# Properties

 Label 600.2.k.d Level $600$ Weight $2$ Character orbit 600.k Analytic conductor $4.791$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.214798336.3 Defining polynomial: $$x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16$$ x^8 - 2*x^7 - 2*x^5 + 9*x^4 - 4*x^3 - 16*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 2\nu^{4} + 3\nu^{3} - 6\nu^{2} - 4\nu ) / 8$$ (-v^7 + 2*v^4 + 3*v^3 - 6*v^2 - 4*v) / 8 $$\beta_{2}$$ $$=$$ $$( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8$$ (-5*v^7 + 2*v^6 + 4*v^5 + 18*v^4 - 21*v^3 - 12*v^2 - 20*v + 56) / 8 $$\beta_{3}$$ $$=$$ $$( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} - 18\nu^{4} + 25\nu^{3} + 10\nu^{2} + 24\nu - 64 ) / 8$$ (5*v^7 - 4*v^6 - 4*v^5 - 18*v^4 + 25*v^3 + 10*v^2 + 24*v - 64) / 8 $$\beta_{4}$$ $$=$$ $$( 3\nu^{7} - 2\nu^{6} - 2\nu^{5} - 10\nu^{4} + 15\nu^{3} + 8\nu^{2} + 10\nu - 36 ) / 4$$ (3*v^7 - 2*v^6 - 2*v^5 - 10*v^4 + 15*v^3 + 8*v^2 + 10*v - 36) / 4 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 2\nu^{6} + 4\nu^{5} + 10\nu^{4} - 15\nu^{3} - 12\nu^{2} - 8\nu + 36 ) / 4$$ (-3*v^7 + 2*v^6 + 4*v^5 + 10*v^4 - 15*v^3 - 12*v^2 - 8*v + 36) / 4 $$\beta_{6}$$ $$=$$ $$( -4\nu^{7} + 3\nu^{6} + 4\nu^{5} + 12\nu^{4} - 18\nu^{3} - 9\nu^{2} - 10\nu + 44 ) / 4$$ (-4*v^7 + 3*v^6 + 4*v^5 + 12*v^4 - 18*v^3 - 9*v^2 - 10*v + 44) / 4 $$\beta_{7}$$ $$=$$ $$( 7\nu^{7} - 4\nu^{6} - 6\nu^{5} - 22\nu^{4} + 31\nu^{3} + 18\nu^{2} + 26\nu - 80 ) / 4$$ (7*v^7 - 4*v^6 - 6*v^5 - 22*v^4 + 31*v^3 + 18*v^2 + 26*v - 80) / 4
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2$$ (b7 + b6 - b4 + b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 ) / 2$$ (b6 - b5 + b4 + b2 - b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 2$$ (b7 + b6 + b4 - b3 + b2 + 2*b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} + 5\beta_{2} + \beta_1 ) / 2$$ (4*b7 + b6 + b5 - b4 - 2*b3 + 5*b2 + b1) / 2 $$\nu^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + 2\beta_{5} + 7\beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 2$$ (-b7 + b6 + 2*b5 + 7*b4 - b3 + b2 - 2*b1 - 1) / 2 $$\nu^{6}$$ $$=$$ $$( 4\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - 8\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 2$$ (4*b7 + 3*b6 + b5 - b4 - 8*b3 - 5*b2 + 5*b1) / 2 $$\nu^{7}$$ $$=$$ $$( 7\beta_{7} - 5\beta_{6} + 8\beta_{5} - \beta_{4} - 11\beta_{3} + 3\beta_{2} - 2\beta _1 + 5 ) / 2$$ (7*b7 - 5*b6 + 8*b5 - b4 - 11*b3 + 3*b2 - 2*b1 + 5) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 1.41216 − 0.0762223i 1.41216 + 0.0762223i −0.565036 − 1.29643i −0.565036 + 1.29643i −1.08003 − 0.912978i −1.08003 + 0.912978i 1.23291 − 0.692769i 1.23291 + 0.692769i
−1.29150 0.576222i 1.00000i 1.33594 + 1.48838i 0 0.576222 1.29150i −1.97676 −0.867721 2.69204i −1.00000 0
301.2 −1.29150 + 0.576222i 1.00000i 1.33594 1.48838i 0 0.576222 + 1.29150i −1.97676 −0.867721 + 2.69204i −1.00000 0
301.3 −1.16863 0.796431i 1.00000i 0.731395 + 1.86147i 0 −0.796431 + 1.16863i 4.72294 0.627801 2.75787i −1.00000 0
301.4 −1.16863 + 0.796431i 1.00000i 0.731395 1.86147i 0 −0.796431 1.16863i 4.72294 0.627801 + 2.75787i −1.00000 0
301.5 0.0591148 1.41298i 1.00000i −1.99301 0.167056i 0 1.41298 + 0.0591148i 1.33411 −0.353863 + 2.80620i −1.00000 0
301.6 0.0591148 + 1.41298i 1.00000i −1.99301 + 0.167056i 0 1.41298 0.0591148i 1.33411 −0.353863 2.80620i −1.00000 0
