# Properties

 Label 285.2.i.f Level $285$ Weight $2$ Character orbit 285.i Analytic conductor $2.276$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(106,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("285.106");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.i (of order $$3$$, 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: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16$$ x^10 - x^9 + 9*x^8 - 2*x^7 + 56*x^6 - 18*x^5 + 125*x^4 + x^3 + 189*x^2 - 52*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{8} + 3\nu^{7} - 9\nu^{6} + 15\nu^{5} - 45\nu^{4} + 135\nu^{3} - 86\nu^{2} + 24\nu - 72 ) / 234$$ (-v^8 + 3*v^7 - 9*v^6 + 15*v^5 - 45*v^4 + 135*v^3 - 86*v^2 + 24*v - 72) / 234 $$\beta_{3}$$ $$=$$ $$( \nu^{9} - 3\nu^{8} + 9\nu^{7} - 15\nu^{6} + 45\nu^{5} - 135\nu^{4} + 86\nu^{3} - 258\nu^{2} + 72\nu - 702 ) / 234$$ (v^9 - 3*v^8 + 9*v^7 - 15*v^6 + 45*v^5 - 135*v^4 + 86*v^3 - 258*v^2 + 72*v - 702) / 234 $$\beta_{4}$$ $$=$$ $$( 9 \nu^{9} - 9 \nu^{8} + 79 \nu^{7} - 12 \nu^{6} + 486 \nu^{5} - 132 \nu^{4} + 1035 \nu^{3} + 279 \nu^{2} + 1529 \nu - 420 ) / 468$$ (9*v^9 - 9*v^8 + 79*v^7 - 12*v^6 + 486*v^5 - 132*v^4 + 1035*v^3 + 279*v^2 + 1529*v - 420) / 468 $$\beta_{5}$$ $$=$$ $$( 28 \nu^{9} - 44 \nu^{8} + 275 \nu^{7} - 229 \nu^{6} + 1713 \nu^{5} - 1551 \nu^{4} + 4223 \nu^{3} - 2497 \nu^{2} + 6334 \nu - 3932 ) / 702$$ (28*v^9 - 44*v^8 + 275*v^7 - 229*v^6 + 1713*v^5 - 1551*v^4 + 4223*v^3 - 2497*v^2 + 6334*v - 3932) / 702 $$\beta_{6}$$ $$=$$ $$( 31 \nu^{9} - 5 \nu^{8} + 275 \nu^{7} + 158 \nu^{6} + 1830 \nu^{5} + 906 \nu^{4} + 4319 \nu^{3} + 2963 \nu^{2} + 6685 \nu + 2332 ) / 702$$ (31*v^9 - 5*v^8 + 275*v^7 + 158*v^6 + 1830*v^5 + 906*v^4 + 4319*v^3 + 2963*v^2 + 6685*v + 2332) / 702 $$\beta_{7}$$ $$=$$ $$( - 34 \nu^{9} - \nu^{8} - 257 \nu^{7} - 248 \nu^{6} - 1740 \nu^{5} - 1176 \nu^{4} - 3254 \nu^{3} - 3479 \nu^{2} - 6541 \nu - 1306 ) / 702$$ (-34*v^9 - v^8 - 257*v^7 - 248*v^6 - 1740*v^5 - 1176*v^4 - 3254*v^3 - 3479*v^2 - 6541*v - 1306) / 702 $$\beta_{8}$$ $$=$$ $$( - 9 \nu^{9} + 9 \nu^{8} - 79 \nu^{7} + 12 \nu^{6} - 486 \nu^{5} + 132 \nu^{4} - 1035 \nu^{3} - 123 \nu^{2} - 1529 \nu + 420 ) / 156$$ (-9*v^9 + 9*v^8 - 79*v^7 + 12*v^6 - 486*v^5 + 132*v^4 - 1035*v^3 - 123*v^2 - 1529*v + 420) / 156 $$\beta_{9}$$ $$=$$ $$( - 49 \nu^{9} + 56 \nu^{8} - 428 \nu^{7} + 163 \nu^{6} - 2595 \nu^{5} + 1389 \nu^{4} - 5228 \nu^{3} + 1423 \nu^{2} - 7909 \nu + 4160 ) / 702$$ (-49*v^9 + 56*v^8 - 428*v^7 + 163*v^6 - 2595*v^5 + 1389*v^4 - 5228*v^3 + 1423*v^2 - 7909*v + 4160) / 702
