# Properties

 Label 120.2.m.b Level $120$ Weight $2$ Character orbit 120.m Analytic conductor $0.958$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 120.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.958204824255$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4$$ x^16 + 24*x^14 + 192*x^12 + 672*x^10 + 1092*x^8 + 880*x^6 + 352*x^4 + 64*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{13}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - \beta_{13} q^{3} - \beta_{11} q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14}) q^{7} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7}) q^{8} + (\beta_{11} + \beta_1 - 1) q^{9}+O(q^{10})$$ q - b3 * q^2 - b13 * q^3 - b11 * q^4 + b7 * q^5 + (-b9 + b5) * q^6 + (-b15 + b14) * q^7 + (b13 - b12 + b8 - b7) * q^8 + (b11 + b1 - 1) * q^9 $$q - \beta_{3} q^{2} - \beta_{13} q^{3} - \beta_{11} q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14}) q^{7} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7}) q^{8} + (\beta_{11} + \beta_1 - 1) q^{9} + ( - \beta_{14} + \beta_{11} + \beta_{10} - \beta_{9} + 1) q^{10} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} - \beta_1) q^{11} + ( - \beta_{15} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{6} - \beta_{2}) q^{12} + (\beta_{15} - \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{13} + (\beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{14} + (\beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{15} + 2 \beta_{9} q^{16} + ( - \beta_{13} + \beta_{12} + \beta_{3} - \beta_{2}) q^{17} + (\beta_{15} + \beta_{12} - \beta_{10} - \beta_{6} + \beta_{3} - \beta_{2}) q^{18} + ( - \beta_{11} + \beta_{10} - 2) q^{19} + ( - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{20} + (\beta_{11} - 2 \beta_{9} - \beta_{5} - \beta_{4}) q^{21} + (\beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{22} + ( - \beta_{8} + \beta_{7}) q^{23} + ( - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{24} + (\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} + \beta_{9} + 1) q^{25} + (2 \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{26} + (\beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{3} + \beta_{2}) q^{27} + (2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{28} + ( - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{29} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{30} + (\beta_{11} + \beta_{10} + 2 \beta_{9}) q^{31} + ( - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{8} - 2 \beta_{7}) q^{32} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{11} - \beta_{10} - \beta_{9} - 2) q^{34} + (\beta_{13} - \beta_{12} - \beta_{9} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{35} + ( - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{36} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{37} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7} + 2 \beta_{3} + 2 \beta_{2}) q^{38} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{3} + \beta_{2}) q^{39} + ( - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{40}+ \cdots + (3 