# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,2,Mod(16,55)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("55.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ x^8 - 3*x^7 + 5*x^6 - 3*x^5 + 4*x^4 + 3*x^3 + 5*x^2 + 3*x + 1 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{9}+O(q^{10})$$ q - b6 * q^2 + (-b6 - b5 - b4 + b3 + b1 - 1) * q^3 + (-b7 + b6 - b2 - b1) * q^4 + (b7 + b4 - b3 + 1) * q^5 + (b6 + b3 - b2 - 1) * q^6 + (b7 - 2*b6 + b4 - b3 + b2 + 2*b1) * q^7 + (b6 + b5 + b4 - 2*b1 + 1) * q^8 + (-2*b7 + 2*b6 - b5 - b4 + b2 - 2) * q^9 $$q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 1) q^{3}+ \cdots + ( - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots - 6) q^{99}+O(q^{100})$$ q - b6 * q^2 + (-b6 - b5 - b4 + b3 + b1 - 1) * q^3 + (-b7 + b6 - b2 - b1) * q^4 + (b7 + b4 - b3 + 1) * q^5 + (b6 + b3 - b2 - 1) * q^6 + (b7 - 2*b6 + b4 - b3 + b2 + 2*b1) * q^7 + (b6 + b5 + b4 - 2*b1 + 1) * q^8 + (-2*b7 + 2*b6 - b5 - b4 + b2 - 2) * q^9 + b5 * q^10 + (2*b7 + b5 + b4 - 2*b3 + 2*b2 - b1 + 1) * q^11 + (b7 + 2*b5 - b3 + 2) * q^12 + (-2*b7 + 3*b6 - 2*b4 - 2) * q^13 + (-2*b4 + b3 - 2*b1 - 2) * q^14 + (-b6 - b4 + b3 + b1) * q^15 + (-b7 + b6 - b5 - b4 - b2 + b1) * q^16 + (2*b7 - b6 + 2*b4 + b3 + b2 - 1) * q^17 + (-2*b7 + b4 - b3) * q^18 + (-b6 - b5 + 5*b3 + 3*b1) * q^19 + (-b6 - b5 + b4 + b2) * q^20 + (-b7 - b5 + b3 - b2 - b1 - 2) * q^21 + (b7 - b6 - b5 - 3*b3 - b2 + 1) * q^22 + (-b2 - b1 + 2) * q^23 + (2*b7 - b6 + b5 - 2*b4 - b2 + 2) * q^24 - b3 * q^25 + (-3*b7 - b6 + b2 + b1) * q^26 + (3*b7 + b6 + b5 + 3*b4 - 4*b3 - b2 - b1 + 4) * q^27 + (4*b7 + b6 + b5 + 4*b4 - 2*b3 - b2 - b1 + 2) * q^28 + (-3*b7 + b6 + 2*b4 - 2*b3 - 2*b2 - b1) * q^29 + (-b6 - b5 - b4 + b3 - 1) * q^30 - 5*b6 * q^31 + (-4*b7 - 2*b5 + 4*b3 + 2*b2 + 2*b1 - 1) * q^32 + (3*b7 - b6 - 2*b4 + b3 - 3*b2 + 2) * q^33 + (b7 - 2*b5 - b3 + 2*b2 + 2*b1) * q^34 + (-b7 + b6 + 2*b5 - b4 - 2*b2 - 1) * q^35 + (-b6 - b5 - 4*b4 + 2*b3 + 3*b1 - 4) * q^36 + (-4*b7 + 3*b6 - 7*b4 + 7*b3 - 3*b2 - 3*b1) * q^37 + (-2*b7 - 6*b6 - 4*b5 - 2*b4 + 3*b3 + 6*b2 + 4*b1 - 3) * q^38 + (2*b7 - b6 + 2*b5 + 2*b4 - 5*b3 + b2 - 2*b1 + 5) * q^39 + (b7 + b6 + b4 - b3 + b2 - b1) * q^40 + (3*b6 + 3*b5 + b4 + b3 - 4*b1 + 1) * q^41 + (b7 + 4*b6 + b5 + 2*b4 - b2 + 1) * q^42 + (-b5 + 2*b2 + 2*b1 - 6) * q^43 + (3*b7 + b6 - b5 + 5*b4 - b3 + b2 + b1) * q^44 + (-2*b7 - 3*b5 + 2*b3 + b2 + b1 - 1) * q^45 + (b7 + b5 + b4 - b2 + 1) * q^46 + (3*b6 + 3*b5 - 3*b3 - 2*b1) * q^47 + (-b7 - b6 + b4 - b3 + 2*b2 + b1) * q^48 + (b6 + 4*b5 - b3 - b2 - 4*b1 + 1) * q^49 + (b6 + b5 - b2 - b1) * q^50 + (-2*b7 + 2*b6 + b4 - b3 + b2 - 2*b1) * q^51 + (-b6 - b5 - 5*b4 + 2*b1 - 5) * q^52 + (b7 + b6 + 5*b5 - 5*b4 - 5*b2 + 1) * q^53 + (4*b5 + 1) * q^54 + (-b7 + 3*b6 + b5 - b4 - b3 - b1) * q^55 + (4*b7 - 2*b5 - 4*b3 + 3*b2 + 3*b1 + 3) * q^56 + (-8*b7 - b6 - 7*b5 + b4 + 7*b2 - 8) * q^57 + (4*b6 + 4*b5 + b4 + b3 - 4*b1 + 1) * q^58 + (7*b7 - 4*b6 + 5*b4 - 5*b3 + b2 + 4*b1) * q^59 + (b7 + 2*b6 + 2*b5 + b4 - 2*b3 - 2*b2 - 2*b1 + 2) * q^60 + (3*b7 - 2*b6 - 4*b5 + 3*b4 - 2*b3 + 2*b2 + 4*b1 + 2) * q^61 + (5*b7 + 5*b6 - 5*b2 - 5*b1) * q^62 + (-2*b6 - 2*b5 + 6*b4 - 3*b3 + b1 + 6) * q^63 + (-4*b7 - 3*b6 - 2*b4 - 4) * q^64 + (-2*b7 - 3*b5 + 2*b3) * q^65 + (b7 + b6 + 2*b5 + 3*b3 - 2*b2 + 3*b1) * q^66 + (-6*b7 + b5 + 6*b3 - 4*b2 - 4*b1 - 5) * q^67 + (4*b7 - 3*b6 - b5 + 4*b4 + b2 + 4) * q^68 + (-b6 - b5 - 2*b4 + 3*b3 + 3*b1 - 2) * q^69 + (-b7 - 2*b4 + 2*b3 + 2*b2) * q^70 + (-b7 + 3*b6 + 3*b5 - b4 + 3*b3 - 3*b2 - 3*b1 - 3) * q^71 + (-6*b7 + b6 - b5 - 6*b4 + 9*b3 - b2 + b1 - 9) * q^72 + (-b7 - 2*b6 + 3*b2 + 2*b1) * q^73 + (-4*b6 - 4*b5 + 3*b4 + 6*b1 + 3) * q^74 + (b7 + b5 - b2 + 1) * q^75 + (2*b7 + b5 - 2*b3 - 7*b2 - 7*b1 + 6) * q^76 + (-4*b7 - 2*b6 - 4*b5 - 6*b4 - b3 + b2 + 3*b1 + 2) * q^77 + (3*b7 + 2*b5 - 3*b3 - 4*b2 - 4*b1 + 2) * q^78 + (6*b7 - 4*b6 - 3*b5 - 3*b4 + 3*b2 + 6) * q^79 + (-2*b6 - 2*b5 + b4 + b1 + 1) * q^80 + (4*b7 - 2*b6 + b4 - b3 + 6*b2 + 2*b1) * q^81 + (b7 - 4*b6 - 4*b5 + b4 - 4*b3 + 4*b2 + 4*b1 + 4) * q^82 + (-7*b7 - b6 - 4*b5 - 7*b4 + b3 + b2 + 4*b1 - 1) * q^83 + (-2*b7 - 3*b6 - 3*b4 + 3*b3 + 3*b2 + 3*b1) * q^84 + (b6 + b5 - 3*b4 + b3 - 3) * q^85 + (-2*b7 + 3*b6 - b5 - b4 + b2 - 2) * q^86 + (4*b7 + 8*b5 - 4*b3 - 3*b2 - 3*b1 + 7) * q^87 + (-2*b7 + b6 + 3*b5 + 2*b4 - 2*b3 + b2 - b1 - 5) * q^88 + (6*b5 - 6*b2 - 6*b1 + 1) * q^89 + (-b7 + 2*b4 - 1) * q^90 + (-2*b6 - 2*b5 + 8*b4 - b3 + 2*b1 + 8) * q^91 + (-4*b7 + b6 - b4 + b3 - b1) * q^92 + (5*b6 + 5*b3 - 5*b2 - 5) * q^93 + (-b7 - b6 - b4 - 2*b3 + b2 + 2) * q^94 + (5*b7 - b6 - 2*b2 + b1) * q^95 + (b6 + b5 + 3*b4 - 9*b3 - 7*b1 + 3) * q^96 + (8*b7 - 2*b5 + 13*b4 + 2*b2 + 8) * q^97 + (3*b7 - 2*b5 - 3*b3 + 4) * q^98 + (-4*b7 - b6 - 5*b5 + b4 + 11*b3 + 3*b2 + b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 5 q^{3} - 2 q^{4} + 2 q^{5} - 7 q^{6} - q^{7} + 4 q^{8} - 5 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 5 * q^3 - 2 * q^4 + 2 * q^5 - 7 * q^6 - q^7 + 4 * q^8 - 5 * q^9 $$8 q - 2 q^{2} - 5 q^{3} - 2 q^{4} + 2 q^{5} - 7 q^{6} - q^{7} + 4 q^{8} - 5 q^{9} + 2 q^{10} + 3 q^{11} + 