# Properties

 Label 224.3.g.b Level 224 Weight 3 Character orbit 224.g 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.292213762624.3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + \beta_{2} q^{7} + ( 6 + \beta_{3} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + \beta_{2} q^{7} + ( 6 + \beta_{3} - \beta_{7} ) q^{9} + ( 4 + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{13} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{15} + ( -10 + 4 \beta_{3} ) q^{17} + ( -7 + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{5} ) q^{21} + ( 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{5} ) q^{23} + ( -2 - \beta_{3} + \beta_{7} ) q^{25} + ( 4 + 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{27} + ( 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{29} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{31} + ( 4 + 4 \beta_{6} + 4 \beta_{7} ) q^{33} + ( -7 + \beta_{3} + \beta_{6} + \beta_{7} ) q^{35} + ( -2 \beta_{1} - 14 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -11 \beta_{1} + 5 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{39} + ( 16 + 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} ) q^{41} + ( 4 \beta_{3} + \beta_{6} - 3 \beta_{7} ) q^{43} + ( -13 \beta_{1} + 13 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} ) q^{45} + ( 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{4} + 4 \beta_{5} ) q^{47} -7 q^{49} + ( 46 - 10 \beta_{3} - 4 \beta_{7} ) q^{51} + ( 4 \beta_{1} - 4 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -6 \beta_{1} - 14 \beta_{2} + 8 \beta_{4} - 4 \beta_{5} ) q^{55} + ( 7 - 7 \beta_{3} - \beta_{7} ) q^{57} + ( -13 - 13 \beta_{3} - 2 \beta_{6} + 6 \beta_{7} ) q^{59} + ( -3 \beta_{1} + 19 \beta_{2} - 5 \beta_{4} ) q^{61} + ( -5 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} ) q^{63} + ( -9 - 11 \beta_{3} - 5 \beta_{7} ) q^{65} + ( -38 + 6 \beta_{3} - \beta_{6} + 3 \beta_{7} ) q^{67} + ( 2 \beta_{1} - 18 \beta_{2} - 10 \beta_{4} - 12 \beta_{5} ) q^{69} + ( -8 \beta_{1} - 16 \beta_{4} ) q^{71} + ( -14 + 4 \beta_{6} + 4 \beta_{7} ) q^{73} + ( -9 - 9 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{75} + ( 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{77} + ( 16 \beta_{1} - 8 \beta_{2} - 12 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 6 + 3 \beta_{3} - 8 \beta_{6} - 3 \beta_{7} ) q^{81} + ( -9 - \beta_{3} - 6 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 10 \beta_{1} - 18 \beta_{2} + 2 \beta_{4} + 8 \beta_{5} ) q^{85} + ( 18 \beta_{1} - 6 \beta_{2} - 24 \beta_{4} - 8 \beta_{5} ) q^{87} + ( -64 - 6 \beta_{3} - 4 \beta_{6} - 6 \beta_{7} ) q^{89} + ( 7 - \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{91} + ( 24 \beta_{2} - 12 \beta_{4} ) q^{93} + ( 7 \beta_{1} - 9 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{95} + ( 8 - 6 \beta_{3} - 8 \beta_{6} + 2 \beta_{7} ) q^{97} + ( -32 - 12 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} + 48q^{9} + O(q^{10})$$ $$8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{6} + 4 \nu^{5} - 6 \nu^{4} + 4 \nu^{3} - 160 \nu^{2} + 96 \nu$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} - 12 \nu^{5} + 6 \nu^{4} + 12 \nu^{3} - 96 \nu$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} - 2 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 8 \nu - 16$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 14 \nu^{4} - 4 \nu^{3} - 128$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{6} - 4 \nu^{5} - 22 \nu^{4} + 84 \nu^{3} + 48 \nu^{2} + 128 \nu - 384$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{5} - 4 \nu^{4} + 22 \nu^{3} - 88 \nu - 64$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + \nu^{6} + 18 \nu^{4} - 20 \nu^{3} - 8 \nu^{2} + 16 \nu + 192$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 7 \beta_{1} + 5$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + \beta_{1} + 13$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} - \beta_{6} + \beta_{5} - 21 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 3 \beta_{1} - 71$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 35 \beta_{3} - 71 \beta_{2} - 15 \beta_{1} - 19$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$12 \beta_{7} + 23 \beta_{6} - 7 \beta_{5} - 61 \beta_{4} + 89 \beta_{3} - 91 \beta_{2} + 29 \beta_{1} - 55$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$-84 \beta_{7} - 21 \beta_{6} - 27 \beta_{5} - 153 \beta_{4} + 21 \beta_{3} + 49 \beta_{2} + 9 \beta_{1} - 27$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 −1.67467 + 1.09337i −1.67467 − 1.09337i 1.85837 + 0.739226i 1.85837 − 0.739226i −1.05468 + 1.69931i −1.05468 − 1.69931i 1.37098 − 1.45617i 1.37098 + 1.45617i
0 −4.56747 0 5.73252i 0 2.64575i 0 11.8618 0
15.2 0 −4.56747 0 5.73252i 0 2.64575i 0 11.8618 0
