# Properties

 Label 224.2.i.b Level 224 Weight 2 Character orbit 224.i Analytic conductor 1.789 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 224.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{15} -5 \zeta_{12}^{2} q^{17} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{19} + ( -4 - \zeta_{12}^{2} ) q^{21} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + 8 q^{29} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{31} + ( -9 + 9 \zeta_{12}^{2} ) q^{33} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{35} + ( 5 - 5 \zeta_{12}^{2} ) q^{37} + 4 q^{41} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{43} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{51} + \zeta_{12}^{2} q^{53} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{55} + 3 q^{57} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{59} + ( -11 + 11 \zeta_{12}^{2} ) q^{61} + ( 7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} -3 q^{69} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{71} -15 \zeta_{12}^{2} q^{73} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{75} + ( -12 - 3 \zeta_{12}^{2} ) q^{77} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{79} + 9 \zeta_{12}^{2} q^{81} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{83} + 5 q^{85} + ( -8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{87} + ( -7 + 7 \zeta_{12}^{2} ) q^{89} + ( 15 - 15 \zeta_{12}^{2} ) q^{93} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{95} + 12 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} - 10q^{17} - 18q^{21} + 8q^{25} + 32q^{29} - 18q^{33} + 10q^{37} + 16q^{41} - 4q^{49} + 2q^{53} + 12q^{57} - 22q^{61} - 12q^{69} - 30q^{73} - 54q^{77} + 18q^{81} + 20q^{85} - 14q^{89} + 30q^{93} + 48q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 −0.500000 + 0.866025i 0 1.73205 2.00000i 0 0 0
65.2 0 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 0 −1.73205 + 2.00000i 0 0 0
193.1 0 −0.866025 + 1.50000i 0 −0.500000 0.866025i 0 1.73205 + 2.00000i 0 0 0
193.2 0 0.866025 1.50000i 0 −0.500000 0.866025i 0 −1.73205 2.00000i 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 224.2.i.b 4
3.b odd 2 1 2016.2.s.r 4
4.b odd 2 1 inner 224.2.i.b 4
7.b odd 2 1 1568.2.i.u 4
7.c even 3 1 inner 224.2.i.b 4
7.c even 3 1 1568.2.a.s 2
7.d odd 6 1 1568.2.a.n 2
7.d odd 6 1 1568.2.i.u 4
8.b even 2 1 448.2.i.i 4
8.d odd 2 1 448.2.i.i 4
12.b even 2 1 2016.2.s.r 4
21.h odd 6 1 2016.2.s.r 4
28.d even 2 1 1568.2.i.u 4
28.f even 6 1 1568.2.a.n 2
28.f even 6 1 1568.2.i.u 4
28.g odd 6 1 inner 224.2.i.b 4
28.g odd 6 1 1568.2.a.s 2
56.j odd 6 1 3136.2.a.bu 2
56.k odd 6 1 448.2.i.i 4
56.k odd 6 1 3136.2.a.bh 2
56.m even 6 1 3136.2.a.bu 2
56.p even 6 1 448.2.i.i 4
56.p even 6 1 3136.2.a.bh 2
84.n even 6 1 2016.2.s.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 1.a even 1 1 trivial
224.2.i.b 4 4.b odd 2 1 inner
224.2.i.b 4 7.c even 3 1 inner
224.2.i.b 4 28.g odd 6 1 inner
448.2.i.i 4 8.b even 2 1
448.2.i.i 4 8.d odd 2 1
448.2.i.i 4 56.k odd 6 1
448.2.i.i 4 56.p even 6 1
1568.2.a.n 2 7.d odd 6 1
1568.2.a.n 2 28.f even 6 1
1568.2.a.s 2 7.c even 3 1
1568.2.a.s 2 28.g odd 6 1
1568.2.i.u 4 7.b odd 2 1
1568.2.i.u 4 7.d odd 6 1
1568.2.i.u 4 28.d even 2 1
1568.2.i.u 4 28.f even 6 1
2016.2.s.r 4 3.b odd 2 1
2016.2.s.r 4 12.b even 2 1
2016.2.s.r 4 21.h odd 6 1
2016.2.s.r 4 84.n even 6 1
3136.2.a.bh 2 56.k odd 6 1
3136.2.a.bh 2 56.p even 6 1
3136.2.a.bu 2 56.j odd 6 1
3136.2.a.bu 2 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} )$$
$5$ $$( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8}$$
$23$ $$1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} )$$
$37$ $$( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 4 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 38 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 50 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 15 T + 152 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 118 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 12 T + 97 T^{2} )^{4}$$