# Properties

 Label 39.4.j.c Level $39$ Weight $4$ Character orbit 39.j Analytic conductor $2.301$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648$$ x^10 + 70*x^8 + 1645*x^6 + 14700*x^4 + 44100*x^2 + 27648 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - 3 \beta_{2} q^{3} + (\beta_{5} - \beta_{4} - 6 \beta_{2} + 6) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{2} + 2) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8} + (9 \beta_{2} - 9) q^{9}+O(q^{10})$$ q + b3 * q^2 - 3*b2 * q^3 + (b5 - b4 - 6*b2 + 6) * q^4 + (-b7 + b6 + 2*b2 - b1 - 1) * q^5 + (-3*b3 + 3*b1) * q^6 + (-b9 + b8 + 2*b2 + 2) * q^7 + (-b9 + b7 - b6 - 2*b5 + b4 + 7*b1) * q^8 + (9*b2 - 9) * q^9 $$q + \beta_{3} q^{2} - 3 \beta_{2} q^{3} + (\beta_{5} - \beta_{4} - 6 \beta_{2} + 6) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{2} + 2) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8} + (9 \beta_{2} - 9) q^{9} + (2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{3} + 8 \beta_{2} - 6 \beta_1) q^{10} + ( - \beta_{8} - \beta_{7} + 2 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - 4 \beta_{2} + 8) q^{11} + (3 \beta_{4} - 18) q^{12} + (\beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{3} + 17 \beta_{2} - 3 \beta_1 - 6) q^{13} + ( - 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - \beta_1 - 6) q^{14} + (3 \beta_{7} + 3 \beta_{3} - 3 \beta_{2} + 6) q^{15} + (2 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} + 6 \beta_{3} - 50 \beta_{2} - 12 \beta_1) q^{16} + (\beta_{9} + \beta_{8} + 4 \beta_{3} - 21 \beta_{2} + 4 \beta_1 + 21) q^{17} - 9 \beta_1 q^{18} + (\beta_{9} - \beta_{8} + 3 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} + \cdots + 12) q^{19}+ \cdots + (9 \beta_{9} + 9 \beta_{7} - 9 \beta_{6} - 36 \beta_{5} + 18 \beta_{4} + 72 \beta_{2} + \cdots - 36) q^{99}+O(q^{100})$$ q + b3 * q^2 - 3*b2 * q^3 + (b5 - b4 - 6*b2 + 6) * q^4 + (-b7 + b6 + 2*b2 - b1 - 1) * q^5 + (-3*b3 + 3*b1) * q^6 + (-b9 + b8 + 2*b2 + 2) * q^7 + (-b9 + b7 - b6 - 2*b5 + b4 + 7*b1) * q^8 + (9*b2 - 9) * q^9 + (2*b9 - b8 - b7 + 2*b6 - b5 + 3*b3 + 8*b2 - 6*b1) * q^10 + (-b8 - b7 + 2*b5 - 4*b4 - 5*b3 - 4*b2 + 8) * q^11 + (3*b4 - 18) * q^12 + (b8 + 2*b7 - b6 - 2*b5 - 3*b3 + 17*b2 - 3*b1 - 6) * q^13 + (-2*b7 - 2*b6 + 3*b4 + 2*b3 - b1 - 6) * q^14 + (3*b7 + 3*b3 - 3*b2 + 6) * q^15 + (2*b7 - 4*b6 + 5*b5 + 6*b3 - 50*b2 - 