# Properties

 Label 145.2.b.b 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

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

## 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
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 1.41421i −1.73205 1.73205 1.41421i −2.73205 2.44949i 0.517638i 1.00000 −2.73205 3.34607i
59.2 0.517638i 1.41421i 1.73205 −1.73205 + 1.41421i 0.732051 2.44949i 1.93185i 1.00000 0.732051 + 0.896575i
59.3 0.517638i 1.41421i 1.73205 −1.73205 1.41421i 0.732051 2.44949i 1.93185i 1.00000 0.732051 0.896575i
59.4 1.93185i 1.41421i −1.73205 1.73205 + 1.41421i −2.73205 2.44949i 0.517638i 1.00000 −2.73205 + 3.34607i
 $$n$$: 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.b 4
3.b odd 2 1 1305.2.c.f 4
4.b odd 2 1 2320.2.d.f 4
5.b even 2 1 inner 145.2.b.b 4
5.c odd 4 2 725.2.a.f 4
15.d odd 2 1 1305.2.c.f 4
15.e even 4 2 6525.2.a.bj 4
20.d odd 2 1 2320.2.d.f 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 4T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(145, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 1$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$T^{4} - 2T^{2} + 25$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$T^{4} + 48T^{2} + 144$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T^{2} - 10 T + 22)^{2}$$
$23$ $$T^{4} + 52T^{2} + 484$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} + 14 T + 46)^{2}$$
$37$ $$T^{4} + 84T^{2} + 36$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$T^{4} + 84T^{2} + 36$$
$47$ $$(T^{2} + 2)^{2}$$
$53$ $$T^{4} + 112T^{2} + 2704$$
$59$ $$(T^{2} - 108)^{2}$$
$61$ $$(T^{2} + 8 T - 32)^{2}$$
$67$ $$(T^{2} + 18)^{2}$$
$71$ $$(T^{2} - 12)^{2}$$
$73$ $$(T^{2} + 54)^{2}$$
$79$ $$(T^{2} + 2 T - 26)^{2}$$
$83$ $$T^{4} + 124T^{2} + 2116$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$T^{4} + 156T^{2} + 4356$$