# Properties

 Label 224.3.d.b Level 224 Weight 3 Character orbit 224.d Analytic conductor 6.104 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1997017344.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{4} q^{5} -\beta_{3} q^{7} + ( -5 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{4} q^{5} -\beta_{3} q^{7} + ( -5 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{9} + ( -2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{11} + ( 4 + \beta_{4} ) q^{13} + ( -2 \beta_{1} - 6 \beta_{3} - \beta_{7} ) q^{15} + ( -2 + 2 \beta_{6} ) q^{17} + ( 3 \beta_{1} + 2 \beta_{5} ) q^{19} + ( -\beta_{2} + \beta_{4} ) q^{21} + ( -6 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{23} + ( 13 - 3 \beta_{2} - 5 \beta_{4} + \beta_{6} ) q^{25} + ( -6 \beta_{1} - 12 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{27} + ( -10 + 2 \beta_{4} ) q^{29} + ( 6 \beta_{1} - 4 \beta_{3} - 2 \beta_{7} ) q^{31} + ( -4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{33} + ( 3 \beta_{1} + \beta_{5} - \beta_{7} ) q^{35} + ( -22 + 6 \beta_{2} - 2 \beta_{6} ) q^{37} + ( 6 \beta_{1} + 6 \beta_{3} + \beta_{7} ) q^{39} + ( 18 + 6 \beta_{2} + 2 \beta_{6} ) q^{41} + ( 10 \beta_{3} + 7 \beta_{5} + 3 \beta_{7} ) q^{43} + ( 32 - 8 \beta_{2} + 3 \beta_{4} - 4 \beta_{6} ) q^{45} + ( 6 \beta_{1} - 4 \beta_{5} + 4 \beta_{7} ) q^{47} -7 q^{49} + ( -14 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - 2 \beta_{7} ) q^{51} + ( 6 + 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{53} + ( 20 \beta_{3} - 12 \beta_{5} + 2 \beta_{7} ) q^{55} + ( -50 + 7 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{57} + ( 3 \beta_{1} - 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{7} ) q^{59} + ( -24 + 7 \beta_{4} - 4 \beta_{6} ) q^{61} + ( 4 \beta_{1} + 5 \beta_{3} + 6 \beta_{5} + \beta_{7} ) q^{63} + ( -38 + 3 \beta_{2} + \beta_{4} - \beta_{6} ) q^{65} + ( 6 \beta_{1} + 18 \beta_{3} - 13 \beta_{5} - \beta_{7} ) q^{67} + ( 72 - 4 \beta_{2} + 4 \beta_{4} - 4 \beta_{6} ) q^{69} + ( 4 \beta_{3} + 4 \beta_{5} - 6 \beta_{7} ) q^{71} + ( 34 + 2 \beta_{4} - 6 \beta_{6} ) q^{73} + ( 3 \beta_{1} - 4 \beta_{3} + 20 \beta_{5} - 6 \beta_{7} ) q^{75} + ( -14 - 3 \beta_{2} - 4 \beta_{4} ) q^{77} + ( -12 \beta_{1} + 8 \beta_{3} + 18 \beta_{5} ) q^{79} + ( 63 - 17 \beta_{2} + 17 \beta_{4} - \beta_{6} ) q^{81} + ( 3 \beta_{1} + 28 \beta_{3} - 14 \beta_{5} + 2 \beta_{7} ) q^{83} + ( -20 - 6 \beta_{2} - 6 \beta_{6} ) q^{85} + ( -6 \beta_{1} + 12 \beta_{3} + 2 \beta_{7} ) q^{87} + ( -10 - 6 \beta_{2} - 4 \beta_{6} ) q^{89} + ( -3 \beta_{1} - 4 \beta_{3} - \beta_{5} + \beta_{7} ) q^{91} + ( -76 + 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{6} ) q^{93} + ( -6 \beta_{1} - 26 \beta_{3} - 7 \beta_{7} ) q^{95} + ( 66 + 6 \beta_{2} - 6 \beta_{4} - 4 \beta_{6} ) q^{97} + ( -8 \beta_{1} + 6 \beta_{3} + 19 \beta_{5} + 5 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 40q^{9} + O(q^{10})$$ $$8q - 40q^{9} + 32q^{13} - 16q^{17} + 104q^{25} - 80q^{29} - 176q^{37} + 144q^{41} + 256q^{45} - 56q^{49} + 48q^{53} - 400q^{57} - 192q^{61} - 304q^{65} + 576q^{69} + 272q^{73} - 112q^{77} + 504q^{81} - 160q^{85} - 80q^{89} - 608q^{93} + 528q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 11 \nu^{4} + 24 \nu^{2} + 8$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 14 \nu^{5} + 51 \nu^{3} + 42 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 13 \nu^{4} + 38 \nu^{2} + 12$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 13 \nu^{5} - 42 \nu^{3} - 32 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$\nu^{4} + 11 \nu^{2} + 16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 16 \nu^{5} - 77 \nu^{3} - 110 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} + \beta_{2} - 14$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{5} - 6 \beta_{3} - 12 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{6} + 11 \beta_{4} - 11 \beta_{2} + 90$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} + 26 \beta_{5} + 70 \beta_{3} + 88 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$53 \beta_{6} - 97 \beta_{4} + 105 \beta_{2} - 686$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-75 \beta_{7} - 262 \beta_{5} - 658 \beta_{3} - 704 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 2.92812i − 1.92812i − 1.27733i − 0.277334i 0.277334i 1.27733i 1.92812i 2.92812i
0 5.85623i 0 −5.78167 0 2.64575i 0 −25.2955 0
127.2 0 3.85623i 0 0.490168 0 2.64575i 0 −5.87054 0
