# Properties

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

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## 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -3 q^{8} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + q^{13} + \zeta_{6} q^{16} + ( 7 - 7 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} -7 \zeta_{6} q^{19} -3 q^{22} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -\zeta_{6} q^{26} -5 q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} -7 q^{34} + 3 q^{36} -8 \zeta_{6} q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} + 2 q^{43} -3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + 7 \zeta_{6} q^{47} -5 q^{50} + ( 1 - \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + 5 \zeta_{6} q^{58} + ( -7 + 7 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} + 7 q^{64} + ( 3 - 3 \zeta_{6} ) q^{67} -7 \zeta_{6} q^{68} -5 q^{71} -9 \zeta_{6} q^{72} + ( 14 - 14 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} -7 q^{76} + 6 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -2 \zeta_{6} q^{86} + ( -9 + 9 \zeta_{6} ) q^{88} + 6 q^{92} + ( 7 - 7 \zeta_{6} ) q^{94} + 14 q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{4} - 6q^{8} + 3q^{9} + 3q^{11} + 2q^{13} + q^{16} + 7q^{17} + 3q^{18} - 7q^{19} - 6q^{22} + 6q^{23} + 5q^{25} - q^{26} - 10q^{29} - 5q^{32} - 14q^{34} + 6q^{36} - 8q^{37} - 7q^{38} + 4q^{43} - 3q^{44} + 6q^{46} + 7q^{47} - 10q^{50} + q^{52} + 3q^{53} + 5q^{58} - 7q^{59} - 7q^{61} + 14q^{64} + 3q^{67} - 7q^{68} - 10q^{71} - 9q^{72} + 14q^{73} - 8q^{74} - 14q^{76} + 6q^{79} - 9q^{81} - 2q^{86} - 9q^{88} + 12q^{92} + 7q^{94} + 28q^{97} + 18q^{99} + O(q^{100})$$

## 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
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −3.00000 1.50000 2.59808i 0
508.1 −0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 −3.00000 1.50000 + 2.59808i 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.a 2
7.b odd 2 1 91.2.e.a 2
7.c even 3 1 637.2.a.d 1
7.c even 3 1 inner 637.2.e.a 2
7.d odd 6 1 91.2.e.a 2
7.d odd 6 1 637.2.a.c 1
21.c even 2 1 819.2.j.b 2
21.g even 6 1 819.2.j.b 2
21.g even 6 1 5733.2.a.c 1
21.h odd 6 1 5733.2.a.d 1
28.d even 2 1 1456.2.r.g 2
28.f even 6 1 1456.2.r.g 2
91.b odd 2 1 1183.2.e.b 2
91.r even 6 1 8281.2.a.e 1
91.s odd 6 1 1183.2.e.b 2
91.s odd 6 1 8281.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 7.b odd 2 1
91.2.e.a 2 7.d odd 6 1
637.2.a.c 1 7.d odd 6 1
637.2.a.d 1 7.c even 3 1
637.2.e.a 2 1.a even 1 1 trivial
637.2.e.a 2 7.c even 3 1 inner
819.2.j.b 2 21.c even 2 1
819.2.j.b 2 21.g even 6 1
1183.2.e.b 2 91.b odd 2 1
1183.2.e.b 2 91.s odd 6 1
1456.2.r.g 2 28.d even 2 1
1456.2.r.g 2 28.f even 6 1
5733.2.a.c 1 21.g even 6 1
5733.2.a.d 1 21.h odd 6 1
8281.2.a.e 1 91.r even 6 1
8281.2.a.f 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$49 - 7 T + T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$49 - 7 T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$49 + 7 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$( 5 + T )^{2}$$
$73$ $$196 - 14 T + T^{2}$$
$79$ $$36 - 6 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$
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