# Properties

 Label 52.2.l.b Level $52$ Weight $2$ Character orbit 52.l Analytic conductor $0.415$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(7,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 11]))

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

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

chi := DirichletCharacter("52.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.l (of order $$12$$, degree $$4$$, 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: $$0.415222090511$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.102930383934669717504.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256$$ x^16 - 4*x^15 + 5*x^14 - 2*x^13 + 5*x^12 - 8*x^11 - 12*x^10 + 32*x^9 - 36*x^8 + 64*x^7 - 48*x^6 - 64*x^5 + 80*x^4 - 64*x^3 + 320*x^2 - 512*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{3}) q^{3} + (\beta_{12} - \beta_{10} + \beta_{8} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{12} + \beta_{10} - \beta_{7} - \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{15} + \beta_{13} + 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + ( - \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{12} - \beta_{10} + \beta_{5} + \beta_{3} + \beta_1) q^{9}+O(q^{10})$$ q - b12 * q^2 + (b14 - b13 + b12 + b3) * q^3 + (b12 - b10 + b8 + b4 + b3) * q^4 + (-b12 + b10 - b7 - b4 - b3 - 1) * q^5 + (b15 + b13 + 2*b9 + b7 + b6 + b3 - b2 + b1 - 1) * q^6 + (-b15 - 2*b14 + b13 - 2*b12 + 2*b10 - b9 - 2*b8 - b6 - b5 - 3*b3 - b1) * q^7 + (b15 + b14 - b13 + b12 + b11 - b10 - b7 + b6 + b3 + b2 - b1) * q^8 + (b12 - b10 + b5 + b3 + b1) * q^9 $$q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{3}) q^{3} + (\beta_{12} - \beta_{10} + \beta_{8} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{12} + \beta_{10} - \beta_{7} - \beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{15} + \beta_{13} + 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + ( - \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{8} + 2 \beta_1) q^{99}+O(q^{100})$$ q - b12 * q^2 + (b14 - b13 + b12 + b3) * q^3 + (b12 - b10 + b8 + b4 + b3) * q^4 + (-b12 + b10 - b7 - b4 - b3 - 1) * q^5 + (b15 + b13 + 2*b9 + b7 + b6 + b3 - b2 + b1 - 1) * q^6 + (-b15 - 2*b14 + b13 - 2*b12 + 2*b10 - b9 - 2*b8 - b6 - b5 - 3*b3 - b1) * q^7 + (b15 + b14 - b13 + b12 + b11 - b10 - b7 + b6 + b3 + b2 - b1) * q^8 + (b12 - b10 + b5 + b3 + b1) * q^9 + (-b15 - b13 + b12 + b7 + b5 + b2 - b1) * q^10 + (-b10 + b8 - b6 + b5 - b3 + b1) * q^11 + (-b15 - 2*b14 + b13 - b12 - 2*b11 + b10 - 2*b9 - b8 - b7 - b6 - 2*b5 - b4 - 3*b3 - b2 - b1 + 1) * q^12 + (b12 - b10 - 2*b9 + 2*b7 - b5 - b3 - b1 - 1) * q^13 + (-b15 + b13 + b10 - 2*b9 + b8 + b7 - 2*b6 + b5 - b4 - b3 + 3*b1) * q^14 + (b15 + b14 + b13 + 2*b12 - b10 + 2*b9 + 2*b8 + 2*b6 + b5 + 3*b3 + b2 + b1) * q^15 + (-b15 + b13 - 2*b12 - 2*b8 + b7 + 2*b4 - 2*b3 + b2 - b1) * q^16 + (b13 - b11 + b9 - b5 + 2*b4 + b3 - b2 + 2) * q^17 + (b15 - b14 + b13 - b12 - b11 + b10 - b7 - b6 - b3 - b2 + b1) * q^18 + (b15 + b6 + 2*b3 - b2 - b1) * q^19 + (b15 - b13 + 2*b11 - b7 - b6 - 2*b4 - b2 + b1 - 1) * q^20 + (-2*b13 - b12 + b11 + b10 + 2*b9 + b4 + b3 + 2*b2 - 2*b1 - 1) * q^21 + (b14 - 2*b13 + b11 + b10 + 2*b9 - 4*b7 + b6 - 2*b4 + b3 + b2 - 2*b1) * q^22 + (b14 - b13 - b12 - 2*b10 + 2*b6 - 2*b5 + b3) * q^23 + (-b15 + b14 - b13 + b12 + b11 + b10 - 2*b9 + b7 + 2*b5 + b3 + b2 + b1 - 1) * q^24 + (b13 + b12 - b10 + b9 - b7 + b5 - b4 + 2*b3 - b2 + 2*b1) * q^25 + (-b15 + b14 - b13 + b11 + b10 - 2*b8 + b7 + b6 - 2*b5 + 2*b4 - b3 - b2 - b1) * q^26 + (-b13 - b12 - b10 + b9 + b6 - b3 - b2 - 2*b1) * q^27 + (2*b15 + 2*b14 - 2*b13 + 2*b12 - 2*b11 + 4*b9 + 2*b8 + 2*b7 + 2*b6 + 4*b3 - 2*b2 + 2) * q^28 + (-2*b13 - 2*b12 + 3*b11 + 2*b10 + 2*b5 - 2*b3 + 2*b2 - 3) * q^29 + (-2*b13 - 2*b11 - 2*b10 - 2*b9 - 2*b8 - b5 - b3 + b2 - 4*b1 + 4) * q^30 + (3*b15 + b13 + 2*b12 - b10 + 3*b8 - 3*b6 + 2*b5 + 2*b3 - 2*b2 + 5*b1) * q^31 + (-b15 - b14 + b13 - b11 + b8 - 3*b7 - 2*b6 - b4 + b2 + 3*b1 + 3) * q^32 + (3*b13 - 3*b11 + b7 - 3*b5 + 3*b4 - 3*b2 + 1) * q^33 + (b15 + b13 - b10 + b8 - b7 + b6 - b4 + 2*b3 + b2 + b1 + 1) * q^34 + (-2*b15 - b14 + b13 - 2*b12 - 3*b9 - b8 - b6 - 2*b3 + 2*b2) * q^35 + (-b15 + b13 + 2*b10 + b7 + b2 - b1) * q^36 + (-b4 - 1) * q^37 + (-2*b14 + 2*b13 - b12 + 2*b11 - 2*b9 + 4*b7 - b6 - b5 + 4*b4 - 2*b3 - b2 - 1) * q^38 + (-4*b15 - b14 - 2*b12 + b10 - 4*b9 - 2*b8 - 2*b6 + 2*b5 - 4*b3 + 3*b2 - 3*b1) * q^39 + (2*b13 - 2*b12 + 2*b10 + 2*b9 + b6 - 2*b3 - 2*b2 - 3) * q^40 + (-3*b4 + 3) * q^41 + (b15 + 2*b14 + b13 + 4*b12 + 2*b11 - 4*b10 + 2*b8 - b7 - 2*b4 + 3*b3 + b1 - 2) * q^42 + (-b14 + 2*b13 + b12 + 4*b10 + 2*b9 - 2*b8 + b6 - 3*b5 + b2 + b1) * q^43 + (2*b11 - 2*b10 + 4*b5 - 2*b4 + 2*b2 - 2) * q^44 + (-b13 - 2*b11 - 2*b9 - 2*b7 - b5 - 2*b4 - 2*b3 + b2 - 2*b1 + 2) * q^45 + (-b15 - b13 + 2*b12 - 4*b11 - 4*b10 - 2*b9 - b7 + b6 + 4*b4 + b3 + b2 - b1 - 1) * q^46 + (b15 + 2*b14 - 3*b13 + 2*b12 + b10 + 4*b9 - b8 + b6 - 2*b2 - 3*b1) * q^47 + (2*b15 - b14 - b11 + 2*b9 + b8 + 6*b7 - b6 + 3*b4 - 2*b2 + 2*b1) * q^48 + (b13 + 2*b12 + b11 - 2*b10 - 3*b9 + b7 + 2*b5 - b3 - b2 + 3*b1 + 1) * q^49 + (2*b15 - 2*b14 + 2*b13 - b12 - 2*b11 + b10 + b8 - 2*b7 - b6 + b5 - b4 - b3 - 2*b2 + 2*b1 + 1) * q^50 + (-2*b14 + 2*b10 + b9 - b6 - 2*b5 - 2*b3 - 2*b2 + b1) * q^51 + (-b15 - 2*b14 + b13 - 3*b12 - 2*b11 + b10 - b8 + b7 - b4 - 3*b3 + b2 + 3*b1 - 2) * q^52 + (-2*b13 + 2*b12 - 2*b10 - b9 + b5 + b3 + 2*b2 - b1 - 2) * q^53 + (-2*b15 - 2*b14 - 2*b13 - 2*b12 + 4*b11 + b10 - 2*b9 - b8 - 4*b7 - b6 - b4 - 3*b3 + 2*b2 - 2*b1 - 3) * q^54 + (-b15 + b14 - 3*b13 + b12 + 2*b10 - b9 - 2*b8 - 2*b5 + b3 - b2 - 2*b1) * q^55 + (-2*b14 - 2*b12 + 2*b11 - 2*b8 + 2*b6 - 4*b5 - 2*b4 - 2*b2 - 4*b1 - 4) * q^56 + (-2*b13 + b12 + b11 - b10 + 4*b5 - b4 + b3 + 2*b2 + 2*b1 - 1) * q^57 + (-2*b15 - 2*b13 + 2*b7 - 2*b6 - 3*b3 + 2*b2 - 2*b1 - 2) * q^58 + (3*b15 + 4*b14 + b13 + 6*b12 - 2*b10 + 3*b9 + 4*b8 + b6 + 5*b5 + 7*b3 + 2*b2 + 3*b1) * q^59 + (-b15 + 3*b13 - 3*b12 + 4*b11 + b10 - b8 - b7 - b6 + 3*b4 + b3 - b2 + 3*b1 - 3) * q^60 + (b13 - 2*b12 + b11 + 2*b10 - b9 - 3*b5 - 3*b3 - b2 - 2*b1) * q^61 + (2*b15 + 2*b14 + 2*b13 - b12 + 4*b10 + 6*b9 + 4*b6 - 3*b5 + 2*b3 - 2*b2 - 4*b1) * q^62 + (-2*b12 + 2*b9 - 2*b8 + 2*b6 - 2*b5 - 2*b3 - 2*b1) * q^63 + (3*b15 + 4*b14 - b13 + 3*b12 - 4*b11 - 3*b10 + 2*b9 + 3*b8 - 3*b7 + 2*b6 + 4*b5 - 3*b4 + 7*b3 + b2 + b1 + 2) * q^64 + (3*b13 - b12 + b10 + 2*b9 - 3*b7 - 3*b5 + 3*b4 + b3 - 3*b2 + 1) * q^65 + (2*b12 + 3*b6 - b5 + 3*b3 + 2*b2 + 3) * q^66 + (-2*b15 - 2*b14 + 2*b13 - 5*b12 - 5*b9 - 3*b8 - 3*b6 - 5*b3 + 2*b2) * q^67 + (b14 + b12 - 3*b11 - 3*b10 - 2*b9 + 2*b7 + 4*b4 + b3 - 2*b1 + 3) * q^68 + (2*b13 - 3*b12 + b11 + 3*b10 + 2*b9 + b5 - 4*b4 - b3 - 2*b2 + 3*b1 - 2) * q^69 + (b15 + 2*b14 - b13 + 4*b12 - 2*b11 - 3*b10 + 3*b8 + b7 - b6 + 2*b5 + 3*b4 + 4*b3 + 3*b1 + 3) * q^70 + (-b15 - 2*b14 - 4*b9 + 2*b8 - 3*b6 + 2*b5 - b2 + b1) * q^71 + (b15 + b14 - b13 + 2*b12 + b11 - 2*b10 + b8 + 3*b7 - b4 + 2*b3 - b2 + b1 + 3) * q^72 + (-b13 - 2*b12 + 2*b10 + 4*b9 - 3*b7 - 3*b4 + 2*b3 + b2 - b1 + 3) * q^73 + (b12 - b2) * q^74 + (-b15 - 2*b14 - 3*b12 - b10 - 4*b9 - 4*b6 - b5 - 4*b3 - b2 + 3*b1) * q^75 + (-b15 + b14 - b13 - b12 + b11 + b10 - 2*b9 + b7 - 4*b5 + 2*b4 - 3*b3 + b2 - b1 + 1) * q^76 + (-4*b11 - b9 + 5*b7 - b5 + 5*b4 - b3 - b1 + 2) * q^77 + (2*b15 + b14 - 2*b13 + 3*b12 + 5*b11 - b10 + 2*b9 + 4*b8 - 3*b6 + 4*b5 + 4*b3 + 6*b1 + 2) * q^78 + (-2*b15 - 2*b13 - 2*b10 - 4*b9 + 2*b8 - 2*b6 + 2*b1) * q^79 + (-3*b15 - 3*b14 - 3*b13 + b11 + 2*b10 - 2*b9 - b8 - b7 - b6 - 3*b4 - 4*b3 + 3*b2 - 3*b1 + 2) * q^80 + (-2*b13 - 5*b11 - 2*b9 + 2*b7 + 4*b4 - 2*b3 + 2*b2 - 2*b1 + 5) * q^81 + (-3*b12 - 3*b2) * q^82 + (-5*b15 - 2*b13 - 3*b12 + 2*b10 - 5*b8 + 5*b6 - 4*b5 - 3*b3 + 3*b2 - 8*b1) * q^83 + (2*b15 + 2*b14 + 4*b9 - 2*b8 - 4*b7 + 4*b6 - 2*b2 - 2*b1 + 4) * q^84 + (-3*b13 + 5*b11 - 2*b7 + 3*b5 - 5*b4 + 3*b2 - 2) * q^85 + (-b15 + b14 + 3*b13 - 9*b11 - b10 - 4*b9 - 2*b8 + 5*b7 - b6 - 4*b4 - 2*b3 - b2 + 3*b1 + 4) * q^86 + (8*b15 + 5*b14 + b13 + 5*b12 - 4*b10 + 7*b9 + 4*b8 + 5*b6 + 2*b5 + 11*b3 - 2*b2 + 3*b1) * q^87 + (-2*b14 - 2*b12 + 2*b11 + 2*b10 - 4*b7 - 4*b6 - 4*b3 - 2*b2 + 2) * q^88 + (3*b12 - 3*b10 + 3*b9 - b4 + 6*b3 - 1) * q^89 + (-2*b15 + 2*b14 - 2*b13 + 2*b11 - 2*b8 + 2*b7 + b6 + 2*b5 + 2*b4 + 2*b3 + 2*b2 - 2*b1 - 1) * q^90 + (7*b15 + 4*b14 - b13 + 4*b12 - 4*b10 + 7*b9 + 4*b8 + 5*b6 + 3*b5 + 9*b3 - 4*b2 - b1) * q^91 + (b15 - b13 + 5*b12 - b10 - 2*b9 - b8 - 3*b7 - b6 + 2*b5 + 3*b4 + 3*b3 + 5*b2 + b1 - 1) * q^92 + (6*b13 - b12 + 3*b11 + b10 + b9 - 3*b7 - 3*b5 + 4*b4 - 6*b2 + 3*b1 - 7) * q^93 + (-2*b15 - 4*b14 + 2*b13 - 7*b12 + 6*b11 + 6*b10 + 2*b9 - 4*b8 - 4*b7 - b5 - 8*b4 - 6*b3 - 6) * q^94 + (-b14 + 2*b13 + 2*b9 + b8 + b6 + b5 - b3 + b2) * q^95 + (-b15 + b13 - b12 + 3*b10 - b8 - 3*b7 + b6 - 8*b5 - 3*b4 - b3 - 3*b2 - 3*b1 - 3) * q^96 + (-b13 + 6*b11 - 2*b9 + 5*b7 - b5 + 6*b4 - 2*b3 + b2 - 2*b1 - 5) * q^97 + (-b15 - 3*b14 - b13 - 8*b12 - 3*b11 + 6*b10 - 3*b8 + b7 - 2*b6 - b5 + 3*b4 - 7*b3 - b1 + 1) * q^98 + (-2*b14 + 2*b13 + 2*b8 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 6 q^{4} - 12 q^{5} - 14 q^{6} + 10 q^{8} + 4 q^{9}+O(q^{10})$$ 16 * q - 2 * q^2 - 6 * q^4 - 12 * q^5 - 14 * q^6 + 10 * q^8 + 4 * q^9 $$16 q - 2 q^{2} - 6 q^{4} - 12 q^{5} - 14 q^{6} + 10 q^{8} + 4 q^{9} - 12 q^{13} + 8 q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{18} + 2 q^{20} - 28 q^{21} + 10 q^{24} + 16 q^{26} + 12 q^{28} - 8 q^{29} + 42 q^{30} + 28 q^{32} - 20 q^{33} + 14 q^{34} - 6 q^{36} - 16 q^{37} - 40 q^{40} + 48 q^{41} - 28 q^{42} - 8 q^{44} + 20 q^{45} - 46 q^{46} - 10 q^{48} + 60 q^{49} + 10 q^{50} - 32 q^{52} - 32 q^{53} - 16 q^{54} - 60 q^{56} + 12 q^{57} - 48 q^{58} - 24 q^{60} + 4 q^{61} - 18 q^{62} - 8 q^{65} + 56 q^{66} + 16 q^{68} - 12 q^{69} + 28 q^{70} + 56 q^{72} + 20 q^{73} + 4 q^{74} + 22 q^{76} + 68 q^{78} + 44 q^{80} + 48 q^{81} + 84 q^{84} + 20 q^{85} + 16 q^{86} + 36 q^{88} - 52 q^{89} - 12 q^{92} - 92 q^{93} - 38 q^{94} - 72 q^{96} - 28 q^{97} - 2 q^{98}+O(q^{100})$$ 16 * q - 2 * q^2 - 6 * q^4 - 12 * q^5 - 14 * q^6 + 10 * q^8 + 4 * q^9 - 12 * q^13 + 8 * q^14 - 2 * q^16 + 12 * q^17 - 6 * q^18 + 2 * q^20 - 28 * q^21 + 10 * q^24 + 16 * q^26 + 12 * q^28 - 8 * q^29 + 42 * q^30 + 28 * q^32 - 20 * q^33 + 14 * q^34 - 6 * q^36 - 16 * q^37 - 40 * q^40 + 48 * q^41 - 28 * q^42 - 8 * q^44 + 20 * q^45 - 46 * q^46 - 10 * q^48 + 60 * q^49 + 10 * q^50 - 32 * q^52 - 32 * q^53 - 16 * q^54 - 60 * q^56 + 12 * q^57 - 48 * q^58 - 24 * q^60 + 4 * q^61 - 18 * q^62 - 8 * q^65 + 56 * q^66 + 16 * q^68 - 12 * q^69 + 28 * q^70 + 56 * q^72 + 20 * q^73 + 4 * q^74 + 22 * q^76 + 68 * q^78 + 44 * q^80 + 48 * q^81 + 84 * q^84 + 20 * q^85 + 16 * q^86 + 36 * q^88 - 52 * q^89 - 12 * q^92 - 92 * q^93 - 38 * q^94 - 72 * q^96 - 28 * q^97 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 19 \nu^{15} + 26 \nu^{14} + 49 \nu^{13} + 12 \nu^{12} - 163 \nu^{11} - 146 \nu^{10} + 172 \nu^{9} + 184 \nu^{8} + 412 \nu^{7} - 408 \nu^{6} - 1072 \nu^{5} - 544 \nu^{4} + 1360 \nu^{3} + \cdots - 1152 ) / 1408$$ (-19*v^15 + 26*v^14 + 49*v^13 + 12*v^12 - 163*v^11 - 146*v^10 + 172*v^9 + 184*v^8 + 412*v^7 - 408*v^6 - 1072*v^5 - 544*v^4 + 1360*v^3 + 2592*v^2 - 2112*v - 1152) / 1408 $$\beta_{3}$$ $$=$$ $$( \nu^{15} - 4 \nu^{14} + 5 \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 32 \nu^{8} - 36 \nu^{7} + 64 \nu^{6} - 48 \nu^{5} - 64 \nu^{4} + 80 \nu^{3} - 64 \nu^{2} + 320 \nu - 512 ) / 128$$ (v^15 - 4*v^14 + 5*v^13 - 2*v^12 + 5*v^11 - 8*v^10 - 12*v^9 + 32*v^8 - 36*v^7 + 64*v^6 - 48*v^5 - 64*v^4 + 80*v^3 - 64*v^2 + 320*v - 512) / 128 $$\beta_{4}$$ $$=$$ $$( - 25 \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 76 \nu^{12} - 61 \nu^{11} - 182 \nu^{10} + 40 \nu^{8} + 580 \nu^{7} + 152 \nu^{6} - 704 \nu^{5} - 672 \nu^{4} - 16 \nu^{3} + 2336 \nu^{2} - 512 \nu - 384 ) / 1408$$ (-25*v^15 + 6*v^14 + 31*v^13 + 76*v^12 - 61*v^11 - 182*v^10 + 40*v^8 + 580*v^7 + 152*v^6 - 704*v^5 - 672*v^4 - 16*v^3 + 2336*v^2 - 512*v - 384) / 1408 $$\beta_{5}$$ $$=$$ $$( 3 \nu^{15} - 62 \nu^{14} + 27 \nu^{13} + 56 \nu^{12} + 167 \nu^{11} - 146 \nu^{10} - 400 \nu^{9} + 96 \nu^{8} - 28 \nu^{7} + 1352 \nu^{6} + 160 \nu^{5} - 1600 \nu^{4} - 1104 \nu^{3} - 224 \nu^{2} + \cdots - 2560 ) / 1408$$ (3*v^15 - 62*v^14 + 27*v^13 + 56*v^12 + 167*v^11 - 146*v^10 - 400*v^9 + 96*v^8 - 28*v^7 + 1352*v^6 + 160*v^5 - 1600*v^4 - 1104*v^3 - 224*v^2 + 5632*v - 2560) / 1408 $$\beta_{6}$$ $$=$$ $$( 10 \nu^{15} - 37 \nu^{14} - 12 \nu^{13} + 7 \nu^{12} + 106 \nu^{11} + 87 \nu^{10} - 266 \nu^{9} - 80 \nu^{8} - 264 \nu^{7} + 612 \nu^{6} + 872 \nu^{5} - 480 \nu^{4} - 800 \nu^{3} - 1744 \nu^{2} + \cdots + 1216 ) / 704$$ (10*v^15 - 37*v^14 - 12*v^13 + 7*v^12 + 106*v^11 + 87*v^10 - 266*v^9 - 80*v^8 - 264*v^7 + 612*v^6 + 872*v^5 - 480*v^4 - 800*v^3 - 1744*v^2 + 2976*v + 1216) / 704 $$\beta_{7}$$ $$=$$ $$( - 25 \nu^{15} + 72 \nu^{14} - 13 \nu^{13} - 34 \nu^{12} - 149 \nu^{11} - 28 \nu^{10} + 396 \nu^{9} - 136 \nu^{8} + 404 \nu^{7} - 992 \nu^{6} - 880 \nu^{5} + 1440 \nu^{4} + 688 \nu^{3} + 1984 \nu^{2} + \cdots + 2432 ) / 1408$$ (-25*v^15 + 72*v^14 - 13*v^13 - 34*v^12 - 149*v^11 - 28*v^10 + 396*v^9 - 136*v^8 + 404*v^7 - 992*v^6 - 880*v^5 + 1440*v^4 + 688*v^3 + 1984*v^2 - 5440*v + 2432) / 1408 $$\beta_{8}$$ $$=$$ $$( - 7 \nu^{15} + 17 \nu^{14} + \nu^{13} + 5 \nu^{12} - 57 \nu^{11} - 9 \nu^{10} + 100 \nu^{9} - 14 \nu^{8} + 152 \nu^{7} - 300 \nu^{6} - 216 \nu^{5} + 232 \nu^{4} + 272 \nu^{3} + 992 \nu^{2} - 1888 \nu + 544 ) / 352$$ (-7*v^15 + 17*v^14 + v^13 + 5*v^12 - 57*v^11 - 9*v^10 + 100*v^9 - 14*v^8 + 152*v^7 - 300*v^6 - 