# Properties

 Label 140.2.e.b Level $140$ Weight $2$ Character orbit 140.e Analytic conductor $1.118$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 i ) q^{5} + i q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 1 + 2 i ) q^{5} + i q^{7} + 3 q^{9} -4 i q^{13} + 4 i q^{17} -4 q^{19} -8 i q^{23} + ( -3 + 4 i ) q^{25} -2 q^{29} -8 q^{31} + ( -2 + i ) q^{35} -8 i q^{37} + 6 q^{41} -8 i q^{43} + ( 3 + 6 i ) q^{45} + 8 i q^{47} - q^{49} + 4 q^{59} -6 q^{61} + 3 i q^{63} + ( 8 - 4 i ) q^{65} + 8 i q^{67} + 12 q^{71} + 4 i q^{73} + 4 q^{79} + 9 q^{81} + ( -8 + 4 i ) q^{85} + 10 q^{89} + 4 q^{91} + ( -4 - 8 i ) q^{95} -12 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 6 q^{9} + O(q^{10})$$ $$2 q + 2 q^{5} + 6 q^{9} - 8 q^{19} - 6 q^{25} - 4 q^{29} - 16 q^{31} - 4 q^{35} + 12 q^{41} + 6 q^{45} - 2 q^{49} + 8 q^{59} - 12 q^{61} + 16 q^{65} + 24 q^{71} + 8 q^{79} + 18 q^{81} - 16 q^{85} + 20 q^{89} + 8 q^{91} - 8 q^{95} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.00000i 1.00000i
0 0 0 1.00000 2.00000i 0 1.00000i 0 3.00000 0
29.2 0 0 0 1.00000 + 2.00000i 0 1.00000i 0 3.00000 0
 $$n$$: 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 140.2.e.b 2
3.b odd 2 1 1260.2.k.b 2
4.b odd 2 1 560.2.g.c 2
5.b even 2 1 inner 140.2.e.b 2
5.c odd 4 1 700.2.a.f 1
5.c odd 4 1 700.2.a.h 1
7.b odd 2 1 980.2.e.a 2
7.c even 3 2 980.2.q.d 4
7.d odd 6 2 980.2.q.e 4
8.b even 2 1 2240.2.g.d 2
8.d odd 2 1 2240.2.g.c 2
12.b even 2 1 5040.2.t.g 2
15.d odd 2 1 1260.2.k.b 2
15.e even 4 1 6300.2.a.g 1
15.e even 4 1 6300.2.a.y 1
20.d odd 2 1 560.2.g.c 2
20.e even 4 1 2800.2.a.o 1
20.e even 4 1 2800.2.a.s 1
35.c odd 2 1 980.2.e.a 2
35.f even 4 1 4900.2.a.l 1
35.f even 4 1 4900.2.a.m 1
35.i odd 6 2 980.2.q.e 4
35.j even 6 2 980.2.q.d 4
40.e odd 2 1 2240.2.g.c 2
40.f even 2 1 2240.2.g.d 2
60.h even 2 1 5040.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 1.a even 1 1 trivial
140.2.e.b 2 5.b even 2 1 inner
560.2.g.c 2 4.b odd 2 1
560.2.g.c 2 20.d odd 2 1
700.2.a.f 1 5.c odd 4 1
700.2.a.h 1 5.c odd 4 1
980.2.e.a 2 7.b odd 2 1
980.2.e.a 2 35.c odd 2 1
980.2.q.d 4 7.c even 3 2
980.2.q.d 4 35.j even 6 2
980.2.q.e 4 7.d odd 6 2
980.2.q.e 4 35.i odd 6 2
1260.2.k.b 2 3.b odd 2 1
1260.2.k.b 2 15.d odd 2 1
2240.2.g.c 2 8.d odd 2 1
2240.2.g.c 2 40.e odd 2 1
2240.2.g.d 2 8.b even 2 1
2240.2.g.d 2 40.f even 2 1
2800.2.a.o 1 20.e even 4 1
2800.2.a.s 1 20.e even 4 1
4900.2.a.l 1 35.f even 4 1
4900.2.a.m 1 35.f even 4 1
5040.2.t.g 2 12.b even 2 1
5040.2.t.g 2 60.h even 2 1
6300.2.a.g 1 15.e even 4 1
6300.2.a.y 1 15.e even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$144 + T^{2}$$