# Properties

 Label 770.2.n.j Level $770$ Weight $2$ Character orbit 770.n Analytic conductor $6.148$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} - 350 x^{3} + 510 x^{2} - 175 x + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} - 350 x^{3} + 510 x^{2} - 175 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$12925652272 \nu^{11} - 689411783229 \nu^{10} + 1158393636547 \nu^{9} - 6500442898522 \nu^{8} + 5089841799003 \nu^{7} - 22235847142081 \nu^{6} + 31959318188308 \nu^{5} - 50181676922288 \nu^{4} - 33033968740543 \nu^{3} - 282417040528075 \nu^{2} + 116322382813330 \nu - 68957262745775$$$$)/ 206669284189945$$ $$\beta_{2}$$ $$=$$ $$($$$$171694051509 \nu^{11} - 3538073194043 \nu^{10} + 6710004699384 \nu^{9} - 33812071962329 \nu^{8} + 24891059999561 \nu^{7} - 116517829857207 \nu^{6} + 96085945450136 \nu^{5} - 255345496994316 \nu^{4} - 189418812627521 \nu^{3} - 1496069668327950 \nu^{2} + 617459400892385 \nu - 872744346462385$$$$)/ 206669284189945$$ $$\beta_{3}$$ $$=$$ $$($$$$1093010781921 \nu^{11} - 1619787283007 \nu^{10} + 10879795136331 \nu^{9} - 6273835566151 \nu^{8} + 37072036936079 \nu^{7} - 29326445729798 \nu^{6} + 88370770295969 \nu^{5} + 78863770949376 \nu^{4} + 555127345742466 \nu^{3} - 180685886350265 \nu^{2} + 557460395900970 \nu - 79252706903350$$$$)/ 206669284189945$$ $$\beta_{4}$$ $$=$$ $$($$$$-1093010781921 \nu^{11} + 1619787283007 \nu^{10} - 10879795136331 \nu^{9} + 6273835566151 \nu^{8} - 37072036936079 \nu^{7} + 29326445729798 \nu^{6} - 88370770295969 \nu^{5} - 78863770949376 \nu^{4} - 555127345742466 \nu^{3} + 180685886350265 \nu^{2} - 350791111711025 \nu + 79252706903350$$$$)/ 206669284189945$$ $$\beta_{5}$$ $$=$$ $$($$$$1185075482262 \nu^{11} - 2450405323424 \nu^{10} + 12479616226722 \nu^{9} - 12495005400282 \nu^{8} + 39249052015903 \nu^{7} - 48099817253426 \nu^{6} + 102821973475408 \nu^{5} + 48983833899492 \nu^{4} + 489284366746582 \nu^{3} - 490703941148390 \nu^{2} + 481004938518940 \nu - 68337530631950$$$$)/ 206669284189945$$ $$\beta_{6}$$ $$=$$ $$($$$$2733501225278 \nu^{11} - 4281926968294 \nu^{10} + 27618108154634 \nu^{9} - 17588897251336 \nu^{8} + 94111542385560 \nu^{7} - 78291500671051 \nu^{6} + 222516804049096 \nu^{5} + 201228017585416 \nu^{4} + 1227122861994310 \nu^{3} - 467441062100718 \nu^{2} + 903381683743390 \nu + 2642224095290$$$$)/ 206669284189945$$ $$\beta_{7}$$ $$=$$ $$($$$$2758290509831 \nu^{11} - 5503655367390 \nu^{10} + 29651783824912 \nu^{9} - 29182801971594 \nu^{8} + 101072886984887 \nu^{7} - 113516650123730 \nu^{6} + 250834913331188 \nu^{5} + 131257776542224 \nu^{4} + 1138641532814873 \nu^{3} - 998435647181393 \nu^{2} + 1124311119485735 \nu - 366378456407095$$$$)/ 206669284189945$$ $$\beta_{8}$$ $$=$$ $$($$$$-3170108276134 \nu^{11} + 5247205770347 \nu^{10} - 33251403754467 \nu^{9} + 23991395901143 \nu^{8} - 117360387203075 \nu^{7} + 99242618937683 \nu^{6} - 284514273607468 \nu^{5} - 202494668236793 \nu^{4} - 1445180437963130 \nu^{3} + 554410550904434 \nu^{2} - 1436069334478075 \nu + 203977836612425$$$$)/ 206669284189945$$ $$\beta_{9}$$ $$=$$ $$($$$$6759751764174 \nu^{11} - 11030588267832 \nu^{10} + 70986300515617 \nu^{9} - 48386040858223 \nu^{8} + 251104164538245 \nu^{7} - 194093475374298 \nu^{6} + 613573282631388 \nu^{5} + 473173927098043 \nu^{4} + 3108320799340980 \nu^{3} - 1169321120091979 \nu^{2} + 3311104547261585 \nu + 7410594838995$$$$)/ 206669284189945$$ $$\beta_{10}$$ $$=$$ $$($$$$-6923728370080 \nu^{11} + 13776035137694 \nu^{10} - 76812432137873 \nu^{9} + 76603457694164 \nu^{8} - 275899334623899 \nu^{7} + 300729036546526 \nu^{6} - 698793852941352 \nu^{5} - 220880731173839 \nu^{4} - 3007480968446996 \nu^{3} + 2367534958049804 \nu^{2} - 3801356284840875 \nu + 1324968684082510$$$$)/ 206669284189945$$ $$\beta_{11}$$ $$=$$ $$($$$$-7100336146293 \nu^{11} + 10864290641274 \nu^{10} - 70862768140539 \nu^{9} + 41778451743806 \nu^{8} - 237405738244780 \nu^{7} + 183243705133141 \nu^{6} - 554905942184836 \nu^{5} - 563812477235096 \nu^{4} - 3185417947781235 \nu^{3} + 1205623397343828 \nu^{2} - 2110762126897310 \nu - 7554307496840$$$$)/ 206669284189945$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} + \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - 3 \beta_{8} - \beta_{7} + \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - 5 \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{11} - 2 \beta_{10} + 9 \beta_{8} + 14 \beta_{7} + 9 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 7 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{10} - 11 \beta_{9} + 15 \beta_{7} - 15 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} - 20 \beta_{2} + 19 \beta_{1} + 19$$ $$\nu^{6}$$ $$=$$ $$-50 \beta_{11} - 22 \beta_{9} - 74 \beta_{8} - 151 \beta_{6} - 24 \beta_{5} + 30 \beta_{4} + 30 \beta_{3} + 24 \beta_{1} + 74$$ $$\nu^{7}$$ $$=$$ $$-74 \beta_{11} + 102 \beta_{10} + 74 \beta_{9} - 64 \beta_{8} - 152 \beta_{7} - 201 \beta_{5} + 96 \beta_{3} + 176 \beta_{2} - 152$$ $$\nu^{8}$$ $$=$$ $$198 \beta_{11} + 198 \beta_{10} + 377 \beta_{9} + 485 \beta_{8} - 485 \beta_{7} + 1076 \beta_{6} - 318 \beta_{4} - 226 \beta_{3} + 377 \beta_{2} - 226 \beta_{1} - 1076$$ $$\nu^{9}$$ $$=$$ $$893 \beta_{11} - 603 \beta_{10} + 1373 \beta_{8} + 700 \beta_{7} + 1373 \beta_{6} + 1453 \beta_{5} - 789 \beta_{4} - 1453 \beta_{3} - 893 \beta_{2} - 664 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-2949 \beta_{10} - 2949 \beta_{9} + 4713 \beta_{7} - 4713 \beta_{6} + 1976 \beta_{5} + 1976 \beta_{4} - 4631 \beta_{2} + 990 \beta_{1} + 8097$$ $$\nu^{11}$$ $$=$$ $$-7597 \beta_{11} - 4925 \beta_{9} - 11843 \beta_{8} - 18471 \beta_{6} - 6395 \beta_{5} + 11046 \beta_{4} + 11046 \beta_{3} + 6395 \beta_{1} + 11843$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$-\beta_{8}$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −0.674672 − 2.07643i 0.289142 + 0.889888i 0.885530 + 2.72538i −0.674672 + 2.07643i 0.289142 − 0.889888i 0.885530 − 2.72538i −1.36475 + 0.991547i 0.198931 − 0.144532i 1.66582 − 1.21029i −1.36475 − 0.991547i 0.198931 + 0.144532i 1.66582 + 1.21029i
