# Properties

 Label 507.4.b.j Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $18$ 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: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + 88861676 x^{4} + 98825392 x^{2} + 26460736$$ x^18 + 97*x^16 + 3906*x^14 + 84743*x^12 + 1077128*x^10 + 8187552*x^8 + 36483705*x^6 + 88861676*x^4 + 98825392*x^2 + 26460736 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$13^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} - 3 q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{12} + 2 \beta_{10}) q^{5} - 3 \beta_{10} q^{6} + ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9}) q^{7} + ( - \beta_{17} + \beta_{16} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9}) q^{8} + 9 q^{9}+O(q^{10})$$ q + b10 * q^2 - 3 * q^3 + (b2 - 4) * q^4 + (-b12 + 2*b10) * q^5 - 3*b10 * q^6 + (-b14 + b11 - b10 - b9) * q^7 + (-b17 + b16 + b13 + b12 + b11 - 2*b10 + b9) * q^8 + 9 * q^9 $$q + \beta_{10} q^{2} - 3 q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{12} + 2 \beta_{10}) q^{5} - 3 \beta_{10} q^{6} + ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9}) q^{7} + ( - \beta_{17} + \beta_{16} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9}) q^{8} + 9 q^{9} + (\beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{3} + 2 \beta_{2} - 24) q^{10} + (\beta_{17} + \beta_{16} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 3 \beta_{10}) q^{11} + ( - 3 \beta_{2} + 12) q^{12} + (\beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 12) q^{14} + (3 \beta_{12} - 6 \beta_{10}) q^{15} + (\beta_{7} - \beta_{6} + 4 \beta_{5} - \beta_{4} - 7 \beta_{2} - \beta_1 + 9) q^{16} + (\beta_{8} + \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 17) q^{17} + 9 \beta_{10} q^{18} + (2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{12} + 5 \beta_{11} - 2 \beta_{10} - 7 \beta_{9}) q^{19} + (\beta_{17} + 5 \beta_{16} + 2 \beta_{13} - 27 \beta_{10} + 2 \beta_{9}) q^{20} + (3 \beta_{14} - 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9}) q^{21} + ( - 4 \beta_{8} - 3 \beta_{7} + 6 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 28) q^{22} + (2 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} - 25) q^{23} + (3 \beta_{17} - 3 \beta_{16} - 3 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{10} - 3 \beta_{9}) q^{24} + ( - 5 \beta_{8} + \beta_{7} - 4 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} + \cdots - 68) q^{25}+ \cdots + (9 \beta_{17} + 9 \beta_{16} + 9 \beta_{14} - 18 \beta_{13} - 18 \beta_{12} - 18 \beta_{11} - 27 \beta_{10}) q^{99}+O(q^{100})$$ q + b10 * q^2 - 3 * q^3 + (b2 - 4) * q^4 + (-b12 + 2*b10) * q^5 - 3*b10 * q^6 + (-b14 + b11 - b10 - b9) * q^7 + (-b17 + b16 + b13 + b12 + b11 - 2*b10 + b9) * q^8 + 9 * q^9 + (b8 - 2*b7 - b6 - b3 + 2*b2 - 24) * q^10 + (b17 + b16 + b14 - 2*b13 - 2*b12 - 2*b11 - 3*b10) * q^11 + (-3*b2 + 12) * q^12 + (b8 + b7 - b6 - 2*b4 - b3 - b2 - 2*b1 + 12) * q^14 + (3*b12 - 6*b10) * q^15 + (b7 - b6 + 4*b5 - b4 - 7*b2 - b1 + 9) * q^16 + (b8 + b6 - b5 + 3*b4 - 2*b3 - 3*b2 - 2*b1 + 17) * q^17 + 9*b10 * q^18 + (2*b16 - 2*b15 + 2*b14 + 2*b12 + 5*b11 - 2*b10 - 7*b9) * q^19 + (b17 + 5*b16 + 2*b13 - 27*b10 + 2*b9) * q^20 + (3*b14 - 3*b11 + 3*b10 + 3*b9) * q^21 + (-4*b8 - 3*b7 + 6*b5 + 2*b3 + 2*b2 + 6*b1 + 28) * q^22 + (2*b8 + 3*b7 - 2*b6 - 2*b5 - 3*b3 - b2 - 25) * q^23 + (3*b17 - 3*b16 - 3*b13 - 3*b12 - 3*b11 + 6*b10 - 3*b9) * q^24 + (-5*b8 + b7 - 4*b6 - 5*b5 + 5*b4 + 4*b3 + 5*b2 - 7*b1 - 68) * q^25 - 27 * q^27 + (17*b17 + b16 + 4*b15 + 11*b14 - 8*b13 - 11*b12 - 15*b11 - 4*b10 - 5*b9) * q^28 + (-5*b8 + 4*b7 + 5*b6 + 3*b5 + b4 + 5*b3 + 3*b2 - 6*b1 - 55) * q^29 + (-3*b8 + 6*b7 + 3*b6 + 3*b3 - 6*b2 + 72) * q^30 + (b17 - 4*b16 + 2*b15 - 6*b13 + 10*b12 + 3*b11 - 8*b10 + 6*b9) * q^31 + (7*b17 - 3*b16 - 3*b15 + 17*b14 - 6*b13 - 4*b12 - 17*b11 + 26*b10 - 8*b9) * q^32 + (-3*b17 - 3*b16 - 3*b14 + 6*b13 + 6*b12 + 6*b11 + 9*b10) * q^33 + (7*b17 + 2*b16 - 5*b15 + 11*b14 - 13*b13 + 7*b12 - 2*b11 + 24*b10 - 19*b9) * q^34 + (-7*b8 + 3*b7 + 2*b6 - 10*b5 - 6*b4 - 8*b3 - 4*b2 - 12*b1 + 14) * q^35 + (9*b2 - 36) * q^36 + (12*b17 - 9*b16 + 10*b15 + 9*b14 - 4*b13 - 2*b12 - 2*b11 - 17*b10 + b9) * q^37 + (5*b8 - 3*b7 - 12*b6 + 2*b5 - 12*b4 + 6*b3 - 15*b2 - b1 + 52) * q^38 + (4*b8 - 7*b7 - 5*b6 + 18*b5 - b3 - 39*b2 + 3*b1 + 170) * q^40 + (b17 - 5*b15 + 3*b14 + 10*b13 - 5*b12 + 6*b11 - 46*b10 + b9) * q^41 + (-3*b8 - 3*b7 + 3*b6 + 6*b4 + 3*b3 + 3*b2 + 6*b1 - 36) * q^42 + (-4*b8 + 2*b7 - 8*b6 - b5 + 7*b4 + 10*b3 + 17*b2 + b1 - 22) * q^43 + (-21*b17 - b16 - 14*b15 - 25*b14 + 9*b13 + 2*b12 + 20*b11 + 23*b10 - 2*b9) * q^44 + (-9*b12 + 18*b10) * q^45 + (17*b17 + b16 + 17*b15 + 21*b14 - 6*b13 - 18*b12 - 17*b11 - 41*b10 + 4*b9) * q^46 + (-16*b17 - 2*b16 + 10*b15 - 2*b14 + 2*b13 - 6*b12 + 17*b11 + 14*b10 - 11*b9) * q^47 + (-3*b7 + 3*b6 - 12*b5 + 3*b4 + 21*b2 + 3*b1 - 27) * q^48 + (-5*b8 + 2*b7 - 5*b6 - 21*b5 + 15*b4 - 5*b3 - 7*b2 - 6*b1 - 74) * q^49 + (3*b17 + 12*b16 + 20*b15 - b14 - 7*b13 - 10*b12 - 32*b11 - 84*b10 - 6*b9) * q^50 + (-3*b8 - 3*b6 + 3*b5 - 9*b4 + 6*b3 + 9*b2 + 6*b1 - 51) * q^51 + (-9*b8 + 6*b7 - 4*b6 - 4*b5 + 4*b4 + 16*b3 - 7*b2 - 4*b1 + 165) * q^53 - 27*b10 * q^54 + (-42*b8 + 19*b7 + 5*b6 + 9*b5 + 3*b4 + 35*b3 + 11*b2 + 6*b1 - 148) * q^55 + (-14*b8 + 6*b7 + 13*b6 + 8*b5 - 2*b4 + 27*b3 + 70*b2 + 19*b1 + 42) * q^56 + (-6*b16 + 6*b15 - 6*b14 - 6*b12 - 15*b11 + 6*b10 + 21*b9) * q^57 + (-8*b17 - 3*b16 - 24*b15 + 11*b14 - 13*b13 + 2*b12 - 6*b11 - 47*b10 - 22*b9) * q^58 + (-7*b17 - 16*b16 + 29*b15 - 14*b14 + 16*b13 - 5*b12 - 6*b11 + 13*b10 - 3*b9) * q^59 + (-3*b17 - 15*b16 - 6*b13 + 81*b10 - 6*b9) * q^60 + (-21*b8 + 4*b7 - 7*b6 + 29*b5 - 11*b4 + 4*b3 + 19*b2 - 2*b1 + 233) * q^61 + (-27*b8 + 9*b7 + b6 + 9*b5 + 8*b3 + 13*b2 - b1 + 32) * q^62 + (-9*b14 + 9*b11 - 9*b10 - 9*b9) * q^63 + (-b8 - 12*b7 + 3*b6 + b5 + 8*b4 + 24*b3 + 46*b2 + 29*b1 - 304) * q^64 + (12*b8 + 9*b7 - 18*b5 - 6*b3 - 6*b2 - 18*b1 - 84) * q^66 + (-31*b17 + 5*b16 - 3*b15 - 30*b14 + 10*b13 - 27*b12 - 10*b11 - 59*b10) * q^67 + (-10*b8 - 3*b7 - 3*b6 - 5*b5 + 4*b4 + 18*b3 + 54*b2 + 12*b1 - 204) * q^68 + (-6*b8 - 9*b7 + 6*b6 + 6*b5 + 9*b3 + 3*b2 + 