# Properties

 Label 224.3.n.a Level $224$ Weight $3$ Character orbit 224.n Analytic conductor $6.104$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 4 q^{7} - 32 q^{9}+O(q^{10})$$ 28 * q + 4 * q^7 - 32 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 4 q^{7} - 32 q^{9} - 28 q^{15} - 6 q^{17} - 30 q^{23} - 32 q^{25} + 6 q^{31} - 6 q^{33} + 20 q^{39} + 294 q^{47} - 20 q^{49} + 124 q^{57} - 432 q^{63} - 52 q^{65} + 136 q^{71} + 234 q^{73} + 162 q^{79} - 18 q^{81} - 48 q^{87} - 150 q^{89} - 290 q^{95}+O(q^{100})$$ 28 * q + 4 * q^7 - 32 * q^9 - 28 * q^15 - 6 * q^17 - 30 * q^23 - 32 * q^25 + 6 * q^31 - 6 * q^33 + 20 * q^39 + 294 * q^47 - 20 * q^49 + 124 * q^57 - 432 * q^63 - 52 * q^65 + 136 * q^71 + 234 * q^73 + 162 * q^79 - 18 * q^81 - 48 * q^87 - 150 * q^89 - 290 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −2.78005 + 4.81519i 0 −1.52921 2.64866i 0 −0.608243 6.97352i 0 −10.9574 18.9787i 0
17.2 0 −1.94818 + 3.37434i 0 4.42985 + 7.67272i 0 6.92329 + 1.03347i 0 −3.09078 5.35338i 0
17.3 0 −1.93494 + 3.35141i 0 2.33882 + 4.05096i 0 −6.95505 + 0.792023i 0 −2.98798 5.17534i 0
17.4 0 −1.70138 + 2.94687i 0 −2.15858 3.73877i 0 1.43197 + 6.85197i 0 −1.28938 2.23327i 0
17.5 0 −1.16781 + 2.02271i 0 −1.55055 2.68563i 0 6.89374 1.21502i 0 1.77242 + 3.06992i 0
17.6 0 −0.455431 + 0.788830i 0 −3.17251 5.49495i 0 −3.79106 + 5.88455i 0 4.08516 + 7.07571i 0
17.7 0 −0.126628 + 0.219326i 0 −1.78589 3.09325i 0 −2.89466 6.37346i 0 4.46793 + 7.73868i 0
17.8 0 0.126628 0.219326i 0 1.78589 + 3.09325i 0 −2.89466 6.37346i 0 4.46793 + 7.73868i 0
17.9 0 0.455431 0.788830i 0 3.17251 + 5.49495i 0 −3.79106 + 5.88455i 0 4.08516 + 7.07571i 0
17.10 0 1.16781 2.02271i 0 1.55055 + 2.68563i 0 6.89374 1.21502i 0 1.77242 + 3.06992i 0
17.11 0 1.70138 2.94687i 0 2.15858 + 3.73877i 0 1.43197 + 6.85197i 0 −1.28938 2.23327i 0
17.12 0 1.93494 3.35141i 0 −2.33882 4.05096i 0 −6.95505 + 0.792023i 0 −2.98798 5.17534i 0
17.13 0 1.94818 3.37434i 0 −4.42985 7.67272i 0 6.92329 + 1.03347i 0 −3.09078 5.35338i 0
17.14 0 2.78005 4.81519i 0 1.52921 + 2.64866i 0 −0.608243 6.97352i 0 −10.9574 18.9787i 0
145.1 0 −2.78005 4.81519i 0 −1.52921 + 2.64866i 0 −0.608243 + 6.97352i 0 −10.9574 + 18.9787i 0
145.2 0 −1.94818 3.37434i 0 4.42985 7.67272i 0 6.92329 1.03347i 0 −3.09078 + 5.35338i 0
145.3 0 −1.93494 3.35141i 0 2.33882 4.05096i 0 −6.95505 0.792023i 0 −2.98798 + 5.17534i 0
145.4 0 −1.70138 2.94687i 0 −2.15858 + 3.73877i 0 1.43197 6.85197i 0 −1.28938 + 2.23327i 0
145.5 0 −1.16781 2.02271i 0 −1.55055 + 2.68563i 0 6.89374 + 1.21502i 0 1.77242 3.06992i 0
145.6 0 −0.455431 0.788830i 0 −3.17251 + 5.49495i 0 −3.79106 5.88455i 0 4.08516 7.07571i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.b even 2 1 inner
56.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.n.a 28
4.b odd 2 1 56.3.j.a 28
7.c even 3 1 1568.3.h.a 28
7.d odd 6 1 inner 224.3.n.a 28
7.d odd 6 1 1568.3.h.a 28
8.b even 2 1 inner 224.3.n.a 28
8.d odd 2 1 56.3.j.a 28
28.d even 2 1 392.3.j.e 28
28.f even 6 1 56.3.j.a 28
28.f even 6 1 392.3.h.a 28
28.g odd 6 1 392.3.h.a 28
28.g odd 6 1 392.3.j.e 28
56.e even 2 1 392.3.j.e 28
56.j odd 6 1 inner 224.3.n.a 28
56.j odd 6 1 1568.3.h.a 28
56.k odd 6 1 392.3.h.a 28
56.k odd 6 1 392.3.j.e 28
56.m even 6 1 56.3.j.a 28
56.m even 6 1 392.3.h.a 28
56.p even 6 1 1568.3.h.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.j.a 28 4.b odd 2 1
56.3.j.a 28 8.d odd 2 1
56.3.j.a 28 28.f even 6 1
56.3.j.a 28 56.m even 6 1
224.3.n.a 28 1.a even 1 1 trivial
224.3.n.a 28 7.d odd 6 1 inner
224.3.n.a 28 8.b even 2 1 inner
224.3.n.a 28 56.j odd 6 1 inner
392.3.h.a 28 28.f even 6 1
392.3.h.a 28 28.g odd 6 1
392.3.h.a 28 56.k odd 6 1
392.3.h.a 28 56.m even 6 1
392.3.j.e 28 28.d even 2 1
392.3.j.e 28 28.g odd 6 1
392.3.j.e 28 56.e even 2 1
392.3.j.e 28 56.k odd 6 1
1568.3.h.a 28 7.c even 3 1
1568.3.h.a 28 7.d odd 6 1
1568.3.h.a 28 56.j odd 6 1
1568.3.h.a 28 56.p even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(224, [\chi])$$.