Properties

Label 160.4.c.b
Level $160$
Weight $4$
Character orbit 160.c
Analytic conductor $9.440$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(\Q(i, \sqrt{29})\)
Defining polynomial: \(x^{4} + 15 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -3 + \beta_{1} ) q^{5} + \beta_{2} q^{7} + 11 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -3 + \beta_{1} ) q^{5} + \beta_{2} q^{7} + 11 q^{9} + \beta_{3} q^{11} + 2 \beta_{1} q^{13} + ( -3 \beta_{2} + \beta_{3} ) q^{15} -4 \beta_{1} q^{17} + 3 \beta_{3} q^{19} -16 q^{21} + 13 \beta_{2} q^{23} + ( -107 - 6 \beta_{1} ) q^{25} + 38 \beta_{2} q^{27} + 158 q^{29} + 4 \beta_{3} q^{31} -16 \beta_{1} q^{33} + ( -3 \beta_{2} + \beta_{3} ) q^{35} + 26 \beta_{1} q^{37} + 2 \beta_{3} q^{39} -170 q^{41} -79 \beta_{2} q^{43} + ( -33 + 11 \beta_{1} ) q^{45} + 61 \beta_{2} q^{47} + 327 q^{49} -4 \beta_{3} q^{51} + 46 \beta_{1} q^{53} + ( -116 \beta_{2} - 3 \beta_{3} ) q^{55} -48 \beta_{1} q^{57} -15 \beta_{3} q^{59} + 82 q^{61} + 11 \beta_{2} q^{63} + ( -232 - 6 \beta_{1} ) q^{65} + 173 \beta_{2} q^{67} -208 q^{69} -22 \beta_{3} q^{71} -40 \beta_{1} q^{73} + ( -107 \beta_{2} - 6 \beta_{3} ) q^{75} -16 \beta_{1} q^{77} -8 \beta_{3} q^{79} -311 q^{81} -235 \beta_{2} q^{83} + ( 464 + 12 \beta_{1} ) q^{85} + 158 \beta_{2} q^{87} -6 q^{89} + 2 \beta_{3} q^{91} -64 \beta_{1} q^{93} + ( -348 \beta_{2} - 9 \beta_{3} ) q^{95} + 100 \beta_{1} q^{97} + 11 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{5} + 44q^{9} + O(q^{10}) \) \( 4q - 12q^{5} + 44q^{9} - 64q^{21} - 428q^{25} + 632q^{29} - 680q^{41} - 132q^{45} + 1308q^{49} + 328q^{61} - 928q^{65} - 832q^{69} - 1244q^{81} + 1856q^{85} - 24q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 15 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 44 \nu \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} - 32 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\( 16 \nu^{2} + 120 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 120\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{2} - 8 \beta_{1}\)\()/4\)

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
3.19258i
2.19258i
2.19258i
3.19258i
0 4.00000i 0 −3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.2 0 4.00000i 0 −3.00000 + 10.7703i 0 4.00000i 0 11.0000 0
129.3 0 4.00000i 0 −3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.4 0 4.00000i 0 −3.00000 + 10.7703i 0 4.00000i 0 11.0000 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 inner
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.b 4
3.b odd 2 1 1440.4.f.h 4
4.b odd 2 1 inner 160.4.c.b 4
5.b even 2 1 inner 160.4.c.b 4
5.c odd 4 1 800.4.a.n 2
5.c odd 4 1 800.4.a.r 2
8.b even 2 1 320.4.c.i 4
8.d odd 2 1 320.4.c.i 4
12.b even 2 1 1440.4.f.h 4
15.d odd 2 1 1440.4.f.h 4
20.d odd 2 1 inner 160.4.c.b 4
20.e even 4 1 800.4.a.n 2
20.e even 4 1 800.4.a.r 2
40.e odd 2 1 320.4.c.i 4
40.f even 2 1 320.4.c.i 4
40.i odd 4 1 1600.4.a.cc 2
40.i odd 4 1 1600.4.a.co 2
40.k even 4 1 1600.4.a.cc 2
40.k even 4 1 1600.4.a.co 2
60.h even 2 1 1440.4.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.b 4 1.a even 1 1 trivial
160.4.c.b 4 4.b odd 2 1 inner
160.4.c.b 4 5.b even 2 1 inner
160.4.c.b 4 20.d odd 2 1 inner
320.4.c.i 4 8.b even 2 1
320.4.c.i 4 8.d odd 2 1
320.4.c.i 4 40.e odd 2 1
320.4.c.i 4 40.f even 2 1
800.4.a.n 2 5.c odd 4 1
800.4.a.n 2 20.e even 4 1
800.4.a.r 2 5.c odd 4 1
800.4.a.r 2 20.e even 4 1
1440.4.f.h 4 3.b odd 2 1
1440.4.f.h 4 12.b even 2 1
1440.4.f.h 4 15.d odd 2 1
1440.4.f.h 4 60.h even 2 1
1600.4.a.cc 2 40.i odd 4 1
1600.4.a.cc 2 40.k even 4 1
1600.4.a.co 2 40.i odd 4 1
1600.4.a.co 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 16 + T^{2} )^{2} \)
$5$ \( ( 125 + 6 T + T^{2} )^{2} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( -1856 + T^{2} )^{2} \)
$13$ \( ( 464 + T^{2} )^{2} \)
$17$ \( ( 1856 + T^{2} )^{2} \)
$19$ \( ( -16704 + T^{2} )^{2} \)
$23$ \( ( 2704 + T^{2} )^{2} \)
$29$ \( ( -158 + T )^{4} \)
$31$ \( ( -29696 + T^{2} )^{2} \)
$37$ \( ( 78416 + T^{2} )^{2} \)
$41$ \( ( 170 + T )^{4} \)
$43$ \( ( 99856 + T^{2} )^{2} \)
$47$ \( ( 59536 + T^{2} )^{2} \)
$53$ \( ( 245456 + T^{2} )^{2} \)
$59$ \( ( -417600 + T^{2} )^{2} \)
$61$ \( ( -82 + T )^{4} \)
$67$ \( ( 478864 + T^{2} )^{2} \)
$71$ \( ( -898304 + T^{2} )^{2} \)
$73$ \( ( 185600 + T^{2} )^{2} \)
$79$ \( ( -118784 + T^{2} )^{2} \)
$83$ \( ( 883600 + T^{2} )^{2} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( ( 1160000 + T^{2} )^{2} \)
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