# Properties

 Label 570.2.m.a Level $570$ Weight $2$ Character orbit 570.m 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16$$ x^20 + 153*x^16 + 6416*x^12 + 78648*x^8 + 19120*x^4 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{4} q^{4} + \beta_{6} q^{5} + q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} - \beta_{11} q^{8} - \beta_{4} q^{9}+O(q^{10})$$ q + b2 * q^2 + b11 * q^3 + b4 * q^4 + b6 * q^5 + q^6 + (-b9 + b6) * q^7 - b11 * q^8 - b4 * q^9 $$q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{4} q^{4} + \beta_{6} q^{5} + q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} - \beta_{11} q^{8} - \beta_{4} q^{9} - \beta_{13} q^{10} - \beta_1 q^{11} + \beta_{2} q^{12} + ( - \beta_{19} - \beta_{13} + \beta_{12}) q^{13} + ( - \beta_{15} - \beta_{13}) q^{14} + \beta_{14} q^{15} - q^{16} + ( - \beta_{19} - \beta_{17} - \beta_{16} - \beta_{7} - \beta_{5} + \beta_{3}) q^{17} + \beta_{11} q^{18} + (\beta_{17} - \beta_{4} - \beta_{3}) q^{19} - \beta_{7} q^{20} + (\beta_{14} + \beta_{12}) q^{21} - \beta_{3} q^{22} + ( - \beta_{9} - \beta_{6} + 2 \beta_{4} + 2) q^{23} + \beta_{4} q^{24} + (\beta_{17} - \beta_{16} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + \beta_1) q^{25} + ( - \beta_{9} - \beta_{7} - \beta_1) q^{26} - \beta_{2} q^{27} + ( - \beta_{7} + \beta_{5}) q^{28} + (\beta_{19} - \beta_{18} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + \beta_{3}) q^{29} + \beta_{6} q^{30} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{8} + \beta_{3}) q^{31} - \beta_{2} q^{32} - \beta_{19} q^{33} + (\beta_{18} - \beta_{14} + \beta_{12} + \beta_{10}) q^{34} + (\beta_{17} - \beta_{16} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 4 \beta_{4} + \beta_1) q^{35} + q^{36} + (\beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{8} + \beta_{3}) q^{37} + ( - \beta_{18} + \beta_{11} - \beta_{10} - \beta_{8}) q^{38} + (\beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_1) q^{39} - \beta_{14} q^{40} + ( - \beta_{19} - \beta_{14} - \beta_{12} - \beta_{11} + \beta_{3} + \beta_{2}) q^{41} + ( - \beta_{9} + \beta_{6}) q^{42} + ( - \beta_{17} + \beta_{16} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 3) q^{43} + ( - \beta_{10} - \beta_{8} + \beta_1) q^{44} + \beta_{7} q^{45} + ( - \beta_{15} + \beta_{13} - 2 \beta_{11} + 2 \beta_{2}) q^{46} + ( - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{47} - \beta_{11} q^{48} + (\beta_{19} + 2 \beta_{17} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 2 \beta_1) q^{49}+ \cdots + (\beta_{10} + \beta_{8} - \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^2 + b11 * q^3 + b4 * q^4 + b6 * q^5 + q^6 + (-b9 + b6) * q^7 - b11 * q^8 - b4 * q^9 - b13 * q^10 - b1 * q^11 + b2 * q^12 + (-b19 - b13 + b12) * q^13 + (-b15 - b13) * q^14 + b14 * q^15 - q^16 + (-b19 - b17 - b16 - b7 - b5 + b3) * q^17 + b11 * q^18 + (b17 - b4 - b3) * q^19 - b7 * q^20 + (b14 + b12) * q^21 - b3 * q^22 + (-b9 - b6 + 2*b4 + 2) * q^23 + b4 * q^24 + (b17 - b16 - b10 - b9 - b8 - b6 + b4 + b1) * q^25 + (-b9 - b7 - b1) * q^26 - b2 * q^27 + (-b7 + b5) * q^28 + (b19 - b18 + b15 + b14 + b13 - b12 - b10 + b3) * q^29 + b6 * q^30 + (-b19 + b18 + b15 + b14 - b13 + b12 + b8 + b3) * q^31 - b2 * q^32 - b19 * q^33 + (b18 - b14 + b12 + b10) * q^34 + (b17 - b16 - b10 - b9 - b8 - b6 - 4*b4 + b1) * q^35 + q^36 + (b15 + b14 - 2*b13 - 2*b12 - b10 + b8 + b3) * q^37 + (-b18 + b11 - b10 - b8) * q^38 + (b10 + b8 + b6 - b5 - b1) * q^39 - b14 * q^40 + (-b19 - b14 - b12 - b11 + b3 + b2) * q^41 + (-b9 + b6) * q^42 + (-b17 + b16 + b9 - b7 + b6 + b5 + 3*b4 + 3) * q^43 + (-b10 - b8 + b1) * q^44 + b7 * q^45 + (-b15 + b13 - 2*b11 + 2*b2) * q^46 + (-b10 + b9 - b8 + 2*b7 - b6 + 2*b5 + 2*b1) * q^47 - b11 * q^48 + (b19 + 2*b17 - 2*b10 - b9 - 2*b8 + b7 - b6 + b5 - b4 - b3 + 2*b1) * q^49 + (b19 - b18 - b15 + b13 - b11 - b8) * q^50 + (b18 + b15 - b13 + b8) * q^51 + (-b15 - b14 - b3) * q^52 + (-2*b18 + 2*b13 - 2*b12 - 2*b11 - b10 - b8) * q^53 - b4 * q^54 + (-b19 - b17 - b16 + b10 + 2*b9 + b8 - 2*b5 + 2*b4 + b3) * q^55 + (-b14 - b12) * q^56 + (b10 - b2 - b1) * q^57 + (-b17 + b16 + b10 + b9 + b8 + b7 + b6 - b5) * q^58 + (b19 - b18 + b14 - b12 + b11 - b10 + b3 + b2) * q^59 - b13 * q^60 + (-b9 - b7 + b6 + b5 + 2) * q^61 + (-b19 - b17 - b16 + b10 - b9 + b8 - b7 + b6 - b5 + b3 - 2*b1) * q^62 + (b7 - b5) * q^63 - b4 * q^64 + (2*b19 + b15 + b13 - 2*b12 - b11 + b3 - 3*b2) * q^65 - b1 * q^66 + (2*b15 + 2*b14 + 2*b13 + 2*b12) * q^67 + (b17 - b16 - b9 - b6) * q^68 + (-b14 + b12 + 2*b11 + 2*b2) * q^69 + (b19 - b18 - b15 + b13 + 4*b11 - b8) * q^70 + (-2*b15 + 2*b13) * q^71 + b2 * q^72 + (-b10 - 2*b9 - b8 - 2*b6 - b4 - 1) * q^73 + (-b19 - 2*b17 + b10 + 2*b9 + b8 - 2*b7 + b6 - b5 + b3 - b1) * q^74 + (b18 - b14 + b12 + b10 - b3 + b2) * q^75 + (b19 + b16 - b3 + 1) * q^76 + (-2*b19 - 2*b17 - 2*b16 + 2*b9 - 2*b7 - 2*b6 - 2*b5 + 2*b4 + 2*b3 - 2) * q^77 + (-b19 - b13 + b12) * q^78 + (-b19 + b18 + b10 - b3) * q^79 - b6 * q^80 - q^81 + (b10 + b9 + b8 - b6 + b4 - 2*b1 - 1) * q^82 + (-b17 + b16 + b10 + b9 + b8 + b6 - 6*b4 - 6) * q^83 + (-b15 - b13) * q^84 + (b17 - b16 + b10 - 2*b9 + b8 - b7 + 3*b5 - 2*b1 - 2) * q^85 + (b18 + b15 - b14 - b13 - b12 - 3*b11 + b8 + 3*b2) * q^86 + (b19 + b17 + b16 - b10 + b9 - b8 + b7 - b6 + b5 - b3 + 2*b1) * q^87 + b19 * q^88 + (b15 - 2*b14 + b13 + 2*b12 - 5*b11 - 5*b2) * q^89 + b14 * q^90 + (b19 - 2*b14 - 2*b12 + 2*b11 - b3 - 2*b2) * q^91 + (b7 + b5 + 2*b4 - 2) * q^92 + (-b17 + b16 + b10 + b9 + b8 + b7 + b6 - b5) * q^93 + (b19 + b15 + 2*b14 + b13 - 2*b12 + b3) * q^94 + (b19 + b18 + b17 + b16 + 2*b15 - b11 - b10 + 2*b9 - b8 + 2*b7 - b6 + 2*b4 + b2 + 2*b1 - 2) * q^95 - q^96 + (-3*b15 - 3*b14 + b13 + b12 + b10 - b8 - 3*b3 - 2*b2) * q^97 + (2*b19 - 2*b18 - b15 + b14 + b13 - b12 + b11 - b10 - b8) * q^98 + (b10 + b8 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 4 q^{5} + 20 q^{6} - 4 q^{7}+O(q^{10})$$ 20 * q - 4 * q^5 + 20 * q^6 - 4 * q^7 $$20 q - 4 q^{5} + 20 q^{6} - 4 q^{7} - 8 q^{11} - 20 q^{16} + 4 q^{17} + 44 q^{23} + 4 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{30} + 4 q^{35} + 20 q^{36} - 4 q^{38} - 4 q^{42} + 52 q^{43} + 4 q^{47} + 16 q^{55} - 4 q^{57} + 8 q^{58} + 32 q^{61} - 8 q^{62} + 4 q^{63} - 8 q^{66} + 4 q^{68} - 20 q^{73} + 20 q^{76} - 24 q^{77} + 4 q^{80} - 20 q^{81} - 24 