ $$( 50\!\cdots\!73 \nu^{18} + \cdots + 27\!\cdots\!00 ) / 19\!\cdots\!60$$ (500049830480365573*v^18 - 26853371300080553627*v^16 + 973611724591644926638*v^14 - 19751957740599355308511*v^12 + 293126096795924998674734*v^10 - 2466731611398723280220171*v^8 + 14058638538916910665154421*v^6 - 3961211896788241491588464*v^4 + 2011613639685262493158656*v^2 + 2767743422950792678481600) / 199205514117741212416160 $$\beta_{8}$$ $$=$$ $$( 61\!\cdots\!51 \nu^{18} + \cdots - 19\!\cdots\!20 ) / 79\!\cdots\!40$$ (6127716449081845851*v^18 - 344402996639488846669*v^16 + 12762368352536966758326*v^14 - 272326913753964234905397*v^12 + 4213229150892882540463638*v^10 - 39531443858717692567018977*v^8 + 252664052696301752872086327*v^6 - 520248013144255336273517988*v^4 + 387823273636419302284853232*v^2 - 195666308402441808644670720) / 796822056470964849664640 $$\beta_{9}$$ $$=$$ $$( - 70\!\cdots\!21 \nu^{18} + \cdots - 58\!\cdots\!32 ) / 59\!\cdots\!80$$ (-7020595794143047621*v^18 + 377056802304029679503*v^16 - 13670794607421825670582*v^14 + 277354178438602119716091*v^12 - 4115877574351396654172726*v^10 + 34636170004228813498624319*v^8 - 197393800963866269876747445*v^6 + 55620647194014479164427696*v^4 - 28245712539215525529417984*v^2 - 58567977236862131399204032) / 597616542353223637248480 $$\beta_{10}$$ $$=$$ $$( - 46\!\cdots\!39 \nu^{19} + \cdots + 43\!\cdots\!12 ) / 47\!\cdots\!40$$ (-4673620406845704839*v^19 - 13544177717164641213*v^18 + 251000160748817658829*v^17 + 761161040777279194563*v^16 - 9100410397211593033826*v^15 - 28203454884126104692722*v^14 + 184626794030849888109441*v^13 + 601732683656537055201507*v^12 - 2739868174959095480155618*v^11 - 9308190964398390544398066*v^10 + 23056696459688555262020317*v^9 + 87318100929154867650941079*v^8 - 131421284595570768195886839*v^7 - 557972048239383109158178233*v^6 + 37025698253797837476717328*v^5 + 1148373019897645404192171876*v^4 - 18802679979478048906758912*v^3 - 856416561871835924445356304*v^2 - 70153048266428608415709824*v + 432080435669437252262406912) / 4780932338825789097987840 $$\beta_{11}$$ $$=$$ $$( - 46\!\cdots\!39 \nu^{19} + \cdots - 43\!\cdots\!12 ) / 47\!\cdots\!40$$ (-4673620406845704839*v^19 + 13544177717164641213*v^18 + 251000160748817658829*v^17 - 761161040777279194563*v^16 - 9100410397211593033826*v^15 + 28203454884126104692722*v^14 + 184626794030849888109441*v^13 - 601732683656537055201507*v^12 - 2739868174959095480155618*v^11 + 9308190964398390544398066*v^10 + 23056696459688555262020317*v^9 - 87318100929154867650941079*v^8 - 131421284595570768195886839*v^7 + 557972048239383109158178233*v^6 + 37025698253797837476717328*v^5 - 1148373019897645404192171876*v^4 - 18802679979478048906758912*v^3 + 856416561871835924445356304*v^2 - 70153048266428608415709824*v - 432080435669437252262406912) / 4780932338825789097987840 $$\beta_{12}$$ $$=$$ $$( - 67\!