# Properties

 Label 1080.2.d.i Level $1080$ Weight $2$ Character orbit 1080.d Analytic conductor $8.624$ 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] = [1080,2,Mod(109,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("1080.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.d (of order $$2$$, degree $$1$$, 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: $$8.62384341830$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024$$ x^20 - 3*x^18 + 8*x^16 - 24*x^14 + 56*x^12 - 92*x^10 + 224*x^8 - 384*x^6 + 512*x^4 - 768*x^2 + 1024 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{2} + \beta_{11} q^{4} + \beta_{14} q^{5} + ( - \beta_{18} + \beta_{12}) q^{7} + ( - \beta_{16} + \beta_{7} - \beta_{2}) q^{8}+O(q^{10})$$ q + b8 * q^2 + b11 * q^4 + b14 * q^5 + (-b18 + b12) * q^7 + (-b16 + b7 - b2) * q^8 $$q + \beta_{8} q^{2} + \beta_{11} q^{4} + \beta_{14} q^{5} + ( - \beta_{18} + \beta_{12}) q^{7} + ( - \beta_{16} + \beta_{7} - \beta_{2}) q^{8} - \beta_{17} q^{10} + (\beta_{14} - \beta_{10} + \beta_{7} - \beta_{5}) q^{11} + ( - \beta_{17} + \beta_{13} - \beta_{11} - \beta_1 + 1) q^{13} + (\beta_{16} - \beta_{6}) q^{14} + (\beta_{12} + \beta_{11} - \beta_{4} + \beta_1 - 2) q^{16} + ( - \beta_{16} - \beta_{7}) q^{17} + ( - \beta_{18} - \beta_{17} - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{9} - 2 \beta_{4} - \beta_{3} + \cdots - 1) q^{19}+ \cdots + (2 \beta_{19} + \beta_{16} - 2 \beta_{14} + 2 \beta_{10} - 3 \beta_{8} + \beta_{6}) q^{98}+O(q^{100})$$ q + b8 * q^2 + b11 * q^4 + b14 * q^5 + (-b18 + b12) * q^7 + (-b16 + b7 - b2) * q^8 - b17 * q^10 + (b14 - b10 + b7 - b5) * q^11 + (-b17 + b13 - b11 - b1 + 1) * q^13 + (b16 - b6) * q^14 + (b12 + b11 - b4 + b1 - 2) * q^16 + (-b16 - b7) * q^17 + (-b18 - b17 - b13 + b12 + 2*b11 - b9 - 2*b4 - b3 - b1 - 1) * q^19 + (b19 + b14 - b10 - b8 - b5) * q^20 + (-b18 - b17 + b12 + 2*b11 - b9 - b4 - b3 - b1 - 2) * q^22 + (b14 - b10 + b7 + b5) * q^23 + (-b18 + b12 - b9 + b3 - b1 + 1) * q^25 + (b19 + b16 + b15 + b14 - b8 - b5 + b2) * q^26 + (-b18 - b12 + b11 - b4 - b3 - b1 + 2) * q^28 + (-b16 - b15 - b8 - b5) * q^29 + (-b12 + b11 - b3 + 1) * q^31 + (-b16 - 2*b15 - 2*b14 + 2*b10 - b8 + b6) * q^32 + (2*b18 - b12 - b11 + b4 + b1) * q^34 + (-b19 + b16 + b15 + 3*b8 - 2*b7 - b6) * q^35 + (-b17 + b13 - b12 + b11 + b9 + b1 - 1) * q^37 + (b19 - b15 + b14 - 2*b10 - 2*b8 + 2*b7 + b6 - b5 + b2) * q^38 + (b18 - b17 + 2*b13 - b12 - b11 + b4 + b1 + 1) * q^40 + (b16 - b15 - 2*b14 + b8 - b7 + 2*b6 + 2*b2) * q^41 + (-b12 - b9 - 2*b3 - 2*b1 + 2) * q^43 + (b19 - b10 - 3*b8 + 2*b7) * q^44 + (-b18 - b17 + b12 - b9 - b4 + b3 - b1) * q^46 + (3*b16 + b15 + b14 - b10 + b8 - b5) * q^47 + (-2*b17 + 2*b13 + b12 - b9 - 2) * q^49 + (2*b16 + b15 + b14 - b10 + b8 + 2*b7 + b6 - b5 - b2) * q^50 + (b18 + 2*b13 - 2*b12 - 2*b11 + 2*b9 + 2*b4 + b3) * q^52 + (-2*b19 - b16 - b14 + b10 + 2*b8 + b5 - 2*b2) * q^53 + (b18 + 2*b4 + b3 + b1 - 2) * q^55 + (2*b14 + b8 + b7 - b6 + 2*b5 + b2) * q^56 + (-b17 + b12 + b11 - b9 - b4 - 2*b3 - b1) * q^58 + (-b16 - 2*b15 - 2*b14 + 2*b10 - 2*b8 - 2*b7 - b5) * q^59 + (b18 + b12 + b11 + b9 - b3 - 1) * q^61 + (-b16 + b8 - b6 + 2*b5) * q^62 + (-b18 + 3*b12 - b4 - b3 - b1 + 2) * q^64 + (-b19 - 2*b15 - 2*b14 + 2*b10 + 2*b8 + b6 + b5 - 2*b2) * q^65 + (b12 + 3*b11 - 2*b9 - b3 + 2*b1 - 2) * q^67 + (-b16 - 2*b14 + b8 - 2*b7 + b6) * q^68 + (b18 + b17 - 2*b13 - 3*b12 + 2*b11 + b4 - b3 + b1) * q^70 + (-2*b19 + b16 - b14 + b10 + 2*b8 - 2*b7 + b5 + 2*b2) * q^71 + (-b18 + b17 + b13 + b12 + b11 - 2*b4 - b1) * q^73 + (b19 - 2*b16 - b15 + b14 + 2*b10 - 2*b8 - b6 + b5 - b2) * q^74 + (-2*b18 - 2*b17 + 2*b13 + 3*b12 - 2*b9 - b4 - 3*b1) * q^76 + (2*b15 + 3*b14 - b10 - 2*b8 - 4*b6) * q^77 + (b17 - b13 - b11 + 2*b9 + 2*b3 + b1 + 1) * q^79 + (b19 - 2*b14 + 3*b10 + b8 - 2*b7 + 2*b5) * q^80 + (-b18 + b17 + 2*b12 - b11 - b9 + 2*b4 + b3 - 2*b1 + 2) * q^82 + (b16 + b15 + 2*b14 - b8 - b7 - 2*b6 + 2*b2) * q^83 + (-2*b18 - b17 + b13 + 2*b12 + 2*b11 - 2*b4 - 2*b1 - 2) * q^85 + (b16 + 2*b15 - 2*b10 + b8 - b6 + 2*b5 + 2*b2) * q^86 + (-b18 + 2*b13 + b12 - 2*b11 - b4 + b3 - b1) * q^88 + (-2*b19 - b16 + b15 - b14 - b10 - b8 + b7 - 2*b2) * q^89 + (3*b18 - b17 - b13 - 3*b12 - b11 + 2*b4 + b1) * q^91 + (b19 + 2*b16 + 2*b14 - b10 - b8 + 2*b7 + 2*b6 - 2*b5) * q^92 + (-2*b18 - b12 + b11 + b4 + b1) * q^94 + (-b16 + 2*b15 - b10 + b7 + 2*b6 - 2*b2) * q^95 + (b18 + b17 + b13 - 3*b12 - 5*b11 + 2*b4 + 2*b3 + 3*b1 + 2) * q^97 + (2*b19 + b16 - 2*b14 + 2*b10 - 3*b8 + b6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 6 q^{4}+O(q^{10})$$ 20 * q + 6 * q^4 $$20 q + 6 q^{4} - 2 q^{10} - 4 q^{13} - 14 q^{16} - 34 q^{22} + 20 q^{25} + 20 q^{28} + 12 q^{31} + 6 q^{34} - 32 q^{37} - 6 q^{40} + 12 q^{43} + 2 q^{46} - 52 q^{49} - 50 q^{52} - 28 q^{55} + 6 q^{58} + 54 q^{64} + 12 