# Properties

 Label 224.3.c.b 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: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81$$ x^8 + 20*x^6 + 116*x^4 + 180*x^2 + 81 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9 $$q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{7} + \beta_{3}) q^{15} + \beta_1 q^{17} + (3 \beta_{7} - 3 \beta_{6} + 2 \beta_{2}) q^{19} + ( - \beta_{5} + 3 \beta_{4} + \beta_1 - 4) q^{21} + (2 \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{23} + ( - \beta_{4} - 7) q^{25} + (\beta_{7} - \beta_{6} + 9 \beta_{2}) q^{27} + (6 \beta_{4} + 2) q^{29} + ( - 5 \beta_{7} + 5 \beta_{6} + 9 \beta_{2}) q^{31} + ( - 6 \beta_{5} - \beta_1) q^{33} + (3 \beta_{7} - 6 \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{35} + ( - 14 \beta_{4} - 6) q^{37} + (3 \beta_{7} + 4 \beta_{6} - \beta_{3} - 4 \beta_{2}) q^{39} + (6 \beta_{5} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}) q^{43} + (\beta_{5} + \beta_1) q^{45} + (3 \beta_{7} - 3 \beta_{6} - 11 \beta_{2}) q^{47} + (6 \beta_{5} + 10 \beta_{4} + \beta_1 + 17) q^{49} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{2}) q^{51} + ( - 8 \beta_{4} + 34) q^{53} + ( - 7 \beta_{7} + 7 \beta_{6} - 23 \beta_{2}) q^{55} + (19 \beta_{4} - 16) q^{57} + ( - 7 \beta_{7} + 7 \beta_{6} - 6 \beta_{2}) q^{59} + 11 \beta_{5} q^{61} + (\beta_{7} - \beta_{6} - \beta_{3} + 3 \beta_{2}) q^{63} + ( - 31 \beta_{4} + 24) q^{65} + ( - 6 \beta_{7} - \beta_{6} - 5 \beta_{3} + \beta_{2}) q^{67} + ( - 10 \beta_{5} + 3 \beta_1) q^{69} + (\beta_{7} + 5 \beta_{6} - 4 \beta_{3} - 5 \beta_{2}) q^{71} + ( - 6 \beta_{5} - 3 \beta_1) q^{73} + (\beta_{7} - \beta_{6} - 8 \beta_{2}) q^{75} + ( - 10 \beta_{5} + 23 \beta_{4} - 4 \beta_1 - 26) q^{77} + ( - 5 \beta_{7} + \beta_{6} - 6 \beta_{3} - \beta_{2}) q^{79} + ( - 11 \beta_{4} - 63) q^{81} + ( - 2 \beta_{7} + 2 \beta_{6} + 23 \beta_{2}) q^{83} + (32 \beta_{4} + 8) q^{85} + ( - 6 \beta_{7} + 6 \beta_{6} + 8 \beta_{2}) q^{87} + ( - 2 \beta_{5} + 5 \beta_1) q^{89} + ( - 2 \beta_{7} + 9 \beta_{6} - 5 \beta_{3} - 34 \beta_{2}) q^{91} + ( - 44 \beta_{4} - 72) q^{93} + ( - 7 \beta_{7} - 12 \beta_{6} + 5 \beta_{3} + 12 \beta_{2}) q^{95} + ( - 4 \beta_{5} + 7 \beta_1) q^{97} + ( - 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{3} + 5 \beta_{2}) q^{99}+O(q^{100})$$ q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9 + (-b6 + b3 + b2) * q^11 + (-b5 - b1) * q^13 + (b7 + b3) * q^15 + b1 * q^17 + (3*b7 - 3*b6 + 2*b2) * q^19 + (-b5 + 3*b4 + b1 - 4) * q^21 + (2*b7 + b6 + b3 - b2) * q^23 + (-b4 - 7) * q^25 + (b7 - b6 + 9*b2) * q^27 + (6*b4 + 2) * q^29 + (-5*b7 + 5*b6 + 9*b2) * q^31 + (-6*b5 - b1) * q^33 + (3*b7 - 