12*b1) * q^16 + (b9 + b8 + 4*b3 - 21*b2 + 4*b1 + 21) * q^17 - 9*b1 * q^18 + (b9 - b8 + 3*b6 + 4*b5 + 4*b4 - 9*b3 + 12*b2 + 9*b1 + 12) * q^19 + (4*b9 - 4*b8 + 2*b6 - 5*b5 - 5*b4 + 18*b3 + 34*b2 - 18*b1 + 34) * q^20 + (3*b9 - 12*b2 + 6) * q^21 + (-b9 - b8 + 6*b7 - 3*b6 - 8*b5 + 8*b4 + 21*b3 + 58*b2 + 21*b1 - 58) * q^22 + (-6*b9 + 3*b8 + b7 - 2*b6 - b3 - 12*b2 + 2*b1) * q^23 + (3*b8 - 3*b7 + 3*b5 - 6*b4 - 21*b3) * q^24 + (-b7 - b6 - 2*b4 - 42*b3 + 21*b1 - 96) * q^25 + (-b9 - 2*b8 - b7 + b6 - 7*b5 + 8*b4 + 28*b3 + 102*b2 - 30*b1 - 54) * q^26 + 27 * q^27 + (b8 - 9*b7 + 4*b5 - 8*b4 - 45*b3 - 10*b2 + 20) * q^28 + (-3*b7 + 6*b6 + 6*b5 + 11*b3 - 99*b2 - 22*b1) * q^29 + (-3*b9 - 3*b8 + 6*b7 - 3*b6 + 3*b5 - 3*b4 + 9*b3 - 24*b2 + 9*b1 + 24) * q^30 + (-2*b9 - 3*b7 + 3*b6 - 12*b5 + 6*b4 - 44*b2 + 9*b1 + 22) * q^31 + (-3*b9 + 3*b8 - 3*b6 - 3*b5 - 3*b4 - 51*b3 + 96*b2 + 51*b1 + 96) * q^32 + (-3*b9 + 3*b8 + 3*b6 + 6*b5 + 6*b4 + 15*b3 - 12*b2 - 15*b1 - 12) * q^33 + (-6*b7 + 6*b6 + 2*b5 - b4 - 100*b2 + 18*b1 + 50) * q^34 + (3*b9 + 3*b8 - 10*b7 + 5*b6 + 18*b5 - 18*b4 + 35*b3 - 12*b2 + 35*b1 + 12) * q^35 + (-9*b5 + 54*b2) * q^36 + (-b8 + 12*b7 + 36*b3 + 27*b2 - 54) * q^37 + (-b9 + 2*b8 + b7 + b6 - 6*b4 - 34*b3 + 17*b1 - 138) * q^38 + (3*b9 - 3*b8 - 3*b7 - 3*b6 + 6*b5 - 6*b4 + 18*b3 - 33*b2 - 9*b1 + 51) * q^39 + (-b9 + 2*b8 + 7*b7 + 7*b6 - 7*b4 + 126*b3 - 63*b1 + 200) * q^40 + (-3*b8 + 8*b7 - 6*b5 + 12*b4 - 32*b3 - 71*b2 + 142) * q^41 + (-6*b7 + 12*b6 - 9*b5 - 3*b3 + 18*b2 + 6*b1) * q^42 + (-b9 - b8 - 20*b7 + 10*b6 + 6*b5 - 6*b4 - 18*b3 + 74*b2 - 18*b1 - 74) * q^43 + (7*b9 + 15*b7 - 15*b6 + 32*b5 - 16*b4 - 500*b2 - 69*b1 + 250) * q^44 + (-9*b6 - 9*b3 - 9*b2 + 9*b1 - 9) * q^45 + (-3*b9 + 3*b8 - 21*b6 + 10*b5 + 10*b4 - 27*b3 - 26*b2 + 27*b1 - 26) * q^46 + (-b9 + 3*b7 - 3*b6 - 8*b5 + 4*b4 + 8*b2 + 47*b1 - 4) * q^47 + (-12*b7 + 6*b6 - 15*b5 + 15*b4 + 18*b3 + 150*b2 + 18*b1 - 150) * q^48 + (4*b9 - 2*b8 + 7*b7 - 14*b6 + 12*b5 - 3*b3 + 155*b2 + 6*b1) * q^49 + (b8 - b7 - 23*b5 + 46*b4 - 84*b3 + 288*b2 - 576) * q^50 + (3*b9 - 6*b8 - 24*b3 + 12*b1 - 63) * q^51 + (b9 - 7*b7 + 17*b6 + 22*b5 - 27*b4 + 18*b3 - 44*b2 - 117*b1 + 316) * q^52 + (4*b9 - 8*b8 + 9*b7 + 9*b6 + 6*b4 - 6*b3 + 3*b1 + 33) * q^53 + 27*b3 * q^54 + (2*b9 - b8 + 11*b7 - 22*b6 - 46*b5 + 33*b3 - 52*b2 - 66*b1) * q^55 + (5*b9 + 5*b8 + 18*b7 - 9*b6 - 36*b5 + 36*b4 + 19*b3 + 534*b2 + 19*b1 - 