# Properties

 Label 63.2.h.b Level $63$ Weight $2$ Character orbit 63.h Analytic conductor $0.503$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,2,Mod(25,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("63.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 63.h (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: $$0.503057532734$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$$ x^10 - 2*x^9 + 9*x^8 - 8*x^7 + 40*x^6 - 36*x^5 + 90*x^4 - 3*x^3 + 36*x^2 - 9*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 2\nu^{9} + 9\nu^{8} - 3\nu^{7} + 95\nu^{6} + 18\nu^{5} + 402\nu^{4} - 87\nu^{3} + 936\nu^{2} + 342\nu + 72 ) / 189$$ (2*v^9 + 9*v^8 - 3*v^7 + 95*v^6 + 18*v^5 + 402*v^4 - 87*v^3 + 936*v^2 + 342*v + 72) / 189 $$\beta_{3}$$ $$=$$ $$( -2\nu^{9} + \nu^{8} - 12\nu^{7} - 8\nu^{6} - 68\nu^{5} - 30\nu^{4} - 123\nu^{3} - 204\nu^{2} - 270\nu - 63 ) / 63$$ (-2*v^9 + v^8 - 12*v^7 - 8*v^6 - 68*v^5 - 30*v^4 - 123*v^3 - 204*v^2 - 270*v - 63) / 63 $$\beta_{4}$$ $$=$$ $$( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639 ) / 567$$ (17*v^9 - 24*v^8 + 159*v^7 - 106*v^6 + 786*v^5 - 417*v^4 + 1893*v^3 - 27*v^2 + 1395*v + 639) / 567 $$\beta_{5}$$ $$=$$ $$( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 ) / 567$$ (16*v^9 - 39*v^8 + 156*v^7 - 176*v^6 + 663*v^5 - 780*v^4 + 1680*v^3 - 351*v^2 + 684*v - 180) / 567 $$\beta_{6}$$ $$=$$ $$( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504 ) / 567$$ (20*v^9 - 24*v^8 + 141*v^7 - 4*v^6 + 624*v^5 - 57*v^4 + 1020*v^3 + 1620*v^2 + 369*v + 504) / 567 $$\beta_{7}$$ $$=$$ $$( 8\nu^{9} - 12\nu^{8} + 69\nu^{7} - 43\nu^{6} + 330\nu^{5} - 219\nu^{4} + 732\nu^{3} - 45\nu^{2} + 477\nu - 306 ) / 189$$ (8*v^9 - 12*v^8 + 69*v^7 - 43*v^6 + 330*v^5 - 219*v^4 + 732*v^3 - 45*v^2 + 477*v - 306) / 189 $$\beta_{8}$$ $$=$$ $$( - 71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234 ) / 567$$ (-71*v^9 + 123*v^8 - 591*v^7 + 403*v^6 - 2604*v^5 + 1794*v^4 - 5214*v^3 - 1458*v^2 - 1476*v - 234) / 567 $$\beta_{9}$$ $$=$$ $$( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 ) / 567$$ (-82*v^9 + 165*v^8 - 732*v^7 + 632*v^6 - 3264*v^5 + 2850*v^4 - 7260*v^3 - 432*v^2 - 2898*v + 720) / 567
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + 3\beta_{6} + \beta_{4} - \beta_{2} - 3$$ b8 + 3*b6 + b4 - b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{9} + 4\beta_{5} - \beta_{3} - 4\beta _1 - 1$$ b9 + 4*b5 - b3 - 4*b1 - 1 $$\nu^{4}$$ $$=$$ $$-5\beta_{8} - 5\beta_{7} - 13\beta_{6} - \beta_{4} - \beta_{3} + 4\beta_{2} - \beta_1$$ -5*b8 - 5*b7 - 13*b6 - b4 - b3 + 4*b2 - b1 $$\nu^{5}$$ $$=$$ $$-7\beta_{9} + \beta_{8} - 2\beta_{7} - 7\beta_{6} - 19\beta_{5} - \beta_{4} + \beta_{2} + 7$$ -7*b9 + b8 - 2*b7 - 7*b6 - 19*b5 - b4 + b2 + 7 $$\nu^{6}$$ $$=$$ $$-10\beta_{9} + 9\beta_{8} + 15\beta_{7} - 10\beta_{5} - 15\beta_{4} + 10\beta_{3} + 9\beta_{2} + 10\beta _1 + 61$$ -10*b9 + 9*b8 + 15*b7 - 10*b5 - 15*b4 + 10*b3 + 9*b2 + 10*b1 + 61 $$\nu^{7}$$ $$=$$ $$11\beta_{8} + 11\beta_{7} + 46\beta_{6} + 19\beta_{4} + 43\beta_{3} + 8\beta_{2} + 94\beta_1$$ 11*b8 + 11*b7 + 46*b6 + 19*b4 + 43*b3 + 8*b2 + 94*b1 $$\nu^{8}$$ $$=$$ $$73\beta_{9} + 56\beta_{8} + 62\beta_{7} + 298\beta_{6} + 76\beta_{5} + 118\beta_{4} - 118\beta_{2} - 298$$ 73*b9 + 56*b8 + 62*b7 + 298*b6 + 76*b5 + 118*b4 - 118*b2 - 298 $$\nu^{9}$$ $$=$$ $$253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta _1 - 295$$ 253*b9 - 135*b8 + 48*b7 + 478*b5 - 48*b4 - 253*b3 - 135*b2 - 478*b1 - 295

