Properties

Label 160.3.b.b
Level $160$
Weight $3$
Character orbit 160.b
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{2} ) q^{9} + ( 5 \beta_{1} + \beta_{3} ) q^{11} + ( -8 + 6 \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{15} + ( -2 + 4 \beta_{2} ) q^{17} + ( 3 \beta_{1} + 5 \beta_{3} ) q^{19} + ( -6 + 2 \beta_{2} ) q^{21} + ( -7 \beta_{1} - 2 \beta_{3} ) q^{23} + 5 q^{25} + ( 8 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 30 - 4 \beta_{2} ) q^{29} + ( 6 \beta_{1} - 8 \beta_{3} ) q^{31} + ( -32 + 8 \beta_{2} ) q^{33} + ( -2 \beta_{1} + \beta_{3} ) q^{35} + ( 12 - 14 \beta_{2} ) q^{37} + ( -20 \beta_{1} + 6 \beta_{3} ) q^{39} -26 \beta_{2} q^{41} + ( 17 \beta_{1} - 16 \beta_{3} ) q^{43} + ( 10 + 3 \beta_{2} ) q^{45} + ( -9 \beta_{1} - 10 \beta_{3} ) q^{47} + ( 43 + 2 \beta_{2} ) q^{49} + ( -10 \beta_{1} + 4 \beta_{3} ) q^{51} + ( -40 - 26 \beta_{2} ) q^{53} + ( -9 \beta_{1} + 7 \beta_{3} ) q^{55} + ( -28 - 4 \beta_{2} ) q^{57} + ( -31 \beta_{1} + 11 \beta_{3} ) q^{59} + ( -48 + 22 \beta_{2} ) q^{61} + ( -\beta_{1} + 2 \beta_{3} ) q^{63} + ( 30 - 8 \beta_{2} ) q^{65} + ( -21 \beta_{1} + 4 \beta_{3} ) q^{67} + ( 46 - 10 \beta_{2} ) q^{69} + ( 8 \beta_{1} - 18 \beta_{3} ) q^{71} + ( -26 - 20 \beta_{2} ) q^{73} + 5 \beta_{1} q^{75} + ( -32 + 8 \beta_{2} ) q^{77} + ( 16 \beta_{1} + 20 \beta_{3} ) q^{79} + ( -25 + 30 \beta_{2} ) q^{81} + ( 31 \beta_{1} - 38 \beta_{3} ) q^{83} + ( 20 - 2 \beta_{2} ) q^{85} + ( 38 \beta_{1} - 4 \beta_{3} ) q^{87} + ( 82 - 16 \beta_{2} ) q^{89} + ( -20 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -20 + 28 \beta_{2} ) q^{93} + ( -\beta_{1} + 13 \beta_{3} ) q^{95} + ( 26 + 48 \beta_{2} ) q^{97} + ( -3 \beta_{1} + 17 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 32q^{13} - 8q^{17} - 24q^{21} + 20q^{25} + 120q^{29} - 128q^{33} + 48q^{37} + 40q^{45} + 172q^{49} - 160q^{53} - 112q^{57} - 192q^{61} + 120q^{65} + 184q^{69} - 104q^{73} - 128q^{77} - 100q^{81} + 80q^{85} + 328q^{89} - 80q^{93} + 104q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{3} + 10 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 5 \beta_{1}\)\()/4\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.61803i
0.618034i
0.618034i
1.61803i
0 3.23607i 0 −2.23607 0 3.23607i 0 −1.47214 0
31.2 0 1.23607i 0 2.23607 0 1.23607i 0 7.47214 0
31.3 0 1.23607i 0 2.23607 0 1.23607i 0 7.47214 0