q^{82} - 116 q^{83} - 60 q^{85} + 8 q^{87} - 44 q^{92} + 8 q^{93} - 32 q^{95} - 20 q^{96}+O(q^{100})$$ 20 * q - 4 * q^5 + 20 * q^6 - 4 * q^7 - 8 * q^11 - 20 * q^16 + 4 * q^17 + 44 * q^23 + 4 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^30 + 4 * q^35 + 20 * q^36 - 4 * q^38 - 4 * q^42 + 52 * q^43 + 4 * q^47 + 16 * q^55 - 4 * q^57 + 8 * q^58 + 32 * q^61 - 8 * q^62 + 4 * q^63 - 8 * q^66 + 4 * q^68 - 20 * q^73 + 20 * q^76 - 24 * q^77 + 4 * q^80 - 20 * q^81 - 24 * q^82 - 116 * q^83 - 60 * q^85 + 8 * q^87 - 44 * q^92 + 8 * q^93 - 32 * q^95 - 20 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -16765\nu^{16} - 2625903\nu^{12} - 115242314\nu^{8} - 1515721480\nu^{4} - 599186688 ) / 174141704$$ (-16765*v^16 - 2625903*v^12 - 115242314*v^8 - 1515721480*v^4 - 599186688) / 174141704 $$\beta_{2}$$ $$=$$ $$( 806911\nu^{17} + 123405389\nu^{13} + 5168924442\nu^{9} + 63121710948\nu^{5} + 12328816792\nu ) / 2089700448$$ (806911*v^17 + 123405389*v^13 + 5168924442*v^9 + 63121710948*v^5 + 12328816792*v) / 2089700448 $$\beta_{3}$$ $$=$$ $$( -604825\nu^{17} - 92597906\nu^{13} - 3887923677\nu^{9} - 47719026672\nu^{5} - 10577293948\nu ) / 522425112$$ (-604825*v^17 - 92597906*v^13 - 3887923677*v^9 - 47719026672*v^5 - 10577293948*v) / 522425112 $$\beta_{4}$$ $$=$$ $$( 54129\nu^{18} + 8282406\nu^{14} + 347394797\nu^{10} + 4261561356\nu^{6} + 1096207364\nu^{2} ) / 31662128$$ (54129*v^18 + 8282406*v^14 + 347394797*v^10 + 4261561356*v^6 + 1096207364*v^2) / 31662128 $$\beta_{5}$$ $$=$$ $$( 11884471 \nu^{18} - 808344 \nu^{16} + 1818129965 \nu^{14} - 123229932 \nu^{12} + 76223127354 \nu^{10} - 5124003060 \nu^{8} + 933673009644 \nu^{6} + \cdots - 7006091376 ) / 4179400896$$ (11884471*v^18 - 808344*v^16 + 1818129965*v^14 - 123229932*v^12 + 76223127354*v^10 - 5124003060*v^8 + 933673009644*v^6 - 61610737104*v^4 + 216220477192*v^2 - 7006091376) / 4179400896 $$\beta_{6}$$ $$=$$ $$( - 11884471 \nu^{18} - 808344 \nu^{16} - 1818129965 \nu^{14} - 123229932 \nu^{12} - 76223127354 \nu^{10} - 5124003060 \nu^{8} + \cdots - 7006091376 ) / 4179400896$$ (-11884471*v^18 - 808344*v^16 - 1818129965*v^14 - 123229932*v^12 - 76223127354*v^10 - 5124003060*v^8 - 933673009644*v^6 - 61610737104*v^4 - 216220477192*v^2 - 7006091376) / 4179400896 $$\beta_{7}$$ $$=$$ $$( 13498293 \nu^{18} - 604298 \nu^{16} + 2064940743 \nu^{14} - 92070010 \nu^{12} + 86560976238 \nu^{10} - 3830938056 \nu^{8} + 1059916431540 \nu^{6} + \cdots - 6281901056 ) / 4179400896$$ (13498293*v^18 - 604298*v^16 + 2064940743*v^14 - 92070010*v^12 + 86560976238*v^10 - 3830938056*v^8 + 1059916431540*v^6 - 46444027032*v^4 + 240878110776*v^2 - 6281901056) / 4179400896 $$\beta_{8}$$ $$=$$ $$( - 12734103 \nu^{18} + 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} + 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896$$ (-12734103*v^18 + 1699264*v^17 - 201180*v^16 - 1948517589*v^14 + 260775248*v^13 - 31510836*v^12 - 81728941602*v^10 + 11011628496*v^9 - 1382907768*v^8 - 1002348912780*v^6 + 137351806272*v^5 - 18188657760*v^4 - 248442762312*v^2 + 64444570240*v - 7190240256) / 4179400896 $$\beta_{9}$$ $$=$$ $$( - 13498293 \nu^{18} - 604298 \nu^{16} - 