# Properties

 Label 637.2.c.d Level $637$ Weight $2$ Character orbit 637.c Analytic conductor $5.086$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{4} ) q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{2} ) q^{4} + ( \beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{6} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{8} + ( -\beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{4} ) q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{2} ) q^{4} + ( \beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{6} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{8} + ( -\beta_{1} + \beta_{2} ) q^{9} -\beta_{1} q^{10} + ( 3 \beta_{3} - \beta_{4} ) q^{11} + ( 4 + 2 \beta_{2} ) q^{12} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{13} + ( 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{15} + ( 2 + 2 \beta_{2} ) q^{16} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( -\beta_{3} - \beta_{4} ) q^{18} + ( \beta_{4} - 4 \beta_{5} ) q^{19} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{20} + ( 5 + \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{23} + ( -6 \beta_{3} + 2 \beta_{4} ) q^{24} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{25} + ( -2 - \beta_{1} + 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{26} + ( 2 + 2 \beta_{1} ) q^{27} + ( -3 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 3 - \beta_{1} + \beta_{2} ) q^{30} + ( 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( -2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{33} + ( -5 \beta_{3} + \beta_{4} + \beta_{5} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} ) q^{36} + ( 3 \beta_{3} + 3 \beta_{4} ) q^{37} + ( 2 + 3 \beta_{1} + 4 \beta_{2} ) q^{38} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{39} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{40} + ( 2 \beta_{3} - 4 \beta_{4} ) q^{41} + ( -5 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -6 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{44} + ( 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{45} + ( -7 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{46} + ( -6 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{47} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{48} + ( -2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{50} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{51} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{52} + ( -\beta_{1} - \beta_{2} ) q^{53} + ( -4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{54} + ( -2 + 3 \beta_{1} ) q^{55} + ( -11 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{57} + ( 9 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{58} + ( -\beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{59} + ( 2 \beta_{3} + 4 \beta_{5} ) q^{60} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -2 - 3 \beta_{1} + 2 \beta_{2} ) q^{62} + 4 \beta_{2} q^{64} + ( -1 - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{65} + ( -6 - 4 \beta_{1} - 4 \beta_{2} ) q^{66} + ( 4 \beta_{3} - 2 \beta_{5} ) q^{67} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{69} + ( -3 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -4 \beta_{3} - 2 \beta_{4} ) q^{72} + ( 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( -13 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{76} + ( 3 + \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{78} + ( 5 + 4 \beta_{1} + 2 \beta_{2} ) q^{79} + 2 \beta_{5} q^{80} + ( -6 + 3 \beta_{1} - 5 \beta_{2} ) q^{81} + ( 10 + 4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -\beta_{3} + 4 \beta_{4} - 5 \beta_{5} ) q^{83} + ( 3 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{85} + ( 7 \beta_{3} - 3 \beta_{4} ) q^{86} + ( 8 + \beta_{1} + 4 \beta_{2} ) q^{87} + ( -8 - 6 \beta_{1} - 4 \beta_{2} ) q^{88} + ( -3 