# Properties

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

# 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{3} q^{13} + ( -5 + 3 \beta_{3} ) q^{15} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{17} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{19} + ( 3 - \beta_{2} - 2 \beta_{3} ) q^{21} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{23} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{27} -2 \beta_{3} q^{29} + ( 7 + \beta_{1} + 7 \beta_{2} ) q^{31} + \beta_{2} q^{33} + ( 8 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{35} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{37} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{39} + ( -4 - 2 \beta_{3} ) q^{41} + ( -4 - 4 \beta_{3} ) q^{43} + ( -8 - 2 \beta_{1} - 8 \beta_{2} ) q^{45} + ( \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{47} + ( -5 - 4 \beta_{1} - 2 \beta_{3} ) q^{49} + ( 5 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{51} + ( 1 + \beta_{2} ) q^{53} + ( -3 - \beta_{3} ) q^{55} + ( 3 - 4 \beta_{3} ) q^{57} + ( 1 - 7 \beta_{1} + \beta_{2} ) q^{59} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 8 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{63} + ( -2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -7 - 3 \beta_{1} - 7 \beta_{2} ) q^{67} + ( 5 - 2 \beta_{3} ) q^{69} + ( -8 - 4 \beta_{3} ) q^{71} + ( 9 - 4 \beta_{1} + 9 \beta_{2} ) q^{73} + ( -8 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{75} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{77} + ( 5 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{79} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( -4 + 8 \beta_{3} ) q^{83} + ( -11 + 8 \beta_{3} ) q^{85} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{87} -9 \beta_{2} q^{89} + ( 4 + 8 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 8 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} ) q^{93} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{95} + ( -4 + 2 \beta_{3} ) q^{97} + ( -4 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 2q^{5} + 4q^{7} + O(q^{10})$$ $$4q + 2q^{3} - 2q^{5} + 4q^{7} - 2q^{11} - 20q^{15} + 6q^{17} + 10q^{19} + 14q^{21} - 2q^{23} - 8q^{25} - 4q^{27} + 14q^{31} - 2q^{33} + 22q^{35} - 6q^{37} - 8q^{39} - 16q^{41} - 16q^{43} - 16q^{45} + 18q^{47} - 20q^{49} - 14q^{51} + 2q^{53} - 12q^{55} + 12q^{57} + 2q^{59} - 6q^{61} + 24q^{63} + 16q^{65} - 14q^{67} + 20q^{69} - 32q^{71} + 18q^{73} + 24q^{75} + 10q^{77} - 2q^{79} + 2q^{81} - 16q^{83} - 44q^{85} + 8q^{87} + 18q^{89} - 18q^{93} + 2q^{95} - 16q^{97} - 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.207107 0.358719i 0 0.914214 1.58346i 0 1.00000 + 2.44949i 0 1.41421 2.44949i 0
65.2 0 1.20711 + 2.09077i 0 −1.91421 + 3.31552i 0 1.00000 2.44949i 0 −1.41421 + 2.44949i 0
193.1 0 −0.207107 + 0.358719i 0 0.914214 + 1.58346i 0 1.00000 2.44949i 0 1.41421 + 2.44949i 0
193.2 0 1.20711 2.09077i 0 −1.91421 3.31552i 0 1.00000 + 2.44949i 0 −1.41421 2.44949i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.d yes 4
3.b odd 2 1 2016.2.s.s 4
4.b odd 2 1 224.2.i.a 4
7.b odd 2 1 1568.2.i.o 4
7.c even 3 1 inner 224.2.i.d yes 4
7.c even 3 1 1568.2.a.l 2
7.d odd 6 1 1568.2.a.u 2
7.d odd 6 1 1568.2.i.o 4
8.b even 2 1 448.2.i.g 4
8.d odd 2 1 448.2.i.j 4
12.b even 2 1 2016.2.s.q 4
21.h odd 6 1 2016.2.s.s 4
28.d even 2 1 1568.2.i.x 4
28.f even 6 1 1568.2.a.j 2
28.f even 6 1 1568.2.i.x 4
28.g odd 6 1 224.2.i.a 4
28.g odd 6 1 1568.2.a.w 2
56.j odd 6 1 3136.2.a.be 2
56.k odd 6 1 448.2.i.j 4
56.k odd 6 1 3136.2.a.bd 2
56.m even 6 1 3136.2.a.bx 2
56.p even 6 1 448.2.i.g 4
56.p even 6 1 3136.2.a.bw 2
84.n even 6 1 2016.2.s.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 4.b odd 2 1
224.2.i.a 4 28.g odd 6 1
224.2.i.d yes 4 1.a even 1 1 trivial
224.2.i.d yes 4 7.c even 3 1 inner
448.2.i.g 4 8.b even 2 1
448.2.i.g 4 56.p even 6 1
448.2.i.j 4 8.d odd 2 1
448.2.i.j 4 56.k odd 6 1
1568.2.a.j 2 28.f even 6 1
1568.2.a.l 2 7.c even 3 1
1568.2.a.u 2 7.d odd 6 1
1568.2.a.w 2 28.g odd 6 1
1568.2.i.o 4 7.b odd 2 1
1568.2.i.o 4 7.d odd 6 1
1568.2.i.x 4 28.d even 2 1
1568.2.i.x 4 28.f even 6 1
2016.2.s.q 4 12.b even 2 1
2016.2.s.q 4 84.n even 6 1
2016.2.s.s 4 3.b odd 2 1
2016.2.s.s 4 21.h odd 6 1
3136.2.a.bd 2 56.k odd 6 1
3136.2.a.be 2 56.j odd 6 1
3136.2.a.bw 2 56.p even 6 1
3136.2.a.bx 2 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8}$$
$5$ $$1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 22 T^{5} - 2057 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 18 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8}$$
$19$ $$1 - 10 T + 39 T^{2} - 230 T^{3} + 1460 T^{4} - 4370 T^{5} + 14079 T^{6} - 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 782 T^{5} - 13225 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$( 1 + 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$1 - 14 T + 87 T^{2} - 658 T^{3} + 4844 T^{4} - 20398 T^{5} + 83607 T^{6} - 417074 T^{7} + 923521 T^{8}$$
$37$ $$1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 66834 T^{5} + 333559 T^{6} - 1868814 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 2 T - 17 T^{2} + 194 T^{3} - 3276 T^{4} + 11446 T^{5} - 59177 T^{6} - 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 2 T - 105 T^{2} - 98 T^{3} + 5324 T^{4} - 7742 T^{5} - 655305 T^{6} + 986078 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2}$$