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{5} - \beta_{4} - \beta_1 + 4) q^{99}+O(q^{100})$$ q - b3 * q^2 - b13 * q^3 - b11 * q^4 + b7 * q^5 + (-b9 + b5) * q^6 + (-b15 + b14) * q^7 + (b13 - b12 + b8 - b7) * q^8 + (b11 + b1 - 1) * q^9 + (-b14 + b11 + b10 - b9 + 1) * q^10 + (-b11 - b10 + b9 - b5 + b4 - b1) * q^11 + (-b15 - b12 + b11 - b9 - b6 - b2) * q^12 + (b15 - b14 - b11 + b10 + b9 - b8 + b7 + 2*b6) * q^13 + (b10 - b8 - b7 + b5 + b4 + b3 + b2 + b1) * q^14 + (b11 - b10 + 2*b8 - b7 - b6 - b5 - b4) * q^15 + 2*b9 * q^16 + (-b13 + b12 + b3 - b2) * q^17 + (b15 + b12 - b10 - b6 + b3 - b2) * q^18 + (-b11 + b10 - 2) * q^19 + (-b10 - b8 + b7 - b5 - b4 - b3 + b2 - b1) * q^20 + (b11 - 2*b9 - b5 - b4) * q^21 + (b13 + b12 - b11 + b10 + b9 - b8 + b7 + 2*b6) * q^22 + (-b8 + b7) * q^23 + (-b11 + b10 - b8 - b7 + b5 + b4 + b3 + b2 - b1 - 2) * q^24 + (b15 + b14 + b13 + b12 - 2*b10 + b9 + 1) * q^25 + (2*b11 - b9 + b8 + b7 - b5 - 3*b4 - b3 - b2 + b1) * q^26 + (b15 + b14 + 2*b13 - b12 - b11 - b10 + b9 - b3 + b2) * q^27 + (2*b14 + b13 + b12 - 2*b10 + b8 - b7 - 2*b6) * q^28 + (-b11 + b10 - b8 - b7 + 2*b5 + 2*b4 + b3 + b2) * q^29 + (b15 - b14 - b13 - b11 - b10 + 2*b9 + b6 + b4 - b1 + 1) * q^30 + (b11 + b10 + 2*b9) * q^31 + (-2*b13 + 2*b12 + 2*b8 - 2*b7) * q^32 + (-b15 - b14 - b13 + b12 + b11 + b10 - b9 - 2*b3 + 2*b2) * q^33 + (b11 - b10 - b9 - 2) * q^34 + (b13 - b12 - b9 + b5 - b4 + 2*b3 - 2*b2 - b1) * q^35 + (-b10 + b8 + b7 - b5 + b4 - b3 - b2 - b1 - 2) * q^36 + (b15 - b14 + b11 - b10 - b9 + b8 - b7 - 2*b6) * q^37 + (b13 - b12 + b8 - b7 + 2*b3 + 2*b2) * q^38 + (-2*b11 - 2*b10 + b9 - b8 - b7 + b3 + b2) * q^39 + (-b13 - b12 - 2*b11 + 2*b10 - b8 + b7 + 2*b6 + 2) * q^40 + (b11 + b10 - 2*b9 + 2*b5 - 2*b4) * q^41 + (b15 + b13 - 2*b12 - b11 + b9 - 3*b8 + 3*b7 + b6 + b2) * q^42 + (-2*b15 - 2*b14 - b13 - b12 + 2*b11 + 2*b10 - 2*b9) * q^43 + (2*b11 - 2*b5 - 2*b4 + 2*b1) * q^44 + (-b15 + b14 + b11 + 2*b10 + b9 + b8 - 2*b7 + b5 + b4) * q^45 + (b10 - b9) * q^46 + (b8 - b7) * q^47 + (-2*b15 + 2*b14 + 2*b3 + 2*b2) * q^48 + (-b11 + b10 + 1) * q^49 + (-b13 + b12 + b9 + 2*b7 - b5 + b4 - 2*b3 - 3*b2 + b1) * q^50 + (b9 - b5 + b4 + b1 + 2) * q^51 + (2*b15 - 2*b14 - 2*b13 - 2*b12) * q^52 + (-3*b8 + 3*b7 - 3*b3 - 3*b2) * q^53 + (-b11 + 2*b10 + 2*b9 + b8 + b7 + b4 - b3 - b2 + b1 + 2) * q^54 + (-b15 + b14 - 2*b10 - 4*b9 + b8 - b7 - 2*b6) * q^55 + (-2*b11 + 2*b9 + 4*b4) * q^56 + (-b15 - b14 + 2*b13 - 2*b12 + b11 + b10 - b9 + b3 - b2) * q^57 + (-2*b15 + 2*b14 + b13 + b12 + b11 - b10 - b9 + b8 - b7 - 2*b6) * q^58 + (b11 + b10 - b9 + b5 - b4 + b1) * q^59 + (-b15 - b13 + 2*b12 + b11 - b9 + b8 + b7 + b6 - 2*b4 - 2*b3 - 3*b2) * q^60 + (-3*b11 - 3*b10) * q^61 + (-3*b13 + 3*b12 + b8 - b7 + 2*b2) * q^62 + (2*b15 - 2*b14 - b11 + b10 + b9 + 2*b6 - 3*b3 - 3*b2) * q^63 - 4*b10 * q^64 + (3*b13 - 3*b12 - b11 - b10 + 2*b9 - 2*b5 + 2*b4 - b3 + b2) * q^65 + (-2*b11 - 2*b10 - b9 - b8 - b7 - b5 - b4 + b3 + b2 - b1 + 4) * q^66 + (b13 + b12) * q^67 + (-2*b8 + 2*b7 + 2*b3 - 2*b2) * q^68 + (-b10 + b8 + b7 - b5 - b4 - b3 - b2) * q^69 + (-2*b15 - 2*b13 - 2*b12 + 3*b11 + 