# Properties

 Label 507.4.b.i Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$10$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} - 6) q^{4} + ( - \beta_{8} - \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} + \beta_{6} - 7 \beta_1) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 + 3 * q^3 + (b4 - 6) * q^4 + (-b8 - b2 + b1) * q^5 + 3*b1 * q^6 + (-b9 + 2*b2) * q^7 + (b9 + b8 + b6 - 7*b1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} - 6) q^{4} + ( - \beta_{8} - \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} + \beta_{6} - 7 \beta_1) q^{8} + 9 q^{9} + (\beta_{7} - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 8) q^{10} + ( - \beta_{9} + \beta_{8} + 2 \beta_{6} - 4 \beta_{2} - 5 \beta_1) q^{11} + (3 \beta_{4} - 18) q^{12} + ( - 2 \beta_{5} + 5 \beta_{4} + \beta_{3} - 6) q^{14} + ( - 3 \beta_{8} - 3 \beta_{2} + 3 \beta_1) q^{15} + (2 \beta_{5} - 7 \beta_{4} - 6 \beta_{3} + 50) q^{16} + ( - \beta_{7} - 4 \beta_{3} - 21) q^{17} + 9 \beta_1 q^{18} + (\beta_{9} + 3 \beta_{8} + 4 \beta_{6} + 12 \beta_{2} + 9 \beta_1) q^{19} + (4 \beta_{9} + 2 \beta_{8} - 5 \beta_{6} + 34 \beta_{2} - 18 \beta_1) q^{20} + ( - 3 \beta_{9} + 6 \beta_{2}) q^{21} + (\beta_{7} - 3 \beta_{5} - 5 \beta_{4} - 21 \beta_{3} + 58) q^{22} + ( - 3 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 12) q^{23} + (3 \beta_{9} + 3 \beta_{8} + 3 \beta_{6} - 21 \beta_1) q^{24} + ( - \beta_{5} - \beta_{4} - 21 \beta_{3} - 96) q^{25} + 27 q^{27} + (\beta_{9} + 9 \beta_{8} + 4 \beta_{6} - 10 \beta_{2} - 45 \beta_1) q^{28} + ( - 3 \beta_{5} - 3 \beta_{4} - 11 \beta_{3} + 99) q^{29} + (3 \beta_{7} - 3 \beta_{5} + 6 \beta_{4} - 9 \beta_{3} - 24) q^{30} + (2 \beta_{9} - 3 \beta_{8} + 6 \beta_{6} + 22 \beta_{2} - 9 \beta_1) q^{31} + ( - 3 \beta_{9} - 3 \beta_{8} - 3 \beta_{6} + 96 \beta_{2} + 51 \beta_1) q^{32} + ( - 3 \beta_{9} + 3 \beta_{8} + 6 \beta_{6} - 12 \beta_{2} - 15 \beta_1) q^{33} + ( - 6 \beta_{8} - \beta_{6} + 50 \beta_{2} - 18 \beta_1) q^{34} + ( - 3 \beta_{7} + 5 \beta_{5} + 13 \beta_{4} - 35 \beta_{3} - 12) q^{35} + (9 \beta_{4} - 54) q^{36} + ( - \beta_{9} - 12 \beta_{8} + 27 \beta_{2} + 36 \beta_1) q^{37} + (\beta_{7} + \beta_{5} - 7 \beta_{4} - 17 \beta_{3} - 138) q^{38} + (\beta_{7} + 7 \beta_{5} - 14 \beta_{4} + 63 \beta_{3} + 200) q^{40} + ( - 3 \beta_{9} - 8 \beta_{8} - 6 \beta_{6} - 71 \beta_{2} - 32 \beta_1) q^{41} + ( - 6 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} - 18) q^{42} + (\beta_{7} + 