# Properties

 Label 75.2.g.b Level $75$ Weight $2$ Character orbit 75.g Analytic conductor $0.599$ 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] = [75,2,Mod(16,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("75.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.26265625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ x^8 - 3*x^7 + 2*x^6 + x^4 + 8*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - \beta_{3}) q^{2} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{8} - \beta_{6} q^{9}+O(q^{10})$$ q + (-b7 - b3) * q^2 + (b6 + b3 + b1 - 1) * q^3 + (b6 - b4 + 2*b3 + 2*b1 - 1) * q^4 + (-b6 - b4 - b1) * q^5 + (b5 - b1) * q^6 + (b7 - 2*b6 - b3 - b2 + 1) * q^7 + (b5 - b3 - b2 - 5*b1 + 1) * q^8 - b6 * q^9 $$q + ( - \beta_{7} - \beta_{3}) q^{2} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{8} - \beta_{6} q^{9} + (\beta_{7} + \beta_{4} - 4 \beta_1 + 1) q^{10} + (3 \beta_{6} + \beta_{4} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{12} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{13} + ( - \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - \beta_1) q^{14} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{15} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 2) q^{16} - \beta_{5} q^{17} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{18} + (2 \beta_{5} + \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{20} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{22} + (\beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{24} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{25} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{26} - \beta_{3} q^{27} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{28} + ( - 4 \beta_{6} + \beta_{4} - 5 \beta_{3} - 5 \beta_1 + 4) q^{29} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{30} + (2 \beta_{3} - \beta_{2} - 5 \beta_1 - 2) q^{31} + (2 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{32}+ \cdots + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q + (-b7 - b3) * q^2 + (b6 + b3 + b1 - 1) * q^3 + (b6 - b4 + 2*b3 + 2*b1 - 1) * q^4 + (-b6 - b4 - b1) * q^5 + (b5 - b1) * q^6 + (b7 - 2*b6 - b3 - b2 + 1) * q^7 + (b5 - b3 - b2 - 5*b1 + 1) * q^8 - b6 * q^9 + (b7 + b4 - 4*b1 + 1) * q^10 + (3*b6 + b4 + 3*b3 + b2 + 3*b1) * q^11 + (-b6 - b5 + b4 + b2 + 2*b1 - 1) * q^12 + (b7 + b6 + b4 + b3 + 2*b1 - 2) * q^13 + (-b6 - 2*b4 - 4*b3 - 2*b2 - b1) * q^14 + (-b5 + b4 - b3 + b2 + b1 + 1) * q^15 + (-2*b7 + b6 - 3*b5 + b4 - 2*b3 + 3*b2 + 5*b1 - 2) * q^16 - b5 * q^17 + (b7 + b3 - b2) * q^18 + (2*b5 + b2 - 2*b1) * q^19 + (-b7 - 3*b6 - 3*b5 + 2*b4 - 2*b3 + 2*b2 + 3*b1) * q^20 + (b6 - b4 - b3 - b1 - 1) * q^21 + (-2*b7 - 2*b6 + 2*b5 + b4 - b3 - b1 + 2) * q^22 + (b7 + 2*b6 + b4 + b3 + b2 + 2*b1) * q^23 + (b7 + b6 + b5 - b4 + 2*b3 - b2 - b1 + 3) * q^24 + (-b6 - b5 + b4 + b3 - b2 + 3*b1 - 3) * q^25 + (b7 - 3*b6 + b5 - b4 - 2*b3 - b2 - b1 + 2) * q^26 - b3 * q^27 + (-b7 + 2*b6 + b5 - 3*b4 - b3 - b1 - 2) * q^28 + (-4*b6 + b4 - 5*b3 - 5*b1 + 4) * q^29 + (b6 - b4 + b3 - b2 + 3) * q^30 + (2*b3 - b2 - 5*b1 - 2) * q^31 + (2*b7 + 7*b6 + 2*b5 - 2*b4 + 9*b3 - 2*b2 - 2*b1 - 1) * q^32 + (3*b3 - b2 + 3*b1 - 3) * q^33 + (3*b6 - b1 + 1) * q^34 + (-2*b7 + 2*b6 + 2*b5 - b4 + 4*b3 + b2 + b1 - 3) * q^35 + (-b7 - b6 - 3*b3 - b1) * q^36 + (-2*b7 - 2*b6 + b5 - 3*b4 - 2*b3 - b2 - 2*b1 + 1) * q^37 + (b7 - 8*b6 - 2*b5 + 3*b4 + b3 + 2*b2 + b1 + 1) * q^38 + (-2*b6 - b4 - b3 - b2 - 2*b1) * q^39 + (5*b7 + 6*b6 + b5 - b4 + 9*b3 - 4*b2 + 2*b1 - 2) * q^40 + (5*b6 + 4*b1 - 4) * q^41 + (-b3 + 2*b2 - 4*b1 + 1) * q^42 + (-b6 + 2*b5 - 2*b4 - b3 - 2*b1 + 5) * q^43 + (-2*b5 + 3*b3 + b2 + 10*b1 - 3) * q^44 + (-b7 + b6 - 1) * q^45 + (-b7 - 5*b6 + b5 - 4*b3 - 4*b1 + 5) * q^46 + (-2*b7 + 2*b5 - b4 - b3 - b1) * q^47 + (-3*b7 - 2*b6 + 2*b4 - 4*b3 + 2*b2 - 2*b1) * q^48 + (-b7 + b6 - 4*b5 + 4*b4 + b2 + 4*b1 + 1) * q^49 + (3*b7 + 5*b6 + 2*b5 - 3*b4 + 4*b3 - 4*b2 + 4*b1 - 3) * q^50 + (-b7 - b3 + b2 + 1) * q^51 - 3*b3 * q^52 + (-2*b7 - b6 + 2*b5 - b4 - 3*b3 - 3*b1 + 1) * q^53 - b4 * q^54 + (-b7 - 3*b6 - 2*b5 - 2*b4 - 7*b3 + 2*b2 - 2*b1 + 6) * q^55 + (2*b5 + 3*b3 - b2 - b1 - 3) * q^56 + (2*b7 - b5 + b4 + 2*b3 - 2*b2 + b1) * q^57 + (-6*b5 + b3 + b2 + 10*b1 - 1) * q^58 + (3*b7 + 3*b6 + 3*b4 + 3*b3 - 3*b1 + 3) * q^59 + (-3*b7 + b5 - 6*b3 + b2 - 2*b1) * q^60 + (-4*b6 - 2*b4 - b3 - 2*b2 - 4*b1) * q^61 + (b7 + 2*b6 - 5*b5 + 6*b4 + b3 + 5*b2 + 8*b1 - 3) * q^62 + (-b6 - b5 + b4 + b2 - b1 + 2) * q^63 + (-2*b7 - 4*b6 + 5*b4 - 10*b3 + 5*b2 - 4*b1) * q^64 + (-2*b7 + 2*b6 + b5 - 2*b3 + 2*b2 + 2*b1 + 3) * q^65 + (2*b7 + 2*b6 + 3*b5 - b4 + 2*b3 - 3*b2 - 3) * q^66 + (-2*b5 - b3 + 1) * q^67 + (-2*b7 - b5 + b4 - 2*b3 + 2*b2 + b1 - 2) * q^68 + (-b5 + 2*b3 - b2 + b1 - 2) * q^69 + (2*b7 - 2*b6 + 2*b5 + 4*b4 + 9*b3 + b2 + b1 - 4) * q^70 + (4*b7 - 2*b6 - 4*b5 + b4 + 4*b3 + 4*b1 + 2) * q^71 + (b7 + 3*b6 - b5 + 5*b3 + 5*b1 - 3) * q^72 + (4*b7 + b6 - 2*b4 - b3 - 2*b2 + b1) * q^73 + (5*b6 + b5 - b4 + 5*b3 - b1 - 7) * q^74 + (-b7 - 3*b6 + 2*b5 - 2*b4 - 5*b3 - 4*b1 + 1) * q^75 + (5*b7 - b6 + 4*b3 - 5*b2 + 8) * q^76 + (5*b7 - 6*b6 - b4 - 5*b3 - b2 - 6*b1) * q^77 + (b7 + 2*b6 - b5 - 2) * q^78 + (-3*b7 + 10*b6 + 3*b5 + 6*b3 + 6*b1 - 10) * q^79 + (-2*b7 - 10*b6 + b5 + 3*b4 - 17*b3 + 2*b2 - 10*b1 + 3) * q^80 - b1 * q^81 + (-b7 + 4*b5 - 4*b4 - b3 + b2 - 4*b1) * q^82 + (-4*b2 - 5*b1) * q^83 + (b7 - 2*b6 - 2*b5 + 3*b4 + b3 + 2*b2 + 2) * q^84 + (b7 + b4 + b1 - 4) * q^85 + (-4*b7 - 6*b6 + b4 - 6*b3 + b2 - 6*b1) * q^86 + (4*b6 + b5 - b4 - b2 - 2*b1 + 1) * q^87 + (4*b5 - 4*b4 - 4*b2 - 15*b1 + 11) * q^88 + (-4*b6 - b4 + 7*b3 - b2 - 4*b1) * q^89 + (3*b6 + 4*b3 + b2 + 4*b1 - 3) * q^90 + (3*b6 - 3*b5 + 3*b4 + 3*b2 + 3) * q^91 + (-2*b5 + b2 + 7*b1) * q^92 + (-2*b6 + b5 - b4 - 2*b3 - b1 + 7) * q^93 + (7*b3 + b2 + 6*b1 - 7) * q^94 + (-b7 - b6 - 3*b4 - 2*b3 - 3*b2 + 8) * q^95 + (2*b7 - b6 - 2*b5 + 8*b3 + 8*b1 + 1) * q^96 + (4*b7 + 5*b6 - 4*b5 + 4*b4 + 7*b3 + 7*b1 - 5) * q^97 + (-b7 + 13*b6 - 3*b4 + 7*b3 - 3*b2 + 13*b1) * q^98 + (-3*b6 + b5 - b4 - 3*b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9}+O(q^{10})$$ 8 * q - q^2 - 2 * q^3 + q^4 - 5 * q^5 - q^6 + 4 * q^7 - 2 * q^9 $$8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9} + 16 q^{11} - 9 q^{12} - 8 q^{13} - 8 q^{14} + 5 q^{15} - 17 q^{16} - q^{17} + 4 q^{18} - 5 q^{19} - 10 q^{20} - 11 q^{21} + 13 q^{22} + 7 q^{23} + 30 q^{24} - 15 q^{25} + 6 q^{26} - 2 q^{27} - 17 q^{28} + 5 q^{29} + 30 q^{30} - 19 q^{31} + 24 q^{32} - 9 q^{33} + 12 q^{34} - 10 q^{35} - 9 q^{36} - q^{37} - 10 q^{38} - 8 q^{39} + 25 q^{40} - 14 q^{41} - 8 q^{42} + 32 q^{43} - 3 q^{44} - 5 q^{45} + 16 q^{46} - q^{47} - 17 q^{48} + 16 q^{49} + 10 q^{50} + 4 q^{51} - 6 q^{52} - 3 q^{53} - q^{54} + 15 q^{55} - 15 q^{56} + 10 q^{57} + 5 q^{58} + 30 q^{59} - 15 q^{60} - 14 q^{61} - 17 q^{62} + 9 q^{63} - 44 q^{64} + 25 q^{65} - 7 q^{66} + 4 q^{67} - 22 q^{68} - 8 q^{69} - 15 q^{70} + 21 q^{71} + 2 q^{73} - 38 q^{74} - 15 q^{75} + 80 q^{76} - 37 q^{77} - 14 q^{78} - 30 q^{79} - 50 q^{80} - 2 q^{81} - 12 q^{82} + 2 q^{83} + 8 q^{84} - 30 q^{85} - 34 q^{86} + 15 q^{87} + 70 q^{88} - 5 q^{90} + 21 q^{91} + 9 q^{92} + 46 q^{93} - 33 q^{94} + 65 q^{95} + 34 q^{96} - 6 q^{97} + 73 q^{98} - 14 q^{99}+O(q^{100})$$ 8 * q - q^2 - 2 * q^3 + q^4 - 5 * q^5 - q^6 + 4 * q^7 - 2 * q^9 + 16 * q^11 - 9 * q^12 - 8 * q^13 - 8 * q^14 + 5 * q^15 - 17 * q^16 - q^17 + 4 * q^18 - 5 * q^19 - 10 * q^20 - 11 * q^21 + 13 * q^22 + 7 * q^23 + 30 * q^24 - 15 * q^25 + 6 * q^26 - 2 * q^27 - 17 * q^28 + 5 * q^29 + 30 * q^30 - 19 * q^31 + 24 * q^32 - 9 * q^33 + 12 * q^34 - 10 * q^35 - 9 * q^36 - q^37 - 10 * q^38 - 8 * q^39 + 25 * q^40 - 14 * q^41 - 8 * q^42 + 32 * q^43 - 3 * q^44 - 5 * q^45 + 16 * q^46 - q^47 - 17 * q^48 + 16 * q^49 + 10 * q^50 + 4 * q^51 - 6 * q^52 - 3 * q^53 - q^54 + 15 * q^55 - 15 * q^56 + 10 * q^57 + 5 * q^58 + 30 * q^59 - 15 * q^60 - 14 * q^61 - 17 * q^62 + 9 * q^63 - 44 * q^64 + 25 * q^65 - 7 * q^66 + 4 * q^67 - 22 * q^68 - 8 * q^69 - 15 * q^70 + 21 * q^71 + 2 * q^73 - 38 * q^74 - 15 * q^75 + 80 * q^76 - 37 * q^77 - 14 * q^78 - 30 * q^79 - 50 * q^80 - 2 * q^81 - 12 * q^82 + 2 * q^83 + 8 * q^84 - 30 * q^85 - 34 * q^86 + 15 * q^87 + 70 * q^88 - 5 * q^90 + 21 * q^91 + 9 * q^92 + 46 * q^93 - 33 * q^94 + 65 * q^95 + 34 * q^96 - 6 * q^97 + 73 * q^98 - 14 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8$$ (v^7 - v^6 + v^3 + 2*v^2 + 4*v - 8) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4$$ (v^7 - v^6 + v^3 - 2*v^2 + 12*v - 12) / 4 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8$$ (-7*v^7 + 9*v^6 + 2*v^5 + 4*v^4 + v^3 - 4*v^2 - 60*v + 64) / 8 $$\beta_{4}$$ $$=$$ $$( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8$$ (11*v^7 - 15*v^6 - 4*v^5 - 8*v^4 + 3*v^3 + 2*v^2 + 92*v - 96) / 8 $$\beta_{5}$$ $$=$$ $$( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8$$ (11*v^7 - 13*v^6 - 6*v^5 - 8*v^4 + 3*v^3 + 4*v^2 + 96*v - 88) / 8 $$\beta_{6}$$ $$=$$ $$( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8$$ (13*v^7 - 21*v^6 - 4*v^5 - 4*v^4 + 5*v^3 + 10*v^2 + 120*v - 144) / 8 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8$$ (19*v^7 - 31*v^6 - 4*v^5 - 8*v^4 + 11*v^3 + 18*v^2 + 180*v - 224) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5$$ (b7 - b6 + b5 - 2*b4 - b3 + b2 - 4*b1 + 4) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 - 4*b4 - 2*b3 - 3*b2 + 2*b1 + 3) / 5 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 + b4 + 8*b3 + 2*b2 + 7*b1 + 3) / 5 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1$$ -b7 + 2*b6 - b4 + b2 + 2*b1 + 1 $$\nu^{5}$$ $$=$$ $$( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5$$ (6*b7 - 11*b6 - 9*b5 + 3*b4 - 11*b3 + 6*b2 + 6*b1 + 19) / 5 $$\nu^{6}$$ $$=$$ $$( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5$$ (2*b7 - 7*b6 + 7*b5 - 9*b4 - 7*b3 + 7*b2 + 12*b1 - 12) / 5 $$\nu^{7}$$ $$=$$ $$( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5$$ (-8*b7 + 3*b6 - 3*b5 + 6*b4 - 7*b3 + 7*b2 + 57*b1 + 3) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1} + \beta_{3} + \beta_{6}$$

## 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.

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
 1.33631 + 0.462894i −0.0272949 − 1.41395i 1.40799 + 0.132563i −1.21700 − 0.720348i 1.40799 − 0.132563i −1.21700 + 0.720348i 1.33631 − 0.462894i −0.0272949 + 1.41395i
−2.18949 1.59076i 0.309017 + 0.951057i 1.64533 + 5.06380i 0.336312 + 2.21063i 0.836312 2.57390i 0.470294 2.78023 8.55667i −0.809017 + 0.587785i 2.78023 5.37515i
16.2 1.38048 + 1.00297i 0.309017 + 0.951057i 0.281722 + 0.867051i −1.02729 1.98612i −0.527295 + 1.62285i −3.94243 0.573870 1.76619i −0.809017 + 0.587785i 0.573870 3.77214i
31.1 −0.346820 1.06740i −0.809017 + 0.587785i 0.598970 0.435177i 0.407987 2.19853i 0.907987 + 0.659691i 1.11373 −2.48822 1.80780i 0.309017 0.951057i −2.48822 + 0.327009i
31.2 0.655837 + 2.01846i −0.809017 + 0.587785i −2.02602 + 1.47199i −2.21700 0.291365i −1.71700 1.24748i 4.35840 −0.865884 0.629102i 0.309017 0.951057i −0.865884 4.66602i
46.1 −0.346820 + 1.06740i −0.809017 0.587785i 0.598970 + 0.435177i 0.407987 + 2.19853i 0.907987 0.659691i 1.11373 −2.48822 + 1.80780i 0.309017 + 0.951057i −2.48822 0.327009i
