# Properties

 Label 1045.2.a.k Level $1045$ Weight $2$ Character orbit 1045.a Self dual yes Analytic conductor $8.344$ Analytic rank $0$ Dimension $9$ 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: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27$$ x^9 - 3*x^8 - 9*x^7 + 29*x^6 + 23*x^5 - 84*x^4 - 23*x^3 + 89*x^2 + 8*x - 27 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu - 1$$ v^3 - 5*v - 1 $$\beta_{4}$$ $$=$$ $$( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 21\nu^{4} - 31\nu^{3} - 12\nu^{2} + 15\nu - 1 ) / 2$$ (v^8 - 2*v^7 - 9*v^6 + 16*v^5 + 21*v^4 - 31*v^3 - 12*v^2 + 15*v - 1) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 32\nu^{5} - 7\nu^{4} - 59\nu^{3} + 24\nu^{2} + 25\nu - 7 ) / 2$$ (v^8 - 4*v^7 - 5*v^6 + 32*v^5 - 7*v^4 - 59*v^3 + 24*v^2 + 25*v - 7) / 2 $$\beta_{6}$$ $$=$$ $$\nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 8\nu^{3} - 45\nu^{2} + 19$$ v^7 - 3*v^6 - 7*v^5 + 24*v^4 + 8*v^3 - 45*v^2 + 19 $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 2\nu^{6} + 9\nu^{5} - 16\nu^{4} - 21\nu^{3} + 31\nu^{2} + 13\nu - 15$$ -v^7 + 2*v^6 + 9*v^5 - 16*v^4 - 21*v^3 + 31*v^2 + 13*v - 15 $$\beta_{8}$$ $$=$$ $$( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 30\nu^{5} - 25\nu^{4} - 47\nu^{3} + 64\nu^{2} + 15\nu - 27 ) / 2$$ (v^8 - 4*v^7 - 3*v^6 + 30*v^5 - 25*v^4 - 47*v^3 + 64*v^2 + 15*v - 27) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta _1 + 1$$ b3 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{6} - \beta_{4} + 7\beta_{2} + 15$$ b8 + b6 - b4 + 7*b2 + 15 $$\nu^{5}$$ $$=$$ $$2\beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + 7\beta_{3} + \beta_{2} + 27\beta _1 + 10$$ 2*b8 + b7 + 2*b6 - b5 - b4 + 7*b3 + b2 + 27*b1 + 10 $$\nu^{6}$$ $$=$$ $$12\beta_{8} + \beta_{7} + 11\beta_{6} - 2\beta_{5} - 10\beta_{4} + \beta_{3} + 44\beta_{2} + 2\beta _1 + 89$$ 12*b8 + b7 + 11*b6 - 2*b5 - 10*b4 + b3 + 44*b2 + 2*b1 + 89 $$\nu^{7}$$ $$=$$ $$26\beta_{8} + 10\beta_{7} + 24\beta_{6} - 13\beta_{5} - 13\beta_{4} + 44\beta_{3} + 16\beta_{2} + 155\beta _1 + 85$$ 26*b8 + 10*b7 + 24*b6 - 13*b5 - 13*b4 + 44*b3 + 16*b2 + 155*b1 + 85 $$\nu^{8}$$ $$=$$ $$107\beta_{8} + 13\beta_{7} + 94\beta_{6} - 28\beta_{5} - 77\beta_{4} + 16\beta_{3} + 277\beta_{2} + 36\beta _1 + 564$$ 107*b8 + 13*b7 + 94*b6 - 28*b5 - 77*b4 + 16*b3 + 277*b2 + 36*b1 + 564

## 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
 −2.35228 −1.35249 −1.24377 −0.730132 0.682920 1.22961 1.92391 2.14345 2.69878