216*v^5 + 232*v^4 + 272*v^3 + 992*v^2 - 1888*v + 544) / 352 $$\beta_{9}$$ $$=$$ $$( - 7 \nu^{15} + 28 \nu^{14} - 21 \nu^{13} - 6 \nu^{12} - 57 \nu^{11} + 24 \nu^{10} + 166 \nu^{9} - 124 \nu^{8} + 152 \nu^{7} - 520 \nu^{6} - 40 \nu^{5} + 672 \nu^{4} + 96 \nu^{3} + 640 \nu^{2} - 2592 \nu + 1600 ) / 352$$ (-7*v^15 + 28*v^14 - 21*v^13 - 6*v^12 - 57*v^11 + 24*v^10 + 166*v^9 - 124*v^8 + 152*v^7 - 520*v^6 - 40*v^5 + 672*v^4 + 96*v^3 + 640*v^2 - 2592*v + 1600) / 352 $$\beta_{10}$$ $$=$$ $$( - 47 \nu^{15} + 78 \nu^{14} + 13 \nu^{13} + 32 \nu^{12} - 191 \nu^{11} - 150 \nu^{10} + 420 \nu^{9} - 160 \nu^{8} + 876 \nu^{7} - 952 \nu^{6} - 1136 \nu^{5} + 992 \nu^{4} + 368 \nu^{3} + 3744 \nu^{2} + \cdots + 3200 ) / 704$$ (-47*v^15 + 78*v^14 + 13*v^13 + 32*v^12 - 191*v^11 - 150*v^10 + 420*v^9 - 160*v^8 + 876*v^7 - 952*v^6 - 1136*v^5 + 992*v^4 + 368*v^3 + 3744*v^2 - 6592*v + 3200) / 704 $$\beta_{11}$$ $$=$$ $$( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + 628 \nu^{8} - 948 \nu^{7} + 2140 \nu^{6} + 528 \nu^{5} - 2912 \nu^{4} - 304 \nu^{3} - 3312 \nu^{2} + \cdots - 8704 ) / 704$$ (53*v^15 - 139*v^14 + 61*v^13 + 3*v^12 + 271*v^11 - 59*v^10 - 748*v^9 + 628*v^8 - 948*v^7 + 2140*v^6 + 528*v^5 - 2912*v^4 - 304*v^3 - 3312*v^2 + 12800*v - 8704) / 704 $$\beta_{12}$$ $$=$$ $$( 34 \nu^{15} - 83 \nu^{14} + 31 \nu^{13} - 7 \nu^{12} + 173 \nu^{11} - \nu^{10} - 467 \nu^{9} + 340 \nu^{8} - 596 \nu^{7} + 1228 \nu^{6} + 508 \nu^{5} - 1648 \nu^{4} - 192 \nu^{3} - 2480 \nu^{2} + \cdots - 4608 ) / 352$$ (34*v^15 - 83*v^14 + 31*v^13 - 7*v^12 + 173*v^11 - v^10 - 467*v^9 + 340*v^8 - 596*v^7 + 1228*v^6 + 508*v^5 - 1648*v^4 - 192*v^3 - 2480*v^2 + 7568*v - 4608) / 352 $$\beta_{13}$$ $$=$$ $$( 73 \nu^{15} - 204 \nu^{14} + 109 \nu^{13} + 6 \nu^{12} + 365 \nu^{11} - 112 \nu^{10} - 1068 \nu^{9} + 960 \nu^{8} - 1252 \nu^{7} + 3072 \nu^{6} + 480 \nu^{5} - 4544 \nu^{4} + 80 \nu^{3} + \cdots - 13568 ) / 704$$ (73*v^15 - 204*v^14 + 109*v^13 + 6*v^12 + 365*v^11 - 112*v^10 - 1068*v^9 + 960*v^8 - 1252*v^7 + 3072*v^6 + 480*v^5 - 4544*v^4 + 80*v^3 - 4160*v^2 + 17728*v - 13568) / 704 $$\beta_{14}$$ $$=$$ $$( - 82 \nu^{15} + 201 \nu^{14} - 52 \nu^{13} + 13 \nu^{12} - 434 \nu^{11} - 51 \nu^{10} + 1138 \nu^{9} - 624 \nu^{8} + 1560 \nu^{7} - 2932 \nu^{6} - 1608 \nu^{5} + 3568 \nu^{4} + 992 \nu^{3} + \cdots + 9728 ) / 704$$ (-82*v^15 + 201*v^14 - 52*v^13 + 13*v^12 - 434*v^11 - 51*v^10 + 1138*v^9 - 624*v^8 + 1560*v^7 - 2932*v^6 - 1608*v^5 + 3568*v^4 + 992*v^3 + 6416*v^2 - 17696*v + 9728) / 704 $$\beta_{15}$$ $$=$$ $$( - 46 \nu^{15} + 129 \nu^{14} - 72 \nu^{13} + 3 \nu^{12} - 230 \nu^{11} + 87 \nu^{10} + 654 \nu^{9} - 686 \nu^{8} + 848 \nu^{7} - 1896 \nu^{6} - 112 \nu^{5} + 2744 \nu^{4} - 400 \nu^{3} + 2496 \nu^{2} + \cdots + 9760 ) / 352$$ (-46*v^15 + 129*v^14 - 72*v^13 + 3*v^12 - 230*v^11 + 87*v^10 + 654*v^9 - 686*v^8 + 848*v^7 - 1896*v^6 - 112*v^5 + 2744*v^4 - 400*v^3 + 2496*v^2 - 11552*v + 9760) / 352
