# Properties

 Label 77.2.i.a Level 77 Weight 2 Character orbit 77.i Analytic conductor 0.615 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 77.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 8 x^{10} + 47 x^{8} - 122 x^{6} + 233 x^{4} - 119 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$13 \nu^{10} + 113 \nu^{8} - 929 \nu^{6} + 6401 \nu^{4} - 11510 \nu^{2} + 16793$$$$)/6363$$ $$\beta_{3}$$ $$=$$ $$($$$$-148 \nu^{10} - 797 \nu^{8} + 1766 \nu^{6} - 27353 \nu^{4} + 2798 \nu^{2} - 55601$$$$)/69993$$ $$\beta_{4}$$ $$=$$ $$($$$$305 \nu^{10} - 3875 \nu^{8} + 25682 \nu^{6} - 83459 \nu^{4} + 133274 \nu^{2} - 13895$$$$)/69993$$ $$\beta_{5}$$ $$=$$ $$($$$$64 \nu^{11} - 376 \nu^{9} + 2209 \nu^{7} - 1864 \nu^{5} + 952 \nu^{3} + 34403 \nu$$$$)/9999$$ $$\beta_{6}$$ $$=$$ $$($$$$656 \nu^{11} - 7187 \nu^{9} + 45140 \nu^{7} - 152426 \nu^{5} + 299729 \nu^{3} - 203147 \nu$$$$)/69993$$ $$\beta_{7}$$ $$=$$ $$($$$$-799 \nu^{11} + 5944 \nu^{9} - 34921 \nu^{7} + 82015 \nu^{5} - 173119 \nu^{3} + 18424 \nu$$$$)/69993$$ $$\beta_{8}$$ $$=$$ $$($$$$799 \nu^{10} - 5944 \nu^{8} + 34921 \nu^{6} - 82015 \nu^{4} + 173119 \nu^{2} - 88417$$$$)/69993$$ $$\beta_{9}$$ $$=$$ $$($$$$-2249 \nu^{10} + 18629 \nu^{8} - 106529 \nu^{6} + 273398 \nu^{4} - 452162 \nu^{2} + 180866$$$$)/69993$$ $$\beta_{10}$$ $$=$$ $$($$$$-2392 \nu^{11} + 17386 \nu^{9} - 96310 \nu^{7} + 202987 \nu^{5} - 325552 \nu^{3} - 3857 \nu$$$$)/69993$$ $$\beta_{11}$$ $$=$$ $$($$$$-916 \nu^{11} + 7048 \nu^{9} - 41407 \nu^{7} + 105004 \nu^{5} - 205273 \nu^{3} + 104839 \nu$$$$)/23331$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 3 \beta_{8} + \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 4 \beta_{7} - \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{9} + 12 \beta_{8} + 6 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 5$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{11} - \beta_{10} - 16 \beta_{7} + 2 \beta_{6} - 17 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{9} + 8 \beta_{8} + 31 \beta_{4} + 15 \beta_{2} - 44$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{10} + 8 \beta_{6} + 31 \beta_{5} - 83 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-59 \beta_{9} - 186 \beta_{8} - 153 \beta_{3} - 47 \beta_{2} - 80$$ $$\nu^{9}$$ $$=$$ $$-153 \beta_{11} + 94 \beta_{10} + 292 \beta_{7} - 47 \beta_{6} + 153 \beta_{5} - 47 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-492 \beta_{9} - 1064 \beta_{8} - 739 \beta_{4} - 739 \beta_{3} - 492 \beta_{2} + 492$$ $$\nu^{11}$$ $$=$$ $$-739 \beta_{11} + 247 \beta_{10} + 1309 \beta_{7} - 494 \beta_{6} + 1556 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$1 + \beta_{8}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −1.87742 − 1.08393i −1.43898 − 0.830794i −0.636099 − 0.367252i 0.636099 + 0.367252i 1.43898 + 0.830794i 1.87742 + 1.08393i −1.87742 + 1.08393i −1.43898 + 0.830794i −0.636099 + 0.367252i 0.636099 − 0.367252i 1.43898 − 0.830794i 1.87742 − 1.08393i
