# Properties

 Label 29.2.e.a Level $29$ Weight $2$ Character orbit 29.e Analytic conductor $0.232$ 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] = [29,2,Mod(4,29)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(29, base_ring=CyclotomicField(14))

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

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

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

chi := DirichletCharacter("29.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 29.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: $$0.231566165862$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: 12.0.7877952219361.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64$$ x^12 - 3*x^11 + 13*x^9 - 18*x^8 - 14*x^7 + 57*x^6 - 28*x^5 - 72*x^4 + 104*x^3 - 96*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + \cdots + 96 ) / 128$$ (v^11 + 7*v^10 - 10*v^9 - 7*v^8 + 40*v^7 - 14*v^6 - 83*v^5 + 102*v^4 + 68*v^3 - 176*v^2 + 32*v + 96) / 128 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} + \cdots - 288 ) / 128$$ (-v^11 + 5*v^10 - 10*v^9 + 7*v^8 + 4*v^7 - 58*v^6 + 59*v^5 + 38*v^4 - 196*v^3 + 96*v^2 + 256*v - 288) / 128 $$\beta_{3}$$ $$=$$ $$( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} + \cdots - 32 ) / 128$$ (-v^11 + 5*v^10 - 10*v^9 + 7*v^8 + 36*v^7 - 58*v^6 - 5*v^5 + 134*v^4 - 100*v^3 - 128*v^2 + 192*v - 32) / 128 $$\beta_{4}$$ $$=$$ $$( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + \cdots + 64 ) / 64$$ (v^11 - 3*v^10 + 4*v^9 + v^8 - 18*v^7 + 22*v^6 + 17*v^5 - 52*v^4 + 44*v^3 + 40*v^2 - 80*v + 64) / 64 $$\beta_{5}$$ $$=$$ $$( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + \cdots + 224 ) / 128$$ (-v^11 + 5*v^10 - 10*v^9 - 25*v^8 + 68*v^7 + 6*v^6 - 165*v^5 + 134*v^4 + 220*v^3 - 288*v^2 + 224) / 128 $$\beta_{6}$$ $$=$$ $$( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + \cdots + 96 ) / 128$$ (3*v^11 + v^10 - 18*v^9 + 11*v^8 + 36*v^7 - 50*v^6 - 49*v^5 + 94*v^4 + 12*v^3 - 128*v^2 + 96) / 128 $$\beta_{7}$$ $$=$$ $$( -\nu^{10} + 3\nu^{9} - 9\nu^{7} + 10\nu^{6} + 6\nu^{5} - 29\nu^{4} + 16\nu^{3} + 20\nu^{2} - 40\nu + 16 ) / 16$$ (-v^10 + 3*v^9 - 9*v^7 + 10*v^6 + 6*v^5 - 29*v^4 + 16*v^3 + 20*v^2 - 40*v + 16) / 16 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} + 9 \nu^{8} - 36 \nu^{7} + 26 \nu^{6} + 53 \nu^{5} - 86 \nu^{4} + \cdots + 32 ) / 64$$ (v^11 - 5*v^10 + 10*v^9 + 9*v^8 - 36*v^7 + 26*v^6 + 53*v^5 - 86*v^4 - 12*v^3 + 96*v^2 - 64*v + 32) / 64 $$\beta_{9}$$ $$=$$ $$( \nu^{11} - 5\nu^{9} + 5\nu^{8} + 9\nu^{7} - 16\nu^{6} - 5\nu^{5} + 31\nu^{4} - 12\nu^{2} + 8\nu + 16 ) / 32$$ (v^11 - 5*v^9 + 5*v^8 + 9*v^7 - 16*v^6 - 5*v^5 + 31*v^4 - 12*v^2 + 8*v + 16) / 32 $$\beta_{10}$$ $$=$$ $$( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + \cdots - 352 ) / 128$$ (3*v^11 - 11*v^10 - 14*v^9 + 59*v^8 - 40*v^7 - 122*v^6 + 199*v^5 + 82*v^4 - 420*v^3 + 176*v^2 + 352*v - 352) / 128 $$\beta_{11}$$ $$=$$ $$( - 15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} + \cdots + 480 ) / 128$$ (-15*v^11 + 31*v^10 + 30*v^9 - 151*v^8 + 80*v^7 + 290*v^6 - 435*v^5 - 146*v^4 + 740*v^3 - 368*v^2 - 416*v + 480) / 128
 $$\nu$$ $$=$$ $$\beta_{8} + \beta_{5} + \beta_{2}$$ b8 + b5 + b2 $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1$$ b9 - b6 - b4 - b3 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1$$ b9 + b7 - b6 + b5 + b4 + b3 + b2 - 1 $$\nu^{4}$$ $$=$$ $$\beta_{10} + \beta_{9} - 2\beta_{8} - 3\beta_{6} - \beta_{5} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 1$$ b10 + b9 - 2*b8 - 3*b6 - b5 - b3 - 2*b2 + 2*b1 + 1 $$\nu^{5}$$ $$=$$ $$-\beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 2\beta _1 - 2$$ -b10 - b8 - b7 - b5 + 5*b4 + 4*b3 - 2*b1 - 2 $$\nu^{6}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4$$ 2*b11 + 2*b10 + 2*b9 + 3*b8 - 3*b7 + 2*b6 - 2*b5 + 2*b4 + 2*b3 - 2*b2 - 2*b1 - 4 $$\nu^{7}$$ $$=$$ $$-5\beta_{10} + \beta_{9} + 6\beta_{8} - 5\beta_{7} + 5\beta_{6} + 5\beta_{3} + \beta_{2} - 10\beta _1 - 5$$ -5*b10 + b9 + 6*b8 - 5*b7 + 5*b6 + 5*b3 + b2 - 10*b1 - 5 $$\nu^{8}$$ $$=$$ $$4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots - 10$$ 4*b11 + 4*b10 + 10*b9 + 11*b8 + 4*b7 + 4*b6 + b5 - 6*b4 + 8*b3 + b2 - 4*b1 - 10 $$\nu^{9}$$ $$=$$ $$- 6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} + \cdots + 5$$ -6*b11 - 10*b10 - 3*b9 + 4*b8 - 11*b6 + 4*b5 - 15*b4 - 5*b3 + 6*b1 + 5 $$\nu^{10}$$ $$=$$ $$2\beta_{11} + 9\beta_{9} + 9\beta_{7} - 7\beta_{6} - 9\beta_{5} + \beta_{4} + 9\beta_{3} + 5\beta_{2} + 18\beta _1 - 1$$ 2*b11 + 9*b9 + 9*b7 - 7*b6 - 9*b5 + b4 + 9*b3 + 5*b2 + 18*b1 - 1 $$\nu^{11}$$ $$=$$ $$- 18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} + 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} + \cdots + 11$$ -18*b11 - 29*b10 - 29*b9 + 8*b8 - 28*b7 - 7*b6 + b5 - 39*b3 + 8*b2 + 36*b1 + 11

