# Properties

 Label 570.2.m.b Level $570$ Weight $2$ Character orbit 570.m Analytic conductor $4.551$ 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] = [570,2,Mod(37,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("570.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.m (of order $$4$$, 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: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} + 108x^{16} + 1318x^{12} + 4652x^{8} + 5057x^{4} + 256$$ x^20 + 108*x^16 + 1318*x^12 + 4652*x^8 + 5057*x^4 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} - \beta_{10} q^{3} + \beta_{2} q^{4} + (\beta_{6} + 1) q^{5} - q^{6} + (\beta_{16} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{7} - \beta_{10} q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b7 * q^2 - b10 * q^3 + b2 * q^4 + (b6 + 1) * q^5 - q^6 + (b16 - b6 - b3 + b2 - 1) * q^7 - b10 * q^8 - b2 * q^9 $$q + \beta_{7} q^{2} - \beta_{10} q^{3} + \beta_{2} q^{4} + (\beta_{6} + 1) q^{5} - q^{6} + (\beta_{16} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{7} - \beta_{10} q^{8} - \beta_{2} q^{9} + ( - \beta_{11} + \beta_{7}) q^{10} + (\beta_{16} + \beta_{9} + \beta_{6} + \beta_{4}) q^{11} - \beta_{7} q^{12} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_1) q^{13} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{5} - \beta_1) q^{14} + ( - \beta_{14} - \beta_{10}) q^{15} - q^{16} + ( - \beta_{17} - \beta_{15} + \beta_{9} + \beta_{6}) q^{17} + \beta_{10} q^{18} + (\beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{12} - \beta_{9} - \beta_{7}) q^{19} + (\beta_{8} + \beta_{2}) q^{20} + (\beta_{10} + \beta_{5} + \beta_1) q^{21} + ( - \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{7} + \beta_{5}) q^{22} + ( - \beta_{9} + 2 \beta_{8} + \beta_{6} + 2 \beta_{4} + \beta_{2} + 1) q^{23} - \beta_{2} q^{24} + ( - \beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots - \beta_1) q^{25}+ \cdots + (\beta_{9} - \beta_{8} - \beta_{3}) q^{99}+O(q^{100})$$ q + b7 * q^2 - b10 * q^3 + b2 * q^4 + (b6 + 1) * q^5 - q^6 + (b16 - b6 - b3 + b2 - 1) * q^7 - b10 * q^8 - b2 * q^9 + (-b11 + b7) * q^10 + (b16 + b9 + b6 + b4) * q^11 - b7 * q^12 + (-b14 + b13 - b12 + b11 - b1) * q^13 + (-b14 + b13 - b12 + b11 - b10 + b5 - b1) * q^14 + (-b14 - b10) * q^15 - q^16 + (-b17 - b15 + b9 + b6) * q^17 + b10 * q^18 + (b19 - b18 + b17 - b16 + b12 - b9 - b7) * q^19 + (b8 + b2) * q^20 + (b10 + b5 + b1) * q^21 + (-b14 - 2*b12 - b11 + b7 + b5) * q^22 + (-b9 + 2*b8 + b6 + 2*b4 + b2 + 1) * q^23 - b2 * q^24 + (-b17 - b16 + b15 - 2*b14 + b13 - 2*b12 + b11 - 2*b10 + b7 + 2*b6 + b5 + b4 + b2 - b1) * q^25 + (b16 - b8 - b6 - b4) * q^26 + b7 * q^27 + (b16 + b9 - b8 - b4 + b3 - b2 - 1) * q^28 + (b19 - b18 - b16 - b14 + b13 + b12 + b11 - b10 - b9 - b7) * q^29 + (-b6 - 1) * q^30 + (-b19 - b18 + b13 - b11 - b10 - b9 + b8 + 2*b7 - b6 + b5 + b4 - b3 + 2*b2 + b1) * q^31 - b7 * q^32 + (-b14 + b13 + b1) * q^33 + (b19 - b18 - b16 - b13 - b11 - b9) * q^34 + (-b17 + 2*b16 + b15 - 2*b14 + b13 - 2*b12 + b11 - 2*b10 - b9 + b7 - 2*b6 + b5 - b4 - b3 + 2*b2 - b1 - 2) * q^35 + q^36 + (2*b18 + b16 - b13 - b11 + 2*b9 - b8 + b7 + b6 - b5 - b4 + b3 - 2*b2) * q^37 + (-b19 + b17 + b16 - b15 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 + b9 - b7 - b6 - b5 + b1 - 1) * q^38 + (b9 - b8 + b6 - b4 + b3) * q^39 + (-b14 - b10) * q^40 + (-b13 + b11) * q^41 + (-b16 + b6 + b3 - b2 + 1) * q^42 + (b17 - b16 - b15 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 - b9 - b7 - b5 - b3 + b1) * q^43 + (-b9 + b8 + b3) * q^44 + (-b8 - b2) * q^45 + (-2*b14 + b13 - 2*b12 - b11 - b10 + b7) * q^46 + (b16 + b9 + b8 - b4 - b3 + 3*b2 - 3) * q^47 + b10 * q^48 + (2*b17 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 + 3*b9 - 3*b8 - b7 - b6 - b5 + b4 - 5*b2 + b1) * q^49 + (b19 + b18 + b14 - b13 - 2*b11 - b10 + b9 - b8 - b7 + b6 - b5 - b4 + b3 - 2*b2) * q^50 + (-b19 - b18 - b14 - b12 - b9 + b8 - b6 + b4 - b3 + 2*b2) * q^51 + (b13 + b11 + b7 + b5) * q^52 + (-2*b19 + b16 + 2*b14 - 2*b13 + 2*b12 - 2*b11 + b8 - b6 + b4 - b3 + 2*b2 + 2*b1) * q^53 + b2 * q^54 + (-3*b17 + b15 - 4*b14 + 2*b13 - 4*b12 + 2*b11 - 4*b10 + 2*b7 + 2*b5 - b2 - 2*b1 + 3) * q^55 + (b10 + b5 + b1) * q^56 + (b18 + b17 + b15 + b9 - b8 + b6 + b3 - 2*b2 + 1) * q^57 + (b17 - b15 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 + b9 - b8 - b7 - b6 - b5 - b4 - b2 + b1 - 1) * q^58 + (b19 - b18 - b16 - 2*b12 - b10 - b9 + b5 - b1) * q^59 + (b11 - b7) * q^60 + (-4*b16 - 3*b9 + b8 + 3*b6 + 3*b4 + 4) * q^61 + (b17 - b16 + b15 + b8 + b6 - b4 + b3 + b2 - 1) * q^62 + (-b16 - b9 + b8 + b4 - b3 + b2 + 1) * q^63 - b2 * q^64 + (-b14 + 3*b13 - 2*b12 + 2*b11 - 2*b10 + 3*b7 + b5 - 2*b1) * q^65 + (-b16 - b9 - b6 - b4) * q^66 + (4*b18 + 2*b16 + 4*b9 - 2*b8 + 2*b6 - 2*b4 + 2*b3 - 4*b2) * q^67 + (b17 - b15 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 + b8 - b7 - b5 + b4 + b1) * q^68 + (-b14 + 2*b13 + b12 + 2*b11 - b10 - b7) * q^69 + (b19 + b18 - 2*b14 + 3*b13 - b12 + 2*b11 - 2*b10 + b9 - b8 + b6 + 2*b5 - b4 + b3 - 2*b2 - b1) * q^70 + (-2*b10 - 2*b5 - 2*b1) * q^71 + b7 * q^72 + (b16 + 2*b9 - 5*b8 - b6 - 5*b4 + b3 - 3*b2 - 3) * q^73 + (-2*b17 - 2*b14 + b13 - 2*b12 + b11 - 2*b10 - b9 + b8 + b7 - b6 + b5 + b4 - b3 + 2*b2 - b1) * q^74 + (b19 - b18 - b16 - 2*b14 + b13 - b12 - b9 - b7 - b1) * q^75 + (-b19 - b18 + b15 - b14 - b12 + b11 - b9 + b8 - b6 + b5 + b4 - b3 + 2*b2) * q^76 + (2*b16 - 2*b9 - 4*b6 - 2*b3 - 2*b2 + 2) * q^77 + (b14 - b13 + b12 - b11 + b1) * q^78 + (b19 - b18 - b16 + b14 + b12 + 5*b10 - b9 + 4*b7 - b5 + b1) * q^79 + (-b6 - 1) * q^80 - q^81 + (-b8 + b4) * q^82 + (b17 + b16 - b15 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 - b7 + b6 - b5 + b3 + 4*b2 + b1 + 4) * q^83 + (b14 - b13 + b12 - b11 + b10 - b5 + b1) * q^84 + (-b17 - 3*b16 + b15 - 2*b14 + b13 - 2*b12 + b11 - 2*b10 - b9 + b8 + b7 + b5 - b4 - b3 - 2*b2 - b1) * q^85 + (-b19 - b18 + b14 + b12 - b9 + b8 - b7 - b6 - b5 + b4 - b3 + 2*b2 - b1) * q^86 + (b17 + b15 + b9 - b8 + b6 + b4 - b2 + 1) * q^87 + (-b14 + b13 + b1) * q^88 + (b14 - 3*b13 + b12 - 3*b11 + 2*b10 + b7 - b5 + b1) * q^89 + (b14 + b10) * q^90 + (-2*b19 - 2*b18 + 2*b14 - 3*b13 + 2*b12 + 3*b11 + 6*b10 - 2*b9 + 2*b8 - 7*b7 - 2*b6 - b5 + 2*b4 - 2*b3 + 4*b2 - b1) * q^91 + (-2*b9 + b8 - 2*b6 - b4 + b2 - 1) * q^92 + (-b17 - b16 + b15 - 2*b14 + b13 - 2*b12 + b11 - 2*b10 + b8 + b7 - b6 + b5 + b4 - b3 - b2 - b1 - 1) * q^93 + (-2*b14 - 3*b10 - 2*b7 + b5 - b1) * q^94 + (b19 + b16 + b13 + 3*b12 + 2*b11 - 2*b10 + 2*b9 - b8 - 4*b7 + b6 - b5 + b4 - b2 - 2*b1 - 1) * q^95 + q^96 + (2*b18 + b16 + 4*b14 - b13 + 3*b12 - 2*b11 + 2*b9 - b8 - 3*b7 + b6 - 5*b5 - b4 + b3 - 2*b2) * q^97 + (-2*b19 + b16 + 3*b14 - 3*b13 - b12 + b11 + 5*b10 + b8 - b6 + b4 - b3 + 2*b2) * q^98 + (b9 - b8 - b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 12 q^{5} - 20 q^{6} - 4 q^{7}+O(q^{10})$$ 20 * q + 12 * q^5 - 20 * q^6 - 4 * q^7 $$20 q + 12 q^{5} - 20 q^{6} - 4 q^{7} - 8 q^{11} - 20 q^{16} - 12 q^{17} - 4 q^{23} - 28 q^{25} + 24 q^{26} - 4 q^{28} - 12 q^{30} + 4 q^{35} + 20 q^{36} - 12 q^{38} + 4 q^{42} - 12 q^{43} - 44 q^{47} + 64 q^{55} + 12 q^{57} - 8 q^{58} - 24 q^{62} + 4 q^{63} + 8 q^{66} - 12 q^{68} - 4 q^{73} + 4 q^{76} + 88 q^{77} - 12 q^{80} - 20 q^{81} - 8 q^{82} + 76 q^{83} - 12 q^{85} + 8 q^{87} + 4 q^{92} - 24 q^{93} - 24 q^{95} + 20 q^{96}+O(q^{100})$$ 20 * q + 12 * q^5 - 20 * q^6 - 4 * q^7 - 8 * q^11 - 20 * q^16 - 12 * q^17 - 4 * q^23 - 28 * q^25 + 24 * q^26 - 4 * q^28 - 12 * q^30 + 4 * q^35 + 20 * q^36 - 12 * q^38 + 4 * q^42 - 12 * q^43 - 44 * q^47 + 64 * q^55 + 12 * q^57 - 8 * q^58 - 24 * q^62 + 4 * q^63 + 8 * q^66 - 12 * q^68 - 4 * q^73 + 4 * q^76 + 88 * q^77 - 12 * q^80 - 20 * q^81 - 8 * q^82 + 76 * q^83 - 12 * q^85 + 8 * q^87 + 4 * q^92 - 24 * q^93 - 24 * q^95 + 20 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 108x^{16} + 1318x^{12} + 4652x^{8} + 5057x^{4} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -1233\nu^{19} - 120336\nu^{15} - 289166\nu^{11} + 6031344\nu^{7} + 9961151\nu^{3} ) / 5112544$$ (-1233*v^19 - 120336*v^15 - 289166*v^11 + 6031344*v^7 + 9961151*v^3) / 5112544 $$\beta_{2}$$ $$=$$ $$( 1843\nu^{18} + 199306\nu^{14} + 2457754\nu^{10} + 8963426\nu^{6} + 11554103\nu^{2} ) / 2556272$$ (1843*v^18 + 199306*v^14 + 2457754*v^10 + 8963426*v^6 + 11554103*v^2) / 2556272 $$\beta_{3}$$ $$=$$ $$( 2595\nu^{18} + 273087\nu^{14} + 2661844\nu^{10} + 4366379\nu^{6} - 932561\nu^{2} ) / 1278136$$ (2595*v^18 + 273087*v^14 + 2661844*v^10 + 4366379*v^6 - 932561*v^2) / 1278136 $$\beta_{4}$$ $$=$$ $$( 4296 \nu^{18} - 7207 \nu^{16} + 477582 \nu^{14} - 770625 \nu^{12} + 7084096 \nu^{10} - 8670841 \nu^{8} + 32921622 \nu^{6} - 24076543 \nu^{4} + 42067188 \nu^{2} + \cdots - 13013696 ) / 5112544$$ (4296*v^18 - 7207*v^16 + 477582*v^14 - 770625*v^12 + 7084096*v^10 - 8670841*v^8 + 32921622*v^6 - 24076543*v^4 + 42067188*v^2 - 13013696) / 5112544 $$\beta_{5}$$ $$=$$ $$( - 3686 \nu^{19} + 4131 \nu^{17} - 398612 \nu^{15} + 450983 \nu^{13} - 4915508 \nu^{11} + 5923921 \nu^{9} - 17926852 \nu^{7} + 21144461 \nu^{5} + \cdots + 8864472 \nu ) / 5112544$$ (-3686*v^19 + 4131*v^17 - 398612*v^15 + 450983*v^13 - 4915508*v^11 + 5923921*v^9 - 17926852*v^7 + 21144461*v^5 - 23108206*v^3 + 8864472*v) / 5112544 $$\beta_{6}$$ $$=$$ $$( - 4296 \nu^{18} - 7207 \nu^{16} - 477582 \nu^{14} - 770625 \nu^{12} - 7084096 \nu^{10} - 8670841 \nu^{8} - 32921622 \nu^{6} - 24076543 \nu^{4} + \cdots - 13013696 ) / 5112544$$ (-4296*v^18 - 7207*v^16 - 477582*v^14 - 770625*v^12 - 7084096*v^10 - 8670841*v^8 - 32921622*v^6 - 24076543*v^4 - 42067188*v^2 - 13013696) / 5112544 $$\beta_{7}$$ $$=$$ $$( -10283\nu^{17} - 1090267\nu^{13} - 11417761\nu^{9} - 27008625\nu^{5} - 12050376\nu ) / 5112544$$ (-10283*v^17 - 1090267*v^13 - 11417761*v^9 - 27008625*v^5 - 12050376*v) / 5112544 $$\beta_{8}$$ $$=$$ $$( - 14579 \nu^{18} - 5890 \nu^{16} - 1567849 \nu^{14} - 610604 \nu^{12} - 18501857 \nu^{10} - 5104050 \nu^{8} - 59930247 \nu^{6} - 3630112 \nu^{4} + \cdots + 9123648 ) / 5112544$$ (-14579*v^18 - 5890*v^16 - 