# Properties

 Label 637.2.e.g Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{5} + 2 q^{6} -2 \beta_{3} q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{5} + 2 q^{6} -2 \beta_{3} q^{8} + ( 1 + \beta_{2} ) q^{9} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{10} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{11} + q^{13} + ( -2 - 3 \beta_{3} ) q^{15} + ( 4 + 4 \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{3} ) q^{18} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{19} -6 q^{22} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{23} -4 \beta_{2} q^{24} + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{25} + \beta_{1} q^{26} -4 \beta_{3} q^{27} + ( 3 + 2 \beta_{3} ) q^{29} + ( 6 - 2 \beta_{1} + 6 \beta_{2} ) q^{30} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{31} + ( 6 + 6 \beta_{2} ) q^{33} + 2 q^{34} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -\beta_{1} - \beta_{3} ) q^{39} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{40} + ( -6 + 2 \beta_{3} ) q^{41} -5 q^{43} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{45} + ( 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{46} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{47} -4 \beta_{3} q^{48} + ( 12 + 6 \beta_{3} ) q^{50} + ( -2 - 2 \beta_{2} ) q^{51} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 8 + 8 \beta_{2} ) q^{54} + ( 6 + 9 \beta_{3} ) q^{55} + ( -6 + 3 \beta_{3} ) q^{57} + ( -4 + 3 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{59} + ( 6 + 6 \beta_{2} ) q^{61} + ( 6 + \beta_{3} ) q^{62} + 8 q^{64} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{65} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{66} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{67} + ( 4 - 3 \beta_{3} ) q^{69} + ( -6 + 5 \beta_{3} ) q^{71} + 2 \beta_{1} q^{72} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{73} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -12 + 6 \beta_{1} - 12 \beta_{2} ) q^{75} + 2 q^{78} + ( -7 - 6 \beta_{1} - 7 \beta_{2} ) q^{79} + ( -4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{80} -5 \beta_{2} q^{81} + ( -4 - 6 \beta_{1} - 4 \beta_{2} ) q^{82} + ( -9 + 3 \beta_{3} ) q^{83} + ( -2 - 3 \beta_{3} ) q^{85} -5 \beta_{1} q^{86} + ( -3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{87} + 12 \beta_{2} q^{88} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{89} + ( 2 + 3 \beta_{3} ) q^{90} + ( -6 + \beta_{1} - 6 \beta_{2} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{94} + ( -6 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{95} + ( 1 + 9 \beta_{3} ) q^{97} + 3 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} + 8q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 6q^{5} + 8q^{6} + 2q^{9} + 4q^{10} + 4q^{13} - 8q^{15} + 8q^{16} - 6q^{19} - 24q^{22} + 6q^{23} + 8q^{24} - 12q^{25} + 12q^{29} + 12q^{30} - 2q^{31} + 12q^{33} + 8q^{34} + 4q^{37} + 12q^{38} - 8q^{40} - 24q^{41} - 20q^{43} - 6q^{45} - 8q^{46} + 6q^{47} + 48q^{50} - 4q^{51} + 6q^{53} + 16q^{54} + 24q^{55} - 24q^{57} - 8q^{58} + 12q^{59} + 12q^{61} + 24q^{62} + 32q^{64} + 6q^{65} + 12q^{67} + 16q^{69} - 24q^{71} - 10q^{73} + 12q^{74} - 24q^{75} + 8q^{78} - 14q^{79} - 24q^{80} + 10q^{81} - 8q^{82} - 36q^{83} - 8q^{85} - 8q^{87} - 24q^{88} + 6q^{89} + 8q^{90} - 12q^{93} + 4q^{94} + 6q^{95} + 4q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i −0.707107 1.22474i 0 2.20711 3.82282i 2.00000 0 −2.82843 0.500000 0.866025i 3.12132 + 5.40629i
79.2 0.707107 1.22474i 0.707107 + 1.22474i 0 0.792893 1.37333i 2.00000 0 2.82843 0.500000 0.866025i −1.12132 1.94218i
508.1 −0.707107 1.22474i −0.707107 + 1.22474i 0 2.20711 + 3.82282i 2.00000 0 −2.82843 0.500000 + 0.866025i 3.12132 5.40629i
508.2 0.707107 + 1.22474i 0.707107 1.22474i 0 0.792893 + 1.37333i 2.00000 0 2.82843 0.500000 + 0.866025i −1.12132 + 1.94218i
 $$n$$: 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.g 4
7.b odd 2 1 637.2.e.f 4
7.c even 3 1 637.2.a.g 2
7.c even 3 1 inner 637.2.e.g 4
7.d odd 6 1 91.2.a.c 2
7.d odd 6 1 637.2.e.f 4
21.g even 6 1 819.2.a.h 2
21.h odd 6 1 5733.2.a.s 2
28.f even 6 1 1456.2.a.q 2
35.i odd 6 1 2275.2.a.j 2
56.j odd 6 1 5824.2.a.bl 2
56.m even 6 1 5824.2.a.bk 2
91.r even 6 1 8281.2.a.v 2
91.s odd 6 1 1183.2.a.d 2
91.bb even 12 2 1183.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 7.d odd 6 1
637.2.a.g 2 7.c even 3 1
637.2.e.f 4 7.b odd 2 1
637.2.e.f 4 7.d odd 6 1
637.2.e.g 4 1.a even 1 1 trivial
637.2.e.g 4 7.c even 3 1 inner
819.2.a.h 2 21.g even 6 1
1183.2.a.d 2 91.s odd 6 1
1183.2.c.d 4 91.bb even 12 2
1456.2.a.q 2 28.f even 6 1
2275.2.a.j 2 35.i odd 6 1
5733.2.a.s 2 21.h odd 6 1
5824.2.a.bk 2 56.m even 6 1
5824.2.a.bl 2 56.j odd 6 1
8281.2.a.v 2 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{2} + 4$$ $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}^{4} - 6 T_{5}^{3} + 29 T_{5}^{2} - 42 T_{5} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$4 + 2 T^{2} + T^{4}$$
$5$ $$49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$324 + 18 T^{2} + T^{4}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$( 1 - 6 T + T^{2} )^{2}$$
$31$ $$289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4}$$
$37$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( 28 + 12 T + T^{2} )^{2}$$
$43$ $$( 5 + T )^{4}$$
$47$ $$49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$( 36 - 6 T + T^{2} )^{2}$$
$67$ $$1296 + 432 T + 180 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$( -14 + 12 T + T^{2} )^{2}$$
$73$ $$49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$529 - 322 T + 219 T^{2} + 14 T^{3} + T^{4}$$
$83$ $$( 63 + 18 T + T^{2} )^{2}$$
$89$ $$49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$( -161 - 2 T + T^{2} )^{2}$$