# Properties

 Label 224.3.r.a Level 224 Weight 3 Character orbit 224.r Analytic conductor 6.104 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{12} q^{3} -9 \zeta_{12}^{2} q^{5} + ( 8 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 16 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + 5 \zeta_{12} q^{3} -9 \zeta_{12}^{2} q^{5} + ( 8 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 16 \zeta_{12}^{2} q^{9} + 3 \zeta_{12} q^{11} + 16 q^{13} -45 \zeta_{12}^{3} q^{15} + ( 7 - 7 \zeta_{12}^{2} ) q^{17} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{19} + ( 15 + 25 \zeta_{12}^{2} ) q^{21} + ( -19 \zeta_{12} + 19 \zeta_{12}^{3} ) q^{23} + ( -56 + 56 \zeta_{12}^{2} ) q^{25} + 35 \zeta_{12}^{3} q^{27} -32 q^{29} + 11 \zeta_{12} q^{31} + 15 \zeta_{12}^{2} q^{33} + ( -27 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{35} + \zeta_{12}^{2} q^{37} + 80 \zeta_{12} q^{39} -40 q^{41} -40 \zeta_{12}^{3} q^{43} + ( 144 - 144 \zeta_{12}^{2} ) q^{45} + ( -85 \zeta_{12} + 85 \zeta_{12}^{3} ) q^{47} + ( 39 + 16 \zeta_{12}^{2} ) q^{49} + ( 35 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{51} + ( -7 + 7 \zeta_{12}^{2} ) q^{53} -27 \zeta_{12}^{3} q^{55} + 55 q^{57} + 53 \zeta_{12} q^{59} -79 \zeta_{12}^{2} q^{61} + ( 48 \zeta_{12} + 80 \zeta_{12}^{3} ) q^{63} -144 \zeta_{12}^{2} q^{65} -11 \zeta_{12} q^{67} -95 q^{69} -48 \zeta_{12}^{3} q^{71} + ( -143 + 143 \zeta_{12}^{2} ) q^{73} + ( -280 \zeta_{12} + 280 \zeta_{12}^{3} ) q^{75} + ( 9 + 15 \zeta_{12}^{2} ) q^{77} + ( -35 \zeta_{12} + 35 \zeta_{12}^{3} ) q^{79} + ( -31 + 31 \zeta_{12}^{2} ) q^{81} -8 \zeta_{12}^{3} q^{83} -63 q^{85} -160 \zeta_{12} q^{87} + 97 \zeta_{12}^{2} q^{89} + ( 128 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{91} + 55 \zeta_{12}^{2} q^{93} -99 \zeta_{12} q^{95} -88 q^{97} + 48 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 18q^{5} + 32q^{9} + O(q^{10})$$ $$4q - 18q^{5} + 32q^{9} + 64q^{13} + 14q^{17} + 110q^{21} - 112q^{25} - 128q^{29} + 30q^{33} + 2q^{37} - 160q^{41} + 288q^{45} + 188q^{49} - 14q^{53} + 220q^{57} - 158q^{61} - 288q^{65} - 380q^{69} - 286q^{73} + 66q^{77} - 62q^{81} - 252q^{85} + 194q^{89} + 110q^{93} - 352q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −4.33013 2.50000i 0 −4.50000 7.79423i 0 −6.92820 1.00000i 0 8.00000 + 13.8564i 0
95.2 0 4.33013 + 2.50000i 0 −4.50000 7.79423i 0 6.92820 + 1.00000i 0 8.00000 + 13.8564i 0
191.1 0 −4.33013 + 2.50000i 0 −4.50000 + 7.79423i 0 −6.92820 + 1.00000i 0 8.00000 13.8564i 0
191.2 0 4.33013 2.50000i 0 −4.50000 + 7.79423i 0 6.92820 1.00000i 0 8.00000 13.8564i 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.r.a 4
4.b odd 2 1 inner 224.3.r.a 4
7.c even 3 1 inner 224.3.r.a 4
7.c even 3 1 1568.3.d.e 2
7.d odd 6 1 1568.3.d.a 2
8.b even 2 1 448.3.r.c 4
8.d odd 2 1 448.3.r.c 4
28.f even 6 1 1568.3.d.a 2
28.g odd 6 1 inner 224.3.r.a 4
28.g odd 6 1 1568.3.d.e 2
56.k odd 6 1 448.3.r.c 4
56.p even 6 1 448.3.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.a 4 1.a even 1 1 trivial
224.3.r.a 4 4.b odd 2 1 inner
224.3.r.a 4 7.c even 3 1 inner
224.3.r.a 4 28.g odd 6 1 inner
448.3.r.c 4 8.b even 2 1
448.3.r.c 4 8.d odd 2 1
448.3.r.c 4 56.k odd 6 1
448.3.r.c 4 56.p even 6 1
1568.3.d.a 2 7.d odd 6 1
1568.3.d.a 2 28.f even 6 1
1568.3.d.e 2 7.c even 3 1
1568.3.d.e 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 25 T_{3}^{2} + 625$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 7 T^{2} - 32 T^{4} - 567 T^{6} + 6561 T^{8}$$
$5$ $$( 1 + 9 T + 56 T^{2} + 225 T^{3} + 625 T^{4} )^{2}$$
$7$ $$1 - 94 T^{2} + 2401 T^{4}$$
$11$ $$1 + 233 T^{2} + 39648 T^{4} + 3411353 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 16 T + 169 T^{2} )^{4}$$
$17$ $$( 1 - 7 T - 240 T^{2} - 2023 T^{3} + 83521 T^{4} )^{2}$$
$19$ $$( 1 - 46 T^{2} + 130321 T^{4} )( 1 + 647 T^{2} + 130321 T^{4} )$$
$23$ $$1 + 697 T^{2} + 205968 T^{4} + 195049177 T^{6} + 78310985281 T^{8}$$
$29$ $$( 1 + 32 T + 841 T^{2} )^{4}$$
$31$ $$1 + 1801 T^{2} + 2320080 T^{4} + 1663261321 T^{6} + 852891037441 T^{8}$$
$37$ $$( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 + 40 T + 1681 T^{2} )^{4}$$
$43$ $$( 1 - 2098 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 2807 T^{2} + 2999568 T^{4} - 13697264567 T^{6} + 23811286661761 T^{8}$$
$53$ $$( 1 + 7 T - 2760 T^{2} + 19663 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 + 4153 T^{2} + 5130048 T^{4} + 50323400233 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 79 T + 2520 T^{2} + 293959 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$1 + 8857 T^{2} + 58295328 T^{4} + 178478478697 T^{6} + 406067677556641 T^{8}$$
$71$ $$( 1 - 7778 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 + 46 T + 5329 T^{2} )^{2}( 1 + 97 T + 5329 T^{2} )^{2}$$
$79$ $$1 + 11257 T^{2} + 87769968 T^{4} + 438461061817 T^{6} + 1517108809906561 T^{8}$$
$83$ $$( 1 - 13714 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 97 T + 1488 T^{2} - 768337 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 + 88 T + 9409 T^{2} )^{4}$$