# Properties

 Label 637.2.g.h Level $637$ Weight $2$ Character orbit 637.g 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.g (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: 8.0.1485512441856.7 Defining polynomial: $$x^{8} + 24 x^{6} + 455 x^{4} + 2904 x^{2} + 14641$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 24 x^{6} + 455 x^{4} + 2904 x^{2} + 14641$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$24 \nu^{6} + 455 \nu^{4} + 10920 \nu^{2} + 69696$$$$)/55055$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 2556$$$$)/455$$ $$\beta_{4}$$ $$=$$ $$($$$$-167 \nu^{6} - 5460 \nu^{4} - 75985 \nu^{2} - 484968$$$$)/55055$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 3011 \nu$$$$)/5005$$ $$\beta_{6}$$ $$=$$ $$($$$$-191 \nu^{7} - 5915 \nu^{5} - 86905 \nu^{3} - 554664 \nu$$$$)/605605$$ $$\beta_{7}$$ $$=$$ $$($$$$24 \nu^{7} + 455 \nu^{5} + 10920 \nu^{3} + 69696 \nu$$$$)/55055$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 12 \beta_{2} - 12$$ $$\nu^{3}$$ $$=$$ $$13 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 13 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-24 \beta_{4} - 167 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-191 \beta_{7} - 264 \beta_{6}$$ $$\nu^{6}$$ $$=$$ $$-455 \beta_{3} + 2556$$ $$\nu^{7}$$ $$=$$ $$5005 \beta_{5} + 3011 \beta_{1}$$

## Character values

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

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

## 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
 2.04914 + 3.54921i −1.34203 − 2.32446i 1.34203 + 2.32446i −2.04914 − 3.54921i 2.04914 − 3.54921i −1.34203 + 2.32446i 1.34203 − 2.32446i −2.04914 + 3.54921i
−0.500000 + 0.866025i −1.41421 0.500000 + 0.866025i −1.34203 2.32446i 0.707107 1.22474i 0 −3.00000 −1.00000 2.68406
263.2 −0.500000 + 0.866025i −1.41421 0.500000 + 0.866025i 2.04914 + 3.54921i 0.707107 1.22474i 0 −3.00000 −1.00000 −4.09827
263.3 −0.500000 + 0.866025i 1.41421 0.500000 + 0.866025i −2.04914 3.54921i −0.707107 + 1.22474i 0 −3.00000 −1.00000 4.09827
263.4 −0.500000 + 0.866025i 1.41421 0.500000 + 0.866025i 1.34203 + 2.32446i −0.707107 + 1.22474i 0 −3.00000 −1.00000 −2.68406
373.1 −0.500000 0.866025i −1.41421 0.500000 0.866025i −1.34203 + 2.32446i 0.707107 + 1.22474i 0 −3.00000 −1.00000 2.68406
373.2 −0.500000 0.866025i −1.41421 0.500000 0.866025i 2.04914 3.54921i 0.707107 + 1.22474i 0 −3.00000 −1.00000 −4.09827
373.3 −0.500000 0.866025i 1.41421 0.500000 0.866025i −2.04914 + 3.54921i −0.707107 1.22474i 0 −3.00000 −1.00000 4.09827
373.4 −0.500000 0.866025i 1.41421 0.500000 0.866025i 1.34203 2.32446i −0.707107 1.22474i 0 −3.00000 −1.00000 −2.68406
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.4 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.g even 3 1 inner
91.m 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.g.h 8
7.b odd 2 1 inner 637.2.g.h 8
7.c even 3 1 637.2.f.g 8
7.c even 3 1 637.2.h.k 8
7.d odd 6 1 637.2.f.g 8
7.d odd 6 1 637.2.h.k 8
13.c even 3 1 637.2.h.k 8
91.g even 3 1 inner 637.2.g.h 8
91.g even 3 1 8281.2.a.bv 4
91.h even 3 1 637.2.f.g 8
91.m odd 6 1 inner 637.2.g.h 8
91.m odd 6 1 8281.2.a.bv 4
91.n odd 6 1 637.2.h.k 8
91.p odd 6 1 8281.2.a.bn 4
91.u even 6 1 8281.2.a.bn 4
91.v odd 6 1 637.2.f.g 8

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

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{3}^{2} - 2$$ $$T_{5}^{8} + 24 T_{5}^{6} + 455 T_{5}^{4} + 2904 T_{5}^{2} + 14641$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{4}$$
$3$ $$( -2 + T^{2} )^{4}$$
$5$ $$14641 + 2904 T^{2} + 455 T^{4} + 24 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( -22 - 2 T + T^{2} )^{4}$$
$13$ $$28561 + 3380 T^{2} + 231 T^{4} + 20 T^{6} + T^{8}$$
$17$ $$2401 + 1568 T^{2} + 975 T^{4} + 32 T^{6} + T^{8}$$
$19$ $$( -8 + T^{2} )^{4}$$
$23$ $$( 196 + 84 T + 50 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$29$ $$( 49 + 56 T + 71 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$37$ $$( 361 + 76 T + 35 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$707281 + 87464 T^{2} + 9975 T^{4} + 104 T^{6} + T^{8}$$
$43$ $$( 196 - 84 T + 50 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$47$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$53$ $$( 6889 + 498 T + 119 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$59$ $$38416 + 30576 T^{2} + 24140 T^{4} + 156 T^{6} + T^{8}$$
$61$ $$( 169 - 72 T^{2} + T^{4} )^{2}$$
$67$ $$( -22 - 2 T + T^{2} )^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{4}$$
$73$ $$28398241 + 1023168 T^{2} + 31535 T^{4} + 192 T^{6} + T^{8}$$
$79$ $$( 676 - 364 T + 170 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$83$ $$( -98 + T^{2} )^{4}$$
$89$ $$1097199376 + 12322128 T^{2} + 105260 T^{4} + 372 T^{6} + T^{8}$$
$97$ $$( 324 + 18 T^{2} + T^{4} )^{2}$$