Properties

 Label 160.3.p.f Level $160$ Weight $3$ Character orbit 160.p Analytic conductor $4.360$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 160.p (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.35968422976$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.3534400.1 Defining polynomial: $$x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 1 + \beta_{2} + \beta_{4} ) q^{5} + ( 2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 1 + \beta_{2} + \beta_{4} ) q^{5} + ( 2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( -4 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{11} + ( -3 + 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{13} + ( 2 - 3 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} ) q^{15} + ( -3 - 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{19} + ( -6 - \beta_{1} - 5 \beta_{2} + \beta_{4} + 5 \beta_{5} ) q^{21} + ( 10 - 2 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} ) q^{23} + ( -3 - \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{25} + ( 8 + 8 \beta_{3} + 4 \beta_{5} ) q^{27} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{29} + ( -36 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{31} + ( 14 - 6 \beta_{1} + 2 \beta_{2} - 14 \beta_{3} ) q^{33} + ( 8 - 2 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{35} + ( 17 + 17 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} ) q^{37} + ( -2 \beta_{1} + 9 \beta_{2} - 36 \beta_{3} - 2 \beta_{4} + 9 \beta_{5} ) q^{39} + ( 18 + 3 \beta_{1} + \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{41} + ( 28 + 4 \beta_{1} + 11 \beta_{2} - 28 \beta_{3} ) q^{43} + ( 4 + 4 \beta_{1} - \beta_{2} - 21 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{45} + ( 14 + 14 \beta_{3} - 10 \beta_{4} - 9 \beta_{5} ) q^{47} + ( -3 \beta_{1} - \beta_{2} + 17 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{49} + ( -64 + 8 \beta_{1} - \beta_{2} - 8 \beta_{4} + \beta_{5} ) q^{51} + ( -19 - 4 \beta_{1} - 2 \beta_{2} + 19 \beta_{3} ) q^{53} + ( 12 + 4 \beta_{1} - 3 \beta_{2} + 44 \beta_{3} - 8 \beta_{4} + 11 \beta_{5} ) q^{55} + ( -36 - 36 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} ) q^{57} + ( -10 \beta_{1} - 60 \beta_{3} - 10 \beta_{4} ) q^{59} + ( -38 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{4} - 5 \beta_{5} ) q^{61} + ( 34 + 6 \beta_{1} + 5 \beta_{2} - 34 \beta_{3} ) q^{63} + ( -23 - \beta_{1} - 13 \beta_{2} + 39 \beta_{3} + 7 \beta_{4} - 9 \beta_{5} ) q^{65} + ( 12 + 12 \beta_{3} - 4 \beta_{4} + 13 \beta_{5} ) q^{67} + ( -5 \beta_{1} - 9 \beta_{2} - 34 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} ) q^{69} + ( -52 - 6 \beta_{1} + 17 \beta_{2} + 6 \beta_{4} - 17 \beta_{5} ) q^{71} + ( 5 + 20 \beta_{1} - 5 \beta_{3} ) q^{73} + ( 4 + 8 \beta_{1} - \beta_{2} + 68 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{75} + ( 38 + 38 \beta_{3} - 18 \beta_{4} + 26 \beta_{5} ) q^{77} + ( 8 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 