2064940743 \nu^{14} - 92070010 \nu^{12} - 86560976238 \nu^{10} - 3830938056 \nu^{8} + \cdots - 6281901056 ) / 4179400896$$ (-13498293*v^18 - 604298*v^16 - 2064940743*v^14 - 92070010*v^12 - 86560976238*v^10 - 3830938056*v^8 - 1059916431540*v^6 - 46444027032*v^4 - 240878110776*v^2 - 6281901056) / 4179400896 $$\beta_{10}$$ $$=$$ $$( - 12734103 \nu^{18} - 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} - 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896$$ (-12734103*v^18 - 1699264*v^17 - 201180*v^16 - 1948517589*v^14 - 260775248*v^13 - 31510836*v^12 - 81728941602*v^10 - 11011628496*v^9 - 1382907768*v^8 - 1002348912780*v^6 - 137351806272*v^5 - 18188657760*v^4 - 248442762312*v^2 - 64444570240*v - 7190240256) / 4179400896 $$\beta_{11}$$ $$=$$ $$( - 45261895 \nu^{19} - 6924865889 \nu^{15} - 290369158398 \nu^{11} - 3558464452956 \nu^{7} - 850240722328 \nu^{3} ) / 4179400896$$ (-45261895*v^19 - 6924865889*v^15 - 290369158398*v^11 - 3558464452956*v^7 - 850240722328*v^3) / 4179400896 $$\beta_{12}$$ $$=$$ $$( - 21963356 \nu^{19} + 518651 \nu^{17} - 3360431092 \nu^{15} + 79345865 \nu^{13} - 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632$$ (-21963356*v^19 + 518651*v^17 - 3360431092*v^15 + 79345865*v^13 - 140922857856*v^11 + 3327761602*v^9 - 1727640346328*v^7 + 40901095068*v^5 - 423306773744*v^3 + 13651993928*v) / 1393133632 $$\beta_{13}$$ $$=$$ $$( 21963356 \nu^{19} + 518651 \nu^{17} + 3360431092 \nu^{15} + 79345865 \nu^{13} + 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632$$ (21963356*v^19 + 518651*v^17 + 3360431092*v^15 + 79345865*v^13 + 140922857856*v^11 + 3327761602*v^9 + 1727640346328*v^7 + 40901095068*v^5 + 423306773744*v^3 + 13651993928*v) / 1393133632 $$\beta_{14}$$ $$=$$ $$( 24345032 \nu^{19} - 518651 \nu^{17} + 3724856956 \nu^{15} - 79345865 \nu^{13} + 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632$$ (24345032*v^19 - 518651*v^17 + 3724856956*v^15 - 79345865*v^13 + 156208228924*v^11 - 3327761602*v^9 + 1915149045992*v^7 - 40901095068*v^5 + 471539897760*v^3 - 12258860296*v) / 1393133632 $$\beta_{15}$$ $$=$$ $$( - 24345032 \nu^{19} - 518651 \nu^{17} - 3724856956 \nu^{15} - 79345865 \nu^{13} - 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632$$ (-24345032*v^19 - 518651*v^17 - 3724856956*v^15 - 79345865*v^13 - 156208228924*v^11 - 3327761602*v^9 - 1915149045992*v^7 - 40901095068*v^5 - 471539897760*v^3 - 12258860296*v) / 1393133632 $$\beta_{16}$$ $$=$$ $$( - 77025497 \nu^{19} - 2419300 \nu^{17} + 1726694 \nu^{16} - 11784791035 \nu^{15} - 370391624 \nu^{13} + 264708058 \nu^{12} + \cdots + 11065189808 ) / 4179400896$$ (-77025497*v^19 - 2419300*v^17 + 1726694*v^16 - 11784791035*v^15 - 370391624*v^13 + 264708058*v^12 - 494177340558*v^11 - 15551694708*v^9 + 11136819804*v^8 - 6057012474372*v^7 - 190876106688*v^5 + 136345642968*v^4 - 1459603333880*v^3 - 42309175792*v + 11065189808) / 4179400896 $$\beta_{17}$$ $$=$$ $$( - 77025497 \nu^{19} - 22270858 \nu^{18} - 2419300 \nu^{17} - 11784791035 \nu^{15} - 3406995518 \nu^{14} - 370391624 \nu^{13} + \cdots - 42309175792 \nu ) / 4179400896$$ (-77025497*v^19 - 22270858*v^18 - 2419300*v^17 - 