# Properties

 Label 117.2.t.c Level $117$ Weight $2$ Character orbit 117.t Analytic conductor $0.934$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(25,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("117.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.t (of order $$6$$, 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: $$0.934249703649$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049$$ x^20 - 6*x^16 + 9*x^14 + 54*x^12 + 81*x^10 + 486*x^8 + 729*x^6 - 4374*x^4 + 59049 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{17} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{6} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{19} + \beta_{16} + \beta_{4}) q^{6} + (\beta_{17} - \beta_{15}) q^{7} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{12} - \beta_{7} - \beta_{4}) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{9}+O(q^{10})$$ q - b17 * q^2 + b10 * q^3 + (b11 - b6 + 1) * q^4 - b4 * q^5 + (b19 + b16 + b4) * q^6 + (b17 - b15) * q^7 + (-b19 + b18 + b15 + b12 - b7 - b4) * q^8 + (-b6 + b5 - b3 + b2) * q^9 $$q - \beta_{17} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{6} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{19} + \beta_{16} + \beta_{4}) q^{6} + (\beta_{17} - \beta_{15}) q^{7} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{12} - \beta_{7} - \beta_{4}) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{9} + (\beta_{13} - \beta_{11} - 2 \beta_{10} + \beta_{5}) q^{10} + (\beta_{19} + \beta_{18} - \beta_{9}) q^{11} + (\beta_{14} - \beta_{13} + \beta_{10} + \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{12} + (\beta_{14} - \beta_{12} + \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{14} - \beta_{11} + 2 \beta_{6} + \beta_{2} - 2) q^{14} + ( - \beta_{17} + \beta_{15} + \beta_{8} - \beta_{7} - \beta_{4}) q^{15} + ( - \beta_{14} + \beta_{6} + \beta_{3}) q^{16} + (\beta_{13} - 2 \beta_{10} + \beta_{5} - \beta_{3}) q^{17} + ( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - \beta_{12} + 2 \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{18}+ \cdots + ( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{15} - \beta_{12} - \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{99}+O(q^{100})$$ q - b17 * q^2 + b10 * q^3 + (b11 - b6 + 1) * q^4 - b4 * q^5 + (b19 + b16 + b4) * q^6 + (b17 - b15) * q^7 + (-b19 + b18 + b15 + b12 - b7 - b4) * q^8 + (-b6 + b5 - b3 + b2) * q^9 + (b13 - b11 - 2*b10 + b5) * q^10 + (b19 + b18 - b9) * q^11 + (b14 - b13 + b10 + b6 - b5 - b2 - 1) * q^12 + (b14 - b12 + b4 - b2) * q^13 + (-b14 - b11 + 2*b6 + b2 - 2) * q^14 + (-b17 + b15 + b8 - b7 - b4) * q^15 + (-b14 + b6 + b3) * q^16 + (b13 - 2*b10 + b5 - b3) * q^17 + (-b19 - b18 + b17 - b16 - b15 - b12 + 2*b9 + b7 + 2*b4) * q^18 + (b19 - b18 - b17 - b16 - b9 + b8 + b7 + b4) * q^19 + (-b19 - b18 + b17 - b15 + b9 + 2*b7) * q^20 + (-b19 + b17 - b15 - b8 - b7 - 2*b4) * q^21 + (b14 - b13 - b11 - b10 + b6 - b5 + b3 - b2 - 1) * q^22 + (-b14 + 2*b11 + b10 - b6 - 2*b5 + b2 - b1 + 1) * q^23 + (-b18 - b16 + b15 - b9 - b8 - b7) * q^24 + (-b13 + b10 - 