# Properties

 Label 637.2.q.f Level $637$ Weight $2$ Character orbit 637.q 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.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{6} + ( 1 - 2 \beta_{2} ) q^{8} + ( -3 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} ) q^{6} + ( 1 - 2 \beta_{2} ) q^{8} + ( -3 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{10} + ( -3 + 2 \beta_{2} - \beta_{3} ) q^{11} -5 q^{12} + ( -3 - \beta_{2} ) q^{13} + ( -3 + \beta_{2} + \beta_{3} ) q^{15} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 3 - 3 \beta_{2} ) q^{17} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{18} + ( 3 - 3 \beta_{1} ) q^{19} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{20} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{22} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 1 - 2 \beta_{2} + 3 \beta_{3} ) q^{24} + ( 2 + \beta_{1} + \beta_{3} ) q^{25} + ( -4 + \beta_{1} + 3 \beta_{3} ) q^{26} -5 q^{27} + ( -1 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{29} + ( -3 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{30} + ( -5 + 10 \beta_{2} ) q^{31} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{32} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{33} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{34} + ( 2 - 4 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{36} + ( 8 - 4 \beta_{2} ) q^{37} + ( 9 - 3 \beta_{1} - 3 \beta_{3} ) q^{38} + ( 4 - 7 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{39} + ( -2 + \beta_{1} + \beta_{3} ) q^{40} + ( -6 + 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -4 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 3 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -\beta_{1} - \beta_{2} ) q^{45} + ( -8 + 6 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 5 - 2 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 6 - \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{48} + ( -1 + \beta_{2} - \beta_{3} ) q^{50} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{51} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} ) q^{52} + ( -5 + 4 \beta_{1} + 4 \beta_{3} ) q^{53} + ( -5 + 5 \beta_{3} ) q^{54} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} + ( 3 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{57} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{58} + ( 3 - 8 \beta_{1} - 5 \beta_{2} ) q^{59} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{60} + ( -4 + 6 \beta_{1} + 10 \beta_{2} - 12 \beta_{3} ) q^{61} + ( 5 - 10 \beta_{1} + 5 \beta_{3} ) q^{62} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( -1 - 3 \beta_{1} + 4 \beta_{3} ) q^{65} + 5 q^{66} + ( 4 + \beta_{2} - 6 \beta_{3} ) q^{67} + ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{68} + ( -8 - 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 1 + 3 \beta_{1} + 4 \beta_{2} ) q^{72} + ( 2 - 4 \beta_{2} ) q^{73} + ( 4 + 4 \beta_{1} - 8 \beta_{3} ) q^{74} + ( -3 + 6 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 12 - 3 \beta_{2} - 6 \beta_{3} ) q^{76} + ( 5 - 4 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{78} -6 q^{79} + ( 3 - \beta_{2} - \beta_{3} ) q^{80} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{82} + ( 3 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -3 + 3 \beta_{1} ) q^{85} + ( 3 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -10 + 5 \beta_{1} + 15 \beta_{2} - 10 \beta_{3} ) q^{87} + ( -1 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{88} + ( 11 - 8 \beta_{2} + 5 \beta_{3} ) q^{89} + q^{90} + ( -14 + 4 \beta_{1} + 4 \beta_{3} ) q^{92} + ( -5 + 10 \beta_{2} - 15 \beta_{3} ) q^{93} + ( -3 + 6 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{94} + ( 6 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{95} + ( -5 + 10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{96} + ( 8 - 3 \beta_{1} + 5 \beta_{2} ) q^{97} + ( 9 - 7 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} + q^{3} + q^{4} - 9q^{6} - 5q^{9} + O(q^{10})$$ $$4q + 3q^{2} + q^{3} + q^{4} - 9q^{6} - 5q^{9} - 5q^{10} - 9q^{11} - 20q^{12} - 14q^{13} - 9q^{15} - q^{16} + 6q^{17} + 9q^{19} - 12q^{20} - q^{22} - 6q^{23} + 3q^{24} + 10q^{25} - 12q^{26} - 20q^{27} + 9q^{29} - 8q^{30} + 21q^{32} - 15q^{33} - 8q^{36} + 24q^{37} + 30q^{38} - 2q^{39} - 6q^{40} - 18q^{41} - 5q^{43} - 3q^{45} - 30q^{46} + 11q^{48} - 3q^{50} + 6q^{51} - 5q^{52} - 12q^{53} - 15q^{54} + q^{55} + 3q^{58} - 6q^{59} - 2q^{61} + 15q^{62} + 32q^{64} - 3q^{65} + 20q^{66} + 12q^{67} - 3q^{68} - 18q^{69} + 6q^{71} + 15q^{72} + 12q^{74} + 13q^{75} + 36q^{76} + 27q^{78} - 24q^{79} + 9q^{80} + 10q^{81} - 2q^{82} - 9q^{85} - 15q^{87} + 9q^{88} + 33q^{89} + 4q^{90} - 48q^{92} - 15q^{93} - 5q^{94} + 15q^{95} + 39q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 1.39564 + 0.228425i −0.895644 − 1.09445i 1.39564 − 0.228425i −0.895644 + 1.09445i
−0.395644 + 0.228425i 1.39564 + 2.41733i −0.895644 + 1.55130i 0.456850i −1.10436 0.637600i 0 1.73205i −2.39564 + 4.14938i −0.104356 0.180750i
491.2 1.89564 1.09445i −0.895644 1.55130i 1.39564 2.41733i 2.18890i −3.39564 1.96048i 0 1.73205i −0.104356 + 0.180750i −2.39564 4.14938i
589.1 −0.395644 0.228425i 1.39564 2.41733i −0.895644 1.55130i 0.456850i −1.10436 + 0.637600i 0 1.73205i −2.39564 4.14938i −0.104356 + 0.180750i
589.2 1.89564 + 1.09445i −0.895644 + 1.55130i 1.39564 + 2.41733i 2.18890i −3.39564 + 1.96048i 0 1.73205i −0.104356 0.180750i −2.39564 + 4.14938i
 $$n$$: 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
13.e 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.q.f yes 4
7.b odd 2 1 637.2.q.e 4
7.c even 3 1 637.2.k.f 4
7.c even 3 1 637.2.u.d 4
7.d odd 6 1 637.2.k.d 4
7.d odd 6 1 637.2.u.e 4
13.e even 6 1 inner 637.2.q.f yes 4
13.f odd 12 2 8281.2.a.bq 4
91.k even 6 1 637.2.u.d 4
91.l odd 6 1 637.2.u.e 4
91.p odd 6 1 637.2.k.d 4
91.t odd 6 1 637.2.q.e 4
91.u even 6 1 637.2.k.f 4
91.bc even 12 2 8281.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.d 4 7.d odd 6 1
637.2.k.d 4 91.p odd 6 1
637.2.k.f 4 7.c even 3 1
637.2.k.f 4 91.u even 6 1
637.2.q.e 4 7.b odd 2 1
637.2.q.e 4 91.t odd 6 1
637.2.q.f yes 4 1.a even 1 1 trivial
637.2.q.f yes 4 13.e even 6 1 inner
637.2.u.d 4 7.c even 3 1
637.2.u.d 4 91.k even 6 1
637.2.u.e 4 7.d odd 6 1
637.2.u.e 4 91.l odd 6 1
8281.2.a.bq 4 13.f odd 12 2
8281.2.a.bs 4 91.bc even 12 2

