Properties

Label 320.3.b.b
Level $320$
Weight $3$
Character orbit 320.b
Analytic conductor $8.719$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 4 \beta_{2} ) q^{7} + ( -5 + 6 \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 4 \beta_{2} ) q^{7} + ( -5 + 6 \beta_{3} ) q^{9} + ( 3 \beta_{1} + \beta_{2} ) q^{11} + ( -16 - 2 \beta_{3} ) q^{13} + ( 2 \beta_{1} - \beta_{2} ) q^{15} + ( -10 + 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{19} + ( 22 - 14 \beta_{3} ) q^{21} + ( \beta_{1} - 10 \beta_{2} ) q^{23} + 5 q^{25} + ( 8 \beta_{1} - 6 \beta_{2} ) q^{27} + ( 18 + 12 \beta_{3} ) q^{29} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{31} + ( 40 - 16 \beta_{3} ) q^{33} + ( -6 \beta_{1} - 7 \beta_{2} ) q^{35} + ( -28 + 2 \beta_{3} ) q^{37} + ( 12 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 40 - 14 \beta_{3} ) q^{41} + ( -\beta_{1} - 8 \beta_{2} ) q^{43} + ( 30 - 5 \beta_{3} ) q^{45} + ( -\beta_{1} + 14 \beta_{2} ) q^{47} + ( -77 - 10 \beta_{3} ) q^{49} + ( 18 \beta_{1} - 4 \beta_{2} ) q^{51} + ( -32 + 14 \beta_{3} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{55} + ( -36 + 12 \beta_{3} ) q^{57} + ( -\beta_{1} - 29 \beta_{2} ) q^{59} + ( -16 + 22 \beta_{3} ) q^{61} + ( -41 \beta_{1} - 22 \beta_{2} ) q^{63} + ( -10 - 16 \beta_{3} ) q^{65} + ( -11 \beta_{1} + 28 \beta_{2} ) q^{67} + ( 34 - 26 \beta_{3} ) q^{69} + ( 16 \beta_{1} + 34 \beta_{2} ) q^{71} + ( -2 + 60 \beta_{3} ) q^{73} -5 \beta_{1} q^{75} + ( -40 + 48 \beta_{3} ) q^{77} + ( 24 \beta_{1} + 36 \beta_{2} ) q^{79} + ( 79 - 6 \beta_{3} ) q^{81} + ( 17 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 20 - 10 \beta_{3} ) q^{85} + ( 6 \beta_{1} - 12 \beta_{2} ) q^{87} -30 q^{89} + ( -4 \beta_{1} + 78 \beta_{2} ) q^{91} + ( -12 - 4 \beta_{3} ) q^{93} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{95} + ( 66 - 24 \beta_{3} ) q^{97} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 20q^{9} + O(q^{10}) \) \( 4q - 20q^{9} - 64q^{13} - 40q^{17} + 88q^{21} + 20q^{25} + 72q^{29} + 160q^{33} - 112q^{37} + 160q^{41} + 120q^{45} - 308q^{49} - 128q^{53} - 144q^{57} - 64q^{61} - 40q^{65} + 136q^{69} - 8q^{73} - 160q^{77} + 316q^{81} + 80q^{85} - 120q^{89} - 48q^{93} + 264q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 6 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 3 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.61803i
0.618034i
0.618034i
1.61803i
0 5.23607i 0 −2.23607 0 10.1803i 0 −18.4164 0
191.2 0 0.763932i 0 2.23607 0 12.1803i 0 8.41641 0
191.3 0 0.763932i 0 2.23607 0 12.1803i 0 8.41641 0
191.4 0 5.23607i 0 −2.23607 0 10.1803i 0 −18.4164 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.b.b 4
3.b odd 2 1 2880.3.e.a 4
4.b odd 2 1 inner 320.3.b.b 4
5.b even 2 1 1600.3.b.n 4
5.c odd 4 1 1600.3.h.d 4
5.c odd 4 1 1600.3.h.m 4
8.b even 2 1 160.3.b.a 4
8.d odd 2 1 160.3.b.a 4
12.b even 2 1 2880.3.e.a 4
16.e even 4 1 1280.3.g.a 4
16.e even 4 1 1280.3.g.d 4
16.f odd 4 1 1280.3.g.a 4
16.f odd 4 1 1280.3.g.d 4
20.d odd 2 1 1600.3.b.n 4
20.e even 4 1 1600.3.h.d 4
20.e even 4 1 1600.3.h.m 4
24.f even 2 1 1440.3.e.b 4
24.h odd 2 1 1440.3.e.b 4
40.e odd 2 1 800.3.b.d 4
40.f even 2 1 800.3.b.d 4
40.i odd 4 1 800.3.h.c 4
40.i odd 4 1 800.3.h.j 4
40.k even 4 1 800.3.h.c 4
40.k even 4 1 800.3.h.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.a 4 8.b even 2 1
160.3.b.a 4 8.d odd 2 1
320.3.b.b 4 1.a even 1 1 trivial
320.3.b.b 4 4.b odd 2 1 inner
800.3.b.d 4 40.e odd 2 1
800.3.b.d 4 40.f even 2 1
800.3.h.c 4 40.i odd 4 1
800.3.h.c 4 40.k even 4 1
800.3.h.j 4 40.i odd 4 1
800.3.h.j 4 40.k even 4 1
1280.3.g.a 4 16.e even 4 1
1280.3.g.a 4 16.f odd 4 1
1280.3.g.d 4 16.e even 4 1
1280.3.g.d 4 16.f odd 4 1
1440.3.e.b 4 24.f even 2 1
1440.3.e.b 4 24.h odd 2 1
1600.3.b.n 4 5.b even 2 1
1600.3.b.n 4 20.d odd 2 1
1600.3.h.d 4 5.c odd 4 1
1600.3.h.d 4 20.e even 4 1
1600.3.h.m 4 5.c odd 4 1
1600.3.h.m 4 20.e even 4 1
2880.3.e.a 4 3.b odd 2 1
2880.3.e.a 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 28 T_{3}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 28 T^{2} + T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 15376 + 252 T^{2} + T^{4} \)
$11$ \( 6400 + 240 T^{2} + T^{4} \)
$13$ \( ( 236 + 32 T + T^{2} )^{2} \)
$17$ \( ( 20 + 20 T + T^{2} )^{2} \)
$19$ \( ( 144 + T^{2} )^{2} \)
$23$ \( 309136 + 1308 T^{2} + T^{4} \)
$29$ \( ( -396 - 36 T + T^{2} )^{2} \)
$31$ \( 256 + 752 T^{2} + T^{4} \)
$37$ \( ( 764 + 56 T + T^{2} )^{2} \)
$41$ \( ( 620 - 80 T + T^{2} )^{2} \)
$43$ \( 15376 + 732 T^{2} + T^{4} \)
$47$ \( 1008016 + 2492 T^{2} + T^{4} \)
$53$ \( ( 44 + 64 T + T^{2} )^{2} \)
$59$ \( 8386816 + 9888 T^{2} + T^{4} \)
$61$ \( ( -2164 + 32 T + T^{2} )^{2} \)
$67$ \( 57456400 + 15260 T^{2} + T^{4} \)
$71$ \( 26050816 + 16688 T^{2} + T^{4} \)
$73$ \( ( -17996 + 4 T + T^{2} )^{2} \)
$79$ \( 119771136 + 24768 T^{2} + T^{4} \)
$83$ \( 2835856 + 7868 T^{2} + T^{4} \)
$89$ \( ( 30 + T )^{4} \)
$97$ \( ( 1476 - 132 T + T^{2} )^{2} \)
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