# Properties

 Label 770.2.n.i 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: $$x^{12} - 3 x^{11} + 11 x^{10} - 21 x^{9} + 61 x^{8} - 34 x^{7} + 141 x^{6} + 192 x^{5} + 289 x^{4} - 55 x^{3} + 222 x^{2} - 24 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 11 x^{10} - 21 x^{9} + 61 x^{8} - 34 x^{7} + 141 x^{6} + 192 x^{5} + 289 x^{4} - 55 x^{3} + 222 x^{2} - 24 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$27218910292 \nu^{11} - 98998119348 \nu^{10} + 577035333512 \nu^{9} - 1678425495360 \nu^{8} + 5469795070560 \nu^{7} - 9123230230654 \nu^{6} + 23389434344437 \nu^{5} - 15346562559640 \nu^{4} + 46794157201284 \nu^{3} + 33168479495342 \nu^{2} + 79065916975275 \nu - 8921537346326$$$$)/ 83041551747559$$ $$\beta_{3}$$ $$=$$ $$($$$$-57839182555 \nu^{11} + 25030916843 \nu^{10} - 944387310072 \nu^{9} + 2545013921522 \nu^{8} - 10184226264031 \nu^{7} + 14365224307255 \nu^{6} - 57563605946245 \nu^{5} + 21366121084050 \nu^{4} - 134890772634922 \nu^{3} - 117665859569765 \nu^{2} + 13883151413978 \nu + 128492152255425$$$$)/ 83041551747559$$ $$\beta_{4}$$ $$=$$ $$($$$$756356413694 \nu^{11} - 1988591540098 \nu^{10} + 7508583879638 \nu^{9} - 13080407782797 \nu^{8} + 41264955531779 \nu^{7} - 11784244862477 \nu^{6} + 104264275203845 \nu^{5} + 168424612102574 \nu^{4} + 304517437522731 \nu^{3} + 7050671601780 \nu^{2} + 131281096073141 \nu - 8905487495815$$$$)/ 83041551747559$$ $$\beta_{5}$$ $$=$$ $$($$$$758466237704 \nu^{11} - 2315285952880 \nu^{10} + 8258213919829 \nu^{9} - 15727724577708 \nu^{8} + 44692827542572 \nu^{7} - 24692296093752 \nu^{6} + 99400462679386 \nu^{5} + 138886238703404 \nu^{4} + 201352647451952 \nu^{3} - 127203131028240 \nu^{2} + 146761805411417 \nu - 6099055983892$$$$)/ 83041551747559$$ $$\beta_{6}$$ $$=$$ $$($$$$-6099055983892 \nu^{11} + 17538701713972 \nu^{10} - 64774329869932 \nu^{9} + 119821961741903 \nu^{8} - 356314690439704 \nu^{7} + 162675075909756 \nu^{6} - 835274597635020 \nu^{5} - 1270419211586650 \nu^{4} - 1901513418048192 \nu^{3} + 134095431662108 \nu^{2} - 1226787297395784 \nu - 384461798009$$$$)/ 83041551747559$$ $$\beta_{7}$$ $$=$$ $$($$$$6115105834403 \nu^{11} - 18315988768907 \nu^{10} + 66840471646303 \nu^{9} - 127090557148760 \nu^{8} + 368695713608312 \nu^{7} - 199015931288349 \nu^{6} + 840198641188894 \nu^{5} + 1192625942025354 \nu^{4} + 1722380650183657 \nu^{3} - 392701453761660 \nu^{2} + 1256468172102788 \nu - 135257465459680$$$$)/ 83041551747559$$ $$\beta_{8}$$ $$=$$ $$($$$$-8905487495815 \nu^{11} + 25960106073751 \nu^{10} - 95971770913867 \nu^{9} + 179506653532477 \nu^{8} - 530154329461918 \nu^{7} + 261521619325931 \nu^{6} - 1243889492047438 \nu^{5} - 1814117874400325 \nu^{4} - 2742110498393109 \nu^{3} + 185284374747094 \nu^{2} - 1984068895672710 \nu + 82450603826419$$$$)/ 83041551747559$$ $$\beta_{9}$$ $$=$$ $$($$$$-13210691808625 \nu^{11} + 37908100871573 \nu^{10} - 140869867980898 \nu^{9} + 260730882276699 \nu^{8} - 777570728389875 \nu^{7} + 360503848319742 \nu^{6} - 1851245080789213 \nu^{5} - 2742920657581274 \nu^{4} - 4267710431577244 \nu^{3} + 181061306906547 \nu^{2} - 3072754072474749 \nu + 127689837420424$$$$)/ 83041551747559$$ $$\beta_{10}$$ $$=$$ $$($$$$-13325947510218 \nu^{11} + 40254045460811 \nu^{10} - 147445262229447 \nu^{9} + 282641481035258 \nu^{8} - 819088036978075 \nu^{7} + 471106367849297 \nu^{6} - 1896145924731036 \nu^{5} - 2512038381582250 \nu^{4} - 3862593773139443 \nu^{3} + 819051642603431 \nu^{2} - 3145716928102473 \nu + 210612483311877$$$$)/ 83041551747559$$ $$\beta_{11}$$ $$=$$ $$($$$$37373296056781 \nu^{11} - 110083091955905 \nu^{10} + 404321849282105 \nu^{9} - 760410883625436 \nu^{8} + 2230223922339278 \nu^{7} - 1134634026352864 \nu^{6} + 5168262871535759 \nu^{5} + 7466421883844421 \nu^{4} + 11143248277518324 \nu^{3} - 1642213775295809 \nu^{2} + 8081111410953742 \nu - 335824354765506$$$$)/ 83041551747559$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} - 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{11} - 6 \beta_{9} - \beta_{8} + 17 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} - 6 \beta_{4} + 6 \beta_{3} + 16 \beta_{2} - 14 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{10} - 8 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 38 \beta_{5} - 38 \beta_{4} + 10 \beta_{3} + 51 \beta_{2} - 51 \beta_{1} - 16$$ $$\nu^{6}$$ $$=$$ $$36 \beta_{11} + 36 \beta_{10} - 5 \beta_{9} + 14 \beta_{8} - 36 \beta_{7} + 93 \beta_{6} - 28 \beta_{4} + 5 \beta_{3} + 64 \beta_{2} - 89 \beta_{1} - 14$$ $$\nu^{7}$$ $$=$$ $$59 \beta_{11} - 25 \beta_{9} + 199 \beta_{8} - 85 \beta_{7} - 249 \beta_{5} - 72 \beta_{4} - 72 \beta_{1} - 85$$ $$\nu^{8}$$ $$=$$ $$72 \beta_{11} - 224 \beta_{10} + 558 \beta_{8} - 558 \beta_{7} - 557 \beta_{6} - 512 \beta_{5} - 388 \beta_{4} - 72 \beta_{3} - 512 \beta_{2} + 72 \beta_{1} - 629$$ $$\nu^{9}$$ $$=$$ $$236 \beta_{11} - 440 \beta_{10} + 236 \beta_{9} + 504 \beta_{8} - 1300 \beta_{7} - 268 \beta_{6} - 1189 \beta_{5} + 236 \beta_{4} - 676 \beta_{3} - 2870 \beta_{2} + 1425 \beta_{1} - 236$$ $$\nu^{10}$$ $$=$$ $$-749 \beta_{10} + 749 \beta_{9} - 21 \beta_{7} - 21 \beta_{6} - 3462 \beta_{5} + 3462 \beta_{4} - 2194 \beta_{3} - 7553 \beta_{2} + 7553 \beta_{1} + 3414$$ $$\nu^{11}$$ $$=$$ $$-2017 \beta_{11} - 2017 \beta_{10} + 3342 \beta_{9} - 4543 \beta_{8} + 2017 \beta_{7} - 5546 \beta_{6} + 8203 \beta_{4} - 3342 \beta_{3} - 10220 \beta_{2} + 19866 \beta_{1} + 4543$$

