# Properties

 Label 637.2.h.f 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} -3 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 1 - 4 \beta_{2} ) q^{8} + ( 1 - 3 \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} -3 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 1 - 4 \beta_{2} ) q^{8} + ( 1 - 3 \beta_{1} + \beta_{3} ) q^{9} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 6 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{12} + ( 4 + 3 \beta_{3} ) q^{13} + ( 2 + 3 \beta_{1} + 2 \beta_{3} ) q^{15} + ( 5 - 3 \beta_{2} ) q^{16} + ( -5 - 4 \beta_{2} ) q^{17} + ( -2 - 5 \beta_{1} - 2 \beta_{3} ) q^{18} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{19} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{20} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 + 4 \beta_{2} ) q^{23} + ( 9 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} ) q^{24} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{25} + ( 4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{26} + ( -1 - 2 \beta_{2} ) q^{27} + ( 1 - 5 \beta_{1} + \beta_{3} ) q^{29} + ( 5 + 8 \beta_{1} + 5 \beta_{3} ) q^{30} + ( 5 - 6 \beta_{1} + 5 \beta_{3} ) q^{31} + ( 6 - 3 \beta_{2} ) q^{32} -3 \beta_{1} q^{33} + ( -1 - 3 \beta_{2} ) q^{34} + ( -9 - 6 \beta_{1} - 9 \beta_{3} ) q^{36} + 4 q^{37} -3 \beta_{1} q^{38} + ( -3 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{39} + ( -9 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} ) q^{40} + ( 4 - 2 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 9 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{43} + 9 \beta_{3} q^{44} + ( -2 + 5 \beta_{2} ) q^{45} + ( -2 + 6 \beta_{2} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} + ( 11 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{48} -3 \beta_{1} q^{50} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{51} + ( 9 \beta_{1} - 3 \beta_{2} ) q^{52} + ( -7 + 2 \beta_{1} - 7 \beta_{3} ) q^{53} + ( 1 - 3 \beta_{2} ) q^{54} + 3 \beta_{1} q^{55} -3 \beta_{2} q^{57} + ( -4 - 9 \beta_{1} - 4 \beta_{3} ) q^{58} + ( -1 - 2 \beta_{2} ) q^{59} + ( 9 + 15 \beta_{1} + 9 \beta_{3} ) q^{60} + ( -6 - 6 \beta_{3} ) q^{61} + ( -1 - 7 \beta_{1} - \beta_{3} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( 3 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{65} + ( -3 - 6 \beta_{1} - 3 \beta_{3} ) q^{66} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 12 + 3 \beta_{2} ) q^{68} + ( -6 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{69} + ( -10 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -11 - 11 \beta_{1} - 11 \beta_{3} ) q^{72} + ( -2 - 2 \beta_{3} ) q^{73} + ( 4 - 4 \beta_{2} ) q^{74} -3 \beta_{2} q^{75} + ( -9 - 9 \beta_{3} ) q^{76} + ( -6 + 3 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{78} + 4 \beta_{3} q^{79} + ( -11 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} ) q^{80} + ( -6 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 2 + 2 \beta_{3} ) q^{82} + ( 3 + 6 \beta_{2} ) q^{83} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{85} + ( 16 \beta_{1} + 16 \beta_{2} + 7 \beta_{3} ) q^{86} + ( 4 - 9 \beta_{2} ) q^{87} + ( 3 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{88} + ( -13 - 5 \beta_{2} ) q^{89} + ( -7 + 12 \beta_{2} ) q^{90} + ( -12 + 6 \beta_{2} ) q^{92} + ( 1 - 7 \beta_{2} ) q^{93} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{94} + 3 \beta_{2} q^{95} + ( 12 \beta_{1} + 12 \beta_{2} + 9 \beta_{3} ) q^{96} + ( -3 \beta_{1} - 3 \beta_{2} - 14 \beta_{3} ) q^{97} + ( 12 + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{2} - 3q^{3} + 6q^{4} + 3q^{5} - 7q^{6} + 12q^{8} - q^{9} + O(q^{10})$$ $$4q + 6q^{2} - 3q^{3} + 