−1.87742 + 1.08393i −0.555632 0.320794i 1.34981 2.33795i −2.93818 + 1.69636i 1.39088 −2.49548 0.878952i 1.51670i −1.29418 2.24159i 3.67747 6.36957i
10.2 −1.43898 + 0.830794i −1.97141 1.13819i 0.380438 0.658939i 2.80150 1.61745i 3.78242 2.23530 1.41542i 2.05891i 1.09097 + 1.88962i −2.68754 + 4.65495i
10.3 −0.636099 + 0.367252i 1.02704 + 0.592963i −0.730252 + 1.26483i 0.136673 0.0789082i −0.871067 1.12959 + 2.39249i 2.54175i −0.796790 1.38008i −0.0579584 + 0.100387i
10.4 0.636099 0.367252i 1.02704 + 0.592963i −0.730252 + 1.26483i 0.136673 0.0789082i 0.871067 −1.12959 2.39249i 2.54175i −0.796790 1.38008i 0.0579584 0.100387i
10.5 1.43898 0.830794i −1.97141 1.13819i 0.380438 0.658939i 2.80150 1.61745i −3.78242 −2.23530 + 1.41542i 2.05891i 1.09097 + 1.88962i 2.68754 4.65495i
10.6 1.87742 1.08393i −0.555632 0.320794i 1.34981 2.33795i −2.93818 + 1.69636i −1.39088 2.49548 + 0.878952i 1.51670i −1.29418 2.24159i −3.67747 + 6.36957i
54.1 −1.87742 1.08393i −0.555632 + 0.320794i 1.34981 + 2.33795i −2.93818 1.69636i 1.39088 −2.49548 + 0.878952i 1.51670i −1.29418 + 2.24159i 3.67747 + 6.36957i
54.2 −1.43898 0.830794i −1.97141 + 1.13819i 0.380438 + 0.658939i 2.80150 + 1.61745i 3.78242 2.23530 + 1.41542i 2.05891i 1.09097 1.88962i −2.68754 4.65495i
54.3 −0.636099 0.367252i 1.02704 0.592963i −0.730252 1.26483i 0.136673 + 0.0789082i −0.871067 1.12959 2.39249i 2.54175i −0.796790 + 1.38008i −0.0579584 0.100387i
54.4 0.636099 + 0.367252i 1.02704 0.592963i −0.730252 1.26483i 0.136673 + 0.0789082i 0.871067 −1.12959 + 2.39249i 2.54175i −0.796790 + 1.38008i 0.0579584 + 0.100387i
54.5 1.43898 + 0.830794i −1.97141 + 1.13819i 0.380438 + 0.658939i 2.80150 + 1.61745i −3.78242 −2.23530 1.41542i 2.05891i 1.09097 1.88962i 2.68754 + 4.65495i
54.6 1.87742 + 1.08393i −0.555632 + 0.320794i 1.34981 + 2.33795i −2.93818 1.69636i −1.39088 2.49548 0.878952i 1.51670i −1.29418 + 2.24159i −3.67747 6.36957i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 54.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.i.a 12
3.b odd 2 1 693.2.bg.a 12
4.b odd 2 1 1232.2.bn.a 12
7.b odd 2 1 539.2.i.c 12
7.c even 3 1 539.2.b.b 12
7.c even 3 1 539.2.i.c 12
7.d odd 6 1 inner 77.2.i.a 12
7.d odd 6 1 539.2.b.b 12
11.b odd 2 1 inner 77.2.i.a 12
11.c even 5 4 847.2.r.b 48
11.d odd 10 4 847.2.r.b 48
21.g even 6 1 693.2.bg.a 12
28.f even 6 1 1232.2.bn.a 12
33.d even 2 1 693.2.bg.a 12
44.c even 2 1 1232.2.bn.a 12
77.b even 2 1 539.2.i.c 12
77.h odd 6 1 539.2.b.b 12
77.h odd 6 1 539.2.i.c 12
