Properties

Label 160.2.c.a
Level $160$
Weight $2$
Character orbit 160.c
Analytic conductor $1.278$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.27760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{5} + 3 q^{9} + 2 \beta q^{13} - 4 \beta q^{17} + (2 \beta - 3) q^{25} - 10 q^{29} - 6 \beta q^{37} - 10 q^{41} + (3 \beta + 3) q^{45} + 7 q^{49} - 2 \beta q^{53} + 10 q^{61} + (2 \beta - 8) q^{65} + 8 \beta q^{73} + 9 q^{81} + ( - 4 \beta + 16) q^{85} + 10 q^{89} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 6 q^{9} - 6 q^{25} - 20 q^{29} - 20 q^{41} + 6 q^{45} + 14 q^{49} + 20 q^{61} - 16 q^{65} + 18 q^{81} + 32 q^{85} + 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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 1.00000 2.00000i 0 0 0 3.00000 0
129.2 0 0 0 1.00000 + 2.00000i 0 0 0 3.00000 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 160.2.c.a 2
3.b odd 2 1 1440.2.f.c 2
4.b odd 2 1 CM 160.2.c.a 2
5.b even 2 1 inner 160.2.c.a 2
5.c odd 4 1 800.2.a.e 1
5.c odd 4 1 800.2.a.f 1
8.b even 2 1 320.2.c.a 2
8.d odd 2 1 320.2.c.a 2
12.b even 2 1 1440.2.f.c 2
15.d odd 2 1 1440.2.f.c 2
15.e even 4 1 7200.2.a.y 1
15.e even 4 1 7200.2.a.bb 1
16.e even 4 1 1280.2.f.c 2
16.e even 4 1 1280.2.f.d 2
16.f odd 4 1 1280.2.f.c 2
16.f odd 4 1 1280.2.f.d 2
20.d odd 2 1 inner 160.2.c.a 2
20.e even 4 1 800.2.a.e 1
20.e even 4 1 800.2.a.f 1
24.f even 2 1 2880.2.f.n 2
24.h odd 2 1 2880.2.f.n 2
40.e odd 2 1 320.2.c.a 2
40.f even 2 1 320.2.c.a 2
40.i odd 4 1 1600.2.a.l 1
40.i odd 4 1 1600.2.a.m 1
40.k even 4 1 1600.2.a.l 1
40.k even 4 1 1600.2.a.m 1
60.h even 2 1 1440.2.f.c 2
60.l odd 4 1 7200.2.a.y 1
60.l odd 4 1 7200.2.a.bb 1
80.k odd 4 1 1280.2.f.c 2
80.k odd 4 1 1280.2.f.d 2
80.q even 4 1 1280.2.f.c 2
80.q even 4 1 1280.2.f.d 2
120.i odd 2 1 2880.2.f.n 2
120.m even 2 1 2880.2.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.a 2 1.a even 1 1 trivial
160.2.c.a 2 4.b odd 2 1 CM
160.2.c.a 2 5.b even 2 1 inner
160.2.c.a 2 20.d odd 2 1 inner
320.2.c.a 2 8.b even 2 1
320.2.c.a 2 8.d odd 2 1
320.2.c.a 2 40.e odd 2 1
320.2.c.a 2 40.f even 2 1
800.2.a.e 1 5.c odd 4 1
800.2.a.e 1 20.e even 4 1
800.2.a.f 1 5.c odd 4 1
800.2.a.f 1 20.e even 4 1
1280.2.f.c 2 16.e even 4 1
1280.2.f.c 2 16.f odd 4 1
1280.2.f.c 2 80.k odd 4 1
1280.2.f.c 2 80.q even 4 1
1280.2.f.d 2 16.e even 4 1
1280.2.f.d 2 16.f odd 4 1
1280.2.f.d 2 80.k odd 4 1
1280.2.f.d 2 80.q even 4 1
1440.2.f.c 2 3.b odd 2 1
1440.2.f.c 2 12.b even 2 1
1440.2.f.c 2 15.d odd 2 1
1440.2.f.c 2 60.h even 2 1
1600.2.a.l 1 40.i odd 4 1
1600.2.a.l 1 40.k even 4 1
1600.2.a.m 1 40.i odd 4 1
1600.2.a.m 1 40.k even 4 1
2880.2.f.n 2 24.f even 2 1
2880.2.f.n 2 24.h odd 2 1
2880.2.f.n 2 120.i odd 2 1
2880.2.f.n 2 120.m even 2 1
7200.2.a.y 1 15.e even 4 1
7200.2.a.y 1 60.l odd 4 1
7200.2.a.bb 1 15.e even 4 1
7200.2.a.bb 1 60.l odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 144 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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