# Properties

 Label 75.2.g.c Level $75$ Weight $2$ Character orbit 75.g Analytic conductor $0.599$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1$$ x^12 + 3*x^10 - 2*x^9 + 34*x^8 - 22*x^7 + 236*x^6 - 179*x^5 + 877*x^4 - 409*x^3 + 96*x^2 - 11*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{8} q^{3} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{5} + \beta_1 - 1) q^{4} + ( - \beta_{11} + \beta_{7} + \beta_{5}) q^{5} + ( - \beta_{6} - \beta_{4} - \beta_{2} - \beta_1) q^{6} + (\beta_{6} - 1) q^{7} + (\beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{8} + \beta_{5} q^{9}+O(q^{10})$$ q - b2 * q^2 + b8 * q^3 + (b11 + b9 - 2*b8 - b5 + b1 - 1) * q^4 + (-b11 + b7 + b5) * q^5 + (-b6 - b4 - b2 - b1) * q^6 + (b6 - 1) * q^7 + (b8 - b7 + 2*b6 - b5 + b4 + 2*b2 + b1) * q^8 + b5 * q^9 $$q - \beta_{2} q^{2} + \beta_{8} q^{3} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{5} + \beta_1 - 1) q^{4} + ( - \beta_{11} + \beta_{7} + \beta_{5}) q^{5} + ( - \beta_{6} - \beta_{4} - \beta_{2} - \beta_1) q^{6} + (\beta_{6} - 1) q^{7} + (\beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{8} + \beta_{5} q^{9} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + \cdots - 2) q^{10}+ \cdots + (2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100})$$ q - b2 * q^2 + b8 * q^3 + (b11 + b9 - 2*b8 - b5 + b1 - 1) * q^4 + (-b11 + b7 + b5) * q^5 + (-b6 - b4 - b2 - b1) * q^6 + (b6 - 1) * q^7 + (b8 - b7 + 2*b6 - b5 + b4 + 2*b2 + b1) * q^8 + b5 * q^9 + (-b11 + b10 + b9 + b8 - 3*b7 - b6 - b5 - 3*b4 - b3 - b2 - b1 - 2) * q^10 + (-b11 + b8 + b3 + b2 - 1) * q^11 + (b11 + b10 - b7 - 3*b5 + b1 - 1) * q^12 + (b11 - b10 - 2*b9 - b6 - b5 + 3*b4 + b3 + b2) * q^13 + (b11 - 2*b10 - b9 - 4*b8 + 5*b7 - b6 + 5*b5 + 2*b4 + b3 + 2*b2 + 4) * q^14 + (-b10 - b8 + b7 + b5) * q^15 + (b11 - b10 - 2*b9 + 4*b7 - b6 + 5*b5 + 2*b4 + b3 - b1 + 4) * q^16 + (-b11 + 2*b10 - 3*b7 + b6 - b4 - 2*b3 - b2) * q^17 - b6 * q^18 + (-b11 + b10 + b9 + 2*b8 + b6 - 2*b5 - b4 - 2*b3 - b2) * q^19 + (-b11 + b10 + 5*b8 - 5*b7 + b6 - 6*b5 - 2*b3 - 2*b1 - 1) * q^20 + (-b8 - b1) * q^21 + (-b11 - b9 + 3*b8 - 2*b6 + b5 - 2*b4 - 3*b1 + 1) * q^22 + (2*b11 - 2*b10 - b9 - 4*b8 + 2*b7 - b6 + 2*b5 + 2*b4 + 4) * q^23 + (b8 - b7 + 2*b6 + b4 + b2) * q^24 + (b10 + b9 - b8 + b7 + b6 - 2*b4 + b3 + b1) * q^25 + (b9 - b8 + b7 + b6 - b4 + b3 - b2 + b1 - 5) * q^26 + (-b8 + b7 + b5 + 1) * q^27 + (-b11 - b9 + 2*b8 - b6 - b4 + 2*b1) * q^28 + (b10 + b8 - b6 - b5 - b4 + b3 + b1 - 1) * q^29 + (-b8 - b7 - b3 + b2 + b1 + 2) * q^30 + (b11 - 2*b10 - b7 + b6 + 2*b4 + 2*b3 + 3*b2 + b1) * q^31 + (2*b11 - b10 - 3*b8 + 3*b7 - 2*b6 - b4 + 2*b3 - 2*b2 + 2*b1 - 2) * q^32 + (-b10 + b9 - b8 + b6 + b5 + b4 + b2 + b1) * q^33 + (-b11 + b10 + 2*b9 - 3*b7 + b6 + b5 + b4 - b3 - 2*b2 - b1 - 3) * q^34 + (b11 + b9 - 2*b8 - b7 - b6 - b2 - b1 - 1) * q^35 + (-b11 + 2*b10 + b9 + 2*b8 - 3*b7 + b6 - 3*b5 - 2*b4 - b3 - b2 - 2) * q^36 + (-b11 + b10 + 2*b9 - b7 + b6 - 4*b5 - b3 + b1 - 1) * q^37 + (-b11 + b10 + 2*b9 - 3*b7 + b6 - 3*b4 - b3 - 3*b2 - 2*b1 - 3) * q^38 + (-b11 + b8 - b7 - b5 + b3 - b2 - 1) * q^39 + (-b10 - 2*b8 + 5*b7 + 4*b5 + 2*b4 + b3 - 2*b2 - 2*b1) * q^40 + (2*b10 + 2*b9 + b6 - b5 - b3 + 2*b2 + 4*b1) * q^41 + (b11 - b10 - b9 - b8 + 5*b7 + b5 + 3*b4 + 2*b3 + 2*b2 + 2*b1) * q^42 + (-b9 + b8 - b7 + b6 - 2*b4 - b3 - 2*b2 - b1) * q^43 + (5*b8 - b7 - 3*b6 - 5*b5 - 2*b4 - 3*b2 - 2*b1) * q^44 + (b11 - b10 - b9 - b8 + b7 - b6 + b4 + b3 + b2) * q^45 + (b11 + b9 - 3*b8 + 3*b6 - 2*b5 + 3*b4 + 7*b1 - 2) * q^46 + (b11 + b9 + 3*b8 - 3*b5 - b1 - 3) * q^47 + (-b11 - b8 + 5*b7 - b6 + 5*b5 + b3 - b2 - b1 + 1) * q^48 + (-b9 + b8 - b7 - 2*b6 - b3 - b1 - 2) * q^49 + (b11 - 2*b10 - 3*b9 - b8 + 3*b7 + 4*b5 + 2*b4 + 3*b2 - b1 + 10) * q^50 + (-2*b11 + b10 + b6 - 3*b4 - 2*b3 - 2*b2 - 2*b1 + 3) * q^51 + (-2*b10 - b9 - 5*b8 + b7 + b5 + 2*b4 + 2*b3 + 6*b2 + b1 + 5) * q^52 + (b11 + b9 + b8 - 2*b6 + 4*b5 - 2*b4 - b1 + 4) * q^53 + b1 * q^54 + (b10 - 6*b8 + 4*b7 + 2*b6 + 3*b5 - b4 - b2 + 4) * q^55 + (-2*b11 + 3*b10 + b9 + 5*b8 - 10*b7 + b6 - 5*b5 - 5*b4 - 4*b3 - 3*b2 - 3*b1) * q^56 + (-b9 + 2*b8 - 2*b7 - b4 - b3 - b2 - b1 - 2) * q^57 + (-b11 + 2*b10 + 5*b8 - 7*b7 + b6 - 5*b5 - 4*b4 - 2*b3 - b2 - 3*b1) * q^58 + (2*b11 - 2*b10 - 4*b9 - b7 - 2*b6 + b5 + 2*b4 + 2*b3 + b2 - b1 - 1) * q^59 + (-b11 - b9 + 5*b8 - 6*b7 + b6 - b5 + b4 - b3 - b2 - b1 - 1) * q^60 + (2*b11 - 2*b10 - b9 + b8 - b7 - 4*b6 - b5 + 2*b4 - 3*b1 - 1) * q^61 + (-4*b10 - 4*b9 + 9*b7 - 2*b6 + 10*b5 + 5*b4 + 2*b3 + 3*b2 - b1 + 9) * q^62 + (-b5 + b4) * q^63 + (2*b10 + b9 - 6*b8 + 3*b6 - 2*b4 - 2*b3 + 3*b2 + 2*b1 + 6) * q^64 + (b11 + 3*b10 + 4*b8 - 3*b7 - 2*b5 - 2*b4 - b3 - 3*b2 + 2*b1 - 5) * q^65 + (-b11 - b10 + b7 + 4*b5 + 2*b4 + 2*b2 + b1 + 1) * q^66 + (b11 - 3*b10 + b9 - 3*b8 + b7 - 2*b6 + 3*b5 + 3*b4 + 2*b3 + 2*b1) * q^67 + (-2*b11 + b10 + 3*b9 - 8*b8 + 8*b7 + b3 + b2 + b1 - 6) * q^68 + (b11 - 2*b9 + 2*b8 + 2*b7 - 2*b6 - 2*b5 + b4 + 2*b3) * q^69 + (2*b11 - b8 + 4*b7 + 2*b6 - 5*b5 + 6*b4 + 3*b3 + 5*b2 + 6*b1 - 5) * q^70 + (-2*b11 - b10 - 2*b9 + 4*b8 - b6 - 2*b5 - b4 - b3 - 5*b1 - 2) * q^71 + (b6 + b5 + b4 - b1 + 1) * q^72 + (-3*b11 + 4*b10 + 2*b9 + b8 + 2*b7 + b6 + 2*b5 - 4*b4 - b3 - 2*b2 - b1 - 1) * q^73 + (-2*b11 + b10 + 2*b9 + 3*b8 - 3*b7 + 5*b6 + b2 - 