# Properties

 Label 637.2.a.m Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.4507648.1 Defining polynomial: $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8$$

## 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
 −1.66745 2.35100 −1.20475 1.90903 −0.146243 0.758419
−2.44785 0.667452 3.99195 −0.910286 −1.63382 0 −4.87599 −2.55451 2.22824
1.2 −1.17619 −3.35100 −0.616586 −3.14862 3.94140 0 3.07759 8.22917 3.70337
1.3 −0.656184 0.204753 −1.56942 1.35996 −0.134356 0 2.34220 −2.95808 −0.892385
1.4 0.264627 −2.90903 −1.92997 1.43515 −0.769807 0 −1.03998 5.46247 0.379780
1.5 1.83237 −0.853757 1.35758 −2.62555 −1.56440 0 −1.17715 −2.27110 −4.81098
1.6 2.18322 −1.75842 2.76645 −2.11065 −3.83901 0 1.67333 0.0920365 −4.60802
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.m 6
3.b odd 2 1 5733.2.a.bu 6
7.b odd 2 1 637.2.a.n yes 6
7.c even 3 2 637.2.e.o 12
7.d odd 6 2 637.2.e.n 12
13.b even 2 1 8281.2.a.cc 6
21.c even 2 1 5733.2.a.br 6
91.b odd 2 1 8281.2.a.cd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 1.a even 1 1 trivial
637.2.a.n yes 6 7.b odd 2 1
637.2.e.n 12 7.d odd 6 2
637.2.e.o 12 7.c even 3 2
5733.2.a.br 6 21.c even 2 1
5733.2.a.bu 6 3.b odd 2 1
8281.2.a.cc 6 13.b even 2 1
8281.2.a.cd 6 91.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{6} - 8 T_{2}^{4} + 14 T_{2}^{2} + 4 T_{2} - 2$$ $$T_{3}^{6} + 8 T_{3}^{5} + 20 T_{3}^{4} + 12 T_{3}^{3} - 12 T_{3}^{2} - 8 T_{3} + 2$$ $$T_{17}^{6} + 16 T_{17}^{5} + 56 T_{17}^{4} - 324 T_{17}^{3} - 2792 T_{17}^{2} - 6792 T_{17} - 5294$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + 4 T + 14 T^{2} - 8 T^{4} + T^{6}$$
$3$ $$2 - 8 T - 12 T^{2} + 12 T^{3} + 20 T^{4} + 8 T^{5} + T^{6}$$
$5$ $$31 + 26 T - 31 T^{2} - 24 T^{3} + 5 T^{4} + 6 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$-562 - 692 T + 186 T^{2} + 156 T^{3} - 38 T^{4} - 4 T^{5} + T^{6}$$
$13$ $$( 1 + T )^{6}$$
$17$ $$-5294 - 6792 T - 2792 T^{2} - 324 T^{3} + 56 T^{4} + 16 T^{5} + T^{6}$$
$19$ $$-73 + 6 T + 83 T^{2} - 16 T^{3} - 17 T^{4} + 2 T^{5} + T^{6}$$
$23$ $$529 + 874 T - 169 T^{2} - 272 T^{3} - 37 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$529 + 230 T - 401 T^{2} - 268 T^{3} - 33 T^{4} + 6 T^{5} + T^{6}$$
$31$ $$-44249 + 10046 T + 4181 T^{2} - 508 T^{3} - 115 T^{4} + 6 T^{5} + T^{6}$$
$37$ $$254 - 716 T + 378 T^{2} + 236 T^{3} - 82 T^{4} + T^{6}$$
$41$ $$28784 - 19744 T + 1720 T^{2} + 968 T^{3} - 120 T^{4} - 8 T^{5} + T^{6}$$
$43$ $$35153 + 29214 T + 6575 T^{2} - 188 T^{3} - 161 T^{4} - 2 T^{5} + T^{6}$$
$47$ $$-135617 - 107350 T - 25621 T^{2} - 1480 T^{3} + 215 T^{4} + 30 T^{5} + T^{6}$$
$53$ $$-1319 + 3386 T + 1199 T^{2} - 680 T^{3} - 37 T^{4} + 14 T^{5} + T^{6}$$
$59$ $$1532 - 760 T - 1236 T^{2} - 44 T^{3} + 146 T^{4} + 24 T^{5} + T^{6}$$
$61$ $$-216584 + 20768 T + 16252 T^{2} - 112 T^{3} - 246 T^{4} + T^{6}$$
$67$ $$6112 - 3712 T - 1984 T^{2} + 800 T^{3} - 16 T^{5} + T^{6}$$
$71$ $$-1206162 - 104028 T + 33274 T^{2} + 1856 T^{3} - 316 T^{4} - 8 T^{5} + T^{6}$$
$73$ $$-142657 - 22234 T + 12011 T^{2} + 808 T^{3} - 241 T^{4} - 6 T^{5} + T^{6}$$
$79$ $$7913 + 9486 T - 11129 T^{2} - 3172 T^{3} - 65 T^{4} + 22 T^{5} + T^{6}$$
$83$ $$-167041 - 13034 T + 26145 T^{2} + 7992 T^{3} + 941 T^{4} + 50 T^{5} + T^{6}$$
$89$ $$9959 + 26 T - 9053 T^{2} - 1188 T^{3} + 131 T^{4} + 26 T^{5} + T^{6}$$
$97$ $$217287 - 193902 T + 13171 T^{2} + 3620 T^{3} - 261 T^{4} - 14 T^{5} + T^{6}$$