# Properties

 Label 145.2.b.a Level $145$ Weight $2$ Character orbit 145.b Analytic conductor $1.158$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [145,2,Mod(59,145)]

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([1, 0]))

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

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

chi := DirichletCharacter("145.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.b (of order $$2$$, degree $$1$$, 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: $$1.15783082931$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3$$ (v^3 + 2*v^2 - 2*v - 6) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3$$ (v^3 - v^2 - 2*v - 3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 1$$ -b3 + b2 + 1 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ 2*b3 + b2 + 2*b1 + 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/145\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$117$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i 1.68614 + 0.396143i −1.18614 − 1.26217i
1.73205i 2.52434i −1.00000 −2.18614 + 0.469882i −4.37228 1.58457i 1.73205i −3.37228 0.813859 + 3.78651i
59.2 1.73205i 0.792287i −1.00000 0.686141 + 2.12819i 1.37228 5.04868i 1.73205i 2.37228 3.68614 1.18843i
59.3 1.73205i 0.792287i −1.00000 0.686141 2.12819i 1.37228 5.04868i 1.73205i 2.37228 3.68614 + 1.18843i
59.4 1.73205i 2.52434i −1.00000 −2.18614 0.469882i −4.37228 1.58457i 1.73205i −3.37228 0.813859 3.78651i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.b.a 4
3.b odd 2 1 1305.2.c.e 4
4.b odd 2 1 2320.2.d.c 4
5.b even 2 1 inner 145.2.b.a 4
5.c odd 4 2 725.2.a.g 4
15.d odd 2 1 1305.2.c.e 4
15.e even 4 2 6525.2.a.bk 4
20.d odd 2 1 2320.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.a 4 1.a even 1 1 trivial
145.2.b.a 4 5.b even 2 1 inner
725.2.a.g 4 5.c odd 4 2
1305.2.c.e 4 3.b odd 2 1
1305.2.c.e 4 15.d odd 2 1
2320.2.d.c 4 4.b odd 2 1
2320.2.d.c 4 20.d odd 2 1
6525.2.a.bk 4 15.e even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$T^{4} + 7T^{2} + 4$$
$5$ $$T^{4} + 3 T^{3} + 4 T^{2} + 15 T + 25$$
$7$ $$T^{4} + 28T^{2} + 64$$
$11$ $$(T^{2} - 7 T + 4)^{2}$$
$13$ $$T^{4} + 19T^{2} + 16$$
$17$ $$T^{4} + 28T^{2} + 64$$
$19$ $$(T + 4)^{4}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} + T - 8)^{2}$$
$37$ $$T^{4} + 112T^{2} + 1024$$
$41$ $$(T^{2} - 2 T - 32)^{2}$$
$43$ $$T^{4} + 151T^{2} + 3844$$
$47$ $$T^{4} + 151T^{2} + 3844$$
$53$ $$T^{4} + 19T^{2} + 16$$
$59$ $$(T^{2} - 10 T - 8)^{2}$$
$61$ $$(T - 6)^{4}$$
$67$ $$T^{4} + 76T^{2} + 256$$
$71$ $$(T^{2} - 2 T - 32)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} + 17 T + 64)^{2}$$
$83$ $$T^{4} + 376 T^{2} + 26896$$
$89$ $$(T^{2} + 10 T - 8)^{2}$$
$97$ $$(T^{2} + 48)^{2}$$