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + 3\beta_{4}$$ b8 + 3*b4 $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{3} + 5\beta_{2}$$ -b7 - b6 - b3 + 5*b2 $$\nu^{4}$$ $$=$$ $$\beta_{9} - 8\beta_{8} - 2\beta_{6} - 14\beta_{4} - 8\beta_{3} + \beta_{2} - 14$$ b9 - 8*b8 - 2*b6 - 14*b4 - 8*b3 + b2 - 14 $$\nu^{5}$$ $$=$$ $$10\beta_{9} - 11\beta_{8} - \beta_{6} + 8\beta_{5} - 11\beta_{4} - 28\beta_{2} - 19\beta_1$$ 10*b9 - 11*b8 - b6 + 8*b5 - 11*b4 - 28*b2 - 19*b1 $$\nu^{6}$$ $$=$$ $$9\beta_{9} + 3\beta_{7} + 12\beta_{6} + 57\beta_{3} - 24\beta_{2} + 76$$ 9*b9 + 3*b7 + 12*b6 + 57*b3 - 24*b2 + 76 $$\nu^{7}$$ $$=$$ $$- 66 \beta_{9} + 96 \beta_{8} + 54 \beta_{7} + 78 \beta_{6} - 54 \beta_{5} + 96 \beta_{4} + 96 \beta_{3} - 12 \beta_{2} + 103 \beta _1 + 42$$ -66*b9 + 96*b8 + 54*b7 + 78*b6 - 54*b5 + 96*b4 + 96*b3 - 12*b2 + 103*b1 + 42 $$\nu^{8}$$ $$=$$ $$-174\beta_{9} + 397\beta_{8} + 66\beta_{6} - 42\beta_{5} + 495\beta_{4} + 156\beta_{2} + 48\beta_1$$ -174*b9 + 397*b8 + 66*b6 - 42*b5 + 495*b4 + 156*b2 + 48*b1 $$\nu^{9}$$ $$=$$ $$-108\beta_{9} - 355\beta_{7} - 463\beta_{6} - 769\beta_{3} + 1181\beta_{2} - 426$$ -108*b9 - 355*b7 - 463*b6 - 769*b3 + 1181*b2 - 426

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/285\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 −1.12375 − 1.94639i −0.690702 − 1.19633i 0.145349 + 0.251751i 0.823305 + 1.42601i 1.34580 + 2.33099i −1.12375 + 1.94639i −0.690702 + 1.19633i 0.145349 − 0.251751i 0.823305 − 1.42601i 1.34580 − 2.33099i
−1.12375 + 1.94639i 0.500000 0.866025i −1.52562 2.64245i −0.500000 + 0.866025i 1.12375 + 1.94639i 3.16638 2.36264 −0.500000 0.866025i −1.12375 1.94639i
106.2 −0.690702 + 1.19633i 0.500000 0.866025i 0.0458624 + 0.0794360i −0.500000 + 0.866025i 0.690702 + 1.19633i −4.36264 −2.88952 −0.500000 0.866025i −0.690702 1.19633i
106.3 0.145349 0.251751i 0.500000 0.866025i 0.957748 + 1.65887i −0.500000 + 0.866025i −0.145349 0.251751i −0.486575 1.13822 −0.500000 0.866025i 0.145349 + 0.251751i
106.4 0.823305 1.42601i 0.500000 0.866025i −0.355663 0.616027i −0.500000 + 0.866025i −0.823305 1.42601i 4.47988 2.12194 −0.500000 0.866025i 0.823305 + 1.42601i
106.5 1.34580 2.33099i 0.500000 0.866025i −2.62233 4.54201i −0.500000 + 0.866025i −1.34580 2.33099i −0.797044 −8.73329 −0.500000 0.866025i 1.34580 + 2.33099i
121.1 −1.12375 1.94639i 0.500000 + 0.866025i −1.52562 + 2.64245i −0.500000 0.866025i 1.12375 1.94639i 3.16638 2.36264 −0.500000 + 0.866025i −1.12375 + 1.94639i
121.2 −0.690702 1.19633i 0.500000 + 0.866025i 0.0458624 0.0794360i −0.500000 0.866025i 0.690702 1.19633i −4.36264 −2.88952 −0.500000 + 0.866025i −0.690702 + 1.19633i
