# Properties

 Label 570.2.q.c Level $570$ Weight $2$ Character orbit 570.q Analytic conductor $4.551$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + \cdots + 1024$$ x^20 - 49*x^16 - 8*x^15 + 72*x^13 + 2145*x^12 - 648*x^11 + 32*x^10 - 7056*x^9 - 11968*x^8 + 10368*x^7 + 9344*x^6 + 18176*x^5 + 56320*x^4 + 28160*x^3 + 8192*x^2 + 4096*x + 1024 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{11} q^{2} + \beta_{11} q^{3} + ( - \beta_{12} + 1) q^{4} + ( - \beta_{17} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_1) q^{5} + (\beta_{12} - 1) q^{6} + ( - \beta_{17} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{5} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{5}) q^{8} + ( - \beta_{12} + 1) q^{9}+O(q^{10})$$ q - b11 * q^2 + b11 * q^3 + (-b12 + 1) * q^4 + (-b17 + b8 + b7 - b5 - b1) * q^5 + (b12 - 1) * q^6 + (-b17 + b11 - b10 + b9 + b5 - b1) * q^7 + (-b11 - b5) * q^8 + (-b12 + 1) * q^9 $$q - \beta_{11} q^{2} + \beta_{11} q^{3} + ( - \beta_{12} + 1) q^{4} + ( - \beta_{17} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_1) q^{5} + (\beta_{12} - 1) q^{6} + ( - \beta_{17} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{5} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{5}) q^{8} + ( - \beta_{12} + 1) q^{9} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} + 1) q^{10} + ( - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4}) q^{11} + (\beta_{11} + \beta_{5}) q^{12} + ( - 2 \beta_{14} - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{15} + \beta_{12} - \beta_{8} + \beta_{3}) q^{14} + (\beta_{15} - \beta_{13} + \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} - 1) q^{15} - \beta_{12} q^{16} + ( - 2 \beta_{18} - \beta_{17} + \beta_{16} - \beta_{11} + \beta_{8} + \beta_{2}) q^{17} + ( - \beta_{11} - \beta_{5}) q^{18} + (\beta_{19} + \beta_{15} + \beta_{8} - \beta_{6}) q^{19} + ( - \beta_{17} - \beta_{2} - \beta_1) q^{20} + (\beta_{15} - \beta_{12} + \beta_{8} - \beta_{3}) q^{21} + ( - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{8} + \beta_{2}) q^{22} + ( - \beta_{9} + \beta_{8} - \beta_{7} + 3 \beta_{5} - \beta_{2} + \beta_1) q^{23} + \beta_{12} q^{24} + (\beta_{19} + \beta_{17} - 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + \cdots + 3) q^{25}+ \cdots + ( - \beta_{19} - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 1) q^{99}+O(q^{100})$$ q - b11 * q^2 + b11 * q^3 + (-b12 + 1) * q^4 + (-b17 + b8 + b7 - b5 - b1) * q^5 + (b12 - 1) * q^6 + (-b17 + b11 - b10 + b9 + b5 - b1) * q^7 + (-b11 - b5) * q^8 + (-b12 + 1) * q^9 + (-b15 + b13 - b12 + b9 + b8 + b6 - b3 + 1) * q^10 + (-b17 + b16 + b15 + b13 - b10 - b9 + b8 + b7 - b6 - b4) * q^11 + (b11 + b5) * q^12 + (-2*b14 - 2*b11 - 2*b10 + b9 - b8 + b7 - b2 - b1) * q^13 + (-b15 + b12 - b8 + b3) * q^14 + (b15 - b13 + b12 - b9 - b8 - b6 + b3 - 1) * q^15 - b12 * q^16 + (-2*b18 - b17 + b16 - b11 + b8 + b2) * q^17 + (-b11 - b5) * q^18 + (b19 + b15 + b8 - b6) * q^19 + (-b17 - b2 - b1) * q^20 + (b15 - b12 + b8 - b3) * q^21 + (-b18 - b17 + b16 + b8 + b2) * q^22 + (-b9 + b8 - b7 + 3*b5 - b2 + b1) * q^23 + b12 * q^24 + (b19 + b17 - 2*b16 - b15 - b14 - b13 - 2*b12 - 2*b11 - b10 + 3*b9 - 3*b8 - b7 + b6 + b4 - b3 - 2*b2 - 2*b1 + 3) * q^25 + (b6 - 2*b4 + 1) * q^26 + (b11 + b5) * q^27 + (-b16 - b11 - b10 + b9 - b8 + b5 - b2 - b1) * q^28 + (-2*b19 - b17 + b16 + 3*b13 - 2*b12 - b10 + 2*b9 + 3*b8 + b7 + b6 - 2*b4 - b3 - b1 + 1) * q^29 + (b17 + b2 + b1) * q^30 + (-b17 + b16 + b15 + b13 - b10 - b9 + b8 + b7 + b6 + b4 + 1) * q^31 - b5 * q^32 + (b18 + b17 - b16 - b8 - b2) * q^33 + (-2*b19 - b17 + b16 + 2*b15 + b13 - b10 - b9 + 2*b8 + b7 - b6 - 2*b4 + b3 - 1) * q^34 + (b19 - b18 - b17 + 2*b16 + b15 - 2*b13 + 2*b12 - b10 - 2*b9 + b5 + 2*b3 + b2 - b1 - 1) * q^35 - b12 * q^36 + (-b18 - 2*b17 + b14 + 3*b11 + b9 + b8 + 2*b5 - b1) * q^37 + (b17 - b14 + b10 - b9 - b7 + b2 + b1) * q^38 + (-b6 + 2*b4 - 1) * q^39 + (-b17 + b16 - b15 + b13 - b12 - b10 + b9 + b8 + b7 - b3 - b1 + 1) * q^40 + (-4*b15 + 3*b13 - 5*b12 + 3*b9 - b8 - 2*b3 + 3) * q^41 + (b16 + b11 + b10 - b9 + b8 - b5 + b2 + b1) * q^42 + (b18 - 2*b16 + b11 + b10 + b7 - 2*b5 - b2) * q^43 + (-b19 - b17 + b16 + b15 + b13 - b10 - b9 + b8 + b7 - b6 - b4 + b3 - 1) * q^44 + (-b17 - b2 - b1) * q^45 + (-2*b17 + 2*b16 - 2*b10 - 2*b9 + 2*b7 - b6 - 2*b1 + 2) * q^46 + (b16 - b14 + b9 - b8 + b7 - 2*b5 + b2 - b1) * q^47 + b5 * q^48 + (b15 + b13 + b8 - 2*b6 + b4 + b1 - 2) * q^49 + (b18 + 2*b17 - b16 - b15 - b14 - b13 - 2*b11 + b10 + b9 - 2*b8 - b7 + b6 - b5 - b4 - b2 + 1) * q^50 + (2*b19 + b17 - b16 - 2*b15 - b13 + b10 + b9 - 2*b8 - b7 + b6 + 2*b4 - b3 + 1) * q^51 + (-2*b18 - b17 - b11 + b8 + b7 - b5) * q^52 + (-3*b16 + 2*b14 - b11 - b10 - b9 + b8 + b7 - 3*b5 + b1) * q^53 + b12 * q^54 + (2*b18 + b17 + b15 + b11 - b10 + b5 - b3 - b2 - 2*b1) * q^55 + (b17 - b16 - b15 - b13 + b10 + b9 - b8 - b7 + b6 + 2) * q^56 + (-b17 + b14 - b10 + b9 + b7 - b2 - b1) * q^57 + (-2*b18 - 2*b17 - b16 + 2*b14 + 2*b10 + 2*b8 - b7 - 2*b5) * q^58 + (b19 + b17 - b16 - b15 + b13 - b12 + b10 + b9 - b8 - b7 - 2*b3 + 2*b1 + 2) * q^59 + (b17 - b16 + b15 - b13 + b12 + b10 - b9 - b8 - b7 + b3 + b1 - 1) * q^60 + (3*b17 - 3*b16 + 3*b15 - 6*b13 + 2*b12 + 3*b10 - 3*b7 + 1) * q^61 + (b18 - 3*b17 + b16 - b11 + 3*b8 + 2*b7 - 2*b5 + b2) * q^62 + (-b16 - b11 - b10 + b9 - b8 + b5 - b2 - b1) * q^63 - q^64 + (-b18 + 2*b17 - 3*b15 + b14 - 3*b13 + b11 + b10 - 2*b9 - 2*b8 + 2*b6 + 2*b4 + 3*b2 + 4) * q^65 + (b19 + b17 - b16 - b15 - b13 + b10 + b9 - b8 - b7 + b6 + b4 - b3 + 1) * q^66 + (2*b16 - b14 + b11 + b10 + 2*b9 - 2*b8 + b2 - 2*b1) * q^67 + (-2*b18 - b17 + 2*b14 + b10 - b9 + 2*b8 - 2*b5 + 2*b2 + b1) * q^68 + (2*b17 - 2*b16 + 2*b10 + 2*b9 - 2*b7 + b6 + 2*b1 - 2) * q^69 + (-b19 - b17 + b16 + 2*b15 - b14 + b12 - b11 - 2*b10 - 4*b9 + 2*b8 + 3*b7 - 2*b6 - b5 - b4 + 2*b3 + b2 + 2*b1 - 2) * q^70 + (-3*b19 + 2*b17 - 2*b16 + 4*b15 - 3*b13 + 5*b12 + 2*b10 - 3*b9 - b8 - 2*b7 + 5*b3 + 4*b1 - 6) * q^71 - b5 * q^72 + (2*b18 - b17 + 4*b16 + 6*b11 - 3*b10 + b8 + 3*b5 + b2) * q^73 + (-b19 - b15 + b13 + 2*b12 + b9) * q^74 + (-b18 - 2*b17 + b16 + b15 + b14 + b13 + 2*b11 - b10 - b9 + 2*b8 + b7 - b6 + b5 + b4 + b2 - 1) * q^75 + (-2*b17 + 2*b16 + b15 + b13 - 2*b10 - b9 + 2*b8 + 2*b7 - b6 - b4 + b3 - 2*b1 - 1) * q^76 + (b18 - 2*b17 + 2*b16 - b14 - 4*b11 - 3*b10 + 3*b9 - b8 + 2*b7 - 3*b5 - b2 - 3*b1) * q^77 + (2*b18 + b17 + b11 - b8 - b7 + b5) * q^78 + (-b19 + 2*b17 - 2*b16 + 2*b13 - b12 + 2*b10 + 2*b9 - 2*b7 - 3*b3 + 4*b1 + 1) * q^79 + (-b8 - b7 + b5 - b2) * q^80 - b12 * q^81 + (-b16 - b11 - b10 + 4*b9 - 4*b8 - 3*b7 + b5 - 4*b2 - 4*b1) * q^82 + (4*b18 + 3*b17 - b16 - 4*b14 - 4*b11 - b10 + b9 - 4*b8 - b7 - 4*b2 - b1) * q^83 + (-b17 + b16 + b15 + b13 - b10 - b9 + b8 + b7 - b6 - 2) * q^84 + (3*b19 + 2*b17 - b16 - 2*b15 + b14 - 3*b13 + b12 + 2*b11 + 4*b10 - 4*b9 - b7 + b6 - 2*b5 + 3*b4 - b3 + 3*b2 + 5*b1 + 1) * q^85 + (b19 + b17 - b16 - 3*b15 + b13 - b12 + b10 + 4*b9 - b7 + 3*b6 + b4 - 3*b3 - b1 + 2) * q^86 + (2*b18 + 2*b17 + b16 - 2*b14 - 2*b10 - 2*b8 + b7 + 2*b5) * q^87 + (-b18 - b17 + b14 + b8 - b5 + b2) * q^88 + (-3*b19 - 2*b17 + 2*b16 + 3*b15 + 2*b13 + 2*b12 - 2*b10 - 3*b9 + 2*b8 + 2*b7 - 2*b6 - 3*b4 + 2*b3 + b1 - 4) * q^89 + (-b17 + b16 - b15 + b13 - b12 - b10 + b9 + b8 + b7 - b3 - b1 + 1) * q^90 + (4*b19 + b15 - 5*b13 + 8*b12 - 7*b9 - 6*b8 - 5*b6 + 4*b4 + 5*b3 + 2*b1 - 8) * q^91 + (-b17 + 2*b16 - 2*b11 - 2*b10 + b8 + b7 + b5) * q^92 + (-b18 + 3*b17 - b16 + b11 - 3*b8 - 2*b7 + 2*b5 - b2) * q^93 + (b17 - b16 + b15 + b13 + b10 + b9 + b8 - b7 - b4 + 2*b1 - 2) * q^94 + (2*b18 - 2*b16 - b15 + b14 - b13 - 2*b12 + 2*b11 + 4*b9 - b8 - 3*b7 + b6 + 2*b5 + b4 - b3 - b2 - 4*b1 - 2) * q^95 + q^96 + (b18 - b16 + 5*b11 + 3*b10 - 2*b7 - b5 + 2*b2) * q^97 + (b18 + b17 + 2*b11 + b10 - b8 - 2*b7 + b5 + b2) * q^98 + (-b19 - b17 + b16 + b15 + b13 - b10 - b9 + b8 + b7 - b6 - b4 + b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 10 q^{4} - 10 q^{6} + 10 q^{9}+O(q^{10})$$ 20 * q + 10 * q^4 - 10 * q^6 + 10 * q^9 $$20 q + 10 q^{4} - 10 q^{6} + 10 q^{9} - 2 q^{10} + 12 q^{11} + 10 q^{14} + 2 q^{15} - 10 q^{16} + 6 q^{19} - 10 q^{21} + 10 q^{24} + 14 q^{25} + 8 q^{29} + 40 q^{31} + 12 q^{34} + 2 q^{35} - 10 q^{36} + 2 q^{40} - 14 q^{41} + 6 q^{44} + 44 q^{46} - 8 q^{49} - 8 q^{50} - 12 q^{51} + 10 q^{54} + 20 q^{56} + 8 q^{59} - 2 q^{60} + 16 q^{61} - 20 q^{64} + 40 q^{65} - 6 q^{66} - 44 q^{69} + 8 q^{70} - 4 q^{71} + 26 q^{74} + 8 q^{75} + 8 q^{79} - 10 q^{81} - 20 q^{84} - 16 q^{85} - 20 q^{86} - 2 q^{89} + 2 q^{90} - 44 q^{91} - 32 q^{94} - 80 q^{95} + 20 q^{96} + 6 q^{99}+O(q^{100})$$ 20 * q + 10 * q^4 - 10 * q^6 + 10 * q^9 - 2 * q^10 + 12 * q^11 + 10 * q^14 + 2 * q^15 - 10 * q^16 + 6 * q^19 - 10 * q^21 + 10 * q^24 + 14 * q^25 + 8 * q^29 + 40 * q^31 + 12 * q^34 + 2 * q^35 - 10 * q^36 + 2 * q^40 - 14 * q^41 + 6 * q^44 + 44 * q^46 - 8 * q^49 - 8 * q^50 - 12 * q^51 + 10 * q^54 + 20 * q^56 + 8 * q^59 - 2 * q^60 + 16 * q^61 - 20 * q^64 + 40 * q^65 - 6 * q^66 - 44 * q^69 + 8 * q^70 - 4 * q^71 + 26 * q^74 + 8 * q^75 + 8 * q^79 - 10 * q^81 - 20 * q^84 - 16 * q^85 - 20 * q^86 - 2 * q^89 + 2 * q^90 - 44 * q^91 - 32 * q^94 - 80 * q^95 + 20 * q^96 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + \cdots + 1024$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 12\!\cdots\!99 \nu^{19} + \cdots + 78\!\cdots\!92 ) / 12\!\cdots\!52$$ (-1247415232500299*v^19 + 14121847713683436*v^18 - 2452242241905736*v^17 - 1345766534397896*v^16 + 65703114624924227*v^15 - 691819782689747060*v^14 + 19186881322460808*v^13 - 7078576395524816*v^12 - 1843195025438242611*v^11 + 31306801912862989668*v^10 - 15100640410954392168*v^9 + 8364294482985111976*v^8 - 76266479634225026904*v^7 - 185075074826781213152*v^6 + 207634332638339783200*v^5 + 54202215092729018944*v^4 + 226095576498151131648*v^3 + 646913736064602024448*v^2 + 74315506995497216*v + 7896690386388411392) / 120410357182372042752 $$\beta_{3}$$ $$=$$ $$( - 33\!\cdots\!57 \nu^{19} + \cdots - 93\!\cdots\!36 ) / 10\!\cdots\!76$$ (-3386282915786266757*v^19 + 1405722860355455696*v^18 + 1691818199538348336*v^17 - 3166847588073419904*v^16 + 159444937358634070533*v^15 - 37371725192928414376*v^14 - 106497758856045419280*v^13 - 90380106099253612008*v^12 - 6831828743877138610549*v^11 + 5163070527522030233720*v^10 + 2884616161494605905520*v^9 + 15222516788927210798288*v^8 + 19762356858230169519952*v^7 - 51018372230969002051328*v^6 - 40605238149096354346208*v^5 + 77784244236226642424704*v^4 - 174407667837172815736320*v^3 - 48014712778533209974528*v^2 - 25197546629123584491008*v - 93198922890801102759936) / 109934656107505675032576 $$\beta_{4}$$ $$=$$ $$( - 42\!\cdots\!01 \nu^{19} + \cdots + 27\!\cdots\!92 ) / 10\!\cdots\!76$$ (-4232101711420771201*v^19 + 2181486017059807400*v^18 - 4124378885648077616*v^17 + 7897757488520220768*v^16 + 203874060607586606945*v^15 - 73137442077379054496*v^14 + 177447029744981331984*v^13 - 646457938616487397544*v^12 - 8781772937239702318673*v^11 + 7146456142840655417264*v^10 - 9377744037444471122096*v^9 + 49144256794412031271536*v^8 + 21103295458928210307824*v^7 - 38635350105783020220544*v^6 - 38809047547892025412960*v^5 - 153076350746732603062912*v^4 - 81038318609659313118720*v^3 - 22795926509410679341824*v^2 - 11496553397312437555712*v + 275149990140827868475392) / 109934656107505675032576 $$\beta_{5}$$ $$=$$ $$( 31\!\cdots\!63 \nu^{19} + \cdots + 22\!\cdots\!44 ) / 54\!\cdots\!88$$ (3133260286595853563*v^19 - 8917971619032791856*v^18 + 4476196673936002298*v^17 - 1384964632967199976*v^16 - 153147956971322109391*v^15 + 411566901687289627400*v^14 - 147180711438880372858*v^13 + 256929796626116377360*v^12 + 6070954064014948604319*v^11 - 20831081326414491767240*v^10 + 15345591852660473358458*v^9 - 28211644710196789844200*v^8 + 27257710650423835000892*v^7 + 107001943669920263298480*v^6 - 105079993696826594907168*v^5 + 31913630923549524706080*v^4 + 39990879319161082027072*v^3 - 366215923858014787519488*v^2 + 4751904279189369378688*v + 2229839295497973177344) / 54967328053752837516288 $$\beta_{6}$$ $$=$$ $$( 16\!\cdots\!58 \nu^{19} + \cdots - 52\!\cdots\!24 ) / 27\!\cdots\!44$$ (1648552185841481858*v^19 - 868072745834387710*v^18 + 1407550487586609139*v^17 - 3367177978530011772*v^16 - 79066071146916754258*v^15 + 29043521716966332070*v^14 - 60745525849893456627*v^13 + 273164235167645503348*v^12 + 3416635977348824906626*v^11 - 2825268522512545867918*v^10 + 3345480027873051449347*v^9 - 19652502825124211122740*v^8 - 7717350402685912928368*v^7 + 15688567928872081773416*v^6 + 16310030431118385830240*v^5 + 62558402718936493084736*v^4 + 33010525034365549456992*v^3 + 9306741062666398339968*v^2 + 4686268323500600734720*v - 52286566420136855591424) / 27483664026876418758144 $$\beta_{7}$$ $$=$$ $$( - 10\!\cdots\!81 \nu^{19} + \cdots - 15\!\cdots\!92 ) / 10\!