# Properties

 Label 245.6.a.e Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 82x^{2} + 58x + 1168$$ x^4 - x^3 - 82*x^2 + 58*x + 1168 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} + \beta_1 - 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} - 25 q^{5} + (11 \beta_{2} + 2 \beta_1 - 29) q^{6} + ( - 6 \beta_{3} + 21 \beta_{2} - 7 \beta_1 + 133) q^{8} + ( - \beta_{3} - 40 \beta_{2} + 33 \beta_1 + 165) q^{9}+O(q^{10})$$ q + (-b1 + 2) * q^2 + (-b3 + b1 - 4) * q^3 + (3*b2 - 5*b1 + 15) * q^4 - 25 * q^5 + (11*b2 + 2*b1 - 29) * q^6 + (-6*b3 + 21*b2 - 7*b1 + 133) * q^8 + (-b3 - 40*b2 + 33*b1 + 165) * q^9 $$q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} + \beta_1 - 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} - 25 q^{5} + (11 \beta_{2} + 2 \beta_1 - 29) q^{6} + ( - 6 \beta_{3} + 21 \beta_{2} - 7 \beta_1 + 133) q^{8} + ( - \beta_{3} - 40 \beta_{2} + 33 \beta_1 + 165) q^{9} + (25 \beta_1 - 50) q^{10} + ( - 9 \beta_{3} + 72 \beta_{2} - 7 \beta_1 + 228) q^{11} + (10 \beta_{3} + 16 \beta_{2} - 30 \beta_1 - 192) q^{12} + ( - 39 \beta_{3} - 41 \beta_1 - 14) q^{13} + (25 \beta_{3} - 25 \beta_1 + 100) q^{15} + ( - 42 \beta_{3} + 51 \beta_{2} - 87 \beta_1 - 117) q^{16} + ( - 9 \beta_{3} + 144 \beta_{2} - 55 \beta_1 - 418) q^{17} + (80 \beta_{3} - 165 \beta_{2} + 49 \beta_1 - 427) q^{18} + (18 \beta_{3} - 24 \beta_{2} - 226 \beta_1 - 92) q^{19} + ( - 75 \beta_{2} + 125 \beta_1 - 375) q^{20} + ( - 144 \beta_{3} + 291 \beta_{2} - 510 \beta_1 - 197) q^{22} + ( - 30 \beta_{3} + 72 \beta_{2} - 274 \beta_1 - 1488) q^{23} + ( - 32 \beta_{3} - 370 \beta_{2} + 40 \beta_1 + 1358) q^{24} + 625 q^{25} + (669 \beta_{2} - 304 \beta_1 + 2593) q^{26} + ( - 191 \beta_{3} + 128 \beta_{2} + 559 \beta_1 + 1736) q^{27} + ( - 141 \beta_{3} - 168 \beta_{2} + 333 \beta_1 + 1814) q^{29} + ( - 275 \beta_{2} - 50 \beta_1 + 725) q^{30} + ( - 48 \beta_{3} - 456 \beta_{2} - 576 \beta_1 + 1368) q^{31} + (90 \beta_{3} + 279 \beta_{2} - 283 \beta_1 - 641) q^{32} + (119 \beta_{3} - 1120 \beta_{2} - 519 \beta_1 + 1632) q^{33} + ( - 288 \beta_{3} + 579 \beta_{2} - 224 \beta_1 - 577) q^{34} + (362 \beta_{3} - 317 \beta_{2} + 413 \beta_1 - 7361) q^{36} + ( - 240 \beta_{3} - 264 \beta_{2} - 528 \beta_1 + 7326) q^{37} + (48 \beta_{3} + 378 \beta_{2} - 424 \beta_1 + 9522) q^{38} + ( - 21 \beta_{3} - 680 \beta_{2} + 1429 \beta_1 + 13664) q^{39} + (150 \beta_{3} - 525 \beta_{2} + 175 \beta_1 - 