# Properties

 Label 3025.2.a.bj Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4$$ x^8 - 3*x^7 - 7*x^6 + 18*x^5 + 16*x^4 - 30*x^3 - 12*x^2 + 12*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 12\nu^{2} + 6\nu - 6 ) / 2$$ (v^5 - 3*v^4 - 5*v^3 + 12*v^2 + 6*v - 6) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 12\nu^{3} - 6\nu^{2} + 8\nu ) / 2$$ (-v^6 + 3*v^5 + 5*v^4 - 12*v^3 - 6*v^2 + 8*v) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 12\nu^{3} + 8\nu^{2} - 10\nu - 4 ) / 2$$ (v^6 - 3*v^5 - 5*v^4 + 12*v^3 + 8*v^2 - 10*v - 4) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 13\nu^{3} - 26\nu^{2} + 18\nu + 12 ) / 2$$ (-v^6 + 2*v^5 + 10*v^4 - 13*v^3 - 26*v^2 + 18*v + 12) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 15\nu^{4} + 11\nu^{3} - 18\nu^{2} - 4\nu + 2 ) / 2$$ (v^7 - 3*v^6 - 6*v^5 + 15*v^4 + 11*v^3 - 18*v^2 - 4*v + 2) / 2 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 13\nu^{4} - 26\nu^{3} + 18\nu^{2} + 12\nu ) / 2$$ (-v^7 + 2*v^6 + 10*v^5 - 13*v^4 - 26*v^3 + 18*v^2 + 12*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta _1 + 2$$ b4 + b3 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 3\beta_{4} + 2\beta_{3} - \beta_{2} + 6\beta _1 + 2$$ b7 + b6 + 3*b4 + 2*b3 - b2 + 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 3\beta_{6} + \beta_{5} + 13\beta_{4} + 9\beta_{3} - 2\beta_{2} + 14\beta _1 + 11$$ 3*b7 + 3*b6 + b5 + 13*b4 + 9*b3 - 2*b2 + 14*b1 + 11 $$\nu^{5}$$ $$=$$ $$14\beta_{7} + 14\beta_{6} + 3\beta_{5} + 42\beta_{4} + 25\beta_{3} - 9\beta_{2} + 54\beta _1 + 25$$ 14*b7 + 14*b6 + 3*b5 + 42*b4 + 25*b3 - 9*b2 + 54*b1 + 25 $$\nu^{6}$$ $$=$$ $$45\beta_{7} + 45\beta_{6} + 14\beta_{5} + 149\beta_{4} + 88\beta_{3} - 25\beta_{2} + 162\beta _1 + 94$$ 45*b7 + 45*b6 + 14*b5 + 149*b4 + 88*b3 - 25*b2 + 162*b1 + 94 $$\nu^{7}$$ $$=$$ $$163\beta_{7} + 165\beta_{6} + 45\beta_{5} + 489\beta_{4} + 275\beta_{3} - 88\beta_{2} + 556\beta _1 + 279$$ 163*b7 + 165*b6 + 45*b5 + 489*b4 + 275*b3 - 88*b2 + 556*b1 + 279

