Properties

Label 160.3.h.b
Level $160$
Weight $3$
Character orbit 160.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + ( -1 - \beta_{1} + \beta_{4} ) q^{5} + ( 2 + \beta_{3} - \beta_{4} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + ( -1 - \beta_{1} + \beta_{4} ) q^{5} + ( 2 + \beta_{3} - \beta_{4} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{9} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{11} + \beta_{5} q^{13} + ( -7 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{15} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} + 2 \beta_{5} ) q^{19} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{21} + ( -12 - 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{25} + ( 32 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{27} + ( 10 + 8 \beta_{1} ) q^{29} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{31} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{33} + ( -19 + \beta_{1} + 5 \beta_{2} + 4 \beta_{4} ) q^{35} + ( 1 + \beta_{1} + 8 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{37} + ( 4 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{39} + ( -13 - 5 \beta_{1} - \beta_{3} + \beta_{4} ) q^{41} + ( -13 - \beta_{1} - 6 \beta_{3} + 6 \beta_{4} ) q^{43} + ( -5 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{45} + ( 42 - 8 \beta_{1} + \beta_{3} - \beta_{4} ) q^{47} + ( -12 - 5 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} ) q^{49} + ( 6 + 6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{51} + ( 8 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{53} + ( -45 - \beta_{1} - 9 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( 7 + 7 \beta_{1} + 12 \beta_{2} - 7 \beta_{3} - 7 \beta_{4} - \beta_{5} ) q^{57} + ( -15 \beta_{2} - 2 \beta_{5} ) q^{59} + ( -17 - \beta_{1} + 7 \beta_{3} - 7 \beta_{4} ) q^{61} + ( -28 + 2 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} ) q^{63} + ( 3 + 11 \beta_{1} - 8 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{65} + ( 47 - 13 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -39 - 25 \beta_{1} - \beta_{3} + \beta_{4} ) q^{69} + ( -6 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{71} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{73} + ( -47 - 7 \beta_{1} + 10 \beta_{2} - 8 \beta_{4} ) q^{75} + ( -11 - 11 \beta_{1} + 24 \beta_{2} + 11 \beta_{3} + 11 \beta_{4} + \beta_{5} ) q^{77} + ( -6 - 6 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{79} + ( 46 + 19 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} ) q^{81} + ( -33 + 3 \beta_{1} + 14 \beta_{3} - 14 \beta_{4} ) q^{83} + ( 5 + 21 \beta_{1} - 16 \beta_{2} - 13 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{85} + ( 106 + 26 \beta_{1} - 8 \beta_{3} + 8 \beta_{4} ) q^{87} + ( 10 - 8 \beta_{1} - 16 \beta_{3} + 16 \beta_{4} ) q^{89} + ( -2 - 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -8 - 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} ) q^{93} + ( -1 + 19 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{95} + ( -5 - 5 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{97} + ( -4 - 4 \beta_{1} - 33 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} + O(q^{10}) \) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} - 36q^{15} + 8q^{21} - 68q^{23} - 10q^{25} + 184q^{27} + 44q^{29} - 108q^{35} - 68q^{41} - 76q^{43} - 6q^{45} + 268q^{47} - 62q^{49} - 288q^{55} - 100q^{61} - 172q^{63} + 308q^{67} - 184q^{69} - 284q^{75} + 238q^{81} - 204q^{83} - 32q^{85} + 584q^{87} + 76q^{89} - 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 14 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 20 \nu^{2} + 23 \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{5} + 40 \nu^{3} + 76 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{5} - 2 \nu^{4} - 70 \nu^{3} - 10 \nu^{2} - 92 \nu + 7 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -8 \nu^{5} + 4 \nu^{4} - 70 \nu^{3} + 30 \nu^{2} - 92 \nu + 21 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{5} + 140 \nu^{3} + 224 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{3} - \beta_{1} - 1\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{3} + 3 \beta_{1} - 11\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_{1} + 2\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{4} - 5 \beta_{3} - 10 \beta_{1} + 32\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(41 \beta_{5} + 21 \beta_{4} + 21 \beta_{3} - 70 \beta_{2} - 21 \beta_{1} - 21\)\()/8\)

