# Properties

 Label 380.3.g.c Level $380$ Weight $3$ Character orbit 380.g Analytic conductor $10.354$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 380.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3542500457$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241$$ x^12 + 20*x^10 + 44*x^8 - 270*x^6 + 36676*x^4 - 71664*x^2 + 687241 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{19}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{3} + (\beta_{3} + 1) q^{5} + \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 6) q^{9}+O(q^{10})$$ q - b5 * q^3 + (b3 + 1) * q^5 + b8 * q^7 + (-b4 + b3 - b2 + 6) * q^9 $$q - \beta_{5} q^{3} + (\beta_{3} + 1) q^{5} + \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 6) q^{9} + ( - \beta_{4} + \beta_{3} - 2) q^{11} + (\beta_{9} + \beta_{5}) q^{13} + (\beta_{9} - \beta_{6} - 2 \beta_{5} + \beta_1) q^{15} + ( - \beta_{8} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{2} - \beta_1 + 6) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{21} + 5 \beta_{8} q^{23} + ( - 2 \beta_{10} + \beta_{8} + 2 \beta_{3} + \beta_{2} - 6) q^{25} + ( - \beta_{11} - 6 \beta_{5}) q^{27} + (\beta_{7} + \beta_{6} + \beta_1) q^{29} + (\beta_{9} + 2 \beta_{7}) q^{31} + \beta_{9} q^{33} + (\beta_{10} + 2 \beta_{8} - \beta_{3} + 2 \beta_{2} - 2) q^{35} + ( - \beta_{11} - \beta_{5}) q^{37} + ( - 4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 21) q^{39} + (\beta_{9} - 2 \beta_{6} - 2 \beta_1) q^{41} + ( - 2 \beta_{10} - 5 \beta_{4} - 5 \beta_{3}) q^{43} + ( - 3 \beta_{10} + 9 \beta_{8} + 5 \beta_{4} + 5 \beta_{3} - \beta_{2} + 18) q^{45} + (6 \beta_{10} - 4 \beta_{8} - 6 \beta_{4} - 6 \beta_{3}) q^{47} + (3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 18) q^{49} + ( - \beta_{9} + \beta_{7} + 3 \beta_{6} - 3 \beta_1) q^{51} + ( - \beta_{9} - 7 \beta_{5}) q^{53} + ( - 2 \beta_{10} + \beta_{8} - 2 \beta_{3} + \beta_{2} + 15) q^{55} + (\beta_{11} + 4 \beta_{10} + \beta_{9} - 7 \beta_{8} + \beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{57} + (3 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} + \beta_1) q^{59} + ( - 7 \beta_{4} + 7 \beta_{3} - 50) q^{61} + ( - 2 \beta_{10} + 12 \beta_{8} + 8 \beta_{4} + 8 \beta_{3}) q^{63} + (\beta_{11} - 2 \beta_{9} + \beta_{7} + 4 \beta_{6} - 4 \beta_{5}) q^{65} + ( - 2 \beta_{11} - 5 \beta_{5}) q^{67} + ( - 5 \beta_{7} - 5 \beta_{6} + 5 \beta_1) q^{69} + ( - \beta_{9} - 2 \beta_{7} - 4 \beta_1) q^{71} + (4 \beta_{10} - \beta_{8} + 5 \beta_{4} + 5 \beta_{3}) q^{73} + (\beta_{11} + 3 \beta_{9} + \beta_{7} - \beta_{6} + 11 \beta_{5} + 5 \beta_1) q^{75} + (2 \beta_{10} - \beta_{4} - \beta_{3}) q^{77} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_1) q^{79} + ( - 12 \beta_{2} + 37) q^{81} + ( - 8 \beta_{10} + 4 \beta_{8} + 4 \beta_{4} + 4 \beta_{3}) q^{83} + (\beta_{10} - 3 \beta_{8} - \beta_{3} - 3 \beta_{2} + 33) q^{85} + ( - 6 \beta_{10} - 7 \beta_{8} + 2 \beta_{4} + 2 \beta_{3}) q^{87} + ( - 3 \beta_{9} - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_1) q^{89} + ( - \beta_{9} - \beta_{7} + \beta_{6} - 5 \beta_1) q^{91} + ( - 2 \beta_{10} - 24 \beta_{8} + 3 \beta_{4} + 3 \beta_{3}) q^{93} + ( - \beta_{11} + \beta_{10} + \beta_{9} - 8 \beta_{8} - \beta_{7} + 2 \beta_{6} - 9 \beta_{5} + \cdots + 11) q^{95}+ \cdots + (4 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 12) q^{99}+O(q^{100})$$ q - b5 * q^3 + (b3 + 1) * q^5 + b8 * q^7 + (-b4 + b3 - b2 + 6) * q^9 + (-b4 + b3 - 2) * q^11 + (b9 + b5) * q^13 + (b9 - b6 - 2*b5 + b1) * q^15 + (-b8 - b4 - b3) * q^17 + (b2 - b1 + 6) * q^19 + (-b7 - b6 + b1) * q^21 + 5*b8 * q^23 + (-2*b10 + b8 + 2*b3 + b2 - 6) * q^25 + (-b11 - 6*b5) * q^27 + (b7 + b6 + b1) * q^29 + (b9 + 2*b7) * q^31 + b9 * q^33 + (b10 + 2*b8 - b3 + 2*b2 - 2) * q^35 + (-b11 - b5) * q^37 + (-4*b4 + 4*b3 + 3*b2 - 21) * q^39 + (b9 - 2*b6 - 2*b1) * q^41 + (-2*b10 - 5*b4 - 5*b3) * q^43 + (-3*b10 + 9*b8 + 5*b4 + 5*b3 - b2 + 18) * q^45 + (6*b10 - 4*b8 - 6*b4 - 6*b3) * q^47 + (3*b4 - 3*b3 + b2 + 18) * q^49 + (-b9 + b7 + 3*b6 - 3*b1) * q^51 + (-b9 - 7*b5) * q^53 + (-2*b10 + b8 - 2*b3 + b2 + 15) * q^55 + (b11 + 4*b10 + b9 - 7*b8 + b5 - 5*b4 - 5*b3) * q^57 + (3*b9 + 3*b7 - 3*b6 + b1) * q^59 + (-7*b4 + 7*b3 - 50) * q^61 + (-2*b10 + 12*b8 + 8*b4 + 8*b3) * q^63 + (b11 - 2*b9 + b7 + 4*b6 - 4*b5) * q^65 + (-2*b11 - 5*b5) * q^67 + (-5*b7 - 5*b6 + 5*b1) * q^69 + (-b9 - 2*b7 - 4*b1) * q^71 + (4*b10 - b8 + 5*b4 + 5*b3) * q^73 + (b11 + 3*b9 + b7 - b6 + 11*b5 + 5*b1) * q^75 + (2*b10 - b4 - b3) * q^77 + (-2*b7 - 2*b6 + 6*b1) * q^79 + (-12*b2 + 37) * q^81 + (-8*b10 + 4*b8 + 4*b4 + 4*b3) * q^83 + (b10 - 3*b8 - b3 - 3*b2 + 33) * q^85 + (-6*b10 - 7*b8 + 2*b4 + 2*b3) * q^87 + (-3*b9 - 2*b7 + 4*b6 - 4*b1) * q^89 + (-b9 - b7 + b6 - 5*b1) * q^91 + (-2*b10 - 24*b8 + 3*b4 + 3*b3) * q^93 + (-b11 + b10 + b9 - 8*b8 - b7 + 2*b6 - 9*b5 - 5*b4 + 7*b3 + 2*b2 - b1 + 11) * q^95 + (b11 - 4*b9 + 15*b5) * q^97 + (4*b4 - 4*b3 + 2*b2 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{5} + 76 q^{9}+O(q^{10})$$ 12 * q + 12 * q^5 + 76 * q^9 $$12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100})$$ 12 * q + 12 * q^5 + 76 * q^9 - 24 * q^11 + 68 * q^19 - 76 * q^25 - 32 * q^35 - 264 * q^39 + 220 * q^45 + 212 * q^49 + 176 * q^55 - 600 * q^61 + 492 * q^81 + 408 * q^85 + 124 * q^95 + 136 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241$$ :

 $$\beta_{1}$$ $$=$$ $$( 28618\nu^{10} + 524952\nu^{8} + 4391039\nu^{6} + 54167333\nu^{4} + 1825258945\nu^{2} - 558303007 ) / 908842237$$ (28618*v^10 + 524952*v^8 + 4391039*v^6 + 54167333*v^4 + 1825258945*v^2 - 558303007) / 908842237 $$\beta_{2}$$ $$=$$ $$( 60512 \nu^{10} + 2443821 \nu^{8} + 32340835 \nu^{6} + 100053868 \nu^{4} + 517724371 \nu^{2} + 21947164823 ) / 1817684474$$ (60512*v^10 + 2443821*v^8 + 32340835*v^6 + 100053868*v^4 + 517724371*v^2 + 21947164823) / 1817684474 $$\beta_{3}$$ $$=$$ $$( - 194997239 \nu^{11} + 17200092 \nu^{10} - 5048505213 \nu^{9} + 7318528889 \nu^{8} - 28805936576 \nu^{7} + \cdots + 187896630501135 ) / 57260696299948$$ (-194997239*v^11 + 17200092*v^10 - 5048505213*v^9 + 7318528889*v^8 - 28805936576*v^7 + 123691327169*v^6 - 162835339255*v^5 - 43476949766*v^4 - 12940529662267*v^3 + 2638688527393*v^2 - 36331199656762*v + 187896630501135) / 57260696299948 $$\beta_{4}$$ $$=$$ $$( - 194997239 \nu^{11} - 17200092 \nu^{10} - 5048505213 \nu^{9} - 7318528889 \nu^{8} - 28805936576 \nu^{7} + \cdots - 187896630501135 ) / 57260696299948$$ (-194997239*v^11 - 17200092*v^10 - 5048505213*v^9 - 7318528889*v^8 - 28805936576*v^7 - 123691327169*v^6 - 162835339255*v^5 + 43476949766*v^4 - 12940529662267*v^3 - 2638688527393*v^2 - 36331199656762*v - 187896630501135) / 57260696299948 $$\beta_{5}$$ $$=$$ $$( - 9729001 \nu^{11} - 195485288 \nu^{9} - 813261775 \nu^{7} - 3883239581 \nu^{5} - 354532580162 \nu^{3} + 2065201213063 \nu ) / 1506860428946$$ (-9729001*v^11 - 195485288*v^9 - 813261775*v^7 - 3883239581*v^5 - 354532580162*v^3 + 2065201213063*v) / 1506860428946 $$\beta_{6}$$ $$=$$ $$( 744934843 \nu^{11} + 1291775986 \nu^{10} + 9572417455 \nu^{9} + 18377506607 \nu^{8} - 122950074222 \nu^{7} + \cdots - 104748473351889 ) / 57260696299948$$ (744934843*v^11 + 1291775986*v^10 + 9572417455*v^9 + 18377506607*v^8 - 122950074222*v^7 - 1354472427*v^6 - 998868406049*v^5 - 4708627602976*v^4 + 29117423544893*v^3 + 12226184429355*v^2 - 