# Properties

 Label 224.3.d.a Level 224 Weight 3 Character orbit 224.d Analytic conductor 6.104 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 224.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} -\beta_{3} q^{7} + 5 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} -\beta_{3} q^{7} + 5 q^{9} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{11} + ( -2 + 3 \beta_{2} ) q^{13} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{15} + ( -14 + 2 \beta_{2} ) q^{17} + ( -5 \beta_{1} + 8 \beta_{3} ) q^{19} + \beta_{2} q^{21} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{23} + ( 7 - 4 \beta_{2} ) q^{25} + 14 \beta_{1} q^{27} + ( 10 + 6 \beta_{2} ) q^{29} -2 \beta_{1} q^{31} + ( 8 - 4 \beta_{2} ) q^{33} + ( -7 \beta_{1} + 2 \beta_{3} ) q^{35} + ( 34 + 2 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 12 \beta_{3} ) q^{39} + ( -30 - 2 \beta_{2} ) q^{41} + ( 6 \beta_{1} - 20 \beta_{3} ) q^{43} + ( -10 + 5 \beta_{2} ) q^{45} + ( -18 \beta_{1} - 8 \beta_{3} ) q^{47} -7 q^{49} + ( -14 \beta_{1} + 8 \beta_{3} ) q^{51} + ( -70 + 4 \beta_{2} ) q^{53} + ( 32 \beta_{1} - 16 \beta_{3} ) q^{55} + ( 20 - 8 \beta_{2} ) q^{57} + ( -9 \beta_{1} - 16 \beta_{3} ) q^{59} + ( -2 - 3 \beta_{2} ) q^{61} -5 \beta_{3} q^{63} + ( 88 - 8 \beta_{2} ) q^{65} + ( 12 \beta_{1} - 20 \beta_{3} ) q^{67} + ( -24 - 4 \beta_{2} ) q^{69} + ( -32 \beta_{1} + 16 \beta_{3} ) q^{71} + ( -38 - 16 \beta_{2} ) q^{73} + ( 7 \beta_{1} - 16 \beta_{3} ) q^{75} + ( 28 - 2 \beta_{2} ) q^{77} -40 \beta_{3} q^{79} -11 q^{81} + ( 31 \beta_{1} + 24 \beta_{3} ) q^{83} + ( 84 - 18 \beta_{2} ) q^{85} + ( 10 \beta_{1} + 24 \beta_{3} ) q^{87} + ( 34 + 8 \beta_{2} ) q^{89} + ( -21 \beta_{1} + 2 \beta_{3} ) q^{91} + 8 q^{93} + ( 66 \beta_{1} - 36 \beta_{3} ) q^{95} + ( -150 - 2 \beta_{2} ) q^{97} + ( -10 \beta_{1} + 20 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} + 20q^{9} + O(q^{10})$$ $$4q - 8q^{5} + 20q^{9} - 8q^{13} - 56q^{17} + 28q^{25} + 40q^{29} + 32q^{33} + 136q^{37} - 120q^{41} - 40q^{45} - 28q^{49} - 280q^{53} + 80q^{57} - 8q^{61} + 352q^{65} - 96q^{69} - 152q^{73} + 112q^{77} - 44q^{81} + 336q^{85} + 136q^{89} + 32q^{93} - 600q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 5 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −1.32288 − 0.500000i 1.32288 − 0.500000i −1.32288 + 0.500000i 1.32288 + 0.500000i
0 2.00000i 0 −7.29150 0 2.64575i 0 5.00000 0
127.2 0 2.00000i 0 3.29150 0 2.64575i 0 5.00000 0
127.3 0 2.00000i 0 −7.29150 0 2.64575i 0 5.00000 0
127.4 0 2.00000i 0 3.29150 0 2.64575i 0 5.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.d.a 4
3.b odd 2 1 2016.3.m.a 4
4.b odd 2 1 inner 224.3.d.a 4
7.b odd 2 1 1568.3.d.h 4
8.b even 2 1 448.3.d.c 4
8.d odd 2 1 448.3.d.c 4
12.b even 2 1 2016.3.m.a 4
16.e even 4 1 1792.3.g.a 4
16.e even 4 1 1792.3.g.c 4
16.f odd 4 1 1792.3.g.a 4
16.f odd 4 1 1792.3.g.c 4
28.d even 2 1 1568.3.d.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.a 4 1.a even 1 1 trivial
224.3.d.a 4 4.b odd 2 1 inner
448.3.d.c 4 8.b even 2 1
448.3.d.c 4 8.d odd 2 1
1568.3.d.h 4 7.b odd 2 1
1568.3.d.h 4 28.d even 2 1
1792.3.g.a 4 16.e even 4 1
1792.3.g.a 4 16.f odd 4 1
1792.3.g.c 4 16.e even 4 1
1792.3.g.c 4 16.f odd 4 1
2016.3.m.a 4 3.b odd 2 1
2016.3.m.a 4 12.b even 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 14 T^{2} + 81 T^{4} )^{2}$$
$5$ $$( 1 + 4 T + 26 T^{2} + 100 T^{3} + 625 T^{4} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$1 - 228 T^{2} + 35110 T^{4} - 3338148 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 + 4 T + 90 T^{2} + 676 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$( 1 + 28 T + 662 T^{2} + 8092 T^{3} + 83521 T^{4} )^{2}$$
$19$ $$1 - 348 T^{2} + 111718 T^{4} - 45351708 T^{6} + 16983563041 T^{8}$$
$23$ $$1 - 1604 T^{2} + 1138374 T^{4} - 448864964 T^{6} + 78310985281 T^{8}$$
$29$ $$( 1 - 20 T + 774 T^{2} - 16820 T^{3} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 1906 T^{2} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 68 T + 3782 T^{2} - 93092 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 + 60 T + 4150 T^{2} + 100860 T^{3} + 2825761 T^{4} )^{2}$$
$43$ $$1 - 1508 T^{2} + 5793318 T^{4} - 5155551908 T^{6} + 11688200277601 T^{8}$$
$47$ $$1 - 5348 T^{2} + 14587206 T^{4} - 26096533988 T^{6} + 23811286661761 T^{8}$$
$53$ $$( 1 + 140 T + 10070 T^{2} + 393260 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 - 9692 T^{2} + 45396006 T^{4} - 117441462812 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 4 T + 7194 T^{2} + 14884 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$1 - 11204 T^{2} + 65233446 T^{4} - 225773159684 T^{6} + 406067677556641 T^{8}$$
$71$ $$1 - 8388 T^{2} + 39052870 T^{4} - 213153180228 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 76 T + 4934 T^{2} + 405004 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 - 1282 T^{2} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 11804 T^{2} + 67754214 T^{4} - 560198021084 T^{6} + 2252292232139041 T^{8}$$
$89$ $$( 1 - 68 T + 15206 T^{2} - 538628 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 + 300 T + 41206 T^{2} + 2822700 T^{3} + 88529281 T^{4} )^{2}$$