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{5} - \beta_{4} + \beta_1$$ b8 + b5 - b4 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta _1 + 1$$ b14 - b13 + 2*b12 - b11 - 2*b10 + b8 + 2*b5 + b4 + 2*b3 + b2 + b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{15} + 2\beta_{12} + \beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{4} + 3\beta_{3} - 2\beta_{2} + 2\beta_1$$ b15 + 2*b12 + b9 + 2*b8 + b7 + 2*b4 + 3*b3 - 2*b2 + 2*b1 $$\nu^{5}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta _1 - 3$$ b15 + 2*b14 - 2*b13 + 4*b12 + 2*b11 + 2*b9 + 2*b8 - 3*b7 + b6 + 3*b5 - 2*b4 + 4*b3 + 2*b2 + b1 - 3 $$\nu^{6}$$ $$=$$ $$3 \beta_{15} + 4 \beta_{14} + 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 7 \beta_{3} + \beta _1 - 2$$ 3*b15 + 4*b14 + 4*b12 + 4*b11 - 4*b10 + b9 + 3*b8 + 3*b7 + 2*b6 + 3*b5 + 3*b4 + 7*b3 + b1 - 2 $$\nu^{7}$$ $$=$$ $$3 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + 4 \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 4 \beta _1 - 2$$ 3*b15 + 3*b14 + 5*b13 + 6*b12 - 3*b11 + 4*b9 + b8 + 3*b7 + b6 - b5 + 5*b4 + 4*b3 - 5*b2 + 4*b1 - 2 $$\nu^{8}$$ $$=$$ $$- 2 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 9 \beta_1$$ -2*b15 + 4*b14 - 4*b13 + 2*b12 + 8*b11 + 4*b10 + 2*b9 - b8 - 2*b7 + 4*b6 - 5*b5 - b4 - 2*b3 - 2*b2 - 9*b1 $$\nu^{9}$$ $$=$$ $$2 \beta_{15} + 7 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 4 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + 5 \beta _1 + 5$$ 2*b15 + 7*b14 + 5*b13 - 2*b12 + 5*b11 + 4*b9 - 5*b8 - 10*b7 + 2*b6 - 5*b4 + 6*b3 - b2 + 5*b1 + 5 $$\nu^{10}$$ $$=$$ $$- 3 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 11 \beta_{9} + 5 \beta_{7} + 4 \beta_{6} - 12 \beta_{5} - 13 \beta_{3} - 12 \beta_{2} + 8 \beta _1 - 6$$ -3*b15 + 2*b14 + 10*b13 - 10*b12 - 6*b11 - 11*b9 + 5*b7 + 4*b6 - 12*b5 - 13*b3 - 12*b2 + 8*b1 - 6 $$\nu^{11}$$ $$=$$ $$- 13 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} - 22 \beta_{11} - 16 \beta_{9} - 2 \beta_{8} - 17 \beta_{7} - 11 \beta_{6} + 15 \beta_{5} - 22 \beta_{4} - 18 \beta_{3} - 2 \beta_{2} + \beta _1 + 17$$ -13*b15 + 2*b14 - 6*b13 - 22*b11 - 16*b9 - 2*b8 - 17*b7 - 11*b6 + 15*b5 - 22*b4 - 18*b3 - 2*b2 + b1 + 17 $$\nu^{12}$$ $$=$$ $$- 7 \beta_{15} - 20 \beta_{13} - 4 \beta_{12} - 20 \beta_{10} - 27 \beta_{9} + 7 \beta_{8} - 23 \beta_{7} - 16 \beta_{6} - 3 \beta_{5} + 23 \beta_{4} - \beta_{3} - 4 \beta_{2} + 7 \beta _1 + 20$$ -7*b15 - 20*b13 - 4*b12 - 20*b10 - 27*b9 + 7*b8 - 23*b7 - 16*b6 - 3*b5 + 23*b4 - b3 - 4*b2 + 7*b1 + 20 $$\nu^{13}$$ $$=$$ $$13 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 14 \beta_{12} - 35 \beta_{11} + 16 \beta_{10} - 30 \beta_{9} + 27 \beta_{8} - 35 \beta_{7} - 27 \beta_{6} + 9 \beta_{5} - 33 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} + 58 \beta _1 + 2$$ 13*b15 - 13*b14 + 13*b13 + 14*b12 - 35*b11 + 16*b10 - 30*b9 + 27*b8 - 35*b7 - 27*b6 + 9*b5 - 33*b4 + 14*b3 - 13*b2 + 58*b1 + 2 $$\nu^{14}$$ $$=$$ $$30 \beta_{14} - 78 \beta_{13} + 54 \beta_{12} + 30 \beta_{11} - 60 \beta_{10} - 78 \beta_{9} + 45 \beta_{8} - 30 \beta_{6} + 77 \beta_{5} - 27 \beta_{4} - 2 \beta_{3} + 56 \beta_{2} + 15 \beta _1 - 60$$ 30*b14 - 78*b13 + 54*b12 + 30*b11 - 60*b10 - 78*b9 + 45*b8 - 30*b6 + 77*b5 - 27*b4 - 2*b3 + 56*b2 + 15*b1 - 60 $$\nu^{15}$$ $$=$$ $$78 \beta_{15} + 15 \beta_{14} + 45 \beta_{13} + 90 \beta_{12} - 27 \beta_{11} - 72 \beta_{10} + 93 \beta_{8} + 34 \beta_{7} - 78 \beta_{6} + 66 \beta_{5} + 61 \beta_{4} + 90 \beta_{3} - 45 \beta_{2} + 111 \beta _1 + 61$$ 78*b15 + 15*b14 + 45*b13 + 90*b12 - 27*b11 - 72*b10 + 93*b8 + 34*b7 - 78*b6 + 66*b5 + 61*b4 + 90*b3 - 45*b2 + 111*b1 + 61

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.31256 + 0.526485i −0.873468 + 1.11223i 1.08916 − 0.902074i −1.39427 − 0.236640i −0.00757716 − 1.41419i 1.41121 − 0.0921725i 1.17605 + 0.785427i −0.713659 + 1.22094i 1.31256 − 0.526485i −0.873468 − 1.11223i 1.08916 + 0.902074i −1.39427 + 0.236640i −0.00757716 + 1.41419i 1.41121 + 0.0921725i 1.17605 − 0.785427i −0.713659 − 1.22094i