2*b6 + b5 + b1) * q^25 + (b19 - b18 - b17 + b15 - b13 + b12 + b11 + 2*b10 + b8 - b5 - b2 - 1) * q^26 + (-b14 - 2*b11 - b5 + 2*b3 + b2 - 1) * q^27 + (2*b17 - 2*b8 + b7 + b4) * q^28 + (-b14 + b13 + b10 + 2*b6 - 2*b5 + b3) * q^29 + (b14 + b13 - b10 - 2*b6 - b5 + b3 - b2 - b1 - 1) * q^30 + (-b19 + b18 + b16 + b12 - 2*b9 - b4) * q^31 + (-2*b12 - b8 + b4) * q^32 + (-b19 + b18 + b17 + 2*b15 + b12 + b9 - 2*b8 - b7 - b4) * q^33 + (-2*b19 - b18 - b16 + 2*b9 + 3*b7) * q^34 + (-b13 + b11 + b10 - 2*b5 - b2 + b1 - 1) * q^35 + (-b14 - b13 + b11 + 2*b5 + b2 + 2*b1 + 2) * q^36 + (b19 - b18 - 2*b17 + 2*b16 + 2*b9 + 2*b8 + 2*b7 + 2*b4) * q^37 + (3*b14 - 2*b10 + b5 - 3*b3 - b1) * q^38 + (b19 - b15 + b14 - b13 - b12 + b11 - b8 + 2*b7 - 2*b6 + b5 + b4 + b1) * q^39 + (-2*b14 + b13 + b10 + b6 + b5 - b3 + 2*b2 - 1) * q^40 + (b12 + b8 + 2*b4) * q^41 + (-2*b14 + 2*b13 - b11 - 2*b10 + b6 + b5 + b2 - b1 + 1) * q^42 + (-b14 - b13 + b10 + b5 + b1) * q^43 + (b19 - b18 + b17 - b8 - 2*b7 - 2*b4) * q^44 + (b19 - 2*b17 - b16 + b15 - b12 + b8 - b7 + b4) * q^45 + (-2*b19 + 2*b18 - 2*b17 - b16 + b15 + b12 - b9 + 2*b8 - 5*b7 - 5*b4) * q^46 + (b18 + b17 - b16 - 2*b15 - b7) * q^47 + (b14 + b13 - b10 + b6 + b5 - 2*b3 - b2 - b1 + 2) * q^48 + (b13 + 2*b10 + b6 - b5 - b3 - b1 - 1) * q^49 + (b19 + b18 + b16 - b12 + b8 + 4*b4) * q^50 + (b14 - b11 - b10 - 2*b6 - b5 + 2*b3 - 2*b2 - 2) * q^51 + (2*b19 + b18 - b17 + b16 + b14 + b13 - b10 - 2*b9 - 2*b7 - b5 - 2*b3 - b1) * q^52 + (b13 - b10 + 2*b5 - b3 - b1 + 6) * q^53 + (b19 + b18 + 4*b17 + b16 - 3*b15 - b12 - 2*b9 - b8 + b7 - b4) * q^54 + (-2*b11 + 2*b3 + b2 + 1) * q^55 + (2*b14 - b13 + b10 + 2*b6 + b5 + b1) * q^56 + (3*b18 - b17 + b16 + b15 + 3*b12 + b8) * q^57 + (b19 + 2*b18 + 2*b16 + b12 - b9 - b8 - 2*b4) * q^58 + (-b19 - b18 - b16 - b12 - 2*b4) * q^59 + (-b18 - b15 + b12 - b9 + 3*b8 - 3*b4) * q^60 + (3*b14 - b13 - 3*b10 - b6 + 3*b5 - b1) * q^61 + (-4*b11 + b10 + b5 + 4*b3 - 2*b2 - b1 - 2) * q^62 + (-b18 - 2*b17 + b16 - b15 - b12 - b9 - b7 - 2*b4) * q^63 + (-b13 + 4*b11 + 2*b10 - b5 - 3*b3 - 1) * q^64 + (-b18 + b16 - b15 + b14 - 2*b10 + b7 - 2*b6 + b5 + b3 - b1) * q^65 + (-b14 + 2*b11 + 2*b6 + b3 + b1 + 3) * q^66 + (b19 - b18 - b16 - 2*b12 + 2*b9 + 2*b8 + b4) * q^67 + (-3*b14 + b13 - b10 - 3*b6 + 5*b5 - b3 + 3*b2 + 2*b1 + 3) * q^68 + (-b14 - b13 + b11 + 3*b10 + 2*b6 - 3*b5 - b3 - b2 - b1 + 2) * q^69 + (b19 + 2*b18 - b17 - b16 + 2*b15 - b9 - 4*b7) * q^70 + (b19 - b18 - b17 + b15 + b12 + b8) * q^71 + (2*b18 - 4*b17 + b16 + b12 - b9 + b8 + 3*b7 + 3*b4) * q^72 + (b19 - b18 + 3*b17 + 2*b16 + 2*b9 - 3*b8) * q^73 + (-b13 + 5*b10 - 6*b6 - b5 + b3 + 3*b1) * q^74 + (-2*b11 - b10 + 3*b6 + b2 + b1 - 2) * q^75 + (-3*b18 - 3*b16 + 3*b9 - 2*b8 - 2*b4) * q^76 + (2*b14 - b13 - b11 - 3*b10 - b6 + 3*b5 + b3 - 2*b2 + 2*b1 + 1) * q^77 + (-b19 + b18 - b17 - b16 + b15 - b13 + b12 - b10 + b9 + 3*b8 - b7 + 2*b5 + 2*b4 - 2*b3 + 2*b1 + 3) * q^78 + (-2*b14 + 2*b13 + 2*b10 - 4*b5 + b3) * q^79 + (-b19 + b18 + b17 - b8 - b7 - b4) * q^80 + (-2*b14 + 3*b13 + 2*b11 + 3*b6 - 2*b5 - 2*b3 + 2*b2 - 3*b1 + 1) * q^81 + (-2*b13 + b11 + 4*b10 - 2*b5 + b3 + b2 - 2) * q^82 + (-2*b19 - 3*b18 - b17 + b16 + b15 + 2*b9 + 4*b7) * q^83 + (-2*b19 + b17 + b12 + 3*b7) * q^84 + (-3*b8 + 3*b4) * q^85 + (b19 + 2*b18 + 2*b16 - 2*b12 - b9 - 2*b8 + 4*b4) * q^86 + (b13 + 3*b11 + b10 + b5 - 4*b3 - 2*b1 - 3) * q^87 + (3*b13 + b10 + 3*b6 - 5*b5 - 2*b3 - b1) * q^88 + (-3*b19 + 3*b18 + 2*b17 - b16 - b15 - b12 - b9 - 2*b8 + b7 + b4) * q^89 + (2*b14 - b13 + 4*b11 - 4*b6 - 3*b5 - b3 - b2 - b1 + 2) * q^90 + (-2*b19 + 2*b18 + 3*b17 - b16 + b15 + b13 + b12 - 2*b10 - b9 - 3*b8 - b7 + b5 - b4 - b3 - 3) * q^91 + (b13 - 3*b10 - 6*b6 + 5*b3 - 2*b1) * q^92 + (-3*b18 - 4*b17 - b16 + 2*b12 + b7 - b4) * q^93 + (-3*b14 - b13 + b11 - b10 - b5 + b3 + 3*b2) * q^94 + (-3*b11 - 3*b6 + 3) * q^95 + (2*b19 - b17 + b16 - b15 - 2*b12 - b8 + 4*b7 + 2*b4) * q^96 + (-2*b19 - b18 + b17 - b16 + 2*b15 + 2*b9 - 6*b7) * q^97 + (2*b19 - 2*b18 + b17 + b16 + b9 - b8) * q^98 + (-b19 - b18 + b17 - b16 + 2*b15 - b12 - b9 + b7 + 2*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 2 q^{3} + 12 q^{4} - 2 q^{9}+O(q^{10})$$ 20 * q + 2 * q^3 + 12 * q^4 - 2 * q^9 $$20 q + 2 q^{3} + 12 q^{4} - 2 q^{9} - 16 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 4 q^{16} - 12 q^{17} - 10 q^{22} + 24 q^{23} - 12 q^{25} - 12 q^{26} - 22 q^{27} + 12 q^{29} - 54 q^{30} - 12 q^{35} + 50 q^{36} + 12 q^{38} - 8 q^{39} - 8 q^{40} + 6 q^{42} + 4 q^{43} + 38 q^{48} - 10 q^{49} - 78 q^{51} + 108 q^{53} + 20 q^{55} + 36 q^{56} - 2 q^{61} - 72 q^{62} + 8 q^{64} - 24 q^{65} + 78 q^{66} + 24 q^{68} + 72 q^{69} - 42 q^{74} - 8 q^{75} - 6 q^{77} + 66 q^{78} - 14 q^{79} + 46 q^{81} - 4 q^{82} - 54 q^{87} + 22 q^{88} + 24 q^{90} - 72 q^{91} - 84 q^{92} + 20 q^{94} + 24 q^{95}+O(q^{100})$$ 20 * q + 2 * q^3 + 12 * q^4 - 2 * q^9 - 16 * q^10 - 2 * q^12 - 4 * q^13 - 18 * q^14 + 4 * q^16 - 12 * q^17 - 10 * q^22 + 24 * q^23 - 12 * q^25 - 12 * q^26 - 22 * q^27 + 12 * q^29 - 54 * q^30 - 12 * q^35 + 50 * q^36 + 12 * q^38 - 8 * q^39 - 8 * q^40 + 6 * q^42 + 4 * q^43 + 38 * q^48 - 10 * q^49 - 78 * q^51 + 108 * q^53 + 20 * q^55 + 36 * q^56 - 2 * q^61 - 72 * q^62 + 8 * q^64 - 24 * q^65 + 78 * q^66 + 24 * q^68 + 72 * q^69 - 42 * q^74 - 8 * q^75 - 6 * q^77 + 66 * q^78 - 14 * q^79 + 46 * q^81 - 4 * q^82 - 54 * q^87 + 22 * q^88 + 24 * q^90 - 72 * q^91 - 84 * q^92 + 20 * q^94 + 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$( -\nu^{18} - 6\nu^{16} - 3\nu^{14} - 54\nu^{10} - 81\nu^{8} - 972\nu^{6} - 2916\nu^{4} - 2187\nu^{2} + 13122 ) / 6561$$ (-v^18 - 6*v^16 - 3*v^14 - 54*v^10 - 81*v^8 - 972*v^6 - 2916*v^4 - 2187*v^2 + 13122) / 6561 $$\beta_{3}$$ $$=$$ $$( 4 \nu^{18} + 21 \nu^{16} + 66 \nu^{14} + 99 \nu^{12} + 189 \nu^{10} + 648 \nu^{8} + 4617 \nu^{6} + 18954 \nu^{4} + 39366 \nu^{2} + 39366 ) / 19683$$ (4*v^18 + 21*v^16 + 66*v^14 + 99*v^12 + 189*v^10 + 648*v^8 + 4617*v^6 + 18954*v^4 + 39366*v^2 + 39366) / 19683 $$\beta_{4}$$ $$=$$ $$( 5 \nu^{19} + 15 \nu^{17} - 12 \nu^{15} - 153 \nu^{13} - 27 \nu^{11} + 324 \nu^{9} + 2673 \nu^{7} + 5103 \nu^{5} - 13122 \nu^{3} - 98415 \nu ) / 59049$$ (5*v^19 + 15*v^17 - 12*v^15 - 153*v^13 - 