\cdots\!76$$ (-10889321835906579181*v^19 + 20480885828464279044*v^18 - 14508099400000947584*v^17 + 6115691526109346312*v^16 + 526434800852081918149*v^15 - 908100348750920697724*v^14 + 535220462230478409792*v^13 - 965811755004771615856*v^12 - 21616254071927221899477*v^11 + 49545471162119392308300*v^10 - 43767528870466275198432*v^9 + 99449244681413964232280*v^8 - 31860523066686255721560*v^7 - 231871919707911102984928*v^6 + 209477736315213849314528*v^5 - 168539132167513140495040*v^4 - 356775643605785589354240*v^3 + 498778830461939754037760*v^2 - 17947123472360399154944*v - 15616434662069267182592) / 109934656107505675032576 $$\beta_{8}$$ $$=$$ $$( - 43\!\cdots\!88 \nu^{19} + \cdots - 16\!\cdots\!96 ) / 27\!\cdots\!44$$ (-4348683772245218088*v^19 + 450224447508307223*v^18 - 250293380222287660*v^17 + 252414601602031042*v^16 + 211936627997478730656*v^15 + 13307367314496253097*v^14 + 8336154124993271236*v^13 - 324187926250202751138*v^12 - 9242242633044145305656*v^11 + 3747665020505397420871*v^10 - 940060277648747221732*v^9 + 31353272552620295570722*v^8 + 46262382297897708509688*v^7 - 46795392525082251159280*v^6 - 35851029569796743766544*v^5 - 73660308958475322375328*v^4 - 225492176012277395225344*v^3 - 112287673008321933193152*v^2 - 32741613241344859875840*v - 16365664137352620048896) / 27483664026876418758144 $$\beta_{9}$$ $$=$$ $$( - 6169005809303 \nu^{19} + 3083823703934 \nu^{18} - 1600056848304 \nu^{17} + 905444482148 \nu^{16} + 301739777308423 \nu^{15} + \cdots - 21\!\cdots\!64 ) / 34\!\cdots\!32$$ (-6169005809303*v^19 + 3083823703934*v^18 - 1600056848304*v^17 + 905444482148*v^16 + 301739777308423*v^15 - 101624356612614*v^14 + 50753283745472*v^13 - 474300220611292*v^12 - 12992549378199047*v^11 + 10490522679024534*v^10 - 5431574304698976*v^9 + 46523908980338900*v^8 + 50324957261568352*v^7 - 88946154925346464*v^6 - 19365916040629696*v^5 - 102172307974395328*v^4 - 295267134460650752*v^3 - 15600348856665216*v^2 - 7910772435618304*v - 21232184143012864) / 34667475668677632 $$\beta_{10}$$ $$=$$ $$( 11\!\cdots\!77 \nu^{19} + \cdots + 22\!\cdots\!04 ) / 27\!\cdots\!44$$ (11647748895677625177*v^19 - 2678420044235495016*v^18 - 1006896450659993441*v^17 + 876149593765785408*v^16 - 570537544771009026203*v^15 + 40308075782636361480*v^14 + 65847772884033547121*v^13 + 806794749290181775616*v^12 + 24771131063838911503947*v^11 - 13481413914046234941392*v^10 + 221704128229926374015*v^9 - 79832070942964433951880*v^8 - 120402255969919478060242*v^7 + 164331049679298755322760*v^6 + 75811190849997163996192*v^5 + 173733884004395553256912*v^4 + 581090240131191728140768*v^3 + 172547640415924211795968*v^2 + 12619980319796627765184*v + 22984806829298006831104) / 27483664026876418758144 $$\beta_{11}$$ $$=$$ $$( 31\!\cdots\!33 \nu^{19} + \cdots + 65\!\cdots\!88 ) / 54\!\cdots\!88$$ (31964187768266836033*v^19 - 8697367544490436176*v^18 + 900448895016614446*v^17 - 500586760444575320*v^16 - 1565740371441870903533*v^15 + 168159753848822773048*v^14 + 26614734628992506194*v^13 + 2318093827565198736848*v^12 + 67914806910431957788509*v^11 - 39197278939925200360696*v^10 + 8518184049595333594798*v^9 - 227419429448188289492312*v^8 - 319840854105376902501500*v^7 + 423929463377185973009520*v^6 + 205082585456520813573792*v^5 + 509279017736424524202720*v^4 + 1652902437191837560627904*v^3 + 449127175529839312238592*v^2 + 37275280180998054396032*v + 65442086616131240639488) / 54967328053752837516288 $$\beta_{12}$$ $$=$$ $$( - 20734554827161 \nu^{19} + 6169005809303 \nu^{18} - 3083823703934 \nu^{17} + 1600056848304 \nu^{16} + \cdots - 42\!\cdots\!20 ) / 34\!\cdots\!32$$ (-20734554827161*v^19 + 6169005809303*v^18 - 3083823703934*v^17 + 1600056848304*v^16 + 1015087742048741*v^15 - 135863338691135*v^14 + 101624356612614*v^13 - 1543641231301064*v^12 - 44001319883649053*v^11 + 26428540906199375*v^10 - 11154028433493686*v^9 + 151734593165146992*v^8 + 201627243191123948*v^7 - 265300821709573600*v^6 - 104797525379645920*v^5 - 357505352497848640*v^4 - 1065597819891312192*v^3 - 288617929472203008*v^2 - 154257124287437696*v - 42350488467755520) / 34667475668677632 $$\beta_{13}$$ $$=$$ $$( - 36\!\cdots\!23 \nu^{19} + \cdots - 17\!\cdots\!48 ) / 49\!