3325) q^{40} + (270 \beta_{3} - 504 \beta_{2} + 1474 \beta_1 - 3842) q^{41} + ( - 210 \beta_{3} - 1464 \beta_{2} + 370 \beta_1 - 2300) q^{43} + ( - 294 \beta_{3} + 1824 \beta_{2} - 2702 \beta_1 + 12752) q^{44} + (25 \beta_{3} + 1000 \beta_{2} - 825 \beta_1 - 4125) q^{45} + ( - 144 \beta_{3} + 1386 \beta_{2} + 300 \beta_1 + 8314) q^{46} + ( - 429 \beta_{3} + 144 \beta_{2} + 1149 \beta_1 - 10304) q^{47} + (420 \beta_{3} - 924 \beta_{2} + 672 \beta_1 + 13764) q^{48} + ( - 625 \beta_1 + 1250) q^{50} + (1221 \beta_{3} - 1528 \beta_{2} - 2245 \beta_1 + 2560) q^{51} + ( - 90 \beta_{3} + 2250 \beta_{2} - 4200 \beta_1 + 8002) q^{52} + ( - 402 \beta_{3} - 960 \beta_{2} - 158 \beta_1 + 4534) q^{53} + ( - 256 \beta_{3} + 1253 \beta_{2} - 1398 \beta_1 - 18411) q^{54} + (225 \beta_{3} - 1800 \beta_{2} + 175 \beta_1 - 5700) q^{55} + (478 \beta_{3} + 3320 \beta_{2} - 398 \beta_1 - 10840) q^{57} + (336 \beta_{3} + 639 \beta_{2} - 1016 \beta_1 - 4901) q^{58} + ( - 288 \beta_{3} - 1344 \beta_{2} + 4128 \beta_1 - 3884) q^{59} + ( - 250 \beta_{3} - 400 \beta_{2} + 750 \beta_1 + 4800) q^{60} + (330 \beta_{3} + 504 \beta_{2} - 186 \beta_1 - 7254) q^{61} + (912 \beta_{3} + 1488 \beta_{2} - 1968 \beta_1 + 35856) q^{62} + (786 \beta_{3} - 1485 \beta_{2} + 2189 \beta_1 + 8187) q^{64} + (975 \beta_{3} + 1025 \beta_1 + 350) q^{65} + (2240 \beta_{3} - 2349 \beta_{2} + 766 \beta_1 + 40883) q^{66} + (372 \beta_{3} - 1968 \beta_{2} + 5404 \beta_1 + 5956) q^{67} + ( - 870 \beta_{3} + 1254 \beta_{2} - 1512 \beta_1 + 18926) q^{68} + (2306 \beta_{3} + 1208 \beta_{2} - 1458 \beta_1 + 10680) q^{69} + ( - 672 \beta_{3} + 4464 \beta_{2} - 1968 \beta_1 - 18424) q^{71} + ( - 1926 \beta_{3} - 1661 \beta_{2} + 9793 \beta_1 - 21709) q^{72} + ( - 1068 \beta_{3} - 768 \beta_{2} + 604 \beta_1 + 18054) q^{73} + (528 \beta_{3} + 4416 \beta_{2} - 9318 \beta_1 + 46860) q^{74} + ( - 625 \beta_{3} + 625 \beta_1 - 2500) q^{75} + ( - 1332 \beta_{3} + 2124 \beta_{2} - 4456 \beta_1 + 33116) q^{76} + (1360 \beta_{3} - 5353 \beta_{2} - 7442 \beta_1 - 22777) q^{78} + ( - 87 \beta_{3} - 2424 \beta_{2} + 295 \beta_1 + 31064) q^{79} + (1050 \beta_{3} - 1275 \beta_{2} + 2175 \beta_1 + 2925) q^{80} + ( - 2544 \beta_{3} - 3632 \beta_{2} - 1136 \beta_1 + 34409) q^{81} + (1008 \beta_{3} - 9210 \beta_{2} + 11126 \beta_1 - 68942) q^{82} + ( - 1392 \beta_{3} + 1536 \beta_{2} - 4704 \beta_1 + 48476) q^{83} + (225 \beta_{3} - 3600 \beta_{2} + 