16 q^{12} - 2 q^{13} - 16 q^{14} + 5 q^{15} + 4 q^{16} - 13 q^{17} + 15 q^{19} - 3 q^{20} - 20 q^{21} - 7 q^{22} + 10 q^{23} + 13 q^{24} - 2 q^{25} + 10 q^{26} + 10 q^{27} - 6 q^{28} - 9 q^{29} - 8 q^{30} - 10 q^{31} + 16 q^{32} + 5 q^{33} + 4 q^{34} - 4 q^{35} - 15 q^{36} + 24 q^{37} + 21 q^{39} - 4 q^{40} + 8 q^{41} + 9 q^{42} - 38 q^{43} - 12 q^{44} + 3 q^{46} + 5 q^{48} + q^{49} - 2 q^{50} + q^{51} - 28 q^{52} + 13 q^{53} + 16 q^{54} + 7 q^{55} + 22 q^{56} - 45 q^{57} + 12 q^{58} - 27 q^{59} + 4 q^{60} + 6 q^{61} - 30 q^{62} + 25 q^{63} - 26 q^{64} + 2 q^{65} + 13 q^{66} - 38 q^{67} + 11 q^{68} - q^{69} + 16 q^{70} - 20 q^{71} - 30 q^{72} + 13 q^{73} + 20 q^{74} + 5 q^{75} + 34 q^{77} - 16 q^{78} + 37 q^{79} + q^{80} + 8 q^{81} + 28 q^{82} + 27 q^{83} + 28 q^{84} - 12 q^{85} - 3 q^{86} + 38 q^{87} - 36 q^{88} - 16 q^{89} - 10 q^{90} + 44 q^{91} + 11 q^{92} - 35 q^{93} + 17 q^{94} - 15 q^{95} - 17 q^{96} + 24 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100})$$ 8 * q - 2 * q^2 - 5 * q^3 - 2 * q^4 + 2 * q^5 - 7 * q^6 - q^7 + 4 * q^8 - 5 * q^9 + 2 * q^10 + 3 * q^11 + 16 * q^12 - 2 * q^13 - 16 * q^14 + 5 * q^15 + 4 * q^16 - 13 * q^17 + 15 * q^19 - 3 * q^20 - 20 * q^21 - 7 * q^22 + 10 * q^23 + 13 * q^24 - 2 * q^25 + 10 * q^26 + 10 * q^27 - 6 * q^28 - 9 * q^29 - 8 * q^30 - 10 * q^31 + 16 * q^32 + 5 * q^33 + 4 * q^34 - 4 * q^35 - 15 * q^36 + 24 * q^37 + 21 * q^39 - 4 * q^40 + 8 * q^41 + 9 * q^42 - 38 * q^43 - 12 * q^44 + 3 * q^46 + 5 * q^48 + q^49 - 2 * q^50 + q^51 - 28 * q^52 + 13 * q^53 + 16 * q^54 + 7 * q^55 + 22 * q^56 - 45 * q^57 + 12 * q^58 - 27 * q^59 + 4 * q^60 + 6 * q^61 - 30 * q^62 + 25 * q^63 - 26 * q^64 + 2 * q^65 + 13 * q^66 - 38 * q^67 + 11 * q^68 - q^69 + 16 * q^70 - 20 * q^71 - 30 * q^72 + 13 * q^73 + 20 * q^74 + 5 * q^75 + 34 * q^77 - 16 * q^78 + 37 * q^79 + q^80 + 8 * q^81 + 28 * q^82 + 27 * q^83 + 28 * q^84 - 12 * q^85 - 3 * q^86 + 38 * q^87 - 36 * q^88 - 16 * q^89 - 10 * q^90 + 44 * q^91 + 11 * q^92 - 35 * q^93 + 17 * q^94 - 15 * q^95 - 17 * q^96 + 24 * q^97 + 16 * q^98 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8$$ (-v^7 + 2*v^6 - 3*v^5 - 4*v^3 - 7*v^2 - 12*v - 7) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8$$ (v^7 - 7*v^5 + 20*v^4 - 16*v^3 + 19*v^2 + 6*v + 9) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8$$ (-v^7 + 4*v^6 - 9*v^5 + 12*v^4 - 16*v^3 + 13*v^2 - 10*v - 1) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8$$ (-3*v^7 + 10*v^6 - 17*v^5 + 8*v^4 - 4*v^3 - 13*v^2 - 8*v - 5) / 8 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8$$ (3*v^7 - 12*v^6 + 23*v^5 - 20*v^4 + 16*v^3 + v^2 + 6*v - 1) / 8 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8$$ (-5*v^7 + 18*v^6 - 35*v^5 + 32*v^4 - 28*v^3 - 11*v^2 - 12*v - 7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ b6 + b5 + b4 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1$$ b7 + 3*b6 + 2*b5 + b4 - 3*b2 - 2*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1$$ 3*b7 + 4*b6 + b4 - b3 - 5*b2 - 4*b1 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1$$ 4*b7 - 6*b5 - 4*b3 - 6*b2 - 6*b1 - 1 $$\nu^{6}$$ $$=$$ $$-16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6$$ -16*b6 - 16*b5 - 6*b4 - 6*b3 - 7*b1 - 6 $$\nu^{7}$$ $$=$$ $$-16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16$$ -16*b7 - 51*b6 - 29*b5 - 23*b4 + 29*b2 - 16

## Character values

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

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$1$$ $$\beta_{7}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 1.69513 + 1.23158i −0.386111 − 0.280526i 0.418926 − 1.28932i −0.227943 + 0.701538i 1.69513 − 1.23158i −0.386111 + 0.280526i 0.418926 + 1.28932i −0.227943 − 0.701538i
−0.647481 1.99274i −1.54765 1.12443i −1.93376 + 1.40496i −0.309017 + 0.951057i −1.23863 + 3.81211i 2.48141 1.80285i 0.661536 + 0.480634i 0.203814 + 0.627276i 2.09529
16.2 0.147481 + 0.453901i −0.261370 0.189896i 1.43376 1.04169i −0.309017 + 0.951057i 0.0476470 0.146642i −2.17239 + 1.57833i 1.45650 + 1.05821i −0.894797 2.75390i −0.477260
26.1 −1.09676 0.796845i 0.177837 0.547326i −0.0501062 0.154211i 0.809017 0.587785i −0.631180 + 0.458579i −1.12773 3.47080i −0.905781 + 2.78771i 2.15911 + 1.56869i −1.35567
26.2 0.596764 + 0.433574i −0.868820 + 2.67395i −0.449894 1.38463i 0.809017 0.587785i −1.67784 + 1.21902i 0.318714 + 0.980901i 0.787747 2.42443i −3.96813 2.88301i 0.737640
31.1 −0.647481 + 1.99274i −1.54765 + 1.12443i −1.93376 1.40496i −0.309017 0.951057i −1.23863 3.81211i 2.48141 + 1.80285i 0.661536 0.480634i 0.203814 0.627276i 2.09529
31.2 0.147481 0.453901i −0.261370 + 0.189896i 1.43376 + 1.04169i −0.309017 0.951057i 0.0476470 + 0.146642i −2.17239 1.57833i 1.45650 1.05821i −0.894797 + 2.75390i −0.477260
36.1 −1.09676 + 0.796845i 0.177837 + 0.547326i −0.0501062 + 0.154211i 0.809017 + 0.587785i −0.631180 0.458579i −1.12773 + 3.47080i −0.905781 2.78771i 2.15911 1.56869i −1.35567
36.2 0.596764 0.433574i −0.868820 2.67395i −0.449894 + 1.38463i 0.809017 + 0.587785i −1.67784 1.21902i 0.318714 0.980901i 0.787747 + 2.42443i −3.96813 + 2.88301i 0.737640
