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

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

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$T^{4} - 289 T^{2} + 83521$$
$13$ $$(T - 24)^{4}$$
$17$ $$(T^{2} + T + 1)^{2}$$
$19$ $$T^{4} - 49T^{2} + 2401$$
$23$ $$T^{4} - 49T^{2} + 2401$$
$29$ $$(T - 24)^{4}$$
$31$ $$T^{4} - 1681 T^{2} + \cdots + 2825761$$
$37$ $$(T^{2} - 49 T + 2401)^{2}$$
$41$ $$(T + 48)^{4}$$
$43$ $$(T^{2} + 576)^{2}$$
$47$ $$T^{4} - 3025 T^{2} + \cdots + 9150625$$
$53$ $$(T^{2} - 25 T + 625)^{2}$$
$59$ $$T^{4} - 289 T^{2} + 83521$$
$61$ $$(T^{2} - T + 1)^{2}$$
$67$ $$T^{4} - 4225 T^{2} + \cdots + 17850625$$
$71$ $$(T^{2} + 9216)^{2}$$
$73$ $$(T^{2} + 95 T + 9025)^{2}$$
$79$ $$T^{4} - 1681 T^{2} + \cdots + 2825761$$
$83$ $$(T^{2} + 5184)^{2}$$
$89$ $$(T^{2} + 95 T + 9025)^{2}$$
$97$ $$(T - 144)^{4}$$