# Properties

 Label 547.2.a.a Level 547 Weight 2 Character orbit 547.a Self dual yes Analytic conductor 4.368 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.36781699056$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 1 - 2 \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} -\beta q^{3} + ( 1 - 2 \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} - q^{9} + ( 2 - \beta ) q^{10} + ( -5 + \beta ) q^{11} + ( 4 - \beta ) q^{12} + 3 q^{13} + ( -2 + 2 \beta ) q^{14} -2 q^{15} + 3 q^{16} + ( -6 - \beta ) q^{17} + ( 1 - \beta ) q^{18} + ( -1 + 3 \beta ) q^{19} + ( -4 + \beta ) q^{20} -2 \beta q^{21} + ( 7 - 6 \beta ) q^{22} + ( -2 - 5 \beta ) q^{23} + ( -2 + 3 \beta ) q^{24} -3 q^{25} + ( -3 + 3 \beta ) q^{26} + 4 \beta q^{27} + ( 2 - 4 \beta ) q^{28} + ( -7 - 2 \beta ) q^{29} + ( 2 - 2 \beta ) q^{30} + ( 2 - 2 \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( -2 + 5 \beta ) q^{33} + ( 4 - 5 \beta ) q^{34} + 2 \beta q^{35} + ( -1 + 2 \beta ) q^{36} + ( -2 + \beta ) q^{37} + ( 7 - 4 \beta ) q^{38} -3 \beta q^{39} + ( 2 - 3 \beta ) q^{40} + ( 4 - 5 \beta ) q^{41} + ( -4 + 2 \beta ) q^{42} -8 q^{43} + ( -9 + 11 \beta ) q^{44} -\beta q^{45} + ( -8 + 3 \beta ) q^{46} + ( 11 - \beta ) q^{47} -3 \beta q^{48} -3 q^{49} + ( 3 - 3 \beta ) q^{50} + ( 2 + 6 \beta ) q^{51} + ( 3 - 6 \beta ) q^{52} + ( -7 + 4 \beta ) q^{53} + ( 8 - 4 \beta ) q^{54} + ( 2 - 5 \beta ) q^{55} + ( -6 + 2 \beta ) q^{56} + ( -6 + \beta ) q^{57} + ( 3 - 5 \beta ) q^{58} + ( -6 + 3 \beta ) q^{59} + ( -2 + 4 \beta ) q^{60} + ( 2 - \beta ) q^{61} + ( -6 + 4 \beta ) q^{62} -2 q^{63} + ( -7 + 2 \beta ) q^{64} + 3 \beta q^{65} + ( 12 - 7 \beta ) q^{66} + ( -5 + 7 \beta ) q^{67} + ( -2 + 11 \beta ) q^{68} + ( 10 + 2 \beta ) q^{69} + ( 4 - 2 \beta ) q^{70} + \beta q^{71} + ( 3 - \beta ) q^{72} + ( -1 + 4 \beta ) q^{73} + ( 4 - 3 \beta ) q^{74} + 3 \beta q^{75} + ( -13 + 5 \beta ) q^{76} + ( -10 + 2 \beta ) q^{77} + ( -6 + 3 \beta ) q^{78} + ( -4 + 5 \beta ) q^{79} + 3 \beta q^{80} -5 q^{81} + ( -14 + 9 \beta ) q^{82} + ( 8 + 5 \beta ) q^{83} + ( 8 - 2 \beta ) q^{84} + ( -2 - 6 \beta ) q^{85} + ( 8 - 8 \beta ) q^{86} + ( 4 + 7 \beta ) q^{87} + ( 17 - 8 \beta ) q^{88} -3 \beta q^{89} + ( -2 + \beta ) q^{90} + 6 q^{91} + ( 18 - \beta ) q^{92} + ( 4 - 2 \beta ) q^{93} + ( -13 + 12 \beta ) q^{94} + ( 6 - \beta ) q^{95} + ( -2 - 3 \beta ) q^{96} + ( 9 - 6 \beta ) q^{97} + ( 3 - 3 \beta ) q^{98} + ( 5 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 4q^{6} + 4q^{7} - 6q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 4q^{6} + 4q^{7} - 6q^{8} - 2q^{9} + 4q^{10} - 10q^{11} + 8q^{12} + 6q^{13} - 4q^{14} - 4q^{15} + 6q^{16} - 12q^{17} + 2q^{18} - 2q^{19} - 8q^{20} + 14q^{22} - 4q^{23} - 4q^{24} - 6q^{25} - 6q^{26} + 4q^{28} - 14q^{29} + 4q^{30} + 4q^{31} + 6q^{32} - 4q^{33} + 8q^{34} - 2q^{36} - 4q^{37} + 14q^{38} + 4q^{40} + 8q^{41} - 8q^{42} - 16q^{43} - 18q^{44} - 16q^{46} + 22q^{47} - 6q^{49} + 6q^{50} + 4q^{51} + 6q^{52} - 14q^{53} + 16q^{54} + 4q^{55} - 12q^{56} - 12q^{57} + 6q^{58} - 12q^{59} - 4q^{60} + 4q^{61} - 12q^{62} - 4q^{63} - 14q^{64} + 24q^{66} - 10q^{67} - 4q^{68} + 20q^{69} + 8q^{70} + 6q^{72} - 2q^{73} + 8q^{74} - 26q^{76} - 20q^{77} - 12q^{78} - 8q^{79} - 10q^{81} - 28q^{82} + 16q^{83} + 16q^{84} - 4q^{85} + 16q^{86} + 8q^{87} + 34q^{88} - 4q^{90} + 12q^{91} + 36q^{92} + 8q^{93} - 26q^{94} + 12q^{95} - 4q^{96} + 18q^{97} + 6q^{98} + 10q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.41421 1.41421 3.82843 −1.41421 −3.41421 2.00000 −4.41421 −1.00000 3.41421
1.2 0.414214 −1.41421 −1.82843 1.41421 −0.585786 2.00000 −1.58579 −1.00000 0.585786
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 547.2.a.a 2
3.b odd 2 1 4923.2.a.h 2
4.b odd 2 1 8752.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.a 2 1.a even 1 1 trivial
4923.2.a.h 2 3.b odd 2 1
8752.2.a.n 2 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$547$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(547))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ $$1 + 4 T^{2} + 9 T^{4}$$
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$1 + 10 T + 45 T^{2} + 110 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 3 T + 13 T^{2} )^{2}$$
$17$ $$1 + 12 T + 68 T^{2} + 204 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 14 T + 99 T^{2} + 406 T^{3} + 841 T^{4}$$
$31$ $$1 - 4 T + 58 T^{2} - 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T + 76 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 8 T + 48 T^{2} - 328 T^{3} + 1681 T^{4}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$1 - 22 T + 213 T^{2} - 1034 T^{3} + 2209 T^{4}$$
$53$ $$1 + 14 T + 123 T^{2} + 742 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 136 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T + 124 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 61 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$1 + 140 T^{2} + 5041 T^{4}$$
$73$ $$1 + 2 T + 115 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T + 124 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 16 T + 180 T^{2} - 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 + 160 T^{2} + 7921 T^{4}$$
$97$ $$1 - 18 T + 203 T^{2} - 1746 T^{3} + 9409 T^{4}$$