# Properties

 Label 189.4.e.e Level $189$ Weight $4$ Character orbit 189.e Analytic conductor $11.151$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(109,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.e (of order $$3$$, 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: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 40x^{10} + 1147x^{8} + 15564x^{6} + 154089x^{4} + 578934x^{2} + 1633284$$ x^12 + 40*x^10 + 1147*x^8 + 15564*x^6 + 154089*x^4 + 578934*x^2 + 1633284 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{4} + \cdots - 6) q^{4}+ \cdots + (\beta_{10} - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{8}+O(q^{10})$$ q - b3 * q^2 + (-b11 + b7 + b4 - 6*b2 - 6) * q^4 - b8 * q^5 + (2*b11 + b7 - b6 - 5*b2 - 5) * q^7 + (b10 - b9 - b8 + b5 + b3 + b1) * q^8 $$q - \beta_{3} q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{4} + \cdots - 6) q^{4}+ \cdots + ( - 9 \beta_{10} - 67 \beta_{9} + \cdots + 256 \beta_1) q^{98}+O(q^{100})$$ q - b3 * q^2 + (-b11 + b7 + b4 - 6*b2 - 6) * q^4 - b8 * q^5 + (2*b11 + b7 - b6 - 5*b2 - 5) * q^7 + (b10 - b9 - b8 + b5 + b3 + b1) * q^8 + (b11 + 2*b7 - 3*b6 - 4*b4 + 5*b2 + 2) * q^10 + (-b9 + 2*b5 + 4*b1) * q^11 + (-5*b11 - 3*b6 - 5*b4 + 6) * q^13 + (-2*b10 + 3*b9 + b8 + b5 - b3 + 4*b1) * q^14 + (-b11 - 7*b7 + 3*b6 - 3*b4 - 24*b2 + 3) * q^16 + (5*b9 - 4*b1) * q^17 + (11*b11 - 11*b7 + 11*b6 - 11*b4 - 3*b2 + 11) * q^19 + (2*b10 - b9 - b8 + 2*b5 - 8*b3 - 8*b1) * q^20 + (-11*b11 + 12*b6 - 11*b4 - 59) * q^22 + (-4*b10 - 6*b8 + 10*b3) * q^23 + (12*b11 - 14*b7 + 2*b6 - 10*b4 + 15*b2 + 17) * q^25 + (7*b10 + 8*b8 - 3*b3) * q^26 + (-12*b11 + 15*b7 - b6 - 7*b4 - 61*b2 - 68) * q^28 + (-2*b10 - 9*b9 - 9*b8 - 2*b5 - 2*b3 - 2*b1) * q^29 + (13*b11 - 2*b7 - 11*b6 - 24*b4 + 26*b2 + 15) * q^31 + (-13*b9 + b5 + 23*b1) * q^32 + (-15*b11 - 24*b6 - 15*b4 + 57) * q^34 + (-2*b10 + 10*b9 + 15*b8 - 6*b5 + 6*b3 + 32*b1) * q^35 + (-13*b11 + 11*b7 - 12*b6 + 12*b4 - 197*b2 - 12) * q^37 + (11*b9 - 11*b5 - 30*b1) * q^38 + (32*b11 - 6*b7 + 19*b6 - 19*b4 - 57*b2 + 19) * q^40 + (-4*b10 - 15*b9 - 15*b8 - 4*b5 - 52*b3 - 52*b1) * q^41 + (8*b11 + 10*b6 + 8*b4 - 255) * q^43 + (18*b10 + 7*b8 + 108*b3) * q^44 + (-20*b11 + 66*b7 - 46*b6 - 26*b4 + 150*b2 + 104) * q^46 + (-14*b10 + 29*b8 - 26*b3) * q^47 + (-47*b11 + 15*b7 - 36*b6 + 35*b4 - 159*b2 - 68) * q^49 + (-14*b10 + 8*b9 + 8*b8 - 14*b5 - 51*b3 - 51*b1) * q^50 + (36*b11 - 69*b7 + 33*b6 - 3*b4 + 25*b2 + 58) * q^52 + (8*b9 - 12*b5 + 26*b1) * q^53 + (22*b11 + 46*b6 + 22*b4 + 60) * q^55 + (9*b10 + 