\cdots\!89 \nu^{19} + \cdots + 36\!\cdots\!08 ) / 66\!\cdots\!60$$ (-67486346787941203089*v^19 - 478689713542141177202*v^18 + 3751896608235712270479*v^17 + 26180208338055005552542*v^16 - 138347169384943412261946*v^15 - 957694024918961938798148*v^14 + 2919141911524443777586191*v^13 + 19842896273477777692149678*v^12 - 44776810570491053808253818*v^11 - 299771086395186672753368644*v^10 + 411263303032271953363593747*v^9 + 2648472778079968379213799286*v^8 - 2579787484236388137424128189*v^7 - 15939923548673103580926052602*v^6 + 4574322468379202368456119828*v^5 + 18367201799593963244704915384*v^4 - 3945422378490772990792705872*v^3 - 13152747195939185541079808736*v^2 + 1989485678929955944796894976*v + 3633080425677446695681988608) / 66933052743561047371829760 $$\beta_{13}$$ $$=$$ $$( 67\!\cdots\!89 \nu^{19} + \cdots + 36\!\cdots\!08 ) / 66\!\cdots\!60$$ (67486346787941203089*v^19 - 478689713542141177202*v^18 - 3751896608235712270479*v^17 + 26180208338055005552542*v^16 + 138347169384943412261946*v^15 - 957694024918961938798148*v^14 - 2919141911524443777586191*v^13 + 19842896273477777692149678*v^12 + 44776810570491053808253818*v^11 - 299771086395186672753368644*v^10 - 411263303032271953363593747*v^9 + 2648472778079968379213799286*v^8 + 2579787484236388137424128189*v^7 - 15939923548673103580926052602*v^6 - 4574322468379202368456119828*v^5 + 18367201799593963244704915384*v^4 + 3945422378490772990792705872*v^3 - 13152747195939185541079808736*v^2 - 1989485678929955944796894976*v + 3633080425677446695681988608) / 66933052743561047371829760 $$\beta_{14}$$ $$=$$ $$( - 26\!\cdots\!97 \nu^{19} + \cdots + 84\!\cdots\!36 \nu ) / 13\!\cdots\!20$$ (-263236347548019989597*v^19 + 14817499956558826471771*v^17 - 549442194320906833530554*v^15 + 11741936858857016445496467*v^13 - 181864738476258888920276122*v^11 + 1711220145516623254473829063*v^9 - 10963554688560645586901691105*v^7 + 22989310049528646118410647452*v^5 - 16836128038949008220463715728*v^3 + 8494818625306887301697979136*v) / 133866105487122094743659520 $$\beta_{15}$$ $$=$$ $$( 50\!\cdots\!83 \nu^{19} + \cdots + 39\!\cdots\!68 \nu ) / 15\!\cdots\!80$$ (5036932901091409883*v^19 - 270505479592574307753*v^17 + 9807606782572817801082*v^15 - 198973012525760278892557*v^13 + 2952784382594195664959226*v^11 - 24848441192394105312136569*v^9 + 141625379934598157477777163*v^7 - 39902979478234075366431696*v^5 + 20263843458508078420233984*v^3 + 39100784296608298770392768*v) / 1593644112941929699329280 $$\beta_{16}$$ $$=$$ $$( 33\!\cdots\!57 \nu^{19} + \cdots - 10\!\cdots\!56 \nu ) / 66\!\cdots\!60$$ (335704657979151737657*v^19 - 18910562076379639860751*v^17 + 701451116690977036172354*v^15 - 15001613764840521888365847*v^13 + 232484038703911769283396322*v^11 - 2190489146793626847804024763*v^9 + 14050583315463804291526765245*v^7 - 29706902076054782804912658652*v^5 + 21581537803036638325569749328*v^3 - 10889518565510857752266381056*v) / 66933052743561047371829760 $$\beta_{17}$$ $$=$$ $$( 75\!