# Properties

 Label 637.2.h.j Level $637$ Weight $2$ Character orbit 637.h Analytic conductor $5.086$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.100088711424.6 Defining polynomial: $$x^{8} - 13 x^{6} + 130 x^{4} - 507 x^{2} + 1521$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{5} + \beta_{6} ) q^{2} + ( -\beta_{3} + \beta_{7} ) q^{3} + ( 2 - \beta_{5} + \beta_{6} ) q^{4} + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{1} - \beta_{4} ) q^{6} + 3 q^{8} + ( 3 \beta_{2} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{5} + \beta_{6} ) q^{2} + ( -\beta_{3} + \beta_{7} ) q^{3} + ( 2 - \beta_{5} + \beta_{6} ) q^{4} + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{1} - \beta_{4} ) q^{6} + 3 q^{8} + ( 3 \beta_{2} - \beta_{5} ) q^{9} + ( \beta_{1} - \beta_{4} ) q^{10} + ( -1 - \beta_{2} - 3 \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{12} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{13} + ( 6 \beta_{2} - \beta_{5} ) q^{15} + ( -1 - \beta_{5} + \beta_{6} ) q^{16} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} + 3 \beta_{5} q^{18} -\beta_{7} q^{19} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{20} + ( -10 - 10 \beta_{2} - \beta_{6} ) q^{22} + ( 2 + 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{24} + ( \beta_{2} - \beta_{5} ) q^{25} + ( -\beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{26} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{27} + ( -\beta_{2} - \beta_{5} ) q^{29} + ( 3 \beta_{2} + 6 \beta_{5} ) q^{30} + ( -\beta_{1} - 2 \beta_{4} + \beta_{7} ) q^{31} + ( -4 + \beta_{5} - \beta_{6} ) q^{32} + ( 3 \beta_{1} + 6 \beta_{4} + 2 \beta_{7} ) q^{33} + ( -4 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{34} + ( 3 \beta_{2} + 2 \beta_{5} ) q^{36} + ( 6 - 2 \beta_{5} + 2 \beta_{6} ) q^{37} + ( \beta_{1} + 2 \beta_{4} ) q^{38} + ( 2 - 5 \beta_{2} - \beta_{5} + 4 \beta_{6} ) q^{39} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{40} + ( -2 \beta_{1} - 4 \beta_{4} ) q^{41} + ( 6 + 6 \beta_{2} + 5 \beta_{6} ) q^{43} + ( -11 - 11 \beta_{2} - 4 \beta_{6} ) q^{44} + ( 2 \beta_{1} + 4 \beta_{3} + \beta_{4} ) q^{45} + ( -4 - 2 \beta_{5} + 2 \beta_{6} ) q^{46} + ( \beta_{1} + 3 \beta_{3} - \beta_{4} - 3 \beta_{7} ) q^{47} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{48} + ( -2 \beta_{2} + \beta_{5} ) q^{50} + ( -9 - 9 \beta_{2} - 5 \beta_{6} ) q^{51} + ( \beta_{1} + 3 \beta_{4} - 2 \beta_{7} ) q^{52} + ( 5 \beta_{2} - 2 \beta_{5} ) q^{53} -3 \beta_{3} q^{54} + ( 3 \beta_{1} + 6 \beta_{4} + 2 \beta_{7} ) q^{55} + ( 6 + \beta_{5} - \beta_{6} ) q^{57} + ( -4 \beta_{2} - \beta_{5} ) q^{58} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{59} + ( 9 \beta_{2} + 5 \beta_{5} ) q^{60} -2 \beta_{7} q^{61} + ( -2 \beta_{1} - 4 \beta_{4} + 3 \beta_{7} ) q^{62} + ( -5 + 6 \beta_{5} - 6 \beta_{6} ) q^{64} + ( 2 - 5 \beta_{2} - \beta_{5} + 4 \beta_{6} ) q^{65} + ( \beta_{1} + 2 \beta_{4} - 9 \beta_{7} ) q^{66} + ( -1 - \beta_{2} ) q^{67} + ( -6 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} ) q^{68} + ( -2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} ) q^{69} + ( -4 - 4 \beta_{2} ) q^{71} + ( 9 \beta_{2} - 3 \beta_{5} ) q^{72} + 2 \beta_{7} q^{73} + ( 12 - 6 \beta_{5} + 6 \beta_{6} ) q^{74} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{75} + ( \beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{76} + ( 14 + 4 \beta_{2} - 7 \beta_{5} + 2 \beta_{6} ) q^{78} + ( 4 + 4 \beta_{2} + 2 \beta_{6} ) q^{79} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{80} + ( 6 + 6 \beta_{2} + 4 \beta_{6} ) q^{81} + ( -2 \beta_{1} - 4 \beta_{4} + 6 \beta_{7} ) q^{82} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{83} + ( -9 - 9 \beta_{2} - 5 \beta_{6} ) q^{85} + ( 21 + 21 \beta_{2} + 6 \beta_{6} ) q^{86} + ( 2 \beta_{1} + \beta_{4} ) q^{87} + ( -3 - 3 \beta_{2} - 9 \beta_{6} ) q^{88} + 3 \beta_{3} q^{89} + ( -6 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} ) q^{90} -2 q^{92} + ( -9 + 5 \beta_{5} - 5 \beta_{6} ) q^{93} + ( -2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{94} + ( 6 + \beta_{5} - \beta_{6} ) q^{95} + ( -\beta_{1} + 3 \beta_{3} + \beta_{4} - 3 \beta_{7} ) q^{96} + ( -\beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{7} ) q^{97} + ( -6 - 11 \beta_{5} + 11 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} + 12 q^{4} + 24 q^{8} - 14 q^{9} + O(q^{10})$$ $$8 q + 4 q^{2} + 12 q^{4} + 24 q^{8} - 14 q^{9} + 2 q^{11} - 26 q^{15} - 12 q^{16} + 6 q^{18} - 38 q^{22} + 24 q^{23} - 6 q^{25} + 2 q^{29} - 28 q^{32} - 8 q^{36} + 40 q^{37} + 26 q^{39} + 14 q^{43} - 36 q^{44} - 40 q^{46} + 10 q^{50} - 26 q^{51} - 24 q^{53} + 52 q^{57} + 14 q^{58} - 26 q^{60} - 16 q^{64} + 26 q^{65} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 72 q^{74} + 78 q^{78} + 12 q^{79} + 16 q^{81} - 26 q^{85} + 72 q^{86} + 6 q^{88} - 16 q^{92} - 52 q^{93} + 52 q^{95} - 92 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 13 x^{6} + 130 x^{4} - 507 x^{2} + 1521$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 10 \nu^{4} + 130 \nu^{2} - 507$$$$)/390$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 70 \nu^{5} - 520 \nu^{3} + 3237 \nu$$$$)/1170$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 10 \nu^{5} - 130 \nu^{3} + 117 \nu$$$$)/390$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{6} + 35 \nu^{4} - 260 \nu^{2} + 1014$$$$)/195$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{6} + 70 \nu^{4} - 520 \nu^{2} + 819$$$$)/390$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{7} + 140 \nu^{5} - 1040 \nu^{3} + 2847 \nu$$$$)/1170$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 7 \beta_{2} + 7$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 7 \beta_{4} - \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$13 \beta_{5} + 52 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{7} - 52 \beta_{4} + 13 \beta_{3} - 52 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-130 \beta_{6} + 130 \beta_{5} - 403$$ $$\nu^{7}$$ $$=$$ $$-130 \beta_{7} + 260 \beta_{3} - 403 \beta_{1}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 1.87694 + 1.08365i −1.87694 − 1.08365i −2.49541 − 1.44073i 2.49541 + 1.44073i 1.87694 − 1.08365i −1.87694 + 1.08365i −2.49541 + 1.44073i 2.49541 − 1.44073i
−1.30278 −1.44073 2.49541i −0.302776 −1.44073 2.49541i 1.87694 + 3.25096i 0 3.00000 −2.65139 + 4.59234i 1.87694 + 3.25096i
165.2 −1.30278 1.44073 + 2.49541i −0.302776 1.44073 + 2.49541i −1.87694 3.25096i 0 3.00000 −2.65139 + 4.59234i −1.87694 3.25096i
165.3 2.30278 −1.08365 1.87694i 3.30278 −1.08365 1.87694i −2.49541 4.32218i 0 3.00000 −0.848612 + 1.46984i −2.49541 4.32218i