−2.35228 0.154676 3.53320 1.00000 −0.363841 1.56958 −3.60652 −2.97608 −2.35228
1.2 −1.35249 −2.73889 −0.170779 1.00000 3.70431 2.99075 2.93595 4.50151 −1.35249
1.3 −1.24377 3.09454 −0.453038 1.00000 −3.84889 3.83759 3.05101 6.57616 −1.24377
1.4 −0.730132 0.820975 −1.46691 1.00000 −0.599420 −1.34825 2.53130 −2.32600 −0.730132
1.5 0.682920 2.48419 −1.53362 1.00000 1.69650 0.856948 −2.41318 3.17118 0.682920
1.6 1.22961 −2.38474 −0.488068 1.00000 −2.93229 1.13367 −3.05934 2.68698 1.22961
1.7 1.92391 0.621031 1.70143 1.00000 1.19481 5.15278 −0.574421 −2.61432 1.92391
1.8 2.14345 2.53942 2.59437 1.00000 5.44311 −1.53582 1.27399 3.44866 2.14345
1.9 2.69878 −1.59120 5.28341 1.00000 −4.29429 0.342738 8.86121 −0.468091 2.69878
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.k 9
3.b odd 2 1 9405.2.a.bh 9
5.b even 2 1 5225.2.a.p 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.k 9 1.a even 1 1 trivial
5225.2.a.p 9 5.b even 2 1
9405.2.a.bh 9 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 29T_{2}^{6} + 23T_{2}^{5} - 84T_{2}^{4} - 23T_{2}^{3} + 89T_{2}^{2} + 8T_{2} - 27$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1045))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - 3 T^{8} - 9 T^{7} + 29 T^{6} + \cdots - 27$$
$3$ $$T^{9} - 3 T^{8} - 15 T^{7} + 46 T^{6} + \cdots + 16$$
$5$ $$(T - 1)^{9}$$
$7$ $$T^{9} - 13 T^{8} + 55 T^{7} - 56 T^{6} + \cdots - 64$$
$11$ $$(T - 1)^{9}$$
$13$ $$T^{9} - 5 T^{8} - 29 T^{7} + 125 T^{6} + \cdots - 64$$
$17$ $$T^{9} - 13 T^{8} + 35 T^{7} + \cdots + 5904$$
$19$ $$(T - 1)^{9}$$
$23$ $$T^{9} - 8 T^{8} - 72 T^{7} + 559 T^{6} + \cdots - 192$$
$29$ $$T^{9} + 3 T^{8} - 81 T^{7} + \cdots - 1680$$
$31$ $$T^{9} + 9 T^{8} - 85 T^{7} - 1163 T^{6} + \cdots + 64$$
$37$ $$T^{9} + 7 T^{8} - 162 T^{7} + \cdots - 32192$$
$41$ $$T^{9} - 9 T^{8} - 164 T^{7} + \cdots + 1696176$$
$43$ $$T^{9} - 23 T^{8} + 85 T^{7} + \cdots + 29632$$
$47$ $$T^{9} - 20 T^{8} + 80 T^{7} + \cdots - 576$$
$53$ $$T^{9} + 5 T^{8} - 195 T^{7} + \cdots + 768192$$
$59$ $$T^{9} - 19 T^{8} - 91 T^{7} + \cdots - 3341760$$
$61$ $$T^{9} - T^{8} - 211 T^{7} + \cdots - 23824$$
$67$ $$T^{9} + 10 T^{8} - 241 T^{7} + \cdots + 46441456$$
$71$ $$T^{9} - 298 T^{7} + \cdots - 34636224$$
$73$ $$T^{9} - 12 T^{8} + \cdots - 160279408$$
$79$ $$T^{9} + 21 T^{8} - 146 T^{7} + \cdots - 71054080$$
$83$ $$T^{9} - 47 T^{8} + 777 T^{7} + \cdots + 909504$$
$89$ $$T^{9} + 2 T^{8} - 369 T^{7} + \cdots + 37547280$$
$97$ $$T^{9} + 32 T^{8} + \cdots - 165140288$$