# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -3 q^{9} -2 q^{11} -\beta_{3} q^{13} + 6 \beta_{1} q^{15} + 2 \beta_{2} q^{17} -\beta_{2} q^{19} + ( 6 \beta_{1} + \beta_{3} ) q^{21} -4 \beta_{1} q^{23} + q^{25} -4 \beta_{1} q^{29} -2 \beta_{3} q^{31} -2 \beta_{2} q^{33} + ( 6 - \beta_{2} ) q^{35} -8 \beta_{1} q^{37} -6 \beta_{1} q^{39} + 6 q^{43} -3 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( 5 - 2 \beta_{2} ) q^{49} -12 q^{51} + 4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + 6 q^{57} -\beta_{2} q^{59} -3 \beta_{3} q^{61} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{63} -6 q^{65} + 2 q^{67} + 4 \beta_{3} q^{69} + 10 \beta_{1} q^{71} -6 \beta_{2} q^{73} + \beta_{2} q^{75} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{77} -6 \beta_{1} q^{79} -9 q^{81} + \beta_{2} q^{83} + 12 \beta_{1} q^{85} + 4 \beta_{3} q^{87} + 6 \beta_{2} q^{89} + ( -6 + \beta_{2} ) q^{91} -12 \beta_{1} q^{93} -6 \beta_{1} q^{95} + 2 \beta_{2} q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} - 8q^{11} + 4q^{25} + 24q^{35} + 24q^{43} + 20q^{49} - 48q^{51} + 24q^{57} - 24q^{65} + 8q^{67} - 36q^{81} - 24q^{91} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i
0 2.44949i 0 −2.44949 0 −2.44949 1.00000i 0 −3.00000 0
111.2 0 2.44949i 0 2.44949 0 2.44949 + 1.00000i 0 −3.00000 0
111.3 0 2.44949i 0 −2.44949 0 −2.44949 + 1.00000i 0 −3.00000 0
111.4 0 2.44949i 0 2.44949 0 2.44949 1.00000i 0 −3.00000 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
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e 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.e.b 4
3.b odd 2 1 2016.2.p.e 4
4.b odd 2 1 56.2.e.b 4
7.b odd 2 1 inner 224.2.e.b 4
7.c even 3 2 1568.2.q.e 8
7.d odd 6 2 1568.2.q.e 8
8.b even 2 1 56.2.e.b 4
8.d odd 2 1 inner 224.2.e.b 4
12.b even 2 1 504.2.p.f 4
16.e even 4 1 1792.2.f.e 4
16.e even 4 1 1792.2.f.f 4
16.f odd 4 1 1792.2.f.e 4
16.f odd 4 1 1792.2.f.f 4
21.c even 2 1 2016.2.p.e 4
24.f even 2 1 2016.2.p.e 4
24.h odd 2 1 504.2.p.f 4
28.d even 2 1 56.2.e.b 4
28.f even 6 2 392.2.m.f 8
28.g odd 6 2 392.2.m.f 8
56.e even 2 1 inner 224.2.e.b 4
56.h odd 2 1 56.2.e.b 4
56.j odd 6 2 392.2.m.f 8
56.k odd 6 2 1568.2.q.e 8
56.m even 6 2 1568.2.q.e 8
56.p even 6 2 392.2.m.f 8
84.h odd 2 1 504.2.p.f 4
112.j even 4 1 1792.2.f.e 4
112.j even 4 1 1792.2.f.f 4
112.l odd 4 1 1792.2.f.e 4
112.l odd 4 1 1792.2.f.f 4
168.e odd 2 1 2016.2.p.e 4
168.i even 2 1 504.2.p.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 4.b odd 2 1
56.2.e.b 4 8.b even 2 1
56.2.e.b 4 28.d even 2 1
56.2.e.b 4 56.h odd 2 1
224.2.e.b 4 1.a even 1 1 trivial
224.2.e.b 4 7.b odd 2 1 inner
224.2.e.b 4 8.d odd 2 1 inner
224.2.e.b 4 56.e even 2 1 inner
392.2.m.f 8 28.f even 6 2
392.2.m.f 8 28.g odd 6 2
392.2.m.f 8 56.j odd 6 2
392.2.m.f 8 56.p even 6 2
504.2.p.f 4 12.b even 2 1
504.2.p.f 4 24.h odd 2 1
504.2.p.f 4 84.h odd 2 1
504.2.p.f 4 168.i even 2 1
1568.2.q.e 8 7.c even 3 2
1568.2.q.e 8 7.d odd 6 2
1568.2.q.e 8 56.k odd 6 2
1568.2.q.e 8 56.m even 6 2
1792.2.f.e 4 16.e even 4 1
1792.2.f.e 4 16.f odd 4 1
1792.2.f.e 4 112.j even 4 1
1792.2.f.e 4 112.l odd 4 1
1792.2.f.f 4 16.e even 4 1
1792.2.f.f 4 16.f odd 4 1
1792.2.f.f 4 112.j even 4 1
1792.2.f.f 4 112.l odd 4 1
2016.2.p.e 4 3.b odd 2 1
2016.2.p.e 4 21.c even 2 1
2016.2.p.e 4 24.f even 2 1
2016.2.p.e 4 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 6$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 + 9 T^{4} )^{2}$$
$5$ $$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{4}$$
$13$ $$( 1 + 20 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 - 6 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 112 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 68 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 2 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 70 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 38 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$