# Properties

 Label 637.2.e.i Level $637$ Weight $2$ Character orbit 637.e 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.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.2696112.1 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} + 8 \nu - 40$$$$)/82$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 10 \nu^{4} + 9 \nu^{3} - 36 \nu^{2} + 16 \nu - 121$$$$)/41$$ $$\beta_{4}$$ $$=$$ $$($$$$-10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72$$$$)/82$$ $$\beta_{5}$$ $$=$$ $$($$$$-26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 454 \nu - 26$$$$)/82$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} + 13 \beta_{4} - 2 \beta_{1} - 13$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 18 \beta_{2} - 18 \beta_{1}$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 1.17146 − 2.02903i 0.235342 − 0.407624i −0.906803 + 1.57063i 1.17146 + 2.02903i 0.235342 + 0.407624i −0.906803 − 1.57063i
−1.17146 + 2.02903i −0.573183 0.992782i −1.74464 3.02181i −0.671462 + 1.16301i 2.68585 0 3.48929 0.842923 1.45999i −1.57318 2.72483i
79.2 −0.235342 + 0.407624i 1.12457 + 1.94781i 0.889229 + 1.54019i 0.264658 0.458402i −1.05863 0 −1.77846 −1.02932 + 1.78283i 0.124570 + 0.215762i
79.3 0.906803 1.57063i −1.55139 2.68708i −0.644584 1.11645i 1.40680 2.43665i −5.62721 0 1.28917 −3.31361 + 5.73933i −2.55139 4.41913i
508.1 −1.17146 2.02903i −0.573183 + 0.992782i −1.74464 + 3.02181i −0.671462 1.16301i 2.68585 0 3.48929 0.842923 + 1.45999i −1.57318 + 2.72483i
508.2 −0.235342 0.407624i 1.12457 1.94781i 0.889229 1.54019i 0.264658 + 0.458402i −1.05863 0 −1.77846 −1.02932 1.78283i 0.124570 0.215762i
508.3 0.906803 + 1.57063i −1.55139 + 2.68708i −0.644584 + 1.11645i 1.40680 + 2.43665i −5.62721 0 1.28917 −3.31361 5.73933i −2.55139 + 4.41913i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 508.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.e.i 6
7.b odd 2 1 637.2.e.j 6
7.c even 3 1 637.2.a.j 3
7.c even 3 1 inner 637.2.e.i 6
7.d odd 6 1 91.2.a.d 3
7.d odd 6 1 637.2.e.j 6
21.g even 6 1 819.2.a.i 3
21.h odd 6 1 5733.2.a.x 3
28.f even 6 1 1456.2.a.t 3
35.i odd 6 1 2275.2.a.m 3
56.j odd 6 1 5824.2.a.by 3
56.m even 6 1 5824.2.a.bs 3
91.r even 6 1 8281.2.a.bg 3
91.s odd 6 1 1183.2.a.i 3
91.bb even 12 2 1183.2.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 7.d odd 6 1
637.2.a.j 3 7.c even 3 1
637.2.e.i 6 1.a even 1 1 trivial
637.2.e.i 6 7.c even 3 1 inner
637.2.e.j 6 7.b odd 2 1
637.2.e.j 6 7.d odd 6 1
819.2.a.i 3 21.g even 6 1
1183.2.a.i 3 91.s odd 6 1
1183.2.c.f 6 91.bb even 12 2
1456.2.a.t 3 28.f even 6 1
2275.2.a.m 3 35.i odd 6 1
5733.2.a.x 3 21.h odd 6 1
5824.2.a.bs 3 56.m even 6 1
5824.2.a.by 3 56.j odd 6 1
8281.2.a.bg 3 91.r even 6 1

## Hecke kernels

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

 $$T_{2}^{6} + T_{2}^{5} + 5 T_{2}^{4} + 18 T_{2}^{2} + 8 T_{2} + 4$$ $$T_{3}^{6} + 2 T_{3}^{5} + 10 T_{3}^{4} + 4 T_{3}^{3} + 52 T_{3}^{2} + 48 T_{3} + 64$$ $$T_{5}^{6} - 2 T_{5}^{5} + 7 T_{5}^{4} + 2 T_{5}^{3} + 13 T_{5}^{2} - 6 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 8 T + 18 T^{2} + 5 T^{4} + T^{5} + T^{6}$$
$3$ $$64 + 48 T + 52 T^{2} + 4 T^{3} + 10 T^{4} + 2 T^{5} + T^{6}$$
$5$ $$4 - 6 T + 13 T^{2} + 2 T^{3} + 7 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$64 + 48 T + 52 T^{2} + 4 T^{3} + 10 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$( 1 + T )^{6}$$
$17$ $$16 + 40 T + 84 T^{2} + 48 T^{3} + 26 T^{4} - 4 T^{5} + T^{6}$$
$19$ $$16 - 4 T + 17 T^{2} + 12 T^{3} + 15 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$18496 - 136 T + 1361 T^{2} + 282 T^{3} + 99 T^{4} + 10 T^{5} + T^{6}$$
$29$ $$( -454 + 185 T - 24 T^{2} + T^{3} )^{2}$$
$31$ $$256 - 304 T + 297 T^{2} - 108 T^{3} + 35 T^{4} + 4 T^{5} + T^{6}$$
$37$ $$15376 - 7192 T + 3364 T^{2} - 248 T^{3} + 58 T^{4} + T^{6}$$
$41$ $$( 8 - 28 T + 2 T^{2} + T^{3} )^{2}$$
$43$ $$( 628 - 71 T - 10 T^{2} + T^{3} )^{2}$$
$47$ $$295936 + 42976 T + 10593 T^{2} + 456 T^{3} + 143 T^{4} + 8 T^{5} + T^{6}$$
$53$ $$484 - 770 T + 1049 T^{2} - 324 T^{3} + 99 T^{4} + 8 T^{5} + T^{6}$$
$59$ $$473344 + 107328 T + 27088 T^{2} + 752 T^{3} + 172 T^{4} + 4 T^{5} + T^{6}$$
$61$ $$( 4 + 2 T + T^{2} )^{3}$$
$67$ $$952576 - 121024 T + 27088 T^{2} - 464 T^{3} + 268 T^{4} - 12 T^{5} + T^{6}$$
$71$ $$( 16 - 22 T + 6 T^{2} + T^{3} )^{2}$$
$73$ $$75076 + 27126 T + 12541 T^{2} - 442 T^{3} + 199 T^{4} + 10 T^{5} + T^{6}$$
$79$ $$256 + 80 T + 249 T^{2} - 102 T^{3} + 191 T^{4} - 14 T^{5} + T^{6}$$
$83$ $$( 3268 - 271 T - 12 T^{2} + T^{3} )^{2}$$
$89$ $$178084 - 40090 T + 9869 T^{2} - 654 T^{3} + 99 T^{4} - 2 T^{5} + T^{6}$$
$97$ $$( -22 + 29 T - 10 T^{2} + T^{3} )^{2}$$