2*b10 - 4) * q^70 + (-2*b11 + 2*b10 + 4*b5 + 4*b4) * q^71 + (-2*b14 + b13 + b12 + 2*b10 - 3*b8 + 3*b7 + 2*b6 + 2*b3) * q^72 + (-2*b13 - 2*b12) * q^73 + (-2*b11 - 2*b10 + b9 + b8 + b7 - b5 + b4 - b3 - b2 - 3*b1) * q^74 + (b15 + b14 - b13 + 2*b12 - 2*b11 + 2*b10 + b5 - b4 - b3 + b2 + b1 + 2) * q^75 + (2*b11 + 2*b9 + 4) * q^76 + (4*b8 - 4*b7 + 2*b3 + 2*b2) * q^77 + (2*b14 + b13 - b12 - b11 - b10 + b9 + 3*b8 - 3*b7 - 4*b2) * q^78 + (3*b11 + 3*b10 - 4*b9) * q^79 + (2*b13 - 2*b12 + 2*b11 + 2*b10 - 2*b9 + 2*b5 - 2*b4 - 2*b3 + 2*b2 + 2*b1) * q^80 + (-b11 + 3*b10 - 2*b9 + 2*b5 - 2*b4 - 3) * q^81 + (-2*b15 - 3*b13 - 3*b12 + 2*b11 - 2*b9 + b8 - b7 - 2*b6) * q^82 + (-b13 + b12 - 4*b3 + 4*b2) * q^83 + (b11 + 5*b10 - 2*b9 + b8 + b7 + b5 - b4 - b3 - b2 + b1 + 2) * q^84 + (-b15 + b14 - 2*b10 + b9 + b8 - b7 - 2*b6) * q^85 + (-b10 - b9 - 2*b8 - 2*b7 - 2*b4 + 2*b3 + 2*b2 - 2*b1) * q^86 + (b15 - b14 - 3*b8 + 3*b7 + 2*b3 + 2*b2) * q^87 + (4*b15 - 2*b11 - 2*b10 + 2*b9) * q^88 + (-2*b11 - 2*b10 + 2*b9 - 2*b5 + 2*b4 - 2*b1) * q^89 + (-b15 + b14 - 2*b13 + 3*b12 - b10 + b9 - b8 - b7 - b6 + b5 + b4 + b3 + 4*b2 + b1 - 1) * q^90 + (4*b11 - 4*b10 - 4) * q^91 + (b13 - b12 - b8 + b7 + 2*b2) * q^92 + (-b15 + b14 - b11 + b10 + b9 + 2*b6 + 3*b3 + 3*b2) * q^93 + (-b10 + b9) * q^94 + (b11 - b10 - 2*b8 - 2*b5 - 2*b4) * q^95 + (2*b11 + 2*b10 - 2*b8 - 2*b7 + 2*b5 + 2*b4 + 2*b3 + 2*b2 + 2*b1 + 4) * q^96 + (2*b15 + 2*b14 - 2*b11 - 2*b10 + 2*b9) * q^97 + (b13 - b12 + b8 - b7 - b3 + 2*b2) * q^98 + (3*b11 - 3*b10 - b9 + b5 - b4 - b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{9}+O(q^{10})$$ 16 * q - 16 * q^9 $$16 q - 16 q^{9} + 16 q^{10} - 32 q^{19} - 32 q^{24} + 16 q^{25} + 16 q^{30} - 32 q^{34} - 32 q^{36} + 32 q^{40} + 16 q^{49} + 32 q^{51} + 32 q^{54} + 64 q^{66} - 64 q^{70} + 32 q^{75} + 64 q^{76} - 48 q^{81} + 32 q^{84} - 16 q^{90} - 64 q^{91} + 64 q^{96} + 64 q^{99}+O(q^{100})$$ 16 * q - 16 * q^9 + 16 * q^10 - 32 * q^19 - 32 * q^24 + 16 * q^25 + 16 * q^30 - 32 * q^34 - 32 * q^36 + 32 * q^40 + 16 * q^49 + 32 * q^51 + 32 * q^54 + 64 * q^66 - 64 * q^70 + 32 * q^75 + 64 * q^76 - 48 * q^81 + 32 * q^84 - 16 * q^90 - 64 * q^91 + 64 * q^96 + 64 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{13} + 69\nu^{11} + 506\nu^{9} + 1488\nu^{7} + 1638\nu^{5} + 594\nu^{3} + 44\nu ) / 8$$ (3*v^13 + 69*v^11 + 506*v^9 + 1488*v^7 + 1638*v^5 + 594*v^3 + 44*v) / 8 $$\beta_{2}$$ $$=$$ $$( - 7 \nu^{14} + 3 \nu^{13} - 164 \nu^{12} + 69 \nu^{11} - 1250 \nu^{10} + 506 \nu^{9} - 3984 \nu^{8} + 1488 \nu^{7} - 5334 \nu^{6} + 1638 \nu^{5} - 3024 \nu^{4} + 594 \nu^{3} - 596 \nu^{2} + \cdots - 16 ) / 16$$ (-7*v^14 + 3*v^13 - 164*v^12 + 69*v^11 - 1250*v^10 + 506*v^9 - 3984*v^8 + 1488*v^7 - 5334*v^6 + 1638*v^5 - 3024*v^4 + 594*v^3 - 596*v^2 + 28*v - 16) / 16 $$\beta_{3}$$ $$=$$ $$( 7 \nu^{14} + 3 \nu^{13} + 164 \nu^{12} + 69 \nu^{11} + 1250 \nu^{10} + 506 \nu^{9} + 3984 \nu^{8} + 1488 \nu^{7} + 5334 \nu^{6} + 1638 \nu^{5} + 3024 \nu^{4} + 594 \nu^{3} + 596 \nu^{2} + 28 \nu + 16 ) / 16$$ (7*v^14 + 3*v^13 + 164*v^12 + 69*v^11 + 1250*v^10 + 506*v^9 + 3984*v^8 + 1488*v^7 + 