# Properties

 Label 441.6.a.q Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 17 + 3 \beta ) q^{4} -36 q^{5} + ( 129 - 9 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 17 + 3 \beta ) q^{4} -36 q^{5} + ( 129 - 9 \beta ) q^{8} + ( -36 - 36 \beta ) q^{10} + ( -236 - 8 \beta ) q^{11} + ( 612 + 72 \beta ) q^{13} + ( -847 + 15 \beta ) q^{16} + ( -576 + 216 \beta ) q^{17} + ( 36 - 288 \beta ) q^{19} + ( -612 - 108 \beta ) q^{20} + ( -620 - 252 \beta ) q^{22} + ( 224 + 56 \beta ) q^{23} -1829 q^{25} + ( 4068 + 756 \beta ) q^{26} + ( -2866 - 640 \beta ) q^{29} + ( 5256 - 576 \beta ) q^{31} + ( -4255 - 529 \beta ) q^{32} + ( 9792 - 144 \beta ) q^{34} + ( 6090 - 1056 \beta ) q^{37} + ( -13788 - 540 \beta ) q^{38} + ( -4644 + 324 \beta ) q^{40} + ( -10368 - 216 \beta ) q^{41} + ( -1932 - 2400 \beta ) q^{43} + ( -5164 - 868 \beta ) q^{44} + ( 2912 + 336 \beta ) q^{46} + ( -2232 - 3456 \beta ) q^{47} + ( -1829 - 1829 \beta ) q^{50} + ( 20772 + 3276 \beta ) q^{52} + ( -1702 + 1184 \beta ) q^{53} + ( 8496 + 288 \beta ) q^{55} + ( -33586 - 4146 \beta ) q^{58} + ( -14436 - 864 \beta ) q^{59} + ( -10980 + 4680 \beta ) q^{61} + ( -22392 + 4104 \beta ) q^{62} + ( -2543 - 5793 \beta ) q^{64} + ( -22032 - 2592 \beta ) q^{65} + ( -13580 + 6480 \beta ) q^{67} + ( 21312 + 2592 \beta ) q^{68} + ( 47416 - 2552 \beta ) q^{71} + ( 29232 - 1872 \beta ) q^{73} + ( -44598 + 3978 \beta ) q^{74} + ( -40860 - 5652 \beta ) q^{76} + ( -24808 - 6480 \beta ) q^{79} + ( 30492 - 540 \beta ) q^{80} + ( -20736 - 10800 \beta ) q^{82} + ( 40428 - 9504 \beta ) q^{83} + ( 20736 - 7776 \beta ) q^{85} + ( -117132 - 6732 \beta ) q^{86} + ( -26988 + 1164 \beta ) q^{88} + ( -63432 + 3672 \beta ) q^{89} + ( 11872 + 1792 \beta ) q^{92} + ( -168120 - 9144 \beta ) q^{94} + ( -1296 + 10368 \beta ) q^{95} + ( 8136 + 19584 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8} + O(q^{10})$$ $$2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8} - 108 q^{10} - 480 q^{11} + 1296 q^{13} - 1679 q^{16} - 936 q^{17} - 216 q^{19} - 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} + 8892 q^{26} - 6372 q^{29} + 9936 q^{31} - 9039 q^{32} + 19440 q^{34} + 11124 q^{37} - 28116 q^{38} - 8964 q^{40} - 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} - 7920 q^{47} - 5487 q^{50} + 44820 q^{52} - 2220 q^{53} + 17280 q^{55} - 71318 q^{58} - 29736 q^{59} - 17280 q^{61} - 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} + 45216 q^{68} + 92280 q^{71} + 56592 q^{73} - 85218 q^{74} - 87372 q^{76} - 56096 q^{79} + 60444 q^{80} - 52272 q^{82} + 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} - 123192 q^{89} + 25536 q^{92} - 345384 q^{94} + 7776 q^{95} + 35856 q^{97} + O(q^{100})$$

## 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
 −6.44622 7.44622
−5.44622 0 −2.33867 −36.0000 0 0 187.016 0 196.064
1.2 8.44622 0 39.3387 −36.0000 0 0 61.9840 0 −304.064
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.q 2
3.b odd 2 1 147.6.a.h 2
7.b odd 2 1 441.6.a.r 2
21.c even 2 1 147.6.a.j yes 2
21.g even 6 2 147.6.e.m 4
21.h odd 6 2 147.6.e.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 3.b odd 2 1
147.6.a.j yes 2 21.c even 2 1
147.6.e.m 4 21.g even 6 2
147.6.e.n 4 21.h odd 6 2
441.6.a.q 2 1.a even 1 1 trivial
441.6.a.r 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3 T_{2} - 46$$ $$T_{5} + 36$$ $$T_{13}^{2} - 1296 T_{13} + 169776$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-46 - 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 36 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$54512 + 480 T + T^{2}$$
$13$ $$169776 - 1296 T + T^{2}$$
$17$ $$-2032128 + 936 T + T^{2}$$
$19$ $$-3990384 + 216 T + T^{2}$$
$23$ $$-87808 - 504 T + T^{2}$$
$29$ $$-9612604 + 6372 T + T^{2}$$
$31$ $$8672832 - 9936 T + T^{2}$$
$37$ $$-22869468 - 11124 T + T^{2}$$
$41$ $$107495424 + 20952 T + T^{2}$$
$43$ $$-268110576 + 6264 T + T^{2}$$
$47$ $$-560613312 + 7920 T + T^{2}$$
$53$ $$-66407452 + 2220 T + T^{2}$$
$59$ $$185038992 + 29736 T + T^{2}$$
$61$ $$-982141200 + 17280 T + T^{2}$$
$67$ $$-1919121200 + 20680 T + T^{2}$$
$71$ $$1814661632 - 92280 T + T^{2}$$
$73$ $$631577088 - 56592 T + T^{2}$$
$79$ $$-1239346496 + 56096 T + T^{2}$$
$83$ $$-3085453296 - 71352 T + T^{2}$$
$89$ $$3143484288 + 123192 T + T^{2}$$
$97$ $$-18184056768 - 35856 T + T^{2}$$