49 - 13 \beta_{1} + 5 \beta_{2} + 13 \beta_{4} - 5 \beta_{5} ) q^{81} + ( 40 - 16 \beta_{1} - 5 \beta_{2} - 40 \beta_{3} ) q^{83} + ( 61 - 13 \beta_{1} - 9 \beta_{2} - 43 \beta_{3} + \beta_{4} - 17 \beta_{5} ) q^{85} + ( 28 + 28 \beta_{3} + 12 \beta_{4} + 4 \beta_{5} ) q^{87} + ( 8 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -88 - 4 \beta_{1} - 17 \beta_{2} + 4 \beta_{4} + 17 \beta_{5} ) q^{91} + ( -14 + 6 \beta_{1} + 38 \beta_{2} + 14 \beta_{3} ) q^{93} + ( 16 - 18 \beta_{1} - 4 \beta_{2} + 52 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} ) q^{95} + ( -35 - 35 \beta_{3} + 20 \beta_{4} + 20 \beta_{5} ) q^{97} + ( 14 \beta_{1} - 13 \beta_{2} - 28 \beta_{3} + 14 \beta_{4} - 13 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 4q^{5} + 8q^{7} + O(q^{10})$$ $$6q + 4q^{5} + 8q^{7} - 32q^{11} - 14q^{13} + 8q^{15} - 14q^{17} - 40q^{21} + 56q^{23} - 14q^{25} + 48q^{27} - 208q^{31} + 72q^{33} + 48q^{35} + 86q^{37} + 120q^{41} + 176q^{43} + 34q^{45} + 104q^{47} - 352q^{51} - 122q^{53} + 96q^{55} - 208q^{57} - 216q^{61} + 216q^{63} - 154q^{65} + 80q^{67} - 336q^{71} + 70q^{73} + 32q^{75} + 264q^{77} + 242q^{81} + 208q^{83} + 338q^{85} + 144q^{87} - 544q^{91} - 72q^{93} + 48q^{95} - 250q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 3 x^{4} + 16 x^{3} + x^{2} - 12 x + 40$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$7 \nu^{5} + 100 \nu^{4} - 681 \nu^{3} + 1100 \nu^{2} + 757 \nu - 3140$$$$)/890$$ $$\beta_{2}$$ $$=$$ $$($$$$17 \nu^{5} - 75 \nu^{4} - \nu^{3} + 510 \nu^{2} + 122 \nu - 760$$$$)/445$$ $$\beta_{3}$$ $$=$$ $$($$$$9 \nu^{5} - 24 \nu^{4} - 11 \nu^{3} + 92 \nu^{2} + 7 \nu + 6$$$$)/178$$ $$\beta_{4}$$ $$=$$ $$($$$$-61 \nu^{5} + 400 \nu^{4} - 677 \nu^{3} - 940 \nu^{2} + 3829 \nu - 1880$$$$)/890$$ $$\beta_{5}$$ $$=$$ $$($$$$39 \nu^{5} - 15 \nu^{4} - 107 \nu^{3} + 280 \nu^{2} + 594 \nu + 560$$$$)/445$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{2} - \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} - \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + \beta_{1} + 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{5} - \beta_{4} - 11 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} - 7$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$28 \beta_{5} - 9 \beta_{4} - 60 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} - 38$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$64 \beta_{5} - 17 \beta_{4} - 26 \beta_{3} - 38 \beta_{2} - \beta_{1} - 184$$$$)/4$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 2.19082 + 1.44755i −1.81837 − 0.301352i 0.627553 − 1.14620i 2.19082 − 1.44755i −1.81837 + 0.301352i 0.627553 + 1.14620i
0 −2.89511 2.89511i 0 4.38164 + 2.40857i 0 5.86818 5.86818i 0 7.76328i 0
33.2 0 0.602705 + 0.602705i 0 −3.63675 + 3.43134i 0 −6.67079 + 6.67079i 0 8.27349i 0
33.3 0 2.29240 + 2.29240i 0 1.25511 4.83991i 0 4.80261 4.80261i 0 1.51021i 0
97.1 0 −2.89511 + 2.89511i 0 4.38164 2.40857i 0 5.86818 + 5.86818i 0 7.76328i 0