75) * q^69 + (75*b17 + 23*b16 - 9*b15 + 66*b14 - 13*b13 - 24*b12 - 55*b11 + 29*b10 + 16*b9) * q^70 + (11*b17 + 23*b16 - 19*b15 + 10*b14 - 8*b13 - 13*b12 - 41*b11 + 67*b10 - 39*b9) * q^71 + (-9*b17 + 9*b16 + 9*b13 + 9*b12 + 9*b11 - 18*b10 + 9*b9) * q^72 + (-6*b17 + 4*b16 - 6*b15 + 22*b14 + 12*b13 + 23*b12 - 16*b11 - 80*b10 - 16*b9) * q^73 + (-20*b8 - 9*b7 + 11*b6 - 9*b5 + 10*b4 + 22*b3 + 56*b2 + 6*b1 + 86) * q^74 + (15*b8 - 3*b7 + 12*b6 + 15*b5 - 15*b4 - 12*b3 - 15*b2 + 21*b1 + 204) * q^75 + (45*b17 - 3*b16 + 32*b15 + 20*b14 - 15*b13 - 51*b12 - 65*b11 + 99*b10 - 13*b9) * q^76 + (-46*b8 + 7*b7 + 24*b6 - 3*b5 + 13*b4 + 26*b3 - 10*b2 - 19*b1 + 206) * q^77 + (40*b8 - 7*b7 + 7*b6 - 16*b5 - 26*b4 + 10*b3 + 34*b2 + 7*b1 - 34) * q^79 + (44*b17 - 12*b16 - 24*b15 + 16*b14 - 33*b13 - 7*b12 - 57*b11 + 154*b10 - 67*b9) * q^80 + 81 * q^81 + (28*b8 - 10*b7 - 14*b6 - 23*b5 + 11*b4 - 74*b2 - 13*b1 + 603) * q^82 + (-37*b17 - 7*b16 - 21*b15 - 37*b14 + 8*b13 + 7*b12 + 38*b11 - 44*b10 - 6*b9) * q^83 + (-51*b17 - 3*b16 - 12*b15 - 33*b14 + 24*b13 + 33*b12 + 45*b11 + 12*b10 + 15*b9) * q^84 + (b17 - 2*b16 + 27*b15 + 7*b14 - 60*b13 - 7*b12 - 28*b11 - 8*b10 - 69*b9) * q^85 + (-46*b17 + 5*b16 + 45*b15 - 43*b14 + 20*b13 - 12*b12 - 3*b11 - 93*b10 + 12*b9) * q^86 + (15*b8 - 12*b7 - 15*b6 - 9*b5 - 3*b4 - 15*b3 - 9*b2 + 18*b1 + 165) * q^87 + (22*b8 - 24*b7 - 39*b6 + 16*b5 - 19*b4 - 50*b3 - 36*b2 - 7*b1 + 49) * q^88 + (7*b17 - 3*b16 + 47*b15 - 20*b14 + 32*b13 - 39*b12 + 20*b11 + 37*b10 - 14*b9) * q^89 + (9*b8 - 18*b7 - 9*b6 - 9*b3 + 18*b2 - 216) * q^90 + (-17*b8 - 8*b7 + 7*b6 + 21*b5 + 14*b4 + 14*b3 + b2 + 45*b1 + 222) * q^92 + (-3*b17 + 12*b16 - 6*b15 + 18*b13 - 30*b12 - 9*b11 + 24*b10 - 18*b9) * q^93 + (29*b8 - 57*b7 - 42*b6 + 14*b5 - 50*b4 - 42*b3 - 59*b2 - 23*b1 - 58) * q^94 + (-25*b8 - 15*b7 - 5*b6 + 9*b5 - 27*b4 - 17*b3 - 38*b2 - 20*b1 + 30) * q^95 + (-21*b17 + 9*b16 + 9*b15 - 51*b14 + 18*b13 + 12*b12 + 51*b11 - 78*b10 + 24*b9) * q^96 + (11*b17 - 7*b16 - 11*b15 + 22*b14 + 4*b13 - 26*b12 - 34*b11 - 71*b10 - 4*b9) * q^97 + (22*b17 + 23*b16 + 50*b15 - 7*b14 - 11*b13 - 16*b12 - 10*b11 - 30*b10 - 16*b9) * q^98 + (9*b17 + 9*b16 + 9*b14 - 18*b13 - 18*b12 - 18*b11 - 27*b10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10})$$ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 $$18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100})$$ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + 88861676 x^{4} + 98825392 x^{2} + 26460736$$ :

 $$\beta_{1}$$ $$=$$ $$( - 337699465 \nu^{16} - 32409147909 \nu^{14} - 1282598456190 \nu^{12} - 27119085320007 \nu^{10} - 331625699797868 \nu^{8} + \cdots - 80\!