## Character values

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

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$\beta_{7}$$ $$1$$ $$1$$

## 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
 0.0547712 + 0.0397936i −1.00857 − 0.732772i 2.26282 + 1.64404i 0.0547712 − 0.0397936i −1.00857 + 0.732772i 2.26282 − 1.64404i 0.701664 + 2.15950i −0.748642 − 2.30408i 0.237961 + 0.732370i 0.701664 − 2.15950i −0.748642 + 2.30408i 0.237961 − 0.732370i
0.309017 + 0.951057i −1.97811 1.43718i −0.809017 + 0.587785i 0.309017 0.951057i 0.755572 2.32541i −0.809017 + 0.587785i −0.809017 0.587785i 0.920387 + 2.83266i 1.00000
71.2 0.309017 + 0.951057i 0.338879 + 0.246210i −0.809017 + 0.587785i 0.309017 0.951057i −0.129440 + 0.398376i −0.809017 + 0.587785i −0.809017 0.587785i −0.872831 2.68630i 1.00000
71.3 0.309017 + 0.951057i 2.13924 + 1.55425i −0.809017 + 0.587785i 0.309017 0.951057i −0.817115 + 2.51482i −0.809017 + 0.587785i −0.809017 0.587785i 1.23360 + 3.79662i 1.00000
141.1 0.309017 0.951057i −1.97811 + 1.43718i −0.809017 0.587785i 0.309017 + 0.951057i 0.755572 + 2.32541i −0.809017 0.587785i −0.809017 + 0.587785i 0.920387 2.83266i 1.00000
141.2 0.309017 0.951057i 0.338879 0.246210i −0.809017 0.587785i 0.309017 + 0.951057i −0.129440 0.398376i −0.809017 0.587785i −0.809017 + 0.587785i −0.872831 + 2.68630i 1.00000
141.3 0.309017 0.951057i 2.13924 1.55425i −0.809017 0.587785i 0.309017 + 0.951057i −0.817115 2.51482i −0.809017 0.587785i −0.809017 + 0.587785i 1.23360 3.79662i 1.00000
421.1 −0.809017 + 0.587785i −0.867835 2.67092i 0.309017 0.951057i −0.809017 0.587785i 2.27202 + 1.65072i 0.309017 0.951057i 0.309017 + 0.951057i −3.95363 + 2.87248i 1.00000
421.2 −0.809017 + 0.587785i 0.361989 + 1.11409i 0.309017 0.951057i −0.809017 0.587785i −0.947700 0.688544i 0.309017 0.951057i 0.309017 + 0.951057i 1.31690 0.956780i 1.00000
421.3 −0.809017 + 0.587785i 1.00585 + 3.09567i 0.309017 0.951057i −0.809017 0.587785i −2.63334 1.91323i 0.309017 0.951057i 0.309017 + 0.951057i −6.14442 + 4.46418i 1.00000
631.1 −0.809017 0.587785i −0.867835 + 2.67092i 0.309017 + 0.951057i −0.809017 + 0.587785i 2.27202 1.65072i 0.309017 + 0.951057i 0.309017 0.951057i −3.95363 2.87248i 1.00000
631.2 −0.809017 0.587785i 0.361989 1.11409i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.947700 + 0.688544i 0.309017 + 0.951057i 0.309017 0.951057i 1.31690 + 0.956780i 1.00000
631.3 −0.809017 0.587785i 1.00585 3.09567i 0.309017 + 0.951057i −0.809017 + 0.587785i −2.63334 + 1.91323i 0.309017 + 0.951057i 0.309017 0.951057i −6.14442 4.46418i 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.i 12
11.c even 5 1 inner 770.2.n.i 12
11.c even 5 1 8470.2.a.de 6
11.d odd 10 1 8470.2.a.cy 6