208524546958680*v - 104748473351889) / 57260696299948 $$\beta_{7}$$ $$=$$ $$( - 744934843 \nu^{11} + 13046909770 \nu^{10} - 9572417455 \nu^{9} + 267705610063 \nu^{8} + 122950074222 \nu^{7} + \cdots - 679853386340457 ) / 57260696299948$$ (-744934843*v^11 + 13046909770*v^10 - 9572417455*v^9 + 267705610063*v^8 + 122950074222*v^7 + 232158453165*v^6 + 998868406049*v^5 - 12531395244576*v^4 - 29117423544893*v^3 + 420227260397139*v^2 + 208524546958680*v - 679853386340457) / 57260696299948 $$\beta_{8}$$ $$=$$ $$( 651724712 \nu^{11} + 15674641213 \nu^{9} + 65363306369 \nu^{7} - 565819505556 \nu^{5} + 17994962875673 \nu^{3} + \cdots + 25942480991957 \nu ) / 28630348149974$$ (651724712*v^11 + 15674641213*v^9 + 65363306369*v^7 - 565819505556*v^5 + 17994962875673*v^3 + 25942480991957*v) / 28630348149974 $$\beta_{9}$$ $$=$$ $$( 39207097 \nu^{11} + 503811445 \nu^{9} - 6471056538 \nu^{7} - 52572021371 \nu^{5} + 1532495976047 \nu^{3} + \cdots - 10974976155720 \nu ) / 1506860428946$$ (39207097*v^11 + 503811445*v^9 - 6471056538*v^7 - 52572021371*v^5 + 1532495976047*v^3 - 10974976155720*v) / 1506860428946 $$\beta_{10}$$ $$=$$ $$( 1021426750 \nu^{11} + 23103082157 \nu^{9} + 96267253819 \nu^{7} - 418256401478 \nu^{5} + 31467200921829 \nu^{3} + \cdots + 61986227495459 \nu ) / 28630348149974$$ (1021426750*v^11 + 23103082157*v^9 + 96267253819*v^7 - 418256401478*v^5 + 31467200921829*v^3 + 61986227495459*v) / 28630348149974 $$\beta_{11}$$ $$=$$ $$( 108734495 \nu^{11} + 992670198 \nu^{9} - 24848763895 \nu^{7} - 240525327873 \nu^{5} + 4527798347604 \nu^{3} + \cdots - 34021640801649 \nu ) / 1506860428946$$ (108734495*v^11 + 992670198*v^9 - 24848763895*v^7 - 240525327873*v^5 + 4527798347604*v^3 - 34021640801649*v) / 1506860428946
 $$\nu$$ $$=$$ $$( \beta_{10} - \beta_{8} + 2\beta_{5} ) / 4$$ (b10 - b8 + 2*b5) / 4 $$\nu^{2}$$ $$=$$ $$( -3\beta_{4} + 3\beta_{3} - \beta_{2} + \beta _1 - 7 ) / 2$$ (-3*b4 + 3*b3 - b2 + b1 - 7) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{11} - 7\beta_{10} - 6\beta_{9} + 5\beta_{8} - 6\beta_{5} - 16\beta_{4} - 16\beta_{3} ) / 4$$ (2*b11 - 7*b10 - 6*b9 + 5*b8 - 6*b5 - 16*b4 - 16*b3) / 4 $$\nu^{4}$$ $$=$$ $$( 11\beta_{9} + \beta_{7} - 21\beta_{6} + 37\beta_{4} - 37\beta_{3} + 9\beta_{2} - \beta _1 + 107 ) / 2$$ (11*b9 + b7 - 21*b6 + 37*b4 - 37*b3 + 9*b2 - b1 + 107) / 2 $$\nu^{5}$$ $$=$$ $$( -52\beta_{11} + 117\beta_{10} + 174\beta_{9} - 193\beta_{8} + 46\beta_{5} + 38\beta_{4} + 38\beta_{3} ) / 