−1.11223 0.873468i −2.16981 1.25274i 0.474107 + 1.94299i −2.19962 2.19962i 1.31910 + 3.28859i 0.152604 0.569525i 1.16983 2.57517i 1.63871 + 2.83834i 0.525184 + 4.36778i
7.2 −0.526485 + 1.31256i 2.16981 + 1.25274i −1.44563 1.38209i −2.19962 2.19962i −2.78667 + 2.18846i −0.152604 + 0.569525i 2.57517 1.16983i 1.63871 + 2.83834i 4.04520 1.72907i
7.3 0.236640 1.39427i 0.736159 + 0.425021i −1.88800 0.659882i −0.166404 0.166404i 0.766801 0.925830i −0.684384 + 2.55416i −1.36683 + 2.47624i −1.13871 1.97231i −0.271390 + 0.192635i
7.4 0.902074 + 1.08916i −0.736159 0.425021i −0.372527 + 1.96500i −0.166404 0.166404i −0.201154 1.18519i 0.684384 2.55416i −2.47624 + 1.36683i −1.13871 1.97231i 0.0311314 0.331348i
11.1 −1.22094 0.713659i 1.40004 0.808315i 0.981383 + 1.74267i −1.52798 1.52798i −2.28623 0.0122495i 1.97429 0.529008i 0.0454612 2.82806i −0.193255 + 0.334727i 0.775114 + 2.95603i
11.2 −0.785427 + 1.17605i 1.81380 1.04720i −0.766209 1.84741i 0.894007 + 0.894007i −0.193046 + 2.95563i −4.37156 + 1.17136i 2.77446 + 0.549903i 0.693255 1.20075i −1.75358 + 0.349224i
11.3 0.0921725 + 1.41121i −1.81380 + 1.04720i −1.98301 + 0.260149i 0.894007 + 0.894007i −1.64500 2.46313i 4.37156 1.17136i −0.549903 2.77446i 0.693255 1.20075i −1.17923 + 1.34403i
11.4 1.41419 0.00757716i −1.40004 + 0.808315i 1.99989 0.0214311i −1.52798 1.52798i −1.97381 + 1.15372i −1.97429 + 0.529008i 2.82806 0.0454612i −0.193255 + 0.334727i −2.17244 2.14928i
15.1 −1.11223 + 0.873468i −2.16981 + 1.25274i 0.474107 1.94299i −2.19962 + 2.19962i 1.31910 3.28859i 0.152604 + 0.569525i 1.16983 + 2.57517i 1.63871 2.83834i 0.525184 4.36778i
15.2 −0.526485 1.31256i 2.16981 1.25274i −1.44563 + 1.38209i −2.19962 + 2.19962i −2.78667 2.18846i −0.152604 0.569525i 2.57517 + 1.16983i 1.63871 2.83834i 4.04520 + 1.72907i
15.3 0.236640 + 1.39427i 0.736159 0.425021i −1.88800 + 0.659882i −0.166404 + 0.166404i 0.766801 + 0.925830i −0.684384 2.55416i −1.36683 2.47624i −1.13871 + 1.97231i −0.271390 0.192635i
15.4 0.902074 1.08916i −0.736159 + 0.425021i −0.372527 1.96500i −0.166404 + 0.166404i −0.201154 + 1.18519i 0.684384 + 2.55416i −2.47624 1.36683i −1.13871 + 1.97231i 0.0311314 + 0.331348i
19.1 −1.22094 + 0.713659i 1.40004 + 0.808315i 0.981383 1.74267i −1.52798 + 1.52798i −2.28623 + 0.0122495i 1.97429 + 0.529008i 0.0454612 + 2.82806i −0.193255 0.334727i 0.775114 2.95603i
19.2 −0.785427 1.17605i 1.81380 + 1.04720i −0.766209 + 1.84741i 0.894007 0.894007i −0.193046 2.95563i −4.37156 1.17136i 2.77446 0.549903i 0.693255 + 1.20075i −1.75358 0.349224i
19.3 0.0921725 1.41121i −1.81380 1.04720i −1.98301 0.260149i 0.894007 0.894007i −1.64500 + 2.46313i 4.37156 + 1.17136i −0.549903 + 2.77446i 0.693255 + 1.20075i −1.17923 1.34403i
19.4 1.41419 + 0.00757716i −1.40004 0.808315i 1.99989 + 0.0214311i −1.52798 + 1.52798i −1.97381 1.15372i −1.97429 0.529008i 2.82806 + 0.0454612i −0.193255 0.334727i −2.17244 + 2.14928i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.f odd 12 1 inner
52.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.l.b 16
3.b odd 2 1 468.2.cb.f 16
4.b odd 2 1 inner 52.2.l.b 16
8.b even 2 1 832.2.bu.n 16
8.d odd 2 1 832.2.bu.n 16
12.b even 2 1 468.2.cb.f 16
13.b even 2 1 676.2.l.k 16
