# Properties

 Label 575.2.a.k Level $575$ Weight $2$ Character orbit 575.a Self dual yes Analytic conductor $4.591$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.59139811622$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 14x^{2} - 49x - 9$$ x^7 - x^6 - 12*x^5 + 9*x^4 + 43*x^3 - 14*x^2 - 49*x - 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 14x^{2} - 49x - 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu - 1$$ v^3 - 5*v - 1 $$\beta_{4}$$ $$=$$ $$( 2\nu^{6} - \nu^{5} - 17\nu^{4} + 7\nu^{3} + 27\nu^{2} - 2\nu + 1 ) / 5$$ (2*v^6 - v^5 - 17*v^4 + 7*v^3 + 27*v^2 - 2*v + 1) / 5 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 2\nu^{5} - 11\nu^{4} - 19\nu^{3} + 31\nu^{2} + 39\nu - 7 ) / 5$$ (v^6 + 2*v^5 - 11*v^4 - 19*v^3 + 31*v^2 + 39*v - 7) / 5 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 3\nu^{5} + 11\nu^{4} - 26\nu^{3} - 31\nu^{2} + 41\nu + 22 ) / 5$$ (-v^6 + 3*v^5 + 11*v^4 - 26*v^3 - 31*v^2 + 41*v + 22) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta _1 + 1$$ b3 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} + 7\beta_{2} + 22$$ b6 - b5 + b4 + 7*b2 + 22 $$\nu^{5}$$ $$=$$ $$\beta_{6} + \beta_{5} + 9\beta_{3} + 29\beta _1 + 6$$ b6 + b5 + 9*b3 + 29*b1 + 6 $$\nu^{6}$$ $$=$$ $$9\beta_{6} - 8\beta_{5} + 11\beta_{4} + \beta_{3} + 46\beta_{2} - 2\beta _1 + 132$$ 9*b6 - 8*b5 + 11*b4 + b3 + 46*b2 - 2*b1 + 132

## 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
 2.53289 2.27220 1.69496 −0.202227 −1.07994 −1.63662 −2.58128
−2.53289 0.345624 4.41555 0 −0.875428 −5.12894 −6.11832 −2.88054 0
1.2 −2.27220 −3.13672 3.16289 0 7.12726 4.34930 −2.64233 6.83902 0
1.3 −1.69496 3.30905 0.872898 0 −5.60872 0.852729 1.91040 7.94982 0
1.4 0.202227 −2.69619 −1.95910 0 −0.545243 −2.81698 −0.800639 4.26945 0
1.5 1.07994 −1.06928 −0.833738 0 −1.15476 2.95289 −3.06026 −1.85663 0
1.6 1.63662 2.46212 0.678510 0 4.02954 4.74965 −2.16277 3.06202 0
1.7 2.58128 0.785406 4.66299 0 2.02735 −1.95865 6.87392 −2.38314 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 575.2.a.k 7
3.b odd 2 1 5175.2.a.cg 7
4.b odd 2 1 9200.2.a.da 7
5.b even 2 1 575.2.a.l yes 7
5.c odd 4 2 575.2.b.f 14
15.d odd 2 1 5175.2.a.cb 7
20.d odd 2 1 9200.2.a.db 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.k 7 1.a even 1 1 trivial
575.2.a.l yes 7 5.b even 2 1
575.2.b.f 14 5.c odd 4 2
5175.2.a.cb 7 15.d odd 2 1
5175.2.a.cg 7 3.b odd 2 1
9200.2.a.da 7 4.b odd 2 1
9200.2.a.db 7 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{7} + T_{2}^{6} - 12T_{2}^{5} - 9T_{2}^{4} + 43T_{2}^{3} + 14T_{2}^{2} - 49T_{2} + 9$$ T2^7 + T2^6 - 12*T2^5 - 9*T2^4 + 43*T2^3 + 14*T2^2 - 49*T2 + 9 $$T_{3}^{7} - 18T_{3}^{5} + 85T_{3}^{3} - 8T_{3}^{2} - 65T_{3} + 20$$ T3^7 - 18*T3^5 + 85*T3^3 - 8*T3^2 - 65*T3 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + T^{6} - 12 T^{5} - 9 T^{4} + \cdots + 9$$
$3$ $$T^{7} - 18 T^{5} + 85 T^{3} - 8 T^{2} + \cdots + 20$$
$5$ $$T^{7}$$
$7$ $$T^{7} - 3 T^{6} - 40 T^{5} + \cdots + 1472$$
$11$ $$T^{7} + T^{6} - 58 T^{5} - 84 T^{4} + \cdots - 4800$$
$13$ $$T^{7} + 3 T^{6} - 46 T^{5} + \cdots - 1637$$
$17$ $$T^{7} - 10 T^{6} - 36 T^{5} + \cdots + 46080$$
$19$ $$T^{7} - 15 T^{6} + 30 T^{5} + \cdots - 1600$$
$23$ $$(T + 1)^{7}$$
$29$ $$T^{7} - 3 T^{6} - 122 T^{5} + \cdots - 3375$$
$31$ $$T^{7} - 14 T^{6} + 14 T^{5} + \cdots - 24350$$
$37$ $$T^{7} + 10 T^{6} - 116 T^{5} + \cdots + 9216$$
$41$ $$T^{7} - 19 T^{6} + 22 T^{5} + \cdots + 14217$$
$43$ $$T^{7} - 5 T^{6} - 144 T^{5} + \cdots - 14400$$
$47$ $$T^{7} + 14 T^{6} - 18 T^{5} + \cdots + 31170$$
$53$ $$T^{7} - 4 T^{6} - 124 T^{5} + \cdots - 3456$$
$59$ $$T^{7} + 16 T^{6} - 91 T^{5} + \cdots + 149340$$
$61$ $$T^{7} - 40 T^{6} + 544 T^{5} + \cdots + 79616$$
$67$ $$T^{7} + 4 T^{6} - 224 T^{5} + \cdots + 46080$$
$71$ $$T^{7} + 14 T^{6} - 18 T^{5} + \cdots + 31170$$
$73$ $$T^{7} + 3 T^{6} - 186 T^{5} + \cdots - 603981$$
$79$ $$T^{7} + T^{6} - 232 T^{5} + \cdots + 22720$$
$83$ $$T^{7} - 17 T^{6} - 40 T^{5} + \cdots + 192$$
$89$ $$T^{7} - 16 T^{6} - 244 T^{5} + \cdots + 2478720$$
$97$ $$T^{7} + 24 T^{6} + 4 T^{5} - 2232 T^{4} + \cdots - 896$$