4$$ (-52*b11 + 117*b10 + 174*b9 - 193*b8 + 46*b5 + 38*b4 + 38*b3) / 4 $$\nu^{6}$$ $$=$$ $$( -205\beta_{9} - 67\beta_{7} + 343\beta_{6} - 152\beta_{4} + 152\beta_{3} + 38\beta_{2} + 196\beta _1 - 1504 ) / 2$$ (-205*b9 - 67*b7 + 343*b6 - 152*b4 + 152*b3 + 38*b2 + 196*b1 - 1504) / 2 $$\nu^{7}$$ $$=$$ $$( 844\beta_{11} + 61\beta_{10} - 3572\beta_{9} + 3\beta_{8} - 4914\beta_{5} + 284\beta_{4} + 284\beta_{3} ) / 4$$ (844*b11 + 61*b10 - 3572*b9 + 3*b8 - 4914*b5 + 284*b4 + 284*b3) / 4 $$\nu^{8}$$ $$=$$ $$( 3663 \beta_{9} + 1169 \beta_{7} - 6157 \beta_{6} - 4487 \beta_{4} + 4487 \beta_{3} - 339 \beta_{2} - 3775 \beta _1 - 25057 ) / 2$$ (3663*b9 + 1169*b7 - 6157*b6 - 4487*b4 + 4487*b3 - 339*b2 - 3775*b1 - 25057) / 2 $$\nu^{9}$$ $$=$$ $$( - 15750 \beta_{11} - 42099 \beta_{10} + 57402 \beta_{9} + 57429 \beta_{8} + 51230 \beta_{5} - 24724 \beta_{4} - 24724 \beta_{3} ) / 4$$ (-15750*b11 - 42099*b10 + 57402*b9 + 57429*b8 + 51230*b5 - 24724*b4 - 24724*b3) / 4 $$\nu^{10}$$ $$=$$ $$( - 56558 \beta_{9} - 13056 \beta_{7} + 100060 \beta_{6} + 226937 \beta_{4} - 226937 \beta_{3} + 47133 \beta_{2} + 40801 \beta _1 + 973351 ) / 2$$ (-56558*b9 - 13056*b7 + 100060*b6 + 226937*b4 - 226937*b3 + 47133*b2 + 40801*b1 + 973351) / 2 $$\nu^{11}$$ $$=$$ $$( 193788 \beta_{11} + 1261457 \beta_{10} - 705598 \beta_{9} - 1471617 \beta_{8} - 613302 \beta_{5} + 1040926 \beta_{4} + 1040926 \beta_{3} ) / 4$$ (193788*b11 + 1261457*b10 - 705598*b9 - 1471617*b8 - 613302*b5 + 1040926*b4 + 1040926*b3) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
189.1
 2.80718 + 1.46321i 2.80718 − 1.46321i 1.63363 + 1.34185i 1.63363 − 1.34185i 0.975196 − 4.19028i 0.975196 + 4.19028i −0.975196 − 4.19028i −0.975196 + 4.19028i −1.63363 + 1.34185i −1.63363 − 1.34185i −2.80718 + 1.46321i −2.80718 − 1.46321i
0 −5.61436 0 1.48938 4.77302i 0 7.03410i 0 22.5210 0
189.2 0 −5.61436 0 1.48938 + 4.77302i 0 7.03410i 0 22.5210 0
189.3 0 −3.26725 0 4.26535 2.60898i 0 6.30368i 0 1.67495 0
189.4 0 −3.26725 0 4.26535 + 2.60898i 0 6.30368i 0 1.67495 0
189.5 0 −1.95039 0 −2.75473 4.17271i 0 2.18749i 0 −5.19597 0
189.6 0 −1.95039 0 −2.75473 + 4.17271i 0 2.18749i 0 −5.19597 0
189.7 0 1.95039 0 −2.75473 4.17271i 0 2.18749i 0 −5.19597 0
189.8 0 1.95039 0 −2.75473 + 4.17271i 0 2.18749i 0 −5.19597 0
189.9 0 3.26725 0 4.26535 2.60898i 0 6.30368i 0 1.67495 0
189.10 0 3.26725 0 4.26535 + 2.60898i 0 6.30368i 0 1.67495 0
189.11 0 5.61436 0 1.48938 4.77302i 0 7.03410i 0 22.5210 0
