# Properties

 Label 140.2.c.a Level $140$ Weight $2$ Character orbit 140.c Analytic conductor $1.118$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(139,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("140.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.c (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: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 9$$ x^4 + 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + 2 q^{4} - \beta_{3} q^{5} + 2 \beta_{3} q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} - 7 q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b2 - 2*b1) * q^3 + 2 * q^4 - b3 * q^5 + 2*b3 * q^6 + (-b2 + b1) * q^7 + 2*b2 * q^8 - 7 * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + 2 q^{4} - \beta_{3} q^{5} + 2 \beta_{3} q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} - 7 q^{9} + (\beta_{2} + 2 \beta_1) q^{10} + ( - 2 \beta_{2} - 4 \beta_1) q^{12} + ( - \beta_{3} - 3) q^{14} + 5 \beta_{2} q^{15} + 4 q^{16} - 7 \beta_{2} q^{18} - 2 \beta_{3} q^{20} + ( - 3 \beta_{3} + 5) q^{21} - \beta_{2} q^{23} + 4 \beta_{3} q^{24} - 5 q^{25} + (4 \beta_{2} + 8 \beta_1) q^{27} + ( - 2 \beta_{2} + 2 \beta_1) q^{28} + 6 q^{29} + 10 q^{30} + 4 \beta_{2} q^{32} + ( - 4 \beta_{2} - 3 \beta_1) q^{35} - 14 q^{36} + (2 \beta_{2} + 4 \beta_1) q^{40} + 2 \beta_{3} q^{41} + (8 \beta_{2} + 6 \beta_1) q^{42} - 9 \beta_{2} q^{43} + 7 \beta_{3} q^{45} - 2 q^{46} + ( - 3 \beta_{2} - 6 \beta_1) q^{47} + ( - 4 \beta_{2} - 8 \beta_1) q^{48} + (3 \beta_{3} + 2) q^{49} - 5 \beta_{2} q^{50} - 8 \beta_{3} q^{54} + ( - 2 \beta_{3} - 6) q^{56} + 6 \beta_{2} q^{58} + 10 \beta_{2} q^{60} - 6 \beta_{3} q^{61} + (7 \beta_{2} - 7 \beta_1) q^{63} + 8 q^{64} + 3 \beta_{2} q^{67} - 2 \beta_{3} q^{69} + (3 \beta_{3} - 5) q^{70} - 14 \beta_{2} q^{72} + (5 \beta_{2} + 10 \beta_1) q^{75} - 4 \beta_{3} q^{80} + 19 q^{81} + ( - 2 \beta_{2} - 4 \beta_1) q^{82} + (3 \beta_{2} + 6 \beta_1) q^{83} + ( - 6 \beta_{3} + 10) q^{84} - 18 q^{86} + ( - 6 \beta_{2} - 12 \beta_1) q^{87} + 8 \beta_{3} q^{89} + ( - 7 \beta_{2} - 14 \beta_1) q^{90} - 2 \beta_{2} q^{92} + 6 \beta_{3} q^{94} + 8 \beta_{3} q^{96} + ( - \beta_{2} - 6 \beta_1) q^{98}+O(q^{100})$$ q + b2 * q^2 + (-b2 - 2*b1) * q^3 + 2 * q^4 - b3 * q^5 + 2*b3 * q^6 + (-b2 + b1) * q^7 + 2*b2 * q^8 - 7 * q^9 + (b2 + 2*b1) * q^10 + (-2*b2 - 4*b1) * q^12 + (-b3 - 3) * q^14 + 5*b2 * q^15 + 4 * q^16 - 7*b2 * q^18 - 2*b3 * q^20 + (-3*b3 + 5) * q^21 - b2 * q^23 + 4*b3 * q^24 - 5 * q^25 + (4*b2 + 8*b1) * q^27 + (-2*b2 + 2*b1) * q^28 + 6 * q^29 + 10 * q^30 + 4*b2 * q^32 + (-4*b2 - 3*b1) * q^35 - 14 * q^36 + (2*b2 + 4*b1) * q^40 + 2*b3 * q^41 + (8*b2 + 6*b1) * q^42 - 9*b2 * q^43 + 7*b3 * q^45 - 2 * q^46 + (-3*b2 - 6*b1) * q^47 + (-4*b2 - 8*b1) * q^48 + (3*b3 + 2) * q^49 - 5*b2 * q^50 - 8*b3 * q^54 + (-2*b3 - 6) * q^56 + 6*b2 * q^58 + 10*b2 * q^60 - 6*b3 * q^61 + (7*b2 - 7*b1) * q^63 + 8 * q^64 + 3*b2 * q^67 - 2*b3 * q^69 + (3*b3 - 5) * q^70 - 14*b2 * q^72 + (5*b2 + 10*b1) * q^75 - 4*b3 * q^80 + 19 * q^81 + (-2*b2 - 4*b1) * q^82 + (3*b2 + 6*b1) * q^83 + (-6*b3 + 10) * q^84 - 18 * q^86 + (-6*b2 - 12*b1) * q^87 + 8*b3 * q^89 + (-7*b2 - 14*b1) * q^90 - 2*b2 * q^92 + 6*b3 * q^94 + 8*b3 * q^96 + (-b2 - 