# Properties

 Label 224.2.e.a Level 224 Weight 2 Character orbit 224.e Analytic conductor 1.789 Analytic rank 0 Dimension 2 CM discriminant -7 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: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{7} + 3 q^{9} +O(q^{10})$$ $$q -\beta q^{7} + 3 q^{9} + 4 q^{11} + 2 \beta q^{23} -5 q^{25} -4 \beta q^{29} + 4 \beta q^{37} -12 q^{43} -7 q^{49} + 4 \beta q^{53} -3 \beta q^{63} -4 q^{67} -2 \beta q^{71} -4 \beta q^{77} + 6 \beta q^{79} + 9 q^{81} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{9} + O(q^{10})$$ $$2q + 6q^{9} + 8q^{11} - 10q^{25} - 24q^{43} - 14q^{49} - 8q^{67} + 18q^{81} + 24q^{99} + O(q^{100})$$

## Character values

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

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

## 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
 0.5 + 1.32288i 0.5 − 1.32288i
0 0 0 0 0 2.64575i 0 3.00000 0
111.2 0 0 0 0 0 2.64575i 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 CM by $$\Q(\sqrt{-7})$$
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.a 2
3.b odd 2 1 2016.2.p.a 2
4.b odd 2 1 56.2.e.a 2
7.b odd 2 1 CM 224.2.e.a 2
7.c even 3 2 1568.2.q.a 4
7.d odd 6 2 1568.2.q.a 4
8.b even 2 1 56.2.e.a 2
8.d odd 2 1 inner 224.2.e.a 2
12.b even 2 1 504.2.p.a 2
16.e even 4 2 1792.2.f.d 4
16.f odd 4 2 1792.2.f.d 4
21.c even 2 1 2016.2.p.a 2
24.f even 2 1 2016.2.p.a 2
24.h odd 2 1 504.2.p.a 2
28.d even 2 1 56.2.e.a 2
28.f even 6 2 392.2.m.a 4
28.g odd 6 2 392.2.m.a 4
56.e even 2 1 inner 224.2.e.a 2
56.h odd 2 1 56.2.e.a 2
56.j odd 6 2 392.2.m.a 4
56.k odd 6 2 1568.2.q.a 4
56.m even 6 2 1568.2.q.a 4
56.p even 6 2 392.2.m.a 4
84.h odd 2 1 504.2.p.a 2
112.j even 4 2 1792.2.f.d 4
112.l odd 4 2 1792.2.f.d 4
168.e odd 2 1 2016.2.p.a 2
168.i even 2 1 504.2.p.a 2

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 7 T^{2}$$
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$29$ $$( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} )$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 + 12 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} )$$
$59$ $$( 1 - 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$
$83$ $$( 1 - 83 T^{2} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$