# Properties

 Label 55.2.g.a Level $55$ Weight $2$ Character orbit 55.g Analytic conductor $0.439$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.g (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.159390625.1 Defining polynomial: $$x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$555 \nu^{7} - 2159 \nu^{6} + 7489 \nu^{5} - 18164 \nu^{4} + 40069 \nu^{3} - 84434 \nu^{2} + 43855 \nu + 375$$$$)/94655$$ $$\beta_{3}$$ $$=$$ $$($$$$-970 \nu^{7} - 1002 \nu^{6} - 6608 \nu^{5} + 9063 \nu^{4} - 14943 \nu^{3} + 27673 \nu^{2} - 68120 \nu + 35160$$$$)/94655$$ $$\beta_{4}$$ $$=$$ $$($$$$-1604 \nu^{7} + 4159 \nu^{6} - 12059 \nu^{5} + 28414 \nu^{4} - 81659 \nu^{3} + 38305 \nu^{2} - 13500 \nu - 13875$$$$)/94655$$ $$\beta_{5}$$ $$=$$ $$($$$$-2052 \nu^{7} + 2252 \nu^{6} - 19912 \nu^{5} + 21007 \nu^{4} - 82042 \nu^{3} + 35785 \nu^{2} - 19395 \nu - 90925$$$$)/94655$$ $$\beta_{6}$$ $$=$$ $$($$$$-2667 \nu^{7} + 6691 \nu^{6} - 17466 \nu^{5} + 50856 \nu^{4} - 82441 \nu^{3} + 72554 \nu^{2} - 4035 \nu - 12035$$$$)/94655$$ $$\beta_{7}$$ $$=$$ $$($$$$4024 \nu^{7} - 1464 \nu^{6} + 21519 \nu^{5} - 26434 \nu^{4} + 59219 \nu^{3} + 22635 \nu^{2} + 54640 \nu + 66675$$$$)/94655$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} + 13 \beta_{3} + 13 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 11 \beta_{6} - 20 \beta_{5} + 20 \beta_{4} - 11 \beta_{2} + 8 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-19 \beta_{7} - 19 \beta_{4} - 68 \beta_{3} - 36 \beta_{2} - 24 \beta_{1} + 36$$ $$\nu^{7}$$ $$=$$ $$111 \beta_{7} + 81 \beta_{6} + 111 \beta_{5} - 55 \beta_{4} + 81 \beta_{3} + 148 \beta_{2} - 56 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2} + \beta_{3} + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 1.43801 + 1.04478i −0.628998 − 0.456994i −0.762262 + 2.34600i 0.453245 − 1.39494i 1.43801 − 1.04478i −0.628998 + 0.456994i −0.762262 − 2.34600i 0.453245 + 1.39494i
−0.579725 1.78421i 1.43801 + 1.04478i −1.22929 + 0.893133i 0.309017 0.951057i 1.03045 3.17141i −3.44479 + 2.50279i −0.729292 0.529862i 0.0492728 + 0.151646i −1.87603
16.2 0.697759 + 2.14748i −0.628998 0.456994i −2.50678 + 1.82128i 0.309017 0.951057i 0.542497 1.66963i −0.100294 + 0.0728678i −2.00678 1.45801i −0.740256 2.27827i 2.25800
26.1 −2.04238 1.48388i −0.762262 + 2.34600i 1.35140 + 4.15918i −0.809017 + 0.587785i 5.03801 3.66033i 0.646930 + 1.99105i 1.85140 5.69802i −2.49563 1.81318i 2.52452
26.2 −0.0756511 0.0549637i 0.453245 1.39494i −0.615332 1.89380i −0.809017 + 0.587785i −0.110960 + 0.0806171i 1.39815 + 4.30308i −0.115332 + 0.354955i 0.686611 + 0.498852i 0.0935099
31.1 −0.579725 + 1.78421i 1.43801 1.04478i −1.22929 0.893133i 0.309017 + 0.951057i 1.03045 + 3.17141i −3.44479 2.50279i −0.729292 + 0.529862i 0.0492728 0.151646i −1.87603
31.2 0.697759 2.14748i −0.628998 + 0.456994i −2.50678 1.82128i 0.309017 + 0.951057i 0.542497 + 1.66963i −0.100294 0.0728678i −2.00678 + 1.45801i −0.740256 + 2.27827i 2.25800
36.1 −2.04238 + 1.48388i −0.762262 2.34600i 1.35140 4.15918i −0.809017 0.587785i 5.03801 + 3.66033i 0.646930 1.99105i 1.85140 + 5.69802i −2.49563 + 1.81318i 2.52452
36.2 −0.0756511 + 0.0549637i 0.453245 + 1.39494i −0.615332 + 1.89380i −0.809017 0.587785i −0.110960 0.0806171i 1.39815 4.30308i −0.115332 0.354955i 0.686611 0.498852i 0.0935099
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 36.2 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 55.2.g.a 8
3.b odd 2 1 495.2.n.f 8
4.b odd 2 1 880.2.bo.e 8
5.b even 2 1 275.2.h.b 8
5.c odd 4 2 275.2.z.b 16