27*v^11 + 324*v^9 + 2673*v^7 + 5103*v^5 - 13122*v^3 - 98415*v) / 59049 $$\beta_{5}$$ $$=$$ $$( - 4 \nu^{18} - 12 \nu^{16} + 15 \nu^{14} + 90 \nu^{12} + 135 \nu^{10} - 162 \nu^{8} - 1701 \nu^{6} - 5832 \nu^{4} + 13122 \nu^{2} + 98415 ) / 19683$$ (-4*v^18 - 12*v^16 + 15*v^14 + 90*v^12 + 135*v^10 - 162*v^8 - 1701*v^6 - 5832*v^4 + 13122*v^2 + 98415) / 19683 $$\beta_{6}$$ $$=$$ $$( 4 \nu^{18} + 3 \nu^{16} - 42 \nu^{14} - 117 \nu^{12} - 54 \nu^{10} - 81 \nu^{8} + 972 \nu^{6} + 1458 \nu^{4} - 32805 \nu^{2} - 98415 ) / 19683$$ (4*v^18 + 3*v^16 - 42*v^14 - 117*v^12 - 54*v^10 - 81*v^8 + 972*v^6 + 1458*v^4 - 32805*v^2 - 98415) / 19683 $$\beta_{7}$$ $$=$$ $$( 5 \nu^{19} + 6 \nu^{17} - 66 \nu^{15} - 99 \nu^{13} - 27 \nu^{11} + 81 \nu^{9} + 486 \nu^{7} + 2916 \nu^{5} - 32805 \nu^{3} - 157464 \nu ) / 59049$$ (5*v^19 + 6*v^17 - 66*v^15 - 99*v^13 - 27*v^11 + 81*v^9 + 486*v^7 + 2916*v^5 - 32805*v^3 - 157464*v) / 59049 $$\beta_{8}$$ $$=$$ $$( 4 \nu^{19} - 6 \nu^{17} - 96 \nu^{15} - 225 \nu^{13} - 297 \nu^{11} - 810 \nu^{9} - 1215 \nu^{7} - 2916 \nu^{5} - 72171 \nu^{3} - 196830 \nu ) / 59049$$ (4*v^19 - 6*v^17 - 96*v^15 - 225*v^13 - 297*v^11 - 810*v^9 - 1215*v^7 - 2916*v^5 - 72171*v^3 - 196830*v) / 59049 $$\beta_{9}$$ $$=$$ $$( 2 \nu^{19} + 24 \nu^{17} + 87 \nu^{15} + 171 \nu^{13} + 378 \nu^{11} + 1053 \nu^{9} + 1944 \nu^{7} + 18225 \nu^{5} + 45927 \nu^{3} + 78732 \nu ) / 59049$$ (2*v^19 + 24*v^17 + 87*v^15 + 171*v^13 + 378*v^11 + 1053*v^9 + 1944*v^7 + 18225*v^5 + 45927*v^3 + 78732*v) / 59049 $$\beta_{10}$$ $$=$$ $$( 7 \nu^{18} + 12 \nu^{16} - 33 \nu^{14} - 63 \nu^{12} + 27 \nu^{10} + 405 \nu^{8} + 3159 \nu^{6} + 8019 \nu^{4} - 26244 \nu^{2} - 98415 ) / 19683$$ (7*v^18 + 12*v^16 - 33*v^14 - 63*v^12 + 27*v^10 + 405*v^8 + 3159*v^6 + 8019*v^4 - 26244*v^2 - 98415) / 19683 $$\beta_{11}$$ $$=$$ $$( - 8 \nu^{18} - 24 \nu^{16} + 3 \nu^{14} + 18 \nu^{12} - 54 \nu^{10} - 324 \nu^{8} - 3402 \nu^{6} - 16038 \nu^{4} + 6561 \nu^{2} + 98415 ) / 19683$$ (-8*v^18 - 24*v^16 + 3*v^14 + 18*v^12 - 54*v^10 - 324*v^8 - 3402*v^6 - 16038*v^4 + 6561*v^2 + 98415) / 19683 $$\beta_{12}$$ $$=$$ $$( - 10 \nu^{19} - 21 \nu^{17} + 51 \nu^{15} + 90 \nu^{13} + 216 \nu^{11} - 405 \nu^{9} - 3888 \nu^{7} - 14580 \nu^{5} + 39366 \nu^{3} + 137781 \nu ) / 59049$$ (-10*v^19 - 21*v^17 + 51*v^15 + 90*v^13 + 216*v^11 - 405*v^9 - 3888*v^7 - 14580*v^5 + 39366*v^3 + 137781*v) / 59049 $$\beta_{13}$$ $$=$$ $$( 7 \nu^{18} + 3 \nu^{16} - 87 \nu^{14} - 171 \nu^{12} - 216 \nu^{10} - 324 \nu^{8} + 3159 \nu^{6} + 3645 \nu^{4} - 59049 \nu^{2} - 196830 ) / 19683$$ (7*v^18 + 3*v^16 - 87*v^14 - 171*v^12 - 216*v^10 - 324*v^8 + 3159*v^6 + 3645*v^4 - 59049*v^2 - 196830) / 19683 $$\beta_{14}$$ $$=$$ $$( 7 \nu^{18} - 6 \nu^{16} - 87 \nu^{14} - 198 \nu^{12} - 297 \nu^{10} - 324 \nu^{8} + 1701 \nu^{6} - 5103 \nu^{4} - 72171 \nu^{2} - 177147 ) / 19683$$ (7*v^18 - 6*v^16 - 87*v^14 - 198*v^12 - 297*v^10 - 324*v^8 + 1701*v^6 - 5103*v^4 - 72171*v^2 - 177147) / 19683 $$\beta_{15}$$ $$=$$ $$( - 10 \nu^{19} - 3 \nu^{17} + 78 \nu^{15} + 225 \nu^{13} - 27 \nu^{11} + 81 \nu^{9} - 3888 \nu^{7} - 