## Hecke kernels

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

 $$T_{2}^{4} - 3 T_{2}^{3} + 2 T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{4} - T_{3}^{3} + 6 T_{3}^{2} + 5 T_{3} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$25 + 5 T + 6 T^{2} - T^{3} + T^{4}$$
$5$ $$1 + 5 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$25 + 45 T + 32 T^{2} + 9 T^{3} + T^{4}$$
$13$ $$( 13 + 7 T + T^{2} )^{2}$$
$17$ $$( 9 - 3 T + T^{2} )^{2}$$
$19$ $$81 + 81 T + 18 T^{2} - 9 T^{3} + T^{4}$$
$23$ $$144 - 72 T + 48 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$225 - 135 T + 66 T^{2} - 9 T^{3} + T^{4}$$
$31$ $$( 75 + T^{2} )^{2}$$
$37$ $$( 48 - 12 T + T^{2} )^{2}$$
$41$ $$400 + 360 T + 128 T^{2} + 18 T^{3} + T^{4}$$
$43$ $$1681 - 205 T + 66 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$1681 + 110 T^{2} + T^{4}$$
$53$ $$( -75 + 6 T + T^{2} )^{2}$$
$59$ $$11881 - 654 T - 97 T^{2} + 6 T^{3} + T^{4}$$
$61$ $$35344 - 376 T + 192 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$2601 + 612 T - 3 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$16 + 24 T + 8 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$( 12 + T^{2} )^{2}$$
$79$ $$( 6 + T )^{4}$$
$83$ $$625 + 62 T^{2} + T^{4}$$
$89$ $$2209 - 1551 T + 410 T^{2} - 33 T^{3} + T^{4}$$
$97$ $$12321 - 4329 T + 618 T^{2} - 39 T^{3} + T^{4}$$