# Properties

 Label 637.2.r.c Level $637$ Weight $2$ Character orbit 637.r Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + 4 \zeta_{12}^{3} q^{6} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} + \zeta_{12} q^{5} + 4 \zeta_{12}^{3} q^{6} + ( -1 + \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12}^{2} q^{10} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( -4 + 4 \zeta_{12}^{2} ) q^{12} + ( -2 + 3 \zeta_{12}^{3} ) q^{13} + 2 \zeta_{12}^{3} q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + 6 \zeta_{12}^{2} q^{17} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{18} -3 \zeta_{12} q^{19} + 2 \zeta_{12}^{3} q^{20} -4 q^{22} + ( 3 - 3 \zeta_{12}^{2} ) q^{23} -4 \zeta_{12}^{2} q^{25} + ( -6 - 4 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{26} + 4 q^{27} + 3 q^{29} + ( -4 + 4 \zeta_{12}^{2} ) q^{30} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -4 \zeta_{12} q^{33} + 12 \zeta_{12}^{3} q^{34} -2 q^{36} -6 \zeta_{12} q^{37} -6 \zeta_{12}^{2} q^{38} + ( -6 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{39} -10 \zeta_{12}^{3} q^{41} + q^{43} -4 \zeta_{12} q^{44} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{45} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{46} -11 \zeta_{12} q^{47} + 8 q^{48} -8 \zeta_{12}^{3} q^{50} + ( -12 + 12 \zeta_{12}^{2} ) q^{51} + ( -6 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{52} + 9 \zeta_{12}^{2} q^{53} + 8 \zeta_{12} q^{54} -2 q^{55} -6 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{59} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( 8 - 8 \zeta_{12}^{2} ) q^{61} + 6 q^{62} + 8 q^{64} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{65} -8 \zeta_{12}^{2} q^{66} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -12 + 12 \zeta_{12}^{2} ) q^{68} + 6 q^{69} + 14 \zeta_{12}^{3} q^{71} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{73} -12 \zeta_{12}^{2} q^{74} + ( 8 - 8 \zeta_{12}^{2} ) q^{75} -6 \zeta_{12}^{3} q^{76} + ( -12 - 8 \zeta_{12}^{3} ) q^{78} + ( 9 - 9 \zeta_{12}^{2} ) q^{79} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{80} + 11 \zeta_{12}^{2} q^{81} + ( 20 - 20 \zeta_{12}^{2} ) q^{82} -11 \zeta_{12}^{3} q^{83} + 6 \zeta_{12}^{3} q^{85} + 2 \zeta_{12} q^{86} + 6 \zeta_{12}^{2} q^{87} + 5 \zeta_{12} q^{89} -2 q^{90} + 6 q^{92} + 6 \zeta_{12} q^{93} -22 \zeta_{12}^{2} q^{94} -3 \zeta_{12}^{2} q^{95} + 16 \zeta_{12} q^{96} + 9 \zeta_{12}^{3} q^{97} -2 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 4q^{4} - 2q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 4q^{4} - 2q^{9} + 4q^{10} - 8q^{12} - 8q^{13} + 8q^{16} + 12q^{17} - 16q^{22} + 6q^{23} - 8q^{25} - 12q^{26} + 16q^{27} + 12q^{29} - 8q^{30} - 8q^{36} - 12q^{38} - 8q^{39} + 4q^{43} + 32q^{48} - 24q^{51} - 8q^{52} + 18q^{53} - 8q^{55} + 16q^{61} + 24q^{62} + 32q^{64} - 6q^{65} - 16q^{66} - 24q^{68} + 24q^{69} - 24q^{74} + 16q^{75} - 48q^{78} + 18q^{79} + 22q^{81} + 40q^{82} + 12q^{87} - 8q^{90} + 24q^{92} - 44q^{94} - 6q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$

## 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
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i 1.00000 1.73205i 1.00000 1.73205i −0.866025 + 0.500000i 4.00000i 0 0 −0.500000 0.866025i 1.00000 1.73205i
116.2 1.73205 1.00000i 1.00000 1.73205i 1.00000 1.73205i 0.866025 0.500000i 4.00000i 0 0 −0.500000 0.866025i 1.00000 1.73205i
324.1 −1.73205 1.00000i 1.00000 + 1.73205i 1.00000 + 1.73205i −0.866025 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
324.2 1.73205 + 1.00000i 1.00000 + 1.73205i 1.00000 + 1.73205i 0.866025 + 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
 $$n$$: 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
13.b even 2 1 inner
91.r 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.r.c 4
7.b odd 2 1 637.2.r.a 4
7.c even 3 1 637.2.c.a 2
7.c even 3 1 inner 637.2.r.c 4
7.d odd 6 1 637.2.c.c yes 2
7.d odd 6 1 637.2.r.a 4
13.b even 2 1 inner 637.2.r.c 4
91.b odd 2 1 637.2.r.a 4
91.r even 6 1 637.2.c.a 2
91.r even 6 1 inner 637.2.r.c 4
91.s odd 6 1 637.2.c.c yes 2
91.s odd 6 1 637.2.r.a 4
91.z odd 12 1 8281.2.a.a 1
91.z odd 12 1 8281.2.a.k 1
91.bb even 12 1 8281.2.a.b 1
91.bb even 12 1 8281.2.a.m 1

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

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ $$T_{3}^{2} - 2 T_{3} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} + T^{4}$$
$3$ $$( 4 - 2 T + T^{2} )^{2}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$16 - 4 T^{2} + T^{4}$$
$13$ $$( 13 + 4 T + T^{2} )^{2}$$
$17$ $$( 36 - 6 T + T^{2} )^{2}$$
$19$ $$81 - 9 T^{2} + T^{4}$$
$23$ $$( 9 - 3 T + T^{2} )^{2}$$
$29$ $$( -3 + T )^{4}$$
$31$ $$81 - 9 T^{2} + T^{4}$$
$37$ $$1296 - 36 T^{2} + T^{4}$$
$41$ $$( 100 + T^{2} )^{2}$$
$43$ $$( -1 + T )^{4}$$
$47$ $$14641 - 121 T^{2} + T^{4}$$
$53$ $$( 81 - 9 T + T^{2} )^{2}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$( 64 - 8 T + T^{2} )^{2}$$
$67$ $$20736 - 144 T^{2} + T^{4}$$
$71$ $$( 196 + T^{2} )^{2}$$
$73$ $$6561 - 81 T^{2} + T^{4}$$
$79$ $$( 81 - 9 T + T^{2} )^{2}$$
$83$ $$( 121 + T^{2} )^{2}$$
$89$ $$625 - 25 T^{2} + T^{4}$$
$97$ $$( 81 + T^{2} )^{2}$$