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/29\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{11}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.639551 + 1.26134i 1.38491 − 0.286410i 1.23295 + 0.692694i −1.41140 + 0.0891373i 1.23295 − 0.692694i −1.41140 − 0.0891373i −1.25719 − 0.647667i 0.911180 + 1.08155i −1.25719 + 0.647667i 0.911180 − 1.08155i 0.639551 − 1.26134i 1.38491 + 0.286410i
−2.21089 0.504621i −1.23248 2.55926i 2.83146 + 1.36356i 0.0128801 0.0564316i 1.43341 + 6.28018i 1.40728 0.677709i −2.02596 1.61565i −3.16036 + 3.96297i −0.0569531 + 0.118264i
4.2 −0.536089 0.122359i 0.855966 + 1.77743i −1.52952 0.736577i 0.610610 2.67526i −0.241390 1.05760i −4.03077 + 1.94112i 1.58965 + 1.26771i −0.556117 + 0.697349i −0.654683 + 1.35946i
5.1 −0.909335 + 0.725171i 0.960118 0.219141i −0.144024 + 0.631009i −1.18424 1.48499i −0.714155 + 0.895521i −0.339509 1.48749i −1.33591 2.77404i −1.82910 + 0.880850i 2.15374 + 0.491577i
5.2 1.21127 0.965958i −2.86109 + 0.653024i 0.0890656 0.390222i 0.283269 + 0.355208i −2.83476 + 3.55468i −0.759522 3.32768i 1.07536 + 2.23300i 5.05647 2.43507i 0.686232 + 0.156628i
6.1 −0.909335 0.725171i 0.960118 + 0.219141i −0.144024 0.631009i −1.18424 + 1.48499i −0.714155 0.895521i −0.339509 + 1.48749i −1.33591 + 2.77404i −1.82910 0.880850i 2.15374 0.491577i
6.2 1.21127 + 0.965958i −2.86109 0.653024i 0.0890656 + 0.390222i 0.283269 0.355208i −2.83476 3.55468i −0.759522 + 3.32768i 1.07536 2.23300i 5.05647 + 2.43507i 0.686232 0.156628i
9.1 −1.12916 2.34472i −0.343489 + 0.273923i −2.97573 + 3.73144i 2.32488 1.11960i 1.03013 + 0.496082i 0.0468435 + 0.0587399i 7.03485 + 1.60566i −0.624612 + 2.73660i −5.25031 4.18698i
9.2 0.0741982 + 0.154074i −0.879032 + 0.701005i 1.22875 1.54080i −2.54740 + 1.22676i −0.173229 0.0834229i −1.82432 2.28763i 0.662012 + 0.151100i −0.386273 + 1.69237i −0.378025 0.301465i
13.1 −1.12916 + 2.34472i −0.343489 0.273923i −2.97573 3.73144i 2.32488 + 1.11960i 1.03013 0.496082i 0.0468435 0.0587399i 7.03485 1.60566i −0.624612 2.73660i −5.25031 + 4.18698i
13.2 0.0741982 0.154074i −0.879032 0.701005i 1.22875 + 1.54080i −2.54740 1.22676i −0.173229 + 0.0834229i −1.82432 + 2.28763i 0.662012 0.151100i −0.386273 1.69237i −0.378025 + 0.301465i
22.1 −2.21089 + 0.504621i −1.23248 + 2.55926i 2.83146 1.36356i 0.0128801 + 0.0564316i 1.43341 6.28018i 1.40728 + 0.677709i −2.02596 + 1.61565i −3.16036 3.96297i −0.0569531 0.118264i
22.2 −0.536089 + 0.122359i 0.855966 1.77743i −1.52952 + 0.736577i 0.610610 + 2.67526i −0.241390 + 1.05760i −4.03077 1.94112i 1.58965 1.26771i −0.556117 0.697349i −0.654683 1.35946i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.2 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 29.2.e.a 12
