# Properties

 Label 231.2.i.f Level $231$ Weight $2$ Character orbit 231.i Analytic conductor $1.845$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,2,Mod(67,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("231.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ 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: $$x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12$$ x^10 + 15*x^8 + 72*x^6 + 120*x^4 + 72*x^2 + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + \beta_{2} q^{9}+O(q^{10})$$ q - b3 * q^2 + (b2 + 1) * q^3 + (b9 - b7 + b6 - b5 - b4 - 2*b2 - 2) * q^4 + (-b8 - b2 - b1) * q^5 + b6 * q^6 + (b9 - b5 - b4) * q^7 + (-2*b6 + b5 + b4 - b1) * q^8 + b2 * q^9 $$q - \beta_{3} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + q^{99}+O(q^{100})$$ q - b3 * q^2 + (b2 + 1) * q^3 + (b9 - b7 + b6 - b5 - b4 - 2*b2 - 2) * q^4 + (-b8 - b2 - b1) * q^5 + b6 * q^6 + (b9 - b5 - b4) * q^7 + (-2*b6 + b5 + b4 - b1) * q^8 + b2 * q^9 + (-b9 + 3*b8 - b7 - b6 - b5 - 2*b3 + b1) * q^10 + (-b2 - 1) * q^11 + (-b7 - b4 - 2*b2) * q^12 + (-b9 - b4 + 2) * q^13 + (2*b8 - b7 + b4 + 2*b2 + b1 - 1) * q^14 + (-b1 + 1) * q^15 + (-b8 + b7 + b4 + 3*b2 - b1) * q^16 + (b6 + b3) * q^17 + (b6 + b3) * q^18 + (-b9 + 2*b8 + 2*b4 - b3 + b1) * q^19 + (2*b5 + 2*b4 + b1 - 3) * q^20 + (b8 - b7 - b5 - b4 - b3 + b1) * q^21 - b6 * q^22 + (-b8 + b7 + b4 + 3*b2 - b1) * q^23 + (b9 - b8 + b7 - 2*b6 + b5 - b3 - b1) * q^24 + (-2*b8 + 2*b6 + 2*b3 - b2 - 1) * q^25 + (3*b8 - 2*b7 - 2*b4 - 4*b3 - 3*b2 + 3*b1) * q^26 - q^27 + (-4*b8 + 2*b7 + b5 + b4 + 3*b3 + 4*b2 - b1 + 2) * q^28 + (-b5 - b4 + 3*b1) * q^29 + (-b9 + 3*b8 - b7 + b4 - 2*b3 + 2*b1) * q^30 + (-b9 + b7 - b6 + b5 + b4 - b2 - 1) * q^31 + (2*b8 + b6 + b3) * q^32 - b2 * q^33 + (-b9 - b6 + b5 + 4) * q^34 + (b9 - b8 - b7 - b4 + 2*b3 - 2*b1 + 2) * q^35 + (-b9 - b6 + b5 + 2) * q^36 + (b8 + 2*b3 + 4*b2 + b1) * q^37 + (-b9 - 3*b8 + b7 + b6 + b5 + b4 + 2*b3 - b2 - 1) * q^38 + (-2*b9 + 2*b8 + b4 + 2*b2 + b1 + 2) * q^39 + (b9 - 5*b8 + 3*b7 + b4 + 2*b3 - 2*b2 - 4*b1) * q^40 + (-b6 - b5 - b4 - b1 - 4) * q^41 + (b9 - b8 - b6 + b5 - b2 - 3) * q^42 + (b9 + 2*b6 - 3*b5 - 2*b4 + b1 + 2) * q^43 + (b7 + b4 + 2*b2) * q^44 + (b8 + b2 + 1) * q^45 + (-2*b9 + 4*b8 - 2*b7 + 5*b6 - 2*b5 + 3*b3 + 2*b1) * q^46 + (b7 + b4 - 4*b3 - 2*b2) * q^47 + (b9 + b6 - b5 - b1 - 3) * q^48 + (-2*b9 + b8 + b7 - 2*b6 + b4 + 2*b2 + 2*b1 + 4) * q^49 + (-2*b9 - 3*b6 - 2*b4 - 2*b1 + 8) * q^50 + b3 * q^51 + (5*b9 - 7*b8 + b7 - 3*b6 + b5 - 2*b4 - 2*b3 - 12*b2 - 3*b1 - 12) * q^52 + (2*b9 - 3*b8 + 2*b7 - 2*b6 + 2*b5 - 3*b2 - 2*b1 - 3) * q^53 + b3 * q^54 + (b1 - 1) * q^55 + (-b9 + b8 + 3*b6 - b5 - 2*b4 + 2*b3 + 5*b2 - b1 + 4) * q^56 + (b9 + b6 + b4) * q^57 + (2*b9 - 5*b8 + b7 - 3*b4 + 2*b3 + b2 - 3*b1) * q^58 + (b9 + b8 - b7 + b6 - b5 - b4 - 3*b2 - 3) * q^59 + (2*b9 - 5*b8 + 2*b7 + 2*b5 + 2*b3 - 3*b2 - 2*b1 - 3) * q^60 + (3*b9 - 3*b8 + 3*b7 - 3*b4) * q^61 + (b6 - b5 - b4 + b1) * q^62 + (-b9 + b8 - b7 - b3 + b1) * q^63 + (b9 + b6 + b5 + 2*b4 - 2) * q^64 + (2*b9 - 4*b8 - 4*b4 + 4*b3 - 6*b2 - 2*b1) * q^65 + (-b6 - b3) * q^66 + (3*b9 - 2*b8 + b7 - b6 + b5 - b4 - 3*b2 - 2*b1 - 3) * q^67 + (b9 - b8 + b7 - b4 - 3*b3) * q^68 + (b9 + b6 - b5 - b1 - 3) * q^69 + (-2*b9 + 3*b8 + 2*b7 - 4*b6 + 2*b5 + 4*b4 - 4*b3 + 9*b2 + 7) * q^70 + (3*b9 + b6 - 3*b5 + 4) * q^71 + (b9 - b8 + b7 - b4 - b3) * q^72 + (-4*b9 + 2*b7 + 2*b5 + 3*b4 + 2*b3 - 2*b2 + b1 - 2) * q^73 + (-b9 - 3*b8 + 3*b7 + 2*b6 + 3*b5 + 2*b4 + 5*b3 + 8*b2 - b1 + 8) * q^74 + (-2*b8 + 2*b3 - b2 - 2*b1) * q^75 + (b6 - 2*b5 - 2*b4 - 2*b1 + 8) * q^76 + (-b8 + b7 + b5 + b4 + b3 - b1) * q^77 + (-2*b9 + 2*b6 + 2*b5 + 3*b1 + 3) * q^78 + (-b9 + 4*b8 - 4*b7 - 2*b4 - 2*b3 + 2*b2 + 3*b1) * q^79 + (-2*b9 + 5*b8 - 2*b7 + 2*b6 - 2*b5 - b2 + 2*b1 - 1) * q^80 + (-b2 - 1) * q^81 + (-2*b9 + 7*b8 - 2*b7 + 2*b4 + 2*b3 + 5*b2 + 5*b1) * q^82 + (2*b9 + 4*b6 - 2*b5 + 2*b1 - 2) * q^83 + (b9 - 3*b8 + b7 - b6 - b5 + b3 + 2*b2 - 3*b1 - 2) * q^84 + (b6 + b5 + b4 + b1) * q^85 + (-2*b9 + 6*b8 - 5*b7 - b4 - 2*b3 - 2*b2 + 4*b1) * q^86 + (-b9 - b8 - b7 - b5 - b3 + b1) * q^87 + (-b9 + b8 - b7 + 2*b6 - b5 + b3 + b1) * q^88 + (-2*b9 - 2*b7 + 2*b4 - 2*b1) * q^89 + (b6 + b5 + b4 + b1) * q^90 + (b8 - b7 - 6*b6 + b4 - 4*b3 - 4*b2 - 2*b1 - 3) * q^91 + (-5*b9 - 5*b6 + 7*b5 + 2*b4 + 2*b1 + 12) * q^92 + (b7 + b4 - b2) * q^93 + (3*b9 + b8 - 5*b7 + 4*b6 - 5*b5 - 4*b4 - b3 - 16*b2 + b1 - 16) * q^94 + (-5*b9 + 5*b8 - b7 - 5*b6 - b5 + 2*b4 - 6*b3 + 4*b2 + 3*b1 + 4) * q^95 + (2*b8 + b3 + 2*b1) * q^96 + (-b9 + b6 + 5*b5 + 4*b4 - 2*b1 - 4) * q^97 + (-b9 - b8 + b7 + 4*b6 - b4 - b3 + 6*b2 + 2*b1 + 2) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 