# Properties

 Label 4205.2.a.q Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.a (trivial)

## 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: yes Analytic conductor: $$33.5770940499$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1$$ x^12 - x^11 - 14*x^10 + 11*x^9 + 72*x^8 - 41*x^7 - 164*x^6 + 62*x^5 + 156*x^4 - 43*x^3 - 46*x^2 + 15*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$( - 30 \nu^{11} + 64 \nu^{10} + 366 \nu^{9} - 717 \nu^{8} - 1542 \nu^{7} + 2644 \nu^{6} + 2498 \nu^{5} + \cdots + 51 ) / 139$$ (-30*v^11 + 64*v^10 + 366*v^9 - 717*v^8 - 1542*v^7 + 2644*v^6 + 2498*v^5 - 3542*v^4 - 953*v^3 + 1360*v^2 - 393*v + 51) / 139 $$\beta_{4}$$ $$=$$ $$( 36 \nu^{11} - 49 \nu^{10} - 467 \nu^{9} + 499 \nu^{8} + 2184 \nu^{7} - 1616 \nu^{6} - 4332 \nu^{5} + \cdots + 50 ) / 139$$ (36*v^11 - 49*v^10 - 467*v^9 + 499*v^8 + 2184*v^7 - 1616*v^6 - 4332*v^5 + 1804*v^4 + 3173*v^3 - 659*v^2 - 418*v + 50) / 139 $$\beta_{5}$$ $$=$$ $$( 20 \nu^{11} + 50 \nu^{10} - 383 \nu^{9} - 634 \nu^{8} + 2557 \nu^{7} + 2778 \nu^{6} - 7318 \nu^{5} + \cdots + 661 ) / 139$$ (20*v^11 + 50*v^10 - 383*v^9 - 634*v^8 + 2557*v^7 + 2778*v^6 - 7318*v^5 - 4635*v^4 + 8651*v^3 + 1688*v^2 - 3491*v + 661) / 139 $$\beta_{6}$$ $$=$$ $$( - 51 \nu^{11} + 81 \nu^{10} + 650 \nu^{9} - 927 \nu^{8} - 2955 \nu^{7} + 3633 \nu^{6} + 5720 \nu^{5} + \cdots - 372 ) / 139$$ (-51*v^11 + 81*v^10 + 650*v^9 - 927*v^8 - 2955*v^7 + 3633*v^6 + 5720*v^5 - 5660*v^4 - 4414*v^3 + 3146*v^2 + 986*v - 372) / 139 $$\beta_{7}$$ $$=$$ $$( - 50 \nu^{11} + 14 \nu^{10} + 749 \nu^{9} - 83 \nu^{8} - 4099 \nu^{7} - 134 \nu^{6} + 9816 \nu^{5} + \cdots - 54 ) / 139$$ (-50*v^11 + 14*v^10 + 749*v^9 - 83*v^8 - 4099*v^7 - 134*v^6 + 9816*v^5 + 1232*v^4 - 9465*v^3 - 1162*v^2 + 2403*v - 54) / 139 $$\beta_{8}$$ $$=$$ $$( - 50 \nu^{11} + 14 \nu^{10} + 749 \nu^{9} - 83 \nu^{8} - 4099 \nu^{7} - 134 \nu^{6} + 9816 \nu^{5} + \cdots - 332 ) / 139$$ (-50*v^11 + 14*v^10 + 749*v^9 - 83*v^8 - 4099*v^7 - 134*v^6 + 9816*v^5 + 1232*v^4 - 9604*v^3 - 1023*v^2 + 2959*v - 332) / 139 $$\beta_{9}$$ $$=$$ $$( 69 \nu^{11} - 36 \nu^{10} - 953 \nu^{9} + 273 \nu^{8} + 4742 \nu^{7} - 271 \nu^{6} - 9971 \nu^{5} + \cdots - 298 ) / 139$$ (69*v^11 - 36*v^10 - 953*v^9 + 273*v^8 + 4742*v^7 - 271*v^6 - 9971*v^5 - 1500*v^4 + 7599*v^3 + 2293*v^2 - 917*v - 298) / 139 $$\beta_{10}$$ $$=$$ $$( 74 \nu^{11} - 93 \nu^{10} - 1014 \nu^{9} + 1018 \nu^{8} + 5138 \nu^{7} - 3677 \nu^{6} - 11731 \nu^{5} + \cdots + 458 ) / 139$$ (74*v^11 - 93*v^10 - 1014*v^9 + 1018*v^8 + 5138*v^7 - 3677*v^6 - 11731*v^5 + 4882*v^4 + 11673*v^3 - 2150*v^2 - 3979*v + 458) / 139 $$\beta_{11}$$ $$=$$ $$( 82 \nu^{11} - 73 \nu^{10} - 1195 \nu^{9} + 820 \nu^{8} + 6411 \nu^{7} - 3094 \nu^{6} - 15242 \nu^{5} + \cdots + 778 ) / 139$$ (82*v^11 - 73*v^10 - 1195*v^9 + 820*v^8 + 6411*v^7 - 3094*v^6 - 15242*v^5 + 4557*v^4 + 15050*v^3 - 2698*v^2 - 4458*v + 778) / 139
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} + \beta_{2} + 4\beta_1$$ -b8 + b7 + b2 + 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{5} - \beta_{3} + 5\beta_{2} + \beta _1 + 8$$ b8 + b5 - b3 + 5*b2 + b1 + 8 $$\nu^{5}$$ $$=$$ $$\beta_{11} + \beta_{9} - 4\beta_{8} + 7\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 7\beta_{2} + 18\beta _1 + 2$$ b11 + b9 - 4*b8 + 7*b7 + b6 + b5 - b3 + 7*b2 + 18*b1 + 2 $$\nu^{6}$$ $$=$$ $$\beta_{10} + \beta_{9} + 8 \beta_{8} + \beta_{7} + \beta_{6} + 8 \beta_{5} - 2 \beta_{4} - 9 \beta_{3} + \cdots + 37$$ b10 + b9 + 8*b8 + b7 + b6 + 8*b5 - 2*b4 - 9*b3 + 25*b2 + 10*b1 + 37 $$\nu^{7}$$ $$=$$ $$9 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 9 \beta_{8} + 40 \beta_{7} + 10 \beta_{6} + 11 \beta_{5} + \cdots + 21$$ 9*b11 + 2*b10 + 10*b9 - 9*b8 + 40*b7 + 10*b6 + 11*b5 - b4 - 10*b3 + 42*b2 + 86*b1 + 21 $$\nu^{8}$$ $$=$$ $$2 \beta_{11} + 10 \beta_{10} + 12 \beta_{9} + 53 \beta_{8} + 13 \beta_{7} + 10 \beta_{6} + 52 \beta_{5} + \cdots + 181$$ 2*b11 + 10*b10 + 12*b9 + 53*b8 + 13*b7 + 10*b6 + 52*b5 - 22*b4 - 61*b3 + 129*b2 + 73*b1 + 181 $$\nu^{9}$$ $$=$$ $$59 \beta_{11} + 24 \beta_{10} + 74 \beta_{9} + 15 \beta_{8} + 217 \beta_{7} + 70 \beta_{6} + 87 \beta_{5} + \cdots + 160$$ 59*b11 + 24*b10 + 74*b9 + 15*b8 + 217*b7 + 70*b6 + 87*b5 - 15*b4 - 75*b3 + 242*b2 + 428*b1 + 160 $$\nu^{10}$$ $$=$$ $$25 \beta_{11} + 74 \beta_{10} + 102 \beta_{9} + 333 \beta_{8} + 113 \beta_{7} + 72 \beta_{6} + \cdots + 915$$ 25*b11 + 74*b10 + 102*b9 + 333*b8 + 113*b7 + 72*b6 + 319*b5 - 172*b4 - 374*b3 + 683*b2 + 477*b1 + 915 $$\nu^{11}$$ $$=$$ $$346 \beta_{11} + 197 \beta_{10} + 491 \beta_{9} + 375 \beta_{8} + 1161 \beta_{7} + 426 \beta_{6} + \cdots + 1074$$ 346*b11 + 197*b10 + 491*b9 + 375*b8 + 1161*b7 + 426*b6 + 604*b5 - 149*b4 - 504*b3 + 1377*b2 + 2196*b1 + 1074

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.40450 2.19433 1.91653 1.18316 0.623326 0.218677 0.0975264 −0.886465 −1.10366 −1.48615 −2.00441 −2.15738
−2.40450 −0.433927 3.78164 −1.00000 1.04338 −1.22316 −4.28397 −2.81171 2.40450
1.2 −2.19433 2.38477 2.81508 −1.00000 −5.23297 1.90240 −1.78854 2.68713 2.19433
1.3 −1.91653 −1.12311 1.67310 −1.00000 2.15247 −1.69505 0.626513 −1.73863 1.91653
1.4 −1.18316 2.97173 −0.600128 −1.00000 −3.51603 −3.71603 3.07637 5.83115 1.18316
1.5 −0.623326 −0.691067 −1.61147 −1.00000 0.430760 2.77627 2.25112 −2.52243 0.623326
