# Properties

 Label 770.2.m.e Level $770$ Weight $2$ Character orbit 770.m Analytic conductor $6.148$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 32 x^{14} + 404 x^{12} + 2600 x^{10} + 9170 x^{8} + 17648 x^{6} + 17180 x^{4} + 6904 x^{2} + 529$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{10} q^{2} + ( -1 - \beta_{3} + \beta_{7} - \beta_{15} ) q^{3} + \beta_{3} q^{4} + ( -1 - \beta_{3} + \beta_{7} ) q^{5} + ( \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{6} + \beta_{10} q^{7} -\beta_{11} q^{8} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{10} q^{2} + ( -1 - \beta_{3} + \beta_{7} - \beta_{15} ) q^{3} + \beta_{3} q^{4} + ( -1 - \beta_{3} + \beta_{7} ) q^{5} + ( \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{6} + \beta_{10} q^{7} -\beta_{11} q^{8} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{9} + ( \beta_{4} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{10} + ( \beta_{1} + \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{11} + ( 1 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{15} ) q^{12} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{13} -\beta_{3} q^{14} + ( 1 - \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{15} - q^{16} + ( \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{17} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{18} + ( -\beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} ) q^{19} + ( 1 - \beta_{3} + \beta_{4} - \beta_{15} ) q^{20} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{21} + ( -1 + \beta_{2} + \beta_{6} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{22} + ( 3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{8} - 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} ) q^{23} + ( -\beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{24} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{25} + ( 1 - \beta_{4} - \beta_{13} + \beta_{14} ) q^{26} + ( -2 + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{14} - 2 \beta_{15} ) q^{27} + \beta_{11} q^{28} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} + 2 \beta_{11} ) q^{29} + ( -\beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{9} - \beta_{10} - 4 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{30} + ( -1 + \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - \beta_{14} ) q^{31} + \beta_{10} q^{32} + ( -1 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{33} + ( -1 + 2 \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{34} + ( -\beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{35} + ( -2 - \beta_{4} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} + \beta_{14} ) q^{36} + ( -4 + 4 \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{15} ) q^{37} + ( -\beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{15} ) q^{38} + ( 4 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{39} + ( -\beta_{5} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{41} + ( -1 + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{15} ) q^{42} + ( -4 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{43} + ( \beta_{1} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{44} + ( 3 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{14} - 4 \beta_{15} ) q^{45} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{46} + ( -1 + \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{15} ) q^{47} + ( 1 + \beta_{3} - \beta_{7} + \beta_{15} ) q^{48} + \beta_{3} q^{49} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{50} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{51} + ( \beta_{3} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{52} + ( -2 + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - 4 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{53} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{54} + ( -3 - 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{55} + q^{56} + ( \beta_{3} - 2 \beta_{4} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{57} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{58} + ( 1 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 6 \beta_{8} + 4 \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{59} + ( -3 - \beta_{2} + \beta_{3} - \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{60} + ( \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + 5 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} ) q^{61} + ( -\beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{62} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{63} -\beta_{3} q^{64} + ( 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{65} + ( -1 + \beta_{1} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{13} + 3 \beta_{14} ) q^{66} + ( 4 - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{15} ) q^{67} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{68} + ( -4 + 8 \beta_{2} - 8 \beta_{3} + \beta_{4} - \beta_{8} + 7 \beta_{9} - 8 \beta_{11} - 4 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{69} + ( -1 + \beta_{3} - \beta_{4} + \beta_{15} ) q^{70} + ( -1 - 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{13} - \beta_{14} ) q^{71} + ( \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{72} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - 5 \beta_{11} - \beta_{13} - \beta_{14} ) q^{73} + ( \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 3 + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} + \beta_{8} + 5 \beta_{9} - 4 \beta_{11} - \beta_{13} - 3 \beta_{14} ) q^{75} + ( \beta_{5} + \beta_{6} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{76} + ( 1 - \beta_{2} - \beta_{6} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{77} + ( 3 - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{11} + 4 \beta_{13} - 2 \beta_{15} ) q^{78} + ( -2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 5 \beta_{11} ) q^{79} + ( 1 + \beta_{3} - \beta_{7} ) q^{80} + ( -2 + \beta_{4} - 6 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{13} + 3 \beta_{14} ) q^{81} + ( 1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{3} - 6 \beta_{5} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{83} + ( \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{84} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{85} + ( 2 - 4 \beta_{4} - 2 \beta_{13} + 2 \beta_{14} ) q^{86} + ( 2 \beta_{3} + 4 \beta_{4} - 8 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{87} + ( -\beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{88} + ( 2 - 4 \beta_{2} - 2 \beta_{8} - 6 \beta_{9} + 4 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{89} + ( 2 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} ) q^{90} + ( -1 + \beta_{4} + \beta_{13} - \beta_{14} ) q^{91} + ( -1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{8} - 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{14} ) q^{92} + ( 2 \beta_{2} - \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{15} ) q^{93} + ( 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 5 \beta_{11} + \beta_{13} + \beta_{14} ) q^{95} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{96} + ( -2 + 