Properties

 Label 637.2.e.d 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( 6 - 6 \zeta_{6} ) q^{10} + ( 6 - 6 \zeta_{6} ) q^{11} + q^{13} + 4 \zeta_{6} q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} + 5 \zeta_{6} q^{19} + 6 q^{20} + 12 q^{22} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} -5 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} + ( -8 + 8 \zeta_{6} ) q^{32} + 8 q^{34} -6 q^{36} + 4 \zeta_{6} q^{37} + ( -10 + 10 \zeta_{6} ) q^{38} + 6 q^{41} - q^{43} + 12 \zeta_{6} q^{44} + ( 9 - 9 \zeta_{6} ) q^{45} + ( 6 - 6 \zeta_{6} ) q^{46} + 7 \zeta_{6} q^{47} -8 q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} -18 q^{55} -10 \zeta_{6} q^{58} + ( 8 - 8 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} -6 q^{62} -8 q^{64} -3 \zeta_{6} q^{65} + ( 6 - 6 \zeta_{6} ) q^{67} + 8 \zeta_{6} q^{68} -8 q^{71} + ( -13 + 13 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} -10 q^{76} -3 \zeta_{6} q^{79} + ( 12 - 12 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + 12 \zeta_{6} q^{82} -15 q^{83} -12 q^{85} -2 \zeta_{6} q^{86} + 3 \zeta_{6} q^{89} + 18 q^{90} + 6 q^{92} + ( -14 + 14 \zeta_{6} ) q^{94} + ( 15 - 15 \zeta_{6} ) q^{95} -7 q^{97} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} - 3q^{5} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} - 3q^{5} + 3q^{9} + 6q^{10} + 6q^{11} + 2q^{13} + 4q^{16} + 4q^{17} - 6q^{18} + 5q^{19} + 12q^{20} + 24q^{22} - 3q^{23} - 4q^{25} + 2q^{26} - 10q^{29} - 3q^{31} - 8q^{32} + 16q^{34} - 12q^{36} + 4q^{37} - 10q^{38} + 12q^{41} - 2q^{43} + 12q^{44} + 9q^{45} + 6q^{46} + 7q^{47} - 16q^{50} - 2q^{52} + 9q^{53} - 36q^{55} - 10q^{58} + 8q^{59} - 10q^{61} - 12q^{62} - 16q^{64} - 3q^{65} + 6q^{67} + 8q^{68} - 16q^{71} - 13q^{73} - 8q^{74} - 20q^{76} - 3q^{79} + 12q^{80} - 9q^{81} + 12q^{82} - 30q^{83} - 24q^{85} - 2q^{86} + 3q^{89} + 36q^{90} + 12q^{92} - 14q^{94} + 15q^{95} - 14q^{97} + 36q^{99} + O(q^{100})$$

Character values

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

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

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
1.00000 1.73205i 0 −1.00000 1.73205i −1.50000 + 2.59808i 0 0 0 1.50000 2.59808i 3.00000 + 5.19615i
508.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −1.50000 2.59808i 0 0 0 1.50000 + 2.59808i 3.00000 5.19615i
 $$n$$: 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.d 2
7.b odd 2 1 637.2.e.e 2
7.c even 3 1 637.2.a.a 1
7.c even 3 1 inner 637.2.e.d 2
7.d odd 6 1 91.2.a.a 1
7.d odd 6 1 637.2.e.e 2
21.g even 6 1 819.2.a.f 1
21.h odd 6 1 5733.2.a.l 1
28.f even 6 1 1456.2.a.g 1
35.i odd 6 1 2275.2.a.h 1
56.j odd 6 1 5824.2.a.s 1
56.m even 6 1 5824.2.a.t 1
91.r even 6 1 8281.2.a.l 1
91.s odd 6 1 1183.2.a.b 1
91.bb even 12 2 1183.2.c.b 2

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$49 - 7 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$36 - 6 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$169 + 13 T + T^{2}$$
$79$ $$9 + 3 T + T^{2}$$
$83$ $$( 15 + T )^{2}$$
$89$ $$9 - 3 T + T^{2}$$
$97$ $$( 7 + T )^{2}$$