13.c even 3 1 676.2.f.h 16
13.c even 3 1 676.2.l.m 16
13.d odd 4 1 676.2.l.i 16
13.d odd 4 1 676.2.l.m 16
13.e even 6 1 676.2.f.i 16
13.e even 6 1 676.2.l.i 16
13.f odd 12 1 inner 52.2.l.b 16
13.f odd 12 1 676.2.f.h 16
13.f odd 12 1 676.2.f.i 16
13.f odd 12 1 676.2.l.k 16
39.k even 12 1 468.2.cb.f 16
52.b odd 2 1 676.2.l.k 16
52.f even 4 1 676.2.l.i 16
52.f even 4 1 676.2.l.m 16
52.i odd 6 1 676.2.f.i 16
52.i odd 6 1 676.2.l.i 16
52.j odd 6 1 676.2.f.h 16
52.j odd 6 1 676.2.l.m 16
52.l even 12 1 inner 52.2.l.b 16
52.l even 12 1 676.2.f.h 16
52.l even 12 1 676.2.f.i 16
52.l even 12 1 676.2.l.k 16
104.u even 12 1 832.2.bu.n 16
104.x odd 12 1 832.2.bu.n 16
156.v odd 12 1 468.2.cb.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.b 16 1.a even 1 1 trivial
52.2.l.b 16 4.b odd 2 1 inner
52.2.l.b 16 13.f odd 12 1 inner
52.2.l.b 16 52.l even 12 1 inner
468.2.cb.f 16 3.b odd 2 1
468.2.cb.f 16 12.b even 2 1
468.2.cb.f 16 39.k even 12 1
468.2.cb.f 16 156.v odd 12 1
676.2.f.h 16 13.c even 3 1
676.2.f.h 16 13.f odd 12 1
676.2.f.h 16 52.j odd 6 1
676.2.f.h 16 52.l even 12 1
676.2.f.i 16 13.e even 6 1
676.2.f.i 16 13.f odd 12 1
676.2.f.i 16 52.i odd 6 1
676.2.f.i 16 52.l even 12 1
676.2.l.i 16 13.d odd 4 1
676.2.l.i 16 13.e even 6 1
676.2.l.i 16 52.f even 4 1
676.2.l.i 16 52.i odd 6 1
676.2.l.k 16 13.b even 2 1
676.2.l.k 16 13.f odd 12 1
676.2.l.k 16 52.b odd 2 1
676.2.l.k 16 52.l even 12 1
676.2.l.m 16 13.c even 3 1
676.2.l.m 16 13.d odd 4 1
676.2.l.m 16 52.f even 4 1
676.2.l.m 16 52.j odd 6 1
832.2.bu.n 16 8.b even 2 1
832.2.bu.n 16 8.d odd 2 1
832.2.bu.n 16 104.u even 12 1
832.2.bu.n 16 104.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 14T_{3}^{14} + 131T_{3}^{12} - 686T_{3}^{10} + 2605T_{3}^{8} - 5824T_{3}^{6} + 9164T_{3}^{4} - 5824T_{3}^{2} + 2704$$ acting on $$S_{2}^{\mathrm{new}}(52, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 2 T^{15} + 5 T^{14} + 4 T^{13} + \cdots + 256$$
$3$ $$T^{16} - 14 T^{14} + 131 T^{12} + \cdots + 2704$$
$5$ $$(T^{8} + 6 T^{7} + 18 T^{6} + 18 T^{5} + \cdots + 4)^{2}$$
$7$ $$T^{16} - 30 T^{14} + 207 T^{12} + \cdots + 43264$$
$11$ $$T^{16} + 18 T^{14} - 57 T^{12} + \cdots + 692224$$
$13$ $$(T^{8} + 6 T^{7} + 33 T^{6} + 102 T^{5} + \cdots + 28561)^{2}$$
$17$ $$(T^{8} - 6 T^{7} + 2 T^{6} + 60 T^{5} + 39 T^{4} + \cdots + 1)^{2}$$
$19$ $$T^{16} - 54 T^{14} + 891 T^{12} + \cdots + 77228944$$
$23$ $$T^{16} + 106 T^{14} + \cdots + 77228944$$
$29$ $$(T^{8} + 4 T^{7} + 62 T^{6} + 304 T^{5} + \cdots + 51529)^{2}$$
$31$ $$T^{16} + 9072 T^{12} + \cdots + 1235663104$$
$37$ $$(T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1)^{4}$$
$41$ $$(T^{4} - 12 T^{3} + 45 T^{2} - 54 T + 81)^{4}$$
$43$ $$T^{16} + 266 T^{14} + \cdots + 5671027857664$$
$47$ $$T^{16} + 20976 T^{12} + \cdots + 5671027857664$$
$53$ $$(T^{4} + 8 T^{3} - 3 T^{2} - 112 T - 128)^{4}$$
$59$ $$T^{16} + \cdots + 171720267307264$$
$61$ $$(T^{8} - 2 T^{7} + 62 T^{6} - 200 T^{5} + \cdots + 6889)^{2}$$
$67$ $$T^{16} - 126 T^{14} + \cdots + 2205735869584$$
$71$ $$T^{16} - 126 T^{14} + \cdots + 5067731344$$
$73$ $$(T^{8} - 10 T^{7} + 50 T^{6} - 6 T^{5} + \cdots + 676)^{2}$$
$79$ $$(T^{8} + 160 T^{6} + 7040 T^{4} + \cdots + 53248)^{2}$$
$83$ $$T^{16} + \cdots + 538409280507904$$
$89$ $$(T^{8} + 26 T^{7} + 215 T^{6} + \cdots + 2896804)^{2}$$
$97$ $$(T^{8} + 14 T^{7} + 323 T^{6} + 1812 T^{5} + \cdots + 2116)^{2}$$