1375 \beta_1 + 10450) q^{85} + (2928 \beta_{3} - 1098 \beta_{2} + 6752 \beta_1 + 7534) q^{86} + ( - 3743 \beta_{3} - 5568 \beta_{2} + 9551 \beta_1 + 53560) q^{87} + (960 \beta_{3} + 6558 \beta_{2} - 11480 \beta_1 + 125278) q^{88} + ( - 510 \beta_{3} + 4872 \beta_{2} - 178 \beta_1 + 9934) q^{89} + ( - 2000 \beta_{3} + 4125 \beta_{2} - 1225 \beta_1 + 10675) q^{90} + ( - 1812 \beta_{3} + 1584 \beta_{2} - 3524 \beta_1 + 32336) q^{92} + ( - 2640 \beta_{3} + 10872 \beta_{2} + 9984 \beta_1 + 4344) q^{93} + ( - 288 \beta_{3} + 2847 \beta_{2} + 11174 \beta_1 - 62881) q^{94} + ( - 450 \beta_{3} + 600 \beta_{2} + 5650 \beta_1 + 2300) q^{95} + (2872 \beta_{3} + 2096 \beta_{2} - 8156 \beta_1 - 39280) q^{96} + (2499 \beta_{3} - 6480 \beta_{2} + 5437 \beta_1 - 49042) q^{97} + ( - 3650 \beta_{3} + 6224 \beta_{2} + 15730 \beta_1 - 106900) q^{99}+O(q^{100})$$ q + (-b1 + 2) * q^2 + (-b3 + b1 - 4) * q^3 + (3*b2 - 5*b1 + 15) * q^4 - 25 * q^5 + (11*b2 + 2*b1 - 29) * q^6 + (-6*b3 + 21*b2 - 7*b1 + 133) * q^8 + (-b3 - 40*b2 + 33*b1 + 165) * q^9 + (25*b1 - 50) * q^10 + (-9*b3 + 72*b2 - 7*b1 + 228) * q^11 + (10*b3 + 16*b2 - 30*b1 - 192) * q^12 + (-39*b3 - 41*b1 - 14) * q^13 + (25*b3 - 25*b1 + 100) * q^15 + (-42*b3 + 51*b2 - 87*b1 - 117) * q^16 + (-9*b3 + 144*b2 - 55*b1 - 418) * q^17 + (80*b3 - 165*b2 + 49*b1 - 427) * q^18 + (18*b3 - 24*b2 - 226*b1 - 92) * q^19 + (-75*b2 + 125*b1 - 375) * q^20 + (-144*b3 + 291*b2 - 510*b1 - 197) * q^22 + (-30*b3 + 72*b2 - 274*b1 - 1488) * q^23 + (-32*b3 - 370*b2 + 40*b1 + 1358) * q^24 + 625 * q^25 + (669*b2 - 304*b1 + 2593) * q^26 + (-191*b3 + 128*b2 + 559*b1 + 1736) * q^27 + (-141*b3 - 168*b2 + 333*b1 + 1814) * q^29 + (-275*b2 - 50*b1 + 725) * q^30 + (-48*b3 - 456*b2 - 576*b1 + 1368) * q^31 + (90*b3 + 279*b2 - 283*b1 - 641) * q^32 + (119*b3 - 1120*b2 - 519*b1 + 1632) * q^33 + (-288*b3 + 579*b2 - 224*b1 - 577) * q^34 + (362*b3 - 317*b2 + 413*b1 - 7361) * q^36 + (-240*b3 - 264*b2 - 528*b1 + 7326) * q^37 + (48*b3 + 378*b2 - 424*b1 + 9522) * q^38 + (-21*b3 - 680*b2 + 1429*b1 + 13664) * q^39 + (150*b3 - 525*b2 + 175*b1 - 3325) * q^40 + (270*b3 - 504*b2 + 1474*b1 - 3842) * q^41 + (-210*b3 - 1464*b2 + 370*b1 - 2300) * q^43 + (-294*b3 + 1824*b2 - 2702*b1 + 12752) * q^44 + (25*b3 + 1000*b2 - 825*b1 - 4125) * q^45 + (-144*b3 + 1386*b2 + 300*b1 + 8314) * q^46 + (-429*b3 + 144*b2 + 1149*b1 - 10304) * q^47 + (420*b3 - 924*b2 + 672*b1 + 13764) * q^48 + (-625*b1 + 1250) * q^50 + (1221*b3 - 1528*b2 - 2245*b1 + 2560) * q^51 + (-90*b3 + 2250*b2 - 4200*b1 + 8002) * q^52 + (-402*b3 - 960*b2 - 158*b1 + 4534) * q^53 + (-256*b3 + 1253*b2 - 1398*b1 - 18411) * q^54 + (225*b3 - 1800*b2 + 175*b1 - 5700) * q^55 + (478*b3 + 3320*b2 - 398*b1 - 10840) * q^57 + (336*b3 + 639*b2 - 1016*b1 - 4901) * q^58 + (-288*b3 - 1344*b2 + 4128*b1 - 3884) * q^59 + (-250*b3 - 400*b2 + 750*b1 + 4800) * q^60 + (330*b3 + 504*b2 - 186*b1 - 7254) * q^61 + (912*b3 + 1488*b2 - 1968*b1 + 35856) * q^62 + (786*b3 - 1485*b2 + 2189*b1 + 8187) * q^64 + (975*b3 + 1025*b1 + 350) * q^65 + (2240*b3 - 2349*b2 + 766*b1 + 40883) * q^66 + (372*b3 - 1968*b2 + 5404*b1 + 5956) * q^67 + (-870*b3 + 1254*b2 - 1512*b1 + 18926) * q^68 + (2306*b3 + 1208*b2 - 1458*b1 + 10680) * q^69 + (-672*b3 + 4464*b2 - 1968*b1 - 18424) * q^71 + (-1926*b3 - 1661*b2 + 9793*b1 - 21709) * q^72 + (-1068*b3 - 768*b2 + 604*b1 + 18054) * q^73 + (528*b3 + 4416*b2 - 9318*b1 + 46860) * q^74 + (-625*b3 + 625*b1 - 2500) * q^75 + (-1332*b3 + 2124*b2 - 4456*b1 + 33116) * q^76 + (1360*b3 - 5353*b2 - 7442*b1 - 22777) * q^78 + (-87*b3 - 2424*b2 + 295*b1 + 31064) * q^79 + (1050*b3 - 1275*b2 + 2175*b1 + 2925) * q^80 + (-2544*b3 - 3632*b2 - 1136*b1 + 34409) * q^81 + (1008*b3 - 9210*b2 + 11126*b1 - 68942) * q^82 + (-1392*b3 + 1536*b2 - 4704*b1 + 48476) * q^83 + (225*b3 - 3600*b2 + 1375*b1 + 10450) * q^85 + (2928*b3 - 1098*b2 + 6752*b1 + 7534) * q^86 + (-3743*b3 - 5568*b2 + 9551*b1 + 53560) * q^87 + (960*b3 + 6558*b2 - 11480*b1 + 125278) * q^88 + (-510*b3 + 4872*b2 - 178*b1 + 9934) * q^89 + (-2000*b3 + 4125*b2 - 1225*b1 + 10675) * q^90 + (-1812*b3 + 1584*b2 - 3524*b1 + 32336) * q^92 + (-2640*b3 + 10872*b2 + 9984*b1 + 4344) * q^93 + (-288*b3 + 2847*b2 + 11174*b1 - 62881) * q^94 + (-450*b3 + 600*b2 + 5650*b1 + 2300) * q^95 + (2872*b3 + 2096*b2 - 8156*b1 - 39280) * q^96 + (2499*b3 - 6480*b2 + 5437*b1 - 49042) * q^97 + (-3650*b3 + 6224*b2 + 15730*b1 - 106900) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9}+O(q^{10})$$ 4 * q + 7 * q^2 - 14 * q^3 + 49 * q^4 - 100 * q^5 - 136 * q^6 + 489 * q^8 + 774 * q^9 $$4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9} - 175 q^{10} + 770 q^{11} - 840 q^{12} - 58 q^{13} + 350 q^{15} - 615 q^{16} - 2006 q^{17} - 1409 q^{18} - 564 q^{19} - 1225 q^{20} - 1736 q^{22} - 6340 q^{23} + 6244 q^{24} + 2500 q^{25} + 8730 q^{26} + 7438 q^{27} + 8066 q^{29} + 3400 q^{30} + 5856 q^{31} - 3495 q^{32} + 8130 q^{33} - 