18*b9 - 8*b8 - b5 + 85*b3 + 73*b1) * q^56 + (30*b11 + 52*b7 - 11*b6 + 11*b4 + 7*b2 - 11) * q^58 + (-23*b9 + 26*b5 - 40*b1) * q^59 + (-64*b11 - 62*b7 - b6 + b4 + 126*b2 - 1) * q^61 + (-2*b10 + 35*b9 + 35*b8 - 2*b5 - 65*b3 - 65*b1) * q^62 + (69*b6 - 131) * q^64 + (-16*b10 - 13*b8 + 94*b3) * q^65 + (19*b11 - 73*b7 + 54*b6 + 35*b4 + 342*b2 + 396) * q^67 + (6*b10 - b8 - 76*b3) * q^68 + (-15*b11 - 4*b7 + 11*b6 + 70*b4 - b2 - 358) * q^70 + (18*b10 - 51*b9 - 51*b8 + 18*b5 - 12*b3 - 12*b1) * q^71 + (55*b11 - 35*b7 - 20*b6 - 75*b4 + 443*b2 + 423) * q^73 + (-14*b9 + 11*b5 + 233*b1) * q^74 + (44*b11 - 8*b6 + 44*b4 + 546) * q^76 + (16*b10 - 17*b9 - 36*b8 + 20*b5 + 64*b3 + 206*b1) * q^77 + (-125*b11 + 3*b7 - 64*b6 + 64*b4 + 101*b2 - 64) * q^79 + (37*b9 + 10*b5 - 64*b1) * q^80 + (146*b7 - 73*b6 + 73*b4 - 673*b2 - 73) * q^82 + (14*b10 - 28*b9 - 28*b8 + 14*b5 - 318*b3 - 318*b1) * q^83 + (22*b11 - 34*b6 + 22*b4 - 604) * q^85 + (-6*b10 - 18*b8 + 271*b3) * q^86 + (79*b11 - 138*b7 + 59*b6 - 20*b4 + 999*b2 + 1058) * q^88 + (-26*b10 - 51*b8 - 174*b3) * q^89 + (60*b11 + 30*b7 + 61*b6 + 28*b4 + 522*b2 - 255) * q^91 + (34*b10 + 24*b9 + 24*b8 + 34*b5 - 10*b3 - 10*b1) * q^92 + (-181*b11 + 192*b7 - 11*b6 + 170*b4 - 579*b2 - 590) * q^94 + (80*b9 - 22*b5 + 88*b1) * q^95 + (72*b11 - 82*b6 + 72*b4 - 117) * q^97 + (-9*b10 - 67*b9 + b8 + 15*b5 + 13*b3 + 256*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 32 q^{4} - 26 q^{7}+O(q^{10})$$ 12 * q - 32 * q^4 - 26 * q^7 $$12 q - 32 q^{4} - 26 q^{7} + 20 q^{10} + 104 q^{13} + 148 q^{16} + 62 q^{19} - 712 q^{22} + 46 q^{25} - 348 q^{28} + 82 q^{31} + 840 q^{34} + 1132 q^{37} + 444 q^{40} - 3132 q^{43} + 888 q^{46} + 366 q^{49} + 72 q^{52} + 448 q^{55} - 4 q^{58} - 886 q^{61} - 1848 q^{64} + 2084 q^{67} - 4460 q^{70} + 2398 q^{73} + 6408 q^{76} - 984 q^{79} + 3892 q^{82} - 7200 q^{85} + 5796 q^{88} - 6492 q^{91} - 2772 q^{94} - 1364 q^{97}+O(q^{100})$$ 12 * q - 32 * q^4 - 26 * q^7 + 20 * q^10 + 104 * q^13 + 148 * q^16 + 62 * q^19 - 712 * q^22 + 46 * q^25 - 348 * q^28 + 82 * q^31 + 840 * q^34 + 1132 * q^37 + 444 * q^40 - 3132 * q^43 + 888 * q^46 + 366 * q^49 + 72 * q^52 + 448 * q^55 - 4 * q^58 - 886 * q^61 - 1848 * q^64 + 2084 * q^67 - 4460 * q^70 + 2398 * q^73 + 6408 * q^76 - 984 * q^79 + 3892 * q^82 - 7200 * q^85 + 5796 * q^88 - 6492 * q^91 - 2772 * q^94 - 1364 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 40x^{10} + 1147x^{8} + 15564x^{6} + 154089x^{4} + 578934x^{2} + 1633284$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 173197 \nu^{10} + 6246280 \nu^{8} + 179112079 \nu^{6} + 2135188674 \nu^{4} + \cdots + 24978356640 ) / 65426239998$$ (173197*v^10 + 6246280*v^8 + 179112079*v^6 + 2135188674*v^4 + 24062075973*v^2 + 24978356640) / 65426239998 $$\beta_{3}$$ $$=$$ $$( - 173197 \nu^{11} - 6246280 \nu^{9} - 179112079 \nu^{7} - 2135188674 \nu^{5} + \cdots - 90404596638 \nu ) / 65426239998$$ (-173197*v^11 - 6246280*v^9 - 179112079*v^7 - 2135188674*v^5 - 24062075973*v^3 - 90404596638*v) / 65426239998 $$\beta_{4}$$ $$=$$ $$( 108403 \nu^{10} + 7516849 \nu^{8} + 245835571 \nu^{6} + 4262422452 \nu^{4} + \cdots + 91892207808 ) / 14539164444$$ (108403*v^10 + 7516849*v^8 + 245835571*v^6 + 4262422452*v^4 + 38255600898*v^2 + 91892207808) / 14539164444 $$\beta_{5}$$ $$=$$ $$( - 12280 \nu^{11} - 352129 \nu^{9} - 6257731 \nu^{7} - 47305323 \nu^{5} + \cdots - 10924950600 \nu ) / 1842992676$$ (-12280*v^11 - 352129*v^9 - 6257731*v^7 - 47305323*v^5 - 177732738*v^3 - 10924950600*v) / 1842992676 $$\beta_{6}$$ $$=$$ $$( -1600\nu^{10} - 45880\nu^{8} - 1315609\nu^{6} - 6163560\nu^{4} - 23157360\nu^{2} + 1332538101 ) / 153582723$$ (-1600*v^10 - 45880*v^8 - 1315609*v^6 - 6163560*v^4 - 23157360*v^2 + 1332538101) / 153582723 $$\beta_{7}$$ $$=$$ $$( - 2443778 \nu^{10} - 113590439 \nu^{8} - 2984596505 \nu^{6} - 37517061303 \nu^{4} + \cdots + 35784531648 ) / 130852479996$$ (-2443778*v^10 - 113590439*v^8 - 2984596505*v^6 - 37517061303*v^4 - 223032383550*v^2 + 35784531648) / 130852479996 $$\beta_{8}$$ $$=$$ $$( - 2193901 \nu^{11} - 105145360 \nu^{9} - 3015043198 \nu^{7} - 45990857349 \nu^{5} + \cdots - 1521805596156 \nu ) / 130852479996$$ (-2193901*v^11 - 105145360*v^9 - 3015043198*v^7 - 45990857349*v^5 - 405043584426*v^3 - 1521805596156*v) / 130852479996 $$\beta_{9}$$ $$=$$ $$( - 31480 \nu^{11} - 902689 \nu^{9} - 22045039 \nu^{7} - 121268043 \nu^{5} + \cdots + 12437477316 \nu ) / 1842992676$$ (-31480*v^11 - 902689*v^9 - 22045039*v^7 - 121268043*v^5 - 455621058*v^3 + 12437477316*v) / 1842992676 $$\beta_{10}$$ $$=$$ $$( 777199 \nu^{11} + 22712800 \nu^{9} + 651289540 \nu^{7} + 7118068149 \nu^{5} + \cdots + 328730303880 \nu ) / 43617493332$$ (777199*v^11 + 22712800*v^9 + 651289540*v^7 + 7118068149*v^5 + 87494815980*v^3 + 328730303880*v) / 43617493332 $$\beta_{11}$$ $$=$$ $$( - 235927 \nu^{10} - 8654518 \nu^{8} - 222205510 \nu^{6} - 2556628047 \nu^{4} + \cdots - 52512120288 ) / 6231070476$$ (-235927*v^10 - 8654518*v^8 - 222205510*v^6 - 2556628047*v^4 - 19137502476*v^2 - 52512120288) / 6231070476