1.6 −0.218677 −1.19342 −1.95218 −1.00000 0.260974 1.69290 0.864252 −1.57575 0.218677
1.7 −0.0975264 2.38706 −1.99049 −1.00000 −0.232802 −1.61211 0.389178 2.69808 0.0975264
1.8 0.886465 −2.17820 −1.21418 −1.00000 −1.93090 −4.18862 −2.84926 1.74457 −0.886465
1.9 1.10366 −1.50704 −0.781928 −1.00000 −1.66326 −1.41268 −3.07031 −0.728845 −1.10366
1.10 1.48615 2.28644 0.208632 −1.00000 3.39798 −0.688115 −2.66224 2.22780 −1.48615
1.11 2.00441 −2.59136 2.01764 −1.00000 −5.19413 −0.000698765 0 0.0353668 3.71514 −2.00441
1.12 2.15738 0.688118 2.65427 −1.00000 1.48453 3.16489 1.41152 −2.52649 −2.15738
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$29$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4205.2.a.q 12
29.b even 2 1 4205.2.a.r 12
29.d even 7 2 145.2.k.a 24
145.n even 14 2 725.2.l.d 24
145.p odd 28 4 725.2.r.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.a 24 29.d even 7 2
725.2.l.d 24 145.n even 14 2
725.2.r.c 48 145.p odd 28 4
4205.2.a.q 12 1.a even 1 1 trivial
4205.2.a.r 12 29.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{12} + T_{2}^{11} - 14 T_{2}^{10} - 11 T_{2}^{9} + 72 T_{2}^{8} + 41 T_{2}^{7} - 164 T_{2}^{6} + \cdots - 1$$ T2^12 + T2^11 - 14*T2^10 - 11*T2^9 + 72*T2^8 + 41*T2^7 - 164*T2^6 - 62*T2^5 + 156*T2^4 + 43*T2^3 - 46*T2^2 - 15*T2 - 1 $$T_{3}^{12} - T_{3}^{11} - 21 T_{3}^{10} + 11 T_{3}^{9} + 169 T_{3}^{8} - 3 T_{3}^{7} - 629 T_{3}^{6} + \cdots - 91$$ T3^12 - T3^11 - 21*T3^10 + 11*T3^9 + 169*T3^8 - 3*T3^7 - 629*T3^6 - 299*T3^5 + 949*T3^4 + 895*T3^3 - 133*T3^2 - 357*T3 - 91

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + T^{11} + \cdots - 1$$
$3$ $$T^{12} - T^{11} + \cdots - 91$$
$5$ $$(T + 1)^{12}$$
$7$ $$T^{12} + 5 T^{11} + \cdots + 1$$
$11$ $$T^{12} + T^{11} + \cdots + 113$$
$13$ $$T^{12} + 5 T^{11} + \cdots - 83$$
$17$ $$T^{12} + 20 T^{11} + \cdots + 147139$$
$19$ $$T^{12} - 8 T^{11} + \cdots - 6833$$
$23$ $$T^{12} + 8 T^{11} + \cdots - 7439083$$
$29$ $$T^{12}$$
$31$ $$T^{12} - 7 T^{11} + \cdots + 54674341$$
$37$ $$T^{12} - 266 T^{10} + \cdots + 478631$$
$41$ $$T^{12} - 4 T^{11} + \cdots - 15490663$$
$43$ $$T^{12} + \cdots - 252553097$$
$47$ $$T^{12} - 39 T^{11} + \cdots + 502207$$
$53$ $$T^{12} + \cdots - 600156257$$
$59$ $$T^{12} + 19 T^{11} + \cdots - 13085407$$
$61$ $$T^{12} + \cdots + 5237693287$$
$67$ $$T^{12} + \cdots + 1486528457$$
$71$ $$T^{12} + 20 T^{11} + \cdots - 44250121$$
$73$ $$T^{12} + \cdots - 5489455061$$
$79$ $$T^{12} + 13 T^{11} + \cdots + 75905257$$
$83$ $$T^{12} + \cdots + 27453317717$$
$89$ $$T^{12} + \cdots + 506469104821$$
$97$ $$T^{12} + \cdots - 102994949681$$