46.2 0.655837 2.01846i −0.809017 0.587785i −2.02602 1.47199i −2.21700 + 0.291365i −1.71700 + 1.24748i 4.35840 −0.865884 + 0.629102i 0.309017 + 0.951057i −0.865884 + 4.66602i
61.1 −2.18949 + 1.59076i 0.309017 0.951057i 1.64533 5.06380i 0.336312 2.21063i 0.836312 + 2.57390i 0.470294 2.78023 + 8.55667i −0.809017 0.587785i 2.78023 + 5.37515i
61.2 1.38048 1.00297i 0.309017 0.951057i 0.281722 0.867051i −1.02729 + 1.98612i −0.527295 1.62285i −3.94243 0.573870 + 1.76619i −0.809017 0.587785i 0.573870 + 3.77214i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.g.b 8
3.b odd 2 1 225.2.h.c 8
5.b even 2 1 375.2.g.b 8
5.c odd 4 2 375.2.i.b 16
25.d even 5 1 inner 75.2.g.b 8
25.d even 5 1 1875.2.a.h 4
25.e even 10 1 375.2.g.b 8
25.e even 10 1 1875.2.a.e 4
25.f odd 20 2 375.2.i.b 16
25.f odd 20 2 1875.2.b.c 8
75.h odd 10 1 5625.2.a.n 4
75.j odd 10 1 225.2.h.c 8
75.j odd 10 1 5625.2.a.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b 8 1.a even 1 1 trivial
75.2.g.b 8 25.d even 5 1 inner
225.2.h.c 8 3.b odd 2 1
225.2.h.c 8 75.j odd 10 1
375.2.g.b 8 5.b even 2 1
375.2.g.b 8 25.e even 10 1
375.2.i.b 16 5.c odd 4 2
375.2.i.b 16 25.f odd 20 2
1875.2.a.e 4 25.e even 10 1
1875.2.a.h 4 25.d even 5 1
1875.2.b.c 8 25.f odd 20 2
5625.2.a.i 4 75.j odd 10 1
5625.2.a.n 4 75.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + T_{2}^{7} + 2T_{2}^{6} + 3T_{2}^{5} + 25T_{2}^{4} - 43T_{2}^{3} + 82T_{2}^{2} - 11T_{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{7} + 2 T^{6} + 3 T^{5} + \cdots + 121$$
$3$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$5$ $$T^{8} + 5 T^{7} + 20 T^{6} + 65 T^{5} + \cdots + 625$$
$7$ $$(T^{4} - 2 T^{3} - 16 T^{2} + 27 T - 9)^{2}$$
$11$ $$T^{8} - 16 T^{7} + 127 T^{6} + \cdots + 11881$$
$13$ $$T^{8} + 8 T^{7} + 43 T^{6} + 131 T^{5} + \cdots + 81$$
$17$ $$T^{8} + T^{7} + 7 T^{6} + 13 T^{5} + \cdots + 121$$
$19$ $$T^{8} + 5 T^{7} + 5 T^{6} + \cdots + 75625$$
$23$ $$T^{8} - 7 T^{7} + 58 T^{6} - 169 T^{5} + \cdots + 361$$
$29$ $$T^{8} - 5 T^{7} + 70 T^{6} + \cdots + 164025$$
$31$ $$T^{8} + 19 T^{7} + 172 T^{6} + \cdots + 505521$$
$37$ $$T^{8} + T^{7} + 77 T^{6} + 153 T^{5} + \cdots + 1$$
$41$ $$(T^{4} + 7 T^{3} + 69 T^{2} + 143 T + 121)^{2}$$
$43$ $$(T^{4} - 16 T^{3} + 61 T^{2} + 24 T - 99)^{2}$$
$47$ $$T^{8} + T^{7} + 32 T^{6} + 63 T^{5} + \cdots + 96721$$
$53$ $$T^{8} + 3 T^{7} + 43 T^{6} + \cdots + 68121$$
$59$ $$T^{8} - 30 T^{7} + 495 T^{6} + \cdots + 13286025$$
$61$ $$T^{8} + 14 T^{7} + 157 T^{6} + \cdots + 81$$
$67$ $$T^{8} - 4 T^{7} + 37 T^{6} + \cdots + 29241$$
$71$ $$T^{8} - 21 T^{7} + 447 T^{6} + \cdots + 829921$$
$73$ $$T^{8} - 2 T^{7} - 17 T^{6} + \cdots + 23707161$$
$79$ $$T^{8} + 30 T^{7} + 605 T^{6} + \cdots + 96924025$$
$83$ $$T^{8} - 2 T^{7} + 43 T^{6} + \cdots + 958441$$
$89$ $$T^{8} + 255 T^{6} + \cdots + 10923025$$
$97$ $$T^{8} + 6 T^{7} + 7 T^{6} + \cdots + 4414201$$