10 \beta_{5} - 4 \beta_{4} + 18 \beta_{3} + 74) q^{43} + ( - 7 \beta_{9} + 15 \beta_{8} - 16 \beta_{6} + 250 \beta_{2} + 69 \beta_1) q^{44} + ( - 9 \beta_{8} - 9 \beta_{2} + 9 \beta_1) q^{45} + ( - 3 \beta_{9} - 21 \beta_{8} + 10 \beta_{6} - 26 \beta_{2} + 27 \beta_1) q^{46} + (\beta_{9} + 3 \beta_{8} + 4 \beta_{6} - 4 \beta_{2} - 47 \beta_1) q^{47} + (6 \beta_{5} - 21 \beta_{4} - 18 \beta_{3} + 150) q^{48} + (2 \beta_{7} + 7 \beta_{5} - 19 \beta_{4} + 3 \beta_{3} - 155) q^{49} + (\beta_{9} + \beta_{8} - 23 \beta_{6} + 288 \beta_{2} - 84 \beta_1) q^{50} + ( - 3 \beta_{7} - 12 \beta_{3} - 63) q^{51} + ( - 4 \beta_{7} + 9 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 33) q^{53} + 27 \beta_1 q^{54} + (\beta_{7} + 11 \beta_{5} + 35 \beta_{4} - 33 \beta_{3} + 52) q^{55} + ( - 5 \beta_{7} - 9 \beta_{5} - 27 \beta_{4} - 19 \beta_{3} + 534) q^{56} + (3 \beta_{9} + 9 \beta_{8} + 12 \beta_{6} + 36 \beta_{2} + 27 \beta_1) q^{57} + (3 \beta_{9} + 3 \beta_{8} - 17 \beta_{6} + 136 \beta_{2} + 135 \beta_1) q^{58} + ( - 4 \beta_{9} + 6 \beta_{8} - 16 \beta_{6} + 52 \beta_{2} - 26 \beta_1) q^{59} + (12 \beta_{9} + 6 \beta_{8} - 15 \beta_{6} + 102 \beta_{2} - 54 \beta_1) q^{60} + (\beta_{7} - 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 275) q^{61} + (9 \beta_{7} - 5 \beta_{5} - 28 \beta_{4} - 36 \beta_{3} + 156) q^{62} + ( - 9 \beta_{9} + 18 \beta_{2}) q^{63} + (10 \beta_{5} + 19 \beta_{4} + 66 \beta_{3} - 314) q^{64} + (3 \beta_{7} - 9 \beta_{5} - 15 \beta_{4} - 63 \beta_{3} + 174) q^{66} + (3 \beta_{9} + 12 \beta_{8} - 14 \beta_{6} - 106 \beta_{2} - 36 \beta_1) q^{67} + ( - 3 \beta_{7} - 5 \beta_{5} - 10 \beta_{4} + 15 \beta_{3} + 120) q^{68} + ( - 9 \beta_{7} + 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 36) q^{69} + (3 \beta_{9} - 15 \beta_{8} - 8 \beta_{6} + 502 \beta_{2} - 135 \beta_1) q^{70} + (5 \beta_{9} - \beta_{8} - 4 \beta_{6} + 108 \beta_{2} - 95 \beta_1) q^{71} + (9 \beta_{9} + 9 \beta_{8} + 9 \beta_{6} - 63 \beta_1) q^{72} + ( - 10 \beta_{6} + 57 \beta_{2} + 72 \beta_1) q^{73} + (12 \beta_{7} - 14 \beta_{5} + 53 \beta_{4} + 2 \beta_{3} - 438) q^{74} + ( - 3 \beta_{5} - 3 \beta_{4} - 63 \beta_{3} - 288) q^{75} + ( - \beta_{9} + 21 \beta_{8} + 6 \beta_{6} + 346 \beta_{2} - 9 \beta_1) q^{76} + (2 \beta_{7} - 8 \beta_{5} - 58 \beta_{4} - 56 \beta_{3} - 432) q^{77} + (2 \beta_{7} - 13 \beta_{5} + 11 \beta_{4} + 63 \beta_{3} + 110) q^{79} + (4 \beta_{9} - 6 \beta_{8} + 13 \beta_{6} - 562 \beta_{2} + 