6) * q^74 + (b10 + 2*b9 + b6 - b5 - b4 - b3 + b2 - 1) * q^75 + (4*b11 - 2*b10 - 3*b9 - 5*b8 + 5*b7 - 2*b6 + 6*b4 + b3 + 4*b2 + b1 - 1) * q^76 + (2*b10 + b9 + 2*b8 - 4*b7 - b6 - 4*b5 - 2*b4 - 2*b3 - 2*b2 - 2*b1 - 2) * q^77 + (b11 + b9 - 5*b8 - b6 - b5 - b4 - b1 - 1) * q^78 + (-2*b11 - 2*b10 - 2*b9 + b8 - b6 + 3*b5 - b4 - 2*b3 - 3*b1 + 3) * q^79 + (3*b11 - 3*b10 - b9 - 5*b8 + 2*b7 - 3*b6 - b5 + 4*b4 + 4*b3 + 3*b2 + 6*b1 - 8) * q^80 + b7 * q^81 + (-6*b11 + 3*b10 + 2*b9 + 14*b8 - 14*b7 + 5*b6 - 5*b4 - 4*b3 - 2*b2 - 4*b1 - 4) * q^82 + (2*b11 - b10 - 3*b9 + 5*b7 - 2*b6 + 5*b4 + 4*b3 + 2*b2 + 3*b1) * q^83 + (-b11 - b10 + 2*b5 - 3*b4 + b2) * q^84 + (2*b11 - 2*b10 - 4*b9 + 5*b8 - 5*b6 + 3*b5 + 2*b4 + 5*b3 + b2 - b1 + 1) * q^85 + (3*b11 - 2*b10 - b9 - 10*b8 + 4*b7 - 2*b6 + 4*b5 + 2*b4 - b3 - b2 - b1 + 10) * q^86 + (-b11 + b10 + 2*b9 - b7 + b6 - 2*b4 - b3 + b1 - 1) * q^87 + (2*b10 + 2*b9 - 5*b7 + b6 - 8*b5 - 2*b4 - b3 - 2*b2 - 5) * q^88 + (-2*b11 + 4*b10 + 2*b9 - 3*b8 - 5*b7 + 5*b6 - 5*b5 - 4*b4 - 2*b3 - 3*b2 + 3*b1 + 3) * q^89 + (-b9 + 2*b8 + b6 - b5 + b2 + b1 + 1) * q^90 + (-2*b11 + 2*b9 + b7 + b6 + 7*b5 - 5*b4 - b3 - 2*b2 - 2*b1 + 1) * q^91 + (-b11 + 2*b9 - 4*b8 - 9*b7 + 5*b6 + 4*b5 - 2*b3 + 3*b2 + b1) * q^92 + (2*b11 - b10 + b6 + 4*b4 + 2*b3 + 3*b2 + 2*b1 + 1) * q^93 + (2*b11 - 2*b10 - 2*b9 - b8 + 9*b7 - b6 + b5 + 2*b4 + 4*b3 + 3*b2) * q^94 + (-2*b11 - b10 + 3*b9 + 5*b8 - b7 + b6 - 3*b4 - 2*b2 - 1) * q^95 + (b11 + b10 + b9 - 2*b8 - 3*b6 - 3*b5 - 3*b4 + b3 - 3) * q^96 + (b11 - 2*b10 + b9 - 3*b8 - 2*b6 + 2*b5 - 2*b4 - 2*b3 + 2) * q^97 + (-2*b11 + 4*b10 + 2*b9 + 8*b8 - 9*b7 + b6 - 9*b5 - 4*b4 - 2*b3 - 2*b2 - b1 - 8) * q^98 + (2*b11 - b10 - b9 - b8 + b7 + 2*b4 + b3 + b2 + b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{3} - 10 q^{4} - 6 q^{5} - 12 q^{7} + 9 q^{8} - 3 q^{9}+O(q^{10})$$ 12 * q + 3 * q^3 - 10 * q^4 - 6 * q^5 - 12 * q^7 + 9 * q^8 - 3 * q^9 $$12 q + 3 q^{3} - 10 q^{4} - 6 q^{5} - 12 q^{7} + 9 q^{8} - 3 q^{9} - 9 q^{10} - 4 q^{11} - 2 q^{13} + 6 q^{14} - 9 q^{15} + 16 q^{16} - q^{17} + 7 q^{19} + 26 q^{20} - 3 q^{21} + 13 q^{22} + 19 q^{23} + 6 q^{24} + 4 q^{25} - 56 q^{26} + 3 q^{27} + q^{28} - q^{29} + 19 q^{30} + 13 q^{31} - 32 q^{32} - q^{33} - 25 q^{34} - 10 q^{35} + 8 q^{37} - 22 q^{38} + 2 q^{39} - 28 q^{40} + 8 q^{41} - 16 q^{42} - 4 q^{43} + 33 q^{44} - 6 q^{45} - 22 q^{46} - 13 q^{47} - 16 q^{48} - 28 q^{49} + 81 q^{50} + 26 q^{51} + 44 q^{52} + 44 q^{53} + 9 q^{55} + 45 q^{56} - 22 q^{57} + 41 q^{58} - 22 q^{59} + 14 q^{60} - 8 q^{61} + 41 q^{62} + 3 q^{63} + 49 q^{64} - 38 q^{65} - 3 q^{66} - 6 q^{67} - 100 q^{68} + 6 q^{69} - 