# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -\zeta_{6} q^{9} -4 \zeta_{6} q^{12} + q^{13} + 6 q^{15} -4 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + 6 q^{20} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 q^{27} -9 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -2 q^{36} -2 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} -6 q^{41} - q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -3 \zeta_{6} q^{47} -8 q^{48} -12 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + 14 q^{57} + ( 12 - 12 \zeta_{6} ) q^{60} + 10 \zeta_{6} q^{61} -8 q^{64} + 3 \zeta_{6} q^{65} + ( -14 + 14 \zeta_{6} ) q^{67} -12 \zeta_{6} q^{68} -6 q^{69} -6 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{75} + 14 q^{76} + \zeta_{6} q^{79} + ( 12 - 12 \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} + 3 q^{83} + 18 q^{85} + ( -18 + 18 \zeta_{6} ) q^{87} -15 \zeta_{6} q^{89} -6 q^{92} + 10 \zeta_{6} q^{93} + ( -21 + 21 \zeta_{6} ) q^{95} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{4} + 3q^{5} - q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{4} + 3q^{5} - q^{9} - 4q^{12} + 2q^{13} + 12q^{15} - 4q^{16} + 6q^{17} + 7q^{19} + 12q^{20} - 3q^{23} - 4q^{25} + 8q^{27} - 18q^{29} - 5q^{31} - 4q^{36} - 2q^{37} + 2q^{39} - 12q^{41} - 2q^{43} + 3q^{45} - 3q^{47} - 16q^{48} - 12q^{51} + 2q^{52} + 9q^{53} + 28q^{57} + 12q^{60} + 10q^{61} - 16q^{64} + 3q^{65} - 14q^{67} - 12q^{68} - 12q^{69} - 12q^{71} - 11q^{73} + 8q^{75} + 28q^{76} + q^{79} + 12q^{80} + 11q^{81} + 6q^{83} + 36q^{85} - 18q^{87} - 15q^{89} - 12q^{92} + 10q^{93} - 21q^{95} - 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 + 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 −0.500000 + 0.866025i 0
508.1 0 1.00000 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.c 2
7.b odd 2 1 637.2.e.b 2
7.c even 3 1 91.2.a.b 1
7.c even 3 1 inner 637.2.e.c 2
7.d odd 6 1 637.2.a.b 1
7.d odd 6 1 637.2.e.b 2
21.g even 6 1 5733.2.a.f 1
21.h odd 6 1 819.2.a.c 1
28.g odd 6 1 1456.2.a.k 1
35.j even 6 1 2275.2.a.d 1
56.k odd 6 1 5824.2.a.f 1
56.p even 6 1 5824.2.a.bd 1
91.r even 6 1 1183.2.a.a 1
91.s odd 6 1 8281.2.a.h 1
91.z odd 12 2 1183.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.b 1 7.c even 3 1
637.2.a.b 1 7.d odd 6 1
637.2.e.b 2 7.b odd 2 1
637.2.e.b 2 7.d odd 6 1
637.2.e.c 2 1.a even 1 1 trivial
637.2.e.c 2 7.c even 3 1 inner
819.2.a.c 1 21.h odd 6 1
1183.2.a.a 1 91.r even 6 1
1183.2.c.a 2 91.z odd 12 2
1456.2.a.k 1 28.g odd 6 1
2275.2.a.d 1 35.j even 6 1
5733.2.a.f 1 21.g even 6 1
5824.2.a.f 1 56.k odd 6 1
5824.2.a.bd 1 56.p even 6 1
8281.2.a.h 1 91.s odd 6 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{5}^{2} - 3 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -3 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$( 1 + T )^{2}$$