# Properties

 Label 441.4.a.x Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 74 x^{6} + 1469 x^{4} - 8828 x^{2} + 2500$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 7^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( 8 - \beta_{3} ) q^{4} + \beta_{1} q^{5} + ( -11 \beta_{2} - 2 \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( 8 - \beta_{3} ) q^{4} + \beta_{1} q^{5} + ( -11 \beta_{2} - 2 \beta_{4} ) q^{8} + ( -3 \beta_{5} - 2 \beta_{7} ) q^{10} + ( -7 \beta_{2} + \beta_{4} ) q^{11} + ( 2 \beta_{5} + 3 \beta_{7} ) q^{13} + ( 96 - 9 \beta_{3} ) q^{16} + ( \beta_{1} - 2 \beta_{6} ) q^{17} + ( 6 \beta_{5} - 2 \beta_{7} ) q^{19} + ( 14 \beta_{1} + 5 \beta_{6} ) q^{20} + ( 120 - 4 \beta_{3} ) q^{22} + ( 7 \beta_{2} + 3 \beta_{4} ) q^{23} + ( 65 - 4 \beta_{3} ) q^{25} + ( -18 \beta_{1} - 5 \beta_{6} ) q^{26} + ( -13 \beta_{2} + 9 \beta_{4} ) q^{29} + ( -2 \beta_{5} + 10 \beta_{7} ) q^{31} + ( -107 \beta_{2} - 2 \beta_{4} ) q^{32} + ( -\beta_{5} + 18 \beta_{7} ) q^{34} + ( 20 + 24 \beta_{3} ) q^{37} + ( -32 \beta_{1} - 4 \beta_{6} ) q^{38} + ( -23 \beta_{5} - 62 \beta_{7} ) q^{40} + ( \beta_{1} + 10 \beta_{6} ) q^{41} + ( 280 + 20 \beta_{3} ) q^{43} + ( -108 \beta_{2} - 16 \beta_{4} ) q^{44} + ( -88 + 16 \beta_{3} ) q^{46} + ( 30 \beta_{1} + 2 \beta_{6} ) q^{47} + ( -109 \beta_{2} - 8 \beta_{4} ) q^{50} + ( 43 \beta_{5} + 62 \beta_{7} ) q^{52} + ( -34 \beta_{2} - 14 \beta_{4} ) q^{53} + ( -26 \beta_{5} + 6 \beta_{7} ) q^{55} + ( 280 + 14 \beta_{3} ) q^{58} + ( -2 \beta_{1} - 10 \beta_{6} ) q^{59} + ( 22 \beta_{5} - 55 \beta_{7} ) q^{61} + ( -8 \beta_{1} - 8 \beta_{6} ) q^{62} + ( 928 - 41 \beta_{3} ) q^{64} + ( 191 \beta_{2} + 17 \beta_{4} ) q^{65} + ( 20 + 40 \beta_{3} ) q^{67} + ( -38 \beta_{1} - \beta_{6} ) q^{68} + ( 163 \beta_{2} - 9 \beta_{4} ) q^{71} + ( -28 \beta_{5} + 23 \beta_{7} ) q^{73} + ( 244 \beta_{2} + 48 \beta_{4} ) q^{74} + ( 52 \beta_{5} + 120 \beta_{7} ) q^{76} + ( 660 - 28 \beta_{3} ) q^{79} + ( 150 \beta_{1} + 45 \beta_{6} ) q^{80} + ( -13 \beta_{5} - 102 \beta_{7} ) q^{82} + ( 52 \beta_{1} - 16 \beta_{6} ) q^{83} + ( 390 + 64 \beta_{3} ) q^{85} + ( -60 \beta_{2} + 40 \beta_{4} ) q^{86} + ( 640 - 124 \beta_{3} ) q^{88} + ( 7 \beta_{1} + 10 \beta_{6} ) q^{89} + ( 208 \beta_{2} + 8 \beta_{4} ) q^{92} + ( -92 \beta_{5} - 80 \beta_{7} ) q^{94} + ( 342 \beta_{2} - 26 \beta_{4} ) q^{95} + ( -28 \beta_{5} - 57 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 68q^{4} + O(q^{10})$$ $$8q + 68q^{4} + 804q^{16} + 976q^{22} + 536q^{25} + 64q^{37} + 2160q^{43} - 768q^{46} + 2184q^{58} + 7588q^{64} + 5392q^{79} + 2864q^{85} + 5616q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 74 x^{6} + 1469 x^{4} - 8828 x^{2} + 2500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 148 \nu^{4} + 4633 \nu^{2} - 18202$$$$)/1416$$ $$\beta_{2}$$ $$=$$ $$($$$$13 \nu^{7} - 862 \nu^{5} + 13147 \nu^{3} - 67414 \nu$$$$)/17700$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{6} - 119 \nu^{4} + 1124 \nu^{2} + 1828$$$$)/177$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 148 \nu^{5} - 6049 \nu^{3} + 60682 \nu$$$$)/1416$$ $$\beta_{5}$$ $$=$$ $$($$$$-28 \nu^{7} + 2197 \nu^{5} - 46357 \nu^{3} + 246634 \nu$$$$)/8850$$ $$\beta_{6}$$ $$=$$ $$($$$$-29 \nu^{6} + 1814 \nu^{4} - 22847 \nu^{2} + 