6q^{4} + 3q^{5} - 7q^{6} + 12q^{8} - q^{9} + 7q^{10} + 3q^{11} - 12q^{12} + 10q^{13} + 7q^{15} + 26q^{16} - 12q^{17} - 9q^{18} + 3q^{19} + 12q^{20} - 3q^{22} - 19q^{24} + 3q^{25} + 15q^{26} - 3q^{29} + 18q^{30} + 4q^{31} + 30q^{32} - 3q^{33} + 2q^{34} - 24q^{36} + 16q^{37} - 3q^{38} - 21q^{39} + 19q^{40} + 6q^{41} - 5q^{43} - 18q^{44} - 18q^{45} - 20q^{46} - 27q^{48} - 3q^{50} - q^{51} + 15q^{52} - 12q^{53} + 10q^{54} + 3q^{55} + 6q^{57} - 17q^{58} + 33q^{60} - 12q^{61} - 9q^{62} + 8q^{64} + 21q^{65} - 12q^{66} + 12q^{67} + 42q^{68} + 10q^{69} + 6q^{71} - 33q^{72} - 4q^{73} + 24q^{74} + 6q^{75} - 18q^{76} - 49q^{78} - 8q^{79} + 27q^{80} - 2q^{81} + 4q^{82} + q^{85} - 30q^{86} + 34q^{87} - 21q^{88} - 42q^{89} - 52q^{90} - 60q^{92} + 18q^{93} - 5q^{94} - 6q^{95} - 30q^{96} + 31q^{97} + 42q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
0.381966 −0.190983 0.330792i −1.85410 0.190983 + 0.330792i −0.0729490 0.126351i 0 −1.47214 1.42705 2.47172i 0.0729490 + 0.126351i
165.2 2.61803 −1.30902 2.26728i 4.85410 1.30902 + 2.26728i −3.42705 5.93583i 0 7.47214 −1.92705 + 3.33775i 3.42705 + 5.93583i
471.1 0.381966 −0.190983 + 0.330792i −1.85410 0.190983 0.330792i −0.0729490 + 0.126351i 0 −1.47214 1.42705 + 2.47172i 0.0729490 0.126351i
471.2 2.61803 −1.30902 + 2.26728i 4.85410 1.30902 2.26728i −3.42705 + 5.93583i 0 7.47214 −1.92705 3.33775i 3.42705 5.93583i
 $$n$$: 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.f 4
7.b odd 2 1 637.2.h.g 4
7.c even 3 1 637.2.f.c 4
7.c even 3 1 637.2.g.c 4
7.d odd 6 1 91.2.f.a 4
7.d odd 6 1 637.2.g.b 4
13.c even 3 1 637.2.g.c 4
21.g even 6 1 819.2.o.c 4
28.f even 6 1 1456.2.s.h 4
91.g even 3 1 637.2.f.c 4
91.h even 3 1 inner 637.2.h.f 4
91.h even 3 1 8281.2.a.bb 2
91.k even 6 1 8281.2.a.n 2
91.l odd 6 1 1183.2.a.c 2
91.m odd 6 1 91.2.f.a 4
91.n odd 6 1 637.2.g.b 4
91.v odd 6 1 637.2.h.g 4
91.v odd 6 1 1183.2.a.g 2
91.ba even 12 2 1183.2.c.c 4
273.bf even 6 1 819.2.o.c 4
364.br even 6 1 1456.2.s.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 7.d odd 6 1
91.2.f.a 4 91.m odd 6 1
637.2.f.c 4 7.c even 3 1
637.2.f.c 4 91.g even 3 1
637.2.g.b 4 7.d odd 6 1
637.2.g.b 4 91.n odd 6 1
637.2.g.c 4 7.c even 3 1
637.2.g.c 4 13.c even 3 1
637.2.h.f 4 1.a even 1 1 trivial
637.2.h.f 4 91.h even 3 1 inner
637.2.h.g 4 7.b odd 2 1
637.2.h.g 4 91.v odd 6 1
819.2.o.c 4 21.g even 6 1
819.2.o.c 4 273.bf even 6 1
1183.2.a.c 2 91.l odd 6 1
1183.2.a.g 2 91.v odd 6 1
1183.2.c.c 4 91.ba even 12 2
1456.2.s.h 4 28.f even 6 1
1456.2.s.h 4 364.br even 6 1
8281.2.a.n 2 91.k even 6 1
8281.2.a.bb 2 91.h even 3 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T + T^{2} )^{2}$$
$3$ $$1 + 3 T + 8 T^{2} + 3 T^{3} + T^{4}$$
$5$ $$1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$81 + 27 T + 18 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$( 13 - 5 T + T^{2} )^{2}$$
$17$ $$( -11 + 6 T + T^{2} )^{2}$$
$19$ $$81 + 27 T + 18 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$841 - 87 T + 38 T^{2} + 3 T^{3} + T^{4}$$
$31$ $$1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$16 - 24 T + 32 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$9025 - 475 T + 120 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$25 + 5 T^{2} + T^{4}$$
$53$ $$961 + 372 T + 113 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$( -5 + T^{2} )^{2}$$
$61$ $$( 36 + 6 T + T^{2} )^{2}$$
$67$ $$81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$13456 + 696 T + 152 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$( 4 + 2 T + T^{2} )^{2}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$( -45 + T^{2} )^{2}$$
$89$ $$( 79 + 21 T + T^{2} )^{2}$$
$97$ $$52441 - 7099 T + 732 T^{2} - 31 T^{3} + T^{4}$$