11784791035*v^15 - 3406995518*v^14 - 370391624*v^13 - 494177340558*v^11 - 142817533980*v^10 - 15551694708*v^9 - 6057012474372*v^7 - 1748182870776*v^6 - 190876106688*v^5 - 1459603333880*v^3 - 379366025296*v^2 - 42309175792*v) / 4179400896 $$\beta_{18}$$ $$=$$ $$( 113542400 \nu^{19} + 12734103 \nu^{18} + 201180 \nu^{16} + 17372793436 \nu^{15} + 1948517589 \nu^{14} + 31510836 \nu^{12} + 728609156748 \nu^{11} + \cdots + 7190240256 ) / 4179400896$$ (113542400*v^19 + 12734103*v^18 + 201180*v^16 + 17372793436*v^15 + 1948517589*v^14 + 31510836*v^12 + 728609156748*v^11 + 81728941602*v^10 + 1382907768*v^8 + 8934778735776*v^7 + 1002348912780*v^6 + 18188657760*v^4 + 2227441426544*v^3 + 248442762312*v^2 + 7190240256) / 4179400896 $$\beta_{19}$$ $$=$$ $$( 77025497 \nu^{19} + 11784791035 \nu^{15} + 494177340558 \nu^{11} + 6057012474372 \nu^{7} + 1459603333880 \nu^{3} ) / 2089700448$$ (77025497*v^19 + 11784791035*v^15 + 494177340558*v^11 + 6057012474372*v^7 + 1459603333880*v^3) / 2089700448
 $$\nu$$ $$=$$ $$( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} ) / 2$$ (b15 + b14 + b13 + b12) / 2 $$\nu^{2}$$ $$=$$ $$( - \beta_{19} - 2 \beta_{17} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_{3} - 2 \beta_1 ) / 2$$ (-b19 - 2*b17 + 2*b10 + b9 + 2*b8 - b7 + b6 - b5 + 8*b4 + b3 - 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$( 2 \beta_{19} - 2 \beta_{18} - 7 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} - \beta_{10} - \beta_{8} ) / 2$$ (2*b19 - 2*b18 - 7*b15 + 7*b14 - 7*b13 + 7*b12 + 4*b11 - b10 - b8) / 2 $$\nu^{4}$$ $$=$$ $$( -9\beta_{19} - 18\beta_{16} - 3\beta_{9} - 3\beta_{7} - 13\beta_{6} - 13\beta_{5} + 9\beta_{3} - 16\beta _1 - 60 ) / 2$$ (-9*b19 - 18*b16 - 3*b9 - 3*b7 - 13*b6 - 13*b5 + 9*b3 - 16*b1 - 60) / 2 $$\nu^{5}$$ $$=$$ $$( - 51 \beta_{15} - 51 \beta_{14} - 59 \beta_{13} - 59 \beta_{12} - 13 \beta_{10} + 13 \beta_{8} + 28 \beta_{3} + 72 \beta_{2} ) / 2$$ (-51*b15 - 51*b14 - 59*b13 - 59*b12 - 13*b10 + 13*b8 + 28*b3 + 72*b2) / 2 $$\nu^{6}$$ $$=$$ $$( 79 \beta_{19} + 158 \beta_{17} - 136 \beta_{10} - 143 \beta_{9} - 136 \beta_{8} + 143 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 496 \beta_{4} - 79 \beta_{3} + 136 \beta_1 ) / 2$$ (79*b19 + 158*b17 - 136*b10 - 143*b9 - 136*b8 + 143*b7 + 11*b6 - 11*b5 - 496*b4 - 79*b3 + 136*b1) / 2 $$\nu^{7}$$ $$=$$ $$( - 312 \beta_{19} + 294 \beta_{18} + 531 \beta_{15} - 531 \beta_{14} + 395 \beta_{13} - 395 \beta_{12} - 888 \beta_{11} + 147 \beta_{10} + 147 \beta_{8} ) / 2$$ (-312*b19 + 294*b18 + 531*b15 - 531*b14 + 395*b13 - 395*b12 - 888*b11 + 147*b10 + 147*b8) / 2 $$\nu^{8}$$ $$=$$ $$( 707 \beta_{19} + 1414 \beta_{16} - 343 \beta_{9} - 343 \beta_{7} + 1463 \beta_{6} + 1463 \beta_{5} - 707 \beta_{3} + 1220 \beta _1 + 4328 ) / 2$$ (707*b19 + 1414*b16 - 343*b9 - 343*b7 + 1463*b6 + 1463*b5 - 707*b3 + 1220*b1 + 4328) / 2 $$\nu^{9}$$ $$=$$ $$( 3235 \beta_{15} + 3235 \beta_{14} + 4947 \beta_{13} + 4947 \beta_{12} + 1563 \beta_{10} - 1563 \beta_{8} - 3220 \beta_{3} - 9664 \beta_{2} ) / 2$$ (3235*b15 + 3235*b14 + 4947*b13 + 4947*b12 + 1563*b10 - 1563*b8 - 3220*b3 - 9664*b2) / 2 $$\nu^{10}$$ $$=$$ $$( - 6455 \beta_{19} - 12910 \beta_{17} + 11308 \beta_{10} + 14507 \beta_{9} + 11308 \beta_{8} - 14507 \beta_{7} - 4723 \beta_{6} + 4723 \beta_{5} + 39168 \beta_{4} + 6455 \beta_{3} - 11308 \beta_1 ) / 2$$ (-6455*b19 - 12910*b17 + 11308*b10 + 14507*b9 + 11308*b8 - 14507*b7 - 4723*b6 + 4723*b5 + 39168*b4 + 6455*b3 - 11308*b1) / 2 $$\nu^{11}$$ $$=$$ $$( 32140 \beta_{19} - 32062 \beta_{18} - 46923 \beta_{15} + 46923 \beta_{14} - 27771 \beta_{13} + 27771 \beta_{12} + 99536 \beta_{11} - 16031 \beta_{10} - 16031 \beta_{8} ) / 2$$ (32140*b19 - 32062*b18 - 46923*b15 + 46923*b14 - 27771*b13 + 27771*b12 + 99536*b11 - 16031*b10 - 16031*b8) / 2 $$\nu^{12}$$ $$=$$ $$( - 59911 \beta_{19} - 119822 \beta_{16} + 54059 \beta_{9} + 54059 \beta_{7} - 141819 \beta_{6} - 141819 \beta_{5} + 59911 \beta_{3} - 106756 \beta _1 - 363056 ) / 2$$ (-59911*b19 - 119822*b16 + 54059*b9 + 54059*b7 - 141819*b6 - 141819*b5 + 59911*b3 - 106756*b1 - 363056) / 2 $$\nu^{13}$$ $$=$$ $$( - 247291 \beta_{15} - 247291 \beta_{14} - 449099 \beta_{13} - 449099 \beta_{12} - 160815 \beta_{10} + 160815 \beta_{8} + 315700 \beta_{3} + 997024 \beta_{2} ) / 2$$ (-247291*b15 - 247291*b14 - 449099*b13 - 449099*b12 - 160815*b10 + 160815*b8 + 315700*b3 + 997024*b2) / 2 $$\nu^{14}$$ $$=$$ $$( 562991 \beta_{19} + 1125982 \beta_{17} - 1018020 \beta_{10} - 1377203 \beta_{9} - 1018020 \beta_{8} + 1377203 \beta_{7} + 572851 \beta_{6} - 572851 \beta_{5} - 3416960 \beta_{4} + \cdots + 1018020 \beta_1 ) / 2$$ (562991*b19 + 1125982*b17 - 1018020*b10 - 1377203*b9 - 1018020*b8 + 1377203*b7 + 572851*b6 - 572851*b5 - 3416960*b4 - 562991*b3 + 1018020*b1) / 2 $$\nu^{15}$$ $$=$$ $$( - 3076036 \beta_{19} + 3181742 \beta_{18} + 4317371 \beta_{15} - 4317371 \beta_{14} + 2261611 \beta_{13} - 2261611 \beta_{12} - 9836256 \beta_{11} + 1590871 \beta_{10} + \cdots + 1590871 \beta_{8} ) / 2$$ (-3076036*b19 + 3181742*b18 + 4317371*b15 - 4317371*b14 + 2261611*b13 - 2261611*b12 - 9836256*b11 + 1590871*b10 + 1590871*b8) / 2 $$\nu^{16}$$ $$=$$ $$( 5337647 \beta_{19} + 10675294 \beta_{16} - 5838259 \beta_{9} - 5838259 \beta_{7} + 13331811 \beta_{6} + 13331811 \beta_{5} - 5337647 \beta_{3} + 9760724 \beta _1 + 32468000 ) / 2$$ (5337647*b19 + 10675294*b16 - 5838259*b9 - 5838259*b7 + 13331811*b6 + 13331811*b5 - 5337647*b3 + 9760724*b1 + 32468000) / 2 $$\nu^{17}$$ $$=$$ $$( 21071035 \beta_{15} + 21071035 \beta_{14} + 41593707 \beta_{13} + 41593707 \beta_{12} + 15598983 \beta_{10} - 15598983 \beta_{8} - 29845364 \beta_{3} - 96201728 \beta_{2} ) / 2$$ (21071035*b15 + 21071035*b14 + 41593707*b13 + 41593707*b12 + 15598983*b10 - 15598983*b8 - 29845364*b3 - 96201728*b2) / 2 $$\nu^{18}$$ $$=$$ $$( - 50916399 \beta_{19} - 101832798 \beta_{17} + 93862708 \beta_{10} + 128862627 \beta_{9} + 93862708 \beta_{8} - 128862627 \beta_{7} - 58227795 \beta_{6} + \cdots - 93862708 \beta_1 ) / 2$$ (-50916399*b19 - 101832798*b17 + 93862708*b10 + 128862627*b9 + 93862708*b8 - 128862627*b7 - 58227795*b6 + 58227795*b5 + 310349696*b4 + 50916399*b3 - 93862708*b1) / 2 $$\nu^{19}$$ $$=$$ $$( 288923220 \beta_{19} - 304181006 \beta_{18} - 401128059 \beta_{15} + 401128059 \beta_{14} - 198779851 \beta_{13} + 198779851 \beta_{12} + 936087104 \beta_{11} + \cdots - 152090503 \beta_{8} ) / 2$$ (288923220*b19 - 304181006*b18 - 401128059*b15 + 401128059*b14 - 198779851*b13 + 198779851*b12 + 936087104*b11 - 152090503*b10 - 152090503*b8) / 2