−0.309017 0.951057i −1.76631 1.28330i −0.809017 + 0.587785i 0.309017 0.951057i −0.674672 + 2.07643i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.545950 + 1.68026i −1.00000
71.2 −0.309017 0.951057i 0.756984 + 0.549981i −0.809017 + 0.587785i 0.309017 0.951057i 0.289142 0.889888i −0.809017 + 0.587785i 0.809017 + 0.587785i −0.656505 2.02052i −1.00000
71.3 −0.309017 0.951057i 2.31835 + 1.68438i −0.809017 + 0.587785i 0.309017 0.951057i 0.885530 2.72538i −0.809017 + 0.587785i 0.809017 + 0.587785i 1.61056 + 4.95678i −1.00000
141.1 −0.309017 + 0.951057i −1.76631 + 1.28330i −0.809017 0.587785i 0.309017 + 0.951057i −0.674672 2.07643i −0.809017 0.587785i 0.809017 0.587785i 0.545950 1.68026i −1.00000
141.2 −0.309017 + 0.951057i 0.756984 0.549981i −0.809017 0.587785i 0.309017 + 0.951057i 0.289142 + 0.889888i −0.809017 0.587785i 0.809017 0.587785i −0.656505 + 2.02052i −1.00000
141.3 −0.309017 + 0.951057i 2.31835 1.68438i −0.809017 0.587785i 0.309017 + 0.951057i 0.885530 + 2.72538i −0.809017 0.587785i 0.809017 0.587785i 1.61056 4.95678i −1.00000
421.1 0.809017 0.587785i −0.521287 1.60436i 0.309017 0.951057i −0.809017 0.587785i −1.36475 0.991547i 0.309017 0.951057i −0.309017 0.951057i 0.124831 0.0906951i −1.00000
421.2 0.809017 0.587785i 0.0759851 + 0.233858i 0.309017 0.951057i −0.809017 0.587785i 0.198931 + 0.144532i 0.309017 0.951057i −0.309017 0.951057i 2.37814 1.72782i −1.00000
421.3 0.809017 0.587785i 0.636285 + 1.95828i 0.309017 0.951057i −0.809017 0.587785i 1.66582 + 1.21029i 0.309017 0.951057i −0.309017 0.951057i −1.00297 + 0.728698i −1.00000
631.1 0.809017 + 0.587785i −0.521287 + 1.60436i 0.309017 + 0.951057i −0.809017 + 0.587785i −1.36475 + 0.991547i 0.309017 + 0.951057i −0.309017 + 0.951057i 0.124831 + 0.0906951i −1.00000
631.2 0.809017 + 0.587785i 0.0759851 0.233858i 0.309017 + 0.951057i −0.809017 + 0.587785i 0.198931 0.144532i 0.309017 + 0.951057i −0.309017 + 0.951057i 2.37814 + 1.72782i −1.00000
631.3 0.809017 + 0.587785i 0.636285 1.95828i 0.309017 + 0.951057i −0.809017 + 0.587785i 1.66582 1.21029i 0.309017 + 0.951057i −0.309017 + 0.951057i −1.00297 0.728698i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.3 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 770.2.n.j 12
11.c even 5 1 inner 770.2.n.j 12
11.c even 5 1 8470.2.a.cw 6
11.d odd 10 1 8470.2.a.dc 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.j 12 1.a even 1 1 trivial
770.2.n.j 12 11.c even 5 1 inner
8470.2.a.cw 6 11.c even 5 1
8470.2.a.dc 6 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$3$ $$25 - 100 T + 535 T^{2} - 695 T^{3} + 501 T^{4} - 191 T^{5} + 199 T^{6} - 32 T^{7} + 29 T^{8} - 4 T^{9} + 6 T^{10} - 3 T^{11} + T^{12}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$11$ $$1771561 + 161051 T - 263538 T^{2} + 15972 T^{3} + 23353 T^{4} - 2178 T^{5} - 2187 T^{6} - 198 T^{7} + 193 T^{8} + 12 T^{9} - 18 T^{10} + T^{11} + T^{12}$$
$13$ $$16 + 576 T + 7640 T^{2} - 16504 T^{3} + 16928 T^{4} - 9964 T^{5} + 5040 T^{6} - 1960 T^{7} + 677 T^{8} - 163 T^{9} + 36 T^{10} - 2 T^{11} + T^{12}$$