97.2 0 0.602705 0.602705i 0 −3.63675 3.43134i 0 −6.67079 6.67079i 0 8.27349i 0
97.3 0 2.29240 2.29240i 0 1.25511 + 4.83991i 0 4.80261 + 4.80261i 0 1.51021i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.p.f yes 6
4.b odd 2 1 160.3.p.e 6
5.b even 2 1 800.3.p.i 6
5.c odd 4 1 inner 160.3.p.f yes 6
5.c odd 4 1 800.3.p.i 6
8.b even 2 1 320.3.p.n 6
8.d odd 2 1 320.3.p.m 6
20.d odd 2 1 800.3.p.j 6
20.e even 4 1 160.3.p.e 6
20.e even 4 1 800.3.p.j 6
40.i odd 4 1 320.3.p.n 6
40.k even 4 1 320.3.p.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.p.e 6 4.b odd 2 1
160.3.p.e 6 20.e even 4 1
160.3.p.f yes 6 1.a even 1 1 trivial
160.3.p.f yes 6 5.c odd 4 1 inner
320.3.p.m 6 8.d odd 2 1
320.3.p.m 6 40.k even 4 1
320.3.p.n 6 8.b even 2 1
320.3.p.n 6 40.i odd 4 1
800.3.p.i 6 5.b even 2 1
800.3.p.i 6 5.c odd 4 1
800.3.p.j 6 20.d odd 2 1
800.3.p.j 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(160, [\chi])$$:

 $$T_{3}^{6} - 16 T_{3}^{3} + 196 T_{3}^{2} - 224 T_{3} + 128$$ $$T_{13}^{6} + 14 T_{13}^{5} + 98 T_{13}^{4} - 4000 T_{13}^{3} + 62500 T_{13}^{2} - 125000 T_{13} + 125000$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$128 - 224 T + 196 T^{2} - 16 T^{3} + T^{6}$$
$5$ $$15625 - 2500 T + 375 T^{2} - 40 T^{3} + 15 T^{4} - 4 T^{5} + T^{6}$$
$7$ $$282752 - 64672 T + 7396 T^{2} - 64 T^{3} + 32 T^{4} - 8 T^{5} + T^{6}$$
$11$ $$( -1600 - 140 T + 16 T^{2} + T^{3} )^{2}$$
$13$ $$125000 - 125000 T + 62500 T^{2} - 4000 T^{3} + 98 T^{4} + 14 T^{5} + T^{6}$$
$17$ $$1445000 - 1037000 T + 372100 T^{2} - 10240 T^{3} + 98 T^{4} + 14 T^{5} + T^{6}$$
$19$ $$40960000 + 435200 T^{2} + 1216 T^{4} + T^{6}$$
$23$ $$282752 - 212064 T + 79524 T^{2} - 15040 T^{3} + 1568 T^{4} - 56 T^{5} + T^{6}$$
$29$ $$46022656 + 882944 T^{2} + 1824 T^{4} + T^{6}$$
$31$ $$( 34400 + 3380 T + 104 T^{2} + T^{3} )^{2}$$
$37$ $$2332445000 - 36199000 T + 280900 T^{2} - 22720 T^{3} + 3698 T^{4} - 86 T^{5} + T^{6}$$
$41$ $$( 1600 + 740 T - 60 T^{2} + T^{3} )^{2}$$
$43$ $$2069560448 + 188504480 T + 8584900 T^{2} - 580016 T^{3} + 15488 T^{4} - 176 T^{5} + T^{6}$$
$47$ $$24436414592 - 432858976 T + 3833764 T^{2} - 17440 T^{3} + 5408 T^{4} - 104 T^{5} + T^{6}$$
$53$ $$85805000 + 26593000 T + 4120900 T^{2} + 234560 T^{3} + 7442 T^{4} + 122 T^{5} + T^{6}$$
$59$ $$25600000000 + 67200000 T^{2} + 17600 T^{4} + T^{6}$$
$61$ $$( 18080 + 2660 T + 108 T^{2} + T^{3} )^{2}$$
$67$ $$7869603968 - 182413024 T + 2114116 T^{2} - 9136 T^{3} + 3200 T^{4} - 80 T^{5} + T^{6}$$
$71$ $$( -628000 + 500 T + 168 T^{2} + T^{3} )^{2}$$
$73$ $$2775125000 + 614625000 T + 68062500 T^{2} + 652000 T^{3} + 2450 T^{4} - 70 T^{5} + T^{6}$$
$79$ $$10485760000 + 18124800 T^{2} + 8576 T^{4} + T^{6}$$
$83$ $$34995996800 + 195245280 T + 544644 T^{2} - 418064 T^{3} + 21632 T^{4} - 208 T^{5} + T^{6}$$
$89$ $$4194304 + 10784768 T^{2} + 7552 T^{4} + T^{6}$$
$97$ $$2269515125000 + 12463425000 T + 34222500 T^{2} + 668000 T^{3} + 31250 T^{4} + 250 T^{5} + T^{6}$$