q^{70} - 24 q^{76} + 36 q^{79} + 32 q^{82} - 44 q^{85} - 30 q^{88} - 22 q^{94}+O(q^{100})$$ 20 * q + 6 * q^4 - 2 * q^10 - 4 * q^13 - 14 * q^16 - 34 * q^22 + 20 * q^25 + 20 * q^28 + 12 * q^31 + 6 * q^34 - 32 * q^37 - 6 * q^40 + 12 * q^43 + 2 * q^46 - 52 * q^49 - 50 * q^52 - 28 * q^55 + 6 * q^58 + 54 * q^64 + 12 * q^70 - 24 * q^76 + 36 * q^79 + 32 * q^82 - 44 * q^85 - 30 * q^88 - 22 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\nu^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$( -\nu^{18} + 7\nu^{16} - 12\nu^{14} - 56\nu^{10} - 68\nu^{8} + 240\nu^{6} + 288\nu^{4} - 128\nu^{2} - 1792 ) / 768$$ (-v^18 + 7*v^16 - 12*v^14 - 56*v^10 - 68*v^8 + 240*v^6 + 288*v^4 - 128*v^2 - 1792) / 768 $$\beta_{4}$$ $$=$$ $$( \nu^{18} - 7\nu^{16} + 12\nu^{14} + 56\nu^{10} + 68\nu^{8} - 240\nu^{6} + 480\nu^{4} - 640\nu^{2} + 2560 ) / 768$$ (v^18 - 7*v^16 + 12*v^14 + 56*v^10 + 68*v^8 - 240*v^6 + 480*v^4 - 640*v^2 + 2560) / 768 $$\beta_{5}$$ $$=$$ $$( - \nu^{19} + \nu^{17} - 6 \nu^{15} + 36 \nu^{13} - 56 \nu^{11} + 172 \nu^{9} - 456 \nu^{7} + 432 \nu^{5} - 704 \nu^{3} + 1664 \nu ) / 768$$ (-v^19 + v^17 - 6*v^15 + 36*v^13 - 56*v^11 + 172*v^9 - 456*v^7 + 432*v^5 - 704*v^3 + 1664*v) / 768 $$\beta_{6}$$ $$=$$ $$( \nu^{19} + 5 \nu^{17} - 12 \nu^{15} + 12 \nu^{13} - 88 \nu^{11} + 164 \nu^{9} - 96 \nu^{7} + 144 \nu^{5} - 832 \nu^{3} - 128 \nu ) / 768$$ (v^19 + 5*v^17 - 12*v^15 + 12*v^13 - 88*v^11 + 164*v^9 - 96*v^7 + 144*v^5 - 832*v^3 - 128*v) / 768 $$\beta_{7}$$ $$=$$ $$( -\nu^{19} + 3\nu^{17} + 32\nu^{13} - 24\nu^{11} + 92\nu^{9} - 160\nu^{7} + 416\nu^{5} + 128\nu^{3} + 1536\nu ) / 512$$ (-v^19 + 3*v^17 + 32*v^13 - 24*v^11 + 92*v^9 - 160*v^7 + 416*v^5 + 128*v^3 + 1536*v) / 512 $$\beta_{8}$$ $$=$$ $$( \nu^{19} - 3 \nu^{17} + 8 \nu^{15} - 24 \nu^{13} + 56 \nu^{11} - 92 \nu^{9} + 224 \nu^{7} - 384 \nu^{5} + 512 \nu^{3} - 768 \nu ) / 512$$ (v^19 - 3*v^17 + 8*v^15 - 24*v^13 + 56*v^11 - 92*v^9 + 224*v^7 - 384*v^5 + 512*v^3 - 768*v) / 512 $$\beta_{9}$$ $$=$$ $$( \nu^{18} + 2\nu^{16} + 3\nu^{14} - 6\nu^{12} + 8\nu^{10} - 4\nu^{8} + 36\nu^{6} + 72\nu^{4} + 32\nu^{2} + 64 ) / 192$$ (v^18 + 2*v^16 + 3*v^14 - 6*v^12 + 8*v^10 - 4*v^8 + 36*v^6 + 72*v^4 + 32*v^2 + 64) / 192 $$\beta_{10}$$ $$=$$ $$( -\nu^{19} - 2\nu^{17} - 3\nu^{15} + 6\nu^{13} - 8\nu^{11} + 4\nu^{9} - 36\nu^{7} - 72\nu^{5} - 32\nu^{3} - 64\nu ) / 384$$ (-v^19 - 2*v^17 - 3*v^15 + 6*v^13 - 8*v^11 + 4*v^9 - 