2 q^{2} + 5 q^{3} - 10 q^{4} + 4 q^{5} - 4 q^{6} - q^{7} + 12 q^{8} - 5 q^{9}+O(q^{10})$$ 10 * q - 2 * q^2 + 5 * q^3 - 10 * q^4 + 4 * q^5 - 4 * q^6 - q^7 + 12 * q^8 - 5 * q^9 $$10 q - 2 q^{2} + 5 q^{3} - 10 q^{4} + 4 q^{5} - 4 q^{6} - q^{7} + 12 q^{8} - 5 q^{9} - 2 q^{10} - 5 q^{11} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 8 q^{15} - 16 q^{16} - 2 q^{17} - 2 q^{18} + 3 q^{19} - 16 q^{20} - 2 q^{21} + 4 q^{22} - 16 q^{23} + 6 q^{24} - 7 q^{25} + 10 q^{26} - 10 q^{27} + 4 q^{28} + 2 q^{30} - 5 q^{31} - 4 q^{32} + 5 q^{33} + 40 q^{34} + 26 q^{35} + 20 q^{36} - 15 q^{37} - 6 q^{38} + 5 q^{39} + 6 q^{40} - 44 q^{41} - 14 q^{42} + 6 q^{43} - 10 q^{44} + 4 q^{45} - 16 q^{46} + 2 q^{47} - 32 q^{48} + 31 q^{49} + 68 q^{50} + 2 q^{51} - 40 q^{52} - 6 q^{53} + 2 q^{54} - 8 q^{55} - 12 q^{56} + 6 q^{57} - 12 q^{58} - 16 q^{59} - 8 q^{60} - 12 q^{61} - 8 q^{62} - q^{63} - 8 q^{64} + 28 q^{65} + 2 q^{66} - 7 q^{67} - 10 q^{68} - 32 q^{69} + 32 q^{70} + 48 q^{71} - 6 q^{72} - 17 q^{73} + 36 q^{74} + 7 q^{75} + 60 q^{76} + 2 q^{77} + 20 q^{78} - 7 q^{79} - 16 q^{80} - 5 q^{81} - 8 q^{82} - 24 q^{83} - 28 q^{84} + 4 q^{85} + 18 q^{86} - 6 q^{88} + 6 q^{89} + 4 q^{90} + 11 q^{91} + 136 q^{92} + 5 q^{93} - 82 q^{94} + 18 q^{95} + 4 q^{96} - 28 q^{97} - 38 q^{98} + 10 q^{99}+O(q^{100})$$ 10 * q - 2 * q^2 + 5 * q^3 - 10 * q^4 + 4 * q^5 - 4 * q^6 - q^7 + 12 * q^8 - 5 * q^9 - 2 * q^10 - 5 * q^11 + 10 * q^12 + 10 * q^13 - 10 * q^14 + 8 * q^15 - 16 * q^16 - 2 * q^17 - 2 * q^18 + 3 * q^19 - 16 * q^20 - 2 * q^21 + 4 * q^22 - 16 * q^23 + 6 * q^24 - 7 * q^25 + 10 * q^26 - 10 * q^27 + 4 * q^28 + 2 * q^30 - 5 * q^31 - 4 * q^32 + 5 * q^33 + 40 * q^34 + 26 * q^35 + 20 * q^36 - 15 * q^37 - 6 * q^38 + 5 * q^39 + 6 * q^40 - 44 * q^41 - 14 * q^42 + 6 * q^43 - 10 * q^44 + 4 * q^45 - 16 * q^46 + 2 * q^47 - 32 * q^48 + 31 * q^49 + 68 * q^50 + 2 * q^51 - 40 * q^52 - 6 * q^53 + 2 * q^54 - 8 * q^55 - 12 * q^56 + 6 * q^57 - 12 * q^58 - 16 * q^59 - 8 * q^60 - 12 * q^61 - 8 * q^62 - q^63 - 8 * q^64 + 28 * q^65 + 2 * q^66 - 7 * q^67 - 10 * q^68 - 32 * q^69 + 32 * q^70 + 48 * q^71 - 6 * q^72 - 17 * q^73 + 36 * q^74 + 7 * q^75 + 60 * q^76 + 2 * q^77 + 20 * q^78 - 7 * q^79 - 16 * q^80 - 5 * q^81 - 8 * q^82 - 24 * q^83 - 28 * q^84 + 4 * q^85 + 18 * q^86 - 6 * q^88 + 6 * q^89 + 4 * q^90 + 11 * q^91 + 136 * q^92 + 5 * q^93 - 82 * q^94 + 18 * q^95 + 4 * q^96 - 28 * q^97 - 38 * q^98 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} + 7\nu^{2} + 5$$ v^4 + 7*v^2 + 5 $$\beta_{2}$$ $$=$$ $$( \nu^{9} + 15\nu^{7} + 70\nu^{5} + 102\nu^{3} + 36\nu - 4 ) / 8$$ (v^9 + 15*v^7 + 70*v^5 + 102*v^3 + 36*v - 4) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{9} + 2\nu^{8} + 13\nu^{7} + 28\nu^{6} + 44\nu^{5} + 118\nu^{4} + 6\nu^{3} + 140\nu^{2} - 36\nu + 40 ) / 8$$ (v^9 + 2*v^8 + 13*v^7 + 28*v^6 + 44*v^5 + 118*v^4 + 6*v^3 + 140*v^2 - 36*v + 40) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{9} + \nu^{8} + 14\nu^{7} + 15\nu^{6} + 59\nu^{5} + 70\nu^{4} + 70\nu^{3} + 100\nu^{2} + 18\nu + 32 ) / 4$$ (v^9 + v^8 + 14*v^7 + 15*v^6 + 59*v^5 + 70*v^4 + 70*v^3 + 100*v^2 + 18*v + 32) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{9} - \nu^{8} - 14\nu^{7} - 13\nu^{6} - 59\nu^{5} - 48\nu^{4} - 70\nu^{3} - 36\nu^{2} - 18\nu + 4 ) / 4$$ (-v^9 - v^8 - 14*v^7 - 13*v^6 - 59*v^5 - 48*v^4 - 70*v^3 - 36*v^2 - 18*v + 4) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{8} - 14\nu^{6} - 59\nu^{4} - 70\nu^{2} - 20 ) / 2$$ (-v^8 - 14*v^6 - 59*v^4 - 70*v^2 - 20) / 2 $$\beta_{7}$$ $$=$$ $$( \nu^{9} - 2\nu^{8} + 15\nu^{7} - 30\nu^{6} + 74\nu^{5} - 140\nu^{4} + 130\nu^{3} - 204\nu^{2} + 60\nu - 76 ) / 8$$ (v^9 - 2*v^8 + 15*v^7 - 30*v^6 + 74*v^5 - 140*v^4 + 130*v^3 - 204*v^2 + 60*v - 76) / 8 $$\beta_{8}$$ $$=$$ $$( -3\nu^{9} - 43\nu^{7} - 188\nu^{5} - 4\nu^{4} - 246\nu^{3} - 28\nu^{2} - 96\nu - 20 ) / 8$$ (-3*v^9 - 43*v^7 - 188*v^5 - 4*v^4 - 246*v^3 - 28*v^2 - 96*v - 20) / 8 $$\beta_{9}$$ $$=$$ $$( -\nu^{9} + \nu^{8} - 14\nu^{7} + 15\nu^{6} - 59\nu^{5} + 70\nu^{4} - 70\nu^{3} + 100\nu^{2} - 18\nu + 32 ) / 4$$ (-v^9 + v^8 - 14*v^7 + 15*v^6 - 59*v^5 + 70*v^4 - 70*v^3 + 100*v^2 - 18*v + 32) / 4
 $$\nu$$ $$=$$ $$( 2\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 3$$ (2*b9 - 2*b8 + 2*b7 + b5 - b4 + 2*b3 - 2*b2 - b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$-\beta_{9} - \beta_{6} + \beta_{5} - 3$$ -b9 - b6 + b5 - 3 $$\nu^{3}$$ $$=$$ $$-3\beta_{9} + 2\beta_{8} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - 4\beta_{3} + 2\beta_{2} + \beta _1 + 1$$ -3*b9 + 2*b8 - 4*b7 - 2*b5 + b4 - 4*b3 + 2*b2 + b1 + 1 $$\nu^{4}$$ $$=$$ $$7\beta_{9} + 7\beta_{6} - 7\beta_{5} + \beta _1 + 16$$ 7*b9 + 7*b6 - 7*b5 + b1 + 16 $$\nu^{5}$$ $$=$$ $$17 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} - \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 24 \beta_{3} + \cdots - 6$$ 17*b9 - 10*b8 + 26*b7 - b6 + 13*b5 - 4*b4 + 24*b3 - 12*b2 - 5*b1 - 6 $$\nu^{6}$$ $$=$$ $$-45\beta_{9} - 45\beta_{6} + 47\beta_{5} + 2\beta_{4} - 11\beta _1 - 98$$ -45*b9 - 45*b6 + 47*b5 + 2*b4 - 11*b1 - 98 $$\nu^{7}$$ $$=$$ $$- 101 \beta_{9} + 58 \beta_{8} - 170 \beta_{7} + 11 \beta_{6} - 85 \beta_{5} + 16 \beta_{4} - 148 \beta_{3} + \cdots + 44$$ -101*b9 + 58*b8 - 170*b7 + 11*b6 - 85*b5 + 16*b4 - 148*b3 + 88*b2 + 29*b1 + 44 $$\nu^{8}$$ $$=$$ $$287\beta_{9} + 285\beta_{6} - 315\beta_{5} - 28\beta_{4} + 95\beta _1 + 618$$ 287*b9 + 285*b6 - 315*b5 - 28*b4 + 95*b1 + 618 $$\nu^{9}$$ $$=$$ $$607 \beta_{9} - 350 \beta_{8} + 1114 \beta_{7} - 95 \beta_{6} + 557 \beta_{5} - 50 \beta_{4} + \cdots - 326$$ 607*b9 - 350*b8 + 1114*b7 - 95*b6 + 557*b5 - 50*b4 + 924*b3 - 652*b2 - 175*b1 - 326

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 − 0.518255i 1.21103i 2.57330i − 2.42024i 0.886226i 0.518255i − 1.21103i − 2.57330i 2.42024i − 0.886226i
−1.29725 2.24690i 0.500000 0.866025i −2.36571 + 4.09752i −1.09601 1.89835i −2.59450 −2.61329 + 0.413178i 7.08664 −0.500000 0.866025i −2.84359 + 4.92525i
67.2 −1.17616 2.03717i 0.500000 0.866025i −1.76671 + 3.06003i 2.05761 + 3.56389i −2.35232 2.51585 0.818848i 3.60708 −0.500000 0.866025i 4.84017 8.38342i
67.3 −0.307468 0.532550i 0.500000 0.866025i 0.810927 1.40457i −0.747986 1.29555i −0.614936 2.59895 + 0.495442i −2.22721 −0.500000 0.866025i −0.459963 + 0.796680i
67.4 0.534421 + 0.925645i 0.500000 0.866025i 0.428788 0.742682i 1.34592 + 2.33120i 1.06884 −0.855706 2.50355i 3.05430 −0.500000 0.866025i −1.43858 + 2.49169i
67.5 1.24646 + 2.15892i 0.500000 0.866025i −2.10730 + 3.64995i 0.440463 + 0.762904i 2.49291 −2.14580 + 1.54775i −5.52081 −0.500000 0.866025i −1.09804 + 1.90185i
100.1 −1.29725 + 2.24690i 0.500000 + 0.866025i −2.36571 4.09752i −1.09601 + 1.89835i −2.59450 −2.61329 0.413178i 7.08664 −0.500000 + 0.866025i −2.84359 4.92525i
100.2 −1.17616 + 2.03717i 0.500000 + 0.866025i −1.76671 3.06003i 2.05761 3.56389i −2.35232 2.51585 + 0.818848i 3.60708 −0.500000 + 0.866025i 4.84017 + 8.38342i