6*b6 + b3 - 3*b2) * q^35 + (-14*b4 - 6) * q^37 + (3*b7 + 4*b6 - b3 - 4*b2) * q^39 + (6*b5 - 2*b1) * q^41 + (-2*b7 - b6 - b3 + b2) * q^43 + (b5 + b1) * q^45 + (3*b7 - 3*b6 - 11*b2) * q^47 + (6*b5 + 10*b4 + b1 + 17) * q^49 + (-4*b7 - 4*b6 + 4*b2) * q^51 + (-8*b4 + 34) * q^53 + (-7*b7 + 7*b6 - 23*b2) * q^55 + (19*b4 - 16) * q^57 + (-7*b7 + 7*b6 - 6*b2) * q^59 + 11*b5 * q^61 + (b7 - b6 - b3 + 3*b2) * q^63 + (-31*b4 + 24) * q^65 + (-6*b7 - b6 - 5*b3 + b2) * q^67 + (-10*b5 + 3*b1) * q^69 + (b7 + 5*b6 - 4*b3 - 5*b2) * q^71 + (-6*b5 - 3*b1) * q^73 + (b7 - b6 - 8*b2) * q^75 + (-10*b5 + 23*b4 - 4*b1 - 26) * q^77 + (-5*b7 + b6 - 6*b3 - b2) * q^79 + (-11*b4 - 63) * q^81 + (-2*b7 + 2*b6 + 23*b2) * q^83 + (32*b4 + 8) * q^85 + (-6*b7 + 6*b6 + 8*b2) * q^87 + (-2*b5 + 5*b1) * q^89 + (-2*b7 + 9*b6 - 5*b3 - 34*b2) * q^91 + (-44*b4 - 72) * q^93 + (-7*b7 - 12*b6 + 5*b3 + 12*b2) * q^95 + (-4*b5 + 7*b1) * q^97 + (-2*b7 - 5*b6 + 3*b3 + 5*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^9 $$8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100})$$ 8 * q + 8 * q^9 - 32 * q^21 - 56 * q^25 + 16 * q^29 - 48 * q^37 + 136 * q^49 + 272 * q^53 - 128 * q^57 + 192 * q^65 - 208 * q^77 - 504 * q^81 + 64 * q^85 - 576 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{7} + 80\nu^{5} + 464\nu^{3} + 828\nu ) / 27$$ (4*v^7 + 80*v^5 + 464*v^3 + 828*v) / 27 $$\beta_{2}$$ $$=$$ $$( 7\nu^{7} + 131\nu^{5} + 659\nu^{3} + 567\nu ) / 27$$ (7*v^7 + 131*v^5 + 659*v^3 + 567*v) / 27 $$\beta_{3}$$ $$=$$ $$( -5\nu^{7} - 9\nu^{6} - 91\nu^{5} - 153\nu^{4} - 427\nu^{3} - 585\nu^{2} - 261\nu - 54 ) / 27$$ (-5*v^7 - 9*v^6 - 91*v^5 - 153*v^4 - 427*v^3 - 585*v^2 - 261*v - 54) / 27 $$\beta_{4}$$ $$=$$ $$( 2\nu^{6} + 40\nu^{4} + 214\nu^{2} + 180 ) / 9$$ (2*v^6 + 40*v^4 + 214*v^2 + 180) / 9 $$\beta_{5}$$ $$=$$ $$( -13\nu^{7} - 251\nu^{5} - 1301\nu^{3} - 1107\nu ) / 27$$ (-13*v^7 - 251*v^5 - 1301*v^3 - 1107*v) / 27 $$\beta_{6}$$ $$=$$ $$( 2\nu^{7} + 15\nu^{6} + 40\nu^{5} + 273\nu^{4} + 232\nu^{3} + 1335\nu^{2} + 306\nu + 1134 ) / 27$$ (2*v^7 + 15*v^6 + 40*v^5 + 273*v^4 + 232*v^3 + 1335*v^2 + 306*v + 1134) / 27 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} + 15\nu^{6} + 91\nu^{5} + 273\nu^{4} + 427\nu^{3} + 1335\nu^{2} + 261\nu + 1134 ) / 27$$ (5*v^7 + 15*v^6 + 91*v^5 + 273*v^4 + 427*v^3 + 1335*v^2 + 261*v + 1134) / 27
