# Properties

 Label 224.3.o.a Level $224$ Weight $3$ Character orbit 224.o Analytic conductor $6.104$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + ( - 3 \zeta_{6} - 3) q^{5} + (8 \zeta_{6} - 3) q^{7} + ( - 8 \zeta_{6} + 8) q^{9}+O(q^{10})$$ q + z * q^3 + (-3*z - 3) * q^5 + (8*z - 3) * q^7 + (-8*z + 8) * q^9 $$q + \zeta_{6} q^{3} + ( - 3 \zeta_{6} - 3) q^{5} + (8 \zeta_{6} - 3) q^{7} + ( - 8 \zeta_{6} + 8) q^{9} + 17 \zeta_{6} q^{11} + (16 \zeta_{6} - 8) q^{13} + ( - 6 \zeta_{6} + 3) q^{15} + 25 \zeta_{6} q^{17} + (7 \zeta_{6} - 7) q^{19} + (5 \zeta_{6} - 8) q^{21} + ( - 3 \zeta_{6} - 3) q^{23} + 2 \zeta_{6} q^{25} + 17 q^{27} + ( - 16 \zeta_{6} + 8) q^{29} + (19 \zeta_{6} - 38) q^{31} + (17 \zeta_{6} - 17) q^{33} + ( - 39 \zeta_{6} + 33) q^{35} + (5 \zeta_{6} + 5) q^{37} + (8 \zeta_{6} - 16) q^{39} + 26 q^{41} - 14 q^{43} + (24 \zeta_{6} - 48) q^{45} + (29 \zeta_{6} + 29) q^{47} + (16 \zeta_{6} - 55) q^{49} + (25 \zeta_{6} - 25) q^{51} + ( - 53 \zeta_{6} + 106) q^{53} + ( - 102 \zeta_{6} + 51) q^{55} - 7 q^{57} - 55 \zeta_{6} q^{59} + (13 \zeta_{6} + 13) q^{61} + (24 \zeta_{6} + 40) q^{63} + ( - 72 \zeta_{6} + 72) q^{65} + 17 \zeta_{6} q^{67} + ( - 6 \zeta_{6} + 3) q^{69} - 119 \zeta_{6} q^{73} + (2 \zeta_{6} - 2) q^{75} + (85 \zeta_{6} - 136) q^{77} + ( - 43 \zeta_{6} - 43) q^{79} - 55 \zeta_{6} q^{81} - 110 q^{83} + ( - 150 \zeta_{6} + 75) q^{85} + ( - 8 \zeta_{6} + 16) q^{87} + (71 \zeta_{6} - 71) q^{89} + (16 \zeta_{6} - 104) q^{91} + ( - 19 \zeta_{6} - 19) q^{93} + ( - 21 \zeta_{6} + 42) q^{95} - 22 q^{97} + 136 q^{99} +O(q^{100})$$ q + z * q^3 + (-3*z - 3) * q^5 + (8*z - 3) * q^7 + (-8*z + 8) * q^9 + 17*z * q^11 + (16*z - 8) * q^13 + (-6*z + 3) * q^15 + 25*z * q^17 + (7*z - 7) * q^19 + (5*z - 8) * q^21 + (-3*z - 3) * q^23 + 2*z * q^25 + 17 * q^27 + (-16*z + 8) * q^29 + (19*z - 38) * q^31 + (17*z - 17) * q^33 + (-39*z + 33) * q^35 + (5*z + 5) * q^37 + (8*z - 16) * q^39 + 26 * q^41 - 14 * q^43 + (24*z - 48) * q^45 + (29*z + 29) * q^47 + (16*z - 55) * q^49 + (25*z - 25) * q^51 + (-53*z + 106) * q^53 + (-102*z + 51) * q^55 - 7 * q^57 - 55*z * q^59 + (13*z + 13) * q^61 + (24*z + 40) * q^63 + (-72*z + 72) * q^65 + 17*z * q^67 + (-6*z + 3) * q^69 - 119*z * q^73 + (2*z - 2) * q^75 + (85*z - 136) * q^77 + (-43*z - 43) * q^79 - 55*z * q^81 - 110 * q^83 + (-150*z + 75) * q^85 + (-8*z + 16) * q^87 + (71*z - 71) * q^89 + (16*z - 104) * q^91 + (-19*z - 19) * q^93 + (-21*z + 42) * q^95 - 22 * q^97 + 136 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 9 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10})$$ 2 * q + q^3 - 9 * q^5 + 2 * q^7 + 8 * q^9 $$2 q + q^{3} - 9 q^{5} + 2 q^{7} + 8 q^{9} + 17 q^{11} + 25 q^{17} - 7 q^{19} - 11 q^{21} - 9 q^{23} + 2 q^{25} + 34 q^{27} - 57 q^{31} - 17 q^{33} + 27 q^{35} + 15 q^{37} - 24 q^{39} + 52 q^{41} - 28 q^{43} - 72 q^{45} + 87 q^{47} - 94 q^{49} - 25 q^{51} + 159 q^{53} - 14 q^{57} - 55 q^{59} + 39 q^{61} + 104 q^{63} + 72 q^{65} + 17 q^{67} - 119 q^{73} - 2 q^{75} - 187 q^{77} - 129 q^{79} - 55 q^{81} - 220 q^{83} + 24 q^{87} - 71 q^{89} - 192 q^{91} - 57 q^{93} + 63 q^{95} - 44 q^{97} + 272 q^{99}+O(q^{100})$$ 2 * q + q^3 - 9 * q^5 + 2 * q^7 + 8 * q^9 + 17 * q^11 + 25 * q^17 - 7 * q^19 - 11 * q^21 - 9 * q^23 + 2 * q^25 + 34 * q^27 - 57 * q^31 - 17 * q^33 + 27 * q^35 + 15 * q^37 - 24 * q^39 + 52 * q^41 - 28 * q^43 - 72 * q^45 + 87 * q^47 - 94 * q^49 - 25 * q^51 + 159 * q^53 - 14 * q^57 - 55 * q^59 + 39 * q^61 + 104 * q^63 + 72 * q^65 + 17 * q^67 - 119 * q^73 - 2 * q^75 - 187 * q^77 - 129 * q^79 - 55 * q^81 - 220 * q^83 + 24 * q^87 - 71 * q^89 - 192 * q^91 - 57 * q^93 + 63 * q^95 - 44 * q^97 + 272 * q^99

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 + 0.866025i 0 −4.50000 2.59808i 0 1.00000 + 6.92820i 0 4.00000 6.92820i 0
207.1 0 0.500000 0.866025i 0 −4.50000 + 2.59808i 0 1.00000 6.92820i 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
56.k 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.o.a 2
4.b odd 2 1 56.3.k.a 2
7.c even 3 1 224.3.o.b 2
7.c even 3 1 1568.3.g.c 2
7.d odd 6 1 1568.3.g.f 2
8.b even 2 1 56.3.k.b yes 2
8.d odd 2 1 224.3.o.b 2
28.d even 2 1 392.3.k.a 2
28.f even 6 1 392.3.g.d 2
28.f even 6 1 392.3.k.c 2
28.g odd 6 1 56.3.k.b yes 2
28.g odd 6 1 392.3.g.e 2
56.h odd 2 1 392.3.k.c 2
56.j odd 6 1 392.3.g.d 2
56.j odd 6 1 392.3.k.a 2
56.k odd 6 1 inner 224.3.o.a 2
56.k odd 6 1 1568.3.g.c 2
56.m even 6 1 1568.3.g.f 2
56.p even 6 1 56.3.k.a 2
56.p even 6 1 392.3.g.e 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} + 9T_{5} + 27$$ T5^2 + 9*T5 + 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 9T + 27$$
$7$ $$T^{2} - 2T + 49$$
$11$ $$T^{2} - 17T + 289$$
$13$ $$T^{2} + 192$$
$17$ $$T^{2} - 25T + 625$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 9T + 27$$
$29$ $$T^{2} + 192$$
$31$ $$T^{2} + 57T + 1083$$
$37$ $$T^{2} - 15T + 75$$
$41$ $$(T - 26)^{2}$$
$43$ $$(T + 14)^{2}$$
$47$ $$T^{2} - 87T + 2523$$
$53$ $$T^{2} - 159T + 8427$$
$59$ $$T^{2} + 55T + 3025$$
$61$ $$T^{2} - 39T + 507$$
$67$ $$T^{2} - 17T + 289$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 119T + 14161$$
$79$ $$T^{2} + 129T + 5547$$
$83$ $$(T + 110)^{2}$$
$89$ $$T^{2} + 71T + 5041$$
$97$ $$(T + 22)^{2}$$