\cdots\!56 ) / 84345031640064$$ (-337699465*v^16 - 32409147909*v^14 - 1282598456190*v^12 - 27119085320007*v^10 - 331625699797868*v^8 - 2361773027576816*v^6 - 9241389068385025*v^4 - 16763165801556616*v^2 - 8026346248675856) / 84345031640064 $$\beta_{2}$$ $$=$$ $$( 121443 \nu^{16} + 11006175 \nu^{14} + 406207162 \nu^{12} + 7852438557 \nu^{10} + 85165499404 \nu^{8} + 515475085464 \nu^{6} + \cdots + 737331060496 ) / 11028377568$$ (121443*v^16 + 11006175*v^14 + 406207162*v^12 + 7852438557*v^10 + 85165499404*v^8 + 515475085464*v^6 + 1625999513091*v^4 + 2222661123176*v^2 + 737331060496) / 11028377568 $$\beta_{3}$$ $$=$$ $$( 7448055 \nu^{16} + 701380891 \nu^{14} + 27024992866 \nu^{12} + 548115320953 \nu^{10} + 6268182584692 \nu^{8} + 40202257268304 \nu^{6} + \cdots + 68875575658864 ) / 499083027456$$ (7448055*v^16 + 701380891*v^14 + 27024992866*v^12 + 548115320953*v^10 + 6268182584692*v^8 + 40202257268304*v^6 + 135335190850367*v^4 + 201330790700376*v^2 + 68875575658864) / 499083027456 $$\beta_{4}$$ $$=$$ $$( 188631741 \nu^{16} + 17499465577 \nu^{14} + 664217725590 \nu^{12} + 13287339141715 \nu^{10} + 150391865567196 \nu^{8} + \cdots + 13\!\cdots\!76 ) / 6488079356928$$ (188631741*v^16 + 17499465577*v^14 + 664217725590*v^12 + 13287339141715*v^10 + 150391865567196*v^8 + 960793610862192*v^6 + 3244115832008405*v^4 + 4772771090255464*v^2 + 1370659636381776) / 6488079356928 $$\beta_{5}$$ $$=$$ $$( 1323696165 \nu^{16} + 123712051601 \nu^{14} + 4721212524742 \nu^{12} + 94654984523819 \nu^{10} + \cdots + 97\!\cdots\!32 ) / 42172515820032$$ (1323696165*v^16 + 123712051601*v^14 + 4721212524742*v^12 + 94654984523819*v^10 + 1068362441422492*v^8 + 6755156424533136*v^6 + 22343189096813149*v^4 + 31988356125916840*v^2 + 9755605735118032) / 42172515820032 $$\beta_{6}$$ $$=$$ $$( - 3050639663 \nu^{16} - 279441123251 \nu^{14} - 10408004776370 \nu^{12} - 202505950969409 \nu^{10} + \cdots - 14\!\cdots\!68 ) / 84345031640064$$ (-3050639663*v^16 - 279441123251*v^14 - 10408004776370*v^12 - 202505950969409*v^10 - 2201649168060372*v^8 - 13283953183925968*v^6 - 41498747524208055*v^4 - 55490691953015384*v^2 - 14900353096451568) / 84345031640064 $$\beta_{7}$$ $$=$$ $$( 3573406425 \nu^{16} + 324617832117 \nu^{14} + 11960996341246 \nu^{12} + 229305534031671 \nu^{10} + \cdots + 13\!\cdots\!92 ) / 84345031640064$$ (3573406425*v^16 + 324617832117*v^14 + 11960996341246*v^12 + 229305534031671*v^10 + 2439521061414028*v^8 + 14231034832637808*v^6 + 42166270146463569*v^4 + 52361748095856680*v^2 + 13515449077320592) / 84345031640064 $$\beta_{8}$$ $$=$$ $$( 992878663 \nu^{16} + 91675637363 \nu^{14} + 3455510144330 \nu^{12} + 68407025915321 \nu^{10} + 761940543186028 \nu^{8} + \cdots + 64\!\cdots\!08 ) / 21086257910016$$ (992878663*v^16 + 91675637363*v^14 + 3455510144330*v^12 + 68407025915321*v^10 + 761940543186028*v^8 + 4746736528581776*v^6 + 15415225572368783*v^4 + 21590892721723936*v^2 + 6412202646705808) / 21086257910016 $$\beta_{9}$$ $$=$$ $$( 3209941781 \nu^{17} - 208707089855 \nu^{15} - 31724147958522 \nu^{13} + \cdots + 53\!\cdots\!60 \nu ) / 10\!\cdots\!04$$ (3209941781*v^17 - 208707089855*v^15 - 31724147958522*v^13 - 1251532031668421*v^11 - 23696031437175140*v^9 - 237678070706066768*v^7 - 1197016910942859347*v^5 - 2212811911313403192*v^3 + 539827731169710160*v) / 108467710689122304 $$\beta_{10}$$ $$=$$ $$( - 59877421 \nu^{17} - 5495758441 \nu^{15} - 205573324326 \nu^{13} - 4029427467139 \nu^{11} - 44299174758284 \nu^{9} + \cdots - 172360206360464 \nu ) / 56729974209792$$ (-59877421*v^17 - 5495758441*v^15 - 205573324326*v^13 - 4029427467139*v^11 - 44299174758284*v^9 - 271203833596304*v^7 - 858748244111397*v^5 - 1138737236947544*v^3 - 172360206360464*v) / 56729974209792 $$\beta_{11}$$ $$=$$ $$( - 558686229419 \nu^{17} - 52805876911935 \nu^{15} + \cdots - 80\!