# Properties

 Label 637.2.r.b Level $637$ Weight $2$ Character orbit 637.r Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ CM discriminant -91 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-13})$$ Defining polynomial: $$x^{4} - 13 x^{2} + 169$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{4} -\beta_{1} q^{5} + ( 3 - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{2} q^{4} -\beta_{1} q^{5} + ( 3 - 3 \beta_{2} ) q^{9} + \beta_{3} q^{13} + ( -4 + 4 \beta_{2} ) q^{16} -\beta_{1} q^{19} + 2 \beta_{3} q^{20} + ( -1 + \beta_{2} ) q^{23} + 8 \beta_{2} q^{25} -5 q^{29} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{31} -6 q^{36} + 2 \beta_{3} q^{41} + 9 q^{43} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{45} -\beta_{1} q^{47} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{52} -11 \beta_{2} q^{53} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{59} + 8 q^{64} + ( 13 - 13 \beta_{2} ) q^{65} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{73} + 2 \beta_{3} q^{76} + ( -15 + 15 \beta_{2} ) q^{79} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{80} -9 \beta_{2} q^{81} -5 \beta_{3} q^{83} -\beta_{1} q^{89} + 2 q^{92} + 13 \beta_{2} q^{95} -5 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 6 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 6 q^{9} - 8 q^{16} - 2 q^{23} + 16 q^{25} - 20 q^{29} - 24 q^{36} + 36 q^{43} - 22 q^{53} + 32 q^{64} + 26 q^{65} - 30 q^{79} - 18 q^{81} + 8 q^{92} + 26 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 13 x^{2} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/13$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$13 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$13 \beta_{3}$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 3.12250 − 1.80278i −3.12250 + 1.80278i 3.12250 + 1.80278i −3.12250 − 1.80278i
0 0 −1.00000 + 1.73205i −3.12250 + 1.80278i 0 0 0 1.50000 + 2.59808i 0
116.2 0 0 −1.00000 + 1.73205i 3.12250 1.80278i 0 0 0 1.50000 + 2.59808i 0
324.1 0 0 −1.00000 1.73205i −3.12250 1.80278i 0 0 0 1.50000 2.59808i 0
324.2 0 0 −1.00000 1.73205i 3.12250 + 1.80278i 0 0 0 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
91.b odd 2 1 CM by $$\Q(\sqrt{-91})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
13.b even 2 1 inner
91.r even 6 1 inner
91.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.b 4
7.b odd 2 1 inner 637.2.r.b 4
7.c even 3 1 637.2.c.b 2
7.c even 3 1 inner 637.2.r.b 4
7.d odd 6 1 637.2.c.b 2
7.d odd 6 1 inner 637.2.r.b 4
13.b even 2 1 inner 637.2.r.b 4
91.b odd 2 1 CM 637.2.r.b 4
91.r even 6 1 637.2.c.b 2
91.r even 6 1 inner 637.2.r.b 4
91.s odd 6 1 637.2.c.b 2
91.s odd 6 1 inner 637.2.r.b 4
91.z odd 12 2 8281.2.a.u 2
91.bb even 12 2 8281.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b 2 7.c even 3 1
637.2.c.b 2 7.d odd 6 1
637.2.c.b 2 91.r even 6 1
637.2.c.b 2 91.s odd 6 1
637.2.r.b 4 1.a even 1 1 trivial
637.2.r.b 4 7.b odd 2 1 inner
637.2.r.b 4 7.c even 3 1 inner
637.2.r.b 4 7.d odd 6 1 inner
637.2.r.b 4 13.b even 2 1 inner
637.2.r.b 4 91.b odd 2 1 CM
637.2.r.b 4 91.r even 6 1 inner
637.2.r.b 4 91.s odd 6 1 inner
8281.2.a.u 2 91.z odd 12 2
8281.2.a.u 2 91.bb even 12 2

## Hecke kernels

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

 $$T_{2}$$ $$T_{3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$169 - 13 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 13 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$169 - 13 T^{2} + T^{4}$$
$23$ $$( 1 + T + T^{2} )^{2}$$
$29$ $$( 5 + T )^{4}$$
$31$ $$13689 - 117 T^{2} + T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 52 + T^{2} )^{2}$$
$43$ $$( -9 + T )^{4}$$
$47$ $$169 - 13 T^{2} + T^{4}$$
$53$ $$( 121 + 11 T + T^{2} )^{2}$$
$59$ $$43264 - 208 T^{2} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$13689 - 117 T^{2} + T^{4}$$
$79$ $$( 225 + 15 T + T^{2} )^{2}$$
$83$ $$( 325 + T^{2} )^{2}$$
$89$ $$169 - 13 T^{2} + T^{4}$$
$97$ $$( 325 + T^{2} )^{2}$$