# 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: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{7} + 80 \nu^{5} + 464 \nu^{3} + 828 \nu$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{7} + 131 \nu^{5} + 659 \nu^{3} + 567 \nu$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{7} - 9 \nu^{6} - 91 \nu^{5} - 153 \nu^{4} - 427 \nu^{3} - 585 \nu^{2} - 261 \nu - 54$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{6} + 40 \nu^{4} + 214 \nu^{2} + 180$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{7} - 251 \nu^{5} - 1301 \nu^{3} - 1107 \nu$$$$)/27$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} + 15 \nu^{6} + 40 \nu^{5} + 273 \nu^{4} + 232 \nu^{3} + 1335 \nu^{2} + 306 \nu + 1134$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 15 \nu^{6} + 91 \nu^{5} + 273 \nu^{4} + 427 \nu^{3} + 1335 \nu^{2} + 261 \nu + 1134$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{4} + \beta_{3} - 20$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-13 \beta_{7} + 13 \beta_{6} + 4 \beta_{5} + 17 \beta_{2} - 7 \beta_{1}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{7} - \beta_{6} + 10 \beta_{4} - 5 \beta_{3} + \beta_{2} + 84$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$123 \beta_{7} - 123 \beta_{6} - 68 \beta_{5} - 215 \beta_{2} + 63 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$133 \beta_{7} + 40 \beta_{6} - 275 \beta_{4} + 93 \beta_{3} - 40 \beta_{2} - 1580$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-1159 \beta_{7} + 1159 \beta_{6} + 896 \beta_{5} + 2535 \beta_{2} - 601 \beta_{1}$$$$)/8$$

## 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 $$T_{3}^{4} + 16 T_{3}^{2} + 56$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 20 T^{2} + 254 T^{4} - 1620 T^{6} + 6561 T^{8} )^{2}$$
$5$ $$( 1 - 36 T^{2} + 1566 T^{4} - 22500 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$1 - 68 T^{2} + 2758 T^{4} - 163268 T^{6} + 5764801 T^{8}$$
$11$ $$( 1 + 12 T^{2} - 12154 T^{4} + 175692 T^{6} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 - 132 T^{2} + 37278 T^{4} - 3770052 T^{6} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 - 644 T^{2} + 270214 T^{4} - 53787524 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 372 T^{2} + 131646 T^{4} - 48479412 T^{6} + 16983563041 T^{8} )^{2}$$
$23$ $$( 1 + 1004 T^{2} + 516774 T^{4} + 280960364 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 - 4 T + 1398 T^{2} - 3364 T^{3} + 707281 T^{4} )^{4}$$
$31$ $$( 1 + 252 T^{2} - 435450 T^{4} + 232727292 T^{6} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 + 12 T + 1206 T^{2} + 16428 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$( 1 - 1988 T^{2} + 1048390 T^{4} - 5617612868 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 6284 T^{2} + 16414854 T^{4} + 21483745484 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 - 5892 T^{2} + 16275078 T^{4} - 28751080452 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$( 1 - 68 T + 6262 T^{2} - 191012 T^{3} + 7890481 T^{4} )^{4}$$
$59$ $$( 1 - 7860 T^{2} + 33271422 T^{4} - 95242457460 T^{6} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 - 7140 T^{2} + 40319454 T^{4} - 98859304740 T^{6} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 + 2892 T^{2} + 19813958 T^{4} + 58277041932 T^{6} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 + 11516 T^{2} + 65977926 T^{4} + 292640918396 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 14980 T^{2} + 104177094 T^{4} - 425405650180 T^{6} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 + 7612 T^{2} + 88816006 T^{4} + 296488016572 T^{6} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 - 18644 T^{2} + 171328126 T^{4} - 884812936724 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 - 18308 T^{2} + 203581510 T^{4} - 1148684948228 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 10628 T^{2} + 166577158 T^{4} - 940889198468 T^{6} + 7837433594376961 T^{8} )^{2}$$