# Properties

 Label 4334.2.a.b Level $4334$ Weight $2$ Character orbit 4334.a Self dual yes Analytic conductor $34.607$ Analytic rank $1$ Dimension $15$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4334 = 2 \cdot 11 \cdot 197$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4334.a (trivial)

## 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: yes Analytic conductor: $$34.6071642360$$ Analytic rank: $$1$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} - 2966 x^{6} + 1621 x^{5} + 2328 x^{4} - 1160 x^{3} - 589 x^{2} + 364 x - 45$$ x^15 - 6*x^14 - 8*x^13 + 94*x^12 - 13*x^11 - 582*x^10 + 295*x^9 + 1814*x^8 - 1056*x^7 - 2966*x^6 + 1621*x^5 + 2328*x^4 - 1160*x^3 - 589*x^2 + 364*x - 45 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} - 2966 x^{6} + 1621 x^{5} + 2328 x^{4} - 1160 x^{3} - 589 x^{2} + 364 x - 45$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 38967 \nu^{14} - 368011 \nu^{13} + 425109 \nu^{12} + 4789161 \nu^{11} - 10763764 \nu^{10} - 23566082 \nu^{9} + 64364521 \nu^{8} + 57985681 \nu^{7} + \cdots + 1206969 ) / 634962$$ (38967*v^14 - 368011*v^13 + 425109*v^12 + 4789161*v^11 - 10763764*v^10 - 23566082*v^9 + 64364521*v^8 + 57985681*v^7 - 162535737*v^6 - 80783623*v^5 + 173150090*v^4 + 57532164*v^3 - 54277538*v^2 - 5195161*v + 1206969) / 634962 $$\beta_{3}$$ $$=$$ $$( - 54577 \nu^{14} + 385559 \nu^{13} + 83031 \nu^{12} - 5421409 \nu^{11} + 5331816 \nu^{10} + 29719730 \nu^{9} - 37764223 \nu^{8} - 83109287 \nu^{7} + \cdots - 6667041 ) / 634962$$ (-54577*v^14 + 385559*v^13 + 83031*v^12 - 5421409*v^11 + 5331816*v^10 + 29719730*v^9 - 37764223*v^8 - 83109287*v^7 + 99457169*v^6 + 129455101*v^5 - 109647076*v^4 - 105733568*v^3 + 41879694*v^2 + 29020637*v - 6667041) / 634962 $$\beta_{4}$$ $$=$$ $$( 98545 \nu^{14} - 474903 \nu^{13} - 1355241 \nu^{12} + 7723627 \nu^{11} + 7561874 \nu^{10} - 48396178 \nu^{9} - 23976157 \nu^{8} + 144520227 \nu^{7} + \cdots + 9614679 ) / 634962$$ (98545*v^14 - 474903*v^13 - 1355241*v^12 + 7723627*v^11 + 7561874*v^10 - 48396178*v^9 - 23976157*v^8 + 144520227*v^7 + 46213081*v^6 - 207141833*v^5 - 38061078*v^4 + 131815598*v^3 - 8677928*v^2 - 37132227*v + 9614679) / 634962 $$\beta_{5}$$ $$=$$ $$( - 80572 \nu^{14} + 418510 \nu^{13} + 889566 \nu^{12} - 6314419 \nu^{11} - 3409969 \nu^{10} + 36652260 \nu^{9} + 6862302 \nu^{8} - 101964871 \nu^{7} + \cdots - 4212711 ) / 317481$$ (-80572*v^14 + 418510*v^13 + 889566*v^12 - 6314419*v^11 - 3409969*v^10 + 36652260*v^9 + 6862302*v^8 - 101964871*v^7 - 12475420*v^6 + 137575398*v^5 + 13025086*v^4 - 82511057*v^3 + 4119481*v^2 + 20801875*v - 4212711) / 317481 $$\beta_{6}$$ $$=$$ $$( - 86087 \nu^{14} + 387355 \nu^{13} + 1322100 \nu^{12} - 6517250 \nu^{11} - 8512356 \nu^{10} + 42707227 \nu^{9} + 30673450 \nu^{8} - 136201942 \nu^{7} + \cdots - 8052534 ) / 317481$$ (-86087*v^14 + 387355*v^13 + 1322100*v^12 - 6517250*v^11 - 8512356*v^10 + 42707227*v^9 + 30673450*v^8 - 136201942*v^7 - 63949346*v^6 + 216656261*v^5 + 63141154*v^4 - 160461943*v^3 - 11155560*v^2 + 46589803*v - 8052534) / 317481 $$\beta_{7}$$ $$=$$ $$( 95161 \nu^{14} - 359889 \nu^{13} - 1717950 \nu^{12} + 5892832 \nu^{11} + 13864988 \nu^{10} - 36106162 \nu^{9} - 63023512 \nu^{8} + 98685486 \nu^{7} + \cdots - 2783562 ) / 317481$$ (95161*v^14 - 359889*v^13 - 1717950*v^12 + 5892832*v^11 + 13864988*v^10 - 36106162*v^9 - 63023512*v^8 + 98685486*v^7 + 156946498*v^6 - 108981575*v^5 - 179832582*v^4 + 28493975*v^3 + 60359008*v^2 - 4272852*v - 2783562) / 317481 $$\beta_{8}$$ $$=$$ $$( 122660 \nu^{14} - 479233 \nu^{13} - 2151627 \nu^{12} + 7881734 \nu^{11} + 16736406 \nu^{10} - 48853678 \nu^{9} - 73469164 \nu^{8} + 137700412 \nu^{7} + \cdots - 327249 ) / 317481$$ (122660*v^14 - 479233*v^13 - 2151627*v^12 + 7881734*v^11 + 16736406*v^10 - 48853678*v^9 - 73469164*v^8 + 137700412*v^7 + 177756743*v^6 - 167007722*v^5 - 196917457*v^4 + 67736473*v^3 + 59640216*v^2 - 15173542*v - 327249) / 317481 $$\beta_{9}$$ $$=$$ $$( 315587 \nu^{14} - 1461863 \nu^{13} - 4408503 \nu^{12} + 22996733 \nu^{11} + 26403326 \nu^{10} - 138607224 \nu^{9} - 96384309 \nu^{8} + 394052573 \nu^{7} + \cdots + 6171147 ) / 634962$$ (315587*v^14 - 1461863*v^13 - 4408503*v^12 + 22996733*v^11 + 26403326*v^10 - 138607224*v^9 - 96384309*v^8 + 394052573*v^7 + 222123893*v^6 - 522481431*v^5 - 250384418*v^4 + 282202990*v^3 + 69913222*v^2 - 61626257*v + 6171147) / 634962 $$\beta_{10}$$ $$=$$ $$( 171261 \nu^{14} - 781019 \nu^{13} - 2530686 \nu^{12} + 12684189 \nu^{11} + 16167988 \nu^{10} - 79521244 \nu^{9} - 61404685 \nu^{8} + 238297865 \nu^{7} + \cdots + 9127485 ) / 317481$$ (171261*v^14 - 781019*v^13 - 2530686*v^12 + 12684189*v^11 + 16167988*v^10 - 79521244*v^9 - 61404685*v^8 + 238297865*v^7 + 140933616*v^6 - 343823822*v^5 - 154808801*v^4 + 219074643*v^3 + 39397850*v^2 - 58103513*v + 9127485) / 317481 $$\beta_{11}$$ $$=$$ $$( - 229297 \nu^{14} + 991991 \nu^{13} + 3660657 \nu^{12} - 16438801 \nu^{11} - 25498530 \nu^{10} + 104958869 \nu^{9} + 102423653 \nu^{8} - 318963461 \nu^{7} + \cdots - 12825138 ) / 317481$$ (-229297*v^14 + 991991*v^13 + 3660657*v^12 - 16438801*v^11 - 25498530*v^10 + 104958869*v^9 + 102423653*v^8 - 318963461*v^7 - 236646475*v^6 + 463558579*v^5 + 255858365*v^4 - 295970219*v^3 - 66568482*v^2 + 81641645*v - 12825138) / 317481 $$\beta_{12}$$ $$=$$ $$( 556079 \nu^{14} - 2558707 \nu^{13} - 7971699 \nu^{12} + 40845473 \nu^{11} + 49263864 \nu^{10} - 250923130 \nu^{9} - 183244921 \nu^{8} + 732657025 \nu^{7} + \cdots + 23916789 ) / 634962$$ (556079*v^14 - 2558707*v^13 - 7971699*v^12 + 40845473*v^11 + 49263864*v^10 - 250923130*v^9 - 183244921*v^8 + 732657025*v^7 + 419414591*v^6 - 1015675733*v^5 - 461061778*v^4 + 600794404*v^3 + 115541358*v^2 - 147425245*v + 23916789) / 634962 $$\beta_{13}$$ $$=$$ $$( - 698419 \nu^{14} + 3263243 \nu^{13} + 