# Properties

 Label 637.2.h.c Level $637$ Weight $2$ Character orbit 637.h 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.h (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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{3} ) q^{4} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( -3 + \beta_{3} ) q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{3} ) q^{4} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( -3 + \beta_{3} ) q^{8} + ( 1 + \beta_{2} ) q^{9} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{10} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{11} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{12} + ( 3 - \beta_{2} ) q^{13} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{15} + 3 q^{16} + ( 3 + 2 \beta_{3} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} ) q^{18} + ( 6 + 6 \beta_{2} ) q^{19} + ( -4 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} ) q^{20} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{22} -\beta_{3} q^{23} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{24} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -3 + \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{26} + 4 \beta_{3} q^{27} + ( -7 - 2 \beta_{1} - 7 \beta_{2} ) q^{29} + ( -6 - 5 \beta_{1} - 6 \beta_{2} ) q^{30} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{31} + ( 3 + \beta_{3} ) q^{32} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 + \beta_{3} ) q^{34} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{36} + ( -1 - 6 \beta_{3} ) q^{37} + ( -6 - 6 \beta_{1} - 6 \beta_{2} ) q^{38} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{39} + ( 7 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{40} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{43} + ( 5 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} ) q^{44} + ( 1 - 2 \beta_{3} ) q^{45} + ( -2 + \beta_{3} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{48} + ( 12 + 8 \beta_{1} + 12 \beta_{2} ) q^{50} + ( 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{51} + ( 3 - 2 \beta_{1} - \beta_{2} - 8 \beta_{3} ) q^{52} + ( 3 + 3 \beta_{2} ) q^{53} + ( 8 - 4 \beta_{3} ) q^{54} + ( 6 + 5 \beta_{1} + 6 \beta_{2} ) q^{55} + 6 \beta_{3} q^{57} + ( 11 + 9 \beta_{1} + 11 \beta_{2} ) q^{58} + ( 6 + 3 \beta_{3} ) q^{59} + ( 8 + 9 \beta_{1} + 8 \beta_{2} ) q^{60} + ( 7 + 2 \beta_{1} + 7 \beta_{2} ) q^{61} + ( -6 - 5 \beta_{1} - 6 \beta_{2} ) q^{62} + ( -7 + 2 \beta_{3} ) q^{64} + ( -1 - 8 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{65} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{66} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( -5 - 4 \beta_{3} ) q^{68} + 2 \beta_{2} q^{69} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{71} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{72} + ( -5 + 4 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -11 + 5 \beta_{3} ) q^{74} + ( 8 - 4 \beta_{3} ) q^{75} + ( 6 + 12 \beta_{1} + 6 \beta_{2} ) q^{76} + ( -2 - 4 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} ) q^{78} + ( 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -6 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{80} -5 \beta_{2} q^{81} + ( -7 - 5 \beta_{1} - 7 \beta_{2} ) q^{82} + ( 6 - 5 \beta_{3} ) q^{83} + ( -4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{3} ) q^{86} + ( 4 - 7 \beta_{3} ) q^{87} + ( -5 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{88} + ( -4 - 8 \beta_{3} ) q^{89} + ( -5 + 3 \beta_{3} ) q^{90} + ( 4 - \beta_{3} ) q^{92} + ( -2 + 4 \beta_{3} ) q^{93} + ( 6 \beta_{1} + 10 \beta_{2} + 6 \beta_{3} ) q^{94} + ( 