\cdots\!49 \nu^{19} + \cdots - 23\!\cdots\!20 \nu ) / 13\!\cdots\!20$$ (759924670999047571849*v^19 - 42539280606409184765231*v^17 + 1573444671952708488898354*v^15 - 33436272015869519964283623*v^13 + 515662928855223007622361362*v^11 - 4801255076498129538398907083*v^9 + 30482250946958342416198355613*v^7 - 59718242154198547271053539212*v^5 + 46727917461784158588609096528*v^3 - 23570856821631134670693758720*v) / 133866105487122094743659520 $$\beta_{18}$$ $$=$$ $$( - 11\!\cdots\!13 \nu^{19} + \cdots - 12\!\cdots\!16 \nu ) / 15\!\cdots\!80$$ (-11960859167027050113*v^19 + 642335086612747595179*v^17 - 23288881103762371835726*v^15 + 472470137549139134712663*v^13 - 7011602925748928564602318*v^11 + 59004444751167922788643867*v^9 - 336327418578130969546997265*v^7 + 94752549256535570434300528*v^5 - 48117981427335398989331712*v^3 - 121239184642480326569108416*v) / 1593644112941929699329280 $$\beta_{19}$$ $$=$$ $$( 61\!\cdots\!19 \nu^{19} + \cdots + 46\!\cdots\!58 \nu ) / 74\!\cdots\!60$$ (616328189080242719*v^19 - 33099972164139458767*v^17 + 1200092182934460383798*v^15 - 24347145911084829541209*v^13 + 361312757944059746258614*v^11 - 3040539929280897646923391*v^9 + 17329416443611148084176815*v^7 - 4882664528589583229120944*v^5 + 2479552929668302557624576*v^3 + 4629521248660253073020858*v) / 74702067794152954656060
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - 11\beta_{3} - \beta_{2}$$ b4 - 11*b3 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{19} + \beta_{16} - 2\beta_{15} + 2\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 17\beta_{6}$$ b19 + b16 - 2*b15 + 2*b14 - b13 + b12 + b11 + b10 + 17*b6 $$\nu^{4}$$ $$=$$ $$2\beta_{11} - 2\beta_{10} - \beta_{8} + 2\beta_{5} + 21\beta_{4} - 201\beta_{3} - 201$$ 2*b11 - 2*b10 - b8 + 2*b5 + 21*b4 - 201*b3 - 201 $$\nu^{5}$$ $$=$$ $$8\beta_{17} + 30\beta_{16} + 80\beta_{14} - 28\beta_{13} + 28\beta_{12} + 317\beta_{6} - 317\beta_1$$ 8*b17 + 30*b16 + 80*b14 - 28*b13 + 28*b12 + 317*b6 - 317*b1 $$\nu^{6}$$ $$=$$ $$72\beta_{13} + 72\beta_{12} + 72\beta_{11} - 72\beta_{10} - 34\beta_{9} + 84\beta_{7} + 449\beta_{2} - 3991$$ 72*b13 + 72*b12 + 72*b11 - 72*b10 - 34*b9 + 84*b7 + 449*b2 - 3991 $$\nu^{7}$$ $$=$$ $$-811\beta_{19} + 296\beta_{18} + 2390\beta_{15} - 695\beta_{11} - 695\beta_{10} - 6257\beta_1$$ -811*b19 + 296*b18 + 2390*b15 - 695*b11 - 695*b10 - 6257*b1 $$\nu^{8}$$ $$=$$ $$1982 \beta_{13} + 1982 \beta_{12} - 875 \beta_{9} + 875 \beta_{8} + 2622 \beta_{7} + \cdots + 9973 \beta_{2}$$ 1982*b13 + 1982*b12 - 875*b9 + 875*b8 + 2622*b7 - 2622*b5 - 9973*b4 + 83261*b3 + 9973*b2 $$\nu^{9}$$ $$=$$ $$- 20928 \beta_{19} + 8336 \beta_{18} - 