36*v^7 - 72*v^5 - 32*v^3 - 64*v) / 384 $$\beta_{11}$$ $$=$$ $$( - \nu^{18} + 3 \nu^{16} - 8 \nu^{14} + 24 \nu^{12} - 56 \nu^{10} + 92 \nu^{8} - 224 \nu^{6} + 384 \nu^{4} - 512 \nu^{2} + 768 ) / 256$$ (-v^18 + 3*v^16 - 8*v^14 + 24*v^12 - 56*v^10 + 92*v^8 - 224*v^6 + 384*v^4 - 512*v^2 + 768) / 256 $$\beta_{12}$$ $$=$$ $$( -5\nu^{18} - \nu^{16} + 48\nu^{12} + 8\nu^{10} - 52\nu^{8} - 480\nu^{6} + 96\nu^{4} + 128\nu^{2} + 2560 ) / 768$$ (-5*v^18 - v^16 + 48*v^12 + 8*v^10 - 52*v^8 - 480*v^6 + 96*v^4 + 128*v^2 + 2560) / 768 $$\beta_{13}$$ $$=$$ $$( 7 \nu^{18} - 13 \nu^{16} + 24 \nu^{14} - 48 \nu^{12} + 104 \nu^{10} - 196 \nu^{8} + 384 \nu^{6} - 672 \nu^{4} + 896 \nu^{2} - 512 ) / 768$$ (7*v^18 - 13*v^16 + 24*v^14 - 48*v^12 + 104*v^10 - 196*v^8 + 384*v^6 - 672*v^4 + 896*v^2 - 512) / 768 $$\beta_{14}$$ $$=$$ $$( - \nu^{19} + \nu^{17} - 9 \nu^{15} + 21 \nu^{13} - 32 \nu^{11} + 76 \nu^{9} - 192 \nu^{7} + 132 \nu^{5} - 416 \nu^{3} + 608 \nu ) / 192$$ (-v^19 + v^17 - 9*v^15 + 21*v^13 - 32*v^11 + 76*v^9 - 192*v^7 + 132*v^5 - 416*v^3 + 608*v) / 192 $$\beta_{15}$$ $$=$$ $$( \nu^{19} - 4 \nu^{17} + 9 \nu^{15} - 30 \nu^{13} + 44 \nu^{11} - 100 \nu^{9} + 204 \nu^{7} - 360 \nu^{5} + 368 \nu^{3} - 704 \nu ) / 192$$ (v^19 - 4*v^17 + 9*v^15 - 30*v^13 + 44*v^11 - 100*v^9 + 204*v^7 - 360*v^5 + 368*v^3 - 704*v) / 192 $$\beta_{16}$$ $$=$$ $$( - \nu^{19} + 2 \nu^{17} - 3 \nu^{15} + 18 \nu^{13} - 24 \nu^{11} + 36 \nu^{9} - 116 \nu^{7} + 168 \nu^{5} - 96 \nu^{3} + 448 \nu ) / 128$$ (-v^19 + 2*v^17 - 3*v^15 + 18*v^13 - 24*v^11 + 36*v^9 - 116*v^7 + 168*v^5 - 96*v^3 + 448*v) / 128 $$\beta_{17}$$ $$=$$ $$( -\nu^{18} + \nu^{16} - 9\nu^{14} + 21\nu^{12} - 32\nu^{10} + 76\nu^{8} - 192\nu^{6} + 132\nu^{4} - 416\nu^{2} + 608 ) / 96$$ (-v^18 + v^16 - 9*v^14 + 21*v^12 - 32*v^10 + 76*v^8 - 192*v^6 + 132*v^4 - 416*v^2 + 608) / 96 $$\beta_{18}$$ $$=$$ $$( -\nu^{18} + 2\nu^{16} - 3\nu^{14} + 18\nu^{12} - 24\nu^{10} + 36\nu^{8} - 116\nu^{6} + 168\nu^{4} - 96\nu^{2} + 512 ) / 64$$ (-v^18 + 2*v^16 - 3*v^14 + 18*v^12 - 24*v^10 + 36*v^8 - 116*v^6 + 168*v^4 - 96*v^2 + 512) / 64 $$\beta_{19}$$ $$=$$ $$( - 7 \nu^{19} + \nu^{17} - 24 \nu^{15} + 12 \nu^{13} - 56 \nu^{11} + 100 \nu^{9} - 336 \nu^{7} - 240 \nu^{5} - 1088 \nu^{3} - 1408 \nu ) / 768$$ (-7*v^19 + v^17 - 24*v^15 + 12*v^13 - 56*v^11 + 100*v^9 - 336*v^7 - 240*v^5 - 1088*v^3 - 1408*v) / 768