6 - 12 \beta_{3} ) q^{95} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{96} + ( -2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} + 2q^{5} + 4q^{6} - 12q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} + 2q^{5} + 4q^{6} - 12q^{8} + 2q^{9} - 10q^{10} - 4q^{11} - 8q^{12} + 14q^{13} + 8q^{15} + 12q^{16} + 12q^{17} - 2q^{18} + 12q^{19} + 18q^{20} + 8q^{22} + 4q^{24} - 8q^{25} - 14q^{26} - 14q^{29} - 12q^{30} + 8q^{31} + 12q^{32} - 4q^{33} + 4q^{34} + 2q^{36} - 4q^{37} - 12q^{38} - 14q^{40} + 6q^{41} - 4q^{43} - 12q^{44} + 4q^{45} - 8q^{46} + 4q^{47} + 24q^{50} + 8q^{51} + 14q^{52} + 6q^{53} + 32q^{54} + 12q^{55} + 22q^{58} + 24q^{59} + 16q^{60} + 14q^{61} - 12q^{62} - 28q^{64} + 4q^{65} + 12q^{66} - 20q^{68} - 4q^{69} + 12q^{71} - 6q^{72} - 10q^{73} - 44q^{74} + 32q^{75} + 12q^{76} + 8q^{78} - 12q^{79} + 6q^{80} + 10q^{81} - 14q^{82} + 24q^{83} - 10q^{85} + 16q^{87} + 16q^{88} - 16q^{89} - 20q^{90} + 16q^{92} - 8q^{93} - 20q^{94} + 24q^{95} + 4q^{96} - 16q^{97} - 8q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
−2.41421 −0.707107 1.22474i 3.82843 1.91421 + 3.31552i 1.70711 + 2.95680i 0 −4.41421 0.500000 0.866025i −4.62132 8.00436i
165.2 0.414214 0.707107 + 1.22474i −1.82843 −0.914214 1.58346i 0.292893 + 0.507306i 0 −1.58579 0.500000 0.866025i −0.378680 0.655892i
471.1 −2.41421 −0.707107 + 1.22474i 3.82843 1.91421 3.31552i 1.70711 2.95680i 0 −4.41421 0.500000 + 0.866025i −4.62132 + 8.00436i
471.2 0.414214 0.707107 1.22474i −1.82843 −0.914214 + 1.58346i 0.292893 0.507306i 0 −1.58579 0.500000 + 0.866025i −0.378680 + 0.655892i
 $$n$$: 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.h 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.h.c 4
7.b odd 2 1 637.2.h.b 4
7.c even 3 1 637.2.f.e 4
7.c even 3 1 637.2.g.g 4
7.d odd 6 1 637.2.f.f yes 4
7.d odd 6 1 637.2.g.f 4
13.c even 3 1 637.2.g.g 4
91.g even 3 1 637.2.f.e 4
91.h even 3 1 inner 637.2.h.c 4
91.h even 3 1 8281.2.a.o 2
91.k even 6 1 8281.2.a.y 2
91.l odd 6 1 8281.2.a.x 2
91.m odd 6 1 637.2.f.f yes 4
91.n odd 6 1 637.2.g.f 4
91.v odd 6 1 637.2.h.b 4
91.v odd 6 1 8281.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.e 4 7.c even 3 1
637.2.f.e 4 91.g even 3 1
637.2.f.f yes 4 7.d odd 6 1
637.2.f.f yes 4 91.m odd 6 1
637.2.g.f 4 7.d odd 6 1
637.2.g.f 4 91.n odd 6 1
637.2.g.g 4 7.c even 3 1
637.2.g.g 4 13.c even 3 1
637.2.h.b 4 7.b odd 2 1
637.2.h.b 4 91.v odd 6 1
637.2.h.c 4 1.a even 1 1 trivial
637.2.h.c 4 91.h even 3 1 inner
8281.2.a.o 2 91.h even 3 1
8281.2.a.p 2 91.v odd 6 1
8281.2.a.x 2 91.l odd 6 1
8281.2.a.y 2 91.k even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}^{4} - 2 T_{5}^{3} + 11 T_{5}^{2} + 14 T_{5} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + 2 T + T^{2} )^{2}$$
$3$ $$4 + 2 T^{2} + T^{4}$$
$5$ $$49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( 13 - 7 T + T^{2} )^{2}$$
$17$ $$( 1 - 6 T + T^{2} )^{2}$$
$19$ $$( 36 - 6 T + T^{2} )^{2}$$
$23$ $$( -2 + T^{2} )^{2}$$
$29$ $$1681 + 574 T + 155 T^{2} + 14 T^{3} + T^{4}$$
$31$ $$196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$( -71 + 2 T + T^{2} )^{2}$$
$41$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$( 9 - 3 T + T^{2} )^{2}$$
$59$ $$( 18 - 12 T + T^{2} )^{2}$$
$61$ $$1681 - 574 T + 155 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$324 + 18 T^{2} + T^{4}$$
$71$ $$16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$49 - 70 T + 107 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$83$ $$( -14 - 12 T + T^{2} )^{2}$$
$89$ $$( -112 + 8 T + T^{2} )^{2}$$
$97$ $$3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4}$$