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{97} -\beta_{11} q^{98} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} - \beta_{11} + 5 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{3} - 8q^{5} + O(q^{10})$$ $$16q - 8q^{3} - 8q^{5} - 8q^{11} + 8q^{12} + 24q^{15} - 16q^{16} + 16q^{20} - 8q^{22} + 32q^{23} + 16q^{25} + 16q^{26} - 32q^{27} - 24q^{33} - 48q^{36} - 48q^{37} - 8q^{38} - 8q^{42} + 72q^{45} + 8q^{48} - 16q^{53} - 48q^{55} + 16q^{56} + 32q^{58} - 56q^{60} - 32q^{66} + 48q^{67} - 16q^{70} - 48q^{71} + 112q^{75} + 8q^{77} + 32q^{78} + 8q^{80} - 80q^{81} - 16q^{82} + 32q^{86} - 8q^{88} - 16q^{91} - 32q^{92} + 32q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 32 x^{14} + 404 x^{12} + 2600 x^{10} + 9170 x^{8} + 17648 x^{6} + 17180 x^{4} + 6904 x^{2} + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-133 \nu^{14} - 3381 \nu^{12} - 29156 \nu^{10} - 88672 \nu^{8} + 38697 \nu^{6} + 580417 \nu^{4} + 550826 \nu^{2} - 104150$$$$)/52140$$ $$\beta_{2}$$ $$=$$ $$($$$$-96 \nu^{15} - 12857 \nu^{14} - 12594 \nu^{13} - 396267 \nu^{12} - 306090 \nu^{11} - 4711090 \nu^{10} - 3095298 \nu^{9} - 27494729 \nu^{8} - 15682752 \nu^{7} - 82357779 \nu^{6} - 41772858 \nu^{5} - 118862275 \nu^{4} - 54822474 \nu^{3} - 64101644 \nu^{2} - 26474442 \nu - 44689$$$$)/2398440$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{15} - 270 \nu^{13} - 4277 \nu^{11} - 35680 \nu^{9} - 165183 \nu^{7} - 404918 \nu^{5} - 445273 \nu^{3} - 136832 \nu$$$$)/30360$$ $$\beta_{4}$$ $$=$$ $$($$$$-29 \nu^{14} - 837 \nu^{12} - 9095 \nu^{10} - 46994 \nu^{8} - 119553 \nu^{6} - 139743 \nu^{4} - 59095 \nu^{2} - 688$$$$)/1580$$ $$\beta_{5}$$ $$=$$ $$($$$$3267 \nu^{15} - 37904 \nu^{14} + 125037 \nu^{13} - 1115040 \nu^{12} + 1853445 \nu^{11} - 12465034 \nu^{10} + 13382919 \nu^{9} - 67257980 \nu^{8} + 48319359 \nu^{7} - 183261516 \nu^{6} + 77777865 \nu^{5} - 238829056 \nu^{4} + 32560869 \nu^{3} - 117442646 \nu^{2} - 12716121 \nu - 5847244$$$$)/4796880$$ $$\beta_{6}$$ $$=$$ $$($$$$2362 \nu^{15} - 9154 \nu^{14} + 97503 \nu^{13} - 239016 \nu^{12} + 1552478 \nu^{11} - 2192015 \nu^{10} + 12086401 \nu^{9} - 8173717 \nu^{8} + 48040770 \nu^{7} - 8382948 \nu^{6} + 91732691 \nu^{5} + 12797338 \nu^{4} + 69111166 \nu^{3} + 20447759 \nu^{2} + 13154201 \nu + 5518597$$$$)/2398440$$ $$\beta_{7}$$ $$=$$ $$($$$$-14119 \nu^{15} + 4554 \nu^{14} - 384441 \nu^{13} + 185196 \nu^{12} - 3770627 \nu^{11} + 2925186 \nu^{10} - 15901507 \nu^{9} + 22601502 \nu^{8} - 23732847 \nu^{7} + 88097130 \nu^{6} + 12283723 \nu^{5} + 159122832 \nu^{4} + 50195657 \nu^{3} + 102463482 \nu^{2} + 23592469 \nu + 12567522$$$$)/4796880$$ $$\beta_{8}$$ $$=$$ $$($$$$14119 \nu^{15} - 51244 \nu^{14} + 384441 \nu^{13} - 1510962 \nu^{12} + 3770627 \nu^{11} - 16928552 \nu^{10} + 15901507 \nu^{9} - 91404484 \nu^{8} + 23732847 \nu^{7} - 248045892 \nu^{6} - 12283723 \nu^{5} - 320383238 \nu^{4} - 50195657 \nu^{3} - 165191152 \nu^{2} - 23592469 \nu - 15822896$$$$)/4796880$$ $$\beta_{9}$$ $$=$$ $$($$$$14119 \nu^{15} + 51244 \nu^{14} + 384441 \nu^{13} + 1510962 \nu^{12} + 3770627 \nu^{11} + 16928552 \nu^{10} + 15901507 \nu^{9} + 91404484 \nu^{8} + 23732847 \nu^{7} + 248045892 \nu^{6} - 12283723 \nu^{5} + 320383238 \nu^{4} - 50195657 \nu^{3} + 165191152 \nu^{2} - 23592469 \nu + 15822896$$$$)/4796880$$ $$\beta_{10}$$ $$=$$ $$($$$$7788 \nu^{15} - 9706 \nu^{14} + 227205 \nu^{13} - 281451 \nu^{12} + 2511069 \nu^{11} - 3082598 \nu^{10} + 13345695 \nu^{9} - 16168057 \nu^{8} + 35747514 \nu^{7} - 42555198 \nu^{6} + 46701897 \nu^{5} - 54678935 \nu^{4} + 27732903 \nu^{3} - 31162930 \nu^{2} + 7716027 \nu - 4272089$$$$)/2398440$$ $$\beta_{11}$$ $$=$$ $$($$$$7788 \nu^{15} + 9706 \nu^{14} + 227205 \nu^{13} + 281451 \nu^{12} + 2511069 \nu^{11} + 3082598 \nu^{10} + 13345695 \nu^{9} + 16168057 \nu^{8} + 35747514 \nu^{7} + 42555198 \nu^{6} + 46701897 \nu^{5} + 54678935 \nu^{4} + 27732903 \nu^{3} + 31162930 \nu^{2} + 7716027 \nu + 4272089$$$$)/2398440$$ $$\beta_{12}$$ $$=$$ $$($$$$16825 \nu^{15} + 44022 \nu^{14} + 477105 \nu^{13} + 1270566 \nu^{12} + 5030831 \nu^{11} + 13806210 \nu^{10} + 24638785 \nu^{9} + 71336892 \nu^{8} + 56736477 \nu^{7} + 181481454 \nu^{6} + 55797785 \nu^{5} + 212129874 \nu^{4} + 24097111 \nu^{3} + 89706210 \nu^{2} + 9667109 \nu + 1044384$$$$)/4796880$$ $$\beta_{13}$$ $$=$$ $$($$$$-40961 \nu^{15} + 33442 \nu^{14} - 1235013 \nu^{13} + 906936 \nu^{12} - 14300983 \nu^{11} + 8818844 \nu^{10} - 81147701 \nu^{9} + 36435082 \nu^{8} - 237928413 \nu^{7} + 50952222 \nu^{6} - 348621037 \nu^{5} - 31697956 \nu^{4} - 224645303 \nu^{3} - 86437496 \nu^{2} - 44519689 \nu - 13463050$$$$)/4796880$$ $$\beta_{14}$$ $$=$$ $$($$$$-40961 \nu^{15} - 21206 \nu^{14} - 1235013 \nu^{13} - 595884 \nu^{12} - 14300983 \nu^{11} - 6136492 \nu^{10} - 81147701 \nu^{9} - 28277258 \nu^{8} - 237928413 \nu^{7} - 54512346 \nu^{6} - 348621037 \nu^{5} - 21700408 \nu^{4} - 224645303 \nu^{3} + 35761504 \nu^{2} - 44519689 \nu + 23044850$$$$)/4796880$$ $$\beta_{15}$$ $$=$$ $$($$$$-40769 \nu^{15} - 44022 \nu^{14} - 1209825 \nu^{13} - 1270566 \nu^{12} - 13688803 \nu^{11} - 13806210 \nu^{10} - 74957105 \nu^{9} - 71336892 \nu^{8} - 206562909 \nu^{7} - 181481454 \nu^{6} - 265075321 \nu^{5} - 212129874 \nu^{4} - 115000355 \nu^{3} - 89706210 \nu^{2} + 3632315 \nu - 1044384$$$$)/4796880$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} - 2 \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$12 \beta_{15} - 5 \beta_{14} - 5 \beta_{13} - 6 \beta_{12} + \beta_{11} + 11 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} - 9 \beta_{4} + 4 \beta_{3} + 10 \beta_{2} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{14} - \beta_{13} - 2 \beta_{11} + 12 \beta_{10} + \beta_{9} - 13 \beta_{8} - 2 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + \beta_{4} + 8 \beta_{1} + 28$$ $$\nu^{5}$$ $$=$$ $$($$$$-94 \beta_{15} + 45 \beta_{14} + 45 \beta_{13} + 100 \beta_{12} - 19 \beta_{11} - 99 \beta_{10} - 45 \beta_{9} + 35 \beta_{8} + 10 \beta_{6} + 10 \beta_{5} + 87 \beta_{4} - 52 \beta_{3} - 70 \beta_{2} + 10 \beta_{1} + 35$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{14} + 18 \beta_{13} + 47 \beta_{11} - 140 \beta_{10} - 13 \beta_{9} + 144 \beta_{8} + 38 \beta_{7} + 93 \beta_{6} - 93 \beta_{5} - 7 \beta_{4} - 67 \beta_{1} - 242$$ $$\nu^{7}$$ $$=$$ $$($$$$828 \beta_{15} - 473 \beta_{14} - 473 \beta_{13} - 1246 \beta_{12} + 147 \beta_{11} + 925 \beta_{10} + 447 \beta_{9} - 331 \beta_{8} - 196 \beta_{6} - 196 \beta_{5} - 841 \beta_{4} + 620 \beta_{3} + 582 \beta_{2} - 182 \beta_{1} - 291$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$246 \beta_{14} - 246 \beta_{13} - 692 \beta_{11} + 1568 \beta_{10} + 134 \beta_{9} - 1558 \beta_{8} - 548 \beta_{7} - 876 \beta_{6} + 876 \beta_{5} + 26 \beta_{4} + 608 \beta_{1} + 2317$$ $$\nu^{9}$$ $$=$$ $$($$$$-7750 \beta_{15} + 5091 \beta_{14} + 5091 \beta_{13} + 14208 \beta_{12} - 925 \beta_{11} - 9031 \beta_{10} - 4539 \beta_{9} + 3567 \beta_{8} + 2748 \beta_{6} + 2748 \beta_{5} + 8231 \beta_{4} - 7112 \beta_{3} - 5358 \beta_{2} + 2412 \beta_{1} + 2679$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-2984 \beta_{14} + 2984 \beta_{13} + 8686 \beta_{11} - 17149 \beta_{10} - 1334 \beta_{9} + 16761 \beta_{8} + 