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \beta_{6}$$ $$-1 + \beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 1.19343 − 2.06709i 0.920620 − 1.59456i 0.247934 − 0.429435i −0.335166 + 0.580525i −1.02682 + 1.77851i 1.19343 + 2.06709i 0.920620 + 1.59456i 0.247934 + 0.429435i −0.335166 − 0.580525i −1.02682 − 1.77851i
−2.38687 −1.61557 + 0.624446i 3.69714 1.46043 2.52954i 3.85615 1.49047i −0.138560 2.64212i −4.05086 2.22013 2.01767i −3.48586 + 6.03769i
25.2 −1.84124 1.39291 + 1.02946i 1.39017 −0.667377 + 1.15593i −2.56469 1.89549i 1.90267 + 1.83844i 1.12285 0.880416 + 2.86790i 1.22880 2.12835i
25.3 −0.495868 −0.221298 1.71786i −1.75411 1.84629 3.19787i 0.109735 + 0.851830i 0.926641 + 2.47817i 1.86155 −2.90205 + 0.760316i −0.915516 + 1.58572i
25.4 0.670333 1.65263 0.518475i −1.55065 −0.712469 + 1.23403i 1.10781 0.347551i −2.36039 1.19522i −2.38012 2.46237 1.71369i −0.477591 + 0.827212i
25.5 2.05365 −1.70867 0.283604i 2.21746 0.0731228 0.126652i −3.50901 0.582422i −2.33035 + 1.25278i 0.446582 2.83914 + 0.969173i 0.150168 0.260099i
58.1 −2.38687 −1.61557 0.624446i 3.69714 1.46043 + 2.52954i 3.85615 + 1.49047i −0.138560 + 2.64212i −4.05086 2.22013 + 2.01767i −3.48586 6.03769i
58.2 −1.84124 1.39291 1.02946i 1.39017 −0.667377 1.15593i −2.56469 + 1.89549i 1.90267 1.83844i 1.12285 0.880416 2.86790i 1.22880 + 2.12835i
58.3 −0.495868 −0.221298 + 1.71786i −1.75411 1.84629 + 3.19787i 0.109735 0.851830i 0.926641 2.47817i 1.86155 −2.90205 0.760316i −0.915516 1.58572i
58.4 0.670333 1.65263 + 0.518475i −1.55065 −0.712469 1.23403i 1.10781 + 0.347551i −2.36039 + 1.19522i −2.38012 2.46237 + 1.71369i −0.477591 0.827212i
58.5 2.05365 −1.70867 + 0.283604i 2.21746 0.0731228 + 0.126652i −3.50901 + 0.582422i −2.33035 1.25278i 0.446582 2.83914 0.969173i 0.150168 + 0.260099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.h.b yes 10
3.b odd 2 1 189.2.h.b 10
4.b odd 2 1 1008.2.q.i 10
7.b odd 2 1 441.2.h.f 10
7.c even 3 1 63.2.g.b 10
7.c even 3 1 441.2.f.e 10
7.d odd 6 1 441.2.f.f 10
7.d odd 6 1 441.2.g.f 10
9.c even 3 1 63.2.g.b 10
9.c even 3 1 567.2.e.f 10
9.d odd 6 1 189.2.g.b 10
9.d odd 6 1 567.2.e.e 10
12.b even 2 1 3024.2.q.i 10
21.c even 2 1 1323.2.h.f 10
21.g even 6 1 1323.2.f.f 10
21.g even 6 1 1323.2.g.f 10
21.h odd 6 1 189.2.g.b 10
21.h odd 6 1 1323.2.f.e 10
28.g odd 6 1 1008.2.t.i 10
36.f odd 6 1 1008.2.t.i 10
36.h even 6 1 3024.2.t.i 10
63.g even 3 1 441.2.f.e 10
63.g even 3 1 567.2.e.f 10
63.h even 3 1 inner 63.2.h.b yes 10
63.h even 3 1 3969.2.a.z 5
63.i even 6 1 1323.2.h.f 10
63.i even 6 1 3969.2.a.bb 5
63.j odd 6 1 189.2.h.b 10
63.j odd 6 1 3969.2.a.bc 5
63.k odd 6 1 441.2.f.f 10
