# Properties

 Label 483.3.f.a Level $483$ Weight $3$ Character orbit 483.f Analytic conductor $13.161$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 483.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.1607967686$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 4q^{2} + 116q^{4} - 24q^{6} - 4q^{8} + 144q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 4q^{2} + 116q^{4} - 24q^{6} - 4q^{8} + 144q^{9} + 16q^{13} + 324q^{16} + 12q^{18} - 4q^{23} - 24q^{24} - 176q^{25} + 136q^{26} - 128q^{29} - 8q^{31} - 252q^{32} - 56q^{35} + 348q^{36} + 96q^{39} - 24q^{41} - 148q^{46} - 408q^{47} - 96q^{48} - 336q^{49} + 236q^{50} - 32q^{52} - 72q^{54} - 24q^{55} - 56q^{58} + 136q^{59} + 184q^{62} + 716q^{64} - 48q^{69} - 112q^{70} + 48q^{71} - 12q^{72} - 224q^{73} - 48q^{75} + 224q^{77} - 96q^{78} + 432q^{81} + 640q^{82} - 424q^{85} + 312q^{87} + 1060q^{92} + 192q^{93} + 216q^{94} + 624q^{95} + 48q^{96} - 28q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −3.89548 −1.73205 11.1748 5.58204i 6.74717 2.64575i −27.9492 3.00000 21.7447i
22.2 −3.89548 −1.73205 11.1748 5.58204i 6.74717 2.64575i −27.9492 3.00000 21.7447i
22.3 −3.83573 1.73205 10.7128 8.86189i −6.64368 2.64575i −25.7486 3.00000 33.9918i
22.4 −3.83573 1.73205 10.7128 8.86189i −6.64368 2.64575i −25.7486 3.00000 33.9918i
22.5 −3.28216 1.73205 6.77257 2.78765i −5.68487 2.64575i −9.10001 3.00000 9.14952i
22.6 −3.28216 1.73205 6.77257 2.78765i −5.68487 2.64575i −9.10001 3.00000 9.14952i
22.7 −3.23657 −1.73205 6.47539 4.61855i 5.60591 2.64575i −8.01179 3.00000 14.9483i
22.8 −3.23657 −1.73205 6.47539 4.61855i 5.60591 2.64575i −8.01179 3.00000 14.9483i
22.9 −2.89983 1.73205 4.40899 7.66563i −5.02265 2.64575i −1.18601 3.00000 22.2290i
22.10 −2.89983 1.73205 4.40899 7.66563i −5.02265 2.64575i −1.18601 3.00000 22.2290i
22.11 −2.49369 1.73205 2.21850 2.66866i −4.31920 2.64575i 4.44250 3.00000 6.65481i
22.12 −2.49369 1.73205 2.21850 2.66866i −4.31920 2.64575i 4.44250 3.00000 6.65481i
22.13 −1.75802 −1.73205 −0.909354 0.711109i 3.04499 2.64575i 8.63076 3.00000 1.25015i
22.14 −1.75802 −1.73205 −0.909354 0.711109i 3.04499 2.64575i 8.63076 3.00000 1.25015i
22.15 −1.65091 −1.73205 −1.27448 4.21491i 2.85947 2.64575i 8.70772 3.00000 6.95846i
22.16 −1.65091 −1.73205 −1.27448 4.21491i 2.85947 2.64575i 8.70772 3.00000 6.95846i
22.17 −0.984102 1.73205 −3.03154 2.95865i −1.70451 2.64575i 6.91976 3.00000 2.91162i
22.18 −0.984102 1.73205 −3.03154 2.95865i −1.70451 2.64575i 6.91976 3.00000 2.91162i
22.19 −0.876983 1.73205 −3.23090 8.50705i −1.51898 2.64575i 6.34138 3.00000 7.46053i
22.20 −0.876983 1.73205 −3.23090 8.50705i −1.51898 2.64575i 6.34138 3.00000 7.46053i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.3.f.a 48
23.b odd 2 1 inner 483.3.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.3.f.a 48 1.a even 1 1 trivial
483.3.f.a 48 23.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(483, [\chi])$$.