# Properties

 Label 1080.2.k.d Level $1080$ Weight $2$ Character orbit 1080.k 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(541,1080)]

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, 0]))

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

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

chi := DirichletCharacter("1080.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.k (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: $$x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024$$ x^20 - x^18 + 5*x^16 + 28*x^12 - 28*x^10 + 112*x^8 + 320*x^4 - 256*x^2 + 1024 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_{4} q^{4} - \beta_{6} q^{5} - \beta_{12} q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q - b3 * q^2 + b4 * q^4 - b6 * q^5 - b12 * q^7 + b2 * q^8 $$q - \beta_{3} q^{2} + \beta_{4} q^{4} - \beta_{6} q^{5} - \beta_{12} q^{7} + \beta_{2} q^{8} - \beta_{8} q^{10} + ( - \beta_{7} + \beta_{2}) q^{11} + (\beta_{19} - \beta_{18} - \beta_{8} - \beta_{4}) q^{13} + (\beta_{17} + \beta_{14} - \beta_{7}) q^{14} + (\beta_{13} + \beta_{10} - 1) q^{16} + (\beta_{17} - \beta_{16} + \beta_{15} - \beta_{7} - \beta_{2}) q^{17} + ( - \beta_{12} + \beta_{10} + \beta_{5} - \beta_{4}) q^{19} - \beta_{16} q^{20} + ( - \beta_{19} + \beta_{18} + \beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} + 1) q^{22} + ( - \beta_{17} - \beta_{16} - \beta_{15} - 2 \beta_{3}) q^{23} - q^{25} + ( - \beta_{15} + \beta_{11} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} + \beta_1) q^{26} + ( - \beta_{19} + \beta_{18} + \beta_{13} - \beta_{12} + \beta_{10} + \beta_{8} - \beta_{5} + 1) q^{28} + ( - \beta_{16} - \beta_{14} + \beta_{11} + \beta_{7} + 2 \beta_{6} - \beta_{3}) q^{29} + (\beta_{19} - 2 \beta_{13} + \beta_{12} + \beta_{9} - 3 \beta_{8} + \beta_{4} + 1) q^{31} + ( - \beta_{17} - \beta_{14} + \beta_{11} - 2 \beta_{6} + \beta_{3}) q^{32} + ( - \beta_{12} - \beta_{9} + \beta_{8} - \beta_{5}) q^{34} + \beta_{14} q^{35} + ( - \beta_{19} + 2 \beta_{18} + \beta_{9} - \beta_{8} - \beta_{4}) q^{37} + (\beta_{15} - 2 \beta_{14} + \beta_{11} - \beta_{7} + \beta_{6} - \beta_{2} + \beta_1) q^{38} - \beta_{18} q^{40} + ( - 2 \beta_{15} + \beta_{14} + 2 \beta_{3} - 2 \beta_1) q^{41} + (\beta_{19} + \beta_{12} + \beta_{9} - \beta_{8} - 2 \beta_{5} - \beta_{4}) q^{43} + ( - \beta_{17} + \beta_{15} - \beta_{14} + \beta_{11} + \beta_{6} - \beta_{3} - \beta_{2} - \beta_1) q^{44} + ( - \beta_{19} - \beta_{18} - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{5} + 2 \beta_{4} + 3) q^{46} + (\beta_{16} + 2 \beta_{15} - \beta_{14} + \beta_{11} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{47} + (\beta_{19} + \beta_{18} - \beta_{12} + \beta_{8} + \beta_{4} + 1) q^{49} + \beta_{3} q^{50} + ( - \beta_{19} - \beta_{18} + \beta_{13} - \beta_{12} - \beta_{10} - \beta_{8} + \beta_{5} + 3) q^{52} + (\beta_{16} - \beta_{11} - 2 \beta_{7} - \beta_{6} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{18} + \beta_{9}) q^{55} + (2 \beta_{14} - 2 \beta_{7} - 2 \beta_{3} - 2 \beta_1) q^{56} + ( - 2 \beta_{18} + 2 \beta_{13} - \beta_{12} - \beta_{9} + 2 \beta_{8} + \beta_{5} + 2 \beta_{4} + \cdots - 4) q^{58}+ \cdots + (\beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} + \beta_{11} - 3 \beta_{6} + \beta_{2} + 3 \beta_1) q^{98}+O(q^{100})$$ q - b3 * q^2 + b4 * q^4 - b6 * q^5 - b12 * q^7 + b2 * q^8 - b8 * q^10 + (-b7 + b2) * q^11 + (b19 - b18 - b8 - b4) * q^13 + (b17 + b14 - b7) * q^14 + (b13 + b10 - 1) * q^16 + (b17 - b16 + b15 - b7 - b2) * q^17 + (-b12 + b10 + b5 - b4) * q^19 - b16 * q^20 + (-b19 + b18 + b13 + b10 - b9 + b8 + 1) * q^22 + (-b17 - b16 - b15 - 2*b3) * q^23 - q^25 + (-b15 + b11 + b7 - b6 + b3 - b2 + b1) * q^26 + (-b19 + b18 + b13 - b12 + b10 + b8 - b5 + 1) * q^28 + (-b16 - b14 + b11 + b7 + 2*b6 - b3) * q^29 + (b19 - 2*b13 + b12 + b9 - 3*b8 + b4 + 1) * q^31 + (-b17 - b14 + b11 - 2*b6 + b3) * q^32 + (-b12 - b9 + b8 - b5) * q^34 + b14 * q^35 + (-b19 + 2*b18 + b9 - b8 - b4) * q^37 + (b15 - 2*b14 + b11 - b7 + b6 - b2 + b1) * q^38 - b18 * q^40 + (-2*b15 + b14 + 2*b3 - 2*b1) * q^41 + (b19 + b12 + b9 - b8 - 2*b5 - b4) * q^43 + (-b17 + b15 - b14 + b11 + b6 - b3 - b2 - b1) * q^44 + (-b19 - b18 - b13 + b12 - b10 + b5 + 2*b4 + 3) * q^46 + (b16 + 2*b15 - b14 + b11 - b3 - b2 + 2*b1) * q^47 + (b19 + b18 - b12 + b8 + b4 + 1) * q^49 + b3 * q^50 + (-b19 - b18 + b13 - b12 - b10 - b8 + b5 + 3) * q^52 + (b16 - b11 - 2*b7 - b6 - 3*b3 + b2 - 2*b1) * q^53 + (-b18 + b9) * q^55 + (2*b14 - 2*b7 - 2*b3 - 2*b1) * q^56 + (-2*b18 + 2*b13 - b12 - b9 + 2*b8 + b5 + 2*b4 - 4) * q^58 + (-b16 + b11 + 3*b3 + b2 + 2*b1) * q^59 + (b19 - b18 - b12 + b10 - b8 + b5 - 2*b4) * q^61 + (-b17 - 2*b16 - b14 + b7 + 4*b6 - b3 + 2*b2 + 2*b1) * q^62 + (-b19 + b13 + b12 - b10 - 2*b9 - b8 + b5 - 1) * q^64 + (b16 + b11 + b3 - b2) * q^65 + (b19 - 2*b13 + b9 + b8 - 3*b4) * q^67 + (b17 + b16 - b15 + 3*b14 + b11 + 3*b6 + b3 - b2 - b1) * q^68 + (b12 + b9 - b5) * q^70 + (-2*b16 - b7 - 2*b3 - b2) * q^71 + (b19 - b18 - 2*b13 + 2*b9 - 3*b8 + b4 - 2) * q^73 + (-2*b16 + 2*b15 - 2*b11 - 2*b7 - 2*b6 - 2*b3) * q^74 + (-b19 + b18 + b13 - b12 - b10 - 4*b9 + 3*b8 + b5 + b4 + 3) * q^76 + (b14 - b7 + 