Properties

Label 160.2.f.a
Level $160$
Weight $2$
Character orbit 160.f
Analytic conductor $1.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.27760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} - q^{9} -2 \beta_{2} q^{11} + ( 2 - \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + 2 \beta_{2} q^{19} + 2 \beta_{2} q^{21} -\beta_{3} q^{23} + ( -1 - 2 \beta_{3} ) q^{25} + 4 \beta_{1} q^{27} -4 q^{31} -2 \beta_{3} q^{33} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{35} + 6 \beta_{1} q^{37} + 3 \beta_{1} q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + 3 \beta_{3} q^{47} + q^{49} + 4 \beta_{2} q^{51} -4 \beta_{1} q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + 2 \beta_{3} q^{57} -6 \beta_{2} q^{59} -2 \beta_{2} q^{61} -\beta_{3} q^{63} -3 \beta_{1} q^{67} -2 \beta_{2} q^{69} + 12 q^{71} + 2 \beta_{3} q^{73} + ( \beta_{1} - 4 \beta_{2} ) q^{75} -6 \beta_{1} q^{77} + 4 q^{79} -5 q^{81} -7 \beta_{1} q^{83} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{85} + 6 q^{89} + 4 \beta_{1} q^{93} + ( 6 + 2 \beta_{3} ) q^{95} -2 \beta_{3} q^{97} + 2 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 8q^{15} - 4q^{25} - 16q^{31} + 4q^{49} - 24q^{55} + 48q^{71} + 16q^{79} - 20q^{81} + 24q^{89} + 24q^{95} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
0.707107 1.22474i
0 −1.41421 0 −1.41421 1.73205i 0 2.44949i 0 −1.00000 0
49.2 0 −1.41421 0 −1.41421 + 1.73205i 0 2.44949i 0 −1.00000 0
49.3 0 1.41421 0 1.41421 1.73205i 0 2.44949i 0 −1.00000 0
49.4 0 1.41421 0 1.41421 + 1.73205i 0 2.44949i 0 −1.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.f.a 4
3.b odd 2 1 1440.2.d.c 4
4.b odd 2 1 40.2.f.a 4
5.b even 2 1 inner 160.2.f.a 4
5.c odd 4 2 800.2.d.f 4
8.b even 2 1 inner 160.2.f.a 4
8.d odd 2 1 40.2.f.a 4
12.b even 2 1 360.2.d.b 4
15.d odd 2 1 1440.2.d.c 4
15.e even 4 2 7200.2.k.l 4
16.e even 4 2 1280.2.c.k 4
16.f odd 4 2 1280.2.c.i 4
20.d odd 2 1 40.2.f.a 4
20.e even 4 2 200.2.d.e 4
24.f even 2 1 360.2.d.b 4
24.h odd 2 1 1440.2.d.c 4
40.e odd 2 1 40.2.f.a 4
40.f even 2 1 inner 160.2.f.a 4
40.i odd 4 2 800.2.d.f 4
40.k even 4 2 200.2.d.e 4
60.h even 2 1 360.2.d.b 4
60.l odd 4 2 1800.2.k.m 4
80.i odd 4 2 6400.2.a.cm 4
80.j even 4 2 6400.2.a.co 4
80.k odd 4 2 1280.2.c.i 4
80.q even 4 2 1280.2.c.k 4
80.s even 4 2 6400.2.a.co 4
80.t odd 4 2 6400.2.a.cm 4
120.i odd 2 1 1440.2.d.c 4
120.m even 2 1 360.2.d.b 4
120.q odd 4 2 1800.2.k.m 4
120.w even 4 2 7200.2.k.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 4.b odd 2 1
40.2.f.a 4 8.d odd 2 1
40.2.f.a 4 20.d odd 2 1
40.2.f.a 4 40.e odd 2 1
160.2.f.a 4 1.a even 1 1 trivial
160.2.f.a 4 5.b even 2 1 inner
160.2.f.a 4 8.b even 2 1 inner
160.2.f.a 4 40.f even 2 1 inner
200.2.d.e 4 20.e even 4 2
200.2.d.e 4 40.k even 4 2
360.2.d.b 4 12.b even 2 1
360.2.d.b 4 24.f even 2 1
360.2.d.b 4 60.h even 2 1
360.2.d.b 4 120.m even 2 1
800.2.d.f 4 5.c odd 4 2
800.2.d.f 4 40.i odd 4 2
1280.2.c.i 4 16.f odd 4 2
1280.2.c.i 4 80.k odd 4 2
1280.2.c.k 4 16.e even 4 2
1280.2.c.k 4 80.q even 4 2
1440.2.d.c 4 3.b odd 2 1
1440.2.d.c 4 15.d odd 2 1
1440.2.d.c 4 24.h odd 2 1
1440.2.d.c 4 120.i odd 2 1
1800.2.k.m 4 60.l odd 4 2
1800.2.k.m 4 120.q odd 4 2
6400.2.a.cm 4 80.i odd 4 2
6400.2.a.cm 4 80.t odd 4 2
6400.2.a.co 4 80.j even 4 2
6400.2.a.co 4 80.s even 4 2
7200.2.k.l 4 15.e even 4 2
7200.2.k.l 4 120.w even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( 25 + 2 T^{2} + T^{4} \)
$7$ \( ( 6 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( 6 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( ( -72 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -18 + T^{2} )^{2} \)
$47$ \( ( 54 + T^{2} )^{2} \)
$53$ \( ( -32 + T^{2} )^{2} \)
$59$ \( ( 108 + T^{2} )^{2} \)
$61$ \( ( 12 + T^{2} )^{2} \)
$67$ \( ( -18 + T^{2} )^{2} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( -98 + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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