# Properties

 Label 936.2.m.h Level $936$ Weight $2$ Character orbit 936.m Analytic conductor $7.474$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$936 = 2^{3} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 936.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.47399762919$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} -\beta_{5} q^{7} + ( -\beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})$$ $$q -\beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} -\beta_{5} q^{7} + ( -\beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{2} + \beta_{4} ) q^{10} + ( -\beta_{10} - \beta_{12} - \beta_{14} ) q^{11} + ( \beta_{2} + \beta_{4} + \beta_{15} ) q^{13} + ( -\beta_{1} - \beta_{6} + \beta_{11} ) q^{14} -2 \beta_{2} q^{16} + ( -2 \beta_{1} + \beta_{7} - \beta_{11} ) q^{17} + ( \beta_{13} - \beta_{15} ) q^{19} + ( -2 \beta_{9} - 2 \beta_{12} ) q^{20} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{22} + ( -2 \beta_{6} + \beta_{11} ) q^{23} + ( -\beta_{3} - \beta_{4} ) q^{25} + ( \beta_{1} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{26} + ( -\beta_{8} + 2 \beta_{13} - \beta_{15} ) q^{28} -\beta_{7} q^{29} + ( -\beta_{5} + 2 \beta_{8} ) q^{31} -4 \beta_{9} q^{32} + ( \beta_{5} - \beta_{8} - \beta_{13} - 2 \beta_{15} ) q^{34} + ( -2 \beta_{7} - \beta_{11} ) q^{35} + ( -2 \beta_{13} - 2 \beta_{15} ) q^{37} + ( -\beta_{1} - \beta_{6} - \beta_{11} ) q^{38} + ( 6 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{40} + ( -3 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} ) q^{41} + ( 1 - \beta_{3} + \beta_{4} ) q^{43} + ( \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{14} ) q^{44} + ( \beta_{5} + 3 \beta_{13} + \beta_{15} ) q^{46} + ( -3 \beta_{9} - 4 \beta_{10} + 4 \beta_{12} ) q^{47} + ( -8 + \beta_{3} + \beta_{4} ) q^{49} + ( \beta_{9} + \beta_{12} - \beta_{14} ) q^{50} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{13} ) q^{52} + ( 2 \beta_{7} - \beta_{11} ) q^{53} + ( -5 + 3 \beta_{3} + 3 \beta_{4} ) q^{55} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -\beta_{8} + \beta_{15} ) q^{58} + ( -3 \beta_{10} - 3 \beta_{12} + \beta_{14} ) q^{59} + ( -3 + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{61} + ( -3 \beta_{1} - \beta_{6} + 4 \beta_{7} - \beta_{11} ) q^{62} + ( 4 - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{6} + \beta_{7} + 3 \beta_{9} + \beta_{10} - \beta_{12} ) q^{65} + ( -3 \beta_{13} + \beta_{15} ) q^{67} + ( 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} ) q^{68} + ( \beta_{5} - 2 \beta_{8} - \beta_{13} + 3 \beta_{15} ) q^{70} + ( -5 \beta_{9} - 4 \beta_{10} + 4 \beta_{12} ) q^{71} + ( -2 \beta_{5} + 2 \beta_{8} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{6} - 2 \beta_{11} ) q^{74} + ( 2 \beta_{5} - \beta_{8} + \beta_{15} ) q^{76} + ( \beta_{7} + 3 \beta_{11} ) q^{77} + ( 2 - 4 \beta_{3} - 4 \beta_{4} ) q^{79} + ( -2 \beta_{9} - 6 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} ) q^{80} + ( 9 - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} ) q^{82} + ( \beta_{10} + \beta_{12} + 3 \beta_{14} ) q^{83} + ( -4 \beta_{13} + 2 \beta_{15} ) q^{85} + ( \beta_{9} + \beta_{10} - 3 \beta_{12} - \beta_{14} ) q^{86} + ( -6 + 4 \beta_{3} + 2 \beta_{4} ) q^{88} + ( 7 \beta_{9} - \beta_{10} + \beta_{12} ) q^{89} + ( -5 + 6 \beta_{2} + 5 \beta_{3} + \beta_{4} - 3 \beta_{13} - \beta_{15} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{92} + ( -5 - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} + \beta_{11} ) q^{95} + 2 \beta_{8} q^{97} + ( -\beta_{9} + 8 \beta_{10} - \beta_{12} + \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + O(q^{10})$$ $$16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$310 \nu^{14} - 2816 \nu^{12} - 9222 \nu^{10} + 18376 \nu^{8} + 130292 \nu^{6} - 53906 \nu^{4} + 290050 \nu^{2} + 217000$$$$)/176275$$ $$\beta_{2}$$ $$=$$ $$($$$$11992 \nu^{14} + 9614 \nu^{12} - 190612 \nu^{10} - 554504 \nu^{8} + 1001932 \nu^{6} + 2217000 \nu^{4} + 3496000 \nu^{2} + 5574000$$$$)/881375$$ $$\beta_{3}$$ $$=$$ $$($$$$16948 \nu^{14} + 5486 \nu^{12} - 299188 \nu^{10} - 779596 \nu^{8} + 1911868 \nu^{6} + 4019270 \nu^{4} + 6074000 \nu^{2} + 6351000$$$$)/881375$$ $$\beta_{4}$$ $$=$$ $$($$$$-17344 \nu^{14} - 7298 \nu^{12} + 306684 \nu^{10} + 805078 \nu^{8} - 1876924 \nu^{6} - 3981500 \nu^{4} - 6068600 \nu^{2} - 8342875$$$$)/881375$$ $$\beta_{5}$$ $$=$$ $$($$$$-10661 \nu^{15} - 14207 \nu^{13} + 225956 \nu^{11} + 888177 \nu^{9} - 521116 \nu^{7} - 5727895 \nu^{5} - 13225300 \nu^{3} - 26244000 \nu$$$$)/4406875$$ $$\beta_{6}$$ $$=$$ $$($$$$-25832 \nu^{14} - 21274 \nu^{12} + 450392 \nu^{10} + 1381764 \nu^{8} - 2270262 \nu^{6} - 7392230 \nu^{4} - 12703750 \nu^{2} - 13499250$$$$)/881375$$ $$\beta_{7}$$ $$=$$ $$($$$$-28806 \nu^{14} - 29832 \nu^{12} + 448156 \nu^{10} + 1496702 \nu^{8} - 2047066 \nu^{6} - 6175530 \nu^{4} - 12354350 \nu^{2} - 13465750$$$$)/881375$$ $$\beta_{8}$$ $$=$$ $$($$$$15392 \nu^{15} + 52754 \nu^{13} - 145257 \nu^{11} - 987344 \nu^{9} - 868348 \nu^{7} + 1700315 \nu^{5} + 6288975 \nu^{3} + 15325500 \nu$$$$)/4406875$$ $$\beta_{9}$$ $$=$$ $$($$$$-1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu$$$$)/400625$$ $$\beta_{10}$$ $$=$$ $$($$$$-22194 \nu^{15} - 11523 \nu^{13} + 339409 \nu^{11} + 1024903 \nu^{9} - 1669899 \nu^{7} - 3769000 \nu^{5} - 11127550 \nu^{3} - 9814875 \nu$$$$)/4406875$$ $$\beta_{11}$$ $$=$$ $$($$$$-44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000$$$$)/881375$$ $$\beta_{12}$$ $$=$$ $$($$$$-46447 \nu^{15} - 41694 \nu^{13} + 802127 \nu^{11} + 2508384 \nu^{9} - 4035597 \nu^{7} - 13232745 \nu^{5} - 19604750 \nu^{3} - 21119125 \nu$$$$)/4406875$$ $$\beta_{13}$$ $$=$$ $$($$$$78923 \nu^{15} + 34611 \nu^{13} - 1339113 \nu^{11} - 3553821 \nu^{9} + 8128668 \nu^{7} + 15701920 \nu^{5} + 26400825 \nu^{3} + 32586750 \nu$$$$)/4406875$$ $$\beta_{14}$$ $$=$$ $$($$$$-108813 \nu^{15} - 85331 \nu^{13} + 1829673 \nu^{11} + 5646091 \nu^{9} - 9386828 \nu^{7} - 27713410 \nu^{5} - 49911275 \nu^{3} - 50144500 \nu$$$$)/4406875$$ $$\beta_{15}$$ $$=$$ $$($$$$-128244 \nu^{15} - 47208 \nu^{13} + 2180589 \nu^{11} + 5658588 \nu^{9} - 13459254 \nu^{7} - 25923185 \nu^{5} - 43314975 \nu^{3} - 53041500 \nu$$$$)/4406875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - \beta_{13} + 2 \beta_{12} - 2 \beta_{9} - \beta_{5}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{14} - \beta_{13} + 5 \beta_{12} + 7 \beta_{10} + 5 \beta_{9} + \beta_{8} + \beta_{5}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 8 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 8$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{15} - 12 \beta_{13} - 3 \beta_{9}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$11 \beta_{11} - 11 \beta_{7} - 6 \beta_{6} + 11 \beta_{3} - 15 \beta_{2} + 5 \beta_{1} + 30$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{15} - 55 \beta_{14} - 4 \beta_{13} + 127 \beta_{12} + 55 \beta_{10} - 55 \beta_{9} - 9 \beta_{8} - 4 \beta_{5}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$32 \beta_{11} - 52 \beta_{6} + 53 \beta_{4} + 18 \beta_{3} + 53 \beta_{2} + 32 \beta_{1} + 17$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-137 \beta_{15} - 57 \beta_{14} - 224 \beta_{13} - 57 \beta_{12} + 247 \beta_{10} + 247 \beta_{9} + 87 \beta_{8} + 50 \beta_{5}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$21 \beta_{11} - 36 \beta_{7} + 275 \beta_{4} + 275 \beta_{3} + 607$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-384 \beta_{15} - 341 \beta_{14} - 623 \beta_{13} + 1421 \beta_{12} - 341 \beta_{10} - 1421 \beta_{9} - 239 \beta_{8} - 145 \beta_{5}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$752 \beta_{11} - 462 \beta_{7} - 752 \beta_{6} - 110 \beta_{4} - 177 \beta_{2} + 462 \beta_{1} + 110$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-395 \beta_{15} - 2315 \beta_{14} - 642 \beta_{13} + 2315 \beta_{12} + 