3645 \nu^{5} + 78732 \nu^{3} + 196830 \nu ) / 59049$$ (-10*v^19 - 3*v^17 + 78*v^15 + 225*v^13 - 27*v^11 + 81*v^9 - 3888*v^7 - 3645*v^5 + 78732*v^3 + 196830*v) / 59049 $$\beta_{16}$$ $$=$$ $$( - 10 \nu^{19} - 48 \nu^{17} - 30 \nu^{15} + 9 \nu^{13} - 270 \nu^{11} - 1134 \nu^{9} - 8262 \nu^{7} - 34263 \nu^{5} - 19683 \nu^{3} + 137781 \nu ) / 59049$$ (-10*v^19 - 48*v^17 - 30*v^15 + 9*v^13 - 270*v^11 - 1134*v^9 - 8262*v^7 - 34263*v^5 - 19683*v^3 + 137781*v) / 59049 $$\beta_{17}$$ $$=$$ $$( - 13 \nu^{19} - 30 \nu^{17} + 15 \nu^{15} + 117 \nu^{13} + 54 \nu^{11} - 891 \nu^{9} - 7533 \nu^{7} - 18954 \nu^{5} + 32805 \nu^{3} + 196830 \nu ) / 59049$$ (-13*v^19 - 30*v^17 + 15*v^15 + 117*v^13 + 54*v^11 - 891*v^9 - 7533*v^7 - 18954*v^5 + 32805*v^3 + 196830*v) / 59049 $$\beta_{18}$$ $$=$$ $$( 13 \nu^{19} + 3 \nu^{17} - 231 \nu^{15} - 522 \nu^{13} - 459 \nu^{11} - 567 \nu^{9} + 1701 \nu^{7} - 7290 \nu^{5} - 183708 \nu^{3} - 492075 \nu ) / 59049$$ (13*v^19 + 3*v^17 - 231*v^15 - 522*v^13 - 459*v^11 - 567*v^9 + 1701*v^7 - 7290*v^5 - 183708*v^3 - 492075*v) / 59049 $$\beta_{19}$$ $$=$$ $$( - 25 \nu^{19} - 57 \nu^{17} + 60 \nu^{15} + 252 \nu^{13} + 54 \nu^{11} - 405 \nu^{9} - 11907 \nu^{7} - 34263 \nu^{5} + 59049 \nu^{3} + 373977 \nu ) / 59049$$ (-25*v^19 - 57*v^17 + 60*v^15 + 252*v^13 + 54*v^11 - 405*v^9 - 11907*v^7 - 34263*v^5 + 59049*v^3 + 373977*v) / 59049
 $$\nu$$ $$=$$ $$( 2\beta_{17} - \beta_{15} - 2\beta_{12} - \beta_{8} - \beta_{7} + \beta_{4} ) / 3$$ (2*b17 - b15 - 2*b12 - b8 - b7 + b4) / 3 $$\nu^{2}$$ $$=$$ $$\beta_1$$ b1 $$\nu^{3}$$ $$=$$ $$-\beta_{18} - \beta_{9} + 2\beta_{7} + \beta_{4}$$ -b18 - b9 + 2*b7 + b4 $$\nu^{4}$$ $$=$$ $$-3\beta_{14} + 2\beta_{13} + 3\beta_{6} + \beta_{3} + 3\beta_{2} - \beta_1$$ -3*b14 + 2*b13 + 3*b6 + b3 + 3*b2 - b1 $$\nu^{5}$$ $$=$$ $$3\beta_{19} - 3\beta_{18} - 3\beta_{16} - 3\beta_{15} - 3\beta_{12} + 6\beta_{8}$$ 3*b19 - 3*b18 - 3*b16 - 3*b15 - 3*b12 + 6*b8 $$\nu^{6}$$ $$=$$ $$3\beta_{14} + 3\beta_{13} + 6\beta_{11} - 6\beta_{6} + 3\beta_{3} - 6\beta_{2} + 3$$ 3*b14 + 3*b13 + 6*b11 - 6*b6 + 3*b3 - 6*b2 + 3 $$\nu^{7}$$ $$=$$ $$3 \beta_{18} - 3 \beta_{17} - 9 \beta_{16} - 3 \beta_{15} + 3 \beta_{12} - 15 \beta_{9} - 12 \beta_{8} - 9 \beta_{7} - 9 \beta_{4}$$ 3*b18 - 3*b17 - 9*b16 - 3*b15 + 3*b12 - 15*b9 - 12*b8 - 9*b7 - 9*b4 $$\nu^{8}$$ $$=$$ $$- 18 \beta_{14} - 6 \beta_{13} + 27 \beta_{11} + 54 \beta_{10} - 9 \beta_{6} - 27 \beta_{5} - 3 \beta_{3} + 18 \beta_{2} - 6 \beta _1 - 27$$ -18*b14 - 6*b13 + 27*b11 + 54*b10 - 9*b6 - 27*b5 - 3*b3 + 18*b2 - 6*b1 - 27 $$\nu^{9}$$ $$=$$ $$45 \beta_{19} - 9 \beta_{18} - 45 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} - 27 \beta_{12} + 9 \beta_{9} - 9 \beta_{8} + 45 \beta_{7} + 36 \beta_{4}$$ 45*b19 - 9*b18 - 45*b17 + 9*b16 - 9*b15 - 27*b12 + 9*b9 - 9*b8 + 45*b7 + 36*b4 $$\nu^{10}$$ $$=$$ $$-9\beta_{14} - 27\beta_{13} + 9\beta_{11} + 126\beta_{6} + 81\beta_{5} + 9\beta_{3} - 36\beta_{2} + 9\beta _1 - 117$$ -9*b14 - 27*b13 + 9*b11 + 126*b6 + 81*b5 + 9*b3 - 36*b2 + 9*b1 - 117 $$\nu^{11}$$ $$=$$ $$- 27 \beta_{19} + 18 \beta_{18} + 117 \beta_{17} - 27 \beta_{16} - 99 \beta_{15} + 72 \beta_{12} + 45 \beta_{9} - 72 \beta_{8} + 54 \beta_{7}$$ -27*b19 + 18*b18 + 117*b17 - 27*b16 - 99*b15 + 72*b12 + 45*b9 - 72*b8 + 54*b7 $$\nu^{12}$$ $$=$$ $$27 \beta_{14} + 72 \beta_{13} - 135 \beta_{11} + 81 \beta_{10} - 405 \beta_{6} + 81 \beta_{5} - 99 \beta_{3} - 63 \beta _1 - 189$$ 27*b14 + 72*b13 - 135*b11 + 81*b10 - 405*b6 + 81*b5 - 99*b3 - 63*b1 - 189 $$\nu^{13}$$ $$=$$ $$- 135 \beta_{19} + 162 \beta_{18} - 162 \beta_{17} + 54 \beta_{16} + 216 \beta_{15} + 135 \beta_{12} + 27 \beta_{9} - 189 \beta_{8} + 27 \beta_{7} - 594 \beta_{4}$$ -135*b19 + 162*b18 - 162*b17 + 54*b16 + 216*b15 + 135*b12 + 27*b9 - 189*b8 + 27*b7 - 594*b4 $$\nu^{14}$$ $$=$$ $$432 \beta_{14} - 432 \beta_{13} - 27 \beta_{11} - 486 \beta_{10} + 432 \beta_{6} + 324 \beta_{3} - 54 \beta_{2} - 297 \beta _1 - 1107$$ 432*b14 - 432*b13 - 27*b11 - 486*b10 + 432*b6 + 324*b3 - 54*b2 - 297*b1 - 1107 $$\nu^{15}$$ $$=$$ $$- 81 \beta_{19} + 27 \beta_{18} - 1161 \beta_{17} + 486 \beta_{16} + 378 \beta_{15} + 999 \beta_{12} + 756 \beta_{9} + 1026 \beta_{8} - 810 \beta_{7} - 81 \beta_{4}$$ -81*b19 + 27*b18 - 1161*b17 + 486*b16 + 378*b15 + 999*b12 + 756*b9 + 1026*b8 - 810*b7 - 81*b4 $$\nu^{16}$$ $$=$$ $$243 \beta_{14} - 216 \beta_{13} - 648 \beta_{11} - 243 \beta_{10} - 1863 \beta_{6} - 972 \beta_{5} - 1242 \beta_{3} - 1620 \beta_{2} - 378 \beta _1 + 3321$$ 243*b14 - 216*b13 - 648*b11 - 243*b10 - 1863*b6 - 972*b5 - 1242*b3 - 1620*b2 - 378*b1 + 3321 $$\nu^{17}$$ $$=$$ $$- 2268 \beta_{19} + 3240 \beta_{18} + 3564 \beta_{17} + 81 \beta_{16} + 2916 \beta_{15} - 81 \beta_{12} + 1215 \beta_{9} - 3402 \beta_{8} - 2754 \beta_{7} + 324 \beta_{4}$$ -2268*b19 + 3240*b18 + 3564*b17 + 81*b16 + 2916*b15 - 81*b12 + 1215*b9 - 3402*b8 - 2754*b7 + 324*b4 $$\nu^{18}$$ $$=$$ $$5022 \beta_{14} - 4212 \beta_{13} - 4536 \beta_{11} - 1458 \beta_{10} + 891 \beta_{6} + 3645 \beta_{5} + 405 \beta_{3} + 891 \beta_{2} + 3888 \beta _1 + 2106$$ 5022*b14 - 4212*b13 - 4536*b11 - 1458*b10 + 891*b6 + 3645*b5 + 405*b3 + 891*b2 + 3888*b1 + 2106 $$\nu^{19}$$ $$=$$ $$- 3645 \beta_{19} - 5184 \beta_{18} - 162 \beta_{17} + 9720 \beta_{16} - 3078 \beta_{15} - 2754 \beta_{12} + 4050 \beta_{9} + 810 \beta_{8} + 8019 \beta_{7} + 4131 \beta_{4}$$ -3645*b19 - 5184*b18 - 162*b17 + 9720*b16 - 3078*b15 - 2754*b12 + 4050*b9 + 810*b8 + 8019*b7 + 4131*b4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{6}$$

## 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
 1.66095 + 0.491165i −1.23798 + 1.21137i 0.651881 + 1.60470i 1.65391 − 0.514376i 0.219737 − 1.71806i −0.219737 + 1.71806i −1.65391 + 0.514376i −0.651881 − 1.60470i 1.23798 − 1.21137i −1.66095 − 0.491165i 1.66095 − 0.491165i −1.23798 − 1.21137i 0.651881 − 1.60470i 1.65391 + 0.514376i 0.219737 + 1.71806i −0.219737 − 1.71806i −1.65391 − 0.514376i −0.651881 + 1.60470i 1.23798 + 1.21137i −1.66095 + 0.491165i
−2.14539 1.23864i −1.72976 + 0.0890572i 2.06847 + 3.58269i −0.771397 + 0.445366i 3.82132 + 1.95149i 0.850723 + 0.491165i 5.29379i 2.98414 0.308095i 2.20660
