# Properties

 Label 224.3.c.a Level $224$ Weight $3$ Character orbit 224.c Analytic conductor $6.104$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4694952902656.3 Defining polynomial: $$x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1$$ x^8 + 20*x^6 + 56*x^4 + 20*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - 7) q^{9}+O(q^{10})$$ q - b6 * q^3 + b7 * q^5 + b2 * q^7 + (-b5 - 7) * q^9 $$q - \beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - 7) q^{9} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{11} + (\beta_{7} - \beta_{4}) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{15} + ( - 2 \beta_{7} - 3 \beta_{4}) q^{17} + (\beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{19} + (5 \beta_{7} - \beta_{5} + \beta_{4} - 4) q^{21} + ( - \beta_{3} - \beta_{2} - 4 \beta_1) q^{23} + ( - \beta_{5} + 1) q^{25} + (8 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{27} + ( - 2 \beta_{5} - 14) q^{29} + (2 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{31} + (12 \beta_{7} + 3 \beta_{4}) q^{33} + ( - 7 \beta_{6} + 7 \beta_1) q^{35} + (2 \beta_{5} + 26) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{39} + ( - 6 \beta_{7} - 2 \beta_{4}) q^{41} + (5 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{43} + ( - 17 \beta_{7} - 7 \beta_{4}) q^{45} + ( - 2 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{47} + (4 \beta_{7} + 2 \beta_{5} + 5 \beta_{4} + 1) q^{49} + (4 \beta_{3} + 4 \beta_{2} - 28 \beta_1) q^{51} - 30 q^{53} + ( - 16 \beta_{6} - \beta_{3} + \beta_{2} + \beta_1) q^{55} + (3 \beta_{5} + 24) q^{57} + ( - 17 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{59} + ( - 5 \beta_{7} + 8 \beta_{4}) q^{61} + (14 \beta_{6} - 7 \beta_{3} - 4 \beta_{2} + 28 \beta_1) q^{63} + (\beta_{5} - 32) q^{65} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{67} + 3 \beta_{4} q^{69} + (5 \beta_{3} + 5 \beta_{2} + 15 \beta_1) q^{71} + ( - 16 \beta_{7} + \beta_{4}) q^{73} + (9 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{75} + (6 \beta_{7} + 3 \beta_{5} + 4 \beta_{4} + 54) q^{77} + ( - 3 \beta_{3} - 3 \beta_{2} - 39 \beta_1) q^{79} + (5 \beta_{5} + 89) q^{81} + (23 \beta_{6} - 6 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{83} + (8 \beta_{5} + 24) q^{85} + (34 \beta_{6} + 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{87} + (4 \beta_{7} + \beta_{4}) q^{89} + ( - 7 \beta_{6} + 7 \beta_{3} + \beta_{2} + 14 \beta_1) q^{91} + ( - 4 \beta_{5} + 8) q^{93} + (2 \beta_{3} + 2 \beta_{2} - 19 \beta_1) q^{95} + ( - 18 \beta_{7} + 3 \beta_{4}) q^{97} + ( - 15 \beta_{3} - 15 \beta_{2} + 60 \beta_1) q^{99}+O(q^{100})$$ q - b6 * q^3 + b7 * q^5 + b2 * q^7 + (-b5 - 7) * q^9 + (b3 + b2 - 2*b1) * q^11 + (b7 - b4) * q^13 + (-2*b3 - 2*b2 + 5*b1) * q^15 + (-2*b7 - 3*b4) * q^17 + (b6 + b3 - b2 - b1) * q^19 + (5*b7 - b5 + b4 - 4) * q^21 + (-b3 - b2 - 4*b1) * q^23 + (-b5 + 1) * q^25 + (8*b6 + 3*b3 - 3*b2 - 3*b1) * q^27 + (-2*b5 - 14) * q^29 + (2*b6 - 3*b3 + 3*b2 + 