# Properties

 Label 637.2.c.e Level $637$ Weight $2$ Character orbit 637.c Analytic conductor $5.086$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 11 x^{6} + 36 x^{4} + 31 x^{2} + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{4} ) q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{6} + \beta_{3} q^{8} + ( 2 + \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{4} ) q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{5} q^{5} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{6} + \beta_{3} q^{8} + ( 2 + \beta_{4} - \beta_{6} ) q^{9} + ( \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{10} + ( \beta_{1} + \beta_{7} ) q^{11} + ( 3 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{12} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{13} + ( 2 \beta_{1} + \beta_{5} ) q^{15} + ( \beta_{2} + \beta_{4} ) q^{16} + ( 1 + \beta_{6} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{18} + ( -2 \beta_{1} - \beta_{3} ) q^{19} + ( 2 \beta_{5} - \beta_{7} ) q^{20} + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{22} + ( 1 - \beta_{4} ) q^{23} + ( 3 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{24} + ( 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{26} + ( -3 + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{27} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{29} + ( -6 + \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{30} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{31} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{32} + ( -\beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{33} + ( -\beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{34} + ( -6 - 5 \beta_{4} + 2 \beta_{6} ) q^{36} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{37} + ( 4 - \beta_{2} - \beta_{4} ) q^{38} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{39} + ( -1 - \beta_{4} + \beta_{6} ) q^{40} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{41} + ( -2 \beta_{4} - \beta_{6} ) q^{43} + ( \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{44} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{45} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{46} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{47} + ( -2 - \beta_{2} ) q^{48} + ( -\beta_{1} + 2 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{50} + ( -1 - 2 \beta_{2} + \beta_{4} ) q^{51} + ( -6 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{52} + ( 3 - 2 \beta_{2} ) q^{53} + ( -6 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{54} + ( -1 + 4 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{55} + ( -\beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{7} ) q^{57} + ( -2 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{58} + ( -3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{59} + ( -4 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{60} + ( -1 - 2 \beta_{2} ) q^{61} + ( -4 + \beta_{2} - 2 \beta_{4} ) q^{62} + ( 7 - 2 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{64} + ( 4 - \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{65} + \beta_{2} q^{66} + ( -3 \beta_{3} + 2 \beta_{5} ) q^{67} + ( 1 + 3 \beta_{2} + 4 \beta_{4} - \beta_{6} ) q^{68} + ( 3 - \beta_{4} - \beta_{6} ) q^{69} + ( 4 \beta_{1} + 3 \beta_{3} - \beta_{5} ) q^{71} + ( -2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} ) q^{72} + ( 2 \beta_{1} + \beta_{5} - 2 \beta_{7} ) q^{73} + ( -6 + 3 \beta_{2} ) q^{74} + ( -2 \beta_{4} - \beta_{6} ) q^{75} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{76} + ( 3 - 5 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{78} + ( -3 - \beta_{4} + \beta_{6} ) q^{79} + ( -2 \beta_{1} + \beta_{5} - \beta_{7} ) q^{80} + ( 5 - 4 \beta_{2} + 2 \beta_{4} ) q^{81} + ( -4 - 4 \beta_{4} + \beta_{6} ) q^{82} + ( 5 \beta_{1} + \beta_{3} - \beta_{7} ) q^{83} + ( 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{85} + ( \beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{86} + ( -8 - 2 \beta_{2} + \beta_{6} ) q^{87} + ( -1 - 2 \beta_{2} - 2 \beta_{4} ) q^{88} + ( 5 \beta_{1} + 3 \beta_{3} - \beta_{5} + \beta_{7} ) q^{89} + ( 16 - 6 \beta_{2} + 2 \beta_{4} ) q^{90} + ( 1 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{93} + ( -6 + \beta_{2} - 5 \beta_{4} + \beta_{6} ) q^{94} + ( 1 - 2 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{95} + ( 5 \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{96} + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{97} + ( -3 \beta_{1} + \beta_{5} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 6q^{4} + 12q^{9} + O(q^{10})$$ $$8q - 4q^{3} - 6q^{4} + 12q^{9} - 6q^{10} + 18q^{12} + 6q^{13} - 2q^{16} + 8q^{17} - 18q^{22} + 12q^{23} - 6q^{26} - 16q^{27} - 8q^{29} - 38q^{30} - 28q^{36} + 34q^{38} - 18q^{39} - 4q^{40} + 8q^{43} - 18q^{48} - 16q^{51} - 42q^{52} + 20q^{53} - 12q^{55} - 12q^{61} - 22q^{62} + 44q^{64} + 30q^{65} + 2q^{66} - 2q^{68} + 28q^{69} - 42q^{74} + 8q^{75} + 10q^{78} - 20q^{79} + 24q^{81} - 16q^{82} - 68q^{87} - 4q^{88} + 108q^{90} + 6q^{92} - 26q^{94} + 16q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 31 x^{2} + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 5 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 6 \nu^{3} + 5 \nu$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} + 8 \nu^{4} + 16 \nu^{2} + 5$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} + 10 \nu^{5} + 29 \nu^{3} + 20 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 5 \beta_{2} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 6 \beta_{3} + 19 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} - 8 \beta_{4} + 24 \beta_{2} - 69$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} - 10 \beta_{5} + 31 \beta_{3} - 94 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
246.1
 − 2.28481i − 2.12549i − 1.07305i − 0.332375i 0.332375i 1.07305i 2.12549i 2.28481i
2.28481i −3.15042 −3.22037 2.12499i 7.19813i 0 2.78832i 6.92516 4.85521
246.2 2.12549i 0.178854 −2.51771 3.60603i 0.380153i 0 1.10038i −2.96801 −7.66457
246.3 1.07305i 2.43140 0.848553 0.625432i 2.60903i 0 3.05665i 2.91173 −0.671123
246.4 0.332375i −1.45984 1.88953 1.44562i 0.485214i 0 1.29278i −0.868875 0.480489
246.5 0.332375i −1.45984 1.88953 1.44562i 0.485214i 0 1.29278i −0.868875 0.480489
246.6 1.07305i 2.43140 0.848553 0.625432i 2.60903i 0 3.05665i 2.91173 −0.671123
246.7 2.12549i 0.178854 −2.51771 3.60603i 0.380153i 0 1.10038i −2.96801 −7.66457
246.8 2.28481i −3.15042 −3.22037 2.12499i 7.19813i 0 2.78832i 6.92516 4.85521
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 246.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.e 8
7.b odd 2 1 637.2.c.f 8
7.c even 3 2 637.2.r.f 16
7.d odd 6 2 91.2.r.a 16
13.b even 2 1 inner 637.2.c.e 8
13.d odd 4 2 8281.2.a.cj 8
21.g even 6 2 819.2.dl.e 16
91.b odd 2 1 637.2.c.f 8
91.i even 4 2 8281.2.a.ck 8
91.r even 6 2 637.2.r.f 16
91.s odd 6 2 91.2.r.a 16
91.bb even 12 4 1183.2.e.i 16
273.ba even 6 2 819.2.dl.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 7.d odd 6 2
91.2.r.a 16 91.s odd 6 2
637.2.c.e 8 1.a even 1 1 trivial
637.2.c.e 8 13.b even 2 1 inner
637.2.c.f 8 7.b odd 2 1
637.2.c.f 8 91.b odd 2 1
637.2.r.f 16 7.c even 3 2
637.2.r.f 16 91.r even 6 2
819.2.dl.e 16 21.g even 6 2
819.2.dl.e 16 273.ba even 6 2
1183.2.e.i 16 91.bb even 12 4
8281.2.a.cj 8 13.d odd 4 2
8281.2.a.ck 8 91.i even 4 2