77.i even 6 1 inner 77.2.i.a 12
77.i even 6 1 539.2.b.b 12
77.n even 30 4 847.2.r.b 48
77.p odd 30 4 847.2.r.b 48
231.k odd 6 1 693.2.bg.a 12
308.m odd 6 1 1232.2.bn.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.i.a 12 1.a even 1 1 trivial
77.2.i.a 12 7.d odd 6 1 inner
77.2.i.a 12 11.b odd 2 1 inner
77.2.i.a 12 77.i even 6 1 inner
539.2.b.b 12 7.c even 3 1
539.2.b.b 12 7.d odd 6 1
539.2.b.b 12 77.h odd 6 1
539.2.b.b 12 77.i even 6 1
539.2.i.c 12 7.b odd 2 1
539.2.i.c 12 7.c even 3 1
539.2.i.c 12 77.b even 2 1
539.2.i.c 12 77.h odd 6 1
693.2.bg.a 12 3.b odd 2 1
693.2.bg.a 12 21.g even 6 1
693.2.bg.a 12 33.d even 2 1
693.2.bg.a 12 231.k odd 6 1
847.2.r.b 48 11.c even 5 4
847.2.r.b 48 11.d odd 10 4
847.2.r.b 48 77.n even 30 4
847.2.r.b 48 77.p odd 30 4
1232.2.bn.a 12 4.b odd 2 1
1232.2.bn.a 12 28.f even 6 1
1232.2.bn.a 12 44.c even 2 1
1232.2.bn.a 12 308.m odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(77, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 3 T^{4} - 6 T^{6} + T^{8} + 25 T^{10} + 37 T^{12} + 100 T^{14} + 16 T^{16} - 384 T^{18} + 768 T^{20} + 4096 T^{22} + 4096 T^{24}$$
$3$ $$( 1 + 3 T + 10 T^{2} + 21 T^{3} + 43 T^{4} + 78 T^{5} + 141 T^{6} + 234 T^{7} + 387 T^{8} + 567 T^{9} + 810 T^{10} + 729 T^{11} + 729 T^{12} )^{2}$$
$5$ $$( 1 + 4 T^{2} - 4 T^{4} + 12 T^{5} - 167 T^{6} + 60 T^{7} - 100 T^{8} + 2500 T^{10} + 15625 T^{12} )^{2}$$
$7$ $$1 - 8 T^{2} + 62 T^{4} - 203 T^{6} + 3038 T^{8} - 19208 T^{10} + 117649 T^{12}$$
$11$ $$1 + 4 T - 5 T^{2} - 44 T^{3} - 46 T^{4} - 20 T^{5} - 29 T^{6} - 220 T^{7} - 5566 T^{8} - 58564 T^{9} - 73205 T^{10} + 644204 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 + 67 T^{2} + 1997 T^{4} + 33649 T^{6} + 337493 T^{8} + 1913587 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$1 - 63 T^{2} + 1818 T^{4} - 45823 T^{6} + 1179927 T^{8} - 23396904 T^{10} + 390708825 T^{12} - 6761705256 T^{14} + 98548682967 T^{16} - 1106055824287 T^{18} + 12681927027738 T^{20} - 127007615728287 T^{22} + 582622237229761 T^{24}$$
$19$ $$1 - 31 T^{2} - 322 T^{4} + 5085 T^{6} + 487627 T^{8} - 4100252 T^{10} - 122324807 T^{12} - 1480190972 T^{14} + 63548038267 T^{16} + 239228304885 T^{18} - 5468707299202 T^{20} - 190063053991831 T^{22} + 2213314919066161 T^{24}$$
$23$ $$( 1 + 10 T + 4 T^{2} - 2 T^{3} + 2198 T^{4} + 7306 T^{5} - 9173 T^{6} + 168038 T^{7} + 1162742 T^{8} - 24334 T^{9} + 1119364 T^{10} + 64363430 T^{11} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 91 T^{2} + 4663 T^{4} - 160283 T^{6} + 3921583 T^{8} - 64362571 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 3 T + 84 T^{2} - 243 T^{3} + 4227 T^{4} - 11454 T^{5} + 151369 T^{6} - 355074 T^{7} + 4062147 T^{8} - 7239213 T^{9} + 77575764 T^{10} - 85887453 