$17$ $$208253761 - 14979378 T + 38818697 T^{2} - 10791341 T^{3} + 3249027 T^{4} - 461083 T^{5} + 91121 T^{6} - 8134 T^{7} + 2423 T^{8} - 142 T^{9} + 36 T^{10} - 7 T^{11} + T^{12}$$
$19$ $$126736 + 93272 T + 208416 T^{2} + 12668 T^{3} + 39737 T^{4} - 14000 T^{5} + 6920 T^{6} + 2014 T^{7} + 1138 T^{8} + 104 T^{9} + 5 T^{10} - 6 T^{11} + T^{12}$$
$23$ $$( -2624 + 2496 T + 2288 T^{2} + 160 T^{3} - 92 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$29$ $$3225616 + 1760080 T + 2554320 T^{2} + 540776 T^{3} + 458360 T^{4} + 76240 T^{5} + 33266 T^{6} - 2760 T^{7} + 535 T^{8} - 529 T^{9} + 170 T^{10} - 20 T^{11} + T^{12}$$
$31$ $$6400 - 76800 T + 370560 T^{2} - 326560 T^{3} + 315376 T^{4} - 122496 T^{5} + 52200 T^{6} - 5344 T^{7} + 664 T^{8} + 36 T^{9} + 20 T^{10} - 6 T^{11} + T^{12}$$
$37$ $$5837056 - 11809408 T + 12103296 T^{2} - 7700608 T^{3} + 3648208 T^{4} - 1286976 T^{5} + 369296 T^{6} - 89272 T^{7} + 17952 T^{8} - 2744 T^{9} + 302 T^{10} - 22 T^{11} + T^{12}$$
$41$ $$531579136 + 440554048 T + 169248352 T^{2} + 22223664 T^{3} + 20341825 T^{4} + 1100492 T^{5} + 529827 T^{6} + 57472 T^{7} + 7250 T^{8} - 516 T^{9} + 72 T^{10} - 2 T^{11} + T^{12}$$
$43$ $$( -45004 - 42150 T - 11235 T^{2} - 202 T^{3} + 255 T^{4} + 30 T^{5} + T^{6} )^{2}$$
$47$ $$19360000 + 22880000 T + 12536000 T^{2} + 3352000 T^{3} + 813600 T^{4} + 161600 T^{5} + 36960 T^{6} + 3900 T^{7} + 1541 T^{8} + 109 T^{9} + 66 T^{10} + 4 T^{11} + T^{12}$$
$53$ $$1478656 - 3677184 T + 7681792 T^{2} - 7162560 T^{3} + 3111824 T^{4} + 487712 T^{5} + 339216 T^{6} + 29176 T^{7} + 5056 T^{8} + 24 T^{9} + 146 T^{10} - 18 T^{11} + T^{12}$$
$59$ $$80656 - 252192 T + 324792 T^{2} + 52386 T^{3} + 215977 T^{4} + 255948 T^{5} + 225256 T^{6} + 103434 T^{7} + 35998 T^{8} + 5262 T^{9} + 541 T^{10} + 32 T^{11} + T^{12}$$
$61$ $$41165056 + 67034368 T + 517689088 T^{2} - 366302816 T^{3} + 129755248 T^{4} - 26458144 T^{5} + 3929752 T^{6} - 325256 T^{7} + 17668 T^{8} + 76 T^{9} + 18 T^{10} - 8 T^{11} + T^{12}$$
$67$ $$( -44 - 994 T - 1783 T^{2} + 682 T^{3} + 15 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$71$ $$161900176 - 668468064 T + 1044217048 T^{2} + 95096552 T^{3} + 69826560 T^{4} + 15341884 T^{5} + 3037928 T^{6} + 478476 T^{7} + 68185 T^{8} + 7433 T^{9} + 628 T^{10} + 34 T^{11} + T^{12}$$
$73$ $$3025 - 17875 T + 45765 T^{2} - 46325 T^{3} + 41051 T^{4} - 36132 T^{5} + 30021 T^{6} - 1996 T^{7} + 7299 T^{8} - 443 T^{9} + 104 T^{10} - 14 T^{11} + T^{12}$$
$79$ $$694638736 - 843708272 T + 983026416 T^{2} - 446489112 T^{3} + 88846488 T^{4} + 4277216 T^{5} + 1419726 T^{6} - 28088 T^{7} + 4107 T^{8} + 349 T^{9} + 202 T^{10} + 12 T^{11} + T^{12}$$
$83$ $$6456283201 + 8345335211 T + 3879219806 T^{2} - 688924130 T^{3} + 192804619 T^{4} - 35801848 T^{5} + 5900087 T^{6} - 720937 T^{7} + 82431 T^{8} - 7441 T^{9} + 571 T^{10} - 30 T^{11} + T^{12}$$
$89$ $$( -2420 + 2310 T + 569 T^{2} - 444 T^{3} - 35 T^{4} + 18 T^{5} + T^{6} )^{2}$$
$97$ $$7943978641 - 5510133038 T + 875955728 T^{2} + 717322385 T^{3} + 280065419 T^{4} + 11503364 T^{5} + 6948009 T^{6} - 988132 T^{7} + 175331 T^{8} - 15357 T^{9} + 1009 T^{10} - 39 T^{11} + T^{12}$$