Properties

 Label 418.2.f.c Level $418$ Weight $2$ Character orbit 418.f Analytic conductor $3.338$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.f (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ 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: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.809017 + 2.48990i −0.309017 + 0.951057i −1.30902 + 0.951057i −0.809017 0.587785i −0.618034 1.90211i −2.61803
191.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.309017 0.224514i 0.809017 0.587785i −0.190983 0.587785i 0.309017 0.951057i 1.61803 + 1.17557i −0.381966
229.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.809017 2.48990i −0.309017 0.951057i −1.30902 0.951057i −0.809017 + 0.587785i −0.618034 + 1.90211i −2.61803
267.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.309017 + 0.224514i 0.809017 + 0.587785i −0.190983 + 0.587785i 0.309017 + 0.951057i 1.61803 1.17557i −0.381966
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.c 4
11.c even 5 1 inner 418.2.f.c 4
11.c even 5 1 4598.2.a.bd 2
11.d odd 10 1 4598.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.c 4 1.a even 1 1 trivial
418.2.f.c 4 11.c even 5 1 inner
4598.2.a.u 2 11.d odd 10 1
4598.2.a.bd 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} + T^{3} + 6 T^{2} - 4 T + 1$$
$7$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$11$ $$T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 121$$
$13$ $$T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81$$
$17$ $$T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1$$
$19$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$23$ $$(T + 1)^{4}$$
$29$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$31$ $$T^{4} - 13 T^{3} + 139 T^{2} + \cdots + 1681$$
$37$ $$T^{4} + 8 T^{3} + 114 T^{2} + \cdots + 11881$$
$41$ $$T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961$$
$43$ $$(T^{2} + 2 T - 44)^{2}$$
$47$ $$T^{4} - 17 T^{3} + 139 T^{2} + \cdots + 1681$$
$53$ $$T^{4} + 19 T^{3} + 141 T^{2} + \cdots + 3721$$
$59$ $$T^{4} - 10 T^{3} + 40 T^{2} - 25 T + 25$$
$61$ $$T^{4} - 18 T^{3} + 324 T^{2} + \cdots + 6561$$
$67$ $$(T^{2} + 9 T + 19)^{2}$$
$71$ $$T^{4} + 17 T^{3} + 159 T^{2} + \cdots + 19321$$
$73$ $$T^{4} - T^{3} + 6 T^{2} + 4 T + 1$$
$79$ $$T^{4} + 15 T^{3} + 225 T^{2} + \cdots + 50625$$
$83$ $$T^{4} - 31 T^{3} + 466 T^{2} + \cdots + 19321$$
$89$ $$(T^{2} - 10 T - 155)^{2}$$
$97$ $$T^{4} - 27 T^{3} + 379 T^{2} + \cdots + 14641$$