 $$\nu$$ $$=$$ $$( \beta_{15} + \beta_{14} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} ) / 2$$ (b15 + b14 - b10 - b8 + b7 - b5) / 2 $$\nu^{2}$$ $$=$$ $$\beta_1$$ b1 $$\nu^{3}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta _1 - 1$$ b4 + b3 + b1 - 1 $$\nu^{5}$$ $$=$$ $$-\beta_{16} - \beta_{15} - \beta_{14} + \beta_{10} - 2\beta_{8} - \beta_{6}$$ -b16 - b15 - b14 + b10 - 2*b8 - b6 $$\nu^{6}$$ $$=$$ $$\beta_{18} - 2\beta_{12} - 2\beta_{11} - \beta_{9} + \beta_{4} + \beta_{3} - \beta _1 + 4$$ b18 - 2*b12 - 2*b11 - b9 + b4 + b3 - b1 + 4 $$\nu^{7}$$ $$=$$ $$-\beta_{16} + 2\beta_{10} - \beta_{8} + 3\beta_{7} + \beta_{6} - 3\beta_{5} - 2\beta_{2}$$ -b16 + 2*b10 - b8 + 3*b7 + b6 - 3*b5 - 2*b2 $$\nu^{8}$$ $$=$$ $$\beta_{18} - 2\beta_{13} - 4\beta_{12} - 2\beta_{11} - \beta_{9} + \beta_{4} - 3\beta_{3} + \beta_1$$ b18 - 2*b13 - 4*b12 - 2*b11 - b9 + b4 - 3*b3 + b1 $$\nu^{9}$$ $$=$$ $$2 \beta_{19} - 3 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{10} + 5 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} + 3 \beta_{5}$$ 2*b19 - 3*b16 - 2*b15 - 2*b14 + 2*b10 + 5*b8 + 5*b7 + 3*b6 + 3*b5 $$\nu^{10}$$ $$=$$ $$\beta_{18} + 4\beta_{17} - 2\beta_{13} - 12\beta_{11} + 5\beta_{9} + 7\beta_{4} + 5\beta_{3} + 3\beta _1 - 12$$ b18 + 4*b17 - 2*b13 - 12*b11 + 5*b9 + 7*b4 + 5*b3 + 3*b1 - 12 $$\nu^{11}$$ $$=$$ $$2\beta_{19} - 5\beta_{16} - 8\beta_{15} - 4\beta_{10} + 13\beta_{8} + \beta_{7} - 7\beta_{6} + 3\beta_{5} - 4\beta_{2}$$ 2*b19 - 5*b16 - 8*b15 - 4*b10 + 13*b8 + b7 - 7*b6 + 3*b5 - 4*b2 $$\nu^{12}$$ $$=$$ $$11 \beta_{18} + 4 \beta_{17} + 10 \beta_{13} - 8 \beta_{12} - 20 \beta_{11} - \beta_{9} + 5 \beta_{4} + 7 \beta_{3} - 11 \beta _1 - 20$$ 11*b18 + 4*b17 + 10*b13 - 8*b12 - 20*b11 - b9 + 5*b4 + 7*b3 - 11*b1 - 20 $$\nu^{13}$$ $$=$$ $$- 10 \beta_{19} + 9 \beta_{16} - 16 \beta_{15} - 8 \beta_{14} + 28 \beta_{10} + 39 \beta_{8} - 13 \beta_{7} + 3 \beta_{6} + 9 \beta_{5} - 16 \beta_{2}$$ -10*b19 + 9*b16 - 16*b15 - 8*b14 + 28*b10 + 39*b8 - 13*b7 + 3*b6 + 9*b5 - 16*b2 $$\nu^{14}$$ $$=$$ $$9 \beta_{18} - 20 \beta_{17} - 2 \beta_{13} - 8 \beta_{12} + 20 \beta_{11} - 11 \beta_{9} - 13 \beta_{4} - 39 \beta_{3} - 45 \beta _1 - 24$$ 9*b18 - 20*b17 - 2*b13 - 8*b12 + 20*b11 - 11*b9 - 13*b4 - 39*b3 - 45*b1 - 24 $$\nu^{15}$$ $$=$$ $$2 \beta_{19} + 23 \beta_{16} + 4 \beta_{15} - 36 \beta_{14} + 16 \beta_{10} + 37 \beta_{8} + \beta_{7} + 21 \beta_{6} + 51 \beta_{5} - 32 \beta_{2}$$ 2*b19 + 