Properties

 Label 58.2.e.a Level $58$ Weight $2$ Character orbit 58.e Analytic conductor $0.463$ Analytic rank $0$ Dimension $12$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [58,2,Mod(5,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([11]))

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

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

chi := DirichletCharacter("58.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 58.e (of order $$14$$, degree $$6$$, 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: $$0.463132331723$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: $$\Q(\zeta_{28})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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 primitive root of unity $$\zeta_{28}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/58\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$\chi(n)$$ $$-\zeta_{28}^{4}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.433884 + 0.900969i 0.433884 − 0.900969i −0.974928 + 0.222521i 0.974928 − 0.222521i −0.974928 − 0.222521i 0.974928 + 0.222521i 0.781831 − 0.623490i −0.781831 + 0.623490i −0.433884 − 0.900969i 0.433884 + 0.900969i 0.781831 + 0.623490i −0.781831 − 0.623490i
−0.781831 + 0.623490i 0.240787 0.0549581i 0.222521 0.974928i 2.13946 + 2.68280i −0.153989 + 0.193096i −0.349501 1.53126i 0.433884 + 0.900969i −2.64795 + 1.27518i −3.34540 0.763565i
5.2 0.781831 0.623490i −0.240787 + 0.0549581i 0.222521 0.974928i −0.892482 1.11914i −0.153989 + 0.193096i 0.904459 + 3.96269i −0.433884 0.900969i −2.64795 + 1.27518i −1.39554 0.318523i
9.1 −0.433884 0.900969i 2.19064 1.74698i −0.623490 + 0.781831i −3.43956 + 1.65640i −2.52446 1.21572i 1.36428 + 1.71075i 0.974928 + 0.222521i 1.07942 4.72923i 2.98474 + 2.38025i
9.2 0.433884 + 0.900969i −2.19064 + 1.74698i −0.623490 + 0.781831i 1.63762 0.788637i −2.52446 1.21572i 0.882702 + 1.10687i −0.974928 0.222521i 1.07942 4.72923i 1.42108 + 1.13327i
13.1 −0.433884 + 0.900969i 2.19064 + 1.74698i −0.623490 0.781831i −3.43956 1.65640i −2.52446 + 1.21572i 1.36428 1.71075i 0.974928 0.222521i 1.07942 + 4.72923i 2.98474 2.38025i
13.2 0.433884 0.900969i −2.19064 1.74698i −0.623490 0.781831i 1.63762 + 0.788637i −2.52446 + 1.21572i 0.882702 1.10687i −0.974928 + 0.222521i 1.07942 + 4.72923i 1.42108 1.13327i
33.1 −0.974928 0.222521i 0.626980 + 1.30194i 0.900969 + 0.433884i −0.308457 + 1.35144i −0.321552 1.40881i 1.78967 0.861862i −0.781831 0.623490i 0.568532 0.712916i 0.601447 1.24892i
33.2 0.974928 + 0.222521i −0.626980 1.30194i 0.900969 + 0.433884i −0.136585 + 0.598418i −0.321552 1.40881i −2.59161 + 1.24805i 0.781831 + 0.623490i 0.568532 0.712916i −0.266321 + 0.553021i
35.1 −0.781831 0.623490i 0.240787 + 0.0549581i 0.222521 + 0.974928i 2.13946 2.68280i −0.153989 0.193096i −0.349501 + 1.53126i 0.433884 0.900969i −2.64795 1.27518i −3.34540 + 0.763565i
35.2 0.781831 + 0.623490i −0.240787 0.0549581i 0.222521 + 0.974928i −0.892482 + 1.11914i −0.153989 0.193096i 0.904459 3.96269i −0.433884 + 0.900969i −2.64795 1.27518i −1.39554 + 0.318523i
51.1 −0.974928 + 0.222521i 0.626980 1.30194i 0.900969 0.433884i −0.308457 1.35144i −0.321552 + 1.40881i 1.78967 + 0.861862i −0.781831 + 0.623490i 0.568532 + 0.712916i 0.601447 + 1.24892i
51.2 0.974928 0.222521i −0.626980 + 1.30194i 0.900969 0.433884i −0.136585 0.598418i −0.321552 + 1.40881i −2.59161 1.24805i 0.781831 0.623490i 0.568532 + 0.712916i −0.266321 0.553021i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.2.e.a 12
3.b odd 2 1 522.2.n.a 12
4.b odd 2 1 464.2.y.c 12
29.d even 7 1 1682.2.b.j 12
29.e even 14 1 inner 58.2.e.a 12
29.e even 14 1 1682.2.b.j 12
29.f odd 28 1 1682.2.a.r 6
29.f odd 28 1 1682.2.a.s 6
87.h odd 14 1 522.2.n.a 12
116.h odd 14 1 464.2.y.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.e.a 12 1.a even 1 1 trivial
58.2.e.a 12 29.e even 14 1 inner
464.2.y.c 12 4.b odd 2 1
464.2.y.c 12 116.h odd 14 1
522.2.n.a 12 3.b odd 2 1
522.2.n.a 12 87.h odd 14 1
1682.2.a.r 6 29.f odd 28 1
1682.2.a.s 6 29.f odd 28 1
1682.2.b.j 12 29.d even 7 1
1682.2.b.j 12 29.e even 14 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(58, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{10} + \cdots + 1$$
$3$ $$T^{12} - T^{10} + \cdots + 1$$
$5$ $$T^{12} + 2 T^{11} + \cdots + 841$$
$7$ $$T^{12} - 4 T^{11} + \cdots + 12769$$
$11$ $$T^{12} - 4 T^{10} + \cdots + 841$$
$13$ $$T^{12} + 26 T^{11} + \cdots + 38809$$
$17$ $$T^{12} + 94 T^{10} + \cdots + 12769$$
$19$ $$T^{12} + 26 T^{10} + \cdots + 175561$$
$23$ $$T^{12} + 16 T^{11} + \cdots + 729$$
$29$ $$T^{12} + \cdots + 594823321$$
$31$ $$T^{12} + \cdots + 5377435561$$
$37$ $$T^{12} + 28 T^{11} + \cdots + 3736489$$
$41$ $$T^{12} + 236 T^{10} + \cdots + 60171049$$
$43$ $$T^{12} + 28 T^{11} + \cdots + 8637721$$
$47$ $$T^{12} + 14 T^{11} + \cdots + 24571849$$
$53$ $$T^{12} + \cdots + 2202143329$$
$59$ $$(T^{6} + 24 T^{5} + \cdots + 35869)^{2}$$
$61$ $$T^{12} + \cdots + 105805126729$$
$67$ $$T^{12} + \cdots + 2228500849$$
$71$ $$T^{12} + \cdots + 15150901921$$
$73$ $$T^{12} - 42 T^{11} + \cdots + 5938969$$
$79$ $$T^{12} + \cdots + 4893282304$$
$83$ $$T^{12} - 35 T^{10} + \cdots + 2019241$$
$89$ $$T^{12} - 14 T^{11} + \cdots + 34656769$$
$97$ $$T^{12} + \cdots + 10446475264$$