1567849*v^14 - 610604*v^12 - 18501857*v^10 - 5104050*v^8 - 59930247*v^6 - 3630112*v^4 - 54117564*v^2 + 9123648) / 5112544 $$\beta_{9}$$ $$=$$ $$( 14579 \nu^{18} - 5890 \nu^{16} + 1567849 \nu^{14} - 610604 \nu^{12} + 18501857 \nu^{10} - 5104050 \nu^{8} + 59930247 \nu^{6} - 3630112 \nu^{4} + 54117564 \nu^{2} + \cdots + 9123648 ) / 5112544$$ (14579*v^18 - 5890*v^16 + 1567849*v^14 - 610604*v^12 + 18501857*v^10 - 5104050*v^8 + 59930247*v^6 - 3630112*v^4 + 54117564*v^2 + 9123648) / 5112544 $$\beta_{10}$$ $$=$$ $$( 30391\nu^{19} + 3256034\nu^{15} + 37292880\nu^{11} + 113829150\nu^{7} + 98273977\nu^{3} ) / 10225088$$ (30391*v^19 + 3256034*v^15 + 37292880*v^11 + 113829150*v^7 + 98273977*v^3) / 10225088 $$\beta_{11}$$ $$=$$ $$( - 14918 \nu^{19} + 1504 \nu^{17} - 1619593 \nu^{15} + 147562 \nu^{13} - 20551431 \nu^{11} + 408180 \nu^{9} - 78087767 \nu^{7} - 9194094 \nu^{5} + \cdots - 24973328 \nu ) / 5112544$$ (-14918*v^19 + 1504*v^17 - 1619593*v^15 + 147562*v^13 - 20551431*v^11 + 408180*v^9 - 78087767*v^7 - 9194094*v^5 - 88088723*v^3 - 24973328*v) / 5112544 $$\beta_{12}$$ $$=$$ $$( 14918 \nu^{19} + 1504 \nu^{17} + 1619593 \nu^{15} + 147562 \nu^{13} + 20551431 \nu^{11} + 408180 \nu^{9} + 78087767 \nu^{7} - 9194094 \nu^{5} + \cdots - 24973328 \nu ) / 5112544$$ (14918*v^19 + 1504*v^17 + 1619593*v^15 + 147562*v^13 + 20551431*v^11 + 408180*v^9 + 78087767*v^7 - 9194094*v^5 + 88088723*v^3 - 24973328*v) / 5112544 $$\beta_{13}$$ $$=$$ $$( 18604 \nu^{19} - 1504 \nu^{17} + 2018205 \nu^{15} - 147562 \nu^{13} + 25466939 \nu^{11} - 408180 \nu^{9} + 96014619 \nu^{7} + 9194094 \nu^{5} + \cdots + 19860784 \nu ) / 5112544$$ (18604*v^19 - 1504*v^17 + 2018205*v^15 - 147562*v^13 + 25466939*v^11 - 408180*v^9 + 96014619*v^7 + 9194094*v^5 + 111196929*v^3 + 19860784*v) / 5112544 $$\beta_{14}$$ $$=$$ $$( - 18604 \nu^{19} - 1504 \nu^{17} - 2018205 \nu^{15} - 147562 \nu^{13} - 25466939 \nu^{11} - 408180 \nu^{9} - 96014619 \nu^{7} + 9194094 \nu^{5} + \cdots + 19860784 \nu ) / 5112544$$ (-18604*v^19 - 1504*v^17 - 2018205*v^15 - 147562*v^13 - 25466939*v^11 - 408180*v^9 - 96014619*v^7 + 9194094*v^5 - 111196929*v^3 + 19860784*v) / 5112544 $$\beta_{15}$$ $$=$$ $$( 10893 \nu^{19} + 3076 \nu^{17} + 13621 \nu^{16} + 1169237 \nu^{15} + 319642 \nu^{13} + 1438589 \nu^{12} + 13586349 \nu^{11} + 2746920 \nu^{9} + 14554471 \nu^{8} + \cdots + 18284064 ) / 5112544$$ (10893*v^19 + 3076*v^17 + 13621*v^16 + 1169237*v^15 + 319642*v^13 + 1438589*v^12 + 13586349*v^11 + 2746920*v^9 + 14554471*v^8 + 42003395*v^7 + 2932082*v^5 + 32174759*v^4 + 31009358*v^3 - 963320*v + 18284064) / 5112544 $$\beta_{16}$$ $$=$$ $$( - 14579 \nu^{18} + 23836 \nu^{16} - 1567849 \nu^{14} + 2504338 \nu^{12} - 18501857 \nu^{10} + 24041672 \nu^{8} - 59930247 \nu^{6} + 37863114 \nu^{4} + \cdots - 8423808 ) / 5112544$$ (-14579*v^18 + 23836*v^16 - 1567849*v^14 + 2504338*v^12 - 18501857*v^10 + 24041672*v^8 - 59930247*v^6 + 37863114*v^4 - 54117564*v^2 - 8423808) / 5112544 $$\beta_{17}$$ $$=$$ $$( - 10893 \nu^{19} - 21341 \nu^{18} - 3076 \nu^{17} - 1169237 \nu^{15} - 2286103 \nu^{14} - 319642 \nu^{13} - 13586349 \nu^{11} - 26164285 \nu^{10} + \cdots + 963320 \nu ) / 5112544$$ (-10893*v^19 - 21341*v^18 - 3076*v^17 - 1169237*v^15 - 2286103*v^14 - 319642*v^13 - 13586349*v^11 - 26164285*v^10 - 2746920*v^9 - 42003395*v^7 - 80789181*v^6 - 2932082*v^5 - 31009358*v^3 - 76262450*v^2 + 963320*v) / 5112544 $$\beta_{18}$$ $$=$$ $$( - 11787 \nu^{18} - 17752 \nu^{17} - 8973 \nu^{16} - 1237829 \nu^{14} - 1889572 \nu^{13} - 946867 \nu^{12} - 11825941 \nu^{10} - 20478392 \nu^{9} - 9468811 \nu^{8} + \cdots - 349920 ) / 5112544$$ (-11787*v^18 - 17752*v^17 - 8973*v^16 - 1237829*v^14 - 1889572*v^13 - 946867*v^12 - 11825941*v^10 - 20478392*v^9 - 9468811*v^8 - 17814531*v^6 - 53319220*v^5 - 17116501*v^4 + 12922952*v^2 - 37373624*v - 349920) / 5112544 $$\beta_{19}$$ $$=$$ $$( - 77979 \nu^{19} - 23574 \nu^{18} + 17946 \nu^{16} - 8384792 \nu^{15} - 2475658 \nu^{14} + 1893734 \nu^{12} - 98874134 \nu^{11} - 23651882 \nu^{10} + \cdots + 699840 ) / 10225088$$ (-77979*v^19 - 23574*v^18 + 17946*v^16 - 8384792*v^15 - 2475658*v^14 + 1893734*v^12 - 98874134*v^11 - 23651882*v^10 + 18937622*v^8 - 323323904*v^7 - 35629062*v^6 + 34233002*v^4 - 316937591*v^3 + 25845904*v^2 + 699840) / 10225088
 $$\nu$$ $$=$$ $$( -\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} ) / 2$$ (-b14 - b13 - b12 - b11) / 2 $$\nu^{2}$$ $$=$$ $$( 2 \beta_{17} + 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 6 \beta_{2} + \beta_1 ) / 2$$ (2*b17 + 2*b14 - b13 + 2*b12 - b11 + 2*b10 + b9 - b8 - b7 + b6 - b5 - b4 + 6*b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( 4 \beta_{19} - 2 \beta_{16} - 5 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} + 5 \beta_{11} + 8 \beta_{10} - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 ) / 2$$ (4*b19 - 2*b16 - 5*b14 + 5*b13 - 5*b12 + 5*b11 + 8*b10 - 2*b8 + 2*b6 - 2*b4 + 2*b3 - 4*b2 + 2*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 4 \beta_{16} + 18 \beta_{15} - 18 \beta_{14} + 9 \beta_{13} - 18 \beta_{12} + 9 \beta_{11} - 18 \beta_{10} + 15 \beta_{9} + 11 \beta_{8} + 9 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 13 \beta_{4} - 9 \beta _1 - 38 ) / 2$$ (4*b16 + 18*b15 - 18*b14 + 9*b13 - 18*b12 + 9*b11 - 18*b10 + 15*b9 + 11*b8 + 9*b7 + 13*b6 + 9*b5 + 13*b4 - 9*b1 - 38) / 2 $$\nu^{5}$$ $$=$$ $$( - 48 \beta_{18} - 24 \beta_{16} + 59 \beta_{14} + 39 \beta_{13} + 63 \beta_{12} + 43 \beta_{11} - 48 \beta_{9} + 24 \beta_{8} + 76 \beta_{7} - 24 \beta_{6} - 20 \beta_{5} + 24 \beta_{4} - 24 \beta_{3} + 48 \beta_{2} ) / 2$$ (-48*b18 - 24*b16 + 59*b14 + 39*b13 + 63*b12 + 43*b11 - 48*b9 + 24*b8 + 76*b7 - 24*b6 - 20*b5 + 24*b4 - 24*b3 + 48*b2) / 2 $$\nu^{6}$$ $$=$$ $$( - 166 \beta_{17} - 166 \beta_{14} + 83 \beta_{13} - 166 \beta_{12} + 83 \beta_{11} - 166 \beta_{10} - 131 \beta_{9} + 131 \beta_{8} + 83 \beta_{7} - 111 \beta_{6} + 83 \beta_{5} + 111 \beta_{4} + 52 \beta_{3} - 330 \beta_{2} - 83 \beta_1 ) / 2$$ (-166*b17 - 166*b14 + 83*b13 - 166*b12 + 83*b11 - 166*b10 - 131*b9 + 131*b8 + 83*b7 - 111*b6 + 83*b5 + 111*b4 + 52*b3 - 330*b2 - 83*b1) / 2 $$\nu^{7}$$ $$=$$ $$( - 492 \beta_{19} + 246 \beta_{16} + 411 \beta_{14} - 411 \beta_{13} + 359 \beta_{12} - 359 \beta_{11} - 976 \beta_{10} + 246 \beta_{8} - 246 \beta_{6} + 246 \beta_{4} - 246 \beta_{3} + 492 \beta_{2} - 186 \beta_1 ) / 2$$ (-492*b19 + 246*b16 + 411*b14 - 411*b13 + 359*b12 - 359*b11 - 976*b10 + 246*b8 - 246*b6 + 246*b4 - 246*b3 + 492*b2 - 186*b1) / 2 $$\nu^{8}$$ $$=$$ $$( - 544 \beta_{16} - 1582 \beta_{15} + 1582 \beta_{14} - 791 \beta_{13} + 1582 \beta_{12} - 791 \beta_{11} + 1582 \beta_{10} - 1637 \beta_{9} - 1093 \beta_{8} - 