# Properties

 Label 637.2.a.b Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} - 2q^{4} + 3q^{5} + q^{9} + O(q^{10})$$ $$q + 2q^{3} - 2q^{4} + 3q^{5} + q^{9} - 4q^{12} - q^{13} + 6q^{15} + 4q^{16} + 6q^{17} + 7q^{19} - 6q^{20} + 3q^{23} + 4q^{25} - 4q^{27} - 9q^{29} - 5q^{31} - 2q^{36} + 2q^{37} - 2q^{39} + 6q^{41} - q^{43} + 3q^{45} - 3q^{47} + 8q^{48} + 12q^{51} + 2q^{52} - 9q^{53} + 14q^{57} - 12q^{60} + 10q^{61} - 8q^{64} - 3q^{65} + 14q^{67} - 12q^{68} + 6q^{69} - 6q^{71} - 11q^{73} + 8q^{75} - 14q^{76} - q^{79} + 12q^{80} - 11q^{81} - 3q^{83} + 18q^{85} - 18q^{87} - 15q^{89} - 6q^{92} - 10q^{93} + 21q^{95} + q^{97} + O(q^{100})$$

## 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
 0
0 2.00000 −2.00000 3.00000 0 0 0 1.00000 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
$$7$$ $$-1$$
$$13$$ $$1$$

## 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.b 1
3.b odd 2 1 5733.2.a.f 1
7.b odd 2 1 91.2.a.b 1
7.c even 3 2 637.2.e.b 2
7.d odd 6 2 637.2.e.c 2
13.b even 2 1 8281.2.a.h 1
21.c even 2 1 819.2.a.c 1
28.d even 2 1 1456.2.a.k 1
35.c odd 2 1 2275.2.a.d 1
56.e even 2 1 5824.2.a.f 1
56.h odd 2 1 5824.2.a.bd 1
91.b odd 2 1 1183.2.a.a 1
91.i even 4 2 1183.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.b 1 7.b odd 2 1
637.2.a.b 1 1.a even 1 1 trivial
637.2.e.b 2 7.c even 3 2
637.2.e.c 2 7.d odd 6 2
819.2.a.c 1 21.c even 2 1
1183.2.a.a 1 91.b odd 2 1
1183.2.c.a 2 91.i even 4 2
1456.2.a.k 1 28.d even 2 1
2275.2.a.d 1 35.c odd 2 1
5733.2.a.f 1 3.b odd 2 1
5824.2.a.f 1 56.e even 2 1
5824.2.a.bd 1 56.h odd 2 1
8281.2.a.h 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}$$ $$T_{3} - 2$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$-6 + T$$
$19$ $$-7 + T$$
$23$ $$-3 + T$$
$29$ $$9 + T$$
$31$ $$5 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$1 + T$$
$47$ $$3 + T$$
$53$ $$9 + T$$
$59$ $$T$$
$61$ $$-10 + T$$
$67$ $$-14 + T$$
$71$ $$6 + T$$
$73$ $$11 + T$$
$79$ $$1 + T$$
$83$ $$3 + T$$
$89$ $$15 + T$$
$97$ $$-1 + T$$