45 q^{70} - 21 q^{71} + 9 q^{72} - 16 q^{73} - 44 q^{74} - 4 q^{75} - 52 q^{76} + q^{77} - 19 q^{78} + 10 q^{79} - 99 q^{80} - 3 q^{81} + 26 q^{82} - 10 q^{83} - 6 q^{84} + 23 q^{85} + 56 q^{86} - 4 q^{87} - 16 q^{88} + 57 q^{89} + 16 q^{90} - 7 q^{91} + 3 q^{92} + 22 q^{93} - 23 q^{94} + 21 q^{95} - 23 q^{96} + 4 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100})$$ 12 * q + 3 * q^3 - 10 * q^4 - 6 * q^5 - 12 * q^7 + 9 * q^8 - 3 * q^9 - 9 * q^10 - 4 * q^11 - 2 * q^13 + 6 * q^14 - 9 * q^15 + 16 * q^16 - q^17 + 7 * q^19 + 26 * q^20 - 3 * q^21 + 13 * q^22 + 19 * q^23 + 6 * q^24 + 4 * q^25 - 56 * q^26 + 3 * q^27 + q^28 - q^29 + 19 * q^30 + 13 * q^31 - 32 * q^32 - q^33 - 25 * q^34 - 10 * q^35 + 8 * q^37 - 22 * q^38 + 2 * q^39 - 28 * q^40 + 8 * q^41 - 16 * q^42 - 4 * q^43 + 33 * q^44 - 6 * q^45 - 22 * q^46 - 13 * q^47 - 16 * q^48 - 28 * q^49 + 81 * q^50 + 26 * q^51 + 44 * q^52 + 44 * q^53 + 9 * q^55 + 45 * q^56 - 22 * q^57 + 41 * q^58 - 22 * q^59 + 14 * q^60 - 8 * q^61 + 41 * q^62 + 3 * q^63 + 49 * q^64 - 38 * q^65 - 3 * q^66 - 6 * q^67 - 100 * q^68 + 6 * q^69 - 45 * q^70 - 21 * q^71 + 9 * q^72 - 16 * q^73 - 44 * q^74 - 4 * q^75 - 52 * q^76 + q^77 - 19 * q^78 + 10 * q^79 - 99 * q^80 - 3 * q^81 + 26 * q^82 - 10 * q^83 - 6 * q^84 + 23 * q^85 + 56 * q^86 - 4 * q^87 - 16 * q^88 + 57 * q^89 + 16 * q^90 - 7 * q^91 + 3 * q^92 + 22 * q^93 - 23 * q^94 + 21 * q^95 - 23 * q^96 + 4 * q^97 - 18 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 78219134 \nu^{11} + 275969449 \nu^{10} + 322545241 \nu^{9} + 688323383 \nu^{8} + 2382459144 \nu^{7} + 7518729011 \nu^{6} + \cdots + 1397761481 ) / 30308822750$$ (78219134*v^11 + 275969449*v^10 + 322545241*v^9 + 688323383*v^8 + 2382459144*v^7 + 7518729011*v^6 + 15194306845*v^5 + 49826077334*v^4 + 45143977692*v^3 + 198535658181*v^2 - 35416016020*v + 1397761481) / 30308822750 $$\beta_{3}$$ $$=$$ $$( 295076203 \nu^{11} - 2737251577 \nu^{10} + 423072977 \nu^{9} - 8708425024 \nu^{8} + 14064780143 \nu^{7} - 98445654553 \nu^{6} + \cdots - 16004299508 ) / 30308822750$$ (295076203*v^11 - 2737251577*v^10 + 423072977*v^9 - 8708425024*v^8 + 14064780143*v^7 - 98445654553*v^6 + 114346528860*v^5 - 685028180602*v^4 + 636101258149*v^3 - 2418682513518*v^2 + 716195569825*v - 16004299508) / 30308822750 $$\beta_{4}$$ $$=$$ $$( - 444010163 \nu^{11} - 328887363 \nu^{10} - 1432734827 \nu^{9} - 101573726 \nu^{8} - 14512620193 \nu^{7} - 1187789257 \nu^{6} + \cdots - 6770591262 ) / 30308822750$$ (-444010163*v^11 - 328887363*v^10 - 1432734827*v^9 - 101573726*v^8 - 14512620193*v^7 - 1187789257*v^6 - 101054177500*v^5 + 3452612852*v^4 - 350578208249*v^3 - 92883364032*v^2 + 22525707095*v - 6770591262) / 30308822750 $$\beta_{5}$$ $$=$$ $$( - 1397761481 \nu^{11} + 78219134 \nu^{10} - 3917314994 \nu^{9} + 3118068203 \nu^{8} - 46835566971 \nu^{7} + 33133211726 \nu^{6} + \cdots - 20040639729 ) / 30308822750$$ (-1397761481*v^11 + 78219134*v^10 - 3917314994*v^9 + 3118068203*v^8 - 46835566971*v^7 + 33133211726*v^6 - 322352980505*v^5 + 265393611944*v^4 - 1176010741503*v^3 + 616828423421*v^2 + 64350556005*v - 20040639729) / 30308822750 $$\beta_{6}$$ $$=$$ $$( 1818269283 \nu^{11} + 240671723 \nu^{10} + 5444121987 \nu^{9} - 2938441244 \nu^{8} + 61232986233 \nu^{7} - 31823778603 \nu^{6} + \cdots - 3685345568 ) / 30308822750$$ (1818269283*v^11 + 240671723*v^10 + 5444121987*v^9 - 2938441244*v^8 + 61232986233*v^7 - 31823778603*v^6 + 424398953870*v^5 - 268840754862*v^4 + 1546641523369*v^3 - 535050339608*v^2 + 65280664295*v - 3685345568) / 30308822750 $$\beta_{7}$$ $$=$$ $$( 3685345568 \nu^{11} + 1818269283 \nu^{10} + 11296708427 \nu^{9} - 1926569149 \nu^{8} + 122363308068 \nu^{7} - 19844616263 \nu^{6} + \cdots + 24741863047 ) / 30308822750$$ (3685345568*v^11 + 1818269283*v^10 + 11296708427*v^9 - 1926569149*v^8 + 122363308068*v^7 - 19844616263*v^6 + 837917775445*v^5 - 235277902802*v^4 + 2963207308274*v^3 + 39335186057*v^2 - 181257165080*v + 24741863047) / 30308822750 $$\beta_{8}$$ $$=$$ $$( - 6770591262 \nu^{11} + 444010163 \nu^{10} - 19982886423 \nu^{9} + 14973917351 \nu^{8} - 230098529182 \nu^{7} + 163465627957 \nu^{6} + \cdots + 51950796787 ) / 30308822750$$ (-6770591262*v^11 + 444010163*v^10 - 19982886423*v^9 + 14973917351*v^8 - 230098529182*v^7 + 163465627957*v^6 - 1596671748575*v^5 + 1312990013398*v^4 - 5941261149626*v^3 + 3119750034407*v^2 - 557093397120*v + 51950796787) / 30308822750 $$\beta_{9}$$ $$=$$ $$( - 5281629612 \nu^{11} + 669745048 \nu^{10} - 15574701453 \nu^{9} + 12663738036 \nu^{8} - 180032467327 \nu^{7} + 138755057022 \nu^{6} + \cdots + 81498022997 ) / 15154411375$$ (-5281629612*v^11 + 669745048*v^10 - 15574701453*v^9 + 12663738036*v^8 - 180032467327*v^7 + 138755057022*v^6 - 1252252255045*v^5 + 1100339980428*v^4 - 4687952077811*v^3 + 2721179313802*v^2 - 557710769585*v + 81498022997) / 15154411375 $$\beta_{10}$$ $$=$$ $$( 2874546749 \nu^{11} + 341056225 \nu^{10} + 8742298783 \nu^{9} - 4686864056 \nu^{8} + 97434415327 \nu^{7} - 51790314045 \nu^{6} + \cdots - 19681913548 ) / 6061764550$$ (2874546749*v^11 + 341056225*v^10 + 8742298783*v^9 - 4686864056*v^8 + 97434415327*v^7 - 51790314045*v^6 + 674811778168*v^5 - 435400581988*v^4 + 2487936597601*v^3 - 890205653232*v^2 + 234556298341*v - 19681913548) / 