\cdots\!24 \nu ) / 10\!\cdots\!04$$ (-558686229419*v^17 - 52805876911935*v^15 - 2044423654485946*v^13 - 41789072921757573*v^11 - 484880217302035364*v^9 - 3196817380717218256*v^7 - 11306587343182945427*v^5 - 18271690600947015544*v^3 - 8046782297326415024*v) / 108467710689122304 $$\beta_{12}$$ $$=$$ $$( 315204832455 \nu^{17} + 29976384016219 \nu^{15} + \cdots + 32\!\cdots\!52 \nu ) / 54\!\cdots\!52$$ (315204832455*v^17 + 29976384016219*v^15 + 1172453572398322*v^13 + 24325107163287433*v^11 + 287691647489885188*v^9 + 1933724913832608336*v^7 + 6876718512538540175*v^5 + 10547456021361190152*v^3 + 3280107605088258352*v) / 54233855344561152 $$\beta_{13}$$ $$=$$ $$( - 11333387329 \nu^{17} - 1052085600443 \nu^{15} - 39989575616904 \nu^{13} - 801712561391267 \nu^{11} + \cdots - 10\!\cdots\!80 \nu ) / 16\!\cdots\!36$$ (-11333387329*v^17 - 1052085600443*v^15 - 39989575616904*v^13 - 801712561391267*v^11 - 9097205719378862*v^9 - 58227877578500516*v^7 - 196976172587530637*v^5 - 296950452372422394*v^3 - 107041765798830680*v) / 1694807979517536 $$\beta_{14}$$ $$=$$ $$( 1101606689877 \nu^{17} + 100119548308961 \nu^{15} + \cdots + 56\!\cdots\!40 \nu ) / 10\!\cdots\!04$$ (1101606689877*v^17 + 100119548308961*v^15 + 3693649529292390*v^13 + 70989668581658555*v^11 + 758707381954452540*v^9 + 4461287830305744432*v^7 + 13400283268170213229*v^5 + 17148017724805012136*v^3 + 5684373809732825040*v) / 108467710689122304 $$\beta_{15}$$ $$=$$ $$( 59877421 \nu^{17} + 5495758441 \nu^{15} + 205573324326 \nu^{13} + 4029427467139 \nu^{11} + 44299174758284 \nu^{9} + \cdots + 233817678421072 \nu ) / 4727497850816$$ (59877421*v^17 + 5495758441*v^15 + 205573324326*v^13 + 4029427467139*v^11 + 44299174758284*v^9 + 271203833596304*v^7 + 858748244111397*v^5 + 1138737236947544*v^3 + 233817678421072*v) / 4727497850816 $$\beta_{16}$$ $$=$$ $$( - 694871382623 \nu^{17} - 64734895114835 \nu^{15} + \cdots - 24\!\cdots\!84 \nu ) / 54\!\cdots\!52$$ (-694871382623*v^17 - 64734895114835*v^15 - 2457315743140546*v^13 - 48821950667583473*v^11 - 542479073677032164*v^9 - 3336298632724181392*v^7 - 10492000269817653159*v^5 - 13554839283108738888*v^3 - 2487752893111838384*v) / 54233855344561152 $$\beta_{17}$$ $$=$$ $$( - 942769046151 \nu^{17} - 86802481993355 \nu^{15} + \cdots - 80\!\cdots\!00 \nu ) / 54\!\cdots\!52$$ (-942769046151*v^17 - 86802481993355*v^15 - 3258010330423586*v^13 - 64109174355542825*v^11 - 708458911096852628*v^9 - 4377908729532953616*v^7 - 14208123848162280847*v^5 - 20613625375714780344*v^3 - 8020854297444374000*v) / 54233855344561152