8336 \beta_{17} - 20928 \beta_{16} + 64180 \beta_{15} + \cdots - 129581 \beta_{6}$$ -20928*b19 + 8336*b18 - 8336*b17 - 20928*b16 + 64180*b15 - 64180*b14 + 16562*b13 - 16562*b12 - 16562*b11 - 16562*b10 - 129581*b6 $$\nu^{10}$$ $$=$$ $$- 49796 \beta_{11} + 49796 \beta_{10} + 20532 \beta_{8} - 72912 \beta_{5} - 227281 \beta_{4} + \cdots + 1804203$$ -49796*b11 + 49796*b10 + 20532*b8 - 72912*b5 - 227281*b4 + 1804203*b3 + 1804203 $$\nu^{11}$$ $$=$$ $$- 213568 \beta_{17} - 525077 \beta_{16} - 1637498 \beta_{14} + 388469 \beta_{13} - 388469 \beta_{12} + \cdots + 2788945 \beta_1$$ -213568*b17 - 525077*b16 - 1637498*b14 + 388469*b13 - 388469*b12 - 2788945*b6 + 2788945*b1 $$\nu^{12}$$ $$=$$ $$- 1204074 \beta_{13} - 1204074 \beta_{12} - 1204074 \beta_{11} + 1204074 \beta_{10} + 465429 \beta_{9} + \cdots + 40206193$$ -1204074*b13 - 1204074*b12 - 1204074*b11 + 1204074*b10 + 465429*b9 - 1910714*b7 - 5263029*b2 + 40206193 $$\nu^{13}$$ $$=$$ $$12937890 \beta_{19} - 5249720 \beta_{18} - 40630072 \beta_{15} + 9067464 \beta_{11} + \cdots + 61797373 \beta_1$$ 12937890*b19 - 5249720*b18 - 40630072*b15 + 9067464*b11 + 9067464*b10 + 61797373*b1 $$\nu^{14}$$ $$=$$ $$- 28634368 \beta_{13} - 28634368 \beta_{12} + 10446758 \beta_{9} - 10446758 \beta_{8} + \cdots - 123053505 \beta_{2}$$ -28634368*b13 - 28634368*b12 + 10446758*b9 - 10446758*b8 - 48370924*b7 + 48370924*b5 + 123053505*b4 - 913941599*b3 - 123053505*b2 $$\nu^{15}$$ $$=$$ $$314988255 \beta_{19} - 126533176 \beta_{18} + 126533176 \beta_{17} + 314988255 \beta_{16} + \cdots + 1398412305 \beta_{6}$$ 314988255*b19 - 126533176*b18 + 126533176*b17 + 314988255*b16 - 991780414*b15 + 991780414*b14 - 211662515*b13 + 211662515*b12 + 211662515*b11 + 211662515*b10 + 1398412305*b6 $$\nu^{16}$$ $$=$$ $$676391382 \beta_{11} - 676391382 \beta_{10} - 234869951 \beta_{8} + 1198431894 \beta_{5} + \cdots - 21061554309$$ 676391382*b11 - 676391382*b10 - 234869951*b8 + 1198431894*b5 + 2893636053*b4 - 21061554309*b3 - 21061554309 $$\nu^{17}$$ $$=$$ $$3020954560 \beta_{17} + 7606844548 \beta_{16} + 23967851036 \beta_{14} - 4951028670 \beta_{13} + \cdots - 32115822125 \beta_1$$ 3020954560*b17 + 7606844548*b16 + 23967851036*b14 - 4951028670*b13 + 4951028670*b12 + 32115822125*b6 - 32115822125*b1 $$\nu^{18}$$ $$=$$ $$15943966460 \beta_{13} + 15943966460 \beta_{12} + 15943966460 \beta_{11} - 15943966460 \beta_{10} + \cdots - 489896458675$$ 15943966460*b13 + 15943966460*b12 + 15943966460*b11 - 15943966460*b10 - 5316167352*b9 + 29279482792*b7 + 68276407697*b2 - 489896458675 $$\nu^{19}$$ $$=$$ $$- 182686621641 \beta_{19} + 71799750416 \beta_{18} + 575555594274 \beta_{15} + \cdots - 745133606321 \beta_1$$ -182686621641*b19 + 71799750416*b18 + 575555594274*b15 - 116112842673*b11 - 116112842673*b10 - 745133606321*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 4.21755 − 2.43500i 3.52149 − 2.03314i 3.14484 − 1.81568i 0.793194 − 0.457951i 0.736336 − 0.425124i −0.736336 + 0.425124i −0.793194 + 0.457951i −3.14484 + 1.81568i −3.52149 + 2.03314i −4.21755 + 2.43500i 4.21755 + 2.43500i 3.52149 + 2.03314i 3.14484 + 1.81568i 0.793194 + 0.457951i 0.736336 + 0.425124i −0.736336 − 0.425124i −0.793194 − 0.457951i −3.14484 − 1.81568i −3.52149 − 2.03314i −4.21755 − 2.43500i