5334*v^6 + 1638*v^5 + 3024*v^4 + 594*v^3 + 596*v^2 + 28*v + 16) / 16 $$\beta_{4}$$ $$=$$ $$( 5 \nu^{15} - 20 \nu^{14} + 107 \nu^{13} - 474 \nu^{12} + 657 \nu^{11} - 3698 \nu^{10} + 1074 \nu^{9} - 12334 \nu^{8} - 1686 \nu^{7} - 18152 \nu^{6} - 4770 \nu^{5} - 12164 \nu^{4} - 3014 \nu^{3} + \cdots - 300 ) / 16$$ (5*v^15 - 20*v^14 + 107*v^13 - 474*v^12 + 657*v^11 - 3698*v^10 + 1074*v^9 - 12334*v^8 - 1686*v^7 - 18152*v^6 - 4770*v^5 - 12164*v^4 - 3014*v^3 - 3428*v^2 - 484*v - 300) / 16 $$\beta_{5}$$ $$=$$ $$( - 5 \nu^{15} - 20 \nu^{14} - 107 \nu^{13} - 474 \nu^{12} - 657 \nu^{11} - 3698 \nu^{10} - 1074 \nu^{9} - 12334 \nu^{8} + 1686 \nu^{7} - 18152 \nu^{6} + 4770 \nu^{5} - 12164 \nu^{4} + \cdots - 300 ) / 16$$ (-5*v^15 - 20*v^14 - 107*v^13 - 474*v^12 - 657*v^11 - 3698*v^10 - 1074*v^9 - 12334*v^8 + 1686*v^7 - 18152*v^6 + 4770*v^5 - 12164*v^4 + 3014*v^3 - 3428*v^2 + 484*v - 300) / 16 $$\beta_{6}$$ $$=$$ $$( 8 \nu^{15} - 3 \nu^{14} + 190 \nu^{13} - 70 \nu^{12} + 1487 \nu^{11} - 530 \nu^{10} + 4970 \nu^{9} - 1674 \nu^{8} + 7248 \nu^{7} - 2198 \nu^{6} + 4556 \nu^{5} - 1156 \nu^{4} + 1086 \nu^{3} - 180 \nu^{2} + \cdots - 20 ) / 16$$ (8*v^15 - 3*v^14 + 190*v^13 - 70*v^12 + 1487*v^11 - 530*v^10 + 4970*v^9 - 1674*v^8 + 7248*v^7 - 2198*v^6 + 4556*v^5 - 1156*v^4 + 1086*v^3 - 180*v^2 + 92*v - 20) / 16 $$\beta_{7}$$ $$=$$ $$( - 19 \nu^{15} + 6 \nu^{14} - 448 \nu^{13} + 138 \nu^{12} - 3460 \nu^{11} + 1012 \nu^{10} - 11326 \nu^{9} + 2976 \nu^{8} - 16094 \nu^{7} + 3276 \nu^{6} - 10328 \nu^{5} + 1188 \nu^{4} + \cdots + 16 ) / 16$$ (-19*v^15 + 6*v^14 - 448*v^13 + 138*v^12 - 3460*v^11 + 1012*v^10 - 11326*v^9 + 2976*v^8 - 16094*v^7 + 3276*v^6 - 10328*v^5 + 1188*v^4 - 2904*v^3 + 72*v^2 - 316*v + 16) / 16 $$\beta_{8}$$ $$=$$ $$( 19 \nu^{15} + 6 \nu^{14} + 454 \nu^{13} + 138 \nu^{12} + 3598 \nu^{11} + 1012 \nu^{10} + 12338 \nu^{9} + 2976 \nu^{8} + 19070 \nu^{7} + 3276 \nu^{6} + 13604 \nu^{5} + 1188 \nu^{4} + 4092 \nu^{3} + \cdots + 16 ) / 16$$ (19*v^15 + 6*v^14 + 454*v^13 + 138*v^12 + 3598*v^11 + 1012*v^10 + 12338*v^9 + 2976*v^8 + 19070*v^7 + 3276*v^6 + 13604*v^5 + 1188*v^4 + 4092*v^3 + 72*v^2 + 372*v + 16) / 16 $$\beta_{9}$$ $$=$$ $$( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 8$$ (11*v^15 + 261*v^13 + 2042*v^11 + 6862*v^9 + 10334*v^7 + 7410*v^5 + 2412*v^3 + 252*v) / 8 $$\beta_{10}$$ $$=$$ $$( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + 7004 \nu^{8} + 8530 \nu^{7} + 9752 \nu^{6} + 4916 \nu^{5} + 6092 \nu^{4} + 1046 \nu^{3} + \cdots + 136 ) / 8$$ (11*v^15 + 12*v^14 + 258*v^13 + 282*v^12 + 1971*v^11 + 2164*v^10 + 6311*v^9 + 7004*v^8 + 8530*v^7 + 9752*v^6 + 4916*v^5 + 6092*v^4 + 1046*v^3 + 1608*v^2 + 38*v + 136) / 8 $$\beta_{11}$$ $$=$$ $$( 11 \nu^{15} - 12 \nu^{14} + 258 \nu^{13} - 282 \nu^{12} + 1971 \nu^{11} - 2164 \nu^{10} + 6311 \nu^{9} - 7004 \nu^{8} + 8530 \nu^{7} - 9752 \nu^{6} + 4916 \nu^{5} - 6092 \nu^{4} + 1046 \nu^{3} + \cdots - 136 ) / 8$$ (11*v^15 - 12*v^14 + 258*v^13 - 282*v^12 + 1971*v^11 - 2164*v^10 + 6311*v^9 - 7004*v^8 + 8530*v^7 - 9752*v^6 + 4916*v^5 - 6092*v^4 + 1046*v^3 - 1608*v^2 + 38*v - 136) / 