2*b6 - 4*b3 + b2 - 2*b1) * q^77 + (b19 + b18 + 2*b12 + b8 + b4 - 1) * q^79 + (-b15 + b6 - b1) * q^80 + (-2*b19 - b12 + b9 + b5 - 2*b4 - 4) * q^82 + (-b16 - b14 + b11 + 2*b7 + b6 + 3*b3 - b2 + 2*b1) * q^83 + (b18 + b10 + b9 - b5 - b4) * q^85 + (-b17 - 2*b15 + 3*b14 + b7 - 2*b6) * q^86 + (2*b12 - 2*b10 - 2*b9 + 2*b8 + 2*b4 - 6) * q^88 + (-2*b17 + b16 - b14 + b11 + b7 + b3) * q^89 + (b18 - 2*b13 + b12 - 2*b10 + b9 - 2*b4) * q^91 + (-b16 - 2*b14 - 2*b11 + 4*b6 - 4*b3 + 2*b2 - 2*b1) * q^92 + (b19 + b18 + b13 + b12 - b10 - 2*b9 + b8 - b5 + 2*b4 + 3) * q^94 + (b17 + b16 - b15 + b14) * q^95 + (b19 - 2*b18 - 2*b13 + 2*b12 + 3*b9 - 3*b8 + b4 + 2) * q^97 + (b17 + 2*b16 + b15 + b14 + b11 - 3*b6 + b2 + 3*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 2 q^{4}+O(q^{10})$$ 20 * q + 2 * q^4 $$20 q + 2 q^{4} - 2 q^{10} - 18 q^{16} + 16 q^{22} - 20 q^{25} + 16 q^{28} + 20 q^{31} - 6 q^{34} - 4 q^{40} + 54 q^{46} + 36 q^{49} + 56 q^{52} - 72 q^{58} - 28 q^{64} - 40 q^{73} + 58 q^{76} - 4 q^{79} - 92 q^{82} - 116 q^{88} + 72 q^{94} + 40 q^{97}+O(q^{100})$$ 20 * q + 2 * q^4 - 2 * q^10 - 18 * q^16 + 16 * q^22 - 20 * q^25 + 16 * q^28 + 20 * q^31 - 6 * q^34 - 4 * q^40 + 54 * q^46 + 36 * q^49 + 56 * q^52 - 72 * q^58 - 28 * q^64 - 40 * q^73 + 58 * q^76 - 4 * q^79 - 92 * q^82 - 116 * q^88 + 72 * q^94 + 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( -\nu^{19} - 3\nu^{17} - \nu^{15} - 20\nu^{13} - 28\nu^{11} - 84\nu^{9} - 448\nu^{5} - 320\nu^{3} - 1024\nu ) / 512$$ (-v^19 - 3*v^17 - v^15 - 20*v^13 - 28*v^11 - 84*v^9 - 448*v^5 - 320*v^3 - 1024*v) / 512 $$\beta_{3}$$ $$=$$ $$( \nu^{19} - \nu^{17} + 5\nu^{15} + 28\nu^{11} - 28\nu^{9} + 112\nu^{7} + 320\nu^{3} - 256\nu ) / 512$$ (v^19 - v^17 + 5*v^15 + 28*v^11 - 28*v^9 + 112*v^7 + 320*v^3 - 256*v) / 512 $$\beta_{4}$$ $$=$$ $$( -\nu^{18} + \nu^{16} - 5\nu^{14} - 28\nu^{10} + 28\nu^{8} - 112\nu^{6} - 320\nu^{2} + 256 ) / 256$$ (-v^18 + v^16 - 5*v^14 - 28*v^10 + 28*v^8 - 112*v^6 - 320*v^2 + 256) / 256 $$\beta_{5}$$ $$=$$ $$( 5 \nu^{18} + 61 \nu^{16} + 63 \nu^{14} + 98 \nu^{12} + 84 \nu^{10} + 1260 \nu^{8} + 280 \nu^{6} + 1792 \nu^{4} + 3392 \nu^{2} + 16256 ) / 896$$ (5*v^18 + 61*v^16 + 63*v^14 + 98*v^12 + 84*v^10 + 1260*v^8 + 280*v^6 + 1792*v^4 + 3392*v^2 + 16256) / 896 $$\beta_{6}$$ $$=$$ $$( - 17 \nu^{19} + 53 \nu^{17} + 7 \nu^{15} + 84 \nu^{13} - 252 \nu^{11} + 1260 \nu^{9} - 224 \nu^{7} + 1344 \nu^{5} - 960 \nu^{3} + 15872 \nu ) / 3584$$ (-17*v^19 + 53*v^17 + 7*v^15 + 84*v^13 - 252*v^11 + 1260*v^9 - 224*v^7 + 1344*v^5 - 960*v^3 + 15872*v) / 3584 $$\beta_{7}$$ $$=$$ $$( 19 \nu^{19} - 79 \nu^{17} - 