158 \beta_1) q^{80} + 81 q^{81} + (2 \beta_{7} - 8 \beta_{5} + 3 \beta_{4} - 36 \beta_{3} + 478) q^{82} + (19 \beta_{9} - 37 \beta_{8} + 10 \beta_{6} + 172 \beta_{2} - 35 \beta_1) q^{83} + (3 \beta_{9} + 27 \beta_{8} + 12 \beta_{6} - 30 \beta_{2} - 135 \beta_1) q^{84} + (21 \beta_{9} + 42 \beta_{8} + 14 \beta_{6} + 35 \beta_{2} + 54 \beta_1) q^{85} + ( - 24 \beta_{9} - 18 \beta_{8} + 21 \beta_{6} - 186 \beta_{2} + 77 \beta_1) q^{86} + ( - 9 \beta_{5} - 9 \beta_{4} - 33 \beta_{3} + 297) q^{87} + ( - 23 \beta_{7} - 7 \beta_{5} + 81 \beta_{4} + 249 \beta_{3} - 634) q^{88} + (10 \beta_{9} + 6 \beta_{8} + 4 \beta_{6} - 136 \beta_{2} + 118 \beta_1) q^{89} + (9 \beta_{7} - 9 \beta_{5} + 18 \beta_{4} - 27 \beta_{3} - 72) q^{90} + (7 \beta_{7} - 29 \beta_{5} + 35 \beta_{4} - 153 \beta_{3} - 174) q^{92} + (6 \beta_{9} - 9 \beta_{8} + 18 \beta_{6} + 66 \beta_{2} - 27 \beta_1) q^{93} + (\beta_{7} + \beta_{5} - 63 \beta_{4} - 33 \beta_{3} + 646) q^{94} + (31 \beta_{7} + 13 \beta_{5} + 47 \beta_{4} + 25 \beta_{3} + 276) q^{95} + ( - 9 \beta_{9} - 9 \beta_{8} - 9 \beta_{6} + 288 \beta_{2} + 153 \beta_1) q^{96} + ( - 22 \beta_{9} - 3 \beta_{8} + 32 \beta_{6} - 250 \beta_{2} - 81 \beta_1) q^{97} + ( - 33 \beta_{9} - 21 \beta_{8} - 15 \beta_{6} + 12 \beta_{2} - 11 \beta_1) q^{98} + ( - 9 \beta_{9} + 9 \beta_{8} + 18 \beta_{6} - 36 \beta_{2} - 45 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + 3 * q^3 + (b4 - 6) * q^4 + (-b8 - b2 + b1) * q^5 + 3*b1 * q^6 + (-b9 + 2*b2) * q^7 + (b9 + b8 + b6 - 7*b1) * q^8 + 9 * q^9 + (b7 - b5 + 2*b4 - 3*b3 - 8) * q^10 + (-b9 + b8 + 2*b6 - 4*b2 - 5*b1) * q^11 + (3*b4 - 18) * q^12 + (-2*b5 + 5*b4 + b3 - 6) * q^14 + (-3*b8 - 3*b2 + 3*b1) * q^15 + (2*b5 - 7*b4 - 6*b3 + 50) * q^16 + (-b7 - 4*b3 - 21) * q^17 + 9*b1 * q^18 + (b9 + 3*b8 + 4*b6 + 12*b2 + 9*b1) * q^19 + (4*b9 + 2*b8 - 5*b6 + 34*b2 - 18*b1) * q^20 + (-3*b9 + 6*b2) * q^21 + (b7 - 3*b5 - 5*b4 - 21*b3 + 58) * q^22 + (-3*b7 + b5 - b4 + b3 + 12) * q^23 + (3*b9 + 3*b8 + 3*b6 - 21*b1) * q^24 + (-b5 - b4 - 21*b3 - 96) * q^25 + 27 * q^27 + (b9 + 9*b8 + 4*b6 - 10*b2 - 45*b1) * q^28 + (-3*b5 - 3*b4 - 11*b3 + 99) * q^29 + (3*b7 - 3*b5 + 6*b4 - 9*b3 - 24) * q^30 + (2*b9 - 3*b8 + 6*b6 + 22*b2 - 9*b1) * q^31 + (-3*b9 - 3*b8 - 3*b6 + 96*b2 + 51*b1) * q^32 + (-3*b9 + 3*b8 + 6*b6 - 12*b2 - 15*b1) * q^33 + (-6*b8 - b6 + 50*b2 - 18*b1) * q^34 + (-3*b7 + 5*b5 + 13*b4 - 35*b3 - 12) * q^35 + (9*b4 - 54) * q^36 + (-b9 - 12*b8 + 27*b2 + 36*b1) * q^37 + (b7 + b5 - 