# Properties

 Label 441.6.a.be Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 146 x^{6} + 5453 x^{4} - 40868 x^{2} + 3844$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 7^{4}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( 3 - \beta_{2} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( 3 - \beta_{2} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{4} + \beta_{6} ) q^{8} + ( -\beta_{5} - 3 \beta_{7} ) q^{10} + ( -8 \beta_{4} - 2 \beta_{6} ) q^{11} + ( 3 \beta_{5} + 4 \beta_{7} ) q^{13} + ( -117 + 27 \beta_{2} ) q^{16} + ( -25 \beta_{1} + 4 \beta_{3} ) q^{17} + ( -4 \beta_{5} + 24 \beta_{7} ) q^{19} + ( -26 \beta_{1} - 13 \beta_{3} ) q^{20} + ( -308 + 20 \beta_{2} ) q^{22} + ( -40 \beta_{4} - 24 \beta_{6} ) q^{23} + ( -339 + 80 \beta_{2} ) q^{25} + ( -8 \beta_{1} + 19 \beta_{3} ) q^{26} + ( -344 \beta_{4} + 69 \beta_{6} ) q^{29} + ( -16 \beta_{5} - 184 \beta_{7} ) q^{31} + ( -841 \beta_{4} - 59 \beta_{6} ) q^{32} + ( 165 \beta_{5} + 103 \beta_{7} ) q^{34} + ( -2524 + 264 \beta_{2} ) q^{37} + ( -48 \beta_{1} + 92 \beta_{3} ) q^{38} + ( -397 \beta_{5} + 83 \beta_{7} ) q^{40} + ( -13 \beta_{1} - 236 \beta_{3} ) q^{41} + ( -5140 - 16 \beta_{2} ) q^{43} + ( -612 \beta_{4} + 44 \beta_{6} ) q^{44} + ( -1736 + 184 \beta_{2} ) q^{46} + ( -100 \beta_{1} + 152 \beta_{3} ) q^{47} + ( -2579 \beta_{4} - 80 \beta_{6} ) q^{50} + ( 577 \beta_{5} + 29 \beta_{7} ) q^{52} + ( -4328 \beta_{4} + 238 \beta_{6} ) q^{53} + ( 804 \beta_{5} - 136 \beta_{7} ) q^{55} + ( -11074 - 70 \beta_{2} ) q^{58} + ( 288 \beta_{1} + 440 \beta_{3} ) q^{59} + ( -2399 \beta_{5} + 116 \beta_{7} ) q^{61} + ( 368 \beta_{1} - 752 \beta_{3} ) q^{62} + ( -26517 + 331 \beta_{2} ) q^{64} + ( -728 \beta_{4} + 95 \beta_{6} ) q^{65} + ( -4892 - 800 \beta_{2} ) q^{67} + ( 594 \beta_{1} + 449 \beta_{3} ) q^{68} + ( -5128 \beta_{4} - 636 \beta_{6} ) q^{71} + ( 2605 \beta_{5} + 784 \beta_{7} ) q^{73} + ( -9916 \beta_{4} - 264 \beta_{6} ) q^{74} + ( 3396 \beta_{5} + 20 \beta_{7} ) q^{76} + ( -30320 - 2128 \beta_{2} ) q^{79} + ( 666 \beta_{1} + 351 \beta_{3} ) q^{80} + ( -8247 \beta_{5} - 1613 \beta_{7} ) q^{82} + ( -420 \beta_{1} - 1168 \beta_{3} ) q^{83} + ( -70042 - 1832 \beta_{2} ) q^{85} + ( -4692 \beta_{4} + 16 \beta_{6} ) q^{86} + ( -10948 - 292 \beta_{2} ) q^{88} + ( -895 \beta_{1} + 340 \beta_{3} ) q^{89} + ( -5608 \beta_{4} + 584 \beta_{6} ) q^{92} + ( 5420 \beta_{5} + 1364 \beta_{7} ) q^{94} + ( -4368 \beta_{4} + 724 \beta_{6} ) q^{95} + ( 5613 \beta_{5} - 1736 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 20 q^{4} + O(q^{10})$$ $$8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 146 x^{6} + 5453 x^{4} - 40868 x^{2} + 3844$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$13 \nu^{6} + 2930 \nu^{4} - 324365 \nu^{2} + 1463438$$$$)/82236$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} + 227 \nu^{4} - 2060 \nu^{2} - 90946$$$$)/2937$$ $$\beta_{3}$$ $$=$$ $$($$$$61 \nu^{6} - 8392 \nu^{4} + 271357 \nu^{2} - 970822$$$$)/11748$$ $$\beta_{4}$$ $$=$$ $$($$$$2911 \nu^{7} - 428788 \nu^{5} + 16393987 \nu^{3} - 135790882 \nu$$$$)/5098632$$ $$\beta_{5}$$ $$=$$ $$($$$$-343 \nu^{7} + 49210 \nu^{5} - 1771861 \nu^{3} + 11849026 \nu$$$$)/364188$$ $$\beta_{6}$$ $$=$$ $$($$$$3931 \nu^{7} - 597424 \nu^{5} + 24375907 \nu^{3} - 205795858 \nu$$$$)/2549316$$ $$\beta_{7}$$ $$=$$ $$($$$$-722 \nu^{7} + 105443 \nu^{5} - 3895061 \nu^{3} + 26260934 \nu$$$$)/182094$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{5} - 6 \beta_{4}$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{3} + 49 \beta_{2} + 14 \beta_{1} + 1764$$$$)/49$$ $$\nu^{3}$$ $$=$$ $$($$$$42 \beta_{7} + 75 \beta_{6} - 208 \beta_{5} - 254 \beta_{4}$$$$)/14$$ $$\nu^{4}$$ $$=$$ $$($$$$460 \beta_{3} + 3969 \beta_{2} + 1988 \beta_{1} + 125538$$$$)/49$$ $$\nu^{5}$$ $$=$$ $$($$$$5110 \beta_{7} + 5965 \beta_{6} - 22868 \beta_{5} - 18346 \beta_{4}$$$$)/14$$ $$\nu^{6}$$ $$=$$ $$($$$$46030 \beta_{3} + 328055 \beta_{2} + 211218 \beta_{1} + 10203466$$$$)/49$$ $$\nu^{7}$$ $$=$$ $$($$$$516166 \beta_{7} + 502907 \beta_{6} - 2290332 \beta_{5} - 1527254 \beta_{4}$$$$)/14$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.13708 0.308653 6.71178 9.54020 −6.71178 −9.54020 −3.13708 −0.308653
−8.12599 0 34.0317 −17.4201 0 0 −16.5098 0 141.556
1.2 −8.12599 0 34.0317 17.4201 0 0 −16.5098 0 −141.556
1.3 −1.72287 0 −29.0317 −73.1337 0 0 105.150 0 126.000
1.4 −1.72287 0 −29.0317 73.1337 0 0 105.150 0 −126.000
1.5 1.72287 0 −29.0317 −73.1337 0 0 −105.150 0 −126.000
1.6 1.72287 0 −29.0317 73.1337 0 0 −105.150 0 126.000
1.7 8.12599 0 34.0317 −17.4201 0 0 16.5098 0 −141.556
1.8 8.12599 0 34.0317 17.4201 0 0 16.5098 0 141.556
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.be 8
3.b odd 2 1 inner 441.6.a.be 8
7.b odd 2 1 inner 441.6.a.be 8
21.c even 2 1 inner 441.6.a.be 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.be 8 1.a even 1 1 trivial
441.6.a.be 8 3.b odd 2 1 inner
441.6.a.be 8 7.b odd 2 1 inner
441.6.a.be 8 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{4} - 69 T_{2}^{2} + 196$$ $$T_{5}^{4} - 5652 T_{5}^{2} + 1623076$$ $$T_{13}^{4} - 65076 T_{13}^{2} + 966836836$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 196 - 69 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 1623076 - 5652 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 486820096 - 50336 T^{2} + T^{4} )^{2}$$
$13$ $$( 966836836 - 65076 T^{2} + T^{4} )^{2}$$
$17$ $$( 1776617078404 - 3671492 T^{2} + T^{4} )^{2}$$
$19$ $$( 1304566014976 - 2297152 T^{2} + T^{4} )^{2}$$
$23$ $$( 4889971523584 - 6663744 T^{2} + T^{4} )^{2}$$
$29$ $$( 71690156736016 - 60939144 T^{2} + T^{4} )^{2}$$
$31$ $$( 4532232295284736 - 134646912 T^{2} + T^{4} )^{2}$$
$37$ $$( -63573584 + 4784 T + T^{2} )^{4}$$
$41$ $$( 8711880319202116 - 376626596 T^{2} + T^{4} )^{2}$$
$43$ $$( 26247376 + 10296 T + T^{2} )^{4}$$
$47$ $$( 4349206235014144 - 217430848 T^{2} + T^{4} )^{2}$$
$53$ $$( 883420160831600896 - 1887460256 T^{2} + T^{4} )^{2}$$
$59$ $$( 753088058960183296 - 1738892928 T^{2} + T^{4} )^{2}$$
$61$ $$( 292848892042281316 - 1189339316 T^{2} + T^{4} )^{2}$$
$67$ $$( -608314736 + 10584 T + T^{2} )^{4}$$
$71$ $$( 10660943416583520256 - 6530375232 T^{2} + T^{4} )^{2}$$
$73$ $$( 342808093021392964 - 3717977508 T^{2} + T^{4} )^{2}$$
$79$ $$( -3517390336 + 62768 T + T^{2} )^{4}$$
$83$ $$( 19247961579756012544 - 10070803008 T^{2} + T^{4} )^{2}$$
$89$ $$( 6562421910834002500 - 5402804900 T^{2} + T^{4} )^{2}$$
$97$ $$( 7657638990259796356 - 18436450116 T^{2} + T^{4} )^{2}$$