Properties

 Label 1045.2.a.d Level $1045$ Weight $2$ Character orbit 1045.a Self dual yes Analytic conductor $8.344$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{22} + \zeta_{22}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4\beta_{2} + 6$$ b4 + 4*b2 + 6

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
 0.284630 −0.830830 −1.68251 1.91899 1.30972
−2.39788 −2.59435 3.74982 1.00000 6.22094 −2.89389 −4.19584 3.73066 −2.39788
1.2 −1.54620 1.51334 0.390736 1.00000 −2.33992 −4.16140 2.48825 −0.709811 −1.54620
1.3 −1.37279 −1.23648 −0.115460 1.00000 1.69742 2.43232 2.90407 −1.47112 −1.37279
1.4 0.0881559 −3.20362 −1.99223 1.00000 −0.282418 −1.95185 −0.351939 7.26315 0.0881559
1.5 2.22871 −1.47889 2.96714 1.00000 −3.29602 −4.42518 2.15546 −0.812880 2.22871
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$-1$$

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 1045.2.a.d 5
3.b odd 2 1 9405.2.a.v 5
5.b even 2 1 5225.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.d 5 1.a even 1 1 trivial
5225.2.a.j 5 5.b even 2 1
9405.2.a.v 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} + 3T_{2}^{4} - 3T_{2}^{3} - 15T_{2}^{2} - 10T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1045))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 3 T^{4} - 3 T^{3} - 15 T^{2} + \cdots + 1$$
$3$ $$T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} + 11 T^{4} + 33 T^{3} + \cdots - 253$$
$11$ $$(T + 1)^{5}$$
$13$ $$T^{5} - T^{4} - 37 T^{3} + 47 T^{2} + \cdots - 529$$
$17$ $$T^{5} + 3 T^{4} - 25 T^{3} - 59 T^{2} + \cdots + 23$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} + 8 T^{4} + 8 T^{3} - 29 T^{2} + \cdots - 1$$
$29$ $$T^{5} - 11 T^{4} - 33 T^{3} + \cdots + 1441$$
$31$ $$T^{5} + 5 T^{4} - 111 T^{3} + \cdots + 10649$$
$37$ $$T^{5} + 9 T^{4} - 38 T^{3} + \cdots + 3917$$
$41$ $$T^{5} - 15 T^{4} + 46 T^{3} + \cdots + 593$$
$43$ $$T^{5} + 13 T^{4} - 115 T^{3} + \cdots + 28753$$
$47$ $$T^{5} + 20 T^{4} + 72 T^{3} + \cdots - 989$$
$53$ $$T^{5} + 5 T^{4} - 155 T^{3} + \cdots + 3917$$
$59$ $$T^{5} + 17 T^{4} + 65 T^{3} + \cdots + 197$$
$61$ $$T^{5} - 3 T^{4} - 47 T^{3} + 59 T^{2} + \cdots + 241$$
$67$ $$T^{5} + 28 T^{4} + 175 T^{3} + \cdots - 11881$$
$71$ $$T^{5} + 6 T^{4} - 12 T^{3} - 109 T^{2} + \cdots - 23$$
$73$ $$T^{5} + 16 T^{4} + 87 T^{3} + 186 T^{2} + \cdots + 23$$
$79$ $$T^{5} - 3 T^{4} - 146 T^{3} + \cdots - 15203$$
$83$ $$T^{5} + 33 T^{4} + 341 T^{3} + \cdots - 737$$
$89$ $$T^{5} + 16 T^{4} - 111 T^{3} + \cdots + 9791$$
$97$ $$T^{5} + 14 T^{4} - 25 T^{3} + \cdots + 3323$$