791 \beta_{7} - 1279 \beta_{6} - 791 \beta_{5} + \cdots + 3122 ) / 2$$ (-544*b16 - 1582*b15 + 1582*b14 - 791*b13 + 1582*b12 - 791*b11 + 1582*b10 - 1637*b9 - 1093*b8 - 791*b7 - 1279*b6 - 791*b5 - 1279*b4 + 791*b1 + 3122) / 2 $$\nu^{9}$$ $$=$$ $$( 4856 \beta_{18} + 2428 \beta_{16} - 5213 \beta_{14} - 3445 \beta_{13} - 5757 \beta_{12} - 3989 \beta_{11} + 4856 \beta_{9} - 2428 \beta_{8} - 7832 \beta_{7} + 2428 \beta_{6} + 1768 \beta_{5} - 2428 \beta_{4} + \cdots - 4856 \beta_{2} ) / 2$$ (4856*b18 + 2428*b16 - 5213*b14 - 3445*b13 - 5757*b12 - 3989*b11 + 4856*b9 - 2428*b8 - 7832*b7 + 2428*b6 + 1768*b5 - 2428*b4 + 2428*b3 - 4856*b2) / 2 $$\nu^{10}$$ $$=$$ $$( 15282 \beta_{17} + 15282 \beta_{14} - 7641 \beta_{13} + 15282 \beta_{12} - 7641 \beta_{11} + 15282 \beta_{10} + 12441 \beta_{9} - 12441 \beta_{8} - 7641 \beta_{7} + 10673 \beta_{6} - 7641 \beta_{5} + \cdots + 7641 \beta_1 ) / 2$$ (15282*b17 + 15282*b14 - 7641*b13 + 15282*b12 - 7641*b11 + 15282*b10 + 12441*b9 - 12441*b8 - 7641*b7 + 10673*b6 - 7641*b5 - 10673*b4 - 5400*b3 + 30150*b2 + 7641*b1) / 2 $$\nu^{11}$$ $$=$$ $$( 47428 \beta_{19} - 23714 \beta_{16} - 38789 \beta_{14} + 38789 \beta_{13} - 33389 \beta_{12} + 33389 \beta_{11} + 93656 \beta_{10} - 23714 \beta_{8} + 23714 \beta_{6} - 23714 \beta_{4} + 23714 \beta_{3} + \cdots + 17050 \beta_1 ) / 2$$ (47428*b19 - 23714*b16 - 38789*b14 + 38789*b13 - 33389*b12 + 33389*b11 + 93656*b10 - 23714*b8 + 23714*b6 - 23714*b4 + 23714*b3 - 47428*b2 + 17050*b1) / 2 $$\nu^{12}$$ $$=$$ $$( 52828 \beta_{16} + 148306 \beta_{15} - 148306 \beta_{14} + 74153 \beta_{13} - 148306 \beta_{12} + 74153 \beta_{11} - 148306 \beta_{10} + 156759 \beta_{9} + 103931 \beta_{8} + 74153 \beta_{7} + \cdots - 292662 ) / 2$$ (52828*b16 + 148306*b15 - 148306*b14 + 74153*b13 - 148306*b12 + 74153*b11 - 148306*b10 + 156759*b9 + 103931*b8 + 74153*b7 + 120981*b6 + 74153*b5 + 120981*b4 - 74153*b1 - 292662) / 2 $$\nu^{13}$$ $$=$$ $$( - 461824 \beta_{18} - 230912 \beta_{16} + 489771 \beta_{14} + 324415 \beta_{13} + 542599 \beta_{12} + 377243 \beta_{11} - 461824 \beta_{9} + 230912 \beta_{8} + 746292 \beta_{7} + \cdots + 461824 \beta_{2} ) / 2$$ (-461824*b18 - 230912*b16 + 489771*b14 + 324415*b13 + 542599*b12 + 377243*b11 - 461824*b9 + 230912*b8 + 746292*b7 - 230912*b6 - 165356*b5 + 230912*b4 - 230912*b3 + 461824*b2) / 2 $$\nu^{14}$$ $$=$$ $$( - 1441366 \beta_{17} - 1441366 \beta_{14} + 720683 \beta_{13} - 1441366 \beta_{12} + 720683 \beta_{11} - 1441366 \beta_{10} - 1176507 \beta_{9} + 1176507 \beta_{8} + 720683 \beta_{7} + \cdots - 720683 \beta_1 ) / 2$$ (-1441366*b17 - 1441366*b14 + 720683*b13 - 1441366*b12 + 720683*b11 - 1441366*b10 - 1176507*b9 + 1176507*b8 + 720683*b7 - 1011151*b6 + 720683*b5 + 1011151*b4 + 514652*b3 - 2844682*b2 - 720683*b1) / 2 $$\nu^{15}$$ $$=$$ $$( - 4492972 \beta_{19} + 2246486 \beta_{16} + 3668827 \beta_{14} - 3668827 \beta_{13} + 3154175 \beta_{12} - 3154175 \beta_{11} - 8868288 \beta_{10} + 2246486 \beta_{8} + \cdots - 1606722 \beta_1 ) / 2$$ (-4492972*b19 + 2246486*b16 + 3668827*b14 - 3668827*b13 + 3154175*b12 - 3154175*b11 - 8868288*b10 + 2246486*b8 - 2246486*b6 + 2246486*b4 - 2246486*b3 + 4492972*b2 - 1606722*b1) / 2 $$\nu^{16}$$ $$=$$ $$( - 5007624 \beta_{16} - 14014766 \beta_{15} + 14014766 \beta_{14} - 7007383 \beta_{13} + 14014766 \beta_{12} - 7007383 \beta_{11} + 14014766 \beta_{10} - 14842429 \beta_{9} + \cdots + 27660770 ) / 2$$ (-5007624*b16 - 14014766*b15 + 14014766*b14 - 7007383*b13 + 14014766*b12 - 7007383*b11 + 14014766*b10 - 14842429*b9 - 9834805*b8 - 7007383*b7 - 11441527*b6 - 7007383*b5 - 11441527*b4 + 7007383*b1 + 27660770) / 2 $$\nu^{17}$$ $$=$$ $$( 43699624 \beta_{18} + 21849812 \beta_{16} - 46294061 \beta_{14} - 30672573 \beta_{13} - 51301685 \beta_{12} - 35680197 \beta_{11} + 43699624 \beta_{9} - 21849812 \beta_{8} + \cdots - 43699624 \beta_{2} ) / 2$$ (43699624*b18 + 21849812*b16 - 46294061*b14 - 30672573*b13 - 51301685*b12 - 35680197*b11 + 43699624*b9 - 21849812*b8 - 70630800*b7 + 21849812*b6 + 15621488*b5 - 21849812*b4 + 21849812*b3 - 43699624*b2) / 2 $$\nu^{18}$$ $$=$$ $$( 136287746 \beta_{17} + 136287746 \beta_{14} - 68143873 \beta_{13} + 136287746 \beta_{12} - 68143873 \beta_{11} + 136287746 \beta_{10} + 111270017 \beta_{9} + \cdots + 68143873 \beta_1 ) / 2$$ (136287746*b17 + 136287746*b14 - 68143873*b13 + 136287746*b12 - 68143873*b11 + 136287746*b10 + 111270017*b9 - 111270017*b8 - 68143873*b7 + 95648529*b6 - 68143873*b5 - 95648529*b4 - 48707248*b3 + 268993286*b2 + 68143873*b1) / 2 $$\nu^{19}$$ $$=$$ $$( 424999300 \beta_{19} - 212499650 \beta_{16} - 346996293 \beta_{14} + 346996293 \beta_{13} - 298289045 \beta_{12} + 298289045 \beta_{11} + 838836392 \beta_{10} + \cdots + 151909234 \beta_1 ) / 2$$ (424999300*b19 - 212499650*b16 - 346996293*b14 + 346996293*b13 - 298289045*b12 + 298289045*b11 + 838836392*b10 - 212499650*b8 + 212499650*b6 - 212499650*b4 + 212499650*b3 - 424999300*b2 + 151909234*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.339574 − 0.339574i 2.20512 − 2.20512i −1.20277 + 1.20277i 0.922947 − 0.922947i −0.850665 + 0.850665i −0.339574 + 0.339574i −2.20512 + 2.20512i 1.20277 − 1.20277i −0.922947 + 0.922947i 0.850665 − 0.850665i 0.339574 + 0.339574i 2.20512 + 2.20512i −1.20277 − 1.20277i 0.922947 + 0.922947i −0.850665 − 0.850665i −0.339574 − 0.339574i −2.20512 − 2.20512i 1.20277 + 1.20277i −0.922947 − 0.922947i 0.850665 + 0.850665i
−0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i 0.707107 + 0.707107i 1.00000i −0.367533 2.20566i
37.2 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i 0.707107 + 0.707107i 1.00000i 1.49802 + 1.66010i
37.3 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i 0.707107 + 0.707107i 1.00000i −1.90987 1.16292i
37.4 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i 0.707107 + 0.707107i 1.00000i 0.215841 + 2.22563i
37.5 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i 0.707107 + 0.707107i 1.00000i −1.55777 + 1.60417i
37.6 0.707107 0.707107i −0.707107 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i −0.707107 0.707107i 1.00000i 0.367533 + 2.20566i
37.7 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i −0.707107 0.707107i 1.00000i −1.49802 1.66010i
37.8 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i −0.707107 0.707107i 1.00000i 1.90987 + 1.16292i
37.9 0.707107 0.707107i −0.707107 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i −0.707107 0.707107i 1.00000i −0.215841 2.22563i
37.10 0.707107 0.707107i −0.707107 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i −0.707107 0.707107i 1.00000i 1.55777 1.60417i
493.1 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i 0.707107 0.707107i 1.00000i −0.367533 + 2.20566i
