# Properties

 Label 140.2.m.a Level $140$ Weight $2$ Character orbit 140.m Analytic conductor $1.118$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.11574317056.3 Defining polynomial: $$x^{8} + 45 x^{4} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} + \beta_{6} ) q^{5} + ( \beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} + \beta_{6} ) q^{5} + ( \beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( -1 + \beta_{4} + \beta_{5} ) q^{11} + \beta_{1} q^{13} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{19} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{21} + ( -2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( -2 + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{27} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{6} ) q^{31} + ( -5 \beta_{1} + \beta_{6} + \beta_{7} ) q^{33} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{35} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + ( -3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -\beta_{1} - \beta_{2} - 2 \beta_{6} ) q^{41} + ( -5 - 5 \beta_{3} ) q^{43} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{45} + ( -5 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{49} + ( 1 + \beta_{4} + \beta_{5} ) q^{51} + ( 5 + 5 \beta_{3} ) q^{53} + ( -\beta_{1} + 4 \beta_{2} + \beta_{6} + \beta_{7} ) q^{55} + ( -5 + 5 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{57} + ( \beta_{1} - \beta_{2} + 2 \beta_{7} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{6} ) q^{61} + ( 5 - \beta_{1} + 5 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{63} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( 5 - 5 \beta_{3} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{7} ) q^{69} + ( 2 + 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( 2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{73} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{7} ) q^{75} + ( 5 - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{77} + ( 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -1 - 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 4 \beta_{1} + \beta_{6} + \beta_{7} ) q^{83} + ( 7 - 6 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{87} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{7} ) q^{89} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{91} + ( -5 - 5 \beta_{3} + 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{93} + ( -5 + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{95} + 3 \beta_{2} q^{97} + ( 10 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{7} + O(q^{10})$$ $$8q - 2q^{7} - 12q^{11} + 20q^{15} - 8q^{21} + 4q^{23} - 12q^{25} - 14q^{35} - 4q^{37} - 40q^{43} + 4q^{51} + 40q^{53} - 44q^{57} + 42q^{63} + 20q^{65} + 40q^{67} + 8q^{71} + 44q^{77} + 48q^{85} - 8q^{91} - 52q^{93} - 44q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 45 x^{4} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 47 \nu^{3}$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 47 \nu^{2}$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} + 2 \nu^{5} + 2 \nu^{4} + 127 \nu^{2} + 94 \nu + 38$$$$)/28$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 2 \nu^{5} + 2 \nu^{4} - 127 \nu^{2} - 94 \nu + 38$$$$)/28$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu$$$$)/28$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu$$$$)/28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 7 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{5} + 7 \beta_{4} - 19$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-47 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 127 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$-47 \beta_{7} + 47 \beta_{6} - 315 \beta_{2}$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −1.83051 + 1.83051i −0.386289 + 0.386289i 0.386289 − 0.386289i 1.83051 − 1.83051i −1.83051 − 1.83051i −0.386289 − 0.386289i 0.386289 + 0.386289i 1.83051 + 1.83051i