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.2 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.b 8
3.b odd 2 1 495.2.n.e 8
4.b odd 2 1 880.2.bo.h 8
5.b even 2 1 275.2.h.a 8
5.c odd 4 2 275.2.z.a 16
11.b odd 2 1 605.2.g.k 8
11.c even 5 1 inner 55.2.g.b 8
11.c even 5 1 605.2.a.j 4
11.c even 5 2 605.2.g.m 8
11.d odd 10 1 605.2.a.k 4
11.d odd 10 2 605.2.g.e 8
11.d odd 10 1 605.2.g.k 8
33.f even 10 1 5445.2.a.bi 4
33.h odd 10 1 495.2.n.e 8
33.h odd 10 1 5445.2.a.bp 4
44.g even 10 1 9680.2.a.cm 4
44.h odd 10 1 880.2.bo.h 8
44.h odd 10 1 9680.2.a.cn 4
55.h odd 10 1 3025.2.a.w 4
55.j even 10 1 275.2.h.a 8
55.j even 10 1 3025.2.a.bd 4
55.k odd 20 2 275.2.z.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 1.a even 1 1 trivial
55.2.g.b 8 11.c even 5 1 inner
275.2.h.a 8 5.b even 2 1
275.2.h.a 8 55.j even 10 1
275.2.z.a 16 5.c odd 4 2
275.2.z.a 16 55.k odd 20 2
495.2.n.e 8 3.b odd 2 1
495.2.n.e 8 33.h odd 10 1
605.2.a.j 4 11.c even 5 1
605.2.a.k 4 11.d odd 10 1
605.2.g.e 8 11.d odd 10 2
605.2.g.k 8 11.b odd 2 1
605.2.g.k 8 11.d odd 10 1
605.2.g.m 8 11.c even 5 2
880.2.bo.h 8 4.b odd 2 1
880.2.bo.h 8 44.h odd 10 1
3025.2.a.w 4 55.h odd 10 1
3025.2.a.bd 4 55.j even 10 1
5445.2.a.bi 4 33.f even 10 1
5445.2.a.bp 4 33.h odd 10 1
9680.2.a.cm 4 44.g even 10 1
9680.2.a.cn 4 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} + 2T_{2}^{5} - T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} + \cdots + 1$$
$3$ $$T^{8} + 5 T^{7} + \cdots + 1$$
$5$ $$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{2}$$
$7$ $$T^{8} + T^{7} + \cdots + 961$$
$11$ $$T^{8} - 3 T^{7} + \cdots + 14641$$
$13$ $$T^{8} + 2 T^{7} + \cdots + 19321$$
$17$ $$T^{8} + 13 T^{7} + \cdots + 361$$
$19$ $$T^{8} - 15 T^{7} + \cdots + 625$$
$23$ $$(T^{4} - 5 T^{3} + 4 T^{2} + \cdots - 11)^{2}$$
$29$ $$T^{8} + 9 T^{7} + \cdots + 203401$$
$31$ $$T^{8} + 10 T^{7} + \cdots + 390625$$
$37$ $$T^{8} - 24 T^{7} + \cdots + 1324801$$
$41$ $$T^{8} - 8 T^{7} + \cdots + 101761$$
$43$ $$(T^{4} + 19 T^{3} + \cdots + 211)^{2}$$
$47$ $$T^{8} + 23 T^{6} + \cdots + 28561$$
$53$ $$T^{8} - 13 T^{7} + \cdots + 885481$$
$59$ $$T^{8} + 27 T^{7} + \cdots + 687241$$
$61$ $$T^{8} - 6 T^{7} + \cdots + 28561$$
$67$ $$(T^{4} + 19 T^{3} + \cdots - 4079)^{2}$$
$71$ $$T^{8} + 20 T^{7} + \cdots + 17161$$
$73$ $$T^{8} - 13 T^{7} + \cdots + 121$$
$79$ $$T^{8} - 37 T^{7} + \cdots + 45954841$$
$83$ $$T^{8} - 27 T^{7} + \cdots + 2886601$$
$89$ $$(T^{4} + 8 T^{3} + \cdots + 1861)^{2}$$
$97$ $$T^{8} - 24 T^{7} + \cdots + 9066121$$