# Properties

 Label 441.6.a.bd Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 187 x^{4} + 9570 x^{2} - 135576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 30 + \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + ( 21 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 30 + \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + ( 21 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{8} + ( 112 + 7 \beta_{3} + 8 \beta_{5} ) q^{10} + ( -24 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{11} + ( -28 + 7 \beta_{3} - 13 \beta_{5} ) q^{13} + ( 302 + 43 \beta_{3} + 14 \beta_{5} ) q^{16} + ( 76 \beta_{1} + 10 \beta_{2} ) q^{17} + ( 1568 + 7 \beta_{3} - \beta_{5} ) q^{19} + ( 249 \beta_{1} + 30 \beta_{2} + 7 \beta_{4} ) q^{20} + ( -1492 - 97 \beta_{3} + 28 \beta_{5} ) q^{22} + ( -64 \beta_{1} + 22 \beta_{2} + 4 \beta_{4} ) q^{23} + ( 1246 + 25 \beta_{3} - 35 \beta_{5} ) q^{25} + ( 68 \beta_{1} - 64 \beta_{2} + 7 \beta_{4} ) q^{26} + ( 250 \beta_{1} + 61 \beta_{2} - 8 \beta_{4} ) q^{29} + ( 3857 + 140 \beta_{3} - 18 \beta_{5} ) q^{31} + ( 689 \beta_{1} + 106 \beta_{2} + 11 \beta_{4} ) q^{32} + ( 4592 + 126 \beta_{3} + 80 \beta_{5} ) q^{34} + ( 3060 - 89 \beta_{3} - 133 \beta_{5} ) q^{37} + ( 1724 \beta_{1} + 8 \beta_{2} + 7 \beta_{4} ) q^{38} + ( 11382 + 483 \beta_{3} - 30 \beta_{5} ) q^{40} + ( 184 \beta_{1} - 104 \beta_{2} - 56 \beta_{4} ) q^{41} + ( -7270 - 33 \beta_{3} - 189 \beta_{5} ) q^{43} + ( -2815 \beta_{1} - 122 \beta_{2} - 33 \beta_{4} ) q^{44} + ( -4296 + 222 \beta_{3} + 168 \beta_{5} ) q^{46} + ( -312 \beta_{1} + 68 \beta_{2} - 56 \beta_{4} ) q^{47} + ( 1646 \beta_{1} - 160 \beta_{2} + 25 \beta_{4} ) q^{50} + ( 5768 - 168 \beta_{3} - 110 \beta_{5} ) q^{52} + ( 1202 \beta_{1} - 175 \beta_{2} ) q^{53} + ( 8281 - 763 \beta_{3} - 153 \beta_{5} ) q^{55} + ( 14896 + 203 \beta_{3} + 504 \beta_{5} ) q^{58} + ( 4440 \beta_{1} - 83 \beta_{2} - 70 \beta_{4} ) q^{59} + ( -2576 - 350 \beta_{3} + 274 \beta_{5} ) q^{61} + ( 6987 \beta_{1} + 172 \beta_{2} + 140 \beta_{4} ) q^{62} + ( 31606 + 327 \beta_{3} + 378 \beta_{5} ) q^{64} + ( -3972 \beta_{1} + 414 \beta_{2} + 88 \beta_{4} ) q^{65} + ( -24004 - 313 \beta_{3} + 119 \beta_{5} ) q^{67} + ( 5458 \beta_{1} + 412 \beta_{2} + 126 \beta_{4} ) q^{68} + ( 3336 \beta_{1} - 264 \beta_{2} - 48 \beta_{4} ) q^{71} + ( -16716 + 119 \beta_{3} + 259 \beta_{5} ) q^{73} + ( 348 \beta_{1} - 976 \beta_{2} - 89 \beta_{4} ) q^{74} + ( 56504 + 1848 \beta_{3} + 82 \beta_{5} ) q^{76} + ( -16761 - 714 \beta_{3} - 812 \beta_{5} ) q^{79} + ( 14373 \beta_{1} - 174 \beta_{2} + 259 \beta_{4} ) q^{80} + ( 13552 - 2800 \beta_{3} - 720 \beta_{5} ) q^{82} + ( 4312 \beta_{1} - 567 \beta_{2} - 14 \beta_{4} ) q^{83} + ( 49982 + 642 \beta_{3} + 98 \beta_{5} ) q^{85} + ( -8974 \beta_{1} - 1200 \beta_{2} - 33 \beta_{4} ) q^{86} + ( -124794 - 1773 \beta_{3} - 1806 \beta_{5} ) q^{88} + ( 492 \beta_{1} + 190 \beta_{2} + 112 \beta_{4} ) q^{89} + ( 3698 \beta_{1} + 748 \beta_{2} + 94 \beta_{4} ) q^{92} + ( -19264 - 2436 \beta_{3} + 656 \beta_{5} ) q^{94} + ( 4056 \beta_{1} + 1602 \beta_{2} + 52 \beta_{4} ) q^{95} + ( 72891 - 2149 \beta_{3} + 7 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 182 q^{4} + O(q^{10})$$ $$6 q + 182 q^{4} + 686 q^{10} - 154 q^{13} + 1898 q^{16} + 9422 q^{19} - 9146 q^{22} + 7526 q^{25} + 23422 q^{31} + 27804 q^{34} + 18182 q^{37} + 69258 q^{40} - 43686 q^{43} - 25332 q^{46} + 34272 q^{52} + 48160 q^{55} + 89782 q^{58} - 16156 q^{61} + 190290 q^{64} - 144650 q^{67} - 100058 q^{73} + 342720 q^{76} - 101994 q^{79} + 75712 q^{82} + 