21 \nu^{15} - 252 \nu^{13} + 420 \nu^{11} - 2212 \nu^{9} - 224 \nu^{7} - 3584 \nu^{5} + 4288 \nu^{3} - 31232 \nu ) / 3584$$ (19*v^19 - 79*v^17 - 21*v^15 - 252*v^13 + 420*v^11 - 2212*v^9 - 224*v^7 - 3584*v^5 + 4288*v^3 - 31232*v) / 3584 $$\beta_{8}$$ $$=$$ $$( - 17 \nu^{18} + 53 \nu^{16} + 7 \nu^{14} + 84 \nu^{12} - 252 \nu^{10} + 1260 \nu^{8} - 224 \nu^{6} + 1344 \nu^{4} - 960 \nu^{2} + 15872 ) / 1792$$ (-17*v^18 + 53*v^16 + 7*v^14 + 84*v^12 - 252*v^10 + 1260*v^8 - 224*v^6 + 1344*v^4 - 960*v^2 + 15872) / 1792 $$\beta_{9}$$ $$=$$ $$( 11 \nu^{18} - 17 \nu^{16} + 21 \nu^{14} - 70 \nu^{12} + 252 \nu^{10} - 364 \nu^{8} + 280 \nu^{6} - 448 \nu^{4} + 3072 \nu^{2} - 4736 ) / 896$$ (11*v^18 - 17*v^16 + 21*v^14 - 70*v^12 + 252*v^10 - 364*v^8 + 280*v^6 - 448*v^4 + 3072*v^2 - 4736) / 896 $$\beta_{10}$$ $$=$$ $$( 29 \nu^{18} + 71 \nu^{16} + 77 \nu^{14} + 84 \nu^{12} + 588 \nu^{10} + 1092 \nu^{8} + 1344 \nu^{6} + 1344 \nu^{4} + 9280 \nu^{2} + 12032 ) / 1792$$ (29*v^18 + 71*v^16 + 77*v^14 + 84*v^12 + 588*v^10 + 1092*v^8 + 1344*v^6 + 1344*v^4 + 9280*v^2 + 12032) / 1792 $$\beta_{11}$$ $$=$$ $$( - 9 \nu^{19} - 23 \nu^{17} - 21 \nu^{15} - 56 \nu^{13} - 196 \nu^{11} - 420 \nu^{9} - 336 \nu^{7} - 1120 \nu^{5} - 3776 \nu^{3} - 5248 \nu ) / 896$$ (-9*v^19 - 23*v^17 - 21*v^15 - 56*v^13 - 196*v^11 - 420*v^9 - 336*v^7 - 1120*v^5 - 3776*v^3 - 5248*v) / 896 $$\beta_{12}$$ $$=$$ $$( \nu^{18} - \nu^{16} + \nu^{14} - 4\nu^{12} + 16\nu^{10} - 36\nu^{8} + 32\nu^{6} - 16\nu^{4} + 160\nu^{2} - 384 ) / 64$$ (v^18 - v^16 + v^14 - 4*v^12 + 16*v^10 - 36*v^8 + 32*v^6 - 16*v^4 + 160*v^2 - 384) / 64 $$\beta_{13}$$ $$=$$ $$( - 9 \nu^{18} - 23 \nu^{16} - 21 \nu^{14} - 56 \nu^{12} - 196 \nu^{10} - 420 \nu^{8} - 336 \nu^{6} - 1120 \nu^{4} - 2880 \nu^{2} - 4352 ) / 448$$ (-9*v^18 - 23*v^16 - 21*v^14 - 56*v^12 - 196*v^10 - 420*v^8 - 336*v^6 - 1120*v^4 - 2880*v^2 - 4352) / 448 $$\beta_{14}$$ $$=$$ $$( 5 \nu^{19} - 23 \nu^{17} - 7 \nu^{15} - 56 \nu^{13} + 98 \nu^{11} - 532 \nu^{9} + 112 \nu^{7} - 616 \nu^{5} + 480 \nu^{3} - 6592 \nu ) / 448$$ (5*v^19 - 23*v^17 - 7*v^15 - 56*v^13 + 98*v^11 - 532*v^9 + 112*v^7 - 616*v^5 + 480*v^3 - 6592*v) / 448 $$\beta_{15}$$ $$=$$ $$( - 16 \nu^{19} - 37 \nu^{17} - 63 \nu^{15} - 77 \nu^{13} - 364 \nu^{11} - 756 \nu^{9} - 644 \nu^{7} - 1568 \nu^{5} - 4896 \nu^{3} - 9280 \nu ) / 896$$ (-16*v^19 - 37*v^17 - 63*v^15 - 77*v^13 - 364*v^11 - 756*v^9 - 644*v^7 - 1568*v^5 - 4896*v^3 - 9280*v) / 896 $$\beta_{16}$$ $$=$$ $$( - 31 \nu^{19} - 3 \nu^{17} - 49 \nu^{15} + 14 \nu^{13} - 700 \nu^{11} + 364 \nu^{9} - 952 \nu^{7} - 448 \nu^{5} - 7232 \nu^{3} + 6016 \nu ) / 1792$$ (-31*v^19 - 3*v^17 - 49*v^15 + 14*v^13 - 700*v^11 + 364*v^9 - 952*v^7 - 448*v^5 - 7232*v^3 + 6016*v) / 1792 $$\beta_{17}$$ $$=$$ $$( - 9 \nu^{19} + 5 \nu^{17} - 9 \nu^{15} + 12 \nu^{13} - 156 \nu^{11} + 204 \nu^{9} - 256 \nu^{7} - 320 \nu^{5} - 1600 \nu^{3} + 2048 \nu ) / 512$$ (-9*v^19 + 5*v^17 - 9*v^15 + 12*v^13 - 156*v^11 + 204*v^9 - 256*v^7 - 320*v^5 - 1600*v^3 + 2048*v) / 512 $$\beta_{18}$$ $$=$$ $$( - 31 \nu^{18} - 3 \nu^{16} - 49 \nu^{14} + 14 \nu^{12} - 700 \nu^{10} + 364 \nu^{8} - 952 \nu^{6} - 448 \nu^{4} - 7232 \nu^{2} + 6016 ) / 896$$ (-31*v^18 - 3*v^16 - 49*v^14 + 14*v^12 - 700*v^10 + 364*v^8 - 952*v^6 - 448*v^4 - 7232*v^2 + 6016) / 896 $$\beta_{19}$$ $$=$$ $$( -41\nu^{18} + \nu^{16} - 77\nu^{14} - 868\nu^{10} + 252\nu^{8} - 1456\nu^{6} - 1120\nu^{4} - 8640\nu^{2} + 4864 ) / 896$$ (-41*v^18 + v^16 - 77*v^14 - 868*v^10 + 252*v^8 - 1456*v^6 - 1120*v^4 - 8640*v^2 + 4864) / 896
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{19} - \beta_{13} + \beta_{9} - \beta_{8} - \beta_{4} ) / 2$$ (b19 - b13 + b9 - b8 - b4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{16} - \beta_{11} + \beta_{7} + \beta_{3} ) / 2$$ (b16 - b11 + b7 + b3) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{13} + 2\beta_{12} - 2\beta_{10} + 2\beta_{8} - 2 ) / 2$$ (-b13 + 2*b12 - 2*b10 + 2*b8 - 2) / 2 $$\nu^{5}$$ $$=$$ $$( -2\beta_{17} + \beta_{16} + \beta_{11} + \beta_{7} + 4\beta_{6} + \beta_{3} - 2\beta_{2} - \beta_1 ) / 2$$ (-2*b17 + b16 + b11 + b7 + 4*b6 + b3 - 2*b2 - b1) / 2 $$\nu^{6}$$ $$=$$ $$( -\beta_{19} + 2\beta_{18} + 2\beta_{12} + 2\beta_{10} - \beta_{9} + 3\beta_{8} - 2\beta_{5} - 3\beta_{4} - 2 ) / 2$$ (-b19 + 2*b18 + 2*b12 + 2*b10 - b9 + 3*b8 - 2*b5 - 3*b4 - 2) / 2 $$\nu^{7}$$ $$=$$ $$( - 2 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{11} - 6 \beta_{7} + 2 \beta_{6} + 4 \beta_{3} + 4 \beta_{2} - 3 \beta_1 ) / 2$$ (-2*b16 + 2*b15 + 2*b14 - 2*b11 - 6*b7 + 2*b6 + 4*b3 + 4*b2 - 3*b1) / 2 $$\nu^{8}$$ $$=$$ $$( - 5 \beta_{19} + 6 \beta_{18} + \beta_{13} - 4 \beta_{12} + 4 \beta_{10} + 3 \beta_{9} + \beta_{8} - 2 \beta_{5} + 5 \beta_{4} - 16 ) / 2$$ (-5*b19 + 6*b18 + b13 - 4*b12 + 4*b10 + 3*b9 + b8 - 2*b5 + 5*b4 - 16) / 2 $$\nu^{9}$$ $$=$$ $$( 6 \beta_{17} + 5 \beta_{16} - 6 \beta_{15} + 10 \beta_{14} - 3 \beta_{11} - 7 \beta_{7} - 2 \beta_{6} - 13 \beta_{3} + 2 \beta_{2} - 14 \beta_1 ) / 2$$ (6*b17 + 5*b16 - 6*b15 + 10*b14 - 3*b11 - 7*b7 - 2*b6 - 13*b3 + 2*b2 - 14*b1) / 2 $$\nu^{10}$$ $$=$$ $$( - 8 \beta_{19} - 8 \beta_{18} + 9 \beta_{13} - 10 \beta_{12} - 2 \beta_{10} - 4 \beta_{9} + 18 \beta_{8} - 4 \beta_{5} + 8 \beta_{4} + 22 ) / 2$$ (-8*b19 - 8*b18 + 9*b13 - 10*b12 - 2*b10 - 4*b9 + 18*b8 - 4*b5 + 8*b4 + 22) / 2 $$\nu^{11}$$ $$=$$ $$( 14 \beta_{17} - 21 \beta_{16} + 8 \beta_{14} + 11 \beta_{11} - 5 \beta_{7} + 20 \beta_{6} - 5 \beta_{3} - 18 \beta_{2} + 13 \beta_1 ) / 2$$ (14*b17 - 21*b16 + 8*b14 + 11*b11 - 5*b7 + 20*b6 - 5*b3 - 18*b2 + 13*b1) / 2 $$\nu^{12}$$ $$=$$ $$( 13 \beta_{19} - 6 \beta_{18} - 12 \beta_{13} - 30 \beta_{12} + 2 \beta_{10} - 7 \beta_{9} - 55 \beta_{8} + 6 \beta_{5} - 29 \beta_{4} + 30 ) / 2$$ (13*b19 - 6*b18 - 12*b13 - 30*b12 + 2*b10 - 7*b9 - 55*b8 + 6*b5 - 29*b4 + 30) / 2 $$\nu^{13}$$ $$=$$ $$( 24 \beta_{17} - 14 \beta_{16} + 14 \beta_{15} - 26 \beta_{14} - 14 \beta_{11} - 2 \beta_{7} - 98 \beta_{6} + 44 \beta_{3} - 20 \beta_{2} + 15 \beta_1 ) / 2$$ (24*b17 - 14*b16 + 14*b15 - 26*b14 - 14*b11 - 2*b7 - 98*b6 + 44*b3 - 20*b2 + 15*b1) / 2 $$\nu^{14}$$ $$=$$ $$( \beta_{19} - 22 \beta_{18} + 27 \beta_{13} - 12 \beta_{12} - 36 \beta_{10} - 7 \beta_{9} - 29 \beta_{8} + 50 \beta_{5} - 33 \beta_{4} - 80 ) / 2$$ (b19 - 22*b18 + 27*b13 - 12*b12 - 36*b10 - 7*b9 - 29*b8 + 50*b5 - 33*b4 - 80) / 2 $$\nu^{15}$$ $$=$$ $$( - 38 \beta_{17} + 63 \beta_{16} - 42 \beta_{15} - 58 \beta_{14} + 63 \beta_{11} + 107 \beta_{7} - 14 \beta_{6} + 129 \beta_{3} - 34 \beta_{2} + 6 \beta_1 ) / 2$$ (-38*b17 + 63*b16 - 42*b15 - 58*b14 + 63*b11 + 107*b7 - 14*b6 + 129*b3 - 34*b2 + 6*b1) / 2 $$\nu^{16}$$ $$=$$ $$( 48 \beta_{19} - 104 \beta_{18} + 43 \beta_{13} + 82 \beta_{12} - 6 \beta_{10} - 84 \beta_{9} + 78 \beta_{8} + 20 \beta_{5} + 16 \beta_{4} - 126 ) / 2$$ (48*b19 - 104*b18 + 43*b13 + 82*b12 - 6*b10 - 84*b9 + 78*b8 + 20*b5 + 16*b4 - 126) / 2 $$\nu^{17}$$ $$=$$ $$( - 102 \beta_{17} - 175 \beta_{16} + 112 \beta_{15} - 152 \beta_{14} + 49 \beta_{11} + 33 \beta_{7} + 140 \beta_{6} + 17 \beta_{3} + 90 \beta_{2} + 31 \beta_1 ) / 2$$ (-102*b17 - 175*b16 + 112*b15 - 152*b14 + 49*b11 + 33*b7 + 140*b6 + 17*b3 + 90*b2 + 31*b1) / 2 $$\nu^{18}$$ $$=$$ $$( - 81 \beta_{19} + 174 \beta_{18} + 4 \beta_{13} + 86 \beta_{12} + 118 \beta_{10} - 61 \beta_{9} - 269 \beta_{8} + 50 \beta_{5} + 241 \beta_{4} - 54 ) / 2$$ (-81*b19 + 174*b18 + 4*b13 + 86*b12 + 118*b10 - 61*b9 - 269*b8 + 50*b5 + 241*b4 - 54) / 2 $$\nu^{19}$$ $$=$$ $$( - 136 \beta_{17} + 142 \beta_{16} - 70 \beta_{15} - 30 \beta_{14} - 114 \beta_{11} - 206 \beta_{7} - 630 \beta_{6} - 596 \beta_{3} + 372 \beta_{2} - 163 \beta_1 ) / 2$$ (-136*b17 + 142*b16 - 70*b15 - 30*b14 - 114*b11 - 206*b7 - 630*b6 - 596*b3 + 372*b2 - 163*b1) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 −1.34522 + 