121.3 0.145349 + 0.251751i 0.500000 + 0.866025i 0.957748 1.65887i −0.500000 0.866025i −0.145349 + 0.251751i −0.486575 1.13822 −0.500000 + 0.866025i 0.145349 0.251751i
121.4 0.823305 + 1.42601i 0.500000 + 0.866025i −0.355663 + 0.616027i −0.500000 0.866025i −0.823305 + 1.42601i 4.47988 2.12194 −0.500000 + 0.866025i 0.823305 1.42601i
121.5 1.34580 + 2.33099i 0.500000 + 0.866025i −2.62233 + 4.54201i −0.500000 0.866025i −1.34580 + 2.33099i −0.797044 −8.73329 −0.500000 + 0.866025i 1.34580 2.33099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 106.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.i.f 10
3.b odd 2 1 855.2.k.i 10
19.c even 3 1 inner 285.2.i.f 10
19.c even 3 1 5415.2.a.y 5
19.d odd 6 1 5415.2.a.z 5
57.h odd 6 1 855.2.k.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.f 10 1.a even 1 1 trivial
285.2.i.f 10 19.c even 3 1 inner
855.2.k.i 10 3.b odd 2 1
855.2.k.i 10 57.h odd 6 1
5415.2.a.y 5 19.c even 3 1
5415.2.a.z 5 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - T_{2}^{9} + 9T_{2}^{8} - 2T_{2}^{7} + 56T_{2}^{6} - 18T_{2}^{5} + 125T_{2}^{4} + T_{2}^{3} + 189T_{2}^{2} - 52T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(285, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + 9 T^{8} - 2 T^{7} + \cdots + 16$$
$3$ $$(T^{2} - T + 1)^{5}$$
$5$ $$(T^{2} + T + 1)^{5}$$
$7$ $$(T^{5} - 2 T^{4} - 23 T^{3} + 36 T^{2} + \cdots + 24)^{2}$$
$11$ $$(T^{5} - 5 T^{4} - 32 T^{3} + 148 T^{2} + \cdots - 992)^{2}$$
$13$ $$T^{10} - 8 T^{9} + 63 T^{8} + \cdots + 16384$$
$17$ $$T^{10} + 10 T^{9} + 135 T^{8} + \cdots + 25240576$$
$19$ $$T^{10} - 5 T^{9} + 18 T^{8} + \cdots + 2476099$$
$23$ $$T^{10} - 2 T^{9} + 72 T^{8} + \cdots + 147456$$
$29$ $$T^{10} - 7 T^{9} + 111 T^{8} + \cdots + 9048064$$
$31$ $$(T^{5} + 9 T^{4} - 30 T^{3} - 222 T^{2} + \cdots + 117)^{2}$$
$37$ $$(T^{5} - 6 T^{4} - 87 T^{3} + 472 T^{2} + \cdots - 9024)^{2}$$
$41$ $$T^{10} + 12 T^{9} + 120 T^{8} + \cdots + 36864$$
$43$ $$T^{10} - 8 T^{9} + 63 T^{8} + \cdots + 16384$$
$47$ $$T^{10} + 6 T^{9} + 189 T^{8} + \cdots + 4562496$$
$53$ $$T^{10} + 8 T^{9} + 159 T^{8} + \cdots + 18731584$$
$59$ $$T^{10} - T^{9} + 177 T^{8} + \cdots + 20647936$$
$61$ $$T^{10} + 7 T^{9} + 222 T^{8} + \cdots + 591267856$$
$67$ $$T^{10} - 14 T^{9} + 363 T^{8} + \cdots + 70157376$$
$71$ $$T^{10} - 27 T^{9} + 507 T^{8} + \cdots + 37161216$$
$73$ $$T^{10} + 26 T^{9} + \cdots + 2135179264$$
$79$ $$T^{10} + 23 T^{9} + \cdots + 11935125504$$
$83$ $$(T^{5} + 12 T^{4} - 21 T^{3} - 660 T^{2} + \cdots - 2304)^{2}$$
$89$ $$T^{10} - 9 T^{9} + 213 T^{8} + \cdots + 126157824$$
$97$ $$(T^{2} + 2 T + 4)^{5}$$