23*b16 + 4*b15 - 36*b14 + 16*b10 + 37*b8 + b7 + 21*b6 + 51*b5 - 32*b2 $$\nu^{16}$$ $$=$$ $$- \beta_{18} + 4 \beta_{17} - 22 \beta_{13} + 16 \beta_{12} - 12 \beta_{11} + 59 \beta_{9} - 35 \beta_{4} - 17 \beta_{3} - 27 \beta _1 + 8$$ -b18 + 4*b17 - 22*b13 + 16*b12 - 12*b11 + 59*b9 - 35*b4 - 17*b3 - 27*b1 + 8 $$\nu^{17}$$ $$=$$ $$22 \beta_{19} + 49 \beta_{16} + 28 \beta_{15} + 36 \beta_{14} - 168 \beta_{10} + 59 \beta_{8} - \beta_{7} + 19 \beta_{6} - 35 \beta_{5} + 8 \beta_{2}$$ 22*b19 + 49*b16 + 28*b15 + 36*b14 - 168*b10 + 59*b8 - b7 + 19*b6 - 35*b5 + 8*b2 $$\nu^{18}$$ $$=$$ $$\beta_{18} + 44\beta_{17} + 118\beta_{13} + 4\beta_{11} + 93\beta_{9} - 21\beta_{4} + 33\beta_{3} + 35\beta _1 - 104$$ b18 + 44*b17 + 118*b13 + 4*b11 + 93*b9 - 21*b4 + 33*b3 + 35*b1 - 104 $$\nu^{19}$$ $$=$$ $$- 118 \beta_{19} + 23 \beta_{16} - 68 \beta_{15} + 20 \beta_{14} + 133 \beta_{8} - 207 \beta_{7} + 21 \beta_{6} + 99 \beta_{5} + 56 \beta_{2}$$ -118*b19 + 23*b16 - 68*b15 + 20*b14 + 133*b8 - 207*b7 + 21*b6 + 99*b5 + 56*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 1.37859 − 0.315404i 1.37859 + 0.315404i 1.35662 − 0.399488i 1.35662 + 0.399488i 0.986501 − 1.01332i 0.986501 + 1.01332i 0.847166 − 1.13239i 0.847166 + 1.13239i 0.564088 − 1.29684i 0.564088 + 1.29684i −0.564088 − 1.29684i −0.564088 + 1.29684i −0.847166 − 1.13239i −0.847166 + 1.13239i −0.986501 − 1.01332i −0.986501 + 1.01332i −1.35662 − 0.399488i −1.35662 + 0.399488i −1.37859 − 0.315404i −1.37859 + 0.315404i
−1.37859 0.315404i 0 1.80104 + 0.869628i 2.03021 + 0.937151i 0 3.36920i −2.20862 1.76692i 0 −2.50325 1.93229i
109.2 −1.37859 + 0.315404i 0 1.80104 0.869628i 2.03021 0.937151i 0 3.36920i −2.20862 + 1.76692i 0 −2.50325 + 1.93229i
109.3 −1.35662 0.399488i 0 1.68082 + 1.08390i −1.49578 + 1.66212i 0 1.43534i −1.84722 2.14191i 0 2.69320 1.65731i
109.4 −1.35662 + 0.399488i 0 1.68082 1.08390i −1.49578 1.66212i 0 1.43534i −1.84722 + 2.14191i 0 2.69320 + 1.65731i
109.5 −0.986501 1.01332i 0 −0.0536316 + 1.99928i −0.569524 2.16232i 0 4.64762i 2.07882 1.91795i 0 −1.62929 + 2.71024i
109.6 −0.986501 + 1.01332i 0 −0.0536316 1.99928i −0.569524 + 2.16232i 0 4.64762i 2.07882 + 1.91795i 0 −1.62929 2.71024i
109.7 −0.847166 1.13239i 0 −0.564618 + 1.91865i 1.82935 + 1.28588i 0 1.09701i 2.65098 0.986045i 0 −0.0936411 3.16089i
109.8 −0.847166 + 1.13239i 0 −0.564618 1.91865i 1.82935 1.28588i 0 1.09701i 2.65098 + 0.986045i 0 −0.0936411 + 3.16089i