31.4 0 3.23607i 0 −2.23607 0 3.23607i 0 −1.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 160.3.b.b 4
3.b odd 2 1 1440.3.e.a 4
4.b odd 2 1 inner 160.3.b.b 4
5.b even 2 1 800.3.b.g 4
5.c odd 4 1 800.3.h.e 4
5.c odd 4 1 800.3.h.h 4
8.b even 2 1 320.3.b.d 4
8.d odd 2 1 320.3.b.d 4
12.b even 2 1 1440.3.e.a 4
16.e even 4 1 1280.3.g.b 4
16.e even 4 1 1280.3.g.c 4
16.f odd 4 1 1280.3.g.b 4
16.f odd 4 1 1280.3.g.c 4
20.d odd 2 1 800.3.b.g 4
20.e even 4 1 800.3.h.e 4
20.e even 4 1 800.3.h.h 4
24.f even 2 1 2880.3.e.h 4
24.h odd 2 1 2880.3.e.h 4
40.e odd 2 1 1600.3.b.u 4
40.f even 2 1 1600.3.b.u 4
40.i odd 4 1 1600.3.h.f 4
40.i odd 4 1 1600.3.h.k 4
40.k even 4 1 1600.3.h.f 4
40.k even 4 1 1600.3.h.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.b 4 1.a even 1 1 trivial
160.3.b.b 4 4.b odd 2 1 inner
320.3.b.d 4 8.b even 2 1
320.3.b.d 4 8.d odd 2 1
800.3.b.g 4 5.b even 2 1
800.3.b.g 4 20.d odd 2 1
800.3.h.e 4 5.c odd 4 1
800.3.h.e 4 20.e even 4 1
800.3.h.h 4 5.c odd 4 1
800.3.h.h 4 20.e even 4 1
1280.3.g.b 4 16.e even 4 1
1280.3.g.b 4 16.f odd 4 1
1280.3.g.c 4 16.e even 4 1
1280.3.g.c 4 16.f odd 4 1
1440.3.e.a 4 3.b odd 2 1
1440.3.e.a 4 12.b even 2 1
1600.3.b.u 4 40.e odd 2 1
1600.3.b.u 4 40.f even 2 1
1600.3.h.f 4 40.i odd 4 1
1600.3.h.f 4 40.k even 4 1
1600.3.h.k 4 40.i odd 4 1
1600.3.h.k 4 40.k even 4 1
2880.3.e.h 4 24.f even 2 1
2880.3.e.h 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 12 T_{3}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 12 T^{2} + T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 16 + 12 T^{2} + T^{4} \)
$11$ \( 30976 + 368 T^{2} + T^{4} \)
$13$ \( ( -116 + 16 T + T^{2} )^{2} \)
$17$ \( ( -76 + 4 T + T^{2} )^{2} \)
$19$ \( 30976 + 928 T^{2} + T^{4} \)
$23$ \( 163216 + 812 T^{2} + T^{4} \)
$29$ \( ( 820 - 60 T + T^{2} )^{2} \)
$31$ \( 774400 + 1840 T^{2} + T^{4} \)
$37$ \( ( -836 - 24 T + T^{2} )^{2} \)
$41$ \( ( -3380 + T^{2} )^{2} \)
$43$ \( 17808400 + 8460 T^{2} + T^{4} \)
$47$ \( 1860496 + 4492 T^{2} + T^{4} \)
$53$ \( ( -1780 + 80 T + T^{2} )^{2} \)
$59$ \( 4393216 + 12192 T^{2} + T^{4} \)
$61$ \( ( -116 + 96 T + T^{2} )^{2} \)
$67$ \( 126736 + 5068 T^{2} + T^{4} \)
$71$ \( 11182336 + 8688 T^{2} + T^{4} \)
$73$ \( ( -1324 + 52 T + T^{2} )^{2} \)
$79$ \( 20647936 + 16832 T^{2} + T^{4} \)
$83$ \( 431808400 + 42540 T^{2} + T^{4} \)
$89$ \( ( 5444 - 164 T + T^{2} )^{2} \)
$97$ \( ( -10844 - 52 T + T^{2} )^{2} \)
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