3402 q^{34} - 28759 q^{36} + 29544 q^{37} + 36860 q^{38} + 57466 q^{39} - 12225 q^{40} - 13156 q^{41} - 5692 q^{43} + 44952 q^{44} - 19350 q^{45} + 30928 q^{46} - 39926 q^{47} + 57156 q^{48} + 4375 q^{50} + 9830 q^{51} + 23398 q^{52} + 20300 q^{53} - 77292 q^{54} - 19250 q^{55} - 50876 q^{57} - 22234 q^{58} - 8432 q^{59} + 21000 q^{60} - 30540 q^{61} + 137568 q^{62} + 37121 q^{64} + 1450 q^{65} + 166756 q^{66} + 32792 q^{67} + 72554 q^{68} + 36540 q^{69} - 83920 q^{71} - 71795 q^{72} + 75424 q^{73} + 168762 q^{74} - 8750 q^{75} + 125092 q^{76} - 89204 q^{78} + 129486 q^{79} + 15375 q^{80} + 146308 q^{81} - 247230 q^{82} + 187520 q^{83} + 50150 q^{85} + 36156 q^{86} + 238670 q^{87} + 475556 q^{88} + 30324 q^{89} + 35225 q^{90} + 124464 q^{92} + 8256 q^{93} - 245756 q^{94} + 14100 q^{95} - 172340 q^{96} - 180270 q^{97} - 420668 q^{99}+O(q^{100})$$ 4 * q + 7 * q^2 - 14 * q^3 + 49 * q^4 - 100 * q^5 - 136 * q^6 + 489 * q^8 + 774 * q^9 - 175 * q^10 + 770 * q^11 - 840 * q^12 - 58 * q^13 + 350 * q^15 - 615 * q^16 - 2006 * q^17 - 1409 * q^18 - 564 * q^19 - 1225 * q^20 - 1736 * q^22 - 6340 * q^23 + 6244 * q^24 + 2500 * q^25 + 8730 * q^26 + 7438 * q^27 + 8066 * q^29 + 3400 * q^30 + 5856 * q^31 - 3495 * q^32 + 8130 * q^33 - 3402 * q^34 - 28759 * q^36 + 29544 * q^37 + 36860 * q^38 + 57466 * q^39 - 12225 * q^40 - 13156 * q^41 - 5692 * q^43 + 44952 * q^44 - 19350 * q^45 + 30928 * q^46 - 39926 * q^47 + 57156 * q^48 + 4375 * q^50 + 9830 * q^51 + 23398 * q^52 + 20300 * q^53 - 77292 * q^54 - 19250 * q^55 - 50876 * q^57 - 22234 * q^58 - 8432 * q^59 + 21000 * q^60 - 30540 * q^61 + 137568 * q^62 + 37121 * q^64 + 1450 * q^65 + 166756 * q^66 + 32792 * q^67 + 72554 * q^68 + 36540 * q^69 - 83920 * q^71 - 71795 * q^72 + 75424 * q^73 + 168762 * q^74 - 8750 * q^75 + 125092 * q^76 - 89204 * q^78 + 129486 * q^79 + 15375 * q^80 + 146308 * q^81 - 247230 * q^82 + 187520 * q^83 + 50150 * q^85 + 36156 * q^86 + 238670 * q^87 + 475556 * q^88 + 30324 * q^89 + 35225 * q^90 + 124464 * q^92 + 8256 * q^93 - 245756 * q^94 + 14100 * q^95 - 172340 * q^96 - 180270 * q^97 - 420668 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 82x^{2} + 58x + 1168$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} + \nu - 43 ) / 3$$ (v^2 + v - 43) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 52\nu - 48 ) / 6$$ (v^3 + v^2 - 52*v - 48) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} - \beta _1 + 43$$ 3*b2 - b1 + 43 $$\nu^{3}$$ $$=$$ $$6\beta_{3} - 3\beta_{2} + 53\beta _1 + 5$$ 6*b3 - 3*b2 + 53*b1 + 5