189.12 0 5.61436 0 1.48938 + 4.77302i 0 7.03410i 0 22.5210 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 189.12 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
19.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.g.c 12
3.b odd 2 1 3420.3.h.e 12
5.b even 2 1 inner 380.3.g.c 12
5.c odd 4 2 1900.3.e.e 12
15.d odd 2 1 3420.3.h.e 12
19.b odd 2 1 inner 380.3.g.c 12
57.d even 2 1 3420.3.h.e 12
95.d odd 2 1 inner 380.3.g.c 12
95.g even 4 2 1900.3.e.e 12
285.b even 2 1 3420.3.h.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.c 12 1.a even 1 1 trivial
380.3.g.c 12 5.b even 2 1 inner
380.3.g.c 12 19.b odd 2 1 inner
380.3.g.c 12 95.d odd 2 1 inner
1900.3.e.e 12 5.c odd 4 2
1900.3.e.e 12 95.g even 4 2
3420.3.h.e 12 3.b odd 2 1
3420.3.h.e 12 15.d odd 2 1
3420.3.h.e 12 57.d even 2 1
3420.3.h.e 12 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 46T_{3}^{4} + 497T_{3}^{2} - 1280$$ acting on $$S_{3}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{6} - 46 T^{4} + 497 T^{2} - 1280)^{2}$$
$5$ $$(T^{6} - 6 T^{5} + 37 T^{4} - 160 T^{3} + \cdots + 15625)^{2}$$
$7$ $$(T^{6} + 94 T^{4} + 2393 T^{2} + \cdots + 9408)^{2}$$
$11$ $$(T^{3} + 6 T^{2} - 38 T - 44)^{4}$$
$13$ $$(T^{6} - 682 T^{4} + 127153 T^{2} + \cdots - 7200000)^{2}$$
$17$ $$(T^{6} + 314 T^{4} + 10793 T^{2} + \cdots + 12288)^{2}$$
$19$ $$(T^{6} - 34 T^{5} + 723 T^{4} + \cdots + 47045881)^{2}$$
$23$ $$(T^{6} + 2350 T^{4} + 1495625 T^{2} + \cdots + 147000000)^{2}$$
$29$ $$(T^{6} + 2306 T^{4} + 1274297 T^{2} + \cdots + 51671040)^{2}$$
$31$ $$(T^{6} + 4652 T^{4} + \cdots + 1345781760)^{2}$$
$37$ $$(T^{6} - 5312 T^{4} + \cdots - 5202247680)^{2}$$
$41$ $$(T^{6} + 7140 T^{4} + 3194532 T^{2} + \cdots + 318504960)^{2}$$
$43$ $$(T^{6} + 7540 T^{4} + 12661412 T^{2} + \cdots + 95158272)^{2}$$
$47$ $$(T^{6} + 14056 T^{4} + \cdots + 75606592512)^{2}$$
$53$ $$(T^{6} - 3130 T^{4} + 2302273 T^{2} + \cdots - 136869120)^{2}$$
$59$ $$(T^{6} + 18086 T^{4} + \cdots + 7228354560)^{2}$$
$61$ $$(T^{3} + 150 T^{2} + 5050 T + 18964)^{4}$$
$67$ $$(T^{6} - 22430 T^{4} + \cdots - 301871934720)^{2}$$
$71$ $$(T^{6} + 11980 T^{4} + \cdots + 23313776640)^{2}$$
$73$ $$(T^{6} + 12714 T^{4} + \cdots + 4401589248)^{2}$$
$79$ $$(T^{6} + 26440 T^{4} + \cdots + 4282122240)^{2}$$
$83$ $$(T^{6} + 19616 T^{4} + \cdots + 216760320000)^{2}$$
$89$ $$(T^{6} + 18508 T^{4} + \cdots + 187730165760)^{2}$$
$97$ $$(T^{6} - 22008 T^{4} + \cdots - 318730752000)^{2}$$