# Properties

 Label 138.8.a.h Level $138$ Weight $8$ Character orbit 138.a Self dual yes Analytic conductor $43.109$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.1091335168$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220$$ x^4 - 2*x^3 - 8367*x^2 - 89140*x + 11077220 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - \beta_1 + 68) q^{5} + 216 q^{6} + (\beta_{3} + \beta_1 + 506) q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10})$$ q + 8 * q^2 + 27 * q^3 + 64 * q^4 + (b2 - b1 + 68) * q^5 + 216 * q^6 + (b3 + b1 + 506) * q^7 + 512 * q^8 + 729 * q^9 $$q + 8 q^{2} + 27 q^{3} + 64 q^{4} + (\beta_{2} - \beta_1 + 68) q^{5} + 216 q^{6} + (\beta_{3} + \beta_1 + 506) q^{7} + 512 q^{8} + 729 q^{9} + (8 \beta_{2} - 8 \beta_1 + 544) q^{10} + (3 \beta_{3} - 7 \beta_{2} - 4 \beta_1 + 1028) q^{11} + 1728 q^{12} + ( - 6 \beta_{3} + 20 \beta_{2} + 7 \beta_1 + 2016) q^{13} + (8 \beta_{3} + 8 \beta_1 + 4048) q^{14} + (27 \beta_{2} - 27 \beta_1 + 1836) q^{15} + 4096 q^{16} + ( - 7 \beta_{3} - 80 \beta_{2} - 19 \beta_1 + 9252) q^{17} + 5832 q^{18} + ( - 32 \beta_{3} + 113 \beta_{2} + 31 \beta_1 + 1466) q^{19} + (64 \beta_{2} - 64 \beta_1 + 4352) q^{20} + (27 \beta_{3} + 27 \beta_1 + 13662) q^{21} + (24 \beta_{3} - 56 \beta_{2} - 32 \beta_1 + 8224) q^{22} - 12167 q^{23} + 13824 q^{24} + ( - 20 \beta_{3} - 230 \beta_{2} - 115 \beta_1 + 30245) q^{25} + ( - 48 \beta_{3} + 160 \beta_{2} + 56 \beta_1 + 16128) q^{26} + 19683 q^{27} + (64 \beta_{3} + 64 \beta_1 + 32384) q^{28} + ( - 198 \beta_{3} - 304 \beta_{2} + 214 \beta_1 + 54178) q^{29} + (216 \beta_{2} - 216 \beta_1 + 14688) q^{30} + (118 \beta_{3} - 200 \beta_{2} + 648 \beta_1 + 55672) q^{31} + 32768 q^{32} + (81 \beta_{3} - 189 \beta_{2} - 108 \beta_1 + 27756) q^{33} + ( - 56 \beta_{3} - 640 \beta_{2} - 152 \beta_1 + 74016) q^{34} + (340 \beta_{3} + 92 \beta_{2} - 247 \beta_1 + 17326) q^{35} + 46656 q^{36} + (287 \beta_{3} - 561 \beta_{2} + 1464 \beta_1 + 121470) q^{37} + ( - 256 \beta_{3} + 904 \beta_{2} + 248 \beta_1 + 11728) q^{38} + ( - 162 \beta_{3} + 540 \beta_{2} + 189 \beta_1 + 54432) q^{39} + (512 \beta_{2} - 512 \beta_1 + 34816) q^{40} + ( - 206 \beta_{3} + 786 \beta_{2} - 448 \beta_1 + 84874) q^{41} + (216 \beta_{3} + 216 \beta_1 + 109296) q^{42} + ( - 490 \beta_{3} - 1201 \beta_{2} + 599 \beta_1 + 181898) q^{43} + (192 \beta_{3} - 448 \beta_{2} - 256 \beta_1 + 65792) q^{44} + (729 \beta_{2} - 729 \beta_1 + 49572) q^{45} - 97336 q^{46} + ( - 60 \beta_{3} - 492 \beta_{2} - 3671 \beta_1 + 84286) q^{47} + 110592 q^{48} + (716 \beta_{3} + 990 \beta_{2} - 439 \beta_1 - 76785) q^{49} + ( - 160 \beta_{3} - 1840 \beta_{2} - 920 \beta_1 + 241960) q^{50} + ( - 189 \beta_{3} - 2160 \beta_{2} - 