# Properties

 Label 4025.2.a.t Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} - 11x^{6} + 9x^{5} + 36x^{4} - 23x^{3} - 30x^{2} + 17x - 2$$ x^8 - x^7 - 11*x^6 + 9*x^5 + 36*x^4 - 23*x^3 - 30*x^2 + 17*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 805) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11x^{6} + 9x^{5} + 36x^{4} - 23x^{3} - 30x^{2} + 17x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 6\nu^{2} - \nu + 4$$ v^4 - 6*v^2 - v + 4 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 7\nu^{3} + 9\nu - 2$$ v^5 - 7*v^3 + 9*v - 2 $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} - 11\nu^{5} + 8\nu^{4} + 36\nu^{3} - 16\nu^{2} - 30\nu + 8$$ v^7 - v^6 - 11*v^5 + 8*v^4 + 36*v^3 - 16*v^2 - 30*v + 8 $$\beta_{7}$$ $$=$$ $$2\nu^{7} - \nu^{6} - 22\nu^{5} + 8\nu^{4} + 72\nu^{3} - 17\nu^{2} - 62\nu + 12$$ 2*v^7 - v^6 - 22*v^5 + 8*v^4 + 72*v^3 - 17*v^2 - 62*v + 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta_1$$ b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 6\beta_{2} + \beta _1 + 14$$ b4 + 6*b2 + b1 + 14 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 7\beta_{3} + 26\beta _1 + 2$$ b5 + 7*b3 + 26*b1 + 2 $$\nu^{6}$$ $$=$$ $$\beta_{7} - 2\beta_{6} + 8\beta_{4} + 33\beta_{2} + 10\beta _1 + 71$$ b7 - 2*b6 + 8*b4 + 33*b2 + 10*b1 + 71 $$\nu^{7}$$ $$=$$ $$\beta_{7} - \beta_{6} + 11\beta_{5} + 41\beta_{3} + \beta_{2} + 138\beta _1 + 21$$ b7 - b6 + 11*b5 + 41*b3 + b2 + 138*b1 + 21

## 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
 2.42061 2.28656 1.18794 0.322285 0.181467 −1.17189 −1.94419 −2.28279
−2.42061 0.490228 3.85934 0 −1.18665 −1.00000 −4.50072 −2.75968 0
1.2 −2.28656 −2.13745 3.22836 0 4.88741 −1.00000 −2.80872 1.56870 0
1.3 −1.18794 −1.57054 −0.588791 0 1.86571 −1.00000 3.07534 −0.533418 0
1.4 −0.322285 1.07802 −1.89613 0 −0.347431 −1.00000 1.25566 −1.83786 0
1.5 −0.181467 −3.25071 −1.96707 0 0.589897 −1.00000 0.719892 7.56714 0
1.6 1.17189 1.97939 −0.626674 0 2.31962 −1.00000 −3.07817 0.917966 0
1.7 1.94419 −3.14297 1.77986 0 −6.11052 −1.00000 −0.427999 6.87827 0
1.8 2.28279 −0.445964 3.21111 0 −1.01804 −1.00000 2.76472 −2.80112 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 4025.2.a.t 8
5.b even 2 1 805.2.a.m 8
15.d odd 2 1 7245.2.a.bp 8
35.c odd 2 1 5635.2.a.bb 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.m 8 5.b even 2 1
4025.2.a.t 8 1.a even 1 1 trivial
5635.2.a.bb 8 35.c odd 2 1
7245.2.a.bp 8 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 9T_{2}^{5} + 36T_{2}^{4} + 23T_{2}^{3} - 30T_{2}^{2} - 17T_{2} - 2$$ T2^8 + T2^7 - 11*T2^6 - 9*T2^5 + 36*T2^4 + 23*T2^3 - 30*T2^2 - 17*T2 - 2 $$T_{3}^{8} + 7T_{3}^{7} + 8T_{3}^{6} - 35T_{3}^{5} - 68T_{3}^{4} + 32T_{3}^{3} + 87T_{3}^{2} - 8T_{3} - 16$$ T3^8 + 7*T3^7 + 8*T3^6 - 35*T3^5 - 68*T3^4 + 32*T3^3 + 87*T3^2 - 8*T3 - 16 $$T_{11}^{8} - 6T_{11}^{7} - 35T_{11}^{6} + 189T_{11}^{5} + 317T_{11}^{4} - 1352T_{11}^{3} - 64T_{11}^{2} + 1904T_{11} - 976$$ T11^8 - 6*T11^7 - 35*T11^6 + 189*T11^5 + 317*T11^4 - 1352*T11^3 - 64*T11^2 + 1904*T11 - 976

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{7} - 11 T^{6} - 9 T^{5} + \cdots - 2$$
$3$ $$T^{8} + 7 T^{7} + 8 T^{6} - 35 T^{5} + \cdots - 16$$
$5$ $$T^{8}$$
$7$ $$(T + 1)^{8}$$
$11$ $$T^{8} - 6 T^{7} - 35 T^{6} + 189 T^{5} + \cdots - 976$$
$13$ $$T^{8} + 8 T^{7} + 2 T^{6} - 81 T^{5} + \cdots - 8$$
$17$ $$T^{8} + 4 T^{7} - 85 T^{6} + \cdots - 7024$$
$19$ $$T^{8} - 63 T^{6} - 25 T^{5} + \cdots + 512$$
$23$ $$(T + 1)^{8}$$
$29$ $$T^{8} + 3 T^{7} - 101 T^{6} + \cdots + 19168$$
$31$ $$T^{8} + 16 T^{7} + 34 T^{6} + \cdots - 126296$$
$37$ $$T^{8} - 7 T^{7} - 188 T^{6} + \cdots + 240256$$
$41$ $$T^{8} - 7 T^{7} - 170 T^{6} + \cdots - 2416$$
$43$ $$T^{8} - 10 T^{7} - 157 T^{6} + \cdots + 30976$$
$47$ $$T^{8} + 19 T^{7} - 172 T^{6} + \cdots + 1514432$$
$53$ $$T^{8} - T^{7} - 240 T^{6} + \cdots - 371648$$
$59$ $$T^{8} - 21 T^{7} - 28 T^{6} + \cdots + 612736$$
$61$ $$T^{8} + 7 T^{7} - 264 T^{6} + \cdots - 62944$$
$67$ $$T^{8} + 11 T^{7} - 262 T^{6} + \cdots + 2473984$$
$71$ $$T^{8} - 8 T^{7} - 388 T^{6} + \cdots - 2504704$$
$73$ $$T^{8} - 3 T^{7} - 315 T^{6} + \cdots - 103216$$
$79$ $$T^{8} + 15 T^{7} - 403 T^{6} + \cdots - 47599856$$
$83$ $$T^{8} + 25 T^{7} - 32 T^{6} + \cdots - 227072$$
$89$ $$T^{8} - T^{7} - 382 T^{6} + \cdots - 1934752$$
$97$ $$T^{8} + 10 T^{7} - 202 T^{6} + \cdots - 46256$$