7*b4 - 17*b3 - 138) * q^38 + (b7 + 7*b5 - 14*b4 + 63*b3 + 200) * q^40 + (-3*b9 - 8*b8 - 6*b6 - 71*b2 - 32*b1) * q^41 + (-6*b5 + 15*b4 + 3*b3 - 18) * q^42 + (b7 + 10*b5 - 4*b4 + 18*b3 + 74) * q^43 + (-7*b9 + 15*b8 - 16*b6 + 250*b2 + 69*b1) * q^44 + (-9*b8 - 9*b2 + 9*b1) * q^45 + (-3*b9 - 21*b8 + 10*b6 - 26*b2 + 27*b1) * q^46 + (b9 + 3*b8 + 4*b6 - 4*b2 - 47*b1) * q^47 + (6*b5 - 21*b4 - 18*b3 + 150) * q^48 + (2*b7 + 7*b5 - 19*b4 + 3*b3 - 155) * q^49 + (b9 + b8 - 23*b6 + 288*b2 - 84*b1) * q^50 + (-3*b7 - 12*b3 - 63) * q^51 + (-4*b7 + 9*b5 - 3*b4 - 3*b3 + 33) * q^53 + 27*b1 * q^54 + (b7 + 11*b5 + 35*b4 - 33*b3 + 52) * q^55 + (-5*b7 - 9*b5 - 27*b4 - 19*b3 + 534) * q^56 + (3*b9 + 9*b8 + 12*b6 + 36*b2 + 27*b1) * q^57 + (3*b9 + 3*b8 - 17*b6 + 136*b2 + 135*b1) * q^58 + (-4*b9 + 6*b8 - 16*b6 + 52*b2 - 26*b1) * q^59 + (12*b9 + 6*b8 - 15*b6 + 102*b2 - 54*b1) * q^60 + (b7 - 3*b5 + 7*b4 + 9*b3 + 275) * q^61 + (9*b7 - 5*b5 - 28*b4 - 36*b3 + 156) * q^62 + (-9*b9 + 18*b2) * q^63 + (10*b5 + 19*b4 + 66*b3 - 314) * q^64 + (3*b7 - 9*b5 - 15*b4 - 63*b3 + 174) * q^66 + (3*b9 + 12*b8 - 14*b6 - 106*b2 - 36*b1) * q^67 + (-3*b7 - 5*b5 - 10*b4 + 15*b3 + 120) * q^68 + (-9*b7 + 3*b5 - 3*b4 + 3*b3 + 36) * q^69 + (3*b9 - 15*b8 - 8*b6 + 502*b2 - 135*b1) * q^70 + (5*b9 - b8 - 4*b6 + 108*b2 - 95*b1) * q^71 + (9*b9 + 9*b8 + 9*b6 - 63*b1) * q^72 + (-10*b6 + 57*b2 + 72*b1) * q^73 + (12*b7 - 14*b5 + 53*b4 + 2*b3 - 438) * q^74 + (-3*b5 - 3*b4 - 63*b3 - 288) * q^75 + (-b9 + 21*b8 + 6*b6 + 346*b2 - 9*b1) * q^76 + (2*b7 - 8*b5 - 58*b4 - 56*b3 - 432) * q^77 + (2*b7 - 13*b5 + 11*b4 + 63*b3 + 110) * q^79 + (4*b9 - 6*b8 + 13*b6 - 562*b2 + 158*b1) * q^80 + 81 * q^81 + (2*b7 - 8*b5 + 3*b4 - 36*b3 + 478) * q^82 + (19*b9 - 37*b8 + 10*b6 + 172*b2 - 35*b1) * q^83 + (3*b9 + 27*b8 + 12*b6 - 30*b2 - 135*b1) * q^84 + (21*b9 + 42*b8 + 14*b6 + 35*b2 + 54*b1) * q^85 + (-24*b9 - 18*b8 + 21*b6 - 186*b2 + 77*b1) * q^86 + (-9*b5 - 9*b4 - 33*b3 + 297) * q^87 + (-23*b7 - 7*b5 + 81*b4 + 249*b3 - 634) * q^88 + (10*b9 + 6*b8 + 4*b6 - 136*b2 + 118*b1) * q^89 + (9*b7 - 9*b5 + 18*b4 - 27*b3 - 72) * q^90 + (7*b7 - 29*b5 + 35*b4 - 153*b3 - 174) * q^92 + (6*b9 - 9*b8 + 18*b6 + 66*b2 - 27*b1) * q^93 + (b7 + b5 - 63*b4 - 33*b3 + 646) * q^94 + (31*b7 + 13*b5 + 47*b4 + 25*b3 + 276) * q^95 + (-9*b9 - 9*b8 - 9*b6 + 288*b2 + 153*b1) * q^96 + (-22*b9 - 3*b8 + 32*b6 - 250*b2 - 81*b1) * q^97 + (-33*b9 - 21*b8 - 15*b6 + 12*b2 - 11*b1) * q^98 + (-9*b9 + 9*b8 + 18*b6 - 36*b2 - 