109.9 −0.564088 1.29684i 0 −1.36361 + 1.46307i −2.22935 0.173169i 0 3.43285i 2.66657 + 0.943090i 0 1.03298 + 2.98881i
109.10 −0.564088 + 1.29684i 0 −1.36361 1.46307i −2.22935 + 0.173169i 0 3.43285i 2.66657 0.943090i 0 1.03298 2.98881i
109.11 0.564088 1.29684i 0 −1.36361 1.46307i 2.22935 0.173169i 0 3.43285i −2.66657 + 0.943090i 0 1.03298 2.98881i
109.12 0.564088 + 1.29684i 0 −1.36361 + 1.46307i 2.22935 + 0.173169i 0 3.43285i −2.66657 0.943090i 0 1.03298 + 2.98881i
109.13 0.847166 1.13239i 0 −0.564618 1.91865i −1.82935 + 1.28588i 0 1.09701i −2.65098 0.986045i 0 −0.0936411 + 3.16089i
109.14 0.847166 + 1.13239i 0 −0.564618 + 1.91865i −1.82935 1.28588i 0 1.09701i −2.65098 + 0.986045i 0 −0.0936411 3.16089i
109.15 0.986501 1.01332i 0 −0.0536316 1.99928i 0.569524 2.16232i 0 4.64762i −2.07882 1.91795i 0 −1.62929 2.71024i
109.16 0.986501 + 1.01332i 0 −0.0536316 + 1.99928i 0.569524 + 2.16232i 0 4.64762i −2.07882 + 1.91795i 0 −1.62929 + 2.71024i
109.17 1.35662 0.399488i 0 1.68082 1.08390i 1.49578 + 1.66212i 0 1.43534i 1.84722 2.14191i 0 2.69320 + 1.65731i
109.18 1.35662 + 0.399488i 0 1.68082 + 1.08390i 1.49578 1.66212i 0 1.43534i 1.84722 + 2.14191i 0 2.69320 1.65731i
109.19 1.37859 0.315404i 0 1.80104 0.869628i −2.03021 + 0.937151i 0 3.36920i 2.20862 1.76692i 0 −2.50325 + 1.93229i
109.20 1.37859 + 0.315404i 0 1.80104 + 0.869628i −2.03021 0.937151i 0 3.36920i 2.20862 + 1.76692i 0 −2.50325 1.93229i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.i 20
3.b odd 2 1 inner 1080.2.d.i 20
4.b odd 2 1 4320.2.d.i 20
5.b even 2 1 1080.2.d.j yes 20
8.b even 2 1 1080.2.d.j yes 20
8.d odd 2 1 4320.2.d.j 20
12.b even 2 1 4320.2.d.i 20
15.d odd 2 1 1080.2.d.j yes 20
20.d odd 2 1 4320.2.d.j 20
24.f even 2 1 4320.2.d.j 20
24.h odd 2 1 1080.2.d.j yes 20
40.e odd 2 1 4320.2.d.i 20
40.f even 2 1 inner 1080.2.d.i 20
60.h even 2 1 4320.2.d.j 20
120.i odd 2 1 inner 1080.2.d.i 20
120.m even 2 1 4320.2.d.i 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.i 20 1.a even 1 1 trivial
1080.2.d.i 20 3.b odd 2 1 inner
1080.2.d.i 20 40.f even 2 1 inner
1080.2.d.i 20 120.i odd 2 1 inner
1080.2.d.j yes 20 5.b even 2 1
1080.2.d.j yes 20 8.b even 2 1
1080.2.d.j yes 20 15.d odd 2 1
1080.2.d.j yes 20 24.h odd 2 1
4320.2.d.i 20 4.b odd 2 1
4320.2.d.i 20 12.b even 2 1
4320.2.d.i 20 40.e odd 2 1
4320.2.d.i 20 120.m even 2 1
4320.2.d.j 20 8.d odd 2 1
