# Properties

 Label 637.2.f.a Level $637$ Weight $2$ Character orbit 637.f Analytic conductor $5.086$ Analytic rank $1$ 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.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

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

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i −3.00000 −1.50000 2.59808i 0 −3.00000 −3.00000 5.19615i 1.50000 2.59808i
393.1 −0.500000 0.866025i −1.50000 2.59808i 0.500000 0.866025i −3.00000 −1.50000 + 2.59808i 0 −3.00000 −3.00000 + 5.19615i 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
13.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.f.a 2
7.b odd 2 1 637.2.f.b 2
7.c even 3 1 637.2.g.a 2
7.c even 3 1 637.2.h.a 2
7.d odd 6 1 91.2.g.a 2
7.d odd 6 1 91.2.h.a yes 2
13.c even 3 1 inner 637.2.f.a 2
13.c even 3 1 8281.2.a.j 1
13.e even 6 1 8281.2.a.g 1
21.g even 6 1 819.2.n.c 2
21.g even 6 1 819.2.s.a 2
91.g even 3 1 637.2.h.a 2
91.h even 3 1 637.2.g.a 2
91.l odd 6 1 1183.2.e.c 2
91.m odd 6 1 91.2.h.a yes 2
91.m odd 6 1 1183.2.e.a 2
91.n odd 6 1 637.2.f.b 2
91.n odd 6 1 8281.2.a.i 1
91.p odd 6 1 1183.2.e.c 2
91.t odd 6 1 8281.2.a.c 1
91.v odd 6 1 91.2.g.a 2
91.v odd 6 1 1183.2.e.a 2
273.r even 6 1 819.2.n.c 2
273.bf even 6 1 819.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 7.d odd 6 1
91.2.g.a 2 91.v odd 6 1
91.2.h.a yes 2 7.d odd 6 1
91.2.h.a yes 2 91.m odd 6 1
637.2.f.a 2 1.a even 1 1 trivial
637.2.f.a 2 13.c even 3 1 inner
637.2.f.b 2 7.b odd 2 1
637.2.f.b 2 91.n odd 6 1
637.2.g.a 2 7.c even 3 1
637.2.g.a 2 91.h even 3 1
637.2.h.a 2 7.c even 3 1
637.2.h.a 2 91.g even 3 1
819.2.n.c 2 21.g even 6 1
819.2.n.c 2 273.r even 6 1
819.2.s.a 2 21.g even 6 1
819.2.s.a 2 273.bf even 6 1
1183.2.e.a 2 91.m odd 6 1
1183.2.e.a 2 91.v odd 6 1
1183.2.e.c 2 91.l odd 6 1
1183.2.e.c 2 91.p odd 6 1
8281.2.a.c 1 91.t odd 6 1
8281.2.a.g 1 13.e even 6 1
8281.2.a.i 1 91.n odd 6 1
8281.2.a.j 1 13.c even 3 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{3}^{2} + 3 T_{3} + 9$$ $$T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$( 3 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$49 + 7 T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$49 - 7 T + T^{2}$$
$47$ $$( 1 + T )^{2}$$
$53$ $$( -3 + T )^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$169 + 13 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$169 + 13 T + T^{2}$$
$73$ $$( -13 + T )^{2}$$
$79$ $$( 3 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$25 + 5 T + T^{2}$$