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 8$$ (b7 - b6 - b2 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{4} + \beta_{3} - 20 ) / 4$$ (b7 - b4 + b3 - 20) / 4 $$\nu^{3}$$ $$=$$ $$( -13\beta_{7} + 13\beta_{6} + 4\beta_{5} + 17\beta_{2} - 7\beta_1 ) / 8$$ (-13*b7 + 13*b6 + 4*b5 + 17*b2 - 7*b1) / 8 $$\nu^{4}$$ $$=$$ $$( -6\beta_{7} - \beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 84 ) / 2$$ (-6*b7 - b6 + 10*b4 - 5*b3 + b2 + 84) / 2 $$\nu^{5}$$ $$=$$ $$( 123\beta_{7} - 123\beta_{6} - 68\beta_{5} - 215\beta_{2} + 63\beta_1 ) / 8$$ (123*b7 - 123*b6 - 68*b5 - 215*b2 + 63*b1) / 8 $$\nu^{6}$$ $$=$$ $$( 133\beta_{7} + 40\beta_{6} - 275\beta_{4} + 93\beta_{3} - 40\beta_{2} - 1580 ) / 4$$ (133*b7 + 40*b6 - 275*b4 + 93*b3 - 40*b2 - 1580) / 4 $$\nu^{7}$$ $$=$$ $$( -1159\beta_{7} + 1159\beta_{6} + 896\beta_{5} + 2535\beta_{2} - 601\beta_1 ) / 8$$ (-1159*b7 + 1159*b6 + 896*b5 + 2535*b2 - 601*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
 3.24994i − 0.923094i − 2.71359i 1.10555i − 1.10555i 2.71359i 0.923094i − 3.24994i
0 3.29066i 0 5.90156i 0 −6.86601 + 1.36303i 0 −1.82843 0
97.2 0 3.29066i 0 5.90156i 0 6.86601 + 1.36303i 0 −1.82843 0
97.3 0 2.27411i 0 5.40107i 0 4.34256 5.49019i 0 3.82843 0
97.4 0 2.27411i 0 5.40107i 0 −4.34256 5.49019i 0 3.82843 0
97.5 0 2.27411i 0 5.40107i 0 −4.34256 + 5.49019i 0 3.82843 0
97.6 0 2.27411i 0 5.40107i 0 4.34256 + 5.49019i 0 3.82843 0
97.7 0 3.29066i 0 5.90156i 0 6.86601 1.36303i 0 −1.82843 0
97.8 0 3.29066i 0 5.90156i 0 −6.86601 1.36303i 0 −1.82843 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.b 8
3.b odd 2 1 2016.3.f.b 8
4.b odd 2 1 inner 224.3.c.b 8
7.b odd 2 1 inner 224.3.c.b 8
8.b even 2 1 448.3.c.h 8
8.d odd 2 1 448.3.c.h 8
12.b even 2 1 2016.3.f.b 8
21.c even 2 1 2016.3.f.b 8
28.d even 2 1 inner 224.3.c.b 8
56.e even 2 1 448.3.c.h 8
56.h odd 2 1 448.3.c.h 8
84.h odd 2 1 2016.3.f.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 16T_{3}^{2} + 56$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 16 T^{2} + 56)^{2}$$
$5$ $$(T^{4} + 64 T^{2} + 1016)^{2}$$
$7$ $$T^{8} - 68 T^{6} + 2758 T^{4} + \cdots + 5764801$$
$11$ $$(T^{4} - 472 T^{2} + 14224)^{2}$$
$13$ $$(T^{4} + 544 T^{2} + 49784)^{2}$$
$17$ $$(T^{4} + 512 T^{2} + 65024)^{2}$$
$19$ $$(T^{4} + 1072 T^{2} + 123704)^{2}$$
$23$ $$(T^{4} - 1112 T^{2} + 14224)^{2}$$
$29$ $$(T^{2} - 4 T - 284)^{4}$$
$31$ $$(T^{4} + 4096 T^{2} + 1895936)^{2}$$
$37$ $$(T^{2} + 12 T - 1532)^{4}$$
$41$ $$(T^{4} + 4736 T^{2} + 16256)^{2}$$
$43$ $$(T^{4} - 1112 T^{2} + 14224)^{2}$$
$47$ $$(T^{4} + 2944 T^{2} + 3584)^{2}$$
$53$ $$(T^{2} - 68 T + 644)^{4}$$
$59$ $$(T^{4} + 6064 T^{2} + 2784824)^{2}$$
$61$ $$(T^{4} + 7744 T^{2} + 14875256)^{2}$$
$67$ $$(T^{4} - 15064 T^{2} + 34151824)^{2}$$
$71$ $$(T^{4} - 8648 T^{2} + 696976)^{2}$$
$73$ $$(T^{4} + 6336 T^{2} + 1316736)^{2}$$
$79$ $$(T^{4} - 17352 T^{2} + 71703184)^{2}$$
$83$ $$(T^{4} + 8912 T^{2} + 9367736)^{2}$$
$89$ $$(T^{4} + 13376 T^{2} + 39030656)^{2}$$
$97$ $$(T^{4} + 27008 T^{2} + \cdots + 143638016)^{2}$$