0.436333i −1.34522 − 0.436333i −1.19357 + 0.758543i −1.19357 − 0.758543i −1.17425 + 0.788128i −1.17425 − 0.788128i −0.662801 + 1.24928i −0.662801 − 1.24928i −0.444539 + 1.34253i −0.444539 − 1.34253i 0.444539 + 1.34253i 0.444539 − 1.34253i 0.662801 + 1.24928i 0.662801 − 1.24928i 1.17425 + 0.788128i 1.17425 − 0.788128i 1.19357 + 0.758543i 1.19357 − 0.758543i 1.34522 + 0.436333i 1.34522 − 0.436333i
−1.34522 0.436333i 0 1.61923 + 1.17393i 1.00000i 0 −3.61492 −1.66599 2.28571i 0 −0.436333 + 1.34522i
541.2 −1.34522 + 0.436333i 0 1.61923 1.17393i 1.00000i 0 −3.61492 −1.66599 + 2.28571i 0 −0.436333 1.34522i
541.3 −1.19357 0.758543i 0 0.849224 + 1.81075i 1.00000i 0 4.63236 0.359923 2.80543i 0 −0.758543 + 1.19357i
541.4 −1.19357 + 0.758543i 0 0.849224 1.81075i 1.00000i 0 4.63236 0.359923 + 2.80543i 0 −0.758543 1.19357i
541.5 −1.17425 0.788128i 0 0.757708 + 1.85091i 1.00000i 0 2.12726 0.569020 2.77060i 0 0.788128 1.17425i
541.6 −1.17425 + 0.788128i 0 0.757708 1.85091i 1.00000i 0 2.12726 0.569020 + 2.77060i 0 0.788128 + 1.17425i
541.7 −0.662801 1.24928i 0 −1.12139 + 1.65605i 1.00000i 0 −1.52861 2.81212 + 0.303298i 0 1.24928 0.662801i
541.8 −0.662801 + 1.24928i 0 −1.12139 1.65605i 1.00000i 0 −1.52861 2.81212 0.303298i 0 1.24928 + 0.662801i
541.9 −0.444539 1.34253i 0 −1.60477 + 1.19361i 1.00000i 0 −1.61609 2.31584 + 1.62384i 0 −1.34253 + 0.444539i
541.10 −0.444539 + 1.34253i 0 −1.60477 1.19361i 1.00000i 0 −1.61609 2.31584 1.62384i 0 −1.34253 0.444539i
541.11 0.444539 1.34253i 0 −1.60477 1.19361i 1.00000i 0 −1.61609 −2.31584 + 1.62384i 0 −1.34253 0.444539i
541.12 0.444539 + 1.34253i 0 −1.60477 + 1.19361i 1.00000i 0 −1.61609 −2.31584 1.62384i 0 −1.34253 + 0.444539i
541.13 0.662801 1.24928i 0 −1.12139 1.65605i 1.00000i 0 −1.52861 −2.81212 + 0.303298i 0 1.24928 + 0.662801i
541.14 0.662801 + 1.24928i 0 −1.12139 + 1.65605i 1.00000i 0 −1.52861 −2.81212 0.303298i 0 1.24928 0.662801i
541.15 1.17425 0.788128i 0 0.757708 1.85091i 1.00000i 0 2.12726 −0.569020 2.77060i 0 0.788128 + 1.17425i
541.16 1.17425 + 0.788128i 0 0.757708 + 1.85091i 1.00000i 0 2.12726 −0.569020 + 2.77060i 0 0.788128 1.17425i
541.17 1.19357 0.758543i 0 0.849224 1.81075i 1.00000i 0 4.63236 −0.359923 2.80543i 0 −0.758543 1.19357i
541.18 1.19357 + 0.758543i 0 0.849224 + 1.81075i 1.00000i 0 4.63236 −0.359923 + 2.80543i 0 −0.758543 + 1.19357i
541.19 1.34522 0.436333i 0 1.61923 1.17393i 1.00000i 0 −3.61492 1.66599 2.28571i 0 −0.436333 1.34522i
541.20 1.34522 + 0.436333i 0 1.61923 + 1.17393i 1.00000i 0 −3.61492 1.66599 + 2.28571i 0 −0.436333 + 1.34522i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.20 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
