# Properties

 Label 637.2.a.e Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.61803 2.23607 4.85410 −2.23607 −5.85410 0 −7.47214 2.00000 5.85410
1.2 −0.381966 −2.23607 −1.85410 2.23607 0.854102 0 1.47214 2.00000 −0.854102
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 637.2.a.e 2
3.b odd 2 1 5733.2.a.w 2
7.b odd 2 1 637.2.a.f 2
7.c even 3 2 637.2.e.h 4
7.d odd 6 2 91.2.e.b 4
13.b even 2 1 8281.2.a.ba 2
21.c even 2 1 5733.2.a.v 2
21.g even 6 2 819.2.j.c 4
28.f even 6 2 1456.2.r.j 4
91.b odd 2 1 8281.2.a.z 2
91.s odd 6 2 1183.2.e.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 7.d odd 6 2
637.2.a.e 2 1.a even 1 1 trivial
637.2.a.f 2 7.b odd 2 1
637.2.e.h 4 7.c even 3 2
819.2.j.c 4 21.g even 6 2
1183.2.e.d 4 91.s odd 6 2
1456.2.r.j 4 28.f even 6 2
5733.2.a.v 2 21.c even 2 1
5733.2.a.w 2 3.b odd 2 1
8281.2.a.z 2 91.b odd 2 1
8281.2.a.ba 2 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{2} - 5$$ $$T_{17}^{2} - 6 T_{17} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + T^{2}$$
$3$ $$-5 + T^{2}$$
$5$ $$-5 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-11 - 6 T + T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$31 + 12 T + T^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$( 5 + T )^{2}$$
$37$ $$-41 + 4 T + T^{2}$$
$41$ $$-20 + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-11 - 6 T + T^{2}$$
$53$ $$-11 + 6 T + T^{2}$$
$59$ $$-11 + 6 T + T^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$( 3 + T )^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$-29 + 8 T + T^{2}$$
$79$ $$-29 - 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$-5 + T^{2}$$
$97$ $$-164 - 8 T + T^{2}$$