45*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10})$$ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 $$10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100})$$ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 35\nu^{3} + 210\nu ) / 96$$ (v^5 + 35*v^3 + 210*v) / 96 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 35\nu^{4} + 210\nu^{2} ) / 96$$ (v^6 + 35*v^4 + 210*v^2) / 96 $$\beta_{4}$$ $$=$$ $$\nu^{2} + 14$$ v^2 + 14 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 51\nu^{4} + 706\nu^{2} + 1792 ) / 32$$ (v^6 + 51*v^4 + 706*v^2 + 1792) / 32 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 2940\nu ) / 96$$ (v^7 + 49*v^5 + 700*v^3 + 2940*v) / 96 $$\beta_{7}$$ $$=$$ $$( \nu^{8} + 61\nu^{6} + 1168\nu^{4} + 7140\nu^{2} + 8064 ) / 96$$ (v^8 + 61*v^6 + 1168*v^4 + 7140*v^2 + 8064) / 96 $$\beta_{8}$$ $$=$$ $$( \nu^{9} + 64\nu^{7} + 1309\nu^{5} + 9030\nu^{3} + 15912\nu ) / 576$$ (v^9 + 64*v^7 + 1309*v^5 + 9030*v^3 + 15912*v) / 576 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} - 70\nu^{7} - 1603\nu^{5} - 12654\nu^{3} - 20304\nu ) / 576$$ (-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 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + \beta_{6} - 23\beta_1$$ b9 + b8 + b6 - 23*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 31\beta_{4} - 6\beta_{3} + 322$$ 2*b5 - 31*b4 - 6*b3 + 322 $$\nu^{5}$$ $$=$$ $$-35\beta_{9} - 35\beta_{8} - 35\beta_{6} + 96\beta_{2} + 595\beta_1$$ -35*b9 - 35*b8 - 35*b6 + 96*b2 + 595*b1 $$\nu^{6}$$ $$=$$ $$-70\beta_{5} + 875\beta_{4} + 306\beta_{3} - 8330$$ -70*b5 + 875*b4 + 306*b3 - 8330 $$\nu^{7}$$ $$=$$ $$1015\beta_{9} + 1015\beta_{8} + 1111\beta_{6} - 4704\beta_{2} - 15995\beta_1$$ 1015*b9 + 1015*b8 + 1111*b6 - 4704*b2 - 15995*b1 $$\nu^{8}$$ $$=$$ $$96\beta_{7} + 1934\beta_{5} - 24307\beta_{4} - 11658\beta_{3} + 223930$$ 96*b7 + 1934*b5 - 24307*b4 - 11658*b3 + 223930 $$\nu^{9}$$ $$=$$ $$-28175\beta_{9} - 27599\beta_{8} - 34319\beta_{6} + 175392\beta_{2} + 436603\beta_1$$ -28175*b9 - 27599*b8 - 34319*b6 + 175392*b2 + 436603*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 5.36472i − 5.04537i − 3.27897i − 2.04224i − 0.917374i 0.917374i 2.04224i 3.27897i 5.04537i 5.36472i
5.36472i 3.00000 −20.7803 2.69631i 16.0942i 15.2025i 68.5626i 9.00000 14.4650
337.2 5.04537i 3.00000 −17.4557 20.1174i 15.1361i 15.4279i 47.7076i 9.00000 −101.500
337.3 3.27897i 3.00000 −2.75167 17.5414i 9.83692i 26.6999i 17.2091i 9.00000 57.5178
337.4 2.04224i 3.00000 3.82924 12.0825i 6.12673i 29.7373i 24.1582i 9.00000 −24.6753
