# Properties

 Label 145.2.a.d Level $145$ Weight $2$ Character orbit 145.a Self dual yes Analytic conductor $1.158$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.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: $$1.15783082931$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 0.311108 2.17009 −1.48119
−1.21432 −2.90321 −0.525428 −1.00000 3.52543 1.52543 3.06668 5.42864 1.21432
1.2 1.53919 1.70928 0.369102 −1.00000 2.63090 0.630898 −2.51026 −0.0783777 −1.53919
1.3 2.67513 −0.806063 5.15633 −1.00000 −2.15633 −4.15633 8.44358 −2.35026 −2.67513
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$29$$ $$-1$$

## 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 145.2.a.d 3
3.b odd 2 1 1305.2.a.o 3
4.b odd 2 1 2320.2.a.s 3
5.b even 2 1 725.2.a.d 3
5.c odd 4 2 725.2.b.d 6
7.b odd 2 1 7105.2.a.p 3
8.b even 2 1 9280.2.a.bu 3
8.d odd 2 1 9280.2.a.bm 3
15.d odd 2 1 6525.2.a.bh 3
29.b even 2 1 4205.2.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.d 3 1.a even 1 1 trivial
725.2.a.d 3 5.b even 2 1
725.2.b.d 6 5.c odd 4 2
1305.2.a.o 3 3.b odd 2 1
2320.2.a.s 3 4.b odd 2 1
4205.2.a.e 3 29.b even 2 1
6525.2.a.bh 3 15.d odd 2 1
7105.2.a.p 3 7.b odd 2 1
9280.2.a.bm 3 8.d odd 2 1
9280.2.a.bu 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(145))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T^{2} - T + 5$$
$3$ $$T^{3} + 2 T^{2} + \cdots - 4$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$11$ $$T^{3} - 8 T^{2} + \cdots - 4$$
$13$ $$T^{3} + 6 T^{2} + \cdots - 8$$
$17$ $$T^{3} - 40T + 76$$
$19$ $$T^{3} - 28T + 52$$
$23$ $$T^{3} - 14 T^{2} + \cdots - 76$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} - 12 T^{2} + \cdots - 4$$
$37$ $$T^{3} - 4 T^{2} + \cdots + 68$$
$41$ $$T^{3} + 10 T^{2} + \cdots - 8$$
$43$ $$T^{3} + 10 T^{2} + \cdots + 20$$
$47$ $$T^{3} - 18 T^{2} + \cdots + 92$$
$53$ $$T^{3} - 10 T^{2} + \cdots + 8$$
$59$ $$T^{3} - 4 T^{2} + \cdots - 80$$
$61$ $$T^{3} - 6 T^{2} + \cdots + 40$$
$67$ $$T^{3} + 10 T^{2} + \cdots + 20$$
$71$ $$T^{3} - 24 T^{2} + \cdots - 368$$
$73$ $$T^{3} + 4 T^{2} + \cdots - 1108$$
$79$ $$T^{3} - 8 T^{2} + \cdots + 20$$
$83$ $$T^{3} + 2 T^{2} + \cdots + 52$$
$89$ $$T^{3} - 22 T^{2} + \cdots - 200$$
$97$ $$T^{3} + 36 T^{2} + \cdots + 452$$