## Character values

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

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −1.53190 + 1.53190i −0.120370 + 0.120370i −1.75036 + 1.75036i 2.19691 − 2.19691i 0.498616 − 0.498616i 1.53190 − 1.53190i 0.120370 − 0.120370i 1.75036 − 1.75036i −2.19691 + 2.19691i −0.498616 + 0.498616i −1.53190 − 1.53190i −0.120370 − 0.120370i −1.75036 − 1.75036i 2.19691 + 2.19691i 0.498616 + 0.498616i 1.53190 + 1.53190i 0.120370 + 0.120370i 1.75036 + 1.75036i −2.19691 − 2.19691i −0.498616 − 0.498616i
−0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −2.23502 + 0.0685835i 1.00000 −2.16643 2.16643i 0.707107 + 0.707107i 1.00000i 1.53190 1.62889i
37.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.66396 + 1.49373i 1.00000 −0.170229 0.170229i 0.707107 + 0.707107i 1.00000i 0.120370 2.23283i
37.3 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −0.253765 2.22162i 1.00000 −2.47539 2.47539i 0.707107 + 0.707107i 1.00000i 1.75036 + 1.39149i
37.4 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.25884 + 1.84806i 1.00000 3.10690 + 3.10690i 0.707107 + 0.707107i 1.00000i −2.19691 0.416642i
37.5 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.89390 1.18875i 1.00000 0.705149 + 0.705149i 0.707107 + 0.707107i 1.00000i −0.498616 + 2.17977i
37.6 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.23502 + 0.0685835i 1.00000 −2.16643 2.16643i −0.707107 0.707107i 1.00000i −1.53190 + 1.62889i
37.7 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.66396 + 1.49373i 1.00000 −0.170229 0.170229i −0.707107 0.707107i 1.00000i −0.120370 + 2.23283i
37.8 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −0.253765 2.22162i 1.00000 −2.47539 2.47539i −0.707107 0.707107i 1.00000i −1.75036 1.39149i
37.9 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.25884 + 1.84806i 1.00000 3.10690 + 3.10690i −0.707107 0.707107i 1.00000i 2.19691 + 0.416642i
37.10 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.89390 1.18875i 1.00000 0.705149 + 0.705149i −0.707107 0.707107i 1.00000i 0.498616 2.17977i
493.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −2.23502 0.0685835i 1.00000 −2.16643 + 2.16643i 0.707107 0.707107i 1.00000i 1.53190 + 1.62889i
493.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.66396 1.49373i 1.00000 −0.170229 + 0.170229i 0.707107 0.707107i 1.00000i 0.120370 + 2.23283i
493.3 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −0.253765 + 2.22162i 1.00000 −2.47539 + 2.47539i 0.707107 0.707107i 1.00000i 1.75036 1.39149i
493.4 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.25884 1.84806i 1.00000 3.10690 3.10690i 0.707107 0.707107i 1.00000i −2.19691 + 0.416642i
493.5 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.89390 + 1.18875i 1.00000 0.705149 0.705149i 0.707107 0.707107i 1.00000i −0.498616 2.17977i