3.b odd 2 1 261.2.o.a 12
4.b odd 2 1 464.2.y.d 12
5.b even 2 1 725.2.q.a 12
5.c odd 4 2 725.2.p.a 24
29.b even 2 1 841.2.e.i 12
29.c odd 4 2 841.2.d.m 24
29.d even 7 1 841.2.b.e 12
29.d even 7 1 841.2.e.a 12
29.d even 7 1 841.2.e.e 12
29.d even 7 1 841.2.e.f 12
29.d even 7 1 841.2.e.h 12
29.d even 7 1 841.2.e.i 12
29.e even 14 1 inner 29.2.e.a 12
29.e even 14 1 841.2.b.e 12
29.e even 14 1 841.2.e.a 12
29.e even 14 1 841.2.e.e 12
29.e even 14 1 841.2.e.f 12
29.e even 14 1 841.2.e.h 12
29.f odd 28 2 841.2.a.k 12
29.f odd 28 4 841.2.d.k 24
29.f odd 28 4 841.2.d.l 24
29.f odd 28 2 841.2.d.m 24
87.h odd 14 1 261.2.o.a 12
87.k even 28 2 7569.2.a.bp 12
116.h odd 14 1 464.2.y.d 12
145.l even 14 1 725.2.q.a 12
145.q odd 28 2 725.2.p.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a 12 1.a even 1 1 trivial
29.2.e.a 12 29.e even 14 1 inner
261.2.o.a 12 3.b odd 2 1
261.2.o.a 12 87.h odd 14 1
464.2.y.d 12 4.b odd 2 1
464.2.y.d 12 116.h odd 14 1
725.2.p.a 24 5.c odd 4 2
725.2.p.a 24 145.q odd 28 2
725.2.q.a 12 5.b even 2 1
725.2.q.a 12 145.l even 14 1
841.2.a.k 12 29.f odd 28 2
841.2.b.e 12 29.d even 7 1
841.2.b.e 12 29.e even 14 1
841.2.d.k 24 29.f odd 28 4
841.2.d.l 24 29.f odd 28 4
841.2.d.m 24 29.c odd 4 2
841.2.d.m 24 29.f odd 28 2
841.2.e.a 12 29.d even 7 1
841.2.e.a 12 29.e even 14 1
841.2.e.e 12 29.d even 7 1
841.2.e.e 12 29.e even 14 1
841.2.e.f 12 29.d even 7 1
841.2.e.f 12 29.e even 14 1
841.2.e.h 12 29.d even 7 1
841.2.e.h 12 29.e even 14 1
841.2.e.i 12 29.b even 2 1
841.2.e.i 12 29.d even 7 1
7569.2.a.bp 12 87.k even 28 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(29, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 7 T^{11} + \cdots + 1$$
$3$ $$T^{12} + 7 T^{11} + \cdots + 64$$
$5$ $$T^{12} + T^{11} + \cdots + 1$$
$7$ $$T^{12} + 11 T^{11} + \cdots + 64$$
$11$ $$T^{12} - 7 T^{11} + \cdots + 10816$$
$13$ $$T^{12} - 9 T^{11} + \cdots + 841$$
$17$ $$T^{12} + 71 T^{10} + \cdots + 53824$$
$19$ $$T^{12} + 7 T^{11} + \cdots + 3136$$
$23$ $$T^{12} + 5 T^{11} + \cdots + 64$$
$29$ $$T^{12} + \cdots + 594823321$$
$31$ $$T^{12} + 21 T^{11} + \cdots + 817216$$
$37$ $$T^{12} + \cdots + 110103049$$
$41$ $$T^{12} + 99 T^{10} + \cdots + 107584$$
$43$ $$T^{12} - 7 T^{11} + \cdots + 24364096$$
$47$ $$T^{12} + 7 T^{11} + \cdots + 11343424$$
$53$ $$T^{12} + 10 T^{11} + \cdots + 9409$$
$59$ $$(T^{6} - 22 T^{5} + \cdots + 1856)^{2}$$
$61$ $$T^{12} + \cdots + 325694209$$
$67$ $$T^{12} + \cdots + 415833664$$
$71$ $$T^{12} + \cdots + 2671649344$$
$73$ $$T^{12} - 14 T^{11} + \cdots + 625681$$
$79$ $$T^{12} + \cdots + 30056463424$$
$83$ $$T^{12} + \cdots + 419758144$$
$89$ $$T^{12} + \cdots + 203946961$$
$97$ $$T^{12} - 14 T^{11} + \cdots + 1697809$$