301176 q^{85} - 752310 q^{88} - 120456 q^{94} + 433048 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 187 x^{4} + 9570 x^{2} - 135576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 139 \nu^{3} + 3318 \nu$$$$)/84$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 62$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 181 \nu^{3} - 6888 \nu$$$$)/42$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{4} - 139 \nu^{2} + 3388$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 62$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2 \beta_{2} + 85 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$14 \beta_{5} + 139 \beta_{3} + 5230$$ $$\nu^{5}$$ $$=$$ $$139 \beta_{4} + 362 \beta_{2} + 8497 \beta_{1}$$

## 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
 −10.6223 −7.08933 −4.88952 4.88952 7.08933 10.6223
−10.6223 0 80.8340 −67.4751 0 0 −518.731 0 716.743
1.2 −7.08933 0 18.2586 82.2041 0 0 97.4172 0 −582.772
1.3 −4.88952 0 −8.09260 −42.7504 0 0 196.034 0 209.029
1.4 4.88952 0 −8.09260 42.7504 0 0 −196.034 0 209.029
1.5 7.08933 0 18.2586 −82.2041 0 0 −97.4172 0 −582.772
1.6 10.6223 0 80.8340 67.4751 0 0 518.731 0 716.743
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.bd 6
3.b odd 2 1 inner 441.6.a.bd 6
7.b odd 2 1 441.6.a.bc 6
7.d odd 6 2 63.6.e.f 12
21.c even 2 1 441.6.a.bc 6
21.g even 6 2 63.6.e.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.f 12 7.d odd 6 2
63.6.e.f 12 21.g even 6 2
441.6.a.bc 6 7.b odd 2 1
441.6.a.bc 6 21.c even 2 1
441.6.a.bd 6 1.a even 1 1 trivial
441.6.a.bd 6 3.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{6} - 187 T_{2}^{4} + 9570 T_{2}^{2} - 135576$$ $$T_{5}^{6} - 13138 T_{5}^{4} + 51437073 T_{5}^{2} - 56228247936$$ $$T_{13}^{3} + 77 T_{13}^{2} - 783976 T_{13} - 60080636$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-135576 + 9570 T^{2} - 187 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$-56228247936 + 51437073 T^{2} - 13138 T^{4} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$-17126803572350976 + 234630567921 T^{2} - 902578 T^{4} + T^{6}$$
$13$ $$( -60080636 - 783976 T + 77 T^{2} + T^{3} )^{2}$$
$17$ $$-143225492794546176 + 1172856312720 T^{2} - 2284392 T^{4} + T^{6}$$
$19$ $$( -3713875508 + 7296404 T - 4711 T^{2} + T^{3} )^{2}$$
$23$ $$-11254029407056041984 + 20238938612496 T^{2} - 10191432 T^{4} + T^{6}$$
$29$ $$-$$$$75\!\cdots\!56$$$$+ 1273417115193153 T^{2} - 66189634 T^{4} + T^{6}$$
$31$ $$( 85806884751 + 5269711 T - 11711 T^{2} + T^{3} )^{2}$$
$37$ $$( 32349623604 - 76541360 T - 9091 T^{2} + T^{3} )^{2}$$
$41$ $$-$$$$13\!\cdots\!64$$$$+ 172110236322742272 T^{2} - 727900480 T^{4} + T^{6}$$
$43$ $$( -1643621929904 - 6827376 T + 21843 T^{2} + T^{3} )^{2}$$
$47$ $$-$$$$19\!\cdots\!64$$$$+ 78904457818511616 T^{2} - 613454880 T^{4} + T^{6}$$
$53$ $$-$$$$16\!\cdots\!64$$$$+ 14111509664477745 T^{2} - 666461298 T^{4} + T^{6}$$
$59$ $$-$$$$31\!\cdots\!04$$$$+ 6958714510238532561 T^{2} - 4695322386 T^{4} + T^{6}$$
$61$ $$( -5346133312352 - 496008496 T + 8078 T^{2} + T^{3} )^{2}$$
$67$ $$( 7112548657724 + 1504134932 T + 72325 T^{2} + T^{3} )^{2}$$
$71$ $$-$$$$30\!\cdots\!44$$$$+ 2282421134301143040 T^{2} - 3481391808 T^{4} + T^{6}$$
$73$ $$( 937824660612 + 483227416 T + 50029 T^{2} + T^{3} )^{2}$$
$79$ $$( -111559545344717 - 3564374073 T + 50997 T^{2} + T^{3} )^{2}$$
$83$ $$-$$$$11\!\cdots\!64$$$$+ 2503294783326306081 T^{2} - 7727019426 T^{4} + T^{6}$$
$89$ $$-$$$$60\!\cdots\!96$$$$+ 2433875883884953104 T^{2} - 2820380296 T^{4} + T^{6}$$
$97$ $$( 546802680855102 + 6003490045 T - 216524 T^{2} + T^{3} )^{2}$$