# Properties

 Label 280.6.a.d Level $280$ Weight $6$ Character orbit 280.a Self dual yes Analytic conductor $44.907$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9074695476$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ Defining polynomial: $$x^{2} - x - 9$$ 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 $$\beta = 2\sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 13 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 74 - 26 \beta ) q^{9} +O(q^{10})$$ $$q + ( 13 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 74 - 26 \beta ) q^{9} + ( -7 - 22 \beta ) q^{11} + ( 157 + 5 \beta ) q^{13} + ( 325 - 25 \beta ) q^{15} + ( -73 + 21 \beta ) q^{17} + ( 1424 + 71 \beta ) q^{19} + ( -637 + 49 \beta ) q^{21} + ( 1996 - 71 \beta ) q^{23} + 625 q^{25} + ( 1651 - 169 \beta ) q^{27} + ( 3003 - 96 \beta ) q^{29} + ( 610 + 3 \beta ) q^{31} + ( 3165 - 279 \beta ) q^{33} -1225 q^{35} + ( 3616 - 825 \beta ) q^{37} + ( 1301 - 92 \beta ) q^{39} + ( 3258 + 611 \beta ) q^{41} + ( -1168 - 509 \beta ) q^{43} + ( 1850 - 650 \beta ) q^{45} + ( 6465 + 1521 \beta ) q^{47} + 2401 q^{49} + ( -4057 + 346 \beta ) q^{51} + ( 1608 - 1232 \beta ) q^{53} + ( -175 - 550 \beta ) q^{55} + ( 8004 - 501 \beta ) q^{57} + ( 3364 + 3036 \beta ) q^{59} + ( -378 + 2157 \beta ) q^{61} + ( -3626 + 1274 \beta ) q^{63} + ( 3925 + 125 \beta ) q^{65} + ( 5872 + 2608 \beta ) q^{67} + ( 36456 - 2919 \beta ) q^{69} + ( -952 - 952 \beta ) q^{71} + ( 11306 + 2554 \beta ) q^{73} + ( 8125 - 625 \beta ) q^{75} + ( 343 + 1078 \beta ) q^{77} + ( 31243 - 3218 \beta ) q^{79} + ( 28493 + 2470 \beta ) q^{81} + ( 7600 - 462 \beta ) q^{83} + ( -1825 + 525 \beta ) q^{85} + ( 53247 - 4251 \beta ) q^{87} + ( 21422 - 3005 \beta ) q^{89} + ( -7693 - 245 \beta ) q^{91} + ( 7486 - 571 \beta ) q^{93} + ( 35600 + 1775 \beta ) q^{95} + ( 41411 + 2365 \beta ) q^{97} + ( 84138 - 1446 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} + O(q^{10})$$ $$2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} - 14 q^{11} + 314 q^{13} + 650 q^{15} - 146 q^{17} + 2848 q^{19} - 1274 q^{21} + 3992 q^{23} + 1250 q^{25} + 3302 q^{27} + 6006 q^{29} + 1220 q^{31} + 6330 q^{33} - 2450 q^{35} + 7232 q^{37} + 2602 q^{39} + 6516 q^{41} - 2336 q^{43} + 3700 q^{45} + 12930 q^{47} + 4802 q^{49} - 8114 q^{51} + 3216 q^{53} - 350 q^{55} + 16008 q^{57} + 6728 q^{59} - 756 q^{61} - 7252 q^{63} + 7850 q^{65} + 11744 q^{67} + 72912 q^{69} - 1904 q^{71} + 22612 q^{73} + 16250 q^{75} + 686 q^{77} + 62486 q^{79} + 56986 q^{81} + 15200 q^{83} - 3650 q^{85} + 106494 q^{87} + 42844 q^{89} - 15386 q^{91} + 14972 q^{93} + 71200 q^{95} + 82822 q^{97} + 168276 q^{99} + O(q^{100})$$

## 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
 3.54138 −2.54138
0 0.834475 0 25.0000 0 −49.0000 0 −242.304 0
1.2 0 25.1655 0 25.0000 0 −49.0000 0 390.304 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 280.6.a.d 2
4.b odd 2 1 560.6.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.d 2 1.a even 1 1 trivial
560.6.a.j 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 26 T_{3} + 21$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(280))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$21 - 26 T + T^{2}$$
$5$ $$( -25 + T )^{2}$$
$7$ $$( 49 + T )^{2}$$
$11$ $$-71583 + 14 T + T^{2}$$
$13$ $$20949 - 314 T + T^{2}$$
$17$ $$-59939 + 146 T + T^{2}$$
$19$ $$1281708 - 2848 T + T^{2}$$
$23$ $$3237948 - 3992 T + T^{2}$$
$29$ $$7654041 - 6006 T + T^{2}$$
$31$ $$370768 - 1220 T + T^{2}$$
$37$ $$-87657044 - 7232 T + T^{2}$$
$41$ $$-44636944 - 6516 T + T^{2}$$
$43$ $$-36979764 + 2336 T + T^{2}$$
$47$ $$-300593043 - 12930 T + T^{2}$$
$53$ $$-222052288 - 3216 T + T^{2}$$
$59$ $$-1352843312 - 6728 T + T^{2}$$
$61$ $$-688449168 + 756 T + T^{2}$$
$67$ $$-972165888 - 11744 T + T^{2}$$
$71$ $$-133226688 + 1904 T + T^{2}$$
$73$ $$-837565932 - 22612 T + T^{2}$$
$79$ $$-556492503 - 62486 T + T^{2}$$
$83$ $$26170288 - 15200 T + T^{2}$$
$89$ $$-877541616 - 42844 T + T^{2}$$
$97$ $$887073621 - 82822 T + T^{2}$$