8.b even 2 1 inner
24.h 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.k.d 20
3.b odd 2 1 inner 1080.2.k.d 20
4.b odd 2 1 4320.2.k.d 20
8.b even 2 1 inner 1080.2.k.d 20
8.d odd 2 1 4320.2.k.d 20
12.b even 2 1 4320.2.k.d 20
24.f even 2 1 4320.2.k.d 20
24.h odd 2 1 inner 1080.2.k.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.k.d 20 1.a even 1 1 trivial
1080.2.k.d 20 3.b odd 2 1 inner
1080.2.k.d 20 8.b even 2 1 inner
1080.2.k.d 20 24.h odd 2 1 inner
4320.2.k.d 20 4.b odd 2 1
4320.2.k.d 20 8.d odd 2 1
4320.2.k.d 20 12.b even 2 1
4320.2.k.d 20 24.f even 2 1

## Hecke kernels

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

 $$T_{7}^{5} - 22T_{7}^{3} - 18T_{7}^{2} + 76T_{7} + 88$$ T7^5 - 22*T7^3 - 18*T7^2 + 76*T7 + 88 $$T_{17}^{10} - 119T_{17}^{8} + 5122T_{17}^{6} - 94474T_{17}^{4} + 633189T_{17}^{2} - 34839$$ T17^10 - 119*T17^8 + 5122*T17^6 - 94474*T17^4 + 633189*T17^2 - 34839

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - T^{18} + 5 T^{16} + 28 T^{12} + \cdots + 1024$$
$3$ $$T^{20}$$
$5$ $$(T^{2} + 1)^{10}$$
$7$ $$(T^{5} - 22 T^{3} - 18 T^{2} + 76 T + 88)^{4}$$
$11$ $$(T^{10} + 72 T^{8} + 1688 T^{6} + \cdots + 7744)^{2}$$
$13$ $$(T^{10} + 80 T^{8} + 2104 T^{6} + \cdots + 45504)^{2}$$
$17$ $$(T^{10} - 119 T^{8} + 5122 T^{6} + \cdots - 34839)^{2}$$
$19$ $$(T^{10} + 147 T^{8} + 7646 T^{6} + \cdots + 5464351)^{2}$$
$23$ $$(T^{10} - 171 T^{8} + 10898 T^{6} + \cdots - 21608791)^{2}$$
$29$ $$(T^{10} + 216 T^{8} + 13572 T^{6} + \cdots + 14400)^{2}$$
$31$ $$(T^{5} - 5 T^{4} - 96 T^{3} + 466 T^{2} + \cdots - 4309)^{4}$$
$37$ $$(T^{10} + 264 T^{8} + 22160 T^{6} + \cdots + 5177344)^{2}$$
$41$ $$(T^{10} - 312 T^{8} + 36356 T^{6} + \cdots - 288803776)^{2}$$
$43$ $$(T^{10} + 284 T^{8} + 29724 T^{6} + \cdots + 154840000)^{2}$$
$47$ $$(T^{10} - 316 T^{8} + 36176 T^{6} + \cdots - 12640000)^{2}$$
$53$ $$(T^{10} + 269 T^{8} + 23902 T^{6} + \cdots + 38626225)^{2}$$
$59$ $$(T^{10} + 280 T^{8} + 16040 T^{6} + \cdots + 732736)^{2}$$
$61$ $$(T^{10} + 243 T^{8} + 21722 T^{6} + \cdots + 1234375)^{2}$$
$67$ $$(T^{10} + 312 T^{8} + 32016 T^{6} + \cdots + 72806400)^{2}$$
$71$ $$(T^{10} - 176 T^{8} + 8520 T^{6} + \cdots - 126400)^{2}$$
$73$ $$(T^{5} + 10 T^{4} - 92 T^{3} - 978 T^{2} + \cdots + 6696)^{4}$$
$79$ $$(T^{5} + T^{4} - 174 T^{3} - 848 T^{2} + \cdots - 745)^{4}$$
$83$ $$(T^{10} + 269 T^{8} + 27954 T^{6} + \cdots + 268992801)^{2}$$
$89$ $$(T^{10} - 284 T^{8} + 29724 T^{6} + \cdots - 154840000)^{2}$$
$97$ $$(T^{5} - 10 T^{4} - 228 T^{3} + \cdots + 12928)^{4}$$