8 $$\beta_{12}$$ $$=$$ $$( - 38 \nu^{15} + \nu^{14} - 896 \nu^{13} + 20 \nu^{12} - 6921 \nu^{11} + 100 \nu^{10} - 22674 \nu^{9} - 4 \nu^{8} - 32332 \nu^{7} - 918 \nu^{6} - 20960 \nu^{5} - 1440 \nu^{4} - 5842 \nu^{3} + \cdots - 72 ) / 16$$ (-38*v^15 + v^14 - 896*v^13 + 20*v^12 - 6921*v^11 + 100*v^10 - 22674*v^9 - 4*v^8 - 32332*v^7 - 918*v^6 - 20960*v^5 - 1440*v^4 - 5842*v^3 - 664*v^2 - 540*v - 72) / 16 $$\beta_{13}$$ $$=$$ $$( - 38 \nu^{15} - \nu^{14} - 896 \nu^{13} - 20 \nu^{12} - 6921 \nu^{11} - 100 \nu^{10} - 22674 \nu^{9} + 4 \nu^{8} - 32332 \nu^{7} + 918 \nu^{6} - 20960 \nu^{5} + 1440 \nu^{4} - 5842 \nu^{3} + \cdots + 72 ) / 16$$ (-38*v^15 - v^14 - 896*v^13 - 20*v^12 - 6921*v^11 - 100*v^10 - 22674*v^9 + 4*v^8 - 32332*v^7 + 918*v^6 - 20960*v^5 + 1440*v^4 - 5842*v^3 + 664*v^2 - 540*v + 72) / 16 $$\beta_{14}$$ $$=$$ $$( 38 \nu^{15} + 15 \nu^{14} + 891 \nu^{13} + 352 \nu^{12} + 6803 \nu^{11} + 2692 \nu^{10} + 21762 \nu^{9} + 8638 \nu^{8} + 29356 \nu^{7} + 11726 \nu^{6} + 16838 \nu^{5} + 6808 \nu^{4} + 3518 \nu^{3} + \cdots + 76 ) / 16$$ (38*v^15 + 15*v^14 + 891*v^13 + 352*v^12 + 6803*v^11 + 2692*v^10 + 21762*v^9 + 8638*v^8 + 29356*v^7 + 11726*v^6 + 16838*v^5 + 6808*v^4 + 3518*v^3 + 1496*v^2 + 92*v + 76) / 16 $$\beta_{15}$$ $$=$$ $$( 38 \nu^{15} - 15 \nu^{14} + 891 \nu^{13} - 352 \nu^{12} + 6803 \nu^{11} - 2692 \nu^{10} + 21762 \nu^{9} - 8638 \nu^{8} + 29356 \nu^{7} - 11726 \nu^{6} + 16838 \nu^{5} - 6808 \nu^{4} + 3518 \nu^{3} + \cdots - 76 ) / 16$$ (38*v^15 - 15*v^14 + 891*v^13 - 352*v^12 + 6803*v^11 - 2692*v^10 + 21762*v^9 - 8638*v^8 + 29356*v^7 - 11726*v^6 + 16838*v^5 - 6808*v^4 + 3518*v^3 - 1496*v^2 + 92*v - 76) / 16
 $$\nu$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 2$$ (-b3 - b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{9} + 2\beta_{8} - 2\beta_{6} - 2\beta_{2} - 6 ) / 2$$ (b15 - b14 + b13 - b12 - b9 + 2*b8 - 2*b6 - 2*b2 - 6) / 2 $$\nu^{3}$$ $$=$$ $$( - 3 \beta_{15} - 3 \beta_{14} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} - 8 \beta_1 ) / 2$$ (-3*b15 - 3*b14 + 2*b11 + 2*b10 + 2*b9 + 2*b8 - 2*b7 - 2*b5 + 2*b4 + 9*b3 + 9*b2 - 8*b1) / 2 $$\nu^{4}$$ $$=$$ $$( - 8 \beta_{15} + 8 \beta_{14} - 20 \beta_{13} + 20 \beta_{12} + 5 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} - 22 \beta_{8} + 2 \beta_{7} + 24 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 22 \beta_{2} + 48 ) / 2$$ (-8*b15 + 8*b14 - 20*b13 + 20*b12 + 5*b11 - 5*b10 + 12*b9 - 22*b8 + 2*b7 + 24*b6 - 8*b5 - 8*b4 - 2*b3 + 22*b2 + 48) / 2 $$\nu^{5}$$ $$=$$ $$( 40 \beta_{15} + 40 \beta_{14} - 10 \beta_{13} - 10 \beta_{12} - 31 \beta_{11} - 31 \beta_{10} - 44 \beta_{9} - 31 \beta_{8} + 31 \beta_{7} + 29 \beta_{5} - 29 \beta_{4} - 95 \beta_{3} - 95 \beta_{2} + 83 \beta_1 ) / 2$$ (40*b15 + 40*b14 - 10*b13 - 10*b12 - 31*b11 - 31*b10 - 44*b9 - 31*b8 + 31*b7 + 29*b5 - 29*b4 - 95*b3 - 95*b2 + 83*b1) / 2 $$\nu^{6}$$ $$=$$ $$37 \beta_{15} - 37 \beta_{14} + 140 \beta_{13} - 140 \beta_{12} - 42 \beta_{11} + 42 \beta_{10} - 72 \beta_{9} + 129 \beta_{8} - 15 \beta_{7} - 144 \beta_{6} + 67 \beta_{5} + 67 \beta_{4} + 10 \beta_{3} - 124 \beta_{2} - 252$$ 37*b15 - 37*b14 + 140*b13 - 140*b12 - 42*b11 + 42*b10 - 72*b9 + 129*b8 - 15*b7 - 144*b6 + 67*b5 + 67*b4 + 10*b3 - 124*b2 - 252 $$\nu^{7}$$ $$=$$ $$( - 497 \beta_{15} - 497 \beta_{14} + 175 \beta_{13} + 175 \beta_{12} + 404 \beta_{11} + 404 \beta_{10} + 631 \beta_{9} + 413 \beta_{8} - 413 \beta_{7} - 372 \beta_{5} + 372 \beta_{4} + 1091 \beta_{3} + 1091 \beta_{2} - 956 \beta_1 ) / 2$$ (-497*b15 - 497*b14 + 175*b13 + 175*b12 + 404*b11 + 404*b10 + 631*b9 + 413*b8 - 413*b7 - 372*b5 + 372*b4 + 1091*b3 + 1091*b2 - 956*b1) / 2 $$\nu^{8}$$ $$=$$ $$- 396 \beta_{15} + 396 \beta_{14} - 1792 \beta_{13} + 1792 \beta_{12} + 564 \beta_{11} - 564 \beta_{10} + 876 \beta_{9} - 1556 \beta_{8} + 196 \beta_{7} + 1752 \beta_{6} - 900 \beta_{5} - 900 \beta_{4} - 104 \beta_{3} + \cdots + 2910$$ -396*b15 + 396*b14 - 1792*b13 + 1792*b12 + 564*b11 - 564*b10 + 876*b9 - 1556*b8 + 196*b7 + 1752*b6 - 900*b5 - 900*b4 - 104*b3 + 1464*b2 + 2910 $$\nu^{9}$$ $$=$$ $$3054 \beta_{15} + 3054 \beta_{14} - 1194 \beta_{13} - 1194 \beta_{12} - 2520 \beta_{11} - 2520 \beta_{10} - 4074 \beta_{9} - 2610 \beta_{8} + 2610 \beta_{7} + 2316 \beta_{5} - 2316 \beta_{4} - 6517 \beta_{3} + \cdots + 5723 \beta_1$$ 3054*b15 + 3054*b14 - 1194*b13 - 1194*b12 - 2520*b11 - 2520*b10 - 4074*b9 - 2610*b8 + 2610*b7 + 2316*b5 - 2316*b4 - 6517*b3 - 6517*b2 + 5723*b1 $$\nu^{10}$$ $$=$$ $$4587 \beta_{15} - 4587 \beta_{14} + 22287 \beta_{13} - 22287 \beta_{12} - 7128 \beta_{11} + 7128 \beta_{10} - 10707 \beta_{9} + 18950 \beta_{8} - 2464 \beta_{7} - 21414 \beta_{6} + 11376 \beta_{5} + \cdots - 34858$$ 4587*b15 - 4587*b14 + 22287*b13 - 22287*b12 - 7128*b11 + 7128*b10 - 10707*b9 + 18950*b8 - 2464*b7 - 21414*b6 + 11376*b5 + 11376*b4 + 1180*b3 - 17666*b2 - 34858 $$\nu^{11}$$ $$=$$ $$- 37433 \beta_{15} - 37433 \beta_{14} + 15180 \beta_{13} + 15180 \beta_{12} + 31030 \beta_{11} + 31030 \beta_{10} + 50850 \beta_{9} + 32346 \beta_{8} - 32346 \beta_{7} - 28542 \beta_{5} + \cdots - 69508 \beta_1$$ -37433*b15 - 37433*b14 + 15180*b13 + 15180*b12 + 31030*b11 + 31030*b10 + 50850*b9 + 32346*b8 - 32346*b7 - 28542*b5 + 28542*b4 + 79079*b3 + 79079*b2 - 69508*b1 $$\nu^{12}$$ $$=$$ $$- 54976 \beta_{15} + 54976 \beta_{14} - 274428 \beta_{13} + 274428 \beta_{12} + 88283 \beta_{11} - 88283 \beta_{10} + 131044 \beta_{9} - 231582 \beta_{8} + 30506 \beta_{7} + 262088 \beta_{6} + \cdots + 423384$$ -54976*b15 + 54976*b14 - 274428*b13 + 274428*b12 + 88283*b11 - 88283*b10 + 131044*b9 - 231582*b8 + 30506*b7 + 262088*b6 - 140896*b5 - 140896*b4 - 14010*b3 + 215086*b2 + 423384 $$\nu^{13}$$ $$=$$ $$458484 \beta_{15} + 458484 \beta_{14} - 188422 \beta_{13} - 188422 \beta_{12} - 380577 \beta_{11} - 380577 \beta_{10} - 627076 \beta_{9} - 397897 \beta_{8} + 397897 \beta_{7} + 350371 \beta_{5} + \cdots + 848621 \beta_1$$ 458484*b15 + 458484*b14 - 188422*b13 - 188422*b12 - 380577*b11 - 380577*b10 - 627076*b9 - 397897*b8 + 397897*b7 + 350371*b5 - 350371*b4 - 965133*b3 - 965133*b2 + 848621*b1 $$\nu^{14}$$ $$=$$ $$667774 \beta_{15} - 667774 \beta_{14} + 3367136 \beta_{13} - 3367136 \beta_{12} - 1085560 \beta_{11} + 1085560 \beta_{10} - 1604464 \beta_{9} + 2833662 \beta_{8} - 375266 \beta_{7} + \cdots - 5168936$$ 667774*b15 - 667774*b14 + 3367136*b13 - 3367136*b12 - 1085560*b11 + 1085560*b10 - 1604464*b9 + 2833662*b8 - 375266*b7 - 3208928*b6 + 1732466*b5 + 1732466*b4 + 169524*b3 - 2627920*b2 - 5168936 $$\nu^{15}$$ $$=$$ $$- 5614479 \beta_{15} - 5614479 \beta_{14} + 2318785 \beta_{13} + 2318785 \beta_{12} + 4662232 \beta_{11} + 4662232 \beta_{10} + 7698833 \beta_{9} + 4880807 \beta_{8} + \cdots - 10380392 \beta_1$$ -5614479*b15 - 5614479*b14 + 2318785*b13 + 2318785*b12 + 4662232*b11 + 4662232*b10 + 7698833*b9 + 4880807*b8 - 4880807*b7 - 4294488*b5 + 4294488*b4 + 11804009*b3 + 11804009*b2 - 10380392*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 − 0.724535i 1.05636i − 1.05636i 0.724535i 0.886177i − 2.08509i 2.08509i − 0.886177i − 3.49930i − 0.528036i 0.528036i 3.49930i − 0.357857i − 2.13875i 2.13875i 0.357857i
−1.30656 0.541196i 0.541196 1.64533i 1.41421 + 1.41421i 1.25928 + 1.84776i −1.59755 + 1.85683i 3.29066 −1.08239 2.61313i −2.41421 1.78089i −0.645329 3.09573i
59.2 −1.30656 0.541196i 0.541196 + 1.64533i 1.41421 + 1.41421i −1.25928 + 1.84776i 0.183339 2.44262i −3.29066 −1.08239 2.61313i −2.41421 + 1.78089i 2.64533 1.73270i
59.3 −1.30656 + 0.541196i 0.541196 1.64533i 1.41421 1.41421i −1.25928 1.84776i 0.183339 + 2.44262i −3.29066 −1.08239 + 2.61313i −2.41421 1.78089i 2.64533 + 1.73270i
59.4 −1.30656 + 0.541196i 0.541196 + 1.64533i 1.41421 1.41421i 1.25928 1.84776i −1.59755 1.85683i 3.29066 −1.08239 + 2.61313i −2.41421 + 1.78089i −0.645329 + 3.09573i
59.5 −0.541196 1.30656i −1.30656 1.13705i −1.41421 + 1.41421i −2.10100 + 0.765367i −0.778527 + 2.32248i −2.27411 2.61313 + 1.08239i 0.414214 + 2.97127i 2.13705 + 2.33088i
59.6 −0.541196 1.30656i −1.30656 + 1.13705i −1.41421 + 1.41421i 2.10100 + 0.765367i 2.19274 + 1.09174i 2.27411 2.61313 + 1.08239i 0.414214 2.97127i −0.137055 3.15931i
59.7 −0.541196 + 1.30656i −1.30656 1.13705i −1.41421 1.41421i 2.10100 0.765367i 2.19274 1.09174i 2.27411 2.61313 1.08239i 0.414214 + 2.97127i −0.137055 + 3.15931i
59.8 −0.541196 + 1.30656i −1.30656 + 1.13705i −1.41421 1.41421i −2.10100 0.765367i −0.778527 2.32248i −2.27411 2.61313 1.08239i 0.414214 2.97127i 2.13705 2.33088i
59.9 0.541196 1.30656i 1.30656 1.13705i −1.41421 1.41421i −2.10100 + 0.765367i −0.778527 2.32248i 2.27411 −2.61313 + 1.08239i 0.414214 2.97127i −0.137055 + 3.15931i
59.10 0.541196 1.30656i 1.30656 + 1.13705i −1.41421 1.41421i 2.10100 + 0.765367i 2.19274 1.09174i −2.27411 −2.61313 + 1.08239i 0.414214 + 2.97127i 2.13705 2.33088i
59.11 0.541196 + 1.30656i 1.30656 1.13705i −1.41421 + 1.41421i 2.10100 0.765367i 2.19274 + 1.09174i −2.27411 −2.61313 1.08239i 0.414214 2.97127i 2.13705 + 2.33088i
59.12 0.541196 + 1.30656i 1.30656 + 1.13705i −1.41421 + 1.41421i −2.10100 0.765367i −0.778527 + 2.32248i 2.27411 −2.61313 1.08239i 0.414214 + 2.97127i −0.137055 3.15931i
59.13 1.30656 0.541196i −0.541196 1.64533i 1.41421 1.41421i 1.25928 + 1.84776i −1.59755 1.85683i −3.29066 1.08239 2.61313i −2.41421 + 1.78089i 2.64533 + 1.73270i
