# Properties

 Label 637.2.q.a Level $637$ Weight $2$ Character orbit 637.q 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.q (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 1.00000 + 1.73205i 0.500000 0.866025i 1.73205i −3.00000 1.73205i 0 1.73205i −0.500000 + 0.866025i 1.50000 + 2.59808i
589.1 −1.50000 0.866025i 1.00000 1.73205i 0.500000 + 0.866025i 1.73205i −3.00000 + 1.73205i 0 1.73205i −0.500000 0.866025i 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.e 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.q.a 2
7.b odd 2 1 13.2.e.a 2
7.c even 3 1 637.2.k.c 2
7.c even 3 1 637.2.u.b 2
7.d odd 6 1 637.2.k.a 2
7.d odd 6 1 637.2.u.c 2
13.e even 6 1 inner 637.2.q.a 2
13.f odd 12 2 8281.2.a.q 2
21.c even 2 1 117.2.q.c 2
28.d even 2 1 208.2.w.b 2
35.c odd 2 1 325.2.n.a 2
35.f even 4 2 325.2.m.a 4
56.e even 2 1 832.2.w.a 2
56.h odd 2 1 832.2.w.d 2
84.h odd 2 1 1872.2.by.d 2
91.b odd 2 1 169.2.e.a 2
91.i even 4 2 169.2.c.a 4
91.k even 6 1 637.2.u.b 2
91.l odd 6 1 637.2.u.c 2
91.n odd 6 1 169.2.b.a 2
91.n odd 6 1 169.2.e.a 2
91.p odd 6 1 637.2.k.a 2
91.t odd 6 1 13.2.e.a 2
91.t odd 6 1 169.2.b.a 2
91.u even 6 1 637.2.k.c 2
91.bc even 12 2 169.2.a.a 2
91.bc even 12 2 169.2.c.a 4
273.u even 6 1 117.2.q.c 2
273.u even 6 1 1521.2.b.a 2
273.bn even 6 1 1521.2.b.a 2
273.ca odd 12 2 1521.2.a.k 2
364.v even 6 1 2704.2.f.b 2
364.bc even 6 1 208.2.w.b 2
364.bc even 6 1 2704.2.f.b 2
364.bv odd 12 2 2704.2.a.o 2
455.be odd 6 1 325.2.n.a 2
455.cn even 12 2 4225.2.a.v 2
455.cz even 12 2 325.2.m.a 4
728.bl odd 6 1 832.2.w.d 2
728.ci even 6 1 832.2.w.a 2
1092.bh odd 6 1 1872.2.by.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 7.b odd 2 1
13.2.e.a 2 91.t odd 6 1
117.2.q.c 2 21.c even 2 1
117.2.q.c 2 273.u even 6 1
169.2.a.a 2 91.bc even 12 2
169.2.b.a 2 91.n odd 6 1
169.2.b.a 2 91.t odd 6 1
169.2.c.a 4 91.i even 4 2
169.2.c.a 4 91.bc even 12 2
169.2.e.a 2 91.b odd 2 1
169.2.e.a 2 91.n odd 6 1
208.2.w.b 2 28.d even 2 1
208.2.w.b 2 364.bc even 6 1
325.2.m.a 4 35.f even 4 2
325.2.m.a 4 455.cz even 12 2
325.2.n.a 2 35.c odd 2 1
325.2.n.a 2 455.be odd 6 1
637.2.k.a 2 7.d odd 6 1
637.2.k.a 2 91.p odd 6 1
637.2.k.c 2 7.c even 3 1
637.2.k.c 2 91.u even 6 1
637.2.q.a 2 1.a even 1 1 trivial
637.2.q.a 2 13.e even 6 1 inner
637.2.u.b 2 7.c even 3 1
637.2.u.b 2 91.k even 6 1
637.2.u.c 2 7.d odd 6 1
637.2.u.c 2 91.l odd 6 1
832.2.w.a 2 56.e even 2 1
832.2.w.a 2 728.ci even 6 1
832.2.w.d 2 56.h odd 2 1
832.2.w.d 2 728.bl odd 6 1
1521.2.a.k 2 273.ca odd 12 2
1521.2.b.a 2 273.u even 6 1
1521.2.b.a 2 273.bn even 6 1
1872.2.by.d 2 84.h odd 2 1
1872.2.by.d 2 1092.bh odd 6 1
2704.2.a.o 2 364.bv odd 12 2
2704.2.f.b 2 364.v even 6 1
2704.2.f.b 2 364.bc even 6 1
4225.2.a.v 2 455.cn even 12 2
8281.2.a.q 2 13.f odd 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$12 - 6 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$75 - 15 T + T^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$12 + T^{2}$$
$53$ $$( 3 + T )^{2}$$
$59$ $$48 + 12 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$12 - 6 T + T^{2}$$
$71$ $$12 - 6 T + T^{2}$$
$73$ $$3 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$192 + T^{2}$$
$89$ $$48 - 12 T + T^{2}$$
$97$ $$48 + 12 T + T^{2}$$