25.2 −1.97712 1.14149i 0.833228 1.51846i 1.60600 + 2.78168i 2.78501 1.60793i −3.38070 + 2.05106i 2.09815 + 1.21137i 2.76698i −1.61146 2.53045i −7.34174
25.3 −1.41717 0.818205i 0.471101 + 1.66675i 0.338918 + 0.587023i −0.950358 + 0.548689i 0.696113 2.74753i 2.77942 + 1.60470i 2.16360i −2.55613 + 1.57042i 1.79576
25.4 −0.929969 0.536918i −0.744247 1.56400i −0.423439 0.733417i −1.10543 + 0.638222i −0.147613 + 1.85407i −0.890926 0.514376i 3.05708i −1.89219 + 2.32800i 1.37069
25.5 −0.784270 0.452798i 1.66968 + 0.460628i −0.589947 1.02182i 1.94254 1.12153i −1.10091 1.11728i −2.97576 1.71806i 2.87970i 2.57564 + 1.53820i −2.03130
25.6 0.784270 + 0.452798i 1.66968 + 0.460628i −0.589947 1.02182i −1.94254 + 1.12153i 1.10091 + 1.11728i 2.97576 + 1.71806i 2.87970i 2.57564 + 1.53820i −2.03130
25.7 0.929969 + 0.536918i −0.744247 1.56400i −0.423439 0.733417i 1.10543 0.638222i 0.147613 1.85407i 0.890926 + 0.514376i 3.05708i −1.89219 + 2.32800i 1.37069
25.8 1.41717 + 0.818205i 0.471101 + 1.66675i 0.338918 + 0.587023i 0.950358 0.548689i −0.696113 + 2.74753i −2.77942 1.60470i 2.16360i −2.55613 + 1.57042i 1.79576
25.9 1.97712 + 1.14149i 0.833228 1.51846i 1.60600 + 2.78168i −2.78501 + 1.60793i 3.38070 2.05106i −2.09815 1.21137i 2.76698i −1.61146 2.53045i −7.34174
25.10 2.14539 + 1.23864i −1.72976 + 0.0890572i 2.06847 + 3.58269i 0.771397 0.445366i −3.82132 1.95149i −0.850723 0.491165i 5.29379i 2.98414 0.308095i 2.20660
103.1 −2.14539 + 1.23864i −1.72976 0.0890572i 2.06847 3.58269i −0.771397 0.445366i 3.82132 1.95149i 0.850723 0.491165i 5.29379i 2.98414 + 0.308095i 2.20660
103.2 −1.97712 + 1.14149i 0.833228 + 1.51846i 1.60600 2.78168i 2.78501 + 1.60793i −3.38070 2.05106i 2.09815 1.21137i 2.76698i −1.61146 + 2.53045i −7.34174
103.3 −1.41717 + 0.818205i 0.471101 1.66675i 0.338918 0.587023i −0.950358 0.548689i 0.696113 + 2.74753i 2.77942 1.60470i 2.16360i −2.55613 1.57042i 1.79576
103.4 −0.929969 + 0.536918i −0.744247 + 1.56400i −0.423439 + 0.733417i −1.10543 0.638222i −0.147613 1.85407i −0.890926 + 0.514376i 3.05708i −1.89219 2.32800i 1.37069
103.5 −0.784270 + 0.452798i 1.66968 0.460628i −0.589947 + 1.02182i 1.94254 + 1.12153i −1.10091 + 1.11728i −2.97576 + 1.71806i 2.87970i 2.57564 1.53820i −2.03130
103.6 0.784270 0.452798i 1.66968 0.460628i −0.589947 + 1.02182i −1.94254 1.12153i 1.10091 1.11728i 2.97576 1.71806i 2.87970i 2.57564 1.53820i −2.03130
103.7 0.929969 0.536918i −0.744247 + 1.56400i −0.423439 + 0.733417i 1.10543 + 0.638222i 0.147613 + 1.85407i 0.890926 0.514376i 3.05708i −1.89219 2.32800i 1.37069
103.8 1.41717 0.818205i 0.471101 1.66675i 0.338918 0.587023i 0.950358 + 0.548689i −0.696113 2.74753i −2.77942 + 1.60470i 2.16360i −2.55613 1.57042i 1.79576
103.9 1.97712 1.14149i 0.833228 + 1.51846i 1.60600 2.78168i −2.78501 1.60793i 3.38070 + 2.05106i −2.09815 + 1.21137i 2.76698i −1.61146 + 2.53045i −7.34174
103.10 2.14539 1.23864i −1.72976 0.0890572i 2.06847 3.58269i 0.771397 + 0.445366i −3.82132 + 1.95149i −0.850723 + 0.491165i 5.29379i 2.98414 + 0.308095i 2.20660
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.10 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