6964 \beta_{7} + 8463 \beta_{6} - 8463 \beta_{5} + 74 \beta_{4} - 5855 \beta_{1} - 23316$$ $$\nu^{11}$$ $$=$$ $$($$$$75340 \beta_{15} - 54743 \beta_{14} - 54743 \beta_{13} - 156442 \beta_{12} + 4671 \beta_{11} + 91013 \beta_{10} + 46635 \beta_{9} - 39707 \beta_{8} - 33968 \beta_{6} - 33968 \beta_{5} - 81923 \beta_{4} + 79276 \beta_{3} + 52374 \beta_{2} - 28556 \beta_{1} - 26187$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$34097 \beta_{14} - 34097 \beta_{13} - 101434 \beta_{11} + 185196 \beta_{10} + 13461 \beta_{9} - 179785 \beta_{8} - 82562 \beta_{7} - 83762 \beta_{6} + 83762 \beta_{5} - 2943 \beta_{4} + 58600 \beta_{1} + 240514$$ $$\nu^{13}$$ $$=$$ $$($$$$-751954 \beta_{15} + 586403 \beta_{14} + 586403 \beta_{13} + 1695148 \beta_{12} - 11285 \beta_{11} - 936233 \beta_{10} - 483587 \beta_{9} + 441361 \beta_{8} + 394914 \beta_{6} + 394914 \beta_{5} + 828637 \beta_{4} - 867300 \beta_{3} - 530034 \beta_{2} + 321386 \beta_{1} + 265017$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$-377518 \beta_{14} + 377518 \beta_{13} + 1140739 \beta_{11} - 1986410 \beta_{10} - 138513 \beta_{9} + 1924046 \beta_{8} + 939862 \beta_{7} + 845671 \beta_{6} - 845671 \beta_{5} + 43585 \beta_{4} - 600625 \beta_{1} - 2513158$$ $$\nu^{15}$$ $$=$$ $$($$$$7648508 \beta_{15} - 6262871 \beta_{14} - 6262871 \beta_{13} - 18218730 \beta_{12} - 156783 \beta_{11} + 9753975 \beta_{10} + 5051013 \beta_{9} - 4859745 \beta_{8} - 4438228 \beta_{6} - 4438228 \beta_{5} - 8495391 \beta_{4} + 9380596 \beta_{3} + 5472530 \beta_{2} - 3526606 \beta_{1} - 2736265$$$$)/2$$

## Character values

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

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.914718i − 1.72791i − 3.26007i 1.24483i 2.32893i − 0.313694i − 1.84585i 2.65904i − 0.914718i 1.72791i 3.26007i − 1.24483i − 2.32893i 0.313694i 1.84585i − 2.65904i
−0.707107 0.707107i −2.14984 2.14984i 1.00000i −1.71500 1.43484i 3.04034i 0.707107 + 0.707107i 0.707107 0.707107i 6.24365i 0.198102 + 2.22728i
43.2 −0.707107 0.707107i −1.40684 1.40684i 1.00000i −0.177857 2.22898i 1.98957i 0.707107 + 0.707107i 0.707107 0.707107i 0.958399i −1.45037 + 1.70189i
43.3 −0.707107 0.707107i −0.668611 0.668611i 1.00000i −2.17741 + 0.508801i 0.945559i 0.707107 + 0.707107i 0.707107 0.707107i 2.10592i 1.89944 + 1.17989i
43.4 −0.707107 0.707107i 2.22529 + 2.22529i 1.00000i 2.07027 0.844975i 3.14704i 0.707107 + 0.707107i 0.707107 0.707107i 6.90387i −2.06139 0.866414i
43.5 0.707107 + 0.707107i −2.14984 2.14984i 1.00000i −1.71500 1.43484i 3.04034i −0.707107 0.707107i −0.707107 + 0.707107i 6.24365i −0.198102 2.22728i
43.6 0.707107 + 0.707107i −1.40684 1.40684i 1.00000i −0.177857 2.22898i 1.98957i −0.707107 0.707107i −0.707107 + 0.707107i 0.958399i 1.45037 1.70189i
43.7 0.707107 + 0.707107i −0.668611 0.668611i 1.00000i −2.17741 + 0.508801i 0.945559i −0.707107 0.707107i −0.707107 + 0.707107i 2.10592i −1.89944 1.17989i
43.8 0.707107 + 0.707107i 2.22529 + 2.22529i 1.00000i 2.07027 0.844975i 3.14704i −0.707107 0.707107i −0.707107 + 0.707107i 6.90387i 2.06139 + 0.866414i
197.1 −0.707107 + 0.707107i −2.14984 + 2.14984i 1.00000i −1.71500 + 1.43484i 3.04034i 0.707107 0.707107i 0.707107 + 0.707107i 6.24365i 0.198102 2.22728i
197.2 −0.707107 + 0.707107i −1.40684 + 1.40684i 1.00000i −0.177857 + 2.22898i 1.98957i 0.707107 0.707107i 0.707107 + 0.707107i 0.958399i −1.45037 1.70189i
197.3 −0.707107 + 0.707107i −0.668611 + 0.668611i 1.00000i −2.17741 0.508801i 0.945559i 0.707107 0.707107i 0.707107 + 0.707107i 2.10592i 1.89944 1.17989i
