# Properties

 Label 140.6.a.c Level $140$ Weight $6$ Character orbit 140.a Self dual yes Analytic conductor $22.454$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 140.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.4537347738$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3101016.1 Defining polynomial: $$x^{3} - x^{2} - 312 x + 1740$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( -23 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -3 - \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( -23 - \beta_{1} + \beta_{2} ) q^{9} + ( 26 - 19 \beta_{1} + \beta_{2} ) q^{11} + ( 55 - 25 \beta_{1} - 8 \beta_{2} ) q^{13} + ( 75 + 25 \beta_{1} ) q^{15} + ( 477 - 55 \beta_{1} + 16 \beta_{2} ) q^{17} + ( 591 - 96 \beta_{1} - 15 \beta_{2} ) q^{19} + ( 147 + 49 \beta_{1} ) q^{21} + ( 1399 + 32 \beta_{1} + 25 \beta_{2} ) q^{23} + 625 q^{25} + ( 1061 + 217 \beta_{1} - 10 \beta_{2} ) q^{27} + ( 1858 - 25 \beta_{1} - 23 \beta_{2} ) q^{29} + ( 871 + 178 \beta_{1} - 25 \beta_{2} ) q^{31} + ( 3983 - 147 \beta_{1} + 8 \beta_{2} ) q^{33} + 1225 q^{35} + ( 5021 + 90 \beta_{1} - 57 \beta_{2} ) q^{37} + ( 4694 + 205 \beta_{1} + 113 \beta_{2} ) q^{39} + ( 2587 + 16 \beta_{1} + 71 \beta_{2} ) q^{41} + ( 8225 + 284 \beta_{1} + 55 \beta_{2} ) q^{43} + ( 575 + 25 \beta_{1} - 25 \beta_{2} ) q^{45} + ( 6793 + 883 \beta_{1} - 88 \beta_{2} ) q^{47} + 2401 q^{49} + ( 11006 - 1417 \beta_{1} - 121 \beta_{2} ) q^{51} + ( 250 - 654 \beta_{1} - 246 \beta_{2} ) q^{53} + ( -650 + 475 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 17703 - 300 \beta_{1} + 261 \beta_{2} ) q^{57} + ( 5032 - 2208 \beta_{1} ) q^{59} + ( -1455 - 896 \beta_{1} + 353 \beta_{2} ) q^{61} + ( 1127 + 49 \beta_{1} - 49 \beta_{2} ) q^{63} + ( -1375 + 625 \beta_{1} + 200 \beta_{2} ) q^{65} + ( 10692 + 3144 \beta_{1} + 24 \beta_{2} ) q^{67} + ( -9649 - 2396 \beta_{1} - 307 \beta_{2} ) q^{69} + ( -7584 + 2840 \beta_{1} - 392 \beta_{2} ) q^{71} + ( -18960 - 3604 \beta_{1} + 22 \beta_{2} ) q^{73} + ( -1875 - 625 \beta_{1} ) q^{75} + ( -1274 + 931 \beta_{1} - 49 \beta_{2} ) q^{77} + ( 20776 - 1053 \beta_{1} + 153 \beta_{2} ) q^{79} + ( -43901 + 500 \beta_{1} - 350 \beta_{2} ) q^{81} + ( -4474 + 2520 \beta_{1} + 762 \beta_{2} ) q^{83} + ( -11925 + 1375 \beta_{1} - 400 \beta_{2} ) q^{85} + ( -1495 - 923 \beta_{1} + 278 \beta_{2} ) q^{87} + ( -48457 + 392 \beta_{1} - 101 \beta_{2} ) q^{89} + ( -2695 + 1225 \beta_{1} + 392 \beta_{2} ) q^{91} + ( -41471 + 966 \beta_{1} + 97 \beta_{2} ) q^{93} + ( -14775 + 2400 \beta_{1} + 375 \beta_{2} ) q^{95} + ( 2873 + 7965 \beta_{1} - 720 \beta_{2} ) q^{97} + ( 13166 - 314 \beta_{1} - 184 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9} + O(q^{10})$$ $$3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9} + 58 q^{11} + 148 q^{13} + 250 q^{15} + 1360 q^{17} + 1692 