301.7 1.40101 0.192769i 1.00000i 1.92568 0.540143i 0 −0.192769 1.40101i −0.0802864 2.59378 1.12796i −1.00000 0
301.8 1.40101 + 0.192769i 1.00000i 1.92568 + 0.540143i 0 −0.192769 + 1.40101i −0.0802864 2.59378 + 1.12796i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.k.d 8
3.b odd 2 1 1800.2.k.t 8
4.b odd 2 1 2400.2.k.d 8
5.b even 2 1 600.2.k.e yes 8
5.c odd 4 1 600.2.d.g 8
5.c odd 4 1 600.2.d.h 8
8.b even 2 1 inner 600.2.k.d 8
8.d odd 2 1 2400.2.k.d 8
12.b even 2 1 7200.2.k.r 8
15.d odd 2 1 1800.2.k.q 8
15.e even 4 1 1800.2.d.s 8
15.e even 4 1 1800.2.d.t 8
20.d odd 2 1 2400.2.k.e 8
20.e even 4 1 2400.2.d.g 8
20.e even 4 1 2400.2.d.h 8
24.f even 2 1 7200.2.k.r 8
24.h odd 2 1 1800.2.k.t 8
40.e odd 2 1 2400.2.k.e 8
40.f even 2 1 600.2.k.e yes 8
40.i odd 4 1 600.2.d.g 8
40.i odd 4 1 600.2.d.h 8
40.k even 4 1 2400.2.d.g 8
40.k even 4 1 2400.2.d.h 8
60.h even 2 1 7200.2.k.s 8
60.l odd 4 1 7200.2.d.s 8
60.l odd 4 1 7200.2.d.t 8
120.i odd 2 1 1800.2.k.q 8
120.m even 2 1 7200.2.k.s 8
120.q odd 4 1 7200.2.d.s 8
120.q odd 4 1 7200.2.d.t 8
120.w even 4 1 1800.2.d.s 8
120.w even 4 1 1800.2.d.t 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 5.c odd 4 1
600.2.d.g 8 40.i odd 4 1
600.2.d.h 8 5.c odd 4 1
600.2.d.h 8 40.i odd 4 1
600.2.k.d 8 1.a even 1 1 trivial
600.2.k.d 8 8.b even 2 1 inner
600.2.k.e yes 8 5.b even 2 1
600.2.k.e yes 8 40.f even 2 1
1800.2.d.s 8 15.e even 4 1
1800.2.d.s 8 120.w even 4 1
1800.2.d.t 8 15.e even 4 1
1800.2.d.t 8 120.w even 4 1
1800.2.k.q 8 15.d odd 2 1
1800.2.k.q 8 120.i odd 2 1
1800.2.k.t 8 3.b odd 2 1
1800.2.k.t 8 24.h odd 2 1
2400.2.d.g 8 20.e even 4 1
2400.2.d.g 8 40.k even 4 1
2400.2.d.h 8 20.e even 4 1
2400.2.d.h 8 40.k even 4 1
2400.2.k.d 8 4.b odd 2 1
2400.2.k.d 8 8.d odd 2 1
2400.2.k.e 8 20.d odd 2 1
2400.2.k.e 8 40.e odd 2 1
7200.2.d.s 8 60.l odd 4 1
7200.2.d.s 8 120.q odd 4 1
7200.2.d.t 8 60.l odd 4 1
7200.2.d.t 8 120.q odd 4 1
7200.2.k.r 8 12.b even 2 1
7200.2.k.r 8 24.f even 2 1
7200.2.k.s 8 60.h even 2 1
7200.2.k.s 8 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4T_{7}^{3} - 6T_{7}^{2} + 12T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} - 4 T^{5} - 6 T^{4} + \cdots + 16$$
$3$ $$(T^{2} + 1)^{4}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 4 T^{3} - 6 T^{2} + 12 T + 1)^{2}$$
$11$ $$T^{8} + 32 T^{6} + 336 T^{4} + \cdots + 1600$$
$13$ $$T^{8} + 44 T^{6} + 502 T^{4} + \cdots + 81$$
$17$ $$(T^{4} - 40 T^{2} + 104 T - 24)^{2}$$
$19$ $$T^{8} + 116 T^{6} + 4662 T^{4} + \cdots + 380689$$
$23$ $$(T^{4} - 4 T^{3} - 56 T^{2} + 152 T - 88)^{2}$$
$29$ $$T^{8} + 144 T^{6} + 6288 T^{4} + \cdots + 627264$$
$31$ $$(T^{4} - 4 T^{3} - 70 T^{2} + 204 T + 673)^{2}$$
$37$ $$T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096$$
$41$ $$(T^{4} - 64 T^{2} + 56 T + 328)^{2}$$
$43$ $$T^{8} + 244 T^{6} + 18614 T^{4} + \cdots + 4363921$$
$47$ $$(T^{4} - 72 T^{2} - 256 T - 176)^{2}$$
$53$ $$T^{8} + 256 T^{6} + 19536 T^{4} + \cdots + 23104$$
$59$ $$T^{8} + 432 T^{6} + \cdots + 31181056$$
$61$ $$T^{8} + 236 T^{6} + 14838 T^{4} + \cdots + 3025$$
$67$ $$T^{8} + 372 T^{6} + \cdots + 25979409$$
$71$ $$(T^{4} + 20 T^{3} + 96 T^{2} - 72 T - 536)^{2}$$
$73$ $$(T^{4} + 8 T^{3} - 168 T^{2} - 864 T - 432)^{2}$$
$79$ $$(T^{4} + 8 T^{3} - 184 T^{2} - 864 T + 8080)^{2}$$
$83$ $$T^{8} + 368 T^{6} + 44256 T^{4} + \cdots + 3873024$$
$89$ $$(T^{4} - 224 T^{2} + 64 T + 10880)^{2}$$
$97$ $$(T^{4} + 4 T^{3} - 202 T^{2} - 348 T + 8881)^{2}$$