# Properties

 Label 637.2.a.k Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.746052.1 Defining polynomial: $$x^{5} - x^{4} - 7 x^{3} + 8 x + 2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 7 x^{3} + 8 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 4 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} + 19$$

## 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
 3.00852 1.19566 −0.265608 −1.21332 −1.72525
−2.00852 −1.75906 2.03417 −0.905722 3.53311 0 −0.0686323 0.0942784 1.81916
1.2 −0.195656 0.259788 −1.96172 −3.93251 −0.0508292 0 0.775135 −2.93251 0.769420
1.3 1.26561 2.62728 −0.398235 2.90260 3.32511 0 −3.03523 3.90260 3.67356
1.4 2.21332 −2.47443 2.89879 2.12280 −5.47671 0 1.98932 3.12280 4.69843
1.5 2.72525 1.34642 5.42699 −2.18716 3.66932 0 9.33940 −1.18716 −5.96057
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.k 5
3.b odd 2 1 5733.2.a.bm 5
7.b odd 2 1 637.2.a.l 5
7.c even 3 2 637.2.e.m 10
7.d odd 6 2 91.2.e.c 10
13.b even 2 1 8281.2.a.bx 5
21.c even 2 1 5733.2.a.bl 5
21.g even 6 2 819.2.j.h 10
28.f even 6 2 1456.2.r.p 10
91.b odd 2 1 8281.2.a.bw 5
91.s odd 6 2 1183.2.e.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 7.d odd 6 2
637.2.a.k 5 1.a even 1 1 trivial
637.2.a.l 5 7.b odd 2 1
637.2.e.m 10 7.c even 3 2
819.2.j.h 10 21.g even 6 2
1183.2.e.f 10 91.s odd 6 2
1456.2.r.p 10 28.f even 6 2
5733.2.a.bl 5 21.c even 2 1
5733.2.a.bm 5 3.b odd 2 1
8281.2.a.bw 5 91.b odd 2 1
8281.2.a.bx 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{5} - 4 T_{2}^{4} - T_{2}^{3} + 17 T_{2}^{2} - 12 T_{2} - 3$$ $$T_{3}^{5} - 9 T_{3}^{3} + 16 T_{3} - 4$$ $$T_{17}^{5} - 5 T_{17}^{4} - 22 T_{17}^{3} + 106 T_{17}^{2} + 93 T_{17} - 429$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - 12 T + 17 T^{2} - T^{3} - 4 T^{4} + T^{5}$$
$3$ $$-4 + 16 T - 9 T^{3} + T^{5}$$
$5$ $$48 + 48 T - 20 T^{2} - 15 T^{3} + 2 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$33 - 45 T - 22 T^{2} + 36 T^{3} - 11 T^{4} + T^{5}$$
$13$ $$( 1 + T )^{5}$$
$17$ $$-429 + 93 T + 106 T^{2} - 22 T^{3} - 5 T^{4} + T^{5}$$
$19$ $$223 + 173 T - 176 T^{2} - 14 T^{3} + 9 T^{4} + T^{5}$$
$23$ $$12 - 12 T - 26 T^{2} + 31 T^{3} - 10 T^{4} + T^{5}$$
$29$ $$-108 + 144 T - 19 T^{2} - 25 T^{3} + 3 T^{4} + T^{5}$$
$31$ $$356 + 508 T + 102 T^{2} - 61 T^{3} - 6 T^{4} + T^{5}$$
$37$ $$-7036 + 660 T + 678 T^{2} - 111 T^{3} - 4 T^{4} + T^{5}$$
$41$ $$1584 - 2544 T - 940 T^{2} - 28 T^{3} + 14 T^{4} + T^{5}$$
$43$ $$64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5}$$
$47$ $$5169 + 2811 T - 26 T^{2} - 124 T^{3} + T^{4} + T^{5}$$
$53$ $$19959 - 12759 T + 2426 T^{2} - 74 T^{3} - 17 T^{4} + T^{5}$$
$59$ $$-33 - 45 T + 22 T^{2} + 36 T^{3} + 11 T^{4} + T^{5}$$
$61$ $$8461 + 5881 T + 766 T^{2} - 122 T^{3} - 11 T^{4} + T^{5}$$
$67$ $$-22699 - 591 T + 2160 T^{2} - 162 T^{3} - 13 T^{4} + T^{5}$$
$71$ $$-6336 - 456 T + 853 T^{2} - 25 T^{3} - 15 T^{4} + T^{5}$$
$73$ $$712 + 700 T - 42 T^{2} - 75 T^{3} + T^{5}$$
$79$ $$-1000 + 1500 T - 190 T^{2} - 137 T^{3} - 2 T^{4} + T^{5}$$
$83$ $$7488 + 2688 T - 308 T^{2} - 124 T^{3} + 6 T^{4} + T^{5}$$
$89$ $$-7692 + 2148 T + 694 T^{2} - 155 T^{3} - 4 T^{4} + T^{5}$$
$97$ $$2384 - 2240 T + 612 T^{2} - 16 T^{3} - 12 T^{4} + T^{5}$$