13.b even 2 1 inner
117.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.t.c 20
3.b odd 2 1 351.2.t.c 20
9.c even 3 1 inner 117.2.t.c 20
9.c even 3 1 1053.2.b.j 10
9.d odd 6 1 351.2.t.c 20
9.d odd 6 1 1053.2.b.i 10
13.b even 2 1 inner 117.2.t.c 20
39.d odd 2 1 351.2.t.c 20
117.n odd 6 1 351.2.t.c 20
117.n odd 6 1 1053.2.b.i 10
117.t even 6 1 inner 117.2.t.c 20
117.t even 6 1 1053.2.b.j 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.t.c 20 1.a even 1 1 trivial
117.2.t.c 20 9.c even 3 1 inner
117.2.t.c 20 13.b even 2 1 inner
117.2.t.c 20 117.t even 6 1 inner
351.2.t.c 20 3.b odd 2 1
351.2.t.c 20 9.d odd 6 1
351.2.t.c 20 39.d odd 2 1
351.2.t.c 20 117.n odd 6 1
1053.2.b.i 10 9.d odd 6 1
1053.2.b.i 10 117.n odd 6 1
1053.2.b.j 10 9.c even 3 1
1053.2.b.j 10 117.t even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$:

 $$T_{2}^{20} - 16 T_{2}^{18} + 165 T_{2}^{16} - 1012 T_{2}^{14} + 4501 T_{2}^{12} - 12987 T_{2}^{10} + 27240 T_{2}^{8} - 35874 T_{2}^{6} + 34002 T_{2}^{4} - 18468 T_{2}^{2} + 6561$$ T2^20 - 16*T2^18 + 165*T2^16 - 1012*T2^14 + 4501*T2^12 - 12987*T2^10 + 27240*T2^8 - 35874*T2^6 + 34002*T2^4 - 18468*T2^2 + 6561 $$T_{5}^{20} - 19 T_{5}^{18} + 249 T_{5}^{16} - 1618 T_{5}^{14} + 7456 T_{5}^{12} - 19407 T_{5}^{10} + 36270 T_{5}^{8} - 43821 T_{5}^{6} + 38394 T_{5}^{4} - 19683 T_{5}^{2} + 6561$$ T5^20 - 19*T5^18 + 249*T5^16 - 1618*T5^14 + 7456*T5^12 - 19407*T5^10 + 36270*T5^8 - 43821*T5^6 + 38394*T5^4 - 19683*T5^2 + 6561

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - 16 T^{18} + 165 T^{16} + \cdots + 6561$$
$3$ $$(T^{10} - T^{9} + T^{8} + 3 T^{7} - 9 T^{6} + \cdots + 243)^{2}$$
$5$ $$T^{20} - 19 T^{18} + 249 T^{16} + \cdots + 6561$$
$7$ $$T^{20} - 30 T^{18} + 591 T^{16} + \cdots + 531441$$
$11$ $$T^{20} - 49 T^{18} + \cdots + 276922881$$
$13$ $$T^{20} + 4 T^{19} + \cdots + 137858491849$$
$17$ $$(T^{5} + 3 T^{4} - 33 T^{3} - 90 T^{2} + \cdots + 81)^{4}$$
$19$ $$(T^{10} + 129 T^{8} + 5727 T^{6} + \cdots + 700569)^{2}$$
$23$ $$(T^{10} - 12 T^{9} + 144 T^{8} + \cdots + 531441)^{2}$$
$29$ $$(T^{10} - 6 T^{9} + 69 T^{8} + 198 T^{7} + \cdots + 6561)^{2}$$
$31$ $$T^{20} + \cdots + 138251528157681$$
$37$ $$(T^{10} + 231 T^{8} + 16365 T^{6} + \cdots + 41641209)^{2}$$
$41$ $$T^{20} - 118 T^{18} + \cdots + 273245607441$$
$43$ $$(T^{10} - 2 T^{9} + 57 T^{8} + 234 T^{7} + \cdots + 32041)^{2}$$
$47$ $$T^{20} - 205 T^{18} + \cdots + 49241109874401$$
$53$ $$(T^{5} - 27 T^{4} + 246 T^{3} - 855 T^{2} + \cdots - 243)^{4}$$
$59$ $$T^{20} - 145 T^{18} + \cdots + 75017234240001$$
$61$ $$(T^{10} + T^{9} + 180 T^{8} + \cdots + 118091689)^{2}$$
$67$ $$T^{20} - 270 T^{18} + \cdots + 282429536481$$
$71$ $$(T^{10} + 97 T^{8} + 3046 T^{6} + \cdots + 178929)^{2}$$
$73$ $$(T^{10} + 369 T^{8} + 46521 T^{6} + \cdots + 56746089)^{2}$$
$79$ $$(T^{10} + 7 T^{9} + 186 T^{8} + \cdots + 1481089)^{2}$$
$83$ $$T^{20} - 580 T^{18} + \cdots + 20\!\cdots\!81$$
$89$ $$(T^{10} + 637 T^{8} + 134917 T^{6} + \cdots + 5851791009)^{2}$$
$97$ $$T^{20} - 612 T^{18} + \cdots + 47048089623921$$