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 2 - \beta_{1} ) q^{90} + ( -7 - 5 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 5 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{93} + ( -6 - \beta_{1} - 8 \beta_{2} ) q^{94} + ( 11 - 8 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -4 \beta_{3} - 4 \beta_{5} ) q^{96} + ( 8 \beta_{3} - 9 \beta_{4} + 2 \beta_{5} ) q^{97} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 4q^{4} - 2q^{9} + O(q^{10})$$ $$6q - 4q^{4} - 2q^{9} + 20q^{12} - 8q^{13} + 8q^{16} + 8q^{17} + 24q^{22} + 6q^{23} + 8q^{25} - 12q^{26} + 12q^{27} - 14q^{29} + 16q^{30} + 4q^{36} + 4q^{38} - 8q^{39} + 20q^{40} - 26q^{43} - 8q^{48} + 8q^{51} + 20q^{52} + 2q^{53} - 12q^{55} - 28q^{61} - 16q^{62} - 8q^{64} - 6q^{65} - 28q^{66} - 20q^{68} + 4q^{69} - 24q^{74} - 44q^{75} + 16q^{78} + 26q^{79} - 26q^{81} + 56q^{82} + 40q^{87} - 40q^{88} + 12q^{90} - 36q^{92} - 20q^{94} + 58q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
246.1
 0.403032 + 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 − 0.403032i
2.48119i −1.67513 −4.15633 0.675131i 4.15633i 0 5.35026i −0.193937 −1.67513
246.2 1.17009i −0.539189 0.630898 0.460811i 0.630898i 0 3.07838i −2.70928 −0.539189
246.3 0.688892i 2.21432 1.52543 3.21432i 1.52543i 0 2.42864i 1.90321 2.21432
246.4 0.688892i 2.21432 1.52543 3.21432i 1.52543i 0 2.42864i 1.90321 2.21432
246.5 1.17009i −0.539189 0.630898 0.460811i 0.630898i 0 3.07838i −2.70928 −0.539189
246.6 2.48119i −1.67513 −4.15633 0.675131i 4.15633i 0 5.35026i −0.193937 −1.67513
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 246.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.d 6
7.b odd 2 1 91.2.c.a 6
7.c even 3 2 637.2.r.d 12
7.d odd 6 2 637.2.r.e 12
13.b even 2 1 inner 637.2.c.d 6
13.d odd 4 1 8281.2.a.be 3
13.d odd 4 1 8281.2.a.bi 3
21.c even 2 1 819.2.c.b 6
28.d even 2 1 1456.2.k.c 6
91.b odd 2 1 91.2.c.a 6
91.i even 4 1 1183.2.a.h 3
91.i even 4 1 1183.2.a.j 3
91.r even 6 2 637.2.r.d 12
91.s odd 6 2 637.2.r.e 12
273.g even 2 1 819.2.c.b 6
364.h even 2 1 1456.2.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 7.b odd 2 1
91.2.c.a 6 91.b odd 2 1
637.2.c.d 6 1.a even 1 1 trivial
637.2.c.d 6 13.b even 2 1 inner
637.2.r.d 12 7.c even 3 2
637.2.r.d 12 91.r even 6 2
637.2.r.e 12 7.d odd 6 2
637.2.r.e 12 91.s odd 6 2
819.2.c.b 6 21.c even 2 1
819.2.c.b 6 273.g even 2 1
1183.2.a.h 3 91.i even 4 1
1183.2.a.j 3 91.i even 4 1
1456.2.k.c 6 28.d even 2 1
1456.2.k.c 6 364.h even 2 1
8281.2.a.be 3 13.d odd 4 1
8281.2.a.bi 3 13.d odd 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 8 T_{2}^{4} + 12 T_{2}^{2} + 4$$ $$T_{3}^{3} - 4 T_{3} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 12 T^{2} + 8 T^{4} + T^{6}$$
$3$ $$( -2 - 4 T + T^{3} )^{2}$$
$5$ $$1 + 7 T^{2} + 11 T^{4} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$100 + 164 T^{2} + 28 T^{4} + T^{6}$$
$13$ $$2197 + 1352 T + 91 T^{2} - 64 T^{3} + 7 T^{4} + 8 T^{5} + T^{6}$$
$17$ $$( 34 - 8 T - 4 T^{2} + T^{3} )^{2}$$
$19$ $$37249 + 3867 T^{2} + 119 T^{4} + T^{6}$$
$23$ $$( 79 - 25 T - 3 T^{2} + T^{3} )^{2}$$
$29$ $$( 5 - 21 T + 7 T^{2} + T^{3} )^{2}$$
$31$ $$4225 + 1791 T^{2} + 83 T^{4} + T^{6}$$
$37$ $$2916 + 1620 T^{2} + 108 T^{4} + T^{6}$$
$41$ $$1600 + 2864 T^{2} + 108 T^{4} + T^{6}$$
$43$ $$( -17 + 35 T + 13 T^{2} + T^{3} )^{2}$$
$47$ $$18769 + 5819 T^{2} + 151 T^{4} + T^{6}$$
$53$ $$( 13 - 9 T - T^{2} + T^{3} )^{2}$$
$59$ $$2704 + 816 T^{2} + 68 T^{4} + T^{6}$$
$61$ $$( -152 + 28 T + 14 T^{2} + T^{3} )^{2}$$
$67$ $$256 + 1408 T^{2} + 80 T^{4} + T^{6}$$
$71$ $$792100 + 27836 T^{2} + 304 T^{4} + T^{6}$$
$73$ $$961 + 7723 T^{2} + 263 T^{4} + T^{6}$$
$79$ $$( 185 - 37 T - 13 T^{2} + T^{3} )^{2}$$
$83$ $$26569 + 13095 T^{2} + 227 T^{4} + T^{6}$$
$89$ $$51529 + 4387 T^{2} + 119 T^{4} + T^{6}$$
$97$ $$4765489 + 96115 T^{2} + 575 T^{4} + T^{6}$$