100.3 −0.307468 + 0.532550i 0.500000 + 0.866025i 0.810927 + 1.40457i −0.747986 + 1.29555i −0.614936 2.59895 0.495442i −2.22721 −0.500000 + 0.866025i −0.459963 0.796680i
100.4 0.534421 0.925645i 0.500000 + 0.866025i 0.428788 + 0.742682i 1.34592 2.33120i 1.06884 −0.855706 + 2.50355i 3.05430 −0.500000 + 0.866025i −1.43858 2.49169i
100.5 1.24646 2.15892i 0.500000 + 0.866025i −2.10730 3.64995i 0.440463 0.762904i 2.49291 −2.14580 1.54775i −5.52081 −0.500000 + 0.866025i −1.09804 1.90185i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.i.f 10
3.b odd 2 1 693.2.i.j 10
7.c even 3 1 inner 231.2.i.f 10
7.c even 3 1 1617.2.a.ba 5
7.d odd 6 1 1617.2.a.bb 5
21.g even 6 1 4851.2.a.bz 5
21.h odd 6 1 693.2.i.j 10
21.h odd 6 1 4851.2.a.ca 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.f 10 1.a even 1 1 trivial
231.2.i.f 10 7.c even 3 1 inner
693.2.i.j 10 3.b odd 2 1
693.2.i.j 10 21.h odd 6 1
1617.2.a.ba 5 7.c even 3 1
1617.2.a.bb 5 7.d odd 6 1
4851.2.a.bz 5 21.g even 6 1
4851.2.a.ca 5 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 2T_{2}^{9} + 12T_{2}^{8} + 12T_{2}^{7} + 81T_{2}^{6} + 78T_{2}^{5} + 264T_{2}^{4} + 6T_{2}^{3} + 261T_{2}^{2} + 110T_{2} + 100$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 2 T^{9} + \cdots + 100$$
$3$ $$(T^{2} - T + 1)^{5}$$
$5$ $$T^{10} - 4 T^{9} + \cdots + 1024$$
$7$ $$T^{10} + T^{9} + \cdots + 16807$$
$11$ $$(T^{2} + T + 1)^{5}$$
$13$ $$(T^{5} - 5 T^{4} + \cdots + 476)^{2}$$
$17$ $$T^{10} + 2 T^{9} + \cdots + 100$$
$19$ $$T^{10} - 3 T^{9} + \cdots + 363609$$
$23$ $$T^{10} + 16 T^{9} + \cdots + 38416$$
$29$ $$(T^{5} - 96 T^{3} + \cdots - 1290)^{2}$$
$31$ $$T^{10} + 5 T^{9} + \cdots + 400$$
$37$ $$T^{10} + 15 T^{9} + \cdots + 40401$$
$41$ $$(T^{5} + 22 T^{4} + \cdots - 11536)^{2}$$
$43$ $$(T^{5} - 3 T^{4} + \cdots - 2973)^{2}$$
$47$ $$T^{10} + \cdots + 783664036$$
$53$ $$T^{10} + 6 T^{9} + \cdots + 42302016$$
$59$ $$T^{10} + 16 T^{9} + \cdots + 38416$$
$61$ $$T^{10} + \cdots + 2361960000$$
$67$ $$T^{10} + 7 T^{9} + \cdots + 9265936$$
$71$ $$(T^{5} - 24 T^{4} + \cdots + 5232)^{2}$$
$73$ $$T^{10} + 17 T^{9} + \cdots + 9265936$$
$79$ $$T^{10} + \cdots + 8999937424$$
$83$ $$(T^{5} + 12 T^{4} + \cdots + 2688)^{2}$$
$89$ $$T^{10} + \cdots + 315417600$$
$97$ $$(T^{5} + 14 T^{4} + \cdots - 38906)^{2}$$