Properties

Label 160.4.c.a
Level $160$
Weight $4$
Character orbit 160.c
Analytic conductor $9.440$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -11 + 2 i ) q^{5} + 27 q^{9} +O(q^{10})\) \( q + ( -11 + 2 i ) q^{5} + 27 q^{9} -92 i q^{13} -104 i q^{17} + ( 117 - 44 i ) q^{25} -130 q^{29} -396 i q^{37} + 230 q^{41} + ( -297 + 54 i ) q^{45} + 343 q^{49} + 572 i q^{53} -830 q^{61} + ( 184 + 1012 i ) q^{65} + 592 i q^{73} + 729 q^{81} + ( 208 + 1144 i ) q^{85} -1670 q^{89} -1816 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 22q^{5} + 54q^{9} + O(q^{10}) \) \( 2q - 22q^{5} + 54q^{9} + 234q^{25} - 260q^{29} + 460q^{41} - 594q^{45} + 686q^{49} - 1660q^{61} + 368q^{65} + 1458q^{81} + 416q^{85} - 3340q^{89} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 −11.0000 2.00000i 0 0 0 27.0000 0
129.2 0 0 0 −11.0000 + 2.00000i 0 0 0 27.0000 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 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
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 160.4.c.a 2
3.b odd 2 1 1440.4.f.e 2
4.b odd 2 1 CM 160.4.c.a 2
5.b even 2 1 inner 160.4.c.a 2
5.c odd 4 1 800.4.a.e 1
5.c odd 4 1 800.4.a.g 1
8.b even 2 1 320.4.c.e 2
8.d odd 2 1 320.4.c.e 2
12.b even 2 1 1440.4.f.e 2
15.d odd 2 1 1440.4.f.e 2
20.d odd 2 1 inner 160.4.c.a 2
20.e even 4 1 800.4.a.e 1
20.e even 4 1 800.4.a.g 1
40.e odd 2 1 320.4.c.e 2
40.f even 2 1 320.4.c.e 2
40.i odd 4 1 1600.4.a.z 1
40.i odd 4 1 1600.4.a.bb 1
40.k even 4 1 1600.4.a.z 1
40.k even 4 1 1600.4.a.bb 1
60.h even 2 1 1440.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.a 2 1.a even 1 1 trivial
160.4.c.a 2 4.b odd 2 1 CM
160.4.c.a 2 5.b even 2 1 inner
160.4.c.a 2 20.d odd 2 1 inner
320.4.c.e 2 8.b even 2 1
320.4.c.e 2 8.d odd 2 1
320.4.c.e 2 40.e odd 2 1
320.4.c.e 2 40.f even 2 1
800.4.a.e 1 5.c odd 4 1
800.4.a.e 1 20.e even 4 1
800.4.a.g 1 5.c odd 4 1
800.4.a.g 1 20.e even 4 1
1440.4.f.e 2 3.b odd 2 1
1440.4.f.e 2 12.b even 2 1
1440.4.f.e 2 15.d odd 2 1
1440.4.f.e 2 60.h even 2 1
1600.4.a.z 1 40.i odd 4 1
1600.4.a.z 1 40.k even 4 1
1600.4.a.bb 1 40.i odd 4 1
1600.4.a.bb 1 40.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 125 + 22 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 8464 + T^{2} \)
$17$ \( 10816 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 130 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 156816 + T^{2} \)
$41$ \( ( -230 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 327184 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 830 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 350464 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 1670 + T )^{2} \)
$97$ \( 3297856 + T^{2} \)
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