165.4 2.30278 1.08365 + 1.87694i 3.30278 1.08365 + 1.87694i 2.49541 + 4.32218i 0 3.00000 −0.848612 + 1.46984i 2.49541 + 4.32218i
471.1 −1.30278 −1.44073 + 2.49541i −0.302776 −1.44073 + 2.49541i 1.87694 3.25096i 0 3.00000 −2.65139 4.59234i 1.87694 3.25096i
471.2 −1.30278 1.44073 2.49541i −0.302776 1.44073 2.49541i −1.87694 + 3.25096i 0 3.00000 −2.65139 4.59234i −1.87694 + 3.25096i
471.3 2.30278 −1.08365 + 1.87694i 3.30278 −1.08365 + 1.87694i −2.49541 + 4.32218i 0 3.00000 −0.848612 1.46984i −2.49541 + 4.32218i
471.4 2.30278 1.08365 1.87694i 3.30278 1.08365 1.87694i 2.49541 4.32218i 0 3.00000 −0.848612 1.46984i 2.49541 4.32218i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 471.4 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.b odd 2 1 inner
91.h even 3 1 inner
91.v odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.h.j 8
7.b odd 2 1 inner 637.2.h.j 8
7.c even 3 1 637.2.f.h 8
7.c even 3 1 637.2.g.i 8
7.d odd 6 1 637.2.f.h 8
7.d odd 6 1 637.2.g.i 8
13.c even 3 1 637.2.g.i 8
91.g even 3 1 637.2.f.h 8
91.h even 3 1 inner 637.2.h.j 8
91.h even 3 1 8281.2.a.bu 4
91.k even 6 1 8281.2.a.bo 4
91.l odd 6 1 8281.2.a.bo 4
91.m odd 6 1 637.2.f.h 8
91.n odd 6 1 637.2.g.i 8
91.v odd 6 1 inner 637.2.h.j 8
91.v odd 6 1 8281.2.a.bu 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h 8 7.c even 3 1
637.2.f.h 8 7.d odd 6 1
637.2.f.h 8 91.g even 3 1
637.2.f.h 8 91.m odd 6 1
637.2.g.i 8 7.c even 3 1
637.2.g.i 8 7.d odd 6 1
637.2.g.i 8 13.c even 3 1
637.2.g.i 8 91.n odd 6 1
637.2.h.j 8 1.a even 1 1 trivial
637.2.h.j 8 7.b odd 2 1 inner
637.2.h.j 8 91.h even 3 1 inner
637.2.h.j 8 91.v odd 6 1 inner
8281.2.a.bo 4 91.k even 6 1
8281.2.a.bo 4 91.l odd 6 1
8281.2.a.bu 4 91.h even 3 1
8281.2.a.bu 4 91.v odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 3$$ $$T_{3}^{8} + 13 T_{3}^{6} + 130 T_{3}^{4} + 507 T_{3}^{2} + 1521$$ $$T_{5}^{8} + 13 T_{5}^{6} + 130 T_{5}^{4} + 507 T_{5}^{2} + 1521$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -3 - T + T^{2} )^{4}$$
$3$ $$1521 + 507 T^{2} + 130 T^{4} + 13 T^{6} + T^{8}$$
$5$ $$1521 + 507 T^{2} + 130 T^{4} + 13 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 841 + 29 T + 30 T^{2} - T^{3} + T^{4} )^{2}$$
$13$ $$28561 + 13 T^{4} + T^{8}$$
$17$ $$( 39 - 52 T^{2} + T^{4} )^{2}$$
$19$ $$1521 + 507 T^{2} + 130 T^{4} + 13 T^{6} + T^{8}$$
$23$ $$( -4 - 6 T + T^{2} )^{4}$$
$29$ $$( 9 + 3 T + 4 T^{2} - T^{3} + T^{4} )^{2}$$
$31$ $$1521 + 2028 T^{2} + 2665 T^{4} + 52 T^{6} + T^{8}$$
$37$ $$( 12 - 10 T + T^{2} )^{4}$$
$41$ $$31539456 + 876096 T^{2} + 18720 T^{4} + 156 T^{6} + T^{8}$$
$43$ $$( 4761 + 483 T + 118 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$47$ $$123201 + 54756 T^{2} + 23985 T^{4} + 156 T^{6} + T^{8}$$
$53$ $$( 529 + 276 T + 121 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$59$ $$( 351 - 52 T^{2} + T^{4} )^{2}$$
$61$ $$389376 + 32448 T^{2} + 2080 T^{4} + 52 T^{6} + T^{8}$$
$67$ $$( 1 + T + T^{2} )^{4}$$
$71$ $$( 16 + 4 T + T^{2} )^{4}$$
$73$ $$389376 + 32448 T^{2} + 2080 T^{4} + 52 T^{6} + T^{8}$$
$79$ $$( 16 + 24 T + 40 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$83$ $$( 351 - 52 T^{2} + T^{4} )^{2}$$
$89$ $$( 3159 - 117 T^{2} + T^{4} )^{2}$$
$97$ $$127035441 + 2783937 T^{2} + 49738 T^{4} + 247 T^{6} + T^{8}$$