Properties

Label 160.4.c.c
Level $160$
Weight $4$
Character orbit 160.c
Analytic conductor $9.440$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + 2 \beta_{3} ) q^{3} -5 \beta_{2} q^{5} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{7} + ( -27 - 14 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + 2 \beta_{3} ) q^{3} -5 \beta_{2} q^{5} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{7} + ( -27 - 14 \beta_{2} ) q^{9} + ( -20 \beta_{1} - 15 \beta_{3} ) q^{15} + ( 118 + 86 \beta_{2} ) q^{21} + ( -64 \beta_{1} + \beta_{3} ) q^{23} + 125 q^{25} + ( -56 \beta_{1} - 42 \beta_{3} ) q^{27} -306 q^{29} + ( -35 \beta_{1} + 80 \beta_{3} ) q^{35} -206 \beta_{2} q^{41} + ( -15 \beta_{1} - 104 \beta_{3} ) q^{43} + ( 350 + 135 \beta_{2} ) q^{45} + ( 190 \beta_{1} - 37 \beta_{3} ) q^{47} + ( -343 - 198 \beta_{2} ) q^{49} -18 \beta_{2} q^{61} + ( 64 \beta_{1} + 359 \beta_{3} ) q^{63} + ( 321 \beta_{1} - 76 \beta_{3} ) q^{67} + ( -154 + 374 \beta_{2} ) q^{69} + ( -125 \beta_{1} + 250 \beta_{3} ) q^{75} + ( 251 + 378 \beta_{2} ) q^{81} + ( 377 \beta_{1} + 100 \beta_{3} ) q^{83} + ( 306 \beta_{1} - 612 \beta_{3} ) q^{87} + 1386 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 108q^{9} + O(q^{10}) \) \( 4q - 108q^{9} + 472q^{21} + 500q^{25} - 1224q^{29} + 1400q^{45} - 1372q^{49} - 616q^{69} + 1004q^{81} + 5544q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.618034i
1.61803i
1.61803i
0.618034i
0 9.23607i 0 −11.1803 0 33.5967i 0 −58.3050 0
129.2 0 4.76393i 0 11.1803 0 15.5967i 0 4.30495 0
129.3 0 4.76393i 0 11.1803 0 15.5967i 0 4.30495 0
129.4 0 9.23607i 0 −11.1803 0 33.5967i 0 −58.3050 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.c.c 4
3.b odd 2 1 1440.4.f.g 4
4.b odd 2 1 inner 160.4.c.c 4
5.b even 2 1 inner 160.4.c.c 4
5.c odd 4 1 800.4.a.l 2
5.c odd 4 1 800.4.a.t 2
8.b even 2 1 320.4.c.f 4
8.d odd 2 1 320.4.c.f 4
12.b even 2 1 1440.4.f.g 4
15.d odd 2 1 1440.4.f.g 4
20.d odd 2 1 CM 160.4.c.c 4
20.e even 4 1 800.4.a.l 2
20.e even 4 1 800.4.a.t 2
40.e odd 2 1 320.4.c.f 4
40.f even 2 1 320.4.c.f 4
40.i odd 4 1 1600.4.a.cb 2
40.i odd 4 1 1600.4.a.cp 2
40.k even 4 1 1600.4.a.cb 2
40.k even 4 1 1600.4.a.cp 2
60.h even 2 1 1440.4.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.c 4 1.a even 1 1 trivial
160.4.c.c 4 4.b odd 2 1 inner
160.4.c.c 4 5.b even 2 1 inner
160.4.c.c 4 20.d odd 2 1 CM
320.4.c.f 4 8.b even 2 1
320.4.c.f 4 8.d odd 2 1
320.4.c.f 4 40.e odd 2 1
320.4.c.f 4 40.f even 2 1
800.4.a.l 2 5.c odd 4 1
800.4.a.l 2 20.e even 4 1
800.4.a.t 2 5.c odd 4 1
800.4.a.t 2 20.e even 4 1
1440.4.f.g 4 3.b odd 2 1
1440.4.f.g 4 12.b even 2 1
1440.4.f.g 4 15.d odd 2 1
1440.4.f.g 4 60.h even 2 1
1600.4.a.cb 2 40.i odd 4 1
1600.4.a.cb 2 40.k even 4 1
1600.4.a.cp 2 40.i odd 4 1
1600.4.a.cp 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 108 T_{3}^{2} + 1936 \) acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1936 + 108 T^{2} + T^{4} \)
$5$ \( ( -125 + T^{2} )^{2} \)
$7$ \( 274576 + 1372 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 235806736 + 48668 T^{2} + T^{4} \)
$29$ \( ( 306 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -212180 + T^{2} )^{2} \)
$43$ \( 302899216 + 318028 T^{2} + T^{4} \)
$47$ \( 699285136 + 415292 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -1620 + T^{2} )^{2} \)
$67$ \( 1628176 + 1203052 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( 1280781104656 + 2287148 T^{2} + T^{4} \)
$89$ \( ( -1386 + T )^{4} \)
$97$ \( T^{4} \)
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