6*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 28 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 - 28 * q^9 $$4 q + 8 q^{4} - 28 q^{9} - 12 q^{14} + 16 q^{16} + 20 q^{21} - 20 q^{25} + 24 q^{29} + 40 q^{30} - 56 q^{36} - 8 q^{46} + 8 q^{49} - 24 q^{56} + 32 q^{64} - 20 q^{70} + 76 q^{81} + 40 q^{84} - 72 q^{86}+O(q^{100})$$ 4 * q + 8 * q^4 - 28 * q^9 - 12 * q^14 + 16 * q^16 + 20 * q^21 - 20 * q^25 + 24 * q^29 + 40 * q^30 - 56 * q^36 - 8 * q^46 + 8 * q^49 - 24 * q^56 + 32 * q^64 - 20 * q^70 + 76 * q^81 + 40 * q^84 - 72 * q^86

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 0.707107 + 1.58114i 0.707107 − 1.58114i −0.707107 + 1.58114i −0.707107 − 1.58114i
−1.41421 3.16228i 2.00000 2.23607i 4.47214i 2.12132 + 1.58114i −2.82843 −7.00000 3.16228i
139.2 −1.41421 3.16228i 2.00000 2.23607i 4.47214i 2.12132 1.58114i −2.82843 −7.00000 3.16228i
139.3 1.41421 3.16228i 2.00000 2.23607i 4.47214i −2.12132 + 1.58114i 2.82843 −7.00000 3.16228i
139.4 1.41421 3.16228i 2.00000 2.23607i 4.47214i −2.12132 1.58114i 2.82843 −7.00000 3.16228i
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.c.a 4
4.b odd 2 1 inner 140.2.c.a 4
5.b even 2 1 inner 140.2.c.a 4
5.c odd 4 2 700.2.g.d 4
7.b odd 2 1 inner 140.2.c.a 4
7.c even 3 2 980.2.s.b 8
7.d odd 6 2 980.2.s.b 8
8.b even 2 1 2240.2.e.a 4
8.d odd 2 1 2240.2.e.a 4
20.d odd 2 1 CM 140.2.c.a 4
20.e even 4 2 700.2.g.d 4
28.d even 2 1 inner 140.2.c.a 4
28.f even 6 2 980.2.s.b 8
28.g odd 6 2 980.2.s.b 8
35.c odd 2 1 inner 140.2.c.a 4
35.f even 4 2 700.2.g.d 4
35.i odd 6 2 980.2.s.b 8
35.j even 6 2 980.2.s.b 8
40.e odd 2 1 2240.2.e.a 4
40.f even 2 1 2240.2.e.a 4
56.e even 2 1 2240.2.e.a 4
56.h odd 2 1 2240.2.e.a 4
140.c even 2 1 inner 140.2.c.a 4
140.j odd 4 2 700.2.g.d 4
140.p odd 6 2 980.2.s.b 8
140.s even 6 2 980.2.s.b 8
280.c odd 2 1 2240.2.e.a 4
280.n even 2 1 2240.2.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 1.a even 1 1 trivial
140.2.c.a 4 4.b odd 2 1 inner
140.2.c.a 4 5.b even 2 1 inner
140.2.c.a 4 7.b odd 2 1 inner
140.2.c.a 4 20.d odd 2 1 CM
140.2.c.a 4 28.d even 2 1 inner
140.2.c.a 4 35.c odd 2 1 inner
140.2.c.a 4 140.c even 2 1 inner
700.2.g.d 4 5.c odd 4 2
700.2.g.d 4 20.e even 4 2
700.2.g.d 4 35.f even 4 2
700.2.g.d 4 140.j odd 4 2
980.2.s.b 8 7.c even 3 2
980.2.s.b 8 7.d odd 6 2
980.2.s.b 8 28.f even 6 2
980.2.s.b 8 28.g odd 6 2
980.2.s.b 8 35.i odd 6 2
980.2.s.b 8 35.j even 6 2
980.2.s.b 8 140.p odd 6 2
980.2.s.b 8 140.s even 6 2
2240.2.e.a 4 8.b even 2 1
2240.2.e.a 4 8.d odd 2 1
2240.2.e.a 4 40.e odd 2 1
2240.2.e.a 4 40.f even 2 1
2240.2.e.a 4 56.e even 2 1
2240.2.e.a 4 56.h odd 2 1
2240.2.e.a 4 280.c odd 2 1
2240.2.e.a 4 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 10$$ acting on $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$(T^{2} + 10)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4} - 4T^{2} + 49$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T - 6)^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 20)^{2}$$
$43$ $$(T^{2} - 162)^{2}$$
$47$ $$(T^{2} + 90)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 180)^{2}$$
$67$ $$(T^{2} - 18)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 90)^{2}$$
$89$ $$(T^{2} + 320)^{2}$$
$97$ $$T^{4}$$