# Properties

 Label 225.2.e.e Level $225$ Weight $2$ Character orbit 225.e Analytic conductor $1.797$ 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] = [225,2,Mod(76,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("225.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.e (of order $$3$$, degree $$2$$, 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.1223810289.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9$$ x^8 - 2*x^7 + 8*x^6 - 2*x^5 + 23*x^4 - 8*x^3 + 37*x^2 + 15*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -5\nu^{7} - 67\nu^{6} + 79\nu^{5} - 664\nu^{4} + 181\nu^{3} - 2269\nu^{2} + 54\nu - 2367 ) / 1233$$ (-5*v^7 - 67*v^6 + 79*v^5 - 664*v^4 + 181*v^3 - 2269*v^2 + 54*v - 2367) / 1233 $$\beta_{3}$$ $$=$$ $$( -76\nu^{7} - 32\nu^{6} - 361\nu^{5} - 722\nu^{4} - 3496\nu^{3} - 1691\nu^{2} - 741\nu - 2934 ) / 6165$$ (-76*v^7 - 32*v^6 - 361*v^5 - 722*v^4 - 3496*v^3 - 1691*v^2 - 741*v - 2934) / 6165 $$\beta_{4}$$ $$=$$ $$( 108\nu^{7} - 279\nu^{6} + 1198\nu^{5} - 1029\nu^{4} + 3598\nu^{3} - 1707\nu^{2} + 5848\nu - 1563 ) / 2055$$ (108*v^7 - 279*v^6 + 1198*v^5 - 1029*v^4 + 3598*v^3 - 1707*v^2 + 5848*v - 1563) / 2055 $$\beta_{5}$$ $$=$$ $$( -326\nu^{7} + 728\nu^{6} - 2576\nu^{5} + 1013\nu^{4} - 6776\nu^{3} + 6104\nu^{2} - 10371\nu + 2016 ) / 6165$$ (-326*v^7 + 728*v^6 - 2576*v^5 + 1013*v^4 - 6776*v^3 + 6104*v^2 - 10371*v + 2016) / 6165 $$\beta_{6}$$ $$=$$ $$( -407\nu^{7} + 1451\nu^{6} - 4502\nu^{5} + 5381\nu^{4} - 10502\nu^{3} + 14063\nu^{2} - 18867\nu + 1647 ) / 6165$$ (-407*v^7 + 1451*v^6 - 4502*v^5 + 5381*v^4 - 10502*v^3 + 14063*v^2 - 18867*v + 1647) / 6165 $$\beta_{7}$$ $$=$$ $$( -33\nu^{7} + 51\nu^{6} - 191\nu^{5} - 108\nu^{4} - 422\nu^{3} - 15\nu^{2} - 356\nu - 744 ) / 411$$ (-33*v^7 + 51*v^6 - 191*v^5 - 108*v^4 - 422*v^3 - 15*v^2 - 356*v - 744) / 411
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} + 3\beta_{5} - \beta_{3} + \beta _1 - 3$$ -b7 - b6 + 3*b5 - b3 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2\beta_{6} - 2\beta_{4} - 4\beta_{3} - \beta_{2} - 3$$ b7 - 2*b6 - 2*b4 - 4*b3 - b2 - 3 $$\nu^{4}$$ $$=$$ $$8\beta_{7} - 2\beta_{6} - 15\beta_{5} - 6\beta_{4} - 8\beta_{2} - 7\beta_1$$ 8*b7 - 2*b6 - 15*b5 - 6*b4 - 8*b2 - 7*b1 $$\nu^{5}$$ $$=$$ $$11\beta_{7} + 11\beta_{6} - 27\beta_{5} + 8\beta_{4} + 22\beta_{3} - 8\beta_{2} - 22\beta _1 + 27$$ 11*b7 + 11*b6 - 27*b5 + 8*b4 + 22*b3 - 8*b2 - 22*b1 + 27 $$\nu^{6}$$ $$=$$ $$-19\beta_{7} + 57\beta_{6} + 57\beta_{4} + 49\beta_{3} + 38\beta_{2} + 90$$ -19*b7 + 57*b6 + 57*b4 + 49*b3 + 38*b2 + 90 $$\nu^{7}$$ $$=$$ $$-144\beta_{7} + 57\beta_{6} + 204\beta_{5} + 87\beta_{4} + 144\beta_{2} + 139\beta_1$$ -144*b7 + 57*b6 + 204*b5 + 87*b4 + 144*b2 + 139*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \beta_{5}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.816862 − 1.41485i −0.236627 − 0.409850i 0.736627 + 1.27588i 1.31686 + 2.28087i −0.816862 + 1.41485i −0.236627 + 0.409850i 0.736627 − 1.27588i 1.31686 − 2.28087i
−0.816862 1.41485i −1.06419 + 1.36657i −0.334526 + 0.579416i 0 2.80278 + 0.389365i −0.252674 0.437645i −2.17440 −0.735010 2.90857i 0
76.2 −0.236627 0.409850i 0.544899 1.64411i 0.888015 1.53809i 0 −0.802776 + 0.165713i −1.28153 2.21967i −1.78702 −2.40617 1.79175i 0
76.3 0.736627 + 1.27588i −1.69629 0.350156i −0.0852394 + 0.147639i 0 −0.802776 2.42219i 1.93291 + 3.34791i 2.69535 2.75478 + 1.18793i 0