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} - \beta_{7} + \beta_{6} - \beta_{4} + 14\beta_{2} + 1$$ b11 - b7 + b6 - b4 + 14*b2 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - 17\beta_{3} - 17\beta_1$$ -b10 + b9 + b8 - b5 - 17*b3 - 17*b1 $$\nu^{4}$$ $$=$$ $$-27\beta_{11} + 31\beta_{7} - 4\beta_{6} + 23\beta_{4} - 248\beta_{2} - 252$$ -27*b11 + 31*b7 - 4*b6 + 23*b4 - 248*b2 - 252 $$\nu^{5}$$ $$=$$ $$31\beta_{10} - 19\beta_{8} + 329\beta_{3}$$ 31*b10 - 19*b8 + 329*b3 $$\nu^{6}$$ $$=$$ $$160\beta_{11} - 467\beta_{6} + 160\beta_{4} + 4389$$ 160*b11 - 467*b6 + 160*b4 + 4389 $$\nu^{7}$$ $$=$$ $$-307\beta_{9} + 787\beta_{5} + 6737\beta_1$$ -307*b9 + 787*b5 + 6737*b1 $$\nu^{8}$$ $$=$$ $$9539\beta_{11} - 18715\beta_{7} + 14127\beta_{6} - 14127\beta_{4} + 99788\beta_{2} + 14127$$ 9539*b11 - 18715*b7 + 14127*b6 - 14127*b4 + 99788*b2 + 14127 $$\nu^{9}$$ $$=$$ $$-18715\beta_{10} + 4951\beta_{9} + 4951\beta_{8} - 18715\beta_{5} - 142169\beta_{3} - 142169\beta_1$$ -18715*b10 + 4951*b9 + 4951*b8 - 18715*b5 - 142169*b3 - 142169*b1 $$\nu^{10}$$ $$=$$ $$-315555\beta_{11} + 431707\beta_{7} - 116152\beta_{6} + 199403\beta_{4} - 2108696\beta_{2} - 2224848$$ -315555*b11 + 431707*b7 - 116152*b6 + 199403*b4 - 2108696*b2 - 2224848 $$\nu^{11}$$ $$=$$ $$431707\beta_{10} - 83251\beta_{8} + 3055361\beta_{3}$$ 431707*b10 - 83251*b8 + 3055361*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 2.35377 − 4.07684i 1.84379 − 3.19353i 1.02968 − 1.78345i −1.02968 + 1.78345i −1.84379 + 3.19353i −2.35377 + 4.07684i 2.35377 + 4.07684i 1.84379 + 3.19353i 1.02968 + 1.78345i −1.02968 − 1.78345i −1.84379 − 3.19353i −2.35377 − 4.07684i
−2.35377 4.07684i 0 −7.08042 + 12.2637i −0.415227 0.719194i 0 16.4314 + 8.54463i 29.0024 0 −1.95469 + 3.38563i
109.2 −1.84379 3.19353i 0 −2.79910 + 4.84819i 5.93594 + 10.2814i 0 −15.1628 10.6344i −8.85681 0 21.8892 37.9133i
109.3 −1.02968 1.78345i 0 1.87953 3.25543i −7.25204 12.5609i 0 −7.76857 + 16.8122i −24.2161 0 −14.9345 + 25.8674i
109.4 1.02968 + 1.78345i 0 1.87953 3.25543i 7.25204 + 12.5609i 0 −7.76857 + 16.8122i 24.2161 0 −14.9345 + 25.8674i
109.5 1.84379 + 3.19353i 0 −2.79910 + 4.84819i −5.93594 10.2814i 0 −15.1628 10.6344i 8.85681 0 21.8892 37.9133i
109.6 2.35377 + 4.07684i 0 −7.08042 + 12.2637i 0.415227 + 0.719194i 0 16.4314 + 8.54463i −29.0024 0 −1.95469 + 3.38563i
163.1 −2.35377 + 4.07684i 0 −7.08042 12.2637i −0.415227 + 0.719194i 0 16.4314 8.54463i 29.0024 0 −1.95469 3.38563i
163.2 −1.84379 + 3.19353i 0 −2.79910 4.84819i 5.93594 10.2814i 0 −15.1628 + 10.6344i −8.85681 0 21.8892 + 37.9133i