63.l odd 6 1 441.2.g.f 10
63.n odd 6 1 567.2.e.e 10
63.n odd 6 1 1323.2.f.e 10
63.o even 6 1 1323.2.g.f 10
63.s even 6 1 1323.2.f.f 10
63.t odd 6 1 441.2.h.f 10
63.t odd 6 1 3969.2.a.ba 5
84.n even 6 1 3024.2.t.i 10
252.u odd 6 1 1008.2.q.i 10
252.bb even 6 1 3024.2.q.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 7.c even 3 1
63.2.g.b 10 9.c even 3 1
63.2.h.b yes 10 1.a even 1 1 trivial
63.2.h.b yes 10 63.h even 3 1 inner
189.2.g.b 10 9.d odd 6 1
189.2.g.b 10 21.h odd 6 1
189.2.h.b 10 3.b odd 2 1
189.2.h.b 10 63.j odd 6 1
441.2.f.e 10 7.c even 3 1
441.2.f.e 10 63.g even 3 1
441.2.f.f 10 7.d odd 6 1
441.2.f.f 10 63.k odd 6 1
441.2.g.f 10 7.d odd 6 1
441.2.g.f 10 63.l odd 6 1
441.2.h.f 10 7.b odd 2 1
441.2.h.f 10 63.t odd 6 1
567.2.e.e 10 9.d odd 6 1
567.2.e.e 10 63.n odd 6 1
567.2.e.f 10 9.c even 3 1
567.2.e.f 10 63.g even 3 1
1008.2.q.i 10 4.b odd 2 1
1008.2.q.i 10 252.u odd 6 1
1008.2.t.i 10 28.g odd 6 1
1008.2.t.i 10 36.f odd 6 1
1323.2.f.e 10 21.h odd 6 1
1323.2.f.e 10 63.n odd 6 1
1323.2.f.f 10 21.g even 6 1
1323.2.f.f 10 63.s even 6 1
1323.2.g.f 10 21.g even 6 1
1323.2.g.f 10 63.o even 6 1
1323.2.h.f 10 21.c even 2 1
1323.2.h.f 10 63.i even 6 1
3024.2.q.i 10 12.b even 2 1
3024.2.q.i 10 252.bb even 6 1
3024.2.t.i 10 36.h even 6 1
3024.2.t.i 10 84.n even 6 1
3969.2.a.z 5 63.h even 3 1
3969.2.a.ba 5 63.t odd 6 1
3969.2.a.bb 5 63.i even 6 1
3969.2.a.bc 5 63.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 9T_{2}^{2} + 3T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{5} + 2 T^{4} - 5 T^{3} - 9 T^{2} + 3 T + 3)^{2}$$
$3$ $$T^{10} + T^{9} - 5 T^{8} - 3 T^{7} + \cdots + 243$$
$5$ $$T^{10} - 4 T^{9} + 21 T^{8} - 16 T^{7} + \cdots + 9$$
$7$ $$T^{10} + 4 T^{9} + 12 T^{8} + \cdots + 16807$$
$11$ $$T^{10} - 4 T^{9} + 24 T^{8} + 2 T^{7} + \cdots + 225$$
$13$ $$T^{10} + 8 T^{9} + 51 T^{8} + 130 T^{7} + \cdots + 25$$
$17$ $$T^{10} - 12 T^{9} + 99 T^{8} - 420 T^{7} + \cdots + 81$$
$19$ $$T^{10} - T^{9} + 42 T^{8} + \cdots + 185761$$
$23$ $$T^{10} - 3 T^{9} + 72 T^{8} + \cdots + 2595321$$
$29$ $$T^{10} - 7 T^{9} + 69 T^{8} - 190 T^{7} + \cdots + 81$$
$31$ $$(T^{5} - 3 T^{4} - 21 T^{3} + 64 T^{2} + \cdots - 285)^{2}$$
$37$ $$T^{10} + 96 T^{8} + 560 T^{7} + \cdots + 82944$$
$41$ $$T^{10} - 5 T^{9} + 69 T^{8} + \cdots + 2025$$
$43$ $$T^{10} + 7 T^{9} + 138 T^{8} + \cdots + 687241$$
$47$ $$(T^{5} + 27 T^{4} + 213 T^{3} + 93 T^{2} + \cdots - 6615)^{2}$$
$53$ $$T^{10} + 21 T^{9} + 306 T^{8} + \cdots + 178929$$
$59$ $$(T^{5} + 30 T^{4} + 306 T^{3} + 1113 T^{2} + \cdots - 5625)^{2}$$
$61$ $$(T^{5} - 14 T^{4} + 34 T^{3} + 7 T^{2} + \cdots + 1)^{2}$$
$67$ $$(T^{5} - 2 T^{4} - 203 T^{3} + 340 T^{2} + \cdots - 7121)^{2}$$
$71$ $$(T^{5} + 3 T^{4} - 168 T^{3} - 567 T^{2} + \cdots - 81)^{2}$$
$73$ $$T^{10} - 15 T^{9} + 231 T^{8} + \cdots + 772641$$
$79$ $$(T^{5} - 4 T^{4} - 95 T^{3} - 224 T^{2} + \cdots + 193)^{2}$$
$83$ $$T^{10} - 9 T^{9} + 267 T^{8} + \cdots + 218123361$$
$89$ $$T^{10} - 28 T^{9} + 549 T^{8} + \cdots + 7080921$$
$97$ $$T^{10} + 12 T^{9} + \cdots + 2307745521$$