\cdots\!08$$ (-3646259655903443123*v^19 + 1049158889148839496*v^18 - 561240697228913042*v^17 + 22333734971175960*v^16 + 178756408928343520931*v^15 - 22334906367696174976*v^14 + 18774799517307073746*v^13 - 258873072705287923424*v^12 - 7748987885156851181299*v^11 + 4576100104703020725408*v^10 - 1981019837387064789970*v^9 + 26180658128977366937560*v^8 + 36346923085589964549856*v^7 - 46661932408703203780016*v^6 - 17174198126279741012192*v^5 - 62665768111016438619008*v^4 - 190137108819849924869184*v^3 - 51452042331087350676992*v^2 - 32528346917521801635840*v - 17696182533796593435648) / 4997029823068439774208 $$\beta_{14}$$ $$=$$ $$( - 46\!\cdots\!25 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 54\!\cdots\!88$$ (-46527052292546473825*v^19 - 13575408670476075672*v^18 + 12754703821981264468*v^17 - 3220161156818882080*v^16 + 2281387719715237522501*v^15 + 1036205686456293995312*v^14 - 521292289525887764148*v^13 - 3292602276348860657320*v^12 - 100844332868165628305397*v^11 + 1979670180196761370848*v^10 + 34682937247625982645972*v^9 + 312741418200420979610192*v^8 + 658987268537276688945012*v^7 - 412980240669076675040848*v^6 - 714625482892316831898208*v^5 - 806608960105343683192864*v^4 - 2821712631754951968739584*v^3 - 1815147994886654017121536*v^2 - 51040974120829335778688*v - 109576270124468940804608) / 54967328053752837516288 $$\beta_{15}$$ $$=$$ $$( 10\!\cdots\!39 \nu^{19} + \cdots + 14\!\cdots\!28 ) / 10\!\cdots\!76$$ (100048750214915279139*v^19 - 26347596329974818268*v^18 + 13559910524286036752*v^17 - 9559785219930873160*v^16 - 4895119672668795944515*v^15 + 486618328506871659076*v^14 - 457981518467565561040*v^13 + 7576611074740505420320*v^12 + 212450896844449840710707*v^11 - 120247059670561064548404*v^10 + 49005250239774782740912*v^9 - 735964318230292003394168*v^8 - 990727124139759613489872*v^7 + 1243650893808756763834432*v^6 + 557854431525641700600928*v^5 + 1797096405692255675904256*v^4 + 5149244069846542601718784*v^3 + 1599418858853573468076032*v^2 + 745812124127402779890176*v + 147877792913541010915328) / 109934656107505675032576 $$\beta_{16}$$ $$=$$ $$( - 59\!\cdots\!77 \nu^{19} + \cdots - 11\!\cdots\!84 ) / 54\!\cdots\!88$$ (-59060093438929888077*v^19 + 22096477805655091752*v^18 - 5150082873762744724*v^17 + 2319697375049917824*v^16 + 2893979547600525960065*v^15 - 610061920292864514288*v^14 + 67430556229633727284*v^13 - 4320321462853326166760*v^12 - 125128149124225422722673*v^11 + 85303995485854728439808*v^10 - 26699430163015910787860*v^9 + 425587997041208138986992*v^8 + 549956425935581348941444*v^7 - 840988015348757728234768*v^6 - 294305508105010452269536*v^5 - 934263483799541782017184*v^4 - 2981676149031596296847872*v^3 - 405251627508347704559872*v^2 - 70048591237586813293440*v - 118495627306460833513984) / 54967328053752837516288 $$\beta_{17}$$ $$=$$ $$( - 15\!\cdots\!31 \nu^{19} + \cdots - 33\!\cdots\!52 ) / 10\!\cdots\!76$$ (-157764561058411272331*v^19 + 66435128268844438016*v^18 - 18010750078952063316*v^17 + 6486893757396609936*v^16 + 7723278107177057031283*v^15 - 1995095069589434426504*v^14 + 344020371171181834228*v^13 - 11533855176405836475208*v^12 - 333355619254332956043779*v^11 + 243588462743034976864600*v^10 - 85846485453464908062868*v^9 + 1140440383014881957128032*v^8 + 1400628750647165447287704*v^7 - 2302457793493065079545824*v^6 - 632305027203562291899296*v^5 - 2459052955740344136274880*v^4 - 7735591835136637047905920*v^3 - 960083458241244776247040*v^2 - 297250893657428925801728*v - 338129999525234653437952) / 109934656107505675032576 $$\beta_{18}$$ $$=$$ $$( - 40\!\cdots\!87 \nu^{19} + \cdots - 82\!\cdots\!84 ) / 27\!\cdots\!44$$ (-40223802192232846787*v^19 + 9890552072338621158*v^18 + 484772892229500262*v^17 - 195047145168744320*v^16 + 1972023170728282320859*v^15 - 162861158695652904014*v^14 - 103384232028996576086*v^13 - 2889399513322311849544*v^12 - 85612809236527723004667*v^11 + 47294703425592575999630*v^10 - 6677456366575321201994*v^9 + 283440629233831139367616*v^8 + 413628553340261730937576*v^7 - 539467094386649320965120*v^6 - 277266319277329450639776*v^5 - 636215367432115864474176*v^4 - 2084826245599925660715712*v^3 - 567132525750879492500736*v^2 - 46149434678891103030016*v - 82433651039597839952384) / 27483664026876418758144 $$\beta_{19}$$ $$=$$ $$( 22\!\cdots\!27 \nu^{19} + \cdots + 21\!\cdots\!60 ) / 13\!\cdots\!72$$ (2227909443849796827*v^19 - 656774505610434658*v^18 + 364647356926557260*v^17 - 213174748204975680*v^16 - 109161387048547456643*v^15 + 14367159029100068202*v^14 - 12743321084439647196*v^13 + 167929933156791916280*v^12 + 4732955902985575909635*v^11 - 2826807527039742448090*v^10 + 1269085700510484763548*v^9 - 16431980908444778161232*v^8 - 21862622741273844294072*v^7 + 28389111128407692155520*v^6 + 10905220420002583240992*v^5 + 40761879076292763448320*v^4 + 114488831094101234698368*v^3 + 30997659803712423966976*v^2 + 16574288883169887321856*v + 2186718120392649538560) / 1324513929006092470272