534) * q^56 + (-3*b9 + 9*b7 - 9*b6 - 24*b5 + 12*b4 - 72*b2 - 27*b1 + 36) * q^57 + (3*b9 - 3*b8 + 3*b6 - 17*b5 - 17*b4 - 135*b3 + 136*b2 + 135*b1 + 136) * q^58 + (-4*b9 + 4*b8 + 6*b6 - 16*b5 - 16*b4 + 26*b3 + 52*b2 - 26*b1 + 52) * q^59 + (-12*b9 + 6*b7 - 6*b6 + 30*b5 - 15*b4 - 204*b2 + 54*b1 + 102) * q^60 + (-b9 - b8 + 6*b7 - 3*b6 + 4*b5 - 4*b4 - 9*b3 + 275*b2 - 9*b1 - 275) * q^61 + (18*b9 - 9*b8 - 5*b7 + 10*b6 + 33*b5 + 36*b3 - 156*b2 - 72*b1) * q^62 + (-9*b8 + 18*b2 - 36) * q^63 + (10*b7 + 10*b6 + 29*b4 + 132*b3 - 66*b1 - 314) * q^64 + (-6*b9 + 17*b8 - 6*b7 - 12*b6 + 14*b5 + 26*b4 - 54*b3 + 229*b2 + 56*b1 + 46) * q^65 + (-3*b9 + 6*b8 - 9*b7 - 9*b6 - 24*b4 - 126*b3 + 63*b1 + 174) * q^66 + (3*b8 - 12*b7 - 14*b5 + 28*b4 - 36*b3 - 106*b2 + 212) * q^67 + (-6*b9 + 3*b8 - 5*b7 + 10*b6 + 15*b5 - 15*b3 - 120*b2 + 30*b1) * q^68 + (9*b9 + 9*b8 - 6*b7 + 3*b6 - 3*b3 + 36*b2 - 3*b1 - 36) * q^69 + (-3*b9 - 15*b7 + 15*b6 + 16*b5 - 8*b4 - 1004*b2 + 135*b1 + 502) * q^70 + (5*b9 - 5*b8 - b6 - 4*b5 - 4*b4 + 95*b3 + 108*b2 - 95*b1 + 108) * q^71 + (9*b9 - 9*b8 + 9*b6 + 9*b5 + 9*b4 + 63*b3 - 63*b1) * q^72 + (20*b5 - 10*b4 - 114*b2 - 72*b1 + 57) * q^73 + (-12*b9 - 12*b8 + 28*b7 - 14*b6 + 39*b5 - 39*b4 - 2*b3 - 438*b2 - 2*b1 + 438) * q^74 + (-3*b7 + 6*b6 + 6*b5 + 63*b3 + 288*b2 - 126*b1) * q^75 + (-b8 - 21*b7 + 6*b5 - 12*b4 - 9*b3 + 346*b2 - 692) * q^76 + (-2*b9 + 4*b8 - 8*b7 - 8*b6 - 66*b4 - 112*b3 + 56*b1 - 432) * q^77 + (-6*b9 + 9*b8 + 3*b7 - 3*b5 - 21*b4 + 6*b3 - 144*b2 + 84*b1 + 306) * q^78 + (-2*b9 + 4*b8 - 13*b7 - 13*b6 - 2*b4 + 126*b3 - 63*b1 + 110) * q^79 + (4*b8 + 6*b7 + 13*b5 - 26*b4 + 158*b3 - 562*b2 + 1124) * q^80 - 81*b2 * q^81 + (-2*b9 - 2*b8 + 16*b7 - 8*b6 - 5*b5 + 5*b4 + 36*b3 + 478*b2 + 36*b1 - 478) * q^82 + (-19*b9 - 37*b7 + 37*b6 - 20*b5 + 10*b4 - 344*b2 + 35*b1 + 172) * q^83 + (3*b9 - 3*b8 + 27*b6 + 12*b5 + 12*b4 + 135*b3 - 30*b2 - 135*b1 - 30) * q^84 + (21*b9 - 21*b8 + 42*b6 + 14*b5 + 14*b4 - 54*b3 + 35*b2 + 54*b1 + 35) * q^85 + (24*b9 - 18*b7 + 18*b6 - 42*b5 + 21*b4 + 372*b2 - 77*b1 - 186) * q^86 + (18*b7 - 9*b6 - 18*b5 + 18*b4 + 33*b3 + 297*b2 + 33*b1 - 297) * q^87 + (-46*b9 + 23*b8 - 7*b7 + 14*b6 - 74*b5 - 249*b3 + 634*b2 + 498*b1) * q^88 + (10*b8 - 6*b7 + 4*b5 - 8*b4 + 118*b3 - 136*b2 + 272) * q^89 + (-9*b9 + 18*b8 - 9*b7 - 9*b6 + 9*b4 - 54*b3 + 27*b1 - 72) * q^90 + (3*b9 - 26*b8 + 50*b7 - 25*b6 - 22*b5 + 32*b4 - 