T^{11} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 - 8 T - 51 T^{2} + 216 T^{3} + 4426 T^{4} - 8192 T^{5} - 153011 T^{6} - 303104 T^{7} + 6059194 T^{8} + 10941048 T^{9} - 95582211 T^{10} - 554751656 T^{11} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 + 147 T^{2} + 11733 T^{4} + 590425 T^{6} + 19723173 T^{8} + 415386867 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 - 182 T^{2} + 16094 T^{4} - 864959 T^{6} + 29757806 T^{8} - 622221782 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 9 T + 154 T^{2} + 1143 T^{3} + 12329 T^{4} + 71940 T^{5} + 665017 T^{6} + 3381180 T^{7} + 27234761 T^{8} + 118669689 T^{9} + 751470874 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 + T - 92 T^{2} - 161 T^{3} + 3593 T^{4} + 4762 T^{5} - 147323 T^{6} + 252386 T^{7} + 10092737 T^{8} - 23969197 T^{9} - 725924252 T^{10} + 418195493 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 + 6 T + 46 T^{2} + 204 T^{3} + 212 T^{4} - 2832 T^{5} - 290339 T^{6} - 167088 T^{7} + 737972 T^{8} + 41897316 T^{9} + 557398606 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$1 - 154 T^{2} + 9581 T^{4} - 329790 T^{6} + 4982890 T^{8} + 1013739166 T^{10} - 115924161659 T^{12} + 3772123436686 T^{14} + 68992302660490 T^{16} - 16990904260514190 T^{18} + 1836747765826949261 T^{20} -$$$$10\!\cdots\!54$$$$T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 + 12 T + 3 T^{2} - 1636 T^{3} - 9534 T^{4} + 61764 T^{5} + 1391979 T^{6} + 4138188 T^{7} - 42798126 T^{8} - 492048268 T^{9} + 60453363 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 5 T + 201 T^{2} - 647 T^{3} + 14271 T^{4} - 25205 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$1 - 370 T^{2} + 76046 T^{4} - 11079522 T^{6} + 1263173620 T^{8} - 118509141896 T^{10} + 9378689711155 T^{12} - 631535217163784 T^{14} + 35871908885602420 T^{16} - 1676710889521953858 T^{18} + 61328064148177283726 T^{20} -$$$$15\!\cdots\!30$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 423 T^{2} + 101070 T^{4} + 16919663 T^{6} + 2188998081 T^{8} + 228833270262 T^{10} + 19801390804881 T^{12} + 1428148439705142 T^{14} + 85261652563794561 T^{16} + 4112957826942809423 T^{18} +$$$$15\!\cdots\!70$$$$T^{20} +$$$$40\!\cdots\!23$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$( 1 + 27 T^{2} + 16191 T^{4} + 314125 T^{6} + 111539799 T^{8} + 1281374667 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 33 T + 712 T^{2} + 11517 T^{3} + 157301 T^{4} + 1850406 T^{5} + 18759895 T^{6} + 164686134 T^{7} + 1245981221 T^{8} + 8119127973 T^{9} + 44672475592 T^{10} + 184273961817 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 519 T^{2} + 117501 T^{4} - 14851811 T^{6} + 1105566909 T^{8} - 45946696839 T^{10} + 832972004929 T^{12} )^{2}$$