59.14 1.30656 0.541196i −0.541196 + 1.64533i 1.41421 1.41421i −1.25928 + 1.84776i 0.183339 + 2.44262i 3.29066 1.08239 2.61313i −2.41421 1.78089i −0.645329 + 3.09573i
59.15 1.30656 + 0.541196i −0.541196 1.64533i 1.41421 + 1.41421i −1.25928 1.84776i 0.183339 2.44262i 3.29066 1.08239 + 2.61313i −2.41421 + 1.78089i −0.645329 3.09573i
59.16 1.30656 + 0.541196i −0.541196 + 1.64533i 1.41421 + 1.41421i 1.25928 1.84776i −1.59755 + 1.85683i −3.29066 1.08239 + 2.61313i −2.41421 1.78089i 2.64533 1.73270i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.m.b 16
3.b odd 2 1 inner 120.2.m.b 16
4.b odd 2 1 480.2.m.b 16
5.b even 2 1 inner 120.2.m.b 16
5.c odd 4 2 600.2.b.i 16
8.b even 2 1 480.2.m.b 16
8.d odd 2 1 inner 120.2.m.b 16
12.b even 2 1 480.2.m.b 16
15.d odd 2 1 inner 120.2.m.b 16
15.e even 4 2 600.2.b.i 16
20.d odd 2 1 480.2.m.b 16
20.e even 4 2 2400.2.b.i 16
24.f even 2 1 inner 120.2.m.b 16
24.h odd 2 1 480.2.m.b 16
40.e odd 2 1 inner 120.2.m.b 16
40.f even 2 1 480.2.m.b 16
40.i odd 4 2 2400.2.b.i 16
40.k even 4 2 600.2.b.i 16
60.h even 2 1 480.2.m.b 16
60.l odd 4 2 2400.2.b.i 16
120.i odd 2 1 480.2.m.b 16
120.m even 2 1 inner 120.2.m.b 16
120.q odd 4 2 600.2.b.i 16
120.w even 4 2 2400.2.b.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.b 16 1.a even 1 1 trivial
120.2.m.b 16 3.b odd 2 1 inner
120.2.m.b 16 5.b even 2 1 inner
120.2.m.b 16 8.d odd 2 1 inner
120.2.m.b 16 15.d odd 2 1 inner
120.2.m.b 16 24.f even 2 1 inner
120.2.m.b 16 40.e odd 2 1 inner
120.2.m.b 16 120.m even 2 1 inner
480.2.m.b 16 4.b odd 2 1
480.2.m.b 16 8.b even 2 1
480.2.m.b 16 12.b even 2 1
480.2.m.b 16 20.d odd 2 1
480.2.m.b 16 24.h odd 2 1
480.2.m.b 16 40.f even 2 1
480.2.m.b 16 60.h even 2 1
480.2.m.b 16 120.i odd 2 1
600.2.b.i 16 5.c odd 4 2
600.2.b.i 16 15.e even 4 2
600.2.b.i 16 40.k even 4 2
600.2.b.i 16 120.q odd 4 2
2400.2.b.i 16 20.e even 4 2
2400.2.b.i 16 40.i odd 4 2
2400.2.b.i 16 60.l odd 4 2
2400.2.b.i 16 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 16T_{7}^{2} + 56$$ acting on $$S_{2}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 16)^{2}$$
$3$ $$(T^{8} + 4 T^{6} + 14 T^{4} + 36 T^{2} + \cdots + 81)^{2}$$
$5$ $$(T^{8} - 4 T^{6} + 22 T^{4} - 100 T^{2} + \cdots + 625)^{2}$$
$7$ $$(T^{4} - 16 T^{2} + 56)^{4}$$
$11$ $$(T^{4} + 24 T^{2} + 112)^{4}$$
$13$ $$(T^{4} - 32 T^{2} + 224)^{4}$$
$17$ $$(T^{4} - 16 T^{2} + 32)^{4}$$
$19$ $$(T^{2} + 4 T - 4)^{8}$$
$23$ $$(T^{4} + 8 T^{2} + 8)^{4}$$
$29$ $$(T^{4} - 40 T^{2} + 112)^{4}$$
$31$ $$(T^{4} + 48 T^{2} + 64)^{4}$$
$37$ $$(T^{4} - 64 T^{2} + 224)^{4}$$
$41$ $$(T^{4} + 80 T^{2} + 448)^{4}$$
$43$ $$(T^{4} + 112 T^{2} + 2744)^{4}$$
$47$ $$(T^{4} + 8 T^{2} + 8)^{4}$$
$53$ $$(T^{4} + 144 T^{2} + 2592)^{4}$$
$59$ $$(T^{4} + 24 T^{2} + 112)^{4}$$
$61$ $$(T^{2} + 72)^{8}$$
$67$ $$(T^{4} + 16 T^{2} + 56)^{4}$$
$71$ $$(T^{4} - 192 T^{2} + 7168)^{4}$$
$73$ $$(T^{4} + 64 T^{2} + 896)^{4}$$
$79$ $$(T^{4} + 272 T^{2} + 64)^{4}$$
$83$ $$(T^{4} - 136 T^{2} + 4232)^{4}$$
$89$ $$(T^{4} + 96 T^{2} + 1792)^{4}$$
$97$ $$(T^{4} + 128 T^{2} + 896)^{4}$$