# Properties

 Label 637.2.a.i Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −1.65544 −0.210756 2.86620
−2.65544 2.39593 5.05137 3.65544 −6.36226 0 −8.10275 2.74049 −9.70682
1.2 −1.21076 −1.74483 −0.534070 2.21076 2.11256 0 3.06814 0.0444180 −2.67669
1.3 1.86620 3.34889 1.48270 −0.866198 6.24970 0 −0.965392 8.21509 −1.61650
 $$n$$: 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.i yes 3
3.b odd 2 1 5733.2.a.bd 3
7.b odd 2 1 637.2.a.h 3
7.c even 3 2 637.2.e.k 6
7.d odd 6 2 637.2.e.l 6
13.b even 2 1 8281.2.a.bk 3
21.c even 2 1 5733.2.a.be 3
91.b odd 2 1 8281.2.a.bh 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 7.b odd 2 1
637.2.a.i yes 3 1.a even 1 1 trivial
637.2.e.k 6 7.c even 3 2
637.2.e.l 6 7.d odd 6 2
5733.2.a.bd 3 3.b odd 2 1
5733.2.a.be 3 21.c even 2 1
8281.2.a.bh 3 91.b odd 2 1
8281.2.a.bk 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} - 6$$ $$T_{3}^{3} - 4 T_{3}^{2} - 2 T_{3} + 14$$ $$T_{17}^{3} - 4 T_{17}^{2} - 2 T_{17} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 - 4 T + 2 T^{2} + T^{3}$$
$3$ $$14 - 2 T - 4 T^{2} + T^{3}$$
$5$ $$7 + 3 T - 5 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-2 + 4 T^{2} + T^{3}$$
$13$ $$( 1 + T )^{3}$$
$17$ $$14 - 2 T - 4 T^{2} + T^{3}$$
$19$ $$63 - 3 T - 7 T^{2} + T^{3}$$
$23$ $$-43 - 41 T - T^{2} + T^{3}$$
$29$ $$-3 - 13 T + 7 T^{2} + T^{3}$$
$31$ $$-49 - 41 T + 3 T^{2} + T^{3}$$
$37$ $$-82 + 8 T + 10 T^{2} + T^{3}$$
$41$ $$504 - 116 T - 6 T^{2} + T^{3}$$
$43$ $$101 - 5 T - 9 T^{2} + T^{3}$$
$47$ $$-147 + 89 T - 17 T^{2} + T^{3}$$
$53$ $$9 + 39 T - 13 T^{2} + T^{3}$$
$59$ $$-252 + 144 T - 22 T^{2} + T^{3}$$
$61$ $$224 + 160 T + 24 T^{2} + T^{3}$$
$67$ $$-648 - 36 T + 14 T^{2} + T^{3}$$
$71$ $$194 - 44 T - 4 T^{2} + T^{3}$$
$73$ $$1561 - 219 T - 5 T^{2} + T^{3}$$
$79$ $$-99 - 69 T - T^{2} + T^{3}$$
$83$ $$-203 + 127 T - 23 T^{2} + T^{3}$$
$89$ $$21 - 55 T + 11 T^{2} + T^{3}$$
$97$ $$-7 - 71 T + 3 T^{2} + T^{3}$$
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