493.6 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.23502 0.0685835i 1.00000 −2.16643 + 2.16643i −0.707107 + 0.707107i 1.00000i −1.53190 1.62889i
493.7 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.66396 1.49373i 1.00000 −0.170229 + 0.170229i −0.707107 + 0.707107i 1.00000i −0.120370 2.23283i
493.8 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.253765 + 2.22162i 1.00000 −2.47539 + 2.47539i −0.707107 + 0.707107i 1.00000i −1.75036 + 1.39149i
493.9 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.25884 1.84806i 1.00000 3.10690 3.10690i −0.707107 + 0.707107i 1.00000i 2.19691 0.416642i
493.10 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.89390 + 1.18875i 1.00000 0.705149 0.705149i −0.707107 + 0.707107i 1.00000i 0.498616 + 2.17977i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.10 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.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.m.a 20
3.b odd 2 1 1710.2.p.d 20
5.c odd 4 1 inner 570.2.m.a 20
15.e even 4 1 1710.2.p.d 20
19.b odd 2 1 inner 570.2.m.a 20
57.d even 2 1 1710.2.p.d 20
95.g even 4 1 inner 570.2.m.a 20
285.j odd 4 1 1710.2.p.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.a 20 1.a even 1 1 trivial
570.2.m.a 20 5.c odd 4 1 inner
570.2.m.a 20 19.b odd 2 1 inner
570.2.m.a 20 95.g even 4 1 inner
1710.2.p.d 20 3.b odd 2 1
1710.2.p.d 20 15.e even 4 1
1710.2.p.d 20 57.d even 2 1
1710.2.p.d 20 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{10} + 2 T_{7}^{9} + 2 T_{7}^{8} + 8 T_{7}^{7} + 320 T_{7}^{6} + 864 T_{7}^{5} + 1120 T_{7}^{4} - 1600 T_{7}^{3} + 1600 T_{7}^{2} + 640 T_{7} + 128$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{5}$$
$3$ $$(T^{4} + 1)^{5}$$
$5$ $$(T^{10} + 2 T^{9} + T^{8} + 8 T^{7} + 22 T^{6} + \cdots + 3125)^{2}$$
$7$ $$(T^{10} + 2 T^{9} + 2 T^{8} + 8 T^{7} + \cdots + 128)^{2}$$
$11$ $$(T^{5} + 2 T^{4} - 32 T^{3} - 8 T^{2} + \cdots - 272)^{4}$$
$13$ $$T^{20} + 2656 T^{16} + 1634400 T^{12} + \cdots + 4096$$
$17$ $$(T^{10} - 2 T^{9} + 2 T^{8} + 16 T^{7} + \cdots + 86528)^{2}$$
$19$ $$T^{20} + 54 T^{18} + \cdots + 6131066257801$$
$23$ $$(T^{10} - 22 T^{9} + 242 T^{8} - 1464 T^{7} + \cdots + 128)^{2}$$
$29$ $$(T^{10} - 136 T^{8} + 7088 T^{6} + \cdots - 9193472)^{2}$$
$31$ $$(T^{10} + 136 T^{8} + 7088 T^{6} + \cdots + 9193472)^{2}$$
$37$ $$T^{20} + 21472 T^{16} + \cdots + 50\!\cdots\!76$$
$41$ $$(T^{10} + 216 T^{8} + 13864 T^{6} + \cdots + 700928)^{2}$$
$43$ $$(T^{10} - 26 T^{9} + 338 T^{8} + \cdots + 12500000)^{2}$$
$47$ $$(T^{10} - 2 T^{9} + 2 T^{8} - 968 T^{7} + \cdots + 1520768)^{2}$$
$53$ $$T^{20} + 33488 T^{16} + \cdots + 77\!\cdots\!76$$
$59$ $$(T^{10} - 144 T^{8} + 6008 T^{6} + \cdots - 2196608)^{2}$$
$61$ $$(T^{5} - 8 T^{4} - 16 T^{3} + 112 T^{2} + \cdots - 320)^{4}$$
$67$ $$T^{20} + 39168 T^{16} + \cdots + 17592186044416$$
$71$ $$(T^{10} + 232 T^{8} + 19456 T^{6} + \cdots + 22151168)^{2}$$
$73$ $$(T^{10} + 10 T^{9} + 50 T^{8} + \cdots + 22957088)^{2}$$
$79$ $$(T^{10} - 184 T^{8} + 9968 T^{6} + \cdots - 8192)^{2}$$
$83$ $$(T^{10} + 58 T^{9} + 1682 T^{8} + \cdots + 14623232)^{2}$$
$89$ $$(T^{10} - 472 T^{8} + 63560 T^{6} + \cdots - 62720000)^{2}$$
$97$ $$T^{20} + 94128 T^{16} + \cdots + 43\!\cdots\!96$$