# Properties

 Label 494.2.g.b Level $494$ Weight $2$ Character orbit 494.g Analytic conductor $3.945$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$494 = 2 \cdot 13 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 494.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.94460985985$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.771147.1 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + ( -1 - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{8} + \beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + ( -1 - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{8} + \beta_{5} q^{9} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{10} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{11} + ( -1 + \beta_{2} ) q^{12} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{13} -\beta_{2} q^{14} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{15} + \beta_{4} q^{16} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 - \beta_{2} - \beta_{3} ) q^{18} + ( -1 - \beta_{4} ) q^{19} + ( -1 - \beta_{4} - \beta_{5} ) q^{20} + ( -3 + \beta_{3} ) q^{21} + ( 3 + 3 \beta_{4} + \beta_{5} ) q^{22} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{4} ) q^{24} + ( -2 + \beta_{3} ) q^{25} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{26} + ( -\beta_{2} + 2 \beta_{3} ) q^{27} + \beta_{1} q^{28} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{29} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{30} + ( -4 + 4 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -1 - \beta_{4} ) q^{32} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{33} -2 \beta_{2} q^{34} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{35} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{37} + q^{38} + ( -2 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{39} + ( \beta_{2} + \beta_{3} ) q^{40} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{42} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{43} + ( -2 - \beta_{2} - \beta_{3} ) q^{44} + ( 3 + \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{45} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{46} + ( -3 - 3 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{48} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{49} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{50} + ( -6 + 2 \beta_{3} ) q^{51} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{52} + ( 4 - 4 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{54} + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} ) q^{55} + ( -\beta_{1} + \beta_{2} ) q^{56} + ( -1 + \beta_{2} ) q^{57} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{58} + ( 5 + 6 \beta_{1} - 6 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} ) q^{59} + ( 2 - \beta_{2} - 2 \beta_{3} ) q^{60} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( 3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{62} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{63} + q^{64} + ( 7 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{65} + ( -\beta_{2} + 2 \beta_{3} ) q^{66} + ( 4 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{67} + 2 \beta_{1} q^{68} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{69} + ( -2 + \beta_{3} ) q^{70} + ( 3 + 3 \beta_{4} ) q^{71} + \beta_{5} q^{72} + ( 9 - 4 \beta_{2} - 8 \beta_{3} ) q^{73} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{74} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{75} + \beta_{4} q^{76} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{77} + ( -1 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{78} + ( 1 - 5 \beta_{2} ) q^{79} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{80} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{81} + ( -6 - \beta_{1} + \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 6 - 4 \beta_{2} - 9 \beta_{3} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{84} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{85} + ( -1 - 2 \beta_{2} ) q^{86} + ( -2 - 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{87} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{88} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} ) q^{89} + ( -3 + \beta_{2} ) q^{90} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{92} + ( -7 + 7 \beta_{2} + 7 \beta_{3} + 6 \beta_{4} - 7 \beta_{5} ) q^{93} + ( 1 + 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{94} + ( -1 - \beta_{4} - \beta_{5} ) q^{95} + ( -1 + \beta_{2} ) q^{96} + ( 6 - 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{4} ) q^{97} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{4} + \beta_{5} ) q^{98} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 2 q^{3} - 3 q^{4} + 4 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} - q^{9} + O(q^{10})$$ $$6 q - 3 q^{2} + 2 q^{3} - 3 q^{4} + 4 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} - q^{9} - 