−4.21755 + 2.43500i 6.94145 + 4.00765i 7.85846 13.6113i 7.35984 + 8.41622i −39.0345 −17.6364 + 5.65324i 37.5814i 18.6225 + 32.2551i −51.5340 17.5746i
4.2 −3.52149 + 2.03314i −3.93660 2.27280i 4.26728 7.39115i 9.62212 5.69340i 18.4836 17.0662 + 7.19331i 2.17369i −3.16878 5.48848i −22.3088 + 39.6124i
4.3 −3.14484 + 1.81568i −0.336480 0.194267i 2.59336 4.49183i −10.9591 2.21318i 1.41090 −2.86629 18.2971i 10.2160i −13.4245 23.2520i 38.4831 12.9381i
4.4 −0.793194 + 0.457951i 7.59866 + 4.38709i −3.58056 + 6.20172i −8.53056 7.22700i −8.03628 13.8805 + 12.2610i 13.8861i 24.9931 + 43.2892i 10.0760 + 1.82583i
4.5 −0.736336 + 0.425124i −3.23521 1.86785i −3.63854 + 6.30214i −1.93130 + 11.0123i 3.17627 −3.68367 + 18.1502i 12.9893i −6.52228 11.2969i −3.25949 8.92977i
4.6 0.736336 0.425124i 3.23521 + 1.86785i −3.63854 + 6.30214i 10.5026 + 3.83358i 3.17627 3.68367 18.1502i 12.9893i −6.52228 11.2969i 9.36316 1.64208i
4.7 0.793194 0.457951i −7.59866 4.38709i −3.58056 + 6.20172i −1.99348 11.0012i −8.03628 −13.8805 12.2610i 13.8861i 24.9931 + 43.2892i −6.61922 7.81315i
4.8 3.14484 1.81568i 0.336480 + 0.194267i 2.59336 4.49183i 3.56288 10.5974i 1.41090 2.86629 + 18.2971i 10.2160i −13.4245 23.2520i −8.03683 39.7963i
4.9 3.52149 2.03314i 3.93660 + 2.27280i 4.26728 7.39115i −9.74169 + 5.48630i 18.4836 −17.0662 7.19331i 2.17369i −3.16878 5.48848i −23.1509 + 39.1262i
4.10 4.21755 2.43500i −6.94145 4.00765i 7.85846 13.6113i 3.60874 + 10.5819i −39.0345 17.6364 5.65324i 37.5814i 18.6225 + 32.2551i 40.9870 + 35.8425i
9.1 −4.21755 2.43500i 6.94145 4.00765i 7.85846 + 13.6113i 7.35984 8.41622i −39.0345 −17.6364 5.65324i 37.5814i 18.6225 32.2551i −51.5340 + 17.5746i
9.2 −3.52149 2.03314i −3.93660 + 2.27280i 4.26728 + 7.39115i 9.62212 + 5.69340i 18.4836 17.0662 7.19331i 2.17369i −3.16878 + 5.48848i −22.3088 39.6124i
9.3 −3.14484 1.81568i −0.336480 + 0.194267i 2.59336 + 4.49183i −10.9591 + 2.21318i 1.41090 −2.86629 + 18.2971i 10.2160i −13.4245 + 23.2520i 38.4831 + 12.9381i
9.4 −0.793194 0.457951i 7.59866 4.38709i −3.58056 6.20172i −8.53056 + 7.22700i −8.03628 13.8805 12.2610i 13.8861i 24.9931 43.2892i 10.0760 1.82583i
9.5 −0.736336 0.425124i −3.23521 + 1.86785i −3.63854 6.30214i −1.93130 11.0123i 3.17627 −3.68367 18.1502i 12.9893i −6.52228 + 11.2969i −3.25949 + 8.92977i
9.6 0.736336 + 0.425124i 3.23521 1.86785i −3.63854 6.30214i 10.5026 3.83358i 3.17627 3.68367 + 18.1502i 12.9893i −6.52228 + 11.2969i 9.36316 + 1.64208i