q^{19} + 490 q^{21} + 4204 q^{23} + 1875 q^{25} + 3410 q^{27} + 5572 q^{29} + 2816 q^{31} + 11794 q^{33} + 3675 q^{35} + 15210 q^{37} + 14174 q^{39} + 7706 q^{41} + 24904 q^{43} + 1775 q^{45} + 21350 q^{47} + 7203 q^{49} + 31722 q^{51} + 342 q^{53} - 1450 q^{55} + 52548 q^{57} + 12888 q^{59} - 5614 q^{61} + 3479 q^{63} - 3700 q^{65} + 35196 q^{67} - 31036 q^{69} - 19520 q^{71} - 60506 q^{73} - 6250 q^{75} - 2842 q^{77} + 61122 q^{79} - 130853 q^{81} - 11664 q^{83} - 34000 q^{85} - 5686 q^{87} - 144878 q^{89} - 7252 q^{91} - 123544 q^{93} - 42300 q^{95} + 17304 q^{97} + 39368 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 312 x + 1740$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 7 \nu - 211$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 7 \beta_{1} + 211$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 14.3069 6.22631 −19.5332
0 −17.3069 0 −25.0000 0 −49.0000 0 56.5286 0
1.2 0 −9.22631 0 −25.0000 0 −49.0000 0 −157.875 0
1.3 0 16.5332 0 −25.0000 0 −49.0000 0 30.3467 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.6.a.c 3
4.b odd 2 1 560.6.a.u 3
5.b even 2 1 700.6.a.j 3
5.c odd 4 2 700.6.e.h 6
7.b odd 2 1 980.6.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.c 3 1.a even 1 1 trivial
560.6.a.u 3 4.b odd 2 1
700.6.a.j 3 5.b even 2 1
700.6.e.h 6 5.c odd 4 2
980.6.a.i 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 10 T_{3}^{2} - 279 T_{3} - 2640$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(140))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-2640 - 279 T + 10 T^{2} + T^{3}$$
$5$ $$( 25 + T )^{3}$$
$7$ $$( 49 + T )^{3}$$
$11$ $$-14472500 - 127135 T - 58 T^{2} + T^{3}$$
$13$ $$266840838 - 1012131 T - 148 T^{2} + T^{3}$$
$17$ $$4796110110 - 3824531 T - 1360 T^{2} + T^{3}$$
$19$ $$8250023232 - 4734396 T - 1692 T^{2} + T^{3}$$
$23$ $$11010917376 - 2614524 T - 4204 T^{2} + T^{3}$$
$29$ $$4810314762 + 3213189 T - 5572 T^{2} + T^{3}$$
$31$ $$19217358464 - 16061584 T - 2816 T^{2} + T^{3}$$
$37$ $$-16558995800 + 30928956 T - 15210 T^{2} + T^{3}$$
$41$ $$285705250976 - 46940896 T - 7706 T^{2} + T^{3}$$
$43$ $$-230990747600 + 143319188 T - 24904 T^{2} + T^{3}$$
$47$ $$3541720333592 - 203405743 T - 21350 T^{2} + T^{3}$$
$53$ $$4215757774880 - 916256064 T - 342 T^{2} + T^{3}$$
$59$ $$-11147794362880 - 1467340608 T - 12888 T^{2} + T^{3}$$
$61$ $$27664467900480 - 1928759552 T + 5614 T^{2} + T^{3}$$
$67$ $$78576282671040 - 2673090576 T - 35196 T^{2} + T^{3}$$
$71$ $$-17256845706240 - 4560200384 T + 19520 T^{2} + T^{3}$$
$73$ $$-157999195437960 - 2852336372 T + 60506 T^{2} + T^{3}$$
$79$ $$5100296528456 + 569709705 T - 61122 T^{2} + T^{3}$$
$83$ $$-249960926161280 - 9403369392 T + 11664 T^{2} + T^{3}$$
$89$ $$102863482208800 + 6808695872 T + 144878 T^{2} + T^{3}$$
$97$ $$1265608607319118 - 27263983803 T - 17304 T^{2} + T^{3}$$