123*b3 + 42*b2 + 93*b1 + 456) * q^91 + (-7*b9 + 14*b8 - 29*b7 - 29*b6 + 6*b4 - 306*b3 + 153*b1 - 174) * q^92 + (6*b8 + 9*b7 + 18*b5 - 36*b4 - 27*b3 + 66*b2 - 132) * q^93 + (2*b9 - b8 + b7 - 2*b6 + 62*b5 + 33*b3 - 646*b2 - 66*b1) * q^94 + (-31*b9 - 31*b8 - 26*b7 + 13*b6 + 60*b5 - 60*b4 - 25*b3 + 276*b2 - 25*b1 - 276) * q^95 + (9*b9 - 9*b7 + 9*b6 + 18*b5 - 9*b4 - 576*b2 - 153*b1 + 288) * q^96 + (-22*b9 + 22*b8 - 3*b6 + 32*b5 + 32*b4 + 81*b3 - 250*b2 - 81*b1 - 250) * q^97 + (-33*b9 + 33*b8 - 21*b6 - 15*b5 - 15*b4 + 11*b3 + 12*b2 - 11*b1 + 12) * q^98 + (9*b9 + 9*b7 - 9*b6 - 36*b5 + 18*b4 + 72*b2 + 45*b1 - 36) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 15 q^{3} + 30 q^{4} + 30 q^{7} - 45 q^{9}+O(q^{10})$$ 10 * q - 15 * q^3 + 30 * q^4 + 30 * q^7 - 45 * q^9 $$10 q - 15 q^{3} + 30 q^{4} + 30 q^{7} - 45 q^{9} + 40 q^{10} + 60 q^{11} - 180 q^{12} + 25 q^{13} - 60 q^{14} + 45 q^{15} - 250 q^{16} + 105 q^{17} + 180 q^{19} + 510 q^{20} - 290 q^{22} - 60 q^{23} - 960 q^{25} - 30 q^{26} + 270 q^{27} + 150 q^{28} - 495 q^{29} + 120 q^{30} + 1440 q^{32} - 180 q^{33} + 60 q^{35} + 270 q^{36} - 405 q^{37} - 1380 q^{38} + 345 q^{39} + 2000 q^{40} + 1065 q^{41} + 90 q^{42} - 370 q^{43} - 135 q^{45} - 390 q^{46} - 750 q^{48} + 775 q^{49} - 4320 q^{50} - 630 q^{51} + 2940 q^{52} + 330 q^{53} - 260 q^{55} - 2670 q^{56} + 2040 q^{58} + 780 q^{59} - 1375 q^{61} - 780 q^{62} - 270 q^{63} - 3140 q^{64} + 1605 q^{65} + 1740 q^{66} + 1590 q^{67} - 600 q^{68} - 180 q^{69} + 1620 q^{71} + 2190 q^{74} + 1440 q^{75} - 5190 q^{76} - 4320 q^{77} + 2340 q^{78} + 1100 q^{79} + 8430 q^{80} - 405 q^{81} - 2390 q^{82} - 450 q^{84} + 525 q^{85} - 1485 q^{87} + 3170 q^{88} + 2040 q^{89} - 720 q^{90} + 4770 q^{91} - 1740 q^{92} - 990 q^{93} - 3230 q^{94} - 1380 q^{95} - 3750 q^{97} + 180 q^{98}+O(q^{100})$$ 10 * q - 15 * q^3 + 30 * q^4 + 30 * q^7 - 45 * q^9 + 40 * q^10 + 60 * q^11 - 180 * q^12 + 25 * q^13 - 60 * q^14 + 45 * q^15 - 250 * q^16 + 105 * q^17 + 180 * q^19 + 510 * q^20 - 290 * q^22 - 60 * q^23 - 960 * q^25 - 30 * q^26 + 270 * q^27 + 150 * q^28 - 495 * q^29 + 120 * q^30 + 1440 * q^32 - 180 * q^33 + 60 * q^35 + 270 * q^36 - 405 * q^37 - 1380 * q^38 + 345 * q^39 + 2000 * q^40 + 1065 * q^41 + 90 * q^42 - 370 * q^43 - 135 * q^45 - 390 * q^46 - 750 * q^48 + 775 * q^49 - 4320 * q^50 - 630 * q^51 + 2940 * q^52 + 330 * q^53 - 260 * q^55 - 2670 * q^56 + 2040 * q^58 + 780 * q^59 - 1375 * q^61 - 780 * q^62 - 270 * q^63 - 3140 * q^64 + 1605 * q^65 + 1740 * q^66 + 1590 * q^67 - 600 * q^68 - 180 * q^69 + 1620 * q^71 + 2190 * q^74 + 1440 * q^75 - 5190 * q^76 - 4320 * q^77 + 2340 * q^78 + 1100 * q^79 + 8430 * q^80 - 405 * q^81 - 2390 * q^82 - 450 * q^84 + 525 * q^85 - 1485 * q^87 + 3170 * q^88 + 2040 * q^89 - 720 * q^90 + 4770 * q^91 - 1740 * q^92 - 990 * q^93 - 3230 * q^94 - 1380 * q^95 - 3750 * q^97 + 180 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 35\nu^{3} + 210\nu + 96 ) / 192$$ (v^5 + 35*v^3 + 210*v + 96) / 192 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 35\nu^{4} + 210\nu^{2} + 96\nu ) / 192$$ (v^6 + 35*v^4 + 210*v^2 + 96*v) / 192 $$\beta_{4}$$ $$=$$ $$\nu^{2} + 14$$ v^2 + 14 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 96\nu^{2} + 2940\nu + 1344 ) / 192$$ (v^7 + 49*v^5 + 700*v^3 + 96*v^2 + 2940*v + 1344) / 192 $$\beta_{6}$$ $$=$$ $$( \nu^{9} + 64\nu^{7} + 18\nu^{6} + 1309\nu^{5} + 918\nu^{4} + 9030\nu^{3} + 12132\nu^{2} + 15912\nu + 24192 ) / 1152$$ (v^9 + 64*v^7 + 18*v^6 + 1309*v^5 + 918*v^4 + 9030*v^3 + 12132*v^2 + 15912*v + 24192) / 1152 $$\beta_{7}$$ $$=$$ $$( -\nu^{9} - 64\nu^{7} + 18\nu^{6} - 1309\nu^{5} + 918\nu^{4} - 9030\nu^{3} + 12132\nu^{2} - 15912\nu + 24192 ) / 1152$$ (-v^9 - 64*v^7 + 18*v^6 - 1309*v^5 + 918*v^4 - 9030*v^3 + 12132*v^2 - 15912*v + 24192) / 1152 $$\beta_{8}$$ $$=$$ $$( - \nu^{9} + 6 \nu^{8} - 70 \nu^{7} + 366 \nu^{6} - 1603 \nu^{5} + 7008 \nu^{4} - 12654 \nu^{3} + 42840 \nu^{2} - 20304 \nu + 48384 ) / 1152$$ (-v^9 + 6*v^8 - 70*v^7 + 366*v^6 - 1603*v^5 + 7008*v^4 - 12654*v^3 + 42840*v^2 - 20304*v + 48384) / 1152 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} - 70\nu^{7} - 1603\nu^{5} - 12654\nu^{3} - 20304\nu ) / 576$$ (-v^9 - 70*v^7 - 1603*v^5 - 12654*v^3 - 20304*v) / 576
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - 14$$ b4 - 14 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - 23\beta_1$$ b9 - b7 + b6 + 2*b5 - b4 - 23*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{7} + 2\beta_{6} - 29\beta_{4} - 12\beta_{3} + 6\beta _1 + 322$$ 2*b7 + 2*b6 - 29*b4 - 12*b3 + 6*b1 + 322 $$\nu^{5}$$ $$=$$ $$-35\beta_{9} + 35\beta_{7} - 35\beta_{6} - 70\beta_{5} + 35\beta_{4} + 192\beta_{2} + 595\beta _1 - 96$$ -35*b9 + 35*b7 - 35*b6 - 70*b5 + 35*b4 + 192*b2 + 595*b1 - 96 $$\nu^{6}$$ $$=$$ $$-70\beta_{7} - 70\beta_{6} + 805\beta_{4} + 612\beta_{3} - 306\beta _1 - 8330$$ -70*b7 - 70*b6 + 805*b4 + 612*b3 - 306*b1 - 8330 $$\nu^{7}$$ $$=$$ $$1015 \beta_{9} - 1015 \beta_{7} + 1015 \beta_{6} + 2222 \beta_{5} - 1111 \beta_{4} - 9408 \beta_{2} - 15995 \beta _1 + 4704$$ 1015*b9 - 1015*b7 + 1015*b6 + 2222*b5 - 1111*b4 - 9408*b2 - 15995*b1 + 4704 $$\nu^{8}$$ $$=$$ $$- 96 \beta_{9} + 192 \beta_{8} + 1934 \beta_{7} + 1934 \beta_{6} - 22373 \beta_{4} - 23316 \beta_{3} + 11658 \beta _1 + 223930$$ -96*b9 + 192*b8 + 1934*b7 + 1934*b6 - 22373*b4 - 23316*b3 + 11658*b1 + 223930 $$\nu^{9}$$ $$=$$ $$- 28175 \beta_{9} + 27599 \beta_{7} - 27599 \beta_{6} - 68638 \beta_{5} + 34319 \beta_{4} + 350784 \beta_{2} + 436603 \beta _1 - 175392$$ -28175*b9 + 27599*b7 - 27599*b6 - 68638*b5 + 34319*b4 + 350784*b2 + 436603*b1 - 175392

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 5.36472i − 2.04224i − 0.917374i 3.27897i 5.04537i 5.36472i 2.04224i 0.917374i − 3.27897i − 5.04537i
−4.64599 2.68236i −1.50000 + 2.59808i 10.3901 + 17.9962i 2.69631i 13.9380 8.04709i 13.1657 7.60123i 68.5626i −4.50000 7.79423i −7.23249 + 12.5270i
4.2 −1.76863 1.02112i −1.50000 + 2.59808i −1.91462 3.31622i 12.0825i 5.30590 3.06336i −25.7533 + 14.8686i 24.1582i −4.50000 7.79423i 12.3377 21.3694i
4.3 −0.794469 0.458687i −1.50000 + 2.59808i −3.57921 6.19938i 15.4704i 2.38341 1.37606i 17.8257 10.2917i 13.9059i −4.50000 7.79423i −7.09608 + 12.2908i
4.4 2.83967 + 1.63949i −1.50000 + 2.59808i 1.37583 + 2.38302i 17.5414i −8.51902 + 4.91846i 23.1228 13.3499i 17.2091i −4.50000 7.79423i −28.7589 + 49.8119i
4.5 4.36942 + 2.52268i −1.50000 + 2.59808i 8.72787 + 15.1171i 20.1174i −13.1082 + 7.56805i −13.3609 + 7.71395i 47.7076i −4.50000 7.79423i 50.7498 87.9013i
10.1 −4.64599 + 2.68236i −1.50000 2.59808i 10.3901 17.9962i 2.69631i 13.9380 + 8.04709i 13.1657 + 7.60123i 68.5626i −4.50000 + 7.79423i −7.23249 12.5270i
10.2 −1.76863 + 1.02112i −1.50000 2.59808i −1.91462 + 3.31622i 12.0825i 5.30590 + 3.06336i −25.7533 14.8686i 24.1582i −4.50000 + 7.79423i 12.3377 + 21.3694i
10.3 −0.794469 + 0.458687i −1.50000 2.59808i −3.57921 + 6.19938i 15.4704i 2.38341 + 1.37606i 17.8257 + 10.2917i 13.9059i −4.50000 + 7.79423i −7.09608 12.2908i
10.4 2.83967 1.63949i −1.50000 2.59808i 1.37583 2.38302i 17.5414i −8.51902 4.91846i 23.1228 + 13.3499i 17.2091i −4.50000 + 7.79423i −28.7589 49.8119i