4320.2.d.j 20 20.d odd 2 1
4320.2.d.j 20 24.f even 2 1
4320.2.d.j 20 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}^{10} + 48T_{7}^{8} + 782T_{7}^{6} + 5068T_{7}^{4} + 11001T_{7}^{2} + 7164$$ T7^10 + 48*T7^8 + 782*T7^6 + 5068*T7^4 + 11001*T7^2 + 7164 $$T_{11}^{10} + 53T_{11}^{8} + 972T_{11}^{6} + 7000T_{11}^{4} + 14928T_{11}^{2} + 144$$ T11^10 + 53*T11^8 + 972*T11^6 + 7000*T11^4 + 14928*T11^2 + 144 $$T_{13}^{5} + T_{13}^{4} - 34T_{13}^{3} + 22T_{13}^{2} + 189T_{13} - 243$$ T13^5 + T13^4 - 34*T13^3 + 22*T13^2 + 189*T13 - 243 $$T_{53}^{10} - 300T_{53}^{8} + 27832T_{53}^{6} - 866704T_{53}^{4} + 9083600T_{53}^{2} - 12239296$$ T53^10 - 300*T53^8 + 27832*T53^6 - 866704*T53^4 + 9083600*T53^2 - 12239296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - 3 T^{18} + 8 T^{16} - 24 T^{14} + \cdots + 1024$$
$3$ $$T^{20}$$
$5$ $$T^{20} - 10 T^{18} + 61 T^{16} + \cdots + 9765625$$
$7$ $$(T^{10} + 48 T^{8} + 782 T^{6} + 5068 T^{4} + \cdots + 7164)^{2}$$
$11$ $$(T^{10} + 53 T^{8} + 972 T^{6} + 7000 T^{4} + \cdots + 144)^{2}$$
$13$ $$(T^{5} + T^{4} - 34 T^{3} + 22 T^{2} + \cdots - 243)^{4}$$
$17$ $$(T^{10} + 69 T^{8} + 788 T^{6} + 3184 T^{4} + \cdots + 576)^{2}$$
$19$ $$(T^{10} + 132 T^{8} + 5798 T^{6} + \cdots + 114624)^{2}$$
$23$ $$(T^{10} + 101 T^{8} + 3516 T^{6} + \cdots + 41616)^{2}$$
$29$ $$(T^{10} + 81 T^{8} + 1608 T^{6} + \cdots + 11664)^{2}$$
$31$ $$(T^{5} - 3 T^{4} - 40 T^{3} + 168 T^{2} + \cdots + 32)^{4}$$
$37$ $$(T^{5} + 8 T^{4} - 48 T^{3} - 566 T^{2} + \cdots - 18)^{4}$$
$41$ $$(T^{10} - 252 T^{8} + 21704 T^{6} + \cdots - 83560896)^{2}$$
$43$ $$(T^{5} - 3 T^{4} - 124 T^{3} + 16 T^{2} + \cdots + 6192)^{4}$$
$47$ $$(T^{10} + 213 T^{8} + 14900 T^{6} + \cdots + 3779136)^{2}$$
$53$ $$(T^{10} - 300 T^{8} + 27832 T^{6} + \cdots - 12239296)^{2}$$
$59$ $$(T^{10} + 308 T^{8} + 30288 T^{6} + \cdots + 60466176)^{2}$$
$61$ $$(T^{10} + 236 T^{8} + 12406 T^{6} + \cdots + 114624)^{2}$$
$67$ $$(T^{5} - 226 T^{3} - 4 T^{2} + 7449 T - 12564)^{4}$$
$71$ $$(T^{10} - 464 T^{8} + 72312 T^{6} + \cdots - 16505856)^{2}$$
$73$ $$(T^{10} + 456 T^{8} + 68430 T^{6} + \cdots + 549686556)^{2}$$
$79$ $$(T^{5} - 9 T^{4} - 154 T^{3} + 1362 T^{2} + \cdots + 547)^{4}$$
$83$ $$(T^{10} - 352 T^{8} + 32808 T^{6} + \cdots - 50944)^{2}$$
$89$ $$(T^{10} - 428 T^{8} + 61968 T^{6} + \cdots - 16505856)^{2}$$
$97$ $$(T^{10} + 708 T^{8} + 180822 T^{6} + \cdots + 3391036416)^{2}$$