# Properties

 Label 140.6.a.b Level $140$ Weight $6$ Character orbit 140.a Self dual yes Analytic conductor $22.454$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1009})$$ Defining polynomial: $$x^{2} - x - 252$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1009})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -8 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 73 + 17 \beta ) q^{9} +O(q^{10})$$ $$q + ( -8 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 73 + 17 \beta ) q^{9} + ( -88 + 9 \beta ) q^{11} + ( 202 + 41 \beta ) q^{13} + ( -200 - 25 \beta ) q^{15} + ( 38 + 23 \beta ) q^{17} + ( -644 - 46 \beta ) q^{19} + ( 392 + 49 \beta ) q^{21} + ( -176 + 62 \beta ) q^{23} + 625 q^{25} + ( -2924 + 17 \beta ) q^{27} + ( -3350 - 259 \beta ) q^{29} + ( -5184 - 112 \beta ) q^{31} + ( -1564 + 7 \beta ) q^{33} -1225 q^{35} + ( -906 + 224 \beta ) q^{37} + ( -11948 - 571 \beta ) q^{39} + ( -11982 - 146 \beta ) q^{41} + ( -9820 - 766 \beta ) q^{43} + ( 1825 + 425 \beta ) q^{45} + ( -60 + 739 \beta ) q^{47} + 2401 q^{49} + ( -6100 - 245 \beta ) q^{51} + ( -10786 - 222 \beta ) q^{53} + ( -2200 + 225 \beta ) q^{55} + ( 16744 + 1058 \beta ) q^{57} + ( 492 + 992 \beta ) q^{59} + ( 10598 + 1418 \beta ) q^{61} + ( -3577 - 833 \beta ) q^{63} + ( 5050 + 1025 \beta ) q^{65} + ( 1716 + 636 \beta ) q^{67} + ( -14216 - 382 \beta ) q^{69} + ( -22616 - 1296 \beta ) q^{71} + ( 12922 + 1252 \beta ) q^{73} + ( -5000 - 625 \beta ) q^{75} + ( 4312 - 441 \beta ) q^{77} + ( -5276 - 1929 \beta ) q^{79} + ( 1369 - 1360 \beta ) q^{81} + ( -48012 - 4064 \beta ) q^{83} + ( 950 + 575 \beta ) q^{85} + ( 92068 + 5681 \beta ) q^{87} + ( 19282 - 4766 \beta ) q^{89} + ( -9898 - 2009 \beta ) q^{91} + ( 69696 + 6192 \beta ) q^{93} + ( -16100 - 1150 \beta ) q^{95} + ( 70598 - 2205 \beta ) q^{97} + ( 32132 - 686 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} + O(q^{10})$$ $$2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} - 167 q^{11} + 445 q^{13} - 425 q^{15} + 99 q^{17} - 1334 q^{19} + 833 q^{21} - 290 q^{23} + 1250 q^{25} - 5831 q^{27} - 6959 q^{29} - 10480 q^{31} - 3121 q^{33} - 2450 q^{35} - 1588 q^{37} - 24467 q^{39} - 24110 q^{41} - 20406 q^{43} + 4075 q^{45} + 619 q^{47} + 4802 q^{49} - 12445 q^{51} - 21794 q^{53} - 4175 q^{55} + 34546 q^{57} + 1976 q^{59} + 22614 q^{61} - 7987 q^{63} + 11125 q^{65} + 4068 q^{67} - 28814 q^{69} - 46528 q^{71} + 27096 q^{73} - 10625 q^{75} + 8183 q^{77} - 12481 q^{79} + 1378 q^{81} - 100088 q^{83} + 2475 q^{85} + 189817 q^{87} + 33798 q^{89} - 21805 q^{91} + 145584 q^{93} - 33350 q^{95} + 138991 q^{97} + 63578 q^{99} + O(q^{100})$$

## 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
 16.3824 −15.3824
0 −24.3824 0 25.0000 0 −49.0000 0 351.500 0
1.2 0 7.38238 0 25.0000 0 −49.0000 0 −188.500 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.b 2
4.b odd 2 1 560.6.a.o 2
5.b even 2 1 700.6.a.g 2
5.c odd 4 2 700.6.e.f 4
7.b odd 2 1 980.6.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.b 2 1.a even 1 1 trivial
560.6.a.o 2 4.b odd 2 1
700.6.a.g 2 5.b even 2 1
700.6.e.f 4 5.c odd 4 2
980.6.a.f 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 17 T_{3} - 180$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(140))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-180 + 17 T + T^{2}$$
$5$ $$( -25 + T )^{2}$$
$7$ $$( 49 + T )^{2}$$
$11$ $$-13460 + 167 T + T^{2}$$
$13$ $$-374526 - 445 T + T^{2}$$
$17$ $$-130990 - 99 T + T^{2}$$
$19$ $$-88872 + 1334 T + T^{2}$$
$23$ $$-948624 + 290 T + T^{2}$$
$29$ $$-4814262 + 6959 T + T^{2}$$
$31$ $$24293376 + 10480 T + T^{2}$$
$37$ $$-12026460 + 1588 T + T^{2}$$
$41$ $$139946064 + 24110 T + T^{2}$$
$43$ $$-43907992 + 20406 T + T^{2}$$
$47$ $$-137663232 - 619 T + T^{2}$$
$53$ $$106312720 + 21794 T + T^{2}$$
$59$ $$-247254000 - 1976 T + T^{2}$$
$61$ $$-379356880 - 22614 T + T^{2}$$
$67$ $$-97896960 - 4068 T + T^{2}$$
$71$ $$117530560 + 46528 T + T^{2}$$
$73$ $$-211854580 - 27096 T + T^{2}$$
$79$ $$-899688752 + 12481 T + T^{2}$$
$83$ $$-1661783280 + 100088 T + T^{2}$$
$89$ $$-5444221000 - 33798 T + T^{2}$$
$97$ $$3603178714 - 138991 T + T^{2}$$