 $$\nu$$ $$=$$ $$( \beta_{15} + 12\beta_{10} ) / 13$$ (b15 + 12*b10) / 13 $$\nu^{2}$$ $$=$$ $$( -2\beta_{8} + 2\beta_{6} + 4\beta_{5} - 2\beta_{4} + 9\beta_{2} - 143 ) / 13$$ (-2*b8 + 2*b6 + 4*b5 - 2*b4 + 9*b2 - 143) / 13 $$\nu^{3}$$ $$=$$ $$( - 7 \beta_{17} + \beta_{16} - 28 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + 16 \beta_{12} + 13 \beta_{11} - 185 \beta_{10} - 2 \beta_{9} ) / 13$$ (-7*b17 + b16 - 28*b15 + 9*b14 + 10*b13 + 16*b12 + 13*b11 - 185*b10 - 2*b9) / 13 $$\nu^{4}$$ $$=$$ $$( 56 \beta_{8} - 7 \beta_{7} - 57 \beta_{6} - 104 \beta_{5} + 27 \beta_{4} + 16 \beta_{3} - 201 \beta_{2} - 13 \beta _1 + 2370 ) / 13$$ (56*b8 - 7*b7 - 57*b6 - 104*b5 + 27*b4 + 16*b3 - 201*b2 - 13*b1 + 2370) / 13 $$\nu^{5}$$ $$=$$ $$( 202 \beta_{17} - 15 \beta_{16} + 657 \beta_{15} - 264 \beta_{14} - 239 \beta_{13} - 433 \beta_{12} - 417 \beta_{11} + 3369 \beta_{10} + 185 \beta_{9} ) / 13$$ (202*b17 - 15*b16 + 657*b15 - 264*b14 - 239*b13 - 433*b12 - 417*b11 + 3369*b10 + 185*b9) / 13 $$\nu^{6}$$ $$=$$ $$( - 1395 \beta_{8} + 411 \beta_{7} + 1554 \beta_{6} + 2363 \beta_{5} - 255 \beta_{4} - 464 \beta_{3} + 4168 \beta_{2} + 312 \beta _1 - 44810 ) / 13$$ (-1395*b8 + 411*b7 + 1554*b6 + 2363*b5 - 255*b4 - 464*b3 + 4168*b2 + 312*b1 - 44810) / 13 $$\nu^{7}$$ $$=$$ $$( - 4822 \beta_{17} + 121 \beta_{16} - 15304 \beta_{15} + 6881 \beta_{14} + 4578 \beta_{13} + 10062 \beta_{12} + 10757 \beta_{11} - 65765 \beta_{10} - 6528 \beta_{9} ) / 13$$ (-4822*b17 + 121*b16 - 15304*b15 + 6881*b14 + 4578*b13 + 10062*b12 + 10757*b11 - 65765*b10 - 6528*b9) / 13 $$\nu^{8}$$ $$=$$ $$( 32087 \beta_{8} - 14276 \beta_{7} - 40359 \beta_{6} - 52468 \beta_{5} + 459 \beta_{4} + 12180 \beta_{3} - 85122 \beta_{2} - 5824 \beta _1 + 896522 ) / 13$$ (32087*b8 - 14276*b7 - 40359*b6 - 52468*b5 + 459*b4 + 12180*b3 - 85122*b2 - 5824*b1 + 896522) / 13 $$\nu^{9}$$ $$=$$ $$( 105570 \beta_{17} + 1662 \beta_{16} + 356170 \beta_{15} - 176667 \beta_{14} - 79473 \beta_{13} - 227711 \beta_{12} - 258916 \beta_{11} + 1333598 \beta_{10} + 186585 \beta_{9} ) / 13$$ (105570*b17 + 1662*b16 + 356170*b15 - 176667*b14 - 79473*b13 - 227711*b12 - 258916*b11 + 1333598*b10 + 186585*b9) / 13 $$\nu^{10}$$ $$=$$ $$( - 711946 \beta_{8} + 413713 \beta_{7} + 1010363 \beta_{6} + 1163967 \beta_{5} + 69984 \beta_{4} - 319728 \beta_{3} + 1735331 \beta_{2} + 91858 \beta _1 - 18500373 ) / 13$$ (-711946*b8 + 413713*b7 + 1010363*b6 + 1163967*b5 + 69984*b4 - 319728*b3 + 1735331*b2 + 91858*b1 - 18500373) / 13 $$\nu^{11}$$ $$=$$ $$( - 2199377 \beta_{17} - 115740 \beta_{16} - 8283196 \beta_{15} + 4502785 \beta_{14} + 1257981 \beta_{13} + 5134237 \beta_{12} + 6035110 \beta_{11} - 27731320 \beta_{10} - 4919299 \beta_{9} ) / 13$$ (-2199377*b17 - 115740*b16 - 8283196*b15 + 4502785*b14 + 1257981*b13 + 5134237*b12 + 6035110*b11 - 27731320*b10 - 4919299*b9) / 13 $$\nu^{12}$$ $$=$$ $$1199519 \beta_{8} - 848722 \beta_{7} - 1898215 \beta_{6} - 1996255 \beta_{5} - 219890 \beta_{4} + 642552 \beta_{3} - 2729022 \beta_{2} - 83357 \beta _1 + 29983267$$ 1199519*b8 - 848722*b7 - 1898215*b6 - 1996255*b5 - 219890*b4 + 642552*b3 - 2729022*b2 - 83357*b1 + 29983267 $$\nu^{13}$$ $$=$$ $$( 44331911 \beta_{17} + 3986983 \beta_{16} + 192525009 \beta_{15} - 113654892 \beta_{14} - 17203901 \beta_{13} - 115867557 \beta_{12} - 138289255 \beta_{11} + \cdots + 124653997 \beta_{9} ) / 13$$ (44331911*b17 + 3986983*b16 + 192525009*b15 - 113654892*b14 - 17203901*b13 - 115867557*b12 - 138289255*b11 + 587478215*b10 + 124653997*b9) / 13 $$\nu^{14}$$ $$=$$ $$( - 340897078 \beta_{8} + 281393164 \beta_{7} + 592974020 \beta_{6} + 582481917 \beta_{5} + 85741823 \beta_{4} - 215398820 \beta_{3} + 728595253 \beta_{2} + \cdots - 8342751040 ) / 13$$ (-340897078*b8 + 281393164*b7 + 592974020*b6 + 582481917*b5 + 85741823*b4 - 215398820*b3 + 728595253*b2 + 1803373*b1 - 8342751040) / 13 $$\nu^{15}$$ $$=$$ $$( - 871319300 \beta_{17} - 115322050 \beta_{16} - 4473577903 \beta_{15} + 2837436494 \beta_{14} + 160656480 \beta_{13} + 2619910452 \beta_{12} + 3139492330 \beta_{11} + \cdots - 3087034420 \beta_{9} ) / 13$$ (-871319300*b17 - 115322050*b16 - 4473577903*b15 + 2837436494*b14 + 160656480*b13 + 2619910452*b12 + 3139492330*b11 - 12628880592*b10 - 3087034420*b9) / 13 $$\nu^{16}$$ $$=$$ $$( 7476471422 \beta_{8} - 6987092704 \beta_{7} - 14096453402 \beta_{6} - 13163855848 \beta_{5} - 2298728158 \beta_{4} + 5468329428 \beta_{3} + \cdots + 180847969527 ) / 13$$ (7476471422*b8 - 6987092704*b7 - 14096453402*b6 - 13163855848*b5 - 2298728158*b4 + 5468329428*b3 - 15040607081*b2 + 455677404*b1 + 180847969527) / 13 $$\nu^{17}$$ $$=$$ $$( 16749505559 \beta_{17} + 3103545651 \beta_{16} + 103949299448 \beta_{15} - 70086310441 \beta_{14} + 962989750 \beta_{13} - 59370912020 \beta_{12} + \cdots + 75325281294 \beta_{9} ) / 13$$ (16749505559*b17 + 3103545651*b16 + 103949299448*b15 - 70086310441*b14 + 962989750*b13 - 59370912020*b12 - 70938721257*b11 + 274726879013*b10 + 75325281294*b9) / 13

## 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
 − 4.48584i − 3.76649i − 4.82618i − 2.37739i − 4.23649i − 1.73419i − 2.86460i − 0.614643i − 2.05129i 2.05129i 0.614643i 2.86460i 1.73419i 4.23649i 2.37739i 4.82618i 3.76649i 4.48584i
5.48584i −3.00000 −22.0945 13.3185i 16.4575i 21.4234i 77.3200i 9.00000 −73.0635
337.2 4.76649i −3.00000 −14.7194 18.8390i 14.2995i 23.8593i 32.0282i 9.00000 −89.7961
337.3 3.82618i −3.00000 −6.63963 0.275426i 11.4785i 0.0981245i 5.20502i 9.00000 1.05383
337.4 3.37739i −3.00000 −3.40677 15.7127i 10.1322i 17.1681i 15.5131i 9.00000 53.0679
337.5 3.23649i −3.00000 −2.47490 13.5815i 9.70948i 1.42933i 17.8820i 9.00000 −43.9566
337.6 2.73419i −3.00000 0.524213 21.1246i 8.20257i 25.8618i 23.3068i 9.00000 −57.7586
337.7 1.86460i −3.00000 4.52327 2.36060i 5.59380i 4.86461i 23.3509i 9.00000 −4.40158
337.8 1.61464i −3.00000 5.39293 1.20859i 4.84393i 28.2769i 21.6248i 9.00000 −1.95145
337.9 1.05129i −3.00000 6.89480 17.8886i 3.15386i 30.1975i 15.6587i 9.00000 18.8061
337.10 1.05129i −3.00000 6.89480 17.8886i 3.15386i 30.1975i 15.6587i 9.00000 18.8061
337.11 1.61464i −3.00000 5.39293 1.20859i 4.84393i 28.2769i 21.6248i 9.00000 −1.95145
337.12 1.86460i −3.00000 4.52327 2.36060i 5.59380i 4.86461i 23.3509i 9.00000 −4.40158
337.13 2.73419i −3.00000 0.524213 21.1246i 8.20257i 25.8618i 23.3068i 9.00000 −57.7586