0 −1.83051 + 1.83051i 0 −1.83051 1.28422i 0 −1.57763 + 2.12393i 0 3.70156i 0
13.2 0 −0.386289 + 0.386289i 0 −0.386289 + 2.20245i 0 2.64515 0.0564123i 0 2.70156i 0
13.3 0 0.386289 0.386289i 0 0.386289 2.20245i 0 0.0564123 2.64515i 0 2.70156i 0
13.4 0 1.83051 1.83051i 0 1.83051 + 1.28422i 0 −2.12393 + 1.57763i 0 3.70156i 0
97.1 0 −1.83051 1.83051i 0 −1.83051 + 1.28422i 0 −1.57763 2.12393i 0 3.70156i 0
97.2 0 −0.386289 0.386289i 0 −0.386289 2.20245i 0 2.64515 + 0.0564123i 0 2.70156i 0
97.3 0 0.386289 + 0.386289i 0 0.386289 + 2.20245i 0 0.0564123 + 2.64515i 0 2.70156i 0
97.4 0 1.83051 + 1.83051i 0 1.83051 1.28422i 0 −2.12393 1.57763i 0 3.70156i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.4 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.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.m.a 8
3.b odd 2 1 1260.2.ba.a 8
4.b odd 2 1 560.2.bj.b 8
5.b even 2 1 700.2.m.c 8
5.c odd 4 1 inner 140.2.m.a 8
5.c odd 4 1 700.2.m.c 8
7.b odd 2 1 inner 140.2.m.a 8
7.c even 3 2 980.2.v.b 16
7.d odd 6 2 980.2.v.b 16
15.e even 4 1 1260.2.ba.a 8
20.e even 4 1 560.2.bj.b 8
21.c even 2 1 1260.2.ba.a 8
28.d even 2 1 560.2.bj.b 8
35.c odd 2 1 700.2.m.c 8
35.f even 4 1 inner 140.2.m.a 8
35.f even 4 1 700.2.m.c 8
35.k even 12 2 980.2.v.b 16
35.l odd 12 2 980.2.v.b 16
105.k odd 4 1 1260.2.ba.a 8
140.j odd 4 1 560.2.bj.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.m.a 8 1.a even 1 1 trivial
140.2.m.a 8 5.c odd 4 1 inner
140.2.m.a 8 7.b odd 2 1 inner
140.2.m.a 8 35.f even 4 1 inner
560.2.bj.b 8 4.b odd 2 1
560.2.bj.b 8 20.e even 4 1
560.2.bj.b 8 28.d even 2 1
560.2.bj.b 8 140.j odd 4 1
700.2.m.c 8 5.b even 2 1
700.2.m.c 8 5.c odd 4 1
700.2.m.c 8 35.c odd 2 1
700.2.m.c 8 35.f even 4 1
980.2.v.b 16 7.c even 3 2
980.2.v.b 16 7.d odd 6 2
980.2.v.b 16 35.k even 12 2
980.2.v.b 16 35.l odd 12 2
1260.2.ba.a 8 3.b odd 2 1
1260.2.ba.a 8 15.e even 4 1
1260.2.ba.a 8 21.c even 2 1
1260.2.ba.a 8 105.k odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 45 T^{4} + T^{8}$$
$5$ $$625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8}$$
$7$ $$2401 + 686 T + 98 T^{2} - 182 T^{3} - 62 T^{4} - 26 T^{5} + 2 T^{6} + 2 T^{7} + T^{8}$$
$11$ $$( -8 + 3 T + T^{2} )^{4}$$
$13$ $$4 + 45 T^{4} + T^{8}$$
$17$ $$2500 + 2109 T^{4} + T^{8}$$
$19$ $$( 800 - 58 T^{2} + T^{4} )^{2}$$
$23$ $$( 400 + 40 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$29$ $$( 16 + 33 T^{2} + T^{4} )^{2}$$
$31$ $$( 5000 + 142 T^{2} + T^{4} )^{2}$$
$37$ $$( 400 - 40 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$41$ $$( 800 + 58 T^{2} + T^{4} )^{2}$$
$43$ $$( 50 + 10 T + T^{2} )^{4}$$
$47$ $$7496644 + 17325 T^{4} + T^{8}$$
$53$ $$( 50 - 10 T + T^{2} )^{4}$$
$59$ $$( 800 - 58 T^{2} + T^{4} )^{2}$$
$61$ $$( 12800 + 232 T^{2} + T^{4} )^{2}$$
$67$ $$( 50 - 10 T + T^{2} )^{4}$$
$71$ $$( -40 - 2 T + T^{2} )^{4}$$
$73$ $$16384 + 420 T^{4} + T^{8}$$
$79$ $$( 4 + 45 T^{2} + T^{4} )^{2}$$
$83$ $$17909824 + 16500 T^{4} + T^{8}$$
$89$ $$( 800 - 188 T^{2} + T^{4} )^{2}$$
$97$ $$26244 + 3645 T^{4} + T^{8}$$