## Hecke kernels

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

 $$T_{2}^{8} + 11 T_{2}^{6} + 36 T_{2}^{4} + 31 T_{2}^{2} + 3$$ $$T_{3}^{4} + 2 T_{3}^{3} - 7 T_{3}^{2} - 10 T_{3} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 31 T^{2} + 36 T^{4} + 11 T^{6} + T^{8}$$
$3$ $$( 2 - 10 T - 7 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$5$ $$48 + 160 T^{2} + 103 T^{4} + 20 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$27 + 340 T^{2} + 596 T^{4} + 52 T^{6} + T^{8}$$
$13$ $$28561 - 13182 T + 4732 T^{2} - 1690 T^{3} + 598 T^{4} - 130 T^{5} + 28 T^{6} - 6 T^{7} + T^{8}$$
$17$ $$( 123 + 52 T - 20 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$19$ $$3267 + 2332 T^{2} + 540 T^{4} + 44 T^{6} + T^{8}$$
$23$ $$( -6 + 10 T + 5 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$29$ $$( 624 - 208 T - 63 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$33708 + 19492 T^{2} + 2091 T^{4} + 80 T^{6} + T^{8}$$
$37$ $$8748 + 49572 T^{2} + 4347 T^{4} + 120 T^{6} + T^{8}$$
$41$ $$292032 + 88192 T^{2} + 5732 T^{4} + 132 T^{6} + T^{8}$$
$43$ $$( -104 - 156 T - 66 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$47$ $$240267 + 243892 T^{2} + 12212 T^{4} + 196 T^{6} + T^{8}$$
$53$ $$( 87 + 130 T - 10 T^{3} + T^{4} )^{2}$$
$59$ $$111747 + 161740 T^{2} + 10476 T^{4} + 188 T^{6} + T^{8}$$
$61$ $$( 223 - 94 T - 24 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$67$ $$257547 + 285148 T^{2} + 21652 T^{4} + 284 T^{6} + T^{8}$$
$71$ $$397488 + 253084 T^{2} + 19829 T^{4} + 292 T^{6} + T^{8}$$
$73$ $$2904768 + 472240 T^{2} + 19975 T^{4} + 260 T^{6} + T^{8}$$
$79$ $$( -8 - 60 T + 9 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$83$ $$5483712 + 959920 T^{2} + 28692 T^{4} + 296 T^{6} + T^{8}$$
$89$ $$7622508 + 2938804 T^{2} + 62899 T^{4} + 440 T^{6} + T^{8}$$
$97$ $$192 + 4816 T^{2} + 2740 T^{4} + 104 T^{6} + T^{8}$$