# Properties

 Label 441.6.a.g Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 28 q^{4} - 11 q^{5} - 120 q^{8}+O(q^{10})$$ q + 2 * q^2 - 28 * q^4 - 11 * q^5 - 120 * q^8 $$q + 2 q^{2} - 28 q^{4} - 11 q^{5} - 120 q^{8} - 22 q^{10} - 269 q^{11} - 308 q^{13} + 656 q^{16} - 1896 q^{17} - 164 q^{19} + 308 q^{20} - 538 q^{22} + 3264 q^{23} - 3004 q^{25} - 616 q^{26} - 2417 q^{29} + 2841 q^{31} + 5152 q^{32} - 3792 q^{34} - 11328 q^{37} - 328 q^{38} + 1320 q^{40} + 16856 q^{41} - 7894 q^{43} + 7532 q^{44} + 6528 q^{46} - 21102 q^{47} - 6008 q^{50} + 8624 q^{52} + 29691 q^{53} + 2959 q^{55} - 4834 q^{58} + 8163 q^{59} + 15166 q^{61} + 5682 q^{62} - 10688 q^{64} + 3388 q^{65} - 32078 q^{67} + 53088 q^{68} + 38274 q^{71} + 34866 q^{73} - 22656 q^{74} + 4592 q^{76} + 13529 q^{79} - 7216 q^{80} + 33712 q^{82} + 68103 q^{83} + 20856 q^{85} - 15788 q^{86} + 32280 q^{88} + 114922 q^{89} - 91392 q^{92} - 42204 q^{94} + 1804 q^{95} + 154959 q^{97}+O(q^{100})$$ q + 2 * q^2 - 28 * q^4 - 11 * q^5 - 120 * q^8 - 22 * q^10 - 269 * q^11 - 308 * q^13 + 656 * q^16 - 1896 * q^17 - 164 * q^19 + 308 * q^20 - 538 * q^22 + 3264 * q^23 - 3004 * q^25 - 616 * q^26 - 2417 * q^29 + 2841 * q^31 + 5152 * q^32 - 3792 * q^34 - 11328 * q^37 - 328 * q^38 + 1320 * q^40 + 16856 * q^41 - 7894 * q^43 + 7532 * q^44 + 6528 * q^46 - 21102 * q^47 - 6008 * q^50 + 8624 * q^52 + 29691 * q^53 + 2959 * q^55 - 4834 * q^58 + 8163 * q^59 + 15166 * q^61 + 5682 * q^62 - 10688 * q^64 + 3388 * q^65 - 32078 * q^67 + 53088 * q^68 + 38274 * q^71 + 34866 * q^73 - 22656 * q^74 + 4592 * q^76 + 13529 * q^79 - 7216 * q^80 + 33712 * q^82 + 68103 * q^83 + 20856 * q^85 - 15788 * q^86 + 32280 * q^88 + 114922 * q^89 - 91392 * q^92 - 42204 * q^94 + 1804 * q^95 + 154959 * 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
2.00000 0 −28.0000 −11.0000 0 0 −120.000 0 −22.0000
 $$n$$: 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.g 1
3.b odd 2 1 147.6.a.c 1
7.b odd 2 1 441.6.a.h 1
7.c even 3 2 63.6.e.a 2
21.c even 2 1 147.6.a.d 1
21.g even 6 2 147.6.e.g 2
21.h odd 6 2 21.6.e.a 2
84.n even 6 2 336.6.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 21.h odd 6 2
63.6.e.a 2 7.c even 3 2
147.6.a.c 1 3.b odd 2 1
147.6.a.d 1 21.c even 2 1
147.6.e.g 2 21.g even 6 2
336.6.q.b 2 84.n even 6 2
441.6.a.g 1 1.a even 1 1 trivial
441.6.a.h 1 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{5} + 11$$ T5 + 11 $$T_{13} + 308$$ T13 + 308

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 11$$
$7$ $$T$$
$11$ $$T + 269$$
$13$ $$T + 308$$
$17$ $$T + 1896$$
$19$ $$T + 164$$
$23$ $$T - 3264$$
$29$ $$T + 2417$$
$31$ $$T - 2841$$
$37$ $$T + 11328$$
$41$ $$T - 16856$$
$43$ $$T + 7894$$
$47$ $$T + 21102$$
$53$ $$T - 29691$$
$59$ $$T - 8163$$
$61$ $$T - 15166$$
$67$ $$T + 32078$$
$71$ $$T - 38274$$
$73$ $$T - 34866$$
$79$ $$T - 13529$$
$83$ $$T - 68103$$
$89$ $$T - 114922$$
$97$ $$T - 154959$$