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

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$3$ $$841 - 3625 T + 6897 T^{2} - 4649 T^{3} + 3749 T^{4} - 246 T^{5} + 459 T^{6} - 118 T^{7} + 87 T^{8} - 17 T^{9} + 14 T^{10} - 2 T^{11} + T^{12}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$11$ $$1771561 - 161051 T + 29282 T^{2} - 15972 T^{3} + 7623 T^{4} + 1078 T^{5} - 367 T^{6} + 98 T^{7} + 63 T^{8} - 12 T^{9} + 2 T^{10} - T^{11} + T^{12}$$
$13$ $$234256 - 42592 T + 195536 T^{2} + 91432 T^{3} + 25952 T^{4} + 4600 T^{5} + 4200 T^{6} + 736 T^{7} + 248 T^{8} + 96 T^{9} + 40 T^{10} + 6 T^{11} + T^{12}$$
$17$ $$121 + 1023 T + 25415 T^{2} - 13713 T^{3} + 34873 T^{4} + 898 T^{5} + 6885 T^{6} + 4170 T^{7} + 967 T^{8} - 149 T^{9} - 6 T^{10} + 6 T^{11} + T^{12}$$
$19$ $$525625 + 1239750 T + 1353600 T^{2} + 776285 T^{3} + 370971 T^{4} + 150416 T^{5} + 52119 T^{6} + 14032 T^{7} + 3419 T^{8} + 669 T^{9} + 111 T^{10} + 13 T^{11} + T^{12}$$
$23$ $$( -1856 + 480 T + 768 T^{2} - 184 T^{3} - 56 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$29$ $$9120400 + 10388800 T + 8715440 T^{2} + 6525120 T^{3} + 3975336 T^{4} + 1778256 T^{5} + 585440 T^{6} + 143904 T^{7} + 26824 T^{8} + 3734 T^{9} + 380 T^{10} + 26 T^{11} + T^{12}$$
$31$ $$80656 - 307856 T + 488256 T^{2} - 207720 T^{3} + 87984 T^{4} - 8872 T^{5} + 2952 T^{6} - 848 T^{7} + 816 T^{8} - 104 T^{9} + 46 T^{10} + T^{12}$$
$37$ $$16 + 912 T + 24016 T^{2} + 217552 T^{3} + 19400688 T^{4} + 373344 T^{5} + 176616 T^{6} - 13172 T^{7} + 7892 T^{8} + 1706 T^{9} + 262 T^{10} + 18 T^{11} + T^{12}$$
$41$ $$13315201 - 16146825 T + 8779033 T^{2} - 1843397 T^{3} + 449139 T^{4} - 95692 T^{5} + 68621 T^{6} - 26454 T^{7} + 7627 T^{8} - 1409 T^{9} + 196 T^{10} - 16 T^{11} + T^{12}$$
$43$ $$( -20719 + 24251 T - 9780 T^{2} + 1415 T^{3} + 20 T^{4} - 19 T^{5} + T^{6} )^{2}$$
$47$ $$2485620736 - 2163351552 T + 749824000 T^{2} + 63140352 T^{3} + 41828608 T^{4} + 13600768 T^{5} + 3264320 T^{6} + 458560 T^{7} + 57872 T^{8} + 5336 T^{9} + 444 T^{10} + 26 T^{11} + T^{12}$$
$53$ $$59536 - 356240 T + 4212720 T^{2} - 3959256 T^{3} + 1609720 T^{4} - 145800 T^{5} + 25976 T^{6} + 2760 T^{7} + 2980 T^{8} - 736 T^{9} + 100 T^{10} + T^{12}$$
$59$ $$9025 + 12350 T + 22740 T^{2} - 2725 T^{3} - 5749 T^{4} + 7024 T^{5} + 54095 T^{6} + 1156 T^{7} + 6319 T^{8} - 1329 T^{9} + 235 T^{10} - 21 T^{11} + T^{12}$$
$61$ $$12578968336 - 1750530848 T + 1911411936 T^{2} + 183958584 T^{3} + 38499736 T^{4} + 5771776 T^{5} + 1106616 T^{6} + 13472 T^{7} + 8544 T^{8} + 540 T^{9} + 22 T^{10} - 4 T^{11} + T^{12}$$
$67$ $$( -5821 + 2633 T + 6014 T^{2} + 1179 T^{3} - 226 T^{4} - 5 T^{5} + T^{6} )^{2}$$
$71$ $$169937296 - 276623920 T + 229830768 T^{2} - 113449192 T^{3} + 35966904 T^{4} - 5798072 T^{5} + 147296 T^{6} + 81336 T^{7} + 27472 T^{8} + 1376 T^{9} + 256 T^{10} + 4 T^{11} + T^{12}$$
$73$ $$5152224841 - 1434072641 T + 1598558503 T^{2} + 197621697 T^{3} - 19089763 T^{4} - 2282166 T^{5} + 2512269 T^{6} + 533258 T^{7} + 62043 T^{8} + 4651 T^{9} + 374 T^{10} + 14 T^{11} + T^{12}$$
$79$ $$719137920400 + 401164341200 T + 104611459440 T^{2} + 14192606600 T^{3} + 1400191376 T^{4} + 105833104 T^{5} + 8828104 T^{6} + 330888 T^{7} + 52964 T^{8} + 116 T^{9} + 306 T^{10} + 2 T^{11} + T^{12}$$
$83$ $$2876069641 + 125062828 T + 248348034 T^{2} - 41072207 T^{3} + 9689417 T^{4} - 848388 T^{5} + 719629 T^{6} - 196966 T^{7} + 49229 T^{8} - 6359 T^{9} + 627 T^{10} - 35 T^{11} + T^{12}$$
$89$ $$( -362975 + 55835 T + 31116 T^{2} - 179 T^{3} - 338 T^{4} - T^{5} + T^{6} )^{2}$$
$97$ $$21613201 - 58940022 T + 73014830 T^{2} - 47882423 T^{3} + 19544003 T^{4} - 6386822 T^{5} + 2471385 T^{6} - 410270 T^{7} + 48197 T^{8} - 4559 T^{9} + 519 T^{10} - 19 T^{11} + T^{12}$$