37214$$$$)/708$$ $$\beta_{7}$$ $$=$$ $$($$$$91 \nu^{7} - 6034 \nu^{5} + 92029 \nu^{3} - 347998 \nu$$$$)/17700$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 7 \beta_{2}$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{6} - 7 \beta_{3} - 4 \beta_{1} + 126$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$($$$$44 \beta_{7} + 21 \beta_{5} - 14 \beta_{4} - 231 \beta_{2}$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$($$$$-92 \beta_{6} - 315 \beta_{3} - 296 \beta_{1} + 4284$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$2034 \beta_{7} + 1295 \beta_{5} - 742 \beta_{4} - 9373 \beta_{2}$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$($$$$-4350 \beta_{6} - 14189 \beta_{3} - 15364 \beta_{1} + 177688$$$$)/7$$ $$\nu^{7}$$ $$=$$ $$($$$$95558 \beta_{7} + 64631 \beta_{5} - 35042 \beta_{4} - 414659 \beta_{2}$$$$)/7$$

## 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
 −6.81412 −3.98569 −0.545636 −3.37406 0.545636 3.37406 6.81412 3.98569
−5.39991 0 21.1590 −15.5768 0 0 −71.0573 0 84.1131
1.2 −5.39991 0 21.1590 15.5768 0 0 −71.0573 0 −84.1131
1.3 −1.95985 0 −4.15899 −11.8897 0 0 23.8298 0 23.3019
1.4 −1.95985 0 −4.15899 11.8897 0 0 23.8298 0 −23.3019
1.5 1.95985 0 −4.15899 −11.8897 0 0 −23.8298 0 −23.3019
1.6 1.95985 0 −4.15899 11.8897 0 0 −23.8298 0 23.3019
1.7 5.39991 0 21.1590 −15.5768 0 0 71.0573 0 −84.1131
1.8 5.39991 0 21.1590 15.5768 0 0 71.0573 0 84.1131
 $$n$$: 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 441.4.a.x 8
3.b odd 2 1 inner 441.4.a.x 8
7.b odd 2 1 inner 441.4.a.x 8
7.c even 3 2 441.4.e.z 16
7.d odd 6 2 441.4.e.z 16
21.c even 2 1 inner 441.4.a.x 8
21.g even 6 2 441.4.e.z 16
21.h odd 6 2 441.4.e.z 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.x 8 1.a even 1 1 trivial
441.4.a.x 8 3.b odd 2 1 inner
441.4.a.x 8 7.b odd 2 1 inner
441.4.a.x 8 21.c even 2 1 inner
441.4.e.z 16 7.c even 3 2
441.4.e.z 16 7.d odd 6 2
441.4.e.z 16 21.g even 6 2
441.4.e.z 16 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{4} - 33 T_{2}^{2} + 112$$ $$T_{5}^{4} - 384 T_{5}^{2} + 34300$$ $$T_{13}^{4} - 5268 T_{13}^{2} + 4900$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 112 - 33 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 34300 - 384 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 1355200 - 2348 T^{2} + T^{4} )^{2}$$
$13$ $$( 4900 - 5268 T^{2} + T^{4} )^{2}$$
$17$ $$( 8134588 - 10496 T^{2} + T^{4} )^{2}$$
$19$ $$( 133079296 - 23080 T^{2} + T^{4} )^{2}$$
$23$ $$( 9033472 - 6012 T^{2} + T^{4} )^{2}$$
$29$ $$( 16601200 - 53088 T^{2} + T^{4} )^{2}$$
$31$ $$( 51380224 - 19464 T^{2} + T^{4} )^{2}$$
$37$ $$( -92240 - 16 T + T^{2} )^{4}$$
$41$ $$( 10038958300 - 233264 T^{2} + T^{4} )^{2}$$
$43$ $$( 8800 - 540 T + T^{2} )^{4}$$
$47$ $$( 19149958912 - 335128 T^{2} + T^{4} )^{2}$$
$53$ $$( 4433997568 - 133376 T^{2} + T^{4} )^{2}$$
$59$ $$( 10756480000 - 231096 T^{2} + T^{4} )^{2}$$
$61$ $$( 3928531684 - 745844 T^{2} + T^{4} )^{2}$$
$67$ $$( -256400 + T^{2} )^{4}$$
$71$ $$( 187904819200 - 959388 T^{2} + T^{4} )^{2}$$
$73$ $$( 55088784100 - 535668 T^{2} + T^{4} )^{2}$$
$79$ $$( 328640 - 1348 T + T^{2} )^{4}$$
$83$ $$( 351934815232 - 1919232 T^{2} + T^{4} )^{2}$$
$89$ $$( 13398437500 - 231776 T^{2} + T^{4} )^{2}$$
$97$ $$( 35588822500 - 1382388 T^{2} + T^{4} )^{2}$$