513 \beta_1 + 249804) q^{51} + ( - 384 \beta_{3} + 1280 \beta_{2} + 448 \beta_1 + 129024) q^{52} + ( - 806 \beta_{3} + 4709 \beta_{2} - 381 \beta_1 - 91924) q^{53} + 157464 q^{54} + (1160 \beta_{3} + 1340 \beta_{2} - 1630 \beta_1 + 107460) q^{55} + (512 \beta_{3} + 512 \beta_1 + 259072) q^{56} + ( - 864 \beta_{3} + 3051 \beta_{2} + 837 \beta_1 + 39582) q^{57} + ( - 1584 \beta_{3} - 2432 \beta_{2} + 1712 \beta_1 + 433424) q^{58} + (2338 \beta_{3} + 5346 \beta_{2} + 3229 \beta_1 + 21690) q^{59} + (1728 \beta_{2} - 1728 \beta_1 + 117504) q^{60} + ( - 2379 \beta_{3} - 4903 \beta_{2} - 3478 \beta_1 + 521650) q^{61} + (944 \beta_{3} - 1600 \beta_{2} + 5184 \beta_1 + 445376) q^{62} + (729 \beta_{3} + 729 \beta_1 + 368874) q^{63} + 262144 q^{64} + ( - 2440 \beta_{3} - 206 \beta_{2} - 484 \beta_1 + 393372) q^{65} + (648 \beta_{3} - 1512 \beta_{2} - 864 \beta_1 + 222048) q^{66} + ( - 306 \beta_{3} - 5799 \beta_{2} + 10509 \beta_1 + 1081238) q^{67} + ( - 448 \beta_{3} - 5120 \beta_{2} - 1216 \beta_1 + 592128) q^{68} - 328509 q^{69} + (2720 \beta_{3} + 736 \beta_{2} - 1976 \beta_1 + 138608) q^{70} + (2430 \beta_{3} - 12294 \beta_{2} - 2388 \beta_1 + 567072) q^{71} + 373248 q^{72} + ( - 212 \beta_{3} + 7240 \beta_{2} + 2072 \beta_1 - 273318) q^{73} + (2296 \beta_{3} - 4488 \beta_{2} + 11712 \beta_1 + 971760) q^{74} + ( - 540 \beta_{3} - 6210 \beta_{2} - 3105 \beta_1 + 816615) q^{75} + ( - 2048 \beta_{3} + 7232 \beta_{2} + 1984 \beta_1 + 93824) q^{76} + (2862 \beta_{3} - 3218 \beta_{2} - 7946 \beta_1 + 1456372) q^{77} + ( - 1296 \beta_{3} + 4320 \beta_{2} + 1512 \beta_1 + 435456) q^{78} + (4019 \beta_{3} - 9176 \beta_{2} - 7373 \beta_1 + 12574) q^{79} + (4096 \beta_{2} - 4096 \beta_1 + 278528) q^{80} + 531441 q^{81} + ( - 1648 \beta_{3} + 6288 \beta_{2} - 3584 \beta_1 + 678992) q^{82} + ( - 2903 \beta_{3} + 13259 \beta_{2} + 11034 \beta_1 + 376544) q^{83} + (1728 \beta_{3} + 1728 \beta_1 + 874368) q^{84} + ( - 780 \beta_{3} + 32494 \beta_{2} - 18529 \beta_1 - 2195098) q^{85} + ( - 3920 \beta_{3} - 9608 \beta_{2} + 4792 \beta_1 + 1455184) q^{86} + ( - 5346 \beta_{3} - 8208 \beta_{2} + 5778 \beta_1 + 1462806) q^{87} + (1536 \beta_{3} - 3584 \beta_{2} - 2048 \beta_1 + 526336) q^{88} + ( - 4239 \beta_{3} + 1350 \beta_{2} + 657 \beta_1 + 369828) q^{89} + (5832 \beta_{2} - 5832 \beta_1 + 396576) q^{90} + ( - 1130 \beta_{3} + 8968 \beta_{2} + 21054 \beta_1 - 720684) q^{91} - 778688 q^{92} + (3186 \beta_{3} - 5400 \beta_{2} + 17496 \beta_1 + 1503144) q^{93} + ( - 480 \beta_{3} - 3936 \beta_{2} - 29368 \beta_1 + 674288) q^{94} + ( - 13140 \beta_{3} - 12272 \beta_{2} + 6407 \beta_1 + 2123594) q^{95} + 884736 q^{96} + (9616 \beta_{3} - 33518 \beta_{2} - 6966 \beta_1 + 