2 q^{10} + 8 q^{11} - 4 q^{12} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} + 2 q^{18} - 3 q^{19} - 2 q^{20} - 16 q^{21} + 8 q^{22} - 2 q^{23} + 2 q^{24} - 10 q^{25} + 2 q^{27} + q^{28} + 14 q^{29} - 3 q^{30} - 10 q^{31} - 3 q^{32} - q^{33} - 4 q^{34} + 5 q^{35} - q^{36} + 6 q^{38} - 13 q^{39} + 4 q^{40} - 15 q^{41} + 8 q^{42} + 5 q^{43} - 16 q^{44} + 8 q^{45} - 2 q^{46} - 22 q^{47} + 2 q^{48} + 12 q^{49} + 5 q^{50} - 32 q^{51} + 10 q^{53} - q^{54} + 14 q^{55} + q^{56} - 4 q^{57} + 14 q^{58} + 4 q^{59} + 6 q^{60} - 2 q^{61} + 5 q^{62} - 4 q^{63} + 6 q^{64} + 13 q^{65} + 2 q^{66} - 6 q^{67} + 2 q^{68} - 16 q^{69} - 10 q^{70} + 9 q^{71} - q^{72} + 30 q^{73} + q^{75} - 3 q^{76} + 14 q^{77} - 13 q^{78} - 4 q^{79} - 2 q^{80} + 21 q^{81} - 15 q^{82} + 10 q^{83} + 8 q^{84} + 10 q^{85} - 10 q^{86} - 5 q^{87} + 8 q^{88} + 27 q^{89} - 16 q^{90} + 13 q^{91} + 4 q^{92} - 25 q^{93} + 11 q^{94} - 2 q^{95} - 4 q^{96} + 20 q^{97} + 12 q^{98} + 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 5 \nu^{4} - 25 \nu^{3} + 15 \nu^{2} + 4 \nu - 20$$$$)/79$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{5} + 25 \nu^{4} - 46 \nu^{3} + 75 \nu^{2} + 20 \nu + 216$$$$)/79$$ $$\beta_{4}$$ $$=$$ $$($$$$20 \nu^{5} - 21 \nu^{4} + 105 \nu^{3} + 95 \nu^{2} + 315 \nu + 5$$$$)/79$$ $$\beta_{5}$$ $$=$$ $$($$$$38 \nu^{5} - 32 \nu^{4} + 160 \nu^{3} + 299 \nu^{2} + 480 \nu + 128$$$$)/79$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 5 \beta_{2} - 4$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 15 \beta_{1} - 5$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{5} + 25 \beta_{4} + 41 \beta_{2} - 41 \beta_{1} + 25$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/494\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 1.32555 − 2.29591i −0.136945 + 0.237196i −0.688601 + 1.19269i 1.32555 + 2.29591i −0.136945 − 0.237196i −0.688601 − 1.19269i
−0.500000 + 0.866025i −0.825547 + 1.42989i −0.500000 0.866025i 1.27389 −0.825547 1.42989i 1.32555 + 2.29591i 1.00000 0.136945 + 0.237196i −0.636945 + 1.10322i
191.2 −0.500000 + 0.866025i 0.636945 1.10322i −0.500000 0.866025i 2.37720 0.636945 + 1.10322i −0.136945 0.237196i 1.00000 0.688601 + 1.19269i −1.18860 + 2.05872i
191.3 −0.500000 + 0.866025i 1.18860 2.05872i −0.500000 0.866025i −1.65109 1.18860 + 2.05872i −0.688601 1.19269i 1.00000 −1.32555 2.29591i 0.825547 1.42989i
419.1 −0.500000 0.866025i −0.825547 1.42989i −0.500000 + 0.866025i 1.27389 −0.825547 + 1.42989i 1.32555 2.29591i 1.00000 0.136945 0.237196i −0.636945 1.10322i
419.2 −0.500000 0.866025i 0.636945 + 1.10322i −0.500000 + 0.866025i 2.37720 0.636945 1.10322i −0.136945 + 0.237196i 1.00000 0.688601 1.19269i −1.18860 2.05872i
419.3 −0.500000 0.866025i 1.18860 + 2.05872i −0.500000 + 0.866025i −1.65109 1.18860 2.05872i −0.688601 + 1.19269i 1.00000 −1.32555 + 2.29591i 0.825547 + 1.42989i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.g.b 6
13.c even 3 1 inner 494.2.g.b 6
13.c even 3 1 6422.2.a.u 3
13.e even 6 1 6422.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.b 6 1.a even 1 1 trivial
494.2.g.b 6 13.c even 3 1 inner
6422.2.a.m 3 13.e even 6 1
6422.2.a.u 3 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(494, [\chi])$$:

 $$T_{3}^{6} - 2 T_{3}^{5} + 7 T_{3}^{4} - 4 T_{3}^{3} + 19 T_{3}^{2} - 15 T_{3} + 25$$ $$T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{3}$$
$3$ $$25 - 15 T + 19 T^{2} - 4 T^{3} + 7 T^{4} - 2 T^{5} + T^{6}$$
$5$ $$( 5 - 3 T - 2 T^{2} + T^{3} )^{2}$$
$7$ $$1 + 4 T + 15 T^{2} + 6 T^{3} + 5 T^{4} - T^{5} + T^{6}$$
$11$ $$25 - 85 T + 249 T^{2} - 126 T^{3} + 47 T^{4} - 8 T^{5} + T^{6}$$
$13$ $$2197 - 65 T^{3} + T^{6}$$
$17$ $$64 + 128 T + 240 T^{2} + 48 T^{3} + 20 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$( 1 + T + T^{2} )^{3}$$
$23$ $$25 + 145 T + 851 T^{2} - 48 T^{3} + 33 T^{4} + 2 T^{5} + T^{6}$$
$29$ $$10609 + 3605 T + 2667 T^{2} - 696 T^{3} + 161 T^{4} - 14 T^{5} + T^{6}$$
$31$ $$( 73 - 48 T + 5 T^{2} + T^{3} )^{2}$$
$37$ $$10816 + 5408 T + 2704 T^{2} + 208 T^{3} + 52 T^{4} + T^{6}$$
$41$ $$5329 + 4526 T + 2749 T^{2} + 784 T^{3} + 163 T^{4} + 15 T^{5} + T^{6}$$
$43$ $$25 - 45 T + 106 T^{2} + 35 T^{3} + 34 T^{4} - 5 T^{5} + T^{6}$$
$47$ $$( 5 - 16 T + 11 T^{2} + T^{3} )^{2}$$
$53$ $$( -73 - 48 T - 5 T^{2} + T^{3} )^{2}$$
$59$ $$1 - 129 T + 16645 T^{2} + 514 T^{3} + 145 T^{4} - 4 T^{5} + T^{6}$$
$61$ $$6889 + 2407 T + 1007 T^{2} + 108 T^{3} + 33 T^{4} + 2 T^{5} + T^{6}$$
$67$ $$64 - 320 T + 1552 T^{2} - 256 T^{3} + 76 T^{4} + 6 T^{5} + T^{6}$$
$71$ $$( 9 - 3 T + T^{2} )^{3}$$
$73$ $$( 1747 - 133 T - 15 T^{2} + T^{3} )^{2}$$
$79$ $$( 229 - 107 T + 2 T^{2} + T^{3} )^{2}$$
$83$ $$( 1825 - 256 T - 5 T^{2} + T^{3} )^{2}$$
$89$ $$97969 - 63852 T + 33165 T^{2} - 4882 T^{3} + 525 T^{4} - 27 T^{5} + T^{6}$$
$97$ $$40000 - 23200 T + 9456 T^{2} - 1920 T^{3} + 284 T^{4} - 20 T^{5} + T^{6}$$