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 $2$

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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:

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

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