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 $2$

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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:

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

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