# Properties

 Label 441.6.a.j Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 6 q^{2} + 4 q^{4} + 78 q^{5} - 168 q^{8}+O(q^{10})$$ q + 6 * q^2 + 4 * q^4 + 78 * q^5 - 168 * q^8 $$q + 6 q^{2} + 4 q^{4} + 78 q^{5} - 168 q^{8} + 468 q^{10} - 444 q^{11} + 442 q^{13} - 1136 q^{16} - 126 q^{17} - 2684 q^{19} + 312 q^{20} - 2664 q^{22} - 4200 q^{23} + 2959 q^{25} + 2652 q^{26} + 5442 q^{29} - 80 q^{31} - 1440 q^{32} - 756 q^{34} - 5434 q^{37} - 16104 q^{38} - 13104 q^{40} + 7962 q^{41} - 11524 q^{43} - 1776 q^{44} - 25200 q^{46} - 13920 q^{47} + 17754 q^{50} + 1768 q^{52} + 9594 q^{53} - 34632 q^{55} + 32652 q^{58} + 27492 q^{59} - 49478 q^{61} - 480 q^{62} + 27712 q^{64} + 34476 q^{65} - 59356 q^{67} - 504 q^{68} - 32040 q^{71} + 61846 q^{73} - 32604 q^{74} - 10736 q^{76} - 65776 q^{79} - 88608 q^{80} + 47772 q^{82} + 40188 q^{83} - 9828 q^{85} - 69144 q^{86} + 74592 q^{88} - 7974 q^{89} - 16800 q^{92} - 83520 q^{94} - 209352 q^{95} + 143662 q^{97}+O(q^{100})$$ q + 6 * q^2 + 4 * q^4 + 78 * q^5 - 168 * q^8 + 468 * q^10 - 444 * q^11 + 442 * q^13 - 1136 * q^16 - 126 * q^17 - 2684 * q^19 + 312 * q^20 - 2664 * q^22 - 4200 * q^23 + 2959 * q^25 + 2652 * q^26 + 5442 * q^29 - 80 * q^31 - 1440 * q^32 - 756 * q^34 - 5434 * q^37 - 16104 * q^38 - 13104 * q^40 + 7962 * q^41 - 11524 * q^43 - 1776 * q^44 - 25200 * q^46 - 13920 * q^47 + 17754 * q^50 + 1768 * q^52 + 9594 * q^53 - 34632 * q^55 + 32652 * q^58 + 27492 * q^59 - 49478 * q^61 - 480 * q^62 + 27712 * q^64 + 34476 * q^65 - 59356 * q^67 - 504 * q^68 - 32040 * q^71 + 61846 * q^73 - 32604 * q^74 - 10736 * q^76 - 65776 * q^79 - 88608 * q^80 + 47772 * q^82 + 40188 * q^83 - 9828 * q^85 - 69144 * q^86 + 74592 * q^88 - 7974 * q^89 - 16800 * q^92 - 83520 * q^94 - 209352 * q^95 + 143662 * q^97

## 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
 0
6.00000 0 4.00000 78.0000 0 0 −168.000 0 468.000
 $$n$$: 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.j 1
3.b odd 2 1 147.6.a.b 1
7.b odd 2 1 63.6.a.d 1
21.c even 2 1 21.6.a.a 1
21.g even 6 2 147.6.e.j 2
21.h odd 6 2 147.6.e.i 2
28.d even 2 1 1008.6.a.c 1
84.h odd 2 1 336.6.a.r 1
105.g even 2 1 525.6.a.d 1
105.k odd 4 2 525.6.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 21.c even 2 1
63.6.a.d 1 7.b odd 2 1
147.6.a.b 1 3.b odd 2 1
147.6.e.i 2 21.h odd 6 2
147.6.e.j 2 21.g even 6 2
336.6.a.r 1 84.h odd 2 1
441.6.a.j 1 1.a even 1 1 trivial
525.6.a.d 1 105.g even 2 1
525.6.d.b 2 105.k odd 4 2
1008.6.a.c 1 28.d even 2 1

## Hecke kernels

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

 $$T_{2} - 6$$ T2 - 6 $$T_{5} - 78$$ T5 - 78 $$T_{13} - 442$$ T13 - 442

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 6$$
$3$ $$T$$
$5$ $$T - 78$$
$7$ $$T$$
$11$ $$T + 444$$
$13$ $$T - 442$$
$17$ $$T + 126$$
$19$ $$T + 2684$$
$23$ $$T + 4200$$
$29$ $$T - 5442$$
$31$ $$T + 80$$
$37$ $$T + 5434$$
$41$ $$T - 7962$$
$43$ $$T + 11524$$
$47$ $$T + 13920$$
$53$ $$T - 9594$$
$59$ $$T - 27492$$
$61$ $$T + 49478$$
$67$ $$T + 59356$$
$71$ $$T + 32040$$
$73$ $$T - 61846$$
$79$ $$T + 65776$$
$83$ $$T - 40188$$
$89$ $$T + 7974$$
$97$ $$T - 143662$$