3*b1) * q^31 + (12*b7 + 3*b4) * q^33 + (-7*b6 + 7*b1) * q^35 + (2*b5 + 26) * q^37 + (-2*b3 - 2*b2 - b1) * q^39 + (-6*b7 - 2*b4) * q^41 + (5*b3 + 5*b2 + 4*b1) * q^43 + (-17*b7 - 7*b4) * q^45 + (-2*b6 - 3*b3 + 3*b2 + 3*b1) * q^47 + (4*b7 + 2*b5 + 5*b4 + 1) * q^49 + (4*b3 + 4*b2 - 28*b1) * q^51 - 30 * q^53 + (-16*b6 - b3 + b2 + b1) * q^55 + (3*b5 + 24) * q^57 + (-17*b6 + 3*b3 - 3*b2 - 3*b1) * q^59 + (-5*b7 + 8*b4) * q^61 + (14*b6 - 7*b3 - 4*b2 + 28*b1) * q^63 + (b5 - 32) * q^65 + (-3*b3 - 3*b2) * q^67 + 3*b4 * q^69 + (5*b3 + 5*b2 + 15*b1) * q^71 + (-16*b7 + b4) * q^73 + (9*b6 + 3*b3 - 3*b2 - 3*b1) * q^75 + (6*b7 + 3*b5 + 4*b4 + 54) * q^77 + (-3*b3 - 3*b2 - 39*b1) * q^79 + (5*b5 + 89) * q^81 + (23*b6 - 6*b3 + 6*b2 + 6*b1) * q^83 + (8*b5 + 24) * q^85 + (34*b6 + 6*b3 - 6*b2 - 6*b1) * q^87 + (4*b7 + b4) * q^89 + (-7*b6 + 7*b3 + b2 + 14*b1) * q^91 + (-4*b5 + 8) * q^93 + (2*b3 + 2*b2 - 19*b1) * q^95 + (-18*b7 + 3*b4) * q^97 + (-15*b3 - 15*b2 + 60*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 56 q^{9}+O(q^{10})$$ 8 * q - 56 * q^9 $$8 q - 56 q^{9} - 32 q^{21} + 8 q^{25} - 112 q^{29} + 208 q^{37} + 8 q^{49} - 240 q^{53} + 192 q^{57} - 256 q^{65} + 432 q^{77} + 712 q^{81} + 192 q^{85} + 64 q^{93}+O(q^{100})$$ 8 * q - 56 * q^9 - 32 * q^21 + 8 * q^25 - 112 * q^29 + 208 * q^37 + 8 * q^49 - 240 * q^53 + 192 * q^57 - 256 * q^65 + 432 * q^77 + 712 * q^81 + 192 * q^85 + 64 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 10\nu^{6} + 196\nu^{4} + 490\nu^{2} + 88 ) / 21$$ (10*v^6 + 196*v^4 + 490*v^2 + 88) / 21 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} + 21\nu^{5} - 21\nu^{4} + 77\nu^{3} - 77\nu^{2} + 97\nu - 48 ) / 7$$ (v^7 - v^6 + 21*v^5 - 21*v^4 + 77*v^3 - 77*v^2 + 97*v - 48) / 7 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} + 7\nu^{6} - 63\nu^{5} + 133\nu^{4} - 231\nu^{3} + 259\nu^{2} - 291\nu - 56 ) / 21$$ (-3*v^7 + 7*v^6 - 63*v^5 + 133*v^4 - 231*v^3 + 259*v^2 - 291*v - 56) / 21 $$\beta_{4}$$ $$=$$ $$( 2\nu^{7} + 38\nu^{5} + 74\nu^{3} - 34\nu ) / 3$$ (2*v^7 + 38*v^5 + 74*v^3 - 34*v) / 3 $$\beta_{5}$$ $$=$$ $$2\nu^{6} + 40\nu^{4} + 110\nu^{2} + 20$$ 2*v^6 + 40*v^4 + 110*v^2 + 20 $$\beta_{6}$$ $$=$$ $$( -13\nu^{7} - 259\nu^{5} - 707\nu^{3} - 197\nu ) / 7$$ (-13*v^7 - 259*v^5 - 707*v^3 - 197*v) / 7 $$\beta_{7}$$ $$=$$ $$( -7\nu^{7} - 139\nu^{5} - 373\nu^{3} - 97\nu ) / 3$$ (-7*v^7 - 139*v^5 - 373*v^3 - 97*v) / 3
 $$\nu$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 8$$ (2*b7 - 2*b6 + b4 - b3 + b2 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} - 2\beta_{3} - 2\beta_{2} + 5\beta _1 - 20 ) / 4$$ (-b5 - 2*b3 - 2*b2 + 5*b1 - 20) / 4 $$\nu^{3}$$ $$=$$ $$( -38\beta_{7} + 42\beta_{6} - 13\beta_{4} + 7\beta_{3} - 7\beta_{2} - 7\beta_1 ) / 8$$ (-38*b7 + 42*b6 - 13*b4 + 7*b3 - 7*b2 - 7*b1) / 8 $$\nu^{4}$$ $$=$$ $$( 10\beta_{5} + 15\beta_{3} + 15\beta_{2} - 48\beta _1 + 144 ) / 2$$ (10*b5 + 15*b3 + 15*b2 - 48*b1 + 144) / 2 $$\nu^{5}$$ $$=$$ $$( 646\beta_{7} - 726\beta_{6} + 197\beta_{4} - 97\beta_{3} + 97\beta_{2} + 97\beta_1 ) / 8$$ (646*b7 - 726*b6 + 197*b4 - 97*b3 + 97*b2 + 97*b1) / 8 $$\nu^{6}$$ $$=$$ $$( -343\beta_{5} - 490\beta_{3} - 490\beta_{2} + 1645\beta _1 - 4700 ) / 4$$ (-343*b5 - 490*b3 - 490*b2 + 1645*b1 - 4700) / 4 $$\nu^{7}$$ $$=$$ $$( -10834\beta_{7} + 12206\beta_{6} - 3233\beta_{4} + 1567\beta_{3} - 1567\beta_{2} - 1567\beta_1 ) / 8$$ (-10834*b7 + 12206*b6 - 3233*b4 + 1567*b3 - 1567*b2 - 1567*b1) / 8