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
2.65109i
2.65109i
1.37720i
1.37720i
0.273891i
0.273891i
0 −2.75441 0 4.30219 2.54778i 0 −3.84997 0 −1.41325 0
159.2 0 −2.75441 0 4.30219 + 2.54778i 0 −3.84997 0 −1.41325 0
159.3 0 −0.547781 0 −3.75441 3.30219i 0 10.0566 0 −8.69994 0
159.4 0 −0.547781 0 −3.75441 + 3.30219i 0 10.0566 0 −8.69994 0
159.5 0 5.30219 0 −1.54778 4.75441i 0 −0.206625 0 19.1132 0
159.6 0 5.30219 0 −1.54778 + 4.75441i 0 −0.206625 0 19.1132 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.h.b yes 6
3.b odd 2 1 1440.3.j.b 6
4.b odd 2 1 160.3.h.a 6
5.b even 2 1 160.3.h.a 6
5.c odd 4 1 800.3.b.h 6
5.c odd 4 1 800.3.b.i 6
8.b even 2 1 320.3.h.f 6
8.d odd 2 1 320.3.h.g 6
12.b even 2 1 1440.3.j.a 6
15.d odd 2 1 1440.3.j.a 6
16.e even 4 1 1280.3.e.f 6
16.e even 4 1 1280.3.e.h 6
16.f odd 4 1 1280.3.e.g 6
16.f odd 4 1 1280.3.e.i 6
20.d odd 2 1 inner 160.3.h.b yes 6
20.e even 4 1 800.3.b.h 6
20.e even 4 1 800.3.b.i 6
40.e odd 2 1 320.3.h.f 6
40.f even 2 1 320.3.h.g 6
40.i odd 4 1 1600.3.b.v 6
40.i odd 4 1 1600.3.b.w 6
40.k even 4 1 1600.3.b.v 6
40.k even 4 1 1600.3.b.w 6
60.h even 2 1 1440.3.j.b 6
80.k odd 4 1 1280.3.e.f 6
80.k odd 4 1 1280.3.e.h 6
80.q even 4 1 1280.3.e.g 6
80.q even 4 1 1280.3.e.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 4.b odd 2 1
160.3.h.a 6 5.b even 2 1
160.3.h.b yes 6 1.a even 1 1 trivial
160.3.h.b yes 6 20.d odd 2 1 inner
320.3.h.f 6 8.b even 2 1
320.3.h.f 6 40.e odd 2 1
320.3.h.g 6 8.d odd 2 1
320.3.h.g 6 40.f even 2 1
800.3.b.h 6 5.c odd 4 1
800.3.b.h 6 20.e even 4 1
800.3.b.i 6 5.c odd 4 1
800.3.b.i 6 20.e even 4 1
1280.3.e.f 6 16.e even 4 1
1280.3.e.f 6 80.k odd 4 1
1280.3.e.g 6 16.f odd 4 1
1280.3.e.g 6 80.q even 4 1
1280.3.e.h 6 16.e even 4 1
1280.3.e.h 6 80.k odd 4 1
1280.3.e.i 6 16.f odd 4 1
1280.3.e.i 6 80.q even 4 1
1440.3.j.a 6 12.b even 2 1
1440.3.j.a 6 15.d odd 2 1
1440.3.j.b 6 3.b odd 2 1
1440.3.j.b 6 60.h even 2 1
1600.3.b.v 6 40.i odd 4 1
1600.3.b.v 6 40.k even 4 1
1600.3.b.w 6 40.i odd 4 1
1600.3.b.w 6 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 2 T_{3}^{2} - 16 T_{3} - 8 \) acting on \(S_{3}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( -8 - 16 T - 2 T^{2} + T^{3} )^{2} \)
$5$ \( 15625 + 1250 T + 175 T^{2} - 100 T^{3} + 7 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( ( -8 - 40 T - 6 T^{2} + T^{3} )^{2} \)
$11$ \( 2560000 + 86784 T^{2} + 560 T^{4} + T^{6} \)
$13$ \( 692224 + 43264 T^{2} + 416 T^{4} + T^{6} \)
$17$ \( 6553600 + 270336 T^{2} + 1088 T^{4} + T^{6} \)
$19$ \( 11505664 + 533248 T^{2} + 1712 T^{4} + T^{6} \)
$23$ \( ( -9160 - 152 T + 34 T^{2} + T^{3} )^{2} \)
$29$ \( ( -2120 - 948 T - 22 T^{2} + T^{3} )^{2} \)
$31$ \( 419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6} \)
$37$ \( 432640000 + 2985216 T^{2} + 5408 T^{4} + T^{6} \)
$41$ \( ( -5000 - 100 T + 34 T^{2} + T^{3} )^{2} \)
$43$ \( ( -6280 - 1408 T + 38 T^{2} + T^{3} )^{2} \)
$47$ \( ( -22216 + 4824 T - 134 T^{2} + T^{3} )^{2} \)
$53$ \( 50319462400 + 49973504 T^{2} + 12960 T^{4} + T^{6} \)
$59$ \( 25416011776 + 41968384 T^{2} + 12464 T^{4} + T^{6} \)
$61$ \( ( -81544 - 1732 T + 50 T^{2} + T^{3} )^{2} \)
$67$ \( ( 24760 + 4768 T - 154 T^{2} + T^{3} )^{2} \)
$71$ \( 14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6} \)
$73$ \( 3114532864 + 6447104 T^{2} + 4416 T^{4} + T^{6} \)
$79$ \( 37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6} \)
$83$ \( ( -483400 - 6880 T + 102 T^{2} + T^{3} )^{2} \)
$89$ \( ( 155000 - 13940 T - 38 T^{2} + T^{3} )^{2} \)
$97$ \( 44302336 + 1384448 T^{2} + 7488 T^{4} + T^{6} \)
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