# Properties

 Label 224.1.r.a Level 224 Weight 1 Character orbit 224.r Analytic conductor 0.112 Analytic rank 0 Dimension 4 Projective image $$A_{4}$$ CM/RM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 224.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.111790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{4}$$ Projective field Galois closure of 4.0.3136.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{5} q^{11} -\zeta_{12}^{3} q^{15} -\zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} -\zeta_{12}^{2} q^{21} -\zeta_{12} q^{23} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{5} q^{31} + \zeta_{12}^{4} q^{33} -\zeta_{12} q^{35} -\zeta_{12}^{4} q^{37} + \zeta_{12} q^{47} - q^{49} + \zeta_{12} q^{51} + \zeta_{12}^{2} q^{53} + \zeta_{12}^{3} q^{55} - q^{57} + \zeta_{12}^{5} q^{59} -\zeta_{12}^{4} q^{61} + \zeta_{12}^{5} q^{67} + q^{69} + \zeta_{12}^{2} q^{73} + \zeta_{12}^{2} q^{77} -\zeta_{12} q^{79} + \zeta_{12}^{2} q^{81} + q^{85} -\zeta_{12}^{4} q^{89} + \zeta_{12}^{4} q^{93} + \zeta_{12}^{5} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} - 2q^{17} - 2q^{21} - 2q^{33} + 2q^{37} - 4q^{49} + 2q^{53} - 4q^{57} + 2q^{61} + 4q^{69} + 2q^{73} + 2q^{77} + 2q^{81} + 4q^{85} + 2q^{89} - 2q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 0.500000i 0 −0.500000 0.866025i 0 1.00000i 0 0 0
95.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 1.00000i 0 0 0
191.1 0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 1.00000i 0 0 0
191.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.1.r.a 4
3.b odd 2 1 2016.1.cd.a 4
4.b odd 2 1 inner 224.1.r.a 4
7.b odd 2 1 1568.1.r.a 4
7.c even 3 1 inner 224.1.r.a 4
7.c even 3 1 1568.1.d.b 2
7.d odd 6 1 1568.1.d.a 2
7.d odd 6 1 1568.1.r.a 4
8.b even 2 1 448.1.r.a 4
8.d odd 2 1 448.1.r.a 4
12.b even 2 1 2016.1.cd.a 4
16.e even 4 1 1792.1.o.a 4
16.e even 4 1 1792.1.o.b 4
16.f odd 4 1 1792.1.o.a 4
16.f odd 4 1 1792.1.o.b 4
21.h odd 6 1 2016.1.cd.a 4
28.d even 2 1 1568.1.r.a 4
28.f even 6 1 1568.1.d.a 2
28.f even 6 1 1568.1.r.a 4
28.g odd 6 1 inner 224.1.r.a 4
28.g odd 6 1 1568.1.d.b 2
56.e even 2 1 3136.1.r.b 4
56.h odd 2 1 3136.1.r.b 4
56.j odd 6 1 3136.1.d.d 2
56.j odd 6 1 3136.1.r.b 4
56.k odd 6 1 448.1.r.a 4
56.k odd 6 1 3136.1.d.b 2
56.m even 6 1 3136.1.d.d 2
56.m even 6 1 3136.1.r.b 4
56.p even 6 1 448.1.r.a 4
56.p even 6 1 3136.1.d.b 2
84.n even 6 1 2016.1.cd.a 4
112.u odd 12 1 1792.1.o.a 4
112.u odd 12 1 1792.1.o.b 4
112.w even 12 1 1792.1.o.a 4
112.w even 12 1 1792.1.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 1.a even 1 1 trivial
224.1.r.a 4 4.b odd 2 1 inner
224.1.r.a 4 7.c even 3 1 inner
224.1.r.a 4 28.g odd 6 1 inner
448.1.r.a 4 8.b even 2 1
448.1.r.a 4 8.d odd 2 1
448.1.r.a 4 56.k odd 6 1
448.1.r.a 4 56.p even 6 1
1568.1.d.a 2 7.d odd 6 1
1568.1.d.a 2 28.f even 6 1
1568.1.d.b 2 7.c even 3 1
1568.1.d.b 2 28.g odd 6 1
1568.1.r.a 4 7.b odd 2 1
1568.1.r.a 4 7.d odd 6 1
1568.1.r.a 4 28.d even 2 1
1568.1.r.a 4 28.f even 6 1
1792.1.o.a 4 16.e even 4 1
1792.1.o.a 4 16.f odd 4 1
1792.1.o.a 4 112.u odd 12 1
1792.1.o.a 4 112.w even 12 1
1792.1.o.b 4 16.e even 4 1
1792.1.o.b 4 16.f odd 4 1
1792.1.o.b 4 112.u odd 12 1
1792.1.o.b 4 112.w even 12 1
2016.1.cd.a 4 3.b odd 2 1
2016.1.cd.a 4 12.b even 2 1
2016.1.cd.a 4 21.h odd 6 1
2016.1.cd.a 4 84.n even 6 1
3136.1.d.b 2 56.k odd 6 1
3136.1.d.b 2 56.p even 6 1
3136.1.d.d 2 56.j odd 6 1
3136.1.d.d 2 56.m even 6 1
3136.1.r.b 4 56.e even 2 1
3136.1.r.b 4 56.h odd 2 1
3136.1.r.b 4 56.j odd 6 1
3136.1.r.b 4 56.m even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$5$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$13$ $$( 1 + T^{2} )^{4}$$
$17$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$19$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$23$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$29$ $$( 1 + T^{2} )^{4}$$
$31$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$37$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$41$ $$( 1 + T^{2} )^{4}$$
$43$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$47$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$53$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$59$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$61$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$67$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$71$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$73$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$79$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$83$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$89$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$97$ $$( 1 + T^{2} )^{4}$$