6061764550 $$\beta_{11}$$ $$=$$ $$( - 8838097791 \nu^{11} + 252041914 \nu^{10} - 26000680229 \nu^{9} + 18549046073 \nu^{8} - 299493301711 \nu^{7} + 202386506296 \nu^{6} + \cdots + 26628527446 ) / 15154411375$$ (-8838097791*v^11 + 252041914*v^10 - 26000680229*v^9 + 18549046073*v^8 - 299493301711*v^7 + 202386506296*v^6 - 2073953008960*v^5 + 1634346533429*v^4 - 7678264693623*v^3 + 3772098373286*v^2 - 531297349755*v + 26628527446) / 15154411375
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} - \beta_{10} + \beta_{7} + 5\beta_{5} - \beta _1 + 1$$ -b11 - b10 + b7 + 5*b5 - b1 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{2} + \beta_1$$ -b7 + b6 - b5 + 6*b2 + b1 $$\nu^{4}$$ $$=$$ $$6 \beta_{11} - 7 \beta_{10} - 5 \beta_{9} - 10 \beta_{8} + 31 \beta_{7} - 7 \beta_{6} + 10 \beta_{5} + 12 \beta_{4} + 12 \beta_{3} + 5 \beta_{2} + 6 \beta_1$$ 6*b11 - 7*b10 - 5*b9 - 10*b8 + 31*b7 - 7*b6 + 10*b5 + 12*b4 + 12*b3 + 5*b2 + 6*b1 $$\nu^{5}$$ $$=$$ $$- 2 \beta_{11} + \beta_{10} + 11 \beta_{8} - 11 \beta_{7} + 38 \beta_{6} + 9 \beta_{4} - 2 \beta_{3} + 10 \beta_{2} - 2 \beta _1 + 2$$ -2*b11 + b10 + 11*b8 - 11*b7 + 38*b6 + 9*b4 - 2*b3 + 10*b2 - 2*b1 + 2 $$\nu^{6}$$ $$=$$ $$37 \beta_{11} + 11 \beta_{10} + 37 \beta_{9} - 116 \beta_{8} - 12 \beta_{6} - 82 \beta_{5} - 12 \beta_{4} + 11 \beta_{3} + 50 \beta _1 - 82$$ 37*b11 + 11*b10 + 37*b9 - 116*b8 - 12*b6 - 82*b5 - 12*b4 + 11*b3 + 50*b1 - 82 $$\nu^{7}$$ $$=$$ $$- 2 \beta_{11} - 26 \beta_{10} - 24 \beta_{9} + 99 \beta_{7} - 12 \beta_{6} + 130 \beta_{5} + 166 \beta_{4} + 12 \beta_{3} - 81 \beta_{2} - 107 \beta _1 + 99$$ -2*b11 - 26*b10 - 24*b9 + 99*b7 - 12*b6 + 130*b5 + 166*b4 + 12*b3 - 81*b2 - 107*b1 + 99 $$\nu^{8}$$ $$=$$ $$- 142 \beta_{11} + 470 \beta_{10} + 235 \beta_{9} + 673 \beta_{8} - 1294 \beta_{7} + 346 \beta_{6} - 1294 \beta_{5} - 470 \beta_{4} - 328 \beta_{3} - 91 \beta_{2} + 111 \beta _1 - 673$$ -142*b11 + 470*b10 + 235*b9 + 673*b8 - 1294*b7 + 346*b6 - 1294*b5 - 470*b4 - 328*b3 - 91*b2 + 111*b1 - 673 $$\nu^{9}$$ $$=$$ $$144 \beta_{11} - 255 \beta_{10} - 33 \beta_{9} - 823 \beta_{8} + 1159 \beta_{7} - 1673 \beta_{6} + 823 \beta_{5} - 527 \beta_{4} + 288 \beta_{3} - 1385 \beta_{2} - 671 \beta_1$$ 144*b11 - 255*b10 - 33*b9 - 823*b8 + 1159*b7 - 1673*b6 + 823*b5 - 527*b4 + 288*b3 - 1385*b2 - 671*b1 $$\nu^{10}$$ $$=$$ $$- 1428 \beta_{11} + 714 \beta_{10} - 815 \beta_{9} + 4529 \beta_{8} - 4529 \beta_{7} + 2017 \beta_{6} - 494 \beta_{4} - 2243 \beta_{3} + 220 \beta_{2} - 2243 \beta _1 + 4085$$ -1428*b11 + 714*b10 - 815*b9 + 4529*b8 - 4529*b7 + 2017*b6 - 494*b4 - 2243*b3 + 220*b2 - 2243*b1 + 4085 $$\nu^{11}$$ $$=$$ $$1303 \beta_{11} + 934 \beta_{10} + 1303 \beta_{9} - 3124 \beta_{8} - 5243 \beta_{6} - 6568 \beta_{5} - 5243 \beta_{4} + 934 \beta_{3} + 7137 \beta _1 - 6568$$ 1303*b11 + 934*b10 + 1303*b9 - 3124*b8 - 5243*b6 - 6568*b5 - 5243*b4 + 934*b3 + 7137*b1 - 6568

