Properties

 Label 224.2.f.a Level $224$ Weight $2$ Character orbit 224.f Analytic conductor $1.789$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{3} + ( -\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + ( 1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{3} + ( -\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + ( 1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{9} -2 \zeta_{16}^{4} q^{11} + ( -3 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{13} + ( -2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{15} + ( -4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{17} + ( 3 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{19} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{21} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{23} + ( 1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{25} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{27} + ( -2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{31} + ( 2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{33} + ( \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{35} + ( -2 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{37} + ( 2 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{39} + ( 6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{41} + ( -4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{43} + ( 3 \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{45} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{47} + ( 1 - 2 \zeta_{16} - 4 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{49} + ( 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{51} -2 q^{53} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( -4 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{57} + ( \zeta_{16} + 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{61} + ( \zeta_{16} - 3 \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 5 \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{63} + ( -4 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{65} + ( 4 \zeta_{16}^{2} - 10 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{67} + ( 2 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{69} + ( -2 \zeta_{16}^{2} + 6 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{71} + ( -6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{73} + ( -5 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{75} + ( -2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{77} + ( 2 \zeta_{16}^{2} - 10 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{79} + ( -3 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{81} + ( 3 \zeta_{16} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{83} + 8 q^{85} + ( 10 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 10 \zeta_{16}^{7} ) q^{87} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{89} + ( -5 \zeta_{16} - 6 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} - 6 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{91} + ( 8 + 8 \zeta_{16}^{2} - 8 \zeta_{16}^{6} ) q^{93} + ( -2 \zeta_{16}^{2} + 8 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{95} + ( -4 \zeta_{16} + 8 \zeta_{16}^{3} + 8 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{97} + ( -4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9} + O(q^{10})$$ $$8 q + 8 q^{9} + 8 q^{25} - 16 q^{29} - 16 q^{37} + 8 q^{49} - 16 q^{53} - 32 q^{57} - 32 q^{65} - 16 q^{77} - 24 q^{81} + 64 q^{85} + 64 q^{93} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1
 0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i −0.382683 − 0.923880i 0.382683 + 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i −0.923880 + 0.382683i
0 −2.61313 0 1.08239i 0 1.08239 + 2.41421i 0 3.82843 0
223.2 0 −2.61313 0 1.08239i 0 1.08239 2.41421i 0 3.82843 0
223.3 0 −1.08239 0 2.61313i 0 −2.61313 + 0.414214i 0 −1.82843 0
223.4 0 −1.08239 0 2.61313i 0 −2.61313 0.414214i 0 −1.82843 0
223.5 0 1.08239 0 2.61313i 0 2.61313 0.414214i 0 −1.82843 0
223.6 0 1.08239 0 2.61313i 0 2.61313 + 0.414214i 0 −1.82843 0
223.7 0 2.61313 0 1.08239i 0 −1.08239 2.41421i 0 3.82843 0
223.8 0 2.61313 0 1.08239i 0 −1.08239 + 2.41421i 0 3.82843 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.8 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
7.b odd 2 1 inner
28.d 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.f.a 8
3.b odd 2 1 2016.2.b.b 8
4.b odd 2 1 inner 224.2.f.a 8
7.b odd 2 1 inner 224.2.f.a 8
7.c even 3 2 1568.2.p.a 16
7.d odd 6 2 1568.2.p.a 16
8.b even 2 1 448.2.f.d 8
8.d odd 2 1 448.2.f.d 8
12.b even 2 1 2016.2.b.b 8
16.e even 4 1 1792.2.e.f 8
16.e even 4 1 1792.2.e.g 8
16.f odd 4 1 1792.2.e.f 8
16.f odd 4 1 1792.2.e.g 8
21.c even 2 1 2016.2.b.b 8
24.f even 2 1 4032.2.b.p 8
24.h odd 2 1 4032.2.b.p 8
28.d even 2 1 inner 224.2.f.a 8
28.f even 6 2 1568.2.p.a 16
28.g odd 6 2 1568.2.p.a 16
56.e even 2 1 448.2.f.d 8
56.h odd 2 1 448.2.f.d 8
84.h odd 2 1 2016.2.b.b 8
112.j even 4 1 1792.2.e.f 8
112.j even 4 1 1792.2.e.g 8
112.l odd 4 1 1792.2.e.f 8
112.l odd 4 1 1792.2.e.g 8
168.e odd 2 1 4032.2.b.p 8
168.i even 2 1 4032.2.b.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.f.a 8 1.a even 1 1 trivial
224.2.f.a 8 4.b odd 2 1 inner
224.2.f.a 8 7.b odd 2 1 inner
224.2.f.a 8 28.d even 2 1 inner
448.2.f.d 8 8.b even 2 1
448.2.f.d 8 8.d odd 2 1
448.2.f.d 8 56.e even 2 1
448.2.f.d 8 56.h odd 2 1
1568.2.p.a 16 7.c even 3 2
1568.2.p.a 16 7.d odd 6 2
1568.2.p.a 16 28.f even 6 2
1568.2.p.a 16 28.g odd 6 2
1792.2.e.f 8 16.e even 4 1
1792.2.e.f 8 16.f odd 4 1
1792.2.e.f 8 112.j even 4 1
1792.2.e.f 8 112.l odd 4 1
1792.2.e.g 8 16.e even 4 1
1792.2.e.g 8 16.f odd 4 1
1792.2.e.g 8 112.j even 4 1
1792.2.e.g 8 112.l odd 4 1
2016.2.b.b 8 3.b odd 2 1
2016.2.b.b 8 12.b even 2 1
2016.2.b.b 8 21.c even 2 1
2016.2.b.b 8 84.h odd 2 1
4032.2.b.p 8 24.f even 2 1
4032.2.b.p 8 24.h odd 2 1
4032.2.b.p 8 168.e odd 2 1
4032.2.b.p 8 168.i even 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 8 - 8 T^{2} + T^{4} )^{2}$$
$5$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$7$ $$2401 - 196 T^{2} - 26 T^{4} - 4 T^{6} + T^{8}$$
$11$ $$( 4 + T^{2} )^{4}$$
$13$ $$( 392 + 40 T^{2} + T^{4} )^{2}$$
$17$ $$( 512 + 64 T^{2} + T^{4} )^{2}$$
$19$ $$( 392 - 40 T^{2} + T^{4} )^{2}$$
$23$ $$( 784 + 72 T^{2} + T^{4} )^{2}$$
$29$ $$( -28 + 4 T + T^{2} )^{4}$$
$31$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$37$ $$( -28 + 4 T + T^{2} )^{4}$$
$41$ $$( 6272 + 160 T^{2} + T^{4} )^{2}$$
$43$ $$( 784 + 72 T^{2} + T^{4} )^{2}$$
$47$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$53$ $$( 2 + T )^{8}$$
$59$ $$( 2312 - 104 T^{2} + T^{4} )^{2}$$
$61$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$67$ $$( 4624 + 264 T^{2} + T^{4} )^{2}$$
$71$ $$( 784 + 88 T^{2} + T^{4} )^{2}$$
$73$ $$( 128 + 160 T^{2} + T^{4} )^{2}$$
$79$ $$( 8464 + 216 T^{2} + T^{4} )^{2}$$
$83$ $$( 392 - 136 T^{2} + T^{4} )^{2}$$
$89$ $$( 128 + 32 T^{2} + T^{4} )^{2}$$
$97$ $$( 25088 + 320 T^{2} + T^{4} )^{2}$$