493.2 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i 0.707107 0.707107i 1.00000i 1.49802 1.66010i
493.3 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i 0.707107 0.707107i 1.00000i −1.90987 + 1.16292i
493.4 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i 0.707107 0.707107i 1.00000i 0.215841 2.22563i
493.5 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i 0.707107 0.707107i 1.00000i −1.55777 1.60417i
493.6 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i −0.707107 + 0.707107i 1.00000i 0.367533 2.20566i
493.7 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i −0.707107 + 0.707107i 1.00000i −1.49802 + 1.66010i
493.8 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i −0.707107 + 0.707107i 1.00000i 1.90987 1.16292i
493.9 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i −0.707107 + 0.707107i 1.00000i −0.215841 + 2.22563i
493.10 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i −0.707107 + 0.707107i 1.00000i 1.55777 + 1.60417i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.m.b 20
3.b odd 2 1 1710.2.p.c 20
5.c odd 4 1 inner 570.2.m.b 20
15.e even 4 1 1710.2.p.c 20
19.b odd 2 1 inner 570.2.m.b 20
57.d even 2 1 1710.2.p.c 20
95.g even 4 1 inner 570.2.m.b 20
285.j odd 4 1 1710.2.p.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.b 20 1.a even 1 1 trivial
570.2.m.b 20 5.c odd 4 1 inner
570.2.m.b 20 19.b odd 2 1 inner
570.2.m.b 20 95.g even 4 1 inner
1710.2.p.c 20 3.b odd 2 1
1710.2.p.c 20 15.e even 4 1
1710.2.p.c 20 57.d even 2 1
1710.2.p.c 20 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{10} + 2 T_{7}^{9} + 2 T_{7}^{8} + 8 T_{7}^{7} + 496 T_{7}^{6} + 1184 T_{7}^{5} + 1408 T_{7}^{4} - 320 T_{7}^{3} + 1600 T_{7}^{2} + 3200 T_{7} + 3200$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{5}$$
$3$ $$(T^{4} + 1)^{5}$$
$5$ $$(T^{10} - 6 T^{9} + 25 T^{8} - 80 T^{7} + \cdots + 3125)^{2}$$
$7$ $$(T^{10} + 2 T^{9} + 2 T^{8} + 8 T^{7} + \cdots + 3200)^{2}$$
$11$ $$(T^{5} + 2 T^{4} - 36 T^{3} - 16 T^{2} + \cdots - 400)^{4}$$
$13$ $$T^{20} + 1248 T^{16} + 203872 T^{12} + \cdots + 4096$$
$17$ $$(T^{10} + 6 T^{9} + 18 T^{8} - 64 T^{7} + \cdots + 12800)^{2}$$
$19$ $$T^{20} - 58 T^{18} + \cdots + 6131066257801$$
$23$ $$(T^{10} + 2 T^{9} + 2 T^{8} - 24 T^{7} + \cdots + 156800)^{2}$$
$29$ $$(T^{10} - 112 T^{8} + 4320 T^{6} + \cdots - 51200)^{2}$$
$31$ $$(T^{10} + 192 T^{8} + 12544 T^{6} + \cdots + 204800)^{2}$$
$37$ $$T^{20} + 8928 T^{16} + \cdots + 362615934976$$
$41$ $$(T^{10} + 64 T^{8} + 1320 T^{6} + \cdots + 51200)^{2}$$
$43$ $$(T^{10} + 6 T^{9} + 18 T^{8} + 224 T^{7} + \cdots + 768800)^{2}$$
$47$ $$(T^{10} + 22 T^{9} + 242 T^{8} + \cdots + 2000000)^{2}$$
$53$ $$T^{20} + 30928 T^{16} + \cdots + 12\!\cdots\!56$$
$59$ $$(T^{10} - 160 T^{8} + 1944 T^{6} + \cdots - 3200)^{2}$$
$61$ $$(T^{5} - 248 T^{3} - 816 T^{2} + \cdots + 56000)^{4}$$
$67$ $$T^{20} + 85248 T^{16} + \cdots + 91\!\cdots\!96$$
$71$ $$(T^{10} + 200 T^{8} + 12288 T^{6} + \cdots + 3276800)^{2}$$
$73$ $$(T^{10} + 2 T^{9} + 2 T^{8} + \cdots + 1468820000)^{2}$$
$79$ $$(T^{10} - 384 T^{8} + 28256 T^{6} + \cdots - 2508800)^{2}$$
$83$ $$(T^{10} - 38 T^{9} + 722 T^{8} + \cdots + 356979200)^{2}$$
$89$ $$(T^{10} - 192 T^{8} + 12680 T^{6} + \cdots - 3699200)^{2}$$
$97$ $$T^{20} + 113680 T^{16} + \cdots + 18\!\cdots\!36$$