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\beta_{8}$$

## 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
 −0.667650 − 2.05481i 0.0437845 + 0.134755i 0.623865 + 1.92006i −2.17386 + 1.57940i 0.199632 − 0.145041i 1.97423 − 1.43436i −2.17386 − 1.57940i 0.199632 + 0.145041i 1.97423 + 1.43436i −0.667650 + 2.05481i 0.0437845 − 0.134755i 0.623865 − 1.92006i
−1.74793 1.26995i −0.309017 0.951057i 0.824463 + 2.53744i −0.460159 2.18821i −0.667650 + 2.05481i −3.16056 0.446002 1.37265i −0.809017 + 0.587785i −1.97458 + 4.40921i
16.2 0.114629 + 0.0832830i −0.309017 0.951057i −0.611830 1.88302i 2.14898 + 0.617963i 0.0437845 0.134755i −0.858311 0.174259 0.536314i −0.809017 + 0.587785i 0.194870 + 0.249810i
16.3 1.63330 + 1.18666i −0.309017 0.951057i 0.641469 + 1.97424i −2.07079 + 0.843702i 0.623865 1.92006i 1.01887 −0.0473123 + 0.145612i −0.809017 + 0.587785i −4.38341 1.07931i
31.1 −0.830342 2.55553i 0.809017 0.587785i −4.22323 + 3.06835i −1.34843 1.78375i −2.17386 1.57940i 1.68704 7.00026 + 5.08599i 0.309017 0.951057i −3.43876 + 4.92706i
31.2 0.0762527 + 0.234682i 0.809017 0.587785i 1.56877 1.13978i −2.09387 + 0.784664i 0.199632 + 0.145041i −1.24676 0.786373 + 0.571334i 0.309017 0.951057i −0.343810 0.431561i
31.3 0.754089 + 2.32085i 0.809017 0.587785i −3.19965 + 2.32468i 0.824264 2.07860i 1.97423 + 1.43436i −3.44028 −3.85959 2.80415i 0.309017 0.951057i 5.44569 + 0.345540i
46.1 −0.830342 + 2.55553i 0.809017 + 0.587785i −4.22323 3.06835i −1.34843 + 1.78375i −2.17386 + 1.57940i 1.68704 7.00026 5.08599i 0.309017 + 0.951057i −3.43876 4.92706i
46.2 0.0762527 0.234682i 0.809017 + 0.587785i 1.56877 + 1.13978i −2.09387 0.784664i 0.199632 0.145041i −1.24676 0.786373 0.571334i 0.309017 + 0.951057i −0.343810 + 0.431561i
46.3 0.754089 2.32085i 0.809017 + 0.587785i −3.19965 2.32468i 0.824264 + 2.07860i 1.97423 1.43436i −3.44028 −3.85959 + 2.80415i 0.309017 + 0.951057i 5.44569 0.345540i
61.1 −1.74793 + 1.26995i −0.309017 + 0.951057i 0.824463 2.53744i −0.460159 + 2.18821i −0.667650 2.05481i −3.16056 0.446002 + 1.37265i −0.809017 0.587785i −1.97458 4.40921i
61.2 0.114629 0.0832830i −0.309017 + 0.951057i −0.611830 + 1.88302i 2.14898 0.617963i 0.0437845 + 0.134755i −0.858311 0.174259 + 0.536314i −0.809017 0.587785i 0.194870 0.249810i