9641889 \nu^{12} - 51272989 \nu^{11} - 56674248 \nu^{10} + 308276840 \nu^{9} + 202934273 \nu^{8} + \cdots - 18878145 ) / 634962$$ (-698419*v^14 + 3263243*v^13 + 9641889*v^12 - 51272989*v^11 - 56674248*v^10 + 308276840*v^9 + 202934273*v^8 - 871959305*v^7 - 461314723*v^6 + 1144422331*v^5 + 509130320*v^4 - 607023698*v^3 - 125603226*v^2 + 134300729*v - 18878145) / 634962 $$\beta_{14}$$ $$=$$ $$( - 1054983 \nu^{14} + 4782031 \nu^{13} + 15465027 \nu^{12} - 76630953 \nu^{11} - 98195870 \nu^{10} + 471467996 \nu^{9} + 371832821 \nu^{8} + \cdots - 38939397 ) / 634962$$ (-1054983*v^14 + 4782031*v^13 + 15465027*v^12 - 76630953*v^11 - 98195870*v^10 + 471467996*v^9 + 371832821*v^8 - 1371894949*v^7 - 850031139*v^6 + 1877086147*v^5 + 921906808*v^4 - 1078771302*v^3 - 227477512*v^2 + 264001171*v - 38939397) / 634962
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} + 4$$ b13 + b9 + b8 + b4 + b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 3 \beta_{4} + 3 \beta_{2} + 5 \beta _1 + 4$$ b14 + 2*b13 + b12 - b11 - b10 + 2*b9 + 3*b8 + 3*b4 + 3*b2 + 5*b1 + 4 $$\nu^{4}$$ $$=$$ $$3 \beta_{14} + 12 \beta_{13} + 5 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 12 \beta_{9} + 15 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} + 13 \beta_{2} + 6 \beta _1 + 27$$ 3*b14 + 12*b13 + 5*b12 - 2*b11 - 3*b10 + 12*b9 + 15*b8 - b7 + b6 + 2*b5 + 15*b4 - 2*b3 + 13*b2 + 6*b1 + 27 $$\nu^{5}$$ $$=$$ $$17 \beta_{14} + 36 \beta_{13} + 22 \beta_{12} - 16 \beta_{11} - 17 \beta_{10} + 36 \beta_{9} + 56 \beta_{8} - 6 \beta_{7} + 5 \beta_{6} + 7 \beta_{5} + 52 \beta_{4} - 6 \beta_{3} + 49 \beta_{2} + 42 \beta _1 + 58$$ 17*b14 + 36*b13 + 22*b12 - 16*b11 - 17*b10 + 36*b9 + 56*b8 - 6*b7 + 5*b6 + 7*b5 + 52*b4 - 6*b3 + 49*b2 + 42*b1 + 58 $$\nu^{6}$$ $$=$$ $$60 \beta_{14} + 155 \beta_{13} + 91 \beta_{12} - 52 \beta_{11} - 61 \beta_{10} + 154 \beta_{9} + 228 \beta_{8} - 29 \beta_{7} + 27 \beta_{6} + 42 \beta_{5} + 212 \beta_{4} - 40 \beta_{3} + 188 \beta_{2} + 109 \beta _1 + 252$$ 60*b14 + 155*b13 + 91*b12 - 52*b11 - 61*b10 + 154*b9 + 228*b8 - 29*b7 + 27*b6 + 42*b5 + 212*b4 - 40*b3 + 188*b2 + 109*b1 + 252 $$\nu^{7}$$ $$=$$ $$258 \beta_{14} + 538 \beta_{13} + 355 \beta_{12} - 251 \beta_{11} - 261 \beta_{10} + 538 \beta_{9} + 871 \beta_{8} - 134 \beta_{7} + 117 \beta_{6} + 156 \beta_{5} + 776 \beta_{4} - 141 \beta_{3} + 718 \beta_{2} + 489 \beta _1 + 746$$ 258*b14 + 538*b13 + 355*b12 - 251*b11 - 261*b10 + 538*b9 + 871*b8 - 134*b7 + 117*b6 + 156*b5 + 776*b4 - 141*b3 + 718*b2 + 489*b1 + 746 $$\nu^{8}$$ $$=$$ $$955 \beta_{14} + 2110 \beta_{13} + 1385 \beta_{12} - 922 \beta_{11} - 981 \beta_{10} + 2102 \beta_{9} + 3368 \beta_{8} - 548 \beta_{7} + 501 \beta_{6} + 689 \beta_{5} + 2996 \beta_{4} - 646 \beta_{3} + 2712 \beta_{2} + \cdots + 2872$$ 955*b14 + 2110*b13 + 1385*b12 - 922*b11 - 981*b10 + 2102*b9 + 3368*b8 - 548*b7 + 501*b6 + 689*b5 + 2996*b4 - 