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 − 4.08932i − 0.244539i 1.69385i 0.590371i − 0.590371i − 1.69385i 0.244539i 4.08932i
0 5.43734i 0 6.12900i 0 6.21005 3.23038i 0 −20.5647 0
97.2 0 5.43734i 0 6.12900i 0 −6.21005 3.23038i 0 −20.5647 0
97.3 0 1.56056i 0 3.23038i 0 3.38162 + 6.12900i 0 6.56466 0
97.4 0 1.56056i 0 3.23038i 0 −3.38162 + 6.12900i 0 6.56466 0
97.5 0 1.56056i 0 3.23038i 0 −3.38162 6.12900i 0 6.56466 0
97.6 0 1.56056i 0 3.23038i 0 3.38162 6.12900i 0 6.56466 0
97.7 0 5.43734i 0 6.12900i 0 −6.21005 + 3.23038i 0 −20.5647 0
97.8 0 5.43734i 0 6.12900i 0 6.21005 + 3.23038i 0 −20.5647 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.8 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.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.c.a 8
3.b odd 2 1 2016.3.f.c 8
4.b odd 2 1 inner 224.3.c.a 8
7.b odd 2 1 inner 224.3.c.a 8
8.b even 2 1 448.3.c.g 8
8.d odd 2 1 448.3.c.g 8
12.b even 2 1 2016.3.f.c 8
21.c even 2 1 2016.3.f.c 8
28.d even 2 1 inner 224.3.c.a 8
56.e even 2 1 448.3.c.g 8
56.h odd 2 1 448.3.c.g 8
84.h odd 2 1 2016.3.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.c.a 8 1.a even 1 1 trivial
224.3.c.a 8 4.b odd 2 1 inner
224.3.c.a 8 7.b odd 2 1 inner
224.3.c.a 8 28.d even 2 1 inner
448.3.c.g 8 8.b even 2 1
448.3.c.g 8 8.d odd 2 1
448.3.c.g 8 56.e even 2 1
448.3.c.g 8 56.h odd 2 1
2016.3.f.c 8 3.b odd 2 1
2016.3.f.c 8 12.b even 2 1
2016.3.f.c 8 21.c even 2 1
2016.3.f.c 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 32T_{3}^{2} + 72$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 32 T^{2} + 72)^{2}$$
$5$ $$(T^{4} + 48 T^{2} + 392)^{2}$$
$7$ $$T^{8} - 4 T^{6} + 1862 T^{4} + \cdots + 5764801$$
$11$ $$(T^{4} - 248 T^{2} + 3600)^{2}$$
$13$ $$(T^{4} + 208 T^{2} + 1800)^{2}$$
$17$ $$(T^{4} + 1152 T^{2} + 320000)^{2}$$
$19$ $$(T^{4} + 256 T^{2} + 16200)^{2}$$
$23$ $$(T^{4} - 440 T^{2} + 1296)^{2}$$
$29$ $$(T^{2} + 28 T - 540)^{4}$$
$31$ $$(T^{4} + 1664 T^{2} + 115200)^{2}$$
$37$ $$(T^{2} - 52 T - 60)^{4}$$
$41$ $$(T^{4} + 1856 T^{2} + 10368)^{2}$$
$43$ $$(T^{4} - 4856 T^{2} + 4717584)^{2}$$
$47$ $$(T^{4} + 2048 T^{2} + 1036800)^{2}$$
$53$ $$(T + 30)^{8}$$
$59$ $$(T^{4} + 9344 T^{2} + 21385800)^{2}$$
$61$ $$(T^{4} + 10672 T^{2} + 342792)^{2}$$
$67$ $$(T^{2} - 828)^{4}$$
$71$ $$(T^{4} - 8200 T^{2} + 250000)^{2}$$
$73$ $$(T^{4} + 12928 T^{2} + 35280000)^{2}$$
$79$ $$(T^{4} - 25992 T^{2} + \cdots + 128595600)^{2}$$
$83$ $$(T^{4} + 19424 T^{2} + 89191368)^{2}$$
$89$ $$(T^{4} + 768 T^{2} + 3200)^{2}$$
$97$ $$(T^{4} + 18432 T^{2} + 83980800)^{2}$$