11.b odd 2 1 605.2.g.n 8
11.c even 5 1 inner 55.2.g.a 8
11.c even 5 1 605.2.a.l 4
11.c even 5 2 605.2.g.j 8
11.d odd 10 1 605.2.a.i 4
11.d odd 10 2 605.2.g.g 8
11.d odd 10 1 605.2.g.n 8
33.f even 10 1 5445.2.a.bu 4
33.h odd 10 1 495.2.n.f 8
33.h odd 10 1 5445.2.a.bg 4
44.g even 10 1 9680.2.a.cv 4
44.h odd 10 1 880.2.bo.e 8
44.h odd 10 1 9680.2.a.cs 4
55.h odd 10 1 3025.2.a.be 4
55.j even 10 1 275.2.h.b 8
55.j even 10 1 3025.2.a.v 4
55.k odd 20 2 275.2.z.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a 8 1.a even 1 1 trivial
55.2.g.a 8 11.c even 5 1 inner
275.2.h.b 8 5.b even 2 1
275.2.h.b 8 55.j even 10 1
275.2.z.b 16 5.c odd 4 2
275.2.z.b 16 55.k odd 20 2
495.2.n.f 8 3.b odd 2 1
495.2.n.f 8 33.h odd 10 1
605.2.a.i 4 11.d odd 10 1
605.2.a.l 4 11.c even 5 1
605.2.g.g 8 11.d odd 10 2
605.2.g.j 8 11.c even 5 2
605.2.g.n 8 11.b odd 2 1
605.2.g.n 8 11.d odd 10 1
880.2.bo.e 8 4.b odd 2 1
880.2.bo.e 8 44.h odd 10 1
3025.2.a.v 4 55.j even 10 1
3025.2.a.be 4 55.h odd 10 1
5445.2.a.bg 4 33.h odd 10 1
5445.2.a.bu 4 33.f even 10 1
9680.2.a.cs 4 44.h odd 10 1
9680.2.a.cv 4 44.g even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 18 T + 127 T^{2} + 90 T^{3} + 71 T^{4} + 30 T^{5} + 13 T^{6} + 4 T^{7} + T^{8}$$
$3$ $$25 + 25 T + 10 T^{2} - 5 T^{3} + 21 T^{4} - 11 T^{5} + 6 T^{6} - T^{7} + T^{8}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$25 + 325 T + 1615 T^{2} - 15 T^{3} + 356 T^{4} + 87 T^{5} + 19 T^{6} + 3 T^{7} + T^{8}$$
$11$ $$14641 + 6655 T + 3146 T^{2} + 495 T^{3} + 191 T^{4} + 45 T^{5} + 26 T^{6} + 5 T^{7} + T^{8}$$
$13$ $$121 - 88 T + 37 T^{2} + 20 T^{3} + 21 T^{4} - 20 T^{5} + 33 T^{6} - 4 T^{7} + T^{8}$$
$17$ $$121 - 187 T + 522 T^{2} - 185 T^{3} + 81 T^{4} + 5 T^{5} - 2 T^{6} - T^{7} + T^{8}$$
$19$ $$625 + 1750 T + 2025 T^{2} + 545 T^{3} + 426 T^{4} + 151 T^{5} + 31 T^{6} + T^{7} + T^{8}$$
$23$ $$( -1669 - 706 T - 54 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$29$ $$3025 + 550 T + 6715 T^{2} - 5825 T^{3} + 2856 T^{4} - 869 T^{5} + 171 T^{6} - 19 T^{7} + T^{8}$$
$31$ $$10201 + 6666 T + 2837 T^{2} + 78 T^{3} + 55 T^{4} - 78 T^{5} + 57 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$22801 - 151 T + 13317 T^{2} + 1747 T^{3} + 2880 T^{4} + 203 T^{5} - 23 T^{6} - 4 T^{7} + T^{8}$$
$41$ $$249001 + 119261 T + 99517 T^{2} + 12353 T^{3} + 2130 T^{4} - 103 T^{5} - 3 T^{6} + 4 T^{7} + T^{8}$$
$43$ $$( -59 - 191 T + 121 T^{2} - 21 T^{3} + T^{4} )^{2}$$
$47$ $$5041 + 15194 T + 15417 T^{2} - 10428 T^{3} + 3105 T^{4} - 352 T^{5} + 87 T^{6} - 4 T^{7} + T^{8}$$
$53$ $$1 - 19 T + 823 T^{2} - 1275 T^{3} + 866 T^{4} - 255 T^{5} + 37 T^{6} - 3 T^{7} + T^{8}$$
$59$ $$9150625 + 196625 T + 211975 T^{2} - 17065 T^{3} + 4926 T^{4} + 869 T^{5} + 201 T^{6} + 19 T^{7} + T^{8}$$
$61$ $$3025 - 7700 T + 6065 T^{2} + 6060 T^{3} + 2336 T^{4} + 288 T^{5} + 74 T^{6} + 2 T^{7} + T^{8}$$
$67$ $$( -101 - 238 T - 82 T^{2} + T^{3} + T^{4} )^{2}$$
$71$ $$60824401 - 13765235 T + 3366929 T^{2} - 718575 T^{3} + 113726 T^{4} - 11955 T^{5} + 869 T^{6} - 40 T^{7} + T^{8}$$
$73$ $$151321 - 285137 T + 183278 T^{2} + 74121 T^{3} + 21755 T^{4} + 3441 T^{5} + 368 T^{6} + 23 T^{7} + T^{8}$$
$79$ $$4644025 - 226275 T + 156610 T^{2} + 10875 T^{3} + 2981 T^{4} - 483 T^{5} + 154 T^{6} - 17 T^{7} + T^{8}$$
$83$ $$841 + 145 T + 2481 T^{2} + 1745 T^{3} + 2886 T^{4} + 1285 T^{5} + 271 T^{6} + 25 T^{7} + T^{8}$$
$89$ $$( 725 - 400 T - 150 T^{2} + T^{4} )^{2}$$
$97$ $$625 - 3000 T + 5775 T^{2} - 1770 T^{3} + 1591 T^{4} - 738 T^{5} + 179 T^{6} - 12 T^{7} + T^{8}$$