# Properties

 Label 637.2.k.b Level $637$ Weight $2$ Character orbit 637.k 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.k (of order $$6$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i −0.500000 0.866025i −1.00000 −1.50000 + 0.866025i −1.50000 + 0.866025i 0 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i
569.1 1.73205i −0.500000 + 0.866025i −1.00000 −1.50000 0.866025i −1.50000 0.866025i 0 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i
 $$n$$: 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
91.k even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.k.b 2
7.b odd 2 1 91.2.k.a 2
7.c even 3 1 637.2.q.b 2
7.c even 3 1 637.2.u.a 2
7.d odd 6 1 91.2.u.a yes 2
7.d odd 6 1 637.2.q.c 2
13.e even 6 1 637.2.u.a 2
21.c even 2 1 819.2.bm.a 2
21.g even 6 1 819.2.do.c 2
91.k even 6 1 inner 637.2.k.b 2
91.l odd 6 1 91.2.k.a 2
91.p odd 6 1 637.2.q.c 2
91.t odd 6 1 91.2.u.a yes 2
91.u even 6 1 637.2.q.b 2
91.w even 12 2 1183.2.e.e 4
91.x odd 12 2 8281.2.a.w 2
91.ba even 12 2 8281.2.a.s 2
91.bc even 12 2 1183.2.e.e 4
273.u even 6 1 819.2.do.c 2
273.br even 6 1 819.2.bm.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 7.b odd 2 1
91.2.k.a 2 91.l odd 6 1
91.2.u.a yes 2 7.d odd 6 1
91.2.u.a yes 2 91.t odd 6 1
637.2.k.b 2 1.a even 1 1 trivial
637.2.k.b 2 91.k even 6 1 inner
637.2.q.b 2 7.c even 3 1
637.2.q.b 2 91.u even 6 1
637.2.q.c 2 7.d odd 6 1
637.2.q.c 2 91.p odd 6 1
637.2.u.a 2 7.c even 3 1
637.2.u.a 2 13.e even 6 1
819.2.bm.a 2 21.c even 2 1
819.2.bm.a 2 273.br even 6 1
819.2.do.c 2 21.g even 6 1
819.2.do.c 2 273.u even 6 1
1183.2.e.e 4 91.w even 12 2
1183.2.e.e 4 91.bc even 12 2
8281.2.a.s 2 91.ba even 12 2
8281.2.a.w 2 91.x odd 12 2

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ $$T_{3}^{2} + T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$27 + 9 T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$3 - 3 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$3 - 3 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$75 + 15 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$12 + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$75 - 15 T + T^{2}$$
$71$ $$3 - 3 T + T^{2}$$
$73$ $$75 + 15 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$27 - 9 T + T^{2}$$