# Properties

 Label 441.6.a.x Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.358541904.1 Defining polynomial: $$x^{4} - 111 x^{2} + 756$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 24 + \beta_{3} ) q^{4} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 32 \beta_{1} + 3 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 24 + \beta_{3} ) q^{4} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 32 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -140 + 2 \beta_{3} ) q^{10} + ( 45 \beta_{1} + \beta_{2} ) q^{11} + ( -210 + 16 \beta_{3} ) q^{13} + ( 1108 + 15 \beta_{3} ) q^{16} + ( 131 \beta_{1} + 31 \beta_{2} ) q^{17} + ( -1708 - 16 \beta_{3} ) q^{19} + ( 36 \beta_{1} - 26 \beta_{2} ) q^{20} + ( 2548 + 50 \beta_{3} ) q^{22} + ( -237 \beta_{1} + 47 \beta_{2} ) q^{23} + ( 711 - 80 \beta_{3} ) q^{25} + ( 430 \beta_{1} + 48 \beta_{2} ) q^{26} + ( -496 \beta_{1} + 32 \beta_{2} ) q^{29} + ( 1204 + 48 \beta_{3} ) q^{31} + ( 684 \beta_{1} - 51 \beta_{2} ) q^{32} + ( 8204 + 286 \beta_{3} ) q^{34} + ( 1634 + 240 \beta_{3} ) q^{37} + ( -2348 \beta_{1} - 48 \beta_{2} ) q^{38} + ( 5768 - 158 \beta_{3} ) q^{40} + ( 809 \beta_{1} + 29 \beta_{2} ) q^{41} + ( 2700 + 128 \beta_{3} ) q^{43} + ( 3108 \beta_{1} + 118 \beta_{2} ) q^{44} + ( -11956 - 2 \beta_{3} ) q^{46} + ( 2582 \beta_{1} - 114 \beta_{2} ) q^{47} + ( -2489 \beta_{1} - 240 \beta_{2} ) q^{50} + ( 32144 + 158 \beta_{3} ) q^{52} + ( 1510 \beta_{1} + 14 \beta_{2} ) q^{53} + ( -2884 + 16 \beta_{3} ) q^{55} + ( -26880 - 336 \beta_{3} ) q^{58} + ( 1642 \beta_{1} - 174 \beta_{2} ) q^{59} + ( -3206 + 240 \beta_{3} ) q^{61} + ( 3124 \beta_{1} + 144 \beta_{2} ) q^{62} + ( 1420 - 51 \beta_{3} ) q^{64} + ( 2358 \beta_{1} - 1010 \beta_{2} ) q^{65} + ( 50828 - 96 \beta_{3} ) q^{67} + ( 15452 \beta_{1} - 134 \beta_{2} ) q^{68} + ( -1287 \beta_{1} + 717 \beta_{2} ) q^{71} + ( -4942 - 1120 \beta_{3} ) q^{73} + ( 11234 \beta_{1} + 720 \beta_{2} ) q^{74} + ( -78176 - 2076 \beta_{3} ) q^{76} + ( 8264 - 672 \beta_{3} ) q^{79} + ( -1704 \beta_{1} + 358 \beta_{2} ) q^{80} + ( 46116 + 954 \beta_{3} ) q^{82} + ( -9296 \beta_{1} + 112 \beta_{2} ) q^{83} + ( 87556 - 2032 \beta_{3} ) q^{85} + ( 7820 \beta_{1} + 384 \beta_{2} ) q^{86} + ( 95816 + 2098 \beta_{3} ) q^{88} + ( 1453 \beta_{1} + 113 \beta_{2} ) q^{89} + ( -4452 \beta_{1} - 1510 \beta_{2} ) q^{92} + ( 141400 + 2012 \beta_{3} ) q^{94} + ( 3396 \beta_{1} - 908 \beta_{2} ) q^{95} + ( -109326 - 224 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 94 q^{4} + O(q^{10})$$ $$4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 111 x^{2} + 756$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 96 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 56$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 56$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + 96 \beta_{1}$$

## 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
 −10.1838 −2.69991 2.69991 10.1838
−10.1838 0 71.7105 4.37743 0 0 −404.405 0 −44.5790
1.2 −2.69991 0 −24.7105 87.9366 0 0 153.113 0 −237.421
1.3 2.69991 0 −24.7105 −87.9366 0 0 −153.113 0 −237.421
1.4 10.1838 0 71.7105 −4.37743 0 0 404.405 0 −44.5790
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.x 4
3.b odd 2 1 inner 441.6.a.x 4
7.b odd 2 1 63.6.a.h 4
21.c even 2 1 63.6.a.h 4
28.d even 2 1 1008.6.a.by 4
84.h odd 2 1 1008.6.a.by 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.a.h 4 7.b odd 2 1
63.6.a.h 4 21.c even 2 1
441.6.a.x 4 1.a even 1 1 trivial
441.6.a.x 4 3.b odd 2 1 inner
1008.6.a.by 4 28.d even 2 1
1008.6.a.by 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 111 T_{2}^{2} + 756$$ $$T_{5}^{4} - 7752 T_{5}^{2} + 148176$$ $$T_{13}^{2} + 436 T_{13} - 547484$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$756 - 111 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$148176 - 7752 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$407299536 - 236424 T^{2} + T^{4}$$
$13$ $$( -547484 + 436 T + T^{2} )^{2}$$
$17$ $$20712534557904 - 9102792 T^{2} + T^{4}$$
$19$ $$( 2294992 + 3400 T + T^{2} )^{2}$$
$23$ $$27015936275664 - 20691912 T^{2} + T^{4}$$
$29$ $$269206953394176 - 32917248 T^{2} + T^{4}$$
$31$ $$( -3962672 - 2360 T + T^{2} )^{2}$$
$37$ $$( -131584604 - 3028 T + T^{2} )^{2}$$
$41$ $$1390182638544 - 80977032 T^{2} + T^{4}$$
$43$ $$( -31132016 - 5272 T + T^{2} )^{2}$$
$47$ $$140373649856998656 - 801721632 T^{2} + T^{4}$$
$53$ $$2170530954982656 - 256630944 T^{2} + T^{4}$$
$59$ $$49715173391498496 - 483850272 T^{2} + T^{4}$$
$61$ $$( -122814524 + 6652 T + T^{2} )^{2}$$
$67$ $$( 2566947088 - 101752 T + T^{2} )^{2}$$
$71$ $$118112942107725264 - 3718687752 T^{2} + T^{4}$$
$73$ $$( -2896337276 + 8764 T + T^{2} )^{2}$$
$79$ $$( -975634112 - 17200 T + T^{2} )^{2}$$
$83$ $$9751577272444256256 - 9574483968 T^{2} + T^{4}$$
$89$ $$8194654066720464 - 341227848 T^{2} + T^{4}$$
$97$ $$( 11811076228 + 218428 T + T^{2} )^{2}$$