5199 \beta_{10} + 2315 \beta_{9} + 1037 \beta_{8} + 642 \beta_{5}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$639 \beta_{7} - 639 \beta_{6} + 5431 \beta_{4} + 3360 \beta_{3} + 3360 \beta_{2} + 400 \beta_{1} + 5431$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-4399 \beta_{15} - 7109 \beta_{13} + 2475 \beta_{12} - 2475 \beta_{10} - 3996 \beta_{9}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$469$$ $$703$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −0.556839 − 1.81878i 1.90184 − 0.0324487i 1.90184 + 0.0324487i −0.556839 + 1.81878i 0.752864 − 0.902863i 0.0783900 + 1.17295i 0.0783900 − 1.17295i 0.752864 + 0.902863i −0.752864 − 0.902863i −0.0783900 + 1.17295i −0.0783900 − 1.17295i −0.752864 + 0.902863i 0.556839 − 1.81878i −1.90184 − 0.0324487i −1.90184 + 0.0324487i 0.556839 + 1.81878i
−1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.2 −1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.3 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.4 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.5 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.6 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.7 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.8 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.9 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.10 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.11 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.12 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.13 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.14 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.15 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
181.16 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.16 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
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.m.h 16
3.b odd 2 1 inner 936.2.m.h 16
4.b odd 2 1 3744.2.m.h 16
8.b even 2 1 inner 936.2.m.h 16
8.d odd 2 1 3744.2.m.h 16
12.b even 2 1 3744.2.m.h 16
13.b even 2 1 inner 936.2.m.h 16
24.f even 2 1 3744.2.m.h 16
24.h odd 2 1 inner 936.2.m.h 16
39.d odd 2 1 inner 936.2.m.h 16
52.b odd 2 1 3744.2.m.h 16
104.e even 2 1 inner 936.2.m.h 16
104.h odd 2 1 3744.2.m.h 16
156.h even 2 1 3744.2.m.h 16
312.b odd 2 1 inner 936.2.m.h 16
312.h even 2 1 3744.2.m.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 1.a even 1 1 trivial
936.2.m.h 16 3.b odd 2 1 inner
936.2.m.h 16 8.b even 2 1 inner
936.2.m.h 16 13.b even 2 1 inner
936.2.m.h 16 24.h odd 2 1 inner
936.2.m.h 16 39.d odd 2 1 inner
936.2.m.h 16 104.e even 2 1 inner
936.2.m.h 16 312.b odd 2 1 inner
3744.2.m.h 16 4.b odd 2 1
3744.2.m.h 16 8.d odd 2 1
3744.2.m.h 16 12.b even 2 1
3744.2.m.h 16 24.f even 2 1
3744.2.m.h 16 52.b odd 2 1
3744.2.m.h 16 104.h odd 2 1
3744.2.m.h 16 156.h even 2 1
3744.2.m.h 16 312.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 10 T_{5}^{2} + 20$$ acting on $$S_{2}^{\mathrm{new}}(936, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 20 - 10 T^{2} + T^{4} )^{4}$$
$7$ $$( 220 + 30 T^{2} + T^{4} )^{4}$$
$11$ $$( 20 - 20 T^{2} + T^{4} )^{4}$$
$13$ $$( 28561 - 2028 T^{2} + 54 T^{4} - 12 T^{6} + T^{8} )^{2}$$
$17$ $$( 880 - 60 T^{2} + T^{4} )^{4}$$
$19$ $$( 44 - 34 T^{2} + T^{4} )^{4}$$
$23$ $$( 880 - 80 T^{2} + T^{4} )^{4}$$
$29$ $$( 176 + 28 T^{2} + T^{4} )^{4}$$
$31$ $$( 5500 + 150 T^{2} + T^{4} )^{4}$$
$37$ $$( 704 - 104 T^{2} + T^{4} )^{4}$$
$41$ $$( 324 + 126 T^{2} + T^{4} )^{4}$$
$43$ $$( 80 + 20 T^{2} + T^{4} )^{4}$$
$47$ $$( 1444 + 84 T^{2} + T^{4} )^{4}$$
$53$ $$( 176 + 128 T^{2} + T^{4} )^{4}$$
$59$ $$( 500 - 100 T^{2} + T^{4} )^{4}$$
$61$ $$( 6480 + 180 T^{2} + T^{4} )^{4}$$
$67$ $$( 5324 - 154 T^{2} + T^{4} )^{4}$$
$71$ $$( 484 + 116 T^{2} + T^{4} )^{4}$$
$73$ $$( 3520 + 200 T^{2} + T^{4} )^{4}$$
$79$ $$( -76 - 4 T + T^{2} )^{8}$$
$83$ $$( 500 - 100 T^{2} + T^{4} )^{4}$$
$89$ $$( 12100 + 230 T^{2} + T^{4} )^{4}$$
$97$ $$( 3520 + 160 T^{2} + T^{4} )^{4}$$