# Properties

 Label 46.2.c.a Level $46$ Weight $2$ Character orbit 46.c Analytic conductor $0.367$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$46 = 2 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 46.c (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.367311849298$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/46\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-\zeta_{22}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.142315 + 0.989821i 0.959493 − 0.281733i 0.654861 + 0.755750i −0.841254 − 0.540641i −0.415415 − 0.909632i −0.415415 + 0.909632i 0.142315 − 0.989821i −0.841254 + 0.540641i 0.654861 − 0.755750i 0.959493 + 0.281733i
−0.142315 0.989821i 0.580699 1.27155i −0.959493 + 0.281733i −1.66741 + 1.92429i −1.34125 0.393828i 1.75667 1.12894i 0.415415 + 0.909632i 0.684944 + 0.790468i 2.14200 + 1.37658i
9.1 −0.959493 + 0.281733i 0.712591 + 0.822373i 0.841254 0.540641i 0.174863 + 1.21620i −0.915415 0.588302i 0.260554 0.570534i −0.654861 + 0.755750i 0.258432 1.79743i −0.510424 1.11767i
13.1 −0.654861 0.755750i −2.71616 1.74557i −0.142315 + 0.989821i 0.985691 2.15836i 0.459493 + 3.19584i 0.381761 + 0.112095i 0.841254 0.540641i 3.08427 + 6.75361i −2.27667 + 0.668491i
25.1 0.841254 + 0.540641i −0.0530529 0.368991i 0.415415 + 0.909632i −3.26024 0.957293i 0.154861 0.339098i −0.297176 + 0.342959i −0.142315 + 0.989821i 2.74514 0.806046i −2.22514 2.56794i
27.1 0.415415 + 0.909632i −0.524075 + 0.153882i −0.654861 + 0.755750i 0.767092 + 0.492980i −0.357685 0.412791i −0.601808 4.18567i −0.959493 0.281733i −2.27279 + 1.46063i −0.129769 + 0.902563i
29.1 0.415415 0.909632i −0.524075 0.153882i −0.654861 0.755750i 0.767092 0.492980i −0.357685 + 0.412791i −0.601808 + 4.18567i −0.959493 + 0.281733i −2.27279 1.46063i −0.129769 0.902563i
31.1 −0.142315 + 0.989821i 0.580699 + 1.27155i −0.959493 0.281733i −1.66741 1.92429i −1.34125 + 0.393828i 1.75667 + 1.12894i 0.415415 0.909632i 0.684944 0.790468i 2.14200 1.37658i
35.1 0.841254 0.540641i −0.0530529 + 0.368991i 0.415415 0.909632i −3.26024 + 0.957293i 0.154861 + 0.339098i −0.297176 0.342959i −0.142315 0.989821i 2.74514 + 0.806046i −2.22514 + 2.56794i
39.1 −0.654861 + 0.755750i −2.71616 + 1.74557i −0.142315 0.989821i 0.985691 + 2.15836i 0.459493 3.19584i 0.381761 0.112095i 0.841254 + 0.540641i 3.08427 6.75361i −2.27667 0.668491i
41.1 −0.959493 0.281733i 0.712591 0.822373i 0.841254 + 0.540641i 0.174863 1.21620i −0.915415 + 0.588302i 0.260554 + 0.570534i −0.654861 0.755750i 0.258432 + 1.79743i −0.510424 + 1.11767i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 41.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 46.2.c.a 10
3.b odd 2 1 414.2.i.f 10
4.b odd 2 1 368.2.m.b 10
23.c even 11 1 inner 46.2.c.a 10
23.c even 11 1 1058.2.a.m 5
23.d odd 22 1 1058.2.a.l 5
69.g even 22 1 9522.2.a.bu 5
69.h odd 22 1 414.2.i.f 10
69.h odd 22 1 9522.2.a.bp 5
92.g odd 22 1 368.2.m.b 10
92.g odd 22 1 8464.2.a.bx 5
92.h even 22 1 8464.2.a.bw 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.a 10 1.a even 1 1 trivial
46.2.c.a 10 23.c even 11 1 inner
368.2.m.b 10 4.b odd 2 1
368.2.m.b 10 92.g odd 22 1
414.2.i.f 10 3.b odd 2 1
414.2.i.f 10 69.h odd 22 1
1058.2.a.l 5 23.d odd 22 1
1058.2.a.m 5 23.c even 11 1
8464.2.a.bw 5 92.h even 22 1
8464.2.a.bx 5 92.g odd 22 1
9522.2.a.bp 5 69.h odd 22 1
9522.2.a.bu 5 69.g even 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + 4T_{3}^{9} + 5T_{3}^{8} - 2T_{3}^{7} + 25T_{3}^{6} + T_{3}^{5} + 4T_{3}^{4} + 16T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(46, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$3$ $$T^{10} + 4 T^{9} + 5 T^{8} - 2 T^{7} + \cdots + 1$$
$5$ $$T^{10} + 6 T^{9} + 14 T^{8} + 29 T^{7} + \cdots + 529$$
$7$ $$T^{10} - 3 T^{9} + 20 T^{8} - 71 T^{7} + \cdots + 1$$
$11$ $$T^{10} + 12 T^{9} + 56 T^{8} + \cdots + 109561$$
$13$ $$T^{10} + 14 T^{9} + 86 T^{8} + 258 T^{7} + \cdots + 1$$
$17$ $$T^{10} - 15 T^{9} + 137 T^{8} + \cdots + 214369$$
$19$ $$T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 4489$$
$23$ $$T^{10} + T^{9} + 78 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} + 8 T^{9} - 2 T^{8} + 259 T^{7} + \cdots + 4489$$
$31$ $$T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 20529961$$
$37$ $$T^{10} - 28 T^{9} + 509 T^{8} + \cdots + 49857721$$
$41$ $$T^{10} + 31 T^{9} + \cdots + 172475689$$
$43$ $$T^{10} - 11 T^{9} + 66 T^{8} + \cdots + 7027801$$
$47$ $$(T^{5} - 9 T^{4} - 82 T^{3} + 922 T^{2} + \cdots - 529)^{2}$$
$53$ $$T^{10} + 21 T^{9} + 144 T^{8} + \cdots + 31236921$$
$59$ $$T^{10} + 5 T^{9} - 41 T^{8} + \cdots + 4489$$
$61$ $$T^{10} - 37 T^{9} + \cdots + 349727401$$
$67$ $$T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 94109401$$
$71$ $$T^{10} - 49 T^{9} + 1180 T^{8} + \cdots + 8300161$$
$73$ $$T^{10} + 8 T^{9} + 42 T^{8} + \cdots + 7921$$
$79$ $$T^{10} - 8 T^{9} + 9 T^{8} + \cdots + 17161$$
$83$ $$T^{10} + 7 T^{9} + 49 T^{8} + \cdots + 667137241$$
$89$ $$T^{10} + 13 T^{9} + 147 T^{8} + \cdots + 26739241$$
$97$ $$T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 5031049$$