337.5 0.917374i 3.00000 7.15843 15.4704i 2.75212i 20.5833i 13.9059i 9.00000 14.1922
337.6 0.917374i 3.00000 7.15843 15.4704i 2.75212i 20.5833i 13.9059i 9.00000 14.1922
337.7 2.04224i 3.00000 3.82924 12.0825i 6.12673i 29.7373i 24.1582i 9.00000 −24.6753
337.8 3.27897i 3.00000 −2.75167 17.5414i 9.83692i 26.6999i 17.2091i 9.00000 57.5178
337.9 5.04537i 3.00000 −17.4557 20.1174i 15.1361i 15.4279i 47.7076i 9.00000 −101.500
337.10 5.36472i 3.00000 −20.7803 2.69631i 16.0942i 15.2025i 68.5626i 9.00000 14.4650
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.10 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.b even 2 1 inner

## Twists

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{10} + 70T_{2}^{8} + 1645T_{2}^{6} + 14700T_{2}^{4} + 44100T_{2}^{2} + 27648$$ T2^10 + 70*T2^8 + 1645*T2^6 + 14700*T2^4 + 44100*T2^2 + 27648 $$T_{5}^{10} + 1105T_{5}^{8} + 441955T_{5}^{6} + 76029795T_{5}^{4} + 4880780280T_{5}^{2} + 31632011568$$ T5^10 + 1105*T5^8 + 441955*T5^6 + 76029795*T5^4 + 4880780280*T5^2 + 31632011568

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 70 T^{8} + 1645 T^{6} + \cdots + 27648$$
$3$ $$(T - 3)^{10}$$
$5$ $$T^{10} + 1105 T^{8} + \cdots + 31632011568$$
$7$ $$T^{10} + 2490 T^{8} + \cdots + 14692478786352$$
$11$ $$T^{10} + 10780 T^{8} + \cdots + 50\!\cdots\!32$$
$13$ $$T^{10}$$
$17$ $$(T^{5} + 105 T^{4} - 555 T^{3} + \cdots - 18224352)^{2}$$
$19$ $$T^{10} + 41340 T^{8} + \cdots + 19\!\cdots\!68$$
$23$ $$(T^{5} - 60 T^{4} - 33810 T^{3} + \cdots + 8153671248)^{2}$$
$29$ $$(T^{5} - 495 T^{4} + 69915 T^{3} + \cdots + 427627836)^{2}$$
$31$ $$T^{10} + 116790 T^{8} + \cdots + 35\!\cdots\!00$$
$37$ $$T^{10} + 237285 T^{8} + \cdots + 48\!\cdots\!32$$
$41$ $$T^{10} + 277405 T^{8} + \cdots + 36\!\cdots\!52$$
$43$ $$(T^{5} - 370 T^{4} + \cdots + 227329236796)^{2}$$
$47$ $$T^{10} + 181660 T^{8} + \cdots + 21\!\cdots\!28$$
$53$ $$(T^{5} - 165 T^{4} + \cdots - 46733997168)^{2}$$
$59$ $$T^{10} + 707800 T^{8} + \cdots + 41\!\cdots\!68$$
$61$ $$(T^{5} - 1375 T^{4} + \cdots - 933851008945)^{2}$$
$67$ $$T^{10} + 935610 T^{8} + \cdots + 13\!\cdots\!88$$
$71$ $$T^{10} + 930220 T^{8} + \cdots + 71\!\cdots\!00$$
$73$ $$T^{10} + 600615 T^{8} + \cdots + 20\!\cdots\!75$$
$79$ $$(T^{5} - 550 T^{4} + \cdots - 920208867136)^{2}$$
$83$ $$T^{10} + 3406900 T^{8} + \cdots + 16\!\cdots\!68$$
$89$ $$T^{10} + 1520560 T^{8} + \cdots + 28\!\cdots\!28$$
$97$ $$T^{10} + 4556430 T^{8} + \cdots + 15\!\cdots\!68$$