163.3 −1.02968 + 1.78345i 0 1.87953 + 3.25543i −7.25204 + 12.5609i 0 −7.76857 16.8122i −24.2161 0 −14.9345 25.8674i
163.4 1.02968 1.78345i 0 1.87953 + 3.25543i 7.25204 12.5609i 0 −7.76857 16.8122i 24.2161 0 −14.9345 25.8674i
163.5 1.84379 3.19353i 0 −2.79910 4.84819i −5.93594 + 10.2814i 0 −15.1628 + 10.6344i 8.85681 0 21.8892 + 37.9133i
163.6 2.35377 4.07684i 0 −7.08042 12.2637i 0.415227 0.719194i 0 16.4314 8.54463i −29.0024 0 −1.95469 3.38563i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.e.e 12
3.b odd 2 1 inner 189.4.e.e 12
7.c even 3 1 inner 189.4.e.e 12
7.c even 3 1 1323.4.a.bd 6
7.d odd 6 1 1323.4.a.be 6
21.g even 6 1 1323.4.a.be 6
21.h odd 6 1 inner 189.4.e.e 12
21.h odd 6 1 1323.4.a.bd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.e 12 1.a even 1 1 trivial
189.4.e.e 12 3.b odd 2 1 inner
189.4.e.e 12 7.c even 3 1 inner
189.4.e.e 12 21.h odd 6 1 inner
1323.4.a.bd 6 7.c even 3 1
1323.4.a.bd 6 21.h odd 6 1
1323.4.a.be 6 7.d odd 6 1
1323.4.a.be 6 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(189, [\chi])$$:

 $$T_{2}^{12} + 40T_{2}^{10} + 1147T_{2}^{8} + 15564T_{2}^{6} + 154089T_{2}^{4} + 578934T_{2}^{2} + 1633284$$ T2^12 + 40*T2^10 + 1147*T2^8 + 15564*T2^6 + 154089*T2^4 + 578934*T2^2 + 1633284 $$T_{13}^{3} - 26T_{13}^{2} - 3211T_{13} - 35818$$ T13^3 - 26*T13^2 - 3211*T13 - 35818

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 40 T^{10} + \cdots + 1633284$$
$3$ $$T^{12}$$
$5$ $$T^{12} + \cdots + 418120704$$
$7$ $$(T^{6} + 13 T^{5} + \cdots + 40353607)^{2}$$
$11$ $$T^{12} + \cdots + 41\!\cdots\!44$$
$13$ $$(T^{3} - 26 T^{2} + \cdots - 35818)^{4}$$
$17$ $$T^{12} + \cdots + 23\!\cdots\!64$$
$19$ $$(T^{6} - 31 T^{5} + \cdots + 5047249936)^{2}$$
$23$ $$T^{12} + \cdots + 29\!\cdots\!04$$
$29$ $$(T^{6} + \cdots - 1224041183712)^{2}$$
$31$ $$(T^{6} + \cdots + 4861932390441)^{2}$$
$37$ $$(T^{6} + \cdots + 18026291164644)^{2}$$
$41$ $$(T^{6} + \cdots - 69555350983392)^{2}$$
$43$ $$(T^{3} + 783 T^{2} + \cdots + 15519697)^{4}$$
$47$ $$T^{12} + \cdots + 30\!\cdots\!44$$
$53$ $$T^{12} + \cdots + 61\!\cdots\!24$$
$59$ $$T^{12} + \cdots + 72\!\cdots\!84$$
$61$ $$(T^{6} + \cdots + 48\!\cdots\!69)^{2}$$
$67$ $$(T^{6} + \cdots + 32\!\cdots\!76)^{2}$$
$71$ $$(T^{6} + \cdots - 17\!\cdots\!28)^{2}$$
$73$ $$(T^{6} + \cdots + 19\!\cdots\!44)^{2}$$
$79$ $$(T^{6} + \cdots + 37\!\cdots\!64)^{2}$$
$83$ $$(T^{6} + \cdots - 28\!\cdots\!92)^{2}$$
$89$ $$T^{12} + \cdots + 26\!\cdots\!24$$
$97$ $$(T^{3} + 341 T^{2} + \cdots - 1582590381)^{4}$$