127.3 0 2.55467i 0 9.86836 0 2.64575i 0 2.47367 0
127.4 0 0.554669i 0 −4.57685 0 2.64575i 0 8.69234 0
127.5 0 0.554669i 0 −4.57685 0 2.64575i 0 8.69234 0
127.6 0 2.55467i 0 9.86836 0 2.64575i 0 2.47367 0
127.7 0 3.85623i 0 0.490168 0 2.64575i 0 −5.87054 0
127.8 0 5.85623i 0 −5.78167 0 2.64575i 0 −25.2955 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.d.b 8
3.b odd 2 1 2016.3.m.c 8
4.b odd 2 1 inner 224.3.d.b 8
7.b odd 2 1 1568.3.d.n 8
8.b even 2 1 448.3.d.e 8
8.d odd 2 1 448.3.d.e 8
12.b even 2 1 2016.3.m.c 8
16.e even 4 1 1792.3.g.d 8
16.e even 4 1 1792.3.g.f 8
16.f odd 4 1 1792.3.g.d 8
16.f odd 4 1 1792.3.g.f 8
28.d even 2 1 1568.3.d.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.b 8 1.a even 1 1 trivial
224.3.d.b 8 4.b odd 2 1 inner
448.3.d.e 8 8.b even 2 1
448.3.d.e 8 8.d odd 2 1
1568.3.d.n 8 7.b odd 2 1
1568.3.d.n 8 28.d even 2 1
1792.3.g.d 8 16.e even 4 1
1792.3.g.d 8 16.f odd 4 1
1792.3.g.f 8 16.e even 4 1
1792.3.g.f 8 16.f odd 4 1
2016.3.m.c 8 3.b odd 2 1
2016.3.m.c 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 56 T_{3}^{6} + 848 T_{3}^{4} + 3584 T_{3}^{2} + 1024$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 16 T^{2} + 92 T^{4} + 272 T^{6} - 8570 T^{8} + 22032 T^{10} + 603612 T^{12} - 8503056 T^{14} + 43046721 T^{16}$$
$5$ $$( 1 + 24 T^{2} - 224 T^{3} + 78 T^{4} - 5600 T^{5} + 15000 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 188977477656 T^{10} + 15618188069660 T^{12} - 928974799509416 T^{14} + 45949729863572161 T^{16}$$
$13$ $$( 1 - 16 T + 696 T^{2} - 7536 T^{3} + 176398 T^{4} - 1273584 T^{5} + 19878456 T^{6} - 77228944 T^{7} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 409224 T^{5} + 35746988 T^{6} + 193100552 T^{7} + 6975757441 T^{8} )^{2}$$
$19$ $$1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 138878579550832 T^{10} + 30492628755072220 T^{12} - 4284977683312087696 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 536484029577912 T^{10} + 185352391597952156 T^{12} - 42251395904935178888 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 + 40 T + 3660 T^{2} + 100632 T^{3} + 4741126 T^{4} + 84631512 T^{5} + 2588648460 T^{6} + 23792932840 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5172694104774 T^{8} - 5454285613703192 T^{10} + 5077686791404993500 T^{12} -$$$$28\!\cdots\!64$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$( 1 + 88 T + 3692 T^{2} + 66024 T^{3} + 510982 T^{4} + 90386856 T^{5} + 6919402412 T^{6} + 225783923992 T^{7} + 3512479453921 T^{8} )^{2}$$
$41$ $$( 1 - 72 T + 4364 T^{2} - 180408 T^{3} + 8880422 T^{4} - 303265848 T^{5} + 12331621004 T^{6} - 342007505352 T^{7} + 7984925229121 T^{8} )^{2}$$
$43$ $$1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 112089803034869784 T^{10} +$$$$18\!\cdots\!80$$$$T^{12} -$$$$17\!\cdots\!92$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 531772641369485336 T^{10} +$$$$81\!\cdots\!04$$$$T^{12} -$$$$77\!\cdots\!36$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 - 24 T + 6108 T^{2} - 242920 T^{3} + 18930726 T^{4} - 682362280 T^{5} + 48195057948 T^{6} - 531944667096 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 15950326795002102384 T^{10} +$$$$31\!\cdots\!88$$$$T^{12} -$$$$38\!\cdots\!52$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 + 96 T + 12728 T^{2} + 895872 T^{3} + 70065870 T^{4} + 3333539712 T^{5} + 176229864248 T^{6} + 4945955938656 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 12011944188428070392 T^{10} +$$$$48\!\cdots\!44$$$$T^{12} -$$$$13\!\cdots\!00$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 50600555550540147768 T^{10} +$$$$15\!\cdots\!80$$$$T^{12} -$$$$34\!\cdots\!00$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$( 1 - 136 T + 21660 T^{2} - 1811512 T^{3} + 170528390 T^{4} - 9653547448 T^{5} + 615105900060 T^{6} - 20581454775304 T^{7} + 806460091894081 T^{8} )^{2}$$
$79$ $$1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 72117759397043305912 T^{10} +$$$$43\!\cdots\!96$$$$T^{12} -$$$$15\!\cdots\!68$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 51197170200775182000 T^{10} +$$$$38\!\cdots\!52$$$$T^{12} -$$$$20\!\cdots\!80$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 + 40 T + 25916 T^{2} + 837912 T^{3} + 285914566 T^{4} + 6637100952 T^{5} + 1626027917756 T^{6} + 19879251638440 T^{7} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 68309565816 T^{5} + 4635747270284 T^{6} - 219904609301256 T^{7} + 7837433594376961 T^{8} )^{2}$$