10.5 4.36942 2.52268i −1.50000 2.59808i 8.72787 15.1171i 20.1174i −13.1082 7.56805i −13.3609 7.71395i 47.7076i −4.50000 + 7.79423i 50.7498 + 87.9013i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 10.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.j.c 10
3.b odd 2 1 117.4.q.e 10
4.b odd 2 1 624.4.bv.h 10
13.c even 3 1 507.4.b.i 10
13.e even 6 1 inner 39.4.j.c 10
13.e even 6 1 507.4.b.i 10
13.f odd 12 2 507.4.a.r 10
39.h odd 6 1 117.4.q.e 10
39.k even 12 2 1521.4.a.bk 10
52.i odd 6 1 624.4.bv.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 1.a even 1 1 trivial
39.4.j.c 10 13.e even 6 1 inner
117.4.q.e 10 3.b odd 2 1
117.4.q.e 10 39.h odd 6 1
507.4.a.r 10 13.f odd 12 2
507.4.b.i 10 13.c even 3 1
507.4.b.i 10 13.e even 6 1
624.4.bv.h 10 4.b odd 2 1
624.4.bv.h 10 52.i odd 6 1
1521.4.a.bk 10 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 35T_{2}^{8} + 1015T_{2}^{6} - 288T_{2}^{5} - 7350T_{2}^{4} + 44100T_{2}^{2} + 60480T_{2} + 27648$$ acting on $$S_{4}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 35 T^{8} + 1015 T^{6} + \cdots + 27648$$
$3$ $$(T^{2} + 3 T + 9)^{5}$$
$5$ $$T^{10} + 1105 T^{8} + \cdots + 31632011568$$
$7$ $$T^{10} - 30 T^{9} + \cdots + 14692478786352$$
$11$ $$T^{10} - 60 T^{9} + \cdots + 50\!\cdots\!32$$
$13$ $$T^{10} - 25 T^{9} + \cdots + 51\!\cdots\!57$$
$17$ $$T^{10} + \cdots + 332127005819904$$
$19$ $$T^{10} - 180 T^{9} + \cdots + 19\!\cdots\!68$$
$23$ $$T^{10} + 60 T^{9} + \cdots + 66\!\cdots\!04$$
$29$ $$T^{10} + 495 T^{9} + \cdots + 18\!\cdots\!96$$
$31$ $$T^{10} + 116790 T^{8} + \cdots + 35\!\cdots\!00$$
$37$ $$T^{10} + 405 T^{9} + \cdots + 48\!\cdots\!32$$
$41$ $$T^{10} - 1065 T^{9} + \cdots + 36\!\cdots\!52$$
$43$ $$T^{10} + 370 T^{9} + \cdots + 51\!\cdots\!16$$
$47$ $$T^{10} + 181660 T^{8} + \cdots + 21\!\cdots\!28$$
$53$ $$(T^{5} - 165 T^{4} + \cdots - 46733997168)^{2}$$
$59$ $$T^{10} - 780 T^{9} + \cdots + 41\!\cdots\!68$$
$61$ $$T^{10} + 1375 T^{9} + \cdots + 87\!\cdots\!25$$
$67$ $$T^{10} - 1590 T^{9} + \cdots + 13\!\cdots\!88$$
$71$ $$T^{10} - 1620 T^{9} + \cdots + 71\!\cdots\!00$$
$73$ $$T^{10} + 600615 T^{8} + \cdots + 20\!\cdots\!75$$
$79$ $$(T^{5} - 550 T^{4} + \cdots - 920208867136)^{2}$$
$83$ $$T^{10} + 3406900 T^{8} + \cdots + 16\!\cdots\!68$$
$89$ $$T^{10} - 2040 T^{9} + \cdots + 28\!\cdots\!28$$
$97$ $$T^{10} + 3750 T^{9} + \cdots + 15\!\cdots\!68$$