# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -2 q^{3} -2 q^{4} + i q^{5} -4 i q^{6} + q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -2 q^{3} -2 q^{4} + i q^{5} -4 i q^{6} + q^{9} -2 q^{10} + 2 i q^{11} + 4 q^{12} + ( -2 - 3 i ) q^{13} -2 i q^{15} -4 q^{16} -6 q^{17} + 2 i q^{18} -3 i q^{19} -2 i q^{20} -4 q^{22} -3 q^{23} + 4 q^{25} + ( 6 - 4 i ) q^{26} + 4 q^{27} + 3 q^{29} + 4 q^{30} -3 i q^{31} -8 i q^{32} -4 i q^{33} -12 i q^{34} -2 q^{36} -6 i q^{37} + 6 q^{38} + ( 4 + 6 i ) q^{39} + 10 i q^{41} + q^{43} -4 i q^{44} + i q^{45} -6 i q^{46} -11 i q^{47} + 8 q^{48} + 8 i q^{50} + 12 q^{51} + ( 4 + 6 i ) q^{52} -9 q^{53} + 8 i q^{54} -2 q^{55} + 6 i q^{57} + 6 i q^{58} + 8 i q^{59} + 4 i q^{60} -8 q^{61} + 6 q^{62} + 8 q^{64} + ( 3 - 2 i ) q^{65} + 8 q^{66} -12 i q^{67} + 12 q^{68} + 6 q^{69} -14 i q^{71} + 9 i q^{73} + 12 q^{74} -8 q^{75} + 6 i q^{76} + ( -12 + 8 i ) q^{78} -9 q^{79} -4 i q^{80} -11 q^{81} -20 q^{82} + 11 i q^{83} -6 i q^{85} + 2 i q^{86} -6 q^{87} + 5 i q^{89} -2 q^{90} + 6 q^{92} + 6 i q^{93} + 22 q^{94} + 3 q^{95} + 16 i q^{96} -9 i q^{97} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} - 4q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} - 4q^{4} + 2q^{9} - 4q^{10} + 8q^{12} - 4q^{13} - 8q^{16} - 12q^{17} - 8q^{22} - 6q^{23} + 8q^{25} + 12q^{26} + 8q^{27} + 6q^{29} + 8q^{30} - 4q^{36} + 12q^{38} + 8q^{39} + 2q^{43} + 16q^{48} + 24q^{51} + 8q^{52} - 18q^{53} - 4q^{55} - 16q^{61} + 12q^{62} + 16q^{64} + 6q^{65} + 16q^{66} + 24q^{68} + 12q^{69} + 24q^{74} - 16q^{75} - 24q^{78} - 18q^{79} - 22q^{81} - 40q^{82} - 12q^{87} - 4q^{90} + 12q^{92} + 44q^{94} + 6q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
246.1
 − 1.00000i 1.00000i
2.00000i −2.00000 −2.00000 1.00000i 4.00000i 0 0 1.00000 −2.00000
246.2 2.00000i −2.00000 −2.00000 1.00000i 4.00000i 0 0 1.00000 −2.00000
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.a 2
7.b odd 2 1 637.2.c.c yes 2
7.c even 3 2 637.2.r.c 4
7.d odd 6 2 637.2.r.a 4
13.b even 2 1 inner 637.2.c.a 2
13.d odd 4 1 8281.2.a.a 1
13.d odd 4 1 8281.2.a.k 1
91.b odd 2 1 637.2.c.c yes 2
91.i even 4 1 8281.2.a.b 1
91.i even 4 1 8281.2.a.m 1
91.r even 6 2 637.2.r.c 4
91.s odd 6 2 637.2.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.a 2 1.a even 1 1 trivial
637.2.c.a 2 13.b even 2 1 inner
637.2.c.c yes 2 7.b odd 2 1
637.2.c.c yes 2 91.b odd 2 1
637.2.r.a 4 7.d odd 6 2
637.2.r.a 4 91.s odd 6 2
637.2.r.c 4 7.c even 3 2
637.2.r.c 4 91.r even 6 2
8281.2.a.a 1 13.d odd 4 1
8281.2.a.b 1 91.i even 4 1
8281.2.a.k 1 13.d odd 4 1
8281.2.a.m 1 91.i even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$13 + 4 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$9 + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$9 + T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$100 + T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$121 + T^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$64 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$196 + T^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( 9 + T )^{2}$$
$83$ $$121 + T^{2}$$
$89$ $$25 + T^{2}$$
$97$ $$81 + T^{2}$$
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