# Properties

 Label 224.2.b.b Level 224 Weight 2 Character orbit 224.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2312.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + q^{7} + ( -2 + \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{13} + ( -1 + \beta_{3} ) q^{15} + 2 q^{17} + \beta_{1} q^{19} + \beta_{1} q^{21} + ( -1 + \beta_{3} ) q^{23} + ( -2 - \beta_{3} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{27} + 2 \beta_{1} q^{29} + ( 2 - 2 \beta_{3} ) q^{31} + 4 q^{33} + \beta_{2} q^{35} -2 \beta_{1} q^{37} + ( 1 - \beta_{3} ) q^{39} + ( -4 - 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -4 \beta_{1} + \beta_{2} ) q^{45} + q^{49} + 2 \beta_{1} q^{51} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + ( -5 + \beta_{3} ) q^{57} + ( \beta_{1} - 2 \beta_{2} ) q^{59} -\beta_{2} q^{61} + ( -2 + \beta_{3} ) q^{63} + ( 7 + \beta_{3} ) q^{65} + ( -\beta_{1} - 5 \beta_{2} ) q^{67} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{69} + 8 q^{71} -6 q^{73} + ( \beta_{1} + 2 \beta_{2} ) q^{75} + ( -\beta_{1} + \beta_{2} ) q^{77} + ( 6 - \beta_{3} ) q^{81} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{83} + 2 \beta_{2} q^{85} + ( -10 + 2 \beta_{3} ) q^{87} + ( -8 + 2 \beta_{3} ) q^{89} -\beta_{2} q^{91} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{93} + ( -1 + \beta_{3} ) q^{95} + ( 4 - 2 \beta_{3} ) q^{97} + ( \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{7} - 8q^{9} - 4q^{15} + 8q^{17} - 4q^{23} - 8q^{25} + 8q^{31} + 16q^{33} + 4q^{39} - 16q^{41} + 4q^{49} - 24q^{55} - 20q^{57} - 8q^{63} + 28q^{65} + 32q^{71} - 24q^{73} + 24q^{81} - 40q^{87} - 32q^{89} - 4q^{95} + 16q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{2} + \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu - 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 2 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1} + 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} - \beta_{1} + 7$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 −0.780776 − 1.17915i 1.28078 + 0.599676i 1.28078 − 0.599676i −0.780776 + 1.17915i
0 3.02045i 0 1.69614i 0 1.00000 0 −6.12311 0
113.2 0 0.936426i 0 3.33513i 0 1.00000 0 2.12311 0
113.3 0 0.936426i 0 3.33513i 0 1.00000 0 2.12311 0
113.4 0 3.02045i 0 1.69614i 0 1.00000 0 −6.12311 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.b.b 4
3.b odd 2 1 2016.2.c.c 4
4.b odd 2 1 56.2.b.b 4
7.b odd 2 1 1568.2.b.d 4
7.c even 3 2 1568.2.t.d 8
7.d odd 6 2 1568.2.t.e 8
8.b even 2 1 inner 224.2.b.b 4
8.d odd 2 1 56.2.b.b 4
12.b even 2 1 504.2.c.d 4
16.e even 4 2 1792.2.a.v 4
16.f odd 4 2 1792.2.a.x 4
24.f even 2 1 504.2.c.d 4
24.h odd 2 1 2016.2.c.c 4
28.d even 2 1 392.2.b.c 4
28.f even 6 2 392.2.p.e 8
28.g odd 6 2 392.2.p.f 8
56.e even 2 1 392.2.b.c 4
56.h odd 2 1 1568.2.b.d 4
56.j odd 6 2 1568.2.t.e 8
56.k odd 6 2 392.2.p.f 8
56.m even 6 2 392.2.p.e 8
56.p even 6 2 1568.2.t.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 4.b odd 2 1
56.2.b.b 4 8.d odd 2 1
224.2.b.b 4 1.a even 1 1 trivial
224.2.b.b 4 8.b even 2 1 inner
392.2.b.c 4 28.d even 2 1
392.2.b.c 4 56.e even 2 1
392.2.p.e 8 28.f even 6 2
392.2.p.e 8 56.m even 6 2
392.2.p.f 8 28.g odd 6 2
392.2.p.f 8 56.k odd 6 2
504.2.c.d 4 12.b even 2 1
504.2.c.d 4 24.f even 2 1
1568.2.b.d 4 7.b odd 2 1
1568.2.b.d 4 56.h odd 2 1
1568.2.t.d 8 7.c even 3 2
1568.2.t.d 8 56.p even 6 2
1568.2.t.e 8 7.d odd 6 2
1568.2.t.e 8 56.j odd 6 2
1792.2.a.v 4 16.e even 4 2
1792.2.a.x 4 16.f odd 4 2
2016.2.c.c 4 3.b odd 2 1
2016.2.c.c 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 10 T_{3}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8}$$
$5$ $$1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8}$$
$13$ $$1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{4}$$
$19$ $$1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8}$$
$31$ $$( 1 - 4 T - 2 T^{2} - 124 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8}$$
$67$ $$1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 79 T^{2} )^{4}$$
$83$ $$1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 16 T + 174 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$