# Properties

 Label 637.2.g.f Level $637$ Weight $2$ Character orbit 637.g 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.g (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: $$\Z[a_1, a_2, a_3]$$ 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.

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

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.207107 + 0.358719i 1.41421 0.914214 + 1.58346i 0.914214 + 1.58346i −0.292893 + 0.507306i 0 −1.58579 −1.00000 −0.757359
263.2 1.20711 2.09077i −1.41421 −1.91421 3.31552i −1.91421 3.31552i −1.70711 + 2.95680i 0 −4.41421 −1.00000 −9.24264
373.1 −0.207107 0.358719i 1.41421 0.914214 1.58346i 0.914214 1.58346i −0.292893 0.507306i 0 −1.58579 −1.00000 −0.757359
373.2 1.20711 + 2.09077i −1.41421 −1.91421 + 3.31552i −1.91421 + 3.31552i −1.70711 2.95680i 0 −4.41421 −1.00000 −9.24264
 $$n$$: 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.g 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.g.f 4
7.b odd 2 1 637.2.g.g 4
7.c even 3 1 637.2.f.f yes 4
7.c even 3 1 637.2.h.b 4
7.d odd 6 1 637.2.f.e 4
7.d odd 6 1 637.2.h.c 4
13.c even 3 1 637.2.h.b 4
91.g even 3 1 inner 637.2.g.f 4
91.g even 3 1 8281.2.a.p 2
91.h even 3 1 637.2.f.f yes 4
91.m odd 6 1 637.2.g.g 4
91.m odd 6 1 8281.2.a.o 2
91.n odd 6 1 637.2.h.c 4
91.p odd 6 1 8281.2.a.y 2
91.u even 6 1 8281.2.a.x 2
91.v odd 6 1 637.2.f.e 4

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$49 - 14 T + 11 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 2 - 4 T + T^{2} )^{2}$$
$13$ $$( 13 + 7 T + T^{2} )^{2}$$
$17$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$( -6 + T )^{4}$$
$23$ $$4 + 2 T^{2} + T^{4}$$
$29$ $$1681 + 574 T + 155 T^{2} + 14 T^{3} + T^{4}$$
$31$ $$196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$5041 + 142 T + 75 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$1 + 6 T + 35 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$( 9 - 3 T + T^{2} )^{2}$$
$59$ $$324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$( 41 - 14 T + T^{2} )^{2}$$
$67$ $$( -18 + T^{2} )^{2}$$
$71$ $$16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$49 + 70 T + 107 T^{2} - 10 T^{3} + T^{4}$$
$79$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$83$ $$( -14 + 12 T + T^{2} )^{2}$$
$89$ $$12544 - 896 T + 176 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$3136 - 896 T + 200 T^{2} - 16 T^{3} + T^{4}$$