# Properties

 Label 224.2.u.a Level 224 Weight 2 Character orbit 224.u Analytic conductor 1.789 Analytic rank 1 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.u (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -2 - 2 \zeta_{8} ) q^{5} + ( 2 + 2 \zeta_{8} ) q^{6} + \zeta_{8} q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -2 + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -2 - 2 \zeta_{8} ) q^{5} + ( 2 + 2 \zeta_{8} ) q^{6} + \zeta_{8} q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -2 + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{9} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{10} + ( -3 + 3 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{15} -4 q^{16} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + ( -1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{19} + ( 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{20} + ( 1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{21} + ( 1 - 5 \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + ( -4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{24} + ( 4 + 3 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{25} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{26} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{27} -2 \zeta_{8}^{3} q^{28} + ( -4 - 3 \zeta_{8} - 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{29} + ( -4 - 8 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{30} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + ( 4 - 4 \zeta_{8}^{2} ) q^{32} + ( 10 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{33} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{35} + ( 4 + 2 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{36} + ( -4 - 4 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{37} + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{38} -4 \zeta_{8} q^{39} + ( -4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{40} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{42} + ( -4 + 4 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{43} + ( 4 + 4 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{44} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{45} -2 \zeta_{8}^{2} q^{46} + ( 2 \zeta_{8} + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{48} + \zeta_{8}^{2} q^{49} + ( -8 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{50} + ( -4 - 4 \zeta_{8}^{3} ) q^{52} + ( 5 - 5 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{54} + ( 10 - 10 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{55} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{56} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{57} + ( 7 + 7 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{58} + ( -5 - 5 \zeta_{8} + 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{59} + ( 8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{60} + ( -4 - 4 \zeta_{8}^{3} ) q^{61} + ( 6 - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{62} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + 8 \zeta_{8}^{2} q^{64} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} + ( -10 + 10 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{66} + ( 4 + 5 \zeta_{8} + 5 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{67} + ( 2 + 2 \zeta_{8} ) q^{69} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{70} -6 \zeta_{8} q^{71} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + ( 6 - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{73} + ( 1 + 7 \zeta_{8} - 7 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{74} + ( 3 - 3 \zeta_{8} - 11 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{75} + ( -6 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{76} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{77} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{78} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{79} + ( 8 + 8 \zeta_{8} ) q^{80} + ( 2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( -1 + 7 \zeta_{8} - 7 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{83} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{84} + ( 3 - 5 \zeta_{8} - 5 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{86} + ( -6 + 6 \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{87} + ( -10 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{88} + ( -8 + 4 \zeta_{8} - 8 \zeta_{8}^{2} ) q^{89} + ( -2 + 2 \zeta_{8} + 10 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{90} + ( 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{91} + ( 2 + 2 \zeta_{8}^{2} ) q^{92} + ( 6 + 10 \zeta_{8} + 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{93} + ( -6 - 4 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{94} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + ( -8 - 8 \zeta_{8} ) q^{96} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{97} + ( -1 - \zeta_{8}^{2} ) q^{98} + ( -1 - 12 \zeta_{8} - 12 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} - 8q^{5} + 8q^{6} + 8q^{8} - 8q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} - 8q^{5} + 8q^{6} + 8q^{8} - 8q^{9} + 8q^{10} - 12q^{11} - 8q^{12} - 16q^{16} - 4q^{19} + 4q^{21} + 4q^{22} - 4q^{23} + 16q^{25} + 8q^{26} + 8q^{27} - 16q^{29} - 16q^{30} - 24q^{31} + 16q^{32} + 40q^{33} + 16q^{36} - 16q^{37} + 16q^{38} - 16q^{40} - 8q^{41} - 16q^{43} + 16q^{44} + 24q^{45} + 16q^{48} - 32q^{50} - 16q^{52} + 20q^{53} + 40q^{55} + 8q^{57} + 28q^{58} - 20q^{59} + 32q^{60} - 16q^{61} + 24q^{62} - 4q^{63} - 40q^{66} + 16q^{67} + 8q^{69} + 8q^{70} - 32q^{72} + 24q^{73} + 4q^{74} + 12q^{75} - 24q^{76} - 8q^{77} + 32q^{80} - 4q^{83} - 8q^{84} + 12q^{86} - 24q^{87} - 40q^{88} - 32q^{89} - 8q^{90} + 8q^{92} + 24q^{93} - 24q^{94} + 16q^{95} - 32q^{96} + 16q^{97} - 4q^{98} - 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.00000 1.00000i −1.00000 0.414214i 2.00000i −0.585786 1.41421i 0.585786 + 1.41421i −0.707107 + 0.707107i 2.00000 2.00000i −1.29289 1.29289i −0.828427 + 2.00000i
85.1 −1.00000 + 1.00000i −1.00000 + 0.414214i 2.00000i −0.585786 + 1.41421i 0.585786 1.41421i −0.707107 0.707107i 2.00000 + 2.00000i −1.29289 + 1.29289i −0.828427 2.00000i
141.1 −1.00000 1.00000i −1.00000 + 2.41421i 2.00000i −3.41421 + 1.41421i 3.41421 1.41421i 0.707107 0.707107i 2.00000 2.00000i −2.70711 2.70711i 4.82843 + 2.00000i
197.1 −1.00000 + 1.00000i −1.00000 2.41421i 2.00000i −3.41421 1.41421i 3.41421 + 1.41421i 0.707107 + 0.707107i 2.00000 + 2.00000i −2.70711 + 2.70711i 4.82843 2.00000i
 $$n$$: 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
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.u.a 4
4.b odd 2 1 896.2.u.a 4
32.g even 8 1 inner 224.2.u.a 4
32.h odd 8 1 896.2.u.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.a 4 1.a even 1 1 trivial
224.2.u.a 4 32.g even 8 1 inner
896.2.u.a 4 4.b odd 2 1
896.2.u.a 4 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4 T_{3}^{3} + 12 T_{3}^{2} + 16 T_{3} + 8$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} )^{2}$$
$3$ $$1 + 4 T + 12 T^{2} + 28 T^{3} + 56 T^{4} + 84 T^{5} + 108 T^{6} + 108 T^{7} + 81 T^{8}$$
$5$ $$1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 160 T^{5} + 600 T^{6} + 1000 T^{7} + 625 T^{8}$$
$7$ $$1 + T^{4}$$
$11$ $$1 + 12 T + 86 T^{2} + 428 T^{3} + 1634 T^{4} + 4708 T^{5} + 10406 T^{6} + 15972 T^{7} + 14641 T^{8}$$
$13$ $$1 + 8 T^{2} + 72 T^{3} + 32 T^{4} + 936 T^{5} + 1352 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$1 + 4 T + 36 T^{2} + 196 T^{3} + 920 T^{4} + 3724 T^{5} + 12996 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 16 T + 162 T^{2} + 1216 T^{3} + 7490 T^{4} + 35264 T^{5} + 136242 T^{6} + 390224 T^{7} + 707281 T^{8}$$
$31$ $$( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 16 T + 162 T^{2} + 1312 T^{3} + 9026 T^{4} + 48544 T^{5} + 221778 T^{6} + 810448 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 14760 T^{5} + 53792 T^{6} + 551368 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 16 T + 114 T^{2} + 632 T^{3} + 3810 T^{4} + 27176 T^{5} + 210786 T^{6} + 1272112 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 100 T^{2} + 5766 T^{4} - 220900 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 20 T + 118 T^{2} + 412 T^{3} - 8478 T^{4} + 21836 T^{5} + 331462 T^{6} - 2977540 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 20 T + 300 T^{2} + 3180 T^{3} + 28600 T^{4} + 187620 T^{5} + 1044300 T^{6} + 4107580 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 15616 T^{5} + 357216 T^{6} + 3631696 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 16 T + 226 T^{2} - 2456 T^{3} + 23362 T^{4} - 164552 T^{5} + 1014514 T^{6} - 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 1154 T^{4} + 25411681 T^{8}$$
$73$ $$1 - 24 T + 288 T^{2} - 3096 T^{3} + 30146 T^{4} - 226008 T^{5} + 1534752 T^{6} - 9336408 T^{7} + 28398241 T^{8}$$
$79$ $$( 1 - 60 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 4 T + 132 T^{2} + 1236 T^{3} + 11608 T^{4} + 102588 T^{5} + 909348 T^{6} + 2287148 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 32 T + 512 T^{2} + 6432 T^{3} + 68258 T^{4} + 572448 T^{5} + 4055552 T^{6} + 22559008 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$