Properties

Label 160.3.p.a
Level $160$
Weight $3$
Character orbit 160.p
Analytic conductor $4.360$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -4 + 3 i ) q^{5} + 9 i q^{9} +O(q^{10})\) \( q + ( -4 + 3 i ) q^{5} + 9 i q^{9} + ( -17 + 17 i ) q^{13} + ( 7 + 7 i ) q^{17} + ( 7 - 24 i ) q^{25} + 40 i q^{29} + ( -47 - 47 i ) q^{37} + 80 q^{41} + ( -27 - 36 i ) q^{45} -49 i q^{49} + ( 17 - 17 i ) q^{53} + 120 q^{61} + ( 17 - 119 i ) q^{65} + ( -103 + 103 i ) q^{73} -81 q^{81} + ( -49 - 7 i ) q^{85} + 160 i q^{89} + ( -7 - 7 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} + O(q^{10}) \) \( 2q - 8q^{5} - 34q^{13} + 14q^{17} + 14q^{25} - 94q^{37} + 160q^{41} - 54q^{45} + 34q^{53} + 240q^{61} + 34q^{65} - 206q^{73} - 162q^{81} - 98q^{85} - 14q^{97} + O(q^{100}) \)

Character values

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

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

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
1.00000i
1.00000i
0 0 0 −4.00000 3.00000i 0 0 0 9.00000i 0
97.1 0 0 0 −4.00000 + 3.00000i 0 0 0 9.00000i 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.c odd 4 1 inner
20.e even 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.a 2
4.b odd 2 1 CM 160.3.p.a 2
5.b even 2 1 800.3.p.b 2
5.c odd 4 1 inner 160.3.p.a 2
5.c odd 4 1 800.3.p.b 2
8.b even 2 1 320.3.p.e 2
8.d odd 2 1 320.3.p.e 2
20.d odd 2 1 800.3.p.b 2
20.e even 4 1 inner 160.3.p.a 2
20.e even 4 1 800.3.p.b 2
40.i odd 4 1 320.3.p.e 2
40.k even 4 1 320.3.p.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.p.a 2 1.a even 1 1 trivial
160.3.p.a 2 4.b odd 2 1 CM
160.3.p.a 2 5.c odd 4 1 inner
160.3.p.a 2 20.e even 4 1 inner
320.3.p.e 2 8.b even 2 1
320.3.p.e 2 8.d odd 2 1
320.3.p.e 2 40.i odd 4 1
320.3.p.e 2 40.k even 4 1
800.3.p.b 2 5.b even 2 1
800.3.p.b 2 5.c odd 4 1
800.3.p.b 2 20.d odd 2 1
800.3.p.b 2 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 \(S_{3}^{\mathrm{new}}(160, [\chi])\):

\( T_{3} \)
\( T_{13}^{2} + 34 T_{13} + 578 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 25 + 8 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 578 + 34 T + T^{2} \)
$17$ \( 98 - 14 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1600 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4418 + 94 T + T^{2} \)
$41$ \( ( -80 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 578 - 34 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -120 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 21218 + 206 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 25600 + T^{2} \)
$97$ \( 98 + 14 T + T^{2} \)
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