9.7 0.793194 + 0.457951i −7.59866 + 4.38709i −3.58056 6.20172i −1.99348 + 11.0012i −8.03628 −13.8805 + 12.2610i 13.8861i 24.9931 43.2892i −6.61922 + 7.81315i
9.8 3.14484 + 1.81568i 0.336480 0.194267i 2.59336 + 4.49183i 3.56288 + 10.5974i 1.41090 2.86629 18.2971i 10.2160i −13.4245 + 23.2520i −8.03683 + 39.7963i
9.9 3.52149 + 2.03314i 3.93660 2.27280i 4.26728 + 7.39115i −9.74169 5.48630i 18.4836 −17.0662 + 7.19331i 2.17369i −3.16878 + 5.48848i −23.1509 39.1262i
9.10 4.21755 + 2.43500i −6.94145 + 4.00765i 7.85846 + 13.6113i 3.60874 10.5819i −39.0345 17.6364 + 5.65324i 37.5814i 18.6225 32.2551i 40.9870 35.8425i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.j.a 20
5.b even 2 1 inner 35.4.j.a 20
5.c odd 4 2 175.4.e.g 20
7.b odd 2 1 245.4.j.d 20
7.c even 3 1 inner 35.4.j.a 20
7.c even 3 1 245.4.b.e 10
7.d odd 6 1 245.4.b.f 10
7.d odd 6 1 245.4.j.d 20
35.c odd 2 1 245.4.j.d 20
35.i odd 6 1 245.4.b.f 10
35.i odd 6 1 245.4.j.d 20
35.j even 6 1 inner 35.4.j.a 20
35.j even 6 1 245.4.b.e 10
35.k even 12 2 1225.4.a.bq 10
35.l odd 12 2 175.4.e.g 20
35.l odd 12 2 1225.4.a.bp 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.j.a 20 1.a even 1 1 trivial
35.4.j.a 20 5.b even 2 1 inner
35.4.j.a 20 7.c even 3 1 inner
35.4.j.a 20 35.j even 6 1 inner
175.4.e.g 20 5.c odd 4 2
175.4.e.g 20 35.l odd 12 2
245.4.b.e 10 7.c even 3 1
245.4.b.e 10 35.j even 6 1
245.4.b.f 10 7.d odd 6 1
245.4.b.f 10 35.i odd 6 1
245.4.j.d 20 7.b odd 2 1
245.4.j.d 20 7.d odd 6 1
245.4.j.d 20 35.c odd 2 1
245.4.j.d 20 35.i odd 6 1
1225.4.a.bp 10 35.l odd 12 2
1225.4.a.bq 10 35.k even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - 55 T^{18} + \cdots + 9834496$$
$3$ $$T^{20} + \cdots + 46352367616$$
$5$ $$T^{20} + \cdots + 93\!\cdots\!25$$
$7$ $$T^{20} + \cdots + 22\!\cdots\!49$$
$11$ $$(T^{10} + \cdots + 68542503321600)^{2}$$
$13$ $$(T^{10} + 3445 T^{8} + \cdots + 3071819776)^{2}$$
$17$ $$T^{20} + \cdots + 20\!\cdots\!96$$
$19$ $$(T^{10} + \cdots + 80\!\cdots\!56)^{2}$$
$23$ $$T^{20} + \cdots + 94\!\cdots\!81$$
$29$ $$(T^{5} - 65 T^{4} + \cdots + 1210995800)^{4}$$
$31$ $$(T^{10} + \cdots + 13\!\cdots\!24)^{2}$$
$37$ $$T^{20} + \cdots + 38\!\cdots\!00$$
$41$ $$(T^{5} + 306 T^{4} + \cdots + 30026757530)^{4}$$
$43$ $$(T^{10} + \cdots + 26\!\cdots\!64)^{2}$$
$47$ $$T^{20} + \cdots + 14\!\cdots\!00$$
$53$ $$T^{20} + \cdots + 59\!\cdots\!56$$
$59$ $$(T^{10} + \cdots + 13\!\cdots\!00)^{2}$$
$61$ $$(T^{10} + \cdots + 31\!\cdots\!16)^{2}$$
$67$ $$T^{20} + \cdots + 83\!\cdots\!00$$
$71$ $$(T^{5} + 736 T^{4} + \cdots - 445136828288)^{4}$$
$73$ $$T^{20} + \cdots + 87\!\cdots\!36$$
$79$ $$(T^{10} + \cdots + 38\!\cdots\!36)^{2}$$
$83$ $$(T^{10} + \cdots + 22\!\cdots\!04)^{2}$$
$89$ $$(T^{10} + \cdots + 57\!\cdots\!76)^{2}$$
$97$ $$(T^{10} + \cdots + 56\!\cdots\!24)^{2}$$