197.4 −0.707107 + 0.707107i 2.22529 2.22529i 1.00000i 2.07027 + 0.844975i 3.14704i 0.707107 0.707107i 0.707107 + 0.707107i 6.90387i −2.06139 + 0.866414i
197.5 0.707107 0.707107i −2.14984 + 2.14984i 1.00000i −1.71500 + 1.43484i 3.04034i −0.707107 + 0.707107i −0.707107 0.707107i 6.24365i −0.198102 + 2.22728i
197.6 0.707107 0.707107i −1.40684 + 1.40684i 1.00000i −0.177857 + 2.22898i 1.98957i −0.707107 + 0.707107i −0.707107 0.707107i 0.958399i 1.45037 + 1.70189i
197.7 0.707107 0.707107i −0.668611 + 0.668611i 1.00000i −2.17741 0.508801i 0.945559i −0.707107 + 0.707107i −0.707107 0.707107i 2.10592i −1.89944 + 1.17989i
197.8 0.707107 0.707107i 2.22529 2.22529i 1.00000i 2.07027 + 0.844975i 3.14704i −0.707107 + 0.707107i −0.707107 0.707107i 6.90387i 2.06139 0.866414i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.8 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
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.e 16
5.c odd 4 1 inner 770.2.m.e 16
11.b odd 2 1 inner 770.2.m.e 16
55.e even 4 1 inner 770.2.m.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.e 16 1.a even 1 1 trivial
770.2.m.e 16 5.c odd 4 1 inner
770.2.m.e 16 11.b odd 2 1 inner
770.2.m.e 16 55.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$:

 $$T_{3}^{8} + \cdots$$ $$T_{17}^{16} + 1040 T_{17}^{12} + 276576 T_{17}^{8} + 21971200 T_{17}^{4} + 303595776$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{4}$$
$3$ $$( 324 + 720 T + 800 T^{2} + 392 T^{3} + 100 T^{4} + 8 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$5$ $$( 625 + 500 T + 100 T^{2} - 40 T^{3} - 32 T^{4} - 8 T^{5} + 4 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$7$ $$( 1 + T^{4} )^{4}$$
$11$ $$( 14641 + 5324 T + 1452 T^{2} + 220 T^{3} + 94 T^{4} + 20 T^{5} + 12 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$13$ $$126247696 + 8197024 T^{4} + 143960 T^{8} + 744 T^{12} + T^{16}$$
$17$ $$303595776 + 21971200 T^{4} + 276576 T^{8} + 1040 T^{12} + T^{16}$$
$19$ $$( 324 - 2144 T^{2} + 516 T^{4} - 40 T^{6} + T^{8} )^{2}$$
$23$ $$( 1336336 - 1165248 T + 508032 T^{2} - 62848 T^{3} + 4248 T^{4} - 304 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$29$ $$( 1763584 - 231168 T^{2} + 10208 T^{4} - 176 T^{6} + T^{8} )^{2}$$
$31$ $$( 132 + 16 T - 28 T^{2} + T^{4} )^{4}$$
$37$ $$( 484416 + 144768 T + 21632 T^{2} + 3392 T^{3} + 5488 T^{4} + 1744 T^{5} + 288 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$41$ $$( 126736 + 68320 T^{2} + 4824 T^{4} + 120 T^{6} + T^{8} )^{2}$$
$43$ $$6553600000000 + 190709760000 T^{4} + 291004416 T^{8} + 33536 T^{12} + T^{16}$$
$47$ $$( 1936 + 7040 T + 12800 T^{2} + 7040 T^{3} + 2024 T^{4} + 160 T^{5} + T^{8} )^{2}$$
$53$ $$( 10086976 - 660608 T + 21632 T^{2} + 4544 T^{3} + 14384 T^{4} - 944 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$59$ $$( 142229476 + 5734480 T^{2} + 81452 T^{4} + 480 T^{6} + T^{8} )^{2}$$
$61$ $$( 1295044 + 1346272 T^{2} + 39804 T^{4} + 360 T^{6} + T^{8} )^{2}$$
$67$ $$( 6718464 + 3151872 T + 739328 T^{2} + 23296 T^{3} - 4160 T^{4} - 448 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$71$ $$( 4444 - 776 T - 108 T^{2} + 12 T^{3} + T^{4} )^{4}$$
$73$ $$218889236736 + 3511600384 T^{4} + 9023840 T^{8} + 6864 T^{12} + T^{16}$$
$79$ $$( 255872016 - 8603744 T^{2} + 103832 T^{4} - 536 T^{6} + T^{8} )^{2}$$
$83$ $$218794862224656 + 4284552215392 T^{4} + 1092023768 T^{8} + 61912 T^{12} + T^{16}$$
$89$ $$( 4596736 + 695296 T^{2} + 28352 T^{4} + 384 T^{6} + T^{8} )^{2}$$
$97$ $$( 278784 + 33792 T + 2048 T^{2} - 3584 T^{3} + 2080 T^{4} - 64 T^{5} + T^{8} )^{2}$$
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