# Properties

 Label 224.3.h.a Level 224 Weight 3 Character orbit 224.h Analytic conductor 6.104 Analytic rank 0 Dimension 2 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 224.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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 $$\beta = 8\sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -7 q^{7} -9 q^{9} +O(q^{10})$$ $$q -7 q^{7} -9 q^{9} -\beta q^{11} -18 q^{23} -25 q^{25} -\beta q^{29} -3 \beta q^{37} + 3 \beta q^{43} + 49 q^{49} + 5 \beta q^{53} + 63 q^{63} -3 \beta q^{67} + 114 q^{71} + 7 \beta q^{77} -94 q^{79} + 81 q^{81} + 9 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{7} - 18q^{9} + O(q^{10})$$ $$2q - 14q^{7} - 18q^{9} - 36q^{23} - 50q^{25} + 98q^{49} + 126q^{63} + 228q^{71} - 188q^{79} + 162q^{81} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.5 + 1.32288i 0.5 − 1.32288i
0 0 0 0 0 −7.00000 0 −9.00000 0
209.2 0 0 0 0 0 −7.00000 0 −9.00000 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.b even 2 1 inner
56.h 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.h.a 2
3.b odd 2 1 2016.3.l.b 2
4.b odd 2 1 56.3.h.b 2
7.b odd 2 1 CM 224.3.h.a 2
8.b even 2 1 inner 224.3.h.a 2
8.d odd 2 1 56.3.h.b 2
12.b even 2 1 504.3.l.b 2
21.c even 2 1 2016.3.l.b 2
24.f even 2 1 504.3.l.b 2
24.h odd 2 1 2016.3.l.b 2
28.d even 2 1 56.3.h.b 2
28.f even 6 2 392.3.j.b 4
28.g odd 6 2 392.3.j.b 4
56.e even 2 1 56.3.h.b 2
56.h odd 2 1 inner 224.3.h.a 2
56.k odd 6 2 392.3.j.b 4
56.m even 6 2 392.3.j.b 4
84.h odd 2 1 504.3.l.b 2
168.e odd 2 1 504.3.l.b 2
168.i even 2 1 2016.3.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.b 2 4.b odd 2 1
56.3.h.b 2 8.d odd 2 1
56.3.h.b 2 28.d even 2 1
56.3.h.b 2 56.e even 2 1
224.3.h.a 2 1.a even 1 1 trivial
224.3.h.a 2 7.b odd 2 1 CM
224.3.h.a 2 8.b even 2 1 inner
224.3.h.a 2 56.h odd 2 1 inner
392.3.j.b 4 28.f even 6 2
392.3.j.b 4 28.g odd 6 2
392.3.j.b 4 56.k odd 6 2
392.3.j.b 4 56.m even 6 2
504.3.l.b 2 12.b even 2 1
504.3.l.b 2 24.f even 2 1
504.3.l.b 2 84.h odd 2 1
504.3.l.b 2 168.e odd 2 1
2016.3.l.b 2 3.b odd 2 1
2016.3.l.b 2 21.c even 2 1
2016.3.l.b 2 24.h odd 2 1
2016.3.l.b 2 168.i even 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ $$( 1 + 25 T^{2} )^{2}$$
$7$ $$( 1 + 7 T )^{2}$$
$11$ $$( 1 - 6 T + 121 T^{2} )( 1 + 6 T + 121 T^{2} )$$
$13$ $$( 1 + 169 T^{2} )^{2}$$
$17$ $$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$
$19$ $$( 1 + 361 T^{2} )^{2}$$
$23$ $$( 1 + 18 T + 529 T^{2} )^{2}$$
$29$ $$( 1 - 54 T + 841 T^{2} )( 1 + 54 T + 841 T^{2} )$$
$31$ $$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$
$37$ $$( 1 - 38 T + 1369 T^{2} )( 1 + 38 T + 1369 T^{2} )$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$( 1 - 58 T + 1849 T^{2} )( 1 + 58 T + 1849 T^{2} )$$
$47$ $$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$
$53$ $$( 1 - 6 T + 2809 T^{2} )( 1 + 6 T + 2809 T^{2} )$$
$59$ $$( 1 + 3481 T^{2} )^{2}$$
$61$ $$( 1 + 3721 T^{2} )^{2}$$
$67$ $$( 1 - 118 T + 4489 T^{2} )( 1 + 118 T + 4489 T^{2} )$$
$71$ $$( 1 - 114 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 - 73 T )^{2}( 1 + 73 T )^{2}$$
$79$ $$( 1 + 94 T + 6241 T^{2} )^{2}$$
$83$ $$( 1 + 6889 T^{2} )^{2}$$
$89$ $$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$
$97$ $$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$