## 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
 8.05190 4.73688 −3.86448 −7.92431
−6.05190 −15.9755 4.62555 −25.0000 96.6824 0 165.668 12.2177 151.298
1.2 −2.73688 28.3358 −24.5095 −25.0000 −77.5516 0 154.660 559.917 68.4220
1.3 5.86448 −26.2268 2.39209 −25.0000 −153.807 0 −173.635 444.847 −146.612
1.4 9.92431 −0.133419 66.4919 −25.0000 −1.32409 0 342.308 −242.982 −248.108
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 245.6.a.e 4
7.b odd 2 1 35.6.a.d 4
21.c even 2 1 315.6.a.l 4
28.d even 2 1 560.6.a.v 4
35.c odd 2 1 175.6.a.f 4
35.f even 4 2 175.6.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.d 4 7.b odd 2 1
175.6.a.f 4 35.c odd 2 1
175.6.b.f 8 35.f even 4 2
245.6.a.e 4 1.a even 1 1 trivial
315.6.a.l 4 21.c even 2 1
560.6.a.v 4 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{4} - 7T_{2}^{3} - 64T_{2}^{2} + 250T_{2} + 964$$ T2^4 - 7*T2^3 - 64*T2^2 + 250*T2 + 964 $$T_{3}^{4} + 14T_{3}^{3} - 775T_{3}^{2} - 11976T_{3} - 1584$$ T3^4 + 14*T3^3 - 775*T3^2 - 11976*T3 - 1584

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 7 T^{3} - 64 T^{2} + 250 T + 964$$
$3$ $$T^{4} + 14 T^{3} - 775 T^{2} + \cdots - 1584$$
$5$ $$(T + 25)^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 770 T^{3} + \cdots - 20510223536$$
$13$ $$T^{4} + 58 T^{3} + \cdots + 430417376452$$
$17$ $$T^{4} + 2006 T^{3} + \cdots + 617989800868$$
$19$ $$T^{4} + 564 T^{3} + \cdots + 5221683964480$$
$23$ $$T^{4} + 6340 T^{3} + \cdots - 19307649395456$$
$29$ $$T^{4} - 8066 T^{3} + \cdots + 831691300$$
$31$ $$T^{4} + \cdots - 738407035699200$$
$37$ $$T^{4} - 29544 T^{3} + \cdots - 57\!\cdots\!96$$
$41$ $$T^{4} + \cdots + 225655412958976$$
$43$ $$T^{4} + 5692 T^{3} + \cdots + 16\!\cdots\!68$$
$47$ $$T^{4} + 39926 T^{3} + \cdots - 16\!\cdots\!88$$
$53$ $$T^{4} - 20300 T^{3} + \cdots - 15\!\cdots\!96$$
$59$ $$T^{4} + 8432 T^{3} + \cdots + 13\!\cdots\!40$$
$61$ $$T^{4} + 30540 T^{3} + \cdots - 25\!\cdots\!36$$
$67$ $$T^{4} - 32792 T^{3} + \cdots - 74\!\cdots\!96$$
$71$ $$T^{4} + 83920 T^{3} + \cdots - 17\!\cdots\!24$$
$73$ $$T^{4} - 75424 T^{3} + \cdots - 32\!\cdots\!52$$
$79$ $$T^{4} - 129486 T^{3} + \cdots + 39\!\cdots\!00$$
$83$ $$T^{4} - 187520 T^{3} + \cdots - 29\!\cdots\!64$$
$89$ $$T^{4} - 30324 T^{3} + \cdots + 21\!\cdots\!20$$
$97$ $$T^{4} + 180270 T^{3} + \cdots - 75\!\cdots\!24$$