471910) q^{97} + (5728 \beta_{3} + 7920 \beta_{2} - 3512 \beta_1 - 614280) q^{98} + (2187 \beta_{3} - 5103 \beta_{2} - 2916 \beta_1 + 749412) q^{99}+O(q^{100})$$ q + 8 * q^2 + 27 * q^3 + 64 * q^4 + (b2 - b1 + 68) * q^5 + 216 * q^6 + (b3 + b1 + 506) * q^7 + 512 * q^8 + 729 * q^9 + (8*b2 - 8*b1 + 544) * q^10 + (3*b3 - 7*b2 - 4*b1 + 1028) * q^11 + 1728 * q^12 + (-6*b3 + 20*b2 + 7*b1 + 2016) * q^13 + (8*b3 + 8*b1 + 4048) * q^14 + (27*b2 - 27*b1 + 1836) * q^15 + 4096 * q^16 + (-7*b3 - 80*b2 - 19*b1 + 9252) * q^17 + 5832 * q^18 + (-32*b3 + 113*b2 + 31*b1 + 1466) * q^19 + (64*b2 - 64*b1 + 4352) * q^20 + (27*b3 + 27*b1 + 13662) * q^21 + (24*b3 - 56*b2 - 32*b1 + 8224) * q^22 - 12167 * q^23 + 13824 * q^24 + (-20*b3 - 230*b2 - 115*b1 + 30245) * q^25 + (-48*b3 + 160*b2 + 56*b1 + 16128) * q^26 + 19683 * q^27 + (64*b3 + 64*b1 + 32384) * q^28 + (-198*b3 - 304*b2 + 214*b1 + 54178) * q^29 + (216*b2 - 216*b1 + 14688) * q^30 + (118*b3 - 200*b2 + 648*b1 + 55672) * q^31 + 32768 * q^32 + (81*b3 - 189*b2 - 108*b1 + 27756) * q^33 + (-56*b3 - 640*b2 - 152*b1 + 74016) * q^34 + (340*b3 + 92*b2 - 247*b1 + 17326) * q^35 + 46656 * q^36 + (287*b3 - 561*b2 + 1464*b1 + 121470) * q^37 + (-256*b3 + 904*b2 + 248*b1 + 11728) * q^38 + (-162*b3 + 540*b2 + 189*b1 + 54432) * q^39 + (512*b2 - 512*b1 + 34816) * q^40 + (-206*b3 + 786*b2 - 448*b1 + 84874) * q^41 + (216*b3 + 216*b1 + 109296) * q^42 + (-490*b3 - 1201*b2 + 599*b1 + 181898) * q^43 + (192*b3 - 448*b2 - 256*b1 + 65792) * q^44 + (729*b2 - 729*b1 + 49572) * q^45 - 97336 * q^46 + (-60*b3 - 492*b2 - 3671*b1 + 84286) * q^47 + 110592 * q^48 + (716*b3 + 990*b2 - 439*b1 - 76785) * q^49 + (-160*b3 - 1840*b2 - 920*b1 + 241960) * q^50 + (-189*b3 - 2160*b2 - 513*b1 + 249804) * q^51 + (-384*b3 + 1280*b2 + 448*b1 + 129024) * q^52 + (-806*b3 + 4709*b2 - 381*b1 - 91924) * q^53 + 157464 * q^54 + (1160*b3 + 1340*b2 - 1630*b1 + 107460) * q^55 + (512*b3 + 512*b1 + 259072) * q^56 + (-864*b3 + 3051*b2 + 837*b1 + 39582) * q^57 + (-1584*b3 - 2432*b2 + 1712*b1 + 433424) * q^58 + (2338*b3 + 5346*b2 + 3229*b1 + 21690) * q^59 + (1728*b2 - 1728*b1 + 117504) * q^60 + (-2379*b3 - 4903*b2 - 3478*b1 + 521650) * q^61 + (944*b3 - 1600*b2 + 5184*b1 + 445376) * q^62 + (729*b3 + 729*b1 + 368874) * q^63 + 262144 * q^64 + (-2440*b3 - 206*b2 - 484*b1 + 393372) * q^65 + (648*b3 - 1512*b2 - 864*b1 + 222048) * q^66 + (-306*b3 - 5799*b2 + 10509*b1 + 1081238) * q^67 + (-448*b3 - 5120*b2 - 1216*b1 + 592128) * q^68 - 328509 * q^69 + (2720*b3 + 736*b2 - 1976*b1 + 138608) * q^70 + (2430*b3 - 12294*b2 - 2388*b1 + 567072) * q^71 + 373248 * q^72 + (-212*b3 + 7240*b2 + 2072*b1 - 273318) * q^73 + (2296*b3 - 4488*b2 + 11712*b1 + 971760) * q^74 + (-540*b3 - 6210*b2 - 3105*b1 + 816615) * q^75 + (-2048*b3 + 7232*b2 + 1984*b1 + 93824) * q^76 + (2862*b3 - 3218*b2 - 7946*b1 + 1456372) * q^77 + (-1296*b3 + 4320*b2 + 1512*b1 + 435456) * q^78 + (4019*b3 - 9176*b2 - 7373*b1 + 12574) * q^79 + (4096*b2 - 4096*b1 + 278528) * q^80 + 531441 * q^81 + (-1648*b3 + 6288*b2 - 3584*b1 + 678992) * q^82 + (-2903*b3 + 13259*b2 + 11034*b1 + 376544) * q^83 + (1728*b3 + 1728*b1 + 874368) * q^84 + (-780*b3 + 32494*b2 - 18529*b1 - 2195098) * q^85 + (-3920*b3 - 9608*b2 + 4792*b1 + 1455184) * q^86 + (-5346*b3 - 8208*b2 + 5778*b1 + 1462806) * q^87 + (1536*b3 - 3584*b2 - 2048*b1 + 526336) * q^88 + (-4239*b3 + 1350*b2 + 657*b1 + 369828) * q^89 + (5832*b2 - 5832*b1 + 396576) * q^90 + (-1130*b3 + 8968*b2 + 21054*b1 - 720684) * q^91 - 778688 * q^92 + (3186*b3 - 5400*b2 + 17496*b1 + 1503144) * q^93 + (-480*b3 - 3936*b2 - 29368*b1 + 674288) * q^94 + (-13140*b3 - 12272*b2 + 6407*b1 + 2123594) * q^95 + 884736 * q^96 + (9616*b3 - 33518*b2 - 6966*b1 + 471910) * q^97 + (5728*b3 + 7920*b2 - 3512*b1 - 614280) * q^98 + (2187*b3 - 5103*b2 - 2916*b1 + 749412) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10})$$ 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 $$4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100})$$ 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 + 2160 * q^10 + 4120 * q^11 + 6912 * q^12 + 8036 * q^13 + 16176 * q^14 + 7290 * q^15 + 16384 * q^16 + 37182 * q^17 + 23328 * q^18 + 5702 * q^19 + 17280 * q^20 + 54594 * q^21 + 32960 * q^22 - 48668 * q^23 + 55296 * q^24 + 121480 * q^25 + 64288 * q^26 + 78732 * q^27 + 129408 * q^28 + 217716 * q^29 + 58320 * q^30 + 222852 * q^31 + 131072 * q^32 + 111240 * q^33 + 297456 * q^34 + 68440 * q^35 + 186624 * q^36 + 486428 * q^37 + 45616 * q^38 + 216972 * q^39 + 138240 * q^40 + 338336 * q^41 + 436752 * q^42 + 730974 * q^43 + 263680 * q^44 + 196830 * q^45 - 389344 * q^46 + 338248 * q^47 + 442368 * q^48 - 310552 * q^49 + 971840 * q^50 + 1003914 * q^51 + 514304 * q^52 - 375502 * q^53 + 629856 * q^54 + 424840 * q^55 + 1035264 * q^56 + 153954 * q^57 + 1741728 * q^58 + 71392 * q^59 + 466560 * q^60 + 2101164 * q^61 + 1782816 * q^62 + 1474038 * q^63 + 1048576 * q^64 + 1578780 * q^65 + 889920 * q^66 + 4337162 * q^67 + 2379648 * q^68 - 1314036 * q^69 + 547520 * q^70 + 2288016 * q^71 + 1492992 * q^72 - 1107328 * q^73 + 3891424 * q^74 + 3279960 * q^75 + 364928 * q^76 + 5826200 * q^77 + 1735776 * q^78 + 60610 * q^79 + 1105920 * q^80 + 2125764 * q^81 + 2706688 * q^82 + 1485464 * q^83 + 3494016 * q^84 - 8843820 * q^85 + 5847792 * q^86 + 5878332 * q^87 + 2109440 * q^88 + 1485090 * q^89 + 1574640 * q^90 - 2898412 * q^91 - 3114752 * q^92 + 6017004 * q^93 + 2705984 * q^94 + 8545200 * q^95 + 3538944 * q^96 + 1935444 * q^97 - 2484416 * q^98 + 3003480 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu - 2$$ 4*v - 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 8\nu^{2} - 5551\nu - 110450 ) / 684$$ (v^3 + 8*v^2 - 5551*v - 110450) / 684 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 68\nu^{2} + 4411\nu - 207382 ) / 76$$ (-v^3 + 68*v^2 + 4411*v - 207382) / 76