## 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
 −1.76541 −1.56247 −0.672032 −0.321622 0.860597 1.20828 1.95882 3.29384
−2.76541 −1.64150 5.64751 0 4.53942 1.89792 −10.0869 −0.305488 0
1.2 −2.56247 1.79260 4.56626 0 −4.59347 −3.80088 −6.57595 0.213399 0
1.3 −1.67203 3.10994 0.795692 0 −5.19992 −3.08998 2.01364 6.67173 0
1.4 −1.32162 −2.02642 −0.253315 0 2.67816 −0.348258 2.97803 1.10639 0
1.5 −0.139403 −2.98582 −1.98057 0 0.416233 −3.56959 0.554904 5.91512 0
1.6 0.208285 1.89427 −1.95662 0 0.394547 1.28881 −0.824104 0.588246 0
1.7 0.958815 0.899342 −1.08067 0 0.862303 0.761645 −2.95380 −2.19118 0
1.8 2.29384 −0.0424059 3.26171 0 −0.0972724 −1.13968 2.89416 −2.99820 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
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 3025.2.a.bj 8
5.b even 2 1 3025.2.a.bm 8
11.b odd 2 1 3025.2.a.bn 8
11.d odd 10 2 275.2.h.e yes 16
55.d odd 2 1 3025.2.a.bi 8
55.h odd 10 2 275.2.h.c 16
55.l even 20 4 275.2.z.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.h.c 16 55.h odd 10 2
275.2.h.e yes 16 11.d odd 10 2
275.2.z.c 32 55.l even 20 4
3025.2.a.bi 8 55.d odd 2 1
3025.2.a.bj 8 1.a even 1 1 trivial
3025.2.a.bm 8 5.b even 2 1
3025.2.a.bn 8 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{8} + 5T_{2}^{7} - 31T_{2}^{5} - 34T_{2}^{4} + 25T_{2}^{3} + 34T_{2}^{2} - 3T_{2} - 1$$ T2^8 + 5*T2^7 - 31*T2^5 - 34*T2^4 + 25*T2^3 + 34*T2^2 - 3*T2 - 1 $$T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 15T_{3}^{5} + 74T_{3}^{4} - 67T_{3}^{3} - 105T_{3}^{2} + 90T_{3} + 4$$ T3^8 - T3^7 - 16*T3^6 + 15*T3^5 + 74*T3^4 - 67*T3^3 - 105*T3^2 + 90*T3 + 4 $$T_{19}^{8} + T_{19}^{7} - 64T_{19}^{6} - 119T_{19}^{5} + 1016T_{19}^{4} + 1735T_{19}^{3} - 5555T_{19}^{2} - 5520T_{19} + 10480$$ T19^8 + T19^7 - 64*T19^6 - 119*T19^5 + 1016*T19^4 + 1735*T19^3 - 5555*T19^2 - 5520*T19 + 10480

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 5 T^{7} - 31 T^{5} - 34 T^{4} + \cdots - 1$$
$3$ $$T^{8} - T^{7} - 16 T^{6} + 15 T^{5} + \cdots + 4$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 8 T^{7} + 10 T^{6} - 50 T^{5} + \cdots - 31$$
$11$ $$T^{8}$$
$13$ $$T^{8} + 9 T^{7} - 4 T^{6} - 241 T^{5} + \cdots + 3475$$
$17$ $$T^{8} + 19 T^{7} + 107 T^{6} + \cdots + 439$$
$19$ $$T^{8} + T^{7} - 64 T^{6} - 119 T^{5} + \cdots + 10480$$
$23$ $$T^{8} - 2 T^{7} - 62 T^{6} + 252 T^{5} + \cdots - 121$$
$29$ $$T^{8} + 7 T^{7} - 62 T^{6} + \cdots - 35495$$
$31$ $$T^{8} + 5 T^{7} - 100 T^{6} + \cdots + 33569$$
$37$ $$T^{8} + 8 T^{7} - 54 T^{6} + \cdots - 4451$$
$41$ $$T^{8} - 197 T^{6} + 396 T^{5} + \cdots + 442576$$
$43$ $$T^{8} + 14 T^{7} - 28 T^{6} + \cdots - 101$$
$47$ $$T^{8} - 11 T^{7} - 108 T^{6} + \cdots - 459469$$
$53$ $$T^{8} - 11 T^{7} - 216 T^{6} + \cdots - 48896$$
$59$ $$T^{8} - 17 T^{7} - 140 T^{6} + \cdots - 78605$$
$61$ $$T^{8} - 2 T^{7} - 282 T^{6} + \cdots + 3319739$$
$67$ $$T^{8} - 7 T^{7} - 323 T^{6} + \cdots + 23994961$$
$71$ $$T^{8} - 17 T^{7} - 99 T^{6} + \cdots + 960859$$
$73$ $$T^{8} + 34 T^{7} + 290 T^{6} + \cdots - 466721$$
$79$ $$T^{8} + 23 T^{7} - 42 T^{6} + \cdots - 524695$$
$83$ $$T^{8} + 41 T^{7} + 568 T^{6} + \cdots - 8329$$
$89$ $$T^{8} + 11 T^{7} - 258 T^{6} + \cdots - 278125$$
$97$ $$T^{8} - 2 T^{7} - 257 T^{6} + \cdots + 85616$$