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{16} + \beta_{14} - 4\beta_{5}$$ -b16 + b14 - 4*b5 $$\nu^{3}$$ $$=$$ $$7\beta_{17} - 6\beta_{16} - \beta_{15} - \beta_{13} + 7\beta_{10} - \beta_{8} - 6\beta_{7} + \beta_{6} + 6\beta _1 + 1$$ 7*b17 - 6*b16 - b15 - b13 + 7*b10 - b8 - 6*b7 + b6 + 6*b1 + 1 $$\nu^{4}$$ $$=$$ $$7 \beta_{19} - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} + 13 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 9 \beta_{3} - 4 \beta _1 + 9$$ 7*b19 - 2*b17 + 2*b16 - 2*b15 + 2*b13 + 13*b12 - 2*b10 + 2*b9 + 2*b8 + 2*b7 - 9*b3 - 4*b1 + 9 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{17} + 18 \beta_{16} + 9 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} - 47 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + 4 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} + 38 \beta _1 - 5$$ -9*b17 + 18*b16 + 9*b13 - 4*b12 + 9*b11 - 47*b9 + 9*b8 + 9*b7 - 9*b6 + 4*b5 + 9*b3 + 9*b2 + 38*b1 - 5 $$\nu^{6}$$ $$=$$ $$- 47 \beta_{18} - 51 \beta_{17} + 22 \beta_{16} + 47 \beta_{14} - 87 \beta_{11} - 22 \beta_{10} + 4 \beta_{9} + 47 \beta_{8} + 22 \beta_{7} - 134 \beta_{5} + 65 \beta_{2} - 4 \beta_1$$ -47*b18 - 51*b17 + 22*b16 + 47*b14 - 87*b11 - 22*b10 + 4*b9 + 47*b8 + 22*b7 - 134*b5 + 65*b2 - 4*b1 $$\nu^{7}$$ $$=$$ $$- 4 \beta_{19} + 4 \beta_{18} + 250 \beta_{17} - 250 \beta_{16} - 65 \beta_{15} - 48 \beta_{12} - 17 \beta_{11} + 181 \beta_{10} - 315 \beta_{8} - 181 \beta_{7} + 69 \beta_{3} - 69 \beta_{2} + 181 \beta _1 - 4$$ -4*b19 + 4*b18 + 250*b17 - 250*b16 - 65*b15 - 48*b12 - 17*b11 + 181*b10 - 315*b8 - 181*b7 + 69*b3 - 69*b2 + 181*b1 - 4 $$\nu^{8}$$ $$=$$ $$311 \beta_{19} + 319 \beta_{17} - 319 \beta_{16} - 441 \beta_{15} - 189 \beta_{13} + 413 \beta_{12} + 319 \beta_{10} + 635 \beta_{9} - 125 \beta_{8} - 319 \beta_{7} + 449 \beta_{6} + 311 \beta_{4} - 449 \beta_{3} - 186 \beta _1 - 94$$ 311*b19 + 319*b17 - 319*b16 - 441*b15 - 189*b13 + 413*b12 + 319*b10 + 635*b9 - 125*b8 - 319*b7 + 449*b6 + 311*b4 - 449*b3 - 186*b1 - 94 $$\nu^{9}$$ $$=$$ $$64 \beta_{18} - 8 \beta_{17} + 505 \beta_{16} + 505 \beta_{15} - 64 \beta_{14} + 505 \beta_{13} + 428 \beta_{11} - 64 \beta_{10} - 2119 \beta_{9} + 441 \beta_{8} + 505 \beta_{7} - 513 \beta_{6} + 492 \beta_{5} - 64 \beta_{4} - 72 \beta_{2} + \cdots - 13$$ 64*b18 - 8*b17 + 505*b16 + 505*b15 - 64*b14 + 505*b13 + 428*b11 - 64*b10 - 2119*b9 + 441*b8 + 505*b7 - 513*b6 + 492*b5 - 64*b4 - 72*b2 + 433*b1 - 13 $$\nu^{10}$$ $$=$$ $$- 2047 \beta_{18} - 2747 \beta_{17} + 3629 \beta_{16} - 3383 \beta_{11} - 1438 \beta_{10} + 2747 \beta_{8} + 556 \beta_{7} + 882 \beta_{5} + 2191 \beta_{2}$$ -2047*b18 - 2747*b17 + 3629*b16 - 3383*b11 - 1438*b10 + 2747*b8 + 556*b7 + 882*b5 + 2191*b2 $$\nu^{11}$$ $$=$$ $$- 700 \beta_{19} - 3629 \beta_{17} + 844 \beta_{15} + 700 \beta_{14} + 3485 \beta_{13} - 3288 \beta_{12} - 2929 \beta_{11} - 6558 \beta_{10} - 988 \beta_{9} - 9994 \beta_{8} + 3773 \beta_{7} - 3773 \beta_{6} + \cdots - 341$$ -700*b19 - 3629*b17 + 844*b15 + 700*b14 + 3485*b13 - 3288*b12 - 2929*b11 - 6558*b10 - 988*b9 - 9994*b8 + 3773*b7 - 3773*b6 - 4276*b5 - 700*b4 + 3773*b3 - 2785*b2 - 2785*b1 - 341 $$\nu^{12}$$ $$=$$ $$25857 \beta_{17} - 25857 \beta_{16} - 15167 \beta_{15} - 15167 \beta_{13} + 25857 \beta_{10} + 25857 \beta_{9} - 15167 \beta_{8} - 25857 \beta_{7} + 21025 \beta_{6} + 13479 \beta_{4} + 10690 \beta _1 - 21751$$ 25857*b17 - 25857*b16 - 15167*b15 - 15167*b13 + 25857*b10 + 25857*b9 - 15167*b8 - 25857*b7 + 21025*b6 + 13479*b4 + 10690*b1 - 21751 $$\nu^{13}$$ $$=$$ $$6520 \beta_{19} + 6520 \beta_{18} + 24169 \beta_{17} - 25857 \beta_{16} + 17649 \beta_{15} + 1688 \beta_{13} + 24524 \beta_{12} + 6875 \beta_{11} + 1688 \beta_{9} - 4832 \beta_{8} + 1688 \beta_{7} - 1688 \beta_{5} + \cdots + 8208$$ 6520*b19 + 6520*b18 + 24169*b17 - 25857*b16 + 17649*b15 + 1688*b13 + 24524*b12 + 6875*b11 + 1688*b9 - 4832*b8 + 1688*b7 - 1688*b5 - 27545*b3 - 25857*b2 - 53773*b1 + 8208 $$\nu^{14}$$ $$=$$ $$105487 \beta_{16} - 89071 \beta_{14} + 16416 \beta_{11} + 16416 \beta_{10} - 55668 \beta_{9} + 55668 \beta_{8} - 38674 \beta_{7} + 285688 \beta_{5} - 22258 \beta_{2} + 55668 \beta_1$$ 105487*b16 - 89071*b14 + 16416*b11 + 16416*b10 - 55668*b9 + 55668*b8 - 38674*b7 + 285688*b5 - 22258*b2 + 55668*b1 $$\nu^{15}$$ $$=$$ $$- 55668 \beta_{18} - 719327 \beta_{17} + 552330 \beta_{16} + 183413 \beta_{15} + 55668 \beta_{14} + 183413 \beta_{13} - 195104 \beta_{11} - 680075 \beta_{10} - 72084 \beta_{9} + 239081 \beta_{8} + \cdots + 67359$$ -55668*b18 - 719327*b17 + 552330*b16 + 183413*b15 + 55668*b14 + 183413*b13 - 195104*b11 - 680075*b10 - 72084*b9 + 239081*b8 + 552330*b7 - 199829*b6 - 250772*b5 - 55668*b4 + 72084*b2 - 480246*b1 + 67359 $$\nu^{16}$$ $$=$$ $$- 591575 \beta_{19} + 561930 \beta_{17} - 561930 \beta_{16} + 111322 \beta_{15} - 255490 \beta_{13} - 818429 \beta_{12} + 561930 \beta_{10} - 255490 \beta_{9} - 706098 \beta_{8} + \cdots - 847065$$ -591575*b19 + 561930*b17 - 561930*b16 + 111322*b15 - 255490*b13 - 818429*b12 + 561930*b10 - 255490*b9 - 706098*b8 - 561930*b7 + 991233*b3 + 1123860*b1 - 847065 $$\nu^{17}$$ $$=$$ $$450608 \beta_{19} + 1297673 \beta_{17} - 2595346 \beta_{16} - 594776 \beta_{15} + 450608 \beta_{14} - 1153505 \beta_{13} + 1286132 \beta_{12} - 847065 \beta_{11} + 450608 \beta_{10} + \cdots + 11541$$ 450608*b19 + 1297673*b17 - 2595346*b16 - 594776*b15 + 450608*b14 - 1153505*b13 + 1286132*b12 - 847065*b11 + 450608*b10 + 4690143*b9 - 1153505*b8 - 1153505*b7 + 1441841*b6 - 2025076*b5 + 450608*b4 - 1441841*b3 - 702897*b2 - 3248302*b1 + 11541 $$\nu^{18}$$ $$=$$ $$3951199 \beta_{18} + 7472267 \beta_{17} - 4025646 \beta_{16} - 3951199 \beta_{14} + 7347879 \beta_{11} + 5215198 \beta_{10} - 3521068 \beta_{9} - 3951199 \beta_{8} - 4025646 \beta_{7} + \cdots + 3521068 \beta_1$$ 3951199*b18 + 7472267*b17 - 4025646*b16 - 3951199*b14 + 7347879*b11 + 5215198*b10 - 3521068*b9 - 3951199*b8 - 4025646*b7 + 11299078*b5 - 5645329*b2 + 3521068*b1 $$\nu^{19}$$ $$=$$ $$3521068 \beta_{19} - 3521068 \beta_{18} - 25606226 \beta_{17} + 26795778 \beta_{16} + 4455777 \beta_{15} + 1189552 \beta_{13} + 9193256 \beta_{12} - 4737479 \beta_{11} - 17629381 \beta_{10} + \cdots + 4710620$$ 3521068*b19 - 3521068*b18 - 25606226*b17 + 26795778*b16 + 4455777*b15 + 1189552*b13 + 9193256*b12 - 4737479*b11 - 17629381*b10 + 1189552*b9 + 31251555*b8 + 16439829*b7 + 1189552*b5 - 10355949*b3 + 9166397*b2 - 17629381*b1 + 4710620

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{12}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.78384 − 0.477979i −1.19457 + 0.320085i 2.34324 − 0.627868i −2.56046 + 0.686074i −0.372041 + 0.0996880i −0.627868 − 2.34324i 0.0996880 + 0.372041i 0.686074 + 2.56046i −0.477979 − 1.78384i 0.320085 + 1.19457i 1.78384 + 0.477979i −1.19457 − 0.320085i 2.34324 + 0.627868i −2.56046 − 0.686074i −0.372041 − 0.0996880i −0.627868 + 2.34324i 0.0996880 − 0.372041i 0.686074 − 2.56046i −0.477979 + 1.78384i 0.320085 − 1.19457i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −2.22489 + 0.223342i −0.500000 0.866025i 1.07560i 1.00000i 0.500000 + 0.866025i 2.03848 + 0.919023i
49.2 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.34502 1.78632i −0.500000 0.866025i 4.03495i 1.00000i 0.500000 + 0.866025i 0.271659 + 2.21950i
49.3 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.384547 + 2.20275i −0.500000 0.866025i 2.51805i 1.00000i 0.500000 + 0.866025i 0.768349 2.09991i
49.4 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.99313 1.01362i −0.500000 0.866025i 2.79875i 1.00000i 0.500000 + 0.866025i −2.23291 0.118742i
49.5 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 2.05825 + 0.873846i −0.500000 0.866025i 2.32136i 1.00000i 0.500000 + 0.866025i −1.34557 1.78590i
49.6 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.09991 + 0.768349i −0.500000 0.866025i 2.51805i 1.00000i 0.500000 + 0.866025i −2.20275 0.384547i
49.7 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.78590 1.34557i −0.500000 0.866025i 2.32136i 1.00000i 0.500000 + 0.866025i −0.873846 2.05825i