646*b3 + 2712*b2 + 1634*b1 + 2872 $$\nu^{9}$$ $$=$$ $$3780 \beta_{14} + 7710 \beta_{13} + 5276 \beta_{12} - 3810 \beta_{11} - 3868 \beta_{10} + 7724 \beta_{9} + 12810 \beta_{8} - 2257 \beta_{7} + 2017 \beta_{6} + 2604 \beta_{5} + 11172 \beta_{4} - 2398 \beta_{3} + \cdots + 9758$$ 3780*b14 + 7710*b13 + 5276*b12 - 3810*b11 - 3868*b10 + 7724*b9 + 12810*b8 - 2257*b7 + 2017*b6 + 2604*b5 + 11172*b4 - 2398*b3 + 10288*b2 + 6479*b1 + 9758 $$\nu^{10}$$ $$=$$ $$14182 \beta_{14} + 29379 \beta_{13} + 20143 \beta_{12} - 14367 \beta_{11} - 14643 \beta_{10} + 29391 \beta_{9} + 48717 \beta_{8} - 8812 \beta_{7} + 8025 \beta_{6} + 10429 \beta_{5} + 42395 \beta_{4} + \cdots + 36509$$ 14182*b14 + 29379*b13 + 20143*b12 - 14367*b11 - 14643*b10 + 29391*b9 + 48717*b8 - 8812*b7 + 8025*b6 + 10429*b5 + 42395*b4 - 9732*b3 + 38756*b2 + 23443*b1 + 36509 $$\nu^{11}$$ $$=$$ $$54451 \beta_{14} + 109320 \beta_{13} + 76175 \beta_{12} - 56135 \beta_{11} - 56123 \beta_{10} + 109726 \beta_{9} + 184344 \beta_{8} - 34528 \beta_{7} + 31194 \beta_{6} + 39498 \beta_{5} + \cdots + 131448$$ 54451*b14 + 109320*b13 + 76175*b12 - 56135*b11 - 56123*b10 + 109726*b9 + 184344*b8 - 34528*b7 + 31194*b6 + 39498*b5 + 159027*b4 - 36631*b3 + 146332*b2 + 89588*b1 + 131448 $$\nu^{12}$$ $$=$$ $$204941 \beta_{14} + 412755 \beta_{13} + 288315 \beta_{12} - 212317 \beta_{11} - 212177 \beta_{10} + 414160 \beta_{9} + 696636 \beta_{8} - 132264 \beta_{7} + 120411 \beta_{6} + 152459 \beta_{5} + \cdots + 490297$$ 204941*b14 + 412755*b13 + 288315*b12 - 212317*b11 - 212177*b10 + 414160*b9 + 696636*b8 - 132264*b7 + 120411*b6 + 152459*b5 + 599934*b4 - 142092*b3 + 550769*b2 + 332500*b1 + 490297 $$\nu^{13}$$ $$=$$ $$777391 \beta_{14} + 1545912 \beta_{13} + 1087084 \beta_{12} - 811500 \beta_{11} - 804392 \beta_{10} + 1553778 \beta_{9} + 2627849 \beta_{8} - 506868 \beta_{7} + 460189 \beta_{6} + \cdots + 1808835$$ 777391*b14 + 1545912*b13 + 1087084*b12 - 811500*b11 - 804392*b10 + 1553778*b9 + 2627849*b8 - 506868*b7 + 460189*b6 + 576355*b5 + 2254358*b4 - 536040*b3 + 2074745*b2 + 1256348*b1 + 1808835 $$\nu^{14}$$ $$=$$ $$2926659 \beta_{14} + 5819694 \beta_{13} + 4099451 \beta_{12} - 3064525 \beta_{11} - 3034548 \beta_{10} + 5849580 \beta_{9} + 9903740 \beta_{8} - 1923895 \beta_{7} + 1752098 \beta_{6} + \cdots + 6761899$$ 2926659*b14 + 5819694*b13 + 4099451*b12 - 3064525*b11 - 3034548*b10 + 5849580*b9 + 9903740*b8 - 1923895*b7 + 1752098*b6 + 2191730*b5 + 8486799*b4 - 2042268*b3 + 7805299*b2 + 4702201*b1 + 6761899

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.22455 −1.92605 −1.55193 −1.31874 −1.15333 −0.956062 0.216655 0.311474 0.594746 1.49640 1.50020 1.97073 2.63464 2.64609 3.75972
1.00000 −3.22455 1.00000 1.46351 −3.22455 −3.27885 1.00000 7.39772 1.46351
1.2 1.00000 −2.92605 1.00000 1.09129 −2.92605 3.40685 1.00000 5.56174 1.09129
1.3 1.00000 −2.55193 1.00000 −3.59659 −2.55193 −3.65108 1.00000 3.51234 −3.59659