 $$\nu$$ $$=$$ $$( \beta _1 + 2 ) / 4$$ (b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} + 36\beta_{2} + 15\beta _1 + 16758 ) / 4$$ (4*b3 + 36*b2 + 15*b1 + 16758) / 4 $$\nu^{3}$$ $$=$$ $$( -32\beta_{3} + 2448\beta_{2} + 5431\beta _1 + 318838 ) / 4$$ (-32*b3 + 2448*b2 + 5431*b1 + 318838) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 33.2734 90.4389 −65.5856 −56.1267
8.00000 27.0000 64.0000 −427.795 216.000 345.429 512.000 729.000 −3422.36
1.2 8.00000 27.0000 64.0000 −10.0669 216.000 971.172 512.000 729.000 −80.5353
1.3 8.00000 27.0000 64.0000 340.987 216.000 1267.12 512.000 729.000 2727.89
1.4 8.00000 27.0000 64.0000 366.876 216.000 −561.721 512.000 729.000 2935.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## 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 138.8.a.h 4
3.b odd 2 1 414.8.a.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.h 4 1.a even 1 1 trivial
414.8.a.i 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 270T_{5}^{3} - 180540T_{5}^{2} + 51728000T_{5} + 538752000$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 8)^{4}$$
$3$ $$(T - 27)^{4}$$
$5$ $$T^{4} - 270 T^{3} + \cdots + 538752000$$
$7$ $$T^{4} - 2022 T^{3} + \cdots - 238777204176$$
$11$ $$T^{4} - 4120 T^{3} + \cdots + 25210679155200$$
$13$ $$T^{4} - 8036 T^{3} + \cdots + 10\!\cdots\!84$$
$17$ $$T^{4} - 37182 T^{3} + \cdots - 69\!\cdots\!08$$
$19$ $$T^{4} - 5702 T^{3} + \cdots + 12\!\cdots\!64$$
$23$ $$(T + 12167)^{4}$$
$29$ $$T^{4} - 217716 T^{3} + \cdots - 19\!\cdots\!68$$
$31$ $$T^{4} - 222852 T^{3} + \cdots - 32\!\cdots\!00$$
$37$ $$T^{4} - 486428 T^{3} + \cdots - 26\!\cdots\!28$$
$41$ $$T^{4} - 338336 T^{3} + \cdots + 11\!\cdots\!52$$
$43$ $$T^{4} - 730974 T^{3} + \cdots - 11\!\cdots\!00$$
$47$ $$T^{4} - 338248 T^{3} + \cdots + 25\!\cdots\!76$$
$53$ $$T^{4} + 375502 T^{3} + \cdots + 12\!\cdots\!20$$
$59$ $$T^{4} - 71392 T^{3} + \cdots + 23\!\cdots\!28$$
$61$ $$T^{4} - 2101164 T^{3} + \cdots + 18\!\cdots\!72$$
$67$ $$T^{4} - 4337162 T^{3} + \cdots + 44\!\cdots\!92$$
$71$ $$T^{4} - 2288016 T^{3} + \cdots + 92\!\cdots\!92$$
$73$ $$T^{4} + 1107328 T^{3} + \cdots - 25\!\cdots\!88$$
$79$ $$T^{4} - 60610 T^{3} + \cdots + 83\!\cdots\!48$$
$83$ $$T^{4} - 1485464 T^{3} + \cdots + 11\!\cdots\!08$$
$89$ $$T^{4} - 1485090 T^{3} + \cdots - 10\!\cdots\!56$$
$97$ $$T^{4} - 1935444 T^{3} + \cdots + 94\!\cdots\!60$$