15.3 0 −0.0974366 0 3.46547i 0 2.64575i 0 −8.99051 0
15.4 0 −0.0974366 0 3.46547i 0 2.64575i 0 −8.99051 0
15.5 0 3.44128 0 4.88287i 0 2.64575i 0 2.84239 0
15.6 0 3.44128 0 4.88287i 0 2.64575i 0 2.84239 0
15.7 0 5.22363 0 6.26788i 0 2.64575i 0 18.2863 0
15.8 0 5.22363 0 6.26788i 0 2.64575i 0 18.2863 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d 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.g.b 8
3.b odd 2 1 2016.3.g.b 8
4.b odd 2 1 56.3.g.b 8
7.b odd 2 1 1568.3.g.m 8
8.b even 2 1 56.3.g.b 8
8.d odd 2 1 inner 224.3.g.b 8
12.b even 2 1 504.3.g.b 8
16.e even 4 2 1792.3.d.j 16
16.f odd 4 2 1792.3.d.j 16
24.f even 2 1 2016.3.g.b 8
24.h odd 2 1 504.3.g.b 8
28.d even 2 1 392.3.g.m 8
28.f even 6 2 392.3.k.n 16
28.g odd 6 2 392.3.k.o 16
56.e even 2 1 1568.3.g.m 8
56.h odd 2 1 392.3.g.m 8
56.j odd 6 2 392.3.k.n 16
56.p even 6 2 392.3.k.o 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b 8 4.b odd 2 1
56.3.g.b 8 8.b even 2 1
224.3.g.b 8 1.a even 1 1 trivial
224.3.g.b 8 8.d odd 2 1 inner
392.3.g.m 8 28.d even 2 1
392.3.g.m 8 56.h odd 2 1
392.3.k.n 16 28.f even 6 2
392.3.k.n 16 56.j odd 6 2
392.3.k.o 16 28.g odd 6 2
392.3.k.o 16 56.p even 6 2
504.3.g.b 8 12.b even 2 1
504.3.g.b 8 24.h odd 2 1
1568.3.g.m 8 7.b odd 2 1
1568.3.g.m 8 56.e even 2 1
1792.3.d.j 16 16.e even 4 2
1792.3.d.j 16 16.f odd 4 2
2016.3.g.b 8 3.b odd 2 1
2016.3.g.b 8 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{3} - 22 T_{3}^{2} + 80 T_{3} + 8$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 4 T + 14 T^{2} - 28 T^{3} + 98 T^{4} - 252 T^{5} + 1134 T^{6} - 2916 T^{7} + 6561 T^{8} )^{2}$$
$5$ $$1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 132472500 T^{10} + 2134375000 T^{12} - 22460937500 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 - 16 T + 328 T^{2} - 4944 T^{3} + 49230 T^{4} - 598224 T^{5} + 4802248 T^{6} - 28344976 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 1009946777684 T^{10} + 116597286336856 T^{12} - 10344349794381564 T^{14} + 665416609183179841 T^{16}$$
$17$ $$( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 9106968 T^{5} + 109245468 T^{6} + 965502760 T^{7} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 + 28 T + 1710 T^{2} + 31332 T^{3} + 975266 T^{4} + 11310852 T^{5} + 222848910 T^{6} + 1317284668 T^{7} + 16983563041 T^{8} )^{2}$$
$23$ $$1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 115315782303280 T^{10} + 99623789791135836 T^{12} - 38219105009443439824 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 6192081614658056 T^{10} + 3279405379878872380 T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 11495461442126552 T^{10} + 7197223366370371420 T^{12} -$$$$31\!\cdots\!84$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 34902090701365128 T^{10} + 32275007558901367228 T^{12} -$$$$23\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$( 1 - 64 T + 4956 T^{2} - 221760 T^{3} + 11848326 T^{4} - 372778560 T^{5} + 14004471516 T^{6} - 304006671424 T^{7} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 4680 T^{2} + 58016 T^{3} + 10251086 T^{4} + 107271584 T^{5} + 15999988680 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 620731022190542712 T^{10} +$$$$94\!\cdots\!28$$$$T^{12} -$$$$97\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 6628727822775456600 T^{10} +$$$$10\!\cdots\!08$$$$T^{12} -$$$$92\!\cdots\!20$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$( 1 + 52 T + 2254 T^{2} - 207508 T^{3} - 19795230 T^{4} - 722335348 T^{5} + 27312531694 T^{6} + 2193387749332 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 11311249557773337684 T^{10} +$$$$26\!\cdots\!20$$$$T^{12} -$$$$43\!\cdots\!36$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 + 152 T + 22224 T^{2} + 2037320 T^{3} + 158433022 T^{4} + 9145529480 T^{5} + 447838513104 T^{6} + 13749674089688 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 7484538771485032376 T^{10} +$$$$33\!\cdots\!24$$$$T^{12} -$$$$16\!\cdots\!24$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$( 1 + 56 T + 18460 T^{2} + 736008 T^{3} + 138223494 T^{4} + 3922186632 T^{5} + 524231528860 T^{6} + 8474716672184 T^{7} + 806460091894081 T^{8} )^{2}$$
$79$ $$1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} -$$$$12\!\cdots\!12$$$$T^{10} +$$$$50\!\cdots\!68$$$$T^{12} -$$$$14\!\cdots\!88$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 1225442876 T^{5} + 782018213438 T^{6} + 11769853441284 T^{7} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 + 256 T + 48252 T^{2} + 6269952 T^{3} + 638304966 T^{4} + 49664289792 T^{5} + 3027438612732 T^{6} + 127227210486016 T^{7} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 13525776224 T^{5} + 1724904511004 T^{6} - 26655104157728 T^{7} + 7837433594376961 T^{8} )^{2}$$