# Properties

 Label 637.2.e.h 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 3 \beta_{2} ) q^{6} + ( -1 + 4 \beta_{2} ) q^{8} + ( -2 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 3 \beta_{2} ) q^{6} + ( -1 + 4 \beta_{2} ) q^{8} + ( -2 - 2 \beta_{3} ) q^{9} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{10} -3 \beta_{3} q^{11} + ( -6 - 3 \beta_{1} - 6 \beta_{3} ) q^{12} + q^{13} -5 q^{15} + ( -5 - 3 \beta_{1} - 5 \beta_{3} ) q^{16} + ( -4 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{18} + ( 3 + 3 \beta_{3} ) q^{19} + ( -6 + 3 \beta_{2} ) q^{20} + ( 3 - 3 \beta_{2} ) q^{22} + ( 5 + 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -6 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} ) q^{24} + ( 1 + \beta_{1} + \beta_{3} ) q^{26} + ( 1 + 2 \beta_{2} ) q^{27} + ( -2 - 4 \beta_{2} ) q^{29} + ( -5 - 5 \beta_{1} - 5 \beta_{3} ) q^{30} -5 \beta_{3} q^{31} + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{32} + ( -3 + 6 \beta_{1} - 3 \beta_{3} ) q^{33} + ( -1 - 3 \beta_{2} ) q^{34} -6 \beta_{2} q^{36} + ( 5 - 6 \beta_{1} + 5 \beta_{3} ) q^{37} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{38} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{39} + ( -7 - 6 \beta_{1} - 7 \beta_{3} ) q^{40} + ( -2 - 4 \beta_{2} ) q^{41} -8 q^{43} + 9 \beta_{1} q^{44} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{45} + ( 9 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} ) q^{46} + ( -1 - 4 \beta_{1} - \beta_{3} ) q^{47} + ( 1 - 13 \beta_{2} ) q^{48} + ( 13 - 6 \beta_{1} + 13 \beta_{3} ) q^{51} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{52} + ( -4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{54} + ( -3 - 6 \beta_{2} ) q^{55} + ( 3 + 6 \beta_{2} ) q^{57} + ( 2 + 6 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{59} + ( -15 \beta_{1} - 15 \beta_{2} ) q^{60} + ( 3 + 3 \beta_{3} ) q^{61} + ( 5 - 5 \beta_{2} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{65} + ( 9 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{66} -3 \beta_{3} q^{67} + ( 12 - 3 \beta_{1} + 12 \beta_{3} ) q^{68} + ( 1 + 12 \beta_{2} ) q^{69} + ( 4 + 8 \beta_{2} ) q^{71} + ( 2 + 8 \beta_{1} + 2 \beta_{3} ) q^{72} + ( 6 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{73} + ( -7 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{74} + 9 \beta_{2} q^{76} + ( -1 + 3 \beta_{2} ) q^{78} + ( -7 + 6 \beta_{1} - 7 \beta_{3} ) q^{79} + ( -13 \beta_{1} - 13 \beta_{2} - \beta_{3} ) q^{80} -11 \beta_{3} q^{81} + ( 2 + 6 \beta_{1} + 2 \beta_{3} ) q^{82} + ( 13 + 6 \beta_{2} ) q^{85} + ( -8 - 8 \beta_{1} - 8 \beta_{3} ) q^{86} + 10 \beta_{3} q^{87} + ( 12 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{88} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{89} + ( 2 - 6 \beta_{2} ) q^{90} + ( -6 + 21 \beta_{2} ) q^{92} + ( -5 + 10 \beta_{1} - 5 \beta_{3} ) q^{93} + ( -9 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} ) q^{94} + ( 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{95} + 15 \beta_{1} q^{96} + ( 10 + 12 \beta_{2} ) q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - 3q^{4} - 10q^{6} - 12q^{8} - 4q^{9} + O(q^{10})$$ $$4q + 3q^{2} - 3q^{4} - 10q^{6} - 12q^{8} - 4q^{9} - 5q^{10} + 6q^{11} - 15q^{12} + 4q^{13} - 20q^{15} - 13q^{16} - 6q^{17} + 6q^{18} + 6q^{19} - 30q^{20} + 18q^{22} + 12q^{23} + 20q^{24} + 3q^{26} - 15q^{30} + 10q^{31} + 15q^{32} + 2q^{34} + 12q^{36} + 4q^{37} - 9q^{38} - 20q^{40} - 32q^{43} + 9q^{44} - 23q^{46} - 6q^{47} + 30q^{48} + 20q^{51} - 3q^{52} + 6q^{53} - 5q^{54} + 10q^{58} + 6q^{59} + 15q^{60} + 6q^{61} + 30q^{62} + 8q^{64} - 15q^{66} + 6q^{67} + 21q^{68} - 20q^{69} + 12q^{72} + 8q^{73} + 9q^{74} - 18q^{76} - 10q^{78} - 8q^{79} + 15q^{80} + 22q^{81} + 10q^{82} + 40q^{85} - 24q^{86} - 20q^{87} - 18q^{88} + 20q^{90} - 66q^{92} + 19q^{94} + 15q^{96} + 16q^{97} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$

## 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
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
0.190983 0.330792i 1.11803 + 1.93649i 0.927051 + 1.60570i −1.11803 + 1.93649i 0.854102 0 1.47214 −1.00000 + 1.73205i 0.427051 + 0.739674i
79.2 1.30902 2.26728i −1.11803 1.93649i −2.42705 4.20378i 1.11803 1.93649i −5.85410 0 −7.47214 −1.00000 + 1.73205i −2.92705 5.06980i
508.1 0.190983 + 0.330792i 1.11803 1.93649i 0.927051 1.60570i −1.11803 1.93649i 0.854102 0 1.47214 −1.00000 1.73205i 0.427051 0.739674i
508.2 1.30902 + 2.26728i −1.11803 + 1.93649i −2.42705 + 4.20378i 1.11803 + 1.93649i −5.85410 0 −7.47214 −1.00000 1.73205i −2.92705 + 5.06980i
 $$n$$: 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.h 4
7.b odd 2 1 91.2.e.b 4
7.c even 3 1 637.2.a.e 2
7.c even 3 1 inner 637.2.e.h 4
7.d odd 6 1 91.2.e.b 4
7.d odd 6 1 637.2.a.f 2
21.c even 2 1 819.2.j.c 4
21.g even 6 1 819.2.j.c 4
21.g even 6 1 5733.2.a.v 2
21.h odd 6 1 5733.2.a.w 2
28.d even 2 1 1456.2.r.j 4
28.f even 6 1 1456.2.r.j 4
91.b odd 2 1 1183.2.e.d 4
91.r even 6 1 8281.2.a.ba 2
91.s odd 6 1 1183.2.e.d 4
91.s odd 6 1 8281.2.a.z 2

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

 $$T_{2}^{4} - 3 T_{2}^{3} + 8 T_{2}^{2} - 3 T_{2} + 1$$ $$T_{3}^{4} + 5 T_{3}^{2} + 25$$ $$T_{5}^{4} + 5 T_{5}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$25 + 5 T^{2} + T^{4}$$
$5$ $$25 + 5 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$( 9 - 3 T + T^{2} )^{2}$$
$23$ $$961 - 372 T + 113 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$( 25 - 5 T + T^{2} )^{2}$$
$37$ $$1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( 9 - 3 T + T^{2} )^{2}$$
$67$ $$( 9 - 3 T + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$841 + 232 T + 93 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$25 + 5 T^{2} + T^{4}$$
$97$ $$( -164 - 8 T + T^{2} )^{2}$$