61.3 1.63330 1.18666i −0.309017 + 0.951057i 0.641469 1.97424i −2.07079 0.843702i 0.623865 + 1.92006i 1.01887 −0.0473123 0.145612i −0.809017 0.587785i −4.38341 + 1.07931i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.3 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.c 12
3.b odd 2 1 225.2.h.d 12
5.b even 2 1 375.2.g.c 12
5.c odd 4 2 375.2.i.d 24
25.d even 5 1 inner 75.2.g.c 12
25.d even 5 1 1875.2.a.j 6
25.e even 10 1 375.2.g.c 12
25.e even 10 1 1875.2.a.k 6
25.f odd 20 2 375.2.i.d 24
25.f odd 20 2 1875.2.b.f 12
75.h odd 10 1 5625.2.a.q 6
75.j odd 10 1 225.2.h.d 12
75.j odd 10 1 5625.2.a.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 1.a even 1 1 trivial
75.2.g.c 12 25.d even 5 1 inner
225.2.h.d 12 3.b odd 2 1
225.2.h.d 12 75.j odd 10 1
375.2.g.c 12 5.b even 2 1
375.2.g.c 12 25.e even 10 1
375.2.i.d 24 5.c odd 4 2
375.2.i.d 24 25.f odd 20 2
1875.2.a.j 6 25.d even 5 1
1875.2.a.k 6 25.e even 10 1
1875.2.b.f 12 25.f odd 20 2
5625.2.a.p 6 75.j odd 10 1
5625.2.a.q 6 75.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8 T_{2}^{10} - 3 T_{2}^{9} + 34 T_{2}^{8} - 8 T_{2}^{7} + 91 T_{2}^{6} - 96 T_{2}^{5} + 852 T_{2}^{4} - 321 T_{2}^{3} + 96 T_{2}^{2} - 14 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 8 T^{10} - 3 T^{9} + 34 T^{8} + \cdots + 1$$
$3$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{3}$$
$5$ $$T^{12} + 6 T^{11} + 16 T^{10} + \cdots + 15625$$
$7$ $$(T^{6} + 6 T^{5} + 4 T^{4} - 25 T^{3} + \cdots + 20)^{2}$$
$11$ $$T^{12} + 4 T^{11} + 11 T^{10} + \cdots + 59536$$
$13$ $$T^{12} + 2 T^{11} + 19 T^{10} + \cdots + 10201$$
$17$ $$T^{12} + T^{11} + 29 T^{10} + \cdots + 21520321$$
$19$ $$T^{12} - 7 T^{11} + 73 T^{10} + \cdots + 144400$$
$23$ $$T^{12} - 19 T^{11} + 156 T^{10} + \cdots + 4080400$$
$29$ $$T^{12} + T^{11} + 12 T^{10} + \cdots + 4431025$$
$31$ $$T^{12} - 13 T^{11} + 44 T^{10} + \cdots + 8410000$$
$37$ $$T^{12} - 8 T^{11} + 44 T^{10} + \cdots + 36300625$$
$41$ $$T^{12} - 8 T^{11} + \cdots + 6831849025$$
$43$ $$(T^{6} + 2 T^{5} - 91 T^{4} - 174 T^{3} + \cdots - 6284)^{2}$$
$47$ $$T^{12} + 13 T^{11} + 178 T^{10} + \cdots + 5216656$$
$53$ $$T^{12} - 44 T^{11} + \cdots + 1189905025$$
$59$ $$T^{12} + 22 T^{11} + 213 T^{10} + \cdots + 15366400$$
$61$ $$T^{12} + 8 T^{11} + \cdots + 28314456361$$
$67$ $$T^{12} + 6 T^{11} + 29 T^{10} + \cdots + 215619856$$
$71$ $$T^{12} + 21 T^{11} + 259 T^{10} + \cdots + 38416$$
$73$ $$T^{12} + 16 T^{11} + \cdots + 493062025$$
$79$ $$T^{12} - 10 T^{11} - 35 T^{10} + \cdots + 64000000$$
$83$ $$T^{12} + 10 T^{11} + \cdots + 1683953296$$
$89$ $$T^{12} - 57 T^{11} + \cdots + 142170473025$$
$97$ $$T^{12} - 4 T^{11} + 139 T^{10} + \cdots + 5755201$$