# Properties

 Label 224.3.r.b 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} -\zeta_{12}^{2} q^{5} -7 \zeta_{12}^{3} q^{7} -8 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{3} -\zeta_{12}^{2} q^{5} -7 \zeta_{12}^{3} q^{7} -8 \zeta_{12}^{2} q^{9} -17 \zeta_{12} q^{11} + 24 q^{13} -\zeta_{12}^{3} q^{15} + ( -1 + \zeta_{12}^{2} ) q^{17} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{19} + ( 7 - 7 \zeta_{12}^{2} ) q^{21} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{23} + ( 24 - 24 \zeta_{12}^{2} ) q^{25} -17 \zeta_{12}^{3} q^{27} + 24 q^{29} -41 \zeta_{12} q^{31} -17 \zeta_{12}^{2} q^{33} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{35} + 49 \zeta_{12}^{2} q^{37} + 24 \zeta_{12} q^{39} -48 q^{41} + 24 \zeta_{12}^{3} q^{43} + ( -8 + 8 \zeta_{12}^{2} ) q^{45} + ( 55 \zeta_{12} - 55 \zeta_{12}^{3} ) q^{47} -49 q^{49} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( 25 - 25 \zeta_{12}^{2} ) q^{53} + 17 \zeta_{12}^{3} q^{55} + 7 q^{57} + 17 \zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + ( -56 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{63} -24 \zeta_{12}^{2} q^{65} + 65 \zeta_{12} q^{67} -7 q^{69} + 96 \zeta_{12}^{3} q^{71} + ( -95 + 95 \zeta_{12}^{2} ) q^{73} + ( 24 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{75} + ( -119 + 119 \zeta_{12}^{2} ) q^{77} + ( 41 \zeta_{12} - 41 \zeta_{12}^{3} ) q^{79} + ( -55 + 55 \zeta_{12}^{2} ) q^{81} + 72 \zeta_{12}^{3} q^{83} + q^{85} + 24 \zeta_{12} q^{87} -95 \zeta_{12}^{2} q^{89} -168 \zeta_{12}^{3} q^{91} -41 \zeta_{12}^{2} q^{93} -7 \zeta_{12} q^{95} + 144 q^{97} + 136 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} - 16q^{9} + O(q^{10})$$ $$4q - 2q^{5} - 16q^{9} + 96q^{13} - 2q^{17} + 14q^{21} + 48q^{25} + 96q^{29} - 34q^{33} + 98q^{37} - 192q^{41} - 16q^{45} - 196q^{49} + 50q^{53} + 28q^{57} + 2q^{61} - 48q^{65} - 28q^{69} - 190q^{73} - 238q^{77} - 110q^{81} + 4q^{85} - 190q^{89} - 82q^{93} + 576q^{97} + O(q^{100})$$

## 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 −0.866025 0.500000i 0 −0.500000 0.866025i 0 7.00000i 0 −4.00000 6.92820i 0
95.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 7.00000i 0 −4.00000 6.92820i 0
191.1 0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 7.00000i 0 −4.00000 + 6.92820i 0
191.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 7.00000i 0 −4.00000 + 6.92820i 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.b 4
4.b odd 2 1 inner 224.3.r.b 4
7.c even 3 1 inner 224.3.r.b 4
7.c even 3 1 1568.3.d.d 2
7.d odd 6 1 1568.3.d.c 2
8.b even 2 1 448.3.r.b 4
8.d odd 2 1 448.3.r.b 4
28.f even 6 1 1568.3.d.c 2
28.g odd 6 1 inner 224.3.r.b 4
28.g odd 6 1 1568.3.d.d 2
56.k odd 6 1 448.3.r.b 4
56.p even 6 1 448.3.r.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 17 T^{2} + 208 T^{4} + 1377 T^{6} + 6561 T^{8}$$
$5$ $$( 1 + T - 24 T^{2} + 25 T^{3} + 625 T^{4} )^{2}$$
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$1 - 47 T^{2} - 12432 T^{4} - 688127 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 24 T + 169 T^{2} )^{4}$$
$17$ $$( 1 + T - 288 T^{2} + 289 T^{3} + 83521 T^{4} )^{2}$$
$19$ $$1 + 673 T^{2} + 322608 T^{4} + 87706033 T^{6} + 16983563041 T^{8}$$
$23$ $$1 + 1009 T^{2} + 738240 T^{4} + 282359569 T^{6} + 78310985281 T^{8}$$
$29$ $$( 1 - 24 T + 841 T^{2} )^{4}$$
$31$ $$1 + 241 T^{2} - 865440 T^{4} + 222568561 T^{6} + 852891037441 T^{8}$$
$37$ $$( 1 - 49 T + 1032 T^{2} - 67081 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 + 48 T + 1681 T^{2} )^{4}$$
$43$ $$( 1 - 3122 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$1 + 1393 T^{2} - 2939232 T^{4} + 6797395633 T^{6} + 23811286661761 T^{8}$$
$53$ $$( 1 - 25 T - 2184 T^{2} - 70225 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 + 6673 T^{2} + 32411568 T^{4} + 80859149953 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 - T - 3720 T^{2} - 3721 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$1 + 4753 T^{2} + 2439888 T^{4} + 95778278113 T^{6} + 406067677556641 T^{8}$$
$71$ $$( 1 - 866 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 + 95 T + 3696 T^{2} + 506255 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$1 + 10801 T^{2} + 77711520 T^{4} + 420699824881 T^{6} + 1517108809906561 T^{8}$$
$83$ $$( 1 - 8594 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 + 95 T + 1104 T^{2} + 752495 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 - 144 T + 9409 T^{2} )^{4}$$