49.8 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −0.118742 2.23291i −0.500000 0.866025i 2.79875i 1.00000i 0.500000 + 0.866025i 1.01362 1.99313i
49.9 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0.919023 + 2.03848i −0.500000 0.866025i 1.07560i 1.00000i 0.500000 + 0.866025i −0.223342 + 2.22489i
49.10 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 2.21950 + 0.271659i −0.500000 0.866025i 4.03495i 1.00000i 0.500000 + 0.866025i 1.78632 + 1.34502i
349.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i −2.22489 0.223342i −0.500000 + 0.866025i 1.07560i 1.00000i 0.500000 0.866025i 2.03848 0.919023i
349.2 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.34502 + 1.78632i −0.500000 + 0.866025i 4.03495i 1.00000i 0.500000 0.866025i 0.271659 2.21950i
349.3 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0.384547 2.20275i −0.500000 + 0.866025i 2.51805i 1.00000i 0.500000 0.866025i 0.768349 + 2.09991i
349.4 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.99313 + 1.01362i −0.500000 + 0.866025i 2.79875i 1.00000i 0.500000 0.866025i −2.23291 + 0.118742i
349.5 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 2.05825 0.873846i −0.500000 + 0.866025i 2.32136i 1.00000i 0.500000 0.866025i −1.34557 + 1.78590i
349.6 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.09991 0.768349i −0.500000 + 0.866025i 2.51805i 1.00000i 0.500000 0.866025i −2.20275 + 0.384547i
349.7 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.78590 + 1.34557i −0.500000 + 0.866025i 2.32136i 1.00000i 0.500000 0.866025i −0.873846 + 2.05825i
349.8 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −0.118742 + 2.23291i −0.500000 + 0.866025i 2.79875i 1.00000i 0.500000 0.866025i 1.01362 + 1.99313i
349.9 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0.919023 2.03848i −0.500000 + 0.866025i 1.07560i 1.00000i 0.500000 0.866025i −0.223342 2.22489i
349.10 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 2.21950 0.271659i −0.500000 + 0.866025i 4.03495i 1.00000i 0.500000 0.866025i 1.78632 1.34502i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.c 20
3.b odd 2 1 1710.2.t.c 20
5.b even 2 1 inner 570.2.q.c 20
15.d odd 2 1 1710.2.t.c 20
19.c even 3 1 inner 570.2.q.c 20
57.h odd 6 1 1710.2.t.c 20
95.i even 6 1 inner 570.2.q.c 20
285.n odd 6 1 1710.2.t.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.c 20 1.a even 1 1 trivial
570.2.q.c 20 5.b even 2 1 inner
570.2.q.c 20 19.c even 3 1 inner
570.2.q.c 20 95.i even 6 1 inner
1710.2.t.c 20 3.b odd 2 1
1710.2.t.c 20 15.d odd 2 1
1710.2.t.c 20 57.h odd 6 1
1710.2.t.c 20 285.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{10} + 37T_{7}^{8} + 486T_{7}^{6} + 2834T_{7}^{4} + 7041T_{7}^{2} + 5041$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{5}$$
$3$ $$(T^{4} - T^{2} + 1)^{5}$$
$5$ $$T^{20} - 7 T^{18} + 5 T^{16} + \cdots + 9765625$$
$7$ $$(T^{10} + 37 T^{8} + 486 T^{6} + 2834 T^{4} + \cdots + 5041)^{2}$$
$11$ $$(T^{5} - 3 T^{4} - 13 T^{3} + 27 T^{2} + \cdots - 20)^{4}$$
$13$ $$T^{20} - 98 T^{18} + 6123 T^{16} + \cdots + 12960000$$
$17$ $$T^{20} - 92 T^{18} + 5992 T^{16} + \cdots + 65536$$
$19$ $$(T^{10} - 3 T^{9} - 10 T^{8} + \cdots + 2476099)^{2}$$
$23$ $$T^{20} - 127 T^{18} + \cdots + 283982410000$$
$29$ $$(T^{10} - 4 T^{9} + 70 T^{8} - 596 T^{7} + \cdots + 501264)^{2}$$
$31$ $$(T^{5} - 10 T^{4} - 63 T^{3} + 880 T^{2} + \cdots - 1648)^{4}$$
$37$ $$(T^{10} + 81 T^{8} + 1358 T^{6} + \cdots + 18769)^{2}$$
$41$ $$(T^{10} + 7 T^{9} + 154 T^{8} + \cdots + 244421956)^{2}$$
$43$ $$T^{20} - 134 T^{18} + \cdots + 3110228525056$$
$47$ $$T^{20} - 140 T^{18} + \cdots + 72699496960000$$
$53$ $$T^{20} - 343 T^{18} + \cdots + 34\!\cdots\!76$$
$59$ $$(T^{10} - 4 T^{9} + 76 T^{8} + 136 T^{7} + \cdots + 6400)^{2}$$
$61$ $$(T^{10} - 8 T^{9} + 303 T^{8} + \cdots + 313219204)^{2}$$
$67$ $$T^{20} - 382 T^{18} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{10} + 2 T^{9} + 216 T^{8} + \cdots + 593994384)^{2}$$
$73$ $$T^{20} - 542 T^{18} + \cdots + 31\!\cdots\!00$$
$79$ $$(T^{10} - 4 T^{9} + 183 T^{8} + \cdots + 263997504)^{2}$$
$83$ $$(T^{10} + 356 T^{8} + 45124 T^{6} + \cdots + 184090624)^{2}$$
$89$ $$(T^{10} + T^{9} + 116 T^{8} + \cdots + 13468900)^{2}$$
$97$ $$T^{20} - 368 T^{18} + \cdots + 18\!\cdots\!16$$