337.14 3.23649i −3.00000 −2.47490 13.5815i 9.70948i 1.42933i 17.8820i 9.00000 −43.9566
337.15 3.37739i −3.00000 −3.40677 15.7127i 10.1322i 17.1681i 15.5131i 9.00000 53.0679
337.16 3.82618i −3.00000 −6.63963 0.275426i 11.4785i 0.0981245i 5.20502i 9.00000 1.05383
337.17 4.76649i −3.00000 −14.7194 18.8390i 14.2995i 23.8593i 32.0282i 9.00000 −89.7961
337.18 5.48584i −3.00000 −22.0945 13.3185i 16.4575i 21.4234i 77.3200i 9.00000 −73.0635
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.18 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.j 18
13.b even 2 1 inner 507.4.b.j 18
13.d odd 4 1 507.4.a.n 9
13.d odd 4 1 507.4.a.q yes 9
39.f even 4 1 1521.4.a.be 9
39.f even 4 1 1521.4.a.bj 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.n 9 13.d odd 4 1
507.4.a.q yes 9 13.d odd 4 1
507.4.b.j 18 1.a even 1 1 trivial
507.4.b.j 18 13.b even 2 1 inner
1521.4.a.be 9 39.f even 4 1
1521.4.a.bj 9 39.f even 4 1

## Hecke kernels

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

 $$T_{2}^{18} + 104 T_{2}^{16} + 4432 T_{2}^{14} + 101051 T_{2}^{12} + 1349342 T_{2}^{10} + 10831309 T_{2}^{8} + 51496229 T_{2}^{6} + 137234244 T_{2}^{4} + 182579488 T_{2}^{2} + 89567296$$ T2^18 + 104*T2^16 + 4432*T2^14 + 101051*T2^12 + 1349342*T2^10 + 10831309*T2^8 + 51496229*T2^6 + 137234244*T2^4 + 182579488*T2^2 + 89567296 $$T_{5}^{18} + 1737 T_{5}^{16} + 1231596 T_{5}^{14} + 456756349 T_{5}^{12} + 93726274635 T_{5}^{10} + 10183773140289 T_{5}^{8} + 477991112299211 T_{5}^{6} + \cdots + 252800110296721$$ T5^18 + 1737*T5^16 + 1231596*T5^14 + 456756349*T5^12 + 93726274635*T5^10 + 10183773140289*T5^8 + 477991112299211*T5^6 + 2993277342270618*T5^4 + 3556793465272632*T5^2 + 252800110296721

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + 104 T^{16} + \cdots + 89567296$$
$3$ $$(T + 3)^{18}$$
$5$ $$T^{18} + \cdots + 252800110296721$$
$7$ $$T^{18} + 3729 T^{16} + \cdots + 17\!\cdots\!41$$
$11$ $$T^{18} + 14639 T^{16} + \cdots + 96\!\cdots\!89$$
$13$ $$T^{18}$$
$17$ $$(T^{9} - 134 T^{8} + \cdots - 7231752990904)^{2}$$
$19$ $$T^{18} + 73518 T^{16} + \cdots + 68\!\cdots\!24$$
$23$ $$(T^{9} + 226 T^{8} + \cdots - 91\!\cdots\!88)^{2}$$
$29$ $$(T^{9} + 547 T^{8} + \cdots - 12\!\cdots\!83)^{2}$$
$31$ $$T^{18} + 221401 T^{16} + \cdots + 46\!\cdots\!21$$
$37$ $$T^{18} + 447596 T^{16} + \cdots + 27\!\cdots\!16$$
$41$ $$T^{18} + 634682 T^{16} + \cdots + 27\!\cdots\!04$$
$43$ $$(T^{9} + 158 T^{8} + \cdots + 14\!\cdots\!52)^{2}$$
$47$ $$T^{18} + 1375726 T^{16} + \cdots + 72\!\cdots\!84$$
$53$ $$(T^{9} - 1399 T^{8} + \cdots + 26\!\cdots\!69)^{2}$$
$59$ $$T^{18} + 2111401 T^{16} + \cdots + 42\!\cdots\!84$$
$61$ $$(T^{9} - 2092 T^{8} + \cdots - 18\!\cdots\!04)^{2}$$
$67$ $$T^{18} + 2788802 T^{16} + \cdots + 15\!\cdots\!96$$
$71$ $$T^{18} + 4208996 T^{16} + \cdots + 23\!\cdots\!56$$
$73$ $$T^{18} + 3574601 T^{16} + \cdots + 25\!\cdots\!29$$
$79$ $$(T^{9} + 115 T^{8} + \cdots - 63\!\cdots\!89)^{2}$$
$83$ $$T^{18} + 3486681 T^{16} + \cdots + 30\!\cdots\!01$$
$89$ $$T^{18} + 5490810 T^{16} + \cdots + 10\!\cdots\!24$$
$97$ $$T^{18} + 3787859 T^{16} + \cdots + 65\!\cdots\!61$$