76.4 1.31686 + 2.28087i 1.71558 0.238330i −2.46825 + 4.27513i 0 2.80278 + 3.59916i −0.898714 1.55662i −7.73393 2.88640 0.817746i 0
151.1 −0.816862 + 1.41485i −1.06419 1.36657i −0.334526 0.579416i 0 2.80278 0.389365i −0.252674 + 0.437645i −2.17440 −0.735010 + 2.90857i 0
151.2 −0.236627 + 0.409850i 0.544899 + 1.64411i 0.888015 + 1.53809i 0 −0.802776 0.165713i −1.28153 + 2.21967i −1.78702 −2.40617 + 1.79175i 0
151.3 0.736627 1.27588i −1.69629 + 0.350156i −0.0852394 0.147639i 0 −0.802776 + 2.42219i 1.93291 3.34791i 2.69535 2.75478 1.18793i 0
151.4 1.31686 2.28087i 1.71558 + 0.238330i −2.46825 4.27513i 0 2.80278 3.59916i −0.898714 + 1.55662i −7.73393 2.88640 + 0.817746i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.e yes 8
3.b odd 2 1 675.2.e.c 8
5.b even 2 1 225.2.e.c 8
5.c odd 4 2 225.2.k.c 16
9.c even 3 1 inner 225.2.e.e yes 8
9.c even 3 1 2025.2.a.q 4
9.d odd 6 1 675.2.e.c 8
9.d odd 6 1 2025.2.a.z 4
15.d odd 2 1 675.2.e.e 8
15.e even 4 2 675.2.k.c 16
45.h odd 6 1 675.2.e.e 8
45.h odd 6 1 2025.2.a.p 4
45.j even 6 1 225.2.e.c 8
45.j even 6 1 2025.2.a.y 4
45.k odd 12 2 225.2.k.c 16
45.k odd 12 2 2025.2.b.n 8
45.l even 12 2 675.2.k.c 16
45.l even 12 2 2025.2.b.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 5.b even 2 1
225.2.e.c 8 45.j even 6 1
225.2.e.e yes 8 1.a even 1 1 trivial
225.2.e.e yes 8 9.c even 3 1 inner
225.2.k.c 16 5.c odd 4 2
225.2.k.c 16 45.k odd 12 2
675.2.e.c 8 3.b odd 2 1
675.2.e.c 8 9.d odd 6 1
675.2.e.e 8 15.d odd 2 1
675.2.e.e 8 45.h odd 6 1
675.2.k.c 16 15.e even 4 2
675.2.k.c 16 45.l even 12 2
2025.2.a.p 4 45.h odd 6 1
2025.2.a.q 4 9.c even 3 1
2025.2.a.y 4 45.j even 6 1
2025.2.a.z 4 9.d odd 6 1
2025.2.b.n 8 45.k odd 12 2
2025.2.b.o 8 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 2T_{2}^{7} + 8T_{2}^{6} - 2T_{2}^{5} + 23T_{2}^{4} - 8T_{2}^{3} + 37T_{2}^{2} + 15T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + 8 T^{6} - 2 T^{5} + \cdots + 9$$
$3$ $$T^{8} + T^{7} - 2 T^{6} - 3 T^{5} + \cdots + 81$$
$5$ $$T^{8}$$
$7$ $$T^{8} + T^{7} + 13 T^{6} + 36 T^{5} + \cdots + 81$$
$11$ $$T^{8} - T^{7} + 26 T^{6} + 107 T^{5} + \cdots + 81$$
$13$ $$T^{8} - 2 T^{7} + 34 T^{6} + \cdots + 11449$$
$17$ $$(T^{4} + 11 T^{3} + 20 T^{2} - 110 T - 303)^{2}$$
$19$ $$(T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25)^{2}$$
$23$ $$T^{8} - 15 T^{7} + 168 T^{6} + \cdots + 59049$$
$29$ $$T^{8} + T^{7} + 41 T^{6} + 244 T^{5} + \cdots + 16641$$
$31$ $$T^{8} - 4 T^{7} + 58 T^{6} + \cdots + 59049$$
$37$ $$(T^{4} - T^{3} - 99 T^{2} + 503 T - 647)^{2}$$
$41$ $$T^{8} - 5 T^{7} + 50 T^{6} + \cdots + 42849$$
$43$ $$T^{8} + 10 T^{7} + 148 T^{6} + \cdots + 452929$$
$47$ $$T^{8} - 20 T^{7} + 293 T^{6} + \cdots + 145161$$
$53$ $$(T^{4} + 20 T^{3} + 86 T^{2} - 179 T - 471)^{2}$$
$59$ $$T^{8} + 17 T^{7} + 287 T^{6} + \cdots + 5349969$$
$61$ $$T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1$$
$67$ $$T^{8} - 17 T^{7} + 253 T^{6} + \cdots + 59049$$
$71$ $$(T^{4} + 8 T^{3} - 40 T^{2} - 263 T + 381)^{2}$$
$73$ $$(T^{4} + 2 T^{3} - 96 T^{2} + 241 T + 113)^{2}$$
$79$ $$T^{8} - 7 T^{7} + 82 T^{6} + \cdots + 42849$$
$83$ $$T^{8} - 30 T^{7} + 612 T^{6} + \cdots + 531441$$
$89$ $$(T^{4} + 9 T^{3} - 99 T^{2} - 405 T + 2025)^{2}$$
$97$ $$T^{8} + 19 T^{7} + 280 T^{6} + \cdots + 908209$$