1.4 1.00000 −2.31874 1.00000 0.745559 −2.31874 −1.91118 1.00000 2.37657 0.745559
1.5 1.00000 −2.15333 1.00000 −3.80220 −2.15333 3.88018 1.00000 1.63685 −3.80220
1.6 1.00000 −1.95606 1.00000 −1.56251 −1.95606 1.47305 1.00000 0.826177 −1.56251
1.7 1.00000 −0.783345 1.00000 1.44668 −0.783345 −0.609332 1.00000 −2.38637 1.44668
1.8 1.00000 −0.688526 1.00000 3.20899 −0.688526 −0.886807 1.00000 −2.52593 3.20899
1.9 1.00000 −0.405254 1.00000 −3.75919 −0.405254 −0.916969 1.00000 −2.83577 −3.75919
1.10 1.00000 0.496403 1.00000 −1.51850 0.496403 −2.22955 1.00000 −2.75358 −1.51850
1.11 1.00000 0.500205 1.00000 −1.03580 0.500205 3.13873 1.00000 −2.74980 −1.03580
1.12 1.00000 0.970726 1.00000 0.208503 0.970726 −0.143082 1.00000 −2.05769 0.208503
1.13 1.00000 1.63464 1.00000 1.52654 1.63464 −4.88584 1.00000 −0.327947 1.52654
1.14 1.00000 1.64609 1.00000 −2.29877 1.64609 −1.96736 1.00000 −0.290376 −2.29877
1.15 1.00000 2.75972 1.00000 −3.11751 2.75972 −2.41878 1.00000 4.61606 −3.11751
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$
$$197$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4334.2.a.b 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.b 15 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{15} + 9 T_{3}^{14} + 13 T_{3}^{13} - 101 T_{3}^{12} - 328 T_{3}^{11} + 188 T_{3}^{10} + 1713 T_{3}^{9} + 785 T_{3}^{8} - 3491 T_{3}^{7} - 2897 T_{3}^{6} + 2876 T_{3}^{5} + 2792 T_{3}^{4} - 876 T_{3}^{3} - 911 T_{3}^{2} + \cdots + 92$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4334))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{15}$$
$3$ $$T^{15} + 9 T^{14} + 13 T^{13} - 101 T^{12} + \cdots + 92$$
$5$ $$T^{15} + 11 T^{14} + 20 T^{13} + \cdots - 1593$$
$7$ $$T^{15} + 11 T^{14} + 7 T^{13} + \cdots + 5139$$
$11$ $$(T - 1)^{15}$$
$13$ $$T^{15} + 21 T^{14} + 123 T^{13} + \cdots - 5407092$$
$17$ $$T^{15} + 4 T^{14} - 118 T^{13} + \cdots + 6801437$$
$19$ $$T^{15} + 22 T^{14} + 95 T^{13} + \cdots + 29155276$$
$23$ $$T^{15} + 16 T^{14} + \cdots - 1535732732$$
$29$ $$T^{15} + 8 T^{14} + \cdots + 1332415089$$
$31$ $$T^{15} + 33 T^{14} + \cdots + 4002626421$$
$37$ $$T^{15} + T^{14} - 246 T^{13} + \cdots - 123232320$$
$41$ $$T^{15} + 10 T^{14} + \cdots + 916477780$$
$43$ $$T^{15} + 8 T^{14} - 248 T^{13} + \cdots + 124197232$$
$47$ $$T^{15} + 31 T^{14} + \cdots + 196235160860$$
$53$ $$T^{15} + 18 T^{14} + \cdots - 215937896796$$
$59$ $$T^{15} + 37 T^{14} + \cdots - 4362371913$$
$61$ $$T^{15} + 31 T^{14} + \cdots + 1433948561$$
$67$ $$T^{15} - T^{14} - 308 T^{13} + \cdots + 6717348$$
$71$ $$T^{15} + 28 T^{14} + \cdots + 142096755597$$
$73$ $$T^{15} + 20 T^{14} + \cdots + 1326652570743$$
$79$ $$T^{15} + 6 T^{14} + \cdots + 15958972420$$
$83$ $$T^